Ionic mechanisms in regulation of C-ﬁber following frequency · Ionic mechanisms in regulation of...

Ionic mechanisms in regulation of C-fiberfollowing frequency

Insights from modeling using single and repetitivestimulation

Michail Pantourakis

Thesis to obtain the Master of Science Degree in

Biotechnology

Supervisors:Professor Erik Fransén

Professor Patrícia Margarida Piedade Figueiredo

Examination CommitteeChairperson: Professor Isabel Maria de Sá Correia

Leite de AlmeidaSupervisor: Professor Patrícia Margarida

Piedade FigueiredoMember of the Committee: Professor Fernando Lopes da Silva

October 2, 2014

Abstract

The sense of pain has evolved as the most direct alarm against harmfulfactors. However, chronic pain cases are prevalent and difficult to relieve.Symptoms are often associated with hyperexcitability and high frequency ac-tion potential (AP) conduction of physiologically slow nociceptive C-fibers.Both of these properties are largely determined by the function of ion chan-nels, potential analgesic drug targets. Therefore, in this thesis I investigatedthe main contributing ion channels to the high-frequency-firing properties ofa previously established C-nociceptor computational model. In this model, Iincorporated a NaV1.6 model with resurgent current, a channel well-knownfor facilitating fast-spiking of other neurons.

Following different experimental stimulation paradigms, our model pre-dicted both following frequencies and AP conduction capabilities underrepetitive activation, as well as inherent discharge rates under prolongedconstant depolarization. Parameter-variation analysis of the model clearlyillustrated that NaV1.6 resurgent current and fast delayed rectifiers (Kdr)synergistically shaped shallow afterhyperpolarizations (AHP) and brief APs,allowing rapid post-spike excitability during repetitive activation. Further-more, the deep AHP generated by KA was advantageous in the context ofprolonged stimulation, delaying depolarization block. KA thus appears vitalfor the generation of prolonged high-frequency discharges. Moreover, modelpredictions suggested that cooperation between increased NaV1.7 and KA

densities could explain fast-spiking C-nociceptors with minimal NaV1.6 ex-pression. Therefore, I conclude that the findings of this thesis may contributeto a better understanding of how multiple ion channel subtypes underlie con-duction differences between C-fiber groups, and I hope it will inspire furtherexperimental research.

Keywords: action potential conduction, high firing rates, sodium resur-gent current, potassium current

2

Resumo

A sensacao de dor evoluiu como o alarme mais directo contra factoresadversos. No entanto, casos de dor cronica sao prevalentes e difıceis dealiviar. Os sintomas estao frequentemente associados com hiperexcitabil-idade e conducao de potenciais de accao (AP) de alta frequencia por fi-bras nociceptivas C fisiologicamente lentas. Nesta tese, investiguei entaoa principal contribuicao dos principais canais ionicos para as propriedadesde disparo de alta frequencia num modelo computacional previamente es-tabelecido para fibras nociceptivas C. Neste modelo, incorporei um modeloNaV1.6 com recorrencia de corrente, um canal bem conhecido por facilitardisparos rapidos de outros neuronios.

Seguindo diferentes paradigmas experimentais de estimulacao, o nossomodelo previu tanto as frequencias de seguimento como as capacidades deconducao de AP sob activacao repetitiva, e ainda as taxas de descarga iner-entes sob despolarizacao constante e prolongada. A analise de variacaode parametros do modelo ilustrou claramente que as correntes recorrentesNaV1.6 e os rectificadores com atraso rapidos (Kdr) determinam de formasinergetica pequenas hiperpolarizacoes pos-AP (AHP) e breves APs, per-mitindo rapida excitabilidade pos-AP durante activacao repetitiva. Alemdisso, a AHP profunda gerada por KA foi vantajosa no contexto de estim-ulacao prolongada, atrasando o bloco de despolarizacao. KA parece portantovital para a producao de descargas de alta frequencia prolongadas. Concluoque os achados desta tese podem contribuir para uma melhor compreensaode como multiplos subtipos de canais ionicos sao a base das diferencas naconducao entre grupos de fibras C.

Palavras-Chave: conducao de potenciais de accao, disparo de altafrequencia, corrente de sodio ressurgente, corrente de potassio

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Preface - Acknowledgements

The submission of this thesis concludes the requirements for the Masterof Science degree in Computational and Systems Biology offered by AaltoUniversity (Finland), and for the Master of Science degree in Biotechnologyoffered by University of Lisbon (Portugal), both within the Erasmus MundusMaster program in Systems Biology (euSYSBIO). The present work was con-ducted under the supervision of Professor Erik Fransen in the Departmentof Computational Biology, KTH Royal Institute of Technology (Sweden).

Having never left my country of origin, Greece, before my participationin this programme, I had suspected that these two years around Europewould offer me an amazing wealth of knowledge. Needless to say that Icould not even conceive how much this amazing ’trip’ would mature me andchange the way I realize the world forever.

This trip would have probably not become a reality, if Professor ErikFransen had not directed me, in reply to a simple email, towards the eu-SYSBIO programme. Three years later, I would like to deeply thank him,not only for offering the present topic and guiding me through my first stepsin the field of Computational Neuroscience, but also for being so accessible,supportive and friendly towards a foreign student.

Furthermore, I would like to thank the Alexander Onassis Public Foun-dation Institution and the Steering Committee on behalf of European Com-mission for financially supporting through scholarships my participation tothis programme.

I am also grateful to Professor Patrıcia Figueiredo and Professor SamuelKaski for taking care of the responsibilities of remote supervision on behalfof University of Lisbon and Aalto University respectively. I would also liketo express my gratitude to my beloved Vaso, my classmates (and best teamever) Andreas, Debdas, George, Kul, Son, Tiago and Tuan, and all myfriends for their continuous support.

Finally, a few words in Greek for my dear mother. Αν και έχεις περάσειπολλά στη ζωή σου, ποτέ δεν έπαψες να παλεύεις για το καλύτερο στη δική

μου ζωή. Αυτή η εργασία είναι αφιερωμένη σε σένα αγαπημένη μου μαμά!

Stockholm, July 30, 2014

Michail Pantourakis

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Abbreviations

DRG Dorsal Root Ganglion (or Ganglia)PNS Peripheral Nervous SystemCNS Central Nervous SystemISI Inter-Spike IntervalCV Conduction VelocityADS Activity Dependent SlowingHH Hodgkin-Huxley (model)Standard C-fiber model variant (see Methods)Updated C-fiber model variant (see Methods)Early-resurgent C-fiber model variant (see Methods)No-resurgent C-fiber model variant (see Methods)CT-fiber or LTM C-Tactile fiber, also Low-Threshold-MechanoreceptorCM-fiber Mechanosensitive C-fiberCMi-fiber Mechanoinsensitive C-fiberNaV1-9 Voltage-gated sodium channel isoformsKV1-12 Voltage-gated potassium channel isoformsKA A-type potassium current/channelsKdr Delayed rectifier potassium current/channelsKM M-type potassium current/channelsHCN Hyperpolarization-activated Cyclic Nucleotide-gated

channelsAP (or spike) Action PotentialAHP Afterhyperpolarizationgi Maximum conductance density of current type iVm Membrane VoltageQ10 Temperature coefficient (see Methods)C1-5 Close states (see Methods)I1-6 Inactivated states (see Methods)O Open state (see Methods)OB Open-blocked state (see Methods)

5

Contents

Abstract 2

Resumo 3

Preface - Acknowledgements 4

Abbreviations 5

1 Introduction 12

2 Ionic profile and modeling of C-fibers 152.1 A- and C-fiber classes . . . . . . . . . . . . . . . . . . . . . . 152.2 Ion channels of peripheral afferents . . . . . . . . . . . . . . . 17

2.2.1 Overview of NaV channels . . . . . . . . . . . . . . . . 172.2.2 NaV1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.3 NaV1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.4 NaV1.6 with resurgent current component . . . . . . . 192.2.5 Overview of potassium-selective channels . . . . . . . 202.2.6 Kdr . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.7 KA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.8 KM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 The spatial C-fiber model . . . . . . . . . . . . . . . . . . . . 222.4 The NaV1.6 resurgent model . . . . . . . . . . . . . . . . . . 25

3 Objectives 29

4 Methods 314.1 Model modifications . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.1 Temperature scaling . . . . . . . . . . . . . . . . . . . 314.1.2 Resting membrane potential differences between the

models . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.1.3 Setting maximal conductance densities . . . . . . . . . 334.1.4 Model variants . . . . . . . . . . . . . . . . . . . . . . 33

4.2 Simulation protocols . . . . . . . . . . . . . . . . . . . . . . . 35

6

4.2.1 Voltage clamp . . . . . . . . . . . . . . . . . . . . . . 364.2.2 Current injection protocols . . . . . . . . . . . . . . . 364.2.3 Differential studies . . . . . . . . . . . . . . . . . . . . 38

4.3 Studied properties . . . . . . . . . . . . . . . . . . . . . . . . 384.3.1 Conductive properties . . . . . . . . . . . . . . . . . . 384.3.2 AP waveform properties . . . . . . . . . . . . . . . . . 39

5 Results 415.1 Differences in action potential waveform and in following the

pulse-protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.1.1 Short APs and shallow AHPs characterize the Stan-

dard fiber . . . . . . . . . . . . . . . . . . . . . . . . . 425.1.2 NaV1.6 resurgent, KA and Kdr shape AP waveform . . 435.1.3 Standard model follows higher frequencies . . . . . . . 45

5.2 Sodium and resurgent current contributions to following highfrequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2.1 Standard and Updated models have different resur-

gent NaV1.6 properties . . . . . . . . . . . . . . . . . . 485.2.2 Late but not early resurgent current contributes to

following frequencies by dampening AHPs . . . . . . . 515.2.3 Sodium channels increase peak following frequencies . 53

5.3 The control of potassium currents on following high frequencies 555.3.1 KA affects AHP similar to KV3.4 current . . . . . . . 555.3.2 The competitive nature of KA and Kdr . . . . . . . . . 565.3.3 KM consistently suppresses fast spiking . . . . . . . . 595.3.4 Matched current pairs: NaV1.6 with Kdr, but NaV1.7

with KA for better conduction . . . . . . . . . . . . . 595.4 Discrepancies between pulse and constant protocol . . . . . . 60

5.4.1 Different long term properties modify the effect ofstimulus strength on output frequency . . . . . . . . . 61

5.4.2 Similar NaV1.7, NaV1.6 resurgent, and KM currentcontributions . . . . . . . . . . . . . . . . . . . . . . . 61

5.4.3 Contrary to the pulse-protocol, KA promotes dischargesin the constant protocol . . . . . . . . . . . . . . . . . 62

6 Discussion 646.1 Functional correspondence of models to natural fibers . . . . 64

6.1.1 Firing modalities . . . . . . . . . . . . . . . . . . . . . 646.1.2 Correlations between AP and conduction properties . 656.1.3 Comparison of following frequencies . . . . . . . . . . 666.1.4 Gradual depolarization . . . . . . . . . . . . . . . . . . 676.1.5 Dynamic temporal patterns . . . . . . . . . . . . . . . 686.1.6 Reasons for discrepancies in predicted following fre-

quencies . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7

6.1.7 General utility of constant and pulse protocols . . . . 706.2 Current mechanisms for fast spiking C-fibers . . . . . . . . . 71

6.2.1 Main sodium currents contribute differently based onactivation and repriming properties . . . . . . . . . . . 71

6.2.2 Multiple resurgent currents may facilitate fast C-fiberresponses . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.2.3 KV-mediated contributing mechanisms . . . . . . . . . 73

7 Conclusions 76

A Basics of cellular electrophysiology 79A.1 Neurons: electrical properties and action potential . . . . . . 79A.2 Hodgkin-Huxley type models . . . . . . . . . . . . . . . . . . 80

A.2.1 Membrane currents . . . . . . . . . . . . . . . . . . . . 80A.2.2 Sign conventions . . . . . . . . . . . . . . . . . . . . . 81A.2.3 Gating variables of conduction . . . . . . . . . . . . . 81

A.3 Markov models for single ion channels . . . . . . . . . . . . . 83A.4 Choosing kinetic schemes over HH-type models . . . . . . . . 84A.5 Propagation and multi-compartmental modeling . . . . . . . 85

8

List of Tables

2.1 Gate variables and original sources of ion channel equationsof the spatial C-fiber model. . . . . . . . . . . . . . . . . . . . 22

4.1 Modified parameters for the fiber model variants studied. . . 35

5.1 AP properties of figure 5.1 the the original C-fiber, the Stan-dard, the Updated, the Early-resurgent, and the No-resurgentmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2 A summary of values for four conduction features as predictedby different model versions that varied on the type of mainsodium, potassium, and resurgent NaV1.6 component. . . . . 63

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List of Figures

2.1 Drawing of a peripheral sensory neuron and the C-fiber modelrepresentation, adopted from Petersson [61]. . . . . . . . . . . 23

2.2 Kinetic scheme of NaV1.6 from Khaliq et al. [50] . . . . . . . 252.3 Example time evolution of state occupancies of Khaliq et al.

[50] NaV1.6 model during an action potential. . . . . . . . . . 27

4.1 Summarizing diagram of all model variants and their corre-sponding modifications. . . . . . . . . . . . . . . . . . . . . . 34

4.2 Summarizing diagram of all current injection stimulation pro-tocols and their corresponding settings. . . . . . . . . . . . . 37

4.3 Action potential properties used in the present study. . . . . 39

5.1 Comparison of AP generated by the Original C-fiber and allpresently studied models. . . . . . . . . . . . . . . . . . . . . 42

5.2 Current densities of the original C-fiber and all presently stud-ied models during the AP of figure 5.1. . . . . . . . . . . . . . 44

5.3 After-hyperpolarization (AHP) and the main contributing cur-rents of the Standard and the Updated model. . . . . . . . . 45

5.4 Response to the strong 40-pulse protocol of the original, Stan-dard , and Updated model. . . . . . . . . . . . . . . . . . . . 46

5.5 Comparison between the normalized current densities of thedifferent Standard and Updated NaV1.6 models. . . . . . . . 48

5.6 Voltage-clamp protocols and the corresponding current den-sities for determining inactivation and resurgence of the Stan-dard and the Updated NaV1.6 model. . . . . . . . . . . . . . 50

5.7 State occupancies of NaV1.6 during the AP of figure 5.1 forthe Standard, Updated, Early-resurgent and No-resurgentmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.8 Increased gNaV 1.6 can give bursting and spontaneous firingproperties to the Standard model. . . . . . . . . . . . . . . . 53

5.9 Latency and ISI properties of the Standard and Updatedmodel as a function of the input frequency of the pulse protocol. 54

5.10 Comparison between the normalized current densities of Kdr,KA, and the newly tested KV3.4 model. . . . . . . . . . . . . 56

10

5.11 The opposing nature of Kdr and KA during the action potentials. 575.12 Initial and sustained response frequency in the pulse protocol

depends on the expression levels of Kdr and KA. . . . . . . . 585.13 KM consistently reduces the number of successfully propa-

gated APs of the Standard model. . . . . . . . . . . . . . . . 595.14 Response to the strong and weak constant stimulation proto-

col of the Standard model. . . . . . . . . . . . . . . . . . . . . 61

6.1 Significant correlations between AP and fiber properties aspredicted by the differential study for both the standard, andthe updated model. . . . . . . . . . . . . . . . . . . . . . . . . 65

A.1 Transition state diagram of a gate . . . . . . . . . . . . . . . 82A.2 Transition state diagram of ion channels . . . . . . . . . . . . 84

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Chapter 1

Introduction

Everyone has undoubtedly experienced the sensation of pain. Although wehave learned to recall pain as an unpleasant part of life, it has withstoodeons of evolution as the main warning system for potentially harmful factors.Harmful thermal, mechanical and chemical stimuli are detected by a specialcategory of peripheral sensory neurons called nociceptors, first discovered bySherrington [1].

Physiologically, nociceptors respond only if the intensity of a stimulusexceeds a (characteristic for each nociceptor subtype) threshold, such a stim-ulus is thus recognized as noxious [2–5]. When the painful sensation isevoked by the response of these primary afferent neurons, it is called noci-ceptive. Nociceptors form the peripheral pain pathway, as they innervatethe skin and body surfaces in general with receptors-equipped peripheralbranches of a long axon (figure 2.1a), lacking dendrites for that purpose,while they possess a large soma called Dorsal Root Ganglion (DRG) (ortrigeminal ganglion for the head) [2, 3]. Notice that due to their size, DRGhave been largely used for intracellular membrane potential measurements,whereas the thinner axons have been inaccessible (obviously in vivo, as it iscrucial to connect fiber function with perception). In the following sectionwe will see that nociceptors are divided into classes and subclasses based onanatomical and functional characteristics.

Information is transmitted by activation of the axonal ion channels tothe other side of the axon, the central terminal. This terminal communicatesvia glutamatergic synapses with interneurons in the superficial laminae (Iand II) of the dorsal horn in the spinal cord, from where the signal is thentransmitted via several pathways to various parts of the brain [2, 5–7]. Eventhough both the peripheral and the central nervous system (PNS and CNSrespectively) integrate the incoming information, influencing each other inboth directions, in this thesis I will exclusively focus on the function ofperipheral fibers.

Nociceptive response is highly plastic as well, being reversibly sensitized(peripheral sensitization) by proximal tissue injury or inflammation (causing

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CHAPTER 1. INTRODUCTION 13

inflammatory pain) [3, 4]. At this state nociceptors are characterized by hy-perexcitability (induced by various signal transduction pathways that affecte.g. receptors and ion channels expression [2, 3]), turning innocuous stimulisuch as cooling and light touch into painful experiences (so-called allodynia),and extravagating the pain from weak harmful stimuli (hyperalgesia).

Although all the above types of pain are part of a normal protectivemechanism, abnormal spontaneous pain can persist for years without a clearstimulus. Chronic pain can have various etiologies, such as cancer, lowback pain, osteoarthritis and peripheral neuropathies (pain caused by nervelesions) [8]. Even though locally induced acute pain can be easily preventedwith anesthetics, chronic pain is usually poorly treated with non-steroidalanti-inflammatory agents and opioids, with two-thirds of patients reportingunsatisfactory treatment [9], while they also cause serious side-effects frominteractions with receptors not implicated in the pain pathway [2, 7, 8,10]. Since chronic pain is highly prevalent (up to 20% of the Europeanand American adult population [4, 9]), resulting in huge financial costs forsociety, it is crucial for the research community to focus on identifying newspecific pharmacological targets and drugs.

As with the inflammatory state, hyperexcitability characterizes nocicep-tors in chronic pain cases. As we will see in the next section, C-type noci-ceptors physiologically transmit at low velocities, mediating the subsequent’milder’ sensations of pain. Furthermore, C-nociceptors can normally onlysustain repetitive activation at low frequencies, exhibiting the lowest actionpotential (AP) firing frequencies among peripheral sensory fibers. However,chronic and neuropathic pain is often associated with C-fibers exhibitingabnormal high-frequency firing and/or spontaneous activity [11–13].

Ion channels (along with passive membrane properties) govern the elec-trical activity of nociceptive fibers, while knowledge regarding their functionsand implications in chronic pain has been accumulated. As a consequence,their (especially of nociceptor-exclusive ion channels) importance as anal-gesic targets have been increasingly recognized in recent years [7, 10, 14, 15].

Facilitating this long-term goal, in the present thesis we seek to investi-gate the ion channels mainly implicated in the high-frequency-firing profileof not only pathological C-nociceptors, but also a normal non-nociceptiveC-fiber subgroup. As their size hinders the task of direct intracellular mea-surements in humans, we choose to utilize our computational C-nociceptoraxon model, which has been validated from in vivo human data [16] andtherefore offers a framework that directly connects ionic mechanisms andextracellular recordings. Using this model as a starting point, I will convertit into a fast-spiking C-fiber variant, and subsequently analyze the role ofprominent ion channel subtypes on this new behavior.

Before following the typical structure of this thesis, I would recommendany reader not familiar with theoretical cellular electrophysiology to referto appendix A. This appendix defines basic concepts that are necessary to

CHAPTER 1. INTRODUCTION 14

follow the terminology of this thesis, and in fact represents the basic materialthat I had to understand myself before being able to work in this field.

In chapter 2, I introduce in more detail both the biological and the mod-eling background of this thesis. Chapter 3 serves as a checkpoint, develop-ing the main hypothesis and stating the objectives of this thesis. Chapter4 describes the steps followed for the model modification, the simulationprotocols and the electrophysiological features that facilitate the analysis.Chapter 5 contains the results and analysis of the simulations. In chapter 6I relate the obtained results with previously reported experimental findings,and finally chapter 7 concludes the main findings of the present thesis.

Chapter 2

Ionic profile and modeling of C-fibers

In this chapter, I will introduce in more detail the different peripheral sensoryfiber types and their excitability properties. Furthermore, in section 2.2 Iwill briefly describe the types of sodium and potassium channels prominentlyexpressed in nociceptors, with special emphasis on the properties of thoseinvolved in my present modeling work. Finally, I will introduce the originalC-nociceptor model, along with a newly incorporated sodium channel model,which is vital for many other fast-spiking neurons.

2.1 A- and C-fiber classes

As I already mentioned, DRG somata and their innervating fibers are cate-gorized based on their anatomy and responses to electrically, mechanicallyand/or thermally stimulating diagnostic protocols. The size of their di-ameter and the level of membrane myelination are the most fundamentaldistinguishing features [2].

Large-diameter axons with thick myelination belong to the Aβ group ofprimary sensory fibers. Aβ-fibers mostly detect innocuous stimuli, such aslight touch with soft objects, therefore they are not nociceptive fibers (theyare capable of central sensitization in certain neuropathies though [4]). Dueto their size and myelination, they exhibit very high conduction velocities(CV), around 50 m/s [17].

True nociceptive fibers stem from medium- and small-diameter DRG.The former, thinly-myelinated Aδ-nociceptors, consist the minority [3]. Theyexhibit the highest CV among nociceptors (up to 30 m/s [2, 18]), and arethus thought to mediate the first ’wave’ of acute pain immediately afterinjury. Furthermore, large-diameter A-fibers are capable of following ex-ternally induced firing frequencies of over 100 Hz (obviously attributed totheir ion channel/pump pool) [19, 20]. Functionally, Aδ-fibers can be furthersubdivided into heat-, cold-, and/or mechano-sensitive units [18].

However, the C-fiber group consists the majority of nociceptors. Theseare unmyelinated, slowly conducting (around 1 m/s [2, 18]) afferents that

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CHAPTER 2. IONIC PROFILE AND MODELING OF C-FIBERS 16

mediate the subsequent slow ’waves’ of mild pain. Contrary to A-fibers,C-fibers can generally follow significantly lower frequencies (around 16 Hz[19, 20]). However, their basic classification is also done based on thermal,mechanical, and exclusively for C-fibers, chemical sensitivity (for examplethey are responsible for our reaction to capsaicin), with most units beingpolymodal [2, 18]. In addition, some C-fibers that are located via electricalstimulation lack both thermal and mechanical sensitivity. These are silentC-fibers and respond to noxious stimuli only after chemical sensitization(e.g. during inflammation) [2, 18, 21].

Certain patterns of changes in CV upon electrical stimulation have alsobeen repeatedly used as a marker of C-fiber subtypes in humans [22], andin animal models such as rats [23, 24] and pigs [25]. Porcine and humansilent C-fibers (also called mechano-insensitive, CMi-fibers) are particularlycharacterized by extensive Activity Dependent Slowing (ADS) of CV, andthis is their only reliable classification criterion [21].

More important for the present work though, conduction failures, whichquickly occur upon repetitive electrical stimulation of C-fibers leading tosmall following frequencies, have been closely associated with enhanced ADSpatterns [26–29]. A recent study in pigs has also illustrated this connectionby separating C-fiber subclasses based on their characteristic maximum fol-lowing frequencies [30]. Furthermore, ADS has been linked to the slowaccumulation of sodium channels in the inactivated states [31]. Combined,these results are again hints for the different ion channel profile of each C-fiber subclass, and its role in shaping the processing of information at thelevel of peripheral axons.

Finally, a special mention should be written for a still relatively unex-plored (in humans) non-nociceptive C-fiber subclass, called C-tactile (CT) orlow-threshold mechanoreceptors (LTM). While the mammalian homologuehas been known for several years [32], it had not been found in humans until50 years later [33], mainly because it is not possible to stimulate exclusivelyCT-fibers in healthy subjects that possess both CT- and Aβ-fibers [17].

Although CT-fibers exhibit similar CV as the other unmyelinated af-ferents, they respond vigorously to light touch (like Aβ-fibers), with burstfrequencies up to 100 Hz, namely the highest reported for C-fibers [17]. De-spite the apparent functional similarities, CTs still differ from Aβ-fibers asonly their activity is correlated with pleasantly-perceived touch [17]. How-ever, the possible participation of CTs to pain sensation as well should notbe rejected, as they also respond to high intensity mechanical stimuli [34].

I believe it is important here not to confuse following frequencies due torepetitive electrical stimuli, and the burst frequencies from a continuous me-chanical stimulus such as soft brush stroking. Interestingly, a group that dis-tinguished between non-nociceptive (LTMs) and nociceptive C-fibers in ratsdid not find significant differences in their maximum following frequencies[35]. Nonetheless, in another report CTs exhibit limited ADS, apparently in


accordance with the aforementioned link between following frequencies andADS pattern [36]. Drawing inspiration from CTs as a physiological exampleof high-frequency-firing C-fibers, in this thesis I will investigate the ionicmechanisms that might contribute to their ability.

2.2 Ion channels of peripheral afferents

Many points have already suggested the importance of an ’optimal’ distri-bution of ion channels (or to be more precise, channel types) in determiningwhether, when, and how the information will be transfered from the periph-eral arborizations to enter the central dorsal horn. As we will see shortly,numerous ion channels are present in primary afferent neurons, some beingnerve-type specific or even location specific (expressed exclusively in axonsor DRG).

Putting aside ligand-gated channels that mainly serve as the actual sen-sors at the receiving end of the fibers, the list of peripheral axon channelsinclude voltage-gated sodium, potassium, and calcium (out of the scope ofthis thesis) channels, and leak current-mediating channels [7, 15]. Theoret-ically, every key ion channel has the potential of becoming a therapeuticdrug target. Lack of specificity however, at the level of target location (i.e.exclusive expression of ion channel at a specific site) and/or at the levelof the drug-target interactions at the majority of cases imposes a seriousobstacle.

Obviously, having a large number of channel types that could collectivelyform a continuum of electrical properties makes the analysis of individualcontributions quite difficult. To make matter worse, modulation of ion chan-nels may depend on nerve-type specific features [15], resulting in conflictingresults even for the same channel type, while every current influences othersubtypes complicating their apparent kinetics. Keeping these limitations inmind, I will provide here a brief overview of major contributing channels tothe function of C-fibers (or DRG and other peripheral fiber types if informa-tion from C-fibers is not available), revealing at the same time the sodiumand potassium channels that will be mostly implicated in this thesis.

2.2.1 Overview of NaV channels

Since the revolutionary work of Hodgkin and Huxley [37], the role of voltage-dependent sodium channels in action potential initiation, cell excitabilityand axonal propagation has been recognized repeatedly. In nociceptors, up-regulation of sodium channel expression occurs during peripheral sensitiza-tion (e.g. due to local inflammation), which enhances axonal excitability andconduction [28]. The wide number of reviews about voltage-gated sodiumchannels for peripheral pain (for example [7, 14, 15, 38, 39]), in contrast


to potassium channels ([7, 10, 40]), indicates not only the far better under-standing of the role of sodium channel isoforms in nociception, but also theirpromise as potential therapeutic targets.

Nine different voltage-gated sodium channels isoforms are expressed (NaV

1.1 to NaV1.9) in humans, all sharing a similar conformation with few aminoacid differences shaping their different kinetics, pharmacological and gat-ing properties [7, 15, 41]. Nevertheless, just five (NaV1.1, NaV1.6, NaV1.7,NaV1.8, and NaV1.9) can be found normally in adult peripheral sensoryneurons , whereas NaV1.3 may be expressed only after nerve injury [15].

As an ideal treatment for peripheral pain would require as less inter-action as possible with cells other than sensory nerves, the PNS-specificNaV1.7, NaV1.8 and NaV1.9 have grabbed most of the attention in re-cent years. NaV1.9, preferentially expressed in largely nociceptive subsetof small-diameter DRG and C-fibers, is responsible for a slowly-activating,non-inactivating (persistent) current Waxman and Zamponi [7], Dib-Hajjet al. [15], Herzog et al. [42]. As a result of its slow kinetics, its contributionto APs is minimal, but as a persistent current (that is active at subthresholdpotentials) it clearly reduces the firing threshold and thus facilitates AP ini-tiation [7, 15, 42]. As the results of my work also show, the role of the othertwo PNS-exclusive isoforms during APs is much more significant, thereforethey are discussed in the following subsections in slightly more detail.

On the other hand, NaV1.1 and NaV1.6 are also prominently expressed inthe CNS [7, 15], hindering their utility as pharmaceutical targets for pain.NaV1.1 exhibits fast activation and inactivation kinetics, resembling theother TTX-sensitive isoforms (NaV1.3, NaV1.6, NaV1.7) [7, 14]. However,as NaV1.1 is present mostly in non-nociceptors [4], without any evidencelinking this subtype to pain mechanisms, it is not considered important inthis context [14]. On the contrary, as we will see in great extent below andin my work, NaV1.6 shows particular significance for high-frequency-firingneurons.

2.2.2 NaV1.7

Probably the most studied NaV isoform in peripheral nerves, NaV1.7 drewmuch of the research attention after a genetic study that clearly illustratedthe complete inability of humans to experience pain when lacking the NaV1.7-encoding gene [43]. NaV1.7 appears to be expressed universally in all pe-ripheral sensory neurons and along their entire fiber and DRG [4, 7, 44],with similar expression levels between small- and large-diameter DRG [45],although data from guinea-pigs have shown highest expression levels in no-ciceptive C-fibers [46].

NaV1.7 generates a fast-activating and fast-inactivating TTX-sensitivecurrent, with considerably slow recovery from inactivation (slow-repriming)[15, 44]. As it is characterized by a relatively hyperpolarized voltage-dependence


and slow closed-state inactivation, it can generate substantial subthresholdcurrent, thereby facilitating successful AP initiation [20, 44]. This role ofNaV1.7 is further verified by fiber hyperexcitability and pain disorders due tomutations that to more hyperpolarized activation or impaired inactivation[3, 44].

2.2.3 NaV1.8

NaV1.8, mainly expressed in nociceptive A- and C-fibers [4], can be ac-counted for the majority of the TTX-resistant inward current during thelater phases of APs in DRG that express it [47]. This is because the cur-rent conducted by NaV1.8 is slow-activating and inactivating, with greatlydepolarized voltage-dependency, making it less important at subthresholdranges [14, 15].

Contrary to NaV1.7, NaV1.8 is also characterized by fast-repriming,which (combined with its depolarized inactivation) enables neurons to prop-agate most of the later spikes under repetitive firing conditions, namelywhen NaV1.7 has been mostly inactivated by gradual (slow timescale) de-polarization [12, 15]. However, its slow kinetics and the accumulated (slow)inactivation of NaV1.7 are presumably responsible for the significant ADSand the lower firing frequencies of C-nociceptors [12, 31, 48]. From theabove, we can clearly infer that pathological hyperexcitability and sponta-neous activity in C-fibers can be achieved: (1) by gain-of-function mutationsor modulations in NaV1.7 and NaV1.8 [7], and (2) by significant upregulationof their expression to abnormal levels [28].

2.2.4 NaV1.6 with resurgent current component

Due to the slow recovery from inactivation of NaV1.7, and therefore largerefractory period, neurons that mainly express this subtype are expected tobe capable of firing at low frequencies [14, 20]. Another fast-activating andinactivating TTX-sensitive channel though, NaV1.6, exhibits five-times morerapid-repriming and more depolarized voltage-dependence [20]. Typically,NaV1.6 is the predominant subtype in the nodes of Ranvier of myelinatedaxons. Consequently, the predominant TTX-sensitive current of myelinatedA-fibers follows the kinetics of NaV1.6, whereas NaV1.7 determines the cor-responding sodium current in unmyelinated C-fibers [20, 45]. Recalling nowthat A-fibers are capable of much larger frequencies than C-fibers, we caninfer that the maximum firing rate of each fiber type is largely determinedby the rate of recovery from sodium channel inactivation, and that NaV1.6is more suitable for high-frequency-firing neurons [20].

NaV1.6 channels have to thank one more property for this high-frequency-firing facilitation: the resurgent current, which was first discovered in spon-taneously bursting Purkinje neurons [49, 50]. As I will illustrate with a


NaV1.6 model in section 2.4, the C-terminus of the β4 subunit acts as enendogenous blocking particle of open NaV1.6 channels [49, 51]. Having theability to dissociate during repolarization, the channels reopen to give riseto a brief, ’resurgent’ current. Although it has been shown that only large-diameter DRG in normally NaV1.6-expressing mice demonstrate resurgentcurrents [52], NaV1.7 mutations have been capable as well [53]. Finally, therole of NaV1.6 in nociceptive C-fibers is currently unclear, there is nonethe-less evidence of its homogeneous expression in unmyelinated C-fibers of miceand its sufficiency in generating APs [54].

2.2.5 Overview of potassium-selective channels

Potassium channels constitute the most abundant and diverse class of ionchannels, expressed from over 80 mammalian genes, with 40 of these encod-ing KV channels in humans [41]. The general role of potassium channels forthe post-AP repolarization is well known, and that role in peripheral fibersis no different, controlling excitability by raising AP threshold and hinder-ing conduction across the axon [10]. Therefore, despite the yet incompleteunderstanding of the contributions of potassium channels in peripheral pain,the vast number of channel subtypes leaves a promising field of therapeuticaltarget research. This belief is further reinforced by early evidence that spe-cific potassium channel openers and blockers can dependably decrease andincrease respectively the (at least short-term) excitability of DRG [40, 55].

KV genes are further classified in 12 families (KV1-KV12), expressingsubunits that eventually form homo- or hetero-tetrameric channels [10].However, electrophysiologists have traditionally distinguished KV channelsaccording to the properties of their conducting currents, often resulting inconflicting terminology [7, 10, 40, 41]. A categorization that I choose forthe purposes of this work (based on the C-fiber model, see section 2.3) isbetween delayed rectifying (Kdr), fast-inactivating (KA), and persistent andslow-inactivating (KM) potassium currents, which consist the main potas-sium currents of small-diameter DRG [56]. I introduce these three currentsin their respective sections. However, notice that the modeled currents maynot entirely represent a particular channel subtype, but rather the collectiveeffects of multiple subunit types.

Based on their gating properties, other K+ channels are classified intocation-activated (calcium- or sodium-activated), two-pore (mediating hyper-polarizing leak currents), atypical inward-rectifying (remember that typicalpotassium currents are outward), and the hyperpolarization-activated cation(HCN) channels [7]. The first mainly contribute to afterhyperpolarization(AHP), subthreshold potentials, and firing rate adaptation to cation con-centration changes [7]. The two-pore leak channels represent approximately85% of the K+ resting current in DRG [4], effectively increasing firing thresh-old and stabilizing their resting potential [10]. Finally, HCN channels medi-


ate both sodium and potassium currents (inward and outward respectively),which are well known pacemaker currents in the heart and Purkinje fibersby regulating subthreshold oscillations and neuronal excitability [41], withsimilar contribution to spontaneous firing of injured peripheral fibers [4].

2.2.6 Kdr

The name of delayed rectifiers originate from Hodgkin and Huxley [37], whorecognized an outward potassium current as a delayed response to depolar-ization. Today we know that multiple tetrameric proteins are responsiblefor currents that functionally belong to the class of rapidly-activating Kdr,giving rise to the majority of the potassium current of short AP-generatingneurons.

KV1.1 and KV1.2 are slowly-inactivating delayed rectifiers that havemoderately-depolarized voltage-dependence, thus being introduced relativelyearly during the AP. Although a report suggested the presence of KV1.1 sub-units in some C-fiber subtypes, KV1 channels are predominantly expressedin A-fibers and various other CNS neurons [10].

On the other hand, KV2.1 appears to mediate most of the non-inactivating(sustained) delayed rectifying current in C-nociceptors, while KV2 isoformsare also expressed in myelinated DRG. Since their voltage-dependence ismore depolarized than KV1, they come late in the falling phase of the AP,also affecting the subsequent AHP. The suppressed expression of KV2 inaxotomized DRG neurons may illustrate its importance on preventing hy-perexcitability [10].

Last but not least, a fast type of delayed rectifier currents are driven byKV3.1 and KV3.2 subtypes. Although they probably contribute only 20%of the aforementioned current type in small nociceptors, their function isto open and close swiftly after the AP, limiting AP amplitude, increasingrepolarization, and thus preventing sodium channel inactivation [56, 57].Therefore, these channels are considered perfect for high-frequency-firingneurons, with many examples from neurons of the CNS present [57, 58].

2.2.7 KA

The transient, fast-activating KA current is mediated by members of variousgene families, including KV1, KV3 and KV4 [10, 59], with slight differencesin their kinetics.

Both KV3.3 and KV3.4 are known to mediate high-voltage activated Acurrents. However, while the first is characterized by slow inactivation,the second is considerably limited by rapid inactivation [58]. Althoughboth have been associated with facilitating high-frequency repetitive firingneurons [58], there is little evidence for the existence of KV3.3 subunits insmall-diameter DRG [56]. Nevertheless, KV3.4 probably conducts the high-


threshold transient KA current of nociceptors [10], which at normal levelsprevents mechanical hypersensitivity [59].

KV4 and KV1.4 channels, in contrast to the aforementioned isoforms, hasa much lower voltage-dependence, being activated by small depolarizations.As a result of this property, their currents can dampen the development ofsubthreshold depolarizations, resulting in broader interspike intervals andtherefore limited firing frequencies [10, 41].

2.2.8 KM

Finally, KV7 channels mainly underly slow, non-inactivating M-currents.KM channels have a greatly hyperpolarized voltage-dependence, mediatinga large resting and subthreshold current that usually limits excitability andrepetitive firing [41, 60]. The M-current of small DRG neurons is mediatedby KV7.2, 7.3, and 7.5 subunits, and they are significantly downregulatedafter nerve injury to allow spontaneous firing and high number of actionpotentials on depolarization [4]. Conversely, a KV7 channel opener reducesmechanical allodynia and thermal hypersensitivity in rat models, highlight-ing the potential of potassium channel openers in pain relief [7].

2.3 The spatial C-fiber model

Currents Gates Sources

NaV1.7 m3hs Sheets et al. [63]

NaV1.8 m3hsu Sheets et al. [63], Maingretet al. [64]

NaV1.9 mhs Herzog et al. [42]

Kdr n4 Sheets et al. [63]

KM ns/4 + 3nf/4 Passmore et al. [55], Main-gret et al. [64]

KA mh Sheets et al. [63]

h ns/2 + nf/2 Kouranova et al. [65]

KNa w (Nain dependent) Bischoff et al. [66]

Table 2.1: Gate variables and original sources of ion channel equations of the spa-tial C-fiber model. Apart from w, all other gate variables are voltage-dependent.Notations are kept as presented in [16] for direct connection. Also note that despitesimilar notations, each gate variable of each channel is independent of others. nand m: activation, nf : fast activation, ns: slow activation, h: inactivation, s: slowinactivation, u: ultra slow inactivation.

Although various C-nociceptor models have been developed, they typi-cally consist of a single DRG compartment. As a consequence, these non-


Figure 2.1: Drawing of a peripheral sensory neuron (A) and the C-fiber model rep-resentation (B), adopted from Petersson [61]. The blue and red colors indicate thetemperature difference between the superficial and deeper side of the axon, whereasgrey parts are omitted from the model. (B) The model consists of a ’distal’ branchaxon and a ’proximal’ parent axon, connected via a cone. (C) Axon cross-sectionshowing the intra-axonal space, the periaxonal space, and the extracellular fluid(with constant concentrations), modeled according to Scriven [62]. D = diameterof axon, Θ = width of periaxonal space.


spatial representations cannot directly establish a link between ion concen-trations and channel dynamics, body temperature, axonal morphology andtheir impact on the aforementioned physiological process of axonal spikepropagation (see also brief review of Petersson [61]). Therefore, to effec-tively utilize and associate extracellular microneurographic data from hu-man C-fibers to channel mechanisms, Tigerholm et al. [16] synthesized thefirst multicompartmental model of a primary afferent C-fibers.

Figure 2.1 (part B) presents the biophysical structure of the model. Itconsists of 2430 compartments, totaling an axon length of 12.5 cm that iscomprised by a single, thin (cylinder with 0.25 µm diameter length) branchaxon and a thick (1 µm diameter) parent axon. However, a sudden changeof diameter length between them might introduce artificially induced prop-agation failures. Therefore, the branch and parent axon are connected by atransitional cone-shaped segment. Artifacts from sealed-end boundary con-ditions are also prevented, as passive (minimal conductance) compartmentsfollow both the proximal (central) and distal (peripheral) end. Furthermore,the model accounts for the difference between cutaneous and body temper-ature (something that cannot be explored in a point-neuron model): 32°Cis set for the superficial branch axon, 37°C for the deeper parent axon, andtheir average value for the intermediate segment. As seen in section A.2,temperature is an important factor directly affecting ion channel dynamics.

Several realistic mechanisms, which involve sodium and potassium ions,have been borrowed from the literature. As also shown in figure 2.1 (part C),Na+ and K+ dynamics are explicitly represented, with varying intra- andperi-axonal concentrations, constant extracellular concentrations, and Na-K-ATPase pump equations according to Scriven [62]. Most ion channels thatwere presented in this chapter have also been integrated as HH-type equa-tions: NaV1.7, NaV1.8, NaV1.9, Kdr, KM, KA and h (HCN). Furthermore,atypical, weaker currents present in the C-fiber include a sodium-dependent(voltage-independent) channel KNa, and leak sodium and potassium cur-rents that are proportional to the negative sum of all other sodium andpotassium currents respectively, assuring that cell rests at Vrest. Table 2.1summarizes the original data and the gating variables of each ion channelmechanism. For the full system of equations of each channel, I refer theinterested readers to Tigerholm et al. [16] and references therein.

This spatial model has not only successfully replicated several indepen-dent experimental data from human, pig, and rat CMi [16] and CM [61]nociceptors. In short, the activity-dependent slowing (ADS) on the minutetimescale during repetitive (pulsed) electrical stimulation has been associ-ated with reduced excitation and increased propagation failures, at least inmyelinated frog axons [67], porcine C-fibers [27], and rat cranial dura affer-ents [68]. Since reduced ADS and hyperexcitability is linked to pathologicalC-fibers of chronic pain patients [11], the model has been invaluable in the insilico disentanglement of the contributions of various intertwined -and thus


hard or impossible to separate experimentally- mechanisms on conductionand excitability [5, 16, 61]. For example, clamping (fixing) of the reversalpotential of only specific ion channels, which in reality is impossible, dissoci-ates ion concentration effects on different mechanisms. This feature allowedthe direct correlation between intra-axonal sodium accumulation and ADS[16].

Closely related to the direction of the present thesis, Petersson [61] haspreviously used the model to investigate possible differences in the ion chan-nel/pump density profile of CMi and mechano-sensitive fibers. To generatethe reduced ADS profile of mechanosensitive fibers, the model suggestedan increased ratio

gNaV 1.7

gNaV 1.8and Na-K-ATPase pump density, and decreased

Kdr. Indeed, modifications from lidocaine injections in axonal properties ofdifferent C-fiber types suggested the lower density of NaV1.8 in CMs [29].

As we mentioned earlier in this chapter, further evidence suggests that:1) NaV1.6 channels better enable neurons to sustain higher firing rates thanNaV1.7 channels [20], 2) NaV1.7 expression levels do not vary significantlybetween small and large DRGs [45], 3) NaV1.6 channels are predominantlyfound in large DRGs (which give rise to thickly myelinated A-fibers) [45, 69–71], 4) large DRGs can indeed follow considerably higher firing frequencies[19], while CT-fibers represent the high-frequency-firing subtype of C-fibers[17]. Based on these observations and extending the aforementioned ratio-nale of Petersson [61], the reader can hopefully infer that a natural extensionof the presented C-fiber model is the inclusion of a NaV1.6 channel mecha-nism and the quantification of its effects.

2.4 The NaV1.6 resurgent model

C1 C2 C3 C4 C5 O OB

I1 I2 I3 I4 I5 I6

4α

β

3α

2β

2α

3β

α

4β

γ

δ

ε

ζ

Coff

Con

Coff·a

Con·b

Coff·a2

Con·b2

Coff·a3

Con·b3

Coff·a4

Con·b4

Ooff

Oon

4α·aβ·b

3α·a2β·b

2α·a3β·b

α·a4β·b

γ

δ

Figure 2.2: Kinetic scheme of NaV1.6 from Khaliq et al. [50]

In this section I briefly present the original NaV1.6-mediated currentmodel that will be modified and incorporated in the C-fiber model (seeMethods). As mentioned before, NaV1.6 channels are known to mediateboth a fast-inactivating current and a slow resurgent component that ap-pears during the repolarization phase of APs [49]. Using data from cerebellar


Purkinje neurons (known for their high-frequency spontaneous firing prop-erties), Raman and Bean [51] (later optimized by the same group [50]) werethe first to propose a Markov kinetic scheme of NaV1.6, which captured boththe main and resurgent current properties.

Figure 2.2 shows all transition mechanisms between 13 channel states:five close (C1-5), six inactivated (I1-6), one open (O), and one open-blocked(OB) state. The multiple close and inactivated states represent the gradualconformational changes towards channel opening. Among them, only thetransitions within the C and I states and that of the reopening from the OBstate have been defined as voltage-dependent, with α, β, and ζ followingsimple exponential functions (equation A.5 withH and F non-zero), whereasall other rate parameters are voltage-independent (constants, all values canbe found in [50]). Microscopic reversibility for the loops of this scheme isensured with parameters a and b, which are physiologically interpreted bythe tighter binding of normal inactivation particles to fully activated (O)and partially activated (C2-5) states.

Probably the best way to understand the NaV1.6 kinetic scheme is bysimulating its dynamics during an action potential sequence of a Purkinjepoint-neuron model. In figure 2.3, using the Purkinje model of Khaliq et al.[50] (downloaded from ModelDB, accession number 48332), I depict all stateoccupancies at six time instances during such an AP. Starting from the rest-ing potential, most channels assume the close states, especially accumulatedin C1. As we enter the subthreshold rising phase of the AP (top left panel),activation kicks in and channels pass through the later C stages. At thesame time we can already notice that as soon as considerable fraction ofchannels is activated, the normal fast inactivation begins to manifest itself.As soon as the membrane hits the firing threshold (top right panel), themajority of channels expectedly pass to the O state. After 0.7 seconds themiddle of the falling phase has already been reached (middle left). At thispoint all the open channels have either enter normal inactivation (mainly I6)or have been blocked (OB). At the end of repolarization (middle right), themain component has been subsided. Nevertheless, the voltage-dependenttransient reopening from the prominent OB fraction of channels permits theso-called resurgent current. Simultaneously, the inactivated channels slowlydeactivate (I6→ ... → I1), and gradual deactivation also promotes deinacti-vation (I1 → C1, etc.). During the undershoot phase (bottom left), the OBoccupancy is gradually depleted, as the hyperpolarized membrane largelyfavors deactivation. Eventually, the resting membrane potential is reachedagain (bottom right), and the highest occupancy occurs again for the initialclose states. At this point, refractoriness has ended and the channels areavailable again for activation.

Although the model has originally been built for Purkinje neurons, itcan potentially be useful modeling NaV1.6 channels of peripheral neuronstoo. Indeed, Sittl et al. [72] utilized the same kinetic scheme to simulate the


Figure 2.3: Example time evolution of state occupancies of Khaliq et al. [50] NaV1.6model during an action potential.


effects of a drug on the resurgent NaV1.6 current of A-fibers, and data wereused to modify some parameter values. In Methods and later chapters, Iwill discuss these changes and their consequences in more detail.

Chapter 3

Objectives

Before entering into the work of this thesis, I believe it is useful to highlightat this point our main hypothesis, based on the aforementioned previousfindings in the literature, and raise the questions that this thesis aim toanswer. Therefore, as we saw C-nociceptors physiologically exhibit conduc-tion failures even at low frequencies of repetitive stimulation. Neverthe-less, chronic pain cases are often closely linked to high-frequency discharges,spontaneous activity and hyperexcitability of C-nociceptors. Furthermore,in spite of being anatomically similar, non-nociceptive CT-fibers have thecapacity to fire at higher frequencies than all other C-fiber groups. Since theunderlying ion channel subtypes can be specifically manipulated by phar-macological opening or blocking, it is thus crucial to first elucidate the roleof each ion channel on excitability and conduction of nociceptors.

As a consequence, in this work our main goal is to utilize our previouslyvalidated C-nociceptor model, turn it into a C-fiber with high-frequency-following capabilities (such as the CT-fibers), and identify the contributionsof the main sodium and potassium currents on this ability. As I introducedbefore, the fast-repriming NaV1.6 channel with its resurgent component hasbeen repeatedly reported as vital for fast-spiking neurons in the CNS, whilein periphery, non-nociceptive mechanosensitive fibers are known to abun-dantly express it. Furthermore, NaV1.8 is believed to be expressed mainlyin nociceptive C-fibers. Consequently, our initial hypothesis is that the mainsodium current of high-frequency-following CT-fibers (and perhaps any un-usually fast firing C-fiber) is NaV1.6 instead of NaV1.8.

Subsequently, the main objectives of the newly modified model can besummarized as follows:

� Does (and if yes how) NaV1.6 and its resurgent current contribute tofaithful conduction and maximal following frequencies?

� Is this behavior robust when switching to a constant input protocol,in agreement with previous works?

29

CHAPTER 3. OBJECTIVES 30

� What is the optimal channel configuration for high following frequen-cies, and alternatively, which ion channel mechanisms increase con-duction failures and limit frequencies?

� How do the main potassium currents, especially the fast-inactivatingKA and the non-inactivating Kdr, interact with the main sodium cur-rents and affect following frequencies?

� Finally, are the effects of such configurations similar under a constantprotocol, and if not why?

By answering these questions, we will hopefully be able to gain crucial in-sights not only for the possible ionic profile of non-nociceptive CT-fibers,but most importantly for the physiologically low-frequency-firing nocicep-tive C-fibers and their hyperexcitable pathological states.

Chapter 4

Methods

As we proposed in the previous chapters, the C-nociceptor model of Tiger-holm et al. [16] and the NaV1.6 resurgent current model of Khaliq et al. [50]can be combined in order to study the limitations of a peripheral C-axon tofaithfully propagate high frequency stimuli (namely as by a CT-fiber). Asour goal is to push these upper limits, the slow NaV1.8 current is completelysubstituted here by NaV1.6. In this chapter I thus elaborate on the modi-fications to the NaV1.6 model, giving rise to the studied model variants ofthis thesis. Subsequently, I describe the simulated experimental protocolsand the electrophysiological properties that aid the analysis of the generatedresults.

4.1 Model modifications

In section 2.4 we saw that the kinetic scheme of Khaliq et al. [50] wasdeveloped for a spatially invariant model of Purkinje neurons at room tem-perature. Consequently, its incorporation into the C-fiber model requiredadjustments for (1) cellular differences in the membrane properties of Purk-inje and peripheral afferent neurons, and for (2) the temperature differencesbetween the two models.

4.1.1 Temperature scaling

The first discrepancy to tackle regards temperatures of the two models.More specifically, (as mentioned in section 2.3) the C-fiber model consid-ers the temperature differences along the axon (skin to body temperature).However, the kinetics of the Khaliq et al. [50] NaV1.6 model has been es-timated at room temperature (22°C). Additionally, it has been reportedthat inactivation has a higher temperature coefficient (Q10) than activationof sodium current in the nodes of Ranvier of myelinated peripheral fibers[72, 73], where NaV1.6 is the most prominent sodium channel [74]. I thus up-scaled all inactivation transition rate parameters (Ooff and Coff, see scheme

31

CHAPTER 4. METHODS 32

2.2) with Q10 = 3, and the rest with Q10 = 2.5.

4.1.2 Resting membrane potential differences between themodels

Subsequently, I accounted for the differences between the resting membranepotential of the Purkinje and C-fiber model. In particular, Khaliq et al. [50]reported the resting potentials for wild type Purkinje neurons to be -62.5mV. In addition, they reported the resting membrane potential of wild typecells that did not have spontaneous firing activity, which rested at -75 mV.Consequently, in their model the resting NaV1.6 current (at -62.5 mV) wasalready large enough to generate spontaneous APs. However, as I mentionedearlier the C-fiber model has a resting potential at -55 mV [16], meaningthat direct incorporation of the (unmodified) NaV1.6 model would result ina spontaneously active fiber.

Although spontaneous activity, multiple spiking and high discharge fre-quencies from C-fibers have been associated with neuropathic pain [11–13],all types of responses (i.e. single or multiple (discharge) APs, spontaneouslyactive or not) have been found from intracellular DRG measurements [19]and microneurographic data [12, 13]. In addition, several factors not in-cluded in our model may elicit discharging phenomena (for example it doesnot contain a second branch axon to model unidirectional block of con-duction [13]). Since we preferred to solely study the contributions of ionicmechanisms, and following the paradigm of single-fiber recordings that onlyobserved single spiking C-fibers [25], we chose to retain the resting potentialof the modified models at -55 mV without discharge responses to single stim-uli. Furthermore, voltage-dependence discrepancies between NaV1.6 kineticsextracted from Purkinje and C cells may be observed both because of differ-ent experimental pipette solutions (thus different liquid junction potentials)and because of cell-specific ion channel modifications (e.g. phosphorylation)that have an effect on voltage-dependence. I therefore introduced an extraparameter, Vshift, in the exponent of all voltage-dependent rate parameterfunctions as follows:

αi = H exp(Vm + Vshift

F) (4.1)

Note that an adequately large negative value for Vshift effectively re-duces the depolarizing NaV1.6 current at -55 mV. Furthermore, the differ-ence between the resting potential of the non-spontaneously active Purkinjeneurons and the C-fiber was 20 mV. Consequently, I manually set Vshift =-18 mV, which removed both multiple spiking (evoked by single stimuli) andspontaneous firing.


4.1.3 Setting maximal conductance densities

The inability to perform intracellular measurements directly from small di-ameter C-fibers and the initial goal of this project (to test the hypothesisthat NaV1.6 contribute to high frequency firing C-fibers) render the maxi-mal conductance density of Nav1.6 (gNaV 1.6) a free parameter. As NaV1.6and NaV1.8 is the most prominent sodium current in non-nociceptive andnociceptive fibers respectively, I therefore chose to manually set gNaV 1.6 sothat a typical (first) action potential of the proximal end of the parent axongenerates the same peak current density amplitude as the respective NaV1.8peak of the original model (which is substituted by Nav1.6 here).

Moreover, since we also aim to study the contributions of other sodiumand potassium currents, changes in the original g values may be considered.Biologically, these changes may correspond to ion channel expression differ-ences or pharmacological channel blocking. As I will describe in section 4.2,to carefully analyze the ion channel profile of C-fiber excitability, differentialeffects of NaV1.7, Kdr, KA and KM will be analyzed by systematically chang-ing their respective g values (recall that these changes represent pharmaco-logical ion channel blocking or opening). However, as seen in [16, 61], theoverall contribution of KA was negligible during action potentials. There-fore, I increased gKA

ten times its original value, which was enough to giveabout the same peak current density (during a single AP) between gKA

andgKdr

of the model variants (Standard) NaV1.6-KA and NaV1.6-Kdr respec-tively (see subsection 4.2.3 for details about the variants). This increasesheds light on any possible interactions (that were latent in the originalC-nociceptor model) between KA and other potassium currents.

4.1.4 Model variants

The aforementioned modified parameters, along with the rest of the originalparameter values, composed the standard model settings used in the presentwork. However, this Standard model was not the only parameterizationtested, as three more variants were developed to answer the questions ofthis thesis. Figure 4.1 schematically summarizes all model variants andtheir corresponding modifications.

In detail, the first variant comes from Sittl et al. [72], who used thesame NaV1.6 model of Khaliq et al. [50] to test the effect of an anticancerdrug (oxaliplatin) on the resurgent NaV1.6. Particularly in relation to mywork, the model was adjusted to fit current recordings from large diameterDRGs. In light of these data, the rate constants ε and Ooff (of kinetic scheme2.2) were modified from 1.75 to 0.02 (ms-1) and from 0.005 to 0.001 (ms-1)respectively. As both A-fibers and C-fibers constitute different types ofafferent nociceptive fibers, these empirical changes might be more suitablefor the new model than the original derived parameters from recordings


originalC-nociceptor

CT-fiber

Standard= NaV1.6-Kdr-KA

= late resurgent

Updated≈ Early-resurgent

Early-resurgent

No-resurgent

NaV1.7-Kdr

NaV1.6-Kdr

NaV1.6-KA

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Figure 4.1: Summarizing diagram of all model variants and their correspondingmodifications. Symbol ’=’ precedes alternative names of the same model variant,while ’≈’ shows that the two models are similar. See text for details.


on Purkinje neurons. Nevertheless, in the present work I chose to inspectthe properties of both the Standard and this Updated variant, since ourinitial goal was to highlight which mechanisms contribute and/or limit thefollowing frequencies of these axons.

The other two model variants were devised in order to dissect the effectof the resurgent component from the total NaV1.6 current. Following amodification made by Khaliq et al. [50], we can hypothetically remove theblocking particle, or in other words prevent the transition O → OB, byreducing the associated rate constant ε to 10-12 ms-1. As I will present inchapter 5, this modification removes the late resurgent current and insteadcreates an apparently early resurgent component, thus creating the Early-resurgent model.

To completely remove the resurgent component and acquire the No-resurgent model, a further adjustment was required, which is again basedon a reasoning of Khaliq et al. [50]. More specifically, I increased the valueof Oon (creating faster open channel inactivation) to 1.8 ms-1, a value man-ually fitted to the current density curve of the Standard model (minus theresurgent component of course, see figure 5.5). All model variants and theirparameter value differences are summarized in table 4.1.

Parameters Upd Std Early rsg No rsg

ε (of NaV1.6) 0.02 ms-1 1.75 ms-1 10-12 ms-1

Ooff (of NaV1.6) 0.001 ms-1 0.005 ms-1

Oon (of NaV1.6) 0.75 ms-1 1.8 ms-1

Vshift (of NaV1.6) -18 mV

Q10 (of NaV1.6) 2.5 (for activation), 3 (for inactivation)

gNaV 1.8 0

gNaV 1.6 50 mS cm-2

gKA127.555 mS cm-2

Table 4.1: Modified parameters for the fiber model variants studied. Parameters notlisted retained the same values as the original C-nociceptor model of Tigerholm et al.[16]. Upd = Updated model, Std = Standard model, Early rsg = Early instead oflate resurgent NaV1.6 component, No rsg = Resurgent current completely removed.See text for detailed explanation of the models.

4.2 Simulation protocols

The present thesis required the implementation of several different simula-tion cases. Therefore, in this section I describe the corresponding stimulationprotocols, divided into three categories: (dynamic) voltage clamp, currentinjection, and differential ion channel density protocols. The input andstructure of all protocols was given in MATLAB. Subsequently, all settings


were passed to NEURON software, in which the model variants were im-plemented. NEURON performed all simulations using its available variabletime-step ODE integration method [75].

4.2.1 Voltage clamp

As I mentioned in the previous section, the effects of four different NaV1.6model variants in the fiber were studied. To get a better understandingof their distinct kinetics, I performed voltage clamp analyses in a single-segment version of the axon model. Since the presented results focus onproperties seen at the central end of the parent axon (see next section), asegment of the parent axon (but obviously without conducting properties)with only NaV1.6 channels present was used here. This setting essentiallyrepresents whole-cell patch clamp recordings in which all channels (apartfrom the channels of interest) are either pharmacologically blocked or ge-netically removed.

First, following the classical paradigm of Hodgkin and Huxley [37], stepvoltage clamp protocols were built. A voltage depolarization protocol wasutilized (method similar to experiments in [20]). Starting from the restingmembrane (-55 mV) with single depolarizing steps to -37, -17, 3 and 23 mVrespectively (see figure 5.6a), I could compare the inactivation propertiesbetween the standard and the updated model. Moreover, as seen in figure5.6b a voltage repolarization protocol was implemented, with steps from 48mV to -67, -47, -27, -7 and 13 mV. This protocol, similar to the procedure ofRaman and Bean [51], illustrated the different resurgent current propertiesof the standard and updated models.

To verify the impact of each NaV1.6 variant during an AP, I utilized adynamic clamp strategy. In detail, I extracted the membrane potential timeseries of the standard model’s first AP, and apply it to the single-segmentmodel each time with a particular NaV1.6 variant (figure 5.5). The nor-malized current densities during the dynamic clamp were compared. Sincethe potential and currents are closely intertwined, imposing the same po-tential curve to all currents allowed direct comparison of the current densitycurves, whereas normalization made comparison of their AP kinetics easier.The same procedure was utilized for the comparison of Kdr, KA, and KV3.4(figure 5.10).

4.2.2 Current injection protocols

The main body of my work consists of analyzing the capabilities of fiber torespond to high frequency trains of electrical stimulation pulses and success-fully propagate the evoked APs. Following the experimental paradigms usedin studies of this context [16, 22, 25, 30, 76], I implemented a 40-pulse cur-rent injection experiment (called the pulse protocol from now on), repeated


stimulationprotocol

constantduration = 150 ms

pulseduration = 0.5 ms

40x at 100, 200, 300 Hz

strongIinj = 0.05 nA

weakIinj = 0.03 nA

strongIinj = 0.2 nA

weakIinj = 0.1 nA

Figure 4.2: Summarizing diagram of all current injection stimulation protocols andtheir corresponding settings. See text for details.

at three different input frequencies: 100, 200 and 300 Hz. An importantreminder here is that although direct intracellular injection is modeled, thisis not the case in any of the experimental stimulation protocols that targetthe extremely thin C-fibers, which use electrodes extracellularly (intracellu-lar recordings are only possible in DRG somata).

Each current pulse was injected at the distal end of the branch axon, withthe first stimulus given at time t = 10 ms. The 10 ms period was enough toallow the neuron to reach its resting state (and check whether the restingstate has been retained as intended, recall subsection 4.1.2). To controlfor effects created by a specific current injection strength, I performed allsimulations twice: once with weak (Iinj = 0.1 nA), and once with strongrectangle-waved pulses (Iinj = 0.2 nA). Note that the duration of each pulse(0.5 ms) was selected so that the weak pulse would be long enough to initiatean AP in all model variants.

Although such pulse stimulation protocols can provide in vivo data, theyare far from resembling the duration of natural stimuli. Therefore, a longerconstant protocol was implemented, both to check how the inferred rela-tionships between the studied ionic mechanisms and following frequencyrates would hold under a simple injected constant current, and to enablediscussion of our results with previous in vitro experimental findings. Simi-larly, both a weak (0.03 nA) and a strong stimulation (0.05 nA) was used.Note that stronger input would result in a single AP response (no repeti-tive discharge, and thus no cycles to study), whereas weaker inputs wouldnot be able to invoke any action potential. Each stimulus was introducedfor 150 ms, a duration close to that of the whole pulse protocol at 300 Hz,making comparisons easier. Alternatively, I could have selected a longerduration based on the total injected charge, that would have not changedthe resulting picture nonetheless (figure 5.14). All stimulations protocolsare schematically presented in figure 4.2.


4.2.3 Differential studies

The aforementioned current injection protocols were performed multipletimes using a single parameter-varying procedure, which helped to eluci-date the contributions of each channel type. More specifically, I varied eachone of NaV1.6, NaV1.7, Kdr, KA, and KM with 0.5, 0.75, 1.25, and 1.5 timesthe respective maximal conductance density parameter (see table 4.1) forboth the standard and the updated model (thus one change per test).

Furthermore, since the most prominent action potential currents is NaV1.6and Kdr, while Kdr and KA oppose each other (see chapter 5), I conductedfollow up simulations (of the same stimulation protocols), in which the be-haviour of the model is tested under different pairwise (sodium-potassiumchannels pair) combinations of main currents during APs. Four differentpairs were tested in particular: NaV1.6-Kdr, NaV1.6-KA, NaV1.7-Kdr, andNaV1.7-KA (see figure 4.1). When Kdr was considered, gKA

was decreased10 times, and vice versa. Additionally, in the case of NaV1.7, gNaV 1.6 wasdecreased with the same factor. These changes ensured that only the re-spective pairs influenced the fiber’s behavior.

In addition to the previous pairwise inspection, I tested the standard(namely with late resurgent current), early resurgent, and no resurgentNaV1.6 variants. This analysis assisted in both verifying the differences be-tween the resurgent components, and in elucidating the interplay betweenthem and the main potassium currents. All combinations of model variantsalong with conduction properties are collectively summarized in table 5.2.In the following section, these properties are explained in more detail.

4.3 Studied properties

Multiple conduction and action potential properties are employed in thefollowing analysis of results. All time series data (membrane potentials,current densities, ion channel states, ion concentrations) generated by NEU-RON were saved and subsequently analyzed in MATLAB. In all stimulationprotocols, data were measured at three spatial locations: at the injectionpoint (distal end of the branch axon), as well as the proximal ends of thebranch and parent axon (see figure 2.1 for the positions). Looking at boththe branch and the parent axon I could control for (if any) filtering effectsof the cone-shaped section of the axon.

4.3.1 Conductive properties

The main focus of the differential studies were on contributions of each cur-rent type to four different properties: (1) number of successfully (evokedand) propagated APs, (2) Inter-spike interval (ISI), (3) AP conduction la-tency, and (4) Output frequency. As conduction failure was determined by


Figure 4.3: Action potential properties used in the present study. 1 = Peak mem-brane potential [mV], 2 = Afterhyperpolarization depth [mV], 3 = AP duration atbase [ms], 4 = Afterhyperpolarization duration at 80% recovery [ms]. See text fordetailed descriptions.

the failure of a pulse to evoke an AP at the end of the parent axon and notearlier, all properties in this report were calculated from there.

As a result, the first property refers to the number of generated spikesat the parent, similarly to recent experimental analysis by Obreja et al. [30].ISIs concern the time interval between either the first and second AP (firstISI, ) or the third and fourth AP (third ISI, depicted only in figure 5.9).While the first ISI is easier to predict, as we see in the results the thirdISI often shows different relationships due to transient activity-dependenteffects. The first conduction latency now is defined as the time requiredfor the first pulse to travel from the point of injection to the proximal endof the parent. The fourth conduction latency is calculated accordingly andshown in figure 5.9. Closely related to conduction latency is conductionvelocity, which is simply calculated by dividing the length of the modeledfiber with the latency. Finally, the output frequency (presented in table5.2) is quantified as the number of generated spikes minus the first spike(number of repeated cycles) divided by the time interval between the firstand last spike. Spike time in all cases is considered the moment the segmentis depolarized above -10 mV.

4.3.2 AP waveform properties

Moreover, many of the phenomena in the next chapter are discussed in termsof changes in the AP waveform. Figure 4.3 illustrates the four AP proper-


ties that are mentioned here: (1) The peak membrane potential is simplythe maximum Vm during the AP, and it is indicative of the amplitude ofthe depolarization. (2) An afterhyperpolarization (AHP) typically followsthe overshoot and repolarization phase. The second property depicts theAHP depth, indicating how weak or strong are the sodium or potassiumcurrents respectively during the falling phase (repolarization). (3) The APduration at base (at resting potential), and (4) the AHP duration at 80%recovery (80% of the difference between the resting potential and the min-imum attained Vm) indicate how fast the fiber can fire and recover fromrefractoriness. All presented AP properties are related to the first evokedAP at the end of the parent axon.

Chapter 5

Results

Now that I have introduced the modeling framework of a CT-type, modifiedC-fiber and the necessary tools for analysis, the time has come to unfold themain findings of this thesis. I first identify the main dissimilarities betweenthe aforementioned (see chapter Methods) fiber variants. The comparisonis initially made in terms of both action potential properties and followingfrequencies in the (40-)pulse protocol (again see Methods). Moreover, themajor determinants among the modeled ionic currents are identified. Thesubsequent analysis is divided into sodium and potassium current mecha-nisms. Section 5.2 covers the contributions of NaV1.6 current (distinguishingresurgent and main transient components) and NaV1.7 current, whereas insection 5.3 I analyze the control by the main potassium currents (KA, Kdr

and KM). Finally, I investigate whether and how qualitative relationshipsvary between the pulse and the constant injection protocol.

5.1 Differences in action potential waveform andin following the pulse-protocol

In this section I begin with a preliminary analysis of the results generatedby the injected train of 40 pulses (pulse protocol) at three different highfrequencies. The goal of this work is to connect the ionic properties of thefiber model with its conductive features (mainly at a high frequency regime).Therefore, I initially characterize each variant based on their first AP (andAHP) attributes. Subsequently, connections between AP attributes and themain contributing currents are drawn. In the last subsection, I describethe relationship of both AP attributes and current density waveforms withthe ability of each variant to conduct AP-generating pulse trains at highfrequencies.

41

CHAPTER 5. RESULTS 42

5.1.1 Short APs and shallow AHPs characterize the Stan-dard fiber

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Figure 5.1: Comparison of AP generated by the Original C-fiber and all presentlystudied models. Each AP was evoked by a single weak pulse (as in the weak pulse-protocol, see chapter 4), and measured at the proximal end of the parent axon. A2s-long time window is shown in all cases. See table 5.1 for the quantified APproperties.

AP properties Org Std Upd Early rsg No rsg

AP duration [ms] 4.3 0.51 0.53 0.54 0.48

80% AHP duration [ms] 6.51 7.25 6.8 6.79 6.93

AHP depth [mV] 1.65 14.64 16.74 16.76 16.65

AP peak [mV] 32.07 28.77 32.66 33.23 28.19

Table 5.1: AP properties of figure 5.1 the the original C-fiber (Org), the Standard(Std), the Updated (Upd), the Early-resurgent (Early rsg), and the No-resurgent(No rsg) model. See text and figure 4.3 for detailed explanation of the properties.

As described in section 4.1.4, five different fiber variants are tested:the (1) Standard, (2) Updated, (3) Early-resurgent, (4) No-resurgent , and(5) the original C-nociceptor model. Figure 5.1 and Table 5.1 collectivelypresents the AP of each fiber, as evoked at the proximal end of the parentaxon, by a single weak stimulation pulse injected at the other end of thebranch axon (recall figure 2.1). Figure 5.1 clearly demonstrated that theoriginal C-fiber model gives rise to significantly wider APs compared to allother modified cases (see values of corresponding properties in Table 5.1),clearly a result of the original NaV1.8 channel. Moreover, the inclusion ofNaV1.6 gives rise to a deeper and longer AHP.


Interesting, though smaller, differences exist between the Standard (whichwill be the main referential point) and the Updated fiber. The readercan clearly see that the Updated version of NaV1.6 generates a higher andbroader AP, with a distinct inflection point during the repolarization phase.Moreover, the Updated model exhibits a very deep, but slightly shorter thanthe Standard, AHP waveform.

Recalling that these two NaV1.6 models differ only in the transitions ofthe open state towards blocking and inactivation, we can already suspectthat variations of its resurgent component reflect directly on the AP wave-form. Even with a 20% reduction of NaV1.6 density (that brings the APpeak to 28.70 mV, at the same level as the Standard parameters), the otherproperties remain far from the Standard values (AHP depth 16.25 mV, APduration 0.57 ms, AHP duration 6.90 ms), suggesting that the difference inthe resurgent component is responsible for this waveform.

Accumulating evidence is also gathered by observing the Early-resurgentand No-resurgent cases. Interestingly, the Early-resurgent variant exhibitsa waveform that closely resembles the Updated case, and this is an obser-vation I follow again later. Furthermore, although the No-resurgent casehas reduced AP peak and duration compared to the Early-resurgent model(remember that NaV1.6 inactivation is faster in the no resurgent case, thusopen-channel times are reduced), with values closer to the Standard ones,the AHP properties are not changed considerably.

Therefore, the most distinct feature of the Standard fiber is its shallowand long AHP (always relative to the other modified cases). These examplesare a first illustration of how the resurgent component of NaV1.6 mightcrucially influence the AHP waveform. This topic is addressed thoroughlyin the upcoming sections.

5.1.2 NaV1.6 resurgent, KA and Kdr shape AP waveform

After the description of the characteristic AP properties per fiber variant,we can now turn our attention to the underlying current densities. Figure5.2 clearly illustrates that NaV1.6 (and NaV1.8 in the original model, alsosee [16]) is the main depolarizing current, with NaV1.7 having a small con-tribution. On the other hand, Kdr consists the main repolarizing agent in allcases. Although KA contributes at a smaller degree in the original C-fiber,we are interested in studying its effects and interplay with the other sodiumand potassium currents. Therefore, in the presented modified cases themaximal conductance density of KA,gKA

, has a 10-fold increase, resultingin close amplitudes between KA and Kdr current densities.

Inspecting the different cases together, we can already see hints of theinteractions between the main action potential currents. In the case ofthe original C-fiber, the interaction of the slower rising (due to depolarizedvoltage-dependency) NaV1.8 with Kdr prolongs the falling phase of the lat-


ter. Conversely, Kdr is activated for a shorter period when the depolarizingcurrent stems from NaV1.6.

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Figure 5.2: Current densities of the original C-fiber and all presently studied modelsduring the AP of figure 5.1. Only the main contributing currents are shown. Starttime is different as different models have different AP conduction velocities. Axesranges are kept constant however for comparative purposes

Interestingly, there are two notable differences between the Standard andthe Updated NaV1.6 model: (1) the Updated model generates a second in-crease of the current density at the beginning of repolarization, whereas (2)the Standard model predicts only slower decline and exclusively a lagging tailcurrent. Obviously, the second current rise slows down repolarization (ap-parent as the aforementioned inflection point of Vm-time curve), increasingthe AP duration of the Updated model. Additionally, the Early-resurgentvariant seems to generate the same current density curves (with slightly re-duced amplitudes), while the No-resurgent model resembles the Standard


case but without the lagging tail current. Hopefully, the reader expects atthis point that this lagging tail current is the resurgent component, with theEarly-resurgent and Updated model exhibiting an apparently early resur-gent component, and the No-resurgent variant lacking both. In section 5.2.2I isolate NaV1.6 to better study its properties.

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Figure 5.3: After-hyperpolarization (AHP) and the main contributing currents ofthe Standard (a) and the Updated (b) model. They both correspond to the AP shownin figure 5.1. Only the main contributing currents are shown. Start time is differentas different models have different AP conduction velocities. Axes ranges are keptconstant however for comparative purposes

The sodium tail current of the Standard model does not seem to shortenthe AHP, but it definitely allows rapid AP decline. A closer inspection onthe contributing currents during the AHP (figure 5.3) also reveals that: (1)The slowly activating and inactivating KM current, the hyperpolarizationactivated h current (Ih) and the ion pump current (Ipump) regulate as ex-pected the subthreshold behavior of the fiber. (2) KA diminishes later thanKdr (which is not the case in the original C-fiber), playing a more prominentrole during the AHP. (3) The stronger KA and Kdr currents of the Updatedmodel lead to a deeper AHP, which again holds true even with a reducedNaV1.6 channel density. In sections 5.2 and 5.3, it will become clear howthese currents directly affect the AHP, and indirectly the overall ability ofthe fiber to follow pulses at high frequencies.

5.1.3 Standard model follows higher frequencies

In this part we close with an initial view on the capability of the original,Standard and Updated models to propagate high frequency spike trains,with or without spike propagation failures. More specifically, I present here


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Figure 5.4: Response to the strong 40-pulse protocol of the original (a), Standard(b), and Updated (c) model, depicting the time evolution of membrane potential(Vm) at the distal end of the branch axon (injection point), and the proximal endof both the branch and the parent axon.


the results of the strong pulse-protocol, with current injected successively at300 Hz at the beginning of the branch axon. Figure 5.4 shows how the pulsesdevelop and propagate along the three aforementioned fibers, looking at thepoint of injection, the proximal end of the branch axon, and the proximalend of the parent axon.

As expected, the original C-fiber model is not able to follow such a highfrequency for long, with only 13 action potentials being evoked out of the 40pulses. Due to its very wide AP and very slow repolarization, many pulsesfail to initiate a new AP at the point of injection (top panel in figure 5.4a).Even when they do manage to initiate an AP, repolarization and the conse-quent repriming (deinactivating) of the main sodium and potassium channels(NaV1.8 and Kdr) is incomplete. Additionally, the sodium/potassium con-centration gradients are slowly depleted with the accumulation of sodiumand potassium ions outside and inside the membrane (not shown). Bothof these factors gradually weaken the sodium and potassium currents, andthey result in a gradual depolarization and loss of AHPs at the proximalend of the axon (bottom panel in figure 5.4a).

As our interest turns to the effects of NaV1.6 in the fiber, we can observethat the Standard model performs much better by managing to initiate 31out of 40 APs, while the Updated model only evoked 11 APs (figure 5.4b-c).Although both models have equally short (shorter than the original C-fiber)propagation latencies and gradually collect similar amounts of depolariza-tion, the Updated model seems to exhaust its available ion channels morequickly (thus faster accumulation of depolarization and failure of furtherspike propagation). Interestingly, looking at the respective current densities(not shown), the rapid second rise of the Updated NaV1.6 current (calledearly-resurgent here) is gradually lost, although the AP duration still re-mains larger than the Standard case. On the other hand, the resurgentcurrent at the beginning of the AHP of the Standard model seems unaf-fected, thereby dampening the respective AHP for longer periods. Finally,it is interesting to note the dynamic output frequencies of the two cases. Atfirst the ISIs are apparently larger than the externally imposed ones, theygradually reduce to reach the external input rate, and in the Standard casethey again generate increasing ISIs due to conduction failures (failure ofsome of the late pulses to evoke APs). In the following sections, differentialstudies using the same protocol allows us to study and verify these responsesin more detail.

5.2 Sodium and resurgent current contributionsto following high frequencies

Previously, the comparison between the responses of modified fibers, eachone using a different NaV1.6 model, hinted that the (late) resurgent com-


ponent of the Standard model enabled the most faithful AP conduction atan extreme frequency (300 Hz). As modeling allows the dissection of theresurgent from the transient sodium components, in this section I focus ona careful study of the underlying mechanism behind the beneficial contri-bution of the former, and on verifying the general contribution of sodiumcurrents.

5.2.1 Standard and Updated models have different resurgentNaV1.6 properties

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Figure 5.5: Comparison between the normalized current densities of the differentStandard (std) and Updated (upd) NaV1.6 models. Dynamic clamp with the AP ofthe Standard model (figure 5.1) was used. The models without temperature scaling(w/o Q10) are also depicted. The Updated case yields almost the same current asthe Early-resurgent model. Notice that the No-resurgent case follows the course ofthe Standard case, but without the late resurgent component.

In this part I turn our attention to the differences between the afore-mentioned NaV1.6 models during an AP. To get a better understanding, Iperformed voltage and dynamic clamp analysis, isolating in a single axonalsegment (thus removing spatial properties) the NaV1.6 variants.

Using the AP time series of the Standard model (figure 5.1), the dynamicclamp (figure 5.5) verifies: (1) the longer lasting tail (resurgent) currentof the Standard model, and (2) the distinctly large duration (width) ofthe Updated NaV1.6 current due to the aforementioned second upstroke.Furthermore, to control for effects of the temperature scaling (recall section5.3.1) on the resurgent properties, I also present the kinetics of the Standardand Updated model at 22°C. Although in both cases the lower temperatureresults in much slower kinetics with considerably longer lasting currents, the


distinct resurgent (tail) current of the Standard case remains unchanged.The consistency of these observations show that (1) the temperature scalingdoes not affect the qualitative properties of the resurgent current (other thanoverall faster kinetics), and (2) the Updated parameters of Sittl et al. [72]changes the properties of the resurgent component.

A further verification comes from the hypothetical removal of the block-ing particle (i.e. Early-resurgent variant), which produces almost the samewaveform as the Updated model. This shows that the Updated ε value ofSittl et al. [72] (0.02 ms-1) is already low enough to practically prevent thetransition O → OB. Furthermore, the small dissimilarity during the fallingphase between the two cases is easily explained if we also recall the reduc-tion of Ooff to 0.001 ms-1 of Sittl et al. [72]. The higher Ooff value (0.005ms-1) of the Early-resurgent case allows more I6 → O transitions, and thusthe corresponding current in figure 5.5 is slightly stronger than the Updatedcase (recall figure 5.2).

To further verify the differences of inactivation and resurgence betweenthe Standard and Updated models, I proceed with the depolarizing andhyperpolarizing voltage clamp protocols of figure 5.6. The depolarizing pro-tocol clearly illustrates the slower inactivation rate of the Updated model.Furthermore, only the Standard model exhibits a resurgent tail current thatkicks in at voltages just below 0 (e.g. green line). The same differencebetween the models appear in the repolarizing steps. Unlike the Standardmodel that briefly exhibits large recovery from inactivation generating astrong resurgent current, the Updated variant has lost this property. Thebehavior of the Standard resurgent component has been retained from theoriginal (unmodified) NaV1.6 of Purkinje neurons [51], in which the resur-gent current is the strongest at specific small depolarization steps below 0mV, and it also declines with time because equilibrium favors I states overOB.

Referring back to the first AP of the stimulation protocols (figure 5.1), Ican now analyze the underlying transient changes of the respective NaV1.6state occupancies (figure 5.7). It is evident here that the Updated AP givesrise to limited OB occupancy, closely resembling the non-existent OB occu-pancy of the Early-resurgent and No-resurgent cases. Instead, the bulk ofthe occupancies either pass to inactivated states (resulting in bigger sum of Istate occupancies), or remain deactivated for longer time (resulting in largerminimum C occupancy). Crucially for the aforementioned “early resurgent”effect (second upstroke) of the updated model, the associated inflection pointcoincides with a fast increase of the deactivated occupancies, which is notprominent in the Standard model.

Therefore, the main cause of the second upstroke in the Updated NaV1.6variant is not a stronger resurgent component, but rather the apparentslower decline. As the considerable reduction of ε prevents the rapid inacti-vation by the blocking particle, NaV1.6 channels remain for longer periods


(a) Depolarizing protocol (b) Repolarizing protocol

Figure 5.6: Voltage-clamp protocols and the corresponding current densities fordetermining inactivation and resurgence of the Standard and the Updated NaV1.6model. (a) Depolarizing steps from -55 mV to -37 (blue), -17 (green), 3 (black)and 23 (red) mV. (b) Repolarizing steps from 48 mV to -67 (brown), -47 (blue),-27 (green), -7 (black) and 13 (red) mV. The arrows indicate both the existence oflate resurgence only in the Standard model, and the slower decline of the Updatedcurrent, responsible for the apparent early resurgence.


in their C (deactivated) and O (open) states, and thus permit more currentduring the AP. In other words, the “early resurgent” component of the Up-dated current stems from C5 → O transitions, unlike the “late resurgent”component of the Standard model that originates from OB→ O transitions.

61 61.5 62 62.5 630

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time [ms]

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te o

ccupancy

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OB

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(c) Early-resurgent model current densities

61.5 62 62.5 630

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1

time [ms]

Sta

te o

ccu

pa

ncy

C1−C5

O

OB

I1−I6

(d) No-resurgent model current densities

Figure 5.7: State occupancies of NaV1.6 during the AP of figure 5.1 for the Standard(a), Updated (b), Early-resurgent (c) and No-resurgent (d) model. Notice that theUpdated model exhibits almost the same time evolution as the Early-resurgent case.

5.2.2 Late but not early resurgent current contributes to fol-lowing frequencies by dampening AHPs

I previously found that the Standard fiber is able to follow high frequencypulse-protocols much better than the other variants. Moreover, in the pre-vious section I explained the distinct resurgent current of each NaV1.6 case:


the late resurgent component of the Standard model, and the early one ofthe Updated and Early-resurgent models.

This NaV1.6 property explains the resistance of the Updated fiber tofollow high frequencies. As much of the NaV1.6 population is activated for alonger period after each AP, more NaV1.6 channels become inactivated andtherefore unable to rapidly open again. Furthermore, the resulting widerAP activates more potassium channels, in turn resulting in deeper AHP.Eventually, the deep AHPs in combination with the gradually weaker NaV1.6currents do not allow the parent axon to follow high frequency spiking.

As the Standard model lacks the rapid secondary upstroke of this earlyresurgence, another question arises: does the late resurgent current assisthigh frequency firing or the elimination of the secondary peak is solely re-sponsible for this improvement? To answer this question, I revisit the re-sults of the pulse-protocol experiments at 300 Hz, drawn from the Standard(corresponding to the late-resurgent case), the Early-resurgent, and the No-resurgent fiber. The importance of the latter variant is obvious here, asit dissects the effect of the late resurgent component from the lack of thesecondary upstroke (see Methods for the modifications).

In this section I refer to the three rows of “NaV1.6, Kdr+KA” resultsin Table 5.2, which correspond to results of the aforementioned three fibervariants. Interestingly, first ISIs consistently increase in the presence of bothlate and early resurgent components. However, this is only true for the firstISIs, as subsequent ISIs are considerably shorter in duration. For example,already from the third AP, the two models exhibit shorter ISIs than theirNo-resurgent counterpart, allowing the axon to propagate the injected pulseswith less failures. This property of the resurgent components is also robustto the main potassium current used.

Regarding the first conduction latency, it is as expected more influencedby the total amplitude of sodium currents. Therefore, the Early-resurgentfiber generates significantly more sodium current during the first AP, andthus shorter latencies. On the contrary, when the resurgent current is totallyremoved, first conduction latencies are bigger. As accumulation/removal ofsodium inactivation and conduction failures underlie changes in conductionlatency, later latencies are the smallest for the fiber that has the least failures(highest number of evoked APs).

Unarguably, the late resurgent component consistently (with both strongand weak input and regardless of the main potassium channel type) returnsthe highest number of evoked APs and Output frequencies. This resultshowcases that the late resurgent current is crucial in rising the membranepotential during the AHPs, allowing the Standard fiber to fire even when asubstantial amount of NaV1.6 inactivation has been accumulated. On theother hand, the early resurgent component does not always facilitate fastspiking, as its contribution varies depending on the repolarizing potassiumcurrent. The interplay between the NaV1.6, Kdr, and KA is analyzed in


section 5.3.4.

5.2.3 Sodium channels increase peak following frequencies

0.5 1 1.50

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[standard maximal conductance density]

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Nav1.6 density and AP count

*

(a) Differential effect on the num-ber of Evoked APs (weak input)

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[m

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Vm


40 pulses at 300 Hz

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Vm

[m

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Vm


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Vm

[m

V]

time [ms]

Vm


(b) Bursting of 1.5x NaV1.6 Standard model (*)

Figure 5.8: (a) Increased NaV1.6 density (1.5x gNaV 1.6, *) can give bursting andspontaneous firing properties to the Standard model. (b) An example (correspondingto the * case of panel a) time evolution of Vm with bursting activity following thepulse protocol.

I complete the analysis of sodium current mechanisms by providing theresults of the differential study (described in Methods) for the two mostprominent sodium currents, NaV1.6 and NaV1.7.

In regards to the role of NaV1.6 on the number of evoked APs, it isstraightforward to realize that it enhances the excitability of the fiber. Whilethe Standard model manages to propagate all pulses given at a frequencyof 100 and 200 Hz and fails only at 300 Hz, the Updated variant alreadyexhibit propagation failures at 200 Hz even with 50% increase of NaV1.6maximal conductance density gNaV 1.6 and strong input (26 APs out of 40pulses). Nevertheless, in both channel versions, increased gNaV 1.6 consis-tently promotes the initiation of more or all 40 APs. Figure 5.8a illustratesthe robustness of this increasing function to input strength for the Standardmodel at 300 Hz input frequency.

A notable phenomenon is present in the case of 1.5x gNaV 1.6. In par-ticular, the model generates additional APs, similarly to how the originalNaV1.6 model gave rise to spontaneous activity (firing APs in the absence


100 200 30040

45

50

55

60

65

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85

Input frequency [Hz]

Late

ncy [m

s]

Latency vs Input Frequency

(a) Differential effect of gNaV 1.6

on latency of fourth AP

100 200 3005

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10


ISI

[ms]

Third ISI vs Input Frequency

0.5x standard

0.5x updated

updated

standard

1.5x standard

1.5x updated

(b) Differential effect of gNaV 1.6

on third ISI (strong protocol)

100 200 3005

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10


ISI

[ms]

Third ISI vs Input Frequency

1.5x standard

0.5x standard

1.5x updated

0.5x updated

(c) Differential effect of gNaV 1.7

on third ISI (strong protocol)

Figure 5.9: Latency and ISI properties of the Standard and Updated model as afunction of the input frequency of the pulse protocol. The effects of NaV1.6 (a-b)and NaV1.7 (c) are also depicted.

of stimuli) [50]. This is depicted in figure 5.8b, showing that the APs at theend of the depicted time window are actually the result of bursting activity(firing with decreasing frequency).

Looking at the generated third ISI (figure 5.9b) and conduction latencyof the fourth AP (figure 5.9a) as a function of input frequency, we can easilysee again the reflection of the ionic underlying mechanisms. When pulsesare given at a faster rate, ISIs are forced to be shorter (given that thereis no conduction failure), recovery from inactivation is limited resulting inweaker NaV1.6 currents (when compared to the corresponding APs at lowfrequencies), and consequently larger conduction latency.

The same trends are followed when considering the changes of latencyand ISI with respect to gNaV 1.6 levels. Namely, with more NaV1.6 channelsavailable (increased gNaV 1.6), latency is naturally decreased. For the samereason, the Updated model generates smaller latencies at the beginning ofthe response, as it gives rise to wider and stronger APs. Similarly, third ISIsare a decreasing function of gNaV 1.6, indicating the capability of the fiber tofire at higher frequencies when the corresponding ISIs are reduced (and alsotheir tendency for bursting for extreme changes).

Until now I have not discussed the role of NaV1.7, as its effects arelargely overshadowed by NaV1.6. Hence, there is no clear indication of apositive or negative influence of NaV1.7 on the number of conducted APs.


Nevertheless, useful insights are obtained from its effects on latency andISI data. Consistent with its role as a depolarizing sodium current, it alsopromotes shorter latencies.

In contrast to NaV1.6, increased gNaV 1.7 has a positive effect on the thirdISI, suggesting that NaV1.7 decreases the output frequency in this regime.However, the effect is reversed when inspecting the first ISI. This discrepancyhighlights the property of NaV1.7 to increase its channel availability (andthus current strength) after the first AHP, speeding the start of the secondAP. In later sections I will address again this pattern and its importance inthe close cooperation of NaV1.7 and KA.

5.3 The control of potassium currents on followinghigh frequencies

During the analysis of the underlying sodium currents, I mentioned thatmany of their contributions to following firing frequencies can be influ-enced by potassium currents. Following the same footsteps, in this sectionI continue with the analysis of the major contributing potassium currents:Kdr, KA and KM. Having completed the current mechanisms individually,I subsequently emphasize on their crucial interactions that determine theresponse of the fibers to the pulse protocol.

5.3.1 KA affects AHP similar to KV3.4 current

In the initial analysis of each fiber’s analysis, among the fast activatingpotassium currents, KA was found to diminish later than Kdr, presumablyinfluencing more the AHP properties. Using the same dynamic clamp pro-cedure as before, I confirm the engagement of these KV channels during theStandard AP (figure 5.10). It is evident here that although both Kdr andKA activate at around the same time (close to AP peak) and rise equallyfast, KA current lasts considerably, lasting well into the AHP.

Furthermore, to test the possibility of a beneficial role from KV3.4 me-diating fast current, I also implemented in NEURON the KV3.4 model ofFineberg et al. [77], which was developed based on small-diameter DRGdata. To incorporate it into my model fibers, I also introduced temperaturescaling (similarly to section ), with a universal temperature coefficient Q10

= 2.5 (the same value used by Tigerholm et al. [16] to scale all potassiumcurrents). In figure 5.10 we can see that KV3.4 very closely resembles KA

during the Standard AP, both with fast rise and slower decline that lastsuntil the beginning of the AHP. A minor difference can be observed be-tween their resting currents. Nevertheless, tests with KV3.4 in place of KA

in all case models show that this difference does not have any impact inthe number of evoked APs, and very small changes overall. Therefore, KA


0 0.5 1 1.5 2 2.5 3

−50

0

50

Vm

[m

V]

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

time [ms]

Norm

aliz

ed c

urr

ent density

K

A

Kdr

KV3.4

Figure 5.10: Comparison between the normalized current densities of Kdr (green),KA (blue), and the newly tested KV3.4 (orange) model. Dynamic clamp with theAP of the Standard model (figure 5.1) was used. Notice the minimal differences(mostly at rest) between KA and KV3.4.

was deemed sufficient for the purposes of a fast activating and inactivatingcurrent, and KV3.4 was not further investigated.

5.3.2 The competitive nature of KA and Kdr

After clarifying the AP dynamics of Kdr and KA, we can now investigatetheir contributions to fast spiking fibers. Most importantly, their effectsappear to be closely intertwined and largely opposing. Figure 5.11a clearlyillustrates that Kdr and KA compete with each other for the role of the mainrepolarizing current: when the conductance density of one is increased, theother’s AP current density is diminished. Obviously, this effect is connectedto the limitation imposed by the intracellular and periaxonal potassiumconcentrations.

Furthermore, the number of APs evoked by 40 weak pulses at 300 Hz, isreversely influenced by the two currents (figure 5.11b, mainly in the Standardfiber). Increased gKdr

permits more APs, whereas increased gKAhas the

reverse outcome. Contrary to the weak protocol though, the correspondingstrong protocol does not show considerable variation with respect to therelative levels of gKdr

and gKA. This is explained by the distinct dynamics

of KA during the initial APs. More specifically, KA current density increasesconsiderably for the second AP, whereas Kdr does not (figure 5.12a). Asmentioned before, increased KA increases the depth of AHPs, and as aresult deeper AHPs are present at the beginning of the spiking response. Asa consequence, the strength of the input current is especially important, as


61 61.5 62 62.5

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Vm

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V]

0.5x Kdr

(1st spike)

61 61.5 62 62.50

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ent density [m

A/c

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A

Kdr

61.5 62 62.5 63 63.50

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nsity [

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/cm

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K

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Kdr

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1.5x Kdr

(1st spike)

(a) Compensatory mechanism between Kdr and KA

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Evoked A

ction P

ote

ntials

KA density and AP count

std const

upd const

std pulse

upd pulse

(b) Differential effect on the number of EvokedAPs (weak protocols)

Figure 5.11: The opposing nature of Kdr and KA during the action potentials.(a) When there is considerable conductance density for both potassium currents, achange in conductance density of one (Kdr in this example) will have the oppositeeffect on the other (KA here). (b) Opposite effects of Kdr and KA on the numberof evoked APs. Interestingly, their effects differ between the constant (const) andthe pulse protocol.

the combination of deep AHPs and weak pulses lead to AP initiation and/orconduction failures. This phenomenon manifests itself when comparing theresponses of the Standard model to the strong (figure 5.4b) and weak (figure5.8b) protocol, as only in the latter case some injected pulses fail to initiateAPs at the distal end of the branch axon, apparently creating larger initialISIs in the parent axon.

While the initial membrane potential behavior of the Standard modelvaries depending on the input strength, the subsequent spiking behavior doesnot differ. The explanation for this discrepancy comes from an illustrationof the current density evolution of KA and reduced Kdr during the pulseprotocol (figure 5.12b Standard fiber used). Due to fast inactivation of KA,the non-inactivating Kdr is left as the main repolarizing agent sooner or laterin the repetitive firing, with the timing of this switch depending on gKdr

andgKA

.Finally, the aftereffects on the initial ISI perplexes its interpretation as

a function of gKA. A clear pattern is observed for Kdr, as it universally

decreases ISIs and thus increases output frequencies. As the reader mayalready predict, KA shows the opposite trend, verifying that it prevents highoutput frequencies (figure 5.12d). However, the increasing function does nothold for the first ISI of the weak pulse protocol (figure 5.12c), illustratingthat input strength can in some occasions mask or alter the effects of certainionic currents.


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60 65 70 75 80 85

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(a) Different initial frequencies between Kdr and KA

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)

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(b) Switch to sustained Kdr due to quickKA inactivation

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(d) Differential effect on duration of first ISI (strong pro-tocols)

Figure 5.12: Initial and sustained response frequency in the pulse protocol dependson the expression levels of Kdr and KA. (a) The NaV1.6-KA model exhibits strongerKA (blue) and NaV1.7 (black) currents after the first AP, resulting in lower initialresponse frequencies. (b) With considerable gKdr

though, only Kdr determines sus-tained firing activity (in the pulse-protocol) due to KA inactivation. (c-d) InitialISI durations plotted against the relative gKdr

and gKAchanges. Notice that the

effect of KA on the first ISI is concealed in the strong pulse protocol (c).


5.3.3 KM consistently suppresses fast spiking

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(a) Differential effect on the num-ber of Evoked APs (weak input)

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Input frequency (Hz)

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updated

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updated 1.5x KM

(b) Evoked APs versus Input frequency (strong pulse-protocol)

Figure 5.13: (a) KM consistently reduces the number of successfully propagated APsof the Standard model. (b) The figures illustrate the number of evoked APs as anon-increasing function of input frequency.

The third studied potassium current, KM, shows a decisive effect on theability of the fiber to conduct highly frequent stimuli. As KM rises laterthan the fast Kdr and KA currents, it mostly affects the AHP properties.Unlike KA though, KM has a considerable persistent component (currentat the resting potential, e.g. recall KM in figure 5.3). Therefore, followingthe gradual depolarization of the membrane under the pulse protocol, KM

increases its current density (figure 5.11b) and becomes a growing limitingfactor.

Similar to KA, a reduction of gKM) results in reduced conduction latency

and smaller ISIs. These findings are consistent with the effect on the numberof propagated APs, which is consistently increased by decreased gKM

(figure5.13). Interestingly, the relationship in the Updated fiber is not as clear, withthe effect of KM being especially sensitive to the timing of the injected pulses,as illustrated by the results of the different frequencies in figure 5.13b.

5.3.4 Matched current pairs: NaV1.6 with Kdr, but NaV1.7with KA for better conduction

The previous results supported the major role of the Standard (late resur-gent) NaV1.6 and Kdr in the successful propagation of injected pulses at


high frequencies, whereas NaV1.7 and KA acted in the opposite direction.However, the previous univariate analysis could only elucidate their contri-butions individually, largely disregarding possible current interactions. Inthis part, I finally investigate the interplay between the NaV1.6 variants,NaV1.7, Kdr, and KA.

Since successful propagation of repetitive pulses requires small ISIs andlarge output frequencies, naturally the combination of main currents thatgive the highest output frequency and shortest ISIs should be the most suit-able (see table 5.2). Indeed, the pair NaV1.6+Kdr (which does not havea prominent KA current) consistently shows the best scores in these cat-egories, and the introduction of the fast-inactivating KA deteriorates thisperformance. The worse performance of NaV1.6+KA is more pronouncedfor weak pulses, which supports the findings in section 5.3.2 about the im-portance of remaining close to threshold with shallow AHPs during ISIs.

The contribution of the early resurgent component varies depending onthe repolarizing potassium current. If Kdr is present, early resurgent currentis better than nothing despite broader APs, suggesting that the beneficialeffect of having shallower AHPs allows more APs at the beginning of thetrain (especially obvious in the weak input case) and it overcomes the limita-tion of accumulating NaV1.6 channel inactivation. However, when KA is themain repolarizing agent, this advantage of early resurgence is lost because ofthe initial AHP deepening effect by KA (difference in the weak input case isthus lost) and its quick inactivation (and consequent faster sodium channelinactivation).

Contrasting how NaV1.6 cooperates with the aforementioned potassiumcurrents, NaV1.7 functions better in conjunction with KA. By inspectingthe first ISI in this case, we can see that its duration is unusually short. Asseen before, although this might seem contradictory to the largely reducedfollowing frequencies, this is explained by the removal of the initial amount ofinactivation that is present for both NaV1.7 and KA at the resting potential.As more NaV1.7 channels become available after the first spike, the secondAP rises faster than before. However, this effect is short-lived since bothchannels quickly inactivate, and subsequent ISIs become broader.

5.4 Discrepancies between pulse and constant pro-tocol

Until now the analysis has revolved around the behavior of the fiber modelsunder the stimulation of sequential short pulses, given at high frequencies.Nevertheless, multiple (mainly in vitro) studies in the literature has beenperformed using a long, single constant current injection. Therefore, in thiscase I turn our attention on the ion current dynamics under the constantprotocol (see Methods for details) and how their effects are compared to the


pulse protocol.

5.4.1 Different long term properties modify the effect of stim-ulus strength on output frequency

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V]

time [ms]

Vm


(b) Weak input constant protocol

Figure 5.14: Response to the strong (a) and weak (b) constant stimulation protocolof the Standard model.

Before highlighting the discrepancies between the current mechanisms,let us first have an overview of the general patterns of the two protocolsand input strengths. Whilst in the pulse protocol the weak input leadsto overall higher conduction failure rates, in the constant protocol weakinput leads to discharges with more APs (as long as the current is strongenough to depolarize the axon above spiking threshold, example in figure5.14). Nevertheless, the rest of the quantified features behave similarly inboth protocols, with weaker input consistently leading to lowered outputfrequencies, as well as higher first ISI and latency (table 5.2). Therefore, acritical property of bursting in the constant protocol is that it requires lowerspiking frequencies (or bigger ISIs) which correspond to slower accumulationof inactivation. On the contrary, as expected an axon that generates loweroutput frequencies is not capable of faithfully conducting pulses at highfrequencies.

5.4.2 Similar NaV1.7, NaV1.6 resurgent, and KM currentcontributions

Reinspecting the differential effects of NaV1.6 current , but this time withrespect to the constant protocol, it is obvious that NaV1.6 increases ex-citability regardless of protocols and model variants (figure 5.8). Moreover,the same relationships between the two protocols are also observed in all


cases for conduction latency and first ISI (obviously later ISIs greatly di-verge from their counterparts in the pulse protocol, thus they cannot becompared). In regards to the resurgent current, although it clearly boostsbursting, the advantage of the late over the early component is less clear(table 5.2).

The same patterns are also retained in regards to the effects of NaV1.7and KM. The number of evoked APs is again not influenced by changesof gNaV 1.7 (remember that NaV1.6 is the dominant depolarizing current inthe Standard and Updated fibers), while the same monotonically decreasingfunction holds for the relationship between first ISI and gNaV 1.7. Finally,the reader can easily observe the similar suppressing effect of KM in boththe pulse and the constant protocol (figure 5.13).

5.4.3 Contrary to the pulse-protocol, KA promotes dischargesin the constant protocol

Although the roles of NaV1.6, NaV1.7, and KM remain the same in theconstant protocol, a clear distinction in their role to create bursting orspontaneous-like activity lies ahead for Kdr and KA. In particular, whileKdr clearly reduces the number of APs yielded by the constant stimulus,KA promotes repetitive spiking (figure 5.11). This is also confirmed bylooking at the results of table 5.2 for when both KA and Kdr are present.

Consequently, shallow AHPs are beneficial for repetitive spiking in thepulsed-protocol, whereas deep AHPs are important for repetitive firing ina constant injection scenario. As KA generates deeper AHPs than Kdr, itallows the removal of more sodium and potassium channel inactivation anddeactivation. Furthermore, the deep AHPs delay the depolarization blockof the membrane at the distal end of the branch axon, resulting in similarbeneficial effect as that of the weak constant protocol.

Despite these discrepancies, the situation is not much different with theconstant protocol when considering first ISI and latency (figures 5.12). Sincewhat changes here is the necessity for a channel pair that can both sustainrepetitive spiking and prevent membrane depolarization, the NaV1.6+KA

pair is expectedly the most suitable for bursting activity (table 5.2). Lastly,following the same trend as in the pulse-protocol, NaV1.7 is able to generatemultiple spikes only when coupled with KA.


Na+

curr

ent

K+ c

urr

ent

resu

rgen

t

curr

ent

Stro

ng

inp

ut

Wea

k

Inp

ut

Stro

ng

inp

ut

Wea

k

Inp

ut

Stro

ng

inp

ut

Wea

k

Inp

ut

Stro

ng

inp

ut

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late

31

30

22

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14

9.2

110

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9.9

50

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27

26

18

51

70

9.0

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26

22

19

01

56

8.8

9.6

650

.250

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late

31

28

21

71

95

9.6

11

0.9

51

.15

1.3

ea

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26

23

18

01

57

9.5

91

149

.85

0

no

24

20

18

51

43

9.3

110

.651

.451

.6

late

29

23

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81

62

10

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1.3

50

.55

0.7

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19

16

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29

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.211

.649

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9.7

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11

48

3.8

7.9

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60

.46

0.6

10

81

09

86

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7.9

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1.3

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late

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9.0

71

0.2

50

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1.3

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9.4

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N/A

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late

47

11

91

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9.9

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0.5

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57

12

61

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10

10

.650

.451

.1

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34

11

21

08

9.6

510

.252

.152

.8

late

91

21

24

10

21

0.5

11

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1.2

51

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91

01

12

10

410

.811

.449

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.7

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58

11

81

00

10

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.552

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N/A

N/A

61

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2

33

10

79

0.3

7.6

8.0

86

2.1

62

.9

NaV1.7

Kd

r

KA

NaV1.6

Kd

r

Kd

r +

KA

KA

NaV1.7

Kd

r

KA

Co

nsta

nt

pro

toco

l

NaV1.6

Kd

r

Kd

r +

KA

KA

40-p

uls

e p

roto

co

l (3

00 H

z)

Evo

ked

AP

sO

utp

ut

Freq

uen

cy

Firs

t IS

I (m

s)La

ten

cy (

ms)

Table 5.2: A summary of values for four conduction features as predicted by differ-ent model versions that differed on the type of main sodium (NaV1.6 or NaV1.7),potassium (Kdr and/or KA), and resurgent NaV1.6 component (late, early and nei-ther). Two protocols, a 40-pulse protocol (at 300 Hz) and a constant protocol, weresimulated twice, with strong and weak input (see methods for details). Yellow blockshighlight the model with the highest value for each attribute (entire row), gray blockscorrespond to the lowest one, green font indicates the resurgent component with thehighest value, while red font the lowest.

Chapter 6

Discussion

In this chapter, I develop the findings of my simulations in relation to C-fiberexperimental results, and when these are not available, to experiments andsimulations on other high-frequency-firing neurons. First, I match proper-ties of the firing patterns of the tested model variants to patterns observedin peripheral fibers, reinforcing the plausibility of our predictions. For dis-crepancies between our model and natural C-fibers, the limitations and ne-glected factors of the modeled system are discussed. Subsequently, I proceedto the aforementioned roles of the AP- and AHP-shaping ion channels in thismodel, seeking evidence for the different resurgent properties between theStandard and Updated NaV1.6 model and proposing sodium and potassiumchannel configurations that fit the profile of specific physiological and patho-logical C-fiber.

6.1 Functional correspondence of models to natu-ral fibers

6.1.1 Firing modalities

As mentioned back in section 4.1.2, I adjusted the original NaV1.6 model ofKhaliq et al. [50] so that each brief pulse could evoke only one AP in the mod-ified C-fiber models. Multiple spiking was still observed however in the dif-ferential analysis for increased levels of gNaV 1.6 (figure 5.8). In fact, both sin-gle and multiple firing small-diameter DRG and C-fibers have been reportedbefore [19], while ongoing spontaneous discharges in C-nociceptors have beenrepeatedly implicated in painful neuropathies [12, 13, 78]. Nevertheless,as experimental studies on repetitive activation, following frequencies andactivity-dependent changes focus on (easier-to-control) non-spontaneouslyactive fibers [19, 21, 30, 35, 76], we also chose to develop non-spontaneouslyactive models. Obviously, this modification suggests that NaV1.6 densitymay indeed be a key regulator for spontaneously active C-fibers.

When considering the effects of prolonged depolarizing currents, Waddell

64

CHAPTER 6. DISCUSSION 65

and Lawson [19] depicted four different responses from DRG somata for areported pulse duration of 40-50 ms: single AP, tonic (continuous) spiking,and phasic firing (few spikes only). These observations are in agreementwith the predictions of our model, in particular with the phasic activity(e.g. figure 5.14) of all variants, and the single spiking of the NaV1.7-Kdr

variant. Considering my previous suggestion and the findings of Wang et al.[79] about the regulation of tonic and phasic firing activity, we could predictagain that a substantial rise of NaV1.6 density (or modulation of its voltagedependence to generate higher current densities at rest, see also section6.2.1) along with reduced main potassium current densities would give riseto continuous, tonic firing. Further analysis of sodium channel contributionscan be found in the following corresponding section.

6.1.2 Correlations between AP and conduction properties

6.6 6.8 7 7.2 7.4 7.6 7.8 80.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

0.62

80% AHP duration (ms)

AP

du

ratio

n (

ms)

(a) AP duration vs 80% AHP duration

11 12 13 14 15 16 17 186.6

6.8

7

7.2

7.4

7.6

7.8

8

80%

AH

P d

ura

tion (

ms)

AHP depth (mV)

(b) 80% AHP duration vs AHP depth

0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.622.4

2.42

2.44

2.46

2.48

2.5

2.52

Conduction v

elo

city (

m/s

)

AP duration (ms)

(c) Conduction Velocity vs AP duration

Figure 6.1: Significant correlations (P << 0.01) between AP and fiber propertiesas predicted by the differential study for both the standard (crosses +), and theupdated (circles o) model. No significant correlations were found when points fromboth models were grouped together. Colors indicate which type of g was varied ineach case. Magenta = no change, blue = varied gKA

, green = varied gKdr, red =

varied gKM, and black = varied gNaV 1.7.

Due to the size of peripheral unmyelinated axons, the underlying ionicmechanisms have mostly been studied indirectly, based on intrinsic patternsand externally induced changes in the AP and conduction properties [60].As a result, similar to my work experimental studies (in rats) have triedto show correlations between the AP (and AHP) waveform and conductionvelocity (CV) [19, 35, 76].

Waddell and Lawson [19] managed to show significant correlations be-tween CV and the maximum rate of depolarization and repolarization, andthe AP duration. A connection between CV and AP waveform would alsoindicate ion channels as the main regulators of CV. In an effort to repli-


cate these observations, I utilized data generated during the aforementionedg-varying differential analysis. Interestingly, whilst it has been repeatedlyshown that CV is negatively correlated with AP duration [19, 35, 76], mydata showed a positive correlation for both the Standard and the Updatedvariants (figure 6.1c). Although this result may appear as a limitation of ourmodel, a closer inspection to how the relation between CV and AP dura-tion varies with respect to a specific g (shown with different colors) explainsthis discrepancy. More specifically, when densities of potassium channelsare modified, both CV and AP duration vary accordingly. However, it isclear that changes in gNaV 1.7 (similarly for the not shown gNaV 1.6) resultin the expected negative correlation. These simulated results indicate thatthe observed differences between C-fibers (and/or A-fibers) may be mostlyunderlain by sodium current density variations and to a much lesser extentby potassium channels.

Moving to the relation between AP and AHP duration (figure 6.1a), itis in agreement with the tendency of wide APs to be followed by wide AHPs[19]. However, the relation between AHP duration and AHP depth (figure6.1b) does not appear to agree with the previously reported positive relation[19, 76]. Nevertheless, using the same reasoning as before, we can see thatvarying gKdr

results in the expected relation, highlighting Kdr as the majorfactor in AHP shape discrepancies between A- and C-fibers. Moreover, theapparent lack of direct correlation between AHP duration and CV [19] maybe explained by the aforementioned discrepancies between AP, AHP, and CVrelations, hinting that their relation is probably under the effect of multipleion channel variations.

Besides correlation analysis, these properties might also be helpful inclassifying fibers, though cells even from the same functional group exhibitgreat variability. An inflection during the falling phase of APs appear inmost (if not all) C-fibers, while the waveform of A-fibers apparently varyaccording to their mechanosensitivity thresholds: high threshold A-fibersexhibit high and long APs with inflections and low threshold A-fibers gen-erate no inflections and faster AP properties [19, 35, 76]. Based on thesedescriptions and the reported CVs, our fiber variants seem to fall in theslower limits of Aδ-fibers, significantly faster than C-nociceptors [16], andalso exceeding the fastest reported CV of CT-fibers (1.3 m s-1) [17]. In-terestingly, contrary to natural C-fibers, the fastest Standard model doesnot display an inflection. Nonetheless, the present work demonstrated thatsodium channel isoform changes alone may be capable of making C-fibersconduct as fast as the lightly myelinated Aδ-fibers.

6.1.3 Comparison of following frequencies

The main focus of this thesis was the performance of the model variants infollowing high frequency stimulations. Following Obreja et al. unpublished


experiments on porcine CT-fibers in vivo, I performed simulations at 100 and200 Hz for 40 pulses. Although stimulation at 300 Hz was not performedexperimentally, I chose to further test our model at this frequency as itspeeds the conduction failure process. The results illustrated that all testedfiber variants displayed maximum following frequencies at least between 100and 200 Hz, while the Standard fiber exhibited even higher peak followingfrequencies, with failures only seen at 300 Hz (except for when gNaV 1.6 wasreduced).

Looking at experimental data in the literature, we can see that (around)200 Hz appears to be close to the upper limits of peak following frequenciesof Aδ-fibers [19, 35], although unclassified A-fibers have been able to followfrequencies of up to 800 Hz, for 20 pulses only though [76]. Regarding C-fibers, the maximum following frequencies of nociceptive units have rarelybeen reported above 50 Hz, with especially CMi-nociceptors exhibiting fail-ures even at 5 Hz [30, 76]. Additionally, even after axotomy, C-fibers neverexceeded following frequencies of 100 Hz [76]. Furthermore, low thresh-old mechanosensitive C-fibers (usually the non-nociceptive CT-fibers) havebeen attested with the maximum observed frequencies of 100 Hz [17], eventhough unpublished results of Obreja et al. demonstrated that CT-fiberscan successfully conduct APs at 200 Hz stimulation, at least for a limitedtime. Therefore, even the slower Updated variant of my analysis apparentlyexceeded the expectations from C-fibers. Nevertheless, this does not consista limitation for the physiological interpretation of our model, because: (1)we still draw considerable information about ionic mechanisms that underliehigh-frequency-following C-fibers, and (2) lowering the peak following fre-quency of the model can be easily done with optimizing g values alone (asthis thesis exemplified multiple times).

6.1.4 Gradual depolarization

In Results we saw that repetitive activity in all model variants led to grad-ual depolarization of the membrane potential. Obviously, this long-termdepolarization also causes gradual sodium channel inactivation and eleva-tion of potassium current levels, reducing excitability of the fiber in the longrun. This phenomenon was also observed in fiber measurements, and as themodel predicted in the absence of conducted APs, Vm gradually returned tothe normal resting potential [19, 76]. Slow depolarization and the accom-panying reduction of successive spike amplitudes has also been observed inother types of axons and cells in the CNS [60, 80].

Interestingly, using a very low frequency protocol at 2 Hz, Tigerholmet al. [16] reported that the CMi-nociceptor model showed activity-depend-ent hyperpolarization, instead of depolarization. This behavior was givenin the model in accordance to similar observations in C-fibers [81]. Earlier,Amir and Devor [82] had reported a similar activity-dependent hyperpolar-


ization under the effect of low-frequency pulse trains in excised rat DRG.Naturally, as any other activity-dependent event, the explanation lies in

the pre-AP state of the membrane, as influenced by any preceding activity.Gemes et al. [76] hypothesized and confirmed that the firing rate of the fiberdecides the fate of the resting membrane potential. More specifically, theypredicted that a hyper-/de-polarized pre-spike Vm would hyper-/de-polarizethe post-spike potential. As I also observed in this work (recall figure 5.4),spike propagation (to the proximal end) for the first pulses start at the endof the AHP. Moreover, while later spikes propagate at a faster instantaneousrate, the initial deep AHP is gradually lost (due to KA inactivation). Thelater results in again depolarized Vm when the following AP starts, progres-sively leading to a depolarized state.

Mechanistically, this might be explained by the depletion of intracellu-lar K+ (as the sodium-potassium pump does not recover it quickly), whichconsequently weakens even the non-inactivating potassium currents, as pre-dicted by our model for Kdr. In addition, as Na+ channel inactivation accu-mulates, AP amplitudes become smaller and K+ currents less activated. Asa consequence, the weak Kdr currents are eventually not able to repolarizethe fiber back to the pre-pulse-train resting potential. In lower frequenciesthough, the intracellular potassium concentration is not a limiting factor,and the increased activity of the pump (due to the accumulated Na+ intra-cellularly) hyperpolarize the axons as shown in multiple cases [60].

6.1.5 Dynamic temporal patterns

A final aspect of the observed response patterns in this thesis is that of thechanges in the propagated ISI during the pulse protocol, or in other wordsthe changes in the instantaneous firing rates. Recalling the pattern observedin the Standard model (figure 5.4b), the fiber initially responded with low in-stantaneous firing rates. Subsequently, as the membrane depolarized, spikesbecame shorter and instantaneous frequencies much faster. Finally, con-duction failures started to appear, giving rise to apparently slower firingfrequencies at the end of the axon, signaling an eventual adaptation of thefiber’s response to repetitive input. This pattern is in agreement with thefindings of Weidner et al. [83] for human C-fibers, which stated that at lowrates of activation ISIs increased (therefore output frequencies decreased),whereas at high rates output frequencies progressively increased.

It is interesting to note that while the instantaneous frequency approachedbut never exceeded the input frequency at 300 Hz, at 100 Hz all variantsbriefly propagated at output frequencies above 100 Hz (ISI less than 10 ms).Indeed, Weidner et al. [83] also reported higher (than the tested input at50 Hz) maximum instantaneous frequencies. This might be suggestive oftheir discharge capacities (easily sustaining at least the corresponding inputfrequencies in our model) something that Weidner et al. [83] also recognized


for estimating the maximum attainable output frequencies.

6.1.6 Reasons for discrepancies in predicted following fre-quencies

According to the present experimental data, we could infer that our modelsexceeded the physiological capacities of C-fibers. However, inference fromthe available data is difficult because of multiple reasons. First and fore-most, each of the aforementioned experimental studies used different defini-tions for the peak following frequency: (1) the maximum frequency at whichat least 80% of pulses could propagate an AP for 200 ms [35], (2) at whichall 20 pulses could propagate an AP [76], and (3) starting from 1 Hz andgradually increasing the input frequency, the maximum frequency at whichno conduction failure will occur [19]. The first definition appears to over-estimate the following ability of C-fibers, as some conduction failures areaccepted and at lower frequencies a 200 ms interval contain fewer than our40 pulses, and this is the main reason Fang et al. [35] even reported 200 Hzfor some C-nociceptors. In contrast, using the third definition excitabilitychanges from preceding input frequencies accumulate, consequently leadingto failures at smaller frequencies. As no preceding activity is included inour pulse protocol, the reported frequencies in this thesis should be muchhigher.

Discrepancies between experiments (even from the same model organ-ism) may also arise due to different experimental setups. For example, onecan excise animal DRG with their attached fibers and study at in vitro so-lutions their responses to intracellular stimulation in DRG, or extracellularstimulation on the thin fibers. In contrast, our model was adjusted basedon extracellular recordings in human C-fibers in vivo, without any direct invivo membrane potential measurements (as intracellular measurements aredifficult) [16]. In addition, big differences may be observed even from thesame neuron depending on whether the measurement comes from DRG orfibers [19], while temperature differences may also slow down rates in vitro.Therefore, inconsistencies due to site and type of stimulation, and influencesfrom the surroundings may also appear that are not present in the model.

Obviously, our model comes with some limitations, as it lacks complexstructures such as DRG soma, its connecting T-junction, or multiple axonalbranches. The T-junction in particular has been shown as an importantfilter for high-frequency activation in sensory fibers [35, 76]. Therefore, ourmodel may overall provide overestimated predictions. As I have alreadydiscussed and will illustrate further though, even if we disregard the directcorrespondence between the presented model variants and natural fibers, themodel can still reflect the functional roles of ion channels in the followingfrequencies of multiple C-fiber groups.


6.1.7 General utility of constant and pulse protocols

As we saw the main focus of the discussion until now have been the repeti-tive pulse stimulation protocol. Such protocols provide both reliable in vivostimulation and better track of fiber excitability changes with a numberof quantitative properties (as seen in this thesis with conduction failures,ISI and instantaneous frequencies, AP and AHP properties). Apart fromstudying following frequencies under repetitive stimulation though, a largeportion of scientific reports in the topic of neuronal firing rates analyze ongo-ing spontaneous firing rates and/or discharge rates under a constant (single,extended square-pulse) stimulus, which is closer (than the pulse protocol)to the electrical response to physiological stimuli but difficult to perform invivo. Furthermore, most information from experiments regarding the under-lying ionic mechanisms is provided by such constant stimulation protocols.Therefore, for the purposes of discussion of my findings, I simulated both arepetitive pulse protocol, and an extended constant protocol. Based on ourand previous paradigms, can we add more about the utility of each method?

As once again the answers lie upon the pre-spike history of the system,the response to each method will also depend on the adaptation propertiesof the fiber. I previously found that the model fiber rapidly adapted toa constant stimulus, propagating a limited number of APs. Obviously, asalso discussed by Waddell and Lawson [19] (they performed both protocolstoo), a rapidly adapting fiber soon stops responding to a constant stimulus,a changing stimulus such as the simulated pulse protocol should be moreinformative, and the converse is true as well.

Nevertheless, as I previously illustrated, the implementation of the con-stant protocol shed light on limitation factors that were latent in the pulse-protocol. In particular, the post-spike AHPs act simultaneously in two op-posing ways: (1) hyperpolarizes the fiber further away from threshold, hin-dering conduction, and (2) promotes recovery from sodium channel inactiva-tion, thereby facilitating conduction. On the one hand, in the pulse protocolsignificant sodium channel recovery is allowed during the inter-pulse inter-vals of no stimulation. On the other hand, a constant stimulus works againstnormal recovery. Therefore, as the present results illustrated, which proto-col we apply determines which of the aforementioned numbered effects willdominate. Indeed, as shown experimentally from blocking AHP-enhancingKV1 channels (like KA in our model), their effect on limiting re-excitationovershadows the promoted recovery from inactivation [60]. Furthermore,the presence of depolarizing current during repolarization ( as in constantprotocol) can also change the interactions between ion currents [84]. Fromthe above I conclude that a constant protocol is probably more appropriatefor elucidating the interplay of sodium and potassium channels and howthey underlie axonal adaptation, whereas a repetitive pulse protocol wouldbe suitable for studying axonal excitability patterns and how ion channels


regulate them.

6.2 Current mechanisms for fast spiking C-fibers

6.2.1 Main sodium currents contribute differently based onactivation and repriming properties

In this thesis I studied the role of two sodium channels, NaV1.6 and NaV1.7,on high frequency conduction fidelity. As NaV1.6 was not present in theoriginal CMi-fiber model, I introduced in place of NaV1.8 a NaV1.6 channelmodel from data at room temperature [50], which I subsequently modified toaccount for the faster kinetics at body temperature. The results consistentlydemonstrated in the context of both stimulation protocols that NaV1.6 wasconsiderably better suited for high frequency responses.

One may argue that in the lack of current-specific data at body tempera-tures the upscaling modification probably made NaV1.6 unrealistically fast.Nevertheless, a wealth of in vitro data has already demonstrated the bet-ter high-frequency capabilities of NaV1.6-expressing neurons, not only overNaV1.7 (in agreement with my results), but also over other TTX-sensitivechannels (NaV1.2-4) [20, 85–88].

As it became obvious in the simulations, the rapid accumulation ofNaV1.7 (and to a lesser extent of Updated NaV1.6) inactivation was themain detrimental factor. In fact, the voltage dependence of NaV1.7 activa-tion and inactivation in DRG has been found 7-9 mV more negative thanNaV1.6 [20]. In addition, recovery from inactivation of NaV1.6 was consider-ably faster [20, 85]. As a consequence, (as we also showed) NaV1.6 currentsare better sustained under high-frequency repetitive stimulation [85].

However, I illustrated that NaV1.6 rapidly enters either the OB or in-activated states after opening, in agreement with observations from DRG[20, 85]. Although out of the scope of the rectangle-waved pulses used in thisthesis, this property does not seem to allow NaV1.6 to activate significantlyby small subthreshold depolarizing currents, a task fulfilled by NaV1.7 andother TTX-sensitive sodium channels [20, 85].

Apart from temperature upscaling of the NaV1.6 model, I also modifiedits original voltage dependence, as it was derived based on spontaneously-firing Purkinje neuron data and not DRG with no spontaneous activity. Thismodification effectively reduced the persistent component of NaV1.6 at theC-nociceptor model resting potential. As it becomes obvious, this persistentdepolarizing current is a major contributor of spontaneous activity.

Interestingly, spontaneous-like activity was also shown (figure 5.8) afterthe pulse protocol when gNaV 1.6 was increased in the Standard fiber (notehowever that the accumulation of periaxonal potassium that weakens out-ward potassium currents may also contribute [60]). Therefore, our model


verified studies that have suggested both increased expression and post-translational modulations (mainly phosphorylations) of NaV1.6 as mecha-nisms that increase excitability after inflammation and injury in nociceptors[4, 88].

However, the functional presence of NaV1.6 in small-diameter DRG isarguable. Although a few studies have shown the presence of NaV1.6 tran-scripts in unmyelinated neurons [54, 88], the consensus of functional studiesis that NaV1.6 is the main TTX-sensitive current of the fast spiking A-fibers,and NaV1.7 that of slower C-fibers [14, 20, 45, 85, 88, 89]. Furthermore,NaV1.6 was included in the original C-nociceptor model in place of NaV1.8.Although there is some evidence that the fast spiking, non-nociceptive CT-fibers lack (at least substantial) NaV1.8 expression in rats [4, 46], we shouldnot disregard both the recent conflicting results [90], and that NaV1.8 is thecurrent responsible for the majority of AP in C-nociceptors [16, 47, 89]. Ifthe much slower NaV1.8 current in CT-fibers is indeed present, its lack inthe Standard model may be responsible for the unusually high frequencyresponses.

6.2.2 Multiple resurgent currents may facilitate fast C-fiberresponses

As previously introduced, a second feature of NaV1.6 that critically con-tributes to fast spiking and discharges is its resurgent current. The compar-ison of the Standard with the No-resurgent variant verified its facilitatingeffect for the first time in a C-fiber model. Furthermore, I showed thatthe (late) resurgent current promotes fast exit from the post-AP refractoryperiod by dampening the depth of AHPs, while keeping APs adequatelyshort (a necessity for fast firing rates). Experiments in normal and NaV1.6-deficient cerebellar granule cells have illustrated this crucial effect on AHPs[87]. Although the significance of resurgent currents in AP conduction wasunclear [60], the higher number of conduction failures (of APs that weresuccessfully initiated in the distal end) by the No-resurgent model clearlyestablished its importance in faithful conduction.

In addition to their presence (similar to NaV1.6), the origin of resurgentcurrents in small-diameter axons is also debated. Although it is acceptedthat pore blocking by β4 subunits is necessary (but not sufficient) for resur-gent currents [45, 49, 51], it appears to be expressed at higher levels in large-diameter DRG than in small [71], with only 20% of NaV1.6-transfected smallDRG being capable of generating resurgent currents [85].

Moreover, it has been reported that β4 subunits can generate resurgentcurrents even in the absence of NaV1.6 [45, 89]. Paroxismal Extreme PainDisorder is known to be caused by mutations in NaV1.7 that slow transitionto inactivated states, thereby allowing an OB mechanism with β4 subunits[53]. Resurgent currents from normal NaV1.7 have also been shown after


addition of β4, indicating that NaV1.7 has lower affinity for this particlethan NaV1.6 [45]. In addition, a recent study illustrated for the first timethe presence of a slower, more depolarized, TTX-resistant (largely NaV1.8)resurgent current in small-diameter DRG [89]. The authors speculated thatthe previously reported slow, large current shoulder during repolarization(1. similar to what the Early-resurgent and Updated variants generated,read ahead for explanation of the enumerated list) might also be part of theresurgent current. Therefore, the possibility of resurgent current by cooper-ation of NaV1.8 and increased β4 subunits in fast-spiking and spontaneouslyactive C-fibers should also be considered.

Finally, in our comparative analysis I included the Updated fiber, withparameters changes based on large-diameter DRG data [72]. Subsequently, Iclearly showed that these changes almost corresponded to the Early-resurgentvariant, namely without substantial OB mechanism. Of course, as Sittl et al.[72] empirically fitted the original NaV1.6 model, other parameter modifi-cations might also be solutions, therefore biophysical interpretation of theUpdated fiber can only be a matter of speculation here.

Nevertheless, although to my knowledge no reports of ’early resurgent’current currently exist (and this is expected, because as I explained thisearly component does not come from the OB state, but rather from theclosed ones), (2) a mechanistically-similar early component due to incom-plete inactivation has been already suggested for fast-spiking neurons in theCNS [91]. Consequently, further considering that: (3) as we saw the Early-resurgent variant performed better than the No-resurgent fiber when AHPdepth was not a severe limiting factor, and (4) the Standard fiber exceededrealistically expected frequencies for C-fibers by far, it is plausible that thisearly component due to incomplete inactivation might explain fast-spikingC-fibers instead of the β4-associated, ’late’ resurgent current.

6.2.3 KV-mediated contributing mechanisms

Due to the high number of KV channel subtypes found in sensory fibers,the problem of conflicting evidence regarding their functional roles is evenmore prevalent compared to sodium channels. The contributions of KV-mediated currents on fiber excitability do not only depend on their ownvoltage-dependent kinetics, but also on how they are matched to inactivationand de-inactivation of the driving sodium channels [60]. Therefore, anydiscussion in regards to potassium channels should be done on the basis ofspecific inward sodium current background.

Taking this into consideration, in the present thesis I demonstrated thatamong the main potassium currents, the fast delayed rectifier Kdr was bet-ter matched (in terms of successful conduction at high frequencies) withNaV1.6, whereas the fast inactivating KA performed better with NaV1.7.As Baranauskas [57] has argued, there would be substantial sodium channel


recovery from inactivation only if its rate was comparable to the closing rateof the potassium channel. Seeing that the (de)inactivation time constant ofNaV1.7 is slower than the time constant of (de)activation of Kdr, and closerto that of KA (for the voltage range achieved during AP repolarization),explains both why KA allows efficient NaV1.7 repriming, and why it offersno advantage to the fast-repriming NaV1.6.

Perhaps also related to the matching of NaV1.7-KA, Obreja et al. [30]recently suggested that a fraction of porcine CMi-fibers with ’very high’ me-chanical threshold could alternatively be distinguished from insensitive CMi-fibers based on their unusually high (compared to slow CMi-nociceptors)maximum following frequencies. Clearly, in C-nociceptors that lack signif-icant NaV1.6 expression, higher ratios

gNaV 1.7

gNaV 1.8and

gKAgKdr

would significantly

increase their following frequency (simulations not shown). Strikingly, Pe-tersson [61] reached to similar conclusions, suggesting increased NaV1.7 andreduced Kdr levels to explain the discrepancies on ADS. Therefore, our modeldo not just propose an ionic profile for nociceptive CM-fibers, but it addi-tionally reinforces the idea that enhanced ADS is closely associated withconduction failures and low following frequencies, both underlain by thesame ion channels.

Regarding the relation between Kdr and KA in the presence of NaV1.6,our model showed that their effects on firing frequencies opposed each other,mainly because of their different effects on AHPs. As I have already dis-cussed in section 6.1.7, an AHP can be either beneficial or detrimental tofiring rates, thus the effects of AHP-shaping potassium currents would alsofollow the same reasoning. Therefore, KV3 subunits of the fast delayed rec-tifier type are indeed important in hyperexcitability and spontaneous firingin DRG neurons after nerve injury [56]. On the contrary, the KA current hasbeen found greatly reduced in rodent small-diameter DRG under a painfulneuropathy, converging to my findings that KA enhances conduction failures[28].

Furthermore, our model indicated that KA was severely limited by fastinactivation, resembling both the reported function of KV3.4 channels [58]and its modeled kinetics [77]. Nevertheless, recent results illustrated thattheir fast inactivation can be slowed by phosphorylation, suggesting anothermechanism by which nociceptive fibers can be hyperexcitable in chronicpain patients [92]. We should keep in mind though that prolonging theactivity of KA in the context of the pulse protocol would also increase theAHP depth (as reported by Ritter et al. [92]). A similar effect should beexpected by KV3.3 subunits, if present. Therefore, the NaV1.6-Kdr shouldremain superior in terms of following frequencies. Collectively, the abovefindings highlight the importance of both a fast-repriming sodium currentwith resurgent properties, and a fast-activating, slow- (or non-)inactivatingpotassium current [60].


In contrast to the pulse protocol, KA promotes rapid discharges undera constant depolarizing protocol. In the lack of considerable KA current,several neurons of the CNS are incapable of high frequency discharges, evenwhen considerable sodium resurgent currents are present [80, 93], well inagreement with my findings. As my work further indicated, KA decreasesgradual depolarization, which would inevitably develop both sodium inac-tivation, and elevation of the slower potassium currents that prevent fur-ther spiking (e.g. KM, which as expected generally suppressed firing in ourmodel). Interestingly, Fernandez et al. [84] earlier found a similar effect froma KV3-mediated current in the electrosensory lobe of electric fish, unfortu-nately however they did not investigate further the exact identity of theseKV3 subunits. All in all, the important role of KA on high discharge rateswas clearly demonstrated by our model, distinguishing it from its detrimen-tal effect on AP conduction under repetitive activation.

Chapter 7

Conclusions

Previous empirical evidence suggested a direct connection between hyperex-citability, reduced conduction failures and high discharge rates of C-nocice-ptors with peripheral pain pathology. These observations motivated theinception of the current thesis, aiming to explore and identify the ion chan-nel subtypes that potentially contribute to the high-frequency-firing non-nociceptive CT-fibers. Drawing information regarding the contributing ionchannels, we could then discuss their implications for low-frequency-firingC-nociceptors.

As NaV1.6 (a channel well-known for its resurgent current that facili-tates high-frequency firing), and not NaV1.8, is considered the main AP-generating sodium current in non-nociceptive fibers, I modified our previ-ously published C-fiber model [16] with the inclusion of a (properly adjusted)NaV1.6 model in place of NaV1.8 [50]. Furthermore, as bursting may origi-nate either from an externally imposed input frequency or from an intrinsicmechanism [94], I tested variants of the new model under two stimulationparadigms respectively: a sequence of 40 short pulses, injected at constantfrequency (pulse protocol), and a single, prolonged constant injection (con-stant protocol).

As it became obvious in the previous chapters, the results from parameter-varying analysis provided a large number of predictions regarding the sodiumand potassium channel profile of high-frequency following CT-fiber. Briefly,our model predicted that the NaV1.6 and its resurgent component is capableof significantly increasing both studied firing rates, and converging to prop-erties seen by faster conducting Aδ-fibers. Our findings elucidated that ioncurrents generally controlled following-frequencies depending on a combina-tion of their inactivation rates and their effects on AP duration and AHPdepth. More interestingly, matched sodium and potassium current prop-erties showed a clear impact on generated following-frequencies, suggestingthat the NaV1.6-Kdr pair is responsible for the highest conduction fidelity offast C-fibers and NaV1.7-KA for high following-frequencies of C-nociceptorslacking NaV1.6. Finally, the model verified the facilitating effect of KA on

76

CHAPTER 7. CONCLUSIONS 77

discharge frequencies under constant depolarizing stimuli (and possibly un-der ongoing spontaneous activity). This result also shed light on the relationbetween the two utilized experimental protocols when performed in vivo, il-lustrating their abilities and limitations to address different aspects of thesame underlying mechanism.

Probably the most important limitation of our model for exploring ionicmechanisms consists the complete lack of calcium ion dynamics and by ex-tent voltage-gated calcium channels and calcium-dependent potassium chan-nels [7]. Naturally, such a choice was made by Tigerholm et al. [16] to reducethe computational complexity of the model. I could argue that at least theshort term effects on the axon have been captured in our model by increaseddensity of similarly functioning sodium and potassium channels. For exam-ple, T-type calcium channels appear to simply facilitate the depolarizingeffect of nearby sodium channels [7], while the calcium-dependent potas-sium channel BK appears to resemble the fast-inactivating KA dynamics byincreasing only the depth of initial AHPs, with minor slower contributionsfrom IK-SK currents that resemble KM and h current (also included in ourmodel) [76]. Calcium influx effects would definitely be more important ifmodeling of synaptic events at central terminals are to be modeled [7].

As nociceptive and non-nociceptive C-fiber categorization becomes moreand more clear based on different conduction velocity [21] and peak followingfrequency features [30], I expect such data to be soon utilized by our model.The work of this thesis has definitely contributed into that direction, notonly by indicating how the major ion channels control output frequencies,but also by proposing possible channel configurations for groups in bothextremely fast and normal range frequencies. Furthermore, the inclusion ofNaV1.6 in our C-nociceptor model has opened the path for further studiesin the context of pathological C-nociceptors, as fiber conduction recordingsfrom spontaneously active fibers [12] can now be studied mechanistically insilico. Obviously, as data will become more available, this fitting process willideally be reduced to an optimization problem for the modeled ion channelcurrent densities [5, 95].

Finally, I believe it is appropriate to close this thesis with a referenceback to our major source of motivation, the view of ion channels as phar-macological targets for pain treatment. The problem of target-based drugdiscovery lies on both the specificity and efficiency of a drug for a particularligand, and on successful drug delivery to the intended target only. For ex-ample, although NaV1.6 is definitely important for at least the excitabilityof large-diameter peripheral fibers, it has not attracted considerable atten-tion of the pain research community because its widespread distribution inthe CNS renders it prone to unwanted side effects [15, 88, 96]. Clearly, ourmodel could contribute to the search of location-specific targets in nocicep-tors, since channel density parameters directly represent drug-induced chan-nel opening or blocking. Similarly, Markov kinetic schemes for ion channels

CHAPTER 7. CONCLUSIONS 78

could naturally be expanded to describe the biochemical drug-target kineticsand their effects on conductances. In the future, as exemplified in Tigerholmand Fransen [97], the combination of such modeling tools could help in thedirection of testing the efficiency of intervening with a target ion channeland predict unwanted effects from the interplay between different ion chan-nels (the coupling of sodium-potassium channels shown in this thesis beinga prime example).

Appendix A

Basics of cellular electrophysiology

In this appendix, my intention is to introduce the reader to the necessaryconcepts and clarify some possible unavoidable misnomers in the field ofelectrophysiology. After introducing the most fundamental term in neuro-physiology, the action potential, I describe the mathematical and physiologi-cal framework of the Hodgkin-Huxley type and Markov ion channel models,both utilized in the present work, and how these building units give rise toa neuronal model. This chapter barely scratches the surface even of cellularelectrophysiology, therefore the keen reader is strongly advised to consultsome of the following excellent sources for ion channels and neurophysiology[41, 98, 99], and computational modeling [75, 100, 101], which were the basisof this introductory chapter.

A.1 Neurons: electrical properties and action po-tential

Neurons, composing the basic units of the nervous system, are cells es-pecially characterized by electrical excitability. These electrical propertiesare explained by viewing the details at the molecular level. In particular,neurons maintain different ion concentrations between their interior and ex-terior environment. These differences exist thanks both to the hydrophobicnature of the cellular membrane that does not permit the hydrophilic ionsto cross it, and to proteins embedded in the phospholipid bilayer, namelyion channels and ion pumps, which selectively allow ions to pass throughpores.

As a result of the function of ion channels and pumps, the neuron at resthas an excess of cations outside its membrane, and an excess of anions intra-cellularly. The charge of these ions generate an electrical potential difference(voltage, Vm) across the lipid membrane. When this electrical force balancesthe opposing chemical force (due to ion concentrations gradient), an elec-trochemical equilibrium, characterized by the resting potential, is reached.

79

APPENDIX A. BASICS OF CELLULAR ELECTROPHYSIOLOGY 80

A particular type of ion channels is voltage-dependent, meaning that theprobability of their pores being open changes as a function of membranevoltage. In direct analogy to an electrical circuit (see next section), thepermeability of ion channels can be thought of as electrical conductance. Ifthe voltage-dependent channels are open, they allow specific ions to movealong their electrochemical gradient, which causes further changes of themembrane voltage. Typically, when Vm exceeds a certain threshold, an all-or-none transient signal is ”fired”, the so-called action potential (or simplyspike).

A nerve cell typically consists of a soma, dendrites and an axon. Thesoma, as its meaning in Greek reveals (body), is the main part of the cellwhere all the usual cellular processes take place. Both the axon and den-drites are structures projected from the soma. The classical view is thatthe generated action potential propagates along the axon, which has sev-eral endings to points of other cells, most often to dendrites, which harbormultiple tiny branches that form joining points of neurons called synapses.As I will describe later, the modeling work of this thesis focuses on isolatedaxonal segments, disregarding the soma and synaptic inputs.

A.2 Hodgkin-Huxley type models

Still considered as the epitome of phenomenologically realistic neuron mod-els, Hodgkin and Huxley [37] (HH) published their seminal work based onprototype voltage clamp recordings from the squid giant axon. By insertingtwo silver wire electrodes into the axon, Hodgkin and Huxley managed toignore spatial variability (axon was viewed as a single isopotential compart-ment, see section A.5), and to analyze ion currents without the influence ofcapacitance and voltage-dependent parameters (both because dVm

dt = 0, seemathematical description below) [99]. This section will offer a brief descrip-tion of the general framework of contemporary HH-type models, while itwill also serve as a conceptual guide for the nomenclature used throughoutmy thesis.

A.2.1 Membrane currents

Generally in the HH framework, the neuron is visualized as an equivalentelectrical circuit with current sources in parallel: the membrane capacitivecurrent (Im), the sum of voltage-dependent ion channel currents (INa,n andIK,m for ion channel types n and m mediating sodium and potassium cationsrespectively), the sum of other current sources (Ioth) including ligand-gated,ion pump and leak currents, and intracellularly injected currents (Iext). Fur-thermore, the electrochemical gradients responsible for ion fluxes across themembrane are viewed as voltage sources. Given the definition of capacitance,Kirchoff’s current law, and Ohm’s law for the voltage-gated currents, the


current Iext passing the membrane is described by the following differentialequation:

Iext = CmdVmdt

+∑n

GNa,n(Vm −ENa) +∑m

GK,m(Vm −EK) + Ioth (A.1)

where Vm is the membrane potential, Cm is the membrane capacitance,GNa,n and GK,m denote total conductances of the respective ion channeltype activities. ENa and EK are the reversal (Nerst) potentials, whichdepend according to the Nerst equation on the intra- and extracellular con-centrations of sodium and potassium ions respectively. Note that GNa,n andGK,m are still undefined functions of voltage and time. This is exactly thetopic of subsection A.2.3.

A.2.2 Sign conventions

It is appropriate now to discuss the set of conventions for ionic currents andthe membrane potential Vm, implicit in equation A.1, as these conventionswill be kept for the remaining of this thesis. In particular, since the measur-able quantity is actually the potential difference between the extracellularand intracellular space, defining a quantity as zero is a matter of convenienceand preference. Nevertheless, the usual modern convention (which equationA.1 follows) sets the extracellular potential at zero. This convention resultsin a negative resting intracellular potential, and in positive (or negative)membrane potential Vm changes during depolarization (or hyperpolarizationrespectively).

In regards to ionic currents, it is hopefully clear to the reader that thesign of each (voltage-dependent) sodium current will be negative, becauseVm−ENa is (usually) negative, and equivalently potassium currents are pos-itive. Therefore, the inward flow of sodium ions corresponds to a negativecurrent that depolarizes the cell membrane (increases Vm if there is no othercurrent in equation A.1), and the outward flow of potassium ions as posi-tive current that hyperpolarizes it (decreases Vm). Obviously, the oppositeconventions apply for external currents Iext, as a positive current injectiondepolarizes the membrane.

A.2.3 Gating variables of conduction

Perhaps the most important contribution of Hodgkin and Huxley [37] was amodel for the voltage dependent ionic conductances [100]. More specifically,they proposed a hypothesis that electrically charged gating particles resideeither outside or inside the channel pore, being permissive or nonpermis-sive to ionic currents (a terminology borrowed from Nelson [101] to avoidconfusion with open/closed channels). Today we know that conformational


changes of the channel protein itself are controlled by the local membranepotential, and thus the original hypothesis was reinterpreted so that gatesare now parts of the pore. When all gates assume a permissive conforma-tion, the channel is considered open, whereas a single nonpermissive gate isenough for a channel to be closed. Therefore, for a particular gate type i wehave the following transition diagram:

Nonpermissive Permissiveαi(Vm)

βi(Vm)

Figure A.1: Transition state diagram of a gate

where αi and βi denote the transition probabilities and are functions of Vm.If we now further simplify our description by considering an ensemble of

ion channels that possess this gate type, we can model the fraction of ionchannels with a permissive gate, yi (thus ranging from zero to one), with afirst-order differential equation of the form:

dyidt

= α(Vm)(1− yi)− β(Vm)yi (A.2)

With simple algebra the expressions for the steady-state yi∞ and the timeconstant τi can easily be found:

yi∞ =αi(Vm)

αi(Vm) + βi(Vm)(A.3)

τi =1

αi(Vm) + βi(Vm)(A.4)

The only remaining part left undetermined is how we can express αi

and βi as functions of voltage. Hodgkin and Huxley [37] managed for thefirst time to utilize steady-state measurements from different voltage clampprotocols to empirically fit them to a continuous function of voltage. Today,the rate constant of HH-type models typically have the following generalizedform with up to 6 free parameters [101]:

αi =A+BVm

C +H exp(Vm+DF )

(A.5)

Notice that the rate parameters may also be found experimentally to de-pend on temperature, meaning that in vivo rate parameters may be un-derestimated (and time constants overestimated) in experiments at roomtemperature. To account for this temperature dependence, Q10 tempera-ture coefficients are employed for scaling the rate parameters according to

QT−T0

1010 [41, 100].


Assuming now that a e.g. sodium channel type n has a certain numberof independent gates, then based on the aforementioned definition of gatesthe macroscopic conductance GNa,n will be given by:

GNa,n = GNa,n

∏i

yi (A.6)

where GNa,n is the maximum conductance possible if all channels are open.A final note should be made about the nomenclature used for different

type of gates. If yi approaches one during depolarization, the gate is calledactivation, whereas it approaches zero this is an inactivation gate. On theother hand, with hyperpolarization both activation and inactivation is re-duced, an event also termed deactivation and deinactivation respectively.A (voltage-gated) current is usually characterized by the time constants ofactivation and/or inactivation gates (termed with indicative adjectives fast,slow, etc.), and the presence or not of inactivation (called inactivating ornoninactivating/persistent current respectively).

There have been more than sixty years since the advent of the Hodgkin-Huxley formalism. Nevertheless, with the straightforward method of fittingHH-type currents to voltage-clamp (steady-state) measurements, it still re-mains a widely popular model in cellular electrophysiology [100]. However,the HH formalism also comes with limitations due to its own simplifica-tions, and a more biophysically accurate description has been utilized overthe years. This will be exactly the topic of the next two sections.

A.3 Markov models for single ion channels

.Voltage and current clamp methods allowed the study of macroscopic

ionic conductances. However, our understanding of the conductive proper-ties of single ion channels had been limited until the development of thepatch-clamp technique [102, 103], which allowed the isolation of very smallmembrane areas, and thus the voltage-clamp of a single embedded ion chan-nel [98, 99]. Indeed, single-channel recordings have revealed that the switch-ing mechanism between permissive and nonpermissive ion channel states isstochastic in nature [98, 99]. Furthermore, as already mentioned in sectionA.2.3, conformational changes (in some cases voltage- or ligand-dependent)are responsible for the transition between states. Therefore, based on tran-sition state diagrams similar to A.1 (but now referring to ion channel statesinstead of gating states), stochastic Markov models have been a suitableframework to accurately describe single channel recordings [104, 105].

Markov models represent a discrete sequence of ion channel states ofthe form shown in figure A.2. As the name implies, the system obeys theMarkov property, namely its time evolution depends only on the present


state of the system and the transition probabilities (pi,j here), which in thecase of voltage-gated channels may or may not directly depend on Vm (seemore below). The full mathematical description of such single channel dy-namics is given by the so called Master equation [106]. Moreover, gatingmechanisms such as the above can (as also seen in the example used inthis thesis) form loops [107, 108]. In this cyclic case the principle of micro-scopic reversibility (or detailed balance) should be fulfilled (in reality, truemicroscopic reversibility applies only if no external energy drives gating)[107, 108], therefore the product of forward pi,j rates inside the loop shouldbe equal to the product of backward pj,i rates.

S1 S2 ... Snp1,2

p2,1

p2,3

p3,2

pn−1,n

pn,n−1

Figure A.2: Transition state diagram of ion channels

At this point if the continuum whole cell behavior is desired (as is the casein the present report), we can use the same argument as for equation A.2,leading to a system of n (number of states) first order differential equationsof the following form [106]:

dsidt

=∑j

pj,isj −∑j

pi,jsi (A.7)

where si is now the fraction of ion channels in state Si. Indeed, stochasticMarkov models has been shown to converge well to their deterministic coun-terparts when describing the macroscopic current from a large number ofidentical ion channels [104]. For voltage-dependent transitions, equation A.5is used for the rate parameters pi,j too. Similarly, the macroscopic conduc-tance at any time point is given by the same equation (A.6) as the HH-typemodels, but this time it is simply the product of the open fraction of channels

O with the maximum conductance G (in other words,∏i

yi = O).

Note that here I no longer talk about a stochastic Markov process, butabout a classical kinetic model, which sometimes in the literature is simplycalled Markov model (being a misnomer), or kinetic scheme [95, 109]. Toavoid such confusion in the rest of the report, I will use the term kineticscheme for the deterministic models, and Markov model for the stochasticcounterparts.

A.4 Choosing kinetic schemes over HH-type mod-els

As the reader hopefully suspects by now, a direct connection between theHH-type models and the Markov kinetic schemes can be drawn. In fact,


the HH formalism is simply a subclass of the Markovian representation,meaning any HH-type model can be expressed in terms of a kinetic scheme[100, 106]. For example, the original sodium HH model can be written withan eight-state kinetic scheme [110]. This example hints that HH-type modelsefficiently economize in the number of states and parameters, whereas esti-mating the large number of parameters in a kinetic scheme is more difficult[95, 100, 109].

However, being simply a subset of kinetic schemes, the HH-type modelscome with a number of limitations. Most importantly, the assumption ofuncoupled activation and inactivation has not been been valid for variousion channels. For example, many voltage-gated sodium channels need to de-activate before achieving substantial recovery from inactivation [111]. Mod-eling such interdependence requires the use of full-fledged kinetic schemes.Furthermore, modeling state- (only) dependent and ligand-dependent transi-tions is much more straightforward with kinetic schemes, as the same formal-ism is suitable for describing any kind of underlying molecular interactions[106]. Kinetic schemes also provide a link between channel gating propertiesand macroscopic whole cell currents as predicted by HH-type models, andeasier interpretability between ion channel mutations and changes mecha-nisms [95, 101]. Finally, Markov models (and their schemes) are equippedwith the inherent advantage of being fitted not only to macroscopic currents(like HH-type models), but also to single-channel recordings [95, 109].

As it becomes clear, the choice of framework is mainly based on the scopeof the modeling study. In a comparative study, Destexhe et al. [106] illus-trated the differences between the original HH model, one large (9 states),and one simplified (3 states) kinetic scheme. Whilst the large kinetic schemewas clearly more accurate than the original HH model in replicating voltageclamp experiments (thus the single channel behavior), whole-cell level ex-citability properties and action potentials were similar in all cases. There-fore, choosing one formalism or the other largely depends on the desiredbalance between the level of accuracy and complexity, as well as on thereplicated experimental data [100].

A.5 Propagation and multi-compartmental mod-eling

.The described HH framework reduces the neuron to a single point,

namely the spatial structure of the neuron is ignored. However, the complexneuronal morphology also plays a significant role in signal propagation fromthe dendrites to the axon, and thus on the firing behaviour of the neuron[112].

The HH framework can again be suitably extended with the application


of cable theory, assuming the parts of the neuron to be cylinders of ap-propriate size, and thereby using partial differential equations (PDE) [113].However, the numerical treatment of the complex PDE is necessary, whichdemands discretization in both time and space. With space discretizationthe neuron is split into multiple segments (or compartments), and eachsegment dynamics is now mathematically described by the simpler ODE,essentially reverting back to the spatially invariant HH model [113]. Thefirst systematic use of this framework by Rall [114] have popularized thesecompartmental modelling approaches, which now compose the basis of thetwo major neuronal simulation environments, NEURON [75] (used in thisthesis) and GENESIS [115].

Briefly, for such cylindrical compartments j with diameter d and length∆x, these ODE take the form:

CmdV j

m

dt+ ijion =

d

4Ri

V j+1m − 2V j

m + V j−1m

∆x2+

Ijextπd∆x

(A.8)

whereV j+1m − 2V j

m + V j−1m

∆x2is obviously the discretized version of

V 2m

dx2, ijion

is the sum of all ionic transmembrane current densities, and Ijinj is theinjected current, both at compartment j. Solutions can then be acquired byemploying a numerical integration method. For details regarding efficientnumerical methods that are used in NEURON, as well as the biophysicaland mathematical reasoning behind this treatment, I refer the reader to theNEURON textbook, written by its developers Carnevale and Hines [75].

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