Dynamics and Complexity of FitzHugh-Nagumo Neuronal Systems

Mada Sanjaya W. S1, Muhammad Yusuf1, Agus Kartono1, Irzaman2

1Theoretical Physics Division, Department of Physics, Faculty of Mathematical and Natural Sciences, Institut Pertanian Bogor, Bogor 16680, Indonesia.

2Applied Physics Division, Department of Physics, Faculty of Mathematical and Natural

Sciences, Institut Pertanian Bogor, Bogor 16680, Indonesia. e-mail : madasws@gmail.com

Abstract

System of signals propagation from one neuron to another represent event of very complex electrochemical mechanism This work adresses the dynamics and complexity of neuron mathematical models. The aim is first the understanding of the biological meaning of existing mathematical systems concerning neurons such as Hodgkin-Huxley or FitzHugh-Nagumo models. The local stability and the numerical asymptotic analysis of FitzHugh-Nagumo model are then developed in order to comprehend the dynamic evolution of a single FitzHugh-Nagumo neuron. This examination naturally comes to the study of neuron networks. The analysis of these networks uses the synchronization theory via connections between neurons and can give rise to emergent properties and self-organization. Our result leads to a classical law which describes many self-organized complex systems like earthquakes, linguistic or urban systems. This has been performed using numerical tools. Keyword: Hodgkin-Huxley models, FitzHugh-Nagumo, action potentials, synchronization, chaos.

Introduction

Understanding the mechanisms of the propagation of the nerve activity is one of the fundamental problems in biophysics. The first detailed quantitative measurements of the ionic currents were performed by Hodgkin and Huxley in the early 50-s (Hodgkin & Huxley, 1952). Using the voltage clamp technique, they were able to measure the kinetics of Na+ and K+ currents in the squid giant axon. This led them to a set of differential equations which describe the dynamics of the action potential. Furthermore, by combining these equations with the cable equation for spreading of current in the axon they were able to calculate the shape and velocity of the propagating action potentials (Hodgkin & Huxley, 1952). The

predictions of their model turned out to be in a remarkably good agreement with the experimental observations.

Hodgkin-Huxley model for the action potential of a space clamped squid axon is de_ned by the four dimensional vector field. with variables (v; m; n; h) that represent membrane potential, activation of a sodium current, activation of a potassium current, and inactivation of the sodium current and a parameter I that represents injected current into the space-clamped axon. Although there are improved models the Hodgkin-Huxley model remains the paradigm for conductance-based models of neural system. FitzHugh was the first investigator to apply qualitative phase-plane methods to understanding the Hodgkin-Huxley model. To make

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headway in gaining analytic insight, FitzHugh first considered the variables that change most rapidly, viewing all others as slowly varying parameters of the system. In this way he derived a reduced two-dimensional system that could be viewed as a phase plane. From the Hodgkin-Huxley equations FitzHugh noticed that the variables V and m change more rapidly than h and n, at least during certain time intervals. By arbitrarily setting h and n to be constant we can isolate a set of two equations which describe a two-dimensional (V; m) phase plane. The elegance of applying phase plane methods and reduced systems of equations to this rather complicated problem should not be underestimated.

Determining the dynamical behavior of an ensemble of coupled neurons is an important problem in computational neuroscience. Commonly used models for the study of individual neuron which display spiking behavior (FitzHugh, 1961; Nagumo, 1962; Hodgkin, 1952). From the very beginning of the research in the field of computational neuroscience, people deal with single neuron and its behavior. Present trends of research include investigation of the behavior of neurons considered in a network and their way to fire synchronously. It is assumed that the activities in the brain are synchronous and underlying interests for synchronization of nonlinear oscillators in physical and biological systems range from novel communication strategies to understand how large and small neural assemblies efficiently and sensitively achieve desired functional goal. In recent years, there has been tremendous interest for the study of the synchronization of chaotic systems. The phenomenon of synchronism gives rise to different dynamical behaviors such as chaotic synchronization etc (Belykh, I & Shilnikov, A., 2008). In (Mishra, 2006), nonlinear dynamical analysis on single

and coupled modified FitzHugh-Nagumo model under steady current stimulation is carried out. Also the effect of parameter variation on its behavior is investigated. Mathematical Models Basics Models of Neuron

To describe the evolution of the membrane potential V in the squid giant axon (Hodgkin, A.L & Huxley,A.F., 1952) developed the following conductance-based model:

eksLLkKNaNa IvvgvvvngvvhmvgCdt

dv )())(()()(1 43

nvnvdtdn

nn )()1)((

mvmvdtdm

mm )()1)(( (1)

hvhvdtdh

hh )()1)((

The peak conductances g, reversal potentials E and rate coefficients and of the individual currents and gating variables m, n and h are given by :

110/)40( )1)(40(1.0)( vm evv ,

18/)65(4)( vm ev

20/)65(07.0)( vh ev ,

110/)35( )1()( vh ev

110/)55( )1)(55(01.0)( vn evv ,

80/)65(125.0)( vn ev

For I = 0, the membrane potential settles

into a resting potential of about V = −65

mV.

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And with the gNa = 120, gK = 36, gL = 0.3

mmho cm-2, vNa = 50 mV, vK = -77 mV

and vL = -54.4 mV.

(a) (b) Figure 1. Dinamical Systems in Hodgkin_huxley models at I = 0 ; (a) Action potentials at

t = 5 ms, (b) variable m, h and n at t = 5 ms. FitzHugh-Nagumo Models of Single Neuron

In 1961 FitzHugh proposed to demonstrate that the Hodgkin-Huxley model belongs to a more general class of systems that exhibit the properties of excitability and oscillations. As a fundamental prototype, the van der Pol oscillator was an example of this class, and FitzHugh therefore used it (after suitable modification). A similar approach was developed independently by Nagumo in 1962. FitzHugh proposed the following equations:

)(

31 3

bwavdtdw

Iwvvdtdv

(2)

In these equations the variable v

represents the excitability of the system and could be identified with voltage (membrane potential in the axon); w is a recovery variable, representing combined

forces that tend to return the state of the axonal membrane to rest. Finally I is the applied stimulus that leads to excitation (such as input current), or rectangular pulses. Local Dynamics and Bifurcation

Let us consider the FitzHugh-Nagumo system (2), Equilibria are given by the following system :

0/,0/ dtdwdtdv (3)

The nature of this equilibria is given by the study of the eigenvalues of the jacobian matrix J of this system :

b

vJ

11 2

(4)

We obtain the following polynomial equation :

0)ˆ()ˆ1( 222 vbbbv (5)

Numerical Analysis

Table 1. Numerical Analysis of Stability Equilibrium Point in FitzHugh-Nagumo Systems

No Variation Ieks Equilibrium Point Eigen value Stability 1 0.00 -1.1994,-0.6243 -0.2513 0.211900 i Spiral sink

2 0.32 -0.9769,-0.3461 -0.009176 0.2774 i Spiral sink

3 0.33 -0.9685,-0.3357 -0.001045 0.2757 i Limit cycle

4 0.50 -0.1311,-0.8048 +0.1441 0.191500 i Spiral source

5 1.25 1.8810, 0.8048 +0.1441 0.191500 i Spiral source

6 1.42 2.0857, 0.9685 -0.001045 0.2757 i Spiral sink

7 1.43 0.9769, 2.0961 -0.009176 0.2774 i Spiral sink

8 1.45 0.9933, 2.1166 -0.02532 0.28020 i Spiral sink

9 1.50 2.1656, 1.0325 -0.06501 0.282800i Spiral sink

10 2.00 1.3341, 2.5426 -0.6412, -0.2026 Stable Node

Obviously, this figures show transitions from stable to periodic or quasiperiodic, then complex and finaly chaotic behaviour.

(a) (b) (c) (d)

Figure 2. Phases portrait for; (a) I = 0, (b) I = 0.33, (c) I = 1.45, (d) I = 2

(e) (f) (g) (h)

Figure 3. Time Series for ; (a) I = 0, (b) I = 0.33, (c) I = 1.45, (d) I = 2

Synchronization Generalities

Synchronization is a phenomenon characteristic of many processes in natural systems and non linear science. It has remained an objective of intensive research and is today considered as one of the basic nonlinear phenomena studied in mathematics, physics, engineering or life science.

This word has a greek root, syn = common and chronos = time, which means to share common time or to occur at the same time, that is correlation or agreement in time of different processes.Thus, synchronization of two dynamical systems generally means that one system somehow traces the motion of another.

Chaotic oscillators are found in many dynamical systems of various origins. The behavior of such systems is characterized by instability and, as a result, limited predictability in time. Roughly speaking, a system is chaotic if it is deterministic, if it has a long-term aperiodic behavior, and if it exhibits sensitive dependence on initial conditions on a closed invariant set.

Consequently, for a chaotic system, trajectories starting arbitrarily close to each other diverge exponentially with time, and quickly become uncorrelated. It follows that two identical chaotic systems cannot synchronize. This means that they cannot produce identical chaotic signals, unless they are initialized at exactly the same point, which is in general physically impossible. Thus, at first sight, synchronization of chaotic systems seems to be rather surprising because one may intuitively expect that the sensitive dependence on initial conditions would lead to an immediate breakdown of any

synchronization of coupled chaotic systems, which led to the belief that chaos is uncontrollable and thus unusable.

Despite this, in the last decades, the search for synchronization has moved to chaotic systems. A lot of research has been carried out and, as a result, showed that two chaotic systems could be synchronized by coupling them : synchronization of chaos is actual and chaos could then be exploited. Ever since, many researchers have discussed the theory, the design or applications of synchronized motion in coupled chaotic systems. A broad variety of applications have emerged, for example to increase the power of lasers, to synchronize the output of electronic circuits, to control oscillations in chemical reactions or to encode electronic messages for secure communications (Aziz-Alaoui, MA., 2006). Modified FitzHugh-Nagumo Models

The modified FitzHugh-Nagumo equations are a set of two coupled differential equations which exhibit the qualitative behavior observed in neurons, viz quiescence, excitability and periodicity (Mishra D et al, 2006). The system can be represented as

)(

)(31 3

bwavdtdw

tIwvvdtdv

Where

)cos()/()( tAtI (6) A represents the magnitude of the stimulus and refers to the frequency of given stimulus.

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(a) (b)

(c) (d)

(e) (f)

Figure 4. FitzHugh-Nagumo Models at A = 0.7 ; (a) Time series at = 0.4 (b) Phases portrait at = 0.4 , (c) Time series at = 0.8145 (d) Phases portrait at = 0.8145 , (e) Time series at = 0.93, (f) Phases portrait at = 0.93

The stimulus frequency is varied while

keeping the magnitude at a fixed value of A = 0.71, since at this particular value of A, modified FitzHugh-Nagumo neuron model gives periodic spiking. Simulation results at different stimulus frequencies are shown in Figure 4. It is observed that

with the variation in stimulus frequency, the neuron shows complex chaotic behavior. Hence the stimulus frequency can be considered as a significant parameter that affects the behavior of neuron.

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Coupling FitzHugh-Nagumo neurons

Let us consider a network composed by n FitzHugh-Nagumo neurons. These neurons are coupled by there first variable v. This network can be modeled by the system (Lange, E.et al, 2005; Belykh, I & Shilnikov A., 2008):

)(

),()(31 3

iii

jiiiii

bwavdt

dw

vvhtIwvvdtdv

(7)

The coupling function h is given by :

n

jjijssiji vcgVvvvh

1)()(),( (8)

in which the synaptic coupling Γ is modeled by a sigmoid function with a threshold :

))(exp(11)(

sjj v

v

(9)

Θs is the threshold reached by every action potential for a neuron. Neurons are supposed to be identical and the synapses are fast and instantaneous. The parameter gs corresponds to the synaptic coupling strength. The synapse is exitatory, that is why the reversal potential Vs must be larger than xi(t) for all i and all t.

Coupling two FitzHugh-Nagumo neurons

The first step here is to adapt the previous method to two FitzHugh-Nagumo neurons. In this case, we use a bidirectional connection since the number of inputs (that is one) has to be the same for all neurons to have a synchronous solution.

0110

2C

The following system represents our two FitzHugh-Nagumo neurons bidirectionally coupled :

)(

))(exp(11)(

31

)(

))(exp(11)(

31

222

122

322

2

111

211

311

1

bwavdt

dwv

gVvIwvvdtdv

bwavdtdw

vgVvIwvv

dtdv

sss

sss

(10)

Parameters are fixed as follows : a = 0.7, b = 0.8, = 0.08, Vs = 1, s = -0.25, =1, A = 0.71 and = 0.95.

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(a) (b)

(c) (d)

Figure 5. v2 according to v1 for the coupling strength; (a) gs = 0.1, (b) gs = 0.125, (c) gs = 0.1255 , dan (d) gs = 0.15

We observe that the synchronization numerically appears for a coupling strength gs ≥ 0.15. This phenomenon is also observed for the variables w, for the same numerical values of parameters. Coupling three FitzHugh-Nagumo neurons

In the case of three neurons, we also use bidirectional connections since the number of inputs (that are two) has to be

the same for all neurons to have a synchronous solution.

100010001

3C

The following system represents our three FitzHugh-Nagumo neurons bidirectionally coupled :

9

)(

)))(exp(1

1))(exp(1

1()(31

)(

)))(exp(1

1))(exp(1

1()(31

)(

)))(exp(1

1))(exp(1

1()(31

333

2133

333

3

222

3122

322

2

111

3211

311

1

bwavdt

dwvv

gVvIwvvdt

dv

bwavdt

dwvv

gVvIwvvdt

dv

bwavdt

dwvv

gVvIwvvdtdv

ssss

ssss

ssss

(11)

(a) (b)

(c) (d)

Figure 6. v2 according to v1 for the coupling strength; (a) gs = 0.01, (b) gs = 0.062, (c) gs = 0.064 , dan (d) gs = 0.069.

Synchronization numerically appears for a coupling strength gs = 0.069. This phenomenon is also observed for the variables w, for the same numerical values of parameters.

Coupling four FitzHugh-Nagumo neurons

There are different possible ways for coupling four neurons with the same number of input from other neurons. Here, we decided to connect each neuron to all the others.

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(a) (b)

(c) (d)

Figure 7. v2 according to v1 for the coupling strength; (a) gs = 0.015, (b) gs = 0.018, (c) gs = 0.019 , dan (d) gs = 0.0195.

Synchronization numerically appears for a coupling strength gs = 0.0195. This phenomenon is also observed for the variables w, for the same numerical values of parameters.

This is an emergent property which comes from the collective dynamics of n neurons. Moreover, as given in figure, as the number of neurons n goes larger, the synchronization threshold g gets smaller.

Conclusion

Since the birth of Hodgkin-Huxley equation in neurophysiological modelling, researchers have made considerable efort to try to analyze this system in diferent way. Extensive eforts have been made to discover chaos in many physical and

biological systems including neural systems. We have investigated bifurcations observed in the FitzHugh-Nagumo neuron model as a simplified models. By calculating bifurcations with changing Iext we have identified the parameter regions in which the FitzHugh-Nagumo neuron exhibits properties of excitability and oscillation as an existence of a couple of stable equilibrium points and one stable limit cycle.

In this paper, the characteristics of two dimensional modified FitzHugh-Nagumo neuron model is studied. Dynamical behavior of the modified FitzHugh-Nagumo system under external electrical stimulation is presented and it is verified that the introduction of periodic stimulation modifies the dynamics of

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biological system by presenting the dynamical behavior for the modified FitzHugh-Nagumo system under external electrical stimulation.

Synchronization phenomenon is exhibited in a neuron network, the transmission of information between these neurons is optimal and the network can develop consciousness. This is an emergent property which comes from the collective dynamics of n neurons. Moreover as the number of neurons n goes larger, the synchronization threshold g gets smaller. Consciousness is more important when the number of neurons is larger. This phenomenon which can describe many self-organized complex systems, like earthquakes, linguistic, urban systems. References [1]. Aziz-Alaoui, MA. (2006), Complex

emergent properties and chaos (De) synchronization, Emergent Properties in Natural and Artificial Dynamical Systems. Heidelberg : Springer p129-147.

[2]. Belykh, I., Shilnikov A. (2008), When Weak Inhibition Synchronizes Strongly Desynchronizing Networks of Bursting Neurons. PRL 101, 078102.

[3]. Edelstein-Keshet, L., (1988), Mathematical Models in Biology. Random House, New York.

[4]. FitzHugh, R. (1961), Impulses and Physiological state in theoretical models of nerve membrane. Biophysics Journal.,I,445-466.

[5]. Hodgkin, A. L., and Huxley, A. F. (1952), A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol.,117.500-544.

[6]. Lange, E., Belykh, I.,Hasler, M. (2005), Synchronization of Bursting

Neurons: What matters in the Network Topology. PRL 94, 188101.

[7]. Mishra D, Yadav A, Ray S, Kalra PK. (2006), Controlling Synchronization of Modified FitzHugh-Nagumo Neurons Under External Electrical Stimulation. NeuroQuantology Issue 1 Page 50-67.

[8]. Muratov, C. B. (2008), A quantitative approximation scheme for the traveling wave solutions in the Hodgkin-Huxley model. arXiv.org/abs/nlin/0209053v1.