Multiscale Modeling of Interface Phenomena in Biologyricsac/NotesCMElBioMath.pdf · Multiscale...

Notes of the Course

Multiscale Modeling of InterfacePhenomena in Biology

Ph.D. School in Mathematical Models and Methods in Engineering

Dipartimento di Matematica “F. Brioschi” Politecnico di MilanoMobility Project “Athens” - Politecnico di Milano

Prof. Riccardo Sacco

Dipartimento di Matematica “F. Brioschi” Politecnico di Milano

Piazza Leonardo da Vinci 32 20133 Milano, Italy

E-mail: [email protected]

Home Page: http://www1.mate.polimi.it/~ricsac/

Contents

1. Structure of the Course 3

I. Cellular Interfaces: Structure and Transport Phenomena 5

2. Introduction to Cellular Interfaces and Ion Electrodiffusion 7

2.1. Cells: structure, membrane and ion transport . . . . . . . . . . . . . . . . 7

2.1.1. The cell membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2. Ionic channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2. Transport of charged particles . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1. Units and conventions . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2. The Nernst-Planck equation . . . . . . . . . . . . . . . . . . . . . . 11

3. ODE-Based Modeling of Transmembrane Ion Flow in Cellular Electrophysiol-

ogy 13

3.1. Membrane electrophysiology . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2. General form of transmembrane ionic current densities . . . . . . . . . . . 14

3.3. The ODE model of ion transport . . . . . . . . . . . . . . . . . . . . . . . 15

3.4. Transmembrane current models . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4.1. The linear resistor model . . . . . . . . . . . . . . . . . . . . . . . 16

3.4.2. The Goldman-Hodgkin-Katz model . . . . . . . . . . . . . . . . . . 16

3.4.3. The Hodgkin-Huxley model . . . . . . . . . . . . . . . . . . . . . . 19

3.5. Thermal equilibrium of a system of monovalent ions . . . . . . . . . . . . 20

II. Multiscale Mathematical Models of Transmembrane Ion Transport 23

4. PDE-Based Multiscale Modeling of Ion Flow in Cellular Biology 25

4.1. Electrochemical description of ion flux . . . . . . . . . . . . . . . . . . . . 25

4.1.1. The electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.2. Ion mass flux density . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1.3. Ion electrical current density . . . . . . . . . . . . . . . . . . . . . 27

4.2. Microscale model of cellular ion flow . . . . . . . . . . . . . . . . . . . . . 27

4.3. Multiscale reduction of cellular ion flow . . . . . . . . . . . . . . . . . . . 30

4.4. Macroscale model of ion flow . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.4.1. Electrostatic model of the membrane . . . . . . . . . . . . . . . . . 31

4.4.2. Electrodynamical model of the membrane . . . . . . . . . . . . . . 33

II Indice

4.5. The PNP system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.6. The Cable Equation model . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5. Mathematical Analysis of the Poisson-Nernst-Planck Model 41

5.1. Summary of model equations . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.1.1. Scaling of the PNP system . . . . . . . . . . . . . . . . . . . . . . 42

5.2. The scaled PNP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.3. The electroneutral PNP model (EN-PNP) . . . . . . . . . . . . . . . . . . 45

III. Functional Techniques and Discretization Methods 47

6. Solution Map for the PNP system 49

6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.2. PNP solution map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.2.1. The Nonlinear Poisson equation . . . . . . . . . . . . . . . . . . . 51

6.2.2. The continuity equations . . . . . . . . . . . . . . . . . . . . . . . 53

7. Unified Framework and Well-Posedness Analysis 55

7.1. Unified framework for the PNP solution map . . . . . . . . . . . . . . . . 55

7.2. Weak formulation and well posedness . . . . . . . . . . . . . . . . . . . . . 57

7.2.1. Multi-domain functional setting . . . . . . . . . . . . . . . . . . . . 57

7.2.2. DAR problem reformulation . . . . . . . . . . . . . . . . . . . . . . 57

7.2.3. Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7.2.4. Well posedness analysis and stability estimate . . . . . . . . . . . . 59

8. Finite Element Approximation of the DAR Problem 61

8.1. Motivation to the use of a DMH method . . . . . . . . . . . . . . . . . . . 61

8.2. Geometric Discretization and Finite Element Spaces . . . . . . . . . . . . 63

8.3. A Mixed–Hybridized Method with Numerical Quadrature . . . . . . . . . 66

8.4. Implementation and Post-Processing of the DMH-FV Method . . . . . . . 74

8.4.1. Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

8.4.2. Post-Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

IV. Simulation Results, Applications and Advanced Topics 77

9. Numerical Validation of the DMH-FV Method 79

9.1. Static condensation CAMBIA TITOLO . . . . . . . . . . . . . . . . . . . 79

9.2. A one-dimensional heterogeneous domain . . . . . . . . . . . . . . . . . . 82

9.3. Stationary profile of a binary electrolyte at a boundary . . . . . . . . . . . 83

9.4. Simulation of a neuro-chip . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

9.5. Action potential propagation in an axon . . . . . . . . . . . . . . . . . . . 87

10.Advanced Topics, A Look at the Future and Concluding Remarks 91

10.1. Applications of the future . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

10.2. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Indice III

Acknowledgments

I wish to gratefully acknowledge the fundamental contribution given by Matteo Porro,

PhD student in ”Mathematical Models and Methods in Engineering” at Politecnico di

Milano, in the development and computer implementation of the various multiscale mod-

els illustrated in the course, for his personal assistance during the hands-on sessions on

the computer and for carefully reading (and proof-checking!!) these Lecture Notes.

I wish also to thank Emanuela Abbate, MSc student in Mathematical Engineering,

Politecnico di Milano, for her precious assistance in developing multiscale models and

algorithms for neural interfaces.

Last but certainly not least, my grateful thanks go also to Prof. Joseph W. Jerome,

Dr. Bice Chini, Dr. Yoichiro Mori and Dr. Paola Causin for all the enlightning and

productive discussions and years of common work in the area of mathematica modeling

and simulation of Ionic Transport in Cellular Biology.

Finally, I wish also to thank Marco Brera and Fabio Manganini, two former MSc bril-

liant students under my supervision at Politecnico di Milano, that helped me developing

and implementing a substantial part of models and methods discussed in these Notes.

Indice 1

Introduction

These notes represent the supporting material for the course entitled

Multiscale Modeling of Interface Phenomena in Biology

to be held within the PhD programme ”Mathematical Models and Methods in Engineer-

ing” and within the Mobility Project ”Athens”. For further information on these two

initiatives, refer to:

- http://www.mate.polimi.it/dottorato/index.php?lg=it

- www.athensprogramme.com (course catalogue POLI13)

The course has a duration of 27 hours, distributed over a period of one week and

organized into lectures and hands-on sessions on the computer. The scope of the course

is to introduce the mathematical modeling aspects of biological interfaces and the com-

putational techniques that can be used for their successful simulation.

These notes are divided into four distinct parts:

• Part I gives a short introduction to cellular biology and electrophysiology, providing

also a first example of mathematical model of ion transport across a cellular mem-

brane. The proposed model is based on an equivalent electrical representation of

the cell and is constituted by a system of Ordinary Differential Equations (ODEs)

derived from the application of Kirchhoff’s current law.

• Part II illustrates the mathematical models of ion transport based on systems

of Partial Differential Equations (PDEs) that generalize to the multi-dimensional

setting the basic ideas developed in Part I.

• Part III describes the functional techniques used to decouple the equation systems

and the corresponding numerical methods used for their temporal and spatial ap-

proximate solution.

• Part IV concludes the course presentation with the simulation of several test studies

of interest in biological applications and with a short overview of a few selected

advanced topics including:

1. advanced bio-hybrid interfaces (example: prototype of artificial retina);

2. cellular mechanotransduction processes (example: calcium sparks induced by

chemical and mechanical stimuli).

1. Structure of the Course

Interfaces are ubiquitous in Biology. They have the function to separate two neighbouring

regions maintaining a condition of local equilibrium through a dynamical control of the

electro-chemical and mechanical processes occurring thereby. The ratio between the

interface thickness and the characteristic size of the system is usually a very small quantity

- of the order of 1/1000 in the case of cellular membrane - so that a hierarchical multiscale

vision of the coupling between interface and surrounding environment seems to be the

right option for an efficient and accurate modeling of the problem.

MATHEMATICAL MODELS Motivated by the above consideration, in this course

we aim to provide an introduction to Computational Modeling of interface phenomena

in biological systems by illustrating in detail a hierarchy of ODE/PDE equation-based

models for the study of cellular electrical activity (CEA). The members of the hierarchy,

in increasing order of complexity, include:

(CEA1) 0D equivalent electrical lumped parameter models;

(CEA2) the Cable Equation (CE) model;

(CEA3) the Poisson-Nernst-Planck (PNP) and the electroneutral PNP (EN-PNP) models

for electrodiffusion of Mion ionic species in an electrolyte.

In all CEA formulations, the dynamical response of the interface to externally applied

stimuli is represented by equivalent electrical models including capacitors and nonlinear

resistors to account for charge accumulation and trans-interface currents (TIC). TICs

are treated mathematically by adopting hierarchical models of increasing complexity

that comprise:

(TIC1) the linear resistor model;

(TIC2) the Goldman-Hodgkin-Katz nonlinear model;

(TIC3) the Hodgkin-Huxley system of ODEs.

LINEARIZATION AND NUMERICAL APPROXIMATION Combining together the

CEA/TIC model hierarchies, a highly nonlinear system of ODE/PDEs is obtained. CEA1

is numerically treated by resorting to ODE solvers for stiff problems, while in CEA2 and

CEA3 temporal semi-discretization is treated using the Backward Euler method. System

linearization is dealt with using a Gummel-type fixed-point iteration. This approach

leads to the successive solution of an heterogeneous-domain advection-diffusion-reaction

4 1. Structure of the Course

(HD-ADR) boundary value model problem in conservative form with appropriate flux

transmission conditions across the interface. The numerical approximation of the HD-

ADR problem is dealt with mixed-hybridized finite element methods (MH-FEM) char-

acterized by strong flux conservation across interelement boundaries and robustness in

treating sharp fronts and/or discontinuities.

MATHEMATICAL AND NUMERICAL ISSUES

a) well-posedness of the linearized boundary-value problems;

b) existence and uniqueness of a fixed point;

c) convergence of the iterations;

d) asymptotic analysis with respect to singular perturbation parameters;

e) stability of the MH-FEM scheme (discrete maximum principle);

f) convergence analysis with respect to mesh size;

g) algebraic structure of the MH method and efficient implementation through the use

of static condensation;

h) derivation of a robust and efficient finite volume MH scheme by lumping of the local

mass flux matrix.

SIMULATION RESULTS AND MODEL VALIDATION

A) study with the PNP model of the stationary profile of a binary electrolyte (Mion = 2)

at a boundary (comparison with exact solution);

B) study with the PNP model of the time-dependent electrodiffusion of a ternary elec-

trolyte (Mion = 3) across a membrane (critical analysis of the electroneutrality

assumption);

C) study with PNP and EN-PNP models of the propagation of an action potential along

a neuronal axon;

D) simulation with PNP and EN-PNP models of cellular interfacing with an electronic

substrate of inorganic/organic type (neurochip).

ADVANCED TOPICS AND FURTHER READING The concluding remarks of the

Course are devoted to a short overview of a few selected advanced topics (listed be-

low), including bibliographical references, with suggestions for further development and

application of ideas and methods addressed in the notes:

1. study and simulation of advanced bio-hybrid interfaces (example: prototype of

artificial retina);

2. study and simulation of cellular mechanotransduction processes (example: calcium

sparks induced by chemical and mechanical stimuli).

Part I.

Cellular Interfaces: Structure and

Transport Phenomena

2. Introduction to Cellular Interfaces and

Ion Electrodiffusion

In this chapter, we give a short introduction to the biological setting object of the course.

In particular, we illustrate the structure of a cell and of its membrane. Then, we introduce

the ionic channels and we discuss their role in the regulation of the cellular electrical

activity. For more detailed information, we refer to [Hil01, GH06, BF01].

2.1. Cells: structure, membrane and ion transport

The basic living unit of the body is the cell : each organ is an aggregate of many different

cells held together by intercellular supporting structures. The entire body contains about

100 trillion cells (∼ 1014 cells). Although the many cells of the body often differ markedly

from one another (since each type of cell is specially adapted to perform one or a few

particular functions), all of them have certain basic characteristics that are alike: a

typical cell, as seen by the light microscope, is shown in Fig. 2.1. Its two major parts

are the nucleus and the cytoplasm. The nucleus is separated from the cytoplasm by a

nuclear membrane, and the cytoplasm is separated from the surrounding fluids by a cell

membrane. The principal fluid medium of the cell is water, which is present in most

cells, except for fat cells, in a concentration of 70 to 85 %. Many cellular chemicals are

dissolved in the water. Others are suspended in the water as solid particulates. Chemical

reactions take place among the dissolved chemicals or at the surfaces of the suspended

particles or membranes.

2.1.1. The cell membrane

The cell membrane (also called the plasma membrane), which envelops the cell, is a

thin, pliable, elastic structure with a thickness tm of only 7.5 to 10 nanometers. The

diameter dc of cells varies from 7.5 to 150 micrometers . Thus, the ratio tm/dc varies

in the range 50 × 10−6 ÷ 1.3 × 10−3. Despite of its relative small dimension, the cell

membrane plays some fundamental roles for the living of the cell: it preserves the cell

integrity separating the intracellular fluid from the extracellular fluid and it regulates

the passage of substances from the outside to the inside of the cell, and viceversa.

The cell membrane consists almost entirely of a lipid bilayer, but it also contains large

numbers of protein molecules in the lipid, many of which penetrate all the way through

the membrane, as shown in Fig. 2.2. Most of these penetrating proteins constitute a

pathway through the cell membrane. Some proteins, called channel proteins, allow free

movement of water as well as selected ions or molecules. The channel proteins are usually

8 2. Introduction to Cellular Interfaces and Ion Electrodiffusion

Figure 2.1.: Structure of the cell as seen with the light microscope.

highly selective with respect to the types of molecules or ions that are allowed to cross

the membrane.

Figure 2.2.: Cross-section of a cell membrane.

Ions provide inorganic chemicals for cellular reactions. Also, they are necessary for

operation of some of the cellular control mechanisms. For instance, ions acting at the

cell membrane are required for transmission of electrochemical impulses in nerve and

muscle fibers. The most important ions in the cell are potassium K+, magnesium Mg++,

phosphate, sulfate SO−−4 , bicarbonate HCO−3 , and smaller quantities of sodium Na+,

chloride Cl−, and calcium Ca++. An ion is called cation if positively charged, anion if

negatively charged.

2.1. Cells: structure, membrane and ion transport 9

2.1.2. Ionic channels

Certain cells, commonly called excitable cells, are unique because of their ability to gen-

erate electrical signals. Some examples are neuron cells, muscle cells, and touch receptor

cells. Like all cells, an excitable cell maintains a different concentration of ions in its cy-

toplasm than in its extracellular environment. Together, these concentration differences

create a small electrical potential across the plasma membrane. Then, when the potential

difference reaches a threshold value, typically 55 millivolts, specialized channels in the

plasma membrane, called ion channels, open and allow rapid ion movement into or out

of the cell, and this movement creates an electrical signal. All of these processes charac-

terize the so-called cellular electrical activity (CEA) that represents the way ion channels

can generate an electrical current flowing between a cell and another, thus providing a

fundamental aspect in the life of every biological system.

Ionic channels are large proteins that reside in the membrane of cells (Fig. 2.3) and

conduct ions through a narrow tunnel of fixed charge formed by the amino acid residues

of the protein.

Channels are ideally placed across the membrane in series with the intracellular en-

vironment to control the cellular biological function [SBT02]. Ion channels should be

viewed as natural nanotubes that relate the electrolyte solutions in and outside the cell

to the electric field that is established across the cell membrane.

Figure 2.3.: Ion channels function as pores to permit the flux of ions down their electro-

chemical potential gradient.

Channels are responsible for signaling in the nervous system, coordination of muscle

contraction, and transport in all tissues. Channels are obvious targets for drugs and

disease [Sch96]: as a matter of fact, many of the drugs used in clinical medicine act

directly or indirectly through channels.


2.2. Transport of charged particles

In biological channels, ions move under the controlled action of gradients of concentration

and electric potential [Hil01]. The (opposite) of the gradient of the electric potential,

which is the electric field, forces a particle to move accordingly to its sign (Coulomb’s

law). This mechanism is called drift (see Fig. 2.4(a)). The gradient of concentration

moves particles from regions at higher concentration to regions at lower concentration

(Fick’s law). This mechanism is called diffusion (see Fig. 2.4(b)). The flow of ions

through the channel generates an electrical current denoted henceforth by the symbol

I. From a practical point of view, we are mainly interested in computing the current

density J , which is the charge that crosses a generic cross-sectional area S in a unit of

time, in such a way that (in terms of modules) we can write

J =I

S. (2.1)

Using linear superposition of the drift and diffusion processes, the current density takes

the following form

J = Jdrift + Jdiffusion. (2.2)

Notice that Jdrift and Jdiffusion might act in opposite directions, as shown in Fig. 2.4.

(a) Drift (b) Diffusion

Figure 2.4.: Drift and diffusion currents. The electric field is denoted by E.

2.2.1. Units and conventions

To characterize in mathematical terms the drift-diffusion model for the current density

we need the following fundamental quantities:

• c the concentration of the ionic species;

• z the valence of the ion (dimensionless);

• D the diffusion coefficient of the ionic species (units: m2 s−1);

• E = −∂ϕ∂x

electric field (units: V m−1), where x denotes the spatial coordinate of

ion flow (units: m) and ϕ is the electric potential (units: V).

2.2. Transport of charged particles 11

The definition of the units for the ion concentration c is not unique, according to the

physical meaning expressed by the variable. In fact, if c is a molar density its appropriate

units are mol m−3, while if c is a number density the right units are #ions m−3 (more

shortly, m−3). The first choice is typical of bio-electrochemical applications while the

second choice is the preferred one in electronics applications. We adopt henceforth the

following convention: in numerical computations and/or input/output tables of data,

the concentration will be expressed in mol m−3, while in all the mathematical modeling

treatment the concentration will be expressed in m−3. Thus, possible change of units

will have to be done only twice, one time before computations, the other as a simple

post-processing.

2.2.2. The Nernst-Planck equation

The number flux density of ions through the membrane along the direction x that is driven

by the simultaneous effect of concentration gradients and electric field is expressed by

the relation

f = −D ∂c

∂x︸︷︷︸Fick diffusion

− zq

KTDc

∂ϕ

∂x︸︷︷︸Drift

(2.3)

where KB is Boltzmann’s constant (units: J K−1) and q is the electron charge (units: C).

The units of the flux are

[f ] = [D] ·m−4 = m2 s−1 ·m−4 = m−2 s−1.

Multiplying this latter expression by qz yields

J = qz × f = −qzD(∂c

∂x+

zq

KTc∂ϕ

∂x

). (2.4)

The units of J are

[J ] = [q]× [f ] = Cm−2 s−1 = A m−2

from which we conclude that J is a current density. Equation (2.4) is known as Nernst-

Planck relation and is the most widely used modeling tool in Electrophysiology to describe

the motion of ions under the effect of diffusive and drift forces.

We conclude this section with some concepts and definitions that will play an impor-

tant role in the discussion to follow.

Remark 2.2.1 (Thermal equilibrium). The special (and important) case where Jdrift =

−Jdiffusion corresponds to J = 0 and is commonly referred to as thermal equilibrium

(see Sect. 3.2). In this situation, the net current flow across a section is null although

the separate drift and diffusion contributions are non-zero. Thermal equilibrium is a dy-

namical state and allows to introduce the so-called Nernst potential that is a fundamental

quantity in the analysis of the electrical activity of every excitable cell.

Definition 2.2.1 (Nernst potential). The Nernst potential (units: V) associated with

the ionic species c is defined as

Ec =KBT

zqln

(c(out)

c(in)

)=RT

zFln

(c(out)

c(in)

)(2.5)


where R is the universal gas constant (units: J mol−1 K−1) and F is Faraday’s constant

(units: C mol−1).

Definition 2.2.2 (Thermal equilibrium).

Thermal equilibrium ⇔ J = 0 ⇔ ϕm = Ec. (2.6)

3. ODE-Based Modeling of Transmembrane

Ion Flow in Cellular Electrophysiology

As anticipated in Chapter 2, the cell membrane is a biological interface that separates

the interior of every cell from the outside environment. The cell membrane is selectively

permeable to ions and organic molecules and controls the movement of substances in and

out of the cell. The basic function of cell membrane is to protect the cell from its sur-

roundings, but it is also involved in a variety of cellular processes such as cell adhesion,

ion conductivity and cell signaling. In this chapter, focus on the membrane electrical

activity developing a biophysically sound description of the ionic currents through the

membrane. The resulting mathematical model is based on an equivalent electrical rep-

resentation of the cell and is constituted by a system of Ordinary Differential Equations

(ODEs) derived from the application of Kirchhoff’s current law. The system must be

solved at each spatial point of the membrane surface and at each time level of the tempo-

ral evolution of the biophysical problem under given initial conditions for the membrane

potential and the ionic concentrations in the intra- and extracellular sites.

3.1. Membrane electrophysiology

In the simplest picture, the electrical properties of cell membrane are represented in terms

of the electrical equivalent circuit illustrated in Fig. 3.1. Several electrical components

can be identified in the scheme. Resistors (of both linear and nonlinear types) are used

to model the various types of ion channels embedded in the membrane. Voltage supply

generators (batteries) are used to represent the electrochemical potentials induced by

different values of intra- and extracellular ion concentrations (Nernst potentials, also

called reversal potentials). Capacitors are used to model the charge storage capacity of

the cell membrane during transient phenomena.

In the equivalent circuit, the current across the membrane has two major components,

one associated with the membrane capacitance and the other associated with the flow of

ions through resistive membrane channels. The behavior of the electrical circuit shown

in Fig. 3.1 can be described by the following Ordinary Differential Equation (ODE)

Cmdϕmdt

+ Itot = 0 (3.1)

where:

• ϕm := ϕin − ϕout is the membrane potential, given by the difference between the

value of the electric potential ϕ in the intracellular space and that in the extracel-

lular space;

143. ODE-Based Modeling of Transmembrane Ion Flow in Cellular

Electrophysiology

Figure 3.1.: Electrical equivalent circuit for cell membrane. The capacitor represents

the capacitance of the cell membrane; the three variable resistors repre-

sent voltage-dependent conductances, the fixed resistor represents a voltage-

independent conductance and the batteries represent reversal potentials for

the corresponding conductances.

• Cm is the cell membrane capacitance (units: F);

• Itot is the sum of the transmembrane ionic currents flowing through the linear and

nonlinear resistors in Fig. 3.1 (units: A = C s−1).

Equation (3.1) is the fundamental Kirchhoff current law relating the time rate of change

of the membrane potential to the currents flowing across the membrane.

Definition 3.1.1 (0D - Lumped parameter model). A model like that schematically rep-

resented in Fig. 3.1 will be henceforth referred to as a 0D model, because the dependence

of the unknown ϕm on the spatial variable is neglected in Eq. (3.1). Using the language

of electrical engineering, the 0D model is also called lumped parameter model, because

the complex bio-physical behaviour of the cell membrane with respect to ion transport

is synthetically represented by the lumped electrical parameters (membrane capacitance,

conductances, batteries and current generators) appearing in Fig. 3.1. More general

mathematical formulations based on the use of a distributed parameter approach will be

considered in the remainder of these notes.

At this point, the main challenge is to characterize the form of Itot on the basis of

definition (2.1) and of the general drift-diffusion relation (2.2).

3.2. General form of transmembrane ionic current densities

Transmembrane ionic currents are currents that flow through ion channels, transporters,

or pumps that are located within the cell membrane. To define in mathematical terms

these currents we adopt the formalism of Hodgkin and Huxley [HH52], generalized here

to allow for nonlinear instantaneous current-voltage relations and ion concentration ef-

fects [Mor06]. In the remainder of these notes we always assume, otherwise differently

stated, that in the considered electrolyte solution a number of Mion ≥ 1 ionic species is

flowing. Each ion has a concentration ci and a ionic valence zi in such a way that the

3.3. The ODE model of ion transport 15

amount of charge carried by the ionic species per unite volume is qzici, q being the ele-

mentary charge of the electron equal to 1.602 · 10−19 Coulomb. In accordance with (2.1),

we introduce the quantity S that represents an arbitrarily chosen cross-sectional area of

the membrane surface across which the considered transmembrane current Ii is flowing.

Then, the transmembrane current density (units: A m−2) associated with the i-th ionic

species has the following expression

Ji = Ji

(x,y(x, t), ϕm(x, t), c(in)(x, t), c(out)(x, t)

)i = 1, . . . ,Mion. (3.2)

In the above relation, x denotes the spatial position vector along the membrane, t is the

time variable while the other parameters are defined as follows:

• y(x, t) = (y1, . . . , yNg) is a vector of gating variables where Ng is the total number of

gating variables in all of the channel types that arise in our system. The individual

components of y are dimensionless variables in the range [0, 1] and describe the

time-dependent activation or inactivation profile of the channel.

• ϕm(x, t) is the transmembrane potential. Keeping the other parameters fixed in

Ji, and letting only ϕm vary, we get the instantaneous current-voltage relationship

for the i-th ion from the extracellular space to the intracellular space at point x at

time t.

• c(in)(x, t) = (c(in)1 , . . . , c

(in)Mion

) (and similarly c(out)) is the vector of ion concentra-

tions in the intracellular (respectively extracellular) space. By including the whole

vector of ion concentrations, we allow for the possibility that the current density

carried by the i-th species of ion is influenced by the concentrations of other ionic

species on the two sides of the membrane.

The functional relation (3.2) expresses the bio-physical fact that the current density

of the i-th ion may be influenced by the transmembrane current density of the other

channels (possibly of more than one type) that carry the i-th species of ion across the

membrane separating the intracellular and the extracellular space. Finally, the explicit

dependence of Ji on x reflects the possible inhomogeneity of the membrane, because the

density of channels may vary from one location to another.

3.3. The ODE model of ion transport

Replacing the general expression of the transmembrane current density (3.2) into (3.1)

we obtain the following ODE to be solved at each time level t > 0 in correspondance of

each spatial position x on the membrane:

cmdϕmdt

(x, t) = −Mion∑i=1

Ji

(x,y(x, t), ϕm(x, t), c(in)(x, t), c(out)(x, t)

)(3.3a)

ϕ(x, 0) = ϕ0(x). (3.3b)

In (3.3), the quantity cm is the specific membrane capacitance (units: F m−2) related

to Cm through the relation cm = Cm/Sm, Sm being the area of the membrane, while

ϕ0(x) is the initial value of the membrane potential at each point of the membrane.


Electrophysiology

The concentrations c(in)(x, t) and c(out)(x, t) of the Mion ionic species are assumed to

be (biophysically suitable) given functions, and the same holds for the gating variables

y(x, t). We shall see that this is not the case with the more general PDE-based models

treated in the remainder of these notes. Under these assumptions, (3.3) is a system of

Cauchy problems for the membrane potential ϕm to be solved (in principle) at each spatial

position x of the membrane. In practice, this cannot be done and the ODE system (3.3)

is solved only at a finite number of points xk suitable selected over the membrane surface.

Examples of this approach will be discussed in Part IV of these notes.

3.4. Transmembrane current models

In this section we present the models for the transmembrane current densities that are

most commonly used in the theoretical and computational description of cellular electrical

activity. For a fully detailed treatment of this complex subject, we refer to [KS98, Hil01,

ET10].

3.4.1. The linear resistor model

This is the simplest current-voltage relationship because the ionic current density Ji of

the i-th ion can be expressed as

Ji = gi (ϕm − Ei) = gi

(ϕin − ϕout −

KBT

zqln

(c

(out)i

c(in)i

)), (3.4)

gi being the specific conductance associated with the ionic species ci. The units of gi are

[gi] = A V−1 m−2 = S m−2.

By inspection, we see that (3.4) is consistent with the thermal equilibrium condition (2.2.2).

The graphical representation of the current-voltage relationship (3.4) in the (ϕm, J) plane

is a straight line whose slope is equal to gi. Despite its simplicity, the linear resistor model

proves to be quite accurate in many cases and it is used for instance in [Fro03].

3.4.2. The Goldman-Hodgkin-Katz model

The Goldman-Hodgkin-Katz (GHK) model is a first, significant, example of improvement

of the linear resistor formulation of the previous section. To derive a realistic model of

fluxes and currents that flow across the cellular membrane it is necessary to make some

simplifying assumptions (see also [KS98, Rub90]):

1. equation (2.4) holds across the membrane;

2. ions move independently;

3. the electric field constant across the membrane;

4. the current density is constant across the membrane.

3.4. Transmembrane current models 17

Figure 3.2.: Schematics of ion electrodiffusion across the cell membrane.

Referring to Fig. 3.4.2 for the notation and indicating by x the spatial coordinate par-

allel to the channel (black arrow in the figure) and by l the membrane thickness, the

application of assumption 3. yields

E = −∂ϕ∂x

= constant = −ϕout − ϕinl

=ϕml.

Replacing this relation in (2.4) we get

Ji = −qziDi

(∂ci∂x− ziq

KTciϕml

). (3.5)

Then, applying assumption 4. to (3.5) we obtain

Ji = constant =⇒ ∂Ji∂x

= 0 =⇒ ∂2ci∂x2

− ziq

KT· ϕml· ∂ci∂x

= 0.

The solution of the above differential equation is given by

ci(x) = A+B exp( ziqKT

ϕmx

l

)(3.6)

where A and B are arbitrary constants that can be found by imposing the following

boundary conditions at the two sides of the channel:

ci(0) = c(in)i

ci(l) = c(out)i

=⇒

A+B = c

(in)i

A+Beziq

KTϕm = c

(out)i

=⇒

A = c

(in)i −

c(out)i − c(in)

i

eziq

KTϕm − 1

B =c

(out)i − c(in)

i

eziq

KTϕm − 1

.

For notational simplicity, we introduce the dimensionless variable

X := ziϕmKT/q

which has the physical meaning of a normalized electric potential. Then, to compute the

constant current density throughout the channel we replace (3.6) into (3.5) to obtain

Ji = −FziDi

(BX

lexp

(Xx

l

)− X

lA−BX

lexp

(Xx

l

)).


Electrophysiology

The first and third term in the braces at the right-hand side mutually cancel out, and

we are left with the constant current density

Ji = FziDiX

lA = −FziDi

1

l

[X

eX − 1c

(out)i − XeX

eX − 1c

(in)i

].

It is useful to introduce the inverse of the Bernoulli function

B(X) :=X

eX − 1. (3.7a)

A plot of B(X) and B(−X) is reported in Fig. 3.3.

Figure 3.3.: Plot of B(X) (solid line) and of B(−X) (dashed line).

This function has several properties:

B(X) > 0 ∀X ∈ R; (3.7b)

B(0) = 1; (3.7c)

eXB(X) = B(−X) = X + B(X); (3.7d)

limX→+∞

B(X) = 0+, limX→−∞

B(X) = −X; (3.7e)

limX→+∞

B(−X) = X, limX→−∞

B(X) = 0+. (3.7f)

Using the definition (3.7) and property (3.7d), the constant current density can be

written as

Ji = −qziDi1

l

[B(ziϕmKT/q

)c

(out)i − B

(− ziϕmKT/q

)c

(in)i

]. (3.8)

This is the celebrated Goldman-Hodgkin-Katz (GHK) equation for the current density

associated with the i-th ion.

Let us check that (3.8) satisfies the thermal equilibrium condition (2.2.2). Using

properties (3.7b) and (3.7d), we have

Ji = 0 =⇒ c(out)i − c(in)

i exp(ziqϕmKT

)= 0

from which we get

ϕm =KT

ziqln

(c

(out)i

c(in)i

)≡ Ei

3.4. Transmembrane current models 19

that is, the membrane potential coincides with the Nernst potential as required at thermal

equilibrium.

The GHK current density enjoys other interesting properties. Assume that ϕm = 0

(i.e., the intra- and extracellular potentials have the same value). Then, using prop-

erty (3.7c) the GHK current density degenerates in

Ji = −qziDic

(out)i − c(in)

i

l.

This formula corresponds to a pure diffusion ion flow across the membrane in agreement

with the Nernst-Planck relation (3.5) in absence of electric field.

Conversely, assume that c(out)i = c

(in)i = ci (i.e., the intra- and extracellular ion

concentrations have the same value ci). Then, using property (3.7d) the GHK current

density degenerates in

Ji = −qziDicil

(−zi

ϕmRT/q

).

This formula corresponds to a pure drift ion flow across the membrane in agreement with

the Nernst-Planck relation (3.5) in absence of a concentration gradient.

The above analysis shows that the GHK expression of the ion current density au-

tomatically adapts itself to all possible transport regimes. This makes it amenable to

numerical computations and gives the reason of its wide success and implementation in

contemporary simulation tools.

3.4.3. The Hodgkin-Huxley model

The Hodgkin-Huxley (HH) model [HH52] is a further extension of the simple linear

resistor theory and accounts for voltage-gating mechanism of the channels, which in turn

permits the simulation of the propagation of an action potential. Four ionic species, Na+,

K+ and Cl− are included in the mathematical description because they are responsible

for the majority of the ionic current in a cellular action potential [KS98, Hil01, ET10].

The HH model requires solving a nonlinear, stiff ODE system, that is:

Jm = Cmdϕmdt

+ JK + JNa + JL

JK = n4 gK(ϕm − EK)

JNa = hm3 gNa(ϕm − ENa)

JL = gL(ϕm − EL).

(3.9)

The membrane current Jm is the sum of two ionic current fluxes, associated with

potassium and sodium, a membrane capacitance contribution Cmdϕmdt

and a leakage

current JL. For this latter term, Hodgkin and Huxley do not specify what ion is carrying

it [HH52]. However, since the contribution of the leakage current is quite small, it may

be neglected or arbitrarily associated with both K+ and Cl−. gK , gNa and gL are the

constant specific conductances, while EK , ENa and EL are the reversal potentials of

each ion. The variables m, n and h are called gating variables, because they describe the


Electrophysiology

opening state of the channel. These variables typically vary between 0 and 1, and each

of them are governed by the following ordinary differential equations at each point of the

membrane∂s

∂t= αs(ϕm)(1− s)− βs(ϕm)s, s = m,n, h, (3.10)

where αs and βs are experimentally determined functions. Hodgkin and Huxley [HH52]

used the following functions:

αm = B((25− ϕm)/10)

βm = 4 exp(−ϕm/18)

αh = 0.07 exp(−ϕm/20)

βh = C((30− ϕm)/10)

αn = 0.1B((10− ϕm)/10)

βn = 0.125 exp(−ϕm/80)

where ϕm is expressed in mV and C(t) := 1/(et + 1). A complete analysis of this ODE

system can be found in [KS98]. It is worth noting that accounting for each considered

type of ion channel would lead to a system of increased complexity compared to the linear

model (3.4) and the GHK model (3.8) in terms of the number of state variables. Since

each of these state variables is a local property of the membrane interface, the overall

complexity introduced by the HH model is significant.

3.5. Thermal equilibrium of a system of monovalent ions

In Sect. 2.2.2 we have addressed the important issue of thermal equilibrium for the

electrodiffusive flow of a single ionic species flowing across the cell membrane. In this

section we consider the more general case of a system of Mion monovalent ions (i.e.,

zi = ±1) that are moving in a biological fluid environment, in such a way that

Mion = M+ +M−

where M+ is the number of ions with zi = +1 and M− the number of ions with zi = −1.

To extend Defns. 2.2.1 and 2.2.2 to this (more realistic) situation we can profitably use

the GHK theory of Sect. 3.4.2 and write the following generalization of Def. 2.2.2 as

Definition 3.5.1 (Thermal equilibrium for a system of Mion ≥ 1 monovalent ions).

Thermal equilibrium ⇔ Jtot,Mion = 0 ⇔ ϕm = Ec,Mion (3.11a)

where

Jtot,Mion =

Mtot∑i=1

Ji (3.11b)

3.5. Thermal equilibrium of a system of monovalent ions 21

Ji being given by (3.8), and

Ec,Mion =RT

Fln

M+∑i=1

P+i c

(+,out)i +

M−∑i=1

P−i c(−,out)i

M+∑i=1

P+i c

(+,in)i +

M−∑i=1

P−i c(−,in)i

. (3.11c)

Formula (3.11c) is the so-called Goldman equation and the quantity P±i := D±i /l is the

membrane permeability with respect to the i-th ion, i = 1, . . . ,Mion. The quantity Ec,Mion

is the reversal (or equilibrium) potential of the whole system of ions and is also called

Goldman potential.

Remark 3.5.1. It is important to notice that Def. 3.5.1 does not require the single ion

current density to be equal to zero in thermal equilibrium, but only the (weaker) condition

that the total current density sums up to zero. It is clear that in the case where the cell is

much more permeable to a specific ion than to all the others (highly selective membrane),

the Goldman relation tends to the classical Nernst potential (2.5) and thermal equilibrium

of the whole system is regulated by the most permeant species.

Part II.

Multiscale Mathematical Models of

Transmembrane Ion Transport

4. PDE-Based Multiscale Modeling of Ion

Flow in Cellular Biology

In this chapter, we propose and investigate a hierarchy of mathematical formulations,

based on Partial Differential Equations (PDEs), for the description of ion flow in a cellular

system. In doing this, we extend to the multi-dimensional setting the basic Nernst-Planck

ion transport model (2.4). We start from a microscale view of the system, where the

membrane is described in detail as well as the extra-and intracellular sites. Then, we

perform an upscaling procedure that allows us to eliminate the geometrical description of

the membrane through the introduction of suitable transmission conditions that regulate

ion flow and charge accumulation at the two sides of the cellular environment. The

resulting formulation is the so-called macroscale model. This latter is characterized by

a considerable reduction of computational complexity compared to the microscale model

without a significant loss of bio-physical accuracy, and for this reason it is the basic

starting choice in numerical simulations.

4.1. Electrochemical description of ion flux

In this section, we give a brief summary of the basic laws describing the flow of a chemical

species in an ionic solution (denoted henceforth as the environment or medium). To this

purpose, let c = c(x, t) denote the concentration of a given ionized species having valence

z at point x and time t. We assume that the units of x, t and c are m, s and m−3,

respectively. We also denote by D (units: m2 s−1) and µ (units: V m−2 s−1) the diffusion

coefficient and mobility of ion c, respectively. Assuming the validity of Einstein’s relation

(cf. [KS98], Chpt. 2), we have

D =µVth|z|

(4.1)

where Vth := KBT/q = RT/F is the thermal voltage (units: V), T being the absolute

temperature of the environment, KB being the Boltzmann constant, q the electron charge,

R the gas constant (Rydberg constant) and F the Faraday constant, respectively.

4.1.1. The electric field

Charged ions in motion throughout the medium experience, at a point x and at time t,

an electric force proportional to the local value of the electric field E = E(x, t) (units:

V m−1). According to the quasi-static assumption in Electrodynamics [Jac99], which is

by far satisfied in the present context, the electric field can be expressed as the gradient

26 4. PDE-Based Multiscale Modeling of Ion Flow in Cellular Biology

of the electric potential ϕ = ϕ(x, t) as

E = −∇ϕ. (4.2)

The electric field cannot, in general, be considered a given function, rather it is self-

consistently determined by the solution of the Poisson equation

divD = ρ (4.3)

where D is the electric displacement vector defined as

D = εE (4.4)

ε being the dielectric constant of the medium in which ion flow takes place, and ρ is the

electrical charge density at each point x of the environment and at each time t, defined

as

ρ = ρ0 + q

Mion∑i=1

zici (4.5)

where Mion is the number of ion species that are simultaneously flowing in the medium

and ρ0 is a given fixed charge density accounting, for instance, for the presence of im-

mobile charged ions localized throughout the environment. The units of D, ρ and ε are

C m−2, C m−3 and F m−1, respectively. In the remainder of these notes we take

ε = εw = ε0εwr = 80ε0 (4.6)

where ε0 is the dielectric constant of vacuum and εwr is the relative dielectric constant of

water. The above formula amounts to assuming that the liquid constituting the biological

environment in which ions flow (intracellular site, extracellular site, membrane) is made

entirely of water.

4.1.2. Ion mass flux density

Flux balance for the ion species c is expressed by the following PDE in conservative form

∂ c

∂ t+ divf = P (4.7)

where P is a production term and the flux of ion c is given by the following DD relation,

that is well known as the Nernst-Planck (NP) equation [Rub90, KS98, Hil01]

f = µz

|z|cE︸︷︷︸

drift flux

−D∇ c︸︷︷︸diffusion flux

. (4.8)

Notice that (4.8) is the multi-dimensional counterpart of (2.4) upon the use of Einstein

relation (4.1) and of the definition (4.2). Using (4.1) into (4.8), we see that the DD ion

flux can be written in the following equivalent purely advective form

f = cvc (4.9)

where vc is the advective (or drift) velocity of ion c, defined as

vc = −µ z

|z|∇ϕec. (4.10)

4.2. Microscale model of cellular ion flow 27

The scalar function ϕec is the electrochemical potential, defined as

ϕec = ϕ+Vthz

ln

(c

cref

)(4.11)

where cref is a reference concentration. The units of ϕec are V. From (4.9), we see that

the units of f are m−3 m s−1 = m−2 s−1, consistently with the fact that f is a number

flux density, that is, a time rate of number of ions per unit area.

4.1.3. Ion electrical current density

Since ions are electrically charged, it is natural to associate with the ion flux f the ion

current density

J := qzf . (4.12)

Again, we notice that (4.12) is the multi-dimensional counterpart of the first relation

in (2.4). The units of J are C m−2 s−1 = A m−2, consistently with the fact that J repre-

sents a time rate of charge per unit area. Replacing (4.9) and (4.10) into (4.12), we see

that the ion current density can be expressed in the form of generalized Ohm’s law

Jc = σEc (4.13)

where

σ := q |z|µ c, Ec := −∇ϕc (4.14)

represent the ionic conductivity and the electrochemical field acting on ion c, respectively.

The units of σ are S m−1. We notice also that if the ion concentration is constant, then

Ec = E and the ion current density reduces to the classical Ohm’s law

J = σE

expressing current flow in an electrolyte with electrical conductivity σ and subject to an

electric field E. However, if a concentration gradient if also present, then ion current

flow is driven by the electrochemical field Ec = −∇ϕc so that the generalized Ohm’s

law (4.13) must be used. The relative weight of diffusion and drift forces thus determines

the flow of ion c in the solution environment.

Having introduced the notion of flux current density, the balance equation (4.7) can

also be consistently modified into the following current continuity equation for the ionic

species c

q z∂ c

∂ t+ divJ = q z P. (4.15)

The generalized Ohm’s form (4.13) of ion current flux and the current continuity equa-

tion (4.15) will be useful for interpreting ion exchange between cell compartments in

terms of equivalent electrical circuit parameters, as discussed in Sects. 4.2 and 4.4.

4.2. Microscale model of cellular ion flow

In this section, we introduce the microscale model for electrochemical ion flow in a cellular

system. To this purpose, we consider the simplified geometrical setting shown in Fig. 4.1.


The computational domain Ω is the union of three open disjoint bounded subsets of Rd,d = 1, 2, 3, denoted Ω1, Ω2 and Ωm. The two subdomains Ω′k, k = 1, 2, represent two

cell compartments while Ωm is a thin membrane of thickness tm that separates Ω1 from

Ω2. The external boundary of Ω, denoted ∂Ω, is the union of the boundary portions

Γ′1, Γ′2 and Γm. The outward unit normal vector on ∂Ω is n. On the internal boundary

Γ1 = ∂Ω′1 ∩ ∂Ωm, the outward unit normal vector from Ω′1 into Ωm is n1 while the unit

outward normal vector from Ωm into Ω′1 is nm = −n1. Similarly, on Γ2 = ∂Ω′2 ∩ ∂Ωm,

the outward unit normal vectors are n2 (from Ω′2 into Ωm) and nm (from Ωm into Ω′2),

respectively.

The main feature of the membrane subdomain Ωm is that the thickness tm, according

to biophysical evidence, is much smaller than the characteristic size of the domain ` :=

diam(Ω). This feature will be exploited to construct a reduced-order model of ion flow

in the cellular system.

Figure 4.1.: Two cellular compartments Ω′1 and Ω′2 separated by a membrane Ωm. The

membrane thickness tm is much smaller than ` := diam(Ω).

Unless otherwise specified, we assume from now on that the production term P is

equal to zero and that d = 3 (three-dimensional case). We also assume that Mion ≥ 1

chemical species are present in the medium, each one being characterized by a concen-

tration ci and a valence zi, i = 1, . . . ,Mion. The flow of the medium is neglected in this

analysis, as the electrolyte fluid velocity with respect an inertial system is typically much

smaller compared to the velocity of each ion species. Should this assumption fail to hold,

the ion flow model would have to be completed by adding the Navier-Stokes system for

fluid velocity (see [Rub90]).

The microscale description of ion motion consists of the following nonlinearly coupled,

incompletely parabolic system of PDEs, for the dependent variables ci, i = 1, . . . ,Mion,

4.2. Microscale model of cellular ion flow 29

and ϕ, to be solved in the space-time cylinder QT := Ω× (0, Tend) [Rub90]:

q zi∂ ci∂ t

+ divJ i = 0 (4.16a)

J i = qµi|zi|ciE − qziDi∇ ci (4.16b)

divD = ρ0 +

Mion∑i=1

qzici (4.16c)

D = εE = −ε∇ϕ, (4.16d)

supplied with the following set of initial and boundary conditions:

ci(x, 0) = c0i (x) in Ω. (4.17a)

−κiJ i · n+ αici = βi on ∂Ω (4.17b)

−κϕD · n+ αϕϕ = βϕ on ∂Ω. (4.17c)

The equation system (4.16) is well known as the Poisson-Nernst-Planck (PNP) model for

ion flow in cellular biology [Rub90]. The diffusion coefficients Di, proportional to the ion

mobilities µi through Einstein’s relation (4.1), are piecewise constant and positive over

the domain Ω, not necessarily assuming the same value in each subdomain Ω1, Ω2 and Ωm.

The same assumption applies to the dielectric permittivity ε. The initial concentrations

c0i are nonnegative given functions belonging to L∞(Ω). The dimensionless coefficients

κi and κϕ may take only the values 0 (corresponding to a Robin boundary condition)

and 1 (corresponding to a Dirichlet boundary condition). The coefficients αi and βi are

nonnegative and belong to L∞(∂Ω), while αϕ and βϕ are given functions in L∞(∂Ω). The

units of αi and βi are Coul m s−1 and Coul C m s−1, respectively, while the units of αϕ

and βϕ are F m−2 and Coul m−2, respectively. It can be verified that any weak solution

of (4.16)- (4.17) satisfies the following transmission properties on each internal boundary

Γk, k = 1, 2:

[[ci]]Γk= 0 (4.18a)

[[J i]]Γk= 0 (4.18b)

[[ϕ]]Γk= 0 (4.18c)

[[D]]Γk= 0. (4.18d)

In (4.18), for any scalar function f : Ω→ R, we let

[[f ]]Γkm:= fknk + fmnm k = 1, 2 (4.19)

be the jump of f across the internal boundaries Γ1 and Γ2, fk and fm being the traces

of f along Γk from the subdomains Ωk and Ωm, respectively. In the case of a vector

function V : Ω→ Rd, we let

[[V ]]Γk:= V k · nk + V m · nm k = 1, 2, (4.20)

be the jump of V across Γ1 and Γ2. Notice that the jump of a scalar is a vector, while the

jump of a vector is a scalar. The transmission conditions (4.18) essentially state that the

concentrations and the related fluxes, the electric potential and the related displacement

vector are continuous across the internal boundaries of the cellular domain.


Remark 4.2.1 (Why microscale model?). The name “microscale model” is motivated

by the fact that system (4.16) is to be solved in the partitioned domain Ω = Ω′1 ∪ Ω′2 ∪Ωm where the membrane is distinguished from the cellular compartments. In view of a

numerical solution of the problem, this means that both the small-size membrane region

and the large-size cellular compartments must be geometrically discretized, with an obvious

increase of meshing effort and memory storage.

Remark 4.2.2 (The Poisson equation). Nonlinear coupling in the equation system (4.16)

is due to the presence of the drift term in the flux constitutive equations (4.16b). As a

matter of fact, at each time level the Poisson equation (4.16c) must be self-consistently

solved to update the electric field inside the domain. This update step represents an

elliptic constraint on the remaining parabolic part of the problem (the Mion continuity

equations for the species ci) and may significantly increase the computational effort of

the solution algorithm, especially in the case of sharp transients and/or long-time system

dynamics.

4.3. Multiscale reduction of cellular ion flow

The principal difficulty in the numerical solution of the microscale model (4.16) is the geo-

metrical discretization of the membrane region, which may in turn give rise to a huge num-

ber of degrees of freedom of the numerical method. To reduce computational complexity,

in this section we propose a two-level procedure that combines the micro-to-macro scale

transition method used in [dFPSV12] in the study of nanostructured organic solar cells

with the effective membrane model proposed and investigated in [Mor06, MJP07, MP09]

in the three-dimensional study of cellular electrical activity. These two approaches share

some similarities with those used in [MJR05] for porous media with thin fractures or

in [LS11] for reaction problems with sharp moving reaction fronts, and, more recently,

in [CSV12] for Tissue Engineering applications in regenerative medicine.

The proposed approach consists of a geometrical level and a modeling level. The

geometrical level is based on the following steps:

(G1) introduce the d − 1-dimensional manifold Γ corresponding to the middle cross-

section of the membrane volume Ωm;

(G2) partition the membrane Ωm into the union of the disjoint subregions Ωm1, Ωm2

and Γ, where Ωm1 and Ωm2 are the two open portions of Ωm in contact with Ω′1and Ω′2, respectively;

(G3) define the two “extended” subdomains Ω1 := Ω′1 ∪ Ωm1 and Ω2 := Ω′2 ∪ Ωm2, in

such a way that Ω = Ω1 ∪ Ω2 ∪ Γ.

The new geometrical partition of the cell structure is shown in Fig. 4.2.

The modeling level is based on the following steps:

(M1) replace equations (4.16) in the membrane subdomain Ωm with suitable transmis-

sion conditions across the d− 1-dimensional manifold Γ;

4.4. Macroscale model of ion flow 31

Figure 4.2.: Two cellular compartments Ω1 and Ω2 separated by an interface Γ with zero

thickness. The original boundaries Γ1 and Γ2 between the membrane and

the cell compartments are drawn in dashed lines, together with the thickness

of the membrane.

(M2) solve equations (4.16) into the new partitioned domain Ω \ Γ = Ω1 ∪ Ω2, subject

to the initial/boundary conditions (4.17) and to the interface conditions as in step

(M1).

Remark 4.3.1 (Intra- and extracellular sites). By convention, we assume throughout

that quantities labeled with the subscript ”1” refer to the intracellular site while those

labeled with subscript ”2” refer to the extracellular site. In such an event, we let nΓ := n1

to indicate a prescribed orientation for ion flow across the interface separating the two

biological environments. According to this choice, a current flowing from Ω1 into Ω2 is

positive otherwise it is negative.

4.4. Macroscale model of ion flow

In this section, we apply the general multiscale modeling reduction procedure illustrated

in Sect. 4.3 to the study of ion flow in a cellular system. This requires, in principle, to

characterize in detail the extracellular and intracellular signals that ultimately determine

the biophysical behaviour of the cellular membrane (see, for instance, [WT06, CL08a]).

As the complexity of such phenomena goes far beyond the scope of this article, here, we

merely focus on the basic electrochemical effects that characterize the dynamical response

of the membrane and that have been discussed in Chapt. 3. The membrane modeling

reduction methodology presented here is based on the approach proposed in [Mor06] to

which we refer for more details on the asymptotical analysis of the formulation.

4.4.1. Electrostatic model of the membrane

Let us consider the schematical picture of the cellular region near the interface shown in

Fig. 4.3.

The principal assumption of the modeling reduction of the membrane is that the electric

potential varies linearly inside Ωm. This assumption agrees with the fact that tm `


Figure 4.3.: Microscale model of the membrane: 1D cross-section of the cellular domain.

The electric potential ϕ is assumed to be a linear function of position in-

side the membrane. In the example depicted in the figure, the extracellular

potential is higher than the intracellular potential so that the membrane

potential ϕm is negative.

and, replaced into the transmission condition (4.18d) at Γ1, yields

D1 · n1

∣∣∣Γ1

= −Dm · nm∣∣∣Γ1

= −Dm · (−n1)∣∣∣Γ1

= −εm∇ϕ · n1

∣∣∣Γ1

= −εmϕ′2 − ϕ′1tm

≡ Cm(ϕ′1 − ϕ′2)

(4.21)

where ϕ′1 and ϕ′2 are the traces of ϕ at Γ1 and Γ2, respectively, while

cm :=εmtm

(4.22)

is the intrinsic membrane capacitance and has units Fm−2. In an analogous manner, the

transmission condition (4.18d) at Γ2 becomes

D2 · n2

∣∣∣Γ2

= −Dm · nm∣∣∣Γ2

= εm∇ϕ · nm∣∣∣Γ2

= εmϕ′2 − ϕ′1tm

≡ cm(ϕ′2 − ϕ′1). (4.23)

At this stage of the procedure, the two transmission conditions (4.21) and (4.23) are

still localized at the internal interfaces Γ1 and Γ2 that were introduced in the microscale

description of the cellular geometry. To construct the electrostatic reduced-order model

of the membrane, we perform steps (G1)-(G3), and enforce conditions (4.21) and (4.23)

directly at the interface Γ, by replacing the trace ϕ1 along Γ with the value ϕ′1 along Γ1,

and the trace ϕ2 along Γ with the value ϕ′2 along Γ2. This is equivalent to replacing the

linear variation of ϕ in the membrane with a jump ϕ′2−ϕ′1 and corresponds to performing

step (M1). This step provides the appropriate electrostatic transmission conditions to

be satisfied by the displacement vector in the electrostatic reduced-order model of the

membrane:

−D1 · n1 + cm(ϕ1 − ϕ2) = 0 on Γ (4.24a)

−D2 · n2 + cm(ϕ2 − ϕ1) = 0 on Γ (4.24b)


where the superscript (·)′ has been omitted for sake of clarity. The electrical equivalent

representation of the reduced model of the membrane is shown in Fig. 4.4.

ϕ2

ϕ1

Ω1 Ω2

_____

+++++

Γ

cm

E, D

Figure 4.4.: Macroscale equivalent electrostatic representation of the membrane. The two

sheets of surface charge (positive and negative) are clearly visible on the two

sides of the specific capacitance cm.

In conclusion, the function pair (ϕ, D) satisfies the following conditions at the inter-

face Γ in the macroscale model:

[[D]]Γ = 0 (4.25a)

[[ϕ]]Γ · n1 = ϕ1 − ϕ2 6= 0. (4.25b)

Relation (4.25a) is an immediate consequence of taking the sum of the two relations (4.24),

and shows that in the macroscale model the electric displacement vector D satisfies the

same compatibility condition as in the macroscale model (cf. (4.18d)). Relation (4.25b),

instead, shows that the a-posteriori effect of upscaling the membrane thickness is that

ϕ in the macroscale model is no longer continuous across Γ (as it was in the microscale

model across Γk, k = 1, 2), but it experiences a finite jump discontinuity whose strength

is equal to |ϕ1 − ϕ2|.

4.4.2. Electrodynamical model of the membrane

Let us consider the schematical picture of the membrane region shown in Fig. 4.5.

For each ionic species i = 1, . . . ,Mion, the following contributions to current flow across

the membrane can be distinguished:

• J1,i, J2,i: current densities flowing into the membrane region Ωm from subdomain

Ω′1 and Ω′2, resp.;

• Jm1,i, Jm2,i: transmembrane current densities from Ω′1 into Ω′2 and viceversa, resp.;

• Jσ1,i, Jσ2,i: surface currents on Γ1 and Γ2, respectively.

The transmembrane current densities Jm1,i and Jm2,i represent the biophysical fact that ion

charge flows throughout ion channels, transporters and pumps that are located within

the membrane region. A biophysical characterization of these currents is far from trivial,

and for a more detailed treatment of this issue we refer to [KS98, Hil01] and to [Mor06].


Γ1 Γ

2

nmn1

tm

nm n2

J1,i

J2,i

Jσ

1.i

mJ

1,i

Jm

2,i

Jσ

2,i

Figure 4.5.: Microscale model of the membrane: 3D view of the membrane and current

flow contributions. Each red and green arrow represents a current density

injected into the membrane from each respective compartment.

Examples of models for Jm1,i, Jm2,i that are practically usable in numerical computations

have been given in Sect. 3.2.

The transmembrane current densities satisfy the following conservation conditions

Jm1,i + Jm2,i = 0 i = 1, . . . ,Mion. (4.26)

As a consequence, also the total transmembrane current densities

Jm1 :=

Mion∑i=1

Jm1,i, Jm2 :=

Mion∑i=1

Jm2,i (4.27)

are such that

Jm1 + Jm2 = 0, (4.28)

so that the total ion flux traversing the membrane thickness through the aid of ion

channels, transporters and pumps is conserved.

The current densities Jσ1,i and Jσ2,i represent the biophysical fact that some of the ion

charge density qzici flowing across the membrane accumulates on the separating surfaces

Γ1 and Γ2. To define such currents, it is convenient to introduce first the total surface

charge densities on Γ1 and Γ2

σm1 := cm(ϕ′1 − ϕ′2), σm2 = −σm1 . (4.29)

Relations (4.29) state that the membrane behaves like a linear capacitor and that the

total charge accumulated on Γ1 is instantaneously counterbalanced by the total charge

accumulated on Γ2, Then, the fraction of the total surface charge densities associated

with ion ci over Γ1 and Γ2 is given by

σm1,i := λ′1,iσm1 , σm2,i := λ′2,iσ

m2 i = 1, . . . ,Mion, (4.30)

where

λ′1,i :=z2i (c′1,i)

2∑Mionj=1 z2

j (c′1,j)2, λ′2,i :=

z2i (c′2,i)

2∑Mionj=1 z2

j (c′2,j)2, (4.31)


(uk)′ denoting again the trace of a function u over Γk, k = 1, 2.

The dimensionless parameters λ′1,i ∈ (0, 1] and λ′2,i ∈ (0, 1] have the biophysical

meaning of membrane charge fraction of the i-th ionic charge density with respect to the

total membrane ion charge. By construction, these parameters satisfy the property of

partition of unityMion∑i=1

λ′1,i = 1,

Mion∑i=1

λ′2,i = 1. (4.32)

The flux transmission condition (4.18b) at Γ1, for i = 1, . . . ,Mion, yields

J1,i · n1 = J1,m · n1 := Jm1,i + Jσ1,i = Jm1,i +∂

∂t

(λ′1,icm(ϕ′1 − ϕ′2)

), (4.33)

while the flux transmission condition (4.18b) at Γ2, for i = 1, . . . ,Mion, becomes

J2,i · n2 = J2,m · n2 := Jm2,i + Jσ2,i = Jm2,i +∂

∂t

(λ′2,icm(ϕ′2 − ϕ′1)

). (4.34)

Let

J :=

Mion∑i=1

J i (4.35)

be the total ion flux, and denote by J1 and J2 the traces of J on Γ1 (from Ω′1) and on Γ2

(from Ω′2), respectively. Then, taking the sum of (4.33) and (4.34) over i = 1, . . . ,Mion,

summing the resulting currents, upon using (4.28) and (4.32), yields

J1 · n1 + J2 · n2 = 0 (4.36)

so that the total ion flux traversing the membrane is conserved.

To construct the electrodynamical reduced-order model of the membrane, we perform

steps (G1)-(G3), and enforce conditions (4.33) and (4.34) directly at the interface Γ, by

replacing the quantities ϕ′1, ϕ′2 with ϕ1, ϕ2, the quantities c′1,i, c′2,i with c1,i, c2,i, and the

quantities λ′1,i, λ′2,i with

λ1,i :=z2i c

21,i∑Mion

j=1 z2j c

21,j

, λ2,i :=z2i c

22,i∑Mion

j=1 z2j c

22,j

, (4.37)

respectively. Notice that the dimensionless parameters λ1,i and λ2,i still satisfy the parti-

tion of unity property (4.32). The above procedure corresponds to performing step (M1),

and provides the appropriate electrodynamical transmission conditions to be satisfied by

the current density of the i-th ion species in the reduced-order model, for i = 1, . . . ,Mion:

−J1,i · n1 + Jm1,i +∂

∂t(λ1,icm(ϕ1 − ϕ2)) = 0 on Γ

−J2,i · n2 + Jm2,i +∂

∂t(λ2,icm(ϕ2 − ϕ1)) = 0 on Γ.

(4.38)

In conclusion, the function pairs (ci, J i), i = 1, . . . ,Mion, satisfy the following condi-

tions at the interface Γ in the macroscale model:

[[J i]]Γ = cm∂

∂t((λ1,i − λ2,i)(ϕ1 − ϕ2)) 6= 0 (4.39a)

[[ci]]Γ · n1 = c1,i − c2,i 6= 0. (4.39b)


ϕ1,c

1ϕ2,c

2

Ω1 Ω2

n1

λ1cm

J1

J2

Γ

gm

Figure 4.6.: Equivalent electrical representation of cellular ion flux and membrane. λ1cm

is the intrinsic (nonlinear) membrane capacitance (in F m−2). gm is the

intrinsic (generally nonlinear) membrane conductance (in Ω−1 m−2) associ-

ated with transmembrane current densities. In the example represented in

the figure, J1 and J2 are ion current densities flowing across the membrane

into the intracellular compartment so that the corresponding currents are

negative according to Remark 4.3.1.

Relation (4.39b) is the analogue of (4.25b), and shows that the a-posteriori effect of up-

scaling the membrane thickness is that ci in the macroscale model is no longer continuous

across Γ (as it was in the microscale model across Γk, k = 1, 2), but it experiences a finite

jump discontinuity whose strength is equal to |c1,i − c2,i|.

Consequently, relation (4.39a), that is the result of taking the sum of the two rela-

tions (4.38), shows that in the macroscale model the current flux of the i-th ion expe-

riences a jump in crossing the membrane. This jump is determined by the unbalanced

charge accumulation associated with the flow of ion charge qzici occurring at the two

sides of the interface Γ. Such unbalanced accumulation of charge actually cancels out

when summing over all ionic species flowing across the membrane. As a matter of fact,

taking the sum over i = 1, . . . ,Mion in (4.39a), upon using (4.28) and (4.32), we obtain

[[J ]]Γ = 0 (4.40)

so that the total ion flux traversing the membrane is conserved, in agreement with (4.36).

The electrical equivalent representation of the electrodynamical behaviour of the mem-

brane in the macroscale model is shown in Fig. 4.6.

4.5. The PNP system

Collecting the results of Sects. 4.4.1 and 4.4.2, the macroscale description of ion motion

consists of the following PNP system to be solved in the space-time cylinder QT :=

4.6. The Cable Equation model 37

(Ω \ Γ)× (0, Tend):

q zi∂ ci∂ t

+ divJ i = 0 (4.41a)

J i = qµi|zi|ciE − qziDi∇ ci (4.41b)

divD = ρ0 +

Mion∑i=1

qzici (4.41c)

D = εE = −ε∇ϕ, (4.41d)


ci(x, 0) = c0i (x) in Ω. (4.42a)


−κϕD · n+ αϕϕ = βϕ on ∂Ω, (4.42c)

and with the following set of interface conditions:

−D1 · n1 + cm(ϕ1 − ϕ2) = 0 on Γ (4.43a)

−D2 · n2 + cm(ϕ2 − ϕ1) = 0 on Γ (4.43b)

−J1,i · n1 + Jm1,i +∂

∂t(λ1,icm(ϕ1 − ϕ2)) = 0 on Γ (4.43c)

−J2,i · n2 + Jm2,i +∂

∂t(λ2,icm(ϕ2 − ϕ1)) = 0 on Γ (4.43d)

where the dimensionless parameters λk,i are defined in (4.37).

Remark 4.5.1 (Why macroscale model?). In a similar manner as in Rem. 4.2.1, sys-

tem (4.41) is called “macroscale model” because it does no longer require the detailed

geometrical representation of the volumetric membrane region and the corresponding so-

lution of the equation system (4.16) into that region, but only to characterize the func-

tional behaviour of the membrane through an equivalent model in terms of the trans-

membrane fluxes jmk,i and of the transmembrane intrinsic capacitances λk,icm, k = 1, 2

and i = 1, . . . ,Mion. Both these two sets of parameters are nonlinear functions of the

interface potentials ϕ1, ϕ2 and of the ion concentrations c1,i, c2,i, i = 1, . . . ,Mion. This

makes the communication between the two cellular compartments Ω1 and Ω2 a strongly

nonlinear interface coupling process, that requires the adoption of appropriate iterative

solution maps. For this latter issue, we refer to [MJP07] and [Bre09], and, in these

notes, to Chapt. 6.

4.6. The Cable Equation model

In this section, we consistently derive, starting from the PNP model introduced in

Sect. 4.5, a simplified description of intra-extracellular ion flow that is well-known as

the Cable Model. This approach is a PDE-based formulation that is characterized by a

significantly reduced computational effort than the PNP model and turns out to be quite

appropriate for studying passive electrical flow in neuronal networks. For a biologically

complete treatment of this subject, we refer to [KS98], Chapt. 8 and to [Hil01].


The principal assumption in the cable equation (CE) model is that the ionic concen-

trations in the intra- and extracellular sites are constant in both space and time. As

a consequence, from the constitutive relation (4.41b) it turns out that current flow is

purely ohmic, i.e., we have

J i = qµi|zi|ciE i = 1, . . . ,Mion, (4.44a)

where ci, i = 1, . . . ,Mion are the values of ion concentrations equal to c(in)i and c

(out)i

in the intra- and extracellular sites, respectively. Let us introduce the total electrical

conductivity of the solution (units: S m−1)

σtot :=

Mion∑i=1

qµi|zi|ci. (4.44b)

Then, summing (4.41a) over i = 1, . . . ,Mion and using (4.44b), we obtain the following

CE system of model equations for the electric potential ϕ to be solved in the space-time

cylinder QT :

divJ tot = 0 (4.45a)

J tot = σtotE = −σtot∇ϕ (4.45b)


ϕ(x, 0) = ϕ0(x) in Ω. (4.46a)

−κϕε

σtotJ tot · n+ αϕϕ = βϕ on ∂Ω, (4.46b)


−J tot,1 · n1 + Jmtot,1 + cm∂

∂t(ϕ1 − ϕ2) = 0 on Γ (4.47a)

−J tot,2 · n2 + Jmtot,2 + cm∂

∂t(ϕ2 − ϕ1) = 0 on Γ. (4.47b)

The initial datum ϕ0 is the solution of the linear Poisson equation (4.41c) at t = 0, while

Jmtot,1 and Jmtot,2 are the total transmembrane current densities injected from side 1 and

side 2 of the membrane, respectively.

Remark 4.6.1 (Time dependence in the CE model). The current continuity equa-

tion (4.45a) is an elliptic constraint to be satisfied at each time level t > 0. Time

evolution of the Cable Equation model is governed by the interface conditions at the

membrane (4.47). Integrating these latter equations across the membrane surface, from

each cellular compartment and using the convention on positive currents introduced in

Remark 4.3.1, we immediately get the following Kirchhoff current laws (KCLs):

Itot,1 = Imtot,1 + Cm∂ϕm∂t

from intracellular site (4.47c)

−Itot,2 = −Imtot,2 − Cm∂ϕm∂t

into extracellular site (4.47d)

4.6. The Cable Equation model 39

where, as usual, ϕm = ϕ1−ϕ2 denotes the membrane potential. Summing the two KCLs,

we get

(Itot,1 − Itot,2) =(Imtot,1 − Imtot,2

). (4.47e)

This latter equation has a clear physical meaning: the net (total) electrodiffusive current

crossing the membrane interface is exactly equal to the net (total) transmembrane current.

In the event where no charge is accumulated at the two membrane sites or trapped inside

the lipid layer then we can conclude that the right-hand side of (4.47e) is equal to zero

and the (total) electrodiffusive current is conserved.

For further details on the CE model, its applications in cellular electrophysiology and

its comparison with the full PNP formulation, we refer to [QS89] and to [Mor06].

5. Mathematical Analysis of the

Poisson-Nernst-Planck Model

In the present chapter we focus our attention on the Poisson-Nernst-Planck (PNP) model

for ion electrodiffusion introduced in Chapter 4. In particular, we consider in detail some

mathematical issues that play a significant role in view of i) the analysis of the properties

of the solutions of the model equations and ii) of their numerical approximation. With

this purpose, we first proceed with the adimensionalization of the PNP system; then,

based on the singularly perturbed character of the Poisson equation, we derive a further

electroneutral (EN) reduction of the system, denoted EN-PNP model, that represents a

very good choice for rapid and sufficiently accurate simulations of ion transport problems.

For both PNP and EN-PNP models in adimensional form, we provide suitable sets of

boundary, interface and initial conditions.

5.1. Summary of model equations

For convenience of exposition, we write below the Poisson-Nernst-Planck equation system

for ion electrodiffusion:

q zi∂ci∂t

+ divJi = 0 i = 1, . . . ,Mion (5.1a)

divD = ρ0 + q

Mion∑i=1

zi ci (5.1b)

Ji = −qziDi

(ziVth

ci∇ϕ+ ∇ci

)i = 1, . . . ,Mion (5.1c)

D = εwE = −εw∇ϕ. (5.1d)

To derive relation (5.1c), we use (4.2) and Einstein’s relation (4.1) into Eq. (4.41b) and

obtain

J i = −qµi|zi|ci∇ϕ− qziDi∇ ci

= −qziDi

(µi|zi|Dizi

ci∇ϕ+ ∇ ci

)= −qziDi

(ziVth

ci∇ϕ+ ∇ ci

).

Boundary, interface and initial conditions for (5.1) have been specified in (4.42) and (4.43).

42 5. Mathematical Analysis of the Poisson-Nernst-Planck Model

5.1.1. Scaling of the PNP system

A closed form solution for system (5.1) is, in general, impossible to obtain and an ap-

proximate solution is therefore required. The first step towards a numerically stable

approximation consists of reformulating the system in order to obtain a scaled set of

equations, where variables are adimensional and normalized. This operation, called scal-

ing, is very useful in view of numerical computations, since the variables of the problem

may have physical values of very different magnitudes. The scaling procedure leads to a

set of PDEs where dimensioneless variables have comparable orders of magnitude.

For any quantity w we set

w = w · w (5.2)

where w is the scaled (dimensionless) variable and w is the corresponding scaling factor.

We introduce the following independent scaling factors: x, c, ϕ and µ for x, ci, ϕ and

µi, and we set:

x = diam(Ω) (5.3a)

c = maxi=1,...,Mion

c

(in)i,eq , c

(out)i,eq

(5.3b)

ϕ = Vth (5.3c)

D = maxi=1,...,Mion

Di (5.3d)

where c(in)i,eq and c

(out)i,eq are the intra- and extracellular values of the ion concentrations

in thermal equilibrium conditions. Characteristic values of the scaling constants are

reported in Tab. 5.1.

Scaling factor Value Units

x 10−6 m

c 150 ·NAV = 150 · 6.023 · 1023 #ions m−3

D 1.87 · 10−9 m2V−1s−1

ϕ = Vth = Kb T / q ∼= 27 · 10−3 V

t = x/ v 5.67 · 10−6 s

λ2 :=εw ϕ

q x2 c2 · 10−4 adimensional

J = q c v 1.7 · 106 Am−2

v = D / x 1.87 · 10−3 ms−1

E = ϕ / x ∼= 2.5 · 104 Vm−1

Table 5.1.: Scaling factors and relevant parameters.

We start by scaling Eq. (5.1a) and obtain

q zi∂(c ci)

∂(t t)+

1

xdiv J i J = 0,

5.1. Summary of model equations 43

where t and J are the scaling factors (yet unspecifed) for the time variable t and the

current density J i, respectively. Defining

J :=q c x

t≡ q c v

where v has the units of a velocity ms−1, we obtain the following scaled form of the

continuity equation

zi∂ ci

∂ t+ div J i = 0 i = 1, . . . ,Mion. (5.3e)

We fix the scaling factor J by using the notion of drift velocity (see (4.10))

v = µE =µϕ

x

so that

J = q c v = q cµ x

ϕ(5.3f)

and, as a consequence, we get

t =x2

µϕ. (5.3g)

The current density reads

J i = −qziDic

x

(zici∇ ϕ+ ∇ ci

).

Introducing the scaling factor D for the diffusivity Di, we get

J i = −qDcxziDi

(zici∇ ϕ+ ∇ ci

):= J J i

from which, using (5.3f) and (5.3g), the scaling factor D turns out to be given by

D =x2

t= µϕ. (5.3h)

and the scaled form of the current density finally reads

J i = −qziDi

(zici∇ ϕ+ ∇ ci

)i = 1, . . . ,Mion. (5.3i)

The scaling of Eq. (4.41c) gives

−εwx2 div

(∇(ϕ ϕ)

)= q(ρ0 c) + q

Mion∑i=1

zi(ci c),

from which, upon introducing the parameter

λ2 :=εw ϕ

q x2 c, (5.3j)

we obtain the following scaled form of Poisson’s equation

− λ2 div ∇ϕ = ρ0 +

Mion∑i=1

zici. (5.3k)


Remark 5.1.1 (The scaled Debye length). The parameter λ is the so-called scaled Debye

length and is defined as

λ =

(εw ϕ

q c

)1/2 1

x≡ λD

x,

where λD ≈ 1nm is the standard definition of the Debye length. This latter quantity gives

a measure of the free mean path that a charged ion can travel inside the electrolyte fluid

before experiencing the screening effect of the electric field produced by the presence of the

surrounding charged ions. It is relevant to observe that if λ2 << 1, then the PNP system

exhibits a singularly perturbed character (see [Mar86] and, for a general presentation,

[RST96]), and the corresponding solutions may exhibit internal and/or boundary layers.

In the applications considered in these notes, a typical value is λ2 = O(10−4), so that

we can conclude that the PNP system is markedly singularly perturbed. This feature of

the problem we are dealing with has several important consequences, both in terms of

mathematical model reduction and of numerical discretization. These critical issues will

be addressed in the remainder of this chapter and in Chapt. 8.

5.2. The scaled PNP model

Continuing to denote each scaled variable with the same symbol as in the unscaled case,

the PNP model in scaled form reads:

zi∂ci∂t

+ divJi = 0 i = 1, . . . ,Mion (5.4a)

λ2divE = ρ0 +

Mion∑i=1

zi ci (5.4b)

Ji = −ziDi (zici∇ϕ+ ∇ci) i = 1, . . . ,Mion (5.4c)

E = −∇ϕ. (5.4d)

Applying the same scaling procedure also to (4.42) and (4.43), we end up with the

following set of initial and boundary conditions:

ci(x, 0) = c0i (x) in Ω. (5.5a)


−κϕE · n+ αϕϕ = βϕ on ∂Ω, (5.5c)


−E1 · n1 + cm(ϕ1 − ϕ2) = 0 on Γ (5.6a)

−E2 · n2 + cm(ϕ2 − ϕ1) = 0 on Γ (5.6b)

−J1,i · n1 + Jm1,i + λ2 ∂

∂t(λ1,icm(ϕ1 − ϕ2)) = 0 on Γ (5.6c)

−J2,i · n2 + Jm2,i + λ2 ∂

∂t(λ2,icm(ϕ2 − ϕ1)) = 0 on Γ (5.6d)

where the dimensionless parameters λk,i are defined in (4.37).

Remark 5.2.1 (Analysis of the PNP system). The analysis of the PNP system (5.4) in

steady-state conditions and in the one-dimensional case can be found in [PJ97, BCEJ97],

5.3. The electroneutral PNP model (EN-PNP) 45

while an extensive discussion of the mathematical properties of solutions of the model

(with special emphasis on their asymptotical behaviour with respect to small parameters)

can be found in [Rub90]. Well-posedness and global estimates in time of the nonlinearly

time-dependent coupled system constituted by the PNP and Navier-Stokes equations have

been thoroughly investigated in [Jer02, JS09] and in [Sch09].

5.3. The electroneutral PNP model (EN-PNP)

The very small value of the singular perturbation parameter λ2 = O(10−8) suggests that

the Poisson equation (5.4b) can be substituted, without appreciable loss of accuracy, by

the following relation

ρ(x, t) = ρ0(x) +

Mion∑i=1

zi ci(x, t) = 0 ∀x ∈ Ω, ∀t ∈ (0, Tend). (5.7)

Relation (5.7) expresses the physical property that the fluid solution containing ions in

motion and fixed charges is electroneutral. For this reason, the reduced model emanating

from the PNP formulation under the previous simplification is known as the Electroneu-

tral PNP system (EN-PNP), and reads:

zi∂ci∂t

+ divJi = 0 i = 1, . . . ,Mion (5.8a)

ρ0 +

Mion∑i=1

zi ci = 0 (5.8b)

Ji = −ziDi (zici∇ϕ+ ∇ci) i = 1, . . . ,Mion (5.8c)

E = −∇ϕ (5.8d)

with the following set of initial and boundary conditions:

ci(x, 0) = c0i (x) in Ω. (5.9a)


−κϕE · n+ αϕϕ = βϕ on ∂Ω, (5.9c)


−E1 · n1 + cm(ϕ1 − ϕ2) = 0 on Γ (5.10a)

−E2 · n2 + cm(ϕ2 − ϕ1) = 0 on Γ (5.10b)

−J1,i · n1 + Jm1,i + λ2 ∂

∂t(λ1,icm(ϕ1 − ϕ2)) = 0 on Γ (5.10c)

−J2,i · n2 + Jm2,i + λ2 ∂

∂t(λ2,icm(ϕ2 − ϕ1)) = 0 on Γ. (5.10d)

The EN-PNP model illustrated above was presented and mathematically investigated

in [MJP07] and recently extended in [MP09]. Compared with the PNP system, the EN-

PNP formulation has the mathematical advantage to neglect a-priori the occurrence of

interior layers in the electric potential spatial distribution at the interface Γ. Such layers

would require, as a matter of fact, heavy mesh refinement in the vicinity of Γ in order to

capture the discontinuity of the potential ϕ2 − ϕ1. Instead, using the model (5.8) such


a problem is removed and the associated computational effort is considerably reduced.

Numerical examples demonstrating the quality of the EN-PNP model compared with

that of the full PNP approach can be found in [Mor06]. Applications of the EN-PNP

system will be considered in Part IV of these notes.

Part III.

Functional Techniques and

Discretization Methods

6. Solution Map for the PNP system

In this chapter, we describe in detail the solution map to treat the PNP model equation

system that has been illustrated in Chapter 5. Similar approaches can be used, with

proper modifications to handle the electroneutrality condition (5.7), for the EN-PNP

model. Time advancing is carried out using the Backward Euler method. Then, a

functional iteration approach that is widely used in the decoupled solution of the Drift-

Diffusion semiconductor device equations, well-known as Gummel map, is applied to

the case of the PNP model. Special care is spent in the linearization of the membrane

transmission conditions within the modular structure of the fixed-point algorithm. It is

shown that such a treatment leads to solving a sequence of linear advection-diffusion-

reaction boundary-value problems in a multi-domain fashion.

6.1. Introduction

The system of PDEs, with the associated initial and boundary conditions, illustrated in

Chap. 5 are in general impossible to solve in closed form, hence an approximate numerical

solution must be found. Toward this end, the simultaneous occurrence of the following

issues must be taken care of:

• time dependence adds a dimension to the solution space and complicates the treat-

ment of the equations;

• non-linearity forces the use of an iterative procedure, where a sequence of linearized

problems must be solved until convergence;

• the interface embodied by the membrane has to be adequately treated by the nu-

merical method.

The solution maps, based on functional iterations, that are used to specifically deal

with the PNP and the EN-PNP models share some common approaches, which will be

discussed in the next subsections. Before entering into the details of the algorithms, let

us shortly address the general treatment of each of the issues listed above.

Time dependence is managed for both the PNP and EN-PNP systems and boundary

and interface conditions by introducing a simple temporal semi-discretization. Consider-

ing the problem having a time-span of [0, Tend], Tend being the final time, we indicate with

tm = m∆t the m-th time level, where m = 0, . . . ,MT − 1, MT ≥ 1, and ∆t = Tend/MT ,

and we use a Backward-Euler (BE) method to approximate all time derivatives in such

a way that

∂ci∂t

(x, tm) 'cm+1i (x)− cmi (x)

∆tm = 0, . . . ,MT − 1, i = 1, . . . ,Mion

50 6. Solution Map for the PNP system

where (·)m indicates that the quantity (·) is evaluated at the time level tm. It is well known

that the BE method is unconditionally stable, is easy to implement and introduces a time

discretization error of order ∆t, which is of the same order as the spatial discretization

error introduced by the finite element scheme discussed in Chapt. 8. At each time level

tm a non-linear system of equations must now be solved.

The non-linearity of the system of PDEs (5.4) is due to the coupling between potential

ϕ and concentrations ci through the drift term in the ionic flux constitutive equation

in (5.4c) and (5.8c)

−z2i Di ci∇ϕ = |zi|µi ciE, i = 1, . . . ,Mion

where µi is the scaled mobility associated to the i-th ionic species.

The nonlinear nature of the equations describing the ionic membrane currents, as

shown in Sect.3.4, is also a problem to be dealt with. Functional iterations provide

a well known approach to translating the original nonlinear system into a sequence of

linear problems [OR70], the solution of which should converge to a corresponding, non-

necessarily unique solution of the original problem. The most relevant example of func-

tional iteration is the so-called fixed-point method, an instance of which is the ubiquitous

Newton method [OR70, Jer96]. While this method has the very appealing property of

being quadratically convergent, some essential drawbacks must be pointed out:

• a “good” initial guess must be provided in order to fully enjoy the property of

quadratic convergence, and in some cases to even simply assure that convergence

can be reached;

• the algebraic system associated with the discrete solution of the linearized problem

may be very large in size, as the system must be solved for all variables (potential

and the Mion ions) simultaneously. This large size often proves to be computa-

tionally intensive, as well as causing the associated coefficient matrix to be often

ill-conditioned.

As illustrated in Sect.6.2, the PNP model will be solved with a staggered algorithm,

where the equations defining the potential and each ion will be solved separately, instead

of using a monolithic algorithm as the Newton method. This approach simplifies the

treatment for the following reasons:

• the decoupling of the potential from the concentration variables renders the conti-

nuity equation for the concentrations linear, hence easily solvable;

• the size of the corresponding algebric system is reduced, hopefully decreasing the

global time required to reach a solution, as well as improving the numerical condi-

tioning of the coefficient matrix.

6.2. PNP solution map

As noted in Chapt. 5, system (5.4) is formally identical to the well-known drift-diffusion

(DD) model for semiconductor devices [Jer96]. This similarity can be profitably exploited

6.2. PNP solution map 51

for both mathematical and computational purposes. Precisely, it is convenient to use the

definition of electrochemical potential given in (4.11) to express the ion concentration ci

(in scaled form) as a nonlinear function of the electric and electrochemical potentials as

ci = exp (zi(ϕec,i − ϕ)) i = 1, . . . ,Mion (6.1)

where ϕec,i is the (scaled) electrochemical potential associated with the ionic concentra-

tion ci. The use of (6.1) into (5.4b) has the effect of transforming the linear Poisson

equation into a nonlinear (more properly, semilinear) equation. This approach is the

same as done in the iterative solution of the DD semiconductor model with the so-called

Gummel map, originally introduced in [Gum64] in one spatial dimension and thoroughly

analyzed in multiple spatial dimensions [Jer96]. Thus, because of the mathematical equiv-

alence between the PNP and DD models, Gummel’s iteration appears to be a natural

choice also for the treatment of the PNP system.

The Gummel map is a decoupled algorithm where each dependent variable of the

problem and its corresponding equation are treated separately, as shown in the flow

chart of Fig.6.1. At each time level tm, the iterations start with a guess for the electric

and electro-chemical potential, ϕ(0) and ϕ(0)eci (or equivalently, with a concentration guess

c(0)i ). By indicating with (k) the generic k-th iteration, a single step of the Gummel

process consists of:

• the solution of a nonlinear Poisson equation (NLP), in order to obtain an updated

potential ϕ(k+1);

• the solution of a linear continuity equation for each electro-chemical potential

ϕ(k+1)eci , i = 1, . . . ,Mion, given the known updated potential ϕ(k+1);

• a check of convergence of the iteration, carried out by controlling whether the

maximum absolute difference between two consecutive iterates (k) and (k+1) is less

than a prescibed tolerance, for each i = 1, . . . ,Mion:

‖ϕ(k+1) − ϕ(k)‖L∞(Ω) < εϕ, ‖ϕ(k+1)eci − ϕ(k)

eci‖L∞(Ω) < εϕeci,

where, for any measurable function f , we set (cf. [QV97], Chapt. 1)

‖f‖L∞(Ω) := infM ≥ 0 | |f(x)| ≤ M almost everywhere in Ω.

A complete analysis of the convergence of the Gummel map is carried out in [Jer96]

in the case Mion = 2. The main result is that as k →∞, the map converges to a unique

solution ϕ∗ and c∗i , i = 1, 2, provided that suitable constraints are enforced on boundary

data and problem coefficients. Minor modifications are expected to extend the previous

result to the case of the PNP model (5.4). For more details, see [Rub90].

6.2.1. The Nonlinear Poisson equation

Each step of the Gummel Map for the PNP system requires solving a NLP equation for

the updated potential ϕ(k+1). The electrochemical potentials ϕ(k)ec,i are given functions and

remain unchanged in the solution process of the NLP equation. Dropping the superscript


Figure 6.1.: PNP solution map.

(k) for notation simplicity and using (6.1), we can write the NLP as the problem of finding

the zero of the abstract operator

F (ϕ) = −λ2div∇ϕ− ρ0 −Mion∑i=1

zi exp(zi(ϕeci − ϕ)), (6.2)

i.e., we solve the nonlinear PDE

F (ϕ) = 0 (6.3)

supplied by suitable boundary and interface conditions. A solution of (6.3) can be ob-

tained by applying the damped Newton method [OR70, Sel84]. This latter method is a

fixed-point iteration that amounts to solving at each step j, j ≥ 0, until convergence the

following homogeneous linear boundary value problem for the Newton update U (j):

F ′(ϕ(j))U (j) = −F (ϕ(j)) in Ω \ Γ (6.4a)

− κϕE(U (j)) · n+ αϕU(j) = 0 on ∂Ω (6.4b)

−E1(U(j)1 ) · n1 + cm(U

(j)1 − U (j)

2 ) = 0 on Γ (6.4c)

−E2(U(j)2 ) · n2 + cm(U

(j)2 − U (j)

1 ) = 0 on Γ (6.4d)

where

F ′(ϕ(j))U (j) = −λ2div∇U (j) +

[Mion∑i=1

z2i exp(zi(ϕeci − ϕ(j)))

]U (j). (6.4e)

6.2. PNP solution map 53

Once problem (6.4) is solved, the new iterate is found by the correction step

ϕ(j+1) = ϕ(j) + τ (j) U (j), (6.4f)

where τ (j) ∈ (0, 1] is a damping parameter to be properly chosen.

Remark 6.2.1 (Boundary conditions). The Newton increment U (j) satisfies the homo-

geneous boundary conditions (6.4b) in such a way that the updated potential ϕ(j+1) satis-

fies, because of (6.4f), the corresponding non-homogeneous conditions (5.5c) (assuming,

of course, that this also holds for ϕ(j) for all j ≥ 0).

Remark 6.2.2 (Frechet derivative). Relation (6.4e) expresses the action of the Frechet

derivative F ′(ϕ), evaluated at ϕ(j), on the increment funtion U (j).

Remark 6.2.3 (Damping). The damping parameter τ (j) is chosen in order to enforce a

norm-reduced residual at each iteration (see [Sel84], Chapt. 7), namely

‖F (ϕ(j+1))‖L∞(Ω) < ‖F (ϕ(j))‖L∞(Ω). (6.4g)

The value of τ (j) is found by a simple iteration loop to equation (6.4f), in which τ (j) is

initially set equal to one and then reduced (e.g., divided by two) until condition (6.4g) is

satisfied.

6.2.2. The continuity equations

Once the updated electric potential ϕ(k+1) has been computed by iteratively solving the

NLP equation as illustrated in Sect. 6.2.1, the next step of the Gummel map consists of the

successive solution of Mion linear continuity equations supplied, however, by (possibly)

nonlinear boundary and interface conditions.

Setting for brevity Ci := c(k+1)i (x) (unknown functions), V := ϕ(k+1)(x) (known

function) and Q := λ2cm(V1 − V2) (known function), the boundary value problems to be

solved to determine the updated ion concentrations Ci (and consequently, the updated

electrochemical potentials ϕ(k+1)ec,i ) for each i = 1, . . . ,Mion, read as follows:

ziCi∆t

+ divJi(Ci;V ) = zicmi∆t

in Ω \ Γ (6.5a)

Ji(Ci;V ) = −ziDi (ziCi∇V + ∇Ci) (6.5b)

− κiJ i(Ci;V ) · n+ αiCi = βi on ∂Ω (6.5c)

− J1,i(Ci;V ) · n1 + J1,mi(Ci;V ) + λ1,iQm+1 −Qm

∆t= 0 on Γ (6.5d)

− J2,i(Ci;V ) · n2 + J2,mi(Ci;V )− λ2,iQm+1 −Qm

∆t= 0 on Γ. (6.5e)

The differential equation (6.5a) and boundary conditions (6.5c) are of linear type. A

possible non-linearity is hidden, instead, in the interface conditions (6.5d)-(6.5e) because

of the definition of the ionic membrane currents Jmi as a function of Ci. Referring

to Sect.3.4, we see that:


• the linear resistor model (3.4), reported below for clarity

Jmi = gi

(Vin − Vout −

1

ziln

(C

(out)i

C(in)i

)),

is of nonlinear type with respect to the dependent variable Ci and therefore requires

an iterative treatment for solution;

• the GHK current equation (3.8) is of linear type with respect to the dependent

variable Ci, hence requires no further processing;

• the HH ionic current equations (3.9)2−4 are similar to that of the linear resistor

model so that they require an iterative treatment for solution.

In view of the numerical implementation of a nonlinear interface condition, a further

iteration scheme has to be devised. The solution of the continuity equation for each

i = 1, . . . ,Mion is then divided into a sequence of (p) steps, each one consisting in:

• linearization of the membrane ionic current Jmi evaluated in correspondance of a

known concentration C(p)i . Indicating with an overline the values of the concentra-

tion C(p)i and with no overline the concentrations for the next iteration (p+1), this

linearization can be expressed as

Jmi |ci ' gi

ϕm − 1

ziln

C(out)i

C(in)i

+ gi

C(in)i

C(in)i

−C

(out)i

C(out)i

.

• solution of the continuity equation for each ion, to obtain the updated concentra-

tions C(p+1)i ;

• convergence check between two consecutive iterates (p) and (p+1)

‖C(p+1)i − C(p)

i ‖L∞(Ω) < εci i = 1, . . . ,Mion.

Remark 6.2.4 (Convergence of inner iteration loop for continuity equations). While

no general mathematical proof is given regarding the convergence of this algorithm, ex-

tensive computational tests show that a fast convergence is obtained in every performed

simulation.

Remark 6.2.5 (Linearized interface conditions). Naming u the dependent variable and

J its corresponding flux, the general form of a linear interface condition can be expressed

as: J · n1 = αu1 − β u2 + σ1

J · n2 = β u2 − αu1 − σ2

(6.6)

where u1 and u2 denote the traces of u on either sides of the membrane, n1 = −n2 is

the outward unit normal to the membrane pointing from side (1) (intracellular, symbol:

(·)(in)) to side (2) (extracellular, symbol: (·)(out)) of the membrane, while α, β, σ1 and σ2

are given coefficients. We see that the linearized forms of the interface conditions (6.5d)

and (6.5e) fit the general form (6.6) by setting:

α =gi

C(in)i

, β =gi

C(out)i

, σ1 = −σ2 = gi

ϕm − 1

ziln

C(out)i

C(in)i

.

7. Unified Framework and Well-Posedness

Analysis

In this chapter, we construct a unified PDE model that includes as special cases all the

linearized differential subproblems that we need to solve at each step of the Gummel

iterative procedure illustrated in Chapt. 6. The model is a linear diffusion-advection-

reaction (DAR) PDE to be solved in a bounded open domain Ω with Lipschitz boundary

Γ := ∂Ω, composed by the union of two open subdomains, Ωi, i = 1, 2, and by one

interface Γm that separates them and that plays the role of the membrane between the

intra- and extracellular environments (see Fig. 7.1).

Here and in the next chapter, we assume Ω to be a bounded open domain in R2 despite

many of the ideas, results and methodologies can be extended to the three dimensional

case. The model problem is complemented by mixed type boundary conditions on the

external boundary Γ and by transmission conditions of the form (6.6) on Γm. Then, upon

specifying suitable functional spaces for the model coefficients and the solution, a weak

formulation of the DAR problem is obtained and well posedness of the corresponding

weak solution is found in a suitable subset of H1(Ω). The numerical discretization of the

DAR equation is the object of Chapt. 8.

7.1. Unified framework for the PNP solution map

All the equations that must be solved in the iteration steps described in Sect.6.2 can be

cast into the Diffusion-Advection-Reaction (DAR) model problem, with gradient advec-

tive field, of finding u : Ω = (Ω1 ∪ Ω2)→ R such that:

divJ(u) + c u = f in Ω (7.1a)

J(u) = −D (∇u+ cϕ u∇ϕ) in Ω (7.1b)

u = uD on ΓD (7.1c)

J(u) · n = γ u+ jR on ΓR (7.1d)

J(u) · n1 = αu1 − β u2 + σ1 on Γm,1 (7.1e)

J(u) · n2 = β u2 − αu1 − σ2 on Γm,2. (7.1f)

In view of the multi-domain formulation of problem (7.1), we denote by Γi the boundaries

of Ωi, i = 1, 2, in such a way that ΓD,i = Γi ∩ ΓD, ΓR,i = Γi ∩ Γi and Γm,i = Γi ∩ Γm

for i = 1, 2. Consistently, as represented in Fig. 7.1, on the external portions of the

subdomain boundaries Γi∩Γ, the outward unit normal vector ni coincides with n, while

on the interface Γm we have two distinct outward unit normal vectors n1 and n2, such

that n2(x) = −n1(x) for all x ∈ Γm.

56 7. Unified Framework and Well-Posedness Analysis

Figure 7.1.: Computational domain for the DAR model problem.

The subscripts in (7.1e)- (7.1f) indicate the restrictions on Γm of functions and quan-

tities defined in the subdomains Ω1 and Ω2, respectively. This means, in particular, that

the function u is not in general continuous across the interface Γm.

The partition of Γ into two mutually disjoint subsets is in accordance with the nu-

merical examples illustrated in Part IV of these notes. In particular, on ΓD the Dirichlet

boundary condition (7.1c) is enforced, on ΓR the Robin boundary condition (7.1d) is en-

forced while on the internal interface Γm the transmission conditions (7.1e) and (7.1f) are

enforced. Clearly, conditions (7.1c)-(7.1f) are special cases of the conditions (6.4b)- (6.4d)

(for the NLP equation) and of (6.5c)- (6.5e) (for the linearized continuity equations).

The diffusion coefficient D is a given function in L∞(Ω) such that

0 < Dmin ≤ D(x) ≤ Dmax a.e. in Ω.

The reaction coefficient c is a given function in L∞(Ω) such that

c(x) > 0 a.e. in Ω.

The potential ϕ is a given piecewise smooth function on Ω, such that ϕi ∈ (H1(Ωi) ∩L∞(Ωi)) for i = 1, 2, with a possible jump discontinuity at Γm. We denote by ϕmin and

ϕmax the mininum and maximum of ϕ over Ω. The quantity cϕ is a constant parameter

over Ω (equal to 0 in the case of the linearization of the NLP equation and equal to zi

in the case of the linearization of the continuity equation for the i-th ionic species). The

production term f is a given function in L2(Ω).

Boundary and interface data uD, γ, jR, α, β, σ1 and σ2 are given functions with

suitable regularity on their respective domains of definition, such that

α(x) > 0, β(x) > 0 a.e. on Γm

and

γ(x) ≥ 0 a.e. on ΓR.

For the definition and properties of the Hilbert function spaces used in the present

chapter, see [QV97] and the references cited therein.

7.2. Weak formulation and well posedness 57

7.2. Weak formulation and well posedness

In this section we construct the weak formulation of the DAR model system (7.1) and

prove that it admits a unique solution depending with continuity on problem data. To

ease the presentation, and without any loss of generality, we assume uD = 0 (to avoid

introducing lifting of boundary data). We also set cϕ = +1 as is the case with the

continuity equation (6.5a) for the sodium ion Na+. The construction consists of two

preliminary steps: 1) introduction of an appropriate functional setting; 2) reformulation

of (7.1) as an equivalent boundary value problem of diffusion-reaction type through the

application of a suitable change of the dependent variable.

7.2.1. Multi-domain functional setting

Since the function u is in general discontinuous across the interface Γm we can not

take H1(Ω) (more precisely, a proper subspace accounting for homogeneous boundary

conditions on ΓD) as ambient space for the weak formulation of the DAR equation because

this choice would contradict Prop. 3.2.1 of [QV97]. Therefore, we adopt a more flexible

approach suggested in a natural manner by the multi-domain structure of the problem

at hand. We start introducing the spaces

Vi =v ∈ H1(Ωi) | v = 0 on ΓD,i i = 1, 2

. (7.2a)

The spaces Vi are Hilbert spaces endowed with the norm

‖v‖Vi = ‖∇v‖L2(Ωi) ∀v ∈ Vi (7.2b)

because of Poincare’s inequality. Then, associated with the spaces Vi, we introduce the

product space

V = V1 × V2 (7.2c)

that is a Hilbert function space endowed with the graph norm

‖q‖V =(‖q1‖2V1 + ‖q2‖2V2

)1/2 ∀q ∈ V (7.2d)

where q1 and q2 are the restrictions to Ω1 and Ω2 of the generic function q : Ω → R,

respectively.

The space V is the appropriate ambient space of the weak formulation of the DAR

boundary value problem (7.1) as we are going to see in the remainder of this section.

7.2.2. DAR problem reformulation

The presence of the drift term

−Du∇ϕ

in the flux density J(u) (7.1b) prevents us from a straightforward application of the

Lax-Milgram Lemma because of the difficulty of proving coerciveness of the bilinear

form emanating from the weak formulation of (7.1). To exit this deadlock we exploit

relation (6.1) and set

u = U exp (−ϕ) (7.3)


where U := exp (ϕec). Replacing (7.3) into system (7.1) we can rewrite this latter in the

equivalent form of finding U : Ω = (Ω1 ∪ Ω2)→ R such that:

divJ(U) + cU e−ϕ = f in Ω (7.4a)

J(u) = −De−ϕ∇U in Ω (7.4b)

U = UD on ΓD (7.4c)

J(U) · n = γ U e−ϕ + jR on ΓR (7.4d)

J(U) · n1 = αU1 e−ϕ1 − β U2 e

−ϕ2 + σ1 on Γm,1 (7.4e)

J(U) · n2 = β U2 e−ϕ2 − αU1 e

−ϕ1 − σ2 on Γm,2. (7.4f)

Clearly, problems (7.1) and (7.4) are completely equivalent so that, should the solution

of (7.4) exist and be unique the same property would immediately hold for that of (7.1)

in virtue of the change of variable (7.3).

Remark 7.2.1 (The exponential change of variable). The change of variable (7.3) is

known as Cole-Hopf transformation [Hop50] and strongly relies on the assumptions that

a) Einstein’s relation holds; and b) the drift term in the transport model is a gradient

field. The same formula was also proposed by Jan Slotboom in [Slo73] and from that time

on it is referred to as Slotboom change of variable. The exponential transformation is

widely used in all analytical and numerical treatments of the DD semiconductor model

(see [Mar86, MRS90, Jer96, Sel84]).

Comparing the two equation systems, we see that the application of the Slotboom

transformation has had the following consequences:

1. the DAR problem has become a DR problem;

2. the drift term is disappeared;

3. the diffusion coeddicient D is now modified in De−ϕ;

4. the same modifications apply to the reaction term c, to the Robin coefficient γ and

to the interface coefficients α and β;

5. all of the above modifications share the property that originally bounded coefficients

remain bounded after the application of (7.3) because ϕ is essentially bounded from

above and from below. This avoids blow-up of the solution of (7.4).

7.2.3. Weak formulation

From now on in this chapter, we denote by v = (v1, v2) any function of V whose restric-

tions on Ω1 and Ω2 are v1 and v2, respectively. Then we proceed in the standard manner,

that is, we multiply (7.4a) by v, integrate over Ω and integrate by parts the terms∫Ωi

vi divJ(Ui) dΩi

7.2. Weak formulation and well posedness 59

to obtain the following integral identity∫Γm,1

v1 J(U1) · n1 ds+

∫Γm,2

v2 J(U2) · n2 ds−∫Ω

J(U) ·∇v dΩ

+

∫Ω

cU e−ϕ v dΩ =

∫Ω

f v dΩ ∀v ∈ V.

Using the fact that vi = 0 in the sense of traces on ΓD,i, the boundary condition (7.4d)

and the interface conditions (7.4e)- (7.4f), the above identity translates into the weak

formulation of problem (7.4):

find U ∈ V such that

B(U, v) = F (v) ∀v ∈ V (7.5a)

where:

B(U, v) =

∫Ω

De−ϕ∇U ·∇v dΩ +

∫Ω

cU e−ϕ v dΩ +

∫ΓR

γ U e−ϕ v ds

+

∫Γm,1

(αU1e−ϕ1 − βU2e

−ϕ2) v1 ds+

∫Γm,2

(βU2e−ϕ2 − αU1e

−ϕ1) v2 ds U, v ∈ V

(7.5b)

F (v) =

∫Ω

f v dΩ−∫

ΓR

jR v ds+

∫Γm,2

σ2 v2 ds−∫

Γm,1

σ2 v2 ds v ∈ V.

(7.5c)

7.2.4. Well posedness analysis and stability estimate

To the sole purpose of simplifying the estimates, we assume that the coefficients α and

β, γ are positive constant quantities, with A := max α, β. We also denote by CP the

largest between the two Poincare’s constants of Ω1 and Ω2 and by C∗ the largest between

the two trace constants of Ω1 and Ω2.

To verify that problem (7.5) admits a unique solution in the distributional sense, we

apply the Lax-Milgram Lemma (see, e.g., [QV97], Chapt. 5), and try to prove that there

exist three positive constants M, K and Λ such that:

|B(U, v)| ≤ M‖U‖V ‖v‖V ∀U, v ∈ V (7.6a)

B(U,U) ≥ K‖U‖2V ∀U ∈ V (7.6b)

|F (v)| ≤ Λ‖v‖V ∀v ∈ V. (7.6c)

Continuity of B Let us check condition (7.6a) that expresses the continuity of B(·, ·).Using the upper bound for e−ϕ and Cauchy-Schwarz, Poincare and trace inequalities we

get (7.6a) with

M = e−ϕmin[Dmax + 2C2

P ‖c‖L∞(Ω) + (2‖γ‖L∞(ΓR) + 4A)(C∗)2].


Coercivity of B Let us check condition (7.6b) that expresses the coercivity of B(·, ·).Using the lower bound for e−ϕ we get

B(U,U) ≥ e−ϕmaxDmin‖U‖2V +

∫Γm

(αU2

1 e−ϕ1 + βU2

2 e−ϕ2 − (αe−ϕ1 + βe−ϕ2)U1U2

)ds.

We need to investigate the sign of the term in the integral at the right-hand side. Set

A := αe−ϕ1 , B := βe−ϕ2 , X :=√AU1 and Y :=

√BU2. Then, requiring the integrand

to be a non-negative quantity amounts to solving the following inequality

X 2 + Y2 −

(√A

B+

√B

A

)XY ≥ 0 ∀X ,Y

A sufficient condition for the above condition to hold is clearly

A = B (7.6d)

because in such a case the integrand becomes

X 2 + Y2 − 2XY = (X − Y)2 ≥ 0 ∀X ,Y.

Condition (7.6d) amounts to requiring that

αe−ϕ1 = βe−ϕ2 . (7.6e)

Remark 7.2.2. It can be checked, using (3.7d), that condition (7.6e) is automatically

satisfied if the GHK model (3.8) is used for the transmembrane current.

Under the (sufficient) condition (7.6d), we immediately obtain (7.6b) with

K = Dmine−ϕmax .

Continuity of F Let us check condition (7.6c) that expresses the continuity of F (·).Using Cauchy-Schwarz, Poincare and trace inequalities we get (7.6c) with

Λ = 2CP ‖f‖L2(Ω) + 2C∗‖jR‖L2(ΓR) + 2C∗Σ

where

Σ := max‖σ1‖L2(Γm), ‖σ2‖L2(Γm)

.

The above analysis yields the following concluding result.

Theorem 7.2.1 (Well posedness and stability). Assume that (7.6e) holds. Then, prob-

lem (7.5) admits a unique solution U ∈ V such that

‖U‖V ≤Λ

K≤ 2

eϕmax

Dmin

[CP ‖f‖L2(Ω) + C∗‖jR‖L2(ΓR) + C∗Σ

]. (7.6f)

As a consequence, in virtue of the equivalence between problem (7.4) and problem (7.1),

also this latter problem admits a unique weak solution u ∈ V .

8. Finite Element Approximation of the

DAR Problem

In this chapter, we discuss the numerical approximation of the linear advection-diffusion-

reaction model problem introduced in Chapt. 7 using a dual mixed–hybridized finite el-

ement (volume) method (DMH-FV) with numerical quadrature of the mass flux matrix.

The resulting method is a conservative finite volume scheme over triangular grids, for

which a discrete maximum principle is proved under the assumption that the mesh is of

Delaunay type in the interior of the domain and of weakly acute type along the domain ex-

ternal boundary and internal interface. Compared to a standard displacement-based ap-

proach, the proposed DMH-FV method has the advantage of being flux-conservative (as

standard FV schemes) and self-equilibrated (as standard mixed methods). Moreover, the

fact that the novel DMH-FV formulation incorporates the classical Scharfetter-Gummel

(SG) exponentially fitted scheme allows to compute sharp fronts without spurious oscilla-

tions even in the presence of dominating convective phenomena. The stability, accuracy

and robustness of the proposed method are validated in Chapt. 9 on several numerical

examples motivated by applications in Biology, Electrophysiology and Neuroelectronics.

8.1. Motivation to the use of a DMH method

For convenience of the presentation, we write below the Diffusion-Advection-Reaction

(DAR) model problem (7.1) with gradient advective field introduced and analyzed in

Chapt. 7, assuming cϕ = +1 in (7.1b). Find u : Ω = (Ω1 ∪ Ω2)→ R such that:


J(u) = −D (∇u+ u∇ϕ) in Ω (8.1b)


J(u) · n = γ u+ jR on ΓR (8.1d)

J(u) · n1 = αu1 − β u2 + σ1 on Γm,1 (8.1e)

J(u) · n2 = β u2 − αu1 − σ2 on Γm,2. (8.1f)

The geometrical description of the problem and the assumptions on model coefficients

and boundary data have been already discussed in detail in Chapt. 7 where the well

posedness of the DAR system has been proved upon introducing the exponential change

of variable (7.3). This approach will be also exploited in the numerical approximation

of (8.1) in Sect. 8.3.

The model problem (8.1) is representative of several important applications, rang-

ing from electrokinetic flows in nanofluidics [Rub90, KBA05] to cell biology [KS98,

62 8. Finite Element Approximation of the DAR Problem

FMWT02]. A common feature of these applications is the presence of active inter-

faces (membranes) whose selective behavior controls mass transport from a subdomain

to the neighbouring one according to the difference between the values of the electro-

static potential ϕ across the membrane. In this chapter, ϕ is assumed to be a given

function, but in realistic situations, for example in the study of current flux across ionic

channels using the so-called Poisson-Nernst-Planck (PNP) model [Rub90], the potential

is itself an unknown of the problem and is dynamically determined by the solution of

Gauss’ law in differential form in the domain Ω as described in detail in Chapt. 6 and

in [Jer96, Jer02, MJP07, MJC+07, JCLS08, JS09].

Equation (8.1a) is a conservation law expressing the balance between the flux of the

advective–diffusive vector field J(u) across an arbitrary control volume B ⊆ Ω and the

production term f − c u within the volume itself. In particular, the jump of the nor-

mal component of J(u) is equal to zero across each segment belonging to the interior

of Ω1 and Ω2, respectively, while it is equal to σ1 − σ2 across the membrane Γm, as it

can be checked by summing (8.1e) and (8.1f). Using the terminology of Computational

Mechanics, where u has the meaning of displacement and J(u) is the stress field, it is

well-known that standard displacement-based finite element methods for the numerical

approximation of (8.1) generally fail at satisfying the above properties, despite the op-

timal convergence of the approximate solution uh to u in the H1-norm (see [QV97]).

An effective alternative is represented by dual mixed (DM) methods, where two inde-

pendent discrete solutions uh and Jh are simultaneously sought for, leading to a linear

system in saddle-point form. DM methods satisfy both local self-equilibrium and con-

servation, and optimal error estimates hold for the pair (uh, Jh) in appropriate function

space norms (see [BF91]). However, there are several drawbacks that make them not

so amenable to realistic computations, namely, the increased computational cost, the

indefinite algebraic character of the system, and a possible failure at satisfying the dis-

crete maximum principle (DMP) for uh in the case of a nonvanishing reaction term c

(see [BMM+05] and references cited therein). A considerable improvement consists of

resorting to the hybridization of the DM formulation (see, [AB85] and [BF91], Chapt.

V; for more recent development in the framework of Discontinuous Gakerkin methods,

see also [CDG+09]). The hybridization procedure is based on the introduction of a

Lagrange multiplier denoted by λh (hybrid variable), which is an approximation of u

along mesh edges and allows one to enforce the interelement continuity of the normal

component of Jh. The local elimination of the variables uh and Jh as functions of λh

(static condensation) leads to a dual mixed–hybridized (DMH) finite element scheme of

displacement–based type, acting on the sole λh, which is completely equivalent to the

original DM approximation but at a much reduced computational effort. Moreover, it

can be shown that the hybrid variable enjoys superconvergence properties. However,

the question of ensuring a numerically stable computed solution in the presence of dom-

inating convection and/or reaction terms still remains an open issue, and appropriate

stabilization techniques must be used (see [BMM+05, BMM+06] and the more recent

work [CDG+09]). To this end, we propose in this chapter a finite volume variation of

the standard DMH method, denoted DMH-FV method, based on the introduction of a

quadrature formula for the diagonalization of the local flux mass matrix. This approach,

8.2. Geometric Discretization and Finite Element Spaces 63

that extends to the heterogeneous transport problem (8.1) previously introduced MFV

formulations [ABMO95, BMO96, MSS01, BMM+05], has three important advantages.

The first is that the resulting numerical scheme has a simple and very compact finite

volume structure where for each element of the grid, the computational stencil consists

of the element itself and, at most, its three neighbours. The second is that the treatment

of the exponentially varying diffusion coefficient across inter-element edges allows, under

mild geometric conditions, a DMP for the computed discrete solution. The third is that

the novel method enjoys the same convergence properties as the standard DMH scheme,

including superconvergence in the L2-norm of the post-processed solution obtained from

λh (see [AB85, dFS11]).

A brief outline of the chapter is as follows. In Sect. 8.2, we introduce the geometric

entities and finite element spaces; in Sect. 8.3, we use the change of variable (7.3) to

write problem (8.1) in symmetric form. Then, we describe the DMH-FV method, while

in Sect. 8.4 we illustrate the computer implementation of the scheme and related post-

processing.

For a thorough validation of the numerical performance of the DMH-FV scheme, we

refer to Chapt. 9.

8.2. Geometric Discretization and Finite Element Spaces

Let Th be a regular family of given partitions of the domain Ω into open triangles K

satisfying the usual admissibility condition (see [QV97], Sect. 3.1 and Def. 3.4.1). For a

given Th, we denote by NT and Ne the total number of triangles and edges, respectively,

by |K| and hK the area and the diameter of K, respectively, and we set h = maxTh hK .

Let x = (x, y)T be the position vector in Ω; then, for each K ∈ Th, we denote (see

Fig.8.1) by xq, q = 1, 2, 3, the three vertices of K ordered according to a counterclockwise

orientation, by eq the edge of ∂K which is opposite to xq, by θKq the angle opposite to

eq and by CK the circumcenter of K. We denote by |eq| the length of eq and by nq

the outward unit normal vector along eq. Moreover, we define sKq as the signed distance

between CK and the midpoint Mq of eq. If θKq < π/2 then sKq > 0, while if K is

Figure 8.1.: Geometrical notation on triangle K.

obtuse in θKq then sKq < 0, and CK falls outside K. Notice also that if θKq = π/2 then


sKq = 0, and CK coincides with Mq. We denote by Eh the set of edges of Th, and by

Eh,int and Eh,Γ those belonging to the interior of Ω and to the boundary Γ, respectively.

For each e ∈ Eh,int, we indicate by K1e and K2

e the pair of elements of Th such that

e = ∂K1e ∩ ∂K2

e . Finally, we let se = sK1

ee + s

K2e

e denote the signed distance between CK1e

and CK2e

(see Fig.8.2). If θK1

ee + θ

K2e

e < π for all e ∈ Eh,int, then se > 0, and Th is called

Figure 8.2.: Two neighbouring elements.

a Delaunay triangulation [Del34]. If the inequality is replaced by an equality, for some

e ∈ Eh,int, we call Th a degenerate Delaunay triangulation. For such an edge, se = 0 and

the two circumcenters CK1e, CK2

ecollapse into the midpoint of e. The Delaunay condition

prevents the occurrence of pairs of obtuse neighbouring elements in Th, still allowing the

possibility of having single obtuse triangles in the computational grid (see [FG08] for

algorithmic details). From now on, we assume that Th is a Delaunay triangulation.

For k ≥ 0 and a given set S, we denote by Pk(S) the space of polynomials of degree

≤ k defined over S. We also denote by RT0(K) := (P0(K))2 ⊕ P0(K)x the Raviart–

Thomas (RT) finite element space of lowest degree [RT77], and by P0 the L2-projection

over constant functions. Then, for g ∈ L2(ΓD), we introduce the following finite element

spaces:

Vh := v ∈ (L2(Ω))2 |v|K ∈ RT0(K) ∀K ∈ Th

Wh := w ∈ L2(Ω) |w|K ∈ P0(K) ∀K ∈ Th

Mh,g := m ∈ L2(Eh) |m|∂K ∈ R0(∂K)∀K ∈ Th,

mK1e |e = mK2

e |e ∀e ∈ Eh,int, m|e = P0g|e,∀e ∈ ΓD,

(8.2)

whereR0(∂K) := v ∈ L2(∂K)|v|e ∈ P0(e)∀e ∈ ∂K, and mK1e , mK2

e are the restrictions

of the generic function m ∈ Mh,g on K1e and K2

e , respectively. For each K ∈ Th, the

basis functions of RT0(K) are τ j(x) = (x − xj)/(2|K|), j = 1, 2, 3, and are such that

div τ j = 1/|K| and τ j ·nj = 1/|ej | for each ej ∈ ∂K, which implies that∫ejτ i · nj dς =

δij , i, j = 1, 2, 3, δij being the Kronecker symbol. Functions belonging to Mh,g are single-

valued on Eh,int∪ΓD∪ΓR, while they admit two distinct values on each edge e ∈ Γm. This

8.2. Geometric Discretization and Finite Element Spaces 65

latter, special, situation reproduces, on the discrete level, the selectivity characteristic of

the membrane Γm, and allows accounting for the occurrence of finite jump discontinuities

across Γm. The local degrees of freedom associated with the finite element spaces (8.2)

are depicted in Fig.8.3. It is also useful to introduce the following global finite element

Figure 8.3.: Degrees of freedom of the finite element spaces (8.2) over the element K.

The black circle is associated with Wh, the arrows with Vh and the black

squares with Mh,g.

space

Λh = vh ∈ L2(Ω) | vh ∈ P1(K) ∀K ∈ Th,

vh(MK1

ee ) = vh(M

K2e

e ) ∀e ∈ Eh,int = span ωee∈Eh ,

where the basis functions ωe are the non-conforming elements of Crouzeix-Raviart [CR73]

(see Fig.8.4). Functions in Λh are piecewise linear over Th, continuous at the midpoint

Figure 8.4.: Basis function ωe. The support of the function is the neighbouring pair K1e ,

K2e .

of each edge e ∈ Eh,int and possibly admitting a finite jump discontinuity at each edge

e ∈ Γm.


8.3. A Mixed–Hybridized Method with Numerical Quadrature

With the aim of constructing a finite element approximation of the DAR model problem

(8.1), we use the change of dependent variable (7.3) is such a way that the original

advection–diffusion–reaction system (8.1) is transformed into the equivalent problem of

finding the solution ρ : Ω → R of the following linear diffusion-reaction model problem

in conservative form:

Lρ = div J(ρ) + c ρ e−ϕ = f in Ω (8.3a)

J(ρ) = −De−ϕ∇ρ in Ω (8.3b)

ρ = ρD on ΓD (8.3c)

J(ρ) · n = γ ρ e−ϕ + jR on ΓR (8.3d)

J(ρ) · n1 = αρ1 e−ϕ1 − β ρ2 e

−ϕ2 + σ1 on Γm,1 (8.3e)

J(ρ) · n2 = β ρ2 e−ϕ2 − αρ1 e

−ϕ1 − σ2 on Γm,2, (8.3f)

where ρD := uD eϕD and ϕD := ϕ|ΓD

. Comparing (8.3) with (8.1), we see that the

use of relation (7.3) has transformed the original DAR problem into a new equivalent

diffusion-reaction problem with an exponentially varying diffusion coefficient De−ϕ and

a new dependent variable ρ. From now on, we assume that D, c and f are piecewise

constant given functions over Th, and that ρD, γ and jR are piecewise constant boundary

data over Eh,Γ, with the same assumption for the transmission coefficients α, β, σ1 and

σ2, and that ϕ ∈ Λh. Moreover, given a function η, we denote by ηK and ηe the constant

values of η over each element K ∈ Th and each edge e ∈ Eh, respectively. Finally, we set

for brevity a := De−ϕ and A := a−1.

The DMH Galerkin approximation of problem (8.3) consists of finding (Jh, ρh, λh) ∈(Vh ×Wh ×Mh,ρD) such that:

(AJh, τ h)Th − (ρh, div τ h)Th + 〈λh, τ h · n〉Eh = 0 ∀τ h ∈ Vh (8.4a)

(divJh + c e−ϕ ρh, qh)Th = (f, qh)Th ∀qh ∈Wh (8.4b)

〈Jh · n, µh〉Eh = 〈γ λh e−ϕ, µh〉ΓR+ 〈jR, µh〉ΓR

+ 〈αλh e−ϕ1 , µh〉Γm,1 − 〈β λh e−ϕ2 , µh〉Γm,1

+ 〈σ1, µh〉Γm,1 + 〈β λh e−ϕ2 , µh〉Γm,2

− 〈αλh e−ϕ1 , µh〉Γm,2 − 〈σ2, µh〉Γm,1 ∀µh ∈Mh,0, (8.4c)

where (·, ·)Th and 〈·, ·〉S denote the elementwise L2 inner products over Th and over any

subset S ⊆ Eh, respectively. The equations in (8.4) have the following interpretation:

(8.4a) expresses the approximate local constitutive law; (8.4b) expresses the approxi-

mate local balance between net flux across K and net production of mass inside K;

(8.4c) expresses the approximate continuity of J · n across each interelement edge, the

Robin boundary condition and the interface transmission condition. The approximate

interelement continuity of ρ and the Dirichlet boundary condition are automatically ex-

pressed by the fact that λh is a single–valued function over Eh,int ∪ ΓD ∪ ΓR. Using the

static condensation procedure allows one to eliminate uh and Jh in favor of the sole

8.3. A Mixed–Hybridized Method with Numerical Quadrature 67

hybrid variable λh and leads to solving a linear algebraic system whose size is of the or-

der of Ne, which makes the DMH formulation a generalized displacement-based method.

Once λh is available, the variables uh and Jh can be recovered by post-processing over

each mesh element. The DMH formulation was originally proposed and theoretically

analyzed in [AB85] in the study of an elliptic model problem with Dirichlet boundary

conditions. Further analysis and extensions can be found in [BF91, RT79, RT91]. Re-

lated approaches in the framework of Discontinuous Galerkin methods have been recently

proposed and analyzed, in the general case of higher order polynomials, in the series of

papers [CG04, CG05, CDG08, CGL09, CGW09, CDG+09].

To construct a DMH scheme with reduced computational effort, we proceed as follows.

For each K ∈ Th, we set

JKh (x) =3∑j=1

ΦKj τ j(x) x ∈ K, (8.5)

where the degree of freedom ΦKj =

∫ej

JKh ·njdζ is the flux of JKh across edge ej , j = 1, 2, 3.

Then, we consider the following quadrature formula∫K

AJKh · τ idK =

3∑j=1

ΦKj

∫K

A τ j · τ idK

' 1

2ΦKj A

Ki cot(θKi )δij = ΦK

j AKi

sKi|ei|

δij i, j = 1, 2, 3,

(8.6)

where AKi :=

∫Mi

CKAK(ζ)dζ/|sKi |. Using the fact that ϕ ∈ P1(K), we have

AKi =

∫Mi

CKD−1(ζ)eϕ(ζ)dζ

|sKi |=

1

DiK

eϕi

Be (ϕK − ϕi)(8.7)

where Be(t) := t/(et−1) is the inverse of the Bernoulli function, and DiK

is the constant

value of the diffusion coefficient along the segment CKMi defined as:

DiK

=

DK if sKi ≥ 0

DKi if sKi < 0.(8.8)

The above definition is consistent with physical intuition, because in the case where

sKi < 0 (i.e., θKi > π/2) the path of the integral in (8.7) lies completely in Ki, so that the

diffusion coefficient that must be used to compute the average AKi is that associated with

triangle Ki (opposite to K with respect to edge ei). Using (8.8) makes the average AKi

always a strictly positive quantity. Moreover, it can be shown that the diagonalization

formula (8.6) is affected by the following quadrature error∣∣∣∫K

A τ j · τ idK −AKi

sKi|ei|

δij

∣∣∣ ≤ ChK‖τ i‖H(div ;K)‖τ j‖H(div ;K), (8.9)

where H(div ;K) = τ ∈ (L2(K))2 |div τ ∈ L2(K)K ∈ Th and C is a positive constant

depending on A and on the mesh regularity (see [MSS01, BMM+05, dFS11] for a proof).

Using (8.6) into (8.4a), we obtain the following discrete equations for the DMH

method with diagonalized local mass flux matrix.


• Equation (8.4a):

AKi ΦK

i

sKi|ei|− ρK + λKi = 0 ∀K ∈ Th i = 1, 2, 3. (8.10)

• Equation (8.4b):

3∑i=1

ΦKi + cKe−ϕ

KρK |K| = fK |K| ∀K ∈ Th. (8.11)

• Equation (8.4c):

ΦK1e

e =

−ΦK2

ee e ∈ Eh,int

(γe λe e−ϕe + jRe) |e| e ∈ ΓR

(αeλe,1 e−ϕe,1 − βeλe,2 e−ϕe,2 + σe,1) |e| e ∈ Γm,1

(βeλe,2 e−ϕe,2 − αeλe,1 e−ϕe,1 − σe,2) |e| e ∈ Γm,2.

(8.12)

Equation (8.11) is already in genuine FV form, so that, to construct a finite volume

approximation starting from system (8.10)–(8.12), we need to express the flux ΦKi as a

function of ρK and ρKi , for each K ∈ Th and i = 1, 2, 3, proceeding as follows.

(Step 1). Consider equation (8.10) and assume that θKi 6= π/2. Then, for each K ∈ Thwe obtain the explicit relation

ΦKi = −(A

Ki )−1 λ

Ki − ρK

sKi|ei| i = 1, 2, 3. (8.13)

In the special case where θKi = π/2, then sKi = 0 and equation (8.10) yields

ρK = λKi irrespective of the (undetermined) value of ΦKi . Such a value can be

recovered by post-processing the computed solution ρh by using (8.10) as

ΦKi = (fK − cKe−ϕK

ρK) |K| −3∑

j=1,j 6=iΦKj . (8.14)

(Step 2). For each e ∈ Eh,int we replace (8.13) into (8.12)1, obtaining the explicit relation

λe =(A

K1e

i sK1

ei )−1ρK

1e + (A

K2e

i sK2

ei )−1ρK

2e

(AK1

ei s

K1e

i )−1 + (AK2

ei s

K2e

i )−1. (8.15)

Let Le be the “lumping region” connecting CK1e, CK2

eand the two endpoints of e

(the shaded area in Fig. 8.5). Then, introducing the harmonic average of a over

Le, defined as

He(a) :=

(∫sea−1(ζ) dζ

se

)−1

=se

AK1

ei s

K1e

i +AK2

ei s

K2e

i

,

we can write (8.15) in a more expressive manner as

λe = He(a)

(AK2

ei

sK2

ei

seρK

1e +A

K1e

i

sK1

ei

seρK

2e

)≡ C1

e ρK1

e + C2e ρ

K2e . (8.16)


K1

e

K2

e

Figure 8.5.: Lumping region Le.

The two constants C1e and C2

e are such that C1e + C2

e = 1. This ensures that the

average (8.16) is consistent, i.e., if we set ρK1e = ρK

2e = ρ, then we get λe = ρ, as

should be expected. Using (8.7) over K1e and K2

e , we have

He(a) = e−ϕese

ζ1e + ζ2

e

, (8.17)

where ζre = sKr

ei /

(Di

Kre Be(∆ϕK

re ))

and ∆ϕKre := (ϕK

re − ϕe), r = 1, 2. The har-

monic average (8.17) is a positive quantity, because se > 0 and(ζ1e + ζ2

e

)> 0 due to

the fact that Th is a Delaunay triangulation and (8.8), respectively. For a discussion

of the use of the harmonic average in the finite element approximation of elliptic

problems, and of its impact on the computational performance of the method, we

refer to [BO83, AB85].

(Step 3). Substituting back (8.16) into (8.13) yields for each K ∈ Th the explicit relation

ΦKi = −(A

Ki )−1 CKi

i

ρKi − ρK

sKi|ei| = −(A

Ki )−1 A

Ki s

Ki

siHei(a)

ρKi − ρK

sKi|ei|

= −Hei(a)ρKi − ρK

si|ei| ∀ei ∈ ∂K ∩ Eh,int.

(8.18)

The finite volume nature of the formulation proposed in the present chapter can be

clearly recognized by comparison of the approximate flux ΦKi with the exact flux∫

ei

− a∇ρ · ni ds, which shows that the effect of the quadrature formula (8.6) is to

replace the term −(∇ρ ·ni)|Lei with the incremental ratio −(ρKi − ρK)/si and the

diffusion coefficient a|Lei with its harmonic average Hei(a). This result extends to

the dual mixed finite element setting the approach proposed in [MW94b] for the

Petrov-Galerkin finite element discretization of the convection-diffusion-reaction

equation.

(Step 4). Equation (8.18) already relates the unknown ρK to the neighbouring unknowns

ρKi , then to complete the derivation of the finite volume scheme, we need to consider

the case where ei ∈ Γ. We have:


ei ∈ ΓD: in this case, combining (8.13) and (8.18) immediately yields

ΦKi = −Hei(a)

ρD − ρK

sKi|ei|. (8.19)

ei ∈ ΓR: in this case, equating (8.13) with (8.12)2 and eliminating the hybrid vari-

able λei , yields

ΦKi =

γi e−ϕi ρK + jRi

AKi γi e

−ϕi sKi + 1|ei|. (8.20)

ei ∈ Γm: in this case, combining relations (8.10) and (8.12)3,4 and eliminating the

hybrid variables λe,1 and λe,2, we get:

ΦK1

eii =

αi e−ϕi,1 ρK

1ei − βi e−ϕi,2 ρK

2ei + σi,1 +A

K2ei

i sK2

eii βie

−ϕi,2(σi,1 − σi,2)

1 +AK1

eii s

K1ei

i αie−ϕi,1 +AK2

eii s

K2ei

i βie−ϕi,2

|ei|

ΦK2

eii =

βi e−ϕi,2 ρK

2ei − αi e−ϕi,1 ρK

1ei − σi,2 +A

K1ei

i sK1

eii αie

−ϕi,1(σi,1 − σi,2)

1 +AK1

eii s

K1ei

i αie−ϕi,1 +AK2

eii s

K2ei

i βie−ϕi,2

|ei|.

(8.21)

Replacing the expression of the flux ΦKi into the equilibrium equation (8.11), we ob-

tain the following linear system of algebraic equations that characterize the DMH-FV

approximation of problem (8.3)

Aρ ρ = f (8.22)

where Aρ ∈ RNE×NE is the stiffness matrix, ρ ∈ RNE is the unknown vector and f ∈ RNE

is the load vector, accounting for the contribution of the source function f and of the

boundary and interface data. To write down the entries of Aρ and f , we indicate by IK

and JKi the global indices of element K and Ki, i = 1, 2, 3. Moreover, for each K ∈ Th,

we introduce the non-negative quantities NKD , NK

R and NKm representing the number of

edges of K which belong to ΓD, ΓR and Γm, respectively. Clearly, these quantities are

all equal to zero if ∂K ∩ Γ = ∅. Then, the diagonal entries of Aρ read:

AρIKIK=

3∑i=1

ξKi + cKe−ϕK |K|

ξKi =

Hei(a)|ei|si

ei ∈ Eh,int

Hei(a)|ei|sKi

ei ∈ ΓD

γi e−ϕi

1 +AKi γi e

−ϕi sKi

|ei| ei ∈ ΓR

αi e−ϕi,1

1 +AK1

eii s

K1ei

i αie−ϕi,1 +AK2

eii s

K2ei

i βie−ϕi,2

|ei| ei ∈ Γm,1

βi e−ϕi,2

1 +AK1

eii s

K1ei

i αie−ϕi,1 +AK2

eii s

K2ei

i βie−ϕi,2

|ei| ei ∈ Γm,2,

(8.23)


the off-diagonal entries of Aρ are:

AρIKJKi=

−Hei(a)|ei|si

ei ∈ Eh,int

− βi e−ϕi,2

1 +AK1

eii s

K1ei

i αie−ϕi,1 +AK2

eii s

K2ei

i βie−ϕi,2

|ei| ei ∈ Γm,1

− αi e−ϕi,1

1 +AK1

eii s

K1ei

i αie−ϕi,1 +AK2

eii s

K2ei

i βie−ϕi,2

|ei| ei ∈ Γm,2,

(8.24)

and the load vector entries are:

fuIK = fK |K|+NK

D∑i=1

ηK,Di +

NKR∑

i=1

ηK,Ri +

NKm∑

i=1

ηK,mi

ηK,Di =uDi

ζKi|ei|

ηK,Ri =jRi

1 + γi ζKi|ei|

η1,mi = −σi,1 + βi ζ

2i (σi,1 − σi,2)

1 + αi ζ1i + βi ζ2

i

|ei|

η2,mi =

σi,2 + αi ζ1i (σi,2 − σi,1)

1 + αi ζ1i + βi ζ2

i

|ei|.

(8.25)

Some remarks about the properties of the numerical formulation illustrated in this

section are in order.

The first remark concerns the algebraic properties of the DMH-FV method. Matrix

Aρ has, at most, four non–zero entries on each row, and is structurally symmetric, i.e.,

if Aρij 6= 0 then also Aρ

ji 6= 0. In particular, denoting for each e ∈ Eh,int by I and J

the indices of the two triangles such that e = ∂KI ∩ ∂KJ, we have that AρIJ = Aρ

JI if

e ∈ Eh,int \ Γm while AρIJ 6= Aρ

JI if e ∈ Γm. The lack of symmetry numerically translates

the nonsymmetric action of the transmission conditions (8.3e) and (8.3f) with respect to

the neighbouring subdomains Ω1 and Ω2. To make this issue more precise, we associate

with each edge ei ∈ Γm the following “transmission” matrix Tρi ∈ RNE×NE

I J

Tρi =

|ei|∆i

. . . 0αie−ϕi,1 −βie−ϕi,2

. . .

−αie−ϕi,1 βie−ϕi,2

0 . . .

,

I

J

(8.26)

where ∆i := 1 + AK1

eii s

K1ei

i αie−ϕi,1 + A

K2ei

i sK2

eii βie

−ϕi,2 . By construction, the non-zero

entries of Tρi are the contributions ξKI

i , ξKJ

i to the diagonal entries of Aρ and the off-

diagonal entries AρIJ, AρJI, from which we see that Tρ

i is a nonsymmetric singular matrix


with zero column sum. The global stiffness matrix Aρ can therefore be partitioned

into the sum of a symmetric positive definite part AρS (associated with all the triangles

belonging to the interior of Ω1 and Ω2) and of a non-symmetric part AρNS =

∑ei∈Γm

Tρi .

By suitably renumbering the mesh elements, we see that the non-zero portion of matrix

AρNS has a block diagonal structure, where each block of 2 × 2 size corresponds to the

triangle pair sharing an edge on Γm (for example, KI, KJ or KP, KQ in Fig.8.6).

Figure 8.6.: Neighbouring triangles across the menbrane.

Having characterized the structure and basic properties of the stiffness matrix Aρ,

let us now investigate the numerical stability of the DMH-FV scheme. In this respect, an

important issue in heterogeneous flow transport problems is that the adopted numerical

scheme is monotone or, equivalently, it satisfies a Discrete Maximum Principle (DMP).

This property is the discrete counterpart of the continuous maximum principle associated

with problem (8.3), and is quite desirable because it prevents ρh from being affected by

spurious oscillations and ensures that each component of ρ is positive if each component

of the load vector f is > 0.

The need of devising a monotone approximation of problem (8.3) (typically studied

under more standard Dirichlet-Neumann boundary conditions, i.e., without the pres-

ence of an internal interface) has driven a considerable interest towards the develop-

ment of a special class of finite element schemes, known as exponentially fitted schemes

(see [RST96] for a detailed analysis and references). Such schemes are based on the so–

called Scharfetter–Gummel (SG) finite difference scheme [SG69], also known as Allen–

Southwell method [dGAS55]. The SG method is an optimal upwind difference scheme, it

is nodally exact in the case of constant problem coefficients [BH82] and satisfies a DMP

irrespective of the relative weight between diffusive and convective terms. Extending

the SG scheme to the two and three–dimensional setting, on triangular and tetrahe-

dral decompositions of the computational domain, has been the object of several works:

mixed-hybrid formulations [BMP87, BMP89b, BMP89a, SS97], Petrov–Galerkin formu-

lations [MW94b, MW94a, MW94c, Ker96], and Galerkin formulations with averaging of

the model coefficients along the element edges [BBFS90, GMS98, BJC98, XZ99, LZ05].

These methods share some common features: (i) they recover the SG approximation if

applied to one-dimensional problems; (ii) they satisfy a DMP under proper assumptions

on the angles of the triangulation Th; (iii) they ensure flux conservation across suitably

defined control volumes. Moreover, as a general trend, the schemes exhibit a common

ability in capturing sharp fronts without spurious oscillations, at the price of introducing

a certain amount of crosswind dissipation if the grid is not favorably aligned with the


advection field (cf. the numerical experiments in [BMP89a, SS97] and [GMS98]). More-

over, in some cases (as in the mixed formulation proposed in [BMP89b, BMP89a]), the

presence of a reaction term in the differential model introduces a difficulty in proving

the DMP for any value of the coefficient and of the mesh size, and requires a suitable

modification of the finite element space to reinforce the desired property [MP90]. Con-

ceptually similar approaches (based on the use of a proper lumping quadrature formula)

are adopted in the case of nodal-based formulations [BRF83, BR87, Ker96]

The following result provides sufficient conditions for the DMH-FV method to satisfy

a DMP.

Proposition 8.3.1. Let Th be a Delaunay triangulation such that for each edge e ∈ Γm

we have θK1

ee ≤ π/2, θ

K2e

e ≤ π/2, and for each edge e ∈ ΓD ∪ ΓR we have θKee ≤ π/2.

Then, Aρ is an irreducible M-matrix with strictly positive inverse [Var62], so that ρ > 0

if f ≥ 0 (DMP).

Proof. Under the above geometric assumptions on Th and the properties of AρS and Aρ

NS ,

it turns out that the stiffness matrix Aρ has zero column sums, with strictly positive

diagonal entries and nonpositive off diagonal entries. Moreover, for each element K with

an edge on ΓD, the matrix is diagonally dominant on the column corresponding to K.

The result then immediately follows by application of Theorem 3.1, p.202 of [RST96].

Prop. 8.3.1 provides a characterization of the numerical stability of the DMH-FV

scheme under proper assumptions on the geometrical discretization. It is important

to notice that the monotonicity of the proposed numerical method does not depend

on the value of the reaction coefficient ce−ϕ in (8.3), as is the case with the standard

dual-mixed method of [BMP89b, BMP89a], because in the FV structure of the scheme

such a term introduces a diagonal non-negative contribution to the stiffness matrix which

increases its diagonal dominance. The requirement of weak acuteness of Th on the domain

external boundary is standard and not restrictive for implementation (see [XZ99] and

the references cited therein). The requirement of weak acuteness of Th along the internal

interface is not strictly necessary, as a sufficient (more general) condition for Prop. 8.3.1

to hold is that ∆i > 0 for each edge ei ∈ Γm. In all the numerical experiments reported

in Chapt. 9 the finite element triangulation is chosen to be weakly acute along Γm and

ΓD ∪ ΓR.

The second remark concerns the relation between the proposed DMH-FV method

and other classical methods for the numerical solution of (8.3). Each row of (8.22) is

the finite volume discretization of the restriction to each element K ∈ Th of the mass

balance equation system (8.3a)- (8.3b). Using Euler’s theorem, we have that Ne→ 3NE/2

as the mesh size is refined, so that we can conclude that the computational effort of the

DMH-FV method is substantially lower than that of the standard DMH formulation.

Comparing the DMH-FV scheme to standard displacement-based methods, we see from

relations (8.19) and (8.20) that in the former approach both Dirichlet and Robin boundary

conditions are accounted for in an essential manner, unlike in the latter where Robin

conditions are accounted for in a weak manner. This indicates the robustness of the DMH-

FV method in treating boundary conditions on the flux variable, which are typically the

most important in the applications we are focusing on in the present chapter.


8.4. Implementation and Post-Processing of the DMH-FV

Method

In this section, we discuss how to implement the DMH-FV method in a numerically stable

manner and how to use the computed discrete solution to obtain a further approximation

of the exact solution u of (8.1) that enjoys a better convergence behavior.

8.4.1. Implementation

The solution of system (8.22) is not convenient from the numerical standpoint because

of the dynamic range of the function e−ϕ. This requires one to go back to the original

variable u using the inverse of (7.3) on each element K ∈ Th (cf. [BMP89b, BMP89a,

BMM+05])

ρK = uK eϕK ∀K ∈ Th. (8.27)

The action of (8.27) is a right diagonal scaling of Aρ which transforms (8.22) into the

equivalent algebraic linear system

Au u = f , (8.28)

where Au = AρDϕ ∈ RNE×NE is the new stiffness matrix and u ∈ RNE is the new unknown

vector, Dϕ being a diagonal matrix such that DϕIKIK

= eϕK

, K ∈ Th.

Proposition 8.4.1. Under the same assumptions as in Prop. 8.3.1, we have that Au is

an M-matrix with strictly positive inverse. This implies that u > 0 if f ≥ 0.

8.4.2. Post-Processing

The approximate flux density Jh can be recovered from the computed solution of (8.28)

by using (8.5) over each element K ∈ Th. With this aim, we need the expression of the

flux ΦKi across each edge ei ∈ ∂K, i = 1, 2, 3, such that ei ∈ Eh,int. A similar treatment

holds for the edges belonging to ΓD, ΓR or Γm. Using (8.27) and (8.17) in (8.18) yields

ΦKi = −e

∆ϕKii uKi − e∆ϕK

i uK

ζKi + ζKii

|ei| ei ∈ ∂K ∩ Eh,int. (8.29)

Proposition 8.4.2. Let ei ∈ ∂K ∩ Eh,int and assume that DK = DKi ≡ D and that

ϕ ∈ C1([CK , CKi ]). Then, the flux approximation (8.29) coincides with the classical

Scharfetter-Gummel (SG) exponentially fitted difference formula [SG69]

ΦKi = −D uKi Be(∆ϕi)− uK Be(−∆ϕi)

si|ei|, ∆ϕi := ϕK − ϕKi . (8.30)

Proof. We have to prove that:

e∆ϕKi

sKiBe(∆ϕ1

i )+

sKii

Be(∆ϕ2i )

=Be(−∆ϕi)

si

e∆ϕKii

sKii

Be(∆ϕ1i )

+sKii

Be(∆ϕ2i )

=Be(∆ϕi)

si.

(8.31)

8.4. Implementation and Post-Processing of the DMH-FV Method 75

Let us consider (8.31)1. Noting that ∆ϕi = ∆ϕKi −∆ϕKii , we have

Be(−∆ϕi)

si=

−∆ϕisi(e−∆ϕi − 1)

=−∆ϕi e

∆ϕKi

si(e∆ϕ

Kii − e∆ϕK

i )=

e∆ϕKi

e∆ϕKi − 1

∆ϕi/si− e∆ϕ

Kii − 1

∆ϕi/si

,

which coincides with the left-hand side of (8.31)1 because ∆ϕKi = ∆ϕi(sKi /si) and

∆ϕKii = −∆ϕi(s

Kii /si). In the same manner, we prove (8.31)2.

Proposition 8.4.2 shows that (8.29) is the consistent generalization of the SG method

to the case where both diffusivity coefficient and advective field are piecewise constant

quantities over the interval se, with a possible finite jump discontinuity in correspondance

of the midpoint Me of the inter-element edge e. This connection between the DMH-FV

formulation and the SG discretization is relevant in view of the analysis of the numerical

performance of the former scheme in the presence of dominating convection, as thoroughly

addressed in Chapt. 9.

Let λh ∈ Λh be the hybrid variable representing the approximation of u over Eh.

To recover λh from the computed solution of (8.28) we need to use (8.16) and then

apply (8.17), (8.7), (8.27) and (7.3) to obtain

λe =ζ2e e

∆ϕK1e uK

1e + ζ1

e e∆ϕK2

e uK2e

ζ1e + ζ2

e

∀e ∈ Eh,int. (8.32)

A similar treatment holds for the edges belonging to ΓD, ΓR and Γm, to yield:

λe =

P0(uD,e) e ∈ ΓD

e∆ϕKuK − ζe jRe

1 + γe ζKee ∈ ΓR,

(8.33)

while on Γm we have:

λe,1 =e∆ϕK1

e(1 + βe ζ

2e

)uK

1e + βe ζ

1e e

∆ϕK2e uK

2e − ζ1

e

(σe,1 + βe ζ

2e (σe,1 − σe,2)

)1 + αe ζ1

e + βe ζ2e

λe,2 =e∆ϕK2

e(1 + αe ζ

1e

)uK

2e + αe ζ

2e e

∆ϕK1e uK

1e + ζ2

e

(σe,2 + αe ζ

1e (σe,2 − σe,1)

)1 + αe ζ1

e + βe ζ2e

.

(8.34)

The above expressions of the degrees of freedom of λh over Eh can be used to construct

the following approximation of the exact solution u of (8.1)

u∗h(x) =∑e∈Eh

λe ωe(x), x ∈ Ω. (8.35)

The function u∗h ∈ Λh is the non-conforming piecewise linear interpolate of λh over

the computational grid Th. A thorough experimental analysis illustrated in Chapt. 9

demonstrates that u∗h satisfies the following convergence result

‖u− u∗h‖L2(Ω) ≤ Ch2, (8.36)

C being a positive constant depending on u and J but independent of the mesh size h. For

a proof of (8.36) in the case of homogeneous Dirichlet boundary conditions, see [dFS11].


Since the expected order of accuracy of uh in the L2-norm is O(h), we conclude

that (8.36) represents the superconvergence of the non-conforming approximation u∗h to

the exact solution u of (7.1), indicating, at least experimentally, that the DMH-FV

method, applied to the heterogeneous transport model, enjoys the same convergence

behavior proved in [AB85] for the standard DMH formulation in the case of the elliptic

model problem with Dirichlet boundary conditions.

Part IV.

Simulation Results, Applications and

Advanced Topics

9. Numerical Validation of the DMH-FV

Method

In this chapter, we perform a thorough numerical validation of the DMH-FV method in

the study of several test problems that represent significant examples of realistic appli-

cations in Biology and Electrophysiology. In particular, we include:

1. an experimental convergence analysis of the DMH-FV formulation in the solution

of problem (8.1) on a single domain;

2. a boundary-layer example of a two-ion PNP electrodiffusive transport;

3. the simulation of a prototype of a neuro-chip device; and

4. the simulation of the propagation of the action potential in an axon.

Geometry, parameters and data used in all of the reported test cases are as in [Bre09,

BJMS10].

9.1. Static condensation CAMBIA TITOLO

In Sect. 8.3 problem (8.4) was reformulated in the FV form (8.21), in which the only

unknown left is ρK and the quantities Φ and λK can be obtained with post-processing

operations. In this section instead, we show an alternative approach in which (8.4) is

reduced to a problem having the trace values λK as the only unknowns. For clarity we

report once again the considered problem: find u : Ω = (Ω1 ∪ Ω2)→ R such that:


J(u) = −D (∇u+ u∇ϕ) in Ω (9.1b)


J(u) · n = γ u+ jR on ΓR (9.1d)

J(u) · n1 = αu1 − β u2 + σ1 on Γm,1 (9.1e)

J(u) · n2 = β u2 − αu1 − σ2 on Γm,2. (9.1f)

80 9. Numerical Validation of the DMH-FV Method

If we choose the finite element spaces as in (8.2) the DMH Galerkin approximation of (9.1)

consists in finding (Jh, uh, λh) ∈ (Vh ×Wh ×Mh,uD) such that:

(D−1 Jh, τ h)Th + (D−1 ∇ϕuh, τ h)Th

− (uh, div τ h)Th + 〈λh · n, τ h〉Eh = 0 ∀τ h ∈ Vh (9.2a)

(divJh, qh)Th + (c uh, qh)Th = (f, qh)Th ∀qh ∈Wh (9.2b)

〈Jh · n, µh〉Eh = 〈γ λh µh〉ΓR+ 〈jR, µh〉ΓR

+ 〈αλh µh〉Γm,1 − 〈β λh µh〉Γm,1 + 〈σ1, µh〉Γm,1

+ 〈β λh µh〉Γm,2 − 〈αλh µh〉Γm,2 − 〈σ2, µh〉Γm,1 ∀µh ∈Mh,0. (9.2c)

If we restrict our analysis to one element K at a time and define the matrices

(AK)ij = (D−1τ j , τ i)K i, j = 1, . . . , d+ 1 (9.3a)

(BK)ij = −(qi, div τ j)K i = 1, j = 1, . . . , d+ 1 (9.3b)

(CK)ij = (D−1∇qj , τ i)K i = 1, . . . , d+ 1, j = 1 (9.3c)

(DK)ij = 〈µi · n, τ j〉∂K i, j = 1, . . . , d+ 1 (9.3d)

(EK)ij = −(c qj , qi)K i, j = 1 (9.3e)

(fK)i = −(f, qi)K i = 1, (9.3f)

where with d we denote the dimensionality of the problem (Ω ⊂ Rd), Eqs. (9.2a)

and (9.2b) read

AKJK + CKuK + (BK)TuK + (DK)Tλ∂K = 0 (9.4a)

−BKJK −EKuK = −fK . (9.4b)

Since AK is symmetric positive definite, we can obtain an explicit form for JK from (9.4a)

JK = −(AK)−1[(CK + (BK)T )uK + (DK)Tλ∂K

](9.5)

and being AK a small matrix (AK ∈ R(d+1)×(d+1)) its inversion is a quite numerically

cheap operation. If we insert (9.5) into (9.4b) we get[−BK(AK)−1(CK + (BK)T ) + EK

]︸︷︷︸UK

uK −BK(AK)−1(DK)Tλ∂K = fK (9.6)

and again it is possible to obtain an explicit expression for uK

uK = (UK)−1[fK + BK(AK)−1(DK)Tλ∂K

](9.7)

which depends only on the values of the traces λ∂K . Substituting (9.7) into the previously

obtained expression for JK (9.5) we get

JK =− (AK)−1(CK + (BK)T )(UK)−1BK(AK)−1(DK)Tλ∂K

− (AK)−1(CK + (BK)T )(UK)−1fK − (AK)−1(DK)Tλ∂K

=−[(AK)−1(CK + (BK)T )(UK)−1BK + I

](AK)−1(DK)Tλ∂K

− (AK)−1(CK + (BK)T )(UK)−1fK

=MKλ∂K + bK (9.8)

9.1. Static condensation CAMBIA TITOLO 81

which again depends solely on λ∂K . If we enforce the continuity of the normal component

of the flux across the edges, i.e.

[[Ji · n]]ei = 0 ∀ei ∈ Eh

we obtain ∑K

〈JK · n, µh〉∂K =∑K

∑j

〈µi · n, JKj τ j〉∂K = 0. (9.9)

In (9.9) we recognize (9.3d) so we can finally retrieve the desired algebraic problem in

the sole unknown λ using (9.8) ∑K

DKJK = 0

∑K

DK(MKλ∂K + bK

)= 0∑

K

−DKMKλ∂K =∑K

DKbK∑K

ΛKλ∂K =∑K

BK . (9.10)

Since the obtained problem is obtained with a sum over all the elements of the triangu-

lation Th, it is possible to construct the corresponding matrix and right hand side vector

block by block, summing up all the local contributions.

As for boundary conditions, their implementation does not present particular com-

plexities. Consider the case a Robin (Neumann) conditions has to be enforced on one of

the edges of the element K

κJK · n = γu− δ on ei ⊂ ∂K. (9.11)

By its definition it holds λ|ei = u|ei and in the (9.9) corresponding to the element K the

contribution of ei is ⟨JK · n, µi

⟩ei

=⟨γκλeiµi, µi

⟩ei−⟨δ

κ, µi

⟩ei

.

In order to include the boundary condition in the algebraic problem (9.10) it is enough

to modify the line corresponding to the edge ei

(ΛK)ii → (ΛK)ii +γ

κ(GK)ii and (BK)i → (BK)i +

(δK)iκ

where we defined GK and δK as

(GK)ij = 〈µi, µj〉∂K i, j = 1, . . . , d+ 1

(δK)i = 〈δ, µi〉∂K i = 1, . . . , d+ 1.

The transmission condition (9.2c) can be implemented using the same procedure and for

this reason we do not report the details here.

In addition, should the flux show jumps in its normal component across the inter-element

interfaces according to

[[Ji · n]]ei = σ


the numerical scheme can be easily modified to take into account that by modifying the

right hand side term corresponding to the elements K1 and K2 that share the edge ei

(K1 ∪K2 = ei)

(BKj )i → (BKj )i +(σKj )i

2for j = 1, 2,

where

(σK)i = 〈σ, µi〉∂K i = 1, . . . , d+ 1.

9.2. A one-dimensional heterogeneous domain

In this section, we consider problem (8.1) in the case where Ω = (0, 1) × (−0.5, 0.5)

and a membrane Γm is located at x = 0.5 to separate the left subdomain Ω1 from the

right subdomain Ω2. We set f = 0 and ∇ϕ = [−5, 0]T , while having two different

constant values in each subdomain for the diffusion constant D1 = 50, D2 = 0.5. The

Dirichlet data are uD = 0 at x = 0, y ∈ [−0.5, 0.5] and uD = 1 at x = 1, y ∈ [−0.5, 0.5],

while homogeneous Neumann conditions are enforced along y = 0.5 and y = −0.5, and

σ1 = σ2 = 0 on the interface. These data correspond to a one-dimensional transmembrane

flow along the x-direction. The following three sets of input data are considered:

1) c1 = c2 = 0 and α = β with α→ +∞;

2) c1 = c2 = 0 and α = β = 10;

3) c1 = 0.1, c2 = 10 while again α = β = 10.

Notice that case 1) corresponds to enforcing that u and J ·n are continuous across Γm.

The computed solutions for cases 1) and 2) are depicted in Fig. 9.1 (left), representing a

section at y = 0 of the post-processed quantity u∗h. The problem is diffusion–dominated

in Ω1, and is advection-dominated in Ω2, with an exact solution u almost linear over Ω1

and exponential over Ω2. In case 2), the solution has a finite jump across Γm because

of the selective behaviour of the membrane, while in case 1), the solution is continuous,

because the membrane is completely transparent to the flow of transported mass since

the interface condition is reduced to u1 = u2 on Γm, which is equivalent to eliminating

the membrane and treating the edges on Γm as belonging to Eh,int. The numerical

implementation of problem (8.1) is achieved using the transmission matrices defined

in (8.26). In all simulated cases, the DMH-FV method captures the solution layer without

introducing spurious oscillations, and it can be checked that the post-processed solution

u∗h is nodally exact up to machine precision.

In case 3), because of the fact that c 6= 0, the variable u∗h is no longer nodally exact;

however, the experimental convergence analysis reported in Fig.9.1 (right) indicates that

u∗h exhibits second order accuracy according to the error estimate (8.36). Fig. 9.2 shows

a three-dimensional plot of u∗h. The finite jump across Γm and the non-conforming

interpolation properties of the finite element space Λh are clearly visible.

9.3. Stationary profile of a binary electrolyte at a boundary 83

Figure 9.1.: Left: u∗h. Solid line: α = β = 10, dash-dotted line: α = β → ∞. Right:

‖u− u∗h‖L2(Ω) as a function of the mesh size h.

Figure 9.2.: Left: post-processed solution u∗h in the case c1 = 0.1, c2 = 10 and α = β =

10. Right: zoom of the solution on Ω2.

9.3. Stationary profile of a binary electrolyte at a boundary

In this section, we apply the DMH-FV to numerically study the stationary Poisson-

Nernst-Planck (PNP) system of partial differential equations introduced in Chapt. 5 (see

also [Rub90]). In stationary conditions (∂ci/∂t = 0), the PNP nonlinear differential

system is treated using the decoupled functional iteration described in Chapt. 6 (see

also [Bre09]).

This approach leads to the successive solution of linearized differential subproblems of

the form (8.1). In the considered case, we study a binary electrolyte (i.e., Mion = 2) with

zi = ±1, and the boundary value problem (5.4) is to be solved in the semi-infinite domain

x ∈ [0,+∞) with an applied external voltage drop ∆Vext = ϕ(0) − ϕ(+∞) = 100mV ,

with ϕ(+∞) = 0 and a surface at x = 0 impermeable to the ions. An analytical solution


of this problem for ϕ and u1,2 can be written as [BB93]:

ϕ = 2VT log

(1 +K exp(−

√2x/LD)

1−K exp(−√

2x/LD)

)ui = N0 exp (−zi ϕ/VT ),

where K = tanh(∆Vext/(4VT )), LD =√ε VT /(q N0) is the Debye length, VT ≈ 25mV is

the thermal voltage (having assumed T = 300K), q is the unit charge, ε ≈ 7·10−10 Fm−1

is the dielectric constant of the medium (water in this case) and N0 = 1mM is the bulk

concentration of both ions. The performed simulation is actually carried out on the two-

dimensional domain Ω = (0, L)2, with L = 5LD. The Debye length gives a measure of

the screening effect of a space charge layer, so that the choice of truncating the semi-

infinite domain to a finite length equal to a positive multiple of LD is a very good

approximation of the decaying behavior of ionic densities far away from the layer. The

boundary conditions for ϕ and u1,2 on y = 0 and y = 5LD are of homogeneous Neumann

type in order to obtain a solution dependent upon x solely, while on x = 0 and x = 5LD

the boundary conditions are obtained from the analytical solution. Fig. 9.3 illustrates a

slice along the x-axis of the computed ion concentrations ci and the discretization error as

a function of h. No spurious oscillations affect the results, and, again, superconvergence

as predicted by (8.36) can be observed for both u1 and u2.

Figure 9.3.: Computed ion concentrations (left) and ‖ui − u∗i,h‖L2(Ω), i = 1, 2 (right).

9.4. Simulation of a neuro-chip

In this section, we carry out a validation of the numerical accuracy and robustness of

the DMH-FV formulation in the simulation, using the PNP differential model, of a basic

configuration of a neuro-chip for neuroscience applications [ZF01, SMF01, Fro03, BF05].

The bio-hybrid device is the EOSFET (Electrolyte Oxide-Semiconductor Field Effect

Transistor) schematically depicted in Fig. 9.4 (left).

The aim of the device is to interface a biological component (a neuronal cell) to an

electrical component (solid-state substrate), in order (i) to transduce a chemical signal

generated by the biological component into an electronically readable signal, or, viceversa,

9.4. Simulation of a neuro-chip 85

(ii) to activate the biological component by the application of an electronic signal. In

operation mode (i), the EOSFET is working as a bio-sensor, while in operation mode (ii)

the EOSFET is working as a neuro-prosthetic device, i.e., playing the role of a neuronal

connection or even of a full neuronal network, thus opening the view for future use of

the neuro-chip as a cure for neuro-degenerative deseases like Alzheimer or Parkinson

[BYTV+05].

Figure 9.4.: Left: schematics of a neuro-chip (by courtesy reprinted from: E. Neher,

Molecular biology meets microelectronics, Nature Biotechnology, 19, 114

(2001)). Right: computational domain for stationary neuro-chip simulation.

The computational domain Ω is depicted in Fig. 9.4 (right), where we can distinguish

a portion Ωcell of the cell cytoplasm, the interstitial electrolyte cleft Ωbath separating cell

from substrate, the cell membrane Γm, the cell-to-chip contacting interface Γel and two

reference contacts Γcell and Γref . Dirichlet boundary conditions as in (8.1c) are enforced

on Γcell and Γref , a Robin boundary condition as in (8.1d) is enforced on Γel, while

interface boundary conditions as in (8.1e)- (8.1f) are enforced on Γm. On the remaining

portions of the domain boundary, ΓA, ΓN and the left vertical side of Ω, a homoge-

neous Neumann condition is enforced (γ = jR = 0 in (8.1d)). The geometrical data

used in computations are L = 0.8µm, H = 0.3µm, δcell = 0.25, µm, rcell = 0.5µm and

δcleft = 50nm. Ionic charge flow includes three species, K+, Na+ and Cl−, whose refer-

ence values are kept fixed respectively at (139, 12, 151)mM on Γcell and (4, 145, 149)mM

on Γref . As for the boundary condition for the electrostatic potential ϕ, we set ϕ = 0V

on Γref and ϕ ∈ [−100,+60]mV on Γcell. On the membrane Γm, the interface condi-

tions (6.4c)- (6.4d) are enforced for the potential while the interface conditions (6.5d)-

(6.5e) are enforced for the concentrations. In this latter case, the Goldman-Hodgkin-Katz

model (3.8) is used to describe the flow of ionic concentrations through the membrane.

On Γel, an homogeneous Neumann condition is enforced for the concentrations ci, while

the following compatibility condition is enforced for the polarization vector D

−εw∇ϕ · nΓel= Cel(ϕ) (ϕ− Vel) ,

where Vel is a fixed external potential and Cel(ϕ) is a MOS (Metal Oxide Semiconductor)

capacitance nonlinearly depending upon ϕ as described, e.g., in [TN98].

Fig. 9.5 (left) shows the computed static current-voltage characteristics, which de-

scribes the behaviour of the average value of Ji · n|Γref(positive if current flows out of


Figure 9.5.: Left: static I-V characteristics. Right: computed Na+ current density.

Γref , negative otherwise) as a function of ϕ|Γcell. The accuracy of the results is demon-

strated by the very good agreement between the estimated reverse potentials and the

physiologically correct values (2.5) computed using values for the intra-and extracellu-

lar ion concentrations typical of electrophysiology measurements [KS98, Hil01]. To help

interpret the results shown in Fig. 9.5 (left), we recall that the Nernst potential of each

ionic species is the value of ϕ|Γcellat which the corresponding ionic current density is

equal to zero, in accordance with Def. 2.2.2.

Figure 9.6.: Variations with respect to reference values of potential (left) and Na+ con-

centration (right).

Fig. 9.5 (right) shows the distribution of sodium current density over the computa-

tional domain. We can notice the higher current density in the cleft region between the

cell membrane and the electrical substrate. Such higher current density in turn causes

the rise of the potential in the cleft region, which can be measured by the field-effect

transistor in the substrate. Computed current value is again in quite a good agreement

with measured data [BF05]. We conclude this discussion by showing in Fig. 9.6 the com-

puted variations over Th of the electric potential ϕ and of the Na+ concentration with

respect to their corrsponding reference values (enforced at the Dirichlet boundary). The

results give an idea of the steep boundary layer effects occurring across the membrane

separating the intracellular region from the electrolyte cleft and at the interface with the

electronic substrate, and demonstrates the effectiveness of the DMH-FV formulation in

9.5. Action potential propagation in an axon 87

capturing the essential phenomena without introducing spurious oscillations that would

otherwise make the simulation prediction completely unreliable and inaccurate.

9.5. Action potential propagation in an axon

In this section, we test the performance of the DMH-FV method on a case of physiological

interest, namely, the Hodgkin-Huxley axon [HH52]. To this purpose, we consider the

same problem numerically investigated in [MP09], that is, the propagation of an action

potential in an unmyelinated neuronal axon. Ion transport in the intra and-extracellular

space is described by the EN-PNP model (5.8), while the full Hodgkin-Huxley system

of ordinary differential equations [HH52] is self-consistently solved at each time level of

signal propagation, to account for voltage gating mechanisms occurring along the axon

membrane.

Figure 9.7.: Computed action potential at two different time levels.

A full simulation of an action potential has been performed over the time interval

[0, 5ms], for the potential and three ionic species, namely K+, Na+ and Cl−. An

artificial increase at time t = 0 s of the Cl− conductance at the center of the axon triggers

the action potential, which propagates along the simulated axon of length 4000µm and

diameter 1µm.

Time snapshots of the electric potential spatial distribution along the axon, at t =

3ms and t = 5ms, shown in Fig. 9.7 clearly demonstrate the spreading of the action po-

tential towards the two ends of the axon. A similar trend can be observed in Figs. 9.8, 9.9

and 9.10, where the variations of each ionic concentration with respect to a reference value

are shown.

Fig. 9.11 depicts the spatial distribution of the gating variables at the two considered

time levels.


Figure 9.8.: Computed potassium concentration variation at two different time levels.

Figure 9.9.: Computed sodium concentration variation at two different time levels.

Figure 9.10.: Computed chlorine concentration variation at two different time levels.

9.5. Action potential propagation in an axon 89

Figure 9.11.: Computed gating variables at two different time levels.

10. Advanced Topics, A Look at the Future

and Concluding Remarks

In this concluding chapter, we provide a short overview of a few selected advanced topics,

with related bibliographical references, giving some suggestions for further development

and application of ideas and methods addressed in the notes.

10.1. Applications of the future

Among the wide variety of frontier applications of biological interfaces existing nowadays,

we mention:

1. the realization of advanced bio-hybrid interfaces, such as the prototype of an ar-

tificial retina for artificial vision. For this subject, we suggest to look at the

works [MDD+09, GADM+11, DRR+13] and all references cited therein. Exam-

ples of modeling and simulation methods based on the same material covered in

these notes can be found in [GPR+12] and [PLdF+14];

2. the study of the cellular mechanotransduction processes that control muscle con-

traction (for example, calcium sparks induced by chemical and mechanical stim-

uli that overview heart beat and pumping) and autoregulation phenomena (for

example, but not limited to, in retinal and brain blood circulation and nutrient

delivery/uptake).

For calcium sparks, we refer to the fundamental review work [CL08b] and, for

specific applications in neuroscience and renal smooth muscle cells, [Fri95, FC08,

BAL+07].

For autoregulation mechanisms in retinal microcirculation, we refer to [AHS+13,

GHC+14, HGA+13] and to all references cited therein, while for further important

aspects of electro-chemo-mechanical effects in the brain and other organs in animals

(and also human body), we refer to [YCBR03a, YCBR03b, YCBR05, WCS+11].

10.2. Concluding remarks

Along these Course, we have overviewed a hierarchy of mathematical models for the

study and simulation of charge transport processes occurring at the interfaces separating

two neighbouring biological environments, such as the intra- and extracellular sites of

a cell. These phenomena concern primarily with the electrodiffusive motion of charged

92 10. Advanced Topics, A Look at the Future and Concluding Remarks

ionic species in a biological solution under the action of an external applied force (of

electro-chemical and/or mechanical type) and are very well represented by the use of

mathematical descriptions based on nonlinearly coupled systems of ODEs and PDEs.

We have considered the mainly adopted approaches, namely, the GHK and Hodgkin-

Huxley current relations for the ODE part, and the PNP system for the PDE part,

and we have also discussed how to iteratively solve the problem and how to numerically

treat the linearized resulting equations using the finite element method. Finally, we have

illustrated in several examples the mathematical and biological accuracy of the proposed

models and methods on a set of significant case studies.

In the study of all of the above referenced issues and scientific articles, the mathe-

matical ODE-based and PDE-based models illustrated and analyzed in these notes, are

extensively adopted for describing and simulating model problems as well as realistic sit-

uations that aim to better interpret and, possibly, predict the occurrence of a pathology

and the way(s) for curing it, or, at least, maintaining under control its dangerous effect

and development.

Mathematics, in this sense, can provide a fundamental framework for constructing

a virtual laboratory in which the scientist can easily verify assumptions, claims and hy-

potheses on the functional behavior of a cellular system, a tissue, an organ or a synthetic

prosthesis of this latter, in order to improve knowledge and, ultimately, the health and

life of millions of people. This is a mission for which all of us should invest time and

resources for getting inside the fascinating world of Computational Biology, and, all in

all, represents the main scope of this Course and of these notes.

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