CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
AN APPROXIMATE ANALYTICAL SOLUTION FOR THE EXCITATION
THRESHOLD IN A ONE-DIMENSIONAL FITZHUGH-NAGUMO SYSTEM
A thesis submitted in partial fulfillment of the requirements
For the degree of Master of Science in Physics
By
D’Artagnan G. Greene
August 2014
The thesis of D’Artagnan G. Greene is approved:
Date
Dr. Miroslav Peric
Date
Dr. Donna Sheng
Date
Dr. Yohannes Shiferaw, Chair
California State University, Northridge
ii
ACKNOWLEDGEMENTS
I would especially like to thank my research advisor Yohannes Shiferaw for his patience
and guidance throughout this entire thesis project. I would also like to thank all of my other
professors at California State University Northridge for providing me with the background
that I needed to be able to handle a project like this. Last, but not least, I would like to thank
my parents. Without their sustained encouragement and support throughout my education,
this thesis surely would not exist.
iii
TABLE OF CONTENTS
SIGNATURE PAGE .
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ACKNOWLEDGEMENTS .
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LIST OF FIGURES .
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vii
ABSTRACT .
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x
INTRODUCTION
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1
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4
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8
Hodgkin and Huxley’s Expressions for the Conductance of the Various Ion Channels
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10
The Diffusion of the Potential Along a Chain of Cells: The Cable Equation
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15
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18
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20
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Moving in the Direction of an Analytical Theory for the Threshold Surface: Projected Dynamics
36
1. BASIC MODELS OF THE ACTION POTENTIAL
Establishing the Resting Potential Across the Plasma Membrane for a Cell
Modeling Using the Equivalent Circuit Approach
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The General Form of an Equation Describing an Action Potential
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A Simplified Model of the Action Potential: The Fitzhugh-Nagumo Model
2. ANALYTICAL APPROACHES TO STUDYING THE EXCITATION
THRESHOLD
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Studying Threshold Phenomena for a Linear Chain of Cells: The McKean and Moll Approach
The Formal Derivation of Projected Dynamics
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iv
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38
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43
Analysis of the Projected Dynamics Using a Gaussian Pulse
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48
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Applying Projected Dynamics Using a Gaussian Pulse
Discussion of the Results of the Projected Dynamics Using a Gaussian Pulse
3. FORMULATION OF THE THRESHOLD EQUATION FOR A SQUARE PULSE
USING INITIAL CONDITIONS
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55
A New Formulation: Changing the Initial Voltage Profile to a Square Pulse
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Applying Projected Dynamics for a Square Pulse Using Initial Conditions
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Analysis of the Threshold Equation for a Square Pulse Using Initial Conditions
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71
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71
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74
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77
4. FORMULATION OF THE THRESHOLD EQUATION FOR A CONSTANT
SQUARE PULSE
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From Using the Initial Conditions for a Square Pulse to a Constant Square Pulse
Reformulating the Projected Dynamics Using a Constant Square Pulse
Analysis of the Threshold Equation for a Constant Square Pulse
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5. FORMULATION OF THE THRESHOLD EQUATION FOR A TIME-DEPENDENT
SQUARE PULSE
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82
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6. A SIMPLIFIED APPROXIMATE SOLUTION FOR THE SQUARE PULSE .
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Generalizing the Approach to find the Threshold for a Time-Dependent Square Pulse
Obtaining the Threshold Equation for a Time-Dependent Square Pulse
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Introducing a Simplified Initial Voltage Profile for the Constant Square Pulse .
v
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The Projected Dynamics and the Threshold Equation for the Simple Constant Square Pulse
The Threshold Equation for the Simple Time-Dependent Square Pulse
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CONCLUSION
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102
REFERENCES
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104
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APPENDIX: NUMERICAL METHODS
vi
LIST OF FIGURES
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14
2. The fast phase plane of the reduced Hodgkin and Huxley system as plotted by Fitzhugh.
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1. The solution of the Hodgkin and Huxley equations displaying an action potential.
3. Plots of the time course of the various gating variables during an action potential.
4. The fast-slow phase plane for the Hodgkin and Huxley model.
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24
5. The equivalent circuit for the Fitzhugh-Nagumo system.
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6. A propagating wave “front”; a propagating wave “pulse.”
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30
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7. The cubic function used for excitation in the Fitzhugh-Nagumo system.
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9. The initial waveform studied by McKean and Moll.
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10. The threshold surface visualized by McKean and Moll.
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8. The piecewise-linear function used by McKean.
11. Numerical plots of the critical amplitude (height) vs. the pulse width (base).
12. The phase plane for the projected dynamics with a Gaussian pulse.
13. Comparison of Figures 10-12.
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14. Comparison of the amplitude nullclines obtained using projected dynamics vs. the
numerical threshold solution.
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15. Plot of the critical amplitude (𝛼𝑐 ) vs. the ion channel conductivity (𝑔) for the projected
dynamics with a Gaussian pulse.
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16. Plot of the critical amplitude (𝛼𝑐 ) vs. the longitudinal conductivity (𝐷) for the projected
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17. A sample plot of a square pulse. .
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dynamics with a Gaussian pulse.
18. Plot of the critical amplitude (𝛼𝑐 ) vs. the pulse width (𝑤) for the projected
dynamics with a square pulse.
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19. Plot of the critical amplitude (𝛼𝑐 ) vs. the potassium ion channel conductivity
(𝑔𝑘 ) for the projected dynamics with a square pulse.
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20. Plot of the critical amplitude (𝛼𝑐 ) vs. the sodium ion channel conductivity
(𝑔𝑁𝑎 ) for the projected dynamics with a square pulse. .
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21. Plot of the critical amplitude (𝛼𝑐 ) vs. the longitudinal conductivity (𝐷) for the projected
dynamics with a square pulse.
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22. A comparison of the numerical results for the critical amplitude vs. the pulse width
from two different approaches to applying a square pulse.
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23. A plot of the critical amplitude vs. pulse width for a constant applied square pulse.
24. A plot of the critical amplitude vs. potassium ion channel conductivity for a constant
applied square pulse.
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25. A plot of the critical amplitude vs. sodium ion channel conductivity for a constant
applied square pulse.
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viii
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26. A plot of the critical amplitude vs. longitudinal conductivity for a constant applied
square pulse.
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27. A plot of the critical amplitude vs. the time that the pulse was applied for a uniformly
applied square pulse.
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28. A plot of the critical amplitude vs. the time that the square pulse was applied is provided
for both a 100 cell pulse width and a 5 cell pulse width.
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29. A plot of the critical amplitude vs. pulse width for the simple constant applied square
pulse.
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30. A plot of the critical amplitude vs. the time that the square pulse was applied is provided
for both a 100 cell pulse width and a 5 cell pulse width. Here, the simple square pulse approach
is compared with the numerical result.
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ix
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ABSTRACT
AN APPROXIMATE ANALYTICAL SOLUTION FOR THE EXCITATION
THRESHOLD IN A ONE-DIMENSIONAL FITZHUGH-NAGUMO SYSTEM
By
D’Artagnan G. Greene
Master of Science in Physics
Understanding the nature of electrical excitation of a group of cells is important both in
examining the onset of a cardiac arrhythmia and in designing the treatment for sudden
cardiac arrest. In the past, several attempts have been made to understand the threshold for
the excitation of a one-dimensional chain of cells from a mathematical viewpoint.
However, obtaining an analytical solution to describe threshold phenomena has proven to
be difficult as the equations in this problem are highly non-linear and resist solution by
standard mathematical techniques. Here, we apply a method developed by Neu et al. where
the time evolution of the width and amplitude of a pulse is approximately described by a
gradient flow on a two-dimensional phase plane. Using this approach, we obtain a
mathematical expression that successfully models the excitation threshold for an applied
square current pulse in a simplified Fitzhugh-Nagumo system. We then analyze our
solution to reveal how the excitation threshold depends on key physiological parameters.
x
INTRODUCTION
Certain cells are termed “excitable” if they respond to an externally applied electric current
in a characteristic way. If the applied electric current is large enough, the membrane
potential for the cell will carry out a characteristic rise and fall pattern that is called an
action potential. If the applied current is too small, an action potential will not be observed,
and the membrane potential will quickly return to rest. Furthermore, many cells can be
grouped together whereby if some of the cells are excited they can induce nearby cells to
excite. This can lead to a propagating electric signal across an entire group of cells which
is referred to as an excitation wave.
The excitation threshold is defined as the minimum amount of electric current that must be
applied to excite the cell(s). This threshold plays a particularly important role in cardiac
systems. For example, cardiac arrhythmias often arise from unintended excitation of
regions of the heart. At the other extreme, medical defibrillators attempt to manually excite
cells in regions of the heart that have become completely inactive during sudden cardiac
arrest. Yet, despite its importance, the threshold phenomenon is not completely understood.
This thesis consists of six sections. In the first section, we will look at two early
mathematical models of excitable systems. First, the pioneering Hodgkin and Huxley
model will be covered in detail [1], and this will be followed by the more mathematically
simple Fitzhugh-Nagumo model [2,3] which is of the form:
𝜕𝑣 ∂2 𝑣
=
− 𝑣(𝛼 − 𝑣)(1 − 𝑣),
𝜕𝑡 ∂𝑥 2
1
where 𝛼 is the threshold parameter. Even using the simplified Fitzhugh-Nagumo system,
the non-linearity of this equation makes analysis of the threshold phenomenon particularly
difficult.
In the second section, we will examine two significant approaches to characterizing the
excitation threshold using simplified versions of the Fitzhugh-Nagumo system for a linear
chain of cells. The first of these was carried out by H.P. McKean and V. Moll who showed
that the excitation threshold can be visualized as a surface in phase space [4]. Their
approach was later used as a starting point by J.C. Neu et al. who developed a method
called projected dynamics. The projected dynamics made use of a Gaussian-shaped current
pulse to project a pair of approximate solutions for the time evolution of the pulse
amplitude and pulse width onto a two-dimensional phase plane. The threshold was
observed as that set of trajectories on the phase plane which did not lead to either an
excitation or a decay back to rest [5].
In the third section, we show that the projected dynamics can be further refined by replacing
the Gaussian pulse with one that more closely resembles a square-shaped physiological
current pulse. Doing this corrects some of the defects in the Gaussian approach such as
the wrong threshold limits being reached in the case of large and small pulse widths. At
the same time, we will obtain an approximate analytical expression for the excitation
threshold by noting its similarity to the amplitude nullcline of the projected dynamics.
In section four, more realism will be added to the model as the applied current pulse is
changed from being specified using initial conditions to an externally applied current pulse
that is held constant throughout the excitation process. In making this change we will
discover that the criteria for the excitation threshold will have to be modified. More
2
importantly, this modification brings the approximate analytical expression for the
threshold into very close agreement with the numerical result obtained directly from the
partial differential equation.
In section five, we ease the restriction on the time duration of the applied current pulse so
that the current pulse can be applied for a finite amount of time. In doing so, we will obtain
the most general solution for the excitation threshold. It will be shown that this general
solution consists of both the solution for the constant square pulse and an additional timedependent term.
Finally, in section six, we will introduce a simplification to the model developed in sections
four and five. The simple constant square pulse yields a threshold solution that, at the
expense of a little accuracy, will make the threshold solution more compact and easier to
work with while retaining the correct limiting behavior of the more complicated model that
came before.
3
1. BASIC MODELS OF THE ACTION POTENTIAL
Establishing the Resting Potential Across the Plasma Membrane for a Cell
We will start by describing the electrical properties of a lone cell such as a neuron. The
mathematical treatment will follow the now standard treatments of the subject such as that
which is found in G.B. Ermentrout and D.H. Terman’s Mathematical Foundations of
Neuroscience [6]. A cell in general terms is essentially a collection of organelles, small
chemical factories that serve some biological purpose for the cell, which are placed in a
mostly aqueous environment that is separated from the aqueous environment on the outer
side of the cell by a non-polar plasma membrane. The electrical properties of such a cell
are localized at this plasma membrane and are brought about by the flow of charged ions
across this membrane.
The plasma membrane itself is impermeable to charged ions, but small protein channels
are inserted into the membrane that allow a specific charged ion to move across the
membrane while excluding other ions from passing through. For example, there are
channels in the membrane that allow for sodium ions to move across the membrane, but do
not allow potassium ions to pass through. And then there are other channels that allow
potassium ions to pass through but block sodium. Once an ion channel is opened, the flow
or diffusion of that ion across the membrane is determined by two competing processes:
one that is based on concentration and another that depends on the electric potential. As an
example, consider a cell with a plasma membrane that separates an electrically neutral
internal aqueous solution containing potassium and chloride ions from an external aqueous
solution that also contains an equivalent amount of potassium and chloride ions. If a
channel protein that selectively allowed passage to potassium ions is now inserted into the
4
plasma membrane, no net change would be observed. This is due to the concentration of
the potassium ions inside and outside being the same when the channel protein is inserted.
On the other hand, suppose that initially the number of potassium ions on the inside of the
cell is greater than the number of potassium ions outside the cell. When the channel protein
is inserted, there would be a net diffusion of potassium ions from the inside of the cell to
the outside as the concentrations tend to spontaneously equalize. This diffusion is described
mathematically by Fick’s law of diffusion:
𝐽𝐷 = −𝐷
𝜕[𝐶]
.
𝜕𝑦
(1.1)
In this expression, the concentration [𝐶] is the number of potassium ions per unit volume,
𝐷 is Fick’s diffusion constant, and 𝐽𝐷 is the flux of ions flowing across the cross-sectional
area of the plasma membrane. The 𝑦 coordinate is measured from the inside of the channel
protein to the outside, and the negative sign implies that the flow tends to be from a high
concentration of ions to a low concentration.
If the potassium ions were uncharged, this description would be sufficient. However,
potassium ions carry an electric charge of +1 along with them. As the potassium ions
diffuse across the membrane in an attempt to equalize their concentrations, a charge
imbalance begins to appear. The outer side of the plasma membrane becomes more
positively charged and the inside of the plasma membrane more negatively charged. Thus
a potential difference starts to build up that resists the flow of the diffusing ions. This
potential flux can be described by:
𝐽𝑉 = −𝜇𝑧[𝐶]
𝜕𝑉
.
𝜕𝑦
5
(1.2)
Here 𝜇 represents the mobility of the ion, 𝑧 is the valence of the ion (in the case of
potassium it would be +1), [𝐶] is the concentration, and 𝑉 is the electric potential. Here,
the negative sign indicates that the flow is in the direction of decreasing potential. The total
flux of ions across the membrane is then described as the sum of these two competing
effects:
𝐽 = 𝐽𝐷 + 𝐽𝑉 = −𝐷
𝜕[𝐶]
𝜕𝑉
− 𝜇𝑧[𝐶] .
𝜕𝑦
𝜕𝑦
(1.3)
Albert Einstein developed a theory that expresses the diffusion constant 𝐷 in terms of its
mobility 𝜇, the fundamental charge 𝑞𝑒 = 1.60 x 10-19 C, Boltzmann’s constant 𝑘 = 1.38 x
10-23 J/K, and the absolute temperature 𝑇:
𝑘𝑇
𝜇.
𝑞𝑒
𝐷=
(1.4)
This can be substituted for 𝐷 in equation (1.3) to give:
𝐽= −
𝑘𝑇 𝜕[𝐶]
𝜕𝑉
𝜇
− 𝜇𝑧[𝐶] .
𝑞𝑒 𝜕𝑦
𝜕𝑦
(1.5)
After some time has passed, an equilibrium may be established whereby it is observed that
the net concentration of potassium ions on either side of the membrane is no longer
changing in time. This condition is satisfied when the flux vanishes and 𝐽 = 0. The
potential at equilibrium is called the Nernst potential, and can be found as follows:
0=
𝑘𝑇 𝑑[𝐶]
𝑑𝑉
+ 𝑧[𝐶] ,
𝑞𝑒 𝑑𝑦
𝑑𝑦
𝑘𝑇 𝑑[𝐶]
,
𝑧𝑞𝑒 [𝐶]
(1.7)
𝑘𝑇 [𝐶𝑖𝑛] 𝑑[𝐶]
∫
,
𝑧𝑞𝑒 [𝐶𝑜𝑢𝑡] [𝐶]
(1.8)
𝑑𝑉 = −
∫
𝑉𝑖𝑛
𝑉𝑜𝑢𝑡
𝑑𝑉 = −
(1.6)
6
𝑉𝑁 = 𝑉𝑖𝑛 − 𝑉𝑜𝑢𝑡 =
𝑘𝑇
[𝐶𝑜𝑢𝑡 ]
ln (
).
𝑧𝑞𝑒
[𝐶𝑖𝑛 ]
(1.9)
Multiplying and dividing the right side by Avogadro’s number 𝑁𝑎 = 6.022 x 1023 mol-1
yields:
𝑉𝑁 = (
𝑘𝑁𝐴 𝑇
[𝐶𝑜𝑢𝑡 ]
) ln (
),
𝑞𝑒 𝑁𝐴 𝑧
[𝐶𝑖𝑛 ]
𝑉𝑁 =
𝑅𝑇
[𝐶𝑜𝑢𝑡 ]
ln (
).
𝑧𝐹
[𝐶𝑖𝑛 ]
(1.10)
(1.11)
In equation (1.11), 𝑅 is the ideal gas constant given as 8.314 J/(mol K), and 𝐹 is Faraday’s
constant which is approximately 96,352 C/mol.
At rest, the concentration differences for sodium and potassium ions between the inside
and the outside of the cell are established by the use of a sodium-potassium pump which
requires an external energy input to move the ions against their concentration gradients.
Typical values of the concentrations for a cell in the squid giant axon at rest are given in J.
Keener and J. Sneyd’s Mathematical Physiology [7] and are listed in Table 1 together with
the Nernst potential for each ion which was calculated using equation (1.11) at an assumed
body temperature of 37℃. The squid giant axon was used in the original studies carried
out by Hodgkin and Huxley [1].
The resting potential of the plasma membrane as a whole for the squid giant axon is listed
as -66 mV. This value is a net result of the contributions made by the various ion channels,
and it depends on the relative number of open channels for each ion while the cell is at rest.
In particular, the large negative value indicates that sodium channels play a much smaller
role in the overall resting potential than the potassium channels do. We will return to this
point shortly.
7
[Cin]
[Cout]
VN
(mmol)
(mmol)
(mV)
K+
397
20
-80
Na+
50
437
58
Cl-
40
556
-70
Table 1. Concentrations and Nernst Potentials for the Squid Giant Axon [7].
Modeling Using the Equivalent Circuit Approach
Once the rest state is established, the next step is to capture the behavior of a nerve cell
when the voltage across the membrane is not at the resting potential. A popular approach
is to use the concept of the equivalent circuit first developed by L. Lapicque [8]. This
approach treats the charge separation between the inside and the outside of the cell as a
capacitor with a potential difference across it. A channel protein is treated as both a battery
at the Nernst potential and a resistor that impedes current flow. In such a setup, charge will
flow from the capacitor through the channel proteins until the Nernst potential is reached.
Consider a plasma membrane that contains channels which only allow potassium ions to
cross. The capacitance of the membrane as a whole can be described by:
𝐶=
𝑄
,
𝑉
(1.12)
where 𝑄 is the net ionic charge that is stored on either side of the membrane while V is the
membrane potential established for that charge. The current for the capacitor can be
obtained as:
8
𝑄 = 𝐶𝑉,
𝐼𝐶 =
(1.13)
𝜕𝑄
𝜕𝑉
=𝐶 .
𝜕𝑡
𝜕𝑡
(1.14)
The other potential changes in the circuit occur as the charge moves through an ion channel
across the resistor and the battery. This can be expressed as a current following Ohm’s law:
𝐼𝐾 =
(𝑉 − 𝑉𝐾 )
.
𝑅𝐾
(1.15)
In (1.15), 𝑉 is the potential difference due to the capacitor, 𝑅𝐾 is the resistance in the
potassium channel, and 𝑉𝐾 is the Nernst potential for potassium. Since there is no external
input of current, Kirchhoff’s loop rules require that the sum of these two currents vanish to
give:
0 = 𝐼𝐶 + 𝐼𝐾 ,
0= 𝐶
𝜕𝑉 (𝑉 − 𝑉𝐾 )
+
,
𝜕𝑡
𝑅𝐾
(1.16)
(1.17)
𝜕𝑉
(1.18)
= −𝐺𝐾 (𝑉 − 𝑉𝐾 ).
𝜕𝑡
Here, 𝐺𝐾 is the conductance which is equal to 1⁄𝑅𝐾 . Other ion channels, such as sodium
𝐶
or chloride channels, can be connected in parallel to this circuit which simply extends the
result to:
𝐶
𝜕𝑉
= −𝐺𝐾 (𝑉 − 𝑉𝐾 ) − 𝐺𝑁𝑎 (𝑉 − 𝑉𝑁𝑎 ) − 𝐺𝐶𝑙 (𝑉 − 𝑉𝐶𝑙 ).
𝜕𝑡
(1.19)
If an external current is now applied to the cell, as would be the case when an electrode is
applied to inject current into the cell as an example, equation (1.16) becomes:
𝐼 = 𝐼𝐶 + 𝐼𝐾 .
(1.20)
Carrying the steps through as before, the expression for the equivalent circuit is then:
9
𝐶
𝜕𝑉
= − 𝐺𝐾 (𝑉 − 𝑉𝐾 ) − 𝐺𝑁𝑎 (𝑉 − 𝑉𝑁𝑎 ) − 𝐺𝐶𝑙 (𝑉 − 𝑉𝐶𝑙 ) + 𝐼.
𝜕𝑡
(1.21)
Hodgkin and Huxley’s Expressions for the Conductance of the Various Ion
Channels
Equation (1.21) can be used as a basis to understand the equivalent circuit of a cell with
various ion channels, but such an equation in and of itself does not clearly describe how an
action potential may come about. It was Hodgkin and Huxley who first developed a theory
that described how the conductance for an ion channel may change with time depending
on changes in the potential difference across the plasma membrane, and they explained
how these changes gave rise to an action potential [1].
They distinguished between two types of channels present in the membrane that could
conduct ions; there were non-gated channels, which were channels that were always open
regardless of the electrical state of the membrane, and there were voltage-gated channels
which could change from being open to closed or vice versa depending on the present value
of the potential difference across the membrane in the vicinity of the channel. All of the
non-gated channels were grouped together into one equivalent resistor, and the current that
ran through it was termed the “leak” current. This equivalent resistor included many
channels that played a role in establishing the resting potential such as the chloride
channels. Introducing this leak current, equation (1.19) became:
𝐶
𝜕𝑉
= − 𝐺𝐾 (𝑉 − 𝑉𝐾 ) − 𝐺𝑁𝑎 (𝑉 − 𝑉𝑁𝑎 ) − 𝐺𝐿 (𝑉 − 𝑉𝐿 ).
𝜕𝑡
(1.22)
Since the channel proteins that comprised the leak current were always open, the
conductance 𝐺𝐿 was found to be a constant independent of the potential difference across
10
the membrane, and it could be found experimentally [1]. On the other hand, the sodium
channels and some of the potassium channels were discovered to have conductance values
that changed as a function of time. Hodgkin and Huxley proposed the following two
equations to describe the conductance for the voltage-gated potassium channels:
𝐺𝐾 = 𝐺̅𝐾 𝑛4 ,
(1.23)
𝑑𝑛
= 𝛼𝑛 (1 − 𝑛) − 𝛽𝑛 𝑛.
𝑑𝑡
(1.24)
In equation (1.23), 𝑛 is a parameter that can take any value between 0 and 1 while 𝐺̅𝐾 is a
constant equal to the maximum value of the potassium conductance. Equation (1.24)
describes the manner by which 𝑛 changes with time. Suppose that there are two states for
the potassium channel, a state 𝑛 that is open (𝑂) and a state (1 − 𝑛) that is closed (𝐶).
Further suppose that a chemical equilibrium is established between the two states as shown
below [6]:
𝐶
𝛼𝑛
⇌
𝛽𝑛
(1.25)
𝑂.
In this context, 𝛼𝑛 determines the rate at which a closed state is converted into an open
state and 𝛽𝑛 determines the rate at which an open state is converted into a closed state.
Both 𝛼𝑛 and 𝛽𝑛 depend on the potential difference of the membrane but are otherwise
independent of time. The rate of conversion from the closed state to the open state (the
forward process) is given by:
𝑑𝑛𝑓
(1.26)
= 𝛼𝑛 (1 − 𝑛),
𝑑𝑡
and the rate of the reverse process, the conversion of the open state back to a closed state,
is given by:
11
𝑑𝑛𝑟
= − 𝛽𝑛 𝑛.
𝑑𝑡
(1.27)
The net change in 𝑛 is the sum of (1.26) and (1.27) which is exactly (1.24). Equation (1.24)
can be cast in a slightly different form. First, the steady state solution of 𝑛 can be found
by setting 𝑑𝑛/𝑑𝑡 = 0 and solving for 𝑛∞ :
0 = 𝛼𝑛 (1 − 𝑛∞ ) − 𝛽𝑛 𝑛∞ ,
𝑛∞ =
𝛼𝑛
.
𝛼𝑛 + 𝛽𝑛
(1.28)
(1.29)
𝑛∞ can then be reintroduced into equation (1.24):
𝑑𝑛
= 𝛼𝑛 − (𝛼𝑛 + 𝛽𝑛 )𝑛,
𝑑𝑡
(1.30)
𝑑𝑛
= (𝛼𝑛 + 𝛽𝑛 )𝑛∞ − (𝛼𝑛 + 𝛽𝑛 )𝑛,
𝑑𝑡
(1.31)
𝑑𝑛 (𝑛∞ − 𝑛)
=
,
𝑑𝑡
𝜏𝑛
(1.32)
where 𝜏𝑛 is a time constant given by:
𝜏𝑛 =
1
.
(𝛼𝑛 + 𝛽𝑛 )
(1.33)
The sodium conductance is similar, but contains two types of voltage dependent
parameters, 𝑚 and ℎ:
𝐺𝑁𝑎 = 𝐺̅𝑁𝑎 𝑚3 ℎ.
(1.34)
The expressions for the time dependence of the parameters 𝑚 and ℎ are obtained in the
same way as before:
𝑑𝑚 (𝑚∞ − 𝑚)
=
,
𝑑𝑡
𝜏𝑚
𝛼𝑚
𝑚∞ =
,
𝛼𝑚 + 𝛽𝑚
12
(1.35)
(1.36)
𝜏𝑚 =
1
,
(𝛼𝑚 + 𝛽𝑚 )
(1.37)
𝑑ℎ (ℎ∞ − ℎ)
=
,
𝑑𝑡
𝜏ℎ
𝛼ℎ
ℎ∞ =
,
𝛼ℎ + 𝛽ℎ
(1.38)
1
.
(𝛼ℎ + 𝛽ℎ )
(1.40)
𝜏ℎ =
(1.39)
At this point the various equations can be solved if each of the 𝛼 and 𝛽 expressions are
given along with numerical values for the capacitance, the maximum values of each
conductance, and values of the Nernst potential for each channel type. Hodgkin and Huxley
obtained the required expressions and numerical values from their experimental data [1,6].
A plot of the action potential and the associated time dependent sodium and potassium
conductance is provided in Figure 1.
Hodgkin and Huxley explained how the action potential arose from the time-dependent
changes in the conductivities. An initial external stimulus applied to the cell raised the
membrane potential to a certain level until suddenly the 𝐺𝑁𝑎 conductance began to rapidly
grow. This growth was determined by the parameter 𝑚 which increased rapidly at first to
signify that the sodium channels were opening and that sodium ions were rushing into the
cell. The membrane voltage then rose towards the Nernst potential for sodium of +58 mV.
However, as the membrane voltage climbed upwards, the parameter ℎ increased and tended
to close the sodium channels by blocking the pathway for the sodium ions. The sodium
conductance overall quickly peaked and then dropped back down to a non-conducting state.
13
Figure 1. The solution of the Hodgkin and Huxley equations displaying an action
potential. The lower graph shows the time dependent sodium and potassium
conductivities [6].
Meanwhile, the potassium channels governed by the parameter 𝑛 tended to open more at
higher values of the potential. With the sodium channels now inactivated by the parameter
ℎ, the potassium conductivity became the dominant effect, and the membrane potential
returned to the resting potential of -66 mV to complete the action potential. After an action
potential is fired, there is a period of time, called the refractory period, where the ion
concentrations are re-established at their original resting levels by the sodium-potassium
pump. Once the refractory period is complete, the cell is ready to fire an action potential
once again.
14
The Diffusion of the Potential Along a Chain of Cells: The Cable Equation
So a question might naturally arise at this point: Why would a cell transmit electric signals
by using an action potential? The procedure is quite complex, and a cell can surely receive
an electric signal and return to rest just as well without undergoing such a cycle. The
advantage lies not with the single cell, but rather it comes into play when an electric signal
is propagated from one cell to another across a long chain of cells.
To see this, consider a linear chain of cells that are attached end to end to one another in
the shape of a cylindrical wire. If we were to stimulate one end of the chain with a current
pulse, the plasma membrane of the cells near the stimulus would be at a higher potential
than the cells much further down the chain. Assuming that charge can flow between cells,
a potential difference would be set up along the chain and charge would flow from the end
near the stimulus towards the distant cells at a lower potential. This longitudinal motion of
charge can be described as a current flowing along the 𝑥-axis in the direction of the wire
[6]. Following Ohm’s law, we have:
𝑉(𝑥 + ∆𝑥, 𝑡) − 𝑉(𝑥, 𝑡) = 𝐼𝐷 𝑅𝐷 .
(1.41)
Since we have assumed a cylindrical shape, we can use an expression for the resistance of
a cylindrical wire with radius 𝑎, resistivity 𝜌, and a small change in length ∆𝑥:
𝑅𝐷 = 𝜌
∆𝑥
,
𝜋𝑎2
(𝑉(𝑥 + ∆𝑥, 𝑡) − 𝑉(𝑥, 𝑡))
𝜌
= 𝐼𝐷 2 .
∆𝑥
𝜋𝑎
(1.42)
(1.43)
Taking the limit of (1.43) as ∆𝑥 → 0 gives an expression for the longitudinal diffusion
current:
15
𝜋𝑎2 𝜕𝑉(𝑥, 𝑡)
𝐼𝐷 =
.
𝜌
𝜕𝑥
(1.44)
Since this is a current that can leave one cell and enter another, Kirchhoff’s loop rules
demand for each individual cell that we have:
𝐼𝐷 = 𝐼𝐶 + 𝐼𝑁𝑎 + 𝐼𝐾 + 𝐼𝐿 .
(1.45)
Now, before going any further, it must be said at this point that both the total capacitance
and the total conductance depend on the total surface area of the wire which increases as
the number of cells in the chain increase. In order to write an expression for the voltage
change per small change in the longitudinal distance, we must make use of the specific
capacitance (𝑐) defined as the capacitance per unit area and the specific conductivity (𝑔)
which is defined as the conductivity per unit area. The total capacitance for the chain as a
whole is then the surface area of the cylinder, 2𝜋𝑎𝑥, times the specific capacitance and
similarly for the total conductance:
𝐶 = 2𝜋𝑎𝑥𝑐,
(1.46)
𝐺 = 2𝜋𝑎𝑥𝑔.
(1.47)
For a small change in 𝑥, these become 𝐶 = 2𝜋𝑎∆𝑥𝑐 and 𝐺 = 2𝜋𝑎∆𝑥𝑔. Using (1.22), (1.44)
and (1.45) gives:
𝜋𝑎2 𝜕𝑉(𝑥 + ∆𝑥, 𝑡) 𝜕𝑉(𝑥, 𝑡)
(
−
)=
𝜌
𝜕𝑥
𝜕𝑥
2𝜋𝑎∆𝑥 (𝑐
𝜕𝑉
+ 𝑔𝐾 (𝑉 − 𝑉𝐾 ) + 𝑔𝑁𝑎 (𝑉 − 𝑉𝑁𝑎 ) + 𝑔𝐿 (𝑉 − 𝑉𝐿 )),
𝜕𝑡
𝑎
𝜕𝑉(𝑥 + ∆𝑥, 𝑡) 𝜕𝑉(𝑥, 𝑡)
(
−
)=
2𝜌∆𝑥
𝜕𝑥
𝜕𝑥
𝑐
(1.48)
𝜕𝑉
+ 𝑔𝐾 (𝑉 − 𝑉𝐾 ) + 𝑔𝑁𝑎 (𝑉 − 𝑉𝑁𝑎 ) + 𝑔𝐿 (𝑉 − 𝑉𝐿 ).
𝜕𝑡
Taking the limit ∆𝑥 → 0 and rearranging yields:
16
(1.49)
𝜕𝑉
𝑎 𝜕 2 𝑉(𝑥, 𝑡)
𝑐
=
− 𝑔𝐾 (𝑉 − 𝑉𝐾 ) − 𝑔𝑁𝑎 (𝑉 − 𝑉𝑁𝑎 ) − 𝑔𝐿 (𝑉 − 𝑉𝐿 ).
𝜕𝑡
2𝜌 𝜕𝑥 2
(1.50)
The spatial derivative in equation (1.50) accounts for the longitudinal diffusion of the
membrane potential from one point along the wire to another. In order to see the advantage
of the action potential, let us consider for a moment that voltage-gated ion channels are not
present in the membrane. Then we can group all of the non-gated ion channels into a single
equivalent channel as was done for the leak current before. Taking this channel to have a
constant conductivity 𝑔 and a Nernst potential of zero for simplicity, equation (1.50) would
then become:
𝜕𝑉
𝑎 𝜕 2 𝑉(𝑥, 𝑡)
𝑐
=
− 𝑔𝑉.
𝜕𝑡
2𝜌 𝜕𝑥 2
(1.51)
Solving for the steady state solution with the boundary conditions 𝑉 = 𝑉0 at 𝑥 = 0, and
𝑉 = 0 at 𝑥 = ∞ gives:
0=
𝑑 2 𝑉(𝑥) 2𝜌𝑔
−
𝑉,
𝑑𝑥 2
𝑎
𝑉 = 𝑉0 𝑒 −𝑥⁄√𝑎⁄2𝜌𝑔 .
(1.52)
(1.53)
At the steady state, the amplitude of the voltage decreases exponentially from the point of
stimulation towards the end of the chain. However, such a decrease in the amplitude with
distance can be largely avoided by the use of action potentials to propagate the signal. As
each new cell along the chain fires an action potential, the next cell reaches the same
voltage peak as the one before it (this is provided that each cell can acquire enough charge
from the prior cell to fire its own action potential). The effect is much like dominoes in that
if the first cell is provided a sufficient kick, each cell will act in turn along the line for great
distances with no additional energy input. The action potential then acts as a signal
amplifier for the electric wave as it travels along a chain of cells. At each cell, the peak
17
voltage is renewed as an action potential fires, and the electric signal can therefore arrive
undiminished at its final destination even if it is initiated at a distant location.
The General Form of an Equation Describing an Action Potential
In the years that followed the introduction of the Hodgkin and Huxley model, several other
models describing action potentials made an appearance. These models varied greatly in
both their complexity and purpose, but all of the models shared the same general form. In
order to obtain this more general form, the expression for the membrane potential in the
Hodgkin and Huxley model can be derived using a different starting point: the conservation
of electric charge.
Consider an extended linear chain of cells. If a small portion of these cells is electrically
stimulated by an external current source, the membrane potential in this region rises to a
voltage 𝑉 as a charge 𝑄 builds up on the surface of the membrane. The capacitance per unit
area in this region can be described by:
𝑄
(1.54)
.
𝑉
Since the potential is higher in this region of the membrane than in the surrounding regions,
𝑐=
the charge 𝑄 that has built up in the vicinity of the stimulus will begin to drift longitudinally
down the chain towards the region of lower potential. The law of conservation of electric
charge states that the decrease in the amount of charge in the small region of the membrane
near the stimulus must equal the increase in the amount of charge that has left the region
and is now drifting down the chain. This law can be represented mathematically by a
continuity equation:
18
𝜕𝑄(𝑥, 𝑡)
+ ∇𝐽(𝑥, 𝑡) = 𝑆.
𝜕𝑡
(1.55)
In (1.55), 𝑄 is interpreted as the amount of charge within the small region of the membrane,
𝐽 is the flux of the charge leaving or entering the boundary of the small region, and 𝑆 is a
term representing any additional sources or sinks of electric charge in the small region such
as that provided by an external stimulus or by ion channels. Instead of working with charge,
it is also possible to write an equation for the continuity of the electric potential 𝑉 by using
(1.54) and substituting for 𝑄 in (1.55):
𝑐𝑉 = 𝑄,
(1.56)
𝜕𝑉(𝑥, 𝑡)
+ ∇𝐽(𝑥, 𝑡) = 𝑆.
𝜕𝑡
𝑐
(1.57)
The flux 𝐽(𝑥, 𝑡) in (1.57) can now be interpreted as a potential flux, and it can be described
using Fick’s law:
𝐽(𝑥, 𝑡) = −𝐷
𝜕𝑉(𝑥, 𝑡)
.
𝜕𝑥
(1.58)
Substituting this expression into (1.57) gives:
𝑐
𝜕𝑉(𝑥, 𝑡)
∂ 𝜕𝑉(𝑥, 𝑡)
−𝐷 (
) = 𝑆,
𝜕𝑡
∂𝑥
𝜕𝑥
(1.59)
(1.60)
𝜕𝑉(𝑥, 𝑡)
∂2 𝑉(𝑥, 𝑡)
=𝐷
+ 𝑆.
2
𝜕𝑡
∂𝑥
If we were to switch off the external stimulus at some point, and also assume for the
𝑐
moment that the membrane does not contain any ion channels, 𝑆 can be taken to be zero,
and (1.60) reduces to:
𝑐
𝜕𝑉(𝑥, 𝑡)
∂2 𝑉(𝑥, 𝑡)
=𝐷
.
𝜕𝑡
∂𝑥 2
(1.61)
Equation (1.61) is equivalent in form to the well-known heat equation. It describes the
spread of voltage with position and time.
19
On the other hand, when an external current or ion channels are added to the plasma
membrane they can be treated as a source or sink of charge as these sources allow for
charged ions to move across the plasma membrane in a direction perpendicular to the
direction of the longitudinal diffusion. In such a case, the main equation in the Hodgkin
and Huxley model can be recovered if the source term 𝑆 in equation (1.60) is taken to be:
𝑆 = −𝑔𝐾 (𝑉 − 𝑉𝐾 ) − 𝑔𝑁𝑎 (𝑉 − 𝑉𝑁𝑎 ) − 𝑔𝐿 (𝑉 − 𝑉𝐿 ) + 𝐼.
(1.62)
A Simplified Model of the Action Potential: The Fitzhugh-Nagumo Model
Equation (1.60) may be taken as the general form for an action potential model. Other
expressions for the source term 𝑆 may be introduced into equation (1.60) to create a new
model. However, not just any function 𝑆 that is inserted into (1.60) can generate an action
potential; successful functions must possess at least some properties that are similar to the
Hodgkin and Huxley model in order to reproduce the same basic behavior.
In the years that followed the introduction of the Hodgkin and Huxley model, the focus
shifted to finding simpler and more mathematically tractable equations to work with that
could describe an action potential without having the complexity of a 4-variable system. It
was Richard Fitzhugh that first succeeded in creating a model that only required two
variables and could still generate an action potential [2,7,9]. The original motivation behind
the reduction was to make the Hodgkin and Huxley model amenable to phase space
analysis on a two-dimensional plane. Fitzhugh noticed that the four variables in the
Hodgkin and Huxley model can be paired into two groups. The 𝑉 and 𝑚 variables were
considered to be “fast variables” in that they were the dominant terms during the initial
20
membrane excitation. On the other hand, the 𝑛 and ℎ variables were considered to be “slow
variables” as they only appreciably changed in value after the excitation process was well
underway and the system was just beginning to return to rest [7].
Fitzhugh first studied the behavior of the fast variables for a single cell by holding 𝑛 and ℎ
constant, as 𝑛0 and ℎ0 , while studying the time dependent changes in 𝑉 and 𝑚. This
reduced the Hodgkin and Huxley system to:
𝑐
𝑑𝑉
= −𝑔̅𝐾 𝑛04 (𝑉 − 𝑉𝐾 ) − 𝑔̅𝑁𝑎 𝑚3 ℎ0 (𝑉 − 𝑉𝑁𝑎 ) − 𝑔̅𝐿 (𝑉 − 𝑉𝐿 ),
𝑑𝑡
𝑑𝑚
𝑚∞ − 𝑚
=
.
𝑑𝑡
𝜏𝑚
(1.63)
(1.64)
The nullclines for such a system can be immediately found by setting 𝑑𝑉/𝑑𝑡 and 𝑑𝑚/𝑑𝑡
equal to zero and solving for 𝑉 and 𝑚. With these in hand, and by using the appropriate
expressions and numerical constants from the Hodgkin and Huxley model, Fitzhugh was
able to obtain the fast phase plane shown in Figure 2. In Figure 2, the 𝑉 and 𝑚 nullclines
intersect at three points labeled A, B, and C. Points A and C are stable fixed points that
correspond to the resting state and the excited state respectively. Point B is on a stable
manifold that acts as a threshold which must be overcome in order for the cell to become
excited. If the initial condition is such that the initial value of the potential does not rise
above this threshold, the trajectory will decay down to the resting state at A. If the initial
condition for the potential is above the stable manifold then the trajectory will proceed to
the excited state at C. A few sample trajectories are provided in Figure 2 to demonstrate
this behavior.
Although the fast phase plane was useful for visualizing the threshold behavior for exciting
a cell, without the time dependence of the recovery variables a full action potential could
21
not be visualized. Systems that were excited would stay excited at point C and never return
to rest as required for a true action potential. To visualize a full action potential on a twodimensional phase plane, it became necessary to work with a system that contained one
fast variable and one slow variable while holding the other variables constant. To do this,
Fitzhugh first treated 𝑚 as if it was always in an instantaneous equilibrium so that 𝑚 =
𝑚∞ . This reduced the number of the fast variables to one.
Figure 2. The fast phase plane of the reduced Hodgkin and Huxley system as plotted by
Fitzhugh [9]. The V and m nullclines are visualized as dashed lines while a few sample
trajectories are shown with solid lines. The stable manifold at B acts as a threshold for the
excitation of the cell.
22
In order to reduce the two slow variables down to one, Fitzhugh exploited a near symmetry
in the gating variables 𝑛 and ℎ. From Figure 3 it appears that ℎ + 𝑛 ≈ 0.8, and ℎ can
therefore be eliminated by replacing it with ℎ = 0.8 − 𝑛. A reduced Hodgkin and Huxley
system for the variables 𝑉 and 𝑛 can then be written as:
𝑐
𝑑𝑉
3 (0.8
= −𝑔̅𝐾 𝑛4 (𝑉 − 𝑉𝐾 ) − 𝑔̅𝑁𝑎 𝑚∞
− 𝑛)(𝑉 − 𝑉𝑁𝑎 ) − 𝑔̅𝐿 (𝑉 − 𝑉𝐿 ),
𝑑𝑡
𝑑𝑛
𝑛∞ − 𝑛
=
.
𝑑𝑡
𝜏𝑛
(1.65)
(1.66)
Figure 3. Plots of the time course of the various gating variables during an action
potential [7].
23
The nullclines for this system can be obtained, and the fast-slow phase plane can be plotted
as in Figure 4. In the fast-slow phase plane, the nullclines intersect at only one stable fixed
point which corresponds to the resting potential. Initial conditions close to this point tend
to return to the resting potential directly. However, when the initial condition for the
potential is sufficiently large, something else happens entirely. Instead of returning to the
stable fixed point directly, the trajectory first moves out to larger and larger values of the
potential before cycling back around and returning to the fixed point as shown in Figure
4A. This detour on the phase plane corresponds to the rise and fall of an action potential.
Figure 4. A: The fast-slow phase plane for the Hodgkin and Huxley model. The two
nullclines are shown together with a trajectory for an action potential. B: A plot of the
time course of the action potential that is shown in A [7].
Now, if the shapes of the nullclines in Figure 4 are examined closely, it would appear that
the 𝑉 nullcline that represents the excitation is a simple cubic shape, and the 𝑛 nullcline
that represents the recovery is approximately linear for much of the range of 𝑛. Using what
he termed the Bonhoeffer-van der Pol (BVP) model, Fitzhugh was able to obtain a much
24
simpler set of equations that captured these phase plane features and could generate an
action potential [2]. This set of equations eventually became known as the FitzhughNagumo model [7]:
𝜖
𝑑𝑣
= 𝑓(𝑣) − 𝑤 + 𝐼,
𝑑𝑡
𝑑𝑤
= 𝑣 − 𝛾𝑤.
𝑑𝑡
(1.67)
(1.68)
In this model, 𝑓(𝑣) is taken to be the cubic function:
𝑓(𝑣) = 𝑣(1 − 𝑣)(𝑣 − 𝛼),
𝑓𝑜𝑟 0 < 𝛼 < 1, 𝜖 ≪ 1.
(1.69)
In this system the excitation variable is 𝑣, the relaxation variable is 𝑤, and 𝐼 corresponds
to an externally applied current. The rest of the parameters are constants, and typical values
as given in Keener and Sneyd’s Mathematical Physiology are reproduced in Table 2. If the
nullclines are obtained for this system, it is seen that the 𝑣 nullcline is cubic, and the 𝑤
nullcline is a linear function of 𝑣 in agreement with the approximate form of the nullclines
in the Hodgkin and Huxley fast-slow phase plane.
Parameter
Value
𝛼
0.1
𝛾
0.5
𝜖
0.01
Table 2. Typical values for the parameters used in the Fitzhugh-Nagumo equations [7].
25
A few years after Fitzhugh’s original discovery, a team of Japanese scientists led by J.
Nagumo built the equivalent circuit for Fitzhugh’s set of equations [3]. This circuit is
diagrammed in Figure 5. The three different parallel paths represent the various currents.
The fast excitation current is represented by 𝐹(𝑉) on the left path where Nagumo used a
tunnel diode to capture the non-linear behavior of 𝐹(𝑉). The middle path represents the
recovery current where a battery, a resistor, and an inductor are connected in series with
one another. The far right path is the capacitor representing the potential across the
plasma membrane.
Figure 5. The equivalent circuit for the Fitzhugh-Nagumo system [7].
Both of Fitzhugh’s equations can be derived directly from the equivalent circuit approach
[7]. To see this, Kirchhoff’s loop rules are used to write expressions for the total current
and the voltage drop across the middle path:
26
𝑑𝑉
+ 𝐹(𝑉) + 𝐼𝑅 = 𝐼0 ,
𝑑𝑡
𝑑𝐼𝑅
𝐿
+ 𝑅𝐼𝑅 + 𝑉0 = 𝑉.
𝑑𝑡
𝑐
(1.70)
(1.71)
In (1.70) and (1.71), 𝑉 represents the potential difference across the membrane given by
𝑉 = 𝑉𝑖 − 𝑉𝑒 where 𝑉𝑖 is the potential inside the membrane and 𝑉𝑒 is the potential outside
the membrane. 𝐼𝑅 represents the current that flows across the middle path that contains a
battery with resting potential 𝑉0, an inductor 𝐿, and a resistor 𝑅 while 𝐼0 is an externally
applied current.
Now, if 𝐹(𝑉) is a cubic function in the form assumed by Fitzhugh in equation (1.69), it
should have three fixed points. Two of them are stable at 𝑉 = 0 and 𝑉 = 1 and correspond
to the rest state 𝑉0 and excited state 𝑉1 respectively while the middle fixed point represents
the value of the threshold for excitation at 𝑉 = 𝛼. Also, a passive resistance for the tunnel
diode can be taken as 𝑅1 which is defined as 𝑅1 = − 1⁄𝐹′(0) where 𝐹′(0) is the derivative
of 𝐹(𝑉) evaluated at 𝑉 = 0. Equations (1.70) and (1.71) can now be expressed using
dimensionless variables by making the substitutions: 𝑣 = 𝑉 ⁄𝑉1, 𝑤 = 𝑅1 𝐼𝑅 ⁄𝑉1, 𝑓(𝑣) =
− 𝑅1 𝐹(𝑉1 𝑣)⁄𝑉1 , and 𝜏 = 𝑅1 𝑡⁄𝐿. Equation (1.70) becomes:
𝑐𝑉1 𝑅1 𝑑𝑣 𝑉1
𝑉1
− 𝑓(𝑣) + 𝑤 = 𝐼0 ,
𝐿 𝑑𝜏 𝑅1
𝑅1
(1.72)
𝑉1 𝑐𝑅1 2 𝑑𝑣
(
− 𝑓(𝑣) + 𝑤) = 𝐼0 ,
𝑅1 𝐿 𝑑𝜏
(1.73)
𝜖
𝑑𝑣
= 𝑓(𝑣) − 𝑤 + 𝐼.
𝑑𝜏
(1.74)
Equation (1.74) is the same as equation (1.67) with 𝜖 = 𝑅12 𝑐⁄𝐿 and 𝐼 = 𝐼0 𝑅1 ⁄𝑉1. Making
similar substitutions in equation (1.71) gives:
27
𝑉1 𝑅1 𝐿 𝑑𝑤 𝑅𝑉1
+
𝑤 + 𝑉0 = 𝑣𝑉1 ,
𝑅1 𝐿 𝑑𝜏
𝑅1
(1.75)
𝑑𝑤 𝑅
𝑉0
+ 𝑤 + = 𝑣,
𝑑𝜏 𝑅1
𝑉1
(1.76)
𝑑𝑤
𝑉0
= 𝑣 − 𝛾𝑤 − .
𝑑𝜏
𝑉1
(1.77)
If 𝛾 = 𝑅 ⁄𝑅1 , and the resting potential is given as 𝑉0 = 0, equation (1.77) reduces to
equation (1.68) as desired [7]. The Fitzhugh-Nagumo model was a major step forward in
the mathematical analysis of excitation waves, and it will be used as the foundation for the
analysis in the sections that follow.
28
2. ANALYTICAL APPROACHES TO STUDYING THE EXCITATION THRESHOLD
Studying Threshold Phenomena for a Linear Chain of Cells: The McKean and Moll
Approach
The threshold phenomena described by Fitzhugh is of practical interest for cardiologists.
For example, when defibrillators are employed to try and revive a patient from sudden
cardiac arrest there are several questions that a doctor or a medical engineer might have
which relate to threshold phenomena. What minimum value of the potential is required to
propagate an electric signal that starts at one end of a tissue and travels across it? When an
external current of a given magnitude is applied, what minimum number of cells must be
excited in order to propagate the signal across the entire tissue? To begin to address such
questions, the description of the threshold must be extended from one cell to many
connected cells.
One of the first attempts at understanding the spatial characteristics of threshold
phenomena for a linear chain of cells was carried out by H.P. McKean and V. Moll [4].
They started with the following form of the Fitzhugh-Nagumo system:
𝜕𝑣(𝑥, 𝑡) ∂2 𝑣(𝑥, 𝑡)
=
+ 𝑣(1 − 𝑣)(𝑣 − 𝛼) + 𝑦,
𝜕𝑡
∂𝑥 2
𝜕𝑦
= 𝛽𝑣 − 𝛾𝑦.
𝜕𝑡
(2.1)
(2.2)
The solutions to equation (2.1) are propagating waveforms. In order to study such
waveforms analytically, two major simplifications were made to these equations. First, the
recovery of the system was neglected by setting 𝑦 = 0, 𝛽 = 0, and 𝛾 = 0. The effect of
neglecting recovery on a propagating waveform is demonstrated in Figure 6.
29
Figure 6 shows that when the recovery variables are present, a current pulse above
Figure 6. A: When recovery is neglected, a propagating wave “front” rises to a set value
as the wave propagates to the right at speed c. B: When recovery is present, a
propagating wave “pulse” travels to the right [7].
threshold will cause cells to excite and then recover back down to rest in sequence as the
wave pulse propagates down the chain at speed c. When recovery is removed, a wave front
propagates down the chain where cells rise to a stable excited state and stay there.
The general expression for a wave front travelling down a linear chain of cells is then:
(2.3)
𝜕𝑣(𝑥, 𝑡) ∂2 𝑣(𝑥, 𝑡)
=
+ 𝑓(𝑣).
2
𝜕𝑡
∂𝑥
Such an equation that describes wavefronts is referred to as the bistable equation [7]. This
is because there are two stable fixed points at 𝑣 = 0 and 𝑣 = 1 in this system. Solutions
that exceed the threshold requirement form an expanding wavefront where the cells rise up
to 𝑣 = 1 and stay there. On the other hand, solutions that do not meet the threshold
requirement eventually return to rest at 𝑣 = 0. For other systems that do contain recovery,
30
the only stable fixed point is at 𝑣 = 0 since each cell will eventually return back to the
resting state whether it had been excited at one point or not.
The second major simplification was made to the excitation term 𝑓(𝑣). In the original
Fitzhugh-Nagumo system this term was the cubic polynomial shown in Figure 7.
Figure 7. The cubic function used for excitation in the Fitzhugh-Nagumo system [4].
McKean and Moll suggested that this excitation term could be replaced by a piecewiselinear function whose shape was similar to the cubic [4]. This function is plotted in Figure
8. The piecewise-linear approach introduced a Heaviside function that represented the
transition from zero to one that took place at 𝛼. This system was expressed as:
𝜕𝑣(𝑥, 𝑡) ∂2 𝑣(𝑥, 𝑡)
=
− 𝑣 + Θ(𝑣 − 𝛼).
𝜕𝑡
∂𝑥 2
(2.4)
In equation (2.4) the Heaviside term is Θ(𝑣 − 𝛼), and it has a value of 1 if 𝑣 > 𝛼 and 0 if
𝑣 < 𝛼. Despite some mathematical difficulties that come with the discontinuity in the
Heaviside term, this model was considered to be more mathematically tractable for the
31
analysis of waveforms.
Figure 8. The piecewise-linear function used by McKean [4].
Next, McKean and Moll specified boundary conditions that determined the initial shape,
or voltage profile, of the wave. In particular they chose a symmetric pulse shape where
𝑣(𝑥, 𝑡) went to zero at ±∞ while the peak at 𝑥 = 0 was a maximum such that 𝑣 ′ (0) = 0.
There were two points on such a graph where 𝑣(𝑥, 𝑡) could cross 𝛼 and these two points
were labeled 𝑚(𝑡) corresponding to the median of 𝑣(𝑥, 𝑡). In addition, the symmetry
allowed them to study the time development of the wave in the positive domain only (0 <
𝑥 < ∞) which further simplified the mathematics. Such a waveform is shown in Figure 9
below.
Studying the time dependence of the median allowed McKean and Moll to classify the
various possible waves based on the asymptotic behavior of the solutions as 𝑡 → ∞. In
addition to solutions where the waveform collapsed to zero or expanded to one as described
32
previously, there was a unique solution (provided that 𝛼 < 1/2) where the value of the
median remained fixed over time. This solution was a standing wave, and it acted as a
threshold between the resting and excited states.
Figure 9. The initial waveform studied by McKean and Moll [4].
An analytical expression for the standing wave 𝑤(𝑥) was obtained by setting
𝜕𝑣(𝑥, 𝑡)⁄𝜕𝑡 = 0 and solving the resulting differential equation by matching the boundary
conditions at the median: 𝑥 = ±𝑚. The result was:
(2.5)
𝑎𝑒 𝑥 + 𝑚
𝑓𝑜𝑟 𝑥 ≤ −𝑚
−𝑚
𝑓𝑜𝑟 𝑥 < 𝑚 },
𝑤(𝑥) = {1 − 𝑒 cosh 𝑥
𝑥−𝑚
𝑎𝑒
𝑓𝑜𝑟 𝑥 ≥ 𝑚
1
(2.6)
𝑎 = (1 − 𝑒 −2𝑚 ).
2
In order to classify the various standing wave solutions based on the initial conditions, a
parameter was introduced that was proportional to the amplitude of the pulse. Given some
arbitrary initial condition for the pulse amplitude, as 𝑡 → ∞ the solution would collapse to
zero if the amplitude was less than 𝛼, and it would expand to one if the amplitude was
greater than 𝛼. In between where the solution neither collapsed nor expanded, the
parameter was termed the critical multiplier, and the critical value of the pulse amplitude
33
that corresponded to it appeared on a threshold surface of codimension 1 which represented
the standing wave solution. This threshold surface is shown in Figure 10.
Figure 10. The threshold surface visualized by McKean and Moll. Initial conditions
below the critical surface tended towards zero, those on the critical surface tended towards
a saddle point, and those above the surface tended towards one [4].
While the existence of the critical surface was established analytically by McKean and
Moll, techniques to calculate the critical multiplier were introduced several years later by
Moll and Rosencrans [10]. First, Moll and Rosencrans used standard numerical methods
to solve the partial differential equation (see Appendix: Numerical Methods). The critical
multiplier could be found by varying the initial condition for the pulse amplitude. They
would start at some value of the multiplier where the amplitude was below threshold so
that the resulting solution would decay to zero in time. They would then increase the value
34
of the amplitude by some small numerical amount and solve the system again to see what
happened. At some point, at a certain critical value of the pulse amplitude, the solution
would expand to one indicating that the threshold had been crossed. This value was then
taken to be the critical amplitude or critical height.
Figure 11. Numerical plots of the critical amplitude (height) vs. the pulse width (base).
The result for the McKean and Moll system is on the left while the result for the FitzhughNagumo system appears on the right [10].
The process of finding the critical amplitude was repeated for various initial pulse widths
which were referred to as the “base” of the waveform depicted in Figure 9. The results for
both the McKean and Moll and Fitzhugh-Nagumo systems were plotted in Figure 11. It
was noted that, for sufficiently large pulse widths, the threshold requirement in either
approach was simply that the critical amplitude needed to exceed the value of 𝛼 in order to
excite either system. The threshold was found to increase substantially in the small pulse
width limit.
35
Moll and Rosencrans also developed an analytical method to calculate the critical
multiplier for the McKean approach. Unfortunately, their solution proved to be very nonlinear and complex. To evaluate it, numerical methods had to be employed. The
characteristics of the threshold surface had been established in Figure 11, but an analytical
description of how these curves came about was still missing. Specifically, the rise in the
requirement for the critical amplitude at small widths had not been accounted for in the
earlier analysis by McKean and Moll. Their description of the threshold requirement only
held in the limit of large pulse widths. Also, the scope of their analysis was limited to
simply a discussion of the amplitude vs. the pulse width; the effect of conductance terms
on the critical amplitude was never addressed.
Moving in the Direction of an Analytical Theory for the Threshold Surface:
Projected Dynamics
The next step was to provide an analytical theory that could reproduce the threshold curves
obtained by Moll and Rosencrans that were shown previously in Figure 11. Moll himself
developed a polygonal approximation method that could reproduce curves resembling the
critical surface [11]. It did not yield a concise analytical formula for the threshold however.
Another approach to this problem was carried out a few years later by J. Neu, R.S. Preissig,
Jr., and W. Krassowska [5]. They called their method “projected dynamics”, and it plays
such a central role in what follows that their approach will be described below in great
detail. The starting point is the Fitzhugh-Nagumo model which originally will be taken to
be:
𝜕𝑣
∂2 𝑣
𝑐
= 𝐷 2 − 𝑔𝑓 ′ (𝑣) − 𝑦,
𝜕𝑡
∂𝑥
36
(2.7)
𝜕𝑦
= 𝛽𝑣 − 𝛾𝑦,
𝜕𝑡
(2.8)
1
𝑓𝑜𝑟 0 < 𝛼 < .
2
𝑓 ′ (𝑣) = 𝑣(𝛼 − 𝑣)(1 − 𝑣)
(2.9)
In equations (2.7)-(2.9) the excitation variable is 𝑣, the source term is 𝑓 ′ (𝑣) (which when
written this way refers to the derivative of some function 𝑓(𝑣) with respect to 𝑣), and the
inactivation variable is 𝑦. 𝑐, 𝐷, 𝑔, 𝛽, 𝛾, and 𝛼 are taken to be constants. The symbols 𝑐, 𝐷,
and 𝑔 are chosen to be in analogy with the coefficients that modify the corresponding terms
in the Hodgkin and Huxley equations. This analogy should not be taken too literally as it
must be remembered that these are not exactly the same quantities in the Fitzhugh system
which uses non-dimensional variables. However, owing to their positions in the equation,
𝑐 acts as an effective capacitance term, 𝐷 and 𝑔 act as effective conductance terms for the
longitudinal and ion channel currents respectively. The constants 𝛽 and 𝛾 control the
recovery process, and the threshold parameter is 𝛼. 𝛼 is limited to the range of values
between zero and one half in order to permit a standing wave solution for the threshold as
required by McKean and Moll [4].
To simplify the analysis, the excitation of wavefronts can be studied by setting 𝑦, 𝛽, and 𝛾
all equal to zero which removes the recovery process from the system. Since the solution
we seek is time independent we can also set 𝑐 = 1 (or alternatively we can think of this as
dividing by 𝑐 and absorbing it into the constants 𝐷 and 𝑔). Unlike in Neu’s original
analysis, the constants 𝐷 and 𝑔 will be kept as free parameters throughout in order to
examine their effects on the threshold at a later point. Taking all of this into account, we
have reduced the original system of equations to:
37
𝜕𝑣
∂2 𝑣
= 𝐷 2 − 𝑔𝑓 ′ (𝑣),
𝜕𝑡
∂𝑥
(2.10)
1
𝑓𝑜𝑟 0 < 𝛼 < .
2
𝑓 ′ (𝑣) = 𝑣(𝛼 − 𝑣)(1 − 𝑣)
(2.11)
For equation (2.10), we can write a functional for the total energy 𝐸:
∞
(2.12)
𝐷 𝑑𝑣 2
𝐸 = ∫ ( ( ) + 𝑔𝑓(𝑣)) 𝑑𝑥.
2 𝑑𝑥
−∞
Given (2.12), we can rewrite (2.10) as the gradient flow of the energy:
𝜕𝑣
𝛿𝐸
=− .
𝜕𝑡
𝛿𝑣
(2.13)
The Formal Derivation of Projected Dynamics
To prove (2.13), we may perturb the potential in (2.12) by some small amount 𝜂(𝑥, 𝑡) to
get:
2
𝐷 𝜕(𝑣(𝑥, 𝑡)) 𝜕(𝜂(𝑥, 𝑡))
(
+
) +
𝐸(𝑣(𝑥, 𝑡) + 𝜂(𝑥, 𝑡)) = ∫ ( 2
).
𝜕𝑥
𝜕𝑥
∞
(2.14)
𝑔𝑓(𝑣(𝑥, 𝑡) + 𝜂(𝑥, 𝑡))𝑑𝑥
−∞
The first term in the integral can be expanded and simplified by taking the product of the
derivative of the infinitesimal term, 𝜂(𝑥, 𝑡), to be zero:
2
𝜕(𝑣(𝑥, 𝑡))
(
) +
𝜕𝑥
2
𝐷 𝜕(𝑣(𝑥, 𝑡)) 𝜕(𝜂(𝑥, 𝑡))
𝐷
(
+
) ≅
2
𝜕𝑥
𝜕𝑥
2
(2.15)
.
𝜕(𝑣(𝑥, 𝑡)) 𝜕(𝜂(𝑥, 𝑡))
𝜕𝑥
𝜕𝑥
(
)
The second term can be Taylor expanded where again all terms containing the products of
2
infinitesimals are discarded:
38
𝑔𝑓(𝑣(𝑥, 𝑡) + 𝜂(𝑥, 𝑡)) ≅ 𝑔 (𝑓(𝑣(𝑥, 𝑡)) + 𝑓 ′ (𝑣(𝑥, 𝑡))𝜂(𝑥, 𝑡)).
(2.16)
Inserting (2.15) and (2.16) into (2.14) gives:
𝐸(𝑣(𝑥, 𝑡) + 𝜂(𝑥, 𝑡))
(2.17)
∞
2
𝜕(𝑣(𝑥, 𝑡)) 𝜕(𝜂(𝑥, 𝑡))
𝐷 𝜕(𝑣(𝑥, 𝑡))
= ∫ ( ((
) + 2
)
2
𝜕𝑥
𝜕𝑥
𝜕𝑥
−∞
+ 𝑔(𝑓(𝑣(𝑥, 𝑡)) + 𝑓′(𝑣(𝑥, 𝑡))𝜂(𝑥, 𝑡))) 𝑑𝑥.
Now, to get the change in the energy, we subtract (2.12) from (2.17):
∆𝐸 = 𝐸(𝑣(𝑥, 𝑡) + 𝜂(𝑥, 𝑡)) − 𝐸(𝑣(𝑥, 𝑡)).
(2.18)
Cancelling terms, we are left with:
∞
𝜕(𝑣(𝑥, 𝑡)) 𝜕(𝜂(𝑥, 𝑡))
∆𝐸 = ∫ (𝐷
+ 𝑔𝑓′(𝑣(𝑥, 𝑡))𝜂(𝑥, 𝑡)) 𝑑𝑥.
𝜕𝑥
𝜕𝑥
(2.19)
−∞
Integration by parts can be done on the first term in the integral, and the boundary term
vanishes to yield:
∞
𝜕 2 (𝑣(𝑥, 𝑡))
∆𝐸 = ∫ − (𝐷
− 𝑔𝑓′(𝑣(𝑥, 𝑡))) 𝜂(𝑥, 𝑡)𝑑𝑥.
𝜕𝑥 2
(2.20)
−∞
The term in parenthesis is the same as 𝜕𝑣(𝑥, 𝑡)/𝜕𝑡 as given by (2.10), and when the integral
in (2.20) is equal to an extremum this term is also equal to the variational derivative so
that:
𝜕𝑣
𝛿𝐸
=− .
𝜕𝑡
𝛿𝑣
39
(2.21)
To tackle equation (2.21), Neu chose a parametric representation for the potential 𝑣(𝑥, 𝑡)
where it was represented as a vector of linearly independent time-dependent parameters,
𝒂(𝑡):
𝑣(𝑥, 𝑡) = 𝑉(𝒂(𝑡), 𝑥).
(2.22)
The motivation for this approach is that the amplitude and width can now be set as
independent parameters, and the solutions for each can be obtained from (2.21). Using
(2.22) to rewrite the left side of (2.21):
𝑁
𝜕𝑣
𝜕𝑉 𝜕𝑎𝑗
𝜕𝑉
=∑
=
𝑎̇ .
𝜕𝑡
𝜕𝑎𝑗 𝜕𝑡
𝜕𝑎𝑗 𝑗
(2.23)
𝑗=1
The summation is carried out for 1,2,...𝑁 parameters, 𝑎𝑗̇ represents the time derivative of
the 𝑗𝑡ℎ parameter, and the summation notation has been replaced with repeated index
notation in the far right representation. Repeated index notation will be used from this point
forward.
This makes (2.21):
𝜕𝑉
𝛿𝐸
𝑎𝑗̇ = − .
𝜕𝑎𝑗
𝛿𝑉
(2.24)
To extract the ordinary differential equations for the various 𝑎𝑗̇ , we multiply both sides by
𝜕𝑉⁄𝜕𝑎𝑖 and take the functional inner product:
∞
∞
−∞
−∞
𝜕𝑉 𝜕𝑉
𝛿𝐸 𝜕𝑉
(∫
𝑑𝑥) 𝑎𝑗̇ = − ∫
𝑑𝑥.
𝜕𝑎𝑖 𝜕𝑎𝑗
𝛿𝑉 𝜕𝑎𝑖
(2.25)
Note that 𝑎𝑗̇ is independent of 𝑥 and can be taken out of the integral on the left side. Now
the claim is made that:
40
∞
(2.26)
𝜕𝐸
𝛿𝐸 𝜕𝑉
= ∫
𝑑𝑥.
𝜕𝑎𝑖
𝛿𝑉 𝜕𝑎𝑖
−∞
The steps to show this are very similar to those that were used to show (2.13). To start with,
we’ll express the energy as a function of the parametric representation of the potential:
∞
(2.27)
2
𝐷 𝜕(𝑉(𝒂(𝑡), 𝑥)
𝐸(𝑉(𝒂(𝑡), 𝑥)) = ∫ ( (
) + 𝑔𝑓(𝑉(𝒂(𝑡), 𝑥))) 𝑑𝑥 .
2
𝜕𝑥
−∞
The time-dependent parameters are perturbed by a small amount ∆𝒂, and the potential is
Taylor expanded to the first order in both terms to yield:
𝐸(𝑉(𝒂(𝑡) + ∆𝒂(𝑡), 𝑥)) =
∞
∫
𝐷 𝜕𝑉(𝒂(𝑡), 𝑥) 𝜕 𝜕𝑉(𝒂(𝑡), 𝑥)
(
+
(
) ∆𝒂(𝑡))
2
𝜕𝑥
𝜕𝑥
𝜕𝒂(𝑡)
(2.28)
2
𝜕𝑉(𝒂(𝑡), 𝑥)
−∞ + (𝑓(𝑉(𝒂(𝑡), 𝑥)) + 𝑓′(𝑉(𝒂(𝑡), 𝑥))
∆𝒂(𝑡))
𝜕𝒂(𝑡)
(
)
𝑑𝑥 .
Expanding the first term in the integral and discarding the product of the infinitesimals
gives:
𝐸(𝑉(𝒂(𝑡) + ∆𝒂, 𝑥)) =
(2.29)
2
𝐷 𝜕𝑉(𝒂(𝑡), 𝑥)
𝜕𝑉(𝒂(𝑡), 𝑥) 𝜕 𝜕𝑉(𝒂(𝑡), 𝑥)
((
) + 2
(
) ∆𝒂(𝑡))
2
𝜕𝑥
𝜕𝑥
𝜕𝑥
𝜕𝒂(𝑡)
∞
∫
−∞
(
+ 𝑔 (𝑓(𝑉(𝒂(𝑡), 𝑥)) + 𝑓′(𝑉(𝒂(𝑡), 𝑥))
𝜕𝑉(𝒂(𝑡), 𝑥)
∆𝒂(𝑡))
𝜕𝒂(𝑡)
𝑑𝑥.
)
The difference in the energy is obtained by subtracting (2.27) from (2.29):
∆𝐸 = 𝐸(𝑉(𝒂(𝑡) + ∆𝒂(𝑡), 𝑥)) − 𝐸(𝑉(𝒂(𝑡), 𝑥)).
After cancelling terms and simplifying, we are left with:
41
(2.30)
(2.31)
𝜕(𝑉(𝒂(𝑡), 𝑥)) 𝜕 𝜕𝑉(𝒂(𝑡), 𝑥)
(
)+
𝜕𝑥
𝜕𝑥
𝜕𝒂(𝑡)
∆𝐸 = ∫
∆𝒂(𝑡)𝑑𝑥.
𝜕𝑉(𝒂(𝑡), 𝑥)
−∞
𝑔𝑓′(𝑉(𝒂(𝑡), 𝑥))
𝜕𝒂(𝑡)
(
)
Performing an integration by parts on the first term in the integral and discarding the
∞
𝐷
boundary term yields:
∞
𝜕 2 (𝑉(𝒂(𝑡), 𝑥))
𝜕𝑉(𝒂(𝑡), 𝑥)
∆𝐸 = ∫ − (𝐷
− 𝑔𝑓′(𝑉(𝒂(𝑡), 𝑥)))
∆𝒂(𝑡)𝑑𝑥,
2
𝜕𝑥
𝜕𝒂(𝑡)
(2.32)
−∞
and from here the desired result follows readily:
𝜕 2 (𝑉(𝒂(𝑡), 𝑥))
𝜕𝑉(𝒂(𝑡), 𝑥)
𝛿𝐸
=−
=𝐷
− 𝑔𝑓′(𝑉(𝒂(𝑡), 𝑥)),
𝜕𝑡
𝛿𝑉
𝜕𝑥 2
∞
∆𝐸
𝛿𝐸 𝜕𝑉(𝒂(𝑡), 𝑥)
= ∫ − (− )
𝑑𝑥 ,
∆𝒂(𝑡)
𝛿𝑉
𝜕𝒂(𝑡)
(2.33)
(2.34)
−∞
∞
𝜕𝐸
𝛿𝐸 𝜕𝑉
= ∫
𝑑𝑥.
𝜕𝑎𝑖
𝛿𝑉 𝜕𝑎𝑖
(2.35)
−∞
With (2.26) now established, we can substitute it into (2.25) to obtain the central equation
for projected dynamics:
∞
(∫
−∞
𝜕𝑉 𝜕𝑉
𝜕𝐸
𝑑𝑥) 𝑎𝑗̇ = −
,
𝜕𝑎𝑖 𝜕𝑎𝑗
𝜕𝑎𝑖
(2.36)
or writing this result in matrix form:
𝑀𝒂̇ = −𝛁𝐸.
(2.37)
Here 𝑀 is a 𝑁 x 𝑁 symmetric matrix whose components are given by:
∞
𝜕𝑉 𝜕𝑉
𝑚𝑖𝑗 = ∫
𝑑𝑥.
𝜕𝑎𝑖 𝜕𝑎𝑗
−∞
42
(2.38)
In replacing the original partial differential equation with equation (2.37), the difficult
problem of solving the partial differential equation has been reduced to finding the motion
for specified parameters in a finite dimensional projected phase space. By analyzing the
dynamics of the parameters in phase space, we can acquire information about which
conditions lead to trajectories that either expand or collapse.
Applying Projected Dynamics Using a Gaussian Pulse
Now that the projected dynamics has been formally derived, it is time to apply the theory
to the Fitzhugh-Nagumo system and reproduce the plot of the critical amplitude vs. pulse
width that was obtained by Moll and Rosencrans. Step one of this procedure involves
choosing a shape for the applied pulse that will serve as an initial voltage profile for
equation (2.10). To simplify the calculations for the projected dynamics, Neu chose to
make 𝑣(𝑥, 𝑡) a Gaussian pulse with time dependent parameters for the amplitude 𝑎1 (𝑡) and
the pulse width 𝑎2 (𝑡):
𝑉(𝑥, 𝑡) = 𝑎1 (𝑡)𝑒 −𝑥
2 /(2𝑎 (𝑡)2 )
2
.
(2.39)
Here the pulse width is interpreted as only the width of the applied pulse to the right of its
initial point of stimulation (at 𝑥 = 0). The reason for this is that we will be examining only
the portion of the wavefront that propagates to the right down the cell chain from the initial
point of stimulation (from symmetry an identical wavefront will also travel down the cell
chain to the left in the negative 𝑥 direction). This is a different measure for the pulse width
than the one originally used by Neu, but it is in agreement with the pulse width as defined
in McKean and Moll. We will follow the latter convention for ease of comparison between
the different approaches.
43
Forming a vector for the derivatives of the two time-dependent parameters gives us:
𝒂̇ = [
𝑎1̇
].
𝑎2̇
(2.40)
The goal is to solve the system:
𝑀𝒂̇ = −𝛁𝐸.
(2.41)
To start with, we will obtain the matrix 𝑀 where the matrix elements are given by:
∞
𝑚𝑖𝑗 = ∫
−∞
𝜕𝑉 𝜕𝑉
𝑑𝑥.
𝜕𝑎𝑖 𝜕𝑎𝑗
(2.42)
For the Gaussian pulse, we have:
𝜕𝑉
2
2
= 𝑒 −𝑥 /(2𝑎2 ) ,
𝜕𝑎1
(2.43)
𝜕𝑉
𝑎1 𝑥 2 −𝑥 2 /(2𝑎 2 )
2 .
=
𝑒
𝜕𝑎2
𝑎2 3
(2.44)
Using standard Gaussian integration techniques, the 𝑚𝑖𝑗 become:
𝑚11 = 𝑎2 √𝜋,
𝑚12 = 𝑚21 =
𝑚22 =
𝑎1 √𝜋
,
2
3𝑎1 2 √𝜋
,
4𝑎2
(2.45)
(2.46)
(2.47)
and the matrix 𝑀 is:
𝑎2
𝑀 = √𝜋 [ 𝑎
1
2
𝑎1
2
3𝑎1 2 ].
4𝑎2
Now we want to find the energy gradient −𝛁𝐸. The energy is given by:
44
(2.48)
∞
𝐷 𝑑𝑉 2
𝐸 = ∫ ( ( ) + 𝑔𝑓(𝑉)) 𝑑𝑥.
2 𝑑𝑥
(2.49)
−∞
The derivative for the left term in the integral is:
𝜕𝑉
𝑎1 𝑥
2
2
= − 2 𝑒 −𝑥 /(2𝑎2 ) .
𝜕𝑥
𝑎2
(2.50)
For the term on the right, Neu argued that it was sufficiently accurate to obtain a more
compact solution by applying the condition that 𝛼 ≪ 1 in (2.11). Since 𝛼 is related to the
median on the voltage profile, a small value of 𝛼 implies that the height of the voltage
profile is also small (𝑣 ≪ 1). With this approximation, we then have (1 − 𝑣) ~1 and 𝑓′(𝑣)
can be expressed as a quadratic:
𝑓 ′ (𝑣) = 𝑣(𝛼 − 𝑣)(1 − 𝑣) ~ 𝛼𝑣 − 𝑣 2 .
(2.51)
Integrating (2.51) gives:
(2.52)
𝛼 2 𝑣3
𝑣 − .
2
3
Although this approximation eventually has a rather large effect on the waveform (instead
𝑓(𝑣) =
of going to a value of one after an action potential, the solution blows up to infinity), it was
confirmed through numerical simulation that this approach had very little effect on the
threshold for excitation. This is because the threshold is reached long before the potential
reaches its maximum value following an action potential. Both (2.11) and (2.51) lead to
acceptable solutions for projected dynamics, but the solution using (2.52) is more compact
than using the full cubic, and so we will adopt this approximation as well.
Substitution of (2.50) and (2.52) into (2.49) and carrying out the integration yields:
45
𝐷𝑎1 2
𝛼𝑎1 2 𝑎2
2 𝑎1 3
𝐸 = √𝜋 (
+𝑔(
−√
𝑎 )),
4𝑎2
2
3 3 2
(2.53)
and then the energy gradient becomes (in matrix form):
(2.54)
𝐷𝑎1
2
+ 𝑔 (𝛼𝑎1 𝑎2 − √ 𝑎1 2 𝑎2 )
2𝑎2
3
−𝛁𝐸 = −√𝜋
.
[
𝐷𝑎1 2
𝛼𝑎1 2
2 𝑎1 3
√
−
+𝑔(
−
)
4𝑎2 2
2
3 3
]
Using the inverse matrix for 𝑀, the system can finally be solved for 𝒂̇ :
𝒂̇ = −𝑀−1 𝛁𝐸,
−
𝒂̇ = [
𝑎1̇
]=
𝑎2̇
[
(2.55)
𝐷𝑎1
7
− 𝑔 (𝛼𝑎1 −
𝑎1 2 )
2
𝑎2
3√6
𝐷
2
− 𝑔√ 𝑎1 𝑎2
𝑎2
27
(2.56)
.
]
Taking the amplitude to be 𝑎1 and the pulse width to be 𝑎2 , we can plot the phase portrait
for 𝑎1 vs. 𝑎2 using the solutions from equation (2.56). The phase portrait for the projected
dynamics is given in Figure 12 below with 𝐷 = 𝑔 = 1.0. Several representative trajectories
and the direction fields are shown to indicate the time evolution for various initial
conditions. From the trajectories, it can be seen that the unstable manifold near the center
of the plane represents the threshold for excitation; initial conditions below this curve decay
to zero in time (as can be seen by following the vector field on the phase plane for any
trajectory starting below the unstable manifold) while those trajectories starting above the
unstable manifold rise to a greater amplitude. The nullclines for the phase plane can be
obtained in the usual way: The first nullcline is plotted in Figure 12 in yellow by setting
𝑎1̇ = 0 and solving for 𝑎1 as a function of 𝑎2 . The second nullcline is plotted in red by
46
setting 𝑎2̇ = 0 and solving for 𝑎2 as a function of 𝑎1 .
Figure 12. The phase plane for the projected dynamics with a Gaussian pulse. The
amplitude nullcine is yellow, the pulse width nullcline is red, the unstable manifold
represents the excitation threshold separatrix, and the stable manifold forms a second
separatrix. The green arrows give the direction of the vector field while sample
trajectories are in blue. This phase portrait was created using PPLANE2005.10 by John
Polking [12].
47
Analysis of the Projected Dynamics Using a Gaussian Pulse
Now that we have obtained the phase plane for the projected dynamics in Figure 12, our
focus shifts to the task of trying to understand this result as completely as possible. From
examining the general features of the phase plane, we can see that the approach that Neu
took does indeed reproduce key features reported previously by McKean and Moll in
Figure 10 and Moll and Rosencrans in Figure 11. Figures 10-12 are collected in Figure 13
for a side by side comparison.
Figure 13. Comparison of Figures 10-12. On the left is the depiction of the threshold
surface given by McKean and Moll, in the center is the numerical threshold surface
provided by Moll and Rosencrans, and on the right is the projected dynamics.
Upon observing the phase plane for the projected dynamics in Figure 12, it appears that the
nullcline for the amplitude is in good qualitative agreement with the actual threshold curve
which is the unstable manifold. This is not by coincidence. Regardless of the initial
condition or the time course for the pulse width, if an initial condition for the amplitude is
above the amplitude nullcline it is guaranteed to fire an action potential. The reason for this
is that the nullcline represents a change in sign for the rate of change of the amplitude.
48
Above the nullcline, the amplitude is always changing in a positive direction, and so the
amplitude only climbs upward in this region. At small pulse widths, there is a region of
initial conditions where the nullcline doesn’t agree with the actual threshold curve. This
represents the set of initial conditions where the amplitude initially decreases before
eventually turning upward and firing an action potential. Taking this into account, the
amplitude nullcline represents a sort of upper bound estimation of the threshold. It
overestimates the requirement placed on the amplitude for very small pulse widths, but
approximates the requirement well for pulses with a sufficiently large initial pulse width.
Owing to this similarity between the amplitude nullcline and the actual threshold curve,
the equation for the nullcline may be used to extract an approximate analytical expression
for the threshold curve. To do this, we start with the solution for 𝑎1̇ from (2.56). Since the
initial conditions play the dominant role when it comes to excitation, we may set the pulse
width 𝑎2 to a constant: the initial pulse width 𝑤. Then we set 𝑎1̇ = 0 and solve for the timeindependent critical amplitude 𝑎1 → 𝑎𝑐 . Doing so, we have:
𝑎𝑐 =
3√6 𝐷
(
+ 𝛼).
7 𝑔𝑤 2
(2.57)
To see how this result compares with the true threshold curve, equation (2.10) was solved
for the threshold numerically (see Appendix: Numerical Methods) for a linear chain of 50
cells with 𝐷 = 𝑔 = 1 and 𝛼 = 0.139. This result was plotted in Figure 14 together with
Equation (2.57). It is seen in Figure 14 that (2.57) agrees well with the numerical result for
the true threshold. Aside from the two-variable projected dynamics that yielded (2.57), it
is also possible to do a one-variable projected dynamics where the width is set to a constant
𝑤 at the start of the procedure, and the only free parameter is 𝑎1 . In a later section, the
49
derivation of the one-variable projected dynamics will be carried out explicitly. In the
meantime, the amplitude nullcline for the one-variable projected dynamics is provided in
Figure 14 to compare the two approaches for the case of the Gaussian pulse.
Figure 14. Comparison of the amplitude nullclines obtained using projected dynamics
vs. the numerical threshold solution.
Aside from describing how the amplitude varies with the initial pulse width, equation
(2.57) also makes predictions as to how the critical amplitude should depend on the
effective ion channel and longitudinal conductivities given by 𝑔 and 𝐷 respectively.
Interestingly, equation (2.57) predicts that increasing the conductivity for the ion channel
current decreases the requirement on the amplitude to fire an action potential while
50
increasing the conductivity for the longitudinal current has the opposite effect. To see if
the predictions made by (2.57) accurately describe the threshold behavior for (2.10), we
first solved equation (2.10) numerically as before to find the threshold curve while varying
the parameter 𝑔 and using the following constant parameters: 𝐷 = 1, 𝑤 = 5, and 𝛼 =
0.139. The result is plotted together with the corresponding solution to (2.57) in Figure 15
below.
Figure 15. Plot of the critical amplitude (𝑎𝑐 ) vs. the ion channel conductivity (𝑔) for the
projected dynamics with a Gaussian pulse.
The agreement between the two is quite good, especially for very large and very small
values of 𝑔. On the other hand, (2.57) predicts that 𝑎𝑐 should increase with an increase in
the longitudinal conductivity 𝐷. Once more we solved (2.10) numerically (for 𝐷 ≥ 0.1)
51
with parameters : 𝑔 = 1, 𝑤 = 5, and 𝛼 = 0.139, and (2.57) was also solved with these
same parameters. The result is plotted in Figure 16 below.
Figure 16. Plot of the critical amplitude (𝑎𝑐 ) vs. the longitudinal conductivity (𝐷) for the
projected dynamics with a Gaussian pulse.
The general upward trend for the threshold requirement with increasing longitudinal
conductivity carries through in (2.57) when compared to the numerical simulation.
However, it is also clear that the actual relationship is not so simple and linear like equation
(2.57) implies. For qualitative purposes the agreement is still quite good provided that 𝐷 is
not too large.
52
Discussion of the Results of the Projected Dynamics Using a Gaussian Pulse
For a one-dimensional chain of cells, we can get a qualitative picture of the excitation
process from the above analysis. The potential is initially raised for a certain portion of the
cell chain, and this is represented by an initial amplitude and an initial width for the current
pulse, and then the system is allowed to evolve in time. There are two possible outcomes
for this system: either the cells fire an action potential or the cells decay to zero in time.
Equation (2.57) predicts that there is a minimum amplitude that is necessary to get the
chain to fire. This minimum amplitude (𝑎𝑐 ) depends on various parameters like the initial
pulse width (𝑤), the ion channel conductivity (𝑔), and the longitudinal conductivity (𝐷).
The effect of increasing the initial pulse width is to decrease the requirement on the initial
amplitude in order to get the action potential to fire. This corresponds to raising a larger
portion of the available cell chain when the initial condition for the potential is applied. For
example, this would simulate an electrode with a larger surface area being used to stimulate
the cell chain. Likewise, increasing the ion channel conductivity also decreases the initial
amplitude that is needed to fire an action potential. Since this parameter relates to the ion
channel that is producing excitation in the membrane model, an increase in this
conductivity would correspond to additional ion channels opening. This would lead to a
faster excitation. Conversely, the effect of increasing the longitudinal conductivity is to
increase the requirement on the initial amplitude. Since the longitudinal conductivity refers
to a diffusion process, larger values of 𝐷 drain current out of a stimulated cell and into
neighboring cells that are at rest. If the value of this parameter is too high then current may
drain out of a particular cell before that cell can reach the necessary threshold for an action
potential to fire. Lowering the longitudinal conductivity has the opposite effect, it keeps
53
the applied current localized for a longer time. This allows the charge to build up easier in
a small region of the membrane which may assist the cell in reaching the threshold
requirement.
Although it provides a nice qualitative description of the threshold curve, equation (2.57)
has a few shortcomings. For instance, the numerical results predict that for very large
widths the threshold should be precisely 𝛼. Equation (2.57) gives a result close to this, but
the factor 3√6⁄7 is a little larger than one, and it overestimates the threshold by about 5%:
𝑎𝑐 =
3√6 𝐷
3√6
( 2 + 𝛼) ~
𝛼 ~ 1.05𝛼
7 𝑔𝑤
7
(𝑓𝑜𝑟 𝑤 → ∞).
(2.58)
Also, as mentioned before, there is a noticeable discrepancy between the actual threshold
curve and (2.57) for small pulse widths. Finally, the Gaussian shape of the initial voltage
profile is not as physiologically accurate as the shape of the voltage profile chosen by
McKean and Moll. The voltage profile they chose is closer in shape to one generated by a
real electrode. Many of these issues can be improved by choosing a different shape for the
initial voltage profile and reformulating the projected dynamics. This task will be
undertaken next.
54
3. FORMULATION OF THE THRESHOLD EQUATION FOR A SQUARE PULSE
USING INITIAL CONDITIONS
A New Formulation: Changing the Initial Voltage Profile to a Square Pulse
Now that the projected dynamics has been established for a one-dimensional cell chain
using the Fitzhugh-Nagumo relations with a Gaussian pulse for an initial voltage profile, it
is natural to try and apply the method towards systems with more realistic physiological
characteristics. The most immediate thing to change would be the initial Gaussian pulse. It
is unphysiological in at least two ways. First off, the shape of the Gaussian pulse is not a
great fit to the more physiological square pulse which is generated by an externally applied
current given by:
𝑓𝑜𝑟 − 𝑙 ⁄2 < 𝑥 < 𝑙 ⁄2,
(3.1)
𝐼(𝑥, 𝑡) = 0 𝑓𝑜𝑟 𝑥 > 𝑙 ⁄2 𝑜𝑟 𝑥 < − 𝑙 ⁄2.
(3.2)
𝐼(𝑥, 𝑡) = 𝐼(𝑡)
An example of a square current pulse is shown below in Figure 17.
Figure 17. A sample plot of a square pulse. The amplitude of the applied pulse is 𝐼,
and the width extends from 𝑙 ⁄2 to − 𝑙 ⁄2.
55
The Gaussian voltage profile was chosen for its desirable mathematical properties, but
presumably a more accurate solution can be obtained if we go with a shape generated by a
square pulse. Nevertheless, a square pulse is a discontinuous function and will have to be
approximated to some extent when modeling it analytically. Notice in Figure 17 that the
full width of the square pulse is 𝑙 but the width that we will use is that portion of the solution
travelling to the right. This width will be given by 𝑙 ⁄2 which corresponds to the median of
the pulse width from the McKean and Moll treatment.
The second aspect is a bit more subtle. In the original projected dynamics, Neu
approximated an initial pulse from an external electrode using initial conditions for the
potential specified at the initial time 𝑡 = 0. The significance of this idealization is that by
specifying this initial pulse using initial conditions for the potential directly, one need not
have to worry about introducing an external current 𝐼(𝑥, 𝑡) into the Fitzhugh-Nagumo
relation. Introducing such a current would make the system that was used previously a bit
more complicated:
𝜕𝑣
∂2 𝑣
= 𝐷 2 − 𝑔𝑓 ′ (𝑣) + 𝐼(𝑥, 𝑡),
𝜕𝑡
∂𝑥
′ (𝑣)
𝑓
= 𝑣(𝛼 − 𝑣)
𝑓𝑜𝑟 0 < 𝛼 < 1⁄2 , 𝛼 ≪ 1.
(3.3)
(3.4)
In contrast to the initial conditions approach, for the actual physiological case, the applied
pulse 𝐼(𝑥, 𝑡) will be applied for some finite time that is typically longer than the time for
an action potential to fire. Eventually, we would like to replace the initial conditions
approach with an applied current 𝐼(𝑥, 𝑡), as in the general expression (3.3), so that a timedependent applied current can be used to drive the system as would be the case with a real
electrode or other external current source. Moving in that general direction, we will start
with (3.3) and attempt to find an acceptable shape for the voltage profile that will fit the
56
criteria for an external square pulse 𝐼(𝑥, 𝑡). If we consider that 𝐼(𝑥, 𝑡) is applied for a time
that is much greater than the time needed to excite the system, then 𝐼(𝑥, 𝑡) can be treated
as being constant in time and can be labeled as 𝐼(𝑥). In doing this, we assume that the
current is still being applied while the threshold is reached. Substitution of equation (3.4)
into (3.3) and expanding the source term gives:
𝜕𝑣
∂2 𝑣
= 𝐷 2 − 𝑔𝛼𝑣 + 𝑔𝑣 2 + 𝐼(𝑥).
𝜕𝑡
∂𝑥
(3.5)
Working in analogy to a real physiological model, we will now label the coefficients for
the excitation terms as:
𝑔𝛼 → 𝑔𝐾 ,
(3.6)
𝑔 → 𝑔𝑁𝑎 ,
(3.7)
to make (3.5):
(3.8)
𝜕𝑣
∂2 𝑣
= 𝐷 2 − 𝑔𝐾 𝑣 + 𝑔𝑁𝑎 𝑣 2 + 𝐼(𝑥).
𝜕𝑡
∂𝑥
Here the negative term that tends to return the system to zero as the potential increases is
labeled with the coefficient 𝑔𝑘 in analogy to the potassium channels in the Hodgkin and
Huxley model. Likewise, the positive term that tends to raise the voltage will be labeled
with the coefficient 𝑔𝑁𝑎 in analogy with the sodium channels. Again, caution must be taken
not to take this analogy too literally here. These terms are mainly used to help us interpret
our results as (3.6) and (3.7) play similar roles in (3.8) to the corresponding terms in the
more complicated Hodgkin and Huxley model.
In particular, it must be kept in mind that equation (3.8) is too simple to fire an action
potential as written. The sodium term is not at least a cubic, which was determined to be a
minimum requirement for firing an action potential by Fitzhugh, and it also lacks an
57
inactivation term to keep the solution from blowing up at high values of the potential.
However, for the study of excitation we can get around this difficulty by working as we
did before under the assumption that 𝛼 ≪ 1 which implies that the system will reach
threshold long before these inaccuracies can appreciably influence the threshold result.
Since 𝑣 ≪ 1 at the time of excitation in this case, we can assume the 𝑔𝑁𝑎 𝑣 2 term is small
and neglect it while we search for the shape of the threshold surface. This gives us:
(3.9)
𝜕𝑣
∂2 𝑣
= 𝐷 2 − 𝑔𝐾 𝑣 + 𝐼(𝑥).
𝜕𝑡
∂𝑥
This equation is now similar to the linear piecewise expression used by McKean and Moll,
and we can solve for the time-independent solution using familiar methods. The system is
given by:
0=𝐷
𝑑 2 𝑣1
− 𝑔𝐾 𝑣1 + 𝐼
𝑑𝑥 2
(3.10)
𝑓𝑜𝑟 − 𝑙 ⁄2 < 𝑥 < 𝑙 ⁄2,
(3.11)
𝑑 2 𝑣2
− 𝑔𝐾 𝑣2
𝑓𝑜𝑟 𝑥 > 𝑙 ⁄2 𝑜𝑟 𝑥 < − 𝑙 ⁄2.
2
𝑑𝑥
Due to the symmetry of the desired solution, we need only solve for the solutions in the
0=𝐷
region where 𝑥 > 0. The solution for (3.11) is solved in the standard way for a linear
second order differential equation. Making the identification:
𝑑2 𝑣2
0=𝐷
− 𝑔𝐾 𝑣2 →
𝑑𝑥 2
𝑑 2 𝑣2
− 𝛾 2 𝑣2 = 0
𝑑𝑥 2
𝑔𝐾
,
𝐷
𝑤ℎ𝑒𝑟𝑒 𝛾 = √
(3.12)
we assume a solution of:
𝑣2 = 𝐶𝑒 𝜆𝑥 ,
(3.13)
where 𝐶 and 𝜆 are constants. Inserting (3.13) into (3.12) and simplifying yields:
𝜆 = ± 𝛾.
(3.14)
The general solution is then given by:
58
𝑣2 = 𝐶1 𝑒 𝛾𝑥 + 𝐶2 𝑒 −𝛾𝑥 .
(3.15)
To satisfy the specified requirement for the voltage profile we must have 𝑣2 → 0 at 𝑥 →
∞ so that 𝐶1 = 0. Also, 𝑣2 must originate at 𝑙 ⁄2 and then exponentially decay from there
as 𝑥 increases. Taking these features into account, (3.15) reduces to:
𝑣2 = 𝐶𝑒 −𝛾(𝑥−𝑙⁄2)
𝑓𝑜𝑟 𝑥 ≥ 𝑙/2.
(3.16)
For the first solution, we have a non-homogeneous second order differential equation:
𝑑 2 𝑣1
𝐼
− 𝛾 2 𝑣1 = − .
2
𝑑𝑥
𝐷
(3.17)
The general solution is given as the sum of the homogeneous solution 𝑣ℎ and a particular
solution 𝑣𝑝 :
𝑣 = 𝑣ℎ + 𝑣𝑝 ,
(3.18)
𝑣ℎ = 𝐸1 𝑒 𝛾𝑥 + 𝐸2 𝑒 −𝛾𝑥 ,
(3.19)
𝑣𝑝 = 𝑈1 (𝑥)𝑦1 (𝑥) + 𝑈2 (𝑥)𝑦2 (𝑥),
(3.20)
𝑦1 (𝑥) = 𝑒 𝛾𝑥 ,
(3.21)
𝑦2 (𝑥) = 𝑒 −𝛾𝑥 .
(3.22)
The solutions for 𝑈1 (𝑥) and 𝑈2 (𝑥) are given by [13]:
𝑈1 (𝑥) = − ∫
𝑈2 (𝑥) = ∫
𝑦2 (𝑥)𝑔(𝑥)
𝑑𝑥,
𝑊(𝑦1 , 𝑦2 )(𝑥)
𝑦1 (𝑥)𝑔(𝑥)
𝑑𝑥,
𝑊(𝑦1 , 𝑦2 )(𝑥)
𝑔(𝑥) = − 𝐼 ⁄𝐷 ,
𝑑𝑦2 (𝑥) 𝑑𝑦1 (𝑥)
−
𝑦2 (𝑥)
𝑑𝑥
𝑑𝑥
= 𝑒 𝛾𝑥 (−𝛾)𝑒 −𝛾𝑥 − (𝛾)𝑒 𝛾𝑥 𝑒 −𝛾𝑥 = −2𝛾.
𝑊(𝑦1 , 𝑦2 )(𝑥) = 𝑦1 (𝑥)
(3.23)
(3.24)
(3.25)
(3.26)
Inserting (3.21), (3.22), (3.25), and (3.26) into (3.23) and (3.24), and performing the
integration gives:
59
𝑈1 (𝑥) =
𝐼
𝑒 −𝛾𝑥 ,
2𝐷𝛾 2
(3.30)
𝑈2 (𝑥) =
𝐼
𝑒 𝛾𝑥 .
2𝐷𝛾 2
(3.31)
Placing (3.21), (3.22), (3.30), and (3.31) in (3.20), we obtain the particular solution 𝑣𝑝 :
𝑣𝑝 =
𝐼
𝐼
𝐼
𝐼
−𝛾𝑥 𝛾𝑥
𝛾𝑥 −𝛾𝑥
𝑒
𝑒
+
𝑒
𝑒
=
=
.
2𝐷𝛾 2
2𝐷𝛾 2
𝐷𝛾 2
𝑔𝐾
(3.32)
The general solution for equation (3.18) becomes:
𝑣1 = 𝑣ℎ + 𝑣𝑝 = 𝐸1 𝑒 𝛾𝑥 + 𝐸2 𝑒 −𝛾𝑥 +
𝐼
.
𝑔𝐾
(3.33)
We can simplify (3.33) by noting that the maximum of the curve must appear at 𝑥 = 0, and
the derivative at this point must be continuous:
𝑑𝑣1 (0)
= 0,
𝑑𝑥
(3.34)
𝑑𝑣1 (0)
= 𝛾𝐸1 − 𝛾𝐸2 = 0 → 𝐸1 = 𝐸2 .
𝑑𝑥
(3.35)
This makes (3.33):
𝑣1 = 𝐸(𝑒 𝛾𝑥 + 𝑒 −𝛾𝑥 ) +
𝐼
.
𝑔𝐾
(3.36)
Now to link the solutions for 𝑣1 and 𝑣2 together, we use boundary conditions that require
continuity at 𝑥 = 𝑙/2 of the two solutions and their first derivatives:
𝑣1 = 𝑣2
𝑎𝑡 𝑥 = 𝑙 ⁄2,
(3.37)
𝑑𝑣1
𝑑𝑣2
=
𝑑𝑥
𝑑𝑥
𝑎𝑡 𝑥 = 𝑙 ⁄2.
(3.38)
𝐼
= 𝐶,
𝑔𝐾
(3.39)
Inserting (3.16) and (3.36), these become:
𝐸(𝑒 𝛾𝑙/2 + 𝑒 −𝛾𝑙/2 ) +
𝐸(𝑒 𝛾𝑙/2 − 𝑒 −𝛾𝑙/2 ) = −𝐶.
60
(3.40)
Adding (3.39) to (3.40) gives an expression for 𝐸 while subtracting the two equations gives
an expression for 𝐶. These are:
𝐸=−
𝐶=
𝐼 −𝛾𝑙⁄2
𝑒
,
2𝑔𝐾
(3.41)
𝐼
(1 − 𝑒 −𝛾𝑙 ).
2𝑔𝐾
(3.42)
The solutions to the differential equation are then:
(𝑒 𝛾𝑥 + 𝑒 −𝛾𝑥 )
𝐼
−𝛾𝑙⁄2
𝑣1 =
(1 − 𝑒
)
𝑔𝐾
2
=
𝐼
(1 − 𝑒 −𝛾𝑙⁄2 cosh(𝛾𝑥))
𝑔𝐾
𝑣2 =
(3.43)
𝑓𝑜𝑟 0 < 𝑥 < 𝑙 ⁄2,
𝐼
(1 − 𝑒 −𝛾𝑙 )𝑒 −𝛾(𝑥−𝑙⁄2)
2𝑔𝐾
𝑓𝑜𝑟 𝑥 > 𝑙/2.
(3.44)
Now, before starting into the projected dynamics, we wish to express the potential as a
vector of parameters:
𝑣(𝑥, 𝑡) = 𝑉(𝒂(𝑡), 𝑥).
(3.45)
Replacing 𝐼/𝑔𝐾 with the time dependent amplitude 𝑎1 (𝑡) and replacing 𝑙 with the time
dependent pulse width 𝑎2 (𝑡), we obtain our voltage profile:
𝑉1 (𝒂(𝑡), 𝑥) = 𝑎1 (𝑡)(1 − 𝑒 −𝛾𝑎2 (𝑡)⁄2 cosh(𝛾𝑥))
𝑓𝑜𝑟 0 < 𝑥 < 𝑎2 (𝑡)⁄2,
(3.46)
𝑎1 (𝑡)
(3.47)
(1 − 𝑒 −𝛾𝑎2 (𝑡) )𝑒 −𝛾(𝑥−𝑎2 (𝑡)⁄2)
𝑓𝑜𝑟 𝑥 > 𝑎2 (𝑡)/2.
2
These equations should be compared with those originally provided by McKean and Moll
𝑉2 (𝒂(𝑡), 𝑥) =
(equations (2.5) and (2.6)). The basic form is nearly identical. The only differences are the
variable amplitude due to the presence of the external current input 𝐼 and the appearance
of 𝛾 which contains conductivity parameters within it. This form will now be introduced
into the projected dynamics.
61
Applying Projected Dynamics for a Square Pulse Using Initial Conditions
As before for the Gaussian, the goal of projected dynamics is to solve:
𝑀𝒂̇ = −𝛁𝐸.
(3.48)
where 𝑀 is given by:
∞
𝜕𝑉 𝜕𝑉
𝑚𝑖𝑗 = ∫
𝑑𝑥.
𝜕𝑎𝑖 𝜕𝑎𝑗
(3.49)
−∞
When the equation for the threshold surface was worked out before, an approximation was
made where the width was set to a constant either at the start of the procedure (for onevariable projected dynamics) or at the end after obtaining the projected dynamics (for twovariable projected dynamics). In the Gaussian section, we carried out the analysis explicitly
for the two-variable projected dynamics. In this section, we will carry out the analysis
explicitly for the one-variable projected dynamics. Doing so allows us to neglect the 𝑀
matrix entirely in the calculation; with 𝑎2 set to a constant 𝑤 at the start, 𝑀 is reduced from
a matrix to a single term as the only surviving term in (3.49) will be 𝑚11 . This then makes
(3.48):
𝑎1̇ =
− ∂𝐸/𝜕𝑎1
.
𝑚11
(3.50)
Here 𝑎1̇ = 0 is satisfied simply when the numerator − ∂𝐸/𝜕𝑎1 is equal to zero, so the
denominator can be neglected in this case. This allows us to do far less calculation to obtain
the solution, and we can skip straight to solving the energy integral:
∞
𝐷 𝑑𝑉 2
𝑉2
𝑉3
𝐸 = ∫ ( ( ) + 𝑔𝐾
− 𝑔𝑁𝑎 ) 𝑑𝑥
2 𝑑𝑥
2
3
−∞
62
(3.51)
∞
𝐷 𝑑𝑉 2
𝑉2
𝑉3
= 2 ∫ ( ( ) + 𝑔𝐾
− 𝑔𝑁𝑎 ) 𝑑𝑥.
2 𝑑𝑥
2
3
0
For the integral on the right, we have exploited the symmetry in 𝑉 to rewrite the integral
over only the positive domain of 𝑥. Since we have two domains for 𝑉, we can further split
up this integral to get:
𝑤
2
2
3
(3.52)
𝑑𝑉1
2𝑉1
𝐸 = ∫ (𝐷 (
) + 𝑔𝐾 𝑉1 2 − 𝑔𝑁𝑎
) 𝑑𝑥
𝑑𝑥
3
0
∞
𝑑𝑉2 2
2𝑉2 3
2
+ ∫ (𝐷 (
) + 𝑔𝐾 𝑉2 − 𝑔𝑁𝑎
) 𝑑𝑥.
𝑑𝑥
3
𝑤/2
The two derivatives are obtained by differentiating (3.46) and (3.47):
𝜕𝑉1
= −𝑎1 𝛾𝑒 −𝛾𝑤⁄2 sinh(𝛾𝑥),
𝜕𝑥
𝑤
𝜕𝑉2
1
= − 𝑎1 𝛾𝑒 −𝛾(𝑥− 2 ) (1 − 𝑒 −𝛾𝑤 ).
𝜕𝑥
2
(3.53)
(3.54)
We now Insert (3.46), (3.47), (3.53), and (3.54) into (3.52), perform the integration, and
simplify to yield:
𝐸=
1
𝑎1 2 𝑒 −2𝛾𝑤 (−𝑎1 𝑔𝑁𝑎 + 𝑒 2𝛾𝑤 (3𝛾 2 + 𝑔𝐾 (−9 + 6𝛾𝑤) + 9𝛾𝑎1 −)
12𝛾
(3.55)
4𝛾𝑔𝑁𝑎 𝑎1 𝑤 + 𝑒 𝛾𝑤 (3𝑔𝐾 (3 + 𝛾𝑤) − 8𝑔𝑁𝑎 𝑎1 − 3𝛾(𝛾 + 𝛾 2 𝑤 + 2𝑔𝑁𝑎 𝑎1 𝑤)).
Calculating the negative of the energy gradient for the time derivative of the amplitude
yields:
𝑎1̇ = −
−
𝜕𝐸
,
𝜕𝑎1
𝜕𝐸
1
=
𝑎 𝑒 −2𝛾𝑤 (𝑎1 𝑔𝑁𝑎 + 2𝑒 𝛾𝑤 (3𝛾 2 + 𝛾 3 𝑤 − 𝑔𝐾 (3 + 𝛾𝑤))
𝜕𝑎1
4𝛾 1
+4𝑎1 𝑔𝑁𝑎 + 3𝛾𝑔𝑁𝑎 𝑎1 𝑤 + 𝑒 2𝛾𝑤 (−2𝛾 2 + 𝑔(6 − 4𝛾𝑤) + 𝑎1 (−9 + 4𝛾𝑤)𝑔𝑁𝑎 )).
63
(3.56)
(3.57)
Finally, we set (3.57) equal to zero and solve for 𝑎1 = 𝑎𝑐 . After simplifying, we arrive at
our approximate solution for the threshold:
𝑎𝑐 =
4𝑒 𝛾𝑤 𝑔𝐾 (1 + 𝑒 𝛾𝑤 (𝛾𝑤 − 1))
.
𝑔𝑁𝑎 (1 + 2𝑒 𝛾𝑤 (4 + 3𝛾𝑤) + 𝑒 2𝛾𝑤 (−9 + 4𝛾𝑤))
(3.58)
Analysis of the Threshold Equation for a Square Pulse Using Initial Conditions
Both the one-variable projected dynamics, with the width set to a constant at the start of
the procedure, and the two-variable projected dynamics, with the width set to a constant
after the procedure, were plotted together with the numerical solution in Figure 18 below.
Figure 18. Plot of the critical amplitude (𝑎𝑐 ) vs. the pulse width (𝑤) for the projected
dynamics with a square pulse.
64
The numerical solution was provided using:
𝜕𝑣
∂2 𝑣
= 𝐷 2 − 𝑔𝐾 𝑣 + 𝑔𝑁𝑎 𝑣 2 ,
𝜕𝑡
∂𝑥
(3.59)
and the parameters chosen were 𝑔𝐾 = 0.139, 𝑔𝑁𝑎 = 1.0, 𝐷 = 1.0.
In this case, the initial pulse was specified using initial conditions for the potential as before
in the Gaussian case (see Appendix: Numerical Methods). The more realistic case, where
a term containing the constant 𝐼 is applied as an external pulse instead of initial conditions,
will be treated separately in the next section.
While examining Figure 18, we see right away that the solution obtained by setting the
pulse width to a constant at the start of the projected dynamics is drastically improved over
the solution obtained under these same conditions for the Gaussian pulse (see Figure 14).
In fact, despite the fact that the curve requires larger amplitudes for a given pulse width,
the shape of the curve is arguably a much better fit for the full range of widths than the
curve obtained by setting the width to a constant after the projected dynamics. From this
point forward, we shall focus our attention on the one-variable projected dynamics where
the width is set to a constant at the start of the procedure. The equation for this curve is:
𝑎𝑐 =
4𝑒 𝛾𝑤 𝑔𝐾 (1 + 𝑒 𝛾𝑤 (𝛾𝑤 − 1))
.
𝑔𝑁𝑎 (1 + 2𝑒 𝛾𝑤 (4 + 3𝛾𝑤) + 𝑒 2𝛾𝑤 (−9 + 4𝛾𝑤))
(3.60)
The first thing we would like to do with equation (3.60) is check the large width limit to
see if the threshold value is improved over the result for the Gaussian pulse. For 𝑤 → ∞,
the dominating terms in (3.60) will be those containing both 𝑒 2𝛾𝑤 and 𝑤. In this limit,
(3.60) reduces to:
65
4𝛾𝑤𝑒 2𝛾𝑤 𝑔𝐾
𝑔𝐾
𝑎𝑐 =
→
𝑎
=
.
𝑐
4𝛾𝑤𝑒 2𝛾𝑤 𝑔𝑁𝑎
𝑔𝑁𝑎
(3.61)
If we would like to compare this result to the older result, we need to briefly reintroduce
Fitzhugh’s notation by substituting (3.6) and (3.7) into (3.61):
𝑎𝑐 =
𝑔𝐾
𝑔𝛼
=
= 𝛼.
𝑔𝑁𝑎
𝑔
(3.62)
We see that, as 𝑤 → ∞, the threshold requirement reduces precisely to the threshold 𝛼 in
complete agreement with the same limit obtained from the numerical simulation. Thus, the
square pulse approach has fixed one of the issues from the Gaussian projected dynamics
where the threshold value at 𝑤 → ∞ was 5% larger than 𝛼.
We may also check equation (3.60) in the other extreme limit of small pulse widths.
Performing a series expansion on (3.60), and keeping terms to the second order, gives:
𝑎𝑐 ≅
6𝑔𝐾
2𝑔𝐾 2𝑔𝐾 𝛾𝑤 3(𝑔𝐾 𝛾 2 )𝑤 2
−
+
−
.
𝛾𝑔𝑁𝑎 𝑤 𝑔𝑁𝑎
𝑔𝑁𝑎
2𝑔𝑁𝑎
(3.63)
In the small width limit, as 𝑤 → 0, the dominant term in the expansion is:
𝑎𝑐 ≅
6𝑔𝐾
.
𝛾𝑔𝑁𝑎 𝑤
(3.64)
The scaling goes as 1⁄𝑤 as opposed to the 1/𝑤 2 scaling in the Gaussian approach (2.57).
It appears that using one-variable projected dynamics qualitatively improves upon the
small pulse limit in comparison to this limit in the two-variable approach (see Figure 18).
This qualitative improvement in the shape of the nullcline compared to the true threshold
is the main reason that we will work on refining the approximate solution obtained from
the one-variable projected dynamics.
Returning to the large pulse width limit, equation (3.61) implies that, in our square pulse
model, the critical amplitude at large widths will not be specified by a single parameter as
66
in the Fitzhugh approach. Rather, the threshold will arise from a competition between the
two independent parameters 𝑔𝐾 and 𝑔𝑁𝑎 . Increasing the parameter 𝑔𝐾 increases the
strength for the term that drives the system back to rest. To overcome this, a greater initial
amplitude must be applied to get the cells to excite. On the other hand, increasing 𝑔𝑁𝑎 has
the opposite effect of causing the potential to rise faster and subsequently lowers the
threshold requirement.
The above interpretation can be visualized by first plotting the critical amplitude as a
function of 𝑔𝐾 . This was done in Figure 19 where (3.60) was plotted with 𝑤 = 5, 𝐷 = 1.0,
𝑔𝑁𝑎 = 1.0, and 𝛾 = √𝑔𝐾 for the theory curve. Equation (3.59) was solved using the
appropriate parameters for the numerical case.
Figure 19. Plot of the critical amplitude (𝑎𝑐 ) vs. the potassium ion channel conductivity
(𝑔𝐾 ) for the projected dynamics with a square pulse.
67
Likewise, the critical amplitude as a function of 𝑔𝑁𝑎 with 𝑤 = 5, 𝐷 = 1.0, 𝑔𝐾 = 0.139,
and 𝛾 = √0.139 ~ 0.373 was plotted alongside the numerical solution in Figure 20.
Figure 20. Plot of the critical amplitude (𝑎𝑐 ) vs. the sodium ion channel conductivity
(𝑔𝑁𝑎 ) for the projected dynamics with a square pulse.
It is worthwhile to note that equation (3.60) reduces to an especially simple form for the
present case:
𝑎𝑐 ≅
0.3895
.
𝑔𝑁𝑎
(3.65)
Comparing equation (3.65) to the Gaussian result in equation (2.57) reveals that both
approaches have the same ~1/𝑔 dependence for the excitation parameter.
68
For the longitudinal conductivity 𝐷, (3.60) was plotted in Figure 21 using 𝑤 = 5, 𝑔𝑁𝑎 =
1.0, 𝑔𝐾 = 0.139, and 𝛾 = √0.139⁄𝐷 ~ 0.373⁄√𝐷 together with the numerical solution.
In comparing Figure 21 to the prior result from the Gaussian projected dynamics (Figure
16 and equation (2.57)), it is seen that the theory equation for the square pulse no longer
predicts a linear relationship for the amplitude vs. the longitudinal conductivity as it did in
the Gaussian case.
Figure 21. Plot of the critical amplitude (𝑎𝑐 ) vs. the longitudinal conductivity (𝐷) for the
projected dynamics with a square pulse.
While observing Figures 18-21, the greatest defect in each plot seems to be that the theory
equation (3.60) consistently predicts larger values for the critical amplitude when
compared to the numerical results. This issue presumably has to do in part with the choice
69
of setting the pulse width equal to a constant at the start of the procedure, but another effect
is that our voltage profile for the square pulse approach was derived from (3.9) with a nonzero applied current 𝐼. This is not reflected in the present analysis where 𝐼 was set to zero
in both the projected dynamics and the numerical simulations. Therefore, the next step in
the procedure is to introduce the applied current 𝐼 and redo the analysis. Although one
might expect this addition to have a trivial effect on the outcome that turns out not to be
the case.
70
4. FORMULATION OF THE THRESHOLD EQUATION USING A CONSTANT
SQUARE PULSE
From Using the Initial Conditions for a Square Pulse to a Constant Square Pulse
In the prior treatments, it had always been assumed that treating the applied pulse using
initial conditions for the potential would be a sufficient approximation to the physiological
case. The physiological case of course requires that the pulse be applied to the cells by an
external current source for some finite period of time. In fact, in many cases a pulse of
approximately constant magnitude is applied to a chain of cells via an electrode for a time
that is much larger than the rather short time that it takes for a chain of cells to excite. To
account for this applied current pulse, a term 𝐼(𝑥) is added to the governing equation:
𝜕𝑣
∂2 𝑣
= 𝐷 2 − 𝑔𝐾 𝑣 + 𝑔𝑁𝑎 𝑣 2 + 𝐼(𝑥).
𝜕𝑡
∂𝑥
(4.1)
In the prior section, we took 𝐼(𝑥) to be zero for all 𝑥 as we treated the external pulse by
specifying the initial condition for the potential 𝑣 over a certain number of cells given by
the initial pulse width 𝑤. Now we will take 𝐼(𝑥) to be non-zero for some initial pulse width.
The initial condition for the potential is taken to be zero, as the external applied current
𝐼(𝑥) takes the place of specifying non-zero initial conditions for the potential. The effect
this change has on the critical amplitude turns out to be rather dramatic. The difference
between the two approaches is shown below in Figure 22. Although the shapes are similar,
the actual values for the critical amplitude can be seen to be significantly lower across all
pulse widths when a constant pulse 𝐼(𝑥) is applied in place of using initial conditions for
the potential.
71
Figure 22. A comparison of the numerical results for the critical amplitude vs. pulse width
from two different approaches to applying a square pulse.
In resolving this discrepancy between the two curves, it was found that the criteria for
exciting the cells is totally different for each of the two cases. To see this, let’s start with
equation (4.1) and look for the “space clamped solution”. This is the solution where a
constant applied pulse 𝐼 is applied to the entire chain in the 𝑤 → ∞ limit. In this limit we
can take ∂2 𝑣⁄∂𝑥 2 = 0 as all cells are stepped up to the same potential by the applied current
simultaneously so that no gradient forms between adjacent cells (alternatively, this limit
can be thought of as the limit for a chain that contains only a single cell within it; in this
case, the spatial derivative is zero as well).
72
Searching for the time-independent threshold, we reduce equation (4.1) to:
0 = −𝑔𝐾 𝑣 + 𝑔𝑁𝑎 𝑣 2 + 𝐼.
(4.2)
This is just a quadratic equation, whose positive root 𝑣 = 𝑎𝑐 is given by:
𝑎𝑐 =
𝑔𝐾 + √𝑔𝐾 2 − 4(𝑔𝑁𝑎 )𝐼
.
2𝑔𝑁𝑎
(4.3)
If we take 𝐼 = 0, the critical amplitude is given by:
𝑔𝐾 + √𝑔𝐾 2
𝑔𝐾
𝑎𝑐 =
→ 𝑎𝑐 =
.
2𝑔𝑁𝑎
𝑔𝑁𝑎
(4.4)
This is just the result that we had earlier from the projected dynamics for the square pulse
using initial conditions for the potential (see equation (3.61)). It shows that the critical
amplitude in the limit of a large pulse width is just the positive root of equation (4.2). If we
take 𝑔𝐾 = 0.139 and 𝑔𝑁𝑎 = 1.0 then we have 𝑎𝑐 = 0.139 which was verified by solving
(4.1) numerically for the threshold in the large pulse width limit.
Now, if we take 𝐼 ≠ 0, the criteria for excitation is completely different. Instead of being
a root of (4.2), the new criteria for excitation is merely that the solution for 𝑎𝑐 be a real
number. In order to ensure this, we must have:
√𝑔𝐾 2 − 4(𝑔𝑁𝑎 )𝐼 ≥ 0.
(4.5)
For this equation to hold true, the minimum requirement is met if the term inside the square
root sign is equal to zero. Thus the critical amplitude (𝛼𝑐 ) in this case turns out to be:
𝑔𝐾 2 − 4(𝑔𝑁𝑎 )𝐼 = 0,
𝐼=
𝛼𝑐 =
𝑔𝐾 2
,
4𝑔𝑁𝑎
𝐼
𝑔𝐾
=
.
𝑔𝐾
4𝑔𝑁𝑎
73
(4.6)
(4.7)
(4.8)
We see that the new threshold requirement 𝛼𝑐 is reduced by a factor of four in comparison
to the threshold for the instantaneous square pulse. Using 𝑔𝐾 = 0.139 and 𝑔𝑁𝑎 = 1.0, we
now have that 𝛼𝑐 ≅ 0.035 which agrees with the result obtained numerically in the limit
of a large pulse width.
Reformulating the Projected Dynamics Using a Constant Square Pulse
Now that we have some idea of how the nature of the threshold curve changes as we apply
a constant square pulse, we have to modify the method used to obtain the projected
dynamics in two ways. First, we need to introduce the external current pulse 𝐼(𝑥) to the
expression for the total energy. Second, after we carry through the procedure and obtain
the expression for 𝑎𝑐 , we will need to extract the true criteria for excitation from underneath
the square root sign, and then solve for 𝛼𝑐 .
We will follow the same basic approach from before, but we will make the appropriate
changes as we go along. Starting with:
𝑀𝒂̇ = −𝛁𝐸,
(4.9)
where 𝑀 is given by:
∞
𝑚𝑖𝑗 = ∫
−∞
𝜕𝑉 𝜕𝑉
𝑑𝑥,
𝜕𝑎𝑖 𝜕𝑎𝑗
(4.10)
we again choose to do one-variable projected dynamics and set the pulse width equal to a
constant 𝑤 at the start of the procedure. This allows us to write (4.9) as:
𝑎1̇ =
− ∂𝐸/𝜕𝑎1
.
𝑚11
74
(4.11)
The condition 𝑎1̇ = 0 is met when the numerator − ∂𝐸/𝜕𝑎1 is equal to zero, and so we
can ignore 𝑚11 and focus on obtaining the energy integral. Now the applied current pulse
𝐼(𝑥) is:
𝐼(𝑥) = 𝐼
𝑓𝑜𝑟 𝑥 < 𝑤 ⁄2,
(4.12)
𝐼(𝑥) = 0
𝑓𝑜𝑟 𝑥 > 𝑤 ⁄2.
(4.13)
Introducing this current pulse to the energy integral, we will arrive at:
𝑤
2
𝑑𝑉1 2
2𝑉1 3
𝐸 = ∫ (𝐷 (
) + 𝑔𝐾 𝑉1 2 − 𝑔𝑁𝑎
− 2𝐼𝑉1 ) 𝑑𝑥
𝑑𝑥
3
(4.14)
0
∞
𝑑𝑉2 2
2𝑉2 3
2
+ ∫ (𝐷 (
) + 𝑔𝐾 𝑉2 − 𝑔𝑁𝑎
) 𝑑𝑥.
𝑑𝑥
3
𝑤/2
To evaluate (4.14), we need the functions 𝑉1 and 𝑉2 and their two derivatives which are
unchanged from before:
𝑉1 = 𝑎1 (1 − 𝑒 −𝛾𝑤⁄2 cosh(𝛾𝑥))
𝑉2 =
𝑓𝑜𝑟 𝑥 < 𝑤 ⁄2,
𝑎1
(1 − 𝑒 −𝛾𝑤 )𝑒 −𝛾(𝑥−𝑤⁄2)
𝑓𝑜𝑟 𝑥 > 𝑤/2,
2
𝜕𝑉1
= −𝑎1 𝛾𝑒 −𝛾𝑤⁄2 sinh(𝛾𝑥),
𝜕𝑥
𝑤
𝜕𝑉2
1
= − 𝑎1 𝛾𝑒 −𝛾(𝑥− 2 ) (1 − 𝑒 −𝛾𝑤 ).
𝜕𝑥
2
(4.15)
(4.16)
(4.17)
(4.18)
Inserting (4.15)-(4.18) into (4.14), performing the integration, and simplifying the result
leads to:
𝐸=
1 −2𝛾𝑤
𝑒
(−𝑎1 3 𝑔𝑁𝑎 − 𝑎1 𝑒 2𝛾𝑤 (12𝐼(−1 + 𝛾𝑤) + 𝑎1 2 𝑔𝑁𝑎 (−9 + 4𝛾𝑤))
12𝛾
−3𝑎1 (𝐷𝛾 2 + 𝑔𝐾 − (3 + 2𝛾𝑤)) − 𝑎1 𝑒 𝛾𝑤 (12𝐼 + 𝑎1 (3𝐷𝛾 2 (1 + 𝛾𝑤)
−3𝑔𝐾 (3 + 𝛾𝑤) + 2𝑎1 𝑔𝑁𝑎 (4 + 3𝛾𝑤)))),
and then:
75
(4.19)
𝑎1̇ = −
𝜕𝐸
1 −2𝛾𝑤 2
=
𝑒
(𝑎1 𝑔𝑁𝑎 + 𝑒 2𝛾𝑤 (6𝑎1 𝑔𝐾 − 9𝑎1 2 𝑔𝑁𝑎 − 4𝐼 +
𝜕𝑎1
4𝛾
(4.20)
4(−𝑎1𝑔𝐾 + 𝑎1 2 𝑔𝑁𝑎 + 𝐼)𝛾𝑤 − 2𝑎1 𝐷𝛾 2 ) + 2𝑒 𝛾𝑤 (2𝐼 + 𝑎1 (𝐷𝛾 2 (1 + 𝛾𝑤) −
𝑔𝐾 (3 + 𝛾𝑤) + 𝑎1 𝑔𝑁𝑎 (4 + 3𝛾𝑤)))).
Setting 𝑎1̇ = 0 and solving for 𝑎1 gives us the following quadratic equation:
𝑒 𝛾𝑤 (−𝐷𝛾 2 (1 + 𝛾𝑤) + 𝑔𝐾 (3 + 𝛾𝑤)) +
𝑒 2𝛾𝑤 (𝐷𝛾 2 + 𝑔𝐾 (−3 + 2𝛾𝑤)) ± √𝜇
𝑎1 =
,
𝑔𝑁𝑎 + 2𝑒 𝛾𝑤 𝑔𝑁𝑎 (4 + 3𝛾𝑤) + 𝑒 2𝛾𝑤 𝑔𝑁𝑎 (−9 + 4𝛾𝑤)
(4.21)
where the term under the square root sign, 𝜇, is given by:
𝜇 = 𝑒 𝛾𝑤 (𝑒 𝛾𝑤 (𝐷𝛾 2 (1 − 𝑒 𝛾𝑤 + 𝛾𝑤) − 𝑔𝐾 (3 + 𝛾𝑤 + 𝑒 𝛾𝑤 (−3 + 2𝛾𝑤)))2
(4.22)
−4𝑔𝑁𝑎 𝐼(1 + 𝑒 𝛾𝑤 (−1 + 𝛾𝑤))(1 + 𝑒 𝛾𝑤 (8 + 6𝛾𝑤 + 𝑒 𝛾𝑤 (−9 + 4𝛾𝑤)))).
The criteria for equation (4.22) to be real is:
𝜇 ≥ 0,
(4.23)
and the minimum criteria for excitation is then:
𝜇 = 0 = 𝑒 𝛾𝑤 (𝑒 𝛾𝑤 (𝐷𝛾 2 (1 − 𝑒 𝛾𝑤 + 𝛾𝑤) − 𝑔𝐾 (3 + 𝛾𝑤 + 𝑒 𝛾𝑤 (−3 + 2𝛾𝑤)))2
(4.24)
−4𝑔𝑁𝑎 𝐼(1 + 𝑒 𝛾𝑤 (−1 + 𝛾𝑤))(1 + 𝑒 𝛾𝑤 (8 + 6𝛾𝑤 + 𝑒 𝛾𝑤 (−9 + 4𝛾𝑤)))).
This can be solved for 𝐼:
𝐼=
𝑒 𝛾𝑤 (𝐷𝛾 2 (1 − 𝑒 𝛾𝑤 + 𝛾𝑤) − 𝑔𝐾 (3 + 𝛾𝑤 + 𝑒 𝛾𝑤 (−3 + 2𝛾𝑤)))2
,
4𝑔𝑁𝑎 (1 + 𝑒 𝛾𝑤 (−1 + 𝛾𝑤))(1 + 𝑒 𝛾𝑤 (8 + 6𝛾𝑤 + 𝑒 𝛾𝑤 (−9 + 4𝛾𝑤)))
(4.25)
and then further simplified to:
𝑔𝐾 2 𝑒 𝛾𝑤 (1 + 𝑒 𝛾𝑤 (𝛾𝑤 − 1))
𝐼=
.
𝑔𝑁𝑎 (1 + 2𝑒 𝛾𝑤 (4 + 3𝛾𝑤) + 𝑒 2𝛾𝑤 (4𝛾𝑤 − 9))
(4.26)
Finally, we can divide (4.26) by 𝑔𝐾 to arrive at our desired expression for the critical
amplitude 𝛼𝑐 :
𝛼𝑐 =
𝑔𝐾 𝑒 𝛾𝑤 (1 + 𝑒 𝛾𝑤 (𝛾𝑤 − 1))
.
𝑔𝑁𝑎 (1 + 2𝑒 𝛾𝑤 (4 + 3𝛾𝑤) + 𝑒 2𝛾𝑤 (4𝛾𝑤 − 9))
76
(4.27)
A comparison with equation (3.58) shows that the two expressions only differ from one
another by a factor of four. While this indicates that the form of the projected dynamics
remains largely the same, the agreement between (4.27) and the numerical threshold
solution has now substantially improved as discussed in the next section.
Analysis of the Threshold Equation for a Constant Square Pulse
Equation (4.27) is plotted against the numerical solution of (4.1) in Figure 23. For the
numerical solution (see Appendix: Numerical Methods) the parameters used were 𝑔𝐾 =
0.139, 𝑔𝑁𝑎 = 1.0, 𝐷 = 1.0, and the amplitude was given by 𝛼𝑐 = 𝐼 ⁄𝑔𝐾 .
Figure 23. A plot of the critical amplitude vs. pulse width for a constant applied square
pulse.
77
The agreement between theory and the numerical solution has been drastically improved.
Since we know that the basis for our approximation is the amplitude nullcline, and since
we took the width as constant at the start of the projected dynamics, it seems reasonable to
attribute the improved accuracy for the threshold curve to a diminished dependence on the
time evolution of the pulse width for the case of a constant square pulse.
At this point, it is important to check and see if equation (4.27) reduces properly to the new
threshold criteria that was established in equation (4.8). The expression for the critical
amplitude 𝛼𝑐 was given in (4.27) as:
𝑔𝐾 𝑒 𝛾𝑤 (1 + 𝑒 𝛾𝑤 (𝛾𝑤 − 1))
𝛼𝑐 =
.
𝑔𝑁𝑎 (1 + 2𝑒 𝛾𝑤 (4 + 3𝛾𝑤) + 𝑒 2𝛾𝑤 (4𝛾𝑤 − 9))
(4.28)
In the limit where 𝑤 → ∞, the largest terms will be those that include both 𝑒 2𝛾𝑤 and 𝑤 as
a product. This reduces (4.28) to:
𝛼𝑐 =
𝛾𝑤𝑔𝐾 𝑒 2𝛾𝑤
𝑔𝐾
→ 𝛼𝑐 =
.
2𝛾𝑤
4𝛾𝑤𝑔𝑁𝑎 𝑒
4𝑔𝑁𝑎
(4.29)
Indeed, equation (4.29) reduces to the proper threshold in the limit of a large pulse width.
The small width limit also checks out where the leading term in the series expansion is:
𝛼𝑐 ~
3𝑔𝐾
,
2𝛾𝑔𝑁𝑎 𝑤
(4.30)
which differs from the result in equation (3.64) only by the constant factor of 1/4. The
strong agreement in Figure 23 provides evidence that the correct scaling at small widths is
indeed 1/𝑤 . Now it is time to examine threshold behavior for the ion channel and
longitudinal conductivities. Figure 24 shows the result for the potassium ion channel
conductivity while Figure 25 shows the result for the sodium ion channel conductivity.
78
Figure 24. A plot of the critical amplitude vs. potassium ion channel conductivity for a
constant applied square pulse.
Figure 25. A plot of the critical amplitude vs. sodium ion channel conductivity for a
constant applied square pulse.
79
Both plots were generated using the same parameter values as in the prior section: 𝑔𝑁𝑎 =
1.0, 𝐷 = 1.0, and 𝑤 = 5 for the potassium ion channel plot, and 𝑔𝐾 = 0.139, 𝐷 = 1.0,
and 𝑤 = 5 for the sodium ion channel plot. For the case of sodium, the solution reduces to
a simple relationship (𝛼𝑐 ~ 0.0973769/𝑔𝑁𝑎 ) as it did before in the instantaneous case with
the only difference being that the value of the constant has now been reduced by a factor
of four as in the prior cases. Although the potassium result begins to diverge from theory
after a certain point, the agreement is still quite good for small values of 𝑔𝐾 . In general, the
same can be said for the longitudinal conductivity, which is plotted below in Figure 26
where the numerical parameters were 𝑔𝐾 = 0.139, 𝑔𝑁𝑎 = 1.0, and 𝑤 = 5.
Figure 26. A plot of the critical amplitude vs. longitudinal conductivity for a constant
applied square pulse.
80
In Figure 26, the trend for the theoretical approach is greatly improved over the prior case,
where initial conditions for the potential were used, but it still starts to become more and
more inaccurate as the longitudinal conductivity increases in value. Perhaps this is to be
expected as the longitudinal conductivity affects the width of the pulse more than any other
parameter, and we have neglected the time dependence of the width in the foregoing
analysis. However, this issue is minimized for small values of the conductivity, and the
constant width approximation is certainly less of an issue in the present case than it was in
the prior case. In comparing Figures 23-26, we see that, in general, equation (4.28) provides
a far more accurate description of the threshold for the constant square pulse in comparison
to the numerical result than what came before.
81
5. FORMULATION OF THE THRESHOLD EQUATION FOR A TIME-DEPENDENT
SQUARE PULSE
Generalizing the Approach to find the Threshold for a Time-Dependent Square
Pulse
In the previous section, an equation for the excitation threshold was obtained for the case
where an external square current pulse was applied for some time that was considered to
be much longer than the time needed to excite the cells. In such a situation the applied
current pulse was treated as being a constant since, by the time the current was switched
off, the cells had either long since been excited or they would not be excited by the current
no matter how long it was applied. But what about the more general case where the external
current can be applied for some finite amount of time 𝜏? Can the projected dynamics be
used to obtain not only the solution for when 𝜏 is very large, as was done previously, but
also the case where 𝜏 can be specified to be any amount of time we wish? Indeed, such a
general solution can be found.
To better understand the approach, we will first start with the simplest case where the entire
cell chain is excited to the same level all at once. In doing so, the system becomes spatially
homogeneous, and we can neglect the spatial derivative in equation (4.1). This reduces
equation (4.1) to:
𝑑𝑣
= −𝑔𝐾 𝑣 + 𝑔𝑁𝑎 𝑣 2 + 𝐼,
𝑑𝑡
which can be rearranged to solve for the differential of time:
𝑑𝑣
= 𝑑𝑡.
−𝑔𝐾 𝑣 + 𝑔𝑁𝑎 𝑣 2 + 𝐼
82
(5.1)
(5.2)
At the point when the current is switched off, at the time 𝑡 = 𝜏, the system instantly reverts
back to the 𝐼 = 0 dynamics, where the threshold requirement is found from:
0 = −𝑔𝐾 𝑣 ∗ + 𝑔𝑁𝑎 𝑣 ∗ 2 ,
𝑣∗ =
(5.3)
𝑔𝐾
.
𝑔𝑁𝑎
(5.4)
Equation (5.4) is in agreement with equation (3.61) that was obtained for the threshold in
the case 𝑤 → ∞ using projected dynamics with 𝐼 = 0. The potential must rise from its
initial value of zero to the value given in equation (5.4) by the time the current is switched
off in order for the system to fire an action potential. The integral can now be expressed
as:
∫
𝑔𝐾 ⁄𝑔𝑁𝑎
0
𝜏
𝑑𝑣
=
∫
𝑑𝑡 = 𝜏,
−𝑔𝐾 𝑣 + 𝑔𝑁𝑎 𝑣 2 + 𝐼
0
(5.5)
and, after performing the integration, (5.5) becomes:
𝑔𝐾
4 tan−1 (
)
√−𝑔𝐾 2 + 4𝑔𝑁𝑎 𝐼
𝜏=
.
√−𝑔𝐾 2 + 4𝑔𝑁𝑎 𝐼
(5.6)
In equation (5.6), we can see that the threshold condition for non-zero values of 𝐼, which
is that the solution merely exist, is present inside the square root sign and is identical in
form to the threshold condition worked out previously in equations (4.5) to (4.8) for this
case. At this point, our goal is to invert equation (5.6) to obtain an expression for the critical
amplitude as a function of 𝜏. Letting 𝑦 = √−𝑔𝐾 2 + 4𝑔𝑁𝑎 𝐼, equation (5.6) becomes:
𝜏=
𝑔
4 tan−1 ( 𝑦𝐾 )
𝑦
83
(5.7)
,
tan (
𝑦𝜏
𝑔𝐾
)=
.
4
𝑦
(5.8)
If we assume that 𝑦𝜏⁄4 ≪ 1, we can make the following approximation:
𝑦𝜏
𝑦𝜏
)≅
,
4
4
(5.9)
4𝑔𝐾
,
𝜏
(5.10)
tan (
which leads to:
𝑦2 =
−𝑔𝐾 2 + 4𝑔𝑁𝑎 𝐼 =
𝐼=
4𝑔𝐾
,
𝜏
(5.11)
𝑔𝐾 2
𝑔𝐾
+
.
4𝑔𝑁𝑎 𝜏𝑔𝑁𝑎
(5.12)
𝑔𝐾
1
+
.
4𝑔𝑁𝑎 𝜏𝑔𝑁𝑎
(5.13)
The critical amplitude is then:
𝛼𝑐 =
Equation (5.13) is plotted in Figure 27 against the numerical result (see Appendix:
Numerical Methods) for a time-dependent current pulse 𝐼(𝑡) applied uniformly to a chain
of cells using 𝐷 = 1.0, 𝑔𝑁𝑎 = 1.0, and 𝑔𝐾 = 0.139 as parameters.
Let us now pause for a moment to examine the features of equation (5.13). First, we note
that, as 𝜏 → ∞, equation (5.13) reduces to:
𝛼𝑐 =
𝑔𝐾
,
4𝑔𝑁𝑎
(5.14)
which is in complete agreement with the threshold for a homogeneous constant square
pulse given by equation (4.8). At the other extreme where 𝜏 → 0, the requirement to excite
becomes infinitely large. Qualitatively this result makes sense as, given less time to excite
the cell, the required potential difference to excite must be greater so that the necessary
84
charge is driven across the membrane in a shorter period of time. This is reflected in the
increase that is required for the critical amplitude 𝛼𝑐 when 𝜏 is small.
Figure 27. A plot of the critical amplitude vs. the time that the pulse was applied for a
uniformly applied square pulse.
Also of physiological interest here is the inverse dependence of the second term in equation
(5.13) on the sodium ion channel conductivity, 𝑔𝑁𝑎 . Other studies and models have
indicated that the most important factor in reducing excitability is a reduced availability of
sodium channels associated with a smaller value of 𝑔𝑁𝑎 [15]. In equation (5.13), we see
that lowering the sodium conductivity raises the critical amplitude requirement in both
terms; not only does it raise the critical amplitude for the case when 𝜏 → ∞, but it also
85
serves to increase the second term for the case of finite 𝜏. Hence, equation (5.13) proves to
be remarkably sensitive to a decrease in the sodium conductance which is in qualitative
agreement with those earlier observations. It also suggests that, to decrease the risk for
excitation failure, the pulse application time 𝜏 for the externally applied current must not
be too small. Although 𝜏 does not affect the threshold requirement from the first term in
(5.13) at all, a small enough value of 𝜏 will cause the second term in equation (5.13) to
completely dominate the threshold requirement.
Obtaining the Threshold Equation for a Time-Dependent Square Pulse
We have obtained a time-dependent threshold solution for the case where the pulse is
uniformly applied to all cells in the chain, but we also wish to obtain the solution where
the applied current pulse may only be applied to a finite number of cells (of pulse width 𝑤)
in the cell chain. For this we take the same approach as above, but here we start with the
equation for the time evolution of the amplitude that we get by applying the method of
projected dynamics.
Much of the procedure to obtain such an equation is exactly the same as it was when
projected dynamics was performed earlier for the constant square pulse. However, one step
at the beginning of the procedure must be modified; for this, let us take a closer look at
equation (4.11):
𝑎1̇ =
− ∂𝐸/𝜕𝑎1
.
𝑚11
(5.15)
Earlier, when we used projected dynamics for the square pulse, we were interested only in
the time-independent solution for the amplitude. Since 𝑎1̇ = 0 in that case, we were able
to ignore the M matrix, and we did not have to evaluate 𝑚11 explicitly. Now, we now seek
86
an expression for 𝑎1̇ itself, and so, we must evaluate 𝑚11 for the present case. The matrix
element 𝑚11 is given by:
𝑤/2
𝑚11
∞
𝜕𝑉1 2
𝜕𝑉2 2
=∫ (
) 𝑑𝑥 + ∫ (
) 𝑑𝑥.
𝜕𝑎1
𝜕𝑎1
0
(5.16)
𝑤/2
To evaluate equation (5.16), we differentiate 𝑉1 and 𝑉2 given in equations (4.15) and (4.16)
with respect to 𝑎1 :
𝑉1 = 𝑎1 (1 − 𝑒 −𝛾𝑤⁄2 cosh(𝛾𝑥))
𝑉2 =
𝑓𝑜𝑟 𝑥 < 𝑤 ⁄2,
𝑎1
(1 − 𝑒 −𝛾𝑤 )𝑒 −𝛾(𝑥−𝑤⁄2)
𝑓𝑜𝑟 𝑥 > 𝑤/2,
2
𝜕𝑉1
= (1 − 𝑒 −𝛾𝑤⁄2 cosh(𝛾𝑥)),
𝜕𝑎1
𝜕𝑉2
1
= (1 − 𝑒 −𝛾𝑤 )𝑒 −𝛾(𝑥−𝑤⁄2) .
𝜕𝑎1
2
(5.17)
(5.18)
(5.19)
(5.20)
After placing (5.19) and (5.20) into (5.16) and performing the integration, we arrive at:
𝑚11
𝑒 −2𝛾𝑤 (𝑒 𝛾𝑤 − 1)2 2𝛾𝑤 − 4 + 𝑒 −𝛾𝑤 (4 + 𝛾𝑤 + sinh(𝛾𝑤))
=
+
.
4𝛾
2𝛾
(5.21)
The negative of the energy gradient, − ∂𝐸/𝜕𝑎1, is given exactly as it was before in equation
(4.20), namely:
−
𝜕𝐸
1 −2𝛾𝑤 2
=
𝑒
(𝑎1 𝑔𝑁𝑎 + 𝑒 2𝛾𝑤 (6𝑎1 𝑔𝐾 − 9𝑎1 2 𝑔𝑁𝑎 − 4𝐼 +
𝜕𝑎1
4𝛾
(5.22)
4(−𝑎1 𝑔𝐾 + 𝑎1 2 𝑔𝑁𝑎 + 𝐼)𝛾𝑤 − 2𝑎1 𝐷𝛾 2 ) + 2𝑒 𝛾𝑤 (2𝐼 + 𝑎1 (𝐷𝛾 2 (1 + 𝛾𝑤) −
𝑔𝐾 (3 + 𝛾𝑤) + 𝑎1 𝑔𝑁𝑎 (4 + 3𝛾𝑤)))).
Substituting (5.21) and (5.22) into equation (5.15) gives us our desired expression for the
time derivative of the amplitude:
87
1 −2𝛾𝑤 2
(𝑎1 𝑔𝑁𝑎 + 𝑒 2𝛾𝑤 (6𝑎1 𝑔𝐾 − 9𝑎1 2 𝑔𝑁𝑎 − 4𝐼 + 4(−𝑎1 𝑔𝐾
4𝛾 𝑒
+𝑎1 2 𝑔𝑁𝑎 + 𝐼)𝛾𝑤 − 2𝑎1 𝐷𝛾 2 ) + 2𝑒 𝛾𝑤 (2𝐼 + 𝑎1 (𝐷𝛾 2 (1 + 𝛾𝑤) −
𝑑𝑎1
𝑔𝐾 (3 + 𝛾𝑤) + 𝑎1 𝑔𝑁𝑎 (4 + 3𝛾𝑤))))
=
,
−2𝛾𝑤
𝛾𝑤
𝑒
(𝑒 − 1)2 2𝛾𝑤 − 4 + 𝑒 −𝛾𝑤 (4 + 𝛾𝑤 + sinh(𝛾𝑤))
𝑑𝑡
+
4𝛾
2𝛾
(5.23)
while some further simplification of equation (5.23) leads to:
(𝑒 −𝛾𝑤 (𝑎1 2 𝑔𝑁𝑎 + 𝑒 2𝛾𝑤 (6𝑎1 𝑔𝐾 − 9𝑎1 2 𝑔𝑁𝑎 − 4𝐼 +
4(−𝑎1 𝑔𝐾 + 𝑎1 2 𝑔𝑁𝑎 + 𝐼)𝛾𝑤 − 2𝑎1 𝑔𝐾 ) + 2𝑒 𝛾𝑤 (2𝐼 + 𝑎1 (𝑔𝐾 (1 + 𝛾𝑤)
𝑑𝑎1
−𝑔𝐾 (3 + 𝛾𝑤) + 𝑎1 𝑔𝑁𝑎 (4 + 3𝛾𝑤)))))
=
.
𝑑𝑡
2(3 + 𝛾𝑤 + 𝑒 𝛾𝑤 (2𝛾𝑤 − 3))
(5.24)
Now, we want to group all the terms in (5.24) containing 𝑔𝐾 , all the terms containing 𝑔𝑁𝑎 ,
and all the terms containing 𝐼 separately; then we can write the right side of the equation
in a way the resembles how we wrote equation (5.1). After doing this, we end up with:
𝑑𝑎1
= −𝐴𝑎1 + 𝐵𝑎1 2 + 𝐶𝐼,
𝑑𝑡
(5.25)
where:
2(1 + 𝑒 𝛾𝑤 (𝛾𝑤 − 1))
,
3 + 𝛾𝑤 + 𝑒 𝛾𝑤 (2𝛾𝑤 − 3)
(5.26)
8 + 𝑒 −𝛾𝑤 + 6𝛾𝑤 + 𝑒 𝛾𝑤 (4𝛾𝑤 − 9)
,
2(3 + 𝛾𝑤 + 𝑒 𝛾𝑤 (2𝛾𝑤 − 3))
(5.27)
𝐴 = 𝑔𝐾
𝐵 = 𝑔𝑁𝑎
𝐶=
2(1 + 𝑒 𝛾𝑤 (𝛾𝑤 − 1))
.
3 + 𝛾𝑤 + 𝑒 𝛾𝑤 (2𝛾𝑤 − 3)
(5.28)
Equation (5.25) can be rearranged to solve for the differential of time:
𝑑𝑎1
= 𝑑𝑡.
−𝐴𝑎1 + 𝐵𝑎1 2 + 𝐶𝐼
(5.29)
Following in the same manner as we did before for the case of an infinite pulse width, we
wish to integrate (5.29) to find an expression for the applied current amplitude as a function
of the time duration of the applied current, 𝜏. At the time the current is switched off, the
88
threshold requirement becomes the same as for the 𝐼 = 0 case, and we find that the
threshold requirement for the potential (𝑎1 ∗ ) can be written as:
0 = −𝐴𝑎1 ∗ + 𝐵𝑎1 ∗ 2 ,
𝑎1 ∗ =
(5.30)
𝐴
.
𝐵
(5.31)
The integral of (5.29) is then:
𝐴/𝐵
∫
0
𝜏
𝑑𝑎1
=
∫
𝑑𝑡,
−𝐴𝑎1 + 𝐵𝑎1 2 + 𝐶𝐼
0
𝜏=
(5.32)
𝐴
4 tan−1 ( 𝑦 )
(5.33)
,
𝑦
with 𝑦 given by:
(5.34)
2
𝑦=
−4𝑔𝐾 2 (1 + 𝑒 𝛾𝑤 (𝛾𝑤 − 1)) +
√4𝑔𝑁𝑎 𝐼(1 + 𝑒 𝛾𝑤 (𝛾𝑤 − 1))(8 + 𝑒 −𝛾𝑤 + 6𝛾𝑤 + 𝑒 𝛾𝑤 (4𝛾𝑤 − 9))
(3 + 𝛾𝑤 + 𝑒 𝛾𝑤 (2𝛾𝑤 − 3))
2
.
The goal once again is to solve for the critical amplitude as a function of 𝜏. Proceeding
exactly as before:
𝑦𝜏
𝐴
(5.35)
)= ,
4
𝑦
𝑦𝜏
𝑦𝜏
(5.36)
tan ( ) ≅
,
4
4
4𝐴
(5.37)
𝑦2 =
.
𝜏
Inserting equations (5.26) and (5.34) into (5.37), simplifying, and rearranging leads to:
tan (
𝐼=
(1 + 𝑒 𝛾𝑤 (𝛾𝑤 − 1))
𝑔𝐾 2
(
)+
𝑔𝑁𝑎 (8 + 𝑒 −𝛾𝑤 + 6𝛾𝑤 + 𝑒 𝛾𝑤 (4𝛾𝑤 − 9))
2(3 + 𝛾𝑤 + 𝑒 𝛾𝑤 (2𝛾𝑤 − 3))
𝑔𝐾
(
).
𝜏𝑔𝑁𝑎 (8 + 𝑒 −𝛾𝑤 + 6𝛾𝑤 + 𝑒 𝛾𝑤 (4𝛾𝑤 − 9))
89
(5.38)
Dividing (5.38) by 𝑔𝐾 gives the expression for the critical amplitude:
(1 + 𝑒 𝛾𝑤 (𝛾𝑤 − 1))
𝑔𝐾
𝛼𝑐 =
(
)+
𝑔𝑁𝑎 (8 + 𝑒 −𝛾𝑤 + 6𝛾𝑤 + 𝑒 𝛾𝑤 (4𝛾𝑤 − 9))
(5.39)
2(3 + 𝛾𝑤 + 𝑒 𝛾𝑤 (2𝛾𝑤 − 3))
1
(
).
𝜏𝑔𝑁𝑎 (8 + 𝑒 −𝛾𝑤 + 6𝛾𝑤 + 𝑒 𝛾𝑤 (4𝛾𝑤 − 9))
Using (5.26) and (5.27), this result can also be written more compactly as:
𝛼𝑐 =
𝐴
1
+
.
4𝐵 𝐵𝜏
(5.40)
In Figure 28, equation (5.39) is plotted against the numerical threshold solution with 𝐷 =
1.0, 𝑔𝑁𝑎 = 1.0, and 𝑔𝐾 = 0.139. The case where the pulse width is 𝑤 = 100 cells
(essentially the homogeneous case) and the case where the pulse width is 𝑤 = 5 cells are
included for comparison.
Figure 28. A plot of the critical amplitude vs. the time that the square pulse was applied
is provided for both a 100 cell pulse width and a 5 cell pulse width.
90
We now wish to check equation (5.39) in both the large pulse width limit and the large
pulse time limit. As 𝑤 → ∞, the only surviving terms contain the product 𝑒 𝛾𝑤 𝛾𝑤, and the
equation for the critical amplitude reduces to:
𝛼𝑐 =
𝑔𝐾
𝑒 𝛾𝑤 𝛾𝑤
1
2(𝑒 𝛾𝑤 2𝛾𝑤)
𝑔𝐾
1
( 𝛾𝑤
)+
( 𝛾𝑤
)=
+
,
𝑔𝑁𝑎 (𝑒 4𝛾𝑤)
𝜏𝑔𝑁𝑎 (𝑒 4𝛾𝑤)
4𝑔𝑁𝑎 𝜏𝑔𝑁𝑎
(5.41)
which is exactly what we found before in equation (5.13) for the large pulse width limit.
As 𝜏 → ∞, we find that the second term in equation (5.39) vanishes to give:
𝛼𝑐 =
(1 + 𝑒 𝛾𝑤 (𝛾𝑤 − 1))
𝑔𝐾
(
).
𝑔𝑁𝑎 (8 + 𝑒 −𝛾𝑤 + 6𝛾𝑤 + 𝑒 𝛾𝑤 (4𝛾𝑤 − 9))
(5.42)
Multiplying the top and bottom of this equation by 𝑒 𝛾𝑤 and rewriting 8 + 6𝛾𝑤 in the
denominator as 2(4 + 3𝛾𝑤) yields:
𝛼𝑐 =
𝑔𝐾 𝑒 𝛾𝑤 (1 + 𝑒 𝛾𝑤 (𝛾𝑤 − 1))
.
𝑔𝑁𝑎 (1 + 2𝑒 𝛾𝑤 (4 + 3𝛾𝑤) + 𝑒 2𝛾𝑤 (4𝛾𝑤 − 9))
(5.43)
which is the same as equation (4.27) which was derived earlier for the case of a constant
square pulse. Equation (5.39) then emerges as our most general approximate solution to
the one-dimensional square pulse threshold problem. (5.39) answers the question of
whether or not an action potential will fire given the width and time of application of an
applied square pulse using a wide range of conductivity parameters.
91
6. A SIMPLIFIED APPROXIMATE SOLUTION FOR THE SQUARE PULSE
Introducing a Simplified Initial Voltage Profile for the Constant Square Pulse
From the foregoing analysis, we have obtained a rather accurate approximate analytical
expression of the excitation threshold for the case of a continuously applied square pulse:
𝛼𝑐 =
𝑔𝐾 𝑒 𝛾𝑤 (1 + 𝑒 𝛾𝑤 (𝛾𝑤 − 1))
.
𝑔𝑁𝑎 (1 + 2𝑒 𝛾𝑤 (4 + 3𝛾𝑤) + 𝑒 2𝛾𝑤 (4𝛾𝑤 − 9))
(6.1)
We also have obtained a less accurate, but easier to understand, model of the excitation
threshold for the case of an applied Gaussian pulse specified by using initial conditions for
the potential:
𝑎𝑐 =
3√6 𝐷
(
+ 𝛼).
7 𝑔𝑤 2
(6.2)
Both results have their strengths and weaknesses. The strength of equation (6.1) lies in how
accurately it is able to predict the results of the numerical solution over a wide range of the
given parameters. This gives us confidence in the method we are using and also sheds some
light on the sources of error in our less accurate earlier attempts. The downside to equation
(6.1) is that it is relatively difficult to analyze, and only a limited amount of useful
information can be extracted from it. Equation (6.2) suffers from several defects in that it
doesn’t give the correct value for the critical amplitude at the large pulse width limit and
differs significantly from the numerical solution at the small pulse width limit. Despite
these drawbacks, the qualitative predictions made by equation (6.2), concerning the
behavior of the system as the various parameters are changed, are still in complete
agreement with the trends predicted by the more accurate threshold equation (6.1). In both
cases, increasing the longitudinal conductivity increases the requirement for a cell to reach
92
the excitation threshold whereas increasing the “sodium” ion channel conductivity and
increasing the width of the initial pulse tend to have the opposite effect. In contrast to
equation (6.1), all of these trends for each parameter are very easily interpreted from the
analysis of equation (6.2).
Ideally, what we would like to have is a threshold solution that provides a compromise
between the two extremes outlined above. Such a solution should incorporate some of the
quantitative improvements gained in going from the Gaussian pulse and equation (6.2) to
the constant square pulse and equation (6.1) while decreasing the mathematical complexity
of equation (6.1). To accomplish this task, we need to introduce a new initial voltage profile
that will then be used with the projected dynamics. Let us start by examining the initial
voltage profile that was used previously for the constant square pulse. This was given for
the one-variable projected dynamics as:
𝑉1 = 𝑎(1 − 𝑒 −𝛾𝑤⁄2 cosh(𝛾𝑥))
𝑉2 =
𝑎
(1 − 𝑒 −𝛾𝑤 )𝑒 −𝛾(𝑥−𝑤⁄2)
2
𝑓𝑜𝑟 𝑥 < 𝑤 ⁄2,
𝑓𝑜𝑟 𝑥 > 𝑤/2.
(6.3)
(6.4)
This initial voltage profile features a relatively flat shape for the central region when 𝑥 <
𝑤/2, and an exponential decrease outside of that region. The overall shape of this voltage
profile then approximates the shape of an applied square pulse. Now, we can introduce
those two key features in a simpler way by introducing the following initial voltage profile:
𝑉1 = 𝑎
𝑓𝑜𝑟 𝑥 < 𝑤 ⁄2,
𝑉2 = 𝑎𝑒 −𝛾(𝑥−𝑤⁄2)
𝑓𝑜𝑟 𝑥 > 𝑤/2.
(6.5)
(6.6)
It should be noted that the simplified initial voltage profile given above still matches the
first boundary condition from solving the partial differential equation, since 𝑉1 (𝑤/2) =
𝑉2 (𝑤/2), but it does not match the condition for the first derivatives to agree at the 𝑤/2
93
boundary. This discontinuity arises from the removal of the curvature in the 𝑉1 solution
that was due to the hyperbolic cosine term. This is the price that is paid for simplifying
equations (6.3) and (6.4) to (6.5) and (6.6) respectively, and we certainly expect it to affect
the accuracy of the threshold solution to some extent. Still, let us carry through the
projected dynamics to find the threshold solution using equations (6.5) and (6.6). We will
then compare the threshold solution we obtain to the numerical result as we have done
before, and, in doing so, we will get a qualitative assessment of the error that is introduced.
The Projected Dynamics and the Threshold Equation for the Simple Constant
Square Pulse
The procedure for obtaining the threshold equation is exactly the same as the procedure
described previously for a constant square pulse. Briefly, for the one-variable projected
dynamics, we need to solve:
− ∂𝐸/𝜕𝑎
,
𝑚11
(6.7)
𝑎̇ = − ∂𝐸/𝜕𝑎,
(6.8)
𝑎̇ =
or simply:
since we can neglect the 𝑚11 term when solving the projected dynamics for a constant
square pulse. The energy integral is of the same form as equation (4.14), that is:
𝑤
2
2
3
(6.9)
𝑑𝑉1
2𝑉1
𝐸 = ∫ (𝐷 (
) + 𝑔𝐾 𝑉1 2 − 𝑔𝑁𝑎
− 2𝐼𝑉1 ) 𝑑𝑥 +
𝑑𝑥
3
0
∞
𝑑𝑉2 2
2𝑉2 3
∫ (𝐷 (
) + 𝑔𝐾 𝑉2 2 − 𝑔𝑁𝑎
) 𝑑𝑥.
𝑑𝑥
3
𝑤/2
The derivatives needed for (6.9) are obtained by differentiating (6.5) and (6.6) to get:
94
𝜕𝑉1
= 0,
𝜕𝑥
(6.10)
𝜕𝑉2
= −𝑎𝛾𝑒 −𝛾(𝑥−𝑤⁄2) .
𝜕𝑥
(6.11)
With (6.5), (6.6), (6.10), and (6.11) in hand, the integral of equation (6.9) becomes:
𝐸=
𝑎(−18𝐼𝛾𝑤 − 2𝑎2 𝑔𝑁𝑎 (2 + 3𝛾𝑤) + 9𝑎𝑔𝐾 (2 + 𝛾𝑤))
,
18𝛾
(6.12)
and the time derivative for the amplitude is then:
𝑎̇ = − ∂𝐸/𝜕𝑎 =
3𝐼𝛾𝑤 + 𝑎2 𝑔𝑁𝑎 (2 + 3𝛾𝑤) − 3𝑎𝑔𝐾 (2 + 𝛾𝑤)
.
3𝛾
(6.13)
Setting 𝑎̇ = 0 and solving for 𝑎 gives us:
𝑎=
𝑔𝐾 3(2 + 𝛾𝑤) + ±√𝜇
,
2𝑔𝑁𝑎 (2 + 3𝛾𝑤)
(6.14)
where 𝜇 is:
𝜇 = −12𝐼𝑔𝑁𝑎 𝛾𝑤(2 + 3𝛾𝑤) + 9𝑔𝐾 2 (2 + 𝛾𝑤)2 .
(6.15)
The minimum criteria for excitation is met when:
𝜇 = 0.
(6.16)
Applying this condition and solving (6.15) for 𝐼 yields:
𝑔𝐾 2 3(2 + 𝛾𝑤)2
𝐼=
.
4𝑔𝑁𝑎 𝛾𝑤(2 + 3𝛾𝑤)
(6.17)
After dividing (6.17) by 𝑔𝐾 , we arrive at our expression for the critical amplitude 𝛼𝑐 :
𝛼𝑐 =
𝑔𝐾 3(2 + 𝛾𝑤)2
.
4𝑔𝑁𝑎 𝛾𝑤(2 + 3𝛾𝑤)
(6.18)
Figure 29 shows how equation (6.18) for the simple constant square pulse compares with
our previous result for the constant square pulse that was given in Figure 23.
95
Figure 29. A plot of the critical amplitude vs. pulse width for the simple constant applied
square pulse. The numerical and constant square pulse results from Figure 23 are provided
for comparison.
It is also instructive to check the behavior of equation (6.18) in the large and small pulse
width limits. At the large pulse width limit, where 𝑤 → ∞, equation (6.18) becomes
approximately:
𝛼𝑐 =
3𝑔𝐾 (𝛾𝑤)2
𝑔𝐾
=
,
12𝑔𝑁𝑎 (𝛾𝑤)2 4𝑔𝑁𝑎
(6.19)
and at the small width limit, where 𝑤 → 0, the leading term in the series expansion of
(6.18) gives:
96
𝛼𝑐 ~
3𝑔𝐾
.
2𝛾𝑔𝑁𝑎 𝑤
(6.20)
Both of these results are exactly the same as those obtained for the constant square pulse
earlier in equations (4.29) and (4.30). Taking that into consideration, the simple constant
square pulse does indeed appear to be an acceptable compromise between the Gaussian
and constant square pulse approaches. Equation (6.18) is considerably more compact than
equation (6.1), and, as Figure 29 reveals, the agreement between the two is quite good
(although, from Figure 29, it is also evident that equation (6.18) does not provide as
accurate a description of the numerical result as equation (6.1)). At the same time, in the
limit of large and small pulse widths, equation (6.18) gives precisely the same results as
the more complicated equation (6.1). This feature still stands as a marked improvement
over the Gaussian approach which is incorrect in both the large and small pulse width
limits.
The Threshold Equation for the Simple Time-Dependent Square Pulse
For the case of the time-dependent square pulse, we must return to equation (6.7) and
evaluate the 𝑚11 term. 𝑚11 was given previously as:
𝑤/2
𝑚11
∞
𝜕𝑉1 2
𝜕𝑉2 2
=∫ (
) 𝑑𝑥 + ∫ (
) 𝑑𝑥 .
𝜕𝑎
𝜕𝑎
0
(6.21)
𝑤/2
The derivatives with respect to 𝑎 are:
𝜕𝑉1
= 1,
𝜕𝑎
(6.22)
𝜕𝑉2
= 𝑒 −𝛾(𝑥−𝑤⁄2) ,
𝜕𝑎
(6.23)
which, when placed in (6.21) and integrated, yields:
97
1
.
𝛾
𝑚11 = 𝑤 +
(6.24)
Dividing (6.13) by (6.24) we have:
∂𝐸/𝜕𝑎 3𝐼𝛾𝑤 + 𝑎2 𝑔𝑁𝑎 (2 + 3𝛾𝑤) − 3𝑎𝑔𝐾 (2 + 𝛾𝑤)
𝑎̇ = −
=
,
𝑚11
3(1 + 𝛾𝑤)
(6.25)
which can be rewritten as:
𝑑𝑎
= −𝐴𝑎 + 𝐵𝑎2 + 𝐶𝐼,
𝑑𝑡
(6.26)
where:
𝐴 = 𝑔𝐾
𝐵 = 𝑔𝑁𝑎
𝐶=
(2 + 𝛾𝑤)
,
(1 + 𝛾𝑤)
(2 + 3𝛾𝑤)
(3(1 + 𝛾𝑤))
(6.27)
,
𝛾𝑤
.
(1 + 𝛾𝑤)
(6.28)
(6.29)
Equation (6.26) is then integrated as before:
𝐴/𝐵
∫
0
𝜏
𝑑𝑎
=
∫
𝑑𝑡,
−𝐴𝑎 + 𝐵𝑎2 + 𝐶𝐼
0
𝜏=
𝐴
4 tan−1 ( 𝑦 )
(6.30)
(6.31)
,
𝑦
𝑦𝜏
𝐴
)= ,
4
𝑦
𝑦𝜏
𝑦𝜏
tan ( ) ≅
,
4
4
4𝐴
𝑦2 =
,
𝜏
(6.32)
4𝑔𝑁𝑎 𝐼𝛾𝑤(2 + 3𝛾𝑤) 𝑔𝐾 2 (2 + 𝛾𝑤)2
√
𝑦=
−
.
(1 + 𝛾𝑤)2
3(1 + 𝛾𝑤)2
(6.35)
tan (
(6.33)
(6.34)
with 𝑦 given as:
98
Inserting (6.35) into (6.34), simplifying, and rearranging brings us to:
𝑔𝐾 2 3(2 + 𝛾𝑤)2
𝑔𝐾 3(1 + 𝛾𝑤)(2 + 𝛾𝑤)
𝐼=
+
,
4𝑔𝑁𝑎 𝛾𝑤(2 + 3𝛾𝑤)
𝜏𝑔𝑁𝑎 𝛾𝑤(2 + 3𝛾𝑤)
(6.36)
which can be divided by 𝑔𝐾 to give the desired expression for the critical amplitude as a
function of 𝜏:
𝛼𝑐 =
𝑔𝐾 3(2 + 𝛾𝑤)2
3(1 + 𝛾𝑤)(2 + 𝛾𝑤)
+
.
4𝑔𝑁𝑎 𝛾𝑤(2 + 3𝛾𝑤) 𝜏𝑔𝑁𝑎 𝛾𝑤(2 + 3𝛾𝑤)
(6.37)
Equation (6.37) is plotted in Figure 30 against the numerical threshold solution with 𝐷 =
1.0, 𝑔𝑁𝑎 = 1.0, and 𝑔𝐾 = 0.139. The case where the pulse width is 𝑤 = 100 cells and the
case where the pulse width is 𝑤 = 5 cells are given to facilitate comparison with the prior
treatment in Figure 28.
Figure 30. A plot of the critical amplitude vs. the time that the square pulse was applied
is provided for both a 100 cell pulse width and a 5 cell pulse width. Here, the simple
square pulse approach is compared with the numerical result.
99
An additional discrepancy between theory and the numerical solution once again shows up
for the simple square pulse in Figure 30 as compared to the square pulse that was used to
produce Figure 28, but the qualitative agreement still seems to be quite good for the simple
square pulse.
Finally, we have to check equation (6.37) in the large pulse width and large pulse time
limits to see if the theory reduces properly. As 𝑤 → ∞, the surviving terms contain the
product 3(𝛾𝑤)2, and the equation for the critical amplitude reduces to:
𝛼𝑐 =
𝑔𝐾 3(𝛾𝑤)2
1
3(𝛾𝑤)2
𝑔𝐾
1
(
)
+
(
)=
+
,
2
2
4𝑔𝑁𝑎 3(𝛾𝑤)
𝜏𝑔𝑁𝑎 3(𝛾𝑤)
4𝑔𝑁𝑎 𝜏𝑔𝑁𝑎
(6.38)
while in the limit that 𝜏 → ∞, equation (6.37) reduces to:
𝛼𝑐 =
𝑔𝐾 3(2 + 𝛾𝑤)2
,
4𝑔𝑁𝑎 𝛾𝑤(2 + 3𝛾𝑤)
(6.39)
which is in complete agreement with equation (6.18) for the simple constant square pulse
as expected. In comparing the general equation (6.37) for the simple square pulse with the
general equation for the square pulse derived previously in equation (5.39):
(1 + 𝑒 𝛾𝑤 (𝛾𝑤 − 1))
𝑔𝐾
𝛼𝑐 =
(
)+
𝑔𝑁𝑎 (8 + 𝑒 −𝛾𝑤 + 6𝛾𝑤 + 𝑒 𝛾𝑤 (4𝛾𝑤 − 9))
(6.40)
2(3 + 𝛾𝑤 + 𝑒 𝛾𝑤 (2𝛾𝑤 − 3))
1
(
),
𝜏𝑔𝑁𝑎 (8 + 𝑒 −𝛾𝑤 + 6𝛾𝑤 + 𝑒 𝛾𝑤 (4𝛾𝑤 − 9))
we see immediately that the simple square pulse completely removes all of the exponential
terms that were present in the prior result. In place of the exponential terms in (6.40), the
leading terms in (6.37), describing the dependence of the critical amplitude on the pulse
width, are now quadratic in 𝛾𝑤. This difference has its root in our choice to replace the
hyperbolic cosine term in equation (6.3) with a constant term for 𝑉1 in equation (6.5) at the
100
beginning of this section. The final result in equation (6.37) is somewhat less complicated
than equation (6.40), and it indeed appears to capture the important qualitative features of
the latter equation as desired.
101
CONCLUSION
From the prior considerations, we have emerged with a methodology that can generate an
approximate analytical threshold solution for a one-dimensional Fitzhugh-Nagumo system.
The method can be modified by choosing a different pulse shape as we did in going from
a Gaussian pulse shape to a square pulse shape; the method can be modified by choosing
to initiate the pulse via initial conditions on the potential or by applying an external current
where the threshold criteria is different for each case; and finally, the method can be
modified to account for the amount of time that the external current is applied.
What has been done to this point represents only the first few steps that need to be taken
towards the ultimate goal of producing a threshold equation that can accurately model
cardiac systems. The next step involves generalizing the method to two- and threedimensions. After that, the source terms in the Fitzhugh-Nagumo equation will have to be
replaced with more physiologically accurate expressions that better represent those found
in real cardiac systems. In particular, the dependence on calcium will have to be introduced
into the model at some point as calcium plays a crucial role in the onset of cardiac action
potentials and the associated arrhythmias [15].
Nevertheless, the threshold solution that we have obtained at present offers an accurate
analyzable solution to the question of whether or not an action potential will fire and
propagate as a wave along a one-dimensional chain of cells in the Fitzhugh-Nagumo
system. That is, given the initial size of the pulse width, the time that the pulse is applied,
and the values of the various conduction parameters relevant to the problem, we can predict
with considerable accuracy whether or not an excitation wave will appear when we observe
the numerical solution to the full partial differential equation in the Fitzhugh-Nagumo
102
system. Although the Fitzhugh-Nagumo system is indeed much simpler than a real
physiological system, these results at least provide us with the hope that an analytical
solution can be found which describes the threshold behavior of the more complicated
systems found in nature.
103
REFERENCES
1. Hodgkin AL and Huxley AF. A quantitative description of membrane current and its
application to excitation and conduction in nerve. J Physiol 117: 500-544, 1952.
2. Fitzhugh R. Impulses and physiological states in theoretical models of the nerve
membrane. Biophysical Journal 1: 445-466, 1961.
3. Nagumo J, Arimoto S, and Yoshizawa S. An active pulse transmission line
simulating nerve axon. Proceedings of the IRE 50: 2061-2070, 1962.
4. McKean HP and Moll V. Stabilization to the standing wave in a simple caricature of
the nerve equation. Comm Pure and Appl Math 39: 485-529, 1986.
5. Neu JC, Preissig Jr. RS, and Krassowska, W. Initiation of propagation in a onedimensional excitable medium. Physica D 102: 285-299, 1997.
6. Ermentrout GB and Terman DH. Mathematical Foundations of Neuroscience.
Springer, 2010.
7. Keener J and Sneyd J. Mathematical Physiology I: Cellular Physiology. Second
Edition. Springer, 2009.
8. Lapicque L, Brunel N, and van Rossum MCW, Trans. Quantitative investigations
of electrical nerve excitation treated as polarization. Biol Cybern 2007.
9. Fitzhugh R. Thresholds and Plateaus in the Hodgkin-Huxley Nerve Equations. J Gen
Physiol 43: 867-896, 1960.
10. Moll V and Rosencrans SI. Calculation of the Threshold Surface for Nerve
Equations. SIAM J Appl Math 50: 1419-1441, 1990.
11. Moll V. Polygonal Approximation to the flow on the critical surface for the bistable
equation. Computers Math Appl 25: 45-51, 1993.
104
12. Polking J. dfield and pplane: the java versions. math.rice.edu/~dfield/dfpp.html,
2002.
13. Boyce WE and DiPrima RC. Elementary Differential Equations. Sixth Edition. John
Wiley and Sons Inc., 1997.
14. Strang G. Mathematical Methods for Engineers II: Heat Equation.
http://www.cosmolearning.com/video-lectures/heat-equation/, Cosmo Learning, 2013.
15. Kleber AG and Rudy Y. Basic Mechanisms of Cardiac Impulse Propagation and
Associated Arrhythmias. Physiol Rev 84: 431-488, 2004.
105
APPENDIX: NUMERICAL METHODS
The method of projected dynamics produces an approximate analytical solution for the
threshold of excitation that must be compared with the threshold solution obtained by
directly solving the appropriate partial differential equation. This direct solution is
generated using numerical methods, and here we use a finite difference approximation to
solve the following partial differential equation:
𝜕𝑣
∂2 𝑣
= 𝐷 2 − 𝑔𝑓 ′ (𝑣),
𝜕𝑡
∂𝑥
𝑓 ′ (𝑣) = 𝑣(𝛼 − 𝑣)
𝑓𝑜𝑟 0 < 𝛼 <
(A.1)
1
, 𝛼 ≪ 1.
2
(A.2)
The simplest finite difference approach uses a forward difference approximation for 𝜕𝑣⁄𝜕𝑡
and a centered difference approximation for 𝜕 2 𝑣/𝜕𝑥 2 [14]. Applying these approximations
to (A.1), we get:
𝑣𝑗,𝑛+1 − 𝑣𝑗,𝑛
∆𝑡
∆𝑡 𝑣
∆𝑥 2 𝑣
= 𝐷
− 𝑔𝛼𝑣 + 𝑔𝑣 2 ,
∆𝑡
∆𝑥 2
𝑣𝑗+1,𝑛 − 2𝑣𝑗,𝑛 + 𝑣𝑗−1,𝑛
2
= 𝐷
− 𝑔𝛼𝑣𝑗,𝑛 + 𝑔𝑣𝑗,𝑛
.
∆𝑥 2
(A.3)
(A.4)
In (A.4), 𝑛 indexes the time, and 𝑗 indexes the position coordinate which we usually refer
to as the “cell number” to maintain some connection to the biology. (A.4) can be rearranged
to:
𝑣𝑗,𝑛+1 = 𝑣𝑗,𝑛 + ∆𝑡 (𝐷
𝑣𝑗+1,𝑛 − 2𝑣𝑗,𝑛 + 𝑣𝑗−1,𝑛
2
− 𝑔𝛼𝑣𝑗,𝑛 + 𝑔𝑣𝑗,𝑛
).
∆𝑥 2
(A.5)
The crucial ratio in (A.5) is ∆𝑡/∆𝑥 2 which must be kept small. Typical values used were
∆𝑡 = 0.01 , 𝑛 = 10,000, and ∆𝑥 = 1.0 for the majority of the numerical simulations. 𝑗
was usually indexed from 1 to 122, although we typically limited our focus to a 50 cell
106
chain length starting with a current pulse centered at 𝑗 = 50 and extending to 𝑗 = 109. The
extra cells were added to avoid end effects at the boundaries where we used the following
non-flux boundary conditions:
𝑣(1) = 𝑣(3),
(A.6)
𝑣(𝑗𝑚𝑎𝑥 ) = 𝑣(𝑗𝑚𝑎𝑥 − 2),
(A.7)
where 𝑗𝑚𝑎𝑥 is the highest indexed number in the range of 𝑗 (for example, in this case 𝑗𝑚𝑎𝑥 is
122).
We now turn our attention to the different approaches that were used for the applied current
pulse. For the Gaussian Pulse, we used the following voltage profile as the initial condition
of the potential:
2
𝑣0 (𝑘) = 𝑎0 𝑒 −𝑏(𝐿𝑘−50) .
(A.8)
In (A.8), 𝑎0 is the initial amplitude of the pulse, and 𝑏 is related to the initial pulse width
𝑤 as:
𝑏=
1
.
2𝑤 2
(A.9)
The index 𝑘 refers to the cell number when the constant 𝐿 is set to 1, and the 50 indicates
that the current pulse has been centered on cell 50.
The approach for the square pulse specified using the initial conditions is much the same
except the initial voltage profile is replaced with a constant initial amplitude 𝑎0 spread over
a specified pulse width 𝑤 that is chosen to be within the 50 cell range that we are observing:
𝑣0 (𝑗) = 𝑎0 (50: 49 + 𝑤),
𝑣0 (𝑗) = 0
(𝑓𝑜𝑟 𝑎𝑛𝑦 𝑗 ≠ 50: 49 + 𝑤).
107
(A.10)
(A.11)
In both of the above cases, (A.8), (A.10), and (A.11) specify the initial conditions on the
potential for a system in the form of equation (A.4). When the approach is changed to
applying a constant square pulse, the governing equation changes to:
𝑉𝑗,𝑛+1 = 𝑉𝑗,𝑛 + ∆𝑡 (𝐷
𝑉𝑗+1,𝑛 − 2𝑉𝑗,𝑛 + 𝑉𝑗−1,𝑛
2
− 𝑔𝐾 𝑉𝑗,𝑛 + 𝑔𝑁𝑎 𝑉𝑗,𝑛
+ 𝐼𝑗 ).
∆𝑥 2
(A.12)
The constant square pulse 𝐼 is given by:
𝐼(𝑗) = 𝐼(50: 49 + 𝑤),
𝐼(𝑗) = 0
(A.13)
(𝑓𝑜𝑟 𝑎𝑛𝑦 𝑗 ≠ 50: 49 + 𝑤).
(A.14)
Unlike in the prior cases, equations (A.13) and (A.14) are not specified as initial conditions
here. Rather the applied pulse is given as a constant that always remains present in the
differential equation even as the system evolves in time. This is to simulate an external
electrode providing a continuously applied current. For the time-dependent square pulse,
𝐼(𝑡), we simply add a time condition to equations (A.13) and (A.14):
(𝑓𝑜𝑟 𝑡 ≤ 𝜏),
(A.15)
𝑓𝑜𝑟 𝑎𝑛𝑦 (𝑗 ≠ 50: 49 + 𝑤)𝑤ℎ𝑒𝑛 𝑡 ≤ 𝜏,
).
𝑎𝑛𝑑 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗 𝑤ℎ𝑒𝑛 𝑡 > 𝜏
(A.16)
𝐼(𝑗) = 𝐼(50: 49 + 𝑤)
𝐼(𝑗) = 0
(
Here 𝜏 represents the time at which the applied square current pulse is switched off.
108