Modelling Neuronal Excitation: The
Hodgkin-Huxley Model
Thierry Mondeel
July 13, 2012
Contents
1 A short introduction to the biology of neurons
4
1.1 Neurons
5
1.2 The cell membrane and ion channels
6
1.2.1 The electrochemical gradient: Why does charge move? 7
1.2.2 The key ions: potassium and sodium . . . . . . . . . .
8
1.3 The action potential
10
1.3.1 The underlying currents . . . . . . . . . . . . . . . . . 10
2 Preliminary work: Membranes, channels and gradients
2.1 Single-compartment models
2.2 Capacitance and resistance
2.3 The membrane potential
2.4 The Nernst potential
2.4.1 The sodium and potassium equilibrium potentials
2.4.2 Ion pumps . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 The Goldman equation . . . . . . . . . . . . . . .
2.5 The form of Iion
2.6 The Hodgkin-Huxley gate model
2.6.1 Persistent conductance . . . . . . . . . . . . . . . .
2.6.2 Transient conductance . . . . . . . . . . . . . . . .
. .
. .
. .
. .
. .
12
12
13
13
14
14
14
15
15
16
17
18
3 The Hodgkin-Huxley model
20
3.1 The work of Hodgkin and Huxley in perspective
20
3.2 Combining the preliminary work
21
3.2.1 The sodium and potassium conductances . . . . . . . 22
3.2.2 Summary of the equations . . . . . . . . . . . . . . . . 23
3.3 The dynamics
24
4 The
4.1
4.2
4.3
4.4
Fitzhugh-Nagumo model
History and perspective
The fast-slow simplification of the HH model
The fast-slow phase plane
The FitzHugh-Nagumo equations
1
26
26
27
28
29
0.0
2
5 Summary
5.1 Review
5.2 Futher studies
34
34
35
6 Sources
37
Preface
This is a bachelor thesis about mathematical models that describe the development and propagation of action potentials. At the core of this subject
is a model developed in the 1950s and named after the two scientists behind
it: Alan Lloyd Hodgkin and Andrew Huxley. The content and usefulness
of this model and others derived from it will hopefully become clear to the
reader of this thesis.
I have been especially drawn to this subject because of its inherent interdisciplinary nature, combining mathematics, physics and biology to create
qualitative and quantitative models of one of the most fundamental processes in higher organisms: the spreading of electrical signals across neuronal axons and dendrites. Because of this interdisciplinary nature studying
the models and their derivation will require many concepts from biology and
physics to motivate the resulting mathematics. In fact, it might be said that
most of the effort will be in motivating the mathematics instead of in the
mathematics itself. I have tried to make these non-mathematical ideas clear
wherever they are used. Therefore the following should be readable without
any upfront knowledge of cell physiology, electricity or other knowledge not
generally known to students of mathematics.
My hope is that this thesis is as interesting to read as it was to work on,
both for my thesis advisor(s) and any other reader who might someday, for
whatever reason, decide to read it.
Finally, I would like to thank my thesis advisor dr. Robert Planqué for
tasking me with finding my own topic instead of handing me (a much less
interesting) one. The process of figuring out which subjects I’m most excited
about has shifted my ideas about what topics to study after my bachelor’s
degree more than anything else.
3
Chapter 1
A short introduction to the biology of
neurons
The work of Hodgkin and Huxley is widely recognized as one of the most
outstanding achievements in modern science; indeed they were awarded the
Nobel prize for Physiology and Medicine in 1961. The bulk of this achievement is concentrated in 1952, the year they published four papers (a,b,c and
d respectively). Of these the last is especially notable and its content will
be the main focal point of this thesis. To start to make sense of this work
we will have to start with the basic concepts of cell biology.
We will be exclusively concerned with neurons in what follows, however
it is wise to start at the most fundamental level of this discussion: the
cell. What is a cell? A cell is the smallest structural unit of an organism
that is capable of independent functioning. Or put differently: a cell is the
basic unit of life. The human body is composed of trillions of cells, each a
microscopic compartment, and these cells are virtually identical to the cells
in other animals: elephants, mice etc.
The word cell means “small chamber”, which is quite apt. The usual
range for cell diameters in the human body is 10 − 20 µm, which is indeed
quite small. What makes cells compartments is their cell membrane.
This barrier serves several purposes but most importantly it separates the
intracellular fluid from the extracellular fluid and can be very selective
about what comes out or goes into the cell. As figure 1.1 points out, the cell
contains many more fascinating components. These are however not very
relevant for the rest of our story so we will not go into further detail about
their function.
The major chemical substances in the extracellular fluid are sodium and
chloride ions, whereas the intracellular fluid contains high concentrations of
potassium ions and ionized molecules, particularly proteins that are usually
unable to cross the membrane.
4
1.1 Neurons
5
Figure 1.1: Structures found in most human cells. From: Vander’s Human
Physiology.
1.1 Neurons
Neurons are a specialized kind of cell. Specifically, they are specialized
to transmit electrical signals to other neurons. There are three functional
parts of the neuron that we have to understand: the cell body, dendrites and
the axon. The cell body is the command center of the neuron. All inputs
are gathered here and processed. Many extensions can sprout from the
cell body; these are generally called processes of which there are two kinds:
dendrites and axons. Dendrites gather the input from other neurons while
the axon (each neuron has only one) sends signals to other neurons.
An electrical signal that carries a command from the brain to, for instance, the hand travels along a sequence of neurons that can be visualized
as in figure 1.2. When the signal arrives at the dendrites on the left side of
the neuron, the stimuli given the dendrites are integrated at the cell body,
which can then generate a nerve impulse. The nerve impulse travels along
the axon to the branches of axons on the right side of the neuron. The impulse then ”jumps” to another set of dendrites and the process just described
is repeated.
This jumping of the electrical signal is also interesting, a lot more goes
on there than just jumping. Neurons that are connected to each other are
not actually touching; there is a space between them. We call this space
the synaptic cleft, and the whole junction between the two neurons the
synapse. At these synapses a release and uptake of ions takes place, that
1.2 The cell membrane and ion channels
6
Figure 1.2: A schematic drawing of a sequence of neurons. From: Vander’s
Human Physiology 8th ed.
results through a process that will be discussed below, in the propagation
of the electrical signal.
1.2 The cell membrane and ion channels
Like all other cells, neurons are filled with a large number and variety of
ions and molecules and collectively we call this medium the cytosol. Most
of these molecules carry positive or negative charges often resulting in an
excess concentration of negative charge inside a neuron.
The cell membrane consists of a double layer of fat molecules, appropriately called the lipid bilayer, of about 3 to 4 nm thick which is essentially impermeable to charged molecules. This impermeability causes the
cell membrane to act as a capacitor (it is able to store charge) by separating the charge lying along its interior and exterior. Many ion-conducting
channels or pores are embedded in the cell membrane allowing ions to move
in and out of the cell; see figure 1.4. These ions carry charge and therefore
ion channels may induce current flows into or out of the cell. Ion channels
can be selective, i.e. allowing only a single ion species to pass through them
and the amount of ions that flows through them can be dependent upon
several different things. For instance the membrane potential (voltage dependence), the shape of the membrane and the binding of certain ions or
1.2 The cell membrane and ion channels
7
Figure 1.3: The charges carried by molecules and ions are separated by the
cell membrane from the charges in the extracellular fluid. From: Vander’s
Human Physiology, 8th ed.
molecules like neurotransmitters.
By convention we define the potential of the extracellular fluid to be
zero. When the neuron is at rest the excess internal negative charge of the
neuron causes the potential inside the cell membrane to be negative with
respect to the outside. We call this inner relative to outside potential the
membrane potential; it is an equilibrium point at which the flow of ions
into the cell matches the flow of ions out of the cell. The potential can
change if the balance of ion flow is disturbed by the opening or closing of
ion channels. Normally, neuronal membrane potentials vary over a range
from about −90 to +50 mV.
1.2.1 The electrochemical gradient: Why does charge move?
What we have not yet addressed is why charges move to the membrane
and through ion channels. The reason for this is two-fold: electric forces
and diffusion. These two processes are responsible for driving ions through
channels and together are called the electrochemical gradient. Voltage
differences between the exterior and interior of the cell produce forces on
ions, negative membrane potentials attract positive ions into the neuron
and repel negative ions and vice versa. Also, ions diffuse through channels
because the ion concentrations differ inside and outside the neuron. Diffusion is basically a random walk followed by ions and molecules because
of their thermal energy. Because of mass action, substances tend to diffuse
from regions of high concentration to regions of low concentration. This can
1.2 The cell membrane and ion channels
8
Figure 1.4: (left) A schematic diagram of the lipid bilayer that forms the
cell membrane with two ion channels embedded within it. From: Dayan &
Abbott. (right) Measuring the membrane potential with a voltmeter, inside
with respect to outside potential. From: Vander’s Human Physiology 8th
ed.
be visualized as in figure 1.5.
Figure 1.5: If there are more ions in compartment C1 than in compartment
C2 the random walks of the ions will cause a net flux from C1 to C2. From:
Vander’s Human Physiology 8th ed.
1.2.2 The key ions: potassium and sodium
One of the key things that Hodgkin and Huxley were able to show is
that the electrical properties of neurons can be successfully described by
considering only two ion species, specifically: sodium (Na+ ) and potassium
(K + ). The concentrations of these two ionic species are summarized in
figure 1.6.
The concentration of Na+ is higher outside the cell than inside, so these
ions are driven into the neuron by diffusion. Also, since the inside of the cell
1.2 The cell membrane and ion channels
Potassium
Sodium
Intracellular
397
50
9
Extracellular
20
437
Figure 1.6: Extracellular and Intracellular Ion Concentrations in mM/L for
the squid axon.
Figure 1.7: The membrane potential is closer to the resting potential with
increasing distance from the depolarization site. From: Vander’s Human
Physiology, 8th ed.
is negatively charged relative to the outside, the sodium ions are pulled into
the cell by electrical forces. On the other hand, K + is more concentrated
inside the neuron than outside, so it tends to diffuse out of the cell. However,
it is also pulled into the cell as a result of the negative membrane potential.
What is most relevant about this for us is that these ion flows generate
changes in the membrane potential. These changes in the membrane potential from its resting level produce electric signals, which basically come in
two flavours: graded potentials and action potentials.
Graded potentials are changes in membrane potential that are confined to a relatively small region of the cell membrane and die out within a
few millimetres of their origin. They are usually produced by a change in
the cell’s environment acting on the membrane, and they are called “graded
potentials because the magnitude of the potential change can vary. Whenever a graded potential occurs, charge flows between the place of origin of
the potential and adjacent regions of the plasma membrane, which are still
at the resting potential; see figure 1.7.
Because the electric signal decreases with distance, graded potentials can
function as signals only over very short distances. So how then do neurons
transmit signals over larger distances?
1.3 The action potential
10
1.3 The action potential
An action potential is a specific pattern of change in the membrane
potential very different from a graded potential. Action potentials are rapid,
all-encompassing large alterations to the membrane potential during which
time the potential may depolarize 100 mV, from −70 to +30 mV, and then
repolarize to its resting membrane potential. Action potentials also have
so-called refractory periods, which we will return to later. Nerve and
muscle cells as well as some other kinds of cells, have cell membranes capable of producing action potentials. These membranes are called excitable
membranes, and their ability to generate action potentials is known as excitability. Whereas all cells are capable of conducting graded potentials,
only excitable membranes can conduct action potentials. The propagation
of action potentials is the mechanism used by the nervous system to communicate over long distances making it of fundamental importance to the
functioning of the organism.
1.3.1 The underlying currents
The magnitude of the resting membrane potential depends upon the
concentration gradients and membrane permeabilities of different ions, particularly sodium and potassium. This is true for the action potential as
well! It results from a transient change in membrane ion permeabilities,
which allows certain ions to move down their electrochemical gradients.
In the resting state, the open channels in the cell membrane are predominantly permeable to potassium ions. Very few sodium ion channels
are open, which results in a membrane potential close to 70mV. During
an action potential, however, the membrane permeabilities for sodium and
potassium ions are strongly altered. See figure 1.8.
Figure 1.8: (left) The excursion of the membrane potential that is the action
potential. (right) The ionic currents of sodium and potassium that underlie
the action potential.
1.3 The action potential
11
The depolarizing phase (upward movement towards zero) of the action
potential is due to the opening of voltage-gated sodium channels, which
increases the membrane permeability to sodium ions. Therefore, more positive charge enters the cell in the form of sodium ions than leaves in the form
of potassium ions, and the membrane depolarizes. It even overshoots, becoming positive on the inside and negative on the outside of the membrane.
In this phase (the top of figure 1.8), the membrane potential approaches but
does not reach the sodium equilibrium potential.
After reaching this peak the potential very quickly repolarizes back to
the resting potential. Two processes are at work during this phase of the
action potential. First, the sodium channels that opened during the depolarization phase are inactivated near the peak of the action potential,
which effectively causes them to close. Secondly, voltage-gated potassium
channels, opening more slowly than sodium channels, now open in response
to the depolarization. Thus potassium flows out of the cell, contributing
to the inside of the cell becoming more negatively charged relative to the
outside of the cell.
Chapter 2
Preliminary work: Membranes, channels and gradients
This chapter will introduce the biological terms mentioned in the previous
chapter in a more exact and mathematical way. It will introduce the relations and formulas we will need to derive the Hodgkin-Huxley model (from
now on we will refer to this model as the HH model). Remember that our
goal is to arrive at a mathematical model based on sound biology that has
the desired behaviour, i.e. produces action potentials. To that end we will
have to consider several subjects: the membrane potential, the Nernst potential, the ionic conductance and the opening and closing of ion channels,
and try to come up with mathematical descriptions for these processes.
2.1 Single-compartment models
Models that describe the membrane potential of a neuron by a single
variable V are called single-compartment models. Only such models
will be considered in this thesis. However, it is interesting to note why
one would be interested in more complex or so called multi-compartment
models. Membrane potentials measured at different places within a neuron
can take different values. For example, the potentials in the cell body,
dendrite, and axon can all be different. This causes ions to flow within the
cell, which will tend to equalize these differences. The cytosol (intracellular
fluid) provides resistance to such flow. This resistance is highest for long
and narrow stretches of a dendrite or axon. Therefore a single-compartment
model, i.e. models where we assume the membrane potential to be the same
in the entire neuron, is realistic where there are few of such long and thin
stretches of dendrite or axon or where the spatial effects are considered to
be irrelevant. As stated before, we will consider only single-compartment
models in this thesis.
12
2.3 The membrane potential
13
2.2 Capacitance and resistance
The product of the membrane capacitance and the membrane resistance is a quantity with units of time called the membrane time constant,
τm = Rm Cm . Because Cm and Rm have inverse dependences on the membrane surface area, the membrane time constant is independent of area.
The membrane time constant sets the basic time scale for changes in the
membrane potential and typically falls in the neighbourhood of 50 ms.
2.3 The membrane potential
Capacitance is defined as charge held divided by the voltage needed to
hold that charge
Q
Cm = .
V
This capacitance Cm is assumed to be constant, i.e. not dependent on
time. Therefore by taking the time derivative we can derive the following
formula for the flux of charge across the membrane
dQ
dV
= Cm
.
dt
dt
(2.1)
This formula is fundamental for all work to be done in this subject, it
determines the membrane potential for a single-compartment model. What
this equation really says is that the rate of change of the membrane potential
is proportional to the rate at which the cell builds up charge. With it we can
derive another fundamental result by considering what processes contribute
to the charge build-up. Charge can only enter or leave the cell through
ion-channels or through an applied current by an experimenter. Thus we
have
dQ
= −Iion (V, t) + Iapp
dt
dV
Cm
= −Iion (V, t) + Iapp .
dt
(2.2)
(2.3)
Here the signs on the right are dictated by convention; positive charge
flowing into the cell is defined as positive.
When there is no applied current we have the following important relationship
dV
Cm
+ Iion (V, t) = 0.
(2.4)
dt
This equation forms the basis for modelling the membrane potential.
Really, all that remains for the derivation of the HH model is to be more
specific about Iion , which is what we will do next.
2.4 The Nernst potential
14
2.4 The Nernst potential
The electrical potential generated across the membrane at electrochemical equilibrium, the equilibrium potential, can be predicted by a simple
formula called the Nernst equation. The Nernst equation gives a formula
for one ion species that relates the numerical values of the concentration
gradient to the electrical gradient that balances it.
For an ion X with given outside concentration Xo and inside concentration Xi the Nernst potential VX is given by
VX =
RT
[X]o
).
ln(
zF
[X]i
(2.5)
Here R is the universal gas constant, T is the temperature in Kelvin, F
is Faraday’s constant and Z is the valence of the ion in question. Since three
of these parameters are environmental parameters one often comes across
different formulas where these constants have been represented by their value
in the problem that is being discussed. For instance at 27 degrees Celsius
RT
F = 25.8.
2.4.1 The sodium and potassium equilibrium potentials
Since we will be relying on sodium and potassium ions to do most of
the work in what follows, it might be instructive to use the just derived
Nernst equation to calculate the sodium and potassium equilibrium potentials. Taking the extracellular and intracellular concentrations for sodium
mentioned in table 1.6, we find:
25.8
437
ln(
) = 56 mV.
1
50
Similarly for potassium
25.8
397
ln(
) = −77 mV.
1
20
2.4.2 Ion pumps
Now, the attentive reader will have noticed that the experimentally measured resting membrane potential is not equal to either equilibrium potentials. The resting membrane potential is close but not equal to the potassium
equilibrium potential, because a small number of sodium channels are open
in the resting state, and some sodium ions continually move into the cell,
cancelling the effect of an equivalent number of potassium ions simultaneously moving out. Thus, there is net movement through ion channels of
sodium into the cell and potassium out. If the resting membrane potential
is to be maintained this cannot be all that is going on, and indeed it is not.
2.5 The form of Iion
15
Active-transport mechanisms in the plasma membrane utilize energy derived from cellular metabolism to pump the sodium back out of the cell and
the potassium back in. Actually, the pumping of these ions is linked because they are both transported by the same pump, appropriately named the
Na,K-ATPase pump. ATPase because it gets its energy from the molecule
ATP.
2.4.3 The Goldman equation
As a short side note, it is interesting to consider that we have used the
Nernst potential in the story above. However, the Nernst potential only
considers a membrane permeable to one ion species, which in reality is of
course not the case. So what happens if multiple ionic species are able to
diffuse across the membrane? In this case a more complex formula is available: the Goldman equation. In the Goldman equation the equilibrium
potential is not determined by equation 2.5, but takes a value between the
equilibrium potentials of the individual ion species that it conducts. Suppose wish to know the membrane potential for a membrane permeable to
sodium, potassium and chloride. The Goldman equation would look like
this
RT
PK [K]o + PNa [Na]o + PCl [Cl]i
Vrest =
ln
,
F
PK [K]i + PNa [Na]i + PCl [Cl]o
where the Pion stand for the permeabilities of the different ion species. The
Goldman equation is “Nernst-like” but has a term for each permeable ion
and is thus an extended version of the Nernst equation that takes into account the relative permeabilities of each of the ions involved. The relationship between the two equations becomes obvious in the situation where the
membrane is permeable only to one ion, say, K + . In this case, the Goldman
expression collapses back to the Nernst equation.
We will however, for the HH model, not use this description of the equilibrium potential, but use the simpler Nernst potential instead.
2.5 The form of Iion
As discussed in the previous section on the Nernst potential, the current
carried by a certain type of ion channel with equilibrium potential Vion ,
vanishes when the membrane potential satisfies V = Vion . For many types
of channels the current changes approximately linearly when the membrane
potential deviates from this equilibrium value. The difference V − Vion is
called the driving force for that ion and the membrane current per unit
area due this ion species is written as:
Iion = gion (V − Vion ),
(2.6)
2.6 The Hodgkin-Huxley gate model
16
Figure 2.1: Movements of sodium and potassium ions across the cell membrane of a resting neuron. The diffusion movements (black arrows) are
exactly balanced by the active transport (red arrows) of the ions in the
opposite direction. From: Vander’s Human Physiology, 8th ed.
where the term gion is the conductance per unit area of the membrane due
to the ion channels for this ion species. We call the relationship in 2.6
a Linear I-V curve. It is important to realize that we have made an
assumption here, the linearity of the I-V curve. For some neurons this is
not a realistic assumption; their ion currents are better described by other
relationships.
2.6 The Hodgkin-Huxley gate model
Now, our final point of interest will be how to describe conductances
of specific ions across the membrane. Again, we are mainly interested in
sodium and potassium and these appear to have two very specific types of
conductance. Potassium has a persistent conductance while sodium has a
transient conductance; see figure 2.2.
2.6.1 Persistent conductance
The opening of the persistent conductance gate (we now know) involves
a number of conformational changes (this was unknown at the time of
Hodgkin and Huxley). For example, the K + channel consists of four identical subunits, and it appears that all four must undergo a structural change
for the channel to open. When k identical but independent events are needed
2.6 The Hodgkin-Huxley gate model
17
Figure 2.2: (A) An example of a persistent conductance gate. The gate
is opened and closed by a kind of sensor that responds to the membrane
potential. (B) An example of a transient conductance gate, the activation
gate is coupled to a voltage sensor that acts like the gate in A. A second
gate, denoted by the ball, can block that channel once it is open. The top
figure shows the channel in a deactivated state. The middle panel shows an
activated ion conducting channel, and the bottom panel shows the channel
in its inactivated state.
for a channel to open, and one of these events happens with probability n,
we can write the conductance as
gion = nk .
(2.7)
If large numbers of channels are present, and if they act independently of
each other (which experimental evidence suggests they do to a good approximation), then, the fraction of channels open at any given time is approximately equal to the probability that any one channel is in an open state.
This is an application of the law of large numbers.
Suppose a percentage n of subunit gates is in the open state at a given
moment in time. Thus the probability of one specific subunit gate to be
open is also n and the probability of it being closed is 1 − n. We can
describe the transition of each subunit gate by a simple scheme in which
the gating transition from closed to open occurs at a voltage-dependent
rate αn (V ), and the reverse transition open to closed occurs at a voltagedependent rate βn (V ). The probability that a subunit gate opens over a
2.6 The Hodgkin-Huxley gate model
18
short interval of time is proportional to the probability of finding the gate
closed, 1−n, multiplied by the opening rate αn (V ). Likewise, the probability
that a subunit gate closes during a short time interval is proportional to the
probability of finding the gate open, n, multiplied by the closing rate βn (V ).
The rate at which the open probability for a subunit gate changes is given
by the difference of these two terms
dn
= αn (ν)(1 − n) − βn (ν)n.
dt
(2.8)
Now, we can mold this equation into another often used form by dividing
by αn + βn , resulting in
τn (ν)
dn
= n∞ − n,
dt
where
(2.9)
τn (V ) =
1
αn (V ) + βn (V )
(2.10)
n∞ (V ) =
αn (V )
.
αn (V ) + βn (V )
(2.11)
and
So what does this equation actually say? It says that for a fixed voltage
V , n approaches the value n∞ (V ) exponentially with time constant τn (V ).
The key elements in the equation for n are the opening and closing rate
functions αn (V ) and βn (V ). These are obtained by fitting experimental
data. The data upon which these fits are based are typically obtained using
a technique called voltage clamping; indeed this is what Hodgkin and Huxley
used in their experiments.
2.6.2 Transient conductance
Some channels only open transiently when the membrane potential is
depolarized because they are gated by two processes (subunits) with opposite
voltage dependences; see figure 2.2. One of the gates behaves exactly like
the gate of the persistent conductance. The probability that it is open is
written as mk where m is an activation variable like n, and k is an integer
that again signifies the number of subunits in the channel (or the number
of events that need to happen for it to open). The probability that the ball
(see figure 2.2) does not block the channel is denoted by h and called the
inactivation variable.
Now, for the transient conducting channel to be conducting both gates
must be open, and if the two gates act independently, the probability of this
occurring is mk h. Thus we can write the transient conductance of a certain
ion species as
gion = mk h.
(2.12)
2.6 The Hodgkin-Huxley gate model
19
More generally, if there is not one process blocking the channel but
several, we could describe this with the following formula for the current
flow due to a certain ion species, sometimes referred to as the HodgkinHuxley gating equation
I = gmα hβ (V − Vion ).
(2.13)
Chapter 3
The Hodgkin-Huxley model
Now that we have found some mathematical descriptions for the various
parts of and processes within neurons we are ready to consider the actual
model derived by Hodgkin and Huxley. In this thesis we are concerned
with the space-clamped dynamics of the system; that is, we consider the
spatially homogeneous dynamics of the membrane. With a real axon this
state can be obtained experimentally by having a wire down the middle of
the axon maintained at a fixed potential difference to the outside.
The spatial propagation of action potentials along the nerve axon through
the concept of travelling waves, is a very interesting and crucial subject
to study, but is not included in this thesis. For the interested reader we
refer to Keener & Sneyd - Mathematical Physiology volume I, chapter 6 or
to Murray’s Mathematical Biology Volume II, chapter 1.
Here, we will derive the Hodgkin-Huxley model and the reduced analytically tractable FitzHugh-Nagumo model (See FitzHugh 1961, Nagumo et
al, 1962) which captures the key phenomena of the HH model.
3.1 The work of Hodgkin and Huxley in perspective
Now that we have some basic notions under our belt we are in a better
position to understand the importance of the work of Hodgkin and Huxley.
In 1952, Hodgkin and Huxley wrote five papers (Hodgkin & Huxley a,
b, c, d and e) that described their experiments that aimed at determining
the laws that govern the movement of ions in a nerve cell during an action
potential.
The first paper examined the function of the neuron membrane under normal conditions and explained the basic experimental set-up used
in each of their subsequent studies. The second paper examined the effects
of changes in sodium concentration on the action potential as well as the
20
3.2 Combining the preliminary work
21
Figure 3.1: The inventors of the model, (left) Hodgkin and (right) Huxley.
partitioning of the ionic current into sodium and potassium currents. The
third paper examined the effects of potential changes on the action potential. The fourth paper outlined how the inactivation process reduces sodium
permeability.
Hodgkin and Huxley’s accomplishment is really two-fold. Using the
already existing (but at the time, relatively novel) technique of voltageclamping they were able to gather large amounts of experimental data from
the giant squid axon. This axon was used because of its size, making it
practical to do experiments on. Secondly they then in their final paper
supplied a mathematical model consisting of four non-linear ordinary differential equations, which they showed captured the gathered data quite well,
meaning that for certain parameter values action potential like excursions
of the variables can be observed.
The influence of the work of Hodgkin and Huxley can easily be seen
when opening any textbook on computational neuroscience or mathematical physiology since this subject will always be treated. However, when
opening a biologically oriented textbook this is somewhat harder. Since the
mathematics of the HH model is considered to complicated, there is often
little mention of Hodgkin and Huxley even though the results of their work
are visible in any chapter on action potentials, ion channels and membranes.
The final paper put together all the insights from the earlier papers and
revealed the mathematical model for which they received the 1962 Nobel
prize for Physiology and medicine. We will now focus on its derivation.
3.2 Combining the preliminary work
We saw in the previous chapter that the membrane potential for a singlecompartment model satisfies
Cm
dV
+ Iion (V, t) = 0,
dt
3.2 Combining the preliminary work
22
or, including an applied current
Cm
dV
= −Iion (V, t) + Iapp .
dt
Figure 3.2: The equivalent circuit for the HH model.
This equation can also be arrived at by considering the equivalent circuit
of the HH model; see figure 3.2. The cell membrane can be modelled as a
capacitor in parallel with an ionic current resulting in the formulas above.
We also saw the concept of linear I-V curves in the previous chapter.
There are numerous ions present in the fluids surrounding a cell, however
the HH model assumes they can be reduced to two ions and one extra
variable in which we lump together all others. This results in
dV
= −gN a (V − VN a ) − gK (V − VK ) − gL (V − VL ).
(3.1)
dt
The previous chapter also covered how to model the conductances in
terms of activating and inactivating mechanisms. This last step will result
in the actual HH model.
Cm
3.2.1 The sodium and potassium conductances
It is important to step back for a moment and revel at the magnitude of
what Hodgkin and Huxley did in the following step of the model’s derivation.
In the previous chapter we deduced equations for the conductance based on
the assumption of subunits that all have to undergo the same conformational
change. However, for Hodgkin and Huxley it was the other way around. This
interpretation was not available at the time; indeed, they came up with it.
To their measurements of sodium and potassium conductances they fitted
functions m, n and h, and in a stroke of genius gave them the interpretation
we now use.
3.2 Combining the preliminary work
23
The specific forms for the conductances they came up with are as follows.
For potassium they found
gK = ḡK n4
(3.2)
where g¯K is a constant. This formula is to be interpreted as a channel
consisting of four identical subunits. And for sodium they found
gN a = ḡN a m3 h
(3.3)
where gN
¯ a is a constant. This formula is to be interpreted as again consisting of four subunits, three of which are activating and one of which is
inactivating (think of the ball-like structure in figure 2.2).
3.2.2 Summary of the equations
Combining the results in this chapter we arrive at the full HodgkinHuxley model of 4 non-linear ordinary differential equations:
Cm
dV
= −ḡN a m3 h(V − VN a ) − ḡK n4 (V − VK ) − gL (V − VL ),
dt
(3.4)
where m, h and n are the gating variables that satisfy:
dm
= m∞ − m
dt
dh
τh (ν)
= h∞ − h
dt
dn
= n∞ − n
τn (ν)
dt
τm (ν)
Figure 3.3: (left) The steady-state functions and (right) the time constants
of the HH model. Note: τm is much smaller than the other two. From:
Keener & Sneyd, 2nd ed.
Remember that the parameters τm , m∞ etc. are defined in terms of αm ,
βm etc. and thus have to be estimated from measurements. For the HH
3.3 The dynamics
24
Figure 3.4: (left) The action potential and (right) the gating variables during
the action potential, both on the same time scale. From: Keener & Sneyd,
2nd edition.
model these functions are given in figure 3.3 and figure 3.4. The interested
reader can find the actual expressions and values of the parameters in, for
instance, Keener & Sneyd - Mathematical Physiology Volume I, 2nd ed. (p.
206).
3.3 The dynamics
In this section we will explain some features of the dynamics the HH
model without any proof since these dynamics are not easily investigated.
Because of the complexity of the system, various simpler mathematical models, which capture the key features of the full system, have been proposed,
the best known and most useful one of which is the FitzHugh-Nagumo model
(FitzHugh 1961, Nagumo et al. 1962), which we will derive in the next chapter.
If Iapp = 0, the model is linearly stable but excitable. That is, if the
perturbation from the steady state is large enough there is a large excursion
of the variables in their phase space before returning to the steady state. If
Iapp 6= 0 there is a range of values where regular repetitive firing occurs, that
is, the system exhibits limit cycles. These two points remind us of course
of the threshold potential and action potential we discussed in the first
chapter. Both types of phenomena have been observed experimentally and
this indicates that the HH model is very effective at describing what actually
happens during the action potential.
The behaviour of periodic orbits as Iapp is increased can be nicely summarized in a bifurcation diagram. For each value of Iapp we plot the value of
V at the resting state, and the maximum and minimum values of V over the
associated periodic orbit, see figure 3.5. The point here is just to notice that
the HH model does indeed produce solutions with the desired behaviour and
that oscillations are possible with increased applied current. Investigation of
3.3 The dynamics
25
Figure 3.5: Bifurcations of the HH model with the applied current Iapp as
the bifurcation parameter. HB denotes a Hopf bifurcation, SNP denotes a
saddle-node of periodics bifurcation, osc max and osc min denote, respectively, the maximum and minimum of an oscillation, and ss denotes a steady
state. Solid lines denote stable branches, dashed or dotted lines denote unstable branches From: Keener & Sneyd, 2nd ed.
the bifurcations would require more time than we have here and is therefore
excluded.
In the following chapter we will focus on explaining the dynamics of
simplified systems that will also still produce action potential behaviour even
though the biological context may have been lost. These simplified systems
allow us to more easily investigate the dynamics and we can therefore draw
conclusions from their analysis.
Chapter 4
The Fitzhugh-Nagumo model
4.1 History and perspective
In 1961, Fitzhugh sought to reduce the Hodgkin-Huxley model to a twovariable model for which phase plane analysis is applicable. His general
observation was that the gating variables n and h have slow kinetics relative
to m. Using this he came up with a two-dimensional system that has the
same qualitative features as the HH model. This model is not only named
after Fitzhugh but also after Nagumo, who in 1962 built the equivalent circuit for the above model (see figure 4.1). In the original papers of FitzHugh,
this model was called the Bonhoeffer-van der Pol oscillator because it
contains the Van Der Pol oscillator as a special case.
Figure 4.1: Circuit diagram of the tunnel-diode nerve model of Nagumo et
al. (1962)
26
4.2 The fast-slow simplification of the HH model
27
4.2 The fast-slow simplification of the HH model
In the previous chapter we did not include a very detailed description
of the dynamics of the HH model. The dynamics would be much easier to
investigate if the model could somehow be reduced to two dimensions. The
crucial observation that allows for this follows from the graphs of the gating
variables of the HH model and their time constants. For ease of reference
these have been included again (see figure 4.2).
Figure 4.2: (left) The gating variables and (right) their associated time
constants. From: Keener & Sneyd 2nd edition.
To go from four to two dimensions we will have to drop some information,
that is unavoidable, but we can try to make reasonable assumptions. The
clearest of which is that the time constant of m, τm , is small or rather fast
compared to the other two constants. Remember that τm signifies the speed
with which m converges on its steady state value m∞ . Our first assumption
could therefore be that that m is instantaneously at this value, i.e. m = m∞ .
This is called a quasi steady-state approximation.
The second reduction is a bit more obscure. Looking at the graph of the
gating variables a symmetry between the graphs of h and n can be seen.
Specifically the graph of 1 − h and n seem very similar. This can be used to
eliminate one of the variables. For instance by setting h = 1 − n. Combining
our two assumptions we come upon the simplified Hodgkin-Huxley model
(without an applied current) that contains one fast variable V and one slow
variable n
dV
= −ḡN a m3∞ (V )(1 − n)(V − VN a ) − ḡK n4 (V − VK ) − gL (V − VL )
dt
(4.1)
dn
τn (ν)
= n∞ − n.
(4.2)
dt
Cm
4.3 The fast-slow phase plane
28
This is often referred to as the fast-slow model and its phase plane as
the fast-slow phase plane.
4.3 The fast-slow phase plane
Since we have now reduced the model to two variables, we can draw the
phase plane for it by considering the nullclines of V and n. Setting dn
dt = 0
we find that n = n∞ (V ) which is the monotonically increasing function that
we already know. Similarly setting dV
dt = 0 we find that
V =
ḡN a m3∞ (1 − n)VNa + ḡK n4 VK + gL VL
,
ḡN a m3∞ (1 − n) + ḡK n4 + gL
which has a cubic shape. The nullclines have been plotted in figure 4.3 along
with an action potential resulting from the dynamics. Notice that for the HH
model parameters there is one intersection of the nullclines which is thus the
dV
only steady-state of this system. From the signs of dn
dt and dt we can explain
how this system responds to a perturbation. V is a fast variable and n is
a slow variable and they are usually called the excitation and recovery
variable, respectively. This makes sense since V causes the rise in potential,
and n causes the return to the resting state. Because of the relative speed
differences the trajectories in the phase plane are nearly horizontal except
on the nullcline of V where the flow is vertical. This cubic curve is called the
slow manifold. On this curve the trajectories move slowly in the direction
determined by the sign of dn
dt , but elsewhere the trajectories move quickly
in a horizontal direction, since V is fast. From the sign of dV
dt it follows
that trajectories flow away from the middle branch of the slow manifold,
meaning in between the local minimum and maximum of the function, and
toward the left or right branches. Thus, the middle branch is called the
unstable branch of the slow manifold. This functions as a threshold.
When the perturbation from the resting state is so small that V does not
cross the unstable manifold, then the trajectories move horizontally toward
the left and return to the resting state, i.e. no action potential is generated.
However, if the perturbation is so large that V crosses the unstable manifold,
then the trajectory moves to the right until it reaches the right branch of
the slow manifold. Here dn
dt > 0, and so the solution moves up and along the
branch (since it is stable) until the start of the middle branch is reached (the
local maximum). There, the flow is nearly horizontal to the left and away
from the middle branch, and so the solution moves over to the left branch of
the slow manifold. Here dn
dt < 0, so the solution moves down and along this
branch until the resting state is reached again, completing the excursion we
have come to know as the action potential.
If we turn on the applied current in this fast-slow model what happens?
As Iapp increases, the V nullcline shifts up, until the two nullclines intersect
4.4 The FitzHugh-Nagumo equations
29
Figure 4.3: The fast-slow phase plane of the Hodgkin-Huxley equations.
Both nullclines and an action potential are drawn for Iapp = 0. From:
Keener & Sneyd, 2nd ed.
on the middle branch of the cubic. This branch is unstable so the trajectory
will never approach the resting state again, continuously switching between
the two stable branches for the same reasons we saw above. This behaviour
is called a relaxation limit cycle or relaxation oscillation, see figure 4.4.
Biologically, this represents a continuous stream of action potentials.
4.4 The FitzHugh-Nagumo equations
What FitzHugh did was look at a simplified two-dimensional system
that has the same qualitative characteristics as the fast-slow phase-plane.
The system can be represented in several forms but is often written in the
abstract form
dv
= f (v) − w + Iapp
dt
dw
= (v − γw).
dt
(4.3)
(4.4)
Here f (v) is understood to be cubic, just like the V nullcline in the reduced
system, and Iapp is the same parameter as in the HH model. One form for
f (v) that is often used is f (v) = v(1 − v)(1 − α), with α < 1, although
other choices are available. The nullcline for w is the linear equation w = γv
which roughly looks the same as the n nullcline in the reduced HH system.
Remember that increasing the Iapp parameter will shift the nullcline of the
v upwards, just like we saw in the previous section. Also remember that
the parameter signifies the time-scale difference between the two variables.
It is custom to take small, thereby making the recovery variable w slow
compared to the excitation variable v, as it should. One of the other ways
4.4 The FitzHugh-Nagumo equations
30
Figure 4.4: Fast-slow phase plane of the Hodgkin-Huxley equations, with
Iapp = 50, showing the nullclines and an oscillation. From: Keener & Sneyd,
2nd ed.
the system can thus be represented is by putting in the other equation,
making dv
dt very large and thus v very fast.
dv
= f (v) − w + Iapp
dt
dw
= v − γw.
dt
(4.5)
(4.6)
The nullclines are assumed to have a single intersection point, which,
without loss of generality, is placed to be at the origin and are drawn in
figure 4.5.
Now, the analysis of the dynamics of the FitzHugh-Nagumo system is
analogous to the analysis of the reduced HH model, i.e. the two phase planes
are basically the same.
When the resting state (the intersection of the nullclines) is on the left
branch V− and not too far from the minimum W∗ , the system is excitable.
This is because, as in the reduced model, a sufficiently large perturbation
from the resting state puts the system on a trajectory that moves away from
the resting state before eventually returning to rest. Such a trajectory, again
just as in the reduced model, goes to the right branch V+ , which it follows
as it moves up. When the maximum W ∗ is reached, it moves to the left
branch V− and then returns to rest, following the branch. See figure 4.6.
As we increased Iapp the reduced HH system produced limit cycles; the
FitzHugh-Nagumo system does too. The v nullcline moves up as Iapp increases. So, when Iapp takes values in some intermediate range (not too big
because then it will lie on the right branch and be stable again), the steady
state lies on the middle branch, V0 , and is unstable. Instead of returning to
4.4 The FitzHugh-Nagumo equations
31
Figure 4.5: A schematic diagram of the FitzHugh-Nagumo phase plane. V− ,
V0 and V+ respectively denote the left, middle and right branch we saw in
the reduced model. V− and V+ are stable and V0 is unstable. We denote the
minimal value of w for which V(w) exists by W∗ , and the maximal value of
w for which V+ (w) exists by W ∗ . From: Keener & Sneyd, 2nd ed.
rest after one action potential, the trajectory alternates periodically between
the left and right branches.
As before with the HH model, the behaviour of periodic orbits as Iapp is
increased can be nicely summarized in a bifurcation diagram. For each value
of Iapp we plot the value of V at the resting state, and the maximum and
minimum values of V over the periodic orbit. As Iapp increases, a branch of
periodic orbits appears in a Hopf bifurcation at Iapp = 0.1 and disappears
again in another Hopf bifurcation at Iapp = 1.24. Between these two points
there is a branch of stable periodic orbits. The bifurcation diagram is drawn
in figure 4.8.
This figure shares a lot of its features with the bifurcation diagram of
the HH model. It helps us understand the dynamics of the HH model just a
little better by making it a lot easier to investigate them. We have lost the
biological interpretation of the variables involved but those are still available
when looking at the closely associated HH model. This system of equation
that FitzHugh found has been a cornerstone model for many other subjects
in this field. Its tractability allows easier mathematical analysis and thus
makes it a very popular model.
4.4 The FitzHugh-Nagumo equations
32
Figure 4.6: Phase portrait for the FitzHugh-Nagumo system, with α = 0.1,
γ = 0.5, = 0.01 and no applied current. For these parameter values the
system has a stable resting state, and is excitable. From: Keener & Sneyd,
2nd ed.
Figure 4.7: Phase portrait for the FitzHugh-Nagumo system, with α = 0.1,
γ = 0.5, = 0.01 and Iapp = 0.5. For these parameter values, the resting
state is unstable and there is a periodic orbit. From: Keener & Sneyd, 2nd
ed.
4.4 The FitzHugh-Nagumo equations
33
Figure 4.8: Bifurcation diagram of the FitzHugh-Nagumo equations, with
α = 0.1, γ = 0.5, = 0.01, with the applied current as the bifurcation
parameter. The steady-state solution is labeled ss, while osc max and osc
min denote, respectively, the maximum and minimum of V over an oscillation. HB denotes a Hopf bifurcation point. From: Keener & Sneyd, 2nd ed.
From: Keener & Sneyd, 2nd ed.
Chapter 5
Summary
5.1 Review
In this thesis we looked at the problem of finding mathematical models
for describing the biological phenomenon of action potential generation.
After a brief introduction to the relevant biology we focused on mathematical formulations of the basic properties and processes of neurons. To
this end we started out with the fundamental circuit equation
Cm
dV
= −Iion (V, t) + Iapp .
dt
Then by assuming that the ionic current is composed of only sodium,
potassium and a leakage current and that the I-V curves are linear, we
stumbled upon
Cm
dV
= −gN a (V − VNa ) − gK (V − VK ) − gL (V − VL ) + Iapp .
dt
After that we considered the specific forms of the ionic conductances
which we modeled with a simple kinetic scheme and curve fitted the resulting
Hodgkin-Huxley gating equation to measurements.
All in all we derived the four-dimensional HH model
dV
dt
dm
τm (ν)
dt
dh
τh (ν)
dt
dn
τn (ν)
dt
Cm
= −gN
¯ a m3 h(V − VN a ) − g¯K n4 (V − VK ) − gL (V − VL )
= m∞ − m
= h∞ − h
= n∞ − n,
34
5.2 Futher studies
35
where Iapp denotes the externally applied current. This model works quite
well, and the desired behaviour (action potentials) is reproduced by this
model very nicely.
From this starting point, research went in two directions: one direction
tried to make the model even more realistic, taking into account more ion
species or other phenomena (like synapses etc.). The other direction is
a simplification of the model, such that it is possible to see clearly how
the spike is produced. The most important of these simplifications is the
FizHugh-Nagumo model that we considered in the next chapter.
We started out by making two assumptions, one about the time constants
(m moves fast) and one about the apparent symmetry in h and n. This
resulted in the fast-slow model
dV
= −gN
¯ a m3∞ (V )(1 − n)(V − VN a ) − g¯K n4 (V − VK ) − gL (V − VL )
dt
(5.1)
dn
τn (ν)
= n∞ − n.
(5.2)
dt
Cm
After the reduction of the HH model from 4 to 2 variables we reduced
even further and found the FitzHugh-Nagumo equations which are a simpler
representation of the dynamics of the fast-slow phase plain
dv
= f (v) − w + Iapp
dt
dw
= (v − γw).
dt
This system in particular appeared to be a very useful model with the
same qualitative features as the HH model and indeed is a standard model
in that virtually every book on the subject of mathematical biology or biological oscillators treats it.
5.2 Futher studies
What we did not get to in this thesis due to a lack of time and the fact
that it is an entire topic in itself, is the propagation of these action potentials
through travelling waves. For this we refer the reader to the standard texts
by Keener & Sneyd and Murray. Or for a very readable introduction to
travelling waves in general to the book by Edelstein-Keshet. Rest assured
though, that it is indeed possible to show that travelling waves exist, at least
for the FitzHugh-Nagumo system.
There are numerous other subjects that are now ready to be explored.
This is one of the reasons why the Hodgkin-Huxley model is so important;
it opens doors to all sorts of other interesting problems. Below we name a
few fun ones and the relevant literature.
5.2 Futher studies
36
Consider networks of spiking neurons. Such neurons can synchronize
and exhibit collective behaviour that is not intrinsic to any individual neuron. For example partial synchrony in cortical networks is believed to generate various brain oscillations, such as the alpha and gamma EEG rhythms.
For more, see the book by Izhikevich, Chapter 10.
Other interesting subjects that are perhaps more biological than mathematical include: synaptic plasticity and learning, for this Gerstner’s
book seems to be a good starting point.
All in all we can conclude that the theory of neuron firing and propagation of nerve action potentials is one of the major successes of mathematical
biology as a discipline. The contents of this thesis are merely the tip of the
iceberg, but a very useful tip nonetheless.
Chapter 6
Sources
• Keener & Sneyd - Mathematical physiology, Volume I: Cellular Physiology
• Vander - Human Physiology
• Purves - Neuroscience
• Cronin - Mathematical aspects of Hodgkin-Huxley neural theory
• Abbott & Dayan - Theoretical neuroscience
• Izhikevich - Dynamical systems in neuroscience
• Gerstner - Spiking neuron models
37