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This question is about the Hodgkin-Huxley model as introduced in Eugene M. Izhikevich, Dynamical Systems in Neuroscience, p.33 ff.

I'm having trouble to understand and interpret the differential equation for the activation variable \$m\$:

\$\$dot{m} = (m_infty(V) - m)/ au(V)\$\$

which enters via

\$\$p = m^a h^b\$\$

into the equation

\$\$I = ar{g} p (V − E)\$\$

for the net current \$I\$ generated by a large population of identical channels where \$p\$ is the average proportion of channels in the open state, \$ar{g}\$ is the maximal conductance of the population, and \$E\$ is the reverse potential of the current.

\$m\$ is the probability of one of \$a\$ activation gates to be open. Interchangeably: \$m^a\$ is the proportion of open activation channels (assuming all of its activation gates must be open simultaneously).

The differential equation might give us \$m\$ as an explicit function of time but it - explicitly - involves \$V\$ which is - implicitly - another function of time, which in turn depends on the number of open gates. Things are horribly complicated!

On the other hand - since it's about voltage-gated channels and there are "immediate" voltage-sensitive regions in the channel protein which presumably don't have memory ("the single channel has no memory about the duration of its own state"), but possibly a time lag - I expected the being-open-probability of an activation gate to be a "pure" (possibly time-lagged) function of \$V\$.

My question comes in two disguises

(1) Given two explicit functions \$m_infty(V)\$ and \$ au(V)\$ like these: together with \$m(0)\$ and \$V(0)\$, how could we ever arrive at an explicit solution for \$m(t)\$, assuming that \$V(t)\$ depends somehow on \$m(t')\$ for \$t'leq t\$, but possibly also on some injected currents.

(2) How can an intuitive and sensible interpretation of the terms in

\$\$dot{m} = (m_infty(V) - m)/ au(V)\$\$

be given? What does the time constant \$ au(V)\$ and its dependence on \$V\$ mean? In which respect and by which hypothesised mechanism is the gate "faster", when \$ au\$ is smaller? How can the voltage-sensitive steady-state activation function \$m_infty(V)\$ be intuitively explained other than by "giving the asymptotic value of \$m\$ when the potential \$V\$ is fixed (voltage-clamp)" or by some complicated description of experiments to determine it? What does \$m_infty(V) - m\$ mean, i.e. "the deviation of the current activation from the steady-state activation"?

1) In practice, no one attempts to obtain explicit solutions for m(t), especially considering the presence of many other ion channel species, nor is there any need to. The standard of art is to use numerical simulations with a sampling interval dT that is sufficiently smaller than any relevant time constants.

2) I'm not sure what intuitive and sensible interpretation you are looking for; it may not make sense to come up with an intuitive explanation for each and every subset of terms. The use of time constants implies that the system is modeled as a linear time invariant system at a given voltage.

What does the time constant τ(V) and its dependence on V mean? In which respect and by which hypothesised mechanism is the gate "faster", when τ is smaller?

The time constant is a scalar that describes the rate at which the system approaches some equilibrium. You could think of the changes in channel state as chemical reactions of the form:

Open <=> Closed

which has some equilibrium and some rate constant - both the rate constant and equilibrium conditions can be a function of voltage, as depicted in the figure you provide. The physical meaning of a rate constant being voltage dependent would be that the activation energy for the transition between states is a function of voltage; when the activation energy is lower, the reaction can progress more quickly (in either direction).

How can the voltage-sensitive steady-state activation function m∞(V) be intuitively explained other than by "giving the asymptotic value of mm when the potential V is fixed (voltage-clamp)" or by some complicated description of experiments to determine it?

That's exactly what it means, and the descriptions are not complicated: the steady state value for a reversible reaction is the point at which the number of units moving from open to closed would be the same as the number moving from closed to open, so on average you see no more change: that's what steady-state means.

What does m∞(V)−m mean, i.e. "the deviation of the current activation from the steady-state activation"?

Yes, "the deviation of the current activation from the steady-state activation" is accurate. The equation is an equation for the change of m, and the change of m is proportional to the "distance from equilibrium" - this is a characteristic of any process that follows linear time invariant dynamics. Note that there are other ways to write this equation that involve a "forward rate", a "backward rate", and the concentration of units in each state (the Wikipedia Hodgkin-Huxley page you linked has these equations), but you can simplify these to the equation you show here because some proportion of the "forward rate" process is cancelled by the "backward rate": your equation is sufficient to describe the net change.

## Summary of the Hodgkin-Huxley model

The Hodgkin-Huxley model of the process by which action potentials are generated in the giant axon of the squid lies at the basis of most neuronal models. Here is a brief summary of the equations and assumptions which went into the model.

The mathematical model is based upon the equivalent circuit for a patch of cell membrane. In your text, this is Figure 9-5. The two variable conductances GK and GNa shown in the diagram represent the average effect of the binary gating of many potasssium and sodium channels, and the constant ``leakage conductance'' GL represents the effect of other channels (primarily chloride) which are always open. Each of these is associated with an equilbrium potential, represented by a battery in series with the conductance.

The net current which flows into the cell through these channels has the effect of charging the membrane capacitance, giving the interior of the cell a membrane potential Vm relative to the exterior. From basic circuit theory, we know that the current which charges a capacitor is equal to the capacitance times the rate of change of the voltage across the capacitor. Ohm's law gives the current through each of the conductances, resulting in the equation

Here, an additional term I_inject has been added to describe any currents which are externally applied during the course of an experiment. In principle, all that is needed in order to find the time course of the membrane potential is to solve this simple differential equation.

The hard part was to model the time and voltage dependence of the Na and K conductances. As you know from reading Chapter 9, their solution was to perform a series of voltage clamp experiments measuring both the total current and the current when the Na conductance was disabled. This enabled them to calculate the K current and, from these currents and the known voltages, calculate the values of the two conductances. By performing the experiments with different values of the clamp voltage, they were able to determine the time dependence and equilibrium value of the conductances at different voltages. Figure 9-6 in the text shows some typical results for the behavior of the K and Na conductances when the clamping voltage is stepped to several different values and then released. From these measurements they were able to fit the the K conductance to an equation of the form

where n is called the ``activation state variable'' and has a simple exponential dependence governed by a single time constant, :

is called the ``steady state activation'', i.e. the value reached by n when it is held at the potential V for a long period of time. Hodgkin and Huxley were able to fit the voltage dependence of and to an analytic function of voltage involving exponentials. In the interest of brevity, we will not give these equations here. However, the plot of , shown further below, reveals that it is a monotonically increasing function of V , reaching a maximum value of 1.

If we are describing the time course of action potentials rather than the behavior during a voltage clamp, and are changing along with the changing membrane potential, so we can't use this equation for n . Instead, we use use a differential equation which has this solution when V is constant,

Their fit for the Na conductance was a little different, because they found that at a fixed voltage, the conductance rose with time and then decreased (as shown in Figure 9-6), so they had to fit it to a product

Here, m is the activation variable for Na, and h is called the ``inactivation state variable'', since it becomes smaller when m (and the membrane potential) becomes larger. m and h obey equations just like the ones for n , but with different voltage dependences for their steady state values and time constants. These voltage dependences are shown in the following plot, derived from Hodgkin and Huxley's fit to their experimental results:

We now have all that was needed by Hodgkin and Huxley to reconstruct the action potential. For a given injection current I_inject , Eq. 1 is solved for Vm , using Eqs. 2 and 5 for the conductances. These two equations must be solved simultaneously with Eq. 4 and the two analogous equations for m and h . These last equations make use of the voltage dependent quantities shown in the plot.

This plot shows that although the time constants vary with voltage, the time constant for the Na activation variable m is about an order of magnitude less than that for the Na inactivation and the K activation throughout the entire range. This means that during an action potential, when the voltage is high and m is large, and h is supposed to be small, it will take a while for h to decrease. Also, it will take n a while to become large and contribute to the opposing K current.

As we will see in a later lecture, the behavior of these quantities is the key to understanding the time course of the action potential, as well as the phenomenon of the ``refractory period'' following the action potential.

## 5.3 The Hodgkin and Huxley model Figure 5.1: Alan Hodgkin (left) and Andrew Huxley (right).

Alan Hodgkin (pictured left) and Andrew Huxley (pictured right) were two Cambridge University undergraduates who eventually found themselves working in a marine biology laboratory with the axon of a giant squid. The two men were able to derive the necessary information for their influential model of an action potential using the massive axon of the giant squid.

Hodgkin and Huxley developed a series of equations that could accurately predict and depict action potentials. Their work is a cornerstone for computational modeling as computer modelling can now be used to mimic the biological properties of a neuron that we are unable to directly observe.

Really the Hodgkin-Huxley Model is just an elaboration on the Integrate and Fire Model. The Integrate and Fire model was generated by French neuroscientist Louis Lapicque, who in 1907 sought to generate a mathematical model that could be used to predict and graph an action potential. In his efforts to understand action potentials, Lapicque chose to model the flow of ions as a single leak current.

Hodgkin and Huxley took the single conductance term from the Integrate and Fire Model is broken up into three separate conductance terms, each relating to a different ion channel. These conductance terms are known as gating variables and are labeled m, n, and h. Voltage-gated sodium channel activation is modeled by the letter ms. Voltage-gated sodium channels have three subunits, as these three subunits are involved in the channels activation, m is raised to the third power. Voltage-gated sodium channel also inactivate at the peak of the action potential and this variable is modeled by the letter h. The combination of m and h gives rise to the conductance of Voltage-gated sodium channel which is modeled below: [ar_m^<3>h(V(t)-E_) ] Voltage-gated potassium channels are modeled by the letter n. Voltage-gated potassium channels have four subunits, and thus the gating variable, n, is raised to the fourth power. The conductance of Voltage-gated potassium channels is modeled below: [ar_n^<4>(V(t)-E_ ] The final conductance taken into account by Hodgkin and Huxley is the leak potential of all the ions. The Leak conductance is taken into account for the instance when all ion channels are open. This conductance is represented below: [ar_(V(t)-E_) ] These three conductance variables are combined together to form the Hodgkin-Huxley equation which is written as follows: [Cfrac

=I_(t)-[(ar_m^<3>h(V(t)-E_))+(ar_n^<4>(V(t)-E_))+(ar_(V(t)-E_))] ]

Expression Meaning
n Potassium gating variable
m Sodium activation gating variable
h Sodium inactivation gating variable
(C_m) Specific membrane capacitance
(I_e) Injected current
(ar_) Maximum Na+ conductance
(ar_) Maximum K+ conductance
(ar_) Maximum leak conductance
(V_m) Membrane potential
(E_) Sodium Nernst potential
(E_) Potassium Nernst potential
(E_) Leak Nernst potential

Additionally, we can calculate the value of each gating variable over different voltages and times: [mfrac

= alpha_(V)(1-m)-eta_(V)m ] Both n and h can be substituted for m in the above equation in order to calculate values for each gating variable. Additionally, note the (alpha) and (eta) in the equation are rate constants that govern the opening and closing (respectively), of their channels. Here are their values:
[ alpha_(V_) = frac<0.01(V_+55)><1-exp(-0.1(V_+55))> ] [alpha_(V_) = frac<0.1(V_+40)><1-exp(-0.1(V_+40))>] [alpha_(V_) = 0.07exp(-0.05(V_+65)) ] [eta_(V_m) = 0.125exp(-0.0125(V_+65)) ] [eta_(V_) = 4exp(-0.0556(V_+65)) ] [eta_(V_) = frac<1><1+exp(-0.1(V_+35))> ]

Expression Meaning
(alpha_n) Rate constant for K+ channel opening
(alpha_m) Rate constant for Na+ activation gate opening
(alpha_h) Rate constant for Na+ inactivation gate opening
(eta_n) Rate constant for K+ channel closing
(eta_m) Rate constant for Na+ activation gate closing
(eta_h) Rate constant for Na+ inactivation gate closing

Worked Example:
Have you ever wondered how anesthesia makes a tooth extraction painless? It’s because anesthesia works by blocking the activation of voltage-dependent Na+ channels. This prevents the propagation of the action potentials that carry that awful pain sensation.

Using the equations below, calculate the maximum conductances of each ion in the resting state.

• (V_m) = -68 mV
• (C_m) = -20.1 nF
• (E_L) = -54 mV
• (E_) = 50 mV
• (E_K) = -77 mV
• (R_L = 1/3 MOmega)
• (g_ = 1200 mS/mm^2)

Equations:
[I_ = g_ cdot m^3h(V_m-E_) + g_K cdot n^4 (V_m-E_K) + g_L cdot (V_m-E_L)] [frac

= alpha_n(V) cdot (1-n) - eta_n(V) cdot(n)] [frac
= alpha_m(V) cdot (1-m) - eta_m(V) cdot(m)] [frac
= alpha_h(V) cdot (1-h) - eta_h(V) cdot(h)]

Step 1: Understand the question The resting potential can be considered to be a steady state because the voltage is not changing. Therefore, (frac

=0) for p= and therefore the last three equations will not be used.

Step 2: Calculate n, m, and h We need to now use the resting potential to solve for the steady state values of the gating variables.
[p_infty = frac] for p =

Therefore, the very first step is to calculate each (alpha) and (eta) .

[alpha_(V_) = 0.07exp(-0.05(V_+65)) ] [alpha_(V_) = 0.07 cdot exp(-0.05 cdot(-68+65)) ] [alpha_(V_) = 0.08]

[eta_(V_m) = 0.125exp(-0.0125(V_+65)) ] [eta_(V_) = 0.125 cdot exp(-0.0125 cdot(-68+65)) ] [eta_(V_) = 0.13]

[eta_(V_) = 4exp(-0.0556(V_+65)) ] [eta_(V_) = 4 cdot exp(-0.0556 cdot(-68+65)) ] [eta_(V_) = 4.73]

Now, with each of our (alpha) and (eta) values, we can calculate our gating variables:

Step 3: Calculate (g_L) from resistance units. Remember that (g = frac<1>) ! [g_L = (1/R) = (1/(1/3)) = 3 mS/mm^2] Step 4: Solve for (g_K) Remember that from Kirchhoff’s Law that the algebraic sum of all the currents entering and leaving a junction must be equal to 0. Therefore:

[0 = g_ cdot m^3h(V_m-E_) + g_K cdot n^4 (V_m-E_K) + g_L cdot (V_m-E_L)] And we may plug in the values that we already have:
[0 = 1200 cdot 0.037^30.69 cdot(-68-50) + g_K cdot 0.274^4 cdot (-68--77) + 3 cdot (-68--54.387)] [0 = 4.956 + g_K cdot 0.051 + (-40.84)] [g_K = 703.6]

Worked Example: The voltage of a neuron is clamped at -20 mV, depolarized from its resting potential of -65 mV. The steady-state values of the gating variables in the two conditions are shown below. Comment on what these changes mean for the neuron’s behavior.

V=-65 mV V=-20mV
m=0.0529 m=0.875
n=0.3177 n=0.820
h=0.5961 h=0.009

Answer The value of m represents the probability of voltage-gated Na+ channels to be open. This probability increases as the cell depolarizes. The n value represents the probability that the voltage-gated K+ channel is open. Like the Na+ channels, this probability increases with depolarization, but not to the same extent. The h values represent the probability of Na+ channel inactivation. This decreases significantly with depolarization because we have not hit the peak of the action potential. It is fair to assume that the h value will increase as we near the peak.

## What is the Hodgkin-Huxley model?¶

From Wikipedia: The Hodgkin–Huxley model is a mathematical model that describes how action potentials in neurons are initiated and propagated.

The model describes represents the electrical properties of excitable membranes as typical electrical circuit components. For instance, the cell’s membrane is modeled as a capacitor, and voltage-dependent conductances stand in for what are now known to be voltage-gated ion channels.

For a detailed run through of the Hodgkin-Huxley model’s electronics, math and biology, take a look at the Electrophysiology page.

After you understand the electronic model there, check out the code walkthrough to see an example implementation of the Hodgkin-Huxley model in Python, using a cell modeled in NeuroML2.

You can look at the current-voltage characteristic page to get an understanding of another biological-electronic equivalence that is useful in describing ion channel and cell models.

There are also some exercises you can complete to get a feel for the model. These can be completed using either the Python or NeuroML versions.

## Understanding Hodgkin-Huxley's model and activation variables - Biology

In 1952, Hodgkin and Huxley wrote a series of five papers that described the experiments they conducted that were 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 outlined the basic experimental method pervasive 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 resolution of the ionic current into sodium and potassium currents. The third paper examined the effect of sudden potential changes on the action potential (including the effect of sudden potential changes on the ionic conductance). The fourth paper outlined how the inactivation process reduces sodium permeability. The final paper put together all of the information from the previous papers and turned them into a mathematical models.

A.L Hodgkin and A.F. Huxley developed a mathematical model to explain the behavior of nerve cells in a squid giant axon in 1952. Their model, which was developed well before the advent of electron microscopes or computer simulations, was able to give scientists a basic understanding of how nerve cells work without having a detailed understanding of how the membrane of a nerve cell looked. To create their mathematical model, Hodgkin and Huxley looked at squid giant axons. They used squid giant axons because squids had axons large enough to manipulate and use their specially built glass electrodes on.(Click here for more information on materials and methods) From their experimentation with a squid axon, they were able to create a circuit model that seemed to match how the squid axon carried an action potential.

Current flowing through the membrane can be carried via the charging and discharging of a capacitor or via ions flowing through variable resistances in parallel with the capacitor. Each of the resistances corresponds to charge being carried by different components. In the nerve cell these components are sodium and potassium ions and a small leakage current that is associated with the movement of other ions, including calcium. Each current (I Na , I K , and I L ) can be determined by a driving force which is represented by a voltage difference and a permeability coefficient, which is represented by a conductance in the circuit diagram. Conductance is the inverse of resistance. These equations can easily be derived using Ohm’s law (V=IR)

g Na and g K are both functions of time and membrane potential. E Na , E K , E L , C m and g L are all constants that are determined via experimentation.(Click here for more information on currents)

The influences of membrane potential on permeability were discovered to perform as follows. Under depolarization conditions, there is a transient increase in sodium conductance and a slower but more sustained increase in potassium conductance. These changes can be reversed during repolarization. The nature of these permeability changes was not fully understood when Hodgkin and Huxley did their work. They did not know what the cellular membrane looked like on the micro scale. They did not know about the existence of ion channels and ion pumps in the membrane. Based off of their finding, however, they were able to conclude that changes in permeability were dependant on membrane potential and not membrane current. Molecules aligning or moving with the electric field cause a change in permeability. Originally they supposed that sodium ions crossed the membrane via lipid carrier molecules that were negatively charged. What they observed however, proved that this was not the case. Rather, they supposed that sodium movement depends on the distribution of charged particles which do not act as carriers in the usual sense but rather allow sodium to pass through the membrane when they occupy particular sites on the membrane. This turned out to be the case. These charged particles are ion channels. In the case of sodium permeability, the carrier molecules (as they are referred to by Hodgkin and Huxley) are inactivated when there is a high potential difference. Potassium permeability is similar to sodium permeability but there are some key differences. The activating carrier molecules have an affinity for potassium, not sodium. They move more slowly and they are not blocked or inactivated. (Click here for more information on conductances)

To build their mathematical model that describes how the membrane current works during the voltage clamp experiment, they used the basic circuit equation

where I is the total membrane current density (inward current positive), Ii is the ionic current density (inward current positive), V is the displacement of membrane potential (depolarization is negative), C m is the membrane capacitance, t is time. They chose to model the capacity current and ionic current in parallel because they found that the ionic current when the derivative was set to zero and the capacity current when the ionic current is set to zero were similar. We can enrich this equation further by realizing that

where I Na is the sodium current, I K is the potassium current and I L is the leakage current. We can further expand on this model by adding the following relationships:

Where E R is the resting potential. When examining the graph of the potassium current versus the potassium potential difference, you can see that in the beginning, it’s or third order equation will describe it. But at the end, during the end, it seems to be more first order. In order to explain this in the conductance formula, we let

where is a constant, and n is a dimensionless variable that varies from 0 to 1. It is the proportion of ion channels that are open. To further understand where n comes from we can derive the equation

where alpha is the rate of closing of the channels and beta is the rate of opening. Together, they give us the total rate of change in the channels during an action potential. The sodium conductance is described by the equation

whereis a constant and m is the proportion of activating carrier molecules (ion channels) and h is the proportion of inactivation carrier molecules (ion channels). M and h can be further described by

where alpha and beta are again rate constants that are similar to the rate constants for the potassium conductance. (Click here for more information on inactivation)

Graph used to determine the values for the potassium conductance rate constants alpha and beta

Graph used to determine the values for the sodium activation conductance rate constants (m), alpha and beta

Graph used to detmine values for the sodium inactivation conductance rate constants (h), alpha and beta

## Analyze the Hodgkin-Huxley Model with a Computational App In a previous blog post, we discussed the physiological basis of generating action potential in the excitable cells of living organisms. We spoke about the simple Fitzhugh-Nagumo model, which emulates the process of depolarization and repolarization in a cell’s membrane potential. Today, we analyze a more advanced model for simulating action potential, the Hodgkin-Huxley model. We also go over how to use a computational app to streamline this type of analysis.

### Exploring Action Potential in Cells with the Hodgkin-Huxley Model

We have already gone over the physical basis of the firing mechanism that generates action potential in cells and we studied the generation of such a waveform using the Fitzhugh-Nagumo (FH) model. The dynamics of the simple Fitzhugh-Nagumo model, featured in a computational app.

Today, we will convert the FH model study into a more rigorous mathematical model, the Hodgkin-Huxley (HH) model. Unlike the Fitzhugh-Nagumo model, which works well as a proof of concept, the Hodgkin-Huxley model is based on cell physiology and the simulation results match well with experiments.

In the HH model, the cell membrane contains gated and nongated channels that allow the passage of ions through them. The nongated channels are always open and the gated channels open under particular conditions. When the cell is at rest, the neurons allow the passage of sodium and potassium ions through the nongated channels. First, let us presume that only the potassium channels exist. For potassium, which is in excess inside the cell, the difference of concentration between the inside and outside of the cell acts as a driving force for the ions to migrate. This is the process of movement of ions by diffusion, or the chemical mechanism that initially drives potassium out of the cells.

This movement process cannot go on indefinitely. This is because the potassium ions are charged. Once they accumulate outside the cell, these ions establish an electrical gradient that drives some potassium ions into the cells. This is the second mechanism (the electrical mechanism) that affects the movement of ions. Eventually, these two mechanisms balance each other and the potassium efflux and outflux balances. The potential at which the balance happens is known as the Nernst potential for that ion. In excitable cells, the Nernst potential value for potassium, E_ , is -77 mV and for sodium ions, E_ , is around 50 mV.

We allow the presence of a few nongated sodium channels in the membrane. Because the sodium ions abound in the extracellular region, an influx of sodium ions into the cell must occur. The incoming sodium ions reduce the electrical gradient, disturb the potassium equilibrium, and result in a net potassium efflux from the cell until the cell reaches its resting potential at around -70 mV. It is important to mention here that the net efflux of potassium and net influx of sodium ions cannot go on forever, otherwise the chemical gradient that causes the movement will eventually cease. Ion pumps bring potassium back into the cell and drive sodium out through active transport and maintain the resting potential of the cells in normal conditions.

Let’s derive an equivalent circuit model of a cell in which we can imitate the effects of the different cellular mechanisms we just described by different commonly found circuit components, such as capacitors, resistors, and batteries. The voltage response of the circuit is the signal that corresponds to the action potential.

Overall, there are four currents that are important for the HH model:

1. The current that flows into the cell membrane
2. A sodium current
3. A potassium current
4. A leak current that accounts for any other current, except for the first three Schematic of the currents in a Hodgkin-Huxley model.

The four currents flow through parallel branches, with the membrane potential V as the driving force (see the figure above the ground denotes extracellular potential). The cell membrane has a capacitive character, which allows it to store charge. In the figure above, this is the left-most branch, modeled with a capacitor of strength Cm. The other branches account for three ionic currents that flow through ion channels. In each branch, the effects of channels are modeled through conductance (shown as resistance in the diagram), and the effect of the concentration gradient is represented by the Nernst potential of the ions, which are represented as batteries.

Thus, when a current is injected in the cell, it gets divided into four parts and the conservation of charges leads us to the following balance equation

What is of paramount importance is that the sodium and potassium channel conductances are not constant rather, they are functions of the cell potential. So how do we model them? Remember that some of the ion channels are gated and they can have multiple gates. Assume that there are voltage-dependent rate functions αρ (V) and βρ (V), which give us the rate constants of a gate going from a closed state to open and open to closed, respectively. If ρ denotes the fraction of gates that are open, a simple balance law yields the following equation for the evolution of ρ

Different gated channels are characterized by their gates. In the HH model, the potassium channel is hypothesized to be composed of four n-type gates. Since the channel conducts when all four are open, the potassium conductance is modeled through the equation

For sodium, the situation is assumed to be more complicated. The sodium-gated channel has four gates, but three m-type gates (activation-type gates that are open when the cell depolarizes) and one h-type gate (a deactivation gate that closes when the cell depolarizes). Therefore, the sodium channel conductance is given by

In the above equations, ar_, ar_ is the maximum potassium and sodium conductance. The functional forms of αρ (V), βΡ (V) for p =m, n, h can be found in any standard reference.

The leak conductance is assumed to be a constant. Therefore, the HH model is completely described by the following set of equations

### Understanding the Dynamics of the HH Model with Simulation

The key to understanding the Hodgkin-Huxley model lies in understanding the gate equations. We can recast the equations for the gates in the following form

This is a very well-known equation in electrical circuits. If we assume ρ is voltage independent, then the equation says that ρ asymptotically approaches ρ as its final value, and Τρ, the time constant, dictates the rate of approach. This means that the smaller the Τρ, the faster the approach. The following figure shows the values of these two quantities for p =m, n, h .  The asymptotic values (left) and time constants (right) for the gate equations of the Hodgkin-Huxley model.

It is easy to conclude from the figures above that n, m increases as the cell depolarizes and h decreases under similar conditions. From the second graph, we find that the activation for sodium is much faster compared to the activation of potassium or the leak current.

When depolarization starts, n, m increases and h decreases. The governing equations of all of these quantities demand that they should approach the steady-state values therefore, n, m increases and h decreases. However, we should also remember the differences in time constants of the gating variables. A comparison says that the activation of sodium gates happens much faster as compared to their deactivation or the opening of potassium channels. Therefore, there is an initial overall increase in the sodium conductance. This results in an increase of the sodium current, which raises the membrane potential and causes V to approach E_ . This is how the HH model accounts for the rising part of the action potential.

However, as this process continues, h ightarrow 0 . Once the value of h goes below a threshold, the sodium channels are effectively closed. Also, the approach of V toward E_ kills the driving force for the sodium current. Meanwhile, the potassium channels, which have a slower time constant, open up to a large extent. This, coupled with the large driving force V-E_ that is available for the potassium current, forces the reverse flow. The potassium ions move out of the cell and eventually the membrane potential settles toward the hyperpolarized state.

### Building and Using a Simulation App to Analyze the Hodgkin-Huxley Model

We can build a computational simulation app to analyze the Hodgkin-Huxley model, which enables us to test various parameters without changing the underlying complex model. We can do this by designing a user-friendly app interface using the Application Builder in the COMSOL Multiphysics® software. As a first step, we create a model of the Hodgkin-Huxley equations using the Model Builder in the COMSOL software. After building the underlying model, we transform it into an app using the Application Builder. By building an app, we can restrict and control the various inputs and outputs of our model. We then pass the app to the end user, who doesn’t need to worry about the model setup process and can focus on extracting and analyzing the results of the simulation.

In our case, we implemented the underlying Hodgkin-Huxley model using the Global ODEs and DAEs interface in COMSOL Multiphysics. This interface is a part of the Mathematics features in the COMSOL software and is capable of solving a system of (ordinary) differential-algebraic equations (Global ODEs and DAEs interface). This interface is often used to construct models for which the equations and their initial boundary conditions are generic. In the interface, we can specify the equations and unknowns and add initial conditions. The interface, with model equations, is shown below. We also create the postprocessing elements, graphs, and animations in the Model Builder. Once the model is ready, we move on to the Application Builder again. We connect the elements of the model to the app’s user interface through various GUI options like input fields, control buttons, display panels, and some coded methods.

Finally, we can design the user interface of the Hodgkin-Huxley app. With the Form Editor in the Application Builder, we can design a custom user interface with a number of different buttons, panels, and displays. This user interface features a Model Parameters section to input the different parameters of the HH model, such as the Nernst potential, maximum gate conductance, and membrane capacitance. We can also provide two types of excitation current to the model: a unit step current or an excitation train. As the parameters change, the app displays the action potential and excitation current, as well as the evolution of gate variables m, n, and h.

With the Reset and Compute buttons, it is easy to run multiple tests after changing the parameters. There are also graphical panels that display visualizations and plots of the model results. The Report button generates a summary of the simulation. The user interface for the Hodgkin-Huxley Model simulation app.

Making buttons work in an app is a simple process. All we have to do is write a few methods using the Method Editor tool that comes with the Application Builder and connect them to the buttons properly. Let me illustrate with an example. We can design the Hodgkin-Huxley Model app so that when it launches, the Report button is inactive (see the figure below). This is because the app user will not need to use this button until after they perform a simulation. The Report button is disabled at the start of the simulation.

To do so, we can write a method that instructs the app to execute certain functions during the launch.

A method that disables the Report button during launch.

Observe that we have disabled the Report button using instructions in lines 7 and 8 in the method. If you are worried about coming up with the syntax for your methods, let me assure you that it is much more simple than it seems. First, the methods execute some actions. If we want to record the code corresponding to these actions, we click on the button called Record Code in the Application Builder ribbon. Then, we can go to the Model Builder, execute the actions, and once done, click on the Stop Recording button. The corresponding code will be placed in the method. If necessary, we can then modify the instructions.

Once a simulation is complete, we would like this button to become active in the app. In another method associated with the Compute button, we insert the following code segment

We then ensure that this segment is executed if the solution is computed successfully. You will see that this enables the button.

To summarize, you can use a simulation app to easily compute and visualize parameter changes when working with a complex model that involves multiple equations and types of physics, such as the Hodgkin-Huxley model discussed here. This simulation app is just one example of how you can design the layout of an app and customize its input parameters to fit your needs. Use this app as inspiration to build your own app, whether you are analyzing the action potential in a cell with a mathematical model or teaching students about complicated math and engineering concepts. No matter what purpose your app serves, it will ensure that your simulation process is simple and intuitive.

## Abstract

It is well known that the classical Hodgkin–Huxley (HH) equations exhibit complex nonlinear dynamic properties. This paper examines the ion conductance equations of 28 modified HH models, calculating the ion channel conductance and simulating neuronal firing patterns. It is interesting to discover and confirm that the ion conductance is exactly the coefficient of the fourth-order term at the origin of Taylor’s formula. It is demonstrated that the equations for these models are closely related to the two-variable Taylor’s formula of the conductance around the origin in terms of the channel parameters. We consider one specific model in great detail as an example. We find that the conductance curves are generally of the same form, but that the conductance peaks differ. Several firing patterns are observed in the modified HH models, including bursting and mixed-mode oscillations. The emergence of mixed-mode oscillations in particular may be of interest for future work.

## Computational Modeling of Cardiac K+ Channels and Channelopathies

### HERG Activators as Prospective Tools for Understanding Channel Structure and Function

A prime example of the need to apply computational modeling and simulation approaches is the case of the NS1643 hERG activator because it has many complex nonlinear effects. 140 In the computational modeling study, NS1643 functional effects were modeled by modifying the Hodgkin–Huxley equations of IKr in the 2004 ten Tusscher human ventricular model. As expected, NS1643 was predicted to decrease APD and triangulation, which is considered antiarrhythmic in the setting of LQTS. It also was predicted to increase postrepolarization refractoriness, but was shown to shorten the absolute refractory period, which could favor arrhythmia initiation. Simulations in one-dimensional tissue models predicted that NS1643 could increase the vulnerable window to reentry, which increases the risk of arrhythmias, but suppressed the development of premature AP and unidirectional block close to the APD, which is an antiarrhythmic property. An additional complication is that the effects of NS1643 are affected by the extracellular potassium concentration: While one of the primary drug effects is to increase IKr conductance during hypokalemia, the drug-induced positive shift of the inactivation curve is the effect that predominates during normokalemia. The modeling studies make prediction about situational dependence of drug efficacy. The NS1643 hERG channel activator are predicted to prove especially effective in preventing arrhythmic episodes related to EADs and long QT intervals or under hypokalemic conditions, but may be less effective in other proarrhythmic situations.

A computational modeling study combined with experimental approaches has also been used to reveal ideal properties of IKr activators to correct the APD prolonging effects of the R190Q-KCNQ1 LQT1 linked mutation. 141 LQT1 was modeled reducing IKs by 30% of control, which was the level of IKs registered in LQT1 iPSC–derived myocytes with the R190Q-KCNQ1 mutation. The authors set out to test the hypothesis that normalization of the APD could be achieved by shifting IKr inactivation to more positive voltages to slow the inactivation and increase channel availability during repolarization similarly to IKs channels. To test it, they undertook virtual experiments wherein the inactivation curve was progressively shifted in WT and LQT1 model cells, which was shown to lead to the progressive shortening of APD in both types of cells. They also tested the potential for increasing IKr via an alternative mechanism by slowing deactivation, which was simulated by increasing the time constant the activation (deactivation) gate. However the latter approach was not predicted to be effective in correcting the prolonged APDs in the LQT1 cardiomyocytes. The model predictions were validated by experiments using the ML-T531 IKr activator, whose main effect is a positive shift of the inactivation curve, which was shown experimentally to produce dose-dependent shortening of the APD and normalize the APD of cardiomyocytes in LQT1 patients. 141

## Concluding Remarks

We live in a time marked by capacity to collect data at ever-increasing speed and resolution. As a result, it is tempting to use these data to construct numerical models of increased dimensionality, making them more and more biologically realistic. To avoid the fallacy of attributing importance to each and every measurable parameter, good practice combines methods that point to functional relations between parameters (35) and formulation of low-dimensional phase diagrams. However, reduction of dimensionality—a Via Regia to formal understanding—also comes at a price. In many cases it is a unidirectional path where measurables are abstracted and compressed to an extent that loses the explicit properties of the physiological data from the abstract representation. Consequently, once an abstract low-dimensional model is constructed, evaluation of impacts and subsequent incorporation of new biological features into the low-dimensional model become challenging, if at all possible.

Here we approached the problem by implementation of a methodology that has a long and successful history in membrane physiology: system identification using closed-loop control (i.e., voltage clamp, patch clamp, and dynamic clamp). We describe an experimental–theoretical hybrid, a framework enabling bidirectional real-time interaction between abstract low-dimensional representation and real biological entities. This is not a post hoc fitting procedure rather, it is a live experiment where the impacts—of a given biological component—on the abstract low-dimension representation are identified by implementing a real-time closed loop design.

Specifically, combining dynamic clamp and heterologous expression of ionic channel proteins in Xenopus oocytes, we constructed an excitable system composed of a mix of biologically and computationally expressed components. This experimental configuration enabled systematic sampling of the Hodgkin–Huxley parameter space. The resulting phase diagram validates a theoretically proposed diagram (4).

A spectrum of Hodgkin–Huxley single-compartment representations exists, extending from concrete and computationally intensive Markov kinetic models of channel state transitions, to abstract models that are computationally efficient yet biophysically less realistic (10, 11, 36). The S–K phase diagram is situated in between, touching both ends. On one hand, its two dimensions are expressed in physiologically accessible parameters, and on the other hand, the two dimensions are intimately related to the abstract nonlinear oscillator inspired models with S linked to the cubic polynomial expression that provides fast positive feedback, and K is related to the recovery variable that introduces slow negative feedback. As such, the S–K phase diagram may serve as a common ground to relate various representations in this spectrum to each other.

The shape of the S–K phase diagram proposed in ref. 4 and experimentally constructed in this report suggests that maintenance of excitability amid parametric variation is a low-dimensional, physiologically tenable control process. Moreover, we show that the basic ingredients for such control—namely, memory and adaptation—are manifested in the phase diagram as a natural outcome of ion channel slow inactivation kinetics.

Many theoretical and experimental analyses show that the wide range of temporal scales involved in slow inactivation is sliced thinly to a degree effectively equivalent to a continuum of scales, indicative of the extensive network of configurations within which the channel protein may diffuse giving rise to slow activity-dependent gating and adaptive firing patterns (28, 32, 37 ⇓ ⇓ ⇓ ⇓ ⇓ ⇓ –44). Indeed, slow activity-dependent gating was suggested as a means for maintenance and control of membrane excitability. Specifically, activity dependence of protein kinetics at relatively slow time scales, entailed by multiplicity of protein states, was pointed at as a general automatic and local means for stabilization of cellular function, independent of protein synthesis, and operates over a wide—minutes and beyond—range of time scales (3, 4, 45, 46). Thus, precisely because these ion channels do not have a single, fixed time constant encoded in their molecular structure but rather slide through multiple states, cells have a built-in mechanism to smoothly function over a larger range of firing patterns and voltages. A similar argument holds for the wide range of time scales contributed by the plethora of different Kv channels (25), which also can expand stable operating ranges. This partially mitigates the control problem that cells face: getting it right may not require the perfect match between channel numbers that might otherwise be necessary. Viewed from another angle, multiple states of channel inactivation and recovery from inactivation necessarily result in hysteresis, and the time scales of that hysteresis become a memory mechanism (32, 34, 38) so that cells can use it to keep track of their recent pattern of activity and inactivity. This again expands the time course over which patterns of activity can influence the way the cell responds to physiological inputs. Interestingly, we usually think of the fastest membrane events (action potentials) having little lasting effect on the cells in which they are seen but, looking only at the fast voltage deflections such as action potentials hides the effects of the slower channel dynamics that influence future events.

It remains to be seen how far the approach described here may be used in system identification of excitable membranes more complicated than the minimal, two-conductance single-compartment Hodgkin–Huxley configuration. Certainly, cells that contain many different types of ion channels will show a range of time scales and history dependence (47). Developing intuition into how a given set of firing properties depends on conductance densities of many channels may require new kinds of principled dimensionality reduction to complement brute force numerical simulations.

## Model

We first review the essential framework of Hodgkin-Huxley type models for action potential generation. The dynamics of the membrane potential of a section of neuron, assumed to be spatially homogeneous, are given by : (1) where Here is the membrane capacitance, is the maximal conductance of channels of type , is the probability that a channel of type is open, is the reversal potential for channel type and the subscripts , and refer to sodium, potassium and M-type potassium channels respectively. A leak current is included with conductance and reversal potential , is the membrane area, while is the current resulting from synaptic background activity . Background activity is typically modeled by assuming synaptic conductances are stochastic and consists of an excitatory conductance with reversal potential and an inhibitory conductance with reversal potential , as found in  so that (2) In  the conductances and are modeled by Ornstein-Uhlenbeck processes with correlation times and , and noise diffusion coefficients and respectively .

We are interested in understanding from this model the relationship between onset span and onset rapidity, as defined by . As described above, the onset rapidity is the rate at which the voltage increases near onset the increase in voltage is exponential and so is given by the slope of a plot of versus . The onset span measures the variability of the voltage threshold for action potential initiation,  defines this threshold as the voltage at which , and takes . Due to the stochastic synaptic background, there is a distribution of voltages at which the voltage threshold is attained the onset span is given by the width of this distribution. We calculate the probability distribution of voltage thresholds, and derive the onset span from the moments of this distribution.

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