Conductance-Based Models

The Hodgkin-Huxley Model

Alan Hodgkin and Andrew Huxley published a series of five papers in 1952, detailing a model for the generation and propagation of the active potential by active sodium and potassium channels. They took many measurements on a squid axon to arrive at their model.

An action potential involves fast positive feedback followed by slow negative feedback. Some terms will help describe relevant concepts,

• Channel activation is a potential-dependent process necessary for opening a channel, usually by depolarization
• Channel deactivation is the opposite of channel activation, by hyperpolarization
• Channel inactivation is a process that occurs with depolarization that prevents certain channels from opening
• Channel deinactivation is the opposite of inactivation

An action potential consists of,

1. Sodium activation m$\uparrow$
2. Slower sodium inactivation h$\downarrow$, potassium activation n$\uparrow$
3. Sodium deactivation m$\downarrow$
4. Slower sodium deinactivation h$\uparrow$, potassium deactivation n$\downarrow$

Negative feedback arises from a few mechanisms,

• The sodium reversal point ensures a maximum spike height
• Sodium channels inactivate with depolarization, meaning sodium causes a transient current, which only activates for a short time
• The sharper return after a spike results from potassium channels, which are called a delayed rectifier current (delayed so that a spike can occur at all)

Voltage clamp and current clamp are used to test feedback responses. This is especially easy to due in simulations.

Hodgkin-Huxley Simulations

The model is based on four variables: $V_m$, sodium activation $m$, sodium inactivation $h$, and potassium activation $n$

$m$, $h$, and $n$ are called gating variables, they sit between $0$ and $1$ and represent the fraction of channels in a particular state. Multiplying together all such variables for a given channel type indicates the fraction of the channels that are open. Each variable has an independent steady-state value if membrane potential is fixed.

Gating Variables

To understand gating variables, it’s helpful to examine two-state systems. Say $A$ and $B$ are two states in a system, and $a$ and $b$ represent the fraction of the system in $A$ and $B$ respectively. The rate constants $k_A$ and $k_B$ represent the rate of switching from $A$ to $B$ and vice versa.

$\frac{da}{dt}=k_A(1-a)-k_{B}a$ and
$\frac{db}{dt}=-k_{A}b+k_B(1-b)$

It’s easy to solve for the steady-state and the time constant can be found by rewriting the system as an ODE and solving for the denominator,
$\frac{da}{dt}=\frac{a_{\infty }-a}{{\tau }_a}$ where
$a_{\infty }=\frac{k_A}{k_A+k_B}$ and
${\tau }_a=\frac{1}{k_A+k_B}$

Now, back to the Hodgkin-Huxley model, these rate constants are now voltage-dependent.

The full equation is $C_m\frac{d{V}_m}{dt}=G_L(E_L-V_m)+G_{\text{Na}}^{(\text{max})}{m}^{3}h(E_{\text{Na}}-V_m)+G_{\text{K}}^{(\text{max})}{n}^4(E_{\text{K}}-V_m)+I_{\text{app}}$
This is like the regular LIF model but with two extra terms for the sodium and potassium conductance. The exponents can be considered empirical fits to observed data. Interestingly, $n^4$ could correspond to the four subunits of the potassium delayed-rectifier channel.

Each gating variable has its own dynamical equation,

• $\frac{dm}{dt}={\alpha }_m(1-m)-{\beta }_{m}m$
• $\frac{dh}{dt}={\alpha }_h(1-h)-{\beta }_{h}h$
• $\frac{dn}{dt}={\alpha }_n(1-n)-{\beta }_{n}n$

These rate constants have different voltage dependencies

• ${\alpha }_m$ increases while ${\beta }_m$ decreases with depolarization
• ${\alpha }_n$ and ${\beta }_n$ have similar voltage-dependencies as sodium activation variables but smaller
• ${\alpha }_h$ decreases while ${\beta }_h$ increases with depolarization, the opposite of the other ones

Type-II Neurons

A type-II neuron has a discontinuous jump in its firing rate curve and can be either inactive or active depending on input currents

Different behaviors here are related to subthreshold oscillations, oscillations insufficient to produce a spike

• Bistability is when multiple stable states of activity exist for the same variables
• An anode break is a spike in the membrane potential produced following release from hyperpolarization
• Resonance is an enhanced response to stimulation at a particular resonant frequency

A level of input is reached at which a single spike (or multiple) is produced followed by subthreshold oscillations but no sustained firing. Once enough current is supplied to cause sustained firing, the rate is no less than the rate of the subthreshold oscillations, so a discontinuity appears on the f-I diagram.

Notably, the neuron’s spiking behavior is history dependent, meaning if it was previously spiking then a lower current could cause sustained firing, even if that current would not be enough on its own. This is possible because sodium channels are more deinactivated and potassium channels are more deactivated following a spike than the steady-state level. This is one reason for bistability.

Anode breaks also occur in this model, where releasing hyperpolarization can cause a single spike. This is because hyperpolarization deinactivates $h$, which lowers the threshold for a spike.

Finally, this model displays resonance, where a current that matches the neuron’s natural frequency produces a greater response. This can be observed in the neuron’s response to a frequency sweep (known as the ZAP protocol, impedance amplitude profile).

Type-I Neurons

A type-I neuron’s firing rate increases from zero without any rate-jump. They also do not exhibit any of the features discussed with type-II neurons.

The Connor-Stevens model is a variant of the Hodgkin-Huxley model with altered parameters and an additional potassium current. It is designed to reproduce the responses of typical mammalian cells which can produce spikes at low firing rates. Its f-I curve increases without discontinuity.

The A-current is an outward, hyperpolarizing potassium current which activates at low membrane potentials and inactivates at high membrane potentials. While it might seem strange that it has a different reversal potential than the regular potassium channels, this can be seen as just a way to model additional unknown factors in an effort to fit the data better.

$C_m\frac{d{V}_m}{dt}=G_L(E_L-V_m)+G_{\text{Na}}^{(\text{max})}{m}^{3}h(E_{\text{Na}}-V_m)+G_{\text{K}}^{(\text{max})}{n}^4(E_{\text{K}}-V_m)+G_{\text{A}}^{\text{max}}{a}^{3}b(E_{\text{A}}-V_m)+I_{\text{app}}$

The gating variables update the same way $\frac{dx}{dt}=a_x(1-x)-b_{x}x$ however the A-current gating variables are fit in terms of their steady-state and time-constant, $\frac{dx}{dt}=\frac{x_{\infty }-x}{{\tau }_x}$

The A-current is slightly active right after a spike which helps prevent the rebound spikes found in Type-II behavior (since it’s hyperpolarizing).

Calcium!

Calcium ions are pretty important in fertilization, the cell cycle, transcriptional regulation, muscle contraction, and also help neurons with synaptic transmission and plasticity.

Calcium is maintained at very low concentrations in the cytosol fluid but can be found more in the endoplasmic reticulum and mitochondria. This means calcium channels opening results in an inward current. Rapidly activating channels near axon terminals react to action potentials to let in calcium in order to initiate the release of neurotransmitters. The key point here is that calcium channels create positive feedback.

A calcium spike is a broad positive spike in membrane potential that can last hundreds of milliseconds, and is often accompanied by shorter sodium spikes (action potentials)

A T-type calcium channel is a voltage-gated calcium channel which deinactivates at membrane potentials below rest, such that it applies current following periods of release

This results in post-inhibitory rebound which can only happen once the T-type calcium channels have been deinactivated (since they naturally depolarize)

This mechanism is partially responsible for the pace making properties of heart tissue

Interestingly, these neurons can switch between a depolarized tonic mode and a hyperpolarized bursting mode

• In the tonic mode, a neuron spikes regularly according to input current
• In the bursting mode, a neuron intermittently emits bursts but does not react much to inputs

Modeling Multiple Compartments

Single point models are powerful but cannot differentiate between currents placed at different locations on a neuron. Multi-compartmental models represent different sections of the neuron as compartments which have individual membrane potentials and resistors connecting them

The Pinsky-Rinzel Model

Intrinsic bursters are able to continue firing bursts of rapid action potentials without external input

The two-compartment Pinsky-Rinzel model is a simplification of a 19-compartment model and separates the somatic membrane potential from the dendritic potential

The dynamical equations for the somatic and dendritic membrane potentials are:

$$C_{\text{S}}\frac{d{V}_{\text{S}}}{dt}=G_{\text{Leak}}^{(\text{S})}(E_L-V_{\text{S}})+G_{\text{Na}}^{(\text{max})}{m}^{2}h(E_{\text{Na}}-V_{\text{S}})+G_{\text{K}}^{(\text{max})}{n}^2(E_{\text{K}}-V_{\text{S}})+G_{\text{Link}}(V_{\text{D}}-V_{\text{S}})+I_{\text{app}}^{(\text{S})}$$ $$C_{\text{D}}\frac{d{V}_{\text{D}}}{dt}=G_{\text{Leak}}^{(\text{D})}(E_L-V_{\text{D}})+G_{\text{Ca}}^{(\text{max})}{m}_{\text{Ca}}^{2}h(E_{\text{Ca}}-V_{\text{D}})+G_{\text{KCa}}^{(\text{max})}{m}_{KCa}X(E_{\text{K}}-V_{\text{D}})+G_{\text{KAHP}}^{(\text{max})}{m}_{\text{KAHP}}(E_{\text{K}}-V_{\text{D}})-G_{\text{Link}}(V_{\text{D}}-V_{\text{S}})+I_{\text{app}}^{(\text{D})}$$

The model includes fitted functions for updating the rate constants as well as an equation for the dynamics of calcium concentration:

$$\frac{d\text{Ca}}{dt}=-\frac{\text{Ca}}{{\tau }_{\text{Ca}}}+k\cdot I_{\text{Ca}}$$

where $I_{\text{Ca}}=G_{\text{Ca}}^{(\text{max})}{m}_{\text{Ca}}^2(E_{\text{Ca}}-V_{\text{D}})$ and $k$ is a geometry-dependent constant for converting total charge into a concentration

Note that with multicompartmental modeling the difference between using specific variables (per unit area) and absolute variables becomes more important to consider, specifically with how the link current is handled

In general, the number of compartments necessary for modeling depends heavily on the question being asked. For the bursting neuron of Pinsky and Rinzel, the interplay between two compartments is crucial. If one wanted to measure the impact of spreading out inputs across dendrite branches then more compartments would be appropriate

Otherwise, there are reasons adding more compartments can be problematic

1. “with four free parameters I can make an elephant, with five I can waggle its trunk” -von Neumann, i.e. one must take care to avoid overfitting
2. More parameters makes it harder to test the relevance of each parameter
3. It is difficult to gain insight into complex dynamical systems

A hyperpolarization-activated current $I_h$ is a mixed-cation (depolarizing) inward current that is activated by hyperpolarization and deactivated by depolarization

$$I_h=G_h^{(\text{max})}{m}_h(E_h-V_{\text{D}})$$

This conductance has no inactivation gating variable. It provides negative feedback but depolarizes the membrane potential, helping with regeneration after a spike (like deinactivation of sodium or calcium channels). It may help with homeostasis by regulating the ISI in this manner

The dendrites can be seen as performing some computation over the various inputs they receive. In simple models, this means summing the inputs together and performing some kind of transformation. However, relative location on the cell membrane is important and inputs closer together generally create a stronger response

In a 4-compartment model which takes into account these observations, a current of 75pA applied to 3 compartments (225pA total) does not trigger dendritic spikes, while a current of 150pA applied directly to one compartment does trigger spikes