Neuron Models-Introduction to Hodgkin Huxley Model
In this blog, we will discuss the Hodgkin-Huxley model. Before delving deeper into the model, let us revise some rudimentary concepts.
An important aspect of a neuron is the membrane potential denoted as Vₘ =(Vin — Vout). The Vin is measured inside the neuron and Vout is measured outside the neuron. Now this difference is not the same if we change the measurement position. That is, the difference Vin — Vout is unique to a particular position (if a neuron is considered in 3-D space and the x, y, and z coordinates of measurement change we will have a different voltage difference (Vₘ) each time). It is interesting to note that the voltage difference allows the synaptic inputs to travel. If the voltage were the same there wouldn’t be a reason for current to flow from dendrites towards the soma.
In general, if we think of a single neuron spiking model, they can be of two types-
1) Point Model — Models that describe the membrane potential of a neuron by a single variable Vₘ are called point models/ single compartmental models. This is the simplest model, and our discussion will be based on such models. We will not discuss how the input from different synapses travels along dendrites to soma. We will consider all inputs to be projecting at a point.
2) Multi-Compartmental Model- When the complexities of real membrane conductance are included, the membrane potential must be computed numerically. This is done by splitting the modeled neuron into separate regions or compartments and approximating the continuous membrane potential Vₘ(x, t) by a discrete set of values representing the potentials within the different compartments. This assumes that each compartment is small enough so there is negligible variation of the membrane potential across it. The precision of such a multi-compartmental description depends on the number of compartments used.
Direction of modeling
The cell membrane is a lipid bilayer that is impermeable to most charged molecules. This insulating feature causes the cell membrane to act as a capacitor by separating the charges lying along its interior and exterior surfaces. The membrane conductance depends on the density and types of ion channels.
Since the membrane is mostly impermeable to ions, there has to be a mechanism that lets them in and out. This is the prime role of ion channels. We can think of them as small holes in the membrane, which sometimes open up allowing the flow of ions.
The equivalent circuit is as :
The ion pumps work at relatively steady rates so the currents they generate can be included in a time-independent leakage conductance. All of the time-independent contributions to the membrane current can be lumped together into a single leakage term Gₗ (V − Eₗ). Eₗ is kept as a free parameter and adjusted to make the resting potential of the model neuron match that of the cell being modeled. Similarly, Gₗ is adjusted to match the membrane conductance at rest. Iₑₓₜ is the current injected by synapses in a neuron. For modeling purposes, we will consider it in our control through a patch-clamp experiment. The capacitor and the leakage parameters ( Voltage and Conductance) are in parallel because the current that is going across the membrane has a path within the membrane itself. This entire model is known as the Leaky Integrated Fire Model.
Equations governing this system are as follows:
Note that at resting potential there should be no net current flow across the membrane. This will happen when Vₘ = Vin — Vout will match Eₗ. Hence this is the potential at which the leaky channels are not allowing any current to flow.
The drawback of this model: Let the neuron be at rest (say). A current Iₑₓₜ = Iₒ is injected for a certain period T, and we see how ΔV changes with time. To generate action potentials in the model, the equation is augmented by the rule that whenever V reaches the threshold value Vth, an action potential is fired and the potential is reset to Vrest. It is seen that the waveform of ΔV is periodic implying that the neuron will fire periodically which is unlikely as most neurons have spike rate adaptation. Have a look at this wonderful tutorial to get a better understanding of LIF models.
This model has a few missing things, one being the ion channels. It is the ion channels and their property that cause action potential. We can prove that there is a true threshold VT for these kinds of neurons and also show the action potential is going to be an all or none event.
Hodgkin Huxley Model
We will see how the Hodgkin-Huxley system of equations describes the spiking behavior of neurons. We learned the Leaky integrate and fire model and we will change the model into a spiking model using Hodgkin-Huxley equations by incorporating voltage-dependent Na⁺ and K⁺ ion channels. To explain the behavior of Action Potential the Na⁺ and K⁺ ion channels are sufficient. Hodgkin Huxley saw this in the Gaint squid experiment. They did a patch clamp experiment on a giant squid axon and with a variety of measurements, came up with the exact property that governs the circuitry with the properties of Na⁺ and K⁺ ion channels.
Outside the cell, the concentration of sodium, chloride, and also calcium is high. Inside, the cell potassium has a higher concentration, which is necessary for many of its life operations. It maintains these ionic gradients with specialized pumps, that exchange sodium for potassium.
Na⁺ and K⁺ ions have some equilibrium potential based on the concentration of Na⁺ and K⁺ outside and inside the neuron. That is known as the reversal potential. If the membrane potential is at ENa (Na reversal potential), then there will be no net Na⁺ ions in the membrane. If the voltage is different from ENa and there are paths available for Na⁺ to flow through the membrane then Na⁺ will flow in such a direction so that the voltage is pulled towards ENa. Similarly, it happens for K⁺ ions. We can write the equations as follows:
For us to simulate the differential equation and see how the voltage changes with time given Iₑₓₜ(t) we need to know how GNa and GK are changing with V and t. We do it by voltage clamp experiments. One of the channels (either Na channel or K Channel ) is blocked and the other channel current is measured.
We repeat this experiment for different values of V say V1<V2<V3 and so on and measure the current for different voltage values. For the experiment, we block Na channels and measure K current. From here we can plot how the conductance GK is changing with time for various values of V (The plot is shown below). It is observed that as Voltage increases the maximum value of K⁺ current, and hence the conductance keeps increasing. Also, the time constant in the plots keeps decreasing with increasing voltage. The plots show how GK varies with time and not Voltage. To plot GK as a function of V, we take the GKmax corresponding to every Voltage and plot it.
Similarly, we repeat the experiments for Na⁺ by blocking the K⁺ channels. We observe similarly that the peak's magnitude increases and time constant decreases as voltage clamp values increase.
The difference in the Na⁺ and K⁺ current(I) plots is that Na⁺ current goes back to 0. As an explanation, Hodgkin Huxley proposed that K⁺ ion channels only have activation gates (As voltage increases, no of gates open more and more) while Na⁺ ion channels have both activation and inactivation gates (No of gates opening increases with a decrease in voltage). It was further found out through experiments that K⁺ has 4 activation gates and Na⁺ has 3 activation and 1 inactivation gate.
It was seen that if we repeat those voltage clamp experiments a stochastic behavior is observed i.e. the ion channels are not always opening or whatever fraction of them are supposed to open on average they don’t always open in every trial. This means there is a probability associated with the opening and closing of these gates or ion channels.
Hodgkin Huxley proposed that:
Probability of Na⁺ channel activation gates being open = m; Probability of Na⁺ channel inactivation gate being open = h; Probability of K⁺ channel activation gates being open = n.
With the assumption that these gates work independently and that they are a function of Voltage then the probability of Na⁺ ion channels being open would be m3h and the conductance is multiplied by m3h
Rewriting the equations:
m, h, and n also change with voltage. Assuming first-order kinetics of the gates Hodgkin and Huxley describe the gating variables m,n,h.
There are 4 variables (V, m,h,n) and 4 differential equations. These system of equations are simulated numerically. n∞(V), Tₙ(V), and the other variables in the above equations are obtained with the help of voltage clamp experiments that we had described earlier. So essentially with fitting the curves, Hodgkin and Huxley estimated empirically what n∞ and other variables would look like as a function of V. The plots are given below for n∞(V) and Tₙ(V).
Note that the time constant of each of the gating variables changes almost an order of magnitude from m to n and then from n to h. This is the most important point in terms of describing action potential using Hodgkin-Huxley equations.
Putting everything together :
The potential from Vrest moves to the right slightly because the current is injected into the system. This small depolarization leads to an increase in m. Now since m is very fast, it reaches m almost instantly. But then h is starting to change which effectively results in more Na⁺channels being open. K⁺ channels is also slow, so the gates haven’t open up yet with the small change in V.
Hence, With the increase in V, m is open and Na⁺ will flow in such a way that potential is pulled towards ENa. So from -60mV to 40mV. We see that the Voltage is moving towards the right, and more Na channels are open. The influx in Na⁺ in a short time shoots up the voltage. By this time when all the m gates are open, K⁺ channels start to open as a result K⁺ conductance increases. Now the K⁺ ions will flow in such a direction that the potential is pulled to EK (-80mV). While the potential drops, the h gates close and Na channels are completely closed. After some time, the entire system returns to rest. The diagram of the action potential is given as follows. You can simulate to understand it fully.
Vth is not mentioned while describing Action Potential. We will show that indeed there is a particular value of voltage that needs to be crossed when the current injection is happening only then there will be an action potential. If depolarisation is not large enough the voltage will come back to rest causing no action potential. This we will discuss in a later blog.
References:
[1] https://neuronaldynamics.epfl.ch/online/Ch2.S2.html
[3]Theoretical Neuroscience; Computational and Mathematical Modeling of Neural Systems. By Laurence F. Abbott and Peter Dayan