### Video: A simple spiking neuron model: sodium channels alone

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In the previous video that gave the biological background for a simple neuron model, we learned three important properties of the sodium channels. First, since the concentration of sodium ions is higher outside the cell, sodium rushes into the cell when the sodium channels open, increasing the voltage inside the cell. Second, since sodium channels tend to open when the voltage increases, they create a positive feedback loop. Increasing the voltage leads to more sodium channels opening, which leads to an increased voltage, etc. This positive feedback loop is fast, which can lead to the voltage shooting up rapidly. However, there is a threshold voltage which the voltage must cross before the positive feedback loop is triggered. If the voltage starts below the threshold, it will stay low or even decrease. Only if the voltage is somehow kicked above threshold, say due to input from other neurons, is the positive feedback loop engaged, and the voltage shoots up fast.

Using the three properties of the sodium channels, let's create a dynamical system model of a neuron in the form an autonomous differential equation. For simplicity, we won't model the sodium channels directly, but just model the resulting dynamics of the neuron's voltage.

For now, our dynamical system model will have one state variable $v(t)$, which will be the voltage of the neuron at time $t$. Well, let's not make it exactly the real voltage of the neuron, but a fake voltage to make the equation easier. Here's why we'll use a fake voltage.

When a neuron is at rest, its voltage is near a resting potential, which could be something like -70 mV. The negative value means that the voltage inside the neuron is lower than the voltage outside. But, to make the model simpler, let's forget about this fact, and just pretend the resting voltage is zero.

Also, if the sodium channels were all the way open, the voltage would go to some high value (something like positive 40 mV). But, let's pretend that this voltage is 1. That way, our voltage will live between 0 and 1. It isn't the real voltage but just a rescaled voltage that tells us where we are between the minimum and maximum voltage. There's one more voltage we need to add to our model: the threshold voltage. It is somewhere between the resting potential and the maximum voltage. (In real life, it's much closer to the resting potential). Let's leave it as an unspecified parameter and call the threshold voltage $a$.To model the influence of the sodium channels on the voltage, here's what we want our model to do. If the voltage is below $a$, then the voltage should decrease back down to the resting potential 0. However, if the voltage somehow gets kicked above $v=a$, then the voltage should increase to 1.

If we write the differential equation model in the form \begin{align*} \diff{v}{t} = f(v), \end{align*} what should the function $f(v)$ look like to have these properties? For starters, what should the phase line look like?

We definitely want stable equilibria at $v=0$ and $v=1$. The threshold voltage $v=a$ should be an unstable equilibrium between them. We want the vector field, or direction arrows pointing away from $a$ and toward both 0 and 1. In this way, the voltage will move upward if it is above $a$ and downward otherwise (assuming we stay between 0 and 1).

To get this behavior in the phase line, what should the function $f$ look like? $f$ should be positive in the interval where $v$ is between $a$ and $1$ and should be negative elsewhere. The simplest form of such an $f$ is a cubic $$f(v) = -v(v-a)(v-1).$$ In this case, the graph of the function $f$ looks like this.

With this function $f$, the differential equation is \begin{align*} \diff{v}{t} = -v(v-a)(v-1). \end{align*}

For this model, can you verify that the equilibria are $0$, $a$, and $1$, that the equilibria $0$ and $1$ are stable and the equilibrium $v=a$ is unstable?

The neuron model has the required behavior. Let's imagine that the neuron starts at rest. If it then receivews an input that bumps its voltage upward, but the voltage stays below $a$, what happens? The voltage decays back down to zero. (Since we aren't modeling the inputs, just imagine that you change the voltage by hand and start it at a new value.)

If the neuron receives a larger input, so that its voltage is kicked above $a$, what happens? The voltage increases toward 1. In this way, the neuron responds with a large deflection of its voltage only if it receives a big enough input.

According to this model, the neuron's voltage will stay at a high value forever, approaching the value of 1 as time increases, since $v=1$ is a stable equilibrium.

In reality, though, the voltage should just go up briefly and then come back down. We want it to go back down so that it can respond to future inputs.

Unfortunately, we cannot get this behavior with our one-dimensional autonomous differential equation model. In such a model, the state variable can never turn around, as it can only move in one direction at each point in the phase line.

The secret to bringing the voltage back down lies with the potassium channels, which we have ignored so far. When we add the potassium channels to our model, we will add a second variable and a second differential equation. With such a two-dimensional state space, we will be able to make the voltage turn around and go back down, and maybe even have other behaviors.

In the next video, we'll show how to create such a model.