So what we need to do is to add on some more stuff to our f ov V.

What we need to do is to give f of V a range where the voltage can, in fact,

increase. So now, what have we done here?

We've added in another fixed point. So we have here, stable fixed point.

Now, what's up with this fixed point? So remember, that here, the voltage heads

toward the fixed point. What's going to happen as we cross this

fixed point. So now with voltages larger than, than

this value, you can see that this dV dt is now positive and we're going to start

heading out to larger and larger values. And so in response to dynamics like this,

what's going to happen is that as one crosses this effective threshold.

So now if you have some effective input that takes you above this value, now the

voltage is just going to, now the voltage is just going to increase.

So that means we still need a couple of extra pieces as we needed for the

integrating of firing neuron. We're going to add a maximal voltage, not

a threshold, now the threshold is determined intrinsically by the crossing

by this unstable fix point of f of V. But now we need some maximal voltage

beyond which the spike can not continue to increase.

And when we reach that voltage, we're going to reset again back to some reset

value. One example of form of a f of V that

works quite well, is simply a quadratic function.

So another example of a choice of f of V that's being shown to fit cortical

neurons very well is the exponential entry of fine neuron.

Now here, we can choose f of V, so that has an exponential piece.

So that part of the dynamics sub threshold are linear and part have this

exponentially increasing part that mimics the rapid rise of the, of the spike.

And again we have to add a maximum and reset.

So this model has an important parameter, delta, which governs how sharply

increasing the nonlinearity is. So here's a strongly related example of a

one dimensional model that gets a lot of use.

This is called the theta neuron. And in the theta neuron, the voltage is

thought of as a phase, theta. When the phase reaches pi, here, we call

that a spike. So what's neat about using a phase

instead of a continuous variable, like voltage as before, is that as soon as you

pass through pi, you wrap around to minus pi and that gives you a built-in reset,

so you don't need to add that extra part into the dynamics.

So the dynamics given by, by this equation here.

This is being shown to actually be equivalent to the one dimensional voltage

model with a quadratic nonlinearity. This model also has a fixed point, V

rest, and an unstable point, V thresh, which acts like a threshold.

Now, because this model fires regularly, even without input, so now, let's imagine

that It is zero. You can see that these dynamics are still

regularly firing. They'll continue to oscillate, the theta

neuron is often used to model periodically firing neurons.

So aesthetically, lets say, we're still a little pained by this construction of the

maximum and the reset, or even the reset on the, on the phase variable.

Is there anything else we can do to improve this simple model?

How might we prevent our spike from increasing to infinity, apart from

putting some maximum on it? So, let's try falling.

So what does that do? Now, there's another fixed point, here.

So that we still have our stable fixed point.

We have an unstable fixed point, which acts as our threshold.

And now, we have antother fixed point. Now, is it stable or unstable?

Let's just, let's just check it. So here we're increasing.

There we're decreasing. Here we're increasing.

And here we're moving back toward that fixed point, this is a stable fixed

point. Hopefully you will sort of intuitive by

now that you can tell whether a fixed point in this one-dimensional

representation is stable or unstable, by just looking at the slope of f of V at

that point. Whenever the slope is negative, that's

the stable fixed point. And if the slope is positive, it's an

unstable fixed point. So now we have this fixed point.

What's the dynamics? Now once we get above our threshold, we

increase. And instead of increasing without bounds,

we go to this fixed point. So that's great.

However, the problem is that it stays there.

The system is called bi-stable. In order to allow the dynamics to come

back from that stable fixed point, let's remember what happened in Hodgin-Huxley.

Actually, two separate mechanisms helped to restore the voltage back to rest.