Good morning.

Today we are going to talk about dynamical systems theory.

This is an important theory to understand at least the basics of if you

are interested in modeling single neurons or especially large networks of neurons.

What is a dynamical system?

As the name implies, it is a system that changes its state in time.

That's the definition of a dynamical system in English, but

we can also describe it in a more mathematically precise way.

In math lingo, a dynamical system is a set of differential equations,

which are usually coupled.

So for example, if our system state is described by x1,

x2, x3 all the way up to xn where each variable

represents some property of the system.

Then the dynamical system would be an equation for

the time derivative of x1, which is a function of all the other variables.

The time derivative of x2, which is a function of all the other variables.

So that would be f1.

And you go all the way down the list, and you have a time derivative of xn.

And that is fn of all the other variables.

So that's all a dynamical system is.

One canonical example of a dynamical system in neuroscience

is the Hodgkin-Huxley equations.

The Hodgin-Huxley equations are a four dimensional dynamical system.

So what did the Hodgkin-Huxley equations look like in terms of a dynamical system?

Well, one variable that changed with time was the voltage, so

we had an equation for dV/dt.

The time derivative of the membrane potential,

the membrane voltage and that was the function of the rest of the variables.

We also had an equation for the activation variable of the potassium channels.

That was n.

And we had a dynamical equation for

the activation variable of the sodium channels.

That was m.

And lastly, we had an equation for

the time derivative of the inactivation variable, the sodium channel.

So our four variables were the memory potential, and

then these three inactivation, or activation variables.

So the Hodgkin-Huxley equations describe a four dimensional dynamical system.

Because the state is described by four different numbers, and

all of those numbers change with time.

And the precise form of these equations lead to the phenomenon of an action

potential.

Dynamical Systems Theory tells us about the behavior of our system of differential

equations without requiring us to solve for the actual equations themselves.

Because of that, it ends up being mostly

just drawing pictures that are informative somehow about the system of interest.

Today we're just going to talk about 1-D and 2-D systems,

but if you're interested in higher dimensional dynamical systems, see

the Steve Strogatz book on dynamical systems theory.

That's a wonderful little text that explains things in a very

intuitive manner.

Okay, let's start with 1-D.

One dimensional dynamical systems.

We only have one dimension so we only have one equation.

Here's an example of a one dimensional

dynamical system, dx dt = x squared- 9.

So first, just to get a feel for things, let's plot this out.

So our x-axis will be x, and our Y axis will be dx, dt.

The y-intercept is -1, 2, 3, 4, 5, 6, 7, 8, 9.

And the x intercepts, you can see, are 3 and -3, and this is just a parabola.

Something like that.

Okay, well we want to draw a picture of our dynamical system, so

we can figure out how it changes in time, but we don't actually

want to go through all the rigmarole of solving this system for x sub t.

So what can we do?

This is a one dimensional system, so we can just draw it on a number line.

And the thing to keep in mind here, is that for

every x on our number line, there is an associated dx/dt.

So we've kind of drawn that out over in this graph right here.

So for example, what if we chose, or let's label our number line.

So that's 3- 3.

What if we chose x to be 1?

So, what is dx/dt when x = 1?

Well, that's pretty easy.

We plug in 1 into our dx/dt equation.

So 1 squared- 9 is, so that is dx/dt when x is 1.

So x = 1 : dx/dt goes -8.

And what we will do now is to draw an arrow

indicating what dx dt is at x = 1.

So since dx dt is negative, our arrow will point to the left.

And the magnitude or the length of the arrow will give

an indication of how big that is, of how big that derivative is.

All right, what about when x = -1?

Well, this is a symmetric equation, right?

So, dxdt is also going to be -8.

So at -1, we'll have the same arrow that we had at.

What about when x = 2?

2 squared- 9 is -5.

So dx dt = -5.

So again we'll draw an arrow pointing to the left but

it won't be quite as long as the arrows at x = 1, x = -1.

What if we had x = 4?

Well if we plug that in, x squared, 4 squared- 9 is 7, 16- 9 is 7.

So dxdt = 7.

So now our derivative is positive, so X = 4, we draw an arrow pointing to the left.

So maybe you're kind of getting the idea.

At every possible X, we have an associative derivative.

So we can draw in all of these arrows, just by going through and

calculating the derivative at all of the x's.

What if x = 3?

Well 3 squared- 9 is 0.

So here dx dt = 0.

Instead of drawing an arrow, we'll just draw a dot.

And what about x = -3?

Well, dx dt = 0 again, so we'll draw another dot right there.

And the point is, now we have a picture of our system.

So just by looking at the collection of arrows on this number line,

we can see how our system will change with time.

If we were to plot x at 0,

it would move to the left because it would follow the direction of the arrows.

If we were to plot x at 4,

it would move to the right, x would keep increasing and increasing.

If we were to plot x at 3, it would stay right where it was.

And same with -3.

So this is what dynamical systems theory does.

It tells us how to draw a picture that shows us how the system

behaves without actually having to solve that differential equation.

Now how would we actually characterize our picture?

If I were to call you on the phone and tell you what picture I just drew for

this dynamical system, how would I tell you what's going on?

Well, I'd probably say that when x is greater than 3,

dx dt is positive, so x moves to the right.

And when x is between -3 and 3, dx dt is negative, so x moves to the left.

And when x is less than -3, x is positive again.

So x moves to the right.

And you could get a pretty good idea of the picture.

And the last thing I would tell you is that at -3 and

at +3, dx dt is 0.

So if x is -3 or x is 3, there won't be any push on x.

And x will stay exactly where it is.

These locations are called fixed points.

There are fixed because if x is there, dx/dt is 0.

So the system will not change in time.

So oftentimes, when you are first confronted with a dynamical system, so

a differential equation or a set of differential equations.

The first thing you find are the fixed points.

You find the locations where the time derivative is equal to 0.

So in our case, those locations are x = 3 and x = -3.

So there and there.

Those are the fixed points of our system.

Because x will not move if x is at a fixed point.

The last thing we can do to describe our system is to classify those fixed points.

And our two main classes, are stable and unstable.

So can you guess which of these six points is stable and which is unstable?

Hopefully you guessed that -3 was stable and

that +3 was unstable.

Now how do we know that?

Well, stable means that if you nudge it just a little bit,

it will return to where it was.

So what happens if you nudge x = -3 just a little bit to the right.

Well if you nudge it just a little to the right,

it's going to feel a push to the left.

And if you nudge it just a little bit to the left,

it's going to feel a push to the right.

So the dynamical system tends to keep x at its fixed point -3.

That is a stable fixed point.

What about an unstable fixed point?

In an unstable fixed point, if you nudge x just a little bit to the left or

right, it will keep going in the direction you nudged it.

So if you nudge x just a little bit to the right here, it will want to move more and

more and more to the right.

And if you nudge it to the left, it will want to keep moving to the left more and

more and more.

So that makes the fixed point at x = 3 unstable.

So finding the fixed points of your system and classifying them as stable or

unstable is very important in trying to characterize your system

Dynamical Systems Theory Intro Part 1: Fixed points (by Rich Pang)

Computational Neuroscience

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