Welcome back! In the last part of this second tutorial
we're going to finalize our study of the Wilson-Cowan model using the framework of 2 dimensional dynamical
systems and in particular we're now going to discuss how one can find the fixed points of this 2 dimensional dynamical system and determine
their stability.
So just as a reminder
what we're trying to find are the fixed points of a two dimensional dynamical system using a
complimentary approach to what we did for the 1 dimensional dynamical system, where we modeled a single population of neurons in the first tutorial.
Here we define the fixed points or equivalently the equilibria or the steady states of the dynamical system
as the set of points where the dynamics of the population do not change. So similarly for this two-dimensional
system of excitatory and inhibitory neurons
that is the Wilson-Cowan model, the fixed points are defined as the points where the dynamics of the excitatory and inhibitory
populations don't change. So that's why we define the two derivatives with respect of the excitatory and inhibitory
population with respect to time and we set them equal to zero.
To understand how one can find these fixed points and determine their stability we're first going to
do this in a more graphical approach by considering
the phase plane. Remember the phase plane is the plane in which we plot one
population,
for example the inhibitory, against the other population, for example
the excitatory population, rather than plotting the activity as a function of time.
Now to determine the fixed points in a graphical way
we remember what we learned about the nullclines.
Remember that the excitatory and inhibitory nullcline shown here in the red and the blue denoted the points where the
set of points in the phase plane where either one or the other population do not change.
So the E nullcline in blue was the set of points where the population of the excitatory
activity does not change and similarly for the inhibitory nullcline.
Because the fixed points are defined as
the set of points were both the excitatory and inhibitory
populations don't change in time, you probably guessed that the fixed points are now defined exactly as
the points where the two nullclines intersect, and so this is denoted here by the three different circles,
so they correspond to the points where the two derivatives are both equal to zero.
So as we saw for the one dimensional dynamical system these fixed points
can sometimes be stable and attract different trajectories in the phase plane and can other times be unstable and
can repel all population trajectories in the phase plane.
So the question remains, how can one
determine the stability of these fixed points?
Of course as we can see graphically in the phase plane
it is very clear which are the fixed points that are stable and which are the fixed points that are unstable.
So the
magenta trajectory in this case is stable because it is attracted,
sorry,
the the fixed point 0,0 in this case is stable because it's attracts the magenta
trajectory, on the other hand the fixed point that corresponds to these two high values of the excitatory inhibitory
population is also stable because it attracts
this
trajectory in the sine. On the other hand this
fixed point in the middle of the phase plane is unstable because as you see locally from
the arrows of the of the vector field
It looks like it ends up repelling the trajectories our population trajectories around it.
So, how can we determine the fixed point stability?
In addition to the graphical way that we just illustrated,
we can also determine the stability of the fixed points using an algebraic way very similar
to the one that we apply for the one dimensional dynamical system, namely
we have to linearize the dynamical system around a given fixed point and
determine how a small perturbation around the fixed point will behave, namely whether it will
blow up, explode therefore determining that the fixed point is unstable or
whether this perturbation will actually decay in time
rendering the fixed point
stable.
So to see how we do this linearization
we take a look at the set of equations that describe the Wilson-Cowan system that tell us how the excitatory and the inhibitory population
evolve in time. We're going to do a slightly rearranging of these equations where we take the
time constant tau E
and tau I on the left hand side of the equation and move it to the right-hand side of the equation.
So now, this is the same set of equations that I have written on the slide as on the equations that we had before
we're going to slightly simplify notation, and we're going to
describe all the terms on the right hand sides of these two equations
with the two nonlinear functions that we denote with GE and GI
which are both functions of the excitatory and inhibitory
population. So this allows us to summarize this dynamical system in two dimensions as in this simplified
form where the dynamics of the excitatory and inhibitory
population is described using these two nonlinear populations GE and GI
which are given by these complex expressions that we're just not going to
carry over for our next analysis.
So we have a compact way to describe this two dimensional dynamical system that represents the Wilson-Cowan model.
So now the question remains, how can we use this notation to
linearize the dynamical system
around the given fixed point and determine the stability of the fixed point?
Let's, for simplicity,
assume that the fixed point is given by RE*  and RI*, so we used
starred asterisk to denote that the excitatory and inhibitory
population are at a given fixed point.
To do the linearization and determine the stability of the fixed point we do,
we follow a similar approach to what we did for the one dimensional dynamical system.
There, we apply the perturbation around the fixed point and we derived the one-dimensional
first order differential equation that told us how this perturbation evolved in time. This was a
differential equation that we could solve the solution was an exponential and
the sign of the exponent the lambda which was the eigen value of the system told us whether the
fixed point was stable if the eigenvalue was
negative, sorry and whether the fixed point was unstable if the eigenvalue was positive.
We do the same approach for the two dimensional dynamical system
but here the linearization has to be done in the phase plane,
so in the two dimensional phase plane. To do this
we have to define the concept of a Jacobian matrix that effectively has us compute the
derivatives of these two populations GE and GI that,
sorry, these nonlinearities that we defined to be the expressions on the right hand side of the equations on the on the previous slide that
effectively
are the linearizations along the excitatory and inhibitory
direction in the phase plane. So very similar to what we did for the 1d system where we just had a single population.
So by computing the derivatives with respect to the E and the I
population and evaluating them at the fixed point we can obtain a numerical expression of this matrix for each of the fixed points.
So as before we can obtain than a
differential equation that has solutions that are now a sum of two exponentials and
the exponents in the two exponentials correspond to the eigenvalues of this Jacobian matrix computed at the fixed point and
as before we can investigate what these eigenvalues are to determine the stability of the fixed points.
For the one-dimensional system, because we had a single exponential, the eigenvalue was always real and it was its sign
that determined the stability of the fixed point. In this case, because we have a two-dimensional matrix,
the eigenvalues can actually be real or imaginary.
This is interesting because it gives rise to much more interesting dynamics in the case of the two dimensional system than in the one-dimensional
system but it also makes the analysis a bit more complicated.
That's why we consider these two cases separately.
First we consider the case of real eigenvalues of the Jacobian and then we consider the case of complex eigenvalues of the Jacobian.
For the case of real eigenvalues let us consider the example we have studied so far.
So we have
the phase plane with the to nullclines shown in red and in blue and the three fixed points, two of which are we know
from the graphical analysis are stable and one is unstable. As you will have a chance to explore in the
notebook, if one evaluates the Jacobian
at each of these three different fixed points, we will see that the eigenvalues are real.
So this corresponds to this first case, the two real eigenvalues we denote them with lambda 1 and lambda 2.
For the two fixed points that
the graphical analysis tells us are stable, if we compute these two
eigenvalues we will find that they're real but they are both real and there the values are negative.
So this means that these two
stable fixed points are known
to attract trajectories in the phase plane. So indeed, that's what we saw with the trajectories that we had earlier.
These fixed points are also known or called stable nodes.
One can also
consider a different scenario and find where the if the eigenvalues are a real
but positive. In this case,
they define a fixed point that is unstable, which is known as an unstable node
we don't show such a fixed points in the example here on the right. And
the last case is a scenario where the two real eigenvalues have opposite signs,
so one of the eigenvalues is positive and the other one is negative.
This also defines an unstable fixed point but
one that is called a saddle point
and so interestingly if you kind of look locally around this unstable fixed point in the phase plane, you see that one of the
directions is
attracting and in another direction the trajectories end up being repelled out of this fixed point
yielding it in general as an unstable fixed point.
But in addition to this scenario one where the eigenvalues are real
we also have a scenario where the eigenvalues can be imaginary.
So in this case, it is the real part of these eigenvalues that determines the stability of the fixed points.
So let us consider the phase plane where we have now modified some of the parameters as you will explore in the tutorial
so that now
the two null points intersect at a single fixed point and now if we start with a trajectory that begins at this particular initial condition
for the excitatory and inhibitory population, we see the trajectory end up
spiraling into this fixed point. So graphically we can derive that this fixed point is stable,
but what is interesting is that the dynamics with which these
trajectory approaches the fixed point is very different than what we saw before rather than the trajectories just decaying in time or growing in time
here this decay
towards the fixed point is approached over the fixed points happens in an oscillatory manner.
So this particular fixed point, because it's attracting,
corresponds to is called a focus or stable focus and
not shown here is a different
alternative, a different scenario, where the real part of the eigenvalues can be
positive in which case the fixed point is unstable and is known as an unstable focus.
Now just like what we did for the other types of
fixed points that were nodes where we didn't have the oscillatory dynamics
we can also visualize this trajectory in the activity versus time plane.
So here we plot the excitatory and the inhibitory activity as a function of time and
what is shown by this
spiraling in
trajectory in the phase plane is illustrated by this oscillatory dynamics in the activity versus time plane. So each of the two populations
oscillate in time and their amplitude
decays in time because this fixed point is attracting. If the fixed points were repelling then what we'll see is oscillatory
dynamics with amplitude that grows in time.
Now in two dimensional dynamical systems, in addition to fixed points,
we can also get other types of equilibria or steady states that are not fixed points.
So these are sort of more interesting and relevant for different
concepts in in neuroscience and one such
equilibrium is a limit cycle.
So in this case, we can visualize the limit cycle in the phase plane by considering say two trajectories one in the
black and one in the yellow, and so we see that the two trajectories
don't converge or get away from a particular fixed point, rather
they converge towards this this limit cycle shown here in the scion.
So if we look at this in the activity versus time plane
and and before we show that let's just conclude that these limit cycles just like fixed points can be stable or unstable,
so here we see a limit cycle that is stable or attracting
they can also be of course unstable and repelling but we can also look at them in the
activity versus time plane, and we see that they are characterized by the fact that they generate these oscillations with fixed and steady amplitude.
So unlike the
focuses that we showed earlier where the activity
oscillates, but with a decaying or a growing amplitude here in the case of the limit cycle the activity oscillates, but with a fixed amplitude.
So indeed, we have finished our study of
the dynamics that can be generated by these coupled populations of
excitatory and inhibitory
neurons known as the Wilson-Cowan model. We have shown
in the last few sets of slides of this tutorial how to find the fixed points, how to determine their stability,
how to visualize the activity in the phase plane but also in the activity versus time plane, and
in the tutorial in a moment, you will have the opportunity to calculate the fixed points,
numerically evaluate the Jacobian matrix at each of the fixed points, and
determine the eigenvalues which will then determine the stability of the fixed points.
You can also change some of the parameters in the Wilson-Cowan model and look at when
limit cycles emerge. If there's time you can also study the implications of
some of the dynamics that we have modeled with this Wilson-Cowan model in a more biologically relevant scenario. For example
the generation of working memory.
