Week 5 - Wilson-Cowan Model

Question 1

what is the difference between excitatory and inhibitory neurons?

Answer 1

EXCITATORY NEURONS
- Excite the activity of other neurons

INHIBITORY NEURONS
- Suppress the activity of other neurons

Question 2

What are the four connections in excitatory-Inhibitory coupling of two neurons?

Answer 2

• The two self connnections, which represent how activity in the neuron evolves on its own
  -Excitatory to inhibitory (how much inhibitory activity increases)
• Inhibitory to excitatory (How much excitatory activity decreases)

Question 3

what is an analogy for excitatory and inhibitory interactions

Answer 3

predator and prey analogy

The more prey, the more predators
The more predators, the less prey

Question 4

What is C1, C2, C3, C4

Answer 4

C1= Excitatory self interaction
C4 = Inhibitory self interaction
C2 = Inhibitory to excitatory
C3 = Excitatory to inhibitory

Question 5

what do the coupled differential equations of the Wilson-Cowan model show?

Answer 5

They specify how E and I change over time

Given the previous values of E and I up to time t, what is the Change in E and I at time t

This can then be used to calculate the actual values of E over time

Question 6

How do you calculate the values from the differential equations?

Answer 6

You use numerical integration of the differential equations

This is based on making a local estimate of the rate of change

Question 7

What is eulers method?

Answer 7

Euler’s method is a way of doing numerical integration to approximate the solution of a differential equation.
It predicts values of
𝑦
y when you only know the value at a starting point, but you want to estimate the whole curve.

Starting from an initial point
(
𝑡
0
,
𝑦
0
)
(t
0
​
,y
0
​
), you approximate the curve by moving in a series of small steps.
At each step:

You calculate the slope at the current point (using the differential equation).

You multiply the slope by the step size to estimate how much
𝑦
y changes.

You add this change to the current value of
𝑦
y to find the next point.

The smaller the step, the better the approximation, because the method assumes the slope stays constant over each step.

Question 8

Why is the a -E and -I in the first part of the Wilson-Cowan equations?

Answer 8

These represent the refractory period where the neurons that previously fired, do not fire

So the activity at each time point starts by subracting the amount of activity that has already fired (-E or -I).

The model assumes a population of neurons. So at each time point, the maximum of neurons fired is limited to the neurons that did not fire previously

Question 9

What is the Wilson-Cowan equation

Answer 9

Write out

Question 10

What does the S term in the Wilson-Cowan represent?

Answer 10

The non-linear activation function

This means that the weigthed combination of external and internal input is passed into the non-linear activation function (Se and Si)

This is often a sigmoidal function. The signmoid function has two nested parameters, a and b that allow you to change the shape of the function

Question 11

What is the output of running the wilson cowan model over time?

Answer 11

You get the amplitude of both the excitatory and inhibitory pool of neurons. Each produce oscillations. The two oscillations are slightly out of phase because one is inhibiting the other and the other is exciting the other

You can plot them against each other in a phase diagram, with some parameter sets you get an oval shaped attractor

Question 12

What happens if you give a short external stimulus with really high inhibition

Answer 12

The activity for both E and I might trail out

Question 13

What is the effect of changing the inhibition

Answer 13

The frequency and amplitude of oscillations change

Question 14

what are applications of Wilson Cowan

Answer 14

It is useful in the micro, meso and macroscales

You can apply it to behvaiour in ambigous visual stimuli, as it can frame competition between visual perceptions

At the macro-scale, you can apply the oscillations to different nodes of networks.

This shows that the same equations can be applied across spatial scales

Question 15

How to apply wilson cowan to macroscale neuroimaging

Answer 15

The equations can be extended to allow for coupling between other regions, with E recieving input from other excitatory nodes

E weights can be based on a connectivity matrix e.g DTI based imaging

Deco et al, 2013

Question 16

How to apply wilson - cowan to visual perception?

Answer 16

E-I weights can be modelled based on human perceptual responses, which calculate probabilities of transition between perceptual responses

Huguet et al, 2014