Wilson-Cowan Model of Network Dynamics with Delay

This is a simulation of the neural network dynamics as governed by the Wilson-Cowan equations. The simulation shows that a network of excitatory and inhibitory neurons follows the oscillatory dynamics predicted by the Wilson-Cowan differential equations with a delay.

Wilson-Cowan Equations

The Wilson-Cowan equations are a system of two coupled differential equations. Following the example of Coombs and Laing, we introduced a heterogenous delay. The modified Wilson-Cowan equations are, then:

Each of the variable U, V represent the proportion of active neurons in a excitatory and inhibitory population of neurons, respectively. The delay tau 1 represents a population acting on itself, while tau 2 represents the delay from acting on the other population. The parameter alpha is nearly always one, while beta is typically between 40 and 60. The choice for parameters a, b, c, d depend on the desired dynamic but always have a magnitude less than or equal to one. Some interesting choices are (-1,-1,-1,-1) for synchrony between the groups of neurons, (-1,1,1,-1) for antiphase oscillation, and (-1,-1,1,-1) for the inhibitory group to have a double phase synchronized with the excitatory group.

Network

Instead of using a differential equation to model each neuron, each neuron in this model uses a weighted random number generator. Neurons that have a random value above a threshold are said to fire and then go into a refractory period. The weight comes from the same parameters a, b, c, d as in the Wilson-Cowan equations. Those neurons that fire are said to be in the "active" group and cannot fire until after the refractory period. Increased activity of a population is plotted in the model, and roughly follows the dynamics predicted by the Wilson-Cowan equations.

How to use this simulation

Choose parameters for the desired dynamic. Hit the play button and watch as the top plot, the Wilson-Cowan equations, matches the activity of the neural network depicted in the bottom plot. The blue and cyan plot lines correspond to inhibitory neuron activity, and the red and orange lines represent excitatory neuronal activity.

Sources

Coombs, Stephen and Carlo Laing. Delays in activity based neural networks.. Royal Society J 2008.

Ermentrout, G. Bard and David H. Terman. Mathematical Foundations of Neuroscience. Springer 2010. New York, NY.

Izheikevich, Eugene M. Dynamical System in Neuroscience: The Geometry of Excitablilty and Bursting. MIT 2010. Cambridge, Mass.

Credits

This simulation was designed in EJS 4.3.7 by Colin Thomson under the guidance of Dr. Wolfgang Christian. This simulation is part of a series of computational neuroscience simulations for an independent study at Davidson College entitled PHY/CSC 397: Advanced Software Design for Science. The source code, as an EJS file, is available on the comPADRE digital library.

Fall 2012