Instructions for experimenting with the Wilson-Cowan extended equations (Adapted from Chapter 6 of Simulating, Analyzing and Animating Dynamical Systems by Ermentrout) Download wilson-cowan-network.ode from the course website and run it in XPP. There is a 1D spatial domain with 101 locations at which two variables u (excitatory) and v (inhibitory) are defined. You can see the actual equations by clicking on the "Eqns" tab in the main window. By default only "u" at two locations is plotted. To see more, open the "ICs" window and use the double-down-arrow to scroll down and click in several boxes (e.g. U100, U110, U120, U130, U140, U150 and V100, V110, V120, V130, V140, V150). Now run the simulation and you will see a wave propagating. Where does the wave start? Scroll back up to look at initial conditions for U0 through U4, the first five locations. To see what's happening in both space and time, in the main window type "VA" (Viewaxes Array). A dialog should appear where you can enter the following: Column 1: U0 NCols: 201 Row 1: 0 NRows: 241 RowSkip: 1 Zmin: 0 Zmax: 1 Autoplot(0/1):0 ColSkip: 1 The array plot feature plots time descending vertically and space extended horizontally. Click "Redraw" to see the activation as a function of space and time. Since tmax = 60 and dt = 0.25 there are 241 time steps. To see both activation and inhibition plotted in time and space, hit "edit" and change NCols to 402 and ColSkip to 2, then OK, then Redraw. As a shortcut, you can open the Initial Data window, click U0, scroll down to the bottom, click V200, then click "arry" to generate the array automatically. To plot U or V as a function of x (position) at different times, we can "transpose" the data using (File Transpose) and entering Column 1: U0 NCols: 402 ColSkip: 1 Row 1: 0 NRows: 12 RowSkip: 20 Now in the Initial Data window click on e.g. U4 and the xvst. U4, the 5th row of data now represents the solution at all values of "x" and time t=(0 + (5-1)*20=80) (since RowSkip is 20). Note that once x reaches 200, the data represents V(x-201) rather than U(x), so there is a second peak 200 points later. Back in the Initial Data window, click on several rows (say U0 through U9) to see snapshots of the profile of U and V at times 0, 20, 40, 60, 80, etc. By what time does the response essentially damp out and return to a uniform steady state? Does this match what you observed using the array plot? To make an animation of the dynamics, create a file called something like "wilson-cowan-network.ani" in the same directory as wilson-cowan-network.ode and put the following lines in it: # animation for the array # wilson-cowan-network.ani vtext .8;.95;t=;t fcircle [0..200]/201;(U[j]+1)/2.2;.02;[j]/201 end I have no idea what this syntax means, but it more or less works (for details see the Ermentrout book). Once this file is in place, go to the main simulation window and click (Viewaxes Toon) and an animation window should appear. Select wilson-cowan-network.ani using the File button. Click OK, and then you should get a message such as "Loaded 202 lines successfully!" in your terminal window. Then click "Go" and watch the animation.