Implement Fitzhugh-Nagumo model via Crank-Nicolson - matlab

Question

For a problem, I need to implement the Fitzhugh-Nagumo model with spatial diffusion via Crank-Nicolson's scheme. Now the problem lays withing the spatial diffusion.
(V_{t}) (DV_{xx} + V(V-a)(1-V) - W + I)
(W_{t}) (epsilon(V - b*W )
whereas DV_{xx} is the spatial diffusion.
Using Matlab, the following function can be given to i.e. an ODE45 solver. However it does not yet implement the spatial diffusion...
function dy = FHN( t, Y, D, a, b, eps, I )
V = Y(1);
W = Y(2);
dY = zeros(2,1);
% FHN-model w/o spatial diffusion
Vxx = 0;
dY(0) = D .* Vxx + V .* (V-a) .* (1-V) - W + I;
dY(1) = eps .* (V-b .* W);
The question: How to implement V_{xx} ?
Besides, what matrix shape does V need to be? Normally V is depending only on t, and is thus a [1 by t] vector. Now V is depending on both x as t, thus i would expect it to be a [x by y] vector, correct?
Thank you

Answer 1

It took long, but hey its not a normal everyday problem.
function f = FN( t, Y, dx, xend, D, a, b, eps, I )
% Fitzhug-Nagumo model with spatial diffusion.
% t = Tijd
% X = [V; W]
% dx = stepsize
% xend = Size van x
% Get the column vectors dV and dW from Y
V = Y( 1:xend/dx );
W = Y( xend/dx+1:end );
% Function
Vxx = (V([2:end 1])+V([end 1:end-1])-2*V)/dx^2;
dVdt = D*Vxx + V .* (V-a) .* (1-V) - W + I ;
dWdt = epsilon .* (V-b*W);
f = [dVdt ; dWdt];
Both V as W are column vectors with a size of 1:(xend/dx)
Method of calling:
V = zeros(xend/dx,1);
W = zeros(xend/dx,1);
% Start Boundaries
% V(x, 0) = 0.8 for 4 < x < 5
% V(x, 0) = 0.1 for 3 < x < 4
V( (4/dx+1):(5/dx-1) ) = 0.8;
V( (3/dx+1):(4/dx-1) ) = 0.1;
Y0 = [V; W];
t = 0:0.1:400;
options = '';
[T, Y] = ode45( #FN, t, Y0, options, dx, xend, D, a, b, eps, I );