Differential Equations with Matrices

Question

I'm trying to implement the differential equation of a Cellular Neural Network in Mathematica as seen below:

A = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}};   
B = {{9, 8, 7}, {6, 5, 4}, {3, 2, 1}};   
u = {{10, 11, 12, 1}, {20, 21, 22, 1}, {66, 77, 88, 1}};
Ioff = 1;

NDSolve[{x'[t] == -x[t] + ListConvolve[A, x[t], 1] + ListConvolve[B, u, 1] + Ioff,
x[0] == u}, x, {t, 0, 2}]

But NDSolve returns two errors:

ListConvolve::kldims: "The kernel {{1,2,3},{4,5,6},{7,8,9}} and list x[t] are not both non-empty lists with the same tensor rank."

NDSolve::ndfdmc: Computed derivatives do not have dimensionality consistent with the initial conditions.

I would really appreciate any help since I'm a beginner in Mathematica and I don't know what I'm doing wrong

Answer 1

You need to define a little helper function. First, however, note that this computes:

NDSolveValue[{x'[t] == -x[t] + ListConvolve[B, u, 1] + Ioff, 
  x[0] == u}, x, {t, 0, 2}]

So, the issue is with the ListConvolve that related to the unknown x[t]. You can feed x[t] back into a function:

 fun[xValAtT_] /; MatrixQ[xValAtT, NumericQ] := 
 ListConvolve[A, xValAtT, 1]

and then call NDSolve with that function:

if = NDSolveValue[{x'[t] == -x[t] + fun[x[t]] + 
     ListConvolve[B, u, 1] + Ioff, x[0] == u}, x, {t, 0, 2}];
if[0.2]
(*
{{193.082, 206.046, 294.78, 182.994}, {151.239, 169.641, 227.922, 
  127.526}, {321.59, 337.303, 480.992, 271.091}}
*)

Note, however, that this system seems to explode for larger values of t; perhaps you need to re-check your model, I am not sure.