Coupled PDE system - problem solving

Hi, I have the following set of equations, which for ease of sight I wrote in mathematica:

I am trying to use pdepe() to solve this system, but receive the infamous error: "Spatial discretization has failed."

My code is as follows:

Consts = [Cm VL VK VNa gL_ gK_ gNa_ Immax Imduration Vm0 a rho_i];
[mout,hout,nout,Vmout,tout,xout] = cableEqn(xAxis,tAxis,Consts);
function [mout,hout,nout,Vmout,tout,xout] = cableEqn(xAxis,tAxis,Consts)
        m=0; 
        x=xAxis;
        t=tAxis;

          % constants
          Cm=Consts(1);
          VL=Consts(2);
          VK=Consts(3);
          VNa=Consts(4);
          gL_=Consts(5);
          gK_=Consts(6);
          gNa_=Consts(7);
          Immax = Consts(8);
          Imduration = Consts(9);
          Vm0=Consts(10);
          a = Consts(11);
          rho_i = Consts(12);

          sol = pdepe(m,@pdefun,@pdeic,@pdebc,x,t);
          mout = sol(:,:,1);
          nout = sol(:,:,2);
          hout = sol(:,:,3);
          Vmout = sol(:,:,4);
          tout=t;
          xout=x;

      function [c,f,s]=pdefun(x,t,u,DuDx)

          mm=u(1);
          nn=u(2);
          hh=u(3);
          Vm=u(4);

          %conductivity functions
          gNa =gNa_ * (mm^3) * hh;
          gK = gK_ * (nn^4);
          gL = gL_;

          % currents
          INa = gNa*(Vm-VNa);
          IK = gK*(Vm-VK);
          IL = gL*(Vm-VL);
          Iion = INa+IK+IL;
          Is = Im(t,Immax,Imduration);

          c=[1; 1; 1; Cm/(a/(2*rho_i))];
          f=[0; 0; 0; 1] .* DuDx;
          s=[alpha_m(Vm)*(1-mm)-mm*beta_m(Vm); % m(t) part
              alpha_h(Vm)*(1-hh)-hh*beta_h(Vm); % h(t) part
              alpha_n(Vm)*(1-nn)-nn*beta_n(Vm); % n(t) part
                  -1*(Iion-Is)/(a/(2*rho_i))]; % Vm(t,x) part
      end

      function u0 = pdeic(x)
         [ m0, n0, h0 ] = initVars( Vm0 );

          u0=[m0; h0; n0; Vm0];
      end

      function [pa, qa, pb, qb] = pdebc(xa, ua, xb, ub, t)
          pa=[0; 0; 0; 0];
          qa=[0; 0; 0; 1];
          pb=[0; 0; 0; 0];
          qb=[0; 0; 0; 1];
      end

end

As far as I can see, the problem might be from the first 3 equations which are basically ODE's, but I'm not sure and I don't really know what to do. The goal: solving the The Propagating Action Potential problem as you might've noticed.

Will be happy for your help. It's the first time I'm trying something with pdepe...

Answer 1

[Torsten]

pdepe is not suited to solve mixtures of PDEs and ODEs.

You will first have to discretize the PDE in x-direction and solve the resulting system of ODEs using ODE15S.

Look up "method of lines" for more details.

Best wishes

Torsten.

Answer 2

[Stanislav Steinberg]

Thanks, I thought so. I will try the method of lines then. Will update if further issues arise.