# How to initiate spirals in this model?

I am trying to reproduce Tang & Othmer paper which is related to excitations and oscillations in G-protein model in Dictyostelium discoideum, an amoeba species. The mathematical model in the paper is like below:

where $w_1,w_2,w_3,w_4,w_5,u_1,u_2,u_4$ are biochemicals and other parameters are constants associated with Dictyostelium discoideum, $sr(.)$ is secretion rate function. I am supposed to simulate a situation in a square mesh where cells are uniformly distributed and interact each other. $w_5$ is the extracellular cAMP (cyclic AMP) which diffuses out from the amoeba cells and creates spiral wave forms in space under some conditions. So I basically simulated this using Euler method in MATLAB. But I am unable to generate spirals in simulation.

Can someone give me some idea to generate $w_5$ spirals in the mesh?

Below is the results (for 5 minutes) considering a single cell and no diffusion content. (Just to show that I have reproduced some results from paper)

One 2D solution looks like below (from the paper) :

And my attempt to this problem is unsuccessful. What I did to initiate output is first define every cells with an initial value set for variables like $w_1, w_2 ...$ etc except a patch where those values are $180°$ shifted (imagesc plot of $w_5$ shown in the figure) . I selected these initial values from the ODE plots. But this arrangement is proving to be unsuccessful.

UPDATE : Now I improved my code so that output is bounded between 2 values. Its not blowing up any more. I can provide my code if necessary. Here is the link to a video output I created.

• I'm confused, how come the question shows a one-time-zero-space-dimensional plot, but the rest is about a 2+1 pde? Also, um, have you tried doing the same thing they did in that paper with the same parameter values? If you have, just put the incorrect results directly in the question. Without details these kinds of debugging questions turn into a kind of guessing game (IME), which isn't much fun, and is also usually considered off-topic. Bandwidth and question space are cheap anyway. – Kirill Jul 1 '15 at 9:01
• I'm more confused now. Are you asking how to solve a differential equation with diffusion in it? Or is it that you have implemented it, but it's not working correctly? What exactly does "unable" mean here? You can't get anything useful in 2d after neglecting the sole diffusion term. – Kirill Jul 1 '15 at 9:26
• Yes, but what do the 2D solution look like? How is it unsuccessful? I.e., did you get no result at all, or did you simply fail to achieve the spirals you expect to see? – Bill Barth Jul 1 '15 at 14:21
• Wait, that's a figure from the paper, not your program. I don't think that's what Bill Barth was asking. – Kirill Jul 1 '15 at 23:19
• Check the sign of your diffusion term. – Bill Barth Jul 2 '15 at 11:57

So I managed to create a spiral finally. The method I used to initiate spiral is by initializing the mesh with a set of initial values expect a small square patch which is initialized with $180°$ phase shift with others and some having $90°$ phase shift. The link to spiral video I created is here.