Introduction

Models

>>Simulator<<

Results

References

The Simulator

The simulator is available as a Java applet. It incorporates both the first and second models, although it does not model the second model with uniformly increasing excitability. Only one set of parameters is used for both models - by setting the parameters of the second model appropriately, the first model can be simulated. The parameters are as follows:

PARAMETER	FIRST MODEL	SECOND MODEL
grid size	100	100
cell density	0.2	0.18
degradation rate	0.5	0.5
diffusion rate	5	5
cAMP threshold min	10	4
cAMP threshold max	10	100
cAMP release time	2	1
cAMP released	1500	300
abs refractory time	20	8
rel refractory time	0	2
initial excitability	0	0
excitability incr	0	0.005
excitability decr	0	0.0005
random firers	1	-10

The values used are those specified in the papers, except in these cases:

• The diffusion constants are not specified for either model, nor is the method of discretizing the diffusion equation described. My discretization is described below.
• For the second model FL98 lists a degradation rate of 8, but the rate must be less than 1 in order to avoid all of the cAMP from disappearing each time step.
• The second model's excitability increase rate is listed as 1.24, but in my simulations this rate must be several orders of magnitude smaller or else all cells very rapidly reach maximum excitability and there is no longer any variability in the system.
• The initial conditions are determined by the "random firers" parameter. With a positive value, this parameter indicates the number of cells that should randomly fire if there is no activity in the system for several time steps. This is the setting for the first model. With a negative value, cells that have not fired for 15 times steps have a probability of -1/"random firers" of firing on each time step thereafter. I added this probability for greater control over the initial conditions; a probability of 1 makes too many of the cells fire together, so the second model uses a 0.1 probability. The cells are initialized with a random time since last firing at the beginning of a simulation.
• The grid size for the second model should be 150, in the applet it is lowered to 100 to decrease loading time.

Both models apparently ignore boundary conditions, and so does my simulator. cAMP simply diffuses off the edges of the grid. Both models also ignore chemotaxis until after the spiral wave pattern has formed, so my simulator ignores it as well.

I discretize my diffusion using the following update rule for each box b(i,j,t) of the grid at location (i,j) and time t:

b(i,j,t+1) = 0.5*b(i,j,t) + 0.125(b(i-1,j,t)+b(i+1,j,t)+b(i,j-1,t)+b(i,j+1,t))

Display

The simulator uses different colors to indicate a bion's state. Red through brown through yellow indicate different levels of excitability in the bion's excitable state. Red means the threshold is at its minimum value, brown an intermediate value, and yellow the maximum value. A bion is green while it is releasing cAMP. Blue means the bion is refractory. Relatively refractory is indicated with the appropriate color for the excitability of the bion. Note that the color schemes and merged-model parameter values end up making the first model's constant-threshold bions yellow but the second model's constant-threshold bions red.

The level of cAMP in a grid location is shown using a log scale of whiteness. The more cAMP, the more white the location is. As the cAMP diffuses and is degraded the location turns gray and finally black if no more cAMP is released in or near it.