Physical Interpretations for the Bifurcation Diagram for c=1

Region A

A periodic solution is a uniform train of action potentials. "Nerve action potentials are the electrical signals sent out by the body to control bodily processes such as muscular movement. They are controlled by ions and their concentrations around the nerve cell. They propagate uniformly along the nerve cell and are governed by the all-or-none phenomena. This means that a nerve action potential will not occur unless the depolarization threshold is met. The depolarization threshold is the potential that must be reached before depolarization of the nerve cell will occur." (Action Potential, Oct. 13th, 1999) This means that no matter what the membrane potential or the recovery variable (x,y) are, it will go to this periodic solution. Physically, this means that no matter what the initial stimulus is, the nerve will keep sending out the same signal until the parameters change. This does not occur in healthy cells.

Physically this behavior would occur if the calcium concentration is reduced dramatically in the nerve system bath.

Region B

In this region is what is happening is that the nerve is returning to a resting state after an impulse. In this case, the rest state is at the fixed point.

Region C

For initial conditions, the solution will approach one of the two equilibrium points. If the initial stimulus is greater than some threshold value, the solution will approach the fixed point to the right. However this is impossible physically because the model assumes a constant resting potential, but the actual resting potential is dynamic. So, the saddle point between the two fixed points represents the threshold seperatrix and the point to the right of the threshold seperatrix is the unphysical solution. The solution on the left is the physical resting state.

Physical Interpretations for the Bifurcation Diagram for c=2

Region D

This is another periodic solution. The physical ramifications of this phases diagram are the same as for Region A.

Region E

The behavior in this region is similar to that in Region B. The spiraling steady sink corresponds to the resting state that the nerve returns to after an impulse.

Region F

Once again, the behavior in this region is similar to that in Region B.

Region G

This is the disturbance of an action potential by anodal shock. This is what happens when the cell is sending an impulse which gets interrupted by another impulse of high enough current to perturb the nerve. The result is a spiraling approach to either of the two steady states. If the current of the second impulse is higher than some threshold value, it will cause the solution to jump to the other steady state.

Unlike the solution in Region C, this is physical because the membrane potential is not assumed to be constant but dynamic.

Region H

This is another example of a periodic solution. The behavior in this region is similar to that in Region A.

Region I

The behavior in this region is similar to that in Region G.


Andrea Sereny
Karen J. Hirst