TWO SPACE DIMENSIONS

Only the activator is pictured, but the inhibitor and control, like the activator, are approximately rotations of the corresponding one-dimensional functions. This MPEG movie shows how the equilibrium spike solution can be obtained by temporarily raising the control parameter above threshold in the location where the spike is desired for the two-dimensional system.

TRAVELING SPIKES IN AN ACTIVE TRANSMISSION LINE

For the active transmission line, solutions decay to a stable traveling spike solution for any initial excitation in the region of attraction of the traveling spike solution. The traveling spikes are excited by initial conditions rather than by locally raising the control parameter. This MPEG movie shows the traveling spikes forming, traveling, and interacting with each other. In a linear transmission line, the oppositely-traveling waves would not change shape as a result of passing through each other. However, for the active transmission line, the interactions temporarily change the spike shapes due to the nonlinearity. However, after traveling a certain distance, the spike shapes decay to the stable traveling spike solution again.

HELICAL SOLUTION FOR THE COMPLEX ACTIVATOR-INHIBITOR EQUATION

The complex activator-inhibitor equation with C=0 has a stable helical equilibrium solution. Since random initial conditions were used, there are fronts between the left-handed and right-handed ideal helixes in the above figure. In the coupled oscillator context, this figure represents the equilibrium phases of the oscillators. This MPEG movie shows the helical equilibrium being reached from random initial conditions.

This MPEG movie shows the evolution of the states of the oscillators, in the coupled oscillator context. The state evolution is simply the phase evolution rotated at the common frequency of the oscillators. Note that right-handed and left-handed helical waves ``travel'' in opposite directions.