We now turn our attention to a brief description of periodically forced and coupled neural oscillators. The behaviors generally involve issues that are very difficult to analyze and we will only touch on them briefly. Before treating a specific example, it is useful to discuss certain important aspects of oscillators. We say that a periodic solution to an autonomous (time does not explicitly appear in the right hand side) differential equation is ( orbitally) asymptotically stable if perturbations from the oscillation return to the oscillation as . The difference between asymptotic stability of an oscillation and that of a steady state solution is that for the oscillation the time course may exhibit a shift. That is, we do not expect the solution of the perturbed oscillation to be the same as the unperturbed, rather there will be a shift (see Figure 12A).
Integrate the equation and freeze the curve so you have a reference cycle. Use the (Graphic stuff) (Freeze) (Freeze) option to freeze it. Now it is permanently in memory. To see the effect of a perturbation, set s0=300 and reintegrate. Not much effect, huh? The parameter ph determines where in the cycle the perturbation occurs. Try ph=40 and integrate again. There is a big phase shift.
This shift is due to the time-translation invariance of the periodic solution. Indeed, in phase-space, the periodic trajectory is unchanged by translation in time. This shift that accompanies the perturbation of the limit cycle can be exploited in order to understand the behavior of the oscillator under external forcing. Suppose that an oscillator has a period, say T. We may let t=0 correspond to the time of peak value of one of the oscillating variables, so that at t=T we are back to the peak. Given that we lie on the periodic solution, if some t is specified then we know precisely the state of each oscillating variable. This allows us to introduce the notion of phase of the periodic solution. Let define the phase of the periodic solution so that all define the same point on the periodic solution. For example, if then we are halfway through the oscillator's ninth cycle.