Control of multistability in ring circuits of oscillators

The essential dynamics of some biological central pattern generators (CPGs) can be captured by a model consisting of N neurons connected in a ring. Th ese circuits, like many oscillatory nonlinear circuits of sufficient comple xity, are capable of multistability, that is, of generating different firin g patterns distinguished by the phasic relationships between the firing in each circuit element (neuron). Moreover, a shift in firing pattern can be i nduced by a transient perturbation. A systematic approach, based on phase-r esponse curve (PRC) theory, was used to determine the optimum timing for pe rturbations that induce a shift in the firing pattern. The first step was t o visualize the solution space of the ring circuit: including the attractiv e basins for each stable firing pattern; this was possible using the relati ve phase of N - 1 oscillators, with respect to an arbitrarily selected refe rence oscillator, as coordinate axes. The trajectories in this phase space were determined using an iterative mapping based only on the PRCs of the un coupled component oscillators; this algorithm was called a circuit emulator . For an accurate mapping of the attractive basin of each pattern exhibited by the ring circuit, the emulator had to take into account the effect of a perturbation or input on the timing of two bursts following the onset of t he perturbation, rather than just one. The visualization of the attractive basins for rings of two, three, and four oscillators enabled the accurate p rediction of the amounts of phase resetting applied to up to N - 1 oscillat ors within a cycle that would induce a transition from any pattern to any a nother pattern. Finally, the timing and synaptic characterization of an inp ut called the switch signal was adjusted to produce the desired amount of p hase resetting.