Reduced Models for Biological Limit-cycle Oscillators

Motivated by biological applications, I will discuss certain extensions to standard model reduction techniques for nonlinear, dissipative oscillators. The standard reduction to a single phase variable is often insufficient, since it discards information about the amplitude of the oscillations, which may be of critical importance in biological applications. In neurons, for example, only oscillations of sufficiently large amplitude produce significant synaptic responses. I introduce an alternative reduced model for “events”: a point process for the timing of oscillations with the amplitudes of interest. In an application to neural systems, the event-based approach captures the variability in neuronal oscillations due to noisy forcing, which is essential in characterizing rhythmogenesis in a reduced model of synaptically-coupled neurons.