Neuronal population dynamics with post inhibitory rebound: a reduction to piecewise linear discontinuous circle maps

Post inhibitory rebound is a nonlinear phenomenon present in a variety of nerve cells. It is an important mechanism underlying central pattern generation for heartbeat, swimming and other motor patterns in many neuronal systems. In this paper we propose an extension of the binary threshold neuron model to incorporate the effects of post inhibitory rebound. For a single neuron, the dynamics can be described by a piecewise linear circle map with two discontinuities. Both frequency-locking and chaos can occur. The Liapunov exponent of the map is evaluated and used to define transitions between these two distinct types of asymptotic behaviour. Hysteresis between periodic orbits is also observed. A small network of theses model neurons, with reciprocal inhibition, is shown to exhibit self-sustained anti-phase oscillations, making post inhibitory rebound a plausible mechanism for central pattern generation in neuronal systems. Unlike coupled oscillator theories, network oscillations emerge naturally as a consequence of the biological description from which the neuronal dynamics is derived. The simplicity of the dynamical model allows for the possibility of large population studies in contrast to other classical models of single neuron dynamics that incorporate active processes.