No abstract

We examine the dynamics of two simple coupled non-linear ordinary di erential equations (ODEs) ® rst introduced by Lazer and McKenna (SIAM Review 32(4), 537± 578, 1990) . Using the numerical continuation package AUTO, we obtain multiple coexistence of periodic motions, period-doubling sequences and the onset of`beats' solutions via torus bifurcations. We discuss the implications of these results for the modelling of suspension bridge dynamics.

No abstract

. Neuronal populations with reciprocal inhibition and rebound currents: effects of synaptic and threshold noise.

Post inhibitory rebound (PZR) is a nonlinear phenomenon present in a variety of nerve cells. It is an important mechanism underlying central pattern generationfor 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 efects of PZR. 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 dejine transitions between these two distinct types of asymptotic behaviour. Hysteresis between periodic orbits is also observed. A small network of these model neurons, with reciprocal inhibition, is shown to exhibit 'self-sustained' anti-phase oscillations, making PZR 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.