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

The analysis of networks of time-summating binary neural networks is relevant to the study of coherent oscillatory behavior in neuronal populations. A class of networks based on a discrete-time version of leaky integrator networks has recently been extended to include the effects of hyperpolarization-activated inward currents [S. Coombes and S. H. Doole, Dyn Stability Syst. 11, 193, (1996)]. Such rebound currents are important for central pattern generation in neuronal circuits with reciprocal inhibition. In this paper, we incorporate models of intrinsic synaptic and threshold noise into the above neural system. The macroscopic behavior of time-summating networks with rebound currents and random thresholds is analyzed in the thermodynamic limit. Mean field equations are derived for the average network activity in a homogeneous network with inhibitory synaptic connections. Periodic and chaotic solutions are shown to exist, together with hysteretic transitions between periodic orbits. This hysteresis is observed between particular periodic orbit branches, as well as more globally with respect to variations in external input or threshold noise. Moreover, rebound currents are shown to suppress chaotic network response to external input, in favor of low order periodic responses, which in turn define well ordered coherent macroscopic oscillatory states for the system. The response characteristic of a single neuron in the presence of synaptic multiplicative noise is also considered and compared to its zero noise limit. In this latter case, the dynamics is reduced to a piecewise linear discontinuous circle map, while the former is expressed in terms of a random iterated function system.