A model for the periodic synaptic inhibition of a neuronal oscillator

We develop a simple, piecewise linear differential equation with discontinuous jumps, which captures the essential characteristics of more complicated equations modelling the dynamics of neuronal oscillators, such as those due to Hodgkin & Huxley (1952), Fitzhugh (1960, 1961), and Nagumo et al. (1962). We investigate the effects of periodically applied stimuli of various durations and compare phase-transition curves or Poincaré maps for our model with numerically computed maps from the 'full' equations. We describe some aspects of the qualitative behaviour and bifurcations of these iterated one-dimensional mappings and attempt to relate them to experimental observations.