Hysteresis in Models of Neuronal Dynamics

The Wilson-Cowan model has described a variety of different statistical behaviors in neocortical dynamics. Hysteresis has been observed in the Wilson-Cowan equations since their conception and modeled as an underlying mechanisms for attention and memory. In a pair of coupled Wilson-Cowan equations, adiabatically varying the external stimulus in a closed loop can transfer the system between fixed point steady states. The role of limit cycles in the hysteresis of a two-neuron system is restricted by the maximum of one stable limit cycle existing for a set of parameters. We demonstrate that as the number of coupled neurons is increased, the number of possible stable limit cycles and the proportion of parameter space they occupy also increases. We then show that the system exhibits hysteresis in transitions between the multiple limit cycle steady states. When the external stimulus is chosen such that the equations exhibit a discrete symmetry, the splitting and merging of stable and unstable limit cycles is observed. These results provide a new perspective on hysteresis-based phenomena in neuronal dynamics.