Analytical predictions of stable and unstable firings to chaos in a Hindmarsh-Rose neuron system

In this paper, analytical predictions of the firing cascades formed by stable and unstable firings in a Hindmarsh-Rose (HR) neuron system are completed via an implicit mapping method. The semi-analytical firing cascades present the route from periodic firings to chaos. For such cascades, the continuous firing flow of the nonlinear neuron system is discretized to form a special mapping structure for nonlinear firing activities. Stability and bifurcation analysis of the nonlinear firings are performed based on resultant eigenvalues of the global mapping structures. Stable and unstable firing solutions in the bifurcation tree exhibit clear period-doubling firing cascades toward chaos. Bifurcations are predicted accurately on the connections. Phase bifurcation trees are observed, which provide deep cognitions of neuronal firings. Harmonic dynamics of the period-doubling firing cascades are obtained and discussed for a better understanding of the contribution of the harmonics in frequency domains. The route into chaos is illustrated by the firing chains from period-1 to period-16 firings and verified by numerical solutions. The applied methods and obtained results provide new perspectives to the complex firing dynamics of the HR neuron system and present a potential strategy to regulate the firings of neurons.