Nonlinear Dynamics In The Neuron Model

A kind of the period-adding sequences associated with the chaotic intermediate states is discovered in the interspike internals of pulses (ISI) through the experimental data on the peripheral nerves. It means that chaos appears during pulse rhythms changing from period-n to period-n+1. And the numerical simulation based upon Chay model with certain parameters can carry out the similar bifurcation diagram. A piece-wise smooth map is coined to describe the ISI dynamics, which bases on ISI first return maps of numerical simulations of Chay model, and the corresponding parameters in it are obtained by fitting to the simulational data. The map changes with control parameter, which can simulate ISI's evolution over time by iteration and generate period-adding sequences associated with the chaotic intermediate states.Border-collision phenomena appear when iteration periods of the map change. It is figured out that border-collision bifurcation is the reason of period-adding sequences. A new mechanism for the border-collision bifurcation in the neural firing is revealed in the analytical and numerical analysis of this mapping model. It includes the instability of arbitrary period-n rhythm due to border-collision bifurcation which induces the transitions from the period-n states to chaos, and the transitions from these chaotic states to the period-n+1 rhythms. Meanwhile, according to stability conditions of border-collision, the critical functions of period-n appearing and disappearing are given, and the critical values for the control parameters are obtained.