Birhythmic oscillations and global stability analysis of a conductance-based neuronal model under ion channel fluctuations

Abstract

By drawing inspiration from existing polynomial models for neurons, we make use of the Moris Lecar system to derive a new two dimensional birhythmic conductance-based neuronal model for nerves. The analysis of fixed points and their stability indicates that its dynamics strongly depends on the parameters of the newly nonlinear terms introduced. Using Lindsted's method, it is observed that the neuronal system can exhibit coexistence of attractors. These coexisting attractors are on the one hand the subthreshold oscillation and on the other hand the spike generation well known in neuronal systems. After introducing the effects of the channel fluctuations in the form of a Gaussian white noise, the global stability of the attractors is analyzed. The effective active energy barrier also called threshold potential is obtained. This threshold potential is the one needed by neuron to switch from one attractor to another. The probability distribution is also studied analytically and numerically, using the Fokker–Planck type equation derived from the new model and the Monte Carlo methods. It is observed that the system physiological parameters and the intensity of the noise plays an important role in the probability of neuron to switch from the subthreshold attractor to the spiking one and vice versa.