State transitions in the Morris-Lecar model under stable Lévy noise

Abstract

This paper considers the state transition of the stochastic Morris-Lecar neuronal model driven by symmetric α-stable Lévy noise. The considered system is bistable: a stable fixed point (resting state) and a stable limit cycle (oscillating state), and there is an unstable limit cycle (borderline state) between them. Small disturbances may cause a transition between the two stable states, thus a deterministic quantity, namely the maximal likely trajectory, is used to analyze the transition phenomena in non-Gaussian stochastic environment. According to the numerical experiment, we find that smaller jumps of the Lévy motion and smaller noise intensity can promote such transition from the sustained oscillating state to the resting state. It also can be seen that larger jumps of the Lévy motion and higher noise intensity are conducive for the transition from the borderline state to the sustained oscillating state. As a comparison, Brownian motion is also taken into account. The results show that whether it is the oscillating state or the borderline state, the system disturbed by Brownian motion will be transferred to the resting state under the selected noise intensity.

Total value adjustment of Bermudan option valuation under pure jump Lévy fluctuations

“…Recently, the fractional models have aroused numerous research interests in various fields, ranging from finance, 28,29 neuroscience, 30,31 physics, 32,33 and so on. [34][35][36][37][38][39] We here focus on the fractional model in option pricing.…”

Control of Hopf Bifurcation Type of a Neuron Model Using Washout Filter

“…Repetitive firing can only be produced when the intensity of stimulation reaches the critical value [9]. Some models can only reproduce the characteristics of type I neurons (such as Hodgkin-Huxley model [10]), and others can only reproduce the characteristics of type II neurons (such as Morris-Lecar model [11]). However, in the Prescott model, these two types of excitability can be reproduced by changing key parameters.…”

Bifurcation Analysis of a Two-Dimensional Neuron Model under Electrical Stimulation

“…At present, most theoretical research on neurons is based on two types of models, namely, phenomenological models (such as Integrate-and-Fire model [4,5] and Izhikevich model [6,7]) and physiological models (such as Hodgkin-Huxley model [8][9][10] and Morris-Lecar model [11]). e phenomenological models only consider the external input-output relationship and do not consider the internal details.…”