Vibrational Resonance and Electrical Activity Behavior of a Fractional-Order FitzHugh–Nagumo Neuron System

Jia-Wei Mao,Dong-Liang Hu

doi:10.3390/math10010087

Abstract

Making use of the numerical simulation method, the phenomenon of vibrational resonance and electrical activity behavior of a fractional-order FitzHugh–Nagumo neuron system excited by two-frequency periodic signals are investigated. Based on the definition and properties of the Caputo fractional derivative, the fractional L1 algorithm is applied to numerically simulate the phenomenon of vibrational resonance in the neuron system. Compared with the integer-order neuron model, the fractional-order neuron model can relax the requirement for the amplitude of the high-frequency signal and induce the phenomenon of vibrational resonance by selecting the appropriate fractional exponent. By introducing the time-delay feedback, it can be found that the vibrational resonance will occur with periods in the fractional-order neuron system, i.e., the amplitude of the low-frequency response periodically changes with the time-delay feedback. The weak low-frequency signal in the system can be significantly enhanced by selecting the appropriate time-delay parameter and the fractional exponent. In addition, the original integer-order model is extended to the fractional-order model, and the neuron system will exhibit rich dynamical behaviors, which provide a broader understanding of the neuron system.

Highlights

Vibrational resonance (VR) has attracted considerable attention in the field of nonlinear sciences in the last twenty years
For F = 0.11, the curve of response amplitude is approximately a straight line, which indicates that VR does not occur with the change of α, while for F = 0.15 or F = 0.2, the response amplitude Q is a nonlinear function of α, and the low-frequency signal can be significantly enhanced by selecting an appropriate fractional-order α compared with the integer-order system
ConTchluesviiobnrsational resonance and electrical activity behaviors in the fractional-order FHN neuTrohne avriebrsatutidoineadl irnestohnisanpcaepearn. dWehlecntrtihcealoaricgtivniatlyinbteehgaevri-oorrsdeinr mthoedferlaicstieoxntaeln-odreddetro FtHhNe fnraecutrionaalr-oersdtuerdmiedodinelt,htihsepraepgeior.nWohf ethnethVeRowrigililnbael iwnitdegeer,rs-ortdheerrmeqoudierleims enxtefnodretdhe toamthpeliftruadcteioonfahli-gohrd-ferreqmuoednecly, sthigenraelgciaonn boef rtehleaxVeRd,wanildl btehewni,dtehre, VsoRtphheerneoqmuierenmonenctanfobre the amplitude of high-frequency signal can be relaxed, and the VR phenomenon can be induced by choosing the appropriate fractional-order α

Summary

Introduction

Vibrational resonance (VR) has attracted considerable attention in the field of nonlinear sciences in the last twenty years. Compared with integer calculus, the power-law characteristic of complex social and physical phenomena can be accurately approximated through fractional calculus. This theory is gradually used in viscoelastic materials [13], electrification process [14], control theory [15], and neuron models [16], etc. Inspired by the above-mentioned ideas, the effects of fractional order and time delay on VR and dynamical behavior are studied in the fractional-order FHN neuron model.

Fractional-Order FitzHugh–Nagumo Neuron Model

Multiple VR in Fractional-Order FHN Neuron Model with Delay

Conclusions