The Study of Generalized Synchronization between Two Identical Neurons Based on the Laplace Transform Method

Bin Zhen,Ran Liu

doi:10.3390/app112411774

Abstract

In this paper, a new method is proposed based on the auxiliary system approach to investigate generalized synchronization between two identical neurons with unidirectional coupling. Different from other studies, the synchronization error system between the response and auxiliary systems is converted into a set of Volterra integral equations according to the Laplace transform method and convolution theorem. By using the successive approximation method in the theory of integral equations, an analytical criterion for the detection of generalized synchronization between two identical neurons is obtained. It is found that there is a time difference between two signals of neurons when the generalized synchronization between them is achieved. Furthermore, the value of the time difference has no relation to the generalized synchronization condition but depends on the coupling function between two neurons. The study in this paper shows that one can construct a coupling function between two identical neurons using the current signal of the drive system to predict its future signal or make its past signal reappear.

Highlights

Over the past few decades, chaos synchronization has received a lot of interest and attention because it plays an important role in understanding the feature of coupled chaotic oscillators [1]
Based on the auxiliary system approach, generalized synchronization (GS) between the drive and response systems occurs if synchronization between the response and auxiliary systems is achieved
Different from other researchers, the synchronization problem is solved in this paper by using the Laplace transform and the convolution theorem, as well as the iterative method in the theory of Volterra integral equations

Summary

Introduction

Over the past few decades, chaos synchronization has received a lot of interest and attention because it plays an important role in understanding the feature of coupled chaotic oscillators [1]. One of most exciting problems to GS is how to analytically detect the existence of a functional relation between the signals of the drive and response systems. Different from other studies, the synchronization error between the response and auxiliary systems is found to satisfy a Volterra integral equation from the Laplace transform method and convolution theorem. It is found that there is a time difference τ between the signals of the drive and response systems when the GS condition obtained by the analytical criterion given in this paper is satisfied. The rest of the paper is organized as follows: In Section 2, the GS between two identical FHN neurons with unidirectional coupling is studied using the Laplace transform method. It can be said that there exists the property of GS in System (3) with a transformation H between u1 and u3

Necessary Conditions for GS between Two FHN Neurons

Analysis for GS between Two FHN Neurons and Numerical Simulations

Conclusions