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

2021 ◽  
Vol 11 (24) ◽  
pp. 11774
Author(s):  
Bin Zhen ◽  
Ran Liu
Keyword(s):  
Laplace Transform ◽  
Integral Equations ◽  
Time Difference ◽  
Coupling Function ◽  
Successive Approximation Method ◽  
Generalized Synchronization ◽  
Laplace Transform Method ◽  
Transform Method ◽  
The Laplace Transform ◽  
Analytical Criterion

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.

scholarly journals A Synchronization Criterion for Two Hindmarsh-Rose Neurons with Linear and Nonlinear Coupling Functions Based on the Laplace Transform Method

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Chunlin Su ◽  
Bin Zhen ◽  
Zigen Song
Keyword(s):  
Laplace Transform ◽  
Chaotic Behavior ◽  
Integral Form ◽  
Nonlinear Coupling ◽  
Laplace Transform Method ◽  
Transform Method ◽  
The Laplace Transform ◽  
Analytical Criterion ◽  
Linear And Nonlinear ◽  
The Stability

In this paper, an analytical criterion is proposed to investigate the synchronization between two Hindmarsh-Rose neurons with linear and nonlinear coupling functions based on the Laplace transform method. Different from previous works, the synchronization error system is expressed in its integral form, which is more convenient to analyze. The synchronization problem of two HR coupled neurons is ultimately converted into the stability problem of roots to a nonlinear algebraic equation. Then, an analytical criterion for synchronization between the two HR neurons can be given by using the Routh-Hurwitz criterion. Numerical simulations show that the synchronization criterion derived in this paper is valid, regardless of the periodic spikes or burst-spike chaotic behavior of the two HR neurons. Furthermore, the analytical results have almost the same accuracy as the conditional Lyapunov method. In addition, the calculation quantities always are small no matter the linear and nonlinear coupling functions, which show that the approach presented in this paper is easy to be developed to study synchronization between a large number of HR neurons.

scholarly journals Numerical Solutions of Fractional Fokker-Planck Equations Using Iterative Laplace Transform Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Limei Yan
Keyword(s):  
Laplace Transform ◽  
Numerical Solutions ◽  
Convergent Series ◽  
Space Time ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Promising Tool ◽  
The Laplace Transform ◽  
Fokker Planck Equations ◽  
Fokker Planck

A relatively new iterative Laplace transform method, which combines two methods; the iterative method and the Laplace transform method, is applied to obtain the numerical solutions of fractional Fokker-Planck equations. The method gives numerical solutions in the form of convergent series with easily computable components, requiring no linearization or small perturbation. The numerical results show that the approach is easy to implement and straightforward when applied to space-time fractional Fokker-Planck equations. The method provides a promising tool for solving space-time fractional partial differential equations.

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