The Study for Synchronization between Two Coupled FitzHugh-Nagumo Neurons Based on the Laplace Transform and the Adomian Decomposition Method

Abstract

The synchronization between two coupled FitzHugh-Nagumo (FHN) neurons with or without external current is studied by using the Laplace transform and the Adomian decomposition method. Different from other researches, the synchronization error system is expressed as sets of Volterra integral equations based on the convolution theorem in the Laplace transform. Then, it is easy to analytically obtain the conditions that synchronization errors disappear based on the successive approximation method in integral equation theorem, the correctness of which is verified by numerical simulations. Furthermore, the synchronous dynamics of the two coupled FHN neurons also can be written in the form of Volterra integral equations, which is more convenient to analytically solve by using the Adomian decomposition method. It is found that the occurrence of synchronization between the two FHN neurons only depends on the coupling strength and is irrelevant to the external current. Only synchronous rest state in the two FHN neurons without external current can be achieved, while synchronous spikes appear if the external current is not zero.

The Impedance of Spiking Neurons Coupled by Time‐Delayed Interaction

“…The dynamical features resulting from the interaction of the paradigmatic FitzHugh-Nagumo (FHN) [13,14] neurons, which is the topic of this paper, have been extensively studied. [12,[15][16][17][18][19] With the introduction of the time delay interaction the coupled FHN neurons display a transition from the original disordered motions to periodic ones, which is accompanied by complex bifurcation properties. [20][21][22][23][24][25] Impedance spectroscopy (IS) is a multipurpose characterization technique widely used in materials science and electrochemistry.…”