Type I vs. type II excitable systems with delayed coupling
The authors analyze the oscillatory dynamics of coupled type I and type II excitable systems. The system of equations has the following form \[ \dot x_i = X(x_i,y_i)+C(x_i,x_{i+1}(t-\tau),x_{i-1}(t-\tau) ), \] \[ \dot y_i = Y(x_i,y_i), \quad i=1,2,\dots,N, \;\tau>0, \] with periodic boundary conditions on \(0\) and \(N\)th unit. Two representative examples are considered: FitzHugh-Nagumo and Terman-Wang models, for which \(X=-x^3+(a+1)x^2-ax-y+I\), \(Y=bx-\gamma y\), and \(X=-x^3+3x-y+2+I\), \(Y=\varepsilon[\alpha (1+\tanh(x/\beta))-y]\), respectively. Two types of coupling are considered: diffusive coupling with \(C(x_i,x_{i+1}(t-\tau),x_{i-1}(t-\tau))=c(-2x_i+x_{i-1}(t-\tau) +x_{i+1}(t-\tau))\) and sigmoid coupling with \(C(x_i,x_{i+1}(t-\tau),x_{i-1}(t-\tau))=c \tan^{-1} (x_{i-1}(t-\tau)-x_0)+c\tan^{-1} (x_{i+1}(t-\tau)-x_0)\), where \((x_0,y_0)\) is a stationary solution. For the above mentioned models, the stability of a synchronized state is studied. One of the authors' main conclusions is that that within the same class, characterized by the excitability type and the coupling, the dynamics qualitatively depends only on whether the number of units is even or odd.
scientific article; zbMATH DE number 2167635