Stability and Hopf bifurcation analysis of a pair of three-neuron loops with time delays
The author presents a stability analysis of a trivial steady state solution in a network with delayed coupling which consists of two identical bidirectional rings of three cells with elementwise, symmetric crosscoupling. The crosscoupling delay \(\tau\) is assumed different from the ring-coupling delay \(\sigma\). The model equations are \[ \dot{x}_i (t) = -x_i(t) + af(x_{i-1}(t-\sigma)) + af(x_{i+1}(t-\sigma)) + bg(y_i(t-\tau)), \] \[ \dot{y}_i (t) = -y_i(t) + af(y_{i-1}(t-\sigma)) + af(y_{i+1}(t-\sigma)) + bg(x_i(t-\tau)) \] for \(i=1,2,3\). The linearized system is studied. Some sufficient conditions for the asymptotic stability of the steady state are given and the occurence of purely imaginary eigenvalues, i.e., Hopf bifurcations, is investigated. Numerical illustration is shown in a final chapter.
scientific article; zbMATH DE number 6057985
scientific article