Hopf bifurcation analysis of two neurons with three delays. (English) Zbl 1149.34046

The authors study linear stability and give conditions and direction for Hopf bifurcations in the following system of coupled delay differential equations (two neurons with three delays) \[ \begin{aligned} \dot{x}(t) & = - x(t) +a_{11}f(x(t-\tau)) +a_{12}f(y(t-\tau_1)) ,\\ \dot{y}(t) & = - y(t) +a_{21}f(x(t-\tau_2)) +a_{22}f(y(t-\tau)). \end{aligned} \] Here \(x\) and \(y\) are scalar variables corresponding to two neurons, \(\tau_j\) denote the transmission delays, \(a_{ij}\) are synaptic weights. Additionally, \(f(0)=0\) so that the zero solution is the equilibrium, which produces Hopf bifurcations studied.


34K18 Bifurcation theory of functional-differential equations
34K20 Stability theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations


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