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Stability, bifurcation and global existence of a Hopf-bifurcating periodic solution for a class of three-neuron delayed network models. (English) Zbl 1133.34036
The paper studies a network consisting of three-neurons described by the following system of delay differential equations \align y_1'(t) &= -ky_1(t) +\beta \operatorname{tanh} (y_1(t-\tau))+ w_{12} \operatorname{tanh} [y_2(t-\tau)] + w_{13} \operatorname{tanh} [y_3(t-\tau)] + I_1, \\ y_2'(t) &= -ky_2(t) +w_{21} \operatorname{tanh} (y_1(t-\tau))+ \beta \operatorname{tanh} [y_2(t-\tau)] + w_{23} \operatorname{tanh} [y_3(t-\tau)] + I_2, \\ y_3'(t) &= -ky_3(t) +w_{31} \operatorname{tanh} (y_1(t-\tau))+ w_{32} \operatorname{tanh} [y_2(t-\tau)] + \beta \operatorname{tanh} [y_3(t-\tau)] + I_3, \endalign where $k>0$, $w_{ij}$ and $\beta$ are weights of synaptic connections, $\tau>0$ is the fixed delay time, $I_i$ are constant inputs. The authors provide sufficient conditions for linear stability and occurrence of Hopf bifurcation. Also, sufficient conditions for the existence of multiple periodic solutions are obtained.

##### MSC:
 34K13 Periodic solutions of functional differential equations 34K18 Bifurcation theory of functional differential equations 92B20 General theory of neural networks (mathematical biology) 34K20 Stability theory of functional-differential equations
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