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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

y 1 ' (t)=-ky 1 (t)+βtanh(y 1 (t-τ))+w 12 tanh[y 2 (t-τ)]+w 13 tanh[y 3 (t-τ)]+I 1 ,y 2 ' (t)=-ky 2 (t)+w 21 tanh(y 1 (t-τ))+βtanh[y 2 (t-τ)]+w 23 tanh[y 3 (t-τ)]+I 2 ,y 3 ' (t)=-ky 3 (t)+w 31 tanh(y 1 (t-τ))+w 32 tanh[y 2 (t-τ)]+βtanh[y 3 (t-τ)]+I 3 ,

where k>0, w ij and β are weights of synaptic connections, τ>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.

34K13Periodic solutions of functional differential equations
34K18Bifurcation theory of functional differential equations
92B20General theory of neural networks (mathematical biology)
34K20Stability theory of functional-differential equations