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Stability switches, oscillatory multistability, and spatio-temporal patterns of nonlinear oscillations in recurrently delay coupled neural networks

Makarov , Valeri A. and Velarde, Manuel G. and Song, Yongli (2009) Stability switches, oscillatory multistability, and spatio-temporal patterns of nonlinear oscillations in recurrently delay coupled neural networks. Biological Cybernetics , 101 (2). 147-167 . ISSN 0340-1200

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Abstract

A model of time-delay recurrently coupled spatially segregated neural assemblies is here proposed. We show that it operates like some of the hierarchical architectures of the brain. Each assembly is a neural network with no delay in the local couplings between the units. The delay appears in the long range feedforward and feedback inter-assemblies communications. Bifurcation analysis of a simple four-units system in the autonomous case shows the richness of the dynamical behaviors in a biophysically plausible parameter region. We find oscillatory multistability, hysteresis, and stability switches of the rest state provoked by the time delay. Then we investigate the spatio-temporal patterns of bifurcating periodic solutions by using the symmetric local Hopf bifurcation theory of delay differential equations and derive the equation describing the flow on the center manifold that enables us determining the direction of Hopf bifurcations and stability of the bifurcating periodic orbits. We also discuss computational properties of the system due to the delay when an external drive of the network mimicks external sensory input.


Item Type:Article
Uncontrolled Keywords:Limit-cycle oscillators; Functional-differential equations; Hopf-bifurcation; Time-delay; Periodic-orbits; Identical cells; Phase-locking; Normal forms; Neurons; Ring; Neural network; Delay; Hopf bifurcation; Stability; Spatio-temporal pattern
Subjects:Sciences > Mathematics > Differential equations
ID Code:16621
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