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

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.

Hopf bifurcation and spatio-temporal patterns in delay-coupled van der Pol oscillators

“…From this, together with the transversality condition (13) and the Hopf bifurcation theorem for functional differential equations [23], we have the following theorem on stability and bifurcation of system (2).…”

“…Thus, the time delay is inevitable in the coupled oscillators, and it is of interest to investigate how the dynamics is affected by its presence. Recently there has been much increasing interests in investigating the effect of time delays on the dynamics of the coupled systems (see, for example, [9][10][11][12][13][14] and references therein).…”

Stability, bifurcation, and synchronization of delay-coupled ring neural networks

“…Since internal time delay is often simply neglected as a first approximation based on the assumption that coupling time delay is significantly longer [29,45], the following results are useful for the coupled network free of internal time delay.…”

Dynamics of Neural Systems with Discrete and Distributed Time Delays

“…In the most general formulation, all neural interactions (both within and between the subnetworks) could be represented by distributed time delays. However, due to the close proximity of neurons inside each subnetwork, it is reasonable to assume that the variation of the time delays in the connection between them is negligibly small compared to the variation of the time delays in the long-range connections between subnetworks [39]. This justifies the choice of a discrete time delay within the subnetworks and distributed time delays in the interactions between them, and makes analytical investigations more tractable.…”

“…The importance of this trivial steady state lies in the fact that it represents a state of background activity, which is fundamental for many neural processes [41,42]. Depending on the signs of the coupling weights/strengths a ij , i, j =1 , 2, and the specific form of the transfer function f , there may exist a number of other nontrivial steady states, but their existence is not guaranteed in general [39,40]. Therefore, we concentrate our analysis on the stability of the trivial steady state, and similar considerations can be made for all other steady states when they are permitted by the model.…”