Dynamics of neural systems with discrete and distributed time delays

Rahman, B, Blyuss, K B and Kyrychko, Y N (2015) Dynamics of neural systems with discrete and distributed time delays. SIAM Journal on Applied Dynamical Systems, 14 (4). pp. 2069-2095. ISSN 1536-0040

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Abstract

In real-world systems, interactions between elements do not happen instantaneously, due to the time
required for a signal to propagate, reaction times of individual elements, and so forth. Moreover,
time delays are normally nonconstant and may vary with time. This means that it is vital to introduce
time delays in any realistic model of neural networks. In order to analyze the fundamental
properties of neural networks with time-delayed connections, we consider a system of two coupled
two-dimensional nonlinear delay differential equations. This model represents a neural network,
where one subsystem receives a delayed input from another subsystem. An exciting feature of the
model under consideration is the combination of both discrete and distributed delays, where distributed
time delays represent the neural feedback between the two subsystems, and the discrete
delays describe the neural interaction within each of the two subsystems. Stability properties are
investigated for different commonly used distribution kernels, and the results are compared to the
corresponding results on stability for networks with no distributed delays. It is shown how approximations
of the boundary of the stability region of a trivial equilibrium can be obtained analytically
for the cases of delta, uniform, and weak gamma delay distributions. Numerical techniques are used
to investigate stability properties of the fully nonlinear system, and they fully confirm all analytical
findings.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics
Depositing User: Yuliya Kyrychko
Date Deposited: 11 Dec 2015 09:41
Last Modified: 14 Mar 2017 16:37
URI: http://sro.sussex.ac.uk/id/eprint/58803

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