Physica D 206 (2005) 32–48
Regular dynamics in a delayed network of two neurons with
all-or-none activation functions
Shangjiang Guo
a, ∗
, Lihong Huang
a
, Jianhong Wu
b
a
CollegeofMathematicsandEconometrics,HunanUniversity,Changsha,Hunan410082,PRChina
b
DepartmentofMathematicsandStatistics,YorkUniversity,Toronto,Ont.,CanadaM3J1P3
Received 14 October 2002; received in revised form 25 May 2003; accepted 27 September 2003
Available online 25 May 2005
Communicated by C.K.R.T. Jones
Abstract
We consider a delayed network of two neurons with both self-feedback and interaction described by an all-or-none threshold
function. The model describes a combination of analog and digital signal processing in the network and takes the form of a system
of delay differential equations with discontinuous nonlinearity. We show that the dynamics of the network can be understood in
terms of the iteration of a one-dimensional map, and we obtain simple criteria for the convergence of solutions, the existence,
multiplicity and attractivity of periodic solutions.
© 2005 Elsevier B.V. All rights reserved.
PACS: 02.30.ks; 87.10.+e
Keywords: Neural networks; Delayed feedback; One-dimensional map; Convergence; Periodic solutions
1. Introduction
We consider the following model for an artificial network of two neurons
˙ x =-μx + a
11
f (x(t - τ )) + a
12
f (y(t - τ )),
˙ y =-μy + a
21
f (x(t - τ )) + a
22
f (y(t - τ )),
(1)
where ˙ x = dx/dt , x(t ) and y(t ) denote the state variables associated with the neurons, μ> 0 is the interact decay
rate, τ> 0 is the synaptic transmission delay, a
11
,a
12
,a
21
and a
22
are the synaptic weights, and f : R → R is the
∗
Corresponding author.
E-mailaddress: shangjguo@etang.com (S. Guo).
0167-2789/$ – see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.physd.2003.09.049