Dynamics of a Three Neurons Network with Excitatory-Inhibitory Interactions - Foundations on Natural and Artificial Computation - page 90

Dynamics of a Three Neurons Network with
Excitatory-Inhibitory Interactions

Carlos Aguirre 1 ,JuanI.Cano 2 , and Eloy Anguiano 2

1 GNB, Escuela Politecnica Superior, Universidad Autonoma de Madrid,
28049 Madrid, Spain
[e-mail]
2 CRL, Escuela Politecnica Superior, Universidad Autonoma de Madrid,
28049 Madrid, Spain
[e-mail], [e-mail]

Abstract. Sets of coupled neurons can generate many different patterns
in response to modulatory or sensory inputs. The study of how these pat-
terns have been generated from the inputs has been object of great inter-
est in the literature. These studies have been mainly performed by means
of computer simulations, based on differential models or phenomenolog-
ical models. However complete descriptions of the behaviour of sets of
coupled neurons are hard to obtain due to the complex behaviour of
the dynamics generated even by the simplest neuron models and for the
high number of parameters involved. Here we present a study of a three
neuron network that appears in models of Central Pattern Generators.
The use of a lineal model allows a complete dynamical description of the
system, identifying the relevant situations and drawing some conclusions
concerning the dynamics of the network.

1 Introduction

Numerical studies of the collective behaviour of ensembles of neurons rely on
models of neurons that describe the neuron behaviour on the basis of differential
equations of phenomenological models. Differential models (the Hodking-Huxley
(HH) model [1], or the Hindmarsh-Rose (HR) model [2]) require high computa-
tional effort to reproduce neuronal behaviour such as spiking or bursting. Re-
cently, some models have solved this drawback of the differential models [3,4,5,6].
These new models are built over a phenomenological basis and use iterative two-
dimensional maps that present similar neuro-computational properties to that
of the differential models such as spiking, bursting or subthreshold oscillations.
These models present a low computational effort that makes possible the sim-
ulation of big ensembles of coupled neurons during relatively long periods of
time, or the exploration of many different regimes determined by inputs, initial
conditions and neuron couplings.
Both differential and phenomenological models are adequate for computer
simulations, however they are too complex for the complete characterization of
the dynamics even for a single neuron, not to say for sets of several coupled