Dynamics of a Three Neurons Network with Excitatory-Inhibitory Interactions - Foundations on Natural and Artificial Computation - page 91

neurons. This makes necessary the introduction of simpler models that allow a
complete study of the dynamics but preserve the main characteristics of single
neuron behaviour. An example of such analytic study is the Kuramoto model [7]
developed for the study of many coupled oscillators, similar to the ones present
in the peacemaker cells.
A complete study of the dynamics of a three neuron network composed by
an excitatory-inhibitory loop that controls a third neuron is presented here. 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 model. These network model has been previously studied by
means of computer simulations in [8] and [9] as it appears frequently in Central
Pattern Generators (CPGs).

2T eNtw kM l

Here we propose the use of a leaky integrate and fire (IF) iterative model obtained
from a differential neuron model presented in eq. 2.1 from [10] . Our model is
implemented as an iterated map obtained from a continuous model by an Euler
approximation. This approach has been used, for example, in [11] to obtain an
iterated map from an continuous neuron model to obtain iterative phenomenolo-
gical models. This model of neuron is often used in simulation studies. Simple
IF neurons have been shown to provide good approximations to the dynamics
of more complex model neurons. For example, similar models has been used to
explore the different connectivity effects in models of cortical networks [12], in
[13] to explore the role of the synaptic background activity in spatio-temporal
integration in single pyramidal cells or in [14] to study the chaotic behaviour
of a model of cortical network. This simple model will allow us to perform a
complete description of the different regimes generated by the dynamics of the
network.
We start with the differential model

=
j = i

C dv i ( t )
dt

I j ( t )+ I ( t ) (1)

now we do the approximation dv i
dt

1
Δt

v i ( t ). For the simulation of
the synaptic current ( I j ( t )), we model input to neuron i from neuron j at time
t as W ij H ( v j ( t )

= v i ( t + Δt ) −

T j ), see eq. 4.19 in [10], where W ji is the coupling strength
between neuron j and neuron i .Thevalue W ji is positive if j is an excitatory
neuron and negative otherwise. H ( x ) is the usual Heaviside function ( H ( x )=1
if x

−

0and H ( x )=0otherwise)and Tj is the spike threshold of neuron j .
Now, grouping constants , we obtain is the following model:
v i ( t +1)= a i v i ( t )+
j = i

≥

W ji H ( v j ( t ) − Tj )+ W i I i ( t ) (2)

where v i ( t +1) means the value of the output of neuron i , v i , at the next time ins-
tant. The value a i is the descent rate and is required to be positive. We will note