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Welcome to Neuromodeling Group
Page
Built byATIN
DAS
email:
atin_das@yahoo.com
Last revised: 31 March, 2002
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We are happy to bring the page relating
to the developments of our research group.
Although not much experienced in
nuero-modelling, we have taking the case of
MATHEMATICAL MODELLING in different
bio-mathematical aspects.
For example, Ecology, Human Hormone
System etc.
John Hopfield proposed model with electronic circuit
implementation where the neurons are simple
input output devices capable of receiving signals
from many others but they produce one response
which is then carried along to others by network
of interconnections of axons and dendrites.
Real neurons are not just simple integrate and
fire mechanisms but exhibit multiple and often multistep
processes beginning with the synapses. The principal
biochemical mechanisms involved in the chemical
synapses are rather complex. The neuro transmitter
is synthesized in the neural cell body and actively
transported to the end of dendrites in the little
membrane spheres (vesicles).
Most interneurons (neurons that synapse with other
neurons) extend across the mid line of the central
nervous system, and attractive and repulsive
factors that regulate the fundamental crossing decision
have been identified .
ANN are attempts to model some of the characteristics
of the brain in order to capture and explore
those qualities of the brain's reasoning power
in which the architecture of the brain is assumed to play
a major part. This has led to models which use
connected local processing elements (neurons) accepting
weighted inputs to give a single output which
in turn are fed to other such processing elements, back to
itself or is given as an output from the system.
The extent to which the input of neuron i is driven by the output of
neuron j is characterized by its synaptic
weight wij.
When the output of neuron j excites neuron i
, synaptic weight wij > 0 .
When the output of neuron j inhibits neuron i
, synaptic weight wij < 0.
When the output of neuron j has no influence
on neuron i , synaptic weight wij = 0
The most typical microcircuits are shown in Fig 1. Illustration A consists
of two excitatory synapses from
the same axon branch(a) projecting onto two different dendrites with
the inhibitory dendrite inhibiting the
common target(b). These types of microcircuits are common in retina
and thalamus. Illustration B shows
a synaptic junction having bi-directional synapses with one being inhibitory
and the other being excitatory.
These are common in mitral and granular neurons in the olfactory bulb.
Complex dynamical evolution that lead to chaotic regimes recently have
been experimentally observed in
the neural system. In contrast, in theoretical modeling of neural systems,
emphasis has been put mainly on
either stable or cyclic behaviors.
Perhaps chaotic behavior at neural level could be responsible for schizophrenia,
insomnia, epilepsy and other disorders.[Guevara,
M. R., Glass, L., Mackey, M.C.,& Shrier, A.(1983) Chaos in neurobiology,
IEEE trans
on System, Man and Cybernetics, SMS-13, 790-798.]
Neurobiologists have found that such low level activity is always present
in the brain, but for along time assumed
that it was just irrelevant electric “noise”. Now some believe that
this activity, far from being random and irreverent,
is chaotic and essential to healthy brain activity. Freeman’s discovery
of chaotic behavior in the electroencephalogram
[EEG] tracing in the olfactory bulbs in rabbits has again confirmed
this. [Freeman, W. J., Yao, Y. & Burke, B. (1988),
Central pattern generating and recognizing in
olfactory bulb: A correlation learning rule, Neural Networks, 1, 277-288.]
Nevertheless some theoretical models have been proposed that illustrate
the existence of chaos. A brief discussion
on this type of model can be found in the work of Chapeau & Chauvet
[1998]. These models rely on complex
architecture of the network to show chaos. Sometimes the quantities
that exhibit chaos in these models have no
direct physiological interpretations. An example of such was considered
by van der Mass [3].
In our previous work on chaos [4], we considered a model with limited
connections. That model even showed
chaos, also some parameters of the model was identified which controlled
the system dynamics and brought the
system from chaotic to stable regime.
For example, a small ANN consisting of three neurons are considered
and their activity are determined by
three Differential Equations [DEs]. We studied the linearised system.
By taking proper values of the synaptic weight,
the system was tested for some known results of our earlier works.
An attempt has been made to give the model a
general shape by taking all possible synaptic connections of the neurons
into consideration. By that way, the proposed
model is nearer to reality than previous works. Numerical simulation
has been done to substantiate the analytical result.
How we construct Mathematical Models
We consider the network consisting of three neurons [see Fig 3] where
the neurons have a sigmoidal
response function as
f i (s)= (1 + exp( -x -t)) -1
Let x(t), y(t) and z(t) be the output activity of the neurons
1, 2 and 3 respectively which is interpretable
as short term average of firing rate of respective neurons.
i is the threshold and slope of response function
of neuron i
The value of this type of response function is always non-negative,
bounded by 0 and 1 .
With the above consideration the following DEs can give the equation
of control.
x(t) = f1( w12y(t) + w13z(t)) -
a1.x(t)
y(t) = f2( w21x(t) + w23z(t))
-a2.y(t)
z(t) = f3( w31x(t) + w32y(t)) -
a3.z(t)
When a1, a2, a3 are the respective decay rates of the 3 neurons.
And wij is the synaptic weight of neuron i to neuron j
and i,j = 1,2,3.
Murray[93] showed that these parameters can also
be taken as given function of s empirically determined by
fitting the results to the data as was done in
many previous works, for example, one done by FitzHugh[93].
In this paper we shall concentrate on synaptic
weight and also try to throw some light on decay rates. It may
be noted that we have excluded self weight terms,
that is , wij = 0 for i=j.
We applied our mathematical tools to study the system dynamics and characterised
a few parameters that
sensibly control the system dynamics, change it from one region of
stabilty to period doubling leading to
chaos. Numerical simulation of such model shows chaos is presented
by this type of model of ANN.
For detatils go to Publications.
