Information Technology Reference
In-Depth Information
Chapter 8
Synchronization of Stochastic Neural Oscillators
Using Lyapunov Methods
Abstract
A neural network with weights described by the position of interacting
Brownian particles is considered. Each weight is taken to correspond to a Gaussian
particle. Neural learning aims at leading a set of M weights (Brownian particles)
with different initial values on the 2-D phase plane, to the desirable final position.
A Lyapunov function describes the evolution of the phase diagram towards the
equilibrium Convergence to the goal state is assured for each particle through the
negative definiteness of the associated Lyapunov function. The update of the weight
(trajectory in the phase plane) is affected by (1) a drift force due to the harmonic
potential, and (2) the interaction with neighboring weights (particles). It is finally
shown that the mean of the particles will converge to the equilibrium while using
LaSalle's theorem it is shown that the individual particles will remain within a small
area encircling the equilibrium.
8.1
Representation of the Neurons' Dynamics as Brownian
Motion
It will be shown that the model of the interacting coupled neurons becomes
equivalent to the model of interacting Brownian particles, and that each weight is
associated with a Wiener process. In such a case, the neural network proposed by
Hopfield can be described by a set of ordinary differential equations of the form
R
i
x
i
.t/ C
P
jD1
T
ij
g
j
.x
j
.t// C I
i
1in
1
C
i
x
i
.t/ D
(8.1)
Variable x
i
.t/ represents the voltage at the membrane of the i-th neuron and I
i
is
the external current input to the i-th neuron. Each neuron is characterized by an
input capacitance C
i
and a transfer function g
i
.
u
/ which represents connection to
neighboring neurons. The connection matrix element T
ij
has a value 1=R
ij
when
the output of the j-th neuron is connected to the input of the i-th neuron through
a resistance R
ij
, and a value 1=R
ij
when the inverting output of the j-th neuron
