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Fig. 4.3 Hopfield neural network: (a) network structure; (b) surface of the network potential
in those years. In fact, the work of Rosenblatt was subjected to criticism in 1969 by
Marvin Minsky—the founder of a scientific approach to artificial intelligence—in
his topic
Perceptrons
, written jointly with Seymour Papert. Only about 20 years
later,
in 1982, American physicist Hopfield again sparked interest
in neural
networks, establishing a neural network model named after him.
The Hopfield neural network is a two-dimensional array of formal neurons,
connected with each other in a pairwise fashion (Fig.
4.3
). Each neuron is consid-
ered as an element with two possible states described by the binary variable
s
. One
of the states corresponds to the “excited” neuron (
s
¼
1), the other state to the
ground state (
s
0). In general, the neuron is characterized by the function
f
(
s
)
determining its dynamics.
The status of a network of neurons
N
at the time
t
is defined as the configuration
of all variables at this moment of time. The evolution of this state in phase space
s
i
is determined by the interaction of neurons. Neurons are connected with each other
by synaptic connections. The strength of the connection between the
i
th and the
j
th
neurons is characterized by the value
T
ij
. This matrix is called weight matrix (also
known as the matrix of long-term memory). The condition of neurons at the initial
moment of time depends on the image presented to the network.
The image is converted by the network in accordance with certain rules. Among
the rules governing the evolution of different network models one property remains
invariant: the change of neuron's state is determined by the total excitation reaching
the neuron from all other neurons in accordance with the synaptic weights.
In the Hopfield network all neurons are mutually connected. The memory matrix
T
is symmetrical (
T
ij
¼
¼
T
ji
) and has zero diagonal elements. The function
f
(
s
)is
binary.
Nonlinear transformation of the original image
A
¼
(
S
1
,
,
S
N
) follows the rule:
...
