Neural Networks Approach - Computational Intelligence in Time Series Forecasting - page 85

do not have. This capability enables the networks to generate time-variable outputs
in response to the static inputs.
Because of incorporating internal feedback loops, the critical issue of recurrent
networks is their stability, determined by the time behaviour of the network energy
function . For a binary Hopfield net with a symmetric weights matrix this function
is defined as

i

nn

E ¦¦ . w x x

ij i j

2

i 11 j

Hopfield Network

w 21

Delay unit-1

X 1

Z -1

w 31

neuron-1

y 1

Delay unit-2

w 12

X 2

Z -1

neuron-2

y 2

w 32

:
:

:
:

:
:

:
:

Delay unit- n

w 1 n

X n

Z -1

neuron- n

y n

w 2 n

Figure 3.7. Configuration of a Hopfield network

In the case of a stable network this function must decrease with time and ultimately
reach its minimum, or it's value remains constant. The minima reached are usually
local minima because there are a number of states corresponding to fixed-point
actuators or stored patterns to which the network must converge. Each finally
reached state of the network has its associated energy defined above.
For the generalized form of binary Hopfield network, in which the sigmoid
function

1

fx ()

e

x

1

is used, the changes in time are continuously described following the equation

du u

j j

N wy U ,

¦

ji i j

dt D

i

j