Sequential (Asynchronous) Hopfield Model

Derivation of Synaptic Weights

Given M binary-valued patterns (i.e., have binary values 0 or 1), the weights of the Hopfield network are derived as
The threshold of the network are given as

Energy functions and convergence

We will use the following notion of energy function, the Liapunov function:
Under the ideal circumstance that the stored vectors are perfectly orthogonal, then every original pattern represents a local (or global) minimum of the energy function. This motivates the design of a network that can iteratively search for a local minimum state. A gradient type technique leads to the sequential Hopfield model. The differnece of the energy functions before and after a state update is
In case of a sequential (asynchronous) update, there is only one bit updated at one time. Without los of generality, let us assume it to be  on the ith bit,
Because ,
Let us introduce a discrete version of the gradient as
In order to guarantee the decrease of the energy function,  should be updated in the negative gradient-descent direction, that is,
This leads to the following sequential Hopfield model.

Algorithm (Sequential Hopfield model)

Contents

Artificial Neural Networks
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