Though our concern is with static relations, we cannot ignore movement. In order to categorize relations, children must be able to segregate the scene into distinct objects. However, before seven months, they are generally unable to segregate static configurations of objects [Spelke, 1990], and Thelen and Smith (1994)
have proposed that they may learn to use static properties to perform object segregation by observing objects move and then come to rest in space.
If we are to deal with movement, we must first have a means of dealing with time. In a generalized Hopfield network, we face the problem that settling itself requires time. The response of the network to temporal inputs or the network's generation of temporal output must take place at a time scale beyond that necessary for settling. Most approaches to time in neural networks incorporate some sort of short-term memory, a means by which units respond not only to the current state of the network but also to a limited record of its previous states. Within the generalized Hopfield framework, this can be accomplished through the inclusion of delays on connections, an idea due originally to Kleinfeld (1986).
Connecting any two units in the network there may be any number of connections, each with its own delay. Input to a unit along a connection is a function of the activation and phase angle of the destination unit at the time before the delay. Equation 4 shows this relationship.
where input, activation, and phase angle are a function of time (t) and m is the maximum delay.
Demo 3 shows the behavior a simple network with two hard-wired connections between each pair of units, one with no delay and one with a delay of one ``primitive time step.''