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Defining Energy

The energy function of interest for Hopfield networks and which we have been using to this point is:
H = - 1/2 _ij w_ij a_ia_j

To see that the stored patterns will be low points in the energy landscape consider the contribution of a single pattern (u) to the energy when that pattern is instantiated:

H =	-1/2 ij wuij auiauj
	= -1/2 ij auiauj auiauj
	= -1/2 n2

Any departure of the current pattern from the stored pattern could lead to a negative term in the sum which would increase the entire energy. Exactly what the energy will be depends on the extent to which the stored patterns overlap. In general, however, the stored patterns will be low in energy.