Exact associative neural memory dynamics utilizing Boolean matrices

The exact dynamics of shallow loaded associative neural memories are generated and characterized. The Boolean matrix analysis approach is employed for the efficient generation of all possible state transition trajectories for parallel updated binary-state dynamic associative memories (DAMs). General expressions for the size of the basin of attraction of fundamental and oscillatory memories and the number of oscillatory and stable states are derived for discrete synchronous Hopfield DAMs loaded with one, two, or three even-dimensionality bipolar memory vectors having the same mutual Hamming distances between them. Spurious memories are shown to occur only if the number of stored patterns exceeds two in an even-dimensionality Hopfield memory. The effects of odd- versus even-dimensionality memory vectors on DAM dynamics and the effects of memory pattern encoding on DAM performance are tested. An extension of the Boolean matrix dynamics characterization technique to other, more complex DAMs is presented.