  Sequence memory is an essential attribute of natural and artificial
intelligence that enables agents to encode, store, and retrieve complex
sequences of stimuli and actions. Computational models of sequence memory have
been proposed where recurrent Hopfield-like neural networks are trained with
temporally asymmetric Hebbian rules. However, these networks suffer from
limited sequence capacity (maximal length of the stored sequence) due to
interference between the memories. Inspired by recent work on Dense Associative
Memories, we expand the sequence capacity of these models by introducing a
nonlinear interaction term, enhancing separation between the patterns. We
derive novel scaling laws for sequence capacity with respect to network size,
significantly outperforming existing scaling laws for models based on
traditional Hopfield networks, and verify these theoretical results with
numerical simulation. Moreover, we introduce a generalized pseudoinverse rule
to recall sequences of highly correlated patterns. Finally, we extend this
model to store sequences with variable timing between states' transitions and
describe a biologically-plausible implementation, with connections to motor
neuroscience.
