  Cortical networks are hypothesized to rely on transient network activity to
support short term memory (STM). In this paper we study the capacity of
randomly connected recurrent linear networks for performing STM when the input
signals are approximately sparse in some basis. We leverage results from
compressed sensing to provide rigorous non asymptotic recovery guarantees,
quantifying the impact of the input sparsity level, the input sparsity basis,
and the network characteristics on the system capacity. Our analysis
demonstrates that network memory capacities can scale superlinearly with the
number of nodes, and in some situations can achieve STM capacities that are
much larger than the network size. We provide perfect recovery guarantees for
finite sequences and recovery bounds for infinite sequences. The latter
analysis predicts that network STM systems may have an optimal recovery length
that balances errors due to omission and recall mistakes. Furthermore, we show
that the conditions yielding optimal STM capacity can be embodied in several
network topologies, including networks with sparse or dense connectivities.
