  In this paper we investigate the equilibrium properties of bidirectional
associative memories (BAMs). Introduced by Kosko in 1988 as a generalization of
the Hopfield model to a bipartite structure, the simplest architecture is
defined by two layers of neurons, with synaptic connections only between units
of different layers: even without internal connections within each layer,
information storage and retrieval are still possible through the reverberation
of neural activities passing from one layer to another. We characterize the
computational capabilities of a stochastic extension of this model in the
thermodynamic limit, by applying rigorous techniques from statistical physics.
A detailed picture of the phase diagram at the replica symmetric level is
provided, both at finite temperature and in the noiseless regimes. Also for the
latter, the critical load is further investigated up to one step of replica
symmetry breaking. An analytical and numerical inspection of the transition
curves (namely critical lines splitting the various modes of operation of the
machine) is carried out as the control parameters - noise, load and asymmetry
between the two layer sizes - are tuned. In particular, with a finite asymmetry
between the two layers, it is shown how the BAM can store information more
efficiently than the Hopfield model by requiring less parameters to encode a
fixed number of patterns. Comparisons are made with numerical simulations of
neural dynamics. Finally, a low-load analysis is carried out to explain the
retrieval mechanism in the BAM by analogy with two interacting Hopfield models.
A potential equivalence with two coupled Restricted Boltmzann Machines is also
discussed.
