  We study with numerical simulation the possible limit behaviors of
synchronous discrete-time deterministic recurrent neural networks composed of N
binary neurons as a function of a network's level of dilution and asymmetry.
The network dilution measures the fraction of neuron couples that are
connected, and the network asymmetry measures to what extent the underlying
connectivity matrix is asymmetric. For each given neural network, we study the
dynamical evolution of all the different initial conditions, thus
characterizing the full dynamical landscape without imposing any learning rule.
Because of the deterministic dynamics, each trajectory converges to an
attractor, that can be either a fixed point or a limit cycle. These attractors
form the set of all the possible limit behaviors of the neural network. For
each network, we then determine the convergence times, the limit cycles'
length, the number of attractors, and the sizes of the attractors' basin. We
show that there are two network structures that maximize the number of possible
limit behaviors. The first optimal network structure is fully-connected and
symmetric. On the contrary, the second optimal network structure is highly
sparse and asymmetric. The latter optimal is similar to what observed in
different biological neuronal circuits. These observations lead us to
hypothesize that independently from any given learning model, an efficient and
effective biologic network that stores a number of limit behaviors close to its
maximum capacity tends to develop a connectivity structure similar to one of
the optimal networks we found.
