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 networks 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.
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