Attractors in the fully asymmetric SK-model

Generalized Lotka-Volterra Equations with Random, Nonreciprocal Interactions: The Typical Number of Equilibria

We compute the typical number of equilibria of the generalized Lotka-Volterra equations describing species-rich ecosystems with random, nonreciprocal interactions using the replicated Kac-Rice method. We characterize the multiple-equilibria phase by determining the average abundance and similarity between equilibria as a function of their diversity (i.e., of the number of coexisting species) and of the variability of the interactions. We show that linearly unstable equilibria are dominant, and that the typical number of equilibria differs with respect to the average number.

Structure of attractors in randomly connected networks

The deterministic dynamics of randomly connected neural networks are studied, where a state of binary neurons evolves according to a discrete-time synchronous update rule. We give theoretical support that the overlap of systems' states between the current and a previous time develops in time according to a Markovian stochastic process in large networks. This Markovian process predicts how often a network revisits one of the previously visited states, depending on the system size. The state concentration probability, i.e., the probability that two distinct states coevolve to the same state, is utilized to analytically derive various characteristics that quantify attractors' structure. The analytical predictions about the total number of attractors, the typical cycle length, and the number of states belonging to all attractive cycles match well with numerical simulations for relatively large system sizes.

Threshold disorder as a source of diverse and complex behavior in random nets

We study the diversity of complex spatio-temporal patterns in the behavior of random synchronous asymmetric neural networks (RSANNs). Special attention is given to the impact of disordered threshold values on limit-cycle diversity and limit-cycle complexity in RSANNs which have 'normal' thresholds by default. Surprisingly, RSANNs exhibit only a small repertoire of rather complex limit-cycle patterns when all parameters are fixed. This repertoire of complex patterns is also rather stable with respect to small parameter changes. These two unexpected results may generalize to the study of other complex systems. In order to reach beyond this seemingly disabling 'stable and small' aspect of the limit-cycle repertoire of RSANNs, we have found that if an RSANN has threshold disorder above a critical level, then there is a rapid increase of the size of the repertoire of patterns. The repertoire size initially follows a power-law function of the magnitude of the threshold disorder. As the disorder increases further, the limit-cycle patterns themselves become simpler until at a second critical level most of the limit cycles become simple fixed points. Nonetheless, for moderate changes in the threshold parameters, RSANNs are found to display specific features of behavior desired for rapidly responding processing systems: accessibility to a large set of complex patterns.