Glauber dynamics of the asymmetric SK-model

From random point processes to hierarchical cavity master equations for stochastic dynamics of disordered systems in random graphs: Ising models and epidemics

We start from the theory of random point processes to derive n-point coupled master equations describing the continuous dynamics of discrete variables in random graphs. These equations constitute a hierarchical set of approximations that generalize and improve the cavity master equation (CME), a recently obtained closure for the usual master equation representing the dynamics. Our derivation clarifies some of the hypotheses and approximations that originally led to the CME, considered now as the first order of a more general technique. We tested the scheme in the dynamics of three models defined over diluted graphs: the Ising ferromagnet, the Viana-Bray spin-glass, and the susceptible-infectious-susceptible model for epidemics. In the first two, the equations perform similarly to the best-known approaches in literature. In the latter, they outperform the well-known pair quenched mean-field approximation.

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.

The dynamics of sparse random networks

Recurrent neural networks with full symmetric connectivity have been extensively studied as associative memories and pattern recognition devices. However, there is considerable evidence that sparse, asymmetrically connected, mainly excitatory networks with broadly directed inhibition are more consistent with biological reality. In this paper, we use the technique of return maps to study the dynamics of random networks with sparse, asymmetric connectivity and nonspecific inhibition. These networks show three qualitatively different kinds of behavior: fixed points, cycles of low period, and extremely long cycles verging on aperiodicity. Using statistical arguments, we relate these behaviors to network parameters and present empirical evidence for the accuracy of this statistical model. The model, in turn, leads to methods for controlling the level of activity in networks. Studying random, untrained networks provides an understanding of the intrinsic dynamics of these systems. Such dynamics could provide a substrate for the much more complex behavior shown when synaptic modification is allowed.