Tuning in to Slow Events

This chapter addresses entrainments in various slow events. It challenges the idea that only slow events that are isochronous are capable of entraining neural oscillations. It tackles entrainments in events that afford quasi-isochronous driving rhythms as well as in events that are markedly non-isochronous (but coherent). Coherent sequences have time patterns as in short-short-long or long-short-short sequences. This is an important chapter as it differentiates two entrainment protocols: traditional mode-locking versus transient mode-locking. Traditional mode-locking is familiar; it describes entrainment when neither the driving rhythm nor the driven rhythm change significantly (fluctuations are all right). Traditional mode-locking is governed by a single (global) attractor. By contrast, transient mode-locking refers to fleeting entrainments to changing driving rhythms, given the persisting period of driven oscillation. This form of mode-locking delivers a series of (local) attractors. This chapter develops these ideas and provides many examples.

Metastability in a class of hyperbolic dynamical systems perturbed by heavy-tailed Lévy type noise

We consider a finite dimensional deterministic dynamical system with finitely many local attractors Kι, each of which supports a unique ergodic probability measure Pι, perturbed by a multiplicative non-Gaussian heavy-tailed Lévy noise of small intensity ε > 0. We show that the random system exhibits a metastable behavior: there exists a unique ε-dependent time scale on which the system reminds of a continuous time Markov chain on the set of the invariant measures Pι. In particular our approach covers the case of dynamical systems of Morse–Smale type, whose attractors consist of points and limit cycles, perturbed by multiplicative α-stable Lévy noise in the Itô, Stratonovich and Marcus sense. As examples we consider α-stable Lévy perturbations of the Duffing equation and Pareto perturbations of a biochemical birhythmic system with two nested limit cycles.

Particle Swarm Optimization Based on Local Attractors of Ordinary Differential Equation System

Particle swarm optimization (PSO) is inspired by sociological behavior. In this paper, we interpret PSO as a finite difference scheme for solving a system of stochastic ordinary differential equations (SODE). In this framework, the position points of the swarm converge to an equilibrium point of the SODE and the local attractors, which are easily defined by the present position points, also converge to the global attractor. Inspired by this observation, we propose a class of modified PSO iteration methods (MPSO) based on local attractors of the SODE. The idea of MPSO is to choose the next update state near the present local attractor, rather than the present position point as in the original PSO, according to a given probability density function. In particular, the quantum-behaved particle swarm optimization method turns out to be a special case of MPSO by taking a special probability density function. The MPSO methods with six different probability density functions are tested on a few benchmark problems. These MPSO methods behave differently for different problems. Thus, our framework not only gives an interpretation for the ordinary PSO but also, more importantly, provides a warehouse of PSO-like methods to choose from for solving different practical problems.

“If You Can't Be With the One You Love, Love the One You're With”: How Individual Habituation of Agent Interactions Improves Global Utility

Simple distributed strategies that modify the behavior of selfish individuals in a manner that enhances cooperation or global efficiency have proved difficult to identify. We consider a network of selfish agents who each optimize their individual utilities by coordinating (or anticoordinating) with their neighbors, to maximize the payoffs from randomly weighted pairwise games. In general, agents will opt for the behavior that is the best compromise (for them) of the many conflicting constraints created by their neighbors, but the attractors of the system as a whole will not maximize total utility. We then consider agents that act as creatures of habit by increasing their preference to coordinate (anticoordinate) with whichever neighbors they are coordinated (anticoordinated) with at present. These preferences change slowly while the system is repeatedly perturbed, so that it settles to many different local attractors. We find that under these conditions, with each perturbation there is a progressively higher chance of the system settling to a configuration with high total utility. Eventually, only one attractor remains, and that attractor is very likely to maximize (or almost maximize) global utility. This counterintuitive result can be understood using theory from computational neuroscience; we show that this simple form of habituation is equivalent to Hebbian learning, and the improved optimization of global utility that is observed results from well-known generalization capabilities of associative memory acting at the network scale. This causes the system of selfish agents, each acting individually but habitually, to collectively identify configurations that maximize total utility.