Type I membrane oscillators such as the Connor model (Connor, Walter, and McKown, 1977) and the Morris-Lecar model (Morris and Lecar, 1981) admit very low frequency oscillations near the critical applied current. Hansel et.al., (1995) have numerically shown that synchrony is difficult to achieve with these models and that the phase resetting curve is strictly positive. We use singular perturbation methods and averaging to show that this is a general property of Type I membrane models. We show in a limited sense that so called type 2 resetting occurs with models that obtain rhythmicity via a Hopf bifurcation. We also show the differences between synapses that act rapidly and those that act slowly and derive a canonical form for the phase interactions. 1 Introduction The behavior of coupled neural oscillators has been the subject of a great deal of recent interest. In general, this behavior is quite difficult to analyze. Most of the results to date are primarily based on simulations of ...

We provide a complete analysis of the Kuramoto model of coupled nonlinear oscillators with uncertain natural frequencies and arbitrary interconnection topology. Our work extends and supersedes existing, partial results for the case of an all-to-all connected network. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value all the oscillators synchronize, resulting in convergence of all phase di#erences to a constant value, both in the case of identical natural frequencies as well as uncertain ones. We further explain the behavior of the system as the number of oscillators grows to infinity.

We study non-trivial firing patterns in small assemblies of pulse-coupled oscillatory maps. We find conditions for the existence of waves in rings of coupled maps that are coupled bi-directionally. We also find conditions for stable synchrony in general all-to-all coupled oscillators. Surprisingly, we find that for maps that are derived from physiological data, the stability of synchrony depends on the number of oscillators. We describe rotating waves in two-dimensional lattices of maps and reduce their existence to a reduced system of algebraic equations which are solved numerically.

We study networks of relaxation oscillators coupled with time delay synapses. A pair of oscillators is analyzed and shown to attain loosely synchronous solutions for a wide range of initial conditions and time delays. Simulations of one and two dimensional oscillator networks indicate that locally coupled oscillators are also loosely synchronous. Desynchronous solutions are possible when system parameters are varied. To characterize loosely synchronous networks, we introduce a measure of synchrony, the maximum time difference between any two oscillators. In locally excitatory globally inhibitory oscillator networks with time delays, we find that desynchronous solutions for different groups of oscillators are maintained, and the number of groups that can be segregated is related to the maximum time difference within each group. To examine the maximum time difference, we display its histograms for oscillator networks in one and two dimensions. Also, a range of initial conditions is given...

interaction

In this paper, we present a survey of the use of graph theoretical techniques in Biology. In particular, we discuss recent work on identifying and modelling the structure of bio-molecular networks, as well as the application of centrality measures to interaction networks and research on the hierarchical structure of such networks and network motifs. Work on the link between structural network properties and dynamics is also described, with emphasis on synchronization and disease propagation. 1

How and why the presence of a person directly affects the perception and action of another person is a phenomenon that has been approached in a limited and piecemeal fashion within psychology. This kind of diffuse strategy has failed to capture the jointness of perception and action within and between people. In contradistinction, the authors offer a perspective that retains both integrally social features (e.g., involves interaction) and yet adequately exploits the current state of knowledge regarding the ecological properties of perception–action, while at the same time drawing on aspects of dynamic systems theory. In this article the authors review the best attempts to examine how one individual affects another’s perceptions and actions in the emergence of a social unit of action. Two important approaches, the individual-level and cognitive dynamics approaches, have yielded insights that derive in significant degree from principles of ecological psychology and/or dynamical systems theory. Prototypic of the individual-level approach is a focus on what can be perceived by coactors with the aim of uncovering how the dispositional qualities (affordances) of another person are informationally specified during social interaction. In contrast, the cognitive dynamics approach simulates dynamical characteristics of cognition and psychological influence with the aim of uncovering how cooperative interaction emerges out of its component parts. The authors argue that these approaches involve, respectively, insufficient mutuality and insufficient embodiment. Consequently, a social synergy perspective is discussed that approaches the problem of socially cooperative interaction at the relational, nonreductive level, using novel methods to examine how social perception and action emerge through self-organizing processes.

Abstract — Fireflies exhibit a fascinating phenomenon of spontaneous synchronization that occurs in nature: at dawn, they gather on trees and synchronize progressively without relying on a central entity. The present article 1 reviews this process by looking at experiments that were made on fireflies and the mathematical model of Mirollo and Strogatz [1], which provides key rules to obtaining a synchronized network in a decentralized manner. This model is then applied to wireless ad hoc networks. To properly apply this model with an accuracy limited only to the propagation delay, a novel synchronization scheme, which is derived from the original firefly synchronization principle, is presented, and simulation results are given. I.

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Summary. Motivated by the needs of multiagent systems in the presence of sensing and communication that is delayed, intermittent and asynchronous, we present here a discrete-time Kuramoto oscillator model that incorporates delays. Analytical results are derived for certain cases of stability of synchronized and balanced equilibrium sets both when delay is not present and when it is. In all cases, oscillators are assumed to be identical and fully connected, with constant and equal network delays. For complex cases, the analysis is supplemented with Monte Carlo simulations, and all results are verified in simulation. 1