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57
Synchronization of pulsecoupled biological oscillators
 SIAM J. Appl. Math
, 1990
"... Abstract. A simple model for synchronous firing of biological oscillators based on Peskin’s model of ..."
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Cited by 194 (0 self)
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Abstract. A simple model for synchronous firing of biological oscillators based on Peskin’s model of
Type I Membranes, Phase Resetting Curves, and Synchrony
 Neural Comput
, 1995
"... Type I membrane oscillators such as the Connor model (Connor, Walter, and McKown, 1977) and the MorrisLecar 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 wit ..."
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Cited by 125 (12 self)
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Type I membrane oscillators such as the Connor model (Connor, Walter, and McKown, 1977) and the MorrisLecar 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 ...
On partial contraction analysis for coupled nonlinear oscillators
 technical Report, Nonlinear Systems Laboratory, MIT
, 2003
"... We describe a simple but general method to analyze networks of coupled identical nonlinear oscillators, and study applications to fast synchronization, locomotion, and schooling. Specifically, we use nonlinear contraction theory to derive exact and global (rather than linearized)results on synchroni ..."
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Cited by 62 (33 self)
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We describe a simple but general method to analyze networks of coupled identical nonlinear oscillators, and study applications to fast synchronization, locomotion, and schooling. Specifically, we use nonlinear contraction theory to derive exact and global (rather than linearized)results on synchronization, antisynchronization and oscillatordeath. The method can be applied to coupled networks of various structures and arbitrary size. For oscillators with positivedefinite diffusion coupling, it can be shown that synchronization always occur globally for strong enough coupling strengths, and an explicit upper bound on the corresponding threshold can be computed through eigenvalue analysis. The discussion also extends to the case when network structure varies abruptly and asynchronously, as in “flocks ” of oscillators or dynamic elements.
On the Stability of the Kuramoto Model of Coupled Nonlinear Oscillators
, 2005
"... We provide an analysis of the classic Kuramoto model of coupled nonlinear oscillators that goes beyond the existing results for alltoall networks of identical oscillators. Our work is applicable to oscillator networks of arbitrary interconnection topology with uncertain natural frequencies. Using ..."
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Cited by 59 (8 self)
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We provide an analysis of the classic Kuramoto model of coupled nonlinear oscillators that goes beyond the existing results for alltoall networks of identical oscillators. Our work is applicable to oscillator networks of arbitrary interconnection topology with uncertain natural frequencies. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value, the synchronized state is locally asymptotically stable, resulting in convergence of all phase differences 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.
Graph theory and networks in biology
 IET Systems Biology, 1:89 – 119
, 2007
"... 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 biomolecular networks, as well as the application of centrality measures to interaction networks and research on the hierarch ..."
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Cited by 21 (0 self)
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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 biomolecular 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
Synchrony, Stability, and Firing Patterns in PulseCoupled Oscillators
 PHYSICA D
, 2002
"... We study nontrivial firing patterns in small assemblies of pulsecoupled oscillatory maps. We find conditions for the existence of waves in rings of coupled maps that are coupled bidirectionally. We also find conditions for stable synchrony in general alltoall coupled oscillators. Surprisingly, ..."
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Cited by 16 (0 self)
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We study nontrivial firing patterns in small assemblies of pulsecoupled oscillatory maps. We find conditions for the existence of waves in rings of coupled maps that are coupled bidirectionally. We also find conditions for stable synchrony in general alltoall 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 twodimensional lattices of maps and reduce their existence to a reduced system of algebraic equations which are solved numerically.
Relaxation Oscillators with Time Delay Coupling
, 1996
"... 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 c ..."
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Cited by 13 (4 self)
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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...
Amplitude expansions for instabilities in populations of globallycoupled oscillators
 J. Stat. Phys
, 1994
"... We analyze the nonlinear dynamics near the incoherent state in a meanfield model of coupled oscillators. The population is described by a FokkerPlanck equation for the distribution of phases, and we apply centermanifold reduction to obtain the amplitude equations for steadystate and Hopf bifurca ..."
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Cited by 12 (4 self)
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We analyze the nonlinear dynamics near the incoherent state in a meanfield model of coupled oscillators. The population is described by a FokkerPlanck equation for the distribution of phases, and we apply centermanifold reduction to obtain the amplitude equations for steadystate and Hopf bifurcation from the equilibrium state with a uniform phase distribution. When the population is described by a native frequency distribution that is reflectionsymmetric about zero, the problem has circular symmetry. In the limit of zero extrinsic noise, although the critical eigenvalues are embedded in the continuous spectrum, the nonlinear coefficients in the amplitude equation remain finite in contrast to the singular behavior found in similar instabilities described by the VlasovPoisson equation. For a bimodal reflectionsymmetric distribution, both types of bifurcation are possible and they coincide at a codimensiontwo Takens Bogdanov point. The steadystate bifurcation may be supercritical or subcritical and produces a timeindependent synchronized state. The Hopf bifurcation produces both supercritical