Frequency Plateaus in a Chain of Weakly Coupled Oscillators, I.

Ermentrout, G. Bard; Kopell, Nancy

doi:10.1137/0515019

Cited by 420 publications

(324 citation statements)

References 24 publications

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“…Using the theory of weakly coupled oscillators (Kuramoto 1984;Malkin 1949;Ermentrout 1984), the dynamics of the HCO can be reduced to the consideration of a "phase model", in which the state of the oscillator is described entirely by its phase θ(t). The phase can be decomposed into two parts: θ(t) = t + φ(t) mod T , where t describes the change in the phase of the oscillator in the absence of the external input, and the relative phase φ(t) quantifies the phase shift that results from the external input.…”

Section: Phase Modelsmentioning

confidence: 99%

“…PRCs measure the phase shift of an oscillator in response to a brief perturbation as a function of the timing of the input. PRCs in conjunction with coupled oscillators theory have been widely used in neuroscience to understand how changes in a neuron's intrinsic properties affect its frequency (Schwemmer and Lewis 2011;Ly and Ermentrout 2011), how neuronal oscillators phase-lock to external input (Brumberg and Gutkin 2007;Gouwens et al 2010), and how networks of neurons synchronize (e.g., Ermentrout 1984;Hansel 1995;Mancilla 2007). In almost all cases, the oscillatory units have been taken to be the individual neurons.…”

Section: Introductionmentioning

confidence: 99%

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Phase response properties of half-center oscillators

Zhang

Lewis

2013

J Comput Neurosci

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We examine the phase response properties of half-center oscillators (HCOs) that are modeled by a pair of Morris-Lecar-type neurons connected by strong fast inhibitory synapses. We find that the two basic mechanisms for half-center oscillations, "release" and "escape", give rise to strikingly different phase response curves (PRCs). Release-type HCOs are most sensitive to perturbations delivered to cells at times when they are about to transition from the active to the suppressed state, and PRCs are dominated by a large negative peak (phase delays) at corresponding phases. On the other hand, escape-type HCOs are most sensitive to perturbations delivered to cells at times when they are about to transition from the suppressed to the active state, and PRCs are dominated by a large positive peak (phase advances) at corresponding phases. By analyzing the phase space structure of Morris-Lecar-type HCO models with fast synaptic dynamics, we identify the dynamical mechanisms underlying the shapes of the PRCs. To demonstrate the significance of the different shapes of the PRCs for the release-type and escape-type HCOs, we link the shapes of the PRCs to the different frequency modulation properties of release-type

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Section: Phase Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Phase response properties of half-center oscillators

Zhang

Lewis

2013

J Comput Neurosci

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“…We now argue that the transition to synchrony in the network including the wedged region can be explained using a simple phase oscillator model (Winfree, 1980;Cohen et al, 1982;Kuramoto, 1984;Ermentrout and Kopell, 1984). Specifically, the boundary between the synchronous and the asynchronous states is highly correlated with the degree of natural frequency mismatch among the individual units within the network.…”

Section: Natural Frequency Mismatchmentioning

confidence: 87%

A Model of the Effects of Applied Electric Fields on Neuronal Synchronization

et al. 2005

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We examine the effects of applied electric fields on neuronal synchronization. Two-compartment model neurons were synaptically coupled and embedded within a resistive array, thus allowing the neurons to interact both chemically and electrically. In addition, an external electric field was imposed on the array. The effects of this field were found to be nontrivial, giving rise to domains of synchrony and asynchrony as a function of the heterogeneity among the neurons. A simple phase oscillator reduction was successful in qualitatively reproducing these domains. The findings form several readily testable experimental predictions, and the model can be extended to a larger scale in which the effects of electric fields on seizure activity may be simulated.

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“…The theory of weak coupling (Ermentrout and Kopell 1984;Kuramoto 1984;Neu 1979;, has been widely used to analyze dynamics in networks of oscillating neurons (e.g., Ermentrout 1996;Hoppensteadt and Izikevich 1997;Lewis and Rinzel 2003;Pfeuty et al 2003;. Consider a network of weakly coupled neurons such that each neuron when isolated from the network displays T -periodic limit cycle oscillations.…”

Section: Theory Of Weak Coupling and Reduction To A Phase Modelmentioning

confidence: 99%

“…The two ball-and-stick neurons are coupled via an electrial synapse located at the distal end of their dendrites. We use the theory of weakly coupled oscillators (Ermentrout and Kopell 1984;Kuramoto 1984;Neu 1979; to derive a single scalar differential equation that governs the dynamics of the phase-difference between the two ball-and-stick neurons. This equation inlcudes an analytical representation of the filtering properties of the dendritic coupling.…”

Section: Introductionmentioning

confidence: 99%

The robustness of phase-locking in neurons with dendro-dendritic electrical coupling

Schwemmer

Lewis

2012

J. Math. Biol.

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We examine the effects of dendritic filtering on the existence, stability, and robustness of phase-locked states to heterogeneity and noise in a pair of electrically coupled ball-and-stick neurons with passive dendrites. We use the theory of weakly coupled oscillators and analytically derived filtering properties of the dendritic coupling to systematically explore how the electrotonic length and diameter of dendrites can alter phase-locking. In the case of a fixed value of the coupling conductance (g c ) taken from the literature, we find that repeated exchanges in stability between the synchronous and anti-phase states can occur as the electrical coupling becomes more distally located on the dendrites. However, the robustness of the phase-locked states in this case decreases rapidly towards zero as the distance between the electrical coupling and the somata increases. Published estimates of g c are calculated from the experimentally measured coupling coefficient (CC) based on a single-compartment description of a neuron, and therefore may be severe underestimates of g c . With this in mind, we re-examine the stability and robustness of phase-locking using a fixed value of CC, which imposes a limit on the maximum distance the electrical coupling can be located away from the somata. In this case, although the phase-locked states remain robust over the entire range of possible coupling locations, no exchanges in stability with changing coupling position are observed except for a single exchange that occurs in the case of a high somatic firing frequency and a large dendritic radius. Thus, our analysis suggests that multiple exchanges in stability with changing coupling location are unlikely to be observed in real neural systems.

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Frequency Plateaus in a Chain of Weakly Coupled Oscillators, I.

Cited by 420 publications

References 24 publications

Phase response properties of half-center oscillators

Phase response properties of half-center oscillators

A Model of the Effects of Applied Electric Fields on Neuronal Synchronization

The robustness of phase-locking in neurons with dendro-dendritic electrical coupling

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