Modeling the network dynamics of pulse-coupled neurons

Chandra, Sarthak; Hathcock, David; Crain, Kimberly; Antonsen, Thomas M.; Girvan, Michelle; Ott, Edward

doi:10.1063/1.4977514

Cited by 32 publications

(46 citation statements)

References 37 publications

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“…(8), or by considering the limit of µ i → ∞ in Eqs. (10) and (11), we see that σ F i = ±ω i . Since the stable fixed points corresponds to ρ · σ F i > 0, each agent will go to a stable fixed point given by [sgn(ρ ·ω i )]ω i .…”

Section: A Coherent States For D =mentioning

confidence: 75%

Continuous versus Discontinuous Transitions in the D -Dimensional Generalized Kuramoto Model: Odd D is Different

2019

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The Kuramoto model, originally proposed to model the dynamics of many interacting oscillators, has been used and generalized for a wide range of applications involving the collective behavior of large heterogeneous groups of dynamical units whose states are characterized by a scalar angle variable. One such application in which we are interested is the alignment of orientation vectors among members of a swarm. Despite being commonly used for this purpose, the Kuramoto model can only describe swarms in 2 dimensions, and hence the results obtained do not apply to the often relevant situation of swarms in 3 dimensions. Partly based on this motivation, as well as on relevance to the classical, mean-field, zero-temperature Heisenberg model with quenched site disorder, in this paper we study the Kuramoto model generalized to D dimensions. We show that in the important case of 3 dimensions, as well as for any odd number of dimensions, the D-dimensional generalized Kuramoto model for heterogeneous units has dynamics that are remarkably different from the dynamics in 2 dimensions. In particular, for odd D the transition to coherence occurs discontinuously as the inter-unit coupling constant K is increased through zero, as opposed to the D = 2 case (and, as we show, also the case of even D) for which the transition to coherence occurs continuously as K increases through a positive critical value Kc. We also demonstrate the qualitative applicability of our results to related models constructed specifically to capture swarming and flocking dynamics in three dimensions.

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Section: A Coherent States For D =mentioning

confidence: 75%

Continuous versus Discontinuous Transitions in the D -Dimensional Generalized Kuramoto Model: Odd D is Different

2019

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“…0.000151095 -0.00036262 -0.00253634 0.001951034 a 10 8.560764307 -0.00020545 -0.00218981 0.001684474 b 1 -0.14512093 0.348290252 -1.29140450 0.993388080 b 2 -0.07327966 0.175871190 -0.22112111 0.170093167 b 3 -0.01561869 0.037484866 -0.07939645 0.061074193 b 4 -0.00449755 0.010794122 -0.03842115 0.029554737 b 5 -0.00168564 0.004045538 -0.02107670 0.016212846 b 6 -0.00073921 0.001774105 -0.01208433 0.009295643 b 7 -0.00034753 0.000834072 -0.00691601 0.005320008 b 8 -0.00016011 0.000384282 -0.00380417 0.002926287 b 9 -6.11449933 0.000146747 -0.00190524 0.001465571 b 10 -5.59280546 1.342273311 -0.00075672 0.000582099…”

Section: Fourier Coefficientsmentioning

confidence: 99%

A multiple timescales approach to bridging spiking- and population-level dynamics

Park

Ermentrout

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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A rigorous bridge between spiking-level and macroscopic quantities is an on-going and well-developed story for asynchronously firing neurons, but focus has shifted to include neural populations exhibiting varying synchronous dynamics. Recent literature has used the Ott-Antonsen ansatz (2008) to great effect, allowing a rigorous derivation of an order parameter for large oscillator populations. The ansatz has been successfully applied using several models including networks of Kuramoto oscillators, theta models, and integrate-and-fire neurons, along with many types of network topologies. In the present study, we take a converse approach: given the mean field dynamics of slow synapses, we predict the synchronization properties of finite neural populations. The slow synapse assumption is amenable to averaging theory and the method of multiple timescales. Our proposed theory applies to two heterogeneous populations of excitatory-dimensional and inhibitory-dimensional oscillators with homogeneous synaptic weights. We then demonstrate our theory using two examples. In the first example, we take a network of excitatory and inhibitory theta neurons and consider the case with and without heterogeneous inputs. In the second example, we use Traub models with calcium for the excitatory neurons and Wang-Buzsáki models for the inhibitory neurons. We accurately predict phase drift and phase locking in each example even when the slow synapses exhibit non-trivial mean-field dynamics.

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“…(1) in the limit of a large number of agents (N → ∞). Subsequently, this reduction has been applied in studies of a wide variety of systems (e.g., Refs.5,7,8,[13][14][15][16]28,and 29). Several flocking models employ the Kuramoto model (e.g., Refs.…”

Section: Introductionmentioning

confidence: 99%

Complexity reduction ansatz for systems of interacting orientable agents: Beyond the Kuramoto model

Chandra

Girvan

Ott

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Previous results have shown that a large class of complex systems consisting of many interacting heterogeneous phase oscillators exhibit an attracting invariant manifold. This result has enabled reduced analytic system descriptions from which all the long term dynamics of these systems can be calculated. Although very useful, these previous results are limited by the restriction that the individual interacting system components have one-dimensional dynamics, with states described by a single, scalar, angle-like variable (e.g., the Kuramoto model). In this paper we consider a generalization to an appropriate class of coupled agents with higherdimensional dynamics. For this generalized class of model systems we demonstrate that the dynamics again contain an invariant manifold, hence enabling previously inaccessible analysis and improved numerical study, allowing a similar simplified description of these systems. We also discuss examples illustrating the potential utility of our results for a wide range of interesting situations.The dynamics of systems with many coupled dynamical agents is a subject of increasing importance with a very broad range of applications. Much of the progress in this field has flowed from the discovery of solvable paradigmatic 'toy' systems. In many such systems, the agents are assumed to have one-dimensional dynamics, and a certain class of such systems has been shown to be in some sense 'solvable' via a novel analytic technique. In our work we consider a more general class of dynamical systems of coupled agents that may have arbitrary dimension. This is a significantly broader class of systems that contains, but is not limited to the previously described class of systems with one-dimensional agents. We then demonstrate that in this broader class of dynamical systems we can construct an analogous technique that can be used to solve such systems. Our method provides analytic techniques to allow previously inaccessible mathematical analyses of these systems. We give significant examples applying our method to large systems of interacting higher-dimensional agents, with a particular focus on the Kuramoto model generalized to higher dimensions.

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Modeling the network dynamics of pulse-coupled neurons

Cited by 32 publications

References 37 publications

Continuous versus Discontinuous Transitions in the D -Dimensional Generalized Kuramoto Model: Odd D is Different

Continuous versus Discontinuous Transitions in the D -Dimensional Generalized Kuramoto Model: Odd D is Different

A multiple timescales approach to bridging spiking- and population-level dynamics

Complexity reduction ansatz for systems of interacting orientable agents: Beyond the Kuramoto model

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