1
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Lu Y, Rinzel J. Firing rate models for gamma oscillations in I-I and E-I networks. J Comput Neurosci 2024:10.1007/s10827-024-00877-z. [PMID: 39160322 DOI: 10.1007/s10827-024-00877-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/15/2024] [Revised: 07/15/2024] [Accepted: 08/05/2024] [Indexed: 08/21/2024]
Abstract
Firing rate models for describing the mean-field activities of neuronal ensembles can be used effectively to study network function and dynamics, including synchronization and rhythmicity of excitatory-inhibitory populations. However, traditional Wilson-Cowan-like models, even when extended to include an explicit dynamic synaptic activation variable, are found unable to capture some dynamics such as Interneuronal Network Gamma oscillations (ING). Use of an explicit delay is helpful in simulations at the expense of complicating mathematical analysis. We resolve this issue by introducing a dynamic variable, u, that acts as an effective delay in the negative feedback loop between firing rate (r) and synaptic gating of inhibition (s). In effect, u endows synaptic activation with second order dynamics. With linear stability analysis, numerical branch-tracking and simulations, we show that our r-u-s rate model captures some key qualitative features of spiking network models for ING. We also propose an alternative formulation, a v-u-s model, in which mean membrane potential v satisfies an averaged current-balance equation. Furthermore, we extend the framework to E-I networks. With our six-variable v-u-s model, we demonstrate in firing rate models the transition from Pyramidal-Interneuronal Network Gamma (PING) to ING by increasing the external drive to the inhibitory population without adjusting synaptic weights. Having PING and ING available in a single network, without invoking synaptic blockers, is plausible and natural for explaining the emergence and transition of two different types of gamma oscillations.
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Affiliation(s)
- Yiqing Lu
- Courant Institute of Mathematical Sciences, New York University, New York, NY, USA
| | - John Rinzel
- Courant Institute of Mathematical Sciences, New York University, New York, NY, USA.
- Center for Neural Science, New York University, New York, NY, USA.
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2
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Wüster S, Bhavna R. Spatial correlations in a finite-range Kuramoto model. Phys Rev E 2020; 101:052210. [PMID: 32575303 DOI: 10.1103/physreve.101.052210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2019] [Accepted: 04/21/2020] [Indexed: 11/07/2022]
Abstract
We study spatial correlations between oscillator phases in the steady state of a Kuramoto model, in which phase oscillators that are randomly distributed in space interact with constant strength but within a limited range. Such a model could be relevant, for example, in the synchronization of gene expression oscillations in cells, where only oscillations of neighboring cells are coupled through cell-cell contacts. We analytically infer spatial phase-phase correlation functions from the known steady-state distribution of oscillators for the case of homogenous frequencies and show that these can contain information about the range and strength of interactions, provided the noise in the system can be estimated. We suggest a method for the latter, and also explore when correlations appear to be ergodic in this model, which would enable an experimental measurement of correlation functions to utilize temporal averages. Simulations show that our techniques also give qualitative results for the model with heterogenous frequencies. We illustrate our results by comparison with experimental data on genetic oscillations in the segmentation clock of zebrafish embryos.
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Affiliation(s)
- Sebastian Wüster
- Department of Physics, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462 023, India
| | - Rajasekaran Bhavna
- Department of Biological Sciences, Tata Institute of Fundamental Research, 400005 Mumbai, India
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3
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Shao Y, Zhang J, Tao L. Dimensional reduction of emergent spatiotemporal cortical dynamics via a maximum entropy moment closure. PLoS Comput Biol 2020; 16:e1007265. [PMID: 32516336 PMCID: PMC7304648 DOI: 10.1371/journal.pcbi.1007265] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/11/2019] [Revised: 06/19/2020] [Accepted: 04/29/2020] [Indexed: 11/22/2022] Open
Abstract
Modern electrophysiological recordings and optical imaging techniques have revealed a diverse spectrum of spatiotemporal neural activities underlying fundamental cognitive processing. Oscillations, traveling waves and other complex population dynamical patterns are often concomitant with sensory processing, information transfer, decision making and memory consolidation. While neural population models such as neural mass, population density and kinetic theoretical models have been used to capture a wide range of the experimentally observed dynamics, a full account of how the multi-scale dynamics emerges from the detailed biophysical properties of individual neurons and the network architecture remains elusive. Here we apply a recently developed coarse-graining framework for reduced-dimensional descriptions of neuronal networks to model visual cortical dynamics. We show that, without introducing any new parameters, how a sequence of models culminating in an augmented system of spatially-coupled ODEs can effectively model a wide range of the observed cortical dynamics, ranging from visual stimulus orientation dynamics to traveling waves induced by visual illusory stimuli. In addition to an efficient simulation method, this framework also offers an analytic approach to studying large-scale network dynamics. As such, the dimensional reduction naturally leads to mesoscopic variables that capture the interplay between neuronal population stochasticity and network architecture that we believe to underlie many emergent cortical phenomena.
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Affiliation(s)
- Yuxiu Shao
- Center for Bioinformatics, National Laboratory of Protein Engineering and Plant Genetic Engineering, School of Life Sciences, Peking University, Beijing, China
| | - Jiwei Zhang
- School of Mathematics and Statistics, and Hubei Key Laboratory of Computational Science, Wuhan University, China
| | - Louis Tao
- Center for Bioinformatics, National Laboratory of Protein Engineering and Plant Genetic Engineering, School of Life Sciences, Peking University, Beijing, China
- Center for Quantitative Biology, Peking University, Beijing, China
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4
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Stapmanns J, Kühn T, Dahmen D, Luu T, Honerkamp C, Helias M. Self-consistent formulations for stochastic nonlinear neuronal dynamics. Phys Rev E 2020; 101:042124. [PMID: 32422832 DOI: 10.1103/physreve.101.042124] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/16/2019] [Accepted: 12/18/2019] [Indexed: 01/28/2023]
Abstract
Neural dynamics is often investigated with tools from bifurcation theory. However, many neuron models are stochastic, mimicking fluctuations in the input from unknown parts of the brain or the spiking nature of signals. Noise changes the dynamics with respect to the deterministic model; in particular classical bifurcation theory cannot be applied. We formulate the stochastic neuron dynamics in the Martin-Siggia-Rose de Dominicis-Janssen (MSRDJ) formalism and present the fluctuation expansion of the effective action and the functional renormalization group (fRG) as two systematic ways to incorporate corrections to the mean dynamics and time-dependent statistics due to fluctuations in the presence of nonlinear neuronal gain. To formulate self-consistency equations, we derive a fundamental link between the effective action in the Onsager-Machlup (OM) formalism, which allows the study of phase transitions, and the MSRDJ effective action, which is computationally advantageous. These results in particular allow the derivation of an OM effective action for systems with non-Gaussian noise. This approach naturally leads to effective deterministic equations for the first moment of the stochastic system; they explain how nonlinearities and noise cooperate to produce memory effects. Moreover, the MSRDJ formulation yields an effective linear system that has identical power spectra and linear response. Starting from the better known loopwise approximation, we then discuss the use of the fRG as a method to obtain self-consistency beyond the mean. We present a new efficient truncation scheme for the hierarchy of flow equations for the vertex functions by adapting the Blaizot, Méndez, and Wschebor approximation from the derivative expansion to the vertex expansion. The methods are presented by means of the simplest possible example of a stochastic differential equation that has generic features of neuronal dynamics.
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Affiliation(s)
- Jonas Stapmanns
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA BRAIN Institute I, Jülich Research Centre, Jülich, Germany.,Institute for Theoretical Solid State Physics, RWTH Aachen University, 52074 Aachen, Germany
| | - Tobias Kühn
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA BRAIN Institute I, Jülich Research Centre, Jülich, Germany.,Institute for Theoretical Solid State Physics, RWTH Aachen University, 52074 Aachen, Germany
| | - David Dahmen
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA BRAIN Institute I, Jülich Research Centre, Jülich, Germany
| | - Thomas Luu
- Institut für Kernphysik (IKP-3), Institute for Advanced Simulation (IAS-4) and Jülich Center for Hadron Physics, Jülich Research Centre, Jülich, Germany
| | - Carsten Honerkamp
- Institute for Theoretical Solid State Physics, RWTH Aachen University, 52074 Aachen, Germany.,JARA-FIT, Jülich Aachen Research Alliance-Fundamentals of Future Information Technology, Germany
| | - Moritz Helias
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA BRAIN Institute I, Jülich Research Centre, Jülich, Germany.,Institute for Theoretical Solid State Physics, RWTH Aachen University, 52074 Aachen, Germany
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5
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Abstract
The Wilson-Cowan equations represent a landmark in the history of computational neuroscience. Along with the insights Wilson and Cowan offered for neuroscience, they crystallized an approach to modeling neural dynamics and brain function. Although their iconic equations are used in various guises today, the ideas that led to their formulation and the relationship to other approaches are not well known. Here, we give a little context to some of the biological and theoretical concepts that lead to the Wilson-Cowan equations and discuss how to extend beyond them.
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Affiliation(s)
- Carson C Chow
- Laboratory of Biological Modeling, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland
| | - Yahya Karimipanah
- Laboratory of Biological Modeling, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland
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6
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Zhang J, Shao Y, Rangan AV, Tao L. A coarse-graining framework for spiking neuronal networks: from strongly-coupled conductance-based integrate-and-fire neurons to augmented systems of ODEs. J Comput Neurosci 2019; 46:211-232. [PMID: 30788694 DOI: 10.1007/s10827-019-00712-w] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/05/2018] [Revised: 01/27/2019] [Accepted: 01/31/2019] [Indexed: 11/29/2022]
Abstract
Homogeneously structured, fluctuation-driven networks of spiking neurons can exhibit a wide variety of dynamical behaviors, ranging from homogeneity to synchrony. We extend our partitioned-ensemble average (PEA) formalism proposed in Zhang et al. (Journal of Computational Neuroscience, 37(1), 81-104, 2014a) to systematically coarse grain the heterogeneous dynamics of strongly coupled, conductance-based integrate-and-fire neuronal networks. The population dynamics models derived here successfully capture the so-called multiple-firing events (MFEs), which emerge naturally in fluctuation-driven networks of strongly coupled neurons. Although these MFEs likely play a crucial role in the generation of the neuronal avalanches observed in vitro and in vivo, the mechanisms underlying these MFEs cannot easily be understood using standard population dynamic models. Using our PEA formalism, we systematically generate a sequence of model reductions, going from Master equations, to Fokker-Planck equations, and finally, to an augmented system of ordinary differential equations. Furthermore, we show that these reductions can faithfully describe the heterogeneous dynamic regimes underlying the generation of MFEs in strongly coupled conductance-based integrate-and-fire neuronal networks.
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Affiliation(s)
- Jiwei Zhang
- School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China.,Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, 430072, China
| | - Yuxiu Shao
- Center for Bioinformatics, National Laboratory of Protein Engineering and Plant Genetic Engineering, School of Life Sciences, Peking University, Beijing, 100871, China.,Center for Quantitative Biology, Peking University, Beijing, 100871, China
| | - Aaditya V Rangan
- Courant Institute of Mathematical Sciences, New York University, New York, NY, USA
| | - Louis Tao
- Center for Bioinformatics, National Laboratory of Protein Engineering and Plant Genetic Engineering, School of Life Sciences, Peking University, Beijing, 100871, China. .,Center for Quantitative Biology, Peking University, Beijing, 100871, China.
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7
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Qiu SW, Chow CC. Finite-size effects for spiking neural networks with spatially dependent coupling. Phys Rev E 2018; 98:062414. [PMID: 32478211 PMCID: PMC7258138 DOI: 10.1103/physreve.98.062414] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
We study finite-size fluctuations in a network of spiking deterministic neurons coupled with nonuniform synaptic coupling. We generalize a previously developed theory of finite-size effects for globally coupled neurons with a uniform coupling function. In the uniform coupling case, mean-field theory is well defined by averaging over the network as the number of neurons in the network goes to infinity. However, for nonuniform coupling it is no longer possible to average over the entire network if we are interested in fluctuations at a particular location within the network. We show that if the coupling function approaches a continuous function in the infinite system size limit, then an average over a local neighborhood can be defined such that mean-field theory is well defined for a spatially dependent field. We then use a path-integral formalism to derive a perturbation expansion in the inverse system size around the mean-field limit for the covariance of the input to a neuron (synaptic drive) and firing rate fluctuations due to dynamical deterministic finite-size effects.
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Affiliation(s)
- Si-Wei Qiu
- Laboratory of Biological Modeling, National Institute of Diabetes and Digestive and Kidney Diseases (NIDDK), National Institutes of Health (NIH), Bethesda, Maryland 20892, USA
| | - Carson C Chow
- Laboratory of Biological Modeling, National Institute of Diabetes and Digestive and Kidney Diseases (NIDDK), National Institutes of Health (NIH), Bethesda, Maryland 20892, USA
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8
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Skardal PS. Low-dimensional dynamics of the Kuramoto model with rational frequency distributions. Phys Rev E 2018; 98:022207. [PMID: 30253541 DOI: 10.1103/physreve.98.022207] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/08/2018] [Indexed: 11/07/2022]
Abstract
The Kuramoto model is a paradigmatic tool for studying the dynamics of collective behavior in large ensembles of coupled dynamical systems. Over the past decade a great deal of progress has been made in analytical descriptions of the macroscopic dynamics of the Kuramoto model, facilitated by the discovery of Ott and Antonsen's dimensionality reduction method. However, the vast majority of these works relies on a critical assumption where the oscillators' natural frequencies are drawn from a Cauchy, or Lorentzian, distribution, which allows for a convenient closure of the evolution equations from the dimensionality reduction. In this paper we investigate the low-dimensional dynamics that emerge from a broader family of natural frequency distributions, in particular, a family of rational distribution functions. We show that, as the polynomials that characterize the frequency distribution increase in order, the low-dimensional evolution equations become more complicated, but nonetheless the system dynamics remain simple, displaying a transition from incoherence to partial synchronization at a critical coupling strength. Using the low-dimensional equations we analytically calculate the critical coupling strength corresponding to the onset of synchronization and investigate the scaling properties of the order parameter near the onset of synchronization. These results agree with calculations from Kuramoto's original self-consistency framework, but we emphasize that the low-dimensional equations approach used here allows for a true stability analysis categorizing the bifurcations.
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9
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Gottwald GA. Finite-size effects in a stochastic Kuramoto model. CHAOS (WOODBURY, N.Y.) 2017; 27:101103. [PMID: 29092442 DOI: 10.1063/1.5004618] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/20/2023]
Abstract
We present a collective coordinate approach to study the collective behaviour of a finite ensemble of N stochastic Kuramoto oscillators using two degrees of freedom: one describing the shape dynamics of the oscillators and one describing their mean phase. Contrary to the thermodynamic limit N → ∞ in which the mean phase of the cluster of globally synchronized oscillators is constant in time, the mean phase of a finite-size cluster experiences Brownian diffusion with a variance proportional to 1/N. This finite-size effect is quantitatively well captured by our collective coordinate approach.
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Affiliation(s)
- Georg A Gottwald
- School of Mathematics and Statistics, The University of Sydney, Sydney, New South Wales 2006, Australia
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10
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Barreiro AK, Ly C. Practical approximation method for firing-rate models of coupled neural networks with correlated inputs. Phys Rev E 2017; 96:022413. [PMID: 28950506 DOI: 10.1103/physreve.96.022413] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/13/2017] [Indexed: 01/18/2023]
Abstract
Rapid experimental advances now enable simultaneous electrophysiological recording of neural activity at single-cell resolution across large regions of the nervous system. Models of this neural network activity will necessarily increase in size and complexity, thus increasing the computational cost of simulating them and the challenge of analyzing them. Here we present a method to approximate the activity and firing statistics of a general firing rate network model (of the Wilson-Cowan type) subject to noisy correlated background inputs. The method requires solving a system of transcendental equations and is fast compared to Monte Carlo simulations of coupled stochastic differential equations. We implement the method with several examples of coupled neural networks and show that the results are quantitatively accurate even with moderate coupling strengths and an appreciable amount of heterogeneity in many parameters. This work should be useful for investigating how various neural attributes qualitatively affect the spiking statistics of coupled neural networks.
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Affiliation(s)
- Andrea K Barreiro
- Department of Mathematics, Southern Methodist University, P.O. Box 750235, Dallas, Texas 75275, USA
| | - Cheng Ly
- Department of Statistical Sciences and Operations Research, Virginia Commonwealth University, 1015 Floyd Avenue, Richmond, Virginia 23284, USA
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11
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Ocker GK, Hu Y, Buice MA, Doiron B, Josić K, Rosenbaum R, Shea-Brown E. From the statistics of connectivity to the statistics of spike times in neuronal networks. Curr Opin Neurobiol 2017; 46:109-119. [PMID: 28863386 DOI: 10.1016/j.conb.2017.07.011] [Citation(s) in RCA: 28] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/08/2017] [Revised: 07/21/2017] [Accepted: 07/27/2017] [Indexed: 10/19/2022]
Abstract
An essential step toward understanding neural circuits is linking their structure and their dynamics. In general, this relationship can be almost arbitrarily complex. Recent theoretical work has, however, begun to identify some broad principles underlying collective spiking activity in neural circuits. The first is that local features of network connectivity can be surprisingly effective in predicting global statistics of activity across a network. The second is that, for the important case of large networks with excitatory-inhibitory balance, correlated spiking persists or vanishes depending on the spatial scales of recurrent and feedforward connectivity. We close by showing how these ideas, together with plasticity rules, can help to close the loop between network structure and activity statistics.
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Affiliation(s)
| | - Yu Hu
- Center for Brain Science, Harvard University, United States
| | - Michael A Buice
- Allen Institute for Brain Science, United States; Department of Applied Mathematics, University of Washington, United States
| | - Brent Doiron
- Department of Mathematics, University of Pittsburgh, United States; Center for the Neural Basis of Cognition, Pittsburgh, United States
| | - Krešimir Josić
- Department of Mathematics, University of Houston, United States; Department of Biology and Biochemistry, University of Houston, United States; Department of BioSciences, Rice University, United States
| | - Robert Rosenbaum
- Department of Mathematics, University of Notre Dame, United States
| | - Eric Shea-Brown
- Allen Institute for Brain Science, United States; Department of Applied Mathematics, University of Washington, United States; Department of Physiology and Biophysics, and University of Washington Institute for Neuroengineering, United States.
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12
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Hahne J, Dahmen D, Schuecker J, Frommer A, Bolten M, Helias M, Diesmann M. Integration of Continuous-Time Dynamics in a Spiking Neural Network Simulator. Front Neuroinform 2017; 11:34. [PMID: 28596730 PMCID: PMC5442232 DOI: 10.3389/fninf.2017.00034] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/22/2016] [Accepted: 05/01/2017] [Indexed: 01/21/2023] Open
Abstract
Contemporary modeling approaches to the dynamics of neural networks include two important classes of models: biologically grounded spiking neuron models and functionally inspired rate-based units. We present a unified simulation framework that supports the combination of the two for multi-scale modeling, enables the quantitative validation of mean-field approaches by spiking network simulations, and provides an increase in reliability by usage of the same simulation code and the same network model specifications for both model classes. While most spiking simulations rely on the communication of discrete events, rate models require time-continuous interactions between neurons. Exploiting the conceptual similarity to the inclusion of gap junctions in spiking network simulations, we arrive at a reference implementation of instantaneous and delayed interactions between rate-based models in a spiking network simulator. The separation of rate dynamics from the general connection and communication infrastructure ensures flexibility of the framework. In addition to the standard implementation we present an iterative approach based on waveform-relaxation techniques to reduce communication and increase performance for large-scale simulations of rate-based models with instantaneous interactions. Finally we demonstrate the broad applicability of the framework by considering various examples from the literature, ranging from random networks to neural-field models. The study provides the prerequisite for interactions between rate-based and spiking models in a joint simulation.
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Affiliation(s)
- Jan Hahne
- School of Mathematics and Natural Sciences, Bergische Universität WuppertalWuppertal, Germany
| | - David Dahmen
- Institute of Neuroscience and Medicine (INM-6), Institute for Advanced Simulation (IAS-6), JARA BRAIN Institute I, Jülich Research CentreJülich, Germany
| | - Jannis Schuecker
- Institute of Neuroscience and Medicine (INM-6), Institute for Advanced Simulation (IAS-6), JARA BRAIN Institute I, Jülich Research CentreJülich, Germany
| | - Andreas Frommer
- School of Mathematics and Natural Sciences, Bergische Universität WuppertalWuppertal, Germany
| | - Matthias Bolten
- School of Mathematics and Natural Sciences, Bergische Universität WuppertalWuppertal, Germany
| | - Moritz Helias
- Institute of Neuroscience and Medicine (INM-6), Institute for Advanced Simulation (IAS-6), JARA BRAIN Institute I, Jülich Research CentreJülich, Germany
- Department of Physics, Faculty 1, RWTH Aachen UniversityAachen, Germany
| | - Markus Diesmann
- Institute of Neuroscience and Medicine (INM-6), Institute for Advanced Simulation (IAS-6), JARA BRAIN Institute I, Jülich Research CentreJülich, Germany
- Department of Physics, Faculty 1, RWTH Aachen UniversityAachen, Germany
- Department of Psychiatry, Psychotherapy and Psychosomatics, Medical Faculty, RWTH Aachen UniversityAachen, Germany
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13
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Bressloff PC. Stochastic Fokker-Planck equation in random environments. Phys Rev E 2016; 94:042129. [PMID: 27841623 DOI: 10.1103/physreve.94.042129] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/31/2016] [Indexed: 06/06/2023]
Abstract
We analyze the stochastic dynamics of a large population of noninteracting particles driven by a common environmental input in the form of an Ornstein-Uhlenbeck (OU) process. The density of particles evolves according to a stochastic Fokker-Planck (FP) equation with respect to different realizations of the OU process. We then exploit the connection with previous work on diffusion in randomly switching environments in order to derive moment equations for the distribution of solutions to the stochastic FP equation. We use perturbation theory and Green's functions to calculate the mean and variance of the distribution when the relaxation rate of the OU process is fast (close to the white-noise limit). Finally, we show how the theory of noise-induced synchronization can be recast into the framework of a stochastic FP equation.
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Affiliation(s)
- Paul C Bressloff
- Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA
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14
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Ottino-Löffler B, Strogatz SH. Kuramoto model with uniformly spaced frequencies: Finite-N asymptotics of the locking threshold. Phys Rev E 2016; 93:062220. [PMID: 27415267 DOI: 10.1103/physreve.93.062220] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/11/2016] [Indexed: 11/07/2022]
Abstract
We study phase locking in the Kuramoto model of coupled oscillators in the special case where the number of oscillators, N, is large but finite, and the oscillators' natural frequencies are evenly spaced on a given interval. In this case, stable phase-locked solutions are known to exist if and only if the frequency interval is narrower than a certain critical width, called the locking threshold. For infinite N, the exact value of the locking threshold was calculated 30 years ago; however, the leading corrections to it for finite N have remained unsolved analytically. Here we derive an asymptotic formula for the locking threshold when N≫1. The leading correction to the infinite-N result scales like either N^{-3/2} or N^{-1}, depending on whether the frequencies are evenly spaced according to a midpoint rule or an end-point rule. These scaling laws agree with numerical results obtained by Pazó [D. Pazó, Phys. Rev. E 72, 046211 (2005)PLEEE81539-375510.1103/PhysRevE.72.046211]. Moreover, our analysis yields the exact prefactors in the scaling laws, which also match the numerics.
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Affiliation(s)
| | - Steven H Strogatz
- Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA
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15
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Hong H, Chaté H, Tang LH, Park H. Finite-size scaling, dynamic fluctuations, and hyperscaling relation in the Kuramoto model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:022122. [PMID: 26382359 DOI: 10.1103/physreve.92.022122] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/22/2015] [Indexed: 06/05/2023]
Abstract
We revisit the Kuramoto model to explore the finite-size scaling (FSS) of the order parameter and its dynamic fluctuations near the onset of the synchronization transition, paying particular attention to effects induced by the randomness of the intrinsic frequencies of oscillators. For a population of size N, we study two ways of sampling the intrinsic frequencies according to the same given unimodal distribution g(ω). In the "random" case, frequencies are generated independently in accordance with g(ω), which gives rise to oscillator number fluctuation within any given frequency interval. In the "regular" case, the N frequencies are generated in a deterministic manner that minimizes the oscillator number fluctuations, leading to quasiuniformly spaced frequencies in the population. We find that the two samplings yield substantially different finite-size properties with clearly distinct scaling exponents. Moreover, the hyperscaling relation between the order parameter and its fluctuations is valid in the regular case, but it is violated in the random case. In this last case, a self-consistent mean-field theory that completely ignores dynamic fluctuations correctly predicts the FSS exponent of the order parameter but not its critical amplitude.
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Affiliation(s)
- Hyunsuk Hong
- Department of Physics and Research Institute of Physics and Chemistry, Chonbuk National University, Jeonju 561-756, Korea
| | - Hugues Chaté
- Service de Physique de l'Etat Condensé, CEA-Saclay, CNRS UMR 3680, 91191 Gif-sur-Yvette, France
- Beijing Computational Science Research Center, Beijing 100084, China
| | - Lei-Han Tang
- Beijing Computational Science Research Center, Beijing 100084, China
- Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China
| | - Hyunggyu Park
- School of Physics and QUC, Korea Institute for Advanced Study, Seoul 130-722, Korea
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16
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Chow CC, Buice MA. Path integral methods for stochastic differential equations. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2015; 5:8. [PMID: 25852983 PMCID: PMC4385267 DOI: 10.1186/s13408-015-0018-5] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 02/12/2015] [Accepted: 02/13/2015] [Indexed: 06/04/2023]
Abstract
Stochastic differential equations (SDEs) have multiple applications in mathematical neuroscience and are notoriously difficult. Here, we give a self-contained pedagogical review of perturbative field theoretic and path integral methods to calculate moments of the probability density function of SDEs. The methods can be extended to high dimensional systems such as networks of coupled neurons and even deterministic systems with quenched disorder.
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Affiliation(s)
- Carson C. Chow
- Mathematical Biology Section, Laboratory of Biological Modeling, NIDDK, NIH, Bethesda, MD 20892 USA
| | - Michael A. Buice
- Mathematical Biology Section, Laboratory of Biological Modeling, NIDDK, NIH, Bethesda, MD 20892 USA
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17
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Zhang JW, Rangan AV. A reduction for spiking integrate-and-fire network dynamics ranging from homogeneity to synchrony. J Comput Neurosci 2015; 38:355-404. [PMID: 25601481 DOI: 10.1007/s10827-014-0543-3] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/18/2014] [Revised: 11/29/2014] [Accepted: 12/09/2014] [Indexed: 10/24/2022]
Abstract
In this paper we provide a general methodology for systematically reducing the dynamics of a class of integrate-and-fire networks down to an augmented 4-dimensional system of ordinary-differential-equations. The class of integrate-and-fire networks we focus on are homogeneously-structured, strongly coupled, and fluctuation-driven. Our reduction succeeds where most current firing-rate and population-dynamics models fail because we account for the emergence of 'multiple-firing-events' involving the semi-synchronous firing of many neurons. These multiple-firing-events are largely responsible for the fluctuations generated by the network and, as a result, our reduction faithfully describes many dynamic regimes ranging from homogeneous to synchronous. Our reduction is based on first principles, and provides an analyzable link between the integrate-and-fire network parameters and the relatively low-dimensional dynamics underlying the 4-dimensional augmented ODE.
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Affiliation(s)
- J W Zhang
- Courant Institute of Mathematical Sciences, New York University, New York, NY, USA
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18
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A coarse-grained framework for spiking neuronal networks: between homogeneity and synchrony. J Comput Neurosci 2013; 37:81-104. [DOI: 10.1007/s10827-013-0488-y] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/24/2013] [Revised: 11/06/2013] [Accepted: 11/11/2013] [Indexed: 10/25/2022]
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19
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Buice MA, Chow CC. Generalized activity equations for spiking neural network dynamics. Front Comput Neurosci 2013; 7:162. [PMID: 24298252 PMCID: PMC3829481 DOI: 10.3389/fncom.2013.00162] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2013] [Accepted: 10/23/2013] [Indexed: 11/25/2022] Open
Abstract
Much progress has been made in uncovering the computational capabilities of spiking neural networks. However, spiking neurons will always be more expensive to simulate compared to rate neurons because of the inherent disparity in time scales-the spike duration time is much shorter than the inter-spike time, which is much shorter than any learning time scale. In numerical analysis, this is a classic stiff problem. Spiking neurons are also much more difficult to study analytically. One possible approach to making spiking networks more tractable is to augment mean field activity models with some information about spiking correlations. For example, such a generalized activity model could carry information about spiking rates and correlations between spikes self-consistently. Here, we will show how this can be accomplished by constructing a complete formal probabilistic description of the network and then expanding around a small parameter such as the inverse of the number of neurons in the network. The mean field theory of the system gives a rate-like description. The first order terms in the perturbation expansion keep track of covariances.
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Affiliation(s)
- Michael A. Buice
- Modeling, Analysis and Theory Team, Allen Institute for Brain ScienceSeattle, WA, USA
| | - Carson C. Chow
- Laboratory of Biological Modeling, NIDDK, National Institutes of HealthBethesda, MD, USA
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20
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Nishikawa I, Tanaka G, Aihara K. Nonstandard scaling law of fluctuations in finite-size systems of globally coupled oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:024102. [PMID: 24032967 DOI: 10.1103/physreve.88.024102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/15/2013] [Indexed: 06/02/2023]
Abstract
Universal scaling laws form one of the central issues in physics. A nonstandard scaling law or a breakdown of a standard scaling law, on the other hand, can often lead to the finding of a new universality class in physical systems. Recently, we found that a statistical quantity related to fluctuations follows a nonstandard scaling law with respect to the system size in a synchronized state of globally coupled nonidentical phase oscillators [I. Nishikawa et al., Chaos 22, 013133 (2012)]. However, it is still unclear how widely this nonstandard scaling law is observed. In the present paper, we discuss the conditions required for the unusual scaling law in globally coupled oscillator systems and validate the conditions by numerical simulations of several different models.
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Affiliation(s)
- Isao Nishikawa
- Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan
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21
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Buice MA, Chow CC. Beyond mean field theory: statistical field theory for neural networks. JOURNAL OF STATISTICAL MECHANICS (ONLINE) 2013; 2013:P03003. [PMID: 25243014 PMCID: PMC4169078 DOI: 10.1088/1742-5468/2013/03/p03003] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/12/2023]
Abstract
Mean field theories have been a stalwart for studying the dynamics of networks of coupled neurons. They are convenient because they are relatively simple and possible to analyze. However, classical mean field theory neglects the effects of fluctuations and correlations due to single neuron effects. Here, we consider various possible approaches for going beyond mean field theory and incorporating correlation effects. Statistical field theory methods, in particular the Doi-Peliti-Janssen formalism, are particularly useful in this regard.
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Affiliation(s)
- Michael A Buice
- Center for Learning and Memory, University of Texas at Austin, Austin, TX, USA
| | - Carson C Chow
- Laboratory of Biological Modeling, NIDDK, NIH, Bethesda, MD, USA
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22
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Buice MA, Chow CC. Dynamic finite size effects in spiking neural networks. PLoS Comput Biol 2013; 9:e1002872. [PMID: 23359258 PMCID: PMC3554590 DOI: 10.1371/journal.pcbi.1002872] [Citation(s) in RCA: 46] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/16/2012] [Accepted: 11/21/2012] [Indexed: 11/19/2022] Open
Abstract
We investigate the dynamics of a deterministic finite-sized network of synaptically coupled spiking neurons and present a formalism for computing the network statistics in a perturbative expansion. The small parameter for the expansion is the inverse number of neurons in the network. The network dynamics are fully characterized by a neuron population density that obeys a conservation law analogous to the Klimontovich equation in the kinetic theory of plasmas. The Klimontovich equation does not possess well-behaved solutions but can be recast in terms of a coupled system of well-behaved moment equations, known as a moment hierarchy. The moment hierarchy is impossible to solve but in the mean field limit of an infinite number of neurons, it reduces to a single well-behaved conservation law for the mean neuron density. For a large but finite system, the moment hierarchy can be truncated perturbatively with the inverse system size as a small parameter but the resulting set of reduced moment equations that are still very difficult to solve. However, the entire moment hierarchy can also be re-expressed in terms of a functional probability distribution of the neuron density. The moments can then be computed perturbatively using methods from statistical field theory. Here we derive the complete mean field theory and the lowest order second moment corrections for physiologically relevant quantities. Although we focus on finite-size corrections, our method can be used to compute perturbative expansions in any parameter. One avenue towards understanding how the brain functions is to create computational and mathematical models. However, a human brain has on the order of a hundred billion neurons with a quadrillion synaptic connections. Each neuron is a complex cell comprised of multiple compartments hosting a myriad of ions, proteins and other molecules. Even if computing power continues to increase exponentially, directly simulating all the processes in the brain on a computer is not feasible in the foreseeable future and even if this could be achieved, the resulting simulation may be no simpler to understand than the brain itself. Hence, the need for more tractable models. Historically, systems with many interacting bodies are easier to understand in the two opposite limits of a small number or an infinite number of elements and most of the theoretical efforts in understanding neural networks have been devoted to these two limits. There has been relatively little effort directed to the very relevant but difficult regime of large but finite networks. In this paper, we introduce a new formalism that borrows from the methods of many-body statistical physics to analyze finite size effects in spiking neural networks.
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Affiliation(s)
- Michael A. Buice
- Laboratory of Biological Modeling, NIDDK, NIH, Bethesda, Maryland, United States of America
- * E-mail: (MAB); (CCC)
| | - Carson C. Chow
- Laboratory of Biological Modeling, NIDDK, NIH, Bethesda, Maryland, United States of America
- * E-mail: (MAB); (CCC)
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23
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Naud R, Gerstner W. Coding and decoding with adapting neurons: a population approach to the peri-stimulus time histogram. PLoS Comput Biol 2012; 8:e1002711. [PMID: 23055914 PMCID: PMC3464223 DOI: 10.1371/journal.pcbi.1002711] [Citation(s) in RCA: 34] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/03/2012] [Accepted: 08/03/2012] [Indexed: 11/18/2022] Open
Abstract
The response of a neuron to a time-dependent stimulus, as measured in a Peri-Stimulus-Time-Histogram (PSTH), exhibits an intricate temporal structure that reflects potential temporal coding principles. Here we analyze the encoding and decoding of PSTHs for spiking neurons with arbitrary refractoriness and adaptation. As a modeling framework, we use the spike response model, also known as the generalized linear neuron model. Because of refractoriness, the effect of the most recent spike on the spiking probability a few milliseconds later is very strong. The influence of the last spike needs therefore to be described with high precision, while the rest of the neuronal spiking history merely introduces an average self-inhibition or adaptation that depends on the expected number of past spikes but not on the exact spike timings. Based on these insights, we derive a ‘quasi-renewal equation’ which is shown to yield an excellent description of the firing rate of adapting neurons. We explore the domain of validity of the quasi-renewal equation and compare it with other rate equations for populations of spiking neurons. The problem of decoding the stimulus from the population response (or PSTH) is addressed analogously. We find that for small levels of activity and weak adaptation, a simple accumulator of the past activity is sufficient to decode the original input, but when refractory effects become large decoding becomes a non-linear function of the past activity. The results presented here can be applied to the mean-field analysis of coupled neuron networks, but also to arbitrary point processes with negative self-interaction. How can information be encoded and decoded in populations of adapting neurons? A quantitative answer to this question requires a mathematical expression relating neuronal activity to the external stimulus, and, conversely, stimulus to neuronal activity. Although widely used equations and models exist for the special problem of relating external stimulus to the action potentials of a single neuron, the analogous problem of relating the external stimulus to the activity of a population has proven more difficult. There is a bothersome gap between the dynamics of single adapting neurons and the dynamics of populations. Moreover, if we ignore the single neurons and describe directly the population dynamics, we are faced with the ambiguity of the adapting neural code. The neural code of adapting populations is ambiguous because it is possible to observe a range of population activities in response to a given instantaneous input. Somehow the ambiguity is resolved by the knowledge of the population history, but how precisely? In this article we use approximation methods to provide mathematical expressions that describe the encoding and decoding of external stimuli in adapting populations. The theory presented here helps to bridge the gap between the dynamics of single neurons and that of populations.
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Affiliation(s)
| | - Wulfram Gerstner
- School of Computer and Communication Sciences and School of Life Sciences, Brain Mind Institute, Ecole Polytechnique Fédérale de Lausanne, Lausanne-EPFL, Lausanne, Switzerland
- * E-mail:
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24
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Nishikawa I, Tanaka G, Horita T, Aihara K. Long-term fluctuations in globally coupled phase oscillators with general coupling: finite size effects. CHAOS (WOODBURY, N.Y.) 2012; 22:013133. [PMID: 22463009 DOI: 10.1063/1.3692966] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
We investigate the diffusion coefficient of the time integral of the Kuramoto order parameter in globally coupled nonidentical phase oscillators. This coefficient represents the deviation of the time integral of the order parameter from its mean value on the sample average. In other words, this coefficient characterizes long-term fluctuations of the order parameter. For a system of N coupled oscillators, we introduce a statistical quantity D, which denotes the product of N and the diffusion coefficient. We study the scaling law of D with respect to the system size N. In other well-known models such as the Ising model, the scaling property of D is D∼O(1) for both coherent and incoherent regimes except for the transition point. In contrast, in the globally coupled phase oscillators, the scaling law of D is different for the coherent and incoherent regimes: D∼O(1/N(a)) with a certain constant a>0 in the coherent regime and D∼O(1) in the incoherent regime. We demonstrate that these scaling laws hold for several representative coupling schemes.
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Affiliation(s)
- Isao Nishikawa
- Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
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25
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Buice MA, Chow CC. Effective stochastic behavior in dynamical systems with incomplete information. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:051120. [PMID: 22181382 PMCID: PMC3457716 DOI: 10.1103/physreve.84.051120] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/25/2011] [Revised: 09/12/2011] [Indexed: 05/26/2023]
Abstract
Complex systems are generally analytically intractable and difficult to simulate. We introduce a method for deriving an effective stochastic equation for a high-dimensional deterministic dynamical system for which some portion of the configuration is not precisely specified. We use a response function path integral to construct an equivalent distribution for the stochastic dynamics from the distribution of the incomplete information. We apply this method to the Kuramoto model of coupled oscillators to derive an effective stochastic equation for a single oscillator interacting with a bath of oscillators and also outline the procedure for other systems.
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Affiliation(s)
- Michael A Buice
- Laboratory of Biological Modeling, NIDDK, NIH, Bethesda, Maryland 20892, USA
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26
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Breakspear M, Heitmann S, Daffertshofer A. Generative models of cortical oscillations: neurobiological implications of the kuramoto model. Front Hum Neurosci 2010; 4:190. [PMID: 21151358 PMCID: PMC2995481 DOI: 10.3389/fnhum.2010.00190] [Citation(s) in RCA: 238] [Impact Index Per Article: 17.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/05/2010] [Accepted: 09/22/2010] [Indexed: 11/21/2022] Open
Abstract
Understanding the fundamental mechanisms governing fluctuating oscillations in large-scale cortical circuits is a crucial prelude to a proper knowledge of their role in both adaptive and pathological cortical processes. Neuroscience research in this area has much to gain from understanding the Kuramoto model, a mathematical model that speaks to the very nature of coupled oscillating processes, and which has elucidated the core mechanisms of a range of biological and physical phenomena. In this paper, we provide a brief introduction to the Kuramoto model in its original, rather abstract, form and then focus on modifications that increase its neurobiological plausibility by incorporating topological properties of local cortical connectivity. The extended model elicits elaborate spatial patterns of synchronous oscillations that exhibit persistent dynamical instabilities reminiscent of cortical activity. We review how the Kuramoto model may be recast from an ordinary differential equation to a population level description using the nonlinear Fokker-Planck equation. We argue that such formulations are able to provide a mechanistic and unifying explanation of oscillatory phenomena in the human cortex, such as fluctuating beta oscillations, and their relationship to basic computational processes including multistability, criticality, and information capacity.
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Affiliation(s)
- Michael Breakspear
- School of Psychiatry, University of New South WalesSydney, NSW, Australia
- The Black Dog Institute, Prince of Wales HospitalSydney, NSW, Australia
- Queensland Institute of Medical ResearchBrisbane, QLD, Australia
- Royal Brisbane and Women's Hospital, BrisbaneQLD, Australia
| | - Stewart Heitmann
- School of Psychiatry, University of New South WalesSydney, NSW, Australia
- The Black Dog Institute, Prince of Wales HospitalSydney, NSW, Australia
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27
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Lu W, Rossoni E, Feng J. On a Gaussian neuronal field model. Neuroimage 2010; 52:913-33. [DOI: 10.1016/j.neuroimage.2010.02.075] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/28/2009] [Revised: 02/09/2010] [Accepted: 02/26/2010] [Indexed: 10/19/2022] Open
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28
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Taylor D, Ott E, Restrepo JG. Spontaneous synchronization of coupled oscillator systems with frequency adaptation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:046214. [PMID: 20481814 DOI: 10.1103/physreve.81.046214] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/19/2010] [Indexed: 05/29/2023]
Abstract
We study the synchronization of Kuramoto oscillators with all-to-all coupling in the presence of slow, noisy frequency adaptation. In this paper, we develop a model for oscillators, which adapt both their phases and frequencies. It is found that this model naturally reproduces some observed phenomena that are not qualitatively produced by the standard Kuramoto model, such as long waiting times before the synchronization of clapping audiences. By assuming a self-consistent steady state solution, we find three stability regimes for the coupling constant k , separated by critical points k{1} and k{2}: (i) for k<k{1} only the stable incoherent state exists; (ii) for k>k{2}, the incoherent state becomes unstable and only the synchronized state exists; and (iii) for k{1}<k<k{2} both the incoherent and synchronized states are stable. In the bistable regime spontaneous transitions between the incoherent and synchronized states are observed for finite ensembles. These transitions are well described as a stochastic process on the order parameter r undergoing fluctuations due to the system's finite size, leading to the following conclusions: (a) in the bistable regime, the average waiting time of an incoherent-->coherent transition can be predicted by using Kramer's escape time formula and grows exponentially with the number of oscillators; (b) when the incoherent state is unstable (k>k{2}), the average waiting time grows logarithmically with the number of oscillators.
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Affiliation(s)
- Dane Taylor
- Department of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, Colorado 80309, USA.
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29
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Abstract
Population rate or activity equations are the foundation of a common approach to modeling for neural networks. These equations provide mean field dynamics for the firing rate or activity of neurons within a network given some connectivity. The shortcoming of these equations is that they take into account only the average firing rate, while leaving out higher-order statistics like correlations between firing. A stochastic theory of neural networks that includes statistics at all orders was recently formulated. We describe how this theory yields a systematic extension to population rate equations by introducing equations for correlations and appropriate coupling terms. Each level of the approximation yields closed equations; they depend only on the mean and specific correlations of interest, without an ad hoc criterion for doing so. We show in an example of an all-to-all connected network how our system of generalized activity equations captures phenomena missed by the mean field rate equations alone.
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30
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Rangan AV. Diagrammatic expansion of pulse-coupled network dynamics in terms of subnetworks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:036101. [PMID: 19905174 DOI: 10.1103/physreve.80.036101] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2008] [Revised: 05/11/2009] [Indexed: 05/28/2023]
Abstract
We introduce a framework wherein various measurements of a pulse-coupled network's stationary dynamics can be expanded in terms of the network's connectivity. Such measurements include the occurrence rate of pulses (e.g., firing rates within a neuronal network) as well as higher-order correlations in activity between various nodes in the network. The various terms in this expansion can be interpreted as diagrams corresponding to subnetworks of the original network, which span both space (in terms of the network's graph) as well as time (in the sense of causality).
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Affiliation(s)
- Aaditya V Rangan
- Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012, USA
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31
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Rangan AV. Diagrammatic expansion of pulse-coupled network dynamics. PHYSICAL REVIEW LETTERS 2009; 102:158101. [PMID: 19518674 DOI: 10.1103/physrevlett.102.158101] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/15/2008] [Indexed: 05/27/2023]
Abstract
We introduce a framework wherein various long-time measurements of a pulse-coupled network's stationary dynamics can be expanded in terms of the network's connectivity. Such measurements include the occurrence rate of pulses as well as higher-order correlations in activity between various nodes in the network. The various terms in this expansion can be interpreted as diagrams corresponding to subnetworks of the original network which span both space (in terms of the network's graph) as well as time (in the sense of causality).
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Affiliation(s)
- Aaditya V Rangan
- Courant Institute of Mathematical Sciences, New York University, New York, New York 10012-1185, USA
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32
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Martens EA, Barreto E, Strogatz SH, Ott E, So P, Antonsen TM. Exact results for the Kuramoto model with a bimodal frequency distribution. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:026204. [PMID: 19391817 DOI: 10.1103/physreve.79.026204] [Citation(s) in RCA: 119] [Impact Index Per Article: 7.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/07/2008] [Indexed: 05/08/2023]
Abstract
We analyze a large system of globally coupled phase oscillators whose natural frequencies are bimodally distributed. The dynamics of this system has been the subject of long-standing interest. In 1984 Kuramoto proposed several conjectures about its behavior; ten years later, Crawford obtained the first analytical results by means of a local center manifold calculation. Nevertheless, many questions have remained open, especially about the possibility of global bifurcations. Here we derive the system's stability diagram for the special case where the bimodal distribution consists of two equally weighted Lorentzians. Using an ansatz recently discovered by Ott and Antonsen, we show that in this case the infinite-dimensional problem reduces exactly to a flow in four dimensions. Depending on the parameters and initial conditions, the long-term dynamics evolves to one of three states: incoherence, where all the oscillators are desynchronized; partial synchrony, where a macroscopic group of phase-locked oscillators coexists with a sea of desynchronized ones; and a standing wave state, where two counter-rotating groups of phase-locked oscillators emerge. Analytical results are presented for the bifurcation boundaries between these states. Similar results are also obtained for the case in which the bimodal distribution is given by the sum of two Gaussians.
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Affiliation(s)
- E A Martens
- Department of Theoretical & Applied Mechanics, Cornell University, Ithaca, New York 14853, USA
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33
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Balance between noise and adaptation in competition models of perceptual bistability. J Comput Neurosci 2009; 27:37-54. [PMID: 19125318 DOI: 10.1007/s10827-008-0125-3] [Citation(s) in RCA: 108] [Impact Index Per Article: 7.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/09/2008] [Revised: 10/23/2008] [Accepted: 11/04/2008] [Indexed: 10/21/2022]
Abstract
Perceptual bistability occurs when a physical stimulus gives rise to two distinct interpretations that alternate irregularly. Noise and adaptation processes are two possible mechanisms for switching in neuronal competition models that describe the alternating behaviors. Either of these processes, if strong enough, could alone cause the alternations in dominance. We examined their relative role in producing alternations by studying models where by smoothly varying the parameters, one can change the rhythmogenesis mechanism from being adaptation-driven to noise-driven. In consideration of the experimental constraints on the statistics of the alternations (mean and shape of the dominance duration distribution and correlations between successive durations) we ask whether we can rule out one of the mechanisms. We conclude that in order to comply with the observed mean of the dominance durations and their coefficient of variation, the models must operate within a balance between the noise and adaptation strength-both mechanisms are involved in producing alternations, in such a way that the system operates near the boundary between being adaptation-driven and noise-driven.
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