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Smirnov LA, Munyayev VO, Bolotov MI, Osipov GV, Belykh I. How synaptic function controls critical transitions in spiking neuron networks: insight from a Kuramoto model reduction. FRONTIERS IN NETWORK PHYSIOLOGY 2024; 4:1423023. [PMID: 39185374 PMCID: PMC11341377 DOI: 10.3389/fnetp.2024.1423023] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 04/25/2024] [Accepted: 07/16/2024] [Indexed: 08/27/2024]
Abstract
The dynamics of synaptic interactions within spiking neuron networks play a fundamental role in shaping emergent collective behavior. This paper studies a finite-size network of quadratic integrate-and-fire neurons interconnected via a general synaptic function that accounts for synaptic dynamics and time delays. Through asymptotic analysis, we transform this integrate-and-fire network into the Kuramoto-Sakaguchi model, whose parameters are explicitly expressed via synaptic function characteristics. This reduction yields analytical conditions on synaptic activation rates and time delays determining whether the synaptic coupling is attractive or repulsive. Our analysis reveals alternating stability regions for synchronous and partially synchronous firing, dependent on slow synaptic activation and time delay. We also demonstrate that the reduced microscopic model predicts the emergence of synchronization, weakly stable cyclops states, and non-stationary regimes remarkably well in the original integrate-and-fire network and its theta neuron counterpart. Our reduction approach promises to open the door to rigorous analysis of rhythmogenesis in networks with synaptic adaptation and plasticity.
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Affiliation(s)
- Lev A. Smirnov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia
| | - Vyacheslav O. Munyayev
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia
| | - Maxim I. Bolotov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia
| | - Grigory V. Osipov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia
| | - Igor Belykh
- Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University, Atlanta, GA, United States
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2
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Pietras B. Pulse Shape and Voltage-Dependent Synchronization in Spiking Neuron Networks. Neural Comput 2024; 36:1476-1540. [PMID: 39028958 DOI: 10.1162/neco_a_01680] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/05/2023] [Accepted: 03/18/2024] [Indexed: 07/21/2024]
Abstract
Pulse-coupled spiking neural networks are a powerful tool to gain mechanistic insights into how neurons self-organize to produce coherent collective behavior. These networks use simple spiking neuron models, such as the θ-neuron or the quadratic integrate-and-fire (QIF) neuron, that replicate the essential features of real neural dynamics. Interactions between neurons are modeled with infinitely narrow pulses, or spikes, rather than the more complex dynamics of real synapses. To make these networks biologically more plausible, it has been proposed that they must also account for the finite width of the pulses, which can have a significant impact on the network dynamics. However, the derivation and interpretation of these pulses are contradictory, and the impact of the pulse shape on the network dynamics is largely unexplored. Here, I take a comprehensive approach to pulse coupling in networks of QIF and θ-neurons. I argue that narrow pulses activate voltage-dependent synaptic conductances and show how to implement them in QIF neurons such that their effect can last through the phase after the spike. Using an exact low-dimensional description for networks of globally coupled spiking neurons, I prove for instantaneous interactions that collective oscillations emerge due to an effective coupling through the mean voltage. I analyze the impact of the pulse shape by means of a family of smooth pulse functions with arbitrary finite width and symmetric or asymmetric shapes. For symmetric pulses, the resulting voltage coupling is not very effective in synchronizing neurons, but pulses that are slightly skewed to the phase after the spike readily generate collective oscillations. The results unveil a voltage-dependent spike synchronization mechanism at the heart of emergent collective behavior, which is facilitated by pulses of finite width and complementary to traditional synaptic transmission in spiking neuron networks.
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Affiliation(s)
- Bastian Pietras
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08018, Barcelona, Spain
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3
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Laing CR, Krauskopf B. Theta neuron subject to delayed feedback: a prototypical model for self-sustained pulsing. Proc Math Phys Eng Sci 2022. [DOI: 10.1098/rspa.2022.0292] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022] Open
Abstract
We consider a single theta neuron with delayed self-feedback in the form of a Dirac delta function in time. Because the dynamics of a theta neuron on its own can be solved explicitly—it is either excitable or shows self-pulsations—we are able to derive algebraic expressions for the existence and stability of the periodic solutions that arise in the presence of feedback. These periodic solutions are characterized by one or more equally spaced pulses per delay interval, and there is an increasing amount of multistability with increasing delay time. We present a complete description of where these self-sustained oscillations can be found in parameter space; in particular, we derive explicit expressions for the loci of their saddle-node bifurcations. We conclude that the theta neuron with delayed self-feedback emerges as a prototypical model: it provides an analytical basis for understanding pulsating dynamics observed in other excitable systems subject to delayed self-coupling.
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Affiliation(s)
- Carlo R. Laing
- School of Natural and Computational Sciences Massey University, Private Bag 102-904, North Shore Mail Centre, Auckland 0745, New Zealand
| | - Bernd Krauskopf
- Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
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4
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Barioni AED, de Aguiar MAM. Ott-Antonsen ansatz for the D-dimensional Kuramoto model: A constructive approach. CHAOS (WOODBURY, N.Y.) 2021; 31:113141. [PMID: 34881619 DOI: 10.1063/5.0069350] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/30/2021] [Accepted: 11/03/2021] [Indexed: 06/13/2023]
Abstract
Kuramoto's original model describes the dynamics and synchronization behavior of a set of interacting oscillators represented by their phases. The system can also be pictured as a set of particles moving on a circle in two dimensions, which allows a direct generalization to particles moving on the surface of higher dimensional spheres. One of the key features of the 2D system is the presence of a continuous phase transition to synchronization as the coupling intensity increases. Ott and Antonsen proposed an ansatz for the distribution of oscillators that allowed them to describe the dynamics of the order parameter with a single differential equation. A similar ansatz was later proposed for the D-dimensional model by using the same functional form of the 2D ansatz and adjusting its parameters. In this article, we develop a constructive method to find the ansatz, similarly to the procedure used in 2D. The method is based on our previous work for the 3D Kuramoto model where the ansatz was constructed using the spherical harmonics decomposition of the distribution function. In the case of motion in a D-dimensional sphere, the ansatz is based on the hyperspherical harmonics decomposition. Our result differs from the previously proposed ansatz and provides a simpler and more direct connection between the order parameter and the ansatz.
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Affiliation(s)
- Ana Elisa D Barioni
- Instituto de Física "Gleb Wataghin," Universidade Estadual de Campinas, Unicamp, 13083-970 Campinas, São Paulo, Brazil
| | - Marcus A M de Aguiar
- Instituto de Física "Gleb Wataghin," Universidade Estadual de Campinas, Unicamp, 13083-970 Campinas, São Paulo, Brazil
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5
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Kadhim KL, Hermundstad AM, Brown KS. Structured patterns of activity in pulse-coupled oscillator networks with varied connectivity. PLoS One 2021; 16:e0256034. [PMID: 34379694 PMCID: PMC8357159 DOI: 10.1371/journal.pone.0256034] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/08/2021] [Accepted: 07/28/2021] [Indexed: 11/25/2022] Open
Abstract
Identifying coordinated activity within complex systems is essential to linking their structure and function. We study collective activity in networks of pulse-coupled oscillators that have variable network connectivity and integrate-and-fire dynamics. Starting from random initial conditions, we see the emergence of three broad classes of behaviors that differ in their collective spiking statistics. In the first class (“temporally-irregular”), all nodes have variable inter-spike intervals, and the resulting firing patterns are irregular. In the second (“temporally-regular”), the network generates a coherent, repeating pattern of activity in which all nodes fire with the same constant inter-spike interval. In the third (“chimeric”), subgroups of coherently-firing nodes coexist with temporally-irregular nodes. Chimera states have previously been observed in networks of oscillators; here, we find that the notions of temporally-regular and chimeric states encompass a much richer set of dynamical patterns than has yet been described. We also find that degree heterogeneity and connection density have a strong effect on the resulting state: in binomial random networks, high degree variance and intermediate connection density tend to produce temporally-irregular dynamics, while low degree variance and high connection density tend to produce temporally-regular dynamics. Chimera states arise with more frequency in networks with intermediate degree variance and either high or low connection densities. Finally, we demonstrate that a normalized compression distance, computed via the Lempel-Ziv complexity of nodal spike trains, can be used to distinguish these three classes of behavior even when the phase relationship between nodes is arbitrary.
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Affiliation(s)
- Kyra L. Kadhim
- Department of Chemical, Biological, and Environmental Engineering, Oregon State University, Corvallis, OR, United States of America
| | - Ann M. Hermundstad
- Janelia Research Campus, Howard Hughes Medical Institute, Ashburn, VA, United States of America
| | - Kevin S. Brown
- Department of Chemical, Biological, and Environmental Engineering, Oregon State University, Corvallis, OR, United States of America
- Department of Pharmaceutical Sciences, Oregon State University, Corvallis, OR, United States of America
- * E-mail:
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6
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Laing CR. Effects of degree distributions in random networks of type-I neurons. Phys Rev E 2021; 103:052305. [PMID: 34134197 DOI: 10.1103/physreve.103.052305] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/01/2021] [Accepted: 04/28/2021] [Indexed: 11/07/2022]
Abstract
We consider large networks of theta neurons and use the Ott-Antonsen ansatz to derive degree-based mean-field equations governing the expected dynamics of the networks. Assuming random connectivity, we investigate the effects of varying the widths of the in- and out-degree distributions on the dynamics of excitatory or inhibitory synaptically coupled networks and gap junction coupled networks. For synaptically coupled networks, the dynamics are independent of the out-degree distribution. Broadening the in-degree distribution destroys oscillations in inhibitory networks and decreases the range of bistability in excitatory networks. For gap junction coupled neurons, broadening the degree distribution varies the values of parameters at which there is an onset of collective oscillations. Many of the results are shown to also occur in networks of more realistic neurons.
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Affiliation(s)
- Carlo R Laing
- School of Natural and Computational Sciences, Massey University, Private Bag 102-904 NSMC, Auckland, New Zealand
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7
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Laing CR, Bläsche C, Means S. Dynamics of Structured Networks of Winfree Oscillators. Front Syst Neurosci 2021; 15:631377. [PMID: 33643004 PMCID: PMC7902706 DOI: 10.3389/fnsys.2021.631377] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/20/2020] [Accepted: 01/18/2021] [Indexed: 01/01/2023] Open
Abstract
Winfree oscillators are phase oscillator models of neurons, characterized by their phase response curve and pulsatile interaction function. We use the Ott/Antonsen ansatz to study large heterogeneous networks of Winfree oscillators, deriving low-dimensional differential equations which describe the evolution of the expected state of networks of oscillators. We consider the effects of correlations between an oscillator's in-degree and out-degree, and between the in- and out-degrees of an “upstream” and a “downstream” oscillator (degree assortativity). We also consider correlated heterogeneity, where some property of an oscillator is correlated with a structural property such as degree. We finally consider networks with parameter assortativity, coupling oscillators according to their intrinsic frequencies. The results show how different types of network structure influence its overall dynamics.
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Affiliation(s)
- Carlo R Laing
- School of Natural and Computational Sciences, Massey University, Auckland, New Zealand
| | - Christian Bläsche
- School of Natural and Computational Sciences, Massey University, Auckland, New Zealand
| | - Shawn Means
- School of Natural and Computational Sciences, Massey University, Auckland, New Zealand
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8
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Laing CR, Bläsche C. The effects of within-neuron degree correlations in networks of spiking neurons. BIOLOGICAL CYBERNETICS 2020; 114:337-347. [PMID: 32124039 DOI: 10.1007/s00422-020-00822-0] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/13/2019] [Accepted: 02/15/2020] [Indexed: 05/20/2023]
Abstract
We consider the effects of correlations between the in- and out-degrees of individual neurons on the dynamics of a network of neurons. By using theta neurons, we can derive a set of coupled differential equations for the expected dynamics of neurons with the same in-degree. A Gaussian copula is used to introduce correlations between a neuron's in- and out-degree, and numerical bifurcation analysis is used determine the effects of these correlations on the network's dynamics. For excitatory coupling, we find that inducing positive correlations has a similar effect to increasing the coupling strength between neurons, while for inhibitory coupling it has the opposite effect. We also determine the propensity of various two- and three-neuron motifs to occur as correlations are varied and give a plausible explanation for the observed changes in dynamics.
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Affiliation(s)
- Carlo R Laing
- School of Natural and Computational Sciences, Massey University, NSMC, Private Bag 102-904, Auckland, New Zealand.
| | - Christian Bläsche
- School of Natural and Computational Sciences, Massey University, NSMC, Private Bag 102-904, Auckland, New Zealand
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9
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Bick C, Goodfellow M, Laing CR, Martens EA. Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2020; 10:9. [PMID: 32462281 PMCID: PMC7253574 DOI: 10.1186/s13408-020-00086-9] [Citation(s) in RCA: 95] [Impact Index Per Article: 23.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2019] [Accepted: 05/07/2020] [Indexed: 05/03/2023]
Abstract
Many biological and neural systems can be seen as networks of interacting periodic processes. Importantly, their functionality, i.e., whether these networks can perform their function or not, depends on the emerging collective dynamics of the network. Synchrony of oscillations is one of the most prominent examples of such collective behavior and has been associated both with function and dysfunction. Understanding how network structure and interactions, as well as the microscopic properties of individual units, shape the emerging collective dynamics is critical to find factors that lead to malfunction. However, many biological systems such as the brain consist of a large number of dynamical units. Hence, their analysis has either relied on simplified heuristic models on a coarse scale, or the analysis comes at a huge computational cost. Here we review recently introduced approaches, known as the Ott-Antonsen and Watanabe-Strogatz reductions, allowing one to simplify the analysis by bridging small and large scales. Thus, reduced model equations are obtained that exactly describe the collective dynamics for each subpopulation in the oscillator network via few collective variables only. The resulting equations are next-generation models: Rather than being heuristic, they exactly link microscopic and macroscopic descriptions and therefore accurately capture microscopic properties of the underlying system. At the same time, they are sufficiently simple to analyze without great computational effort. In the last decade, these reduction methods have become instrumental in understanding how network structure and interactions shape the collective dynamics and the emergence of synchrony. We review this progress based on concrete examples and outline possible limitations. Finally, we discuss how linking the reduced models with experimental data can guide the way towards the development of new treatment approaches, for example, for neurological disease.
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Affiliation(s)
- Christian Bick
- Centre for Systems, Dynamics, and Control, University of Exeter, Exeter, UK.
- Department of Mathematics, University of Exeter, Exeter, UK.
- EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter, UK.
- Mathematical Institute, University of Oxford, Oxford, UK.
- Institute for Advanced Study, Technische Universität München, Garching, Germany.
| | - Marc Goodfellow
- Department of Mathematics, University of Exeter, Exeter, UK
- EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter, UK
- Living Systems Institute, University of Exeter, Exeter, UK
- Wellcome Trust Centre for Biomedical Modelling and Analysis, University of Exeter, Exeter, UK
| | - Carlo R Laing
- School of Natural and Computational Sciences, Massey University, Auckland, New Zealand
| | - Erik A Martens
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kgs. Lyngby, Denmark.
- Department of Biomedical Science, University of Copenhagen, Copenhagen N, Denmark.
- Centre for Translational Neuroscience, University of Copenhagen, Copenhagen N, Denmark.
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10
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Vanag VK. Hierarchical network of pulse coupled chemical oscillators with adaptive behavior: Chemical neurocomputer. CHAOS (WOODBURY, N.Y.) 2019; 29:083104. [PMID: 31472522 DOI: 10.1063/1.5099979] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/13/2019] [Accepted: 06/26/2019] [Indexed: 06/10/2023]
Abstract
We consider theoretically a network of pulse coupled oscillators with time delays. Each oscillator is described by the Oregonator-like model for the Belousov-Zhabotinsky (BZ) reaction. Different groups of oscillators constitute five functional units: (1) a central pattern generator (CPG), (2) a "reader" unit that can identify dynamical modes of the CPG, (3) an antenna (A) unit that receives external signals and responds on them by generating different dynamical modes, (4) another reader unit for identification of the dynamical modes in the A unit, and (5) a decision making unit that switches the current dynamical mode of the CPG to the mode that is similar to the current mode in the A unit. We call this network a chemical neurocomputer, since chemical BZ reaction occurs in each micro-oscillator, while pulse connectivity of these cells is inspired by the brain.
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Affiliation(s)
- Vladimir K Vanag
- Center for Nonlinear Chemistry, Immanuel Kant Baltic Federal University, 14 A. Nevskogo Str., Kaliningrad 236041, Russia
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11
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Chandra S, Girvan M, Ott E. Complexity reduction ansatz for systems of interacting orientable agents: Beyond the Kuramoto model. CHAOS (WOODBURY, N.Y.) 2019; 29:053107. [PMID: 31154774 DOI: 10.1063/1.5093038] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/16/2019] [Accepted: 04/19/2019] [Indexed: 06/09/2023]
Abstract
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 higher-dimensional 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.
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Affiliation(s)
- Sarthak Chandra
- Department of Physics, University of Maryland, College Park, Maryland 20740, USA
| | - Michelle Girvan
- Department of Physics, University of Maryland, College Park, Maryland 20740, USA
| | - Edward Ott
- Department of Physics, University of Maryland, College Park, Maryland 20740, USA
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12
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Observable for a Large System of Globally Coupled Excitable Units. MATHEMATICAL AND COMPUTATIONAL APPLICATIONS 2019. [DOI: 10.3390/mca24020037] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
The study of large arrays of coupled excitable systems has largely benefited from a technique proposed by Ott and Antonsen, which results in a low dimensional system of equations for the system’s order parameter. In this work, we show how to explicitly introduce a variable describing the global synaptic activation of the network into these family of models. This global variable is built by adding realistic synaptic time traces. We propose that this variable can, under certain conditions, be a good proxy for the local field potential of the network. We report experimental, in vivo, electrophysiology data supporting this claim.
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13
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Keeley S, Byrne Á, Fenton A, Rinzel J. Firing rate models for gamma oscillations. J Neurophysiol 2019; 121:2181-2190. [PMID: 30943833 DOI: 10.1152/jn.00741.2018] [Citation(s) in RCA: 17] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/26/2023] Open
Abstract
Gamma oscillations are readily observed in a variety of brain regions during both waking and sleeping states. Computational models of gamma oscillations typically involve simulations of large networks of synaptically coupled spiking units. These networks can exhibit strongly synchronized gamma behavior, whereby neurons fire in near synchrony on every cycle, or weakly modulated gamma behavior, corresponding to stochastic, sparse firing of the individual units on each cycle of the population gamma rhythm. These spiking models offer valuable biophysical descriptions of gamma oscillations; however, because they involve many individual neuronal units they are limited in their ability to communicate general network-level dynamics. Here we demonstrate that few-variable firing rate models with established synaptic timescales can account for both strongly synchronized and weakly modulated gamma oscillations. These models go beyond the classical formulations of rate models by including at least two dynamic variables per population: firing rate and synaptic activation. The models' flexibility to capture the broad range of gamma behavior depends directly on the timescales that represent recruitment of the excitatory and inhibitory firing rates. In particular, we find that weakly modulated gamma oscillations occur robustly when the recruitment timescale of inhibition is faster than that of excitation. We present our findings by using an extended Wilson-Cowan model and a rate model derived from a network of quadratic integrate-and-fire neurons. These biophysical rate models capture the range of weakly modulated and coherent gamma oscillations observed in spiking network models, while additionally allowing for greater tractability and systems analysis. NEW & NOTEWORTHY Here we develop simple and tractable models of gamma oscillations, a dynamic feature observed throughout much of the brain with significant correlates to behavior and cognitive performance in a variety of experimental contexts. Our models depend on only a few dynamic variables per population, but despite this they qualitatively capture features observed in previous biophysical models of gamma oscillations that involve many individual spiking units.
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Affiliation(s)
- Stephen Keeley
- Center for Neural Science, New York University , New York, New York.,Princeton Neuroscience Institute , Princeton, New Jersey
| | - Áine Byrne
- Center for Neural Science, New York University , New York, New York
| | - André Fenton
- Center for Neural Science, New York University , New York, New York.,Neuroscience Institute at the NYU Langone Medical Center , New York, New York.,Robert F. Furchgott Center for Neural and Behavioral Science, SUNY Downstate Medical Center , Brooklyn, New York
| | - John Rinzel
- Center for Neural Science, New York University , New York, New York.,Courant Institute of Mathematical Sciences , New York, New York.,Neuroscience Institute at the NYU Langone Medical Center , New York, New York
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14
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Park Y, Ermentrout GB. A multiple timescales approach to bridging spiking- and population-level dynamics. CHAOS (WOODBURY, N.Y.) 2018; 28:083123. [PMID: 30180602 DOI: 10.1063/1.5029841] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/15/2018] [Accepted: 08/10/2018] [Indexed: 06/08/2023]
Abstract
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 N excitatory n-dimensional and N inhibitory m-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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Affiliation(s)
- Youngmin Park
- Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
| | - G Bard Ermentrout
- Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
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15
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Devalle F, Roxin A, Montbrió E. Firing rate equations require a spike synchrony mechanism to correctly describe fast oscillations in inhibitory networks. PLoS Comput Biol 2017; 13:e1005881. [PMID: 29287081 PMCID: PMC5764488 DOI: 10.1371/journal.pcbi.1005881] [Citation(s) in RCA: 51] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2017] [Revised: 01/11/2018] [Accepted: 11/15/2017] [Indexed: 12/25/2022] Open
Abstract
Recurrently coupled networks of inhibitory neurons robustly generate oscillations in the gamma band. Nonetheless, the corresponding Wilson-Cowan type firing rate equation for such an inhibitory population does not generate such oscillations without an explicit time delay. We show that this discrepancy is due to a voltage-dependent spike-synchronization mechanism inherent in networks of spiking neurons which is not captured by standard firing rate equations. Here we investigate an exact low-dimensional description for a network of heterogeneous canonical Class 1 inhibitory neurons which includes the sub-threshold dynamics crucial for generating synchronous states. In the limit of slow synaptic kinetics the spike-synchrony mechanism is suppressed and the standard Wilson-Cowan equations are formally recovered as long as external inputs are also slow. However, even in this limit synchronous spiking can be elicited by inputs which fluctuate on a time-scale of the membrane time-constant of the neurons. Our meanfield equations therefore represent an extension of the standard Wilson-Cowan equations in which spike synchrony is also correctly described. Population models describing the average activity of large neuronal ensembles are a powerful mathematical tool to investigate the principles underlying cooperative function of large neuronal systems. However, these models do not properly describe the phenomenon of spike synchrony in networks of neurons. In particular, they fail to capture the onset of synchronous oscillations in networks of inhibitory neurons. We show that this limitation is due to a voltage-dependent synchronization mechanism which is naturally present in spiking neuron models but not captured by traditional firing rate equations. Here we investigate a novel set of macroscopic equations which incorporate both firing rate and membrane potential dynamics, and that correctly generate fast inhibition-based synchronous oscillations. In the limit of slow-synaptic processing oscillations are suppressed, and the model reduces to an equation formally equivalent to the Wilson-Cowan model.
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Affiliation(s)
- Federico Devalle
- Center for Brain and Cognition, Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain
- Department of Physics, Lancaster University, Lancaster, United Kingdom
| | - Alex Roxin
- Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, Bellaterra, Barcelona, Spain
| | - Ernest Montbrió
- Center for Brain and Cognition, Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain
- * E-mail:
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16
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Esnaola-Acebes JM, Roxin A, Avitabile D, Montbrió E. Synchrony-induced modes of oscillation of a neural field model. Phys Rev E 2017; 96:052407. [PMID: 29347806 DOI: 10.1103/physreve.96.052407] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/28/2017] [Indexed: 05/01/2023]
Abstract
We investigate the modes of oscillation of heterogeneous ring networks of quadratic integrate-and-fire (QIF) neurons with nonlocal, space-dependent coupling. Perturbations of the equilibrium state with a particular wave number produce transient standing waves with a specific temporal frequency, analogously to those in a tense string. In the neuronal network, the equilibrium corresponds to a spatially homogeneous, asynchronous state. Perturbations of this state excite the network's oscillatory modes, which reflect the interplay of episodes of synchronous spiking with the excitatory-inhibitory spatial interactions. In the thermodynamic limit, an exact low-dimensional neural field model describing the macroscopic dynamics of the network is derived. This allows us to obtain formulas for the Turing eigenvalues of the spatially homogeneous state and hence to obtain its stability boundary. We find that the frequency of each Turing mode depends on the corresponding Fourier coefficient of the synaptic pattern of connectivity. The decay rate instead is identical for all oscillation modes as a consequence of the heterogeneity-induced desynchronization of the neurons. Finally, we numerically compute the spectrum of spatially inhomogeneous solutions branching from the Turing bifurcation, showing that similar oscillatory modes operate in neural bump states and are maintained away from onset.
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Affiliation(s)
- Jose M Esnaola-Acebes
- Center for Brain and Cognition, Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08018 Barcelona, Spain
| | - Alex Roxin
- Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona), Spain
| | - Daniele Avitabile
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham NG2 7RD, United Kingdom
| | - Ernest Montbrió
- Center for Brain and Cognition, Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08018 Barcelona, Spain
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Gallego R, Montbrió E, Pazó D. Synchronization scenarios in the Winfree model of coupled oscillators. Phys Rev E 2017; 96:042208. [PMID: 29347589 DOI: 10.1103/physreve.96.042208] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/24/2017] [Indexed: 11/07/2022]
Abstract
Fifty years ago Arthur Winfree proposed a deeply influential mean-field model for the collective synchronization of large populations of phase oscillators. Here we provide a detailed analysis of the model for some special, analytically tractable cases. Adopting the thermodynamic limit, we derive an ordinary differential equation that exactly describes the temporal evolution of the macroscopic variables in the Ott-Antonsen invariant manifold. The low-dimensional model is then thoroughly investigated for a variety of pulse types and sinusoidal phase response curves (PRCs). Two structurally different synchronization scenarios are found, which are linked via the mutation of a Bogdanov-Takens point. From our results, we infer a general rule of thumb relating pulse shape and PRC offset with each scenario. Finally, we compare the exact synchronization threshold with the prediction of the averaging approximation given by the Kuramoto-Sakaguchi model. At the leading order, the discrepancy appears to behave as an odd function of the PRC offset.
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Affiliation(s)
- Rafael Gallego
- Departamento de Matemáticas, Universidad de Oviedo, Campus de Viesques, 33203 Gijón, Spain
| | - Ernest Montbrió
- Center for Brain and Cognition, Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08018 Barcelona, Spain
| | - Diego Pazó
- Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, 39005 Santander, Spain
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Ott E, Antonsen TM. Frequency and phase synchronization in large groups: Low dimensional description of synchronized clapping, firefly flashing, and cricket chirping. CHAOS (WOODBURY, N.Y.) 2017; 27:051101. [PMID: 28576094 DOI: 10.1063/1.4983470] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
A common observation is that large groups of oscillatory biological units often have the ability to synchronize. A paradigmatic model of such behavior is provided by the Kuramoto model, which achieves synchronization through coupling of the phase dynamics of individual oscillators, while each oscillator maintains a different constant inherent natural frequency. Here we consider the biologically likely possibility that the oscillatory units may be capable of enhancing their synchronization ability by adaptive frequency dynamics. We propose a simple augmentation of the Kuramoto model which does this. We also show that, by the use of a previously developed technique [Ott and Antonsen, Chaos 18, 037113 (2008)], it is possible to reduce the resulting dynamics to a lower dimensional system for the macroscopic evolution of the oscillator ensemble. By employing this reduction, we investigate the dynamics of our system, finding a characteristic hysteretic behavior and enhancement of the quality of the achieved synchronization.
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Affiliation(s)
- Edward Ott
- University of Maryland, College Park, Maryland 20742-3511, USA
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