1
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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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2
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Lee S, Braun L, Bönisch F, Schröder M, Thümler M, Timme M. Complexified synchrony. CHAOS (WOODBURY, N.Y.) 2024; 34:053141. [PMID: 38814675 DOI: 10.1063/5.0205897] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2024] [Accepted: 05/06/2024] [Indexed: 05/31/2024]
Abstract
The Kuramoto model and its generalizations have been broadly employed to characterize and mechanistically understand various collective dynamical phenomena, especially the emergence of synchrony among coupled oscillators. Despite almost five decades of research, many questions remain open, in particular, for finite-size systems. Here, we generalize recent work [Thümler et al., Phys. Rev. Lett. 130, 187201 (2023)] on the finite-size Kuramoto model with its state variables analytically continued to the complex domain and also complexify its system parameters. Intriguingly, systems of two units with purely imaginary coupling do not actively synchronize even for arbitrarily large magnitudes of the coupling strengths, |K|→∞, but exhibit conservative dynamics with asynchronous rotations or librations for all |K|. For generic complex coupling, both traditional phase-locked states and asynchronous states generalize to complex locked states, fixed points off the real subspace that exist even for arbitrarily weak coupling. We analyze a new collective mode of rotations exhibiting finite, yet arbitrarily large rotation numbers. Numerical simulations for large networks indicate a novel form of discontinuous phase transition. We close by pointing to a range of exciting questions for future research.
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Affiliation(s)
- Seungjae Lee
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (CFAED) and Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
| | - Lucas Braun
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (CFAED) and Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
- Schülerforschungszentrum Südwürttemberg (SFZ), 88348 Bad Saulgau, Germany
- Gymnasium Wilhelmsdorf, Pfrunger Straße 4/2, 88271 Wilhelmsdorf, Germany
| | - Frieder Bönisch
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (CFAED) and Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
| | - Malte Schröder
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (CFAED) and Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
| | - Moritz Thümler
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (CFAED) and Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
| | - Marc Timme
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (CFAED) and Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
- Cluster of Excellence Physics of Life, TU Dresden, 01062 Dresden, Germany
- Lakeside Labs, 9020 Klagenfurt, Austria
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3
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Laing CR, Omel’chenko OE. Periodic solutions in next generation neural field models. BIOLOGICAL CYBERNETICS 2023; 117:259-274. [PMID: 37535104 PMCID: PMC10600056 DOI: 10.1007/s00422-023-00969-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/24/2023] [Accepted: 07/12/2023] [Indexed: 08/04/2023]
Abstract
We consider a next generation neural field model which describes the dynamics of a network of theta neurons on a ring. For some parameters the network supports stable time-periodic solutions. Using the fact that the dynamics at each spatial location are described by a complex-valued Riccati equation we derive a self-consistency equation that such periodic solutions must satisfy. We determine the stability of these solutions, and present numerical results to illustrate the usefulness of this technique. The generality of this approach is demonstrated through its application to several other systems involving delays, two-population architecture and networks of Winfree oscillators.
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Affiliation(s)
- Carlo R. Laing
- School of Mathematical and Computational Sciences, Massey University, Private Bag 102-904 NSMC, Auckland, New Zealand
| | - Oleh E. Omel’chenko
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam, Germany
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4
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Pietras B, Cestnik R, Pikovsky A. Exact finite-dimensional description for networks of globally coupled spiking neurons. Phys Rev E 2023; 107:024315. [PMID: 36932479 DOI: 10.1103/physreve.107.024315] [Citation(s) in RCA: 6] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/02/2022] [Accepted: 02/10/2023] [Indexed: 06/18/2023]
Abstract
We consider large networks of globally coupled spiking neurons and derive an exact low-dimensional description of their collective dynamics in the thermodynamic limit. Individual neurons are described by the Ermentrout-Kopell canonical model that can be excitable or tonically spiking and interact with other neurons via pulses. Utilizing the equivalence of the quadratic integrate-and-fire and the theta-neuron formulations, we first derive the dynamical equations in terms of the Kuramoto-Daido order parameters (Fourier modes of the phase distribution) and relate them to two biophysically relevant macroscopic observables, the firing rate and the mean voltage. For neurons driven by Cauchy white noise or for Cauchy-Lorentz distributed input currents, we adapt the results by Cestnik and Pikovsky [Chaos 32, 113126 (2022)1054-150010.1063/5.0106171] and show that for arbitrary initial conditions the collective dynamics reduces to six dimensions. We also prove that in this case the dynamics asymptotically converges to a two-dimensional invariant manifold first discovered by Ott and Antonsen. For identical, noise-free neurons, the dynamics reduces to three dimensions, becoming equivalent to the Watanabe-Strogatz description. We illustrate the exact six-dimensional dynamics outside the invariant manifold by calculating nontrivial basins of different asymptotic regimes in a bistable situation.
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Affiliation(s)
- Bastian Pietras
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Tànger 122-140, 08018 Barcelona, Spain
| | - Rok Cestnik
- Department of Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Arkady Pikovsky
- Department of Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
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5
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Zhang Y, Hoveijn I, Efstathiou K. Entrainment degree of globally coupled Winfree oscillators under external forcing. CHAOS (WOODBURY, N.Y.) 2022; 32:103121. [PMID: 36319288 DOI: 10.1063/5.0113961] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/25/2022] [Accepted: 09/21/2022] [Indexed: 06/16/2023]
Abstract
We consider globally connected coupled Winfree oscillators under the influence of an external periodic forcing. Such systems exhibit many qualitatively different regimes of collective dynamics. Our aim is to understand this collective dynamics and, in particular, the system's capability of entrainment to the external forcing. To quantify the entrainment of the system, we introduce the entrainment degree, that is, the proportion of oscillators that synchronize to the forcing, as the main focus of this paper. Through a series of numerical simulations, we study the entrainment degree for different inter-oscillator coupling strengths, external forcing strengths, and distributions of natural frequencies of the Winfree oscillators, and we compare the results for the different cases. In the case of identical oscillators, we give a precise description of the parameter regions where oscillators are entrained. Finally, we use a mean-field method, based on the Ott-Antonsen ansatz, to obtain a low-dimensional description of the collective dynamics and to compute an approximation of the entrainment degree. The mean-field results turn out to be strikingly similar to the results obtained through numerical simulations of the full system dynamics.
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Affiliation(s)
- Yongjiao Zhang
- Bernoulli Institute for Mathematics, Computer Science, and Artificial Intelligence, University of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands
| | - Igor Hoveijn
- Bernoulli Institute for Mathematics, Computer Science, and Artificial Intelligence, University of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands
| | - Konstantinos Efstathiou
- Division of Natural and Applied Sciences and Zu Chongzhi Center for Mathematics and Computational Science, Duke Kunshan University, No. 8 Duke Avenue, Kunshan 215316, Jiangsu Province, China
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6
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Manoranjani M, Gopal R, Senthilkumar DV, Chandrasekar VK. Role of phase-dependent influence function in the Winfree model of coupled oscillators. Phys Rev E 2021; 104:064206. [PMID: 35030866 DOI: 10.1103/physreve.104.064206] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2021] [Accepted: 11/24/2021] [Indexed: 06/14/2023]
Abstract
We consider a globally coupled Winfree model comprised of a phase-dependent influence function and sensitive function, and unravel the impact of offset and integer parameters, characterizing the shape of the influence function, on the phase diagram of the Winfree model. The decreasing value of the offset parameter decreases the degree of positive phase shift among the oscillators by promoting the negative phase shift, which indeed favors the onset of multistability among the synchronous oscillatory state and asynchronous stable steady states in a large region of the phase diagram. Further, large integer parameters lead to brief pulses of the influence function, which again enhances the effect of the offset parameter. There is an explosive transition to a synchronous oscillatory state from an asynchronous steady state via a Hopf bifurcation. Dynamical transitions and multistability emerge through saddle-node, pitchfork, and homoclinic bifurcations in the phase diagram. We deduce two ordinary differential equations corresponding to the two macroscopic variables from the population of globally coupled Winfree oscillators using the Ott-Antonsen ansatz. We also deduce various bifurcation curves analytically from the reduced low-dimensional macroscopic variables for the exactly solvable case. The analytical curves exactly match the simulation boundaries in the phase diagram.
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Affiliation(s)
- M Manoranjani
- Department of Physics, Centre for Nonlinear Science and Engineering, School of Electrical and Electronics Engineering, SASTRA Deemed University, Thanjavur 613401, India
| | - R Gopal
- Department of Physics, Centre for Nonlinear Science and Engineering, School of Electrical and Electronics Engineering, SASTRA Deemed University, Thanjavur 613401, India
| | - D V Senthilkumar
- School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram 695016, India
| | - V K Chandrasekar
- Department of Physics, Centre for Nonlinear Science and Engineering, School of Electrical and Electronics Engineering, SASTRA Deemed University, Thanjavur 613401, India
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7
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Omel'chenko OE, Ocampo-Espindola JL, Kiss IZ. Asymmetry-induced isolated fully synchronized state in coupled oscillator populations. Phys Rev E 2021; 104:L022202. [PMID: 34525593 DOI: 10.1103/physreve.104.l022202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/26/2021] [Accepted: 07/04/2021] [Indexed: 11/07/2022]
Abstract
A symmetry-breaking mechanism is investigated that creates bistability between fully and partially synchronized states in oscillator networks. Two populations of oscillators with unimodal frequency distribution and different amplitudes, in the presence of weak global coupling, are shown to simplify to a modular network with asymmetrical coupling. With increasing the coupling strength, a synchronization transition is observed with an isolated fully synchronized state. The results are interpreted theoretically in the thermodynamic limit and confirmed in experiments with chemical oscillators.
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Affiliation(s)
- Oleh E Omel'chenko
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam, Germany
| | | | - István Z Kiss
- Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St. Louis, Missouri 63103, USA
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8
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Goldobin DS. Mean-field models of populations of quadratic integrate-and-fire neurons with noise on the basis of the circular cumulant approach. CHAOS (WOODBURY, N.Y.) 2021; 31:083112. [PMID: 34470229 DOI: 10.1063/5.0061575] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/28/2021] [Accepted: 07/20/2021] [Indexed: 06/13/2023]
Abstract
We develop a circular cumulant representation for the recurrent network of quadratic integrate-and-fire neurons subject to noise. The synaptic coupling is global or macroscopically equivalent to it. We assume a Lorentzian distribution of the parameter controlling whether the isolated individual neuron is periodically spiking or excitable. For the infinite chain of circular cumulant equations, a hierarchy of smallness is identified; on the basis of it, we truncate the chain and suggest several two-cumulant neural mass models. These models allow one to go beyond the Ott-Antonsen Ansatz and describe the effect of noise on hysteretic transitions between macroscopic regimes of a population with inhibitory coupling. The accuracy of two-cumulant models is analyzed in detail.
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Affiliation(s)
- Denis S Goldobin
- Institute of Continuous Media Mechanics, UB RAS, Academician Korolev Street 1, 614013 Perm, Russia
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9
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Manoranjani M, Gupta S, Chandrasekar VK. The Sakaguchi-Kuramoto model in presence of asymmetric interactions that break phase-shift symmetry. CHAOS (WOODBURY, N.Y.) 2021; 31:083130. [PMID: 34470257 DOI: 10.1063/5.0055664] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/01/2021] [Accepted: 08/07/2021] [Indexed: 06/13/2023]
Abstract
The celebrated Kuramoto model provides an analytically tractable framework to study spontaneous collective synchronization and comprises globally coupled limit-cycle oscillators interacting symmetrically with one another. The Sakaguchi-Kuramoto model is a generalization of the basic model that considers the presence of a phase lag parameter in the interaction, thereby making it asymmetric between oscillator pairs. Here, we consider a further generalization by adding an interaction that breaks the phase-shift symmetry of the model. The highlight of our study is the unveiling of a very rich bifurcation diagram comprising of both oscillatory and non-oscillatory synchronized states as well as an incoherent state: There are regions of two-state as well as an interesting and hitherto unexplored three-state coexistence arising from asymmetric interactions in our model.
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Affiliation(s)
- M Manoranjani
- Centre for Nonlinear Science and Engineering, School of Electrical and Electronics Engineering, SASTRA Deemed University, Thanjavur 613 401, India
| | - Shamik Gupta
- Department of Physics, Ramakrishna Mission Vivekananda Educational and Research Institute, Belur Math, Howrah 711202, India
| | - V K Chandrasekar
- Centre for Nonlinear Science and Engineering, School of Electrical and Electronics Engineering, SASTRA Deemed University, Thanjavur 613 401, India
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10
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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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11
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Abstract
Out of equilibrium, a lack of reciprocity is the rule rather than the exception. Non-reciprocity occurs, for instance, in active matter1-6, non-equilibrium systems7-9, networks of neurons10,11, social groups with conformist and contrarian members12, directional interface growth phenomena13-15 and metamaterials16-20. Although wave propagation in non-reciprocal media has recently been closely studied1,16-20, less is known about the consequences of non-reciprocity on the collective behaviour of many-body systems. Here we show that non-reciprocity leads to time-dependent phases in which spontaneously broken continuous symmetries are dynamically restored. We illustrate this mechanism with simple robotic demonstrations. The resulting phase transitions are controlled by spectral singularities called exceptional points21. We describe the emergence of these phases using insights from bifurcation theory22,23 and non-Hermitian quantum mechanics24,25. Our approach captures non-reciprocal generalizations of three archetypal classes of self-organization out of equilibrium: synchronization, flocking and pattern formation. Collective phenomena in these systems range from active time-(quasi)crystals to exceptional-point-enforced pattern formation and hysteresis. Our work lays the foundation for a general theory of critical phenomena in systems whose dynamics is not governed by an optimization principle.
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12
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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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13
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Pazó D, Gallego R. The Winfree model with non-infinitesimal phase-response curve: Ott-Antonsen theory. CHAOS (WOODBURY, N.Y.) 2020; 30:073139. [PMID: 32752623 DOI: 10.1063/5.0015131] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/25/2020] [Accepted: 07/08/2020] [Indexed: 06/11/2023]
Abstract
A novel generalization of the Winfree model of globally coupled phase oscillators, representing phase reduction under finite coupling, is studied analytically. We consider interactions through a non-infinitesimal (or finite) phase-response curve (PRC), in contrast to the infinitesimal PRC of the original model. For a family of non-infinitesimal PRCs, the global dynamics is captured by one complex-valued ordinary differential equation resorting to the Ott-Antonsen ansatz. The phase diagrams are thereupon obtained for four illustrative cases of non-infinitesimal PRC. Bistability between collective synchronization and full desynchronization is observed in all cases.
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Affiliation(s)
- Diego Pazó
- Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, 39005 Santander, Spain
| | - Rafael Gallego
- Departamento de Matemáticas, Universidad de Oviedo, Campus de Viesques, 33203 Gijón, Spain
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14
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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: 91] [Impact Index Per Article: 22.8] [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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15
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Omel'chenko OE, Sebek M, Kiss IZ. Universal relations of local order parameters for partially synchronized oscillators. Phys Rev E 2018; 97:062207. [PMID: 30011585 DOI: 10.1103/physreve.97.062207] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/02/2018] [Indexed: 11/07/2022]
Abstract
Interactions among discrete oscillatory units (e.g., cells) can result in partially synchronized states when some of the units exhibit phase locking and others phase slipping. Such states are typically characterized by a global order parameter that expresses the extent of synchrony in the system. Here we show that such states carry data-rich information of the system behavior, and a local order parameter analysis reveals universal relations through a semicircle representation. The universal relations are derived from thermodynamic limit analysis of a globally coupled Kuramoto-type phase oscillator model. The relations are confirmed with the partially synchronized states in numerical simulations with a model of circadian cells and in laboratory experiments with chemical oscillators. The application of the theory allows direct approximation of coupling strength, the natural frequency of oscillations, and the phase lag parameter without extensive nonlinear fits as well as a self-consistency check for presence of network interactions and higher harmonic components in the phase model.
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Affiliation(s)
| | - Michael Sebek
- Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St. Louis, Missouri 63103, USA
| | - István Z Kiss
- Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St. Louis, Missouri 63103, USA
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16
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Montbrió E, Pazó D. Kuramoto Model for Excitation-Inhibition-Based Oscillations. PHYSICAL REVIEW LETTERS 2018; 120:244101. [PMID: 29956946 DOI: 10.1103/physrevlett.120.244101] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/04/2018] [Revised: 04/10/2018] [Indexed: 06/08/2023]
Abstract
The Kuramoto model (KM) is a theoretical paradigm for investigating the emergence of rhythmic activity in large populations of oscillators. A remarkable example of rhythmogenesis is the feedback loop between excitatory (E) and inhibitory (I) cells in large neuronal networks. Yet, although the EI-feedback mechanism plays a central role in the generation of brain oscillations, it remains unexplored whether the KM has enough biological realism to describe it. Here we derive a two-population KM that fully accounts for the onset of EI-based neuronal rhythms and that, as the original KM, is analytically solvable to a large extent. Our results provide a powerful theoretical tool for the analysis of large-scale neuronal oscillations.
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Affiliation(s)
- 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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17
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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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