1
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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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2
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Carrillo JA, Holden H, Solem S. Noise-driven bifurcations in a neural field system modelling networks of grid cells. J Math Biol 2022; 85:42. [PMID: 36166151 PMCID: PMC9515060 DOI: 10.1007/s00285-022-01811-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/16/2021] [Revised: 02/01/2022] [Accepted: 05/04/2022] [Indexed: 11/27/2022]
Abstract
The activity generated by an ensemble of neurons is affected by various noise sources. It is a well-recognised challenge to understand the effects of noise on the stability of such networks. We demonstrate that the patterns of activity generated by networks of grid cells emerge from the instability of homogeneous activity for small levels of noise. This is carried out by analysing the robustness of network activity patterns with respect to noise in an upscaled noisy grid cell model in the form of a system of partial differential equations. Inhomogeneous network patterns are numerically understood as branches bifurcating from unstable homogeneous states for small noise levels. We show that there is a phase transition occurring as the level of noise decreases. Our numerical study also indicates the presence of hysteresis phenomena close to the precise critical noise value.
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Affiliation(s)
- José A. Carrillo
- Mathematical Institute, University of Oxford, Oxford, OX2 6GG UK
| | - Helge Holden
- Department of Mathematical Sciences, NTNU Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
| | - Susanne Solem
- Department of Mathematics, Norwegian University of Life Sciences, 1433 Ås, Norway
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3
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Omel'chenko O, Laing CR. Collective states in a ring network of theta neurons. Proc Math Phys Eng Sci 2022; 478:20210817. [PMID: 35280327 PMCID: PMC8908473 DOI: 10.1098/rspa.2021.0817] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/22/2021] [Accepted: 02/08/2022] [Indexed: 11/26/2022] Open
Abstract
We consider a ring network of theta neurons with non-local homogeneous coupling. We analyse the corresponding continuum evolution equation, analytically describing all possible steady states and their stability. By considering a number of different parameter sets, we determine the typical bifurcation scenarios of the network, and put on a rigorous footing some previously observed numerical results.
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Affiliation(s)
- Oleh Omel'chenko
- University of Potsdam, Institute of Physics and Astronomy, Karl-Liebknecht-Str. 24/25, Potsdam 14476, Germany
| | - Carlo R Laing
- School of Natural and Computational Sciences, Massey University, Private Bag 102-904 NSMC, Auckland, New Zealand
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4
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Franović I, Eydam S, Semenova N, Zakharova A. Unbalanced clustering and solitary states in coupled excitable systems. CHAOS (WOODBURY, N.Y.) 2022; 32:011104. [PMID: 35105111 DOI: 10.1063/5.0077022] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2021] [Accepted: 12/28/2021] [Indexed: 06/14/2023]
Abstract
We discover the mechanisms of emergence and the link between two types of symmetry-broken states, the unbalanced periodic two-cluster states and solitary states, in coupled excitable systems with attractive and repulsive interactions. The prevalent solitary states in non-locally coupled arrays, whose self-organization is based on successive (order preserving) spiking of units, derive their dynamical features from the corresponding unbalanced cluster states in globally coupled networks. Apart from the states with successive spiking, we also find cluster and solitary states where the interplay of excitability and local multiscale dynamics gives rise to so-called leap-frog activity patterns with an alternating order of spiking between the units. We show that the noise affects the system dynamics by suppressing the multistability of cluster states and by inducing pattern homogenization, transforming solitary states into patterns of patched synchrony.
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Affiliation(s)
- Igor Franović
- Scientific Computing Laboratory, Center for the Study of Complex Systems, Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia
| | - Sebastian Eydam
- Neural Circuits and Computations Unit, RIKEN Center for Brain Science, 2-1 Hirosawa, 351-0106 Wako, Japan
| | - Nadezhda Semenova
- Institute of Physics and Department of Fundamental Medicine and Medical Technology, Saratov State University, Astrakhanskaya str. 83, Saratov 410012, Russia
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
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5
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Franović I, Omel'chenko OE, Wolfrum M. Bumps, chimera states, and Turing patterns in systems of coupled active rotators. Phys Rev E 2021; 104:L052201. [PMID: 34942776 DOI: 10.1103/physreve.104.l052201] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2021] [Accepted: 10/20/2021] [Indexed: 11/07/2022]
Abstract
Self-organized coherence-incoherence patterns, called chimera states, have first been reported in systems of Kuramoto oscillators. For coupled excitable units, similar patterns where coherent units are at rest are called bump states. Here, we study bumps in an array of active rotators coupled by nonlocal attraction and global repulsion. We demonstrate how they can emerge in a supercritical scenario from completely coherent Turing patterns: a single incoherent unit appears in a homoclinic bifurcation, undergoing subsequent transitions to quasiperiodic and chaotic behavior, which eventually transforms into extensive chaos with many incoherent units. We present different types of transitions and explain the formation of coherence-incoherence patterns according to the classical paradigm of short-range activation and long-range inhibition.
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Affiliation(s)
- Igor Franović
- Scientific Computing Laboratory, Center for the Study of Complex Systems, Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia
| | - Oleh E Omel'chenko
- University of Potsdam, Institute of Physics and Astronomy, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam-Golm, Germany
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6
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Bhattacharya S, Cauchois MBL, Iglesias PA, Chen ZS. The impact of a closed-loop thalamocortical model on the spatiotemporal dynamics of cortical and thalamic traveling waves. Sci Rep 2021; 11:14359. [PMID: 34257333 PMCID: PMC8277909 DOI: 10.1038/s41598-021-93618-6] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/13/2021] [Accepted: 06/21/2021] [Indexed: 12/23/2022] Open
Abstract
Propagation of activity in spatially structured neuronal networks has been observed in awake, anesthetized, and sleeping brains. How these wave patterns emerge and organize across brain structures, and how network connectivity affects spatiotemporal neural activity remains unclear. Here, we develop a computational model of a two-dimensional thalamocortical network, which gives rise to emergent traveling waves similar to those observed experimentally. We illustrate how spontaneous and evoked oscillatory activity in space and time emerge using a closed-loop thalamocortical architecture, sustaining smooth waves in the cortex and staggered waves in the thalamus. We further show that intracortical and thalamocortical network connectivity, cortical excitation/inhibition balance, and thalamocortical or corticothalamic delay can independently or jointly change the spatiotemporal patterns (radial, planar and rotating waves) and characteristics (speed, direction, and frequency) of cortical and thalamic traveling waves. Computer simulations predict that increased thalamic inhibition induces slower cortical frequencies and that enhanced cortical excitation increases traveling wave speed and frequency. Overall, our results provide insight into the genesis and sustainability of thalamocortical spatiotemporal patterns, showing how simple synaptic alterations cause varied spontaneous and evoked wave patterns. Our model and simulations highlight the need for spatially spread neural recordings to uncover critical circuit mechanisms for brain functions.
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Affiliation(s)
- Sayak Bhattacharya
- Department of Electrical and Computer Engineering, Whiting School of Engineering, Johns Hopkins University, Baltimore, MD, 21218, USA
| | - Matthieu B L Cauchois
- Department of Mechanical Engineering, Whiting School of Engineering, Johns Hopkins University, Baltimore, MD, 21218, USA
| | - Pablo A Iglesias
- Department of Electrical and Computer Engineering, Whiting School of Engineering, Johns Hopkins University, Baltimore, MD, 21218, USA.
| | - Zhe Sage Chen
- Department of Psychiatry, Department of Neuroscience and Physiology, Neuroscience Institute, New York University Grossman School of Medicine, New York, NY, 10016, USA.
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7
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Morrison CL, Greenwood PE, Ward LM. Plastic systemic inhibition controls amplitude while allowing phase pattern in a stochastic neural field model. Phys Rev E 2021; 103:032311. [PMID: 33862754 DOI: 10.1103/physreve.103.032311] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/09/2020] [Accepted: 02/19/2021] [Indexed: 11/07/2022]
Abstract
We investigate oscillatory phase pattern formation and amplitude control for a linearized stochastic neuron field model by simulating Mexican-hat-coupled stochastic processes. We find, for several choices of parameters, that spatial pattern formation in the temporal phases of the coupled processes occurs if and only if their amplitudes are allowed to grow unrealistically large. Stimulated by recent work on homeostatic inhibitory plasticity, we introduce static and plastic (adaptive) systemic inhibitory mechanisms to keep the amplitudes stochastically bounded. We find that systems with static inhibition exhibited bounded amplitudes but no sustained phase patterns. With plastic systemic inhibition, on the other hand, the resulting systems exhibit both bounded amplitudes and sustained phase patterns. These results demonstrate that plastic inhibitory mechanisms in neural field models can dynamically control amplitudes while allowing patterns of phase synchronization to develop. Similar mechanisms of plastic systemic inhibition could play a role in regulating oscillatory functioning in the brain.
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Affiliation(s)
- Conor L Morrison
- Department of Statistics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4
| | - Priscilla E Greenwood
- Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
| | - Lawrence M Ward
- Department of Psychology and Djavad Mowafaghian Centre for Brain Health, 2136 West Mall, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4
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8
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Jüttner B, Henriksen C, Martens EA. Birth and destruction of collective oscillations in a network of two populations of coupled type 1 neurons. CHAOS (WOODBURY, N.Y.) 2021; 31:023141. [PMID: 33653075 DOI: 10.1063/5.0031630] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/02/2020] [Accepted: 01/12/2021] [Indexed: 06/12/2023]
Abstract
We study the macroscopic dynamics of large networks of excitable type 1 neurons composed of two populations interacting with disparate but symmetric intra- and inter-population coupling strengths. This nonuniform coupling scheme facilitates symmetric equilibria, where both populations display identical firing activity, characterized by either quiescent or spiking behavior, or asymmetric equilibria, where the firing activity of one population exhibits quiescent but the other exhibits spiking behavior. Oscillations in the firing rate are possible if neurons emit pulses with non-zero width but are otherwise quenched. Here, we explore how collective oscillations emerge for two statistically identical neuron populations in the limit of an infinite number of neurons. A detailed analysis reveals how collective oscillations are born and destroyed in various bifurcation scenarios and how they are organized around higher codimension bifurcation points. Since both symmetric and asymmetric equilibria display bistable behavior, a large configuration space with steady and oscillatory behavior is available. Switching between configurations of neural activity is relevant in functional processes such as working memory and the onset of collective oscillations in motor control.
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Affiliation(s)
- Benjamin Jüttner
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Christian Henriksen
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Erik A Martens
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
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9
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Montbrió E, Pazó D. Exact Mean-Field Theory Explains the Dual Role of Electrical Synapses in Collective Synchronization. PHYSICAL REVIEW LETTERS 2020; 125:248101. [PMID: 33412049 DOI: 10.1103/physrevlett.125.248101] [Citation(s) in RCA: 18] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/01/2020] [Revised: 11/18/2020] [Accepted: 11/18/2020] [Indexed: 06/12/2023]
Abstract
Electrical synapses play a major role in setting up neuronal synchronization, but the precise mechanisms whereby these synapses contribute to synchrony are subtle and remain elusive. To investigate these mechanisms mean-field theories for quadratic integrate-and-fire neurons with electrical synapses have been recently put forward. Still, the validity of these theories is controversial since they assume that the neurons produce unrealistic, symmetric spikes, ignoring the well-known impact of spike shape on synchronization. Here, we show that the assumption of symmetric spikes can be relaxed in such theories. The resulting mean-field equations reveal a dual role of electrical synapses: First, they equalize membrane potentials favoring the emergence of synchrony. Second, electrical synapses act as "virtual chemical synapses," which can be either excitatory or inhibitory depending upon the spike shape. Our results offer a precise mathematical explanation of the intricate effect of electrical synapses in collective synchronization. This reconciles previous theoretical and numerical works, and confirms the suitability of recent low-dimensional mean-field theories to investigate electrically coupled neuronal networks.
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Affiliation(s)
- Ernest Montbrió
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08003 Barcelona, Spain
| | - Diego Pazó
- Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, 39005 Santander, Spain
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10
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Cofré R, Maldonado C, Cessac B. Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics. ENTROPY (BASEL, SWITZERLAND) 2020; 22:E1330. [PMID: 33266513 PMCID: PMC7712217 DOI: 10.3390/e22111330] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 10/09/2020] [Revised: 11/13/2020] [Accepted: 11/15/2020] [Indexed: 12/04/2022]
Abstract
The Thermodynamic Formalism provides a rigorous mathematical framework for studying quantitative and qualitative aspects of dynamical systems. At its core, there is a variational principle that corresponds, in its simplest form, to the Maximum Entropy principle. It is used as a statistical inference procedure to represent, by specific probability measures (Gibbs measures), the collective behaviour of complex systems. This framework has found applications in different domains of science. In particular, it has been fruitful and influential in neurosciences. In this article, we review how the Thermodynamic Formalism can be exploited in the field of theoretical neuroscience, as a conceptual and operational tool, in order to link the dynamics of interacting neurons and the statistics of action potentials from either experimental data or mathematical models. We comment on perspectives and open problems in theoretical neuroscience that could be addressed within this formalism.
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Affiliation(s)
- Rodrigo Cofré
- CIMFAV-Ingemat, Facultad de Ingeniería, Universidad de Valparaíso, Valparaíso 2340000, Chile
| | - Cesar Maldonado
- IPICYT/División de Matemáticas Aplicadas, San Luis Potosí 78216, Mexico;
| | - Bruno Cessac
- Inria Biovision team and Neuromod Institute, Université Côte d’Azur, 06901 CEDEX Inria, France;
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11
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Churilov AN, Milton J, Salakhova ER. An integrate-and-fire model for pulsatility in the neuroendocrine system. CHAOS (WOODBURY, N.Y.) 2020; 30:083132. [PMID: 32872840 DOI: 10.1063/5.0010553] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/13/2020] [Accepted: 07/08/2020] [Indexed: 06/11/2023]
Abstract
A model for pulsatility in neuroendocrine regulation is proposed which combines Goodwin-type feedback control with impulsive input from neurons located in the hypothalamus. The impulsive neural input is modeled using an integrate-and-fire mechanism; namely, inputs are generated only when the membrane potential crosses a threshold, after which it is reset to baseline. The resultant model takes the form of a functional-differential equation with continuous and impulsive components. Despite the impulsive nature of the inputs, realistic hormone profiles are generated, including ultradian and circadian rhythms, pulsatile secretory patterns, and even chaotic dynamics.
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Affiliation(s)
- Alexander N Churilov
- Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetsky av. 28, Stary Peterhof, 198504 St. Petersburg, Russia
| | - John Milton
- Keck Science Department, The Claremont Colleges, 925 North Mills Ave., Claremont, California 91711, USA
| | - Elvira R Salakhova
- Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetsky av. 28, Stary Peterhof, 198504 St. Petersburg, Russia
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12
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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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13
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Pietras B, Devalle F, Roxin A, Daffertshofer A, Montbrió E. Exact firing rate model reveals the differential effects of chemical versus electrical synapses in spiking networks. Phys Rev E 2020; 100:042412. [PMID: 31771022 DOI: 10.1103/physreve.100.042412] [Citation(s) in RCA: 21] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/03/2019] [Indexed: 01/09/2023]
Abstract
Chemical and electrical synapses shape the dynamics of neuronal networks. Numerous theoretical studies have investigated how each of these types of synapses contributes to the generation of neuronal oscillations, but their combined effect is less understood. This limitation is further magnified by the impossibility of traditional neuronal mean-field models-also known as firing rate models or firing rate equations-to account for electrical synapses. Here, we introduce a firing rate model that exactly describes the mean-field dynamics of heterogeneous populations of quadratic integrate-and-fire (QIF) neurons with both chemical and electrical synapses. The mathematical analysis of the firing rate model reveals a well-established bifurcation scenario for networks with chemical synapses, characterized by a codimension-2 cusp point and persistent states for strong recurrent excitatory coupling. The inclusion of electrical coupling generally implies neuronal synchrony by virtue of a supercritical Hopf bifurcation. This transforms the cusp scenario into a bifurcation scenario characterized by three codimension-2 points (cusp, Takens-Bogdanov, and saddle-node separatrix loop), which greatly reduces the possibility for persistent states. This is generic for heterogeneous QIF networks with both chemical and electrical couplings. Our results agree with several numerical studies on the dynamics of large networks of heterogeneous spiking neurons with electrical and chemical couplings.
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Affiliation(s)
- Bastian Pietras
- Faculty of Behavioural and Movement Sciences, Amsterdam Movement Sciences & Institute of Brain and Behavior Amsterdam, Vrije Universiteit Amsterdam, van der Boechorststraat 9, Amsterdam 1081 BT, The Netherlands.,Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom.,Institute of Mathematics, Technical University Berlin, 10623 Berlin, Germany.,Bernstein Center for Computational Neuroscience Berlin, 10115 Berlin, Germany
| | - Federico Devalle
- Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom.,Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08003 Barcelona, Spain
| | - Alex Roxin
- Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona), Spain.,Barcelona Graduate School of Mathematics, 08193 Barcelona, Spain
| | - Andreas Daffertshofer
- Faculty of Behavioural and Movement Sciences, Amsterdam Movement Sciences & Institute of Brain and Behavior Amsterdam, Vrije Universiteit Amsterdam, van der Boechorststraat 9, Amsterdam 1081 BT, The Netherlands
| | - Ernest Montbrió
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08003 Barcelona, Spain
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14
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Laing CR, Omel'chenko O. Moving bumps in theta neuron networks. CHAOS (WOODBURY, N.Y.) 2020; 30:043117. [PMID: 32357659 DOI: 10.1063/1.5143261] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/20/2019] [Accepted: 03/30/2020] [Indexed: 05/20/2023]
Abstract
We consider large networks of theta neurons on a ring, synaptically coupled with an asymmetric kernel. Such networks support stable "bumps" of activity, which move along the ring if the coupling kernel is asymmetric. We investigate the effects of the kernel asymmetry on the existence, stability, and speed of these moving bumps using continuum equations formally describing infinite networks. Depending on the level of heterogeneity within the network, we find complex sequences of bifurcations as the amount of asymmetry is varied, in strong contrast to the behavior of a classical neural field model.
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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
| | - Oleh Omel'chenko
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam-Golm, Germany
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15
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Schmidt H, Avitabile D. Bumps and oscillons in networks of spiking neurons. CHAOS (WOODBURY, N.Y.) 2020; 30:033133. [PMID: 32237760 DOI: 10.1063/1.5135579] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2019] [Accepted: 03/03/2020] [Indexed: 06/11/2023]
Abstract
We study localized patterns in an exact mean-field description of a spatially extended network of quadratic integrate-and-fire neurons. We investigate conditions for the existence and stability of localized solutions, so-called bumps, and give an analytic estimate for the parameter range, where these solutions exist in parameter space, when one or more microscopic network parameters are varied. We develop Galerkin methods for the model equations, which enable numerical bifurcation analysis of stationary and time-periodic spatially extended solutions. We study the emergence of patterns composed of multiple bumps, which are arranged in a snake-and-ladder bifurcation structure if a homogeneous or heterogeneous synaptic kernel is suitably chosen. Furthermore, we examine time-periodic, spatially localized solutions (oscillons) in the presence of external forcing, and in autonomous, recurrently coupled excitatory and inhibitory networks. In both cases, we observe period-doubling cascades leading to chaotic oscillations.
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Affiliation(s)
- Helmut Schmidt
- Max Planck Institute for Human Cognitive and Brain Sciences, Stephanstrasse 1a, 04103 Leipzig, Germany
| | - Daniele Avitabile
- Department of Mathematics, Faculteit der Exacte Wetenschappen, Vrije Universiteit (VU University Amsterdam), De Boelelaan 1081a, 1081 HV Amsterdam, Netherlands and Mathneuro Team, Inria Sophia Antipolis, 2004 Rue des Lucioles, Sophia Antipolis, 06902 Cedex, France
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16
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Byrne Á, O'Dea RD, Forrester M, Ross J, Coombes S. Next-generation neural mass and field modeling. J Neurophysiol 2019; 123:726-742. [PMID: 31774370 DOI: 10.1152/jn.00406.2019] [Citation(s) in RCA: 26] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/01/2023] Open
Abstract
The Wilson-Cowan population model of neural activity has greatly influenced our understanding of the mechanisms for the generation of brain rhythms and the emergence of structured brain activity. As well as the many insights that have been obtained from its mathematical analysis, it is now widely used in the computational neuroscience community for building large-scale in silico brain networks that can incorporate the increasing amount of knowledge from the Human Connectome Project. Here, we consider a neural population model in the spirit of that originally developed by Wilson and Cowan, albeit with the added advantage that it can account for the phenomena of event-related synchronization and desynchronization. This derived mean-field model provides a dynamic description for the evolution of synchrony, as measured by the Kuramoto order parameter, in a large population of quadratic integrate-and-fire model neurons. As in the original Wilson-Cowan framework, the population firing rate is at the heart of our new model; however, in a significant departure from the sigmoidal firing rate function approach, the population firing rate is now obtained as a real-valued function of the complex-valued population synchrony measure. To highlight the usefulness of this next-generation Wilson-Cowan style model, we deploy it in a number of neurobiological contexts, providing understanding of the changes in power spectra observed in electro- and magnetoencephalography neuroimaging studies of motor cortex during movement, insights into patterns of functional connectivity observed during rest and their disruption by transcranial magnetic stimulation, and to describe wave propagation across cortex.
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Affiliation(s)
- Áine Byrne
- Center for Neural Science, New York University, New York, New York.,School of Mathematics and Statistics, University College Dublin, Dublin, Ireland
| | - Reuben D O'Dea
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom
| | - Michael Forrester
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom
| | - James Ross
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom
| | - Stephen Coombes
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom
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17
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Schwalger T, Chizhov AV. Mind the last spike - firing rate models for mesoscopic populations of spiking neurons. Curr Opin Neurobiol 2019; 58:155-166. [PMID: 31590003 DOI: 10.1016/j.conb.2019.08.003] [Citation(s) in RCA: 24] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/01/2019] [Accepted: 08/25/2019] [Indexed: 02/07/2023]
Abstract
The dominant modeling framework for understanding cortical computations are heuristic firing rate models. Despite their success, these models fall short to capture spike synchronization effects, to link to biophysical parameters and to describe finite-size fluctuations. In this opinion article, we propose that the refractory density method (RDM), also known as age-structured population dynamics or quasi-renewal theory, yields a powerful theoretical framework to build rate-based models for mesoscopic neural populations from realistic neuron dynamics at the microscopic level. We review recent advances achieved by the RDM to obtain efficient population density equations for networks of generalized integrate-and-fire (GIF) neurons - a class of neuron models that has been successfully fitted to various cell types. The theory not only predicts the nonstationary dynamics of large populations of neurons but also permits an extension to finite-size populations and a systematic reduction to low-dimensional rate dynamics. The new types of rate models will allow a re-examination of models of cortical computations under biological constraints.
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Affiliation(s)
- Tilo Schwalger
- Bernstein Center for Computational Neuroscience, 10115 Berlin, Germany; Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany.
| | - Anton V Chizhov
- Ioffe Institute, 194021 Saint-Petersburg, Russia; Sechenov Institute of Evolutionary Physiology and Biochemistry of the Russian Academy of Sciences, 194223 Saint-Petersburg, Russia
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18
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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: 18] [Impact Index Per Article: 3.6] [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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