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Song D, Chung DW, Ermentrout GB. Mean-field analysis of synaptic alterations underlying deficient cortical gamma oscillations in schizophrenia. RESEARCH SQUARE 2024:rs.3.rs-3938805. [PMID: 38410475 PMCID: PMC10896366 DOI: 10.21203/rs.3.rs-3938805/v1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/28/2024]
Abstract
Deficient gamma oscillations in the prefrontal cortex (PFC) of individuals with schizophrenia (SZ) are proposed to arise from alterations in the excitatory drive to fast-spiking interneurons ( E → I ) and in the inhibitory drive from these interneurons to excitatory neurons ( I → E ) . Consistent with this idea, prior postmortem studies showed lower levels of molecular and structural markers for the strength of E → I and I → E synapses and also greater variability in E → I synaptic strength in PFC of SZ. Moreover, simulating these alterations in a network of quadratic integrate-and-fire (QIF) neurons revealed a synergistic effect of their interactions on reducing gamma power. In this study, we aimed to investigate the dynamical nature of this synergistic interaction at macroscopic level by deriving a mean-field description of the QIF model network that consists of all-to-all connected excitatory neurons and fast-spiking interneurons. Through a series of numerical simulations and bifurcation analyses, findings from our mean-field model showed that the macroscopic dynamics of gamma oscillations are synergistically disrupted by the interactions among lower strength of E → I and I → E synapses and greater variability in E → I synaptic strength. Furthermore, the two-dimensional bifurcation analyses showed that this synergistic interaction is primarily driven by the shift in Hopf bifurcation due to lower E → I synaptic strength. Together, these simulations predict the nature of dynamical mechanisms by which multiple synaptic alterations interact to robustly reduce PFC gamma power in SZ, and highlight the utility of mean-field model to study macroscopic neural dynamics and their alterations in the illness.
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Affiliation(s)
- Deying Song
- Joint Program in Neural Computation and Machine Learning, Neuroscience Institute, and Machine Learning Department, Carnegie Mellon University, Pittsburgh, PA, USA, 15213
- Center for the Neural Basis of Cognition, University of Pittsburgh, Pittsburgh, PA, USA, 15213
| | - Daniel W. Chung
- Translational Neuroscience Program, Department of Psychiatry, University of Pittsburgh, Pittsburgh, PA, USA, 15213
| | - G. Bard Ermentrout
- Center for the Neural Basis of Cognition, University of Pittsburgh, Pittsburgh, PA, USA, 15213
- Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA, 15213
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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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Clusella P, Köksal-Ersöz E, Garcia-Ojalvo J, Ruffini G. Comparison between an exact and a heuristic neural mass model with second-order synapses. BIOLOGICAL CYBERNETICS 2023; 117:5-19. [PMID: 36454267 PMCID: PMC10160168 DOI: 10.1007/s00422-022-00952-7] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/14/2022] [Accepted: 10/23/2022] [Indexed: 05/05/2023]
Abstract
Neural mass models (NMMs) are designed to reproduce the collective dynamics of neuronal populations. A common framework for NMMs assumes heuristically that the output firing rate of a neural population can be described by a static nonlinear transfer function (NMM1). However, a recent exact mean-field theory for quadratic integrate-and-fire (QIF) neurons challenges this view by showing that the mean firing rate is not a static function of the neuronal state but follows two coupled nonlinear differential equations (NMM2). Here we analyze and compare these two descriptions in the presence of second-order synaptic dynamics. First, we derive the mathematical equivalence between the two models in the infinitely slow synapse limit, i.e., we show that NMM1 is an approximation of NMM2 in this regime. Next, we evaluate the applicability of this limit in the context of realistic physiological parameter values by analyzing the dynamics of models with inhibitory or excitatory synapses. We show that NMM1 fails to reproduce important dynamical features of the exact model, such as the self-sustained oscillations of an inhibitory interneuron QIF network. Furthermore, in the exact model but not in the limit one, stimulation of a pyramidal cell population induces resonant oscillatory activity whose peak frequency and amplitude increase with the self-coupling gain and the external excitatory input. This may play a role in the enhanced response of densely connected networks to weak uniform inputs, such as the electric fields produced by noninvasive brain stimulation.
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Affiliation(s)
- Pau Clusella
- Department of Medicine and Life Sciences, Universitat Pompeu Fabra, Barcelona Biomedical Research Park, 08003, Barcelona, Spain.
| | - Elif Köksal-Ersöz
- LTSI - UMR 1099, INSERM, Univ Rennes, Campus Beaulieu, 35000, Rennes, France
| | - Jordi Garcia-Ojalvo
- Department of Medicine and Life Sciences, Universitat Pompeu Fabra, Barcelona Biomedical Research Park, 08003, Barcelona, Spain
| | - Giulio Ruffini
- Brain Modeling Department, Neuroelectrics, Av. Tibidabo, 47b, 08035, Barcelona, Spain.
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Thiele M, Berner R, Tass PA, Schöll E, Yanchuk S. Asymmetric adaptivity induces recurrent synchronization in complex networks. CHAOS (WOODBURY, N.Y.) 2023; 33:023123. [PMID: 36859232 DOI: 10.1063/5.0128102] [Citation(s) in RCA: 8] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/26/2022] [Accepted: 01/18/2023] [Indexed: 06/18/2023]
Abstract
Rhythmic activities that alternate between coherent and incoherent phases are ubiquitous in chemical, ecological, climate, or neural systems. Despite their importance, general mechanisms for their emergence are little understood. In order to fill this gap, we present a framework for describing the emergence of recurrent synchronization in complex networks with adaptive interactions. This phenomenon is manifested at the macroscopic level by temporal episodes of coherent and incoherent dynamics that alternate recurrently. At the same time, the dynamics of the individual nodes do not change qualitatively. We identify asymmetric adaptation rules and temporal separation between the adaptation and the dynamics of individual nodes as key features for the emergence of recurrent synchronization. Our results suggest that asymmetric adaptation might be a fundamental ingredient for recurrent synchronization phenomena as seen in pattern generators, e.g., in neuronal systems.
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Affiliation(s)
- Max Thiele
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
| | - Rico Berner
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
| | - Peter A Tass
- Department of Neurosurgery, Stanford University, Stanford, California 94305, USA
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
| | - Serhiy Yanchuk
- Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany
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Laing CR. Chimeras on annuli. CHAOS (WOODBURY, N.Y.) 2022; 32:083105. [PMID: 36049938 DOI: 10.1063/5.0103669] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2022] [Accepted: 07/08/2022] [Indexed: 06/15/2023]
Abstract
Chimeras occur in networks of coupled oscillators and are characterized by the coexistence of synchronous and asynchronous groups of oscillators in different parts of the network. We consider a network of nonlocally coupled phase oscillators on an annular domain. The Ott/Antonsen ansatz is used to derive a continuum level description of the oscillators' expected dynamics in terms of a complex-valued order parameter. The equations for this order parameter are numerically analyzed in order to investigate solutions with the same symmetry as the domain and chimeras which are analogous to the "multi-headed" chimeras observed on one-dimensional domains. Such solutions are stable only for domains with widths that are neither too large nor too small. We also study rotating waves with different winding numbers, which are similar to spiral wave chimeras seen in two-dimensional domains. We determine ranges of parameters, such as the size of the domain for which such solutions exist and are stable, and the bifurcations by which they lose stability. All of these bifurcations appear subcritical.
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Affiliation(s)
- Carlo R Laing
- School of Mathematical and Computational Sciences, Massey University, Private Bag 102-904, North Shore Mail Centre, Auckland, New Zealand
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Exact mean-field models for spiking neural networks with adaptation. J Comput Neurosci 2022; 50:445-469. [PMID: 35834100 DOI: 10.1007/s10827-022-00825-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2022] [Accepted: 06/15/2022] [Indexed: 10/17/2022]
Abstract
Networks of spiking neurons with adaption have been shown to be able to reproduce a wide range of neural activities, including the emergent population bursting and spike synchrony that underpin brain disorders and normal function. Exact mean-field models derived from spiking neural networks are extremely valuable, as such models can be used to determine how individual neurons and the network they reside within interact to produce macroscopic network behaviours. In the paper, we derive and analyze a set of exact mean-field equations for the neural network with spike frequency adaptation. Specifically, our model is a network of Izhikevich neurons, where each neuron is modeled by a two dimensional system consisting of a quadratic integrate and fire equation plus an equation which implements spike frequency adaptation. Previous work deriving a mean-field model for this type of network, relied on the assumption of sufficiently slow dynamics of the adaptation variable. However, this approximation did not succeed in establishing an exact correspondence between the macroscopic description and the realistic neural network, especially when the adaptation time constant was not large. The challenge lies in how to achieve a closed set of mean-field equations with the inclusion of the mean-field dynamics of the adaptation variable. We address this problem by using a Lorentzian ansatz combined with the moment closure approach to arrive at a mean-field system in the thermodynamic limit. The resulting macroscopic description is capable of qualitatively and quantitatively describing the collective dynamics of the neural network, including transition between states where the individual neurons exhibit asynchronous tonic firing and synchronous bursting. We extend the approach to a network of two populations of neurons and discuss the accuracy and efficacy of our mean-field approximations by examining all assumptions that are imposed during the derivation. Numerical bifurcation analysis of our mean-field models reveals bifurcations not previously observed in the models, including a novel mechanism for emergence of bursting in the network. We anticipate our results will provide a tractable and reliable tool to investigate the underlying mechanism of brain function and dysfunction from the perspective of computational neuroscience.
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Hua N, He X, Feng J, Lu W. Analytic Investigation for Synchronous Firing Patterns Propagation in Spiking Neural Networks. Neural Process Lett 2022. [DOI: 10.1007/s11063-022-10792-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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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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A Roadmap for Computational Modelling of M/EEG. Brain Topogr 2022; 35:1-3. [PMID: 35084621 PMCID: PMC8813794 DOI: 10.1007/s10548-022-00889-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/10/2022] [Accepted: 01/12/2022] [Indexed: 11/05/2022]
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Clusella P, Pietras B, Montbrió E. Kuramoto model for populations of quadratic integrate-and-fire neurons with chemical and electrical coupling. CHAOS (WOODBURY, N.Y.) 2022; 32:013105. [PMID: 35105122 DOI: 10.1063/5.0075285] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/14/2021] [Accepted: 12/13/2021] [Indexed: 06/14/2023]
Abstract
We derive the Kuramoto model (KM) corresponding to a population of weakly coupled, nearly identical quadratic integrate-and-fire (QIF) neurons with both electrical and chemical coupling. The ratio of chemical to electrical coupling determines the phase lag of the characteristic sine coupling function of the KM and critically determines the synchronization properties of the network. We apply our results to uncover the presence of chimera states in two coupled populations of identical QIF neurons. We find that the presence of both electrical and chemical coupling is a necessary condition for chimera states to exist. Finally, we numerically demonstrate that chimera states gradually disappear as coupling strengths cease to be weak.
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Affiliation(s)
- Pau Clusella
- Department of Experimental and Health Sciences, Universitat Pompeu Fabra, Barcelona Biomedical Research Park, 08003 Barcelona, Spain
| | - Bastian Pietras
- Institute of Mathematics, Technical University Berlin, 10623 Berlin, Germany
| | - Ernest Montbrió
- Neuronal Dynamics Group, Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08018 Barcelona, Spain
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