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Vinci GV, Benzi R, Mattia M. Self-Consistent Stochastic Dynamics for Finite-Size Networks of Spiking Neurons. PHYSICAL REVIEW LETTERS 2023; 130:097402. [PMID: 36930929 DOI: 10.1103/physrevlett.130.097402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/17/2022] [Revised: 12/23/2022] [Accepted: 02/09/2023] [Indexed: 06/18/2023]
Abstract
Despite the huge number of neurons composing a brain network, ongoing activity of local cell assemblies is intrinsically stochastic. Fluctuations in their instantaneous rate of spike firing ν(t) scale with the size of the assembly and persist in isolated networks, i.e., in the absence of external sources of noise. Although deterministic chaos due to the quenched disorder of the synaptic couplings underlies this seemingly stochastic dynamics, an effective theory for the network dynamics of a finite assembly of spiking neurons is lacking. Here, we fill this gap by extending the so-called population density approach including an activity- and size-dependent stochastic source in the Fokker-Planck equation for the membrane potential density. The finite-size noise embedded in this stochastic partial derivative equation is analytically characterized leading to a self-consistent and nonperturbative description of ν(t) valid for a wide class of spiking neuron networks. Power spectra of ν(t) are found in excellent agreement with those from detailed simulations both in the linear regime and across a synchronization phase transition, when a size-dependent smearing of the critical dynamics emerges.
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Affiliation(s)
- Gianni V Vinci
- Natl. Center for Radiation Protection and Computational Physics, Istituto Superiore di Sanità, 00161 Roma, Italy
- PhD Program in Physics, Dept. of Physics, "Tor Vergata" University of Rome, 00133 Roma, Italy
| | - Roberto Benzi
- Dept. of Physics and INFN, "Tor Vergata" University of Rome, 00133 Roma, Italy
- Centro Ricerche "E. Fermi," 00184, Roma, Italy
| | - Maurizio Mattia
- Natl. Center for Radiation Protection and Computational Physics, Istituto Superiore di Sanità, 00161 Roma, Italy
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Layer M, Senk J, Essink S, van Meegen A, Bos H, Helias M. NNMT: Mean-Field Based Analysis Tools for Neuronal Network Models. Front Neuroinform 2022; 16:835657. [PMID: 35712677 PMCID: PMC9196133 DOI: 10.3389/fninf.2022.835657] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/14/2021] [Accepted: 03/17/2022] [Indexed: 11/13/2022] Open
Abstract
Mean-field theory of neuronal networks has led to numerous advances in our analytical and intuitive understanding of their dynamics during the past decades. In order to make mean-field based analysis tools more accessible, we implemented an extensible, easy-to-use open-source Python toolbox that collects a variety of mean-field methods for the leaky integrate-and-fire neuron model. The Neuronal Network Mean-field Toolbox (NNMT) in its current state allows for estimating properties of large neuronal networks, such as firing rates, power spectra, and dynamical stability in mean-field and linear response approximation, without running simulations. In this article, we describe how the toolbox is implemented, show how it is used to reproduce results of previous studies, and discuss different use-cases, such as parameter space explorations, or mapping different network models. Although the initial version of the toolbox focuses on methods for leaky integrate-and-fire neurons, its structure is designed to be open and extensible. It aims to provide a platform for collecting analytical methods for neuronal network model analysis, such that the neuroscientific community can take maximal advantage of them.
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Affiliation(s)
- Moritz Layer
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA-Institute Brain Structure-Function Relationships (INM-10), Jülich Research Centre, Jülich, Germany
- RWTH Aachen University, Aachen, Germany
| | - Johanna Senk
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA-Institute Brain Structure-Function Relationships (INM-10), Jülich Research Centre, Jülich, Germany
| | - Simon Essink
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA-Institute Brain Structure-Function Relationships (INM-10), Jülich Research Centre, Jülich, Germany
- RWTH Aachen University, Aachen, Germany
| | - Alexander van Meegen
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA-Institute Brain Structure-Function Relationships (INM-10), Jülich Research Centre, Jülich, Germany
- Institute of Zoology, Faculty of Mathematics and Natural Sciences, University of Cologne, Cologne, Germany
| | - Hannah Bos
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA-Institute Brain Structure-Function Relationships (INM-10), Jülich Research Centre, Jülich, Germany
| | - Moritz Helias
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA-Institute Brain Structure-Function Relationships (INM-10), Jülich Research Centre, Jülich, Germany
- Department of Physics, Faculty 1, RWTH Aachen University, Aachen, Germany
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Pietras B, Gallice N, Schwalger T. Low-dimensional firing-rate dynamics for populations of renewal-type spiking neurons. Phys Rev E 2021; 102:022407. [PMID: 32942450 DOI: 10.1103/physreve.102.022407] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2020] [Accepted: 06/29/2020] [Indexed: 11/07/2022]
Abstract
The macroscopic dynamics of large populations of neurons can be mathematically analyzed using low-dimensional firing-rate or neural-mass models. However, these models fail to capture spike synchronization effects and nonstationary responses of the population activity to rapidly changing stimuli. Here we derive low-dimensional firing-rate models for homogeneous populations of neurons modeled as time-dependent renewal processes. The class of renewal neurons includes integrate-and-fire models driven by white noise and has been frequently used to model neuronal refractoriness and spike synchronization dynamics. The derivation is based on an eigenmode expansion of the associated refractory density equation, which generalizes previous spectral methods for Fokker-Planck equations to arbitrary renewal models. We find a simple relation between the eigenvalues characterizing the timescales of the firing rate dynamics and the Laplace transform of the interspike interval density, for which explicit expressions are available for many renewal models. Retaining only the first eigenmode already yields a reliable low-dimensional approximation of the firing-rate dynamics that captures spike synchronization effects and fast transient dynamics at stimulus onset. We explicitly demonstrate the validity of our model for a large homogeneous population of Poisson neurons with absolute refractoriness and other renewal models that admit an explicit analytical calculation of the eigenvalues. The eigenmode expansion presented here provides a systematic framework for alternative firing-rate models in computational neuroscience based on spiking neuron dynamics with refractoriness.
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Affiliation(s)
- Bastian Pietras
- Institute of Mathematics, Technical University Berlin, 10623 Berlin, Germany.,Bernstein Center for Computational Neuroscience Berlin, 10115 Berlin, Germany
| | - Noé Gallice
- Brain Mind Institute, École polytechnique fédérale de Lausanne (EPFL), Station 15, CH-1015 Lausanne, Switzerland
| | - Tilo Schwalger
- Institute of Mathematics, Technical University Berlin, 10623 Berlin, Germany.,Bernstein Center for Computational Neuroscience Berlin, 10115 Berlin, Germany
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Lodi M, Della Rossa F, Sorrentino F, Storace M. Analyzing synchronized clusters in neuron networks. Sci Rep 2020; 10:16336. [PMID: 33004897 PMCID: PMC7530773 DOI: 10.1038/s41598-020-73269-9] [Citation(s) in RCA: 8] [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: 07/16/2020] [Accepted: 09/15/2020] [Indexed: 11/08/2022] Open
Abstract
The presence of synchronized clusters in neuron networks is a hallmark of information transmission and processing. Common approaches to study cluster synchronization in networks of coupled oscillators ground on simplifying assumptions, which often neglect key biological features of neuron networks. Here we propose a general framework to study presence and stability of synchronous clusters in more realistic models of neuron networks, characterized by the presence of delays, different kinds of neurons and synapses. Application of this framework to two examples with different size and features (the directed network of the macaque cerebral cortex and the swim central pattern generator of a mollusc) provides an interpretation key to explain known functional mechanisms emerging from the combination of anatomy and neuron dynamics. The cluster synchronization analysis is carried out also by changing parameters and studying bifurcations. Despite some modeling simplifications in one of the examples, the obtained results are in good agreement with previously reported biological data.
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Affiliation(s)
- Matteo Lodi
- DITEN, University of Genoa, Via Opera Pia 11a, 16145, Genova, Italy
| | - Fabio Della Rossa
- Mechanical Engineering Department, University of New Mexico, Albuquerque, NM, 87131, USA
- Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, 20133, Milan, Italy
| | - Francesco Sorrentino
- Mechanical Engineering Department, University of New Mexico, Albuquerque, NM, 87131, USA
| | - Marco Storace
- DITEN, University of Genoa, Via Opera Pia 11a, 16145, Genova, Italy.
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Hasanzadeh N, Rezaei M, Faraz S, Popovic MR, Lankarany M. Necessary Conditions for Reliable Propagation of Slowly Time-Varying Firing Rate. Front Comput Neurosci 2020; 14:64. [PMID: 32848685 PMCID: PMC7405925 DOI: 10.3389/fncom.2020.00064] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2020] [Accepted: 05/28/2020] [Indexed: 11/13/2022] Open
Abstract
Reliable propagation of slow-modulations of the firing rate across multiple layers of a feedforward network (FFN) has proven difficult to capture in spiking neural models. In this paper, we explore necessary conditions for reliable and stable propagation of time-varying asynchronous spikes whose instantaneous rate of changes-in fairly short time windows [20-100] msec-represents information of slow fluctuations of the stimulus. Specifically, we study the effect of network size, level of background synaptic noise, and the variability of synaptic delays in an FFN with all-to-all connectivity. We show that network size and the level of background synaptic noise, together with the strength of synapses, are substantial factors enabling the propagation of asynchronous spikes in deep layers of an FFN. In contrast, the variability of synaptic delays has a minor effect on signal propagation.
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Affiliation(s)
- Navid Hasanzadeh
- Clinical and Computational Neuroscience, Krembil Research Institute, University Health Network, Toronto, ON, Canada.,School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran
| | - Mohammadreza Rezaei
- Clinical and Computational Neuroscience, Krembil Research Institute, University Health Network, Toronto, ON, Canada.,KITE Research Institute, Toronto Rehabilitation Institute, University Health Network, Toronto, ON, Canada
| | - Sayan Faraz
- Clinical and Computational Neuroscience, Krembil Research Institute, University Health Network, Toronto, ON, Canada
| | - Milos R Popovic
- KITE Research Institute, Toronto Rehabilitation Institute, University Health Network, Toronto, ON, Canada.,Institute of Biomedical Engineering, University of Toronto, Toronto, ON, Canada
| | - Milad Lankarany
- Clinical and Computational Neuroscience, Krembil Research Institute, University Health Network, Toronto, ON, Canada.,KITE Research Institute, Toronto Rehabilitation Institute, University Health Network, Toronto, ON, Canada.,Institute of Biomedical Engineering, University of Toronto, Toronto, ON, Canada
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Shao Y, Zhang J, Tao L. Dimensional reduction of emergent spatiotemporal cortical dynamics via a maximum entropy moment closure. PLoS Comput Biol 2020; 16:e1007265. [PMID: 32516336 PMCID: PMC7304648 DOI: 10.1371/journal.pcbi.1007265] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/11/2019] [Revised: 06/19/2020] [Accepted: 04/29/2020] [Indexed: 11/22/2022] Open
Abstract
Modern electrophysiological recordings and optical imaging techniques have revealed a diverse spectrum of spatiotemporal neural activities underlying fundamental cognitive processing. Oscillations, traveling waves and other complex population dynamical patterns are often concomitant with sensory processing, information transfer, decision making and memory consolidation. While neural population models such as neural mass, population density and kinetic theoretical models have been used to capture a wide range of the experimentally observed dynamics, a full account of how the multi-scale dynamics emerges from the detailed biophysical properties of individual neurons and the network architecture remains elusive. Here we apply a recently developed coarse-graining framework for reduced-dimensional descriptions of neuronal networks to model visual cortical dynamics. We show that, without introducing any new parameters, how a sequence of models culminating in an augmented system of spatially-coupled ODEs can effectively model a wide range of the observed cortical dynamics, ranging from visual stimulus orientation dynamics to traveling waves induced by visual illusory stimuli. In addition to an efficient simulation method, this framework also offers an analytic approach to studying large-scale network dynamics. As such, the dimensional reduction naturally leads to mesoscopic variables that capture the interplay between neuronal population stochasticity and network architecture that we believe to underlie many emergent cortical phenomena.
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Affiliation(s)
- Yuxiu Shao
- Center for Bioinformatics, National Laboratory of Protein Engineering and Plant Genetic Engineering, School of Life Sciences, Peking University, Beijing, China
| | - Jiwei Zhang
- School of Mathematics and Statistics, and Hubei Key Laboratory of Computational Science, Wuhan University, China
| | - Louis Tao
- Center for Bioinformatics, National Laboratory of Protein Engineering and Plant Genetic Engineering, School of Life Sciences, Peking University, Beijing, China
- Center for Quantitative Biology, Peking University, Beijing, China
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