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      A mean-field approach to the dynamics of networks of complex neurons, from nonlinear Integrate-and-Fire to Hodgkin–Huxley models

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          Abstract

          We present a mean-field formalism able to predict the collective dynamics of large networks of conductance-based interacting spiking neurons. We apply this formalism to several neuronal models, from the simplest Adaptive Exponential Integrate-and-Fire model to the more complex Hodgkin–Huxley and Morris–Lecar models. We show that the resulting mean-field models are capable of predicting the correct spontaneous activity of both excitatory and inhibitory neurons in asynchronous irregular regimes, typical of cortical dynamics. Moreover, it is possible to quantitatively predict the population response to external stimuli in the form of external spike trains. This mean-field formalism therefore provides a paradigm to bridge the scale between population dynamics and the microscopic complexity of the individual cells physiology.

          NEW & NOTEWORTHY Population models are a powerful mathematical tool to study the dynamics of neuronal networks and to simulate the brain at macroscopic scales. We present a mean-field model capable of quantitatively predicting the temporal dynamics of a network of complex spiking neuronal models, from Integrate-and-Fire to Hodgkin–Huxley, thus linking population models to neurons electrophysiology. This opens a perspective on generating biologically realistic mean-field models from electrophysiological recordings.

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          Most cited references 39

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          Neurons with graded response have collective computational properties like those of two-state neurons.

           John Hopfield (1984)
          A model for a large network of "neurons" with a graded response (or sigmoid input-output relation) is studied. This deterministic system has collective properties in very close correspondence with the earlier stochastic model based on McCulloch - Pitts neurons. The content- addressable memory and other emergent collective properties of the original model also are present in the graded response model. The idea that such collective properties are used in biological systems is given added credence by the continued presence of such properties for more nearly biological "neurons." Collective analog electrical circuits of the kind described will certainly function. The collective states of the two models have a simple correspondence. The original model will continue to be useful for simulations, because its connection to graded response systems is established. Equations that include the effect of action potentials in the graded response system are also developed.
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            Voltage oscillations in the barnacle giant muscle fiber.

             C. Morris,  H Lecar (1981)
            Barnacle muscle fibers subjected to constant current stimulation produce a variety of types of oscillatory behavior when the internal medium contains the Ca++ chelator EGTA. Oscillations are abolished if Ca++ is removed from the external medium, or if the K+ conductance is blocked. Available voltage-clamp data indicate that the cell's active conductance systems are exceptionally simple. Given the complexity of barnacle fiber voltage behavior, this seems paradoxical. This paper presents an analysis of the possible modes of behavior available to a system of two noninactivating conductance mechanisms, and indicates a good correspondence to the types of behavior exhibited by barnacle fiber. The differential equations of a simple equivalent circuit for the fiber are dealt with by means of some of the mathematical techniques of nonlinear mechanics. General features of the system are (a) a propensity to produce damped or sustained oscillations over a rather broad parameter range, and (b) considerable latitude in the shape of the oscillatory potentials. It is concluded that for cells subject to changeable parameters (either from cell to cell or with time during cellular activity), a system dominated by two noninactivating conductances can exhibit varied oscillatory and bistable behavior.
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              Reconstruction and Simulation of Neocortical Microcircuitry.

              We present a first-draft digital reconstruction of the microcircuitry of somatosensory cortex of juvenile rat. The reconstruction uses cellular and synaptic organizing principles to algorithmically reconstruct detailed anatomy and physiology from sparse experimental data. An objective anatomical method defines a neocortical volume of 0.29 ± 0.01 mm(3) containing ~31,000 neurons, and patch-clamp studies identify 55 layer-specific morphological and 207 morpho-electrical neuron subtypes. When digitally reconstructed neurons are positioned in the volume and synapse formation is restricted to biological bouton densities and numbers of synapses per connection, their overlapping arbors form ~8 million connections with ~37 million synapses. Simulations reproduce an array of in vitro and in vivo experiments without parameter tuning. Additionally, we find a spectrum of network states with a sharp transition from synchronous to asynchronous activity, modulated by physiological mechanisms. The spectrum of network states, dynamically reconfigured around this transition, supports diverse information processing strategies.
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                Author and article information

                Journal
                J Neurophysiol
                J. Neurophysiol
                jn
                J Neurophysiol
                JN
                Journal of Neurophysiology
                American Physiological Society (Bethesda, MD )
                0022-3077
                1522-1598
                1 March 2020
                18 December 2019
                18 December 2019
                : 123
                : 3
                : 1042-1051
                Affiliations
                1Department of Integrative and Computational Neuroscience, Paris-Saclay Institute of Neuroscience, Centre National de la Recherche Scientifique , Gif sur Yvette, France
                2Ecole Normale Superieure Paris-Saclay, France
                3Institut d’Investigacions Biomèdiques August Pi i Sunyer , Barcelona, Spain
                4Strathclyde Institute of Pharmacy and Biomedical Sciences, Glasgow, Scotland, United Kingdom
                5Université Grenoble Alpes, Grenoble Institut des Neurosciences and Institut National de la Santé et de la Recherche Médicale (INSERM), U1216, France
                6INSERM, U1099, Rennes, France
                7MathNeuro Team, Inria Sophia Antipolis Méditerranée, Sophia Antipolis, France
                8Physics Department, Sapienza University , Rome, Italy
                9Université Côte d’Azur, Inria Sophia Antipolis Méditerranée, France
                10Laboratoire de Physique Théorique et Modelisation, Université de Cergy-Pontoise , Cergy-Pontoise, France
                Author notes
                [*]

                M. Carlu, O. Chehab, L. Dalla Porta, D. Depannemaecker, C. Hérice, M. Jedynak, E. Köksal Ersöz, P. Muratore, and S. Souihel contributed equally.

                Address for reprint requests and other correspondence: M. di Volo, Laboratoire de Physique Théorique et Modelisation, Université de Cergy-Pontoise, 95302 Cergy-Pontoise cedex, France (e-mail: matteo.di-volo@ 123456u-cergy.fr ).
                Article
                JN-00399-2019 JN-00399-2019
                10.1152/jn.00399.2019
                7099478
                31851573
                Copyright © 2020 the American Physiological Society

                Licensed under Creative Commons Attribution CC-BY 4.0: © the American Physiological Society.

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                Funding
                Funded by: Human Brain project
                Award ID: H2020-720270 & H2020-785907
                Funded by: European Reseach Council
                Award ID: 616268
                Categories
                Research Article
                50 Years of Modeling Neural Activity: Celebrating Jack Cowan's Career
                Custom metadata
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                Neurology

                population models, spiking networks, mean field, cortical dynamics, asynchronous irregular

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