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      A Low Dimensional Description of Globally Coupled Heterogeneous Neural Networks of Excitatory and Inhibitory Neurons

      research-article
      1 , 1 , 2 , 3 , *
      PLoS Computational Biology
      Public Library of Science

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          Abstract

          Neural networks consisting of globally coupled excitatory and inhibitory nonidentical neurons may exhibit a complex dynamic behavior including synchronization, multiclustered solutions in phase space, and oscillator death. We investigate the conditions under which these behaviors occur in a multidimensional parametric space defined by the connectivity strengths and dispersion of the neuronal membrane excitability. Using mode decomposition techniques, we further derive analytically a low dimensional description of the neural population dynamics and show that the various dynamic behaviors of the entire network can be well reproduced by this reduced system. Examples of networks of FitzHugh-Nagumo and Hindmarsh-Rose neurons are discussed in detail.

          Author Summary

          Nowadays we know that most cognitive functions are not represented in the brain by the activation of a single area but rather by a complex and rich behavior of brain networks distributed over various cortical and subcortical areas. The communication between brain areas is not instantaneous but also undergoes significant signal transmission delays of up to 100 ms, which increase the computation time for brain network models enormously. In order to allow the efficient investigation of brain network models and their associated cognitive capabilities, we report here a novel, computationally parsimonious, mathematical representation of clusters of neurons. Such reduced clusters are called “neural masses” and serve as nodes in the brain networks. Traditional neural mass descriptions so far allowed only for a very limited repertoire of behaviors, which ultimately rendered their description biologically unrealistic. The neural mass model presented here overcomes this limitation and captures a wide range of dynamic behaviors, but in a computationally efficient reduced form. The integration of novel neural mass models into brain networks represents a step closer toward a computational and biologically realistic realization of brain function.

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          Most cited references43

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          Impulses and Physiological States in Theoretical Models of Nerve Membrane

          Van der Pol's equation for a relaxation oscillator is generalized by the addition of terms to produce a pair of non-linear differential equations with either a stable singular point or a limit cycle. The resulting "BVP model" has two variables of state, representing excitability and refractoriness, and qualitatively resembles Bonhoeffer's theoretical model for the iron wire model of nerve. This BVP model serves as a simple representative of a class of excitable-oscillatory systems including the Hodgkin-Huxley (HH) model of the squid giant axon. The BVP phase plane can be divided into regions corresponding to the physiological states of nerve fiber (resting, active, refractory, enhanced, depressed, etc.) to form a "physiological state diagram," with the help of which many physiological phenomena can be summarized. A properly chosen projection from the 4-dimensional HH phase space onto a plane produces a similar diagram which shows the underlying relationship between the two models. Impulse trains occur in the BVP and HH models for a range of constant applied currents which make the singular point representing the resting state unstable.
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            Interneurons of the neocortical inhibitory system.

            Mammals adapt to a rapidly changing world because of the sophisticated cognitive functions that are supported by the neocortex. The neocortex, which forms almost 80% of the human brain, seems to have arisen from repeated duplication of a stereotypical microcircuit template with subtle specializations for different brain regions and species. The quest to unravel the blueprint of this template started more than a century ago and has revealed an immensely intricate design. The largest obstacle is the daunting variety of inhibitory interneurons that are found in the circuit. This review focuses on the organizing principles that govern the diversity of inhibitory interneurons and their circuits.
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              From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators

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                Author and article information

                Contributors
                Role: Editor
                Journal
                PLoS Comput Biol
                plos
                ploscomp
                PLoS Computational Biology
                Public Library of Science (San Francisco, USA )
                1553-734X
                1553-7358
                November 2008
                November 2008
                14 November 2008
                : 4
                : 11
                : e1000219
                Affiliations
                [1 ]Department of Physics, Florida Atlantic University, Boca Raton, Florida, United States of America
                [2 ]Theoretical Neuroscience Group, Movement Science Institute, CNRS, UMR6233, Marseille, France
                [3 ]Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, Florida, United States of America
                University College London, United Kingdom
                Author notes

                Conceived and designed the experiments: RAS VKJ. Performed the experiments: RAS. Analyzed the data: RAS VKJ. Wrote the paper: RAS VKJ.

                Article
                08-PLCB-RA-0349R3
                10.1371/journal.pcbi.1000219
                2574034
                19008942
                ae613791-969f-4b72-9e76-482e7ea19496
                Stefanescu, Jirsa. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
                History
                : 7 May 2008
                : 30 September 2008
                Page count
                Pages: 17
                Categories
                Research Article
                Biophysics/Theory and Simulation
                Computational Biology
                Computational Biology/Computational Neuroscience
                Computational Biology/Signaling Networks
                Computational Biology/Systems Biology
                Neuroscience
                Neuroscience/Cognitive Neuroscience
                Neuroscience/Neuronal Signaling Mechanisms
                Neuroscience/Theoretical Neuroscience
                Physics/Interdisciplinary Physics

                Quantitative & Systems biology
                Quantitative & Systems biology

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