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      Fast–Slow Bursters in the Unfolding of a High Codimension Singularity and the Ultra-slow Transitions of Classes

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          Abstract

          Bursting is a phenomenon found in a variety of physical and biological systems. For example, in neuroscience, bursting is believed to play a key role in the way information is transferred in the nervous system. In this work, we propose a model that, appropriately tuned, can display several types of bursting behaviors. The model contains two subsystems acting at different time scales. For the fast subsystem we use the planar unfolding of a high codimension singularity. In its bifurcation diagram, we locate paths that underlie the right sequence of bifurcations necessary for bursting. The slow subsystem steers the fast one back and forth along these paths leading to bursting behavior. The model is able to produce almost all the classes of bursting predicted for systems with a planar fast subsystem. Transitions between classes can be obtained through an ultra-slow modulation of the model’s parameters. A detailed exploration of the parameter space allows predicting possible transitions. This provides a single framework to understand the coexistence of diverse bursting patterns in physical and biological systems or in models.

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          Most cited references 39

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          NEURAL EXCITABILITY, SPIKING AND BURSTING

          Bifurcation mechanisms involved in the generation of action potentials (spikes) by neurons are reviewed here. We show how the type of bifurcation determines the neuro-computational properties of the cells. For example, when the rest state is near a saddle-node bifurcation, the cell can fire all-or-none spikes with an arbitrary low frequency, it has a well-defined threshold manifold, and it acts as an integrator; i.e. the higher the frequency of incoming pulses, the sooner it fires. In contrast, when the rest state is near an Andronov–Hopf bifurcation, the cell fires in a certain frequency range, its spikes are not all-or-none, it does not have a well-defined threshold manifold, it can fire in response to an inhibitory pulse, and it acts as a resonator; i.e. it responds preferentially to a certain (resonant) frequency of the input. Increasing the input frequency may actually delay or terminate its firing. We also describe the phenomenon of neural bursting, and we use geometric bifurcation theory to extend the existing classification of bursters, including many new types. We discuss how the type of burster defines its neuro-computational properties, and we show that different bursters can interact, synchronize and process information differently.
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            Bursts as a unit of neural information: making unreliable synapses reliable.

             J Lisman (1996)
            Several lines of evidence indicate that brief (< 25 ms) bursts of high-frequency firing have special importance in brain function. Recent work shows that many central synapses are surprisingly unreliable at signaling the arrival of single presynaptic action potentials to the postsynaptic neuron. However, bursts are reliably signaled because transmitter release is facilitated. Thus, these synapses can be viewed as filters that transmit bursts, but filter out single spikes. Bursts appear to have a special role in synaptic plasticity and information processing. In the hippocampus, a single burst can produce long-term synaptic modifications. In brain structures whose computational role is known, action potentials that arrive in bursts provide more-precise information than action potentials that arrive singly. These results, and the requirement for multiple inputs to fire a cell suggest that the best stimulus for exciting a cell (that is, a neural code) is coincident bursts.
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                Author and article information

                Contributors
                viktor.jirsa@univ-amu.fr
                Journal
                J Math Neurosci
                J Math Neurosci
                Journal of Mathematical Neuroscience
                Springer Berlin Heidelberg (Berlin/Heidelberg )
                2190-8567
                25 July 2017
                25 July 2017
                2017
                : 7
                Affiliations
                ISNI 0000 0001 2176 4817, GRID grid.5399.6, INS, Institut de Neurosciences des Systèmes, Inserm, , Aix Marseille Univ, ; Marseille, France
                Article
                50
                10.1186/s13408-017-0050-8
                5526832
                28744735
                © The Author(s) 2017

                Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

                Funding
                Funded by: FundRef http://dx.doi.org/10.13039/100000913, James S. McDonnell Foundation;
                Funded by: FundRef http://dx.doi.org/10.13039/501100007601, Horizon 2020;
                Award ID: 720270
                Award Recipient :
                Categories
                Research
                Custom metadata
                © The Author(s) 2017

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