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Mathematical Biosciences and Engineering

2010, Volume 7, Issue 2: 421-442. doi: 10.3934/mbe.2010.7.421
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Wave fronts in neuronal fields with nonlocal post-synaptic axonal connections and delayed nonlocal feedback connections

  • 1.
    Department of Mathematics and Statistics, McGill University, Montreal, Quebec
  • 2.
    Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7
  • Received: 01 July 2009 Accepted: 29 June 2018 Published: 01 April 2010

  • MSC : 35B25, 35R10, 92B20, 92C20.

  • We consider a neuronal network model with both axonal connections (in the form of synaptic coupling) and delayed non-local feedback connections. The kernel in the feedback channel is assumed to be a standard non-local one, while for the kernel in the synaptic coupling, four types are considered. The main concern is the existence of travelling wave front. By employing the speed index function, we are able to obtain the existence of a travelling wave front for each of these four types within certain range of model parameters. We are also able to describe how the feedback coupling strength and the magnitude of the delay affect the wave speed. Some particular kernel functions for these four cases are chosen to numerically demonstrate the theoretical results.

    Citation: Felicia Maria G. Magpantay, Xingfu Zou. Wave fronts in neuronal fields with nonlocalpost-synaptic axonal connections and delayed nonlocal feedbackconnections[J]. Mathematical Biosciences and Engineering, 2010, 7(2): 421-442. doi: 10.3934/mbe.2010.7.421

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  • We consider a neuronal network model with both axonal connections (in the form of synaptic coupling) and delayed non-local feedback connections. The kernel in the feedback channel is assumed to be a standard non-local one, while for the kernel in the synaptic coupling, four types are considered. The main concern is the existence of travelling wave front. By employing the speed index function, we are able to obtain the existence of a travelling wave front for each of these four types within certain range of model parameters. We are also able to describe how the feedback coupling strength and the magnitude of the delay affect the wave speed. Some particular kernel functions for these four cases are chosen to numerically demonstrate the theoretical results.


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    4. Lijun Zhang, Linghai Zhang, Jie Yuan, C. M. Khalique, Existence of Wave Front Solutions of an Integral Differential Equation in Nonlinear Nonlocal Neuronal Network, 2014, 2014, 1085-3375, 1, 10.1155/2014/753614
    5. Tobias Brett, Tobias Galla, Generating functionals and Gaussian approximations for interruptible delay reactions, 2015, 92, 1539-3755, 10.1103/PhysRevE.92.042105
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    7. Lijun Zhang, Existence and uniqueness of wave fronts in neuronal network with nonlocal post-synaptic axonal and delayed nonlocal feedback connections, 2013, 2013, 1687-1847, 10.1186/1687-1847-2013-243
    8. Alan Dyson, Existence and Uniqueness of Traveling Fronts in Lateral Inhibition Neural Fields with Sigmoidal Firing Rates, 2020, 19, 1536-0040, 2194, 10.1137/20M1311697
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  • © 2010 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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