Numerical continuation of travelling waves and pulses in neural fields

Hil Gaétan Ellart Meijer, Stephen Coombes

    Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademic

    47 Downloads (Pure)

    Abstract

    We study travelling waves and pulses in neural fields. Neural fields are a macroscopic description of the activity of brain tissue, which mathematically are formulated as integro-differential equations. While linear and weakly nonlinear analysis can describe instabilities and small amplitude patterns, numerical techniques are needed to study the nonlinear properties of travelling waves and pulses. Here we report on progress of such an approach using an excitatory neural field with linear adaptation. First, we find and analyse an anti-pulse, where the whole field is active except for a moving region with lowered activity. Such an anti-pulse may be relevant in modelling spreading depression. Second, we consider dynamics for relatively shallow, smooth activation functions where the neural field behaves as an excitable medium. We compute numerically the dispersion curves for travelling waves, i.e. the wavespeed as a function of the spatial period. This allows a kinematic analysis, see [3], that may be useful for analysing spreading epileptiform activity. Third, we present a numerical continuation method for periodic orbits of integro-differential equations based on fast Fourier transforms. Hence, neural fields with biophysically realistic mechanisms may be analysed beyond linearisation.
    Original languageUndefined
    Title of host publicationAbstracts from the Twenty Second Annual Computational Neuroscience Meeting: CNS*2013
    PublisherBioMed Central Ltd.
    PagesP70
    Number of pages1
    DOIs
    Publication statusPublished - 8 Jul 2013
    Event22nd Annual Computational Neuroscience Meeting CNS 2013 - Paris, France
    Duration: 13 Jul 201318 Jul 2013
    Conference number: 22

    Publication series

    Name
    PublisherBioMed Central
    NumberSuppl. 1
    Volume14(Suppl 1
    ISSN (Print)1471-2202
    ISSN (Electronic)1471-2202

    Conference

    Conference22nd Annual Computational Neuroscience Meeting CNS 2013
    Abbreviated titleCNS*2013
    CountryFrance
    CityParis
    Period13/07/1318/07/13

    Keywords

    • EWI-24068
    • METIS-302558
    • IR-88241

    Cite this

    Meijer, H. G. E., & Coombes, S. (2013). Numerical continuation of travelling waves and pulses in neural fields. In Abstracts from the Twenty Second Annual Computational Neuroscience Meeting: CNS*2013 (pp. P70). BioMed Central Ltd.. https://doi.org/10.1186/1471-2202-14-S1-P70
    Meijer, Hil Gaétan Ellart ; Coombes, Stephen. / Numerical continuation of travelling waves and pulses in neural fields. Abstracts from the Twenty Second Annual Computational Neuroscience Meeting: CNS*2013. BioMed Central Ltd., 2013. pp. P70
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    Meijer, HGE & Coombes, S 2013, Numerical continuation of travelling waves and pulses in neural fields. in Abstracts from the Twenty Second Annual Computational Neuroscience Meeting: CNS*2013. BioMed Central Ltd., pp. P70, 22nd Annual Computational Neuroscience Meeting CNS 2013, Paris, France, 13/07/13. https://doi.org/10.1186/1471-2202-14-S1-P70

    Numerical continuation of travelling waves and pulses in neural fields. / Meijer, Hil Gaétan Ellart; Coombes, Stephen.

    Abstracts from the Twenty Second Annual Computational Neuroscience Meeting: CNS*2013. BioMed Central Ltd., 2013. p. P70.

    Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademic

    TY - GEN

    T1 - Numerical continuation of travelling waves and pulses in neural fields

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    N2 - We study travelling waves and pulses in neural fields. Neural fields are a macroscopic description of the activity of brain tissue, which mathematically are formulated as integro-differential equations. While linear and weakly nonlinear analysis can describe instabilities and small amplitude patterns, numerical techniques are needed to study the nonlinear properties of travelling waves and pulses. Here we report on progress of such an approach using an excitatory neural field with linear adaptation. First, we find and analyse an anti-pulse, where the whole field is active except for a moving region with lowered activity. Such an anti-pulse may be relevant in modelling spreading depression. Second, we consider dynamics for relatively shallow, smooth activation functions where the neural field behaves as an excitable medium. We compute numerically the dispersion curves for travelling waves, i.e. the wavespeed as a function of the spatial period. This allows a kinematic analysis, see [3], that may be useful for analysing spreading epileptiform activity. Third, we present a numerical continuation method for periodic orbits of integro-differential equations based on fast Fourier transforms. Hence, neural fields with biophysically realistic mechanisms may be analysed beyond linearisation.

    AB - We study travelling waves and pulses in neural fields. Neural fields are a macroscopic description of the activity of brain tissue, which mathematically are formulated as integro-differential equations. While linear and weakly nonlinear analysis can describe instabilities and small amplitude patterns, numerical techniques are needed to study the nonlinear properties of travelling waves and pulses. Here we report on progress of such an approach using an excitatory neural field with linear adaptation. First, we find and analyse an anti-pulse, where the whole field is active except for a moving region with lowered activity. Such an anti-pulse may be relevant in modelling spreading depression. Second, we consider dynamics for relatively shallow, smooth activation functions where the neural field behaves as an excitable medium. We compute numerically the dispersion curves for travelling waves, i.e. the wavespeed as a function of the spatial period. This allows a kinematic analysis, see [3], that may be useful for analysing spreading epileptiform activity. Third, we present a numerical continuation method for periodic orbits of integro-differential equations based on fast Fourier transforms. Hence, neural fields with biophysically realistic mechanisms may be analysed beyond linearisation.

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    Meijer HGE, Coombes S. Numerical continuation of travelling waves and pulses in neural fields. In Abstracts from the Twenty Second Annual Computational Neuroscience Meeting: CNS*2013. BioMed Central Ltd. 2013. p. P70 https://doi.org/10.1186/1471-2202-14-S1-P70