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Heaviside World: Excitation and Self-Organization of Neural Fields

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Abstract

Mathematical treatments of the dynamics of neural fields become much simpler when the Heaviside function is used as an activation function. This is because the dynamics of an excited or active region reduce to the dynamics of the boundary. We call this regime the Heaviside world. Here, we visit the Heaviside world and briefly review bump dynamics in the 1D, 1D two-layer, and 2D cases. We further review the dynamics of forming topological maps by self-organization. The Heaviside world is useful for studying the learning or self-organization equation of receptive fields. The stability analysis shows the formation of a continuous map or the emergence of a block structure responsible for columnar microstructures. The stability of the Kohonen map is also discussed.

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Authors and Affiliations

  1. RIKEN Brain Science Institute, Hirosawa 2-1, Wako-shi, Saitama, 351-0198, Japan

    Shun-ichi Amari

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  1. Shun-ichi Amari
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Correspondence to Shun-ichi Amari .

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Editors and Affiliations

  1. School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom

    Stephen Coombes

  2. Department of German Studies and Linguistics, Humboldt-Universität zu Berlin, Berlin, Germany

    Peter beim Graben

  3. Department of Mathematics, University of Reading, Reading, United Kingdom

    Roland Potthast

  4. School of Medicine, University of Auckland, Auckland, New Zealand

    James Wright

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Amari, Si. (2014). Heaviside World: Excitation and Self-Organization of Neural Fields. In: Coombes, S., beim Graben, P., Potthast, R., Wright, J. (eds) Neural Fields. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54593-1_3

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