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Universal Neural Field Computation

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Neural Fields

Abstract

Turing machines and Gödel numbers are important pillars of the theory of computation. Thus, any computational architecture needs to show how it could relate to Turing machines and how stable implementations of Turing computation are possible. In this chapter, we implement universal Turing computation in a neural field environment. To this end, we employ the canonical symbologram representation of a Turing machine obtained from a Gödel encoding of its symbolic repertoire and generalized shifts. The resulting nonlinear dynamical automaton (NDA) is a piecewise affine-linear map acting on the unit square that is partitioned into rectangular domains. Instead of looking at point dynamics in phase space, we then consider functional dynamics of probability distribution functions (p.d.f.s) over phase space. This is generally described by a Frobenius-Perron integral transformation that can be regarded as a neural field equation over the unit square as feature space of a Dynamic Field Theory (DFT). Solving the Frobenius-Perron equation yields that uniform p.d.f.s with rectangular support are mapped onto uniform p.d.f.s with rectangular support, again. We call the resulting representation dynamic field automaton.

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Acknowledgements

We thank Slawomir Nasuto and Serafim Rodrigues for helpful comments improving this chapter. This research was supported by a DFG Heisenberg fellowship awarded to PbG (GR 3711/1-2).

Author information

Authors and Affiliations

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

    Peter beim Graben

  2. Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universität zu Berlin, Berlin, Germany

    Peter beim Graben

  3. Department of Mathematics and Statistics, University of Reading, Reading, Berkshire, UK

    Roland Potthast

  4. Deutscher Wetterdienst, Offenbach, Germany

    Roland Potthast

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  1. Peter beim Graben
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Correspondence to Peter beim Graben .

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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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beim Graben, P., Potthast, R. (2014). Universal Neural Field Computation. 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_11

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