Neural Fields

1. Front Matter

   Pages i-x

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2. Tutorial on Neural Field Theory

   • Stephen Coombes, Peter beim Graben, Roland Potthast

   Pages 1-43

3. Theory of Neural Fields

   1. Front Matter

      Pages 45-45

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   2. A Personal Account of the Development of the Field Theory of Large-Scale Brain Activity from 1945 Onward

      • Jack Cowan

      Pages 47-96

   3. Heaviside World: Excitation and Self-Organization of Neural Fields

      • Shun-ichi Amari

      Pages 97-118

   4. Spatiotemporal Pattern Formation in Neural Fields with Linear Adaptation

      • G. Bard Ermentrout, Stefanos E. Folias, Zachary P. Kilpatrick

      Pages 119-151

   5. PDE Methods for Two-Dimensional Neural Fields

      • Carlo R. Laing

      Pages 153-173

   6. Numerical Simulation Scheme of One- and Two Dimensional Neural Fields Involving Space-Dependent Delays

      • Axel Hutt, Nicolas Rougier

      Pages 175-185

   7. Spots: Breathing, Drifting and Scattering in a Neural Field Model

      • Stephen Coombes, Helmut Schmidt, Daniele Avitabile

      Pages 187-211

   8. Heterogeneous Connectivity in Neural Fields: A Stochastic Approach

      • Chris A. Brackley, Matthew S. Turner

      Pages 213-234

   9. Stochastic Neural Field Theory

      • Paul C. Bressloff

      Pages 235-268

   10. On the Electrodynamics of Neural Networks

       • Peter beim Graben, Serafim Rodrigues

       Pages 269-296

4. Applications of Neural Fields

   1. Front Matter

      Pages 297-297

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   2. Universal Neural Field Computation

      • Peter beim Graben, Roland Potthast

      Pages 299-318

   3. A Neural Approach to Cognition Based on Dynamic Field Theory

      • Jonas Lins, Gregor Schöner

      Pages 319-339

   4. A Dynamic Neural Field Approach to Natural and Efficient Human-Robot Collaboration

      • Wolfram Erlhagen, Estela Bicho

      Pages 341-365

   5. Neural Field Modelling of the Electroencephalogram: Physiological Insights and Practical Applications

      • David T. J. Liley

      Pages 367-392

   6. Equilibrium and Nonequilibrium Phase Transitions in a Continuum Model of an Anesthetized Cortex

      • D. Alistair Steyn-Ross, Moira L. Steyn-Ross, Jamie W. Sleigh

      Pages 393-416

   7. Large Scale Brain Networks of Neural Fields

      • Viktor Jirsa

      Pages 417-432

   8. Neural Fields, Masses and Bayesian Modelling

      • Dimitris A. Pinotsis, Karl J. Friston

      Pages 433-455

Neural field theory has a long-standing tradition in the mathematical and computational neurosciences. Beginning almost 50 years ago with seminal work by Griffiths and culminating in the 1970ties with the models of Wilson and Cowan, Nunez and Amari, this important research area experienced a renaissance during the 1990ties by the groups of Ermentrout, Robinson, Bressloff, Wright and Haken. Since then, much progress has been made in both, the development of mathematical and numerical techniques and in physiological refinement und understanding. In contrast to large-scale neural network models described by huge connectivity matrices that are computationally expensive in numerical simulations, neural field models described by connectivity kernels allow for analytical treatment by means of methods from functional analysis. Thus, a number of rigorous results on the existence of bump and wave solutions or on inverse kernel construction problems are nowadays available. Moreover, neural fields provide an important interface for the coupling of neural activity to experimentally observable data, such as the electroencephalogram (EEG) or functional magnetic resonance imaging (fMRI). And finally, neural fields over rather abstract feature spaces, also called dynamic fields, found successful applications in the cognitive sciences and in robotics. Up to now, research results in neural field theory have been disseminated across a number of distinct journals from mathematics, computational neuroscience, biophysics, cognitive science and others. There is no comprehensive collection of results or reviews available yet. With our proposed book Neural Field Theory, we aim at filling this gap in the market. We received consent from some of the leading scientists in the field, who are willing to write contributions for the book, among them are two of the founding-fathers of neural field theory: Shun-ichi Amari and Jack Cowan.

• cognition and robotics
• integro-differential equations
• neural field theory
• neurophysics
• synaptic weight kernels
• systems biology

• School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom

  Stephen Coombes

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

  Peter beim Graben

• Department of Mathematics, University of Reading, Reading, United Kingdom

  Roland Potthast

• School of Medicine, University of Auckland, Auckland, New Zealand

  James Wright

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