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Neural field equations with neuron-dependent Heaviside-type activation function and spatial-dependent delay
  • Evgenii Burlakov,
  • Evgeny Zhukovskiy,
  • Vitaly Verkhlyutov
Evgenii Burlakov
Tyumen State University
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Evgeny Zhukovskiy
Derzhavin Tambov State University
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Vitaly Verkhlyutov
Institute of Higher Nervous Activity and Neurophysiology RAS
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Abstract

We introduce a neural field equation with a neuron-dependent Heaviside-type activation function and spatial-dependent delay. The basic object of the study is represented by a Volterra Hammerstein integral equation involving a discontinuous nonlinearity with respect to the state variable that is both time- and space-dependent. We replace the discontinuous nonlinearity by its multi-valued convexification and obtain the corresponding Volterra Hammerstein integral inclusion. We investigate the solvability of this inclusion using the properties of upper semi-continuous multi-valued mappings with convex closed values. Based on these results, we study the solvability of an initial-prehistory problem for the former neural field equation with the Heaviside-type activation function. The application of multi-valued analysis techniques allowed us to avoid some restrictive assumptions standardly used in the investigations of the solutions to neural field equations involving Heaviside-type activation functions.

Peer review status:ACCEPTED

31 Jan 2020Submitted to Mathematical Methods in the Applied Sciences
01 Feb 2020Submission Checks Completed
01 Feb 2020Assigned to Editor
08 Feb 2020Reviewer(s) Assigned
09 Jun 2020Review(s) Completed, Editorial Evaluation Pending
09 Jun 2020Editorial Decision: Revise Minor
12 Jun 20201st Revision Received
13 Jun 2020Submission Checks Completed
13 Jun 2020Assigned to Editor
13 Jun 2020Editorial Decision: Accept