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Traveling wave solutions to a neural field model with oscillatory synaptic coupling types
- Source :
- Mathematical biosciences and engineering : MBE. 16(2)
- Publication Year :
- 2019
-
Abstract
- In this paper, we investigate the existence, uniqueness, and spectral stability of traveling waves arising from a single threshold neural field model with one spatial dimension, a Heaviside firing rate function, axonal propagation delay, and biologically motivated oscillatory coupling types. Neuronal tracing studies show that long-ranged excitatory connections form stripe-like patterns throughout the mammalian cortex; thus, we aim to generalize the notions of pure excitation, lateral inhibition, and lateral excitation by allowing coupling types to spatially oscillate between excitation and inhibition. In turn, we hope to analyze traveling fronts and pulses with novel shapes. With fronts as our main focus, we exploit Heaviside firing rate functions in order to establish existence and utilize speed index functions with at most one critical point as a tool for showing uniqueness of wave speed. We are able to construct Evans functions, the so-called stability index functions, in order to provide positive spectral stability results. Finally, we show that by incorporating slow linear feedback, we can compute fast pulses numerically with phase space dynamics that are similar to their corresponding singular homoclinical orbits; hence, our work answers open problems and provides insight into new ones.<br />Comment: 30 pages, 11 figures
- Subjects :
- Models, Neurological
Dynamical Systems (math.DS)
02 engineering and technology
Stability (probability)
Synaptic Transmission
Feedback
symbols.namesake
Lateral inhibition
Biological Clocks
Oscillometry
0502 economics and business
FOS: Mathematics
0202 electrical engineering, electronic engineering, information engineering
Animals
Humans
Computer Simulation
Uniqueness
Mathematics - Dynamical Systems
Electrodes
Coupling
Physics
Neurons
Quantitative Biology::Neurons and Cognition
Heaviside step function
Applied Mathematics
05 social sciences
Mathematical analysis
Brain
General Medicine
Computational Mathematics
Modeling and Simulation
Phase space
symbols
35B25, 92C20
020201 artificial intelligence & image processing
Nerve Net
General Agricultural and Biological Sciences
Rate function
050203 business & management
Excitation
Algorithms
Subjects
Details
- ISSN :
- 15510018
- Volume :
- 16
- Issue :
- 2
- Database :
- OpenAIRE
- Journal :
- Mathematical biosciences and engineering : MBE
- Accession number :
- edsair.doi.dedup.....ee0f3513bfc33d3943b28bce689faab0