Transient Turing patterns in a neural field model

Elvin, Amanda; Laing, Carlo R.; Roberts, Michael C.

doi:10.1103/physreve.79.011911

Cited by 7 publications

(5 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We have previously shown the existence of both stable and transient spatially-periodic patterns beyond a Turing instability in (1)-(3) in one and two spatial dimensions [22]. Both the non-trivial spatially uniform steady states, spatially-localised solutions and spatially-periodic solutions depend upon the parameter b.…”

Section: The Model and Its Propertiesmentioning

confidence: 99%

Exploiting the Hamiltonian structure of a neural field model

Elvin

Laing

McLachlan

et al. 2010

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

We study the unexpected disappearance of stable homoclinic orbits in regions of parameter space in a neural field model with one spatial dimension. The usual approach of using numerical continuation techniques and local bifurcation theory is insufficient to explain the qualitative change in the model's behaviour. The lack of robustness of the model to small perturbations in parameters is surprising and the phenomenon may be of broader significance than just our model. By exploiting the Hamiltonian structure of the timeindependent system, we develop a numerical technique with which we discover that a small, separate solution curve exists for a range of parameter values. As the firing rate function steepens, the small curve causes the main curve to break and stable homoclinic orbits are destroyed in a region of parameter space. Numerically, we use level set analysis to find that a codimension-one heteroclinic bifurcation occurs at the terminating ends of the solution curves.By replacing the firing rate function with a step function, we show analytically that the bifurcation is related to the value of the firing threshold. We also show the existence of heteroclinic orbits at the breakpoints using a travelling front analysis in the time-dependent system.

show abstract

Section: The Model and Its Propertiesmentioning

confidence: 99%

Exploiting the Hamiltonian structure of a neural field model

Elvin

Laing

McLachlan

et al. 2010

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

show abstract

“…instabilities of spatially-uniform states with respect to patterns with finite spatial wave-numbers. However, experience with neural field models suggests that non-monotonic coupling functions may well result in Turing bifurcations [46,45], providing another route to spatio-temporal pattern formation in spatially-extended Kuramoto networks.…”

Section: Discussionmentioning

confidence: 99%

Fronts and bumps in spatially extended Kuramoto networks

Laing

2011

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

We consider moving fronts and stationary "bumps" in networks of non-locally coupled phase oscillators. Fronts connect regions of high local synchrony with regions of complete asynchrony, while bumps consist of spatially-localised regions of partially-synchronous oscillators surrounded by complete asynchrony. Using the Ott-Antonsen ansatz we derive non-local differential equations which describe the network dynamics in the continuum limit. Front and bump solutions of these equations are studied by either "freezing" them in a travelling coordinate frame or analysing them as homoclinic or heteroclinic orbits. Numerical continuation is used to determine parameter regions in which such solutions exist and are stable.

show abstract

“…Turing patterns (Turing, 1952 ) have been reported in neural field models (Atay and Hutt, 2006 ; Elvin et al, 2009 ; Steyn-Ross et al, 2010 ). The Neural Field Simulator is able to compute and display noisy neural field activity evolving into Turing patterns.…”

Section: Applicationsmentioning

confidence: 98%

Neural field simulator: two-dimensional spatio-temporal dynamics involving finite transmission speed

Nichols

Hutt

2015

Front. Neuroinform.

View full text Add to dashboard Cite

Neural Field models (NFM) play an important role in the understanding of neural population dynamics on a mesoscopic spatial and temporal scale. Their numerical simulation is an essential element in the analysis of their spatio-temporal dynamics. The simulation tool described in this work considers scalar spatially homogeneous neural fields taking into account a finite axonal transmission speed and synaptic temporal derivatives of first and second order. A text-based interface offers complete control of field parameters and several approaches are used to accelerate simulations. A graphical output utilizes video hardware acceleration to display running output with reduced computational hindrance compared to simulators that are exclusively software-based. Diverse applications of the tool demonstrate breather oscillations, static and dynamic Turing patterns and activity spreading with finite propagation speed. The simulator is open source to allow tailoring of code and this is presented with an extension use case.

show abstract

Transient Turing patterns in a neural field model

Cited by 7 publications

References 29 publications

Exploiting the Hamiltonian structure of a neural field model

Exploiting the Hamiltonian structure of a neural field model

Fronts and bumps in spatially extended Kuramoto networks

Neural field simulator: two-dimensional spatio-temporal dynamics involving finite transmission speed

Contact Info

Product

Resources

About