A Space-Time Finite Element Method for Neural Field Equations with Transmission Delays

Polner, Mónika; Vegt, J.J.W. van der; Gils, Stephan A. van

doi:10.1137/16m1085024

Cited by 8 publications

(6 citation statements)

References 13 publications

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“…(Polner et al. 2017 ; Arqub 2017 ; Faugeras and Inglis 2015 ; Rankin et al. 2014 ), with axonal conduction delay (Fang and Faye 2016 ; Breakspear 2017 ; Pinto and Ermentrout 2001 ).…”

Section: The Model and The Equilibrium Solutionmentioning

confidence: 99%

Dynamics of neural fields with exponential temporal kernel

Shamsara,

Yamakou,

Atay

et al. 2024

Theory Biosci.

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We consider the standard neural field equation with an exponential temporal kernel. We analyze the time-independent (static) and time-dependent (dynamic) bifurcations of the equilibrium solution and the emerging spatiotemporal wave patterns. We show that an exponential temporal kernel does not allow static bifurcations such as saddle-node, pitchfork, and in particular, static Turing bifurcations. However, the exponential temporal kernel possesses the important property that it takes into account the finite memory of past activities of neurons, which Green’s function does not. Through a dynamic bifurcation analysis, we give explicit bifurcation conditions. Hopf bifurcations lead to temporally non-constant, but spatially constant solutions, but Turing–Hopf bifurcations generate spatially and temporally non-constant solutions, in particular, traveling waves. Bifurcation parameters are the coefficient of the exponential temporal kernel, the transmission speed of neural signals, the time delay rate of synapses, and the ratio of excitatory to inhibitory synaptic weights.

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“…(Polner et al. 2017 ; Arqub 2017 ; Faugeras and Inglis 2015 ; Rankin et al. 2014 ), with axonal conduction delay (Fang and Faye 2016 ; Breakspear 2017 ; Pinto and Ermentrout 2001 ).…”

Section: The Model and The Equilibrium Solutionmentioning

confidence: 99%

Dynamics of neural fields with exponential temporal kernel

Shamsara,

Yamakou,

Atay

et al. 2024

Theory Biosci.

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show abstract

“…When the space is one-dimensional, Hopf bifurcations were studied in [16] and along with other types of bifurcations also in [5,17]. On two-dimensional domains, numerical experiments were conducted in [7,12]. In this section we study an example of Hopf bifurcation in the twodimensional case based on our analytical results.…”

Section: Characterisation Of the Spectrum And Resolvent Set Of The Ddementioning

confidence: 99%

“…[3] and references therein. Numerical methods developed for the efficient and accurate time simulation of neural fields on higher dimensional domains can be found in [7,9,12]. Moreover, numerical studies of the non-essential spectrum of abstract delay differential equations are also available, [19].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of delayed neural field models in two-dimensional spatial domains

Spek,

Polner,

Dijkstra

et al. 2020

Preprint

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Delayed neural field models can be viewed as a dynamical system in an appropriate functional analytic setting. On two dimensional rectangular space domains, and for a special class of connectivity and delay functions, we describe the spectral properties of the linearized equation. We transform the characteristic integral equation for the delay differential equation (DDE) into a linear partial differential equation (PDE) with boundary conditions. We demonstrate that finding eigenvalues and eigenvectors of the DDE is equivalent with obtaining nontrivial solutions of this boundary value problem (BVP). When the connectivity kernel consists of a single exponential, we construct a basis of the solutions of this BVP that forms a complete set in L 2 . This gives a complete characterization of the spectrum and is used to construct a solution to the resolvent problem. As an application we give an example of a Hopf bifurcation and compute the first Lyapunov coefficient.

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“…We consider a neural field model represented by an infinite-dimensional dynamical system in the form of an integro-differential equation [21][22][23][24], with axonal conduction delay [25][26][27]. In this equation, the position of a neuron at time t is given by a spatial variable x, in the literature usually considered to be continuous in R or R 2 .…”

Section: The Model and The Equilibrium Solutionmentioning

confidence: 99%

Dynamics of neural fields with exponential temporal kernel

Shamsara¹,

Yamakou²,

Atay³

et al. 2019

Preprint

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In this paper, we consider the standard neural field equation with an exponential temporal kernel. We analyze the time-independent (static) and time-dependent (dynamic) bifurcations of the equilibrium solution and the emerging spatio-temporal wave patterns. We show that an exponential temporal kernel does not allow static bifurcations such as saddle-node, pitchfork, and in particular, static Turing bifurcations, in contrast to the Green's function used by Atay and Hutt (SIAM J. Appl. Math. 65: 644-666, 2004). However, the exponential temporal kernel possesses the important property that it takes into account finite memory of past activities of neurons, which the Green's function does not. Through a dynamic bifurcation analysis we give explicit Hopf (temporally nonconstant, but spatially constant solutions) and Turing-Hopf (spatially and temporally non-constant solutions) bifurcation conditions on the parameter space which consists of the internal input current, the time delay rate of synapses, the ratio of excitatory to inhibitory synaptic weights, the coefficient of the exponential temporal kernel, and the transmission speed of neural signals.

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A Space-Time Finite Element Method for Neural Field Equations with Transmission Delays

Cited by 8 publications

References 13 publications

Dynamics of neural fields with exponential temporal kernel

Dynamics of neural fields with exponential temporal kernel

Dynamics of delayed neural field models in two-dimensional spatial domains

Dynamics of neural fields with exponential temporal kernel

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