A one-population Amari model with periodic microstructure

Svanstedt, Nils; Wyller, John; Malyutina, Elena

doi:10.1088/0951-7715/27/6/1391

Cited by 9 publications

(38 citation statements)

References 30 publications

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“…Here we give an outline of the derivation of the homogenized version of (1) by means of two-scale convergence techniques. The detailed derivation of (4) will be presented in a complementary paper [30]. We write the original model in the form…”

Section: Consider the Modelmentioning

confidence: 99%

Waves in random neural media

Coombes¹,

Schmidt²,

Laing³

et al. 2012

Discrete &Amp; Continuous Dynamical Systems - A

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Translationally invariant integro-differential equations are a common choice of model in neuroscience for describing the coarse-grained dynamics of cortical tissue. Here we analyse the propagation of travelling wavefronts in models of neural media that incorporate some form of modulation or randomness such that translational invariance is broken. We begin with a study of neural architectures in which there is a periodic modulation of the neuronal connections. Recent techniques from two-scale convergence analysis are used to construct a homogenized model in the limit that the spatial modulation is rapid compared with the scale of the long range connections. For the special case that the neuronal firing rate is a Heaviside we calculate the speed of a travelling homogenized front exactly and find how the wave speed changes as a function of the amplitude of the modulation. For this special case we further show how to obtain more accurate results about wave speed and the conditions for propagation failure by using an interface dynamics approach that circumvents the requirement of fast modulation. Next we turn our attention to forms of disorder that arise via the variation of firing rate properties across the tissue. To model this we draw parameters of the firing rate function from a distribution with prescribed spatial correlations and analyse the corresponding fluctuations in the wave speed. Finally we consider generalisations of the model to incorporate adaptation and stochastic forcing and show how recent numerical techniques developed for stochastic partial differential equations can be used to determine the wave speed by minimising the L 2 norm of a travelling disordered activity profile against a fixed test function.2000 Mathematics Subject Classification. Primary: 45J05; Secondary: 92C20.

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Section: Consider the Modelmentioning

confidence: 99%

Waves in random neural media

Coombes¹,

Schmidt²,

Laing³

et al. 2012

Discrete &Amp; Continuous Dynamical Systems - A

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“…The multiscale approach of Nguetseng has successfully been applied to homogenization of partial differential equations (see for example Lukkassen et al [20] and the references therein). Apart from the works [21,22,23], it seems not to be very wellknown in the mathematical neuroscience community.…”

Section: Introductionmentioning

confidence: 99%

“…In Svanstedt et al [21,22] it is proved that the one -parameter family of one population Wilson-Cowan models with periodic microstructure…”

Section: Introductionmentioning

confidence: 99%

“…The connectivity kernel ω is periodic in the second variable y = x/ε with Y as a period cell. In Coombes et al [23] the model (3) was used in a study of traveling fronts in a medium with periodic structure, while in Svanstedt et al [22] existence and stability of y -independent single bump solutions of the homogenized Wilson -Cowan model (3) are studied when the firing rate function is given by means of the Heaviside function. In the latter paper the existence and uniqueness issue is studied by means of the pinning function technique using the heterogeneity parameter as a control parameter.…”

Section: Introductionmentioning

confidence: 99%

“…In the latter paper the existence and uniqueness issue is studied by means of the pinning function technique using the heterogeneity parameter as a control parameter. In Svanstedt et al [22] the stability problem is resolved by means of the spectral properties of a Hilbert -Schmidt integral operator in the microvariable y. The spectrum of this operator is computed by means of a Fourier decomposition method in a way analogous to single bump solutions of the homogeneous translational invariant Wilson -Cowan model in two spatial dimensions [24] and the spectral stability of vortex solutions to the Gross -Pitaevski equation in a two dimensional spatial configuration [25].…”

Section: Introductionmentioning

confidence: 99%

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Two bump solutions of a homogenized Wilson–Cowan model with periodic microstructure

Malyutina

Wyller

Ponosov

2014

Physica D: Nonlinear Phenomena

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We study existence and stability of 2-bump solutions of the one-population homogenized Wilson-Cowan model, where the heterogeneity is built in the connectivity functions by assuming periodic modulations in both the synaptic footprint and in the spatial scale. The existence analysis reveals that the generic picture consists of two bumps states for each admissible threshold value for the case when the solutions are independent of the local variable and the firing rate function is modeled as a Heaviside function. A framework for analyzing the stability of 2-bumps is formulated, based on spectral theory for Fredholm integral operators. The stability method deforms to the standard Evans function approach for the translationally invariant case in the limit of no heterogeneity, in a way analogous to the single bump case for the homogenized model. Numerical study of the stability problem reveals that both the broad and narrow bumps are unstable just as in the translationally invariant case when the connectivity function is modeled by means of a wizard hat function. For the damped oscillating connectivity kernel, we give a concrete example of a 2-bump solution which is stable for all admissible values of the heterogeneity parameter.

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Two-scale, non-local diffusion in homogenised heterogeneous media

2021

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We study how and to what extent the existence of non-local diffusion affects the transport of chemical species in a composite medium. For our purposes, we prescribe the mass flux to obey a two-scale, non-local constitutive law featuring derivatives of fractional order, and we employ the asymptotic homogenisation technique to obtain an overall description of the species’ evolution. As a result, the non-local effects at the micro-scale are ciphered in the effective diffusivity, while at the macro-scale the homogenised problem features an integro-differential equation of fractional type. In particular, we prove that in the limit case in which the non-local interactions are neglected, classical results of asymptotic homogenisation theory are re-obtained. Finally, we perform numerical simulations to show the impact of the fractional approach on the overall diffusion of species in a composite medium. To this end, we consider two simplified benchmark problems, and report some details of the numerical schemes based on finite element methods.

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A one-population Amari model with periodic microstructure

Cited by 9 publications

References 30 publications

Waves in random neural media

Waves in random neural media

Two bump solutions of a homogenized Wilson–Cowan model with periodic microstructure

Two-scale, non-local diffusion in homogenised heterogeneous media

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