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

Spek, Len; Polner, Mónika; Dijkstra, Koen; Gils, Stephan A. van

doi:10.1016/j.jde.2022.02.002

Cited by 4 publications

(3 citation statements)

References 19 publications

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“…Some recent and significant contributions to neural field modeling with transmission delays are presented in (Atay and Hutt 2004 , 2006 ; Hutt and Atay 2005 , 2006 ; Veltz and Faugeras 2011 ; Spek et al. 2022 ; van Gils et al. 2013 ).…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

“…The work in (Spek et al. 2022 ) investigated a neural field model that incorporates transmission delays and a connectivity kernel consisting of a linear combination of exponentials. The authors examined the dynamics and stability of this model within a two-dimensional spatial domain.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of neural fields with exponential temporal kernel

Shamsara,

Yamakou,

Atay

et al. 2024

Theory Biosci.

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“…(b) In the special case when ℓ = 0, i.e., τ j,ν ∈ [0, ∆t), only the three values u h i,ν (0), u h i,ν (1) and uh i,ν (1) are known, hence we can only use a quadratic polynomial approximation…”

Section: Discretisation Of the Nonlinear Termmentioning

confidence: 99%

Analysis of dynamics of neural fields and neural networks

Spek¹

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We consider abstract delay differential equations which take values in arbitrary Banach spaces. Here the right-hand side can be broken down into a bounded, non-linear, compact operator which depends on the delay and a linear differential operator which does not need to be bounded. We extend the results of Janssens [2019] on the sun-star calculus for these equations, specifically we describe the generator of the semigroup corresponding to the linearisation of the delay equation. One key result is that the space of continuous functions on a bounded interval taking values in some arbitrary Banach space Y is sun-reflexive with respect to the shift semigroup if and only if Y itself is reflexive. Finally, we develop a method to analytically compute the normal form coefficients of certain bifurcations for these abstract delay differential equations.

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