Brain-wave equation incorporating axodendritic connectivity

Ross, James C.; Margetts, Michelle; Bojak, Ingo; Nicks, Rachel; Avitabile, Daniele; Coombes, Stephen

doi:10.1103/physreve.101.022411

Cited by 2 publications

(4 citation statements)

References 35 publications

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“…1 < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 y Y S + 4 q z b 7 e M A 4 is posed on R 2 , then the model supports solutions for which V (x, 0, t) is a travelling front V (x, 0, t) = ϕ(x − v * t), with ϕ(±∞) = V ± , whose speed v * satisfies the implicit equation [17] (3.3)…”

Section: B T O O a I E X E M D 7 Q A B T W A W H G D 4 H T D H O I / ...mentioning

confidence: 87%

“…It is also important to address the role of axonal delays on the generation of large scale brain rhythms. A recent paper [17] has explored how this might be done in a purely PDE setting, generalising the Nunez brain-wave equation to include dendrites. A natural extension of the work in this paper is to consider a more general numerical treatment of a model with both dendritic processing and space-dependent axonal delays in an integro-differential framework.…”

Section: Discussionmentioning

confidence: 99%

“…where β, θ, κ are positive constants. If S(V ) = H(V − θ), H being the Heaviside function, δ ε is replaced by the Dirac delta distribution, and the evolution equation is posed on R 2 , then the model supports solutions for which V (x, 0, t) is a travelling front V (x, 0, t) = ϕ(x−v * t), with ϕ(±∞) = V ± , whose speed v * satisfies the implicit equation [17] κ…”

mentioning

confidence: 92%

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Numerical investigation of a neural field model including dendritic processing

Avitabile¹,

Coombes²,

Lima³

2020

Journal of Computational Dynamics

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We consider a simple neural field model in which the state variable is dendritic voltage, and in which somas form a continuous one-dimensional layer. This neural field model with dendritic processing is formulated as an integro-differential equation. We introduce a computational method for approximating solutions to this nonlocal model, and use it to perform numerical simulations for neuro-biologically realistic choices of anatomical connectivity and nonlinear firing rate function. For the time discretisation we adopt an Implicit-Explicit (IMEX) scheme; the space discretisation is based on a finitedifference scheme to approximate the diffusion term and uses the trapezoidal rule to approximate integrals describing the nonlocal interactions in the model. We prove that the scheme is of first-order in time and second order in space, and can be efficiently implemented if the factorisation of a small, banded matrix is precomputed. By way of validation we compare the outputs of a numerical realisation to theoretical predictions for the onset of a Turing pattern, and to the speed and shape of a travelling front for a specific choice of Heaviside firing rate. We find that theory and numerical simulations are in excellent agreement.

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Section: B T O O a I E X E M D 7 Q A B T W A W H G D 4 H T D H O I / ...mentioning

confidence: 87%

Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 92%

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Numerical investigation of a neural field model including dendritic processing

Avitabile¹,

Coombes²,

Lima³

2020

Journal of Computational Dynamics

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“…This alternative focus on the synapses rather than the cells was also seen in the models of Freeman [69], Lopes da Silva et al [129], van Rotterdam et al [209], and became a dominant concept in the development of modern neural field modelling. Nonetheless, the wave equation derived by Nunez was a significant theoretical development in neural field modelling, and continues to be a source of discussion throughout the literature (Ross et al [179]). We will see in practice throughout the next Section how essential this synaptic viewpoint is to the construction of modern neural field models.…”

Section: Synaptic Population Dynamicsmentioning

confidence: 99%

Neural Field Models: A mathematical overview and unifying framework

Cook¹,

Peterson²,

Woldman³

et al. 2022

Mathematical Neuroscience and Applications

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Mathematical modelling of the macroscopic electrical activity of the brain is highly non-trivial and requires a detailed understanding of not only the associated mathematical techniques, but also the underlying physiology and anatomy. Neural field theory is a population-level approach to modelling the non-linear dynamics of large populations of neurons, while maintaining a degree of mathematical tractability. This class of models provides a solid theoretical perspective on fundamental processes of neural tissue such as state transitions between different brain activities as observed during epilepsy or sleep. Various anatomical, physiological, and mathematical assumptions are essential for deriving a minimal set of equations that strike a balance between biophysical realism and mathematical tractability. However, these assumptions are not always made explicit throughout the literature. Even though neural field models (NFMs) first appeared in the literature in the early 1970's, the relationships between them have not been systematically addressed. This may partially be explained by the fact that the inter-dependencies between these models are often implicit and non-trivial. Herein we provide a review of key stages of the history and development of neural field theory and contemporary uses of this branch of mathematical neuroscience. First, the principles of the theory are summarised throughout a discussion of the pioneering models by Wilson and Cowan, Amari and Nunez. Upon thorough review of these models, we then present a unified mathematical framework in which all neural field models can be derived by applying different assumptions. We then use this framework to i) derive contemporary models by Robinson, Jansen and Rit, Wendling, Liley, and Steyn-Ross, and ii) make explicit the many significant inherited assumptions that exist in the current literature.

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Brain-wave equation incorporating axodendritic connectivity

Cited by 2 publications

References 35 publications

Numerical investigation of a neural field model including dendritic processing

Numerical investigation of a neural field model including dendritic processing

Neural Field Models: A mathematical overview and unifying framework

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