Standing and travelling waves in a spherical brain model: The Nunez model revisited

Visser, Sid; Nicks, Rachel; Faugeras, Olivier; Coombes, Stephen

doi:10.1016/j.physd.2017.02.017

Cited by 43 publications

(44 citation statements)

References 53 publications

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“…The double-digits numerical subscripts in Eq (27) refer to the row-column element of the respective matrix. Eq (27) can be used to compute the separate harmonic and temporal power spectra of the model, as well as the correlation and functional connectivity matrices (Methods Eq (77)).…”

Section: Wilson-cowan Model Chaossmentioning

confidence: 99%

“…Eq (27) and (28) represent a general result that does not only apply to the Wilson-Cowan model. In fact, these equations describe the harmonic-temporal and harmonic power spectra of stochastic equilibrium fluctuations for the first population of any graph neural field model with two interacting populations and a first-order temporal differential operator.…”

Section: Wilson-cowan Model Chaossmentioning

confidence: 99%

“…and, as a consequence, the integrals in Eq (57) and (59) reduce to convolutions so that Fourier methods can be used in the models' analysis (however, see [27] for a detailed theoretical study of a neural field model on the sphere). Note that in Euclidean space, the kernel K is symmetric, i.e.…”

Section: Continuous Neural Fieldsmentioning

confidence: 99%

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Graph neural fields: a framework for spatiotemporal dynamical models on the human connectome

Aqil¹,

Atasoy

Kringelbach

et al. 2020

Preprint

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Tools from the field of graph signal processing, in particular the graph Laplacian operator, have recently been successfully applied to the investigation of structure-function relationships in the human brain. The eigenvectors of the human connectome graph Laplacian, dubbed “connectome harmonics”, have been shown to relate to the functionally relevant resting-state networks. Whole-brain modelling of brain activity combines structural connectivity with local dynamical models to provide insight into the large-scale functional organization of the human brain. In this study, we employ the graph Laplacian and its properties to define and implement a large class of neural activity models directly on the human connectome. These models, consisting of systems of stochastic integrodifferential equations on graphs, are dubbed graph neural fields, in analogy with the well-established continuous neural fields. We obtain analytic predictions for harmonic and temporal power spectra, as well as functional connectivity and coherence matrices, of graph neural fields, with a technique dubbed CHAOSS (shorthand for Connectome-Harmonic Analysis Of Spatiotemporal Spectra). Combining graph neural fields with appropriate observation models allows for estimating model parameters from experimental data as obtained from electroencephalography (EEG), magnetoencephalography (MEG), or functional magnetic resonance imaging (fMRI); as an example application, we study a stochastic Wilson-Cowan graph neural field model on a high-resolution connectome, and show that the model equilibrium fluctuations can reproduce the empirically observed harmonic power spectrum of BOLD fMRI data. Graph neural fields natively allow the inclusion of important features of cortical anatomy and fast computations of observable quantities for comparison with multimodal empirical data. They thus appear particularly suitable for modelling whole-brain activity at mesoscopic scales, and opening new potential avenues for connectome-graph-based investigations of structure-function relationships.Author summaryThe human brain can be seen as an interconnected network of many thousands neuronal “populations”; in turn, each population contains thousands of neurons, and each is connected both to its neighbors on the cortex, and crucially also to distant populations thanks to long-range white matter fibers. This extremely complex network, unique to each of us, is known as the “human connectome graph”. In this work, we develop a novel approach to investigate how the neural activity that is necessary for our life and experience of the world arises from an individual human connectome graph. For the first time, we implement a mathematical model of neuronal activity directly on a high-resolution connectome graph, and show that it can reproduce the spatial patterns of activity observed in the real brain with magnetic resonance imaging. This new kind of model, made of equations implemented directly on connectome graphs, could help us better understand how brain function is shaped by computational principles and anatomy, but also how it is affected by pathology and lesions.

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Section: Wilson-cowan Model Chaossmentioning

confidence: 99%

Section: Wilson-cowan Model Chaossmentioning

confidence: 99%

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Graph neural fields: a framework for spatiotemporal dynamical models on the human connectome

Aqil¹,

Atasoy

Kringelbach

et al. 2020

Preprint

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show abstract

“…In particular, it is not clear that our results will carry over to the full system (6). In principle we could linearise (6) about a steady state of (14) to determine its stability, but the resulting calculations seem significantly more difficult than those in [64,42,41] (where only one spatial dimension is considered) and [61] (in which only the stability of a spatially uniform state is considered) and we do not pursue this further. However, numerical solutions on the full sphere (using the scheme presented below) indicate that the double spot solution in Fig.…”

Section: Numerical Continuationmentioning

confidence: 99%

Chimeras in Two-Dimensional Domains: Heterogeneity and the Continuum Limit

Laing¹

2017

SIAM J. Appl. Dyn. Syst.

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We consider three different two-dimensional networks of nonlocally coupled heterogeneous phase oscillators. These networks were previously studied with identical oscillators and a number of spatiotemporal patterns found, mostly as a result of direct numerical simulation. Here we take the continuum limit of an infinite number of oscillators and use the Ott/Antonsen ansatz to derive continuum level evolution equations for order parameter-like quantities. Most of the patterns previously found in these networks correspond to relative fixed points of these evolution equations, and we show here the results of extensive numerical investigations of these fixed points: their existence and stability, and the bifurcations involved in their loss of stability as parameters are varied. Our results answer a number of questions posed by previous authors who studied these networks, and provide a better understanding of these networks' dynamics.

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“…For example in the Nunez model [10] for the generation of the α-rhythm neuronal cell types are arranged in sheets, with no representation of dendritic processing. A recent study of standing and traveling waves in the Nunez model posed on a sphere can be found in [11]. Interestingly the Nunez model has a representation of space-dependent delays arising from the finite speed of action potential propagation along axons, and for certain patterns of decaying strength of non-local connectivity it can be formulated as a damped inhomogeneous wave equation [12].…”

Section: Introductionmentioning

confidence: 99%

Brain-wave equation incorporating axodendritic connectivity

et al. 2020

Self Cite

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We introduce an integral model of a two-dimensional neural field that includes a third dimension representing space along a dendritic tree that can incorporate realistic patterns of axo-dendritic connectivity. For natural choices of this connectivity we show how to construct an equivalent brainwave partial differential equation that allows for efficient numerical simulation of the model. This is used to highlight the effects that passive dendritic properties can have on the speed and shape of large scale traveling cortical waves.

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Standing and travelling waves in a spherical brain model: The Nunez model revisited

Cited by 43 publications

References 53 publications

Graph neural fields: a framework for spatiotemporal dynamical models on the human connectome

Graph neural fields: a framework for spatiotemporal dynamical models on the human connectome

Chimeras in Two-Dimensional Domains: Heterogeneity and the Continuum Limit

Brain-wave equation incorporating axodendritic connectivity

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