Collocation techniques for solving neural field models on complex cortical geometries

Martin, R., 2018. Collocation techniques for solving neural field models on complex cortical geometries. PhD, Nottingham Trent University.

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Neural models are deployed in order to gain an insight into the function and behaviour of the brain at a range of different scales, ranging from the micro-scale modelling of individual neurons, to the meso- and macro- scale modelling of large populations of neurons. Neural field models provide a continuous approach to modelling at this larger scale, and typically take the form of a nonlinear partial integrodi-differential equation. Such equations are capable of supporting a variety of patterns and have been linked to neurological phenomena, such as, for example, bumps in models of working memory, and thus play an important role in the interpretation and understanding of the complex, dynamic patterns of brain activity observed via modern brain imaging techniques such as EEG, MEG and fMRI.

In this thesis, we present an approach for solving neural field equations on surfaces more akin to the cortical geometries typically obtained from neuroimaging data. Our approach involves solving the integral of the partial integro-differential equation directly using collocation techniques, alongside efficient numerical procedures for determining geodesic distances between neural units. To illustrate our methods we study localised activity patterns in two different neural field models; namely, the Amari equation, for which we consider stationary bump solutions, and an extended version of the Amari equation that admits both stationary and travelling bump solutions. We solve both equations on a variety of domains, including a flat periodic domain, the curved surface of the torus and the folded surface of the rat cortex. Importantly, we find that collocation techniques are able to replicate solutions obtained using more standard Fourier based methods on a flat, periodic domain, independent of the underlying mesh. This result is particularly significant given the highly irregular nature of the type of meshes derived from modern neuroimaging data.

One of the key contributions of this thesis is our ability to solve neural models on curved geometries for which no analytic formula for the geodesic distance exists. Indeed, by deploying efficient numerical schemes to compute geodesics, our approach is not only capable of modelling macroscopic pattern formation on realistic cortical geometries, such as the rat brain considered herein, but can also be extended to include cortical architectures of more physiological relevance. Importantly, such an approach provides a means by which to investigate the influence of cortical geometry upon the nucleation and propagation of spatially localised neural activity and beyond, and thus promises to provide model-based insights into disorders like epilepsy, or spreading depression, as well as healthy cognitive processes like working memory
or attention.

Item Type: Thesis
Creators: Martin, R.
Date: August 2018
Rights: This work is the intellectual property of the author. You may copy up to 5% of this work for private study, or personal, non-commercial research. Any re-use of the information contained within this document should be fully referenced, quoting the author, title, university, degree level and pagination. Queries or requests for any other use, or if a more substantial copy is required, should be directed to the owner of the Intellectual Property rights.
Divisions: Schools > School of Science and Technology
Record created by: Linda Sullivan
Date Added: 22 May 2019 08:22
Last Modified: 22 May 2019 08:34

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