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Rendering neuronal state equations compatible with the principle of stationary action

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Title: Rendering neuronal state equations compatible with the principle of stationary action
Authors: Fagerholm, ED
Foulkes, W
Friston, KJ
Moran, RJ
Leech, R
Item Type: Journal Article
Abstract: The principle of stationary action is a cornerstone of modern physics, providing a powerful framework for investigating dynamical systems found in classical mechanics through to quantum field theory. However, computational neuroscience, despite its heavy reliance on concepts in physics, is anomalous in this regard as its main equations of motion are not compatible with a Lagrangian formulation and hence with the principle of stationary action. Taking the Dynamic Causal Modelling (DCM) neuronal state equation as an instructive archetype of the first-order linear differential equations commonly found in computational neuroscience, we show that it is possible to make certain modifications to this equation to render it compatible with the principle of stationary action. Specifically, we show that a Lagrangian formulation of the DCM neuronal state equation is facilitated using a complex dependent variable, an oscillatory solution, anc a Hermitian intrinsic connectivity matrix. We first demonstrate proof of principle by using Bayesian model inversion to show that both the original and modified models can be correctly identified via in silico data generated directly from their respective equations of motion. We then provide motivation for adopting the modified models in neuroscience by using three different types of publicly available in vivo neuroimaging datasets, together with open source MATLAB code, to show that the modified (oscillatory) model provides a more parsimonious explanation for some of these empirical timeseries. It is our hope that this work will, in combination with existing techniques, allow people to explore the symmetries and associated conservation laws within neural systems – and to exploit the computational expediency facilitated by direct variational techniques.
Issue Date: 12-Aug-2021
Date of Acceptance: 23-Jul-2021
URI: http://hdl.handle.net/10044/1/90599
DOI: 10.1186/s13408-021-00108-0
ISSN: 2190-8567
Publisher: SpringerOpen
Start Page: 1
End Page: 15
Journal / Book Title: Journal of Mathematical Neuroscience
Volume: 11
Issue: 10
Copyright Statement: © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Keywords: 0102 Applied Mathematics
Publication Status: Published
Online Publication Date: 2021-08-12
Appears in Collections:Condensed Matter Theory
Faculty of Natural Sciences

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