UCL Discovery
UCL home » Library Services » Electronic resources » UCL Discovery

Rendering neuronal state equations compatible with the principle of stationary action

Fagerholm, ED; Foulkes, WMC; Friston, KJ; Moran, RJ; Leech, R; (2021) Rendering neuronal state equations compatible with the principle of stationary action. The Journal of Mathematical Neuroscience , 11 , Article 10. 10.1186/s13408-021-00108-0. Green open access

[thumbnail of Friston_Rendering neuronal state equations compatible with the principle of stationary action_VoR.pdf]
Preview
Text
Friston_Rendering neuronal state equations compatible with the principle of stationary action_VoR.pdf - Published Version

Download (1MB) | Preview

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, and 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.

Type: Article
Title: Rendering neuronal state equations compatible with the principle of stationary action
Open access status: An open access version is available from UCL Discovery
DOI: 10.1186/s13408-021-00108-0
Publisher version: https://doi.org/10.1186/s13408-021-00108-0
Language: English
Additional information: 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: Stationary action, Lagrangian, Computational neuroscience, Neural state equations
UCL classification: UCL
UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences
UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences > Faculty of Brain Sciences
UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences > Faculty of Brain Sciences > UCL Queen Square Institute of Neurology
UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences > Faculty of Brain Sciences > UCL Queen Square Institute of Neurology > Imaging Neuroscience
URI: https://discovery.ucl.ac.uk/id/eprint/10133741
Downloads since deposit
28Downloads
Download activity - last month
Download activity - last 12 months
Downloads by country - last 12 months

Archive Staff Only

View Item View Item