Event Abstract Back to Event Bifurcation analysis of neural mass equations The neural mass (NM) equations, also called the Wilson and Cowan or Hammerstein equations, are important because they provide models of the activity of several neural populations in a macroscopic patch of the cortex. They are rate models and feature a spatio-temporal connectivity matrix function relating different populations at different places in a continuum and a nonlinearity to convert average membrane potentials into firing rates. Previous theoretical studies have described several time-varying solutions consistent with optical imaging experiments. The steady states solutions are also interesting because they may account for the memory holding tasks which have been demonstrated by experimentalists in primates. We study the dependency of the NM solutions with respect to the stiffness of the nonlinearity. This choice is motivated by two reasons. First, it shows many differences with the well-studied, less biologically plausible, infinite stiffness case. Second, the study is generic and the dependency of the solutions with respect to any other parameter can be studied with similar tools. We assume that the cortex patch is finite although its dimension may be arbitrary. Beside of its biological relevance this is crucial for our analysis because it always guarantees the existence of a solution to the NM equations. The bifurcation analysis of the NM equations uses mainly two tools: Turing patterns and reduction to Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs). The first method is used in many papers (Attay-Hutt 05, Blomquist 07, Venkov-Coombes 07) and leads to the description of a lot of behaviors: traveling waves, breathers, persistent states. The second method is to reduce the NM equations to find a homoclinic orbit to ODEs/PDEs (Laing 02, 03) allowing the use of finite-dimensional tools or PDE methods for the bifurcation analysis. We use infinite dimensional methods recently developed for fluid dynamics (Iooss 08 to appear, Kielhoffer 04) which encompass the previous methods. Coupled to a numerical continuation method, they allow to compute numerically the steady states independently of their stability and provide a certified description of the local dynamics. This is unique to our method. We consider a very general type of connectivity matrix function which reduces exactly the dynamics to a system of ODEs. One thrust of the paper is that we use infinite dimensional tools to make general statements on the behaviour of solutions of the NM equations (boundedness, number of solutions, types of bifurcations) and use finite dimensional tools to do the actual computation, in a certified manner. This is achieved through a new method of approximation of the connectivity matrix functions that uses a combination of the Picherle-Goursat kernels and the Gegenbauer polynomials. Because our new method allows to exhaustively explore the set of solutions of the NM equations when varying the parameters, and to characterize their behaviours qualitatively as well as quantitatively, it can be very useful to relate a "cortical behaviour" observed, e.g., in optical imaging, to ranges of parameters values in the NM equations which are known, through the bifurcation study, to feature similar behaviours. Conference: Computational and systems neuroscience 2009, Salt Lake City, UT, United States, 26 Feb - 3 Mar, 2009. Presentation Type: Poster Presentation Topic: Poster Presentations Citation: (2009). Bifurcation analysis of neural mass equations. Front. Syst. Neurosci. Conference Abstract: Computational and systems neuroscience 2009. doi: 10.3389/conf.neuro.06.2009.03.064 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 30 Jan 2009; Published Online: 30 Jan 2009. Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Google Google Scholar PubMed Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.