Population dynamics under the Laplace assumption ☆ André C. Marreiros ⁎, Stefan J. Kiebel, Jean Daunizeau, Lee M. Harrison, Karl J. Friston The Wellcome Trust Centre for Neuroimaging, Institute of Neurology, University College London, 12 Queen Square, London, WC1N 3BG, UK abstract article info Article history: Received 4 August 2008 Revised 30 September 2008 Accepted 10 October 2008 Available online 25 October 2008 Keywords: Neural-mass models Nonlinear Modelling Laplace assumption Mean-field Neuronal In this paper, we describe a generic approach to modelling dynamics in neuronal populations. This approach models a full density on the states of neuronal populations but finesses this high-dimensional problem by re- formulating density dynamics in terms of ordinary differential equations on the sufficient statistics of the densities considered (c.f., the method of moments). The particular form for the population density we adopt is a Gaussian density (c.f., the Laplace assumption). This means population dynamics are described by equations governing the evolution of the population's mean and covariance. We derive these equations from the Fokker-Planck formalism and illustrate their application to a conductance-based model of neuronal exchanges. One interesting aspect of this formulation is that we can uncouple the mean and covariance to furnish a neural-mass model, which rests only on the populations mean. This enables us to compare equivalent mean-field and neural-mass models of the same populations and evaluate, quantitatively, the contribution of population variance to the expected dynamics. The mean-field model presented here will form the basis of a dynamic causal model of observed electromagnetic signals in future work. © 2008 Elsevier Inc. All rights reserved. Introduction Mean-field models of neuronal dynamics have a long history, spanning a half-century (e.g., Beurle 1956). Models are essential for neuroscience, in the sense that most interesting questions about the brain pertain to neuronal mechanisms and processes that are not directly observable (Tass 2003; Breakspear et al., 2006). This means that questions about neuronal function are generally addressed by inference on models or their parameters; where the model links hidden neuronal processes to our observations and questions (Valdes et al., 1999). Broadly speaking, models are used to generate data to study emergent behaviours. Alternatively, they can be used as forward or observation models (e.g., dynamic causal models), which are inverted given empirical data (David et al., 2006, Kiebel et al., 2006). This inversion allows one to select the best model (i.e., hypothesis), given some data and make probabilistic statements about the parameters of that model (e.g., Penny et al., 2004). In particular, mean-field models are appropriate for data that reflect the behaviour of neuronal populations, such as the electro- encephalogram (EEG), magnetoencephalogram (MEG) and functional magnetic resonance imaging (fMRI) data. The most prevalent models of neuronal populations or ensembles are based upon the so-called mean-field approximation. This approximation replaces the time- averaged discharge rate of individual neurons with a common time- dependent population activity (ensemble average; Knight 2000; Haskell et al., 2001). The mean-field approximation is used extensively in statistical physics for otherwise computationally or analytically intractable problems. An exemplary approach, owing to Boltzmann and Maxwell, is the approximation of the motion of molecules in a gas by mean-field terms such as temperature and pressure. Similarly, evoked response potentials (ERPs) represent the average response over millions of neurons, where the mean-field approximation describes the time-dependent distribution of the average population response. This is possible because the dynamics of the mean of the density are much less stochastic than the response of a single neuron. This makes it feasible to develop algorithms that use Bayesian inference to infer neuronal parameters given measured responses, using mean-field models (e.g., Harrison et al., 2005). Usually, neural-mass models are used to model the evolution of the mean response or the response at steady state. Mean-field approxima- tions go further and model the full distribution of the population response. However, mean-field models can be computationally expensive, because one has to consider the density at all points in neuronal state-space as opposed to a single quantity (e.g., the mean). In this paper, we present an approach that simplifies the mean-field model by using the Laplace approximation: Under the Laplace approximation, the population or ensemble density assumes a Gaussian form, whose sufficient statistics comprise the conditional mean and covariance. In contrast to neural-mass models, this allows one to model interactions between the first two moments (i.e., mean and variance) of neuronal states. In a subsequent paper, we will use the Laplace and neural mass approximations presented here as generative NeuroImage 44 (2009) 701–714 ☆ Software Note. Matlab demonstration and modelling routines referred to in this paper are available as academic freeware as part of the SPM software from http://www. fil.ion.ucl.ac.uk/spm (neural models toolbox). ⁎ Corresponding author. Fax: +44 207 813 1445. E-mail address: a.marreiros@fil.ion.ucl.ac.uk (A.C. Marreiros). 1053-8119/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.neuroimage.2008.10.008 Contents lists available at ScienceDirect NeuroImage journal homepage: www.elsevier.com/locate/ynimg