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Parkinson's disease is a common and debilitating condition, caused by aberrant activity in a complex basal ganglia-thalamocortical circuit. Therapeutic advances rely on characterising interactions in this circuit. However, recording electrophysiological responses over the entire circuit is impractical. Dynamic causal modelling offers large-scale models of predictive value based on a limited or partial sampling of complex networks. Using dynamic causal modelling, we determined the network changes underlying the pathological excess of beta oscillations that characterise the Parkinsonian state. We modelled data from five patients undergoing surgery for deep brain stimulation of more than one target. We found that connections to and from the subthalamic nucleus were strengthened and promoted beta synchrony, in the untreated compared to the treated Parkinsonian state. Dynamic causal modelling was able to replicate the effects of lesioning this nucleus and may provide a new means of directing the search for therapeutic targets., (Copyright © 2012 Elsevier Inc. All rights reserved.)

In this paper, we compare mean-field and neural-mass models of electrophysiological responses using Bayesian model comparison. In previous work, we presented a mean-field model of neuronal dynamics as observed with magnetoencephalography and electroencephalography. Unlike neural-mass models, which consider only the mean activity of neuronal populations, mean-field models track the distribution (e.g., mean and dispersion) of population activity. This can be important if the mean affects the dispersion or vice versa. Here, we introduce a dynamical causal model based on mean-field (i.e., population density) models of neuronal activity, and use it to assess the evidence for a coupling between the mean and dispersion of hidden neuronal states using observed electromagnetic responses. We used Bayesian model comparison to compare homologous mean-field and neural-mass models, asking whether empirical responses support a role for population variance in shaping neuronal dynamics. We used the mismatch negativity (MMN) and somatosensory evoked potentials (SEP) as representative neuronal responses in physiological and non-physiological paradigms respectively. Our main conclusion was that although neural-mass models may be sufficient for cognitive paradigms, there is clear evidence for an effect of dispersion at the high levels of depolarization evoked in SEP paradigms. This suggests that (i) the dispersion of neuronal states within populations generating evoked brain signals can be manifest in observed brain signals and that (ii) the evidence for their effects can be accessed with dynamic causal model comparison., (Copyright (c) 2010 Elsevier Inc. All rights reserved.)

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.

This paper demonstrates how the sigmoid activation function of neural-mass models can be understood in terms of the variance or dispersion of neuronal states. We use this relationship to estimate the probability density on hidden neuronal states, using non-invasive electrophysiological (EEG) measures and dynamic casual modelling. The importance of implicit variance in neuronal states for neural-mass models of cortical dynamics is illustrated using both synthetic data and real EEG measurements of sensory evoked responses.