We have described a variant of dynamic causal modelling for event-related potentials or fields as measured with EEG and MEG. We motivated this DCM as an approximation to a continuous neural-field model, using a mixture of overlapping patches, with compact spatial support, on the cortical surface. Time-varying activity in this mixture, caused by activity in other sources and experimental inputs, is propagated through appropriate lead-field or gain-matrices to generate observed channel data. In comparison to ECD variants of DCM, this distributed DCM has three advantages; it has greater face validity, the degrees of freedom of the spatial model can be specified (and therefore optimised using model selection) and the model is linear in the spatial parameters (which finesses computational load). Both our simulations and the application to an EEG auditory mismatch negativity dataset demonstrated the superiority of distributed DCMs, when compared to their ECD homologues.

The greater face validity of spatially distributed DCMs is similar to that of imaging source reconstruction solutions, when compared to ECD-like solutions: the spatial extent of each regional source must be modelled properly when inverting such models (see below). Furthermore, the neural-mass models we use (

Jansen and Rit 1995) were designed originally to model mesoscopic electrocortical activity, at a spatial scale finer than that of EEG/MEG. Using simple approximations of neural-field models, we have proposed a simple modification of neural-mass models that render them able to emulate macroscopic spatiotemporal dynamics. Specifically, these modifications allow us to account for the spatial deployment of sources, which appears to be necessary to explain EEG/MEG data (

see MMN results section).

Although not pursued here, the number of basis functions or different sizes of cortical regions could be optimised. One would repeat the inversion using different basis functions and evaluate the model evidences (as for the analysis of cortical sources in ). This would allow one to optimise the degrees of freedom of the spatial model, in relation to the spatial information supported by the data; similarly for the size of the cortical patches used to model source-specific activity.

Note that there is a formal link between the spatially distributed DCM proposed in this work and EEG/MEG source reconstruction techniques (see e.g.

Daunizeau et al., 2006;
Friston et al., 2007b). The key difference between these two approaches rests on the formal constraints used by DCM. These constrain the temporal expression of source activity to conform to a biologically plausible time-course (

Scherg and Von Cramon 1985). The interpretation of a DCM analysis is not usually concerned with the spatial profile of source activity but focuses on the coupling parameters and how they change with experimental manipulations. However, it is interesting to regard the DCM inversion as a biophysically and neurobiologically informed imaging source reconstruction (see

Kiebel et al., 2006). In other words, one can regard the Bayesian inversion of spatially distributed DCM as a generalisation of classical forward model inversion used to reconstruct source activity from observed EEG or MEG data. The only difference between classical inversion and DCM is that the source activity has to conform to a biophysically plausible model. Generally, this model entails interactions among sources so that activity in one source is caused by activity in others. Classical forward models focus exclusively on the spatial observer function of the hidden states and ignore formal constraints on the temporal expression of source activity. The resulting spatial models are either ECD-based models or distributed source models of the sort used in image reconstruction (

Baillet and Garnero 1997; Pascual-Marqui 2002; Phillips et al., 2005). Exactly the same distinction between ECD and distributed reconstructions can be applied in the context of DCM. In this note, we have described a distributed spatial model that complements existing ECD dynamic causal models.

In the future, it is possible that DCMs will be based on models that are closer to full neural-field models. These models might be more appropriate for EEG and MEG data because they account for continuous lateral interactions within each cortical region. Neural-field models can generate time-dependent dynamics that are expressed as bumps or propagating waves over the cortical surface. In this work, we truncated our space-time decomposition to the fundamental mode (a zero-order approximation). As a consequence, the neural-fields behave as interacting standing-waves; i.e. regionally specific invariant patterns of activity oscillating in response to mutual influence. This space-time separation is a simplified variant of the sort of the spatiotemporal behaviours that could be obtained using a more realistic wave-equation (c.f.

Eq. 6). Our zero-order approximation could be relaxed to increase the complexity of the neural-field model. This can be done by including more modes (see

Eqs. 7 and

8 and

Appendix 2). This would allow one to replace a full PDE to a set of coupled ODEs.

Two additional comments should be made: first, the derivation of the 2D neural field PDE relies on the assumption that lateral (isotropic) interactions are deployed over a small spatial scale (see

Appendix 1). As a consequence, only long spatial wavelengths (relative to the spatial decay of lateral interactions) can be expressed in the 2D cortical neural field. This means that mesoscale phenomena like patchy feature maps (e.g. orientation preference or ocular dominance) in V1 might not be captured accurately (see

Bressloff 2003 for a recent discussion of isotropic connectivity and

Coombes et al., 2007 for an extension of the long-wavelength approximation to

*patchy propagators*).

Second, we motivated our standing wave (fundamental mode) approximation to the neural field by noting that at high propagation velocity, higher harmonics will dissipate quickly. This is consistent with more realistic models (including axonal propagation), which also suggest that higher harmonics are damped more heavily (

Nunez 1995). However, our standing wave approximation to experimentally manipulated (excited) neural fields is different in nature from the emergence of

*global* standing-waves as proposed in

Nunez and Srinivasan (2006). The latter global waves are thought to underlie global coherence of cortical activity in the absence of stimulation (e.g. eyes-closed resting alpha-band activity). Global standing-waves can be thought of as a resonance phenomenon, whose wavelength is related to the size of the brain. Nunez points out that mental tasks “enhance cell assembly activity [i.e. functional segregation], thereby reducing global field behaviour”. This is in contradistinction to the present work, which postulates that local standing-waves emerge from the interaction of segregated neural ensembles. According to this view, segregation is

*necessary* for the standing-waves to emerge, in the sense that it prevents activity spreading over the cortical mantle. In turn, this makes extrinsic functional integration (i.e. between region top-down and bottom-up effects, as opposed to within region lateral interactions) the principal mechanism responsible for sustained large-scale cortical activity.