We have described a dynamic causal model (DCM) of steady-state responses that are summarised in terms of cross-spectral densities. These spectral data-features are generated by a biologically plausible, neural-mass model of coupled electromagnetic sources. Under linearity and stationarity assumptions, inversion of the DCM provides conditional probabilities on both the models and the synaptic parameters of any particular model. The model employed here has previously been shown to produce oscillatory activity at all standard EEG frequency bands, in its linear approximation (Moran et al., 2007
). A nonlinear model analysis could uncover interesting dynamics in some of these bands and will be the subject of further research. This would call for a relaxation of the linearization assumption and present an interesting challenge for model inversion (c.f
., Valdes et al., 1999
Recently, a number of studies have established the utility neural mass models for interrogating EEG data. The motivations behind this approach are varied. In Riera et al., (2006)
neural masses are used to investigate local electrovascular coupling and their multi-modal time domain expression in EEG and fMRI data; while Valdes et al. (1999)
employ neural masses to examine the emergent dynamic properties of alpha-band activity. Closer to the work presented here, Robinson et al., (2004)
have developed a frequency domain description of EEG activity that highlights the importance of corticothalamic interactions, using neural field models. As in Robinson et al., (2004)
, the goal of DCM for steady-state responses is to make inferences about, regionally-specific neurotransmitter and neuromodulatory action that unfolds in a connected but distributed network. The DCM presented in this paper assumes a network of point sources (c.f., equivalent current dipoles) that may be usefully extended to cover neural field models of the sort considered by Robinson et al., (2004)
. DCM enables inference about synaptic physiology and changes induced by pharmacological or behavioural manipulations both within and between neural ensembles; furthermore, the methodology can be applied to the cross-spectral density of invasive or non-invasive electrophysiological recordings.
Usually, in Dynamic Causal Modelling, data prediction involves the integration of a dynamical system to produce a time-series. In the current application, the prediction is over frequencies; however, the form of the inversion remains exactly the same. This is because in DCM for deterministic systems (i.e., models with no system or state noise) the time-series prediction is treated as a finite-length static observation, which is replaced here with a prediction over frequencies. The only difference between DCM for time-series and DCM for cross-spectral density is that the data-features are represented by a three dimensional array, covering c
channels and b
frequency-bins. In conventional time-series analysis the data-features correspond to a two-dimensional array covering c
channels and b
time-bins. The spectral summary used for data inversion comprises the magnitude of cross-spectra, which is a sufficient data-feature, under quasi-stationarity assumptions. Information regarding instantaneous phase or phase-coupling among sources are not considered in this treatment. In some settings, phase-coupling has been used in linear and nonlinear settings to model information exchange across discrete brain sources (e.g., Brovelli et al., 2004, Rosenblum et al., 1996
). The DCM presented here represents a complement to this approach by offering a biophysically meaningful, mechanistic description of neuronal interactions. An alternative DCM approach for M/EEG analysis has been developed to describe (time-dependent) phenomenological coupling among frequencies at different brain sources that occur through both linear and nonlinear mechanisms (Chen et al., 2008
). However, neither DCM model the instantaneous phase. Other recent developments in M/EEG data analysis have tackled this issue: Approaches involving ICA (Anemüller et al., 2003
) have been used to describe the phases of induced responses on a trial by trial basis, and make use of complex lead-field distributions to retain the imaginary parts of the source signals at the scalp level. However this approach studies independent components of brain activity and as such, is not directly comparable to DCM. DCM for phase responses is an active area of research (Penny et al., 2008
) and will receive a full treatment elsewhere.
Our simulation studies provide some face validity for DCM, in terms of internal consistency. DCM was able to identify the correct model and, under one model, parameter values were recovered reliably in settings of high observation noise. Changes in the postsynaptic responsiveness, encoded by the population maximum EPSP, were estimated veridically at levels below prior threshold, with a conditional confidence of more than 74%; even for the highest levels of noise. Similarly, inter-area connection strength estimates were reasonably accurate under high levels of noise. With noisy data, parameter estimates tend to shrink towards their prior expectation, reflecting the adaptive nature of the weights afforded to prior and data information in Bayesian schemes.
We have presented an analysis of empirical LFP data, obtained by invasive recordings in rat CA1 and LA during a fear conditioning paradigm. A previous analysis of these data (Seidenbecher et al., 2003
) showed prominent theta band activity in CA1 during both CS+ and CS- conditions, whereas LA expresses significant theta activity during CS+ trials only. Using an analysis of functional connectivity6
, based on cross-correlograms of LA/CA1 activity in the theta range, Seidenbecher et al., (2003)
demonstrated an increase in connectivity between these two brain regions during CS+ trials. This is consistent with a trial-specific enabling or gating of the CA1 → LA connection during retrieval of conditioned fear in the CS+ condition, leading to a transient coupling of LA responses to the condition-independent theta activity in CA1. However, this analysis of functional connectivity was unable to provide direct evidence for directed or causal interactions. This sort of evidence requires a model of effective connectivity like DCM. The DCM analysis in the present study confirmed the hypothesis based on the cross-correlogram results of Seidenbecher et al., (2003)
. The DCM analysis showed a selective increase in CA1 → LA connectivity during CS+ trials, accompanied by a decrease in LA → CA1 connection strength. An additional finding was the increase in the amplitude of postsynaptic responses in LA during CS+ trials. This result may represent the correlate of long term potentiation of LA neurons following fear conditioning (Rodrigues et al., 2004; LeDoux, 2000
). In summary, one could consider these results as a demonstration of construct validity for DCM, in relation to the previous analyses of functional connectivity using cross-correlograms.
The analysis of parameter estimates was performed only after Bayesian model selection. In the search for an optimum model, we asked (i) which connection type was most plausible, (ii) whether neuronal inputs drive CA1, LA or both regions; and (iii) which extrinsic connectivity pattern was most likely to have generated the observed data (directed CA1 → LA or LA → CA1 or reciprocal connections). The results of sequential model comparisons showed that there was a very strong evidence for a model in which (i) extrinsic connections targeted excitatory neurons, (ii) neuronal inputs drove both CA1 and LA and (iii) the two regions were linked by reciprocal connections. While there is, to our knowledge, no decisive empirical data concerning the first two issues, the last conclusion from our model comparisons is supported strongly by neuroanatomic data from tract-tracing studies. These have demonstrated prominent and reciprocal connections between CA1 and LA (see Pitkänen et al., 2000
for a review). This correspondence between neuroanatomic findings and our model structure, which was inferred from the LFP data, provides further construct validity, in relation to neuroanatomy.
In conclusion, this study has introduced a novel variant of DCM that provides mechanistic explanations, at the level of synaptic physiology, for the cross-spectral density of invasive (LFP) or non-invasive (EEG) electrophysiological recordings. We have demonstrated how this approach can be used to investigate hypotheses about directed interactions among brain regions that cannot be addressed by conventional analyses of functional connectivity. A previous (single-source) DCM study (Moran et al., 2008
) of invasive LFP recordings in rats demonstrated the consistency of model parameter estimates with concurrent microdialysis measurements. The current study is another step towards establishing the validity of models, which we hope will be useful for deciphering the neurophysiological mechanisms that underlie pharmacological effects and pathophysiological processes (Stephan et al., 2006b