Elsevier

NeuroImage

Volume 19, Issue 4, August 2003, Pages 1273-1302
NeuroImage

Regular article
Dynamic causal modelling

https://doi.org/10.1016/S1053-8119(03)00202-7Get rights and content

Abstract

In this paper we present an approach to the identification of nonlinear input–state–output systems. By using a bilinear approximation to the dynamics of interactions among states, the parameters of the implicit causal model reduce to three sets. These comprise (1) parameters that mediate the influence of extrinsic inputs on the states, (2) parameters that mediate intrinsic coupling among the states, and (3) [bilinear] parameters that allow the inputs to modulate that coupling. Identification proceeds in a Bayesian framework given known, deterministic inputs and the observed responses of the system. We developed this approach for the analysis of effective connectivity using experimentally designed inputs and fMRI responses. In this context, the coupling parameters correspond to effective connectivity and the bilinear parameters reflect the changes in connectivity induced by inputs. The ensuing framework allows one to characterise fMRI experiments, conceptually, as an experimental manipulation of integration among brain regions (by contextual or trial-free inputs, like time or attentional set) that is revealed using evoked responses (to perturbations or trial-bound inputs, like stimuli). As with previous analyses of effective connectivity, the focus is on experimentally induced changes in coupling (cf., psychophysiologic interactions). However, unlike previous approaches in neuroimaging, the causal model ascribes responses to designed deterministic inputs, as opposed to treating inputs as unknown and stochastic.

Introduction

This paper is about modelling interactions among neuronal populations, at a cortical level, using neuroimaging (hemodynamic or electromagnetic) time series. It presents the motivation and procedures for dynamic causal modelling of evoked brain responses. The aim of this modelling is to estimate, and make inferences about, the coupling among brain areas and how that coupling is influenced by changes in experimental context. Dynamic causal modelling represents a fundamental departure from existing approaches to effective connectivity because it employs a more plausible generative model of measured brain responses that embraces their nonlinear and dynamic nature.

The basic idea is to construct a reasonably realistic neuronal model of interacting cortical regions. This model is then supplemented with a forward model of how neuronal or synaptic activity is transformed into a measured response. This enables the parameters of the neuronal model (i.e., effective connectivity) to be estimated from observed data. These supplementary models may be forward models of electromagnetic measurements or hemodynamic models of fMRI measurements. In this paper we will focus on fMRI. Responses are evoked by known deterministic inputs that embody designed changes in stimulation or context. This is accomplished by using a dynamic input–state–output model with multiple inputs and outputs. The inputs correspond to conventional stimulus functions that encode experimental manipulations. The state variables cover both the neuronal activities and other neurophysiological or biophysical variables needed to form the outputs. The outputs are measured electromagnetic or hemodynamic responses over the brain regions considered.

Intuitively, this scheme regards an experiment as a designed perturbation of neuronal dynamics that are promulgated and distributed throughout a system of coupled anatomical nodes to change region-specific neuronal activity. These changes engender, through a measurement-specific forward model, responses that are used to identify the architecture and time constants of the system at the neuronal level. This represents a departure from conventional approaches (e.g., structural equation modelling and autoregression models; McIntosh and Gonzalez-Lima 1994, Büchel and Friston 1997; Harrison et al., in press), in which one assumes the observed responses are driven by endogenous or intrinsic noise (i.e., innovations). In contradistinction, dynamic causal models assume the responses are driven by designed changes in inputs. An important conceptual aspect of dynamic causal models, for neuroimaging, pertains to how the experimental inputs enter the model and cause neuronal responses. Experimental variables can elicit responses in one of two ways. First, they can elicit responses through direct influences on specific anatomical nodes. This would be appropriate, for example, in modelling sensory-evoked responses in early visual cortices. The second class of input exerts its effect vicariously, through a modulation of the coupling among nodes. These sorts of experimental variables would normally be more enduring; for example, attention to a particular attribute or the maintenance of some perceptual set. These distinctions are seen most clearly in relation to existing analyses and experimental designs.

The central idea behind dynamic causal modelling (DCM) is to treat the brain as a deterministic nonlinear dynamic system that is subject to inputs and produces outputs. Effective connectivity is parameterised in terms of coupling among unobserved brain states (e.g., neuronal activity in different regions). The objective is to estimate these parameters by perturbing the system and measuring the response. This is in contradistinction to established methods for estimating effective connectivity from neurophysiological time series, which include structural equation modelling and models based on multivariate autoregressive processes. In these models, there is no designed perturbation and the inputs are treated as unknown and stochastic. Multivariate autoregression models and their spectral equivalents like coherence analysis not only assume the system is driven by stochastic innovations, but are restricted to linear interactions. Structural equation modelling assumes the interactions are linear and, furthermore, instantaneous in the sense that structural equation models are not time-series models. In short, DCM is distinguished from alternative approaches not just by accommodating the nonlinear and dynamic aspects of neuronal interactions, but by framing the estimation problem in terms of perturbations that accommodate experimentally designed inputs. This is a critical departure from conventional approaches to causal modelling in neuroimaging and brings the analysis of effective connectivity much closer to the conventional analysis of region-specific effects. DCM calls upon the same experimental design principles to elicit region-specific interactions that we use in experiments to elicit region-specific activations. In fact, as shown later, the convolution model, used in the standard analysis of fMRI time series, is a special and simple case of DCM that ensues when the coupling among regions is discounted. In DCM the causal or explanatory variables that compose the conventional design matrix become the inputs and the parameters become measures of effective connectivity. Although DCM can be framed as a generalisation of the linear models used in conventional analyses to cover bilinear models (see below), it also represents an attempt to embed more plausible forward models of how neuronal dynamics respond to inputs and produces measured responses. This reflects the growing appreciation of the role that neuronal models may have to play in understanding measured brain responses (see Horwitz et al., 2001, for a discussion).

This paper can be regarded as an extension of our previous work on the Bayesian identification of hemodynamic models (Friston, 2002) to cover multiple regions. In Friston (2002) we focussed on the biophysical parameters of a hemodynamic response in a single region. The most important parameter was the efficacy with which experimental inputs could elicit an activity-dependent vasodilatory signal. In this paper neuronal activity is modelled explicitly, allowing for interactions among the neuronal states of multiple regions in generating the observed hemodynamic response. The estimation procedure employed for DCM is formally identical to that described in Friston (2002).

DCM is used to test the specific hypothesis that motivated the experimental design. It is not an exploratory technique; as with all analyses of effective connectivity the results are specific to the tasks and stimuli employed during the experiment. In DCM designed inputs can produce responses in one of two ways. Inputs can elicit changes in the state variables (i.e., neuronal activity) directly. For example, sensory input could be modelled as causing direct responses in primary visual or auditory areas. The second way in which inputs affect the system is through changing the effective connectivity or interactions. Useful examples of this sort of effect would be the attentional modulation of connections between parietal and extrastriate areas. Another ubiquitous example of this second sort of contextual input would be time. Time-dependent changes in connectivity correspond to plasticity. It is useful to regard experimental factors as inputs that belong to the class that produces evoked responses or to the class of contextual factors that induces changes in coupling (although, in principle, all inputs could do both). The first class comprises trial- or stimulus-bound perturbations whereas the second establishes a context in which effects of the first sort evoke responses. This second class is typically trial-free and established by task instructions or other contextual changes. Measured responses in high-order cortical areas are mediated by interactions among brain areas elicited by trial-bound perturbations. These interactions can be modulated by other set-related or contextual factors that modulate the latent or intrinsic coupling among areas. Fig. 1 illustrates this schematically. The important implication here for experimental design in DCM is that it should be multifactorial, with at least one factor controlling sensory perturbation and another factor manipulating the context in which the sensory-evoked responses are promulgated throughout the system (cf., psychophysiological interaction studies; Friston et al., 1997).

In this paper we use bilinear approximations to any DCM. The bilinear approximation reduces the parameters to three sets that control three distinct things: first, the direct or extrinsic influence of inputs on brain states in any particular area; second, the intrinsic or latent connections that couple responses in one area to the state of others; and, finally, changes in this intrinsic coupling induced by inputs. Although, in some instances, the relative strengths of intrinsic connections may be of interest, most analyses of DCMs focus on the changes in connectivity embodied in the bilinear parameters. The first set of parameters is generally of little interest in the context of DCM but is the primary focus in classical analyses of regionally specific effects. In classical analyses the only way experimental effects can be expressed is through a direct or extrinsic influence on each voxel because mass-univariate models (e.g., SPM) preclude connections and their modulation.

We envisage that DCM will be used primarily to answer questions about the modulation of effective connectivity through inferences about the third set of parameters described above. These will be referred to as bilinear in the sense that an input-dependent change in connectivity can be construed as a second-order interaction between the input and activity in a source region when causing a response in a target region. The key role of bilinear terms reflects the fact that the more interesting applications of effective connectivity address changes in connectivity induced by cognitive set or time. In short, DCM with a bilinear approximation allows one to claim that an experimental manipulation has “activated a pathway” as opposed to a cortical region. Bilinear terms correspond to psychophysiologic interaction terms in classical regression analyses of effective connectivity (Friston et al., 1997) and those formed by moderator variables (Kenny and Judd, 1984) in structural equation modelling (Büchel and Friston, 1997). This bilinear aspect speaks again to the importance of multifactorial designs that allow these interactions to be measured and the central role of the context in which region-specific responses are formed (see McIntosh, 2000).

Because DCMs are not restricted to linear or instantaneous systems they are necessarily complicated and, potentially, need a large number of free parameters. This is why they have greater biological plausibility in relation to alternative approaches. However, this makes the estimation of the parameters more dependent upon constraints. A natural way to embody the requisite constraints is within a Bayesian framework. Consequently, dynamic causal models are estimated using Bayesian or conditional estimators and inferences about particular connections are made using the posterior or conditional density. In other words, the estimation procedure provides the probability distribution of a coupling parameter in terms of its mean and standard deviation. Having established this posterior density, the probability that the connection exceeds some specified threshold is easily computed. Bayesian inferences like this are more straightforward and interpretable than corresponding classical inferences and furthermore eschew the multiple comparison problem. The posterior density is computed using the likelihood and prior densitites. The likelihood of the data, given some parameters, is specified by the DCM (in one sense all models are simply ways of specifying the likelihood of an observation). The prior densities on the connectivity parameters offer suitable constraints to ensure robust and efficient estimation. These priors harness some natural constraints about the dynamics of coupled systems (see below) but also allow the user to specify which connections are likely to be present and which are not. An important use of prior constraints of this sort is the restriction of where inputs can elicit extrinsic responses. It is interesting to reflect that conventional analyses suppose that all inputs have unconstrained access to all brain regions. This is because classical models assume activations are caused directly by experimental factors, as opposed to being mediated by afferents from other brain areas.

Additional constraints on the intrinsic connections and their modulation by contextual inputs can also be specified but they are not necessary. These additional constraints can be used to finesse a model by making it more parsimonious, allowing one to focus on a particular connection. We will provide examples of this below. Unlike structural equation modelling, there are no limits on the number of connections that can be modelled because the assumptions and estimations schemes used by dynamic causal modelling are completely different, relying upon known inputs.

This paper comprises a theoretical section and three validation sections. In the theoretical section we present the conceptual and mathematical fundaments that are used in the remaining sections. The later sections address the face, predictive, and construct validity of DCM, respectively. Face validity ensures that the estimation and inference procedure identifies what it is supposed to. We have tried to establish face validity, using model systems and simulated data, to explore the performance of DCM over a range of hyperparameters (e.g., error variance, serial correlations among errors, etc). Some of these manipulations deliberately violate the assumptions of the model, embedded in priors, to establish that the estimation procedure remains robust in these circumstances. The subsequent section on predictive validity uses empirical data from an fMRI study of single word processing at different rates. These data were obtained consecutively in a series of contiguous sessions. This allowed us to repeat the DCM using independent realisations of the same paradigm. Predictive validity, over the multiple sessions, was assessed in terms of the consistency of the effective connectivity estimates and their posterior densities. The final section on construct validity revisits changes in connection strengths among parietal and extrastriate areas induced by attention to optic flow stimuli. We have established previously attentionally mediated increases in effective connectivity using both structural equation modelling and a Volterra formulation of effective connectivity Büchel and Friston 1997, Friston and Büchel 2000. Our aim here is to show that DCM leads to the same conclusions. This paper ends with a brief discussion of DCM, its limitations and potential applications. This paper is primarily theoretical and hopes to introduce the concepts of DCM and establish its validity, at least provisionally.

Section snippets

Theory

In this section we present the theoretical motivation and operational details upon which DCM rests. In brief, DCM is a fairly standard nonlinear system identification procedure using Bayesian estimation of the parameters of deterministic input–state–output dynamic systems. In this paper the system can be construed as a number of interacting brain regions. We will focus on a particular form for the dynamics that corresponds to a bilinear approximation to any analytic system. However, the idea

Conclusion

In this paper we have presented dynamic causal modelling. DCM is a causal modelling procedure for dynamic systems in which causality is inherent in the differential equations that specify the model. The basic idea is to treat the system of interest, in this case the brain, as an input–state–output system. By perturbing the system with known inputs, measured responses are used to estimate various parameters that govern the evolution of brain states. Although there are no restrictions on the

Software implementation note

The theory and estimation procedures described in this paper have been implemented in the SPM2 version of the statistical parametric mapping software (http://www.fil.ion.ucl.ac.uk/spm). Following a conventional analysis, a library of volumes of interest (VOI) structures can be assembled, usually based on maxima in the SPM{T} or SPM{F}. These VOI structures contain information about the original data, analysis and, critically, the region’s first eigenvariate. Selecting from this list specifies

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