Elsevier

NeuroImage

Volume 22, Issue 3, July 2004, Pages 1157-1172
NeuroImage

Comparing dynamic causal models

https://doi.org/10.1016/j.neuroimage.2004.03.026Get rights and content

Abstract

This article describes the use of Bayes factors for comparing dynamic causal models (DCMs). DCMs are used to make inferences about effective connectivity from functional magnetic resonance imaging (fMRI) data. These inferences, however, are contingent upon assumptions about model structure, that is, the connectivity pattern between the regions included in the model. Given the current lack of detailed knowledge on anatomical connectivity in the human brain, there are often considerable degrees of freedom when defining the connectional structure of DCMs. In addition, many plausible scientific hypotheses may exist about which connections are changed by experimental manipulation, and a formal procedure for directly comparing these competing hypotheses is highly desirable. In this article, we show how Bayes factors can be used to guide choices about model structure, both concerning the intrinsic connectivity pattern and the contextual modulation of individual connections. The combined use of Bayes factors and DCM thus allows one to evaluate competing scientific theories about the architecture of large-scale neural networks and the neuronal interactions that mediate perception and cognition.

Introduction

Human brain mapping has been used extensively to provide functional maps showing which regions are specialized for which functions (Frackowiak et al., 1997). A classic example is the study by Zeki et al. (1991) who identified V4 and V5 as being specialized for the processing of color and motion, respectively. More recently, these analyses have been augmented by functional integration studies that describe how functionally specialized areas interact and how these interactions depend on changes of context. These studies make use of the concept of effective connectivity defined as the influence one region exerts over another as instantiated in a statistical model. A classic example is the study by Buchel and Friston (1997) who used structural equation modeling (SEM) to show that attention to motion modulates connectivity in the dorsal stream of the visual system.

In a recent paper (Friston et al., 2003), we have proposed the use of dynamic causal models (DCMs) for the analysis of effective connectivity. DCM posits a causal model whereby neuronal activity in a given region causes changes in neuronal activity in other regions, via interregional connections, and in its own activity, via self-connections. Additionally, any of these connections can be modulated by contextual variables like cognitive set or attention. The resulting neurodynamics of the modeled system then cause functional magnetic resonance imaging (fMRI) time series via local hemodynamics that are characterized by an extended Balloon model Buxton et al., 1998, Friston, 2002.

A DCM is fitted to data by tuning the neurodynamic and hemodynamic parameters so as to minimize the discrepancy between predicted and observed fMRI time series. Importantly, however, the parameters are constrained to agree with a priori specifications of what range the parameters are likely to lie within. These constraints, which take the form of a prior distribution, are then combined with data via a likelihood distribution to form a posterior distribution according to Bayes' rule. Changes in effective connectivity can then be inferred using Bayesian inference based on the posterior densities.

In this paper, we apply Bayesian inference not just to the parameters of DCMs, as in Friston et al. (2003), but to the models themselves. This allows us to make inferences about model structure, that is, which of several alternative models is optimal given the data. Such decisions are of great practical relevance because we still lack detailed knowledge about the anatomical connectivity of the human brain (Passingham et al., 2002). Decisions about the intrinsic connectivity of DCMs are therefore usually based on inferring connections from supposedly equivalent areas in the Macaque brain for which the anatomical connectivity is well known (Stephan et al., 2001). This procedure has many pitfalls, however, including a multitude of incompatible parcellation schemes and frequent uncertainties about the homology and functional equivalence of areas in the brains of man and monkey. This problem may be less severe in sensory systems, but is of particular importance for areas involved in higher cognitive processes like language (Aboitiz and Garcia, 1997). Thus, there are often considerable degrees of freedom when defining the connectional structure of DCMs of the human brain. In this paper, we show how Bayes factors can be used to guide the modeller in making such choices.

A second question concerning model structure is which of the connections included in the model are modulated by experimentally controlled contextual variables (e.g., attention). This choice reflects the modeller's hypothesis about where context-dependent changes of effective connectivity occur in the modeled system. We will also demonstrate how Bayesian model selection can be used to distinguish between competing models that represent the many plausible hypotheses.

The paper is structured as follows. In the Neurobiological issues section, we introduce briefly the neurobiological context in which DCM is usually applied. We focus particularly on hierarchical models and the distinction between anatomical and functional characterizations. In the Theory section, we review dynamic causal modeling from a theoretical perspective describing the model priors and likelihood functions that are used in a Bayesian parameter estimation algorithm. We also describe the Bayes factors that are used to weigh evidence for and against competing scientific hypotheses. Results on simulated and experimental data are presented in the Applications section.

We use uppercase letters to denote matrices and lowercase to denote vectors. N(m, Σ) denotes a uni- or multivariate Gaussian with mean m and variance or covariance Σ. iK denotes the K × K identity matrix, 1K is a 1 × K vector of 1 s, 0K is a 1 × K vector of zeros, if X is a matrix, Xij denotes the i, jth element, XT denotes the matrix transpose, and vec(X) returns a column vector comprising its columns, diag(x) returns a diagonal matrix with leading diagonal elements given by the vector x, denotes the Kronecker product, and log x denotes the natural logarithm.

Section snippets

Neurobiological issues

Many applications of DCM, both in this article and in previous work Friston et al., 2003, Mechelli et al., 2003, refer to “bottom-up” and “top-down” processes, and we envisage that a large number of future applications of DCM will rest on this distinction. Some of the possible DCM architectures for modeling these processes may, at first glance, seem at odds with traditional cognitive theories that relate bottom-up processes to so-called “forward” connections and top-down processes to “backward”

Theory

In the first part of this section, we briefly review the mathematical model underlying DCM, focussing on the specification of the priors and model likelihood. Readers unfamiliar with DCM are advised to first read Friston et al. (2003). We then show how the prior and likelihoods are combined via Bayes rule to form the posterior distribution and how this is computed iteratively using an Expectation-Maximization (EM) algorithm.

The second part of the theory section, starting in the Model evidence,

Applications

In this section, we describe fitting DCMs to fMRI data from an attention to motion experiment and a visual object categorization experiment. We also describe fitting models to simulated data to demonstrate the face validity of the model comparison approach. These data were generated so as to have similar signal to noise ratios (SNRs) to the fMRI data, where SNR is defined as the ratio of signal amplitude to noise amplitude (Papoulis, 1991). For regions receiving driving input, the SNRs were

Discussion

We have described Bayesian inference procedures in the context of dynamic causal models. DCMs are used in the analysis of effective connectivity, and posterior distributions can be used, for example, to assess changes in effective connectivity caused by experimental manipulation. These inferences, however, are contingent on assumptions about the intrinsic and modulatory architecture of the model, that is, which regions are connected to which other regions and which inputs can modulate which

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