Disentangling causal webs in the brain using functional magnetic resonance imaging: A review of current approaches

Abstract

In the past two decades, functional Magnetic Resonance Imaging (fMRI) has been used to relate neuronal network activity to cognitive processing and behavior. Recently this approach has been augmented by algorithms that allow us to infer causal links between component populations of neuronal networks. Multiple inference procedures have been proposed to approach this research question but so far, each method has limitations when it comes to establishing whole-brain connectivity patterns. In this paper, we discuss eight ways to infer causality in fMRI research: Bayesian Nets, Dynamical Causal Modelling, Granger Causality, Likelihood Ratios, Linear Non-Gaussian Acyclic Models, Patel's Tau, Structural Equation Modelling, and Transfer Entropy. We finish with formulating some recommendations for the future directions in this area.

Keywords: Bayesian Nets; Causal inference; Directed Acyclic Graphs; Dynamic Causal Modeling; Effective connectivity; Functional Magnetic Resonance Imaging; Granger Causality; Large-scale brain networks; Pairwise inference; Structural Equation Modeling.

Figure 1.

Causal research in fMRI. The discussed methods can be divided into two families: Network Inference Methods, which are based on a one-step multivariate procedure, and Pairwise Inference Methods, which are based on a two-step pairwise inference procedures. As pairwise methods by definition establish causal connections on a connection-by-connection basis, they do not require any assumptions on the structure of the network, but also do not reveal the structure of the network.

Figure 2.

The full pipeline for the DCM forward model. The model involves three node network stimulated during the cognitive experiment (i). The parameter set describing the dynamics in this network includes a fixed connectivity matrix (A), modulatory connections (B), and inputs to the nodes (C) (ii). In the equation describing the fast neuronal dynamics, z denotes the dynamics in the nodes, and u is an experiment-related input. Red: excitatory connections. Blue: inhibitory connections. The dynamics in this network can be described with use of ordinary differential equations. The outcome is the fast neuronal dynamics (iii). The neuronal time series is then convolved with the hemodynamic response function (HRF) (iv) in order to obtain the BOLD response (v), which may be then subsampled (vertical bars). This is the original, bilinear implementation of DCM (K. J. Friston et al., 2003). Now, more complex versions of DCM with additional features are available, such as spectral DCM (K. J. Friston et al., 2011), stochastic DCM (Daunizeau et al., 2012), nonlinear DCM (Stephan et al., 2008), two-state DCM (Marreiros et al., 2008), large DCMs (Frässle et al., ; Frässle, Lomakina-Rumyantseva, et al., ; Seghier & Friston, 2013) and so on.

Figure 3.

The Linear Non-Gaussian Acyclic Model (LiNGAM). A: The noisy time series $\widehat{X}(t)$ consists of signal X(t) and noise σ_X(t). Y (t) thus becomes a function of both the signal and the noise in $\widehat{X}(t)$. B: Causal inference through the analysis of the noise residuals (figure reprinted from http://videolectures.net/bbci2014_grosse_wentrup_causal_inference/). The causal link from age to length in a population of fish can be inferred from the properties of the residual noise in the system. If fish length is expressed in a function of fish age (upper panel), the residual noise in the dependent variable (length) is uncorrelated with the independent variable (age): the noise variance is constant over a large range of fish age (red bars). On the contrary, once the variables are flipped and fish age becomes a function of fish length (lower panel), the noise variance becomes dependent on the independent variable (length): it is small for small values of fish length and large for the large values of fish length (red bars).

Figure 4.

Bayesian nets. A: Model-based versus model-free approach. β: a regressor coefficient fitted in the modeling procedure. σ(t): additive noise. Both model-based and model-free approach contain a measure of confidence. In a model-based approach, a model is fitted to the data, and p-values associated with this fit are a measure of confidence that the causal link exists (i.e., is a true positive, left panel). In a model-free approach, this confidence is quantified directly by expressing causal relationships in terms of conditional probabilities (right panel). B: Conditional probability for continuous variables. Since BOLD fMRI is a continuous variables, the joint probability distribution for variables X and Y is a two-dimensional distribution. Therefore, conditional probability of P(Y|X = x) becomes a distribution. C: (i) An exemplary Bayesian Net. X₁, X₂, X₃: parents, X₄, X₅: children. (ii) Competitive Bayesian Nets: one can define competitive models (causal structures) in the network and compare their joint probability derived from the data. (iii) Cyclic belief propagation: if there was a cycle in the network, the expression for the joint probability would convert into an infinite series of conditional probabilities.