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A tutorial on group effective connectivity analysis, part 1: first level analysis with DCM for fMRI

Peter Zeidman, Amirhossein Jafarian, Nadège Corbin, Mohamed L. Seghier, Adeel Razi, Cathy J. Price, Karl J. Friston

arXiv:1902.10597v1q-bio.QMq-bio.NC

TL;DR

DCM provides a framework for specifying effective-connectivity models, but its hypothesis-driven use depends on an efficient experimental design. This tutorial works through a DCM analysis of previously published fMRI data, surveys current fMRI implementations, and details the theory and methodology behind them.

  • Problem

    DCM is a framework for specifying models of effective connectivity, whose hypothesis-driven use depends on having an efficient experimental design.

  • Method

    The tutorial analyzes previously published fMRI data to illustrate DCM methodology and theory, working through each analysis step and surveying current fMRI-specific models.

  • Results

    The paper provides a detailed worked analysis and survey of the current implementation and models used for DCM for fMRI.

  • Takeaways & Limitations

    The tutorial uses existing fMRI data to demonstrate how DCM analysis can be applied without claiming new insights into semantic processing or laterality.

  • Takeaways & Limitations

    The analysis does not attempt to provide new insights into laterality or semantic processing.

Abstract

from arXiv · show

Dynamic Causal Modelling (DCM) is the predominant method for inferring effective connectivity from neuroimaging data. In the 15 years since its introduction, the neural models and statistical routines in DCM have developed in parallel, driven by the needs of researchers in cognitive and clinical neuroscience. In this tutorial, we step through an exemplar fMRI analysis in detail, reviewing the current implementation of DCM and demonstrating recent developments in group-level connectivity analysis. In the first part of the tutorial (current paper), we focus on issues specific to DCM for fMRI, unpacking the relevant theory and highlighting practical considerations. In particular, we clarify the assumptions (i.e., priors) used in DCM for fMRI and how to interpret the model parameters. This tutorial is accompanied by all the necessary data and instructions to reproduce the analyses using the SPM software. In the second part (in a companion paper), we move from subject-level to group-level modelling using the Parametric Empirical Bayes framework, and illustrate how to test for commonalities and differences in effective connectivity across subjects, based on imaging data from any modality.

1 Introduction

DCM specifies biologically informed generative models of effective connectivity, estimates their parameters from neuroimaging timeseries, and compares competing architectures. This tutorial uses an exemplar fMRI dataset to explain the theory, implementation, and practical steps of first-level DCM analysis.

  • DCM framework: DCM specifies, estimates, and tests models of directed effective connectivity among brain regions.It is primarily used in human neuroimaging, with applications also reported in rodents and zebrafish.
  • Generative model: A DCM forward model generates neuroimaging timeseries from causes such as neural fluctuations, experimental stimuli, and connection strengths.For fMRI, the model links neural activity to neurovascular coupling, haemodynamics, BOLD signals, and measurement noise.
  • Model inversion: Model inversion estimates parameters by balancing timeseries accuracy against the complexity implied by departures from prior values.This accuracy–complexity trade-off is quantified by model evidence.
  • Model comparison: Model comparison tests competing network architectures by comparing their evidence after accounting for unknown parameters.These comparisons can be performed at the single-subject or group level.
  • Tutorial example: In task-based fMRI, DCM asks how localized experimental effects arose through changes in underlying neural circuitry.The tutorial analyzes interactions between left and right frontal regions during semantic relative to perceptual processing without claiming new laterality findings.
  • Tutorial contribution: The tutorial reviews current fMRI-specific DCM models and provides technical detail, SPM implementation guidance, reproducible data, and step-by-step analyses.It is intended to complement previous DCM reviews and tutorials.

2 Notation

The notation section defines conventions for vectors, matrices, derivatives, multiplication, matrix elements, variables, dimensions, and SPM function names.

  • Notation conventions: Vectors use lowercase bold italics, matrices use uppercase bold italics, and other variables and function names use plain italics.These conventions distinguish mathematical objects throughout the tutorial.
  • Notation conventions: A dot denotes multiplication when standalone and a time derivative when placed above a variable.For example, ż denotes the derivative of a variable with respect to time.
  • Notation conventions: Matrix element A_mn denotes the element in row m and column n of matrix A.Variables and their dimensions are listed in Table 1.
  • SPM implementation: MATLAB function names corresponding to SPM implementation are shown in bold text, such as spm_dcm_fit.m.This convention connects the theoretical methods to their software implementation.

3 Experimental design

The tutorial emphasizes that DCM is hypothesis-driven and therefore requires an efficient design with clearly articulated within- and between-subject hypotheses. Its example varies stimulus type and task to test predicted semantic effects and their connectivity modulation.

  • Experimental design: DCM is a hypothesis-driven approach whose success depends on an efficient experimental design and clearly articulated hypotheses.Hypotheses may concern within-subject effects, between-subject effects, or both.
  • Hypotheses: The within-subject hypothesis predicts greater left than right frontal responses when participants process the meaning of familiar words.This concerns semantic content rather than stimulus presentation alone.
  • Hypotheses: The between-subject hypothesis predicts that the Laterality Index varies across subjects and can be explained by the strength of specific connections.The Laterality Index quantifies the difference in hemispheric responses.
  • DCM effects: Driving effects reflect stimulus manipulations, whereas modulatory effects reflect task-demand or contextual influences on neural responses.The analysis makes this distinction explicit.
  • Experimental design: The example uses two within-subject factors: stimulus type, Words or Pictures, and task, Semantic or Perceptual reasoning.Together they form a balanced factorial design with four experimental conditions.
  • Hypotheses: The design hypothesizes a greater response to words than pictures specifically during the semantic task.DCM is used to test the interaction between stimulus type and task.

4 Region selection and fMRI timeseries extraction

DCM models connectivity among selected regions of interest, which task-based studies typically identify from an initial mass-univariate SPM analysis. The tutorial selects four frontal ROIs and extracts representative timeseries for analysis.

  • ROI selection: DCM models connectivity among brain regions of interest, with ROI-selection criteria depending on the experimental setting.Resting-state studies may use ICA, stereotaxic coordinates, or literature-based masks.
  • ROI selection: For task-based experiments, ROIs are usually selected from an initial mass-univariate SPM analysis that localizes experimental effects.DCM then seeks a simple wiring diagram accounting for those results.
  • Example ROIs: The example identifies four frontal ROIs: left ventral, left dorsal, right ventral, and right dorsal frontal cortex.These regions were identified using an SPM contrast for the main effect of task.
  • Timeseries extraction: Relevant timeseries are extracted, pre-processed, and summarized within each ROI by the first principal component.Figure 2 illustrates the experimental timing and timeseries for an example subject.

5 Neural model specification

DCM for fMRI models hidden neural states driven by experimental inputs, transforms them into BOLD observations, and estimates connectivity through a bilinear neural model. The tutorial specifies task, picture, and word inputs, accounts for slice timing, and uses a network model in which experimental conditions modulate connectivity.

  • Forward model: DCM separates hidden neural dynamics from haemodynamic observation, nuisance GLM effects, and measurement noise.The neural model maps stimuli, current state, and connectivity parameters to changing neural activity; the haemodynamic model maps neural activity to BOLD.
  • Forward model: The observation equation models BOLD data as g(z, θ(h)) plus known effects X₀β₀ and zero-mean observation noise ϵ.The noise variance is estimated from the data.
  • Input specification: Three experimental conditions enter the design: Task includes all semantic trials, while Pictures and Words are condition-specific subsets; control and incorrect trials form the implicit baseline.SPM imports trial onsets from the initial GLM, with block inputs coded as one during presentation and zero otherwise.
  • Slice timing: Slice timing aligns discrete fMRI acquisitions with continuous neural states by modelling each region’s acquisition time.The approach assumes slice acquisition times are known, which can be problematic after spatial normalization and for interleaved or multiband sequences.
  • Bilinear or nonlinear: The default bilinear neural model uses a Taylor approximation to represent effective connectivity and its changes under experimental inputs.The model was selected instead of a nonlinear model because nonlinear effects were not needed in this analysis.
  • Connections: Task drives all regions through C, whereas Pictures and Words modulate regional self-connections through B matrices.Restricting modulation to self-connections was intended to improve biological interpretability and parameter identifiability.

Appendix 4: The neural parameters.

The neural parameters in DCM describe intrinsic self-inhibition, between-region connectivity, and condition-specific modulation. Their signs, units, and constraints determine how estimated parameters should be interpreted, while model choices impose assumptions about dynamics and inputs.

  • Intrinsic connectivity: Each region has an inhibitory self-connection that controls self-inhibition, gain, and sensitivity to network inputs.These parameters can be interpreted biologically through the excitatory-inhibitory balance between pyramidal cells and inhibitory interneurons.
  • Parameter structure: Average and modulatory matrices are partitioned into intrinsic diagonal self-connections and extrinsic off-diagonal between-region connections.The model’s sparsity structure determines which parameters are estimated and which remain fixed at their prior expectation of zero.
  • Parameter interpretation: Self-connection parameters are unitless log-scaling parameters: more positive values indicate stronger inhibition and reduced responsiveness to network inputs.Between-region connectivity parameters are rates of change measured in Hz.
  • Parameter interpretation: A_E 3,1 denotes the connection from region 1 to region 3; positive values are excitatory, negative values inhibitory, and B_E 3,1^(k) captures condition-specific change.The matrices encode directed connectivity among regions.
  • States per region: The tutorial uses a one-state DCM in which each region has one activity state, rather than a two-state model with excitatory and inhibitory populations.Two-state models require positivity and negativity constraints on all connections, which affects interpretation.
  • Deterministic versus stochastic: The selected task model is deterministic, so experimental stimuli drive the neural dynamics and stochastic effects are not included.Stochastic DCM instead estimates time-varying fluctuations in neural activity and measurements, creating a more challenging estimation problem.
  • Centre input: Mean-centering inputs makes A represent average connectivity across conditions, whereas uncentered inputs make A describe the implicit baseline.Mean-centering can improve model evidence and keep parameters closer to their zero prior expectation.
  • Timeseries or CSD: CSD-DCM fits second-order frequency-domain statistics for efficient resting-state analysis while assuming connection strengths remain constant during acquisition.The tutorial instead fits timeseries because it targets condition-specific, time-varying connectivity during a task.

6 Haemodynamic model specification

The DCM haemodynamic and BOLD signal models describe how neural activity produces measured fMRI timeseries through neurovascular and blood-vessel dynamics. The response unfolds through deoxygenated, oxygenated, and sustained phases, with model parameters estimated for each region alongside neural parameters.

  • Forward model: The DCM forward model predicts fMRI timeseries from neural activity by modelling haemodynamic and BOLD signal processes.The haemodynamic and BOLD signal model parameters are estimated on a per-region basis and concatenated with neural-model parameters.
  • Response phases: The BOLD response comprises deoxygenated, oxygenated, and sustained response phases linked to neuronal activity, neurovascular coupling, and blood-vessel dynamics.These phases summarize the temporal evolution of the signal following experimental stimulation.
  • Response phases: During the deoxygenated phase, neurons consume oxygen before blood flow changes, increasing the ratio of deoxygenated to oxygenated blood.This produces a lag in the BOLD response.
  • Response phases: Without further stimulation, neuronal activity returns to its resting state and the BOLD signal gradually decreases toward baseline.The baseline depends on net oxygen extraction exchange between neurons and blood vessels, as well as cerebral blood flow.

7 Model estimation

Model estimation in DCM inverts each subject’s forward model using Bayesian inference, balancing data accuracy against model complexity. Priors encode assumptions about connectivity, and variational free energy provides an approximation to model evidence for evaluating fitted models.

  • Model inversion: DCM inversion estimates parameters that best explain each subject’s fMRI data while keeping them close to their prior or starting values.Bayesian inference quantifies parameter uncertainty before and after observing the data.
  • Priors: Priors restrict parameters to reasonable ranges and are combined with observed fMRI data to produce posterior beliefs.Both priors and posteriors are represented as probability densities.
  • Priors: Switched-on connectivity parameters have zero expectation with non-zero variance, allowing positive or negative values when supported by data.Their variance determines uncertainty about the parameter being zero.
  • Priors: Switched-off parameters have expectation zero and variance close to zero, effectively fixing them at zero regardless of the data.Both switched-on and switched-off specifications are shrinkage priors.
  • Priors: By default, models with more than eight brain regions receive functional-connectivity-based sparsity-inducing priors to reduce estimation time.This automatic constraint was not applied in the experiment described here.
  • Variational Laplace: Variational free energy approximates log model evidence as accuracy minus complexity, rewarding accurate and parsimonious explanations.Accuracy reflects correspondence between predicted and observed timeseries, while complexity is the Kullback-Leibler divergence between priors and posteriors.
  • Model inversion: The first-level analysis ends with subject-specific free-energy estimates and posterior probability densities over the parameters that maximized free energy.These outputs are obtained after each subject’s model is inverted.

8 Results

Subject-level DCM diagnostics showed task-related dynamics and neural parameters departing from their zero priors, while explained variance was modest because control conditions were not modelled. Parameter interpretation linked self-inhibition and extrinsic connections to condition-specific changes in regional sensitivity and response duration.

  • 8.1 Diagnostics: Many parameters moved away from their zero prior, with 90% credible intervals excluding zero in the example subject.
  • 8.1 Diagnostics: 18.85% of variance was explained for the example subject, compared with 17.27% (SD 9.37%) across subjects.
  • 8.1 Diagnostics: The relatively low explained variance reflects that control conditions, including perceptual matching and baseline rest periods, were not modelled.
  • 8.1 Diagnostics: Most subjects showed nontrivial neural parameters with 90% confidence intervals excluding zero, indicating experimental information in the data.
  • 8.2 Interpretation of parameters: Picture stimuli increased self-inhibition in ldF and decreased it in lvF, shifting responses from dorsal to ventral left frontal cortex.
  • 8.2 Interpretation of parameters: In lvF, self-inhibition was −0.17 Hz during pictures versus −4.40 Hz during words, yielding time constants of 5.88 s and 0.23 s, respectively.The corresponding half-lives were 4.08 s for pictures and 0.16 s for words, indicating a more sustained picture response.
  • 8.2 Interpretation of parameters: The positive rvF-to-lvF connection estimate of 0.43 Hz indicates an excitatory between-region connection.
  • 8.2 Interpretation of parameters: Inspecting parameter signs and magnitudes clarifies connection strengths and experimental effects, but does not formally test hypotheses.

9 Discussion

This tutorial reviews DCM for fMRI through a detailed first-level analysis, clarifying its priors, parameter interpretations, and practical workflow. Subject-level inversion estimates connectivity, condition-dependent changes, external-input sensitivities, and free energy, while group-level comparisons are addressed separately.

  • The tutorial reviews the current implementation of DCM for fMRI by working through an exemplar analysis in detail.
  • The first-level analysis identifies regions showing experimental effects before testing conserved and differing neural effects across subjects in companion group-level analyses.
  • Subject-level model inversion estimates baseline connectivity A, condition-dependent changes B, external-input sensitivity C, and free energy F.
  • DCM expresses key assumptions as priors, including expectations about signal-to-noise, neural decay, haemodynamics, observation noise, and model evidence.
  • The basic neural model assumes intrinsic responses decay over seconds, with a 90% prior interval of 1.63s–2.46s without modulation and 0.38s–10.49s with modulation.
  • Free energy is treated as a proxy for log model evidence for linear and weakly nonlinear models, but highly nonlinear models can present local-optimum challenges.

10 Appendix 1: Timeseries extraction

The appendix describes a staged procedure for selecting ROIs and extracting representative fMRI timeseries for DCM. It emphasizes retaining subject information rather than excluding participants solely because one region lacks a strong response.

  • Region selection: ROIs are selected from experimental effects using subject-level GLMs, group-level contrast maps, and subject-specific peak coordinates.
  • Region selection: Subject-level peaks were constrained within 8mm of group peaks and required p < 0.05 uncorrected task-contrast significance.
  • Subject inclusion: Hierarchical modelling may make excluding subjects with noisy or missing data in particular regions unnecessary, because they can still inform other regions and intersubject variability.
  • ROI definition: Each ROI includes voxels within a 4mm-radius sphere around the individual peak that exceed p < 0.05 uncorrected.
  • ROI extraction: Representative ROI timeseries are pre-whitened, high-pass filtered, nuisance-adjusted, and reduced to the first principal component.
  • ROI extraction: PCA is preferred to averaging because opposing responses can cancel and means are more affected by extreme values.

11 Appendix 2: Observation noise specification

DCM models region-specific observation noise with a multi-component precision model. The approach uses pre-whitened timeseries and estimates log precision scalings from the data.

  • DCM estimates the precision, or inverse variance, of zero-mean additive white noise separately for each brain region.
  • The vectorised observations use a precision matrix formed as a linear mixture of one precision component per region.
  • Each component matrix has ones for observations from its region and zeros elsewhere, while exp(λ_i) keeps the estimated precision non-negative.
  • The experiment contained 792 observations per subject, based on 198 fMRI volumes across 4 brain regions.
  • The prior expected precision was 403.43 with 90% credible interval [348.84 466.56], reflecting an expected high signal-to-noise ratio.

12 Appendix 3: Derivation of the fMRI neural model

The appendices derive the DCM neural model by approximating neural dynamics with a Taylor expansion and interpreting its parameters as effective connectivity, modulation, and driving input. They then explain the haemodynamic transformation from neural activity to BOLD.

  • Neural model: A Taylor-series approximation reduces the complicated neural response function to the bilinear DCM model.
  • Neural model: The matrices A, B, and C represent effective connectivity, input-dependent connectivity changes, and responses to external input, respectively.
  • Neural parameters: For a single region, neural activity follows z(t) = z(0) ∙exp(at), so negative self-connections produce exponential decay.
  • Neural parameters: With default self-connection a = -0.5Hz, the time constant is τ = 2s and the isolated-region prior has median τ = 2s with 90% interval [1.63s 2.46s].
  • Neural parameters: Modulation changes the decay rate through a + b∙u, producing a 90% prior interval of [0.38s 10.49s] when u = 1.
  • Haemodynamic model: The haemodynamic model maps neural activity to BOLD through vasoactive signalling, blood-flow-driven changes in blood volume and deoxyhaemoglobin, and nonlinear signal combination.
  • Haemodynamic model: The haemodynamic parameters affect BOLD peak height, response duration, and post-stimulus undershoot through signal decay, feedback, vessel stiffness, and transit dynamics.
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