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Dynamic causal modeling

Dynamic causal modeling (DCM) is a Bayesian framework for inferring the causal architecture of coupled dynamical systems from observed time-series data, particularly in neuroscience to estimate effective connectivity between brain regions.^[1] It employs generative models based on stochastic differential equations to describe how neuronal states evolve and interact under experimental inputs, linking biophysical mechanisms to measured signals like blood-oxygen-level-dependent (BOLD) responses in functional magnetic resonance imaging (fMRI).^[1] Originally formulated as a bilinear approximation to nonlinear dynamics, DCM allows for the quantification of context-dependent modulations in connectivity, such as those induced by cognitive tasks or pharmacological interventions.^[1] Introduced by Karl Friston and colleagues in 2003, DCM was initially developed for evoked responses in fMRI data, building on earlier work in system identification and dynamical modeling in neuroimaging.^[1] The approach uses variational Bayesian inference to estimate posterior distributions of model parameters, including intrinsic connectivity (baseline coupling between regions) and exogenous influences from stimuli.^[2] A key strength lies in its emphasis on model comparison via Bayesian evidence, enabling researchers to select among competing hypotheses about network structures without overfitting.^[3] This probabilistic formulation distinguishes DCM from correlational methods like functional connectivity analysis, as it explicitly models directed influences and their perturbations. Since its inception, DCM has been extended to other neuroimaging modalities, including electroencephalography (EEG) and magnetoencephalography (MEG), where it accounts for spatiotemporal dynamics of evoked and induced responses. For EEG/MEG, the framework incorporates electromagnetic forward models to map neuronal sources to sensor data, facilitating inferences about oscillatory coupling and phase interactions.^[4] Applications span cognitive domains such as attention, language processing, and motor control, and has been used in numerous studies to test theories of brain function. Recent advancements as of 2025 include nonlinear extensions for dense connectivity graphs, integrations with machine learning for group-level analyses, and applications in probabilistic programming languages and modeling complex neural networks.^[5]^[6]

Introduction

Definition and Principles

Dynamic causal modeling (DCM) is a Bayesian framework designed to infer effective connectivity among brain regions from neuroimaging data, such as functional magnetic resonance imaging (fMRI) or electroencephalography (EEG), by employing generative models that simulate directed influences between neuronal systems. This approach treats the brain as a nonlinear dynamic system perturbed by experimental inputs, generating observable responses through a forward model that links hidden neural states to measured signals.^[7] Unlike purely correlational methods, DCM explicitly models causal interactions, enabling the estimation of how activity in one region influences another under specific conditions. The core principles of DCM revolve around forward modeling, Bayesian inversion, and the clear demarcation from other forms of connectivity analysis. In forward modeling, neural dynamics are first specified as differential equations describing how hidden states evolve in response to inputs, which are then transformed into predicted observations via a biophysical observation model, such as a hemodynamic response function for fMRI.^[7] Bayesian inversion follows, where observed data are used to update prior beliefs about model parameters, yielding posterior distributions that quantify uncertainty in connectivity estimates. This distinguishes DCM from functional connectivity, which relies on undirected correlations without causal inference, and from structural connectivity, which maps anatomical pathways but ignores dynamic interactions.^[7] Effective connectivity in DCM captures context-dependent coupling between brain regions, where the strength of directed influences can be modulated by experimental or endogenous inputs, allowing for the investigation of task-specific or state-dependent network changes. At its foundation, the generative model posits that observed data y arise from hidden neural states x according to the equation

y = g(x, \theta) + \epsilon,

where g is the observation function, \theta represents the parameters governing connectivity (e.g., intrinsic coupling matrices), and \epsilon is additive measurement noise.^[7] The evolution of hidden states x is driven by a state equation incorporating inputs, enabling DCM to model bilinear modulations that reflect how experimental factors alter inter-regional influences.

History and Evolution

Dynamic causal modeling (DCM) was introduced in 2003 by Karl Friston and colleagues as a Bayesian framework for inferring effective connectivity from functional magnetic resonance imaging (fMRI) data, treating the brain as a nonlinear dynamical system perturbed by external inputs.^[8] This seminal work extended prior hemodynamic modeling approaches by incorporating bilinear approximations to capture context-dependent interactions among brain regions.^[9] Initial extensions to electroencephalography (EEG) and magnetoencephalography (MEG) occurred in 2006, with David et al. developing DCM for evoked responses using neural mass models to simulate cortical dynamics and forward models for electromagnetic fields.^[10] In 2006, further refinements included parametric empirical Bayes for lead field parameterization, enabling more robust inferences on hierarchical networks.^[11] Nonlinear DCM emerged in 2008, allowing second-order interactions at the neuronal level to model modulatory effects like attention on connectivity.^[12] In the 2010s, DCM evolved to address steady-state responses, with Moran et al. (2009) proposing spectral formulations based on Fokker-Planck equations for frequency-domain analyses of ongoing brain activity.^[13] Resting-state DCM was formalized in 2014 by Friston et al., adapting the framework to infer intrinsic connectivity fluctuations without external tasks, using stochastic inputs to model endogenous dynamics.^[14] As of 2025, recent advancements include integration with probabilistic programming languages, as detailed by Baldy et al., enabling scalable Bayesian inference via tools like Stan and Pyro for complex neural models.^[15] Multi-scale parcellation schemes have been proposed by Zarghami et al. on bioRxiv, facilitating hierarchical region definitions in DCM to bridge meso- and macro-scale brain organization.^[16] These developments underscore DCM's enduring influence, highlighted in the 2025 commemoration of the Statistical Parametric Mapping (SPM) software's 30-year milestone, where DCM remains a cornerstone for connectivity analyses.^[17]

Theoretical Foundations

Bayesian Framework

Dynamic causal modeling (DCM) employs a Bayesian framework to infer the parameters of generative models from observed neuroimaging data, treating model parameters θ as random variables.^[9] The posterior distribution over these parameters, given the data y and model m, is computed according to Bayes' theorem as p(θ|y, m) ∝ p(y|θ, m) p(θ|m), where p(y|θ, m) is the likelihood and p(θ|m) is the prior distribution.^[9] This approach enables the estimation of effective connectivity by integrating prior beliefs with the evidence provided by the data, facilitating robust inference even with noisy measurements typical in functional magnetic resonance imaging (fMRI) or electroencephalography (EEG). Priors in DCM play a crucial role in regularizing the inference process, particularly through hierarchical structures that encode anatomical and physiological knowledge about brain connectivity.^[18] For connectivity parameters, such as intrinsic coupling matrices, Gaussian priors are often specified with means centered at zero and variances tuned to ensure system stability, while hierarchical extensions allow subject-specific parameters to be drawn from group-level hyperpriors informed by diffusion tensor imaging tractography or known neuroanatomy.^[18] These priors prevent overfitting and incorporate domain-specific constraints, such as sparsity in long-range connections, thereby improving the biological plausibility of the estimated directed influences.^[7] To approximate the intractable posterior, DCM utilizes the variational free-energy principle, which provides a lower bound on the model evidence ln p(y|m). The free energy F is defined as
F = \ln p(y|m) - D_{\text{KL}}[q(\theta) \| p(\theta|y,m)],
where D_{\text{KL}} is the Kullback-Leibler divergence between an approximate variational density q(θ) and the true posterior, and F is minimized with respect to q(θ) to tighten the bound.^[19] This principle underpins model selection by approximating the evidence, balancing model fit and complexity. For posterior covariance estimation, the Laplace approximation assumes a Gaussian form around the maximum a posteriori (MAP) estimate, yielding the covariance matrix
\Sigma = \left( \frac{\partial^2 \ln p(\theta|y)}{\partial \theta^2} \right)^{-1},
computed as the inverse Hessian of the negative log-posterior at the mode, enabling efficient characterization of parameter uncertainty.^[19]

Generative Models of Brain Dynamics

Dynamic causal modeling (DCM) employs generative models to simulate the evolution of hidden neural states and their mapping to observed neuroimaging data, enabling inferences about effective connectivity in the brain. These models treat the brain as a nonlinear dynamical system perturbed by external inputs, where the forward process generates synthetic data that can be compared to empirical measurements. The core of the generative framework consists of neural state equations describing the temporal dynamics of neuronal activity and observation equations linking these states to measurable signals.00202-7) The neural dynamics are formalized through ordinary differential equations of the form \dot{x} = f(x, u, \theta), where x represents the vector of hidden neural states (such as membrane potentials or population activities across brain regions), u denotes exogenous inputs (e.g., sensory stimuli or tasks), and \theta encompasses model parameters governing the system's behavior. In DCM, \theta includes connectivity matrices: the intrinsic connectivity matrix A, which captures baseline coupling between regions; modulatory matrices B_j, which encode context- or input-dependent changes in connectivity; and the driving matrix C, which specifies direct influences of inputs on states. For fMRI applications, these dynamics are often approximated using a bilinear form derived from a first-order Taylor expansion around a steady-state point, \dot{x} = (A + \sum_j u_j B_j) x + C u, which linearizes nonlinear interactions while preserving essential modulatory effects. This approximation facilitates tractable simulations of regional interactions under experimental perturbations.00202-7)^[20] For EEG and MEG, DCM generative models incorporate canonical microcircuit architectures to represent local cortical dynamics, typically comprising interconnected excitatory and inhibitory neuronal populations. These models, often based on neural mass formulations, simulate four key subpopulations per source region: spiny stellate cells (excitatory input layer), inhibitory interneurons, superficial pyramidal cells (excitatory feedback), and deep pyramidal cells (excitatory output). Connectivity within and between these populations is parameterized to reflect hierarchical processing, with parameters in \theta modulating synaptic gains and delays to generate spatiotemporal patterns of electrical activity. This structure allows the model to capture oscillatory phenomena and directed influences at the circuit level.^[21] The observation component of the generative model bridges hidden states to data via y = g(x, \theta, \lambda) + z, where y is the observed signal, g is a nonlinear mapping function parameterized by \theta and observation-specific parameters \lambda (e.g., lead fields or hemodynamic responses), and z represents measurement noise. This equation encapsulates the forward process from neural activity to sensor measurements, ensuring the overall model predicts empirical time series under specified priors on parameters.00202-7)

Experimental Design

Task-Based Paradigms

Task-based paradigms in dynamic causal modeling (DCM) utilize controlled experimental manipulations to probe how specific stimuli and cognitive factors influence effective connectivity among brain regions. These paradigms typically involve presenting exogenous sensory inputs or tasks that drive neural activity, allowing researchers to test hypotheses about context-dependent interactions, such as how attention alters signal propagation in sensory hierarchies. By design, task-based approaches enable the isolation of driving effects from modulatory influences, providing a framework to infer causal mechanisms underlying observed neuroimaging signals. Factorial designs are particularly suited for DCM in task-based settings, as they facilitate the examination of main effects and interactions on connectivity parameters. For instance, a 2x2 factorial setup might cross sensory stimuli (e.g., visual vs. auditory) with cognitive modulators (e.g., attention vs. no attention), enabling the assessment of how attentional load changes coupling strengths between regions like the visual cortex and prefrontal areas. This design structure supports Bayesian model comparison to evaluate competing hypotheses, such as whether modulation occurs via top-down enhancement or bottom-up gating. Such approaches have been shown to enhance the sensitivity of DCM to detect subtle connectivity changes induced by experimental factors.^[22] In DCM, driving inputs, represented by the C matrix, capture the direct influence of exogenous stimuli on specific brain regions, modeling how task onsets perturb neural states from baseline. Conversely, modulatory inputs, encoded in the B matrix, reflect experimental factors that nonlinearly alter intrinsic connections, such as increased gain on forward connections during selective attention. These distinctions allow DCM to disentangle baseline coupling (A matrix) from task-induced perturbations, ensuring that inferred effective connectivity reflects experimentally controlled variations rather than endogenous fluctuations. The bilinear approximation underlying these matrices provides a computationally tractable way to estimate how stimuli propagate through networks. A representative application involves face-processing tasks, where DCM models differential responses to faces versus houses to infer top-down versus bottom-up influences in the ventral visual stream. In such paradigms, stimuli like faces drive activity in early visual areas (e.g., V1), while attentional instructions modulate connectivity from higher regions like the fusiform face area to infer hierarchical processing. Bayesian inversion of these models reveals, for example, strengthened backward connections under top-down conditions, supporting theories of predictive coding in perception. This task-based setup has demonstrated robust identifiability of parameters when contrasts are optimized to maximize signal variance across conditions. Design efficiency in task-based DCM paradigms is achieved by optimizing experimental contrasts to improve parameter identifiability and reduce estimation uncertainty. Efficient designs prioritize high-variance inputs that uniquely perturb targeted connections, such as rapid event-related sequences that deconvolve overlapping responses. Simulations indicate that balanced factorial layouts, with sufficient trials per condition, yield posterior estimates with narrow credible intervals, particularly for modulatory effects. This optimization ensures that DCM inferences are reliable, minimizing collinearity among parameters and enhancing the generalizability of findings across subjects.^[22]

Resting-State Approaches

Resting-state dynamic causal modeling (rsDCM) extends the standard DCM framework to analyze intrinsic brain fluctuations in the absence of external tasks or stimuli, treating these fluctuations as stochastic inputs that drive neuronal dynamics. Introduced in 2014, rsDCM models the endogenous variability observed in resting-state fMRI data by incorporating noise terms into the generative model, enabling inferences about effective connectivity from cross-spectral densities rather than evoked responses.^[14] This approach shifts the focus from task-induced changes to the baseline architecture of brain networks, capturing the slow fluctuations (typically 0.01–0.1 Hz) characteristic of resting states.^[14] At its core, rsDCM augments the deterministic state equations of classical DCM with Gaussian noise to represent endogenous perturbations. The neuronal dynamics are described by the stochastic differential equation:

\frac{dx}{dt} = f(x, \theta) + w,

where x denotes the state variables (e.g., neuronal activity), f(x, \theta) captures the deterministic evolution governed by parameters \theta, and w is zero-mean Gaussian noise with a specified covariance, often assuming a power-law spectral density to match the 1/f-like characteristics of resting-state signals.^[14] This formulation allows rsDCM to generate predicted functional connectivity through the propagation of these fluctuations across coupled regions, with model inversion performed using variational Bayes to estimate coupling parameters and their uncertainties.^[14] A prominent application of rsDCM involves inferring effective connectivity within the default mode network (DMN), a set of regions including the medial prefrontal cortex (mPFC), posterior cingulate cortex (PCC), and inferior parietal lobules that exhibit coordinated activity during rest. For instance, analyses of resting-state fMRI data have revealed directed influences such as from mPFC to PCC and from right inferior parietal lobule to both mPFC and PCC, highlighting asymmetric right-hemisphere dominance in DMN interactions.^[23] These findings demonstrate rsDCM's utility in elucidating the causal structure underlying intrinsic network coherence, with Bayesian model selection used to compare competing network topologies.^[23] Unlike task-based DCM, which emphasizes modulatory effects on connectivity induced by experimental inputs, rsDCM primarily estimates the baseline coupling matrix (A) that governs unconditional interactions between regions, without bilinear modulations from external drivers.^[14] This distinction allows rsDCM to probe the intrinsic repertoire of brain states, providing a complementary perspective to the context-dependent inferences from task paradigms.^[14]

Model Specification

Neural Models

In dynamic causal modeling (DCM), neural models begin with the specification of regions of interest (ROIs) that serve as nodes in the network, representing key brain areas whose interactions are hypothesized to underlie observed data. These ROIs are typically selected based on prior anatomical or functional knowledge, often derived from standardized atlases such as the Automated Anatomical Labeling (AAL) atlas or probabilistic functional parcellations from meta-analytic databases like the BrainMap or Neurosynth. For instance, in visual processing studies, ROIs might include primary visual cortex (V1) and motion-sensitive area V5/MT, defined by spherical volumes centered on peak coordinates from group-level activations or atlas labels to ensure reproducibility across subjects. This selection constrains the model to a manageable number of nodes, usually 4–8, to balance biological plausibility with computational feasibility.^[6]^[21] The core of the neural model is captured by connectivity matrices that parameterize the directed influences among ROIs, governed by a system of ordinary differential equations describing the evolution of hidden neural states. The matrix A encodes baseline or intrinsic connectivity, representing the fixed coupling strengths between regions in the absence of experimental perturbations, such as the default forward and backward connections in hierarchical sensory systems. The matrix B specifies modulatory effects, where experimental conditions (e.g., attentional tasks) alter the strengths in A, allowing context-dependent changes in effective connectivity. The matrix C defines direct input effects, modeling how exogenous stimuli (e.g., sensory inputs) drive specific ROIs without intermediary connections. To promote neurobiologically realistic sparsity—reflecting that not all regions are densely interconnected—prior distributions on these matrices impose shrinkage, often using Gaussian priors with zero means for absent connections or hierarchical priors informed by anatomical connectivity atlases like the CoCoMac database. This sparsity regularization prevents overfitting and favors parsimonious models aligned with known cortical hierarchies.^[3]^[21] Nonlinear extensions to these linear formulations enable the modeling of context-sensitive interactions, particularly through bilinear approximations that approximate higher-order dynamics without full nonlinearity. In bilinear DCM, activity in one region acts as a gating factor on connections between others, parameterized by an additional matrix D that captures multiplicative interactions (e.g., prefrontal activity modulating sensory gain in V1-to-V5 pathways during attention). This approach, introduced to handle phenomena like top-down modulation in perceptual inference, expands the state equations to second-order terms while maintaining tractability for Bayesian inference, and has been validated in applications such as face perception where nonlinear gating outperforms linear models in explaining evoked responses. Such extensions are particularly useful for capturing emergent behaviors in distributed systems without resorting to computationally intensive full nonlinear simulations.^[20] To explore uncertainty in neural architecture, DCM employs a hierarchical model space where families of models are defined by varying structural assumptions, such as the presence/absence of specific connections or input regimes. Each family partitions the parameter space—for example, one family assuming only forward connectivity versus another including bidirectional links—and Bayesian model selection or averaging is applied across families to identify the most plausible architecture at individual or group levels. This hierarchical approach, often using priors like the Dirichlet distribution for model probabilities, facilitates inference on overarching hypotheses (e.g., hierarchical versus parallel processing) by pooling evidence from multiple competing specifications, enhancing robustness in group studies.^[24]^[3]

Observation Models for fMRI

In dynamic causal modeling (DCM) for functional magnetic resonance imaging (fMRI), the observation model specifies the generative process that transforms hidden neural states into observable blood oxygenation level-dependent (BOLD) signals. This forward mapping is essential for inferring effective connectivity, as it accounts for the convoluted and nonlinear relationship between neuronal activity and measured hemodynamic responses. The model assumes that regional neural activity, derived from the underlying state equations, drives localized changes in cerebral blood flow, which in turn modulate vascular volume and deoxyhemoglobin concentration to produce the BOLD contrast.^[1] The core of this observation model is the balloon model of hemodynamics, which conceptualizes the cerebral vasculature—particularly postcapillary venules—as compliant balloons that expand in response to neural-induced vasodilation. Introduced by Buxton et al. and extended for nonlinear fMRI analyses, the model links neural input to blood flow dynamics, blood volume changes, and deoxyhemoglobin dissipation, with the BOLD signal emerging as a monotonic, nonlinear function of the intravascular-to-extravascular signal ratio. This framework captures key physiological features, such as flow-volume coupling via Grubb's law (where blood volume scales as flow raised to an exponent α ≈ 0.38) and oxygen extraction fraction adjustments during activation.^[1] The dynamics of the balloon model are described by a system of ordinary differential equations for the hemodynamic state variables: normalized blood flow f, blood volume v, and normalized deoxyhemoglobin content q. Neural activity n (typically the activity of excitatory neuronal populations) serves as the driving input. A key equation governs the rate of change in blood flow, incorporating neural drive, a decay term, and an autoregulatory feedback term:

\frac{df}{dt} = n - \kappa f - \gamma \left( f - \frac{1}{v} \right)

Here, \kappa is the decay rate of the vasodilatory signal (~0.4 s^{-1}), and \gamma is the autoregulation rate (~0.2 s^{-1}), which stabilizes flow around its resting value (normalized to 1) adjusted for volume. Complementary equations model volume and deoxyhemoglobin evolution, incorporating Grubb's law:

\frac{dv}{dt} = f v^{\alpha - 1} - \frac{v}{\tau}

\frac{dq}{dt} = \frac{ f E_0 / v - q f^{\alpha} / v }{\tau}

where \tau is the hemodynamic transit time (typically ~0.9 s), E_0 is the baseline oxygen extraction fraction (~0.34), and \alpha ≈ 0.38 from Grubb's law. The observed BOLD signal y for each region is then given by a weighted sum of hemodynamic states:

y = k_1 (1 - q) + k_2 \left(1 - \frac{q}{v}\right) + k_3 (1 - v) + \epsilon

with parameters k_1, k_2, k_3 reflecting magnetic field strength and tissue properties (e.g., for 1.5 T, k_1 = 7 \times 10^{-3}, etc.), and \epsilon as measurement noise. These equations ensure the model reproduces canonical hemodynamic response functions (HRFs) peaking ~5-6 s post-stimulus.^[1] To handle nonlinear interactions, such as supralinear flow-volume coupling or history-dependent responses, the balloon model employs Volterra kernels for convolution of neural activity with the HRF. The first-order kernel approximates the standard linear HRF, while higher-order (second- and third-order) kernels capture interactions between successive neural inputs, enabling DCM to model phenomena like post-stimulus undershoot or refractory effects without region-specific tuning. These kernels are derived analytically from the differential equations, providing a kernel expansion of the input-output mapping up to second order in most implementations. Bayesian estimation in DCM imposes priors on hemodynamic parameters to ensure physiological plausibility and stationarity across brain regions, reflecting the assumption that neurovascular coupling is regionally invariant. Specifically, parameters like \kappa (signal decay rate, prior ~0.4 s^{-1}), \gamma (autoregulation rate, ~0.2 s^{-1}), \tau (~0.9 s), \alpha (0.32), E_0 (0.34), and \epsilon (intravascular weight, ~0.34) receive Gaussian or log-normal priors centered on empirical values from invasive animal studies and human fMRI validations, with variances allowing modest deviation (~10-20% coefficient of variation). This stationarity simplifies model inversion while enabling group-level inferences on connectivity parameters. Neural states from the core DCM model are convolved with these hemodynamic dynamics to generate predicted BOLD time series for each region.^[1]

Observation Models for EEG/MEG

Observation models in dynamic causal modeling (DCM) for electroencephalography (EEG) and magnetoencephalography (MEG) describe the biophysical processes linking underlying neural activity to measured scalp signals, emphasizing the high temporal resolution of electromagnetic recordings compared to the slower vascular responses in functional magnetic resonance imaging (fMRI).^[4] These models integrate neural mass approximations of cortical dynamics with electromagnetic forward solutions to generate observable data, enabling inference on effective connectivity at millisecond scales.^[25] The core structure posits that scalp potentials or magnetic fields arise from synchronized postsynaptic currents in pyramidal cell populations, modeled as equivalent current dipoles.^[26] Central to these observation models are neural mass models (NMMs), which approximate the collective behavior of neuronal ensembles without resolving single-neuron spiking. The canonical NMM in DCM for EEG/MEG is the Jansen-Rit model, originally developed to simulate alpha rhythms and later adapted for evoked responses.^[25] This model represents each cortical source as three interconnected subpopulations: superficial pyramidal cells (excitatory output), spiny stellate cells (excitatory input), and inhibitory interneurons, with dynamics governed by mean membrane potentials and firing rates via sigmoid functions.^[26] The excitatory-inhibitory balance captures local amplification and suppression, producing oscillatory patterns observed in EEG/MEG, such as event-related potentials or induced rhythms.^[4] The link between these dipole sources and sensor measurements is provided by lead-field matrices, which encode volume conduction and magnetic induction effects. The observation equation is given by:

\mathbf{y}(t) = \mathbf{L} \mathbf{J}(t) + \boldsymbol{\epsilon}(t)

where \mathbf{y}(t) denotes the vector of EEG/MEG channel data at time t, \mathbf{L} is the lead-field matrix (derived from head geometry), \mathbf{J}(t) represents the dipole moments from neural sources, and \boldsymbol{\epsilon}(t) is additive sensor noise.^[25] The lead-field \mathbf{L} is computed using boundary element methods or realistic head models from structural MRI, projecting source activity onto sensor space while accounting for tissue conductivities.^[4] This forward model assumes quasi-static approximations, suitable for the frequencies (up to ~100 Hz) relevant to EEG/MEG.^[26] An advancement in these models is the canonical microcircuit (CMC), introduced in 2017 to incorporate layer-specific cortical dynamics for more biologically plausible simulations of laminar EEG/MEG signals.^[27] The CMC extends the Jansen-Rit framework to four populations—spiny stellate, superficial pyramidal, deep pyramidal, and inhibitory interneurons—reflecting the canonical cortical column with feedforward and feedback connections across layers.^[27] This structure allows DCM to estimate layer-resolved effective connectivity, such as excitatory inputs to granular layers and inhibitory modulation in supragranular regions, enhancing interpretations of source-specific contributions to scalp data.^[4] Source locations in these models are informed by priors derived from structural MRI, ensuring anatomical plausibility. Priors typically constrain dipoles to cortical gray matter surfaces segmented from individual MRIs, with initial positions guided by functional localizations from task data or atlases.^[28] This Bayesian approach incorporates uncertainty in location estimates, often tightening variances based on co-registered fMRI peaks or prior source reconstructions to improve model identifiability.^[29] Such priors mitigate ill-posedness in the inverse problem, facilitating robust inference on neural causes of observed electromagnetic fields.^[4]

Model Estimation

Variational Bayes Approximation

In dynamic causal modeling (DCM), the posterior distribution over model parameters given observed data and the model structure, p(\theta | y, m), is approximated using variational Bayes (VB) under a mean-field assumption. This approach factorizes the approximate posterior q(\theta) into independent marginals over parameter subsets, q(\theta) = \prod_i q_i(\theta_i), to render inference tractable for complex nonlinear generative models. By minimizing the Kullback-Leibler (KL) divergence between q(\theta) and the true posterior, VB provides a deterministic scheme for approximate Bayesian inference that balances model fit and complexity. The objective function in this framework is the variational free energy, defined as

F = \left\langle \ln p(y, \theta | m) \right\rangle_q - \left\langle \ln q(\theta) \right\rangle_q,

where the expectation \langle \cdot \rangle_q is taken with respect to q(\theta). This free energy serves as a lower bound on the log model evidence \ln p(y | m), and its maximization equates to minimizing the KL divergence \mathrm{KL}[q(\theta) || p(\theta | y, m)]. The first term captures the expected log joint density of data and parameters under the generative model, while the second is the entropy of the approximate posterior, promoting parsimonious approximations.^[30] Optimization proceeds iteratively through gradient-based updates on the free energy with respect to the sufficient statistics of q(\theta), typically the mean \mu and precision \Pi (inverse covariance). These updates employ a Gauss-Newton scheme, akin to an extended Kalman filter, to adjust \mu and \Pi until convergence:

\mu \leftarrow \mu - \Pi^{-1} \nabla_\mu F, \quad \Pi \leftarrow -\nabla_\mu^2 F.

This process inverts the DCM by yielding point estimates and uncertainties for parameters, with convergence guaranteed under the variational framework.^[30] The Laplace assumption underpins the Gaussian form of the approximate posteriors, approximating q_i(\theta_i) \sim \mathcal{N}(\mu_i, \Sigma_i) via a second-order Taylor expansion of the log joint around its mode. This local approximation handles the nonlinearities in DCM's generative models efficiently, providing conditional covariances that encode parameter uncertainties without requiring Monte Carlo sampling.

Parameter and Uncertainty Estimation

In dynamic causal modeling (DCM), the variational Bayes (VB) approximation provides posterior estimates of the model parameters, which characterize the effective connectivity among brain regions. These parameters include the intrinsic connectivity matrix \mathbf{A}, which encodes baseline coupling; the modulatory input matrix \mathbf{B}, which captures context-dependent changes; and the direct input matrix \mathbf{C}, which specifies exogenous influences on regional activity. The posterior distribution is approximated as a multivariate Gaussian, with the mean providing point estimates of these matrices and the covariance matrix quantifying parameter uncertainties. The conditional uncertainties arise from the posterior covariance, reflecting the precision of the estimates given the observed data and prior beliefs. Diagonal elements of this covariance matrix yield variances for individual parameters, such as the strength of a specific connection, while off-diagonal elements indicate correlations between parameters. To assess statistical significance, 95% credible intervals are computed from this Gaussian approximation, typically spanning \mu \pm 1.96 \sqrt{\Sigma_{ii}}, where \mu is the posterior mean and \Sigma_{ii} is the variance for parameter i. These intervals allow researchers to determine whether a connection strength differs reliably from zero or a prior expectation. DCM addresses inherent non-identifiability in connectivity estimation—where multiple parameter sets can produce similar observations—through informative priors and structural model constraints. Priors, such as Wishart distributions on \mathbf{A} to promote sparse or physiologically plausible connectivity, regularize the posterior and mitigate overfitting. Model constraints, like fixing certain connections to zero based on anatomical knowledge, further reduce ambiguity, ensuring interpretable estimates.00202-7) For instance, in attention tasks, DCM estimates the modulation strength in \mathbf{B} for connections like V1 to V5, where posterior means might indicate a positive attentional gain (e.g., 0.2 Hz increase in coupling), with 95% credible intervals excluding zero to confirm task-specific enhancement. Uncertainties here are often smaller for well-constrained modulatory parameters due to strong experimental designs that isolate attentional effects.00202-7)

Model Comparison

Bayesian Model Selection

In dynamic causal modeling (DCM), Bayesian model selection (BMS) enables the comparison of competing models to infer the most plausible causal architecture underlying neuroimaging data at the single-subject level. The core quantity for this comparison is the model evidence, denoted as p(y \mid m), which represents the probability of observing the data y given a specific model m. This evidence quantifies how well the model predicts the data while accounting for model complexity, allowing researchers to select among alternative hypotheses, such as different connectivity structures or input regimes. The model evidence is computationally approximated using the variational free energy F, such that p(y \mid m) \approx \exp(F), where F provides a lower bound on the log-evidence derived from variational Bayes inference. This approximation facilitates efficient model comparison by balancing the accuracy of the model's fit to the data against its complexity, as expressed in the decomposition \log p(y \mid m) = \text{Accuracy}(m) - \text{Complexity}(m). For comparing families of models—groups sharing common features like serial versus parallel processing—the evidences are aggregated to yield family-level posteriors, enabling robust inference even when individual model selection is sensitive to priors or data noise. Bayes factors provide a direct metric for pairwise model comparisons, defined as the ratio of evidences B_{ij} = p(y \mid m_i) / p(y \mid m_j) for nested models, where values greater than 3 indicate substantial evidence favoring model i over j. For non-nested models or families, this ratio extends naturally to assess relative support. Posterior probabilities over models are then obtained by applying a softmax function to the log-evidences, assuming equal prior probabilities: p(m_k \mid y) = \frac{\exp(\log p(y \mid m_k))}{\sum_j \exp(\log p(y \mid m_j))}, yielding probabilities that sum to 1 and reflect the updated belief in each model after observing the data. A key advantage of this Bayesian approach is its embodiment of Occam's razor, which automatically penalizes overly complex models through the complexity term in the evidence. This term incorporates the volume of the prior distribution over parameters, such as \frac{1}{2} \log |C_p| where C_p is the prior covariance, effectively favoring parsimonious models that explain the data without unnecessary parameters. In DCM applications, this prevents overfitting in scenarios with sparse connectivity hypotheses, ensuring selected models generalize well to the observed brain dynamics.

Group-Level Inference

Group-level inference in dynamic causal modeling (DCM) addresses the need to generalize findings from individual subjects to populations, accounting for inter-subject variability through hierarchical Bayesian frameworks.^[31] Unlike single-subject analyses, which compute log-model evidences for each participant, group-level methods pool these evidences or posterior parameters to infer population-level effects, such as the prevalence of specific models or differences in connectivity parameters between clinical and control groups.^[32] This approach is essential for studying heterogeneous populations, enabling robust conclusions about neural mechanisms at the cohort level.^[33] Random-effects Bayesian model selection (RE-BMS) is a primary method for group-level model comparison in DCM, treating log-model evidences from individual subjects as random samples from a group distribution rather than assuming uniformity (as in fixed-effects BMS).^[31] In RE-BMS, a hierarchical model is fitted to the subjects' log-evidences using variational Bayes, estimating the posterior probability that a particular model is the most frequent in the population.^[34] Exceedance probabilities, derived from this posterior, quantify the likelihood that one model exceeds all others in prevalence across the group, providing a protected measure against multiple comparisons when evaluating families of models.^[24] For instance, RE-BMS has been applied to compare connectivity models in healthy versus patient cohorts, revealing group-specific patterns without requiring identical model fits for every subject.^[31] Parametric empirical Bayes (PEB) complements RE-BMS by enabling hierarchical inference on model parameters rather than entire models, treating individual posterior means as observations in a second-level linear model with group priors.^[33] PEB estimates group mean parameters and between-subject variances using empirical Bayes, allowing tests for differences in effective connectivity or modulation strengths across populations via posterior probabilities or t-contrasts.^[32] This method is particularly suited for quantifying subtle group effects, such as altered synaptic gains, while incorporating uncertainty from first-level estimations.^[35] In clinical applications, such as schizophrenia research, group-level DCM inference has identified disrupted effective connectivity. For example, DCM analyses have revealed attenuated fronto-thalamic coupling in patients with delusions, with posterior probabilities exceeding 0.99 indicating significant group differences.^[36] Recent advancements as of 2025 include transformer-aided approaches for scalable group-level estimation in large-scale networks.^[37] These findings underscore DCM's utility in delineating disorder-specific network alterations at the population level.^[36]

Validation and Applications

Validation Techniques

Face validity in dynamic causal modeling (DCM) is established through simulations where known neuronal architectures and parameters are used to generate synthetic data, allowing assessment of whether DCM can recover the imposed coupling strengths and connectivity patterns. In early validation efforts, Bayesian estimation procedures successfully identified predefined bilinear modulations and intrinsic connections in three-region models, demonstrating robustness to added noise levels up to 2 units and temporal misalignments of ±1 second. These simulations confirm that DCM's inference mechanism accurately detects what it is designed to estimate, providing a foundational check on the method's internal consistency.^[7] Construct validity assesses DCM's alignment with independent anatomical and experimental evidence, such as diffusion-weighted imaging tractography or targeted neural perturbations. Tractography-derived priors have been integrated into DCM to inform plausible connectivity graphs, enhancing model realism by constraining effective connectivity estimates to respect white-matter pathways observed in probabilistic fiber tracking.^[38] For instance, comparisons with structural equation modeling and Volterra kernel analyses in attentional tasks have shown DCM to reliably capture backward connection modulations, consistent with established neurophysiological principles of hierarchical processing. Such convergences validate DCM's theoretical framework against complementary techniques, ensuring inferences reflect biologically plausible mechanisms.^[2] Predictive validity evaluates DCM's ability to forecast unobserved data, such as held-out responses in neuroimaging time series. Applied to fMRI data from repeated single-word reading tasks, DCM yielded stable estimates of forward connectivity hierarchies (e.g., from auditory to word-form areas), with predicted responses matching empirical signals at noise levels of 0.8-1%. Further support comes from cross-validation with invasive measures, where DCM predictions of synaptic gain changes aligned with microdialysis and electrophysiological recordings in animal models.^[2] This capacity to anticipate brain responses beyond fitted data underscores DCM's utility for hypothesis testing in effective connectivity.^[7] Post-2017 studies have emphasized sensitivity analyses to evaluate prior influences on DCM inferences, particularly in Bayesian parameter estimation. In analyses of alpha power modulation in visual cortex using EEG, priors for intrinsic connectivity (e.g., log-normal distributions with mean 0 and variance 0.25) were perturbed via Jacobian computations, revealing that local inhibitory parameters exerted the strongest effects on spectral features around 10 Hz.^[39] These assessments, averaging impacts across subjects by incrementing parameters (e.g., by e^{-6}), demonstrated robustness of extrinsic versus intrinsic modulations, informing prior selection for reliable group-level inferences.^[39] Such techniques highlight how prior specifications shape uncertainty quantification without altering core model predictions.^[40]

Key Applications in Neuroscience

Dynamic causal modeling (DCM) has been extensively applied to investigate sensory processing in the visual cortex, particularly to infer the directionality of feedforward and feedback connections during perceptual tasks. In studies of visual perception, DCM analyses of fMRI data have revealed that feedforward connections from primary visual cortex (V1) to higher areas like V4 are strengthened during stimulus-driven processing, while feedback from lateral occipital cortex to V1 modulates object recognition and predictive coding mechanisms. For instance, during silent reading tasks, DCM demonstrated top-down predictive signals from higher cortical areas suppressing sensory responses in V1, supporting hierarchical models of perception. Similarly, in electrocorticographic recordings, spectral asymmetries in forward and backward connections were quantified, showing faster forward propagation for exogenous stimuli and slower feedback for endogenous attention.^[41]^[42]^[43] In cognitive control, DCM has elucidated modulation within prefrontal networks during decision-making, highlighting dynamic interactions that underpin adaptive behavior. Analyses of lateral prefrontal cortex (LPFC) activity during inhibitory control tasks using transcranial magnetic stimulation combined with DCM showed that transient disruptions in LPFC connectivity impair downstream control signals to motor areas, providing causal evidence for its role in suppressing impulsive responses. In value-based decision-making, DCM revealed context-dependent strengthening of frontostriatal connections, where prefrontal regions exert top-down influence on striatal valuation processes to resolve conflicts between options. Recent extensions in 2025 have further demonstrated frontostriatal dynamics in cognitive control, with DCM identifying task-specific modulations that enhance flexibility in uncertain environments, including predictions of individual differences in response speed and age from task-evoked effective connectivity.^[44]^[45]^[46]^[47] Clinically, DCM has uncovered altered connectivity patterns in neurodegenerative and psychiatric disorders, offering insights into pathological mechanisms. In Alzheimer's disease, DCM of fMRI data from patients at different progression stages showed reduced effective connectivity from the hippocampus to posterior cingulate cortex, reflecting impaired memory retrieval and default mode network disruption; as of 2025, longitudinal magnetoencephalography (MEG) analyses have further revealed neurophysiological progression through disrupted oscillatory coupling.^[48]^[49] For schizophrenia, spectral DCM analyses of resting-state networks indicated dysconnections in frontotemporal circuits, with weakened feedback inhibition contributing to predictive coding deficits and auditory hallucinations. These findings align with broader evidence of hippocampal-prefrontal decoupling in early psychosis, where DCM quantified diminished top-down regulation.^[50]^[51] Recent applications in 2025 have expanded DCM to social neuroscience, particularly examining cerebello-cerebral interactions during action observation. A large-scale DCM study involving 99 participants across four datasets revealed enhanced effective connectivity from Crus II in the cerebellum to prefrontal areas during social navigation tasks, with modulations triggered by social norm violations, underscoring the cerebellum's role in mentalizing. Extensions of active vision frameworks using DCM have also progressed, integrating real-time fMRI to model predictive eye movements, where feedback loops between parietal and visual cortices adapt to dynamic scenes, building on earlier empirical validations.^[52]^[53]

Limitations and Future Directions

Methodological Constraints

Dynamic causal modeling (DCM) is inherently hypothesis-driven, requiring researchers to specify models a priori based on theoretical assumptions about neural interactions and experimental manipulations, which precludes its use for purely exploratory analyses of brain connectivity.^[54] This approach ensures mechanistic interpretability but limits DCM to testing predefined causal hypotheses rather than discovering novel network structures from data.^[55] The framework relies on bilinear approximations to model neuronal dynamics, which linearize nonlinear interactions into extrinsic inputs, intrinsic couplings, and modulatory effects, potentially failing to capture highly nonlinear or multistable cortical processes.^[54] Such approximations assume instantaneous and deterministic interactions, oversimplifying complex dynamics like those involving recurrent or chaotic neural states.^[55] For instance, in multistable systems, DCM may inadequately evaluate switching between stable states due to these linear constraints. DCM exhibits sensitivity to the selection of regions of interest (ROIs), where outcomes depend on how ROIs are defined—typically using eigenvariates from activation peaks—which can introduce variability if spatial precision or homogeneity assumptions are violated.^[54] Parameter estimation is further influenced by prior distributions, such as Gaussian shrinkage priors on coupling strengths that enforce sparsity and stability but may bias results if priors do not align with true underlying physiology.^[55] Scalability poses a significant constraint for standard DCM, as the number of possible connections grows quadratically with the number of regions, leading to exponential increases in computational demands and parameter estimation challenges for networks exceeding 10 regions.^[56] Pre-2025 critiques highlighted this as a barrier to whole-brain analyses, restricting applications to small-scale circuits despite the method's flexibility in connectivity specification.^[57] Recent extensions have begun addressing these issues through scalable approximations, though core methodological limits persist.^[55]

Emerging Challenges and Extensions

One prominent challenge in advancing dynamic causal modeling (DCM) lies in integrating multi-modal data, such as combining calcium imaging, voltage-sensitive dye imaging, and blood-oxygen-level-dependent (BOLD) signals, which differ in spatial coverage, resolution, and temporal dynamics. This integration is complicated by partial observability and heterogeneous representations, often leading to inaccurate inferences of neural circuits without iterative refinement of priors across scales. To address this, a multi-modal multi-scale DCM (mms-DCM) framework has been proposed, employing shared neural state models with modality-specific observations and reciprocal Bayesian optimization to enhance connectivity estimation accuracy in virtual experiments.^[58] Another key hurdle is handling non-stationarity in brain signals, where time-varying connectivity and degeneracy in biological systems hinder model convergence and reliable causal inference. Non-stationarity arises from fluctuating neural dynamics, complicating the assumption of stable parameters in traditional DCM formulations. Recent approaches mitigate this by modeling time-varying effective connectivity through parametric expansions of DCM, enabling the capture of slow fluctuations in neuroimaging data without assuming ergodicity.^[59] Extensions to DCM have incorporated probabilistic programming languages (PPLs) like Stan, PyMC, and NumPyro to facilitate Bayesian inference on nonlinear ordinary differential equations describing brain dynamics. These PPLs leverage gradient-based Markov chain Monte Carlo (e.g., NUTS) and variational methods (e.g., ADVI) for faster posterior estimation, achieving up to 50% effective sample sizes and reducing computation times to minutes, while addressing multi-modality via hyperparameter tuning and chain stacking. Deep generative models have further extended DCM by enabling hypothesis generation for dynamic causal graphs, modeling time-varying interactions as superpositions of static graphs to handle non-stationarity and nonlinearity beyond linear assumptions. This approach improves F1-scores by 22-28% over baselines in synthetic and real brain data, uncovering state-dependent relationships linked to behavior.^[60] For whole-brain applications, multi-scale parcellation techniques have been developed using top-down Bayesian model comparison on DCMs for hierarchical partitioning. Innovations include naïve Bayesian model reduction for scalability to thousands of regions, revealing modular structures invariant across scales in empirical fMRI data.^[61] Looking ahead, real-time DCM holds promise for neurofeedback applications, where connectivity-based feedback could enable volitional control of brain networks during fMRI sessions, building on foundational implementations to support therapeutic interventions in disorders like anxiety. Recent advances in real-time fMRI neurofeedback underscore its potential for self-regulation, with ongoing efforts to integrate DCM for more precise, causal-targeted training, such as in co-adaptive EEG-fMRI fusion protocols as of May 2025.^[62]

Software Implementations

SPM Integration

Dynamic Causal Modeling (DCM) is primarily implemented within the Statistical Parametric Mapping (SPM) software suite, starting from SPM12, providing a comprehensive framework for Bayesian inference on effective connectivity in neuroimaging data.^[63] This integration enables users to model directed interactions among brain regions using generative models that link neuronal dynamics to observed signals like fMRI or EEG/MEG time series.^[63] Key features include a graphical user interface (GUI) for specifying model structures, such as defining regions of interest (ROIs), endogenous and exogenous connections, and input perturbations, which streamlines the setup process without requiring extensive scripting.^[63] Parameter estimation employs variational Bayes (VB) methods to approximate posterior distributions efficiently, balancing model fit and complexity through free-energy minimization.^[63] For model comparison, Bayesian model selection (BMS) and parametric empirical Bayes (PEB) frameworks facilitate hierarchical inference, allowing evaluation of competing hypotheses at individual and group levels.^[63] These tools, introduced in SPM8 and refined in SPM12, support both deterministic and stochastic formulations of DCM. The workflow for DCM in SPM is tightly integrated with standard preprocessing and general linear model (GLM) pipelines, ensuring seamless analysis from raw data to connectivity estimates. For fMRI, users first perform spatial realignment, slice-timing correction, normalization to a standard space, and smoothing, followed by GLM specification to identify task-relevant activations.^[63] Time series are then extracted from volumes of interest (VOIs) centered on GLM-derived coordinates, formatted as .mat files for DCM input.^[63] Similar pipelines apply to EEG/MEG data, involving source reconstruction and sensor-level preprocessing before DCM fitting.^[64] This end-to-end integration minimizes data handling errors and leverages SPM's batch system for automation across subjects.^[63] Recent versions of SPM, including SPM25 (released in 2025), support resting-state DCM (rsDCM), also known as spectral DCM (spDCM), for modeling intrinsic fluctuations using stochastic differential equations and cross-spectral densities in the frequency domain (0.01–0.1 Hz), integrated with SPM's fMRI preprocessing for group-level effective connectivity analyses without task inputs.^[65]^[14] Canonical microcircuit (CMC) models, refinements of Jansen-Rit neural mass models with four subpopulations (excitatory pyramidal cells, inhibitory interneurons), are available for cross-spectral density analyses in M/EEG, enabling detailed simulations of laminar-specific interactions.^[66]^[43] Bayesian inversion via variational Laplace approximations supports scalability for large datasets.^[67] Tutorials and community resources for SPM DCM are extensively provided by the Wellcome Centre for Human Neuroimaging (FIL) at UCL, including step-by-step guides for first- and second-level analyses using exemplar datasets like the "attention to visual motion" fMRI data.^[68] These materials cover GUI-based model specification, estimation, and inference, with scripts available on the SPM GitHub repository and annual courses at FIL UCL.^[69] Community support occurs via the SPM mailing list and workshops, fostering adoption in neuroscience research.^[70]

Alternative Toolboxes

The TAPAS toolbox, developed by the Translational Neuromodeling Unit at ETH Zurich, provides specialized tools for dynamic causal modeling (DCM) with a particular emphasis on resting-state functional magnetic resonance imaging (fMRI) data through its regression DCM (rDCM) implementation.^[71] This approach enables efficient inference on effective connectivity by regressing regional time series against noise and endogenous fluctuations, incorporating advanced priors such as hierarchical shrinkage to handle inter-regional variability and improve model stability in low-signal scenarios.^[72] rDCM supports massively parallel computations for group-level analyses, making it suitable for large datasets while maintaining Bayesian model inversion via variational Laplace methods.^[73] The VBA-M toolbox offers a general framework for variational Bayesian inference applicable to nonlinear models, including customizable implementations for DCM across neuroimaging and behavioral data.^[74] Hosted on GitHub, it allows users to define DCM structures through modular functions for model specification, simulation, and posterior estimation, emphasizing flexibility in prior selection and observation models without being tied to specific imaging modalities.^[75] This generality facilitates extensions to non-standard DCM variants, such as those incorporating stochastic differential equations for dynamic noise processes.^[76] In 2025, integrations of DCM with probabilistic programming languages (PPLs) have emerged, providing wrappers for frameworks like Stan, PyMC, and NumPyro to enable Hamiltonian Monte Carlo sampling and more flexible hierarchical modeling.^[5] These implementations, exemplified by open-source repositories for event-related potential (ERP) DCM, allow declarative specification of causal models in Python or R, supporting advanced features like automatic differentiation for gradient-based inference and posterior predictive checks. Benchmarks indicate that PyMC and Stan achieve comparable sampling efficiency to traditional variational methods for standard DCMs, with NumPyro offering advantages in scalability for high-dimensional connectomes due to its JAX backend.^[77] Compared to SPM's workflows tailored for neuroimaging-specific pipelines, VBA-M prioritizes algorithmic flexibility for bespoke model adaptations, while PPL wrappers enhance accessibility for interdisciplinary users beyond traditional MATLAB environments.^[74]^[5] TAPAS, in contrast, bridges specificity and efficiency for resting-state analyses, often outperforming general tools in computational speed for parallel fMRI inversions.^[72]

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