Dynamic causal modeling

This sandbox is in the article namespace. Either move this page into your userspace, or remove the {{User sandbox}} template. Dynamic Causal Modelling (DCM) is a method and software framework for specifying models of neural dynamics, estimating their parameters and comparing their evidence ^[1]. It enables testing hypotheses about the interaction of neural populations (effective connectivity) using functional neuroimaging data e.g., functional magnetic resonance imaging (fMRI), magnetoencephalography (MEG), electroencephalography (EEG) or intracranial Local Field Potentials (LFP).

Procedure

DCM is used to estimate the coupling among brain regions and the changes in coupling due to experimental changes (e.g., time or context). The basic idea is to construct reasonably realistic models of interacting brain regions. These models are then supplemented with a forward model of how the hidden states of each brain region (e.g., neuronal activity) cause the measured responses. This enables the best model(s) and their parameters (i.e., effective connectivity) to be identified from observed data. Bayesian model comparison is used to compare models based on their their evidence, which can then be characterised in terms of their parameters (e.g. connection strengths).

Experiments using DCM typically involve the following stages ^[2]:

1. Experimental design. Specific hypotheses are formulated and a neuroimaging experiment is conducted to test them.
2. Data preparation. The acquired data are pre-processed (e.g. select relevant data features and remove confounds).
3. Model specification. One or more forward models (DCMs) are specified for each subject's data.
4. Model estimation. The model(s) are fitted to the data to determine their evidence and parameters.
5. Model comparison. Compare the evidence for the models using Bayesian Model Comparison, at the single-subject level or at the group level, and inspect the parameters of the model(s).

The key steps are briefly reviewed below.

Experimental design

Functional neuroimaging experiments are typically task-based or resting state. In task-based experiments, brain responses are evoked by known deterministic inputs (experimentally controlled stimuli) that embody designed changes in sensory stimulation or cognitive set. These experimental or exogenous variables can change neural activity in one of two ways. First, they can elicit responses through direct influences on specific brain regions. This would include, for example, sensory evoked responses in the early visual cortex. The second class of inputs exerts their effects vicariously, through a modulation of the coupling among nodes, for example, the influence of attention on the processing of sensory information. These two types of input - driving and modulatory - are parameterized separately in DCM. To enable efficient estimation of driving and modulatory effects, a 2x2 factorial experimental design is often used - with one factor modelled as the driving input and the other as the modulatory input ^[2].

Resting state experiments have no experimental manipulations within the period of the neuroimaging recording. Instead, the interest is in the endogenous fluctuations in brain connectivity during the scan, or in the differences in connectivity between scans or subjects. The DCM framework includes models and procedures for resting state data, described below.

Model specification

Dynamic Causal Models (DCMs) are nonlinear state-space models in continuous time, parameterized in terms of directed effective connectivity between brain regions. Unlike Bayesian Networks, DCMs can be cyclic, and unlike Structural Equation modelling and Granger causality, DCM does not depend on the theory of Martingales, i.e., it does not assume that random fluctuations' are serially uncorrelated.

All models in DCM have the following basic form:

${\displaystyle {\begin{aligned}{\dot {z}}&=f(z,u,\theta ^{(n)})\\y&=g(z,\theta ^{(h)})+\epsilon \end{aligned}}}$ 

The first line describes the change in neural activity ${\displaystyle z}$  with respect to time ${\displaystyle {\dot {z}}}$ . This is the hidden state of the brain, which cannot be directly observed using non-invasive functional imaging. The evolution of neural activity over time is controlled by neural function ${\displaystyle f}$  with parameters ${\displaystyle \theta ^{(n)}}$  and experimental inputs ${\displaystyle u}$ . The neural activity in turn causes the timeseries ${\displaystyle y}$ , written on the second line. This is controlled by observation function ${\displaystyle g}$  with parameters ${\displaystyle \theta ^{(h)}}$ . Observation noise ${\displaystyle \epsilon }$  completes the model. Of key interest to experimenters are the neural parameters ${\displaystyle \theta ^{(n)}}$  which, for example, represent the change in connection strengths due to experimental conditions.

Model specification requires selecting models ${\displaystyle f}$  and ${\displaystyle g}$  and setting appropriate priors on the parameters - e.g. selecting which connections should be switched on or off. The choice of model depends on the hypotheses to be tested and the type of data which is available. For example, with fMRI, ${\displaystyle f}$  is a simple differential equation model of neural coupling and ${\displaystyle g}$  is a detailed biophysical model of the BOLD response. The rest of this section surveys the models which have been developed for the DCM framework.

Functional MRI

The neural model in DCM for fMRI uses a simple mathematical device - a Taylor approximation - to capture the gross causal causal influences between brain regions and their change due to experimental inputs. This is coupled with a detailed biophysical model of the generation of the BOLD response and the MRI signal, based on the Balloon model of Buxton et al.^[3] which was extended for use in DCM for fMRI ^[4][5]. Extensions to the basic neural model enable the inclusion of interactions between excitatory and inhibitory neural populations ^[6] and non-linear influences of neural populations on the coupling between other populations^[7].

Support for resting state analysis was first introduced in Stochastic DCM^[8], which estimates both neural fluctuations and connectivity parameters in the time ___domain using a procedure called Generalized Filtering. A faster and more accurate solution for resting state data was introduced which operates in the frequency ___domain, called DCM for Cross-Spectral Densities (CSD) ^[9][10]. Both of these can be applied to large-scale brain networks by using priors based on functional connectivity^[11][12]. Another recent development is Regression DCM^[13] implemented in the Tapas software collection (see Software implementations). Regression DCM operates in the frequency ___domain, but linearizes the model under certain simplifications, such as having a fixed (canonical) haemodynamic response function. The enables the model to be inverted rapidly as a General Linear Model and so can be applied to large-scale brain networks.

EEG / MEG / LFP

EEG and MEG data can support the estimation of more biologically detailed neural models than fMRI, as their higher temporal resolution provide access to richer neural dynamics. The predominant model is DCM for evoked responses (DCM for ERP)^[14][15]. It is a biologically plausible neural mass model, building on the work of several earlier authors especially Jansen and Rit ^[16]. It emulates the activity of a cortical area using three neuronal sub-populations, each of which rests on two operators. The first transforms the pre-synaptic firing rate into a Post-Synaptic Potential (PSP), by convolving a synaptic response function (kernel) by the pre-synaptic input. As a result, this is referred to as a convolution model. The second operator, a sigmoid function, transforms the membrane potential into a firing rate of action potentials. A subsequent extension to this model, DCM for LFP (Local Field Potentials), added the effects of specific ion channels on spike generation ^[17].

A short paragraph on the CMC model please? We can then ask Rosalyn to add a paragraph on conductance based-models.

Model estimation

Model inversion or estimation is implemented in DCM using a variational Bayesian optimisation scheme^[18]. It provides two useful quantities. The log marginal likelihood or model evidence ${\displaystyle \ln {p(y|m)}}$  is the probability of observing of the given data under the model. This cannot be calculated exactly and in DCM it is approximated by a quantity called the negative variational free energy ${\displaystyle F}$  , referred to in machine learning as the Evidence Lower Bound (ELBO). Hypotheses are tested by comparing the evidence for different models based on their free energy, a procedure named Bayesian model comparison. Model estimation also provides estimates of the parameters ${\displaystyle p(\theta |y)}$ , for example the connection strengths, which maximise the free energy. Where models differ only in their priors, Bayesian Model Reduction can be used to rapidly the derive the evidence and parameters for nested or reduced models from a full model.

Model comparison

Neuroimaging studies typically investigate effects which are conserved at the group level, or which differ between subjects. There are two predominant approaches for group-level analysis: random effects Bayesian Model Selection (BMS) ^[19] and Parametric Empirical Bayes (PEB) ^[20]. Random effects BMS posits that subjects differ in terms of which model generated their data - e.g. drawing a random subject from the population, there would be a 25% chance their data were generated by model 1 and a 75% chance their data were generated by model 2. The analysis pipeline for the BMS approach procedure follows a series of steps:

1. Specify and estimate multiple DCMs per subject, where each DCM (or set of DCMs) embodies a hypothesis.
2. Perform random effects BMS to estimate the proportion of subjects whose data were generated by each model
3. Perform Bayesian Model Averaging, which is a weighted average over the parameters of the DCMs. This means that models greater probability contribute more to the average than those with lower probability.

The most recently developed PEB approach^[20] is a hierarchical model over parameters (connection strengths). It eschews the notion of different models at the level of individual subjects, and posits that people differ in the (continuous) strength of their individual connections. The PEB approach separates sources of variability in connection strengths across subjects into hypothesised covariates and uninteresting between-subject variability (random effects). The PEB procedure is as follows:

1. Specify a single 'full' DCM per subject which contains all connectivity parameters of interest.
2. Model the estimated parameters (the full posterior density) from all subjects using a Bayesian General Linear Model at the group level.
3. Test hypotheses by comparing the full group-level model to reduced group-level models where certain combinations of connections have been switched off.

Validation

Developments in DCM have been validated using three approaches.

• Face validity establishes whether the parameters of a model can be recovered from simulated data. This has been performed with the development of each new model (E.g. ^[1][7]).
• Construct validity assesses consistency with other analytical methods - for example Structural Equation Modelling ^[21] and other neurobiological computational models ^[22].
• Predictive validity assesses the ability to predict known or expected effects. This has included testing against iEEG / EEG / stimulation ^{[23][24][25][26]} and against known pharmacological treatments ^[27][28].

Limitations / drawbacks

DCM is a hypothesis-driven approach for investigating the interactions among pre-defined regions of interest. It is not ideally suited for exploratory analyses ^[2]. Although methods have been implemented for automatically searching over reduced models (Bayesian Model Reduction) and for modelling large-scale brain networks^[12], these methods expect clear hypotheses. Other approaches such as psycho-physical interactions (PPI) analysis may be more appropriate in contexts with less strong hypotheses.

The variational Bayesian methods used for model estimation used approximations based on the Laplace approximation that the parameters are normally distributed. This approximation can break down in the context of highly non-linear models, such as those used in EEG / MEG analysis, where local minima can preclude the free energy from serving as a close lower bound on log model evidence. Sampling approaches provide the gold standard, however are time consuming to run, and have been used to validate variational approximations in DCM ^[29].

Software implementations

DCM is implemented in the Statistical Parametric Mapping software package, where it serves as the canonical or reference implementation (http://www.fil.ion.ucl.ac.uk/spm/software/spm12/). It has been re-implemented and developed in the Tapas software collection (https://www.tnu.ethz.ch/en/software/tapas.html) and the VBA toolbox (http://mbb-team.github.io/VBA-toolbox/).

References

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18. Friston, K.; Mattout, J.; Trujillo-Barreto, N.; Ashburner, J.; Penny, W. (2007), "Variational Bayes under the Laplace approximation", Statistical Parametric Mapping, Elsevier, pp. 606–618, ISBN 9780123725608
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26. Daunizeau, J.; Lemieux, L.; Vaudano, A. E.; Friston, K. J.; Stephan, K. E. (2013). "An electrophysiological validation of stochastic DCM for fMRI". Frontiers in Computational Neuroscience. 6. doi:10.3389/fncom.2012.00103. ISSN 1662-5188.
27. Moran, Rosalyn J.; Symmonds, Mkael; Stephan, Klaas E.; Friston, Karl J.; Dolan, Raymond J. (2011-08). "An In Vivo Assay of Synaptic Function Mediating Human Cognition". Current Biology. 21 (15): 1320–1325. doi:10.1016/j.cub.2011.06.053. ISSN 0960-9822. {{cite journal}}: Check date values in: |date= (help); no-break space character in |first3= at position 6 (help); no-break space character in |first4= at position 5 (help); no-break space character in |first5= at position 8 (help); no-break space character in |first= at position 8 (help); no-break space character in |title= at position 6 (help)
28. Moran, Rosalyn J.; Jung, Fabienne; Kumagai, Tetsuya; Endepols, Heike; Graf, Rudolf; Dolan, Raymond J.; Friston, Karl J.; Stephan, Klaas E.; Tittgemeyer, Marc (2011-08-02). "Dynamic Causal Models and Physiological Inference: A Validation Study Using Isoflurane Anaesthesia in Rodents". PLoS ONE. 6 (8): e22790. doi:10.1371/journal.pone.0022790. ISSN 1932-6203.
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