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 methodology and software framework for specifying models of neural dynamics, estimating their parameters and comparing their evidence. It enables hypotheses to be tested about the interaction of neural populations (effective connectivity) using functional neuroimaging data e.g., functional magnetic resonance imaging (fMRI), magnetoencephalography (MEG) or electroencephalography (EEG).

Motivation

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 and its parameters (i.e., effective connectivity) to be identified from observed data. Bayesian model comparison is used to select the best model(s) based on their their evidence, which can then be characterised in terms of their parameters (e.g. connection strengths). This enables one to test hypotheses about how brain regions communicate; e.g., whether an experimental manipulation modulates the coupling between neural populations.

Procedure

Experiments using DCM typically involve the following stages:

Experimental design. Formulate specific hypotheses and conduct a neuroimaging experiment to test those hypotheses.
Data preparation. Pre-process the acquired data (e.g. select relevant data features and remove confounds).
Model specification. Specify one or more forward models (DCMs) of how the data were caused.
Model estimation. Fit the model(s) to the data to determine their evidence and parameters.
Model comparison. Compare the evidence for the models using Bayesian Model Comparison, at the single-subject or group level, and inspect the parameters of the model(s).

Each of these steps is briefly reviewed below.

1. 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.

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

2. Data preparation

For fMRI analysis, summary timeseries are generated for each brain region of interest. For MEG or EEG analysis, the desired data features are selected - e.g. evoked potentials or induced responses.

3. 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. Various models have been developed for use with DCM and the experimenter selects their preferred model based on the types of hypothesis they wish to address and the type of data they have collected.

DCM for fMRI

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 driving and modulatory 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. and later extended by Friston et al. and Stephan et al. The neural model was extended to include the interaction of excitatory and inhibitory neural populations and non-linear influences of neural populations on the coupling between other populations.

Support for resting state analysis was first introduced in Stochastic DCM, which estimates both neural fluctuations and connectivity parameters in the time ___domain using a procedure called Generalized Filtering. A faster and more accurate solution was introduced which operates in the frequency ___domain, called DCM for Cross-Spectral Densities (CSD). Both of these can be applied to large-scale brain networks by using priors based on functional connectivity. Another recent development is Regression DCM. This also operates in the frequency ___domain, but linearizes the model under certain simplifications, such as a fixed (canonical) haemodynamic response function. The means that the model can be inverted rapidly as a General Linear Model and can be applied to large-scale brain networks.

DCM for EEG / MEG

EEG and MEG data can support the estimation of more biologically detailed neural models than fMRI, as they have richer dynamics with higher temporal resolution.

The predominant model is DCM for evoked responses. It is a biologically plausible neural mass model, which emulates the activity of a cortical area using three neuronal subpopulations assigned to granular and agranular layers. A population of excitatory pyramidal (output) cells receive inputs from inhibitory and excitatory populations of interneurons, via intrinsic connections (which are confined to the cortical sheet). Within this model, excitatory interneurons can be regarded as spiny stellate cells found predominantly in layer four and in receipt of forward connections. Excitatory pyramidal cells and inhibitory interneurons are considered to occupy agranular layers and receive backward and lateral inputs.

A short paragraph on on the CMC model please? Same length as the one above.

5. Model comparison

Model inversion or estimation is implemented in DCM using variational Bayesian methods and provides two useful quantities. The log marginal likelihood or model evidence $\ln {p(y|m)}$ 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 $F$ . Hypotheses are tested using Bayesian model comparison, which involves comparing the evidence for different models based on their free energy. Model estimation also provides estimates of the parameters $p(\theta |y)$ , for example the connection strengths, which maximise the free energy.

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) and Parametric Empirical Bayes (PEB). 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 PEB approach 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.

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