Dynamic Causal Modelling is a framework for fitting differential equation models of neuronal activity to brain imaging data using Bayesian inference. The DCM approach can be applied to functional Magnetic Resonance Imaging (fMRI), Electroencephalographic (EEG), Magnetoencephalographic (MEG), and Local Field Potential (LFP) data [22] (link). The empirical work in this paper uses DCM for fMRI. DCMs for fMRI comprise a bilinear model for the neurodynamics and an extended Balloon model [23] (link) for the hemodynamics. The neurodynamics are described by the following multivariate differential equation where indexes continuous time and the dot notation denotes a time derivative. The th entry in corresponds to neuronal activity in the th region, and is the th experimental input.
A DCM is characterised by a set of ‘exogenous connections’, , that specify which regions are connected and whether these connections are unidirectional or bidirectional. We also define a set of input connections, , that specify which inputs are connected to which regions, and a set of modulatory connections, , that specify which intrinsic connections can be changed by which inputs. The overall specification of input, intrinsic and modulatory connectivity comprise our assumptions about model structure. This in turn represents a scientific hypothesis about the structure of the large-scale neuronal network mediating the underlying cognitive function. A schematic of a DCM is shown inFigure 1 .
In DCM, neuronal activity gives rise to fMRI activity by a dynamic process described by an extended Balloon model [24] for each region. This specifies how changes in neuronal activity give rise to changes in blood oxygenation that are measured with fMRI. It involves a set of hemodynamic state variables, state equations and hemodynamic parameters, . In brief, for the th region, neuronal activity causes an increase in vasodilatory signal that is subject to autoregulatory feedback. Inflow responds in proportion to this signal with concomitant changes in blood volume and deoxyhemoglobin content . Outflow is related to volume through Grubb's exponent
[20] (link). The oxygen extraction is a function of flow where is resting oxygen extraction fraction. The Blood Oxygenation Level Dependent (BOLD) signal is then taken to be a static nonlinear function of volume and deoxyhemoglobin that comprises a volume-weighted sum of extra- and intra-vascular signals [20] (link)
where is resting blood volume fraction. The hemodynamic parameters comprise and are specific to each brain region. Together these equations describe a nonlinear hemodynamic process that converts neuronal activity in the th region to the fMRI signal (which is additionally corrupted by additive Gaussian noise). Full details are given in [20] (link),[23] (link).
In DCM, model parameters are estimated using Bayesian methods. Usually, the parameters are of greatest interest as these describe how connections between brain regions are dependent on experimental manipulations. For a given DCM indexed by , a prior distribution, is specified using biophysical and dynamic constraints [20] (link). The likelihood, can be computed by numerically integrating the neurodynamic (equation 1) and hemodynamic processes (equation 2). The posterior density is then estimated using a nonlinear variational approach described in [23] (link),[25] (link). Other Bayesian estimation algorithms can, of course, be used to approximate the posterior density. Reassuringly, posterior confidence regions found using the nonlinear variational approach have been found to be very similar to those obtained using a computationally more expensive sample-based algorithm [26] (link).
A DCM is characterised by a set of ‘exogenous connections’, , that specify which regions are connected and whether these connections are unidirectional or bidirectional. We also define a set of input connections, , that specify which inputs are connected to which regions, and a set of modulatory connections, , that specify which intrinsic connections can be changed by which inputs. The overall specification of input, intrinsic and modulatory connectivity comprise our assumptions about model structure. This in turn represents a scientific hypothesis about the structure of the large-scale neuronal network mediating the underlying cognitive function. A schematic of a DCM is shown in
In DCM, neuronal activity gives rise to fMRI activity by a dynamic process described by an extended Balloon model [24] for each region. This specifies how changes in neuronal activity give rise to changes in blood oxygenation that are measured with fMRI. It involves a set of hemodynamic state variables, state equations and hemodynamic parameters, . In brief, for the th region, neuronal activity causes an increase in vasodilatory signal that is subject to autoregulatory feedback. Inflow responds in proportion to this signal with concomitant changes in blood volume and deoxyhemoglobin content . Outflow is related to volume through Grubb's exponent
[20] (link). The oxygen extraction is a function of flow where is resting oxygen extraction fraction. The Blood Oxygenation Level Dependent (BOLD) signal is then taken to be a static nonlinear function of volume and deoxyhemoglobin that comprises a volume-weighted sum of extra- and intra-vascular signals [20] (link)
where is resting blood volume fraction. The hemodynamic parameters comprise and are specific to each brain region. Together these equations describe a nonlinear hemodynamic process that converts neuronal activity in the th region to the fMRI signal (which is additionally corrupted by additive Gaussian noise). Full details are given in [20] (link),[23] (link).
In DCM, model parameters are estimated using Bayesian methods. Usually, the parameters are of greatest interest as these describe how connections between brain regions are dependent on experimental manipulations. For a given DCM indexed by , a prior distribution, is specified using biophysical and dynamic constraints [20] (link). The likelihood, can be computed by numerically integrating the neurodynamic (equation 1) and hemodynamic processes (equation 2). The posterior density is then estimated using a nonlinear variational approach described in [23] (link),[25] (link). Other Bayesian estimation algorithms can, of course, be used to approximate the posterior density. Reassuringly, posterior confidence regions found using the nonlinear variational approach have been found to be very similar to those obtained using a computationally more expensive sample-based algorithm [26] (link).
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