Dynamic causal modeling

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Author: Dr. Andre Marreiros, Wellcome Trust Centre for Neuroimaging, London, UK
Author: Dr. Klaas Enno Stephan, Wellcome Trust Centre for Neuroimaging, London, UK
Author: Dr. Karl J. Friston, Wellcome Department of Imaging Neuroscience, London, UK

1 Motivation
2 DCM for fMRI
3 DCM for evoked responses
4 Model evidence and selection
5 Applications: fMRI
6 Applications: Evoked responses
7 Hierarchical model comparison
8 DCM developments
9 Recommended reading
10 References
11 External links
12 See also

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Motivation

The aim of dynamic causal modelling (DCM) is to make inferences and estimate the causal architecture of distributed dynamical systems. It rests on comparing models of how data were generated, where these dynamic causal models are formulated in terms of stochastic or ordinary differential equations (i.e., nonlinear state-space models in continuous time). These equations model the dynamics of hidden states in the nodes of a probabilistic graphical model, where conditional dependences are parameterised in terms of directed effective connectivity. Unlike Bayesian Networks the graphs used in DCM can be cyclic. Unlike Structural Equation modelling and Granger causality, DCM does not depend on the theory of Martingales (i.e., does not assume random fluctuations are serially uncorrelated).

DCM has been developed and applied principally to coupling among brain regions and how that coupling is influenced by experimental changes (e.g. time or cognitive set). The basic idea is to construct reasonably realistic models of interacting (cortical) regions or nodes. These models are then supplemented with a forward model of how the hidden states of each node (e.g., neuronal activity) map to 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 in terms of its evidence (inference on model-space), which can then be characterised in terms of its parameters (inference on parameter-space). This enables one to test hypotheses about how nodes communicate; e.g., whether a neuronal populations exerts top-down modulation on another, in a task-specific fashion.

In functional neuroimaging, the data may be functional magnetic resonance (fMRI) measurements or electrophysiological (e.g., in magnetoencephalography or electroencephalography; MEG/EEG). Brain responses are evoked by known deterministic inputs (experimentally controlled stimuli) that embody designed changes in sensory stimulation or cognitive set. In the model, exogenous variables can change hidden states in one of two ways. First, they can elicit responses through direct influences on specific network nodes. This would be appropriate, for example, in modelling sensory evoked responses in early visual cortex. The second class of input exerts its effect vicariously, through a modulation of the coupling among nodes, for example, the influence of attention on processing of sensory information. The hidden states cover both neuronal activity and other neurophysiological or biophysical variables needed to form observed outputs. These outputs are measured hemodynamic or electromagnetic responses over brain regions or sensors considered. Bayesian inversion furnishes the marginal likelihood (evidence) of the model and the posterior distribution of its parameters (e.g. neuronal coupling strengths). The evidence is used for Bayesian model selection (BMS) to disambiguate between competing models and the posterior distribution of the parameters is used to characterise the model selected.

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DCM for fMRI

DCM for fMRI uses a simple (deterministic) model of neural dynamics in a network or graph of n interacting brain regions or nodes (Friston et al. 2003). It models the change of a neuronal state-vector x in time, where each region is represented by a single hidden state variable, using the following bilinear differential equation:

$\dot{x}=f(x,u,\theta)= Ax + \sum_{j=1}^m u_j B^{(j)} x + Cu$

$\text{[math]}$

$A= \frac{\partial f}{\partial x}\bigg|_{u=0} \; \quad\; B= \frac{\partial^2 f}{\partial x\,\partial u} \; \quad\; C= \frac{\partial f}{\partial u}\bigg|_{x=0}$

where $\dot{x}= dx/dt$ . This equation, resulting from a bilinear Taylor expansion, is an approximation to any model of how changes in neuronal activity in one region $x_i$ are caused by activity in the other regions. The matrix A represents the fixed coupling among the regions in the absence of exogenous input $u(t)$ . This can be thought of as the latent coupling in the absence of experimental perturbations. The matrices are effectively the change in latent coupling induced by the j-th input. They encode context-sensitive changes in A or, equivalently, the modulation of coupling by experimental manipulations. Because $B^{(j)}$ are second-order derivatives they are referred to as bilinear. Finally, the matrix C embodies the influences of exogenous input on neuronal activity. The parameters $\theta \supset \{A,B,C\}$ are the connectivity or coupling matrices that we wish to identify and define the functional architecture and interactions among brain regions at a neuronal level. Fig.1 summarises this bilinear state-equation and shows the model in graphical form.

Figure 1: (A) The bilinear state equation of DCM for fMRI. (B) An example of a DCM describing the dynamics in a hierarchical system of visual areas. This system consists of two areas, each represented by a single state variable $(x_1, x_2)$ . Black arrows represent connections, grey arrows represent exogenous inputs and thin dotted arrows indicate the transformation from neural states (blue colour) into hemodynamic observations (red colour); see Fig.2 for the hemodynamic forward model. The state equation system for this particular model is shown on the right. Adapted from (Stephan et al., 2007a).

DCM for fMRI combines this model of neural dynamics with an empirically validated hemodynamic model that describes the transformation of neuronal activity into a BOLD response. This so-called “Balloon model” was initially formulated by (Buxton et al., 1998) and later extended by (Friston et al., 2000) and (Stephan et al., 2007c). Briefly, it consists of a set of differential equations that describe the relations between four hemodynamic state variables, using six parameters $\vartheta\subset\theta$ . More specifically, changes in neural activity elicit a vasodilatory signal that leads to increases in blood flow and subsequently to changes in blood volume and deoxyhemoglobin content. The predicted BOLD signal is a non-linear function of blood volume and deoxyhemoglobin content. This hemodynamic model is summarised in Fig.2 and described in detail in (Friston et al., 2000).

Figure 2: Summary of the hemodynamic model used by DCM for fMRI. Neuronal activity induces a vasodilatory and activity-dependent signal s that increases blood flow f. Blood flow causes changes in volume and deoxyhemoglobin ( $v$ and $q$ ). These two hemodynamic states enter an output nonlinearity, which results in a predicted BOLD response y. In its most recent version, this model has six hemodynamic parameters (Stephan et al., 2007c): the rate constant of the vasodilatory signal decay ( $\kappa$ ), the rate constant for auto-regulatory feedback by blood flow ( $\gamma$ ), transit time ( $\tau$ ), Grubb’s vessel stiffness exponent ( $\alpha$ ), capillary resting net oxygen extraction ( $E_0$ ), and ratio of intra-extravascular BOLD signal ( $\epsilon$ ). $E$ is the oxygen extraction function. This figure encodes graphically the transformation from neuronal states to hemodynamic responses; adapted from (Friston et al., 2003).

Together, the neuronal and hemodynamic state equations yield a deterministic DCM. For any given combination of parameters $\theta$ and inputs $u$ , the measured BOLD response $y$ is modelled as the predicted BOLD signal (the generalised convolution of inputs; $h(x,u,\theta)$ ) plus a linear mixture of confounds $X\beta$ (e.g. signal drift) and Gaussian observation error $\epsilon$ :

$y=h(x,u,\theta) + X\beta + \epsilon$

The combined neural and hemodynamic parameters $\vartheta \supseteq \{A,B,C,\vartheta\}$ are estimated from the measured BOLD data, using a Bayesian approach with empirical priors for the hemodynamic parameters and conservative shrinkage priors for the coupling parameters (see below). Once the parameters of a DCM have been estimated from measured BOLD data, the posterior distributions of the parameter estimates can be used to test hypotheses about connection strengths (e.g., Ethofer et al., 2006; Fairhall and Ishai, 2007; Grol et al., 2007; Kumar et al., 2007; Posner et al., 2006; Stephan et al., 2006; Stephan et al., 2007b; Stephan et al., 2005).

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DCM for evoked responses

DCM for evoked responses was developed as a biologically plausible model to understand how event-related responses result from the dynamics of coupled neural populations. It rests on neural mass models, which use established connectivity rules in hierarchical brain systems to describe the dynamics of a network of coupled neuronal sources (David and Friston, 2003; David et al., 2005; Jansen and Rit, 1995).

The DCM model developed by (David et al., 2006) uses the connectivity rules described in (Felleman and Van Essen, 1991) to assemble a network of coupled sources. Each source is modelled using a neural mass model described in (David and Friston, 2003), based on the model of (Jansen and Rit, 1995). This model emulates the activity of a cortical area using three neuronal subpopulations, assigned to granular and agranular layers. A population of excitatory pyramidal (output) cells receives inputs from inhibitory and excitatory populations of interneurons, via intrinsic connections (intrinsic connections 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.

Figure 3: Schematic of the DCM used to model electrophysiological responses. This schematic shows the state equations describing the dynamics of sources or regions. Each neuronal source is modelled with three subpopulations (pyramidal, spiny stellate and inhibitory interneurons) which are connected by four intrinsic connections with weights $\gamma_{1,2,3,4}$ , as described in (Jansen and Rit, 1995) and (David and Friston, 2003). These have been assigned to granular and agranular cortical layers which receive forward $A^{F}$ , backward $A^B$ and lateral $A^L$ connections respectively. Adapted from (Kiebel et al., 2008).

To model event-related responses, the network receives inputs via input connections. These connections are exactly the same as forward connections and deliver inputs to the spiny stellate cells. In the present context, inputs $u(t)$ model sub-cortical auditory inputs. The vector $C\subset\theta$ controls the influence of the input on each source. The lower, upper and leading diagonal matrices $A^{F},A^{B},A^{L}\subset\theta$ encode forward, backward and lateral connections, respectively. The DCM here is specified in terms of the state equations and a linear output equation

$\dot{x}=f(x,u,\theta)$

$\text{[math]}$

$y= L(\theta)x_0+\epsilon$

where $x_0$ represents the trans-membrane potential of pyramidal cells and $L(\theta)$ is a lead field matrix coupling electrical sources to the EEG channels (Kiebel et al., 2006).

Within each subpopulation, the evolution of neuronal states rests on two operators. The first transforms the average density of pre-synaptic inputs into the average postsynaptic membrane potential. This is modelled by a linear transformation with excitatory and inhibitory kernels parameterised by $H_{e,i}$ and $\tau_{e,i}$ . $H_{e,i}\subset\theta$ control the maximum post-synaptic potential, and $\tau_{e,i}\subset\theta$ represent a lumped rate-constant. The second operator S transforms the average potential of each subpopulation into an average firing rate. This is assumed to be instantaneous and is a sigmoid function (Marreiros et al., 2008a). Interactions, among the subpopulations, depend on constants $\gamma_{1,2,3,4}$ , which control the strength of intrinsic connections and reflect the total number of synapses expressed by each subpopulation.

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Model evidence and selection

Bayesian model selection (BMS) is a powerful method for determining the most likely among a set of competing hypotheses about the mechanisms that generated observed data. In the context of DCM, BMS is used to distinguish between different system architectures. Model comparison and selection rests on the model evidence $p(y|m)$ ; i.e. the probability of observing the data y under a particular model m. The model evidence is obtained by integrating out dependencies on the model parameters

$p(y|m)=\int p(y|\theta,m)p(\theta|m)d\theta$

In many cases, this integration is analytically intractable and numerically difficult to compute. Usually, it is therefore necessary to use computationally tractable approximations to the model evidence (or the log-evidence). Commonly used approximations include the Akaike information criterion (AIC), Bayesian information criterion (BIC), and the (negative) free-energy (F). The latter has several advantages and is the preferred choice for most applications (Stephan et al., 2009). One particularly useful aspect is that F is the objective function optimised during model inversion. For a given DCM, say model m, inversion corresponds to approximating the moments of the posterior or conditional distribution given by Bayes rule

$p(\theta|y,m)= \frac{ p(y|\theta,m)p(\theta|m)}{p(y|m)}$

The estimation procedure employed in DCM is described in (Friston et al., 2003). The posterior moments (mean and covariance) are updated iteratively using Variational Bayes under a fixed-form Laplace, (i.e., Gaussian), approximation to the conditional density, $q(\theta)$ . This can be regarded as an Expectation-Maximization algorithm; EM (Dempster et al., 1977) that employs a local linear approximation of the predicted responses around the current conditional expectation. The estimation scheme can be summarized as follows:

$\ \ E-Step:q \leftarrow \min_{q} F(q,\lambda,m)$

$\ M-Step:\lambda \leftarrow \min_{\lambda} F(q,\lambda,m)$

$\text{[math]}$

$F(q,\lambda,m)= \Big \langle lnq(\theta)-lnp(y|\theta,\lambda)-lnp(\theta|m) \Big \rangle_q$

$\qquad \qquad \ \ =KL \Big(q||p(\theta|y,\lambda)\Big) - ln \Big(p(y|\lambda,m)\Big)$

The free-energy is the Kullback–Leibler divergence (denoted by KL), between the real and approximate conditional density minus the log-evidence. This means that when the free-energy is minimised, the discrepancy between the true and approximate conditional density is suppressed. At this point the free-energy approximates the negative log-evidence: $F = -ln \Big ( p(y|\lambda,m) \Big )$ (Friston et al., 2007; Penny et al., 2004). Model selection is based on this approximation; where the best model is characterised by the greatest log-evidence (i.e. the smallest free-energy). Pairwise model comparisons can be conveniently described by Bayes factors (Kass and Raftery, 1995):

$BF_{i,j} = \frac {p(y|m_i)}{p(y|m_j)}$

For example, (Raftery, 1995) suggests interpretation of the BF as providing weak (BF < 3), positive (3 ≤ BF < 20), strong (20 ≤ BF < 150) or very strong (BF ≥ 150) evidence for preferring one model over another. From the equations above, it can be seen that the Bayes factor is simply the exponential of the difference in log-evidences.

The search for the best model precedes (and is often more important than) inference on the parameters of the model selected. Many studies have used BMS to adjudicate among competing DCMs for fMRI (Acs and Greenlee, 2008; Allen et al., 2008; Grol et al., 2007; Heim et al., 2009; Kumar et al., 2007; Leff et al., 2008; Smith et al., 2006; Stephan et al., 2007c; Summerfield and Koechlin, 2008) and EEG data (Garrido et al., 2008; Garrido et al., 2007).

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Applications: fMRI

Here, we briefly describe, as a practical example, the use of DCM for fMRI by analysing data acquired under a study of attentional modulation during visual motion processing (Büchel and Friston, 1997). These data have been used previously to validate DCM (Friston et al., 2003) and are available from http://www.fil.ion.ucl.ac.uk/spm/data. The experimental manipulations were encoded as three exogenous inputs: A ‘photic stimulation’ input indicated when dots were presented on a screen, a ‘motion’ variable indicated that the dots were moving and the ‘attention’ variable indicated that the subject was attending to possible velocity changes. The activity was modelled in three regions V1, V5 and superior parietal cortex (SPC). Three different DCMs are specified, each of which embody different assumptions about how attention modulates connectivity between V1 and V5. Model 1 assumes that attention modulates the forward connection from V1 to V5, model 2 assumes that attention modulates the backward connection from SPC to V5 and model 3 assumes attention modulates both connections. Each model assumes that the effect of motion is to modulate the connection from V1 to V5 and uses the same reciprocal hierarchical intrinsic connectivity. The models were fitted and the Bayes factors provided consistent evidence in favour of the hypothesis embodied in model 1, that attention modulates solely the forward connection from V1 to V5.

Figure 4: DCM applied to data from a study on attention to visual motion by (Büchel and Friston, 1997). In all models, photic stimulation enters V1 and motion modulates the connection from V1 to V5. All models have reciprocal and hierarchically organised connectivity. They differ in how attention (red colour) modulates the connectivity to V5; with model 1 assuming modulation of the forward connection (V1 to V5), model 2 assuming modulation of the backward connection (SPC to V5) and model 3 assuming both. The broken lines indicate the modulatory connections, adapted from (Penny et al., 2004).

Figure 5: Nonlinear DCM for fMRI applied to the attention to motion paradigm. Left panel: Numbers alongside the connections indicate the maximum a posteriori (MAP) parameter estimates. Right panel: Posterior density of the estimate for the nonlinear modulation parameter for the V1→V5 connection. Given the mean and variance of this posterior density, we can be 99.1% confident that the true parameter value is larger than zero or, in other words, that there is an increase in gain of V5 responses to V1 inputs that is mediated by parietal activity. Adapted from (Stephan et al., 2008).

Note that this model does not specify the source of the attentional top-down effect. This becomes possible with nonlinear dynamic causal models (Stephan et al. 2008). Nonlinear DCM for fMRI enables one to model how activity in one population gates connection strengths among others. Fig.5 shows an application to the previous example where parietal activity, induced by attention to motion, modulates the connection from V1 to V5.

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Applications: Evoked responses

To illustrate DCM for event-related responses (ERPs) we will use data acquired under a mismatch negativity (MMN) paradigm (http://www.fil.ion.ucl.ac.uk/spm/data). In this example, various models over twelve subjects are compared. The results shown are a part of a program that considered the MMN and its underlying mechanisms (Garrido et al., 2007). Three plausible models were specified under an architecture motivated by electrophysiological and neuroimaging MMN studies (Doeller et al., 2003; Opitz et al., 2002). Each has five sources, modelled as Equivalent Current Dipole (ECDs); (Kiebel et al., 2006), over left and right primary auditory cortex (A1), left and right superior temporal gyrus (STG) and right inferior frontal gyrus (IFG). An exogenous (auditory) input enters bilaterally to A1, which are connected to their ipsilateral STG. Right STG is connected with the right IFG. Inter-hemispheric (lateral) connections are placed between left and right STG. All connections are reciprocal (i.e., connected with forward and backward connections or with bilateral connections).

Three models were tested, which differed in the connections which could show putative repetition-dependent changes, i.e., differences between listening to standard or deviant tones. Models F, B and FB allowed changes in forward, backward and both, respectively. All three models were compared against a baseline or null model, which had the same architecture but precluded any coupling changes between standard and deviant trials.

Figure 6: Model specification. Sources are connected with forward (dark grey), backward (grey) or lateral (light grey) connections. A1: primary auditory cortex, STG: superior temporal gyrus, IFG: inferior temporal gyrus. Three different models were tested within the same architecture, allowing for repetition-related changes in forward F, backward B and forward and backward FB connections, respectively. The broken lines indicate the connections we allowed to change, adapted from (Garrido et al., 2007).

Figure 7: Bayesian model selection among DCMs for the three models, F, B and FB, expressed relative to a null model in which no connections changed. The graphs show the negative free-energy approximation to the log-evidence. (Left) Log-evidence for models F , B and FB for each subject (relative to the null). The diamond attributed to each subject identifies the best model on the basis of the subject’s highest log-evidence. (Right) Log-evidence at the group level, i.e., pooled over subjects, for the three models, adapted from (Garrido et al., 2007).

Bayesian model selection based on the increase in log-evidence over the null model was performed for all subjects. Log-evidence for the three models, relative to the null model (for each subject) reveals that they were significantly better in all subjects. The FB-model was significantly better in seven out of eleven subjects. The log-evidence for the group (which is the sum of the log-evidences over each subject’s independent data) showed that there was very strong evidence in favour of model FB.

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Hierarchical model comparison

Comparison at the between-subject level has been used extensively in previous group studies through group Bayes factors (GBF). GBF is simply the product of Bayes factors over subjects and constitutes a fixed-effects analysis. It has been used to decide between competing DCMs for fMRI (Acs and Greenlee, 2008; Allen et al., 2008; Grol et al., 2007; Heim et al., 2009; Kumar et al., 2007; Leff et al., 2008; Smith et al., 2006; Stephan et al., 2007c; Summerfield and Koechlin, 2008) and EEG data (Garrido et al., 2008; Garrido et al., 2007).

When the functional architecture is unlikely to differ across subjects, the conventional GBF is both sufficient and appropriate. However, when subjects exhibit different models or functional architectures, there is a random effect from models and a hierarchical procedure is required (Stephan et al., 2009). This rests on treating the model as a random variable and estimating the parameters of a Dirichlet distribution describing the probabilities of all models considered. These probabilities then define a multinomial distribution over model-space, allowing one to compute how likely it is that a specific model generated the data of a randomly chosen subject (and the exceedance probability of one model being more likely than any other).

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DCM developments

In contrast to many causal models, DCM does not operate on measured time-series directly. Instead, it combines a biophysical model of the hidden (latent) dynamics with a forward model that translates hidden states into predicted measurements; to furnish an explicit generative model how observed data were caused (Friston, 2009). This means the exact form of the DCM changes with each application and speaks to their progressive refinement:

Since its inception (Friston et al., 2003), a number of developments have improved and extended the DCMs; for fMRI, models of precise temporal sampling (Kiebel et al., 2007), multiple hidden states per region (Marreiros et al., 2008b), a refined hemodynamic model (Stephan et al., 2007c) and a nonlinear neuronal model (Stephan et al., 2008). DCM for EEG/MEG (David et al., 2006) has also seen rapid developments; DCM with lead-field parameterization (Kiebel et al., 2006), DCM for induced responses (Chen et al., 2008), DCM for neural-mass and mean-field models (Marreiros et al., 2009), DCM for spectral responses (Moran et al., 2009) and stochastic DCM (Daunizeau et al., 2009). A current review on developments for M/EEG data can be found in (Kiebel et al., 2009).

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Stephan, K.E., Marshall, J.C., Penny, W.D., Friston, K.J., Fink, G.R., 2007b. Interhemispheric integration of visual processing during task-driven lateralization. J Neurosci 27, 3512-3522.

Stephan, K.E., Weiskopf, N., Drysdale, P.M., Robinson, P.A., Friston, K.J., 2007c. Comparing hemodynamic models with DCM. Neuroimage 38, 387-401.

Stephan, K.E., Kasper, L., Harrison, L.M., Daunizeau, J., den Ouden, H.E., Breakspear, M., Friston, K.J., 2008. Nonlinear dynamic causal models for fMRI. Neuroimage 42, 649-662.

Stephan, K.E., Penny, W.D., Daunizeau, J., Moran, R.J., Friston, K.J., 2009. Bayesian model selection for group studies. Neuroimage 46, 1004-1017.

Summerfield, C., Koechlin, E., 2008. A neural representation of prior information during perceptual inference. Neuron 59, 336-347.

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External links

http://www.fil.ion.ucl.ac.uk/spm/

http://www.fmrib.ox.ac.uk/fsl/

http://www.sccn.ucsd.edu/eeglab/

http://afni.nimh.nih.gov/afni/

http://www.humanbrainmapping.org/

http://www.elsevier.com/wps/find/journaldescription.cws_home/622925/description#description

http://www3.interscience.wiley.com/cgi-bin/jhome/38751

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Invited by:	Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia
Action editor:	Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia

Dynamic causal modeling

From Scholarpedia

Contents

Motivation

DCM for fMRI

DCM for evoked responses

Model evidence and selection

Applications: fMRI

Applications: Evoked responses

Hierarchical model comparison

DCM developments

Recommended reading

References

External links

See also

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