Encyclopedia of Computational Neuroscience

Living Edition
| Editors: Dieter Jaeger, Ranu Jung

Dynamic Causal Modeling with Neural Population Models

  • Rosalyn MoranEmail author
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DOI: https://doi.org/10.1007/978-1-4614-7320-6_57-2

Keywords

Neuronal State Neural Mass Model Dynamic Causal Model Bayesian Inversion Average Membrane Potential 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Definition

Dynamic causal models with neural populations are state-space models that impose a specified dynamical systems form to the generation of neuroimaging and electrophysiological data. The forms of these models constitute neurobiologically motivated and identifiable parameterizations, where empirical observations offer conditional descriptions of parameter space following application of a Bayesian inversion scheme. They comprise separate generative processes at the neuronal level and at the observation level through a set of deterministic or stochastic differential equations and an observer functional, respectively.

Detailed Description

Dynamic causal models were first invented in 2003 for the purpose of estimating human brain connectivity and task-dependent functional integration using functional magnetic resonance imaging (fMRI) (Friston et al. 2003). The framework was extended for electrophysiological domains including noninvasive electroencephalography (EEG) and magnetoencephalography (MEG) and later for invasive local field potential recordings. Application domains share a common generative and inversion framework (Stephan et al. 2010). At the neuronal level, a network of “nodes” are connected with intrinsic (within-region) and extrinsic (region-to-region) connections. These effective or model-based connections constrain the output of a set of differential equations and can be modulated by experimental perturbation. These are modality-dependent, mesoscale descriptions of neuronal ensemble activity and are summarized through mean-field reductions. Different connectivity architectures can be considered as different models, embodying competing hypotheses to be formally tested. At the observation level, contributing neuronal states form inputs to a second dynamic process or static function (lead field), transforming neuronal activity to the required measurement space, e.g., blood oxygen level dependent – BOLD fMRI responses. Together, these form a complete forward model that is inverted, given real data, using Bayesian approaches.

Modality-Dependent Population Models

The neural population model describes the time evolution of unobservable neuronal states. For fMRI, x is an n × 1 column vector representing neuronal activity at n brain regions. This activity is under the control of inhibitory rate parameters, a ij j = i, describing the influence of regional activation on its own output activity and the excitatory or inhibitory influence, a ij j ≠ i, from other brain regions. This n × n matrix A is referred to as the endogenous or intrinsic connectivity matrix. A bilinear approximation B (j) is an n × n matrix used to capture effects from m, experimental input modulations, and C, an n x k matrix, represents the direct perturbation from k experimental inputs, giving
$$ \dot{x}=\left(A+{\displaystyle \sum_{j=1}^m{u}_j{B}^{(j)}}\right)\kern0.24em x+ Cu $$
(1)
Here, each state is a one-dimensional neural mass model representing the mean neuronal activity within a region encompassing approximately 10–100 voxels. Such a mean-field reduction is also applied to electrophysiological modalities where multiple-state models are used to represent several dimensions of unobservable dynamics within a single brain region (e.g., membrane depolarization changes at different inhibitory and excitatory cell ensembles), owing to their higher temporal resolution. For example, DCM for EEG employs (among other models) the Jansen and Rit formulation (Jansen and Rit 1995) to describe layer-specific synaptic responses, where inputs, u, are convolved with postsynaptic kernels, h, which can depolarize (e) or hyperpolarize (i) the postsynaptic neuronal ensemble to produce an average membrane potential, v:
$$ \begin{array}{l}\upsilon ={h}_{e/i}\otimes u\\ {}\;{h}_{e/i}={H}_{e/i}{\kappa}_{e/i}\;t \exp \left(-t{\kappa}_{e/i}\right)\end{array} $$
(2)
The parameter H tunes the maximum amplitude of PSPs and κ is a lumped representation of the sum of the rate constants of passive membrane and other spatially distributed delays in the dendritic tree. The convolution leads to a second-order differential equation (David and Friston 2003) that can be decomposed into normal form as
$$ \begin{array}{l}\dot{\upsilon}(t)=i(t)\\ {}\dot{i}(t)={H}_{e/i}{\kappa}_{e/i}\;u(t)-2{\kappa}_{e/i}x(t)-{\kappa}_{e/i}^2\;\upsilon (t)\end{array} $$
(3)
A sigmoidal operator, S, then transforms the average membrane potential into a population firing rate and allows us to substitute direct (thalamic) input, u, with afferent input from other cell populations. These inputs are parameterized with a connectivity parameter γ and can arise from within the local layered neural mass or from pyramidal cells of other connected brain regions (both having identical mathematical form), giving, for example,
$$ \begin{array}{l}\dot{\upsilon}=i\\ {}\dot{i}={\kappa}_e{H}_e\left(\upgamma S\;\left({\upsilon}_{aff}\right)+u\right)-2{\kappa}_ei-{\kappa}_e^2\upsilon \end{array} $$
(4)

Modality-Dependent Observers

Feature extraction from these neural population models is specified through an appropriate transform from the neural state space to measurement space. In the case of fMRI, this transform can be parameterized by a nonlinear “balloon model” (Buxton et al. 1998) describing the neuronal activity-induced dynamic alterations in vasodilation, blood flow, and subsequent volume and deoxyhemoglobin content – to predict the observed BOLD time series. For electrophysiological modalities, a static lead field transforms a mixture of cell membrane potentials from each source to scalp-level-evoked, scalp-level-induced, or correlational data features (Kiebel et al. 2006):
$$ h=L. Kx $$
(5)
where K describes the contribution from different cell ensemble potentials x, which is used to weigh, primarily, pyramidal cell contributions, and L is a lead-field matrix that accounts for passive conduction of their electromagnetic field and can be constrained along three orthogonal dimensions, when dipole orientations are not known.

Together with the neuronal state space, these transforms produce a full generative model with which mechanistic hypotheses regarding the physiological processes underlying empirical data observations can be performed.

Bayesian Inversion, Parameter Estimation, and Model Selection

The models described above produce the data likelihood and when combined with a set of priors on parameter space allow for a full Bayesian inversion. The inversions employed in DCM mostly use a variational expectation maximization routine (Šmídl and Quinn 2006) (stochastic routines employ generalized filtering). Here, a partition between the multivariate Gaussian model parameter space and the observation noise parameter (known as the hyperparameter) provides a scheme over which gradient ascent on a free energy objective function is amenable:
$$ F= \ln .25\mathrm{emp}\left(y\Big|m\right)- KL\left(q\left(\theta \right)\left|\right|p\left(\theta \Big|y,m\right)\right) $$
(6)
This quantity F thence returns an approximation to the model evidence, which is exact for linear models as well as an approximate conditional parameter density q. These two quantities can then be used to assess competing model architectures and the particular conditional (data: y, dependent) distributions on parameters, respectively.

Applications of DCM

The DCM approach has been applied to many experimental preparations and imaging modalities (a PubMed search reveals ~400 articles associated with dynamic causal modeling). As well as offering insights into the fundamental role of effective brain connections, e.g., how callosal connections support interhemispheric integration in a task-dependent manner (Stephan et al. 2005) or how feedback loops are necessary to support prototypical event-related potentials (Garrido et al. 2007), specific network topologies have been elucidated for cognitive and motor phenomena in health and disease. DCM for fMRI was used in one study to ascertain which regions in a prefrontal-subcortical network drive the so-called “status quo” motor-bias behavior (Fleming et al. 2010), finding a prefrontal-causality the most likely. During learning, similar networks have been shown to adapt along bottom-up axes, with changes in connections that support motor responsivity driven by modulation from subcortical structures (den Ouden et al. 2010). In disease, support for disconnection-related mechanisms has been observed in patients with schizophrenia (Dima et al. 2009) but also in neurological disorders more traditionally associated with focal lesions, such as Parkinson’s disease (Marreiros et al. 2012). As a “mathematical microscope,” electrophysiological DCMs have been applied to small, one-node networks to uncover particular synapses and receptors subtending memory performance (Moran et al. 2011). Overall, the methodology provides a principled Bayesian framework to address the mechanisms of functional brain architectures that support our mental repertoire.

Cross-References

References

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Virginia Tech Carilion Research InstituteRoanokeUSA
  2. 2.Bradley Department of Electrical & Computer EngineeringVirginia TechBlacksburgUSA
  3. 3.Department of Psychiatry and Behavioral MedicineVirginia Tech Carilion School of MedicineRoanokeUSA