In this work, we describe a general variational Bayesian approach to make approximate inference on nonlinear stochastic dynamic causal models [Daunizeau 2009]. This scheme extends established deterministic system identification [Friston 2003] to cope with random effects on the system's dynamics. This might be critical for efficiently estimating the network effective connectivity structure, which is central to any DCM analysis of neuroimaging data.

Model identification or inversion entails the estimation of the marginal likelihood or evidence of a model. Due to non-linearities in the evolution and observation functions, this is a difficult integration problem. Nevertheless, it can be finessed by optimising a free-energy bound on the evidence using results from variational calculus (cf [Friston 2007] for a comprehensive work on the Laplace approximation). This furnishes a deterministic update scheme that optimises an approximation to the posterior density on the unknown model variables. We derive such a variational Bayesian scheme in the context of nonlinear stochastic dynamic hierarchical models, for both model identification and time-series prediction.

The computational complexity of the scheme is comparable to that of an extended Kalman filter algorithm, which is critical when inverting high dimensional models and/or long time- series. Using Monte-Carlo simulations, we assess the asymptotic efficiency of this variational Bayesian approach. We also illustrate the self-consistency of the method and its long-term prediction power. Finally, we demonstrate the added-value of the method for assessing effective connectivity in structured spontaneous brain activity and its robustness in the context of the "missing region" problem.

This work is the first methodological attempt to extend existing DCM in order to partially cope with necessary simplifying assumptions when analyzing effective connectivity in the brain (e.g. summarizing the active network by a set of few connected regions). The results showed that the method performs well; attaining asymptotic efficiency for both the system's states and its parameters. Reaching asymptotic efficiency for the stochastic innovations per se might be of critical importance when accounting for unknown exogenous input to the system. In brief, the approach was proven able to retrieve the correct effective connectivity structure when the random effects do not induce phase transitions in the system's dynamics. It also allows for identifying effective connectivity in the context of spontaneous (endogenous) brain activity.

- Daunizeau J., Friston K. J., Kiebel S. J. ; Variational Bayesian identification and prediction of stochastic nonlinear dynamic causal models, Physica D (2009) 238: 2089-2118.
- Friston K. J., Harrison L., Penny W. ; Dynamic causal modelling, NeuroImage (2003) 19: 1273- 1302.
- Friston K. J., Mattout J., Trujillo-Barreto N., Ashburner J., Penny W.; Variational free energy and the Laplace approximation, NeuroImage (2007) 34: 220-234.