Abstract
Dynamic causal modelling (DCM) was introduced to study the effective connectivity among brain regions using neuroimaging data. Until recently, DCM relied on deterministic models of distributed neuronal responses to external perturbation (e.g., sensory stimulation or task demands). However, accounting for stochastic fluctuations in neuronal activity and their interaction with task-specific processes may be of particular importance for studying state-dependent interactions. Furthermore, allowing for random neuronal fluctuations may render DCM more robust to model misspecification and finesse problems with network identification. In this article, we examine stochastic dynamic causal models (sDCM) in relation to their deterministic counterparts (dDCM) and highlight questions that can only be addressed with sDCM. We also compare the network identification performance of deterministic and stochastic DCM, using Monte Carlo simulations and an empirical case study of absence epilepsy. For example, our results demonstrate that stochastic DCM can exploit the modelling of neural noise to discriminate between direct and mediated connections. We conclude with a discussion of the added value and limitations of sDCM, in relation to its deterministic homologue.
Highlights
This article is about modelling distributed neuronal activity in the brain that is mediated by connections among different brain areas or sources
We reviewed the theoretical properties of stochastic dynamical systems, in terms of the impact that state noise can have on brain network dynamics
We reported a comprehensive evaluation of the respective system identification ability of stochastic and deterministic Dynamic causal modelling (DCM)
Summary
This article is about modelling distributed neuronal activity in the brain that is mediated by connections among different brain areas or sources. Note that standard stochastic differential equations (and related Ito calculus) rely upon π being a mixture of Wiener processes, which are non differentiable functions of time (Kloeden and Platen, 1999) This corresponds to a special case of generalized state noise π~ whose high order motion π_ ; π€ ; ... The effect of random fluctuations on the dynamical behaviour of a system depends upon the nature of the system itself: it turns out that some systems are so simple that any trajectory of states – in their stochastic form – resembles the deterministic path when the state noise variance is sufficiently small (in the limit Ψπ → 0). For two-dimensional systems, the local stability matrix J * can be derived analytically and has the following form (see Equation 14 in Ali and Menzinger, 1999):
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