Bayesian Estimation of Dynamical Systems: An Application to fMRI

By Karl J. Friston. NeuroImage, 2002

332 citations recorded by the source corpus.

This paper presents a method for estimating the conditional or posterior distribution of the parameters of deterministic dynamical systems. The procedure conforms to an EM implementation of a Gauss-Newton search for the maximum of the conditional or posterior density. The inclusion of priors in the estimation procedure ensures robust and rapid convergence and the resulting conditional densities enable Bayesian inference about the model parameters. The method is demonstrated using an input-state-output model of the hemodynamic coupling between experimentally designed causes or factors in fMRI studies and the ensuing BOLD response. This example represents a generalization of current fMRI analysis models that accommodates nonlinearities and in which the parameters have an explicit physical interpretation. Second, the approach extends classical inference, based on the likelihood of the data given a null hypothesis about the parameters, to more plausible inferences about the parameters of the model given the data. This inference provides for confidence intervals based on the conditional density.

Subfields: Cognitive Neuroscience, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/9d35747ac893f696189bd7206c8f6b5193d8afe4

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In Cognitive Neuroscience

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Citations in the corpus

Dynamic causal modelling (2003, methodology): “The hemodynamic priors are based on those used in Friston (2002).”
A neural mass model for MEG/EEG: : coupling and neuronal dynamics (2003, unknown): “Friston / NeuroImage 20 (2003) 1743–1755”
Comparing Dynamic Causal Models using AIC, BIC and Free Energy (2012, methodology): “This paper uses DCM for fMRI with bilinear neurodynamics and an extended Balloon model (Friston, 2002) for the hemodynamics.”
Understanding DCM: Ten simple rules for the clinician (2013, unknown): “The intricacies of this estimation process are beyond the scope of this report; for those with a (strong) mathematical background, see (Supplementary Table 1 & Do and Batzoglou, 2008; Friston, 2002; Friston et al., 2003).”
Classical and Bayesian Inference in Neuroimaging: Applications (2002, methodology): “(GLS) scheme described in Worsley and Friston (1995). In this scheme one starts with any observation model that is premultiplied by some weighting or convolution matrix S to give”
Variational free energy and the Laplace approximation (2007, methodology): “This ensuing scheme is that described in Friston (2002) for nonlinear system identification and is implemented in spm _ nlsi.m.”
Learning and inference in the brain (2003, background): “(23) in the posterior mode analysis of non-linear dynamical systems is discussed in Friston (2002c).”
Comparing dynamic causal models (2004, methodology): “DCMs comprise a bilinear model for the neurodynamics and an extended Balloon model (Friston, 2002; Buxton et al., 1998) for the hemodynamics.”
Coupling functions:universal insights into dynamical interaction mechanisms (2017, background): “There are a number of methods which have inferred models that include amplitude coupling functions inherently (Friston, 2002; Smelyanskiy et al., 2005; Voss et al., 2004) or have preestimated most probable models (Berger and Pericchi, 1996), but without including explicit assessment of the coupling…”
Constrained linear basis sets for HRF modelling using Variational Bayes (2004, methodology): “Basis sets used previously range from a single canonical HRF plus its temporal derivative to a set of Gamma functions (Friston et al., 1998a). These basis functions are then separately convolved with the known stimulus to give the same number of regressors as there are basis functions for use in the”
DEM: A variational treatment of dynamic systems (2008, unknown): “In Friston (2002) we described such a scheme for deterministic systems that is almost identical to DEM but eschews the D-step (and generalised coordinates).”
Hierarchical Models in the Brain (2008, methodology): “The ensuing scheme is described in detail in [24], where it is applied to nonlinear hemodynamic models of fMRI time-series.”
Dynamic causal modelling of evoked responses in EEG/MEG with lead field parameterization (2006, unknown): “…and model comparison For a given DCM, say model m, parameter estimation corresponds to approximating the moments of the posterior distribution given by Bayes rule p h j y; m ð Þ¼ p yj h; m ð Þ p h; m ð Þ p yj m ð Þ : ð7Þ The estimation procedure employed in DCM is described in Friston (2002).”
Multilevel linear modelling for FMRI group analysis using Bayesian inference (2004, methodology): “…Bayesian analysis mean that we may make inference on physiological parameters of the haemodynamic response in the complex nonlinear balloon model (Friston, 2002), on spatial noise relationships in multivariate spatial autoregressive models of FMRI data (Woolrich et al., in press) or, in this…”
Modeling the hemodynamic response to brain activation. (2004, unknown): “The balloon model has been refined and compared with experimental data (Feng et al., 2001; Friston, 2002; Friston et al., 2000; Mildner et al., 2001; Obata et al., 2004; Toronov et al., 2003), and some errors in the original parameter estimates were recently corrected (Obata et al.”
Comparing hemodynamic models with DCM (2007, unknown): “The fMRI data are from the study by Stephan et al. (2003) and were obtained at 1.5 T and a TE of 66 ms.”
Nonlinear Dynamic Causal Models for fMRI (2008, unknown): “In the original bilinear DCM for fMRI, the state equations are integrated using a computationally efficient integration scheme based on the matrix exponential of the system's Jacobian, J(x) = ∂f / ∂x (for details, see Friston, 2002b; Friston et al., 2003).”