  Dynamic causal modeling (DCM) is a Bayesian framework to infer directed
connections between compartments, and has been used to describe the
interactions between underlying neural populations based on functional
neuroimaging data. DCM is typically analyzed with the expectation-maximization
(EM) algorithm. However, because the inversion of a large-scale continuous
system is difficult when noisy observations are present, DCM by EM is typically
limited to a small number of compartments ($<10$). Another drawback with the
current method is its complexity; when the forward model changes, the posterior
mean changes, and we need to re-derive the algorithm for optimization. In this
project, we propose the Multiple-Shooting Adjoint (MSA) method to address these
limitations. MSA uses the multiple-shooting method for parameter estimation in
ordinary differential equations (ODEs) under noisy observations, and is
suitable for large-scale systems such as whole-brain analysis in functional MRI
(fMRI). Furthermore, MSA uses the adjoint method for accurate gradient
estimation in the ODE; since the adjoint method is generic, MSA is a generic
method for both linear and non-linear systems, and does not require
re-derivation of the algorithm as in EM. We validate MSA in extensive
experiments: 1) in toy examples with both linear and non-linear models, we show
that MSA achieves better accuracy in parameter value estimation than EM;
furthermore, MSA can be successfully applied to large systems with up to 100
compartments; and 2) using real fMRI data, we apply MSA to the estimation of
the whole-brain effective connectome and show improved classification of autism
spectrum disorder (ASD) vs. control compared to using the functional
connectome. The package is provided
\url{https://jzkay12.github.io/TorchDiffEqPack}
