Efficient gradient computation for dynamical models.
Bottom Line: This is particularly true for systems where the number of parameters is greater than the number of states.For such systems, integrating several sensitivity equations - as required with forward sensitivities - proves to be most expensive, while finite-difference approximations have an intermediate efficiency.In the context of neuroimaging, adjoint based inversion of dynamical causal models (DCMs) can, in principle, enable the study of models with large numbers of nodes and parameters.
Affiliation: Wellcome Trust Centre for Neuroimaging, Institute of Neurology, University College London, 12 Queen Square, London WC1N 3BG, UK. Electronic address: email@example.com.Show MeSH
Mentions: First we consider the linear models(25)x˙=Axwhere x is a D-dimensional state vector with initial value x0 = 1, and A is a D × D connectivity matrix. Readers familiar with DCM for fMRI will recognize A as the endogenous or average connectivity matrix. A model with D states therefore has P = D2 parameters (Fig. 2(A)). The system is integrated from time 0 to T. To ensure stability, we constructed A using the linear expansion(26)A=∑d=1DqdvdvdTwhere vd ∼ (vd; 0, 1) are standard D-dimensional Gaussian random vectors, which are serially orthogonalized. The scalars qd are negative real numbers so that the corresponding eigenstates are exponentially decaying modes. The values of qd were set so that the corresponding time constants were between T/5 and T. Fig. 2(B) shows the time series for five such eigenstates.
Affiliation: Wellcome Trust Centre for Neuroimaging, Institute of Neurology, University College London, 12 Queen Square, London WC1N 3BG, UK. Electronic address: firstname.lastname@example.org.