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Efficient gradient computation for dynamical models.

Sengupta B, Friston KJ, Penny WD - Neuroimage (2014)

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.

View Article: PubMed Central - PubMed

Affiliation: Wellcome Trust Centre for Neuroimaging, Institute of Neurology, University College London, 12 Queen Square, London WC1N 3BG, UK. Electronic address: b.sengupta@ucl.ac.uk.

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Linear System (A) The 5-dimensional state-space model and (B) the linear evolution of its eigenstates.
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f0010: Linear System (A) The 5-dimensional state-space model and (B) the linear evolution of its eigenstates.

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.


Efficient gradient computation for dynamical models.

Sengupta B, Friston KJ, Penny WD - Neuroimage (2014)

Linear System (A) The 5-dimensional state-space model and (B) the linear evolution of its eigenstates.
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4120812&req=5

f0010: Linear System (A) The 5-dimensional state-space model and (B) the linear evolution of its eigenstates.
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.

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.

View Article: PubMed Central - PubMed

Affiliation: Wellcome Trust Centre for Neuroimaging, Institute of Neurology, University College London, 12 Queen Square, London WC1N 3BG, UK. Electronic address: b.sengupta@ucl.ac.uk.

Show MeSH