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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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Forward Sensitivity The solid path indicates a trajectory of points xn, with n = 1…5, for a dynamical system with parameters p. The dotted path indicates the trajectory  for the same dynamical system but with parameters . The dotted path can be reached from the solid path via the total derivative . The Forward Sensitivity approach provides a method for computing this derivative.
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f0005: Forward Sensitivity The solid path indicates a trajectory of points xn, with n = 1…5, for a dynamical system with parameters p. The dotted path indicates the trajectory for the same dynamical system but with parameters . The dotted path can be reached from the solid path via the total derivative . The Forward Sensitivity approach provides a method for computing this derivative.

Mentions: This method is illustrated in Fig. 1 where the solid path indicates a trajectory of points xn for a dynamical system with parameters p and the dotted path indicates the trajectory for the same dynamical system but with parameters . The dotted path can be obtained from the solid path via the total derivative in the direction of the perturbation, δi. The FS method provides a method for computing this derivative. Under a first order Euler approach for integrating the dynamics, this is implemented using the above recursion.


Efficient gradient computation for dynamical models.

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

Forward Sensitivity The solid path indicates a trajectory of points xn, with n = 1…5, for a dynamical system with parameters p. The dotted path indicates the trajectory  for the same dynamical system but with parameters . The dotted path can be reached from the solid path via the total derivative . The Forward Sensitivity approach provides a method for computing this derivative.
© Copyright Policy
Related In: Results  -  Collection

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

f0005: Forward Sensitivity The solid path indicates a trajectory of points xn, with n = 1…5, for a dynamical system with parameters p. The dotted path indicates the trajectory for the same dynamical system but with parameters . The dotted path can be reached from the solid path via the total derivative . The Forward Sensitivity approach provides a method for computing this derivative.
Mentions: This method is illustrated in Fig. 1 where the solid path indicates a trajectory of points xn for a dynamical system with parameters p and the dotted path indicates the trajectory for the same dynamical system but with parameters . The dotted path can be obtained from the solid path via the total derivative in the direction of the perturbation, δi. The FS method provides a method for computing this derivative. Under a first order Euler approach for integrating the dynamics, this is implemented using the above recursion.

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