An Implementation Of Dynamic-Causal-Modelling
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An Implementation Of Dynamic-Causal-Modelling
Christian [email protected]
WWU MünsterInstitute for Computational and Applied Mathematics
27.07.2012
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Overview
Contents:1 About2 Capabilities3 Extensions4 Open Issues5 Sample Report
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About
IOD ( = Implentation Of Dynamic-Causal-Modelling)
Version 1.0 (2011, Diploma Thesis)Version 2.0 (planned for Q4/2012)Open Source (zlib/libpng License)Written in C++11Parallelization using OpenMPNo required dependenciesOptional: gnuplot, graphwiz, tcmalloc, mutt
together with:Prof. Dr. Mario Ohlberger, Dr. Thomas Seidenbecher, Dr. Jörg Lesting
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Capabilities
Scientific:DCM for fMRI (Linear, Bilinear, Nonlinear)DCM for EEG (Default, Extended, Adaption, Habituation,Linearized)Simulations of Systems
Technical:Modular Dynamic and Forward ModelsOrder 1, 2, 5 Runge-Kutta Solver (optionally adaptive)Remote Execution (optionally mailing results)
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Capabilities
Scientific:DCM for fMRI (Linear, Bilinear, Nonlinear)DCM for EEG (Default, Extended, Adaption, Habituation,Linearized)Simulations of Systems
Technical:Modular Dynamic and Forward ModelsOrder 1, 2, 5 Runge-Kutta Solver (optionally adaptive)Remote Execution (optionally mailing results)
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Extension
Major:EM-Algorithm OptimizationDrift Filter
Minor:Positive (Definite) Temporal CorrelationFast Model Evidence CalculationsBandpass FilterXHTML/SVG Reporting
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EM-Algorithm Optimization
Using the following linear algebra lemma:1 (AB)T = BTAT
2 tr(AB) =∑
i∑
j aijbji
3 tr(ABC ) = tr(BCA) = tr(CAB)
reduces significantly the number of flops,as well as the memory consumptions,especially inside the M-step;even more when recycling the E-step.
For more info see: http://j.mp/himpe (p.30-36).
![Page 8: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/8.jpg)
EM-Algorithm Optimization
Using the following linear algebra lemma:1 (AB)T = BTAT
2 tr(AB) =∑
i∑
j aijbji
3 tr(ABC ) = tr(BCA) = tr(CAB)
reduces significantly the number of flops,
as well as the memory consumptions,especially inside the M-step;even more when recycling the E-step.
For more info see: http://j.mp/himpe (p.30-36).
![Page 9: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/9.jpg)
EM-Algorithm Optimization
Using the following linear algebra lemma:1 (AB)T = BTAT
2 tr(AB) =∑
i∑
j aijbji
3 tr(ABC ) = tr(BCA) = tr(CAB)
reduces significantly the number of flops,as well as the memory consumptions,
especially inside the M-step;even more when recycling the E-step.
For more info see: http://j.mp/himpe (p.30-36).
![Page 10: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/10.jpg)
EM-Algorithm Optimization
Using the following linear algebra lemma:1 (AB)T = BTAT
2 tr(AB) =∑
i∑
j aijbji
3 tr(ABC ) = tr(BCA) = tr(CAB)
reduces significantly the number of flops,as well as the memory consumptions,especially inside the M-step;
even more when recycling the E-step.
For more info see: http://j.mp/himpe (p.30-36).
![Page 11: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/11.jpg)
EM-Algorithm Optimization
Using the following linear algebra lemma:1 (AB)T = BTAT
2 tr(AB) =∑
i∑
j aijbji
3 tr(ABC ) = tr(BCA) = tr(CAB)
reduces significantly the number of flops,as well as the memory consumptions,especially inside the M-step;even more when recycling the E-step.
For more info see: http://j.mp/himpe (p.30-36).
![Page 12: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/12.jpg)
EM-Algorithm Optimization
Using the following linear algebra lemma:1 (AB)T = BTAT
2 tr(AB) =∑
i∑
j aijbji
3 tr(ABC ) = tr(BCA) = tr(CAB)
reduces significantly the number of flops,as well as the memory consumptions,especially inside the M-step;even more when recycling the E-step.
For more info see: http://j.mp/himpe (p.30-36).
![Page 13: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/13.jpg)
Drift Filter
Drift Term Xβ:Additional set of parameters β,reflecting unrelated oscillations,modelled by a discrete cosine set X .
The drift matrix X can be customized to a high-pass filter.It is not advisable, though possible, to use as low-pass filter.
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Drift Filter
Drift Term Xβ:Additional set of parameters β,reflecting unrelated oscillations,modelled by a discrete cosine set X .The drift matrix X can be customized to a high-pass filter.
It is not advisable, though possible, to use as low-pass filter.
![Page 15: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/15.jpg)
Drift Filter
Drift Term Xβ:Additional set of parameters β,reflecting unrelated oscillations,modelled by a discrete cosine set X .The drift matrix X can be customized to a high-pass filter.It is not advisable, though possible, to use as low-pass filter.
![Page 16: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/16.jpg)
Open Issues
Major:1 Post-Hoc Model Selection (2.0)2 EEG Model Restructuring (2.0)3 Model Reduction (3.0)4 Optimal Maps (3.0)
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Post-Hoc Model Selection
An implementation of “Post-hoc selection of dynamic causalmodels”, M.J. Rosa, K. Friston, W. Penny, Journal ofNeuroscience Methods, Volume 208, Issue 1, 30 June 2012,Pages 66-78http://j.mp/posthoc
Compute posterior of multiple reduced (connectivity) models,by estimating the full (connectivity) modeland relate to the reduced models priors.
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Post-Hoc Model Selection
An implementation of “Post-hoc selection of dynamic causalmodels”, M.J. Rosa, K. Friston, W. Penny, Journal ofNeuroscience Methods, Volume 208, Issue 1, 30 June 2012,Pages 66-78http://j.mp/posthoc
Compute posterior of multiple reduced (connectivity) models,
by estimating the full (connectivity) modeland relate to the reduced models priors.
![Page 19: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/19.jpg)
Post-Hoc Model Selection
An implementation of “Post-hoc selection of dynamic causalmodels”, M.J. Rosa, K. Friston, W. Penny, Journal ofNeuroscience Methods, Volume 208, Issue 1, 30 June 2012,Pages 66-78http://j.mp/posthoc
Compute posterior of multiple reduced (connectivity) models,by estimating the full (connectivity) model
and relate to the reduced models priors.
![Page 20: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/20.jpg)
Post-Hoc Model Selection
An implementation of “Post-hoc selection of dynamic causalmodels”, M.J. Rosa, K. Friston, W. Penny, Journal ofNeuroscience Methods, Volume 208, Issue 1, 30 June 2012,Pages 66-78http://j.mp/posthoc
Compute posterior of multiple reduced (connectivity) models,by estimating the full (connectivity) modeland relate to the reduced models priors.
![Page 21: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/21.jpg)
EEG Model Restructuring
Split EEG model into:linearnonlinear
submodels,
to:1 improve performance and2 prepare for model reduction.
![Page 22: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/22.jpg)
EEG Model Restructuring
Split EEG model into:linearnonlinear
submodels, to:1 improve performance
and2 prepare for model reduction.
![Page 23: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/23.jpg)
EEG Model Restructuring
Split EEG model into:linearnonlinear
submodels, to:1 improve performance and2 prepare for model reduction.
![Page 24: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/24.jpg)
Model Reduction
Considering many (>100) regions...
the estimation of connectivity parameters consumes the bulkof computation time,because the differentiation (requiring integrations) dominates(except for exponential maps, but they are too limitinganyway).Thus the current model structure is not viable here.
Model Reduction to the Rescue!Find a surrogate model, with a low dimensional parameter space.Two approaches are considered:
1 Projection (Complex, Precise)2 Truncation (Simple, Coarse)
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Model Reduction
Considering many (>100) regions...
the estimation of connectivity parameters consumes the bulkof computation time,
because the differentiation (requiring integrations) dominates(except for exponential maps, but they are too limitinganyway).Thus the current model structure is not viable here.
Model Reduction to the Rescue!Find a surrogate model, with a low dimensional parameter space.Two approaches are considered:
1 Projection (Complex, Precise)2 Truncation (Simple, Coarse)
![Page 26: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/26.jpg)
Model Reduction
Considering many (>100) regions...
the estimation of connectivity parameters consumes the bulkof computation time,because the differentiation (requiring integrations) dominates
(except for exponential maps, but they are too limitinganyway).Thus the current model structure is not viable here.
Model Reduction to the Rescue!Find a surrogate model, with a low dimensional parameter space.Two approaches are considered:
1 Projection (Complex, Precise)2 Truncation (Simple, Coarse)
![Page 27: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/27.jpg)
Model Reduction
Considering many (>100) regions...
the estimation of connectivity parameters consumes the bulkof computation time,because the differentiation (requiring integrations) dominates(except for exponential maps, but they are too limitinganyway).
Thus the current model structure is not viable here.
Model Reduction to the Rescue!Find a surrogate model, with a low dimensional parameter space.Two approaches are considered:
1 Projection (Complex, Precise)2 Truncation (Simple, Coarse)
![Page 28: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/28.jpg)
Model Reduction
Considering many (>100) regions...
the estimation of connectivity parameters consumes the bulkof computation time,because the differentiation (requiring integrations) dominates(except for exponential maps, but they are too limitinganyway).Thus the current model structure is not viable here.
Model Reduction to the Rescue!Find a surrogate model, with a low dimensional parameter space.Two approaches are considered:
1 Projection (Complex, Precise)2 Truncation (Simple, Coarse)
![Page 29: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/29.jpg)
Model Reduction
Considering many (>100) regions...
the estimation of connectivity parameters consumes the bulkof computation time,because the differentiation (requiring integrations) dominates(except for exponential maps, but they are too limitinganyway).Thus the current model structure is not viable here.
Model Reduction to the Rescue!Find a surrogate model, with a low dimensional parameter space.
Two approaches are considered:
1 Projection (Complex, Precise)2 Truncation (Simple, Coarse)
![Page 30: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/30.jpg)
Model Reduction
Considering many (>100) regions...
the estimation of connectivity parameters consumes the bulkof computation time,because the differentiation (requiring integrations) dominates(except for exponential maps, but they are too limitinganyway).Thus the current model structure is not viable here.
Model Reduction to the Rescue!Find a surrogate model, with a low dimensional parameter space.Two approaches are considered:
1 Projection (Complex, Precise)
2 Truncation (Simple, Coarse)
![Page 31: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/31.jpg)
Model Reduction
Considering many (>100) regions...
the estimation of connectivity parameters consumes the bulkof computation time,because the differentiation (requiring integrations) dominates(except for exponential maps, but they are too limitinganyway).Thus the current model structure is not viable here.
Model Reduction to the Rescue!Find a surrogate model, with a low dimensional parameter space.Two approaches are considered:
1 Projection (Complex, Precise)2 Truncation (Simple, Coarse)
![Page 32: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/32.jpg)
Optimal Maps
An implementation of “Bayesian Inference with OptimalMaps”,T. A. El Moselhy, Y. M. Marzouk, arXivhttp://j.mp/optimalmaps
Replacing the EM-algorithm using optimal maps.Find a map that transforms the prior into the posteriordistribution,using for example low-order polynomials,until the variance drops below some threshold.
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Optimal Maps
An implementation of “Bayesian Inference with OptimalMaps”,T. A. El Moselhy, Y. M. Marzouk, arXivhttp://j.mp/optimalmaps
Replacing the EM-algorithm using optimal maps.
Find a map that transforms the prior into the posteriordistribution,using for example low-order polynomials,until the variance drops below some threshold.
![Page 34: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/34.jpg)
Optimal Maps
An implementation of “Bayesian Inference with OptimalMaps”,T. A. El Moselhy, Y. M. Marzouk, arXivhttp://j.mp/optimalmaps
Replacing the EM-algorithm using optimal maps.Find a map that transforms the prior into the posteriordistribution,
using for example low-order polynomials,until the variance drops below some threshold.
![Page 35: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/35.jpg)
Optimal Maps
An implementation of “Bayesian Inference with OptimalMaps”,T. A. El Moselhy, Y. M. Marzouk, arXivhttp://j.mp/optimalmaps
Replacing the EM-algorithm using optimal maps.Find a map that transforms the prior into the posteriordistribution,using for example low-order polynomials,
until the variance drops below some threshold.
![Page 36: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/36.jpg)
Optimal Maps
An implementation of “Bayesian Inference with OptimalMaps”,T. A. El Moselhy, Y. M. Marzouk, arXivhttp://j.mp/optimalmaps
Replacing the EM-algorithm using optimal maps.Find a map that transforms the prior into the posteriordistribution,using for example low-order polynomials,until the variance drops below some threshold.
![Page 38: An Implementation Of Dynamic-Causal-Modelling](https://reader034.fdocuments.in/reader034/viewer/2022042813/5479d6d25906b530358b45fa/html5/thumbnails/38.jpg)
tl;dl
Modular Implementation ( http://j.mp/himpe )Replace the EM algorithm ( http://j.mp/optimalmaps )Include Model Reduction
Thank You