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Network discovery with DCM.

Friston KJ, Li B, Daunizeau J, Stephan KE - Neuroimage (2010)

Bottom Line: The scheme furnishes a network description of distributed activity in the brain that is optimal in the sense of having the greatest conditional probability, relative to other networks.The networks are characterised in terms of their connectivity or adjacency matrices and conditional distributions over the directed (and reciprocal) effective connectivity between connected nodes or regions.We envisage that this approach will provide a useful complement to current analyses of functional connectivity for both activation and resting-state studies.

View Article: PubMed Central - PubMed

Affiliation: The Wellcome Trust Centre for Neuroimaging, Institute of Neurology, University College London, London, UK. k.firston@fil.ion.ucl.ac.uk

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Related in: MedlinePlus

The slaving principle and centre manifolds: This schematic illustrates the basic idea behind the slaving principle. In this example, there are two states, whose flows bring them to an attracting invariant set (the centre manifold); h(ζ1). Once the states have been attracted to this manifold they remain on (or near) it. This means the flow of states can be decomposed into a tangential component (on the manifold) and a transverse component (that draws states to the manifold). This decomposition can be described in terms of a change of coordinates, which implicitly separate fast (stable) transverse dynamics  from slow (unstable) tangential flow  on the centre manifold. We exploit this decomposition to motivate the separation of dynamics into a slow, low-dimensional flow on an attracting manifold and a fast (analytic) fluctuating part that describes perturbations away from (and back to) the manifold. Please see the main text for a full description of the equations.
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f0005: The slaving principle and centre manifolds: This schematic illustrates the basic idea behind the slaving principle. In this example, there are two states, whose flows bring them to an attracting invariant set (the centre manifold); h(ζ1). Once the states have been attracted to this manifold they remain on (or near) it. This means the flow of states can be decomposed into a tangential component (on the manifold) and a transverse component (that draws states to the manifold). This decomposition can be described in terms of a change of coordinates, which implicitly separate fast (stable) transverse dynamics from slow (unstable) tangential flow on the centre manifold. We exploit this decomposition to motivate the separation of dynamics into a slow, low-dimensional flow on an attracting manifold and a fast (analytic) fluctuating part that describes perturbations away from (and back to) the manifold. Please see the main text for a full description of the equations.

Mentions: Put simply, all this means is that the dynamics of any system comprising many elements can be decomposed into a mixture of (orthogonal) patterns over variables describing its state. By necessity, some of these patterns dissipate more quickly than others. Generally, some patterns decay so slowly that they predominate over others that disappear as soon as they are created. Mathematically, this means that P (principal) eigenvalues λp → 0 : p ≤ P are nearly zero and the associated eigenvectors or modes Up(ξ) are slow and unstable. In this case, ζp = Up−ξ : p ≤ P are known as order parameters. Order parameters are mixtures of states encoding the amplitude of the slow (unstable) modes that determine macroscopic behaviour. Other fast (stable) modes ζq = Uq−ξ : q > P have large negative eigenvalues, which means that they decay or dissipate quickly to an invariant attracting set or manifold, h(ζp), such that . In other words, the invariant (centre) manifold h(ζp) attracts trajectories and contains the solutions to Eq. (1). When there is only one order parameter or principal mode, this manifold is a line or curve in state-space and ζ1 could represent the distance along that curve (see Fig. 1). The unstable fast modes decay quickly because the eigenvalue is effectively their rate of decay. One can see this easily by taking a first-order Taylor expansion of Eq. (1) about the centre manifold:(2)ζ˙q≈fqζp,hζp+Uq−IUqζq−hqζp=λqζq−hqζp.


Network discovery with DCM.

Friston KJ, Li B, Daunizeau J, Stephan KE - Neuroimage (2010)

The slaving principle and centre manifolds: This schematic illustrates the basic idea behind the slaving principle. In this example, there are two states, whose flows bring them to an attracting invariant set (the centre manifold); h(ζ1). Once the states have been attracted to this manifold they remain on (or near) it. This means the flow of states can be decomposed into a tangential component (on the manifold) and a transverse component (that draws states to the manifold). This decomposition can be described in terms of a change of coordinates, which implicitly separate fast (stable) transverse dynamics  from slow (unstable) tangential flow  on the centre manifold. We exploit this decomposition to motivate the separation of dynamics into a slow, low-dimensional flow on an attracting manifold and a fast (analytic) fluctuating part that describes perturbations away from (and back to) the manifold. Please see the main text for a full description of the equations.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3094760&req=5

f0005: The slaving principle and centre manifolds: This schematic illustrates the basic idea behind the slaving principle. In this example, there are two states, whose flows bring them to an attracting invariant set (the centre manifold); h(ζ1). Once the states have been attracted to this manifold they remain on (or near) it. This means the flow of states can be decomposed into a tangential component (on the manifold) and a transverse component (that draws states to the manifold). This decomposition can be described in terms of a change of coordinates, which implicitly separate fast (stable) transverse dynamics from slow (unstable) tangential flow on the centre manifold. We exploit this decomposition to motivate the separation of dynamics into a slow, low-dimensional flow on an attracting manifold and a fast (analytic) fluctuating part that describes perturbations away from (and back to) the manifold. Please see the main text for a full description of the equations.
Mentions: Put simply, all this means is that the dynamics of any system comprising many elements can be decomposed into a mixture of (orthogonal) patterns over variables describing its state. By necessity, some of these patterns dissipate more quickly than others. Generally, some patterns decay so slowly that they predominate over others that disappear as soon as they are created. Mathematically, this means that P (principal) eigenvalues λp → 0 : p ≤ P are nearly zero and the associated eigenvectors or modes Up(ξ) are slow and unstable. In this case, ζp = Up−ξ : p ≤ P are known as order parameters. Order parameters are mixtures of states encoding the amplitude of the slow (unstable) modes that determine macroscopic behaviour. Other fast (stable) modes ζq = Uq−ξ : q > P have large negative eigenvalues, which means that they decay or dissipate quickly to an invariant attracting set or manifold, h(ζp), such that . In other words, the invariant (centre) manifold h(ζp) attracts trajectories and contains the solutions to Eq. (1). When there is only one order parameter or principal mode, this manifold is a line or curve in state-space and ζ1 could represent the distance along that curve (see Fig. 1). The unstable fast modes decay quickly because the eigenvalue is effectively their rate of decay. One can see this easily by taking a first-order Taylor expansion of Eq. (1) about the centre manifold:(2)ζ˙q≈fqζp,hζp+Uq−IUqζq−hqζp=λqζq−hqζp.

Bottom Line: The scheme furnishes a network description of distributed activity in the brain that is optimal in the sense of having the greatest conditional probability, relative to other networks.The networks are characterised in terms of their connectivity or adjacency matrices and conditional distributions over the directed (and reciprocal) effective connectivity between connected nodes or regions.We envisage that this approach will provide a useful complement to current analyses of functional connectivity for both activation and resting-state studies.

View Article: PubMed Central - PubMed

Affiliation: The Wellcome Trust Centre for Neuroimaging, Institute of Neurology, University College London, London, UK. k.firston@fil.ion.ucl.ac.uk

Show MeSH
Related in: MedlinePlus