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On nodes and modes in resting state fMRI.

Friston KJ, Kahan J, Razi A, Stephan KE, Sporns O - Neuroimage (2014)

Bottom Line: We first demonstrate that the eigenmodes of functional connectivity--or covariance among regions or nodes--are the same as the eigenmodes of the underlying effective connectivity, provided we limit ourselves to symmetrical connections.Crucially, the principal modes of functional connectivity correspond to the dynamically unstable modes of effective connectivity that decay slowly and show long term memory.In this model, effective connectivity is parameterised in terms of eigenmodes and their Lyapunov exponents--that can also be interpreted as locations in a multidimensional scaling space.

View Article: PubMed Central - PubMed

Affiliation: The Wellcome Trust Centre for Neuroimaging, University College London, Queen Square, London WC1N 3BG, UK. Electronic address: k.friston@ucl.ac.uk.

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This schematic summarises the generative model for the spectral DCM described in this paper. A generative model generates observations from hidden causes. Here, we generate observed complex cross spectra by first sampling log time constants (inverse negative Lyapunov exponents) from a Gaussian distribution and using them to reconstitute an effective connectivity matrix among hidden neuronal states. When combined with regional haemodynamics (lower panel) this effective connectivity (together with other haemodynamic parameters) specifies the transfer functions mapping endogenous fluctuations to expected haemodynamic responses. The cross spectra of these responses are generated from the transfer functions given the spectral density of endogenous neuronal fluctuations and observation noise. These are generated from log amplitude and power law exponents sampled from a normal distribution. The final observations are generated with Gaussian sampling errors with a log precision sampled from a relatively informative (prior) Gaussian distribution. The key simplicity afforded by this generative model is that the eigenmodes required to generate the effective connectivity can be identified with the eigenmodes of the functional connectivity of the measured timeseries. The functions E(x) and F(x) correspond to an oxygen extraction fraction and flow functions respectively.
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f0010: This schematic summarises the generative model for the spectral DCM described in this paper. A generative model generates observations from hidden causes. Here, we generate observed complex cross spectra by first sampling log time constants (inverse negative Lyapunov exponents) from a Gaussian distribution and using them to reconstitute an effective connectivity matrix among hidden neuronal states. When combined with regional haemodynamics (lower panel) this effective connectivity (together with other haemodynamic parameters) specifies the transfer functions mapping endogenous fluctuations to expected haemodynamic responses. The cross spectra of these responses are generated from the transfer functions given the spectral density of endogenous neuronal fluctuations and observation noise. These are generated from log amplitude and power law exponents sampled from a normal distribution. The final observations are generated with Gaussian sampling errors with a log precision sampled from a relatively informative (prior) Gaussian distribution. The key simplicity afforded by this generative model is that the eigenmodes required to generate the effective connectivity can be identified with the eigenmodes of the functional connectivity of the measured timeseries. The functions E(x) and F(x) correspond to an oxygen extraction fraction and flow functions respectively.

Mentions: Fig. 2 shows the form of the generative model in terms of a Bayesian graph. A generative model is simply a model of how data are generated. In this case the data are complex cross spectra of sampled timeseries. The model starts with the spatial eigenmodes μ = eig(Σy) of the sample covariance matrix. Although the number of hidden states exceeds the number of regional timeseries, we can still use the eigenmodes of the sample covariance of regional responses as proxies for the eigenmodes of hidden (neuronal) states — because there is only one neuronal state per region. The remaining hidden states model local haemodynamics, which effectively smooth or convolve the neural activity to produce a BOLD response.


On nodes and modes in resting state fMRI.

Friston KJ, Kahan J, Razi A, Stephan KE, Sporns O - Neuroimage (2014)

This schematic summarises the generative model for the spectral DCM described in this paper. A generative model generates observations from hidden causes. Here, we generate observed complex cross spectra by first sampling log time constants (inverse negative Lyapunov exponents) from a Gaussian distribution and using them to reconstitute an effective connectivity matrix among hidden neuronal states. When combined with regional haemodynamics (lower panel) this effective connectivity (together with other haemodynamic parameters) specifies the transfer functions mapping endogenous fluctuations to expected haemodynamic responses. The cross spectra of these responses are generated from the transfer functions given the spectral density of endogenous neuronal fluctuations and observation noise. These are generated from log amplitude and power law exponents sampled from a normal distribution. The final observations are generated with Gaussian sampling errors with a log precision sampled from a relatively informative (prior) Gaussian distribution. The key simplicity afforded by this generative model is that the eigenmodes required to generate the effective connectivity can be identified with the eigenmodes of the functional connectivity of the measured timeseries. The functions E(x) and F(x) correspond to an oxygen extraction fraction and flow functions respectively.
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4121089&req=5

f0010: This schematic summarises the generative model for the spectral DCM described in this paper. A generative model generates observations from hidden causes. Here, we generate observed complex cross spectra by first sampling log time constants (inverse negative Lyapunov exponents) from a Gaussian distribution and using them to reconstitute an effective connectivity matrix among hidden neuronal states. When combined with regional haemodynamics (lower panel) this effective connectivity (together with other haemodynamic parameters) specifies the transfer functions mapping endogenous fluctuations to expected haemodynamic responses. The cross spectra of these responses are generated from the transfer functions given the spectral density of endogenous neuronal fluctuations and observation noise. These are generated from log amplitude and power law exponents sampled from a normal distribution. The final observations are generated with Gaussian sampling errors with a log precision sampled from a relatively informative (prior) Gaussian distribution. The key simplicity afforded by this generative model is that the eigenmodes required to generate the effective connectivity can be identified with the eigenmodes of the functional connectivity of the measured timeseries. The functions E(x) and F(x) correspond to an oxygen extraction fraction and flow functions respectively.
Mentions: Fig. 2 shows the form of the generative model in terms of a Bayesian graph. A generative model is simply a model of how data are generated. In this case the data are complex cross spectra of sampled timeseries. The model starts with the spatial eigenmodes μ = eig(Σy) of the sample covariance matrix. Although the number of hidden states exceeds the number of regional timeseries, we can still use the eigenmodes of the sample covariance of regional responses as proxies for the eigenmodes of hidden (neuronal) states — because there is only one neuronal state per region. The remaining hidden states model local haemodynamics, which effectively smooth or convolve the neural activity to produce a BOLD response.

Bottom Line: We first demonstrate that the eigenmodes of functional connectivity--or covariance among regions or nodes--are the same as the eigenmodes of the underlying effective connectivity, provided we limit ourselves to symmetrical connections.Crucially, the principal modes of functional connectivity correspond to the dynamically unstable modes of effective connectivity that decay slowly and show long term memory.In this model, effective connectivity is parameterised in terms of eigenmodes and their Lyapunov exponents--that can also be interpreted as locations in a multidimensional scaling space.

View Article: PubMed Central - PubMed

Affiliation: The Wellcome Trust Centre for Neuroimaging, University College London, Queen Square, London WC1N 3BG, UK. Electronic address: k.friston@ucl.ac.uk.

Show MeSH
Related in: MedlinePlus