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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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Related in: MedlinePlus

This figure illustrates the (Lorentzian) form of auto spectra induced by the eigenmodes of a dynamical system. The upper panel shows exemplar spectral densities produced by increasing the Lyapunov exponent from − 2 to − 0.25 Hz. The lower left panel shows the spectral density of mixtures of Lorentzian spectra produced by modes with Lyapunov exponents sampled from a power law distribution in the interval [− 4, − 1/128]. The plot of the logarithm of this spectral density against the logarithm of frequency should be linear — over the power law scaling regime (lower right panel). The blue line corresponds to the numerical estimate and the green line to the theoretical prediction, when the smallest real eigenvalue tends to zero. The ranges of frequencies and exponents were chosen arbitrarily for illustrative purposes.
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f0005: This figure illustrates the (Lorentzian) form of auto spectra induced by the eigenmodes of a dynamical system. The upper panel shows exemplar spectral densities produced by increasing the Lyapunov exponent from − 2 to − 0.25 Hz. The lower left panel shows the spectral density of mixtures of Lorentzian spectra produced by modes with Lyapunov exponents sampled from a power law distribution in the interval [− 4, − 1/128]. The plot of the logarithm of this spectral density against the logarithm of frequency should be linear — over the power law scaling regime (lower right panel). The blue line corresponds to the numerical estimate and the green line to the theoretical prediction, when the smallest real eigenvalue tends to zero. The ranges of frequencies and exponents were chosen arbitrarily for illustrative purposes.

Mentions: In fact, power law scaling over ranges of frequencies emerges with the superposition of a relatively small number of modes that can be sampled from a finite interval (see also Watanabe (2005)). Fig. 1 shows an example where the time constants were restricted to the range and the integrals above were evaluated numerically. We are not supposing that fMRI signals necessarily show a classical power law scaling behaviour — the aim of this analysis is to show that power law scaling, indicative of nonequilibrium steady-state fluctuations, can be explained by a spectrum of Lyapunov exponents in which there are a small number of exponents that approach zero from below and a large number of large negative exponents λi ≈ − 1/ε, characterising modes of activity that dissipate quickly.


On nodes and modes in resting state fMRI.

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

This figure illustrates the (Lorentzian) form of auto spectra induced by the eigenmodes of a dynamical system. The upper panel shows exemplar spectral densities produced by increasing the Lyapunov exponent from − 2 to − 0.25 Hz. The lower left panel shows the spectral density of mixtures of Lorentzian spectra produced by modes with Lyapunov exponents sampled from a power law distribution in the interval [− 4, − 1/128]. The plot of the logarithm of this spectral density against the logarithm of frequency should be linear — over the power law scaling regime (lower right panel). The blue line corresponds to the numerical estimate and the green line to the theoretical prediction, when the smallest real eigenvalue tends to zero. The ranges of frequencies and exponents were chosen arbitrarily for illustrative purposes.
© Copyright Policy
Related In: Results  -  Collection

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

f0005: This figure illustrates the (Lorentzian) form of auto spectra induced by the eigenmodes of a dynamical system. The upper panel shows exemplar spectral densities produced by increasing the Lyapunov exponent from − 2 to − 0.25 Hz. The lower left panel shows the spectral density of mixtures of Lorentzian spectra produced by modes with Lyapunov exponents sampled from a power law distribution in the interval [− 4, − 1/128]. The plot of the logarithm of this spectral density against the logarithm of frequency should be linear — over the power law scaling regime (lower right panel). The blue line corresponds to the numerical estimate and the green line to the theoretical prediction, when the smallest real eigenvalue tends to zero. The ranges of frequencies and exponents were chosen arbitrarily for illustrative purposes.
Mentions: In fact, power law scaling over ranges of frequencies emerges with the superposition of a relatively small number of modes that can be sampled from a finite interval (see also Watanabe (2005)). Fig. 1 shows an example where the time constants were restricted to the range and the integrals above were evaluated numerically. We are not supposing that fMRI signals necessarily show a classical power law scaling behaviour — the aim of this analysis is to show that power law scaling, indicative of nonequilibrium steady-state fluctuations, can be explained by a spectrum of Lyapunov exponents in which there are a small number of exponents that approach zero from below and a large number of large negative exponents λi ≈ − 1/ε, characterising modes of activity that dissipate quickly.

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