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Spherical Harmonics Reveal Standing EEG Waves and Long-Range Neural Synchronization during Non-REM Sleep - PubMed

  • ️Fri Jan 01 2016

Spherical Harmonics Reveal Standing EEG Waves and Long-Range Neural Synchronization during Non-REM Sleep

Siddharth S Sivakumar et al. Front Comput Neurosci. 2016.

Abstract

Previous work from our lab has demonstrated how the connectivity of brain circuits constrains the repertoire of activity patterns that those circuits can display. Specifically, we have shown that the principal components of spontaneous neural activity are uniquely determined by the underlying circuit connections, and that although the principal components do not uniquely resolve the circuit structure, they do reveal important features about it. Expanding upon this framework on a larger scale of neural dynamics, we have analyzed EEG data recorded with the standard 10-20 electrode system from 41 neurologically normal children and adolescents during stage 2, non-REM sleep. We show that the principal components of EEG spindles, or sigma waves (10-16 Hz), reveal non-propagating, standing waves in the form of spherical harmonics. We mathematically demonstrate that standing EEG waves exist when the spatial covariance and the Laplacian operator on the head's surface commute. This in turn implies that the covariance between two EEG channels decreases as the inverse of their relative distance; a relationship that we corroborate with empirical data. Using volume conduction theory, we then demonstrate that superficial current sources are more synchronized at larger distances, and determine the characteristic length of large-scale neural synchronization as 1.31 times the head radius, on average. Moreover, consistent with the hypothesis that EEG spindles are driven by thalamo-cortical rather than cortico-cortical loops, we also show that 8 additional patients with hypoplasia or complete agenesis of the corpus callosum, i.e., with deficient or no connectivity between cortical hemispheres, similarly exhibit standing EEG waves in the form of spherical harmonics. We conclude that spherical harmonics are a hallmark of spontaneous, large-scale synchronization of neural activity in the brain, which are associated with unconscious, light sleep. The analogy with spherical harmonics in quantum mechanics suggests that the variances (eigenvalues) of the principal components follow a Boltzmann distribution, or equivalently, that standing waves are in a sort of "thermodynamic" equilibrium during non-REM sleep. By extension, we speculate that consciousness emerges as the brain dynamics deviate from such equilibrium.

Keywords: agenesis; consciousness; sigma waves; spindles; stage 2 sleep; volume conduction theory.

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Figures

Figure 1
Figure 1

EEG data acquisition and characteristics. (A) Electrode grid and axis orientations for 10–20 system; positive z-axis points out of the page, through the Cz electrode. (B) Power spectrum for raw EEG signals averaged across channels from stage 2 non-REM sleep; a prominent peak is visible in the 10–16 Hz frequency range, corresponding to sigma waves. (C) Sigma-wave-filtered EEG traces; spindles are observed to be highly synchronous and present in all channels. (D) Eigenvalue (variance) distribution of the principal components of EEG data. The noise level is determined by extrapolating the linear trend of the tail.

Figure 2
Figure 2

Rotated spherical harmonics (left) and principal components of sigma waves (right) from a representative normal subject. The theoretical and empirical waves are strikingly similar, confirming that EEG spindles during stage 2 non-REM sleep are standing waves. Spherical harmonics are manually rotated according to the axes rotation operators (denoted by R), based on the orientation of the empirical waves. Principal components are shown from top to bottom in order of decreasing eigenvalue (percentage of total variance). SNR: signal-to-noise ratio. Blue and red correspond to opposing signs of the waves' amplitude. Color scale covers whole range from minimum to maximum amplitude.

Figure 3
Figure 3

MRI data for a normal subject (top) and for a subject with partial agenesis (bottom). A rift in the corpus callosum (arrows) is clearly visible in the coronal MRI image for the patient with agenesis, which is not seen in the normal subject. White matter tracts connect below and through the cortex in the normal subject, but do not do so in the patient with agenesis.

Figure 4
Figure 4

Rotated spherical harmonics (left) and principal components of sigma waves (right) for a representative subject with agenesis of corpus callosum. As in Figure 2, theoretical and empirical waves are strikingly similar, confirming that standing waves are also present in patients with defective inter-hemispherical connections. Spherical harmonics are manually rotated according to the axes rotation operators (denoted by R), based on the orientation of the empirical waves. Principal components are shown from top to bottom in order of decreasing eigenvalue (percentage of total variance). SNR: signal-to-noise ratio. Blue and red correspond to opposing signs of the waves' amplitude. Color scale covers whole range from minimum to maximum amplitude.

Figure 5
Figure 5

Large-scale neural synchronization. (A) As predicted by our theory (see Methods), the covariance of the EEG potential is proportional to the inverse of the relative distance. This proportionality is direct, i.e., the abscissa intercept of regression is essentially zero. (B) Empirically, the covariance of the current sources also shows a similar correlation with inverse distance, but with an additional horizontal offset which represents the inverse of the characteristic length of large-scale neural synchronization (normalized to the head radius). (C) Across all subjects, the characteristic length, D, clusters tightly around a mean of 1.31 (normalized to the head radius). (D) The angular length, or geodesic arc of the characteristic length, γ, clusters around a mean of 82.4°.

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References

    1. Andrillon T., Nir Y., Staba R. J., Ferrarelli F., Cirelli C., Tononi G., et al. . (2011). Sleep spindles in humans: insights from intracranial EEG and unit recordings. J. Neurosci. 31, 17821–17834. 10.1523/JNEUROSCI.2604-11.2011 - DOI - PMC - PubMed
    1. Atkins P. W., Friedman R. S. (2011). Molecular Quantum Mechanics. New York, NY: OUP Oxford.
    1. Ayoub A., Aumann D., Horschelmann A., Kouchekmanesch A., Paul P., Born J., et al. . (2013). Differential effects on fast and slow spindle activity, and the sleep slow oscillation in humans with carbamazepine and flunarizine to antagonize voltage-dependent Na+ and Ca2+ channel activity. Sleep 36, 905–911. 10.5665/sleep.2722 - DOI - PMC - PubMed
    1. Babiloni C., Stella G., Buffo P., Vecchio F., Onorati P., Muratori C., et al. . (2012). Cortical sources of resting state EEG rhythms are abnormal in dyslexic children. Clin. Neurophysiol. 123, 2384–2391. 10.1016/j.clinph.2012.05.002 - DOI - PubMed
    1. Babiloni C., Visser P. J., Frisoni G., De Deyn P. P., Bresciani L., Jelic V., et al. . (2010). Cortical sources of resting EEG rhythms in mild cognitive impairment and subjective memory complaint. Neurobiol. Aging 31, 1787–1798. 10.1016/j.neurobiolaging.2008.09.020 - DOI - PubMed

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