Chaos in the brain Jan Kříž 5th Workshop on Quantum Chaos and Localisation Phenomena Warszawa 5th...
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Transcript of Chaos in the brain Jan Kříž 5th Workshop on Quantum Chaos and Localisation Phenomena Warszawa 5th...
Chaos in the brain
Jan Kříž
5th Workshop on Quantum Chaos and Localisation Phenomena Warszawa
May 22, 2011
University of Hradec Králové,Doppler Institute for mathematical physics
and applied mathematics
Czech Republic
Human EEGmeasures electric potentials on the scalp (generated
by neuronal activity in the brain)
„The analysis of EEG has a long history. Being used
as a diagnostic tool for 80 years it still resists to be a subject of strict and objective analysis.“
Richard P. Feynman (1918 -1988))
I can safely say that nobody understands quantum mechanics
Quantum Mechanics
EEG & quantum mechanics I
- EEG signal = interference of electric signals produced by activity of huge number of neurons
Superposition principle
F. Wolf and T. Geisel. Nature, 395 (1998), 73-74.
M. Schnabel, M. Kaschube, S. Lowel and F. Wolf, Eur. Phys. J. Special Topics, 145 (2007), 137-157.
Structures emerging in the visual cortex are described by random Gaussian fields
(known from quantum chaotic systems)
Example 1: Ocular dominance & nodal domains
P. A. Anderson, J. Olavarria and R. C. Van Sluyters, Journal of Neuroscience, 8 (1988), 2183-2200.
Example 2: Directional selectivity & phase
N. P. Issa, C. Trepel and M. P. Stryker, Journal of Neuroscience, 20 (2000), 8504-8514.
EEG (biomedical signals) & quantum mechanics II
- not only biomedical signals (RADAR, geophysics, speech and image analysis, …)
- most real world signals are non-stationary, i.e. have complex time-varying (spectral) characteristics
- it is not possible to have a “good” information on the frequency spectrum and its time evolution
.constft
Heisenberg uncertainty relations …
S. Krishnan, Conference “Biosignal 2008”, Brno, Czech Republic, Opening Ceremony Keynote Lecture.
EEG (biomedical signals) & quantum mechanics III
- we use mathematical (statistical) tools known from quantum mechanics (chaos):
• Random matrix theory:T. Guhr, A. Müller-Groeling, H. A. Weidenmüller, Physics Reports 299 (1998), 189-425.
• Maximum likelihood estimation:S.T. Merkel, C.A. Riofrío, S.T. Flammia, I.H.Deutsch, Phys. Rev. A 81 (2010), ArtNo. 032126(implementation of QSR to quantum kicked top)B.Dietz, T. Friedrich, H.L. Harney, M. Misky-Oglu, A. Richter, F. Schäfer, H. A. Weidenmüller, Phys. Rev. E 78 (2008), ArtNo. 055204(MLE & chaotic scattering in overlapping resonators)
Human EEG & Random matrix theory
P. Šeba, Random Matrix Analysis of Human EEG Data, Phys. Rev. Lett. 91 (2003), ArtNo 198104.
- demonstration of the existence of universal, subject independent, features of human EEG
- statistical properties of spectra of EEG cross-channel correlations matrices compared with the predictions of RMT
Human EEG & Random matrix theory
xl(tj) … EEG channel l at time tj
N1, N2 chosen such that for Δ=150 ms
- Experiment: clinical19 channel EEG device15 – 20 minutes per measurements90 volunteers
measured without and with visual stimulation
-ensemble of 7000 matrices per one measure
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TTt j ,
Human EEG & Random matrix theory
Eigenvalue density function (log-log scale)
Small eigenvalues:subject dependent
Large eigenvalues:subj. independent tail of the same form as Random Lévy matrics
Z. Burda, J. Jurkiewicz, M.A.Nowak, G. Papp, I. Zahed,Phys. Rev. E 65 (2002), ArtNo 021106 .
Human EEG & Random matrix theory
Level spacing distribution (compared with Wigner formula for GOE)
□ ... visually stimulated+ … no stimulation
Human EEG & Random matrix theory
Number variance (compared with prediction for GOE)
□ ... visually stimulated+ … no stimulation
Human EEG & Random matrix theory
Summary
- Level spacing distribution: very good agreement with the RMT predictions => universal behaviour
- Number variance: sensitive when the subject is visually stimulated
- It is reasonable to assume that also some pathological processes can influence the number variance
Evoked response potentials
- responses to external stimulus (auditory, visual, ...)- sensory and cognitive processing in the brain
low „SNR“ … noise (everything what we are not interested in including background activity of neurons)
Commonly used methods: Filtering + averaging, PCA
Our method: MAXIMUM LIKELIHOOD ESTIMATION
Evoked response potentials
- standard tool of statistical estimation theory- by R. A. Fisher- dating back to 1920’s
Corner stone:mathematical model
Basic concept of MLE (R.A. Fisher in 1920’s)
• assume pdf f of random vector y depending on a parameter set w, i.e. f(y|w)
• it determines the probability of observing the data vector y (in dependence on the parameters w)
• however, we are faced with inverse problem: we have given data vector and we do not know parameters
• MLE: given the observed data (and a model of interest = set of possible pdfs), find the pdf, that is most likely to produce the given data.
MLE & human multiepoch EEG
MLE & human multiepoch EEG
[1] Baryshnikov, B.V., Van Veen, B.D., Wakai R.T., IEEE Trans. Biomed. Eng. 51 ( 2004), p. 1981–1993.
[2] de Munck, J.C., Bijma, F., Gaura, P., Sieluzycki, C.A., Branco, M.I., Heethaar, R.M., IEEE Trans. Biomed. Eng. 51 ( 2004), p. 2123 – 2128.Xj =S +Wj
S=HθCT
C … known matrix of temporal basis vectors, known frequency band is used to construct C
H … unknown matrix of spatial basis vectors θ … unknown matrix of coefficients
MLE & human multiepoch EEG
[2] de Munck, J.C., Bijma, F., Gaura, P., Sieluzycki, C.A., Branco, M.I., Heethaar, R.M., IEEE Trans. Biomed. Eng. 51 ( 2004), p. 2123 – 2128.
Xj =kjS+Wj
Xj=kjH θ CTRxj+Wj
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EEG & quantum mechanics IV
… shift operator in matrix quantum mechanics:
A. K. Kwasniewski, W. Bajguz and I. Jaroszewski, Adv. Appl. Clifford Algebras 8 (1998), 417-432.
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