Chaos in the brainJan 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

What has the human brain in common with quantum

mechanics?

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 & phaseN. 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

2

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N

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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 theoryLevel spacing distribution (compared with Wigner

formula for GOE)

□ ... visually stimulated+ … no stimulation

Human EEG & Random matrix theoryNumber variance (compared with prediction for GOE)

□ ... visually stimulated+ … no stimulation

Human EEG & Random matrix theorySummary

- 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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Experiment: Pattern reversalMLE & human multiepoch EEG

MLE & human multiepoch EEGOur MLE method

Baryshnikov et al MLE method

Averaging method

MLE & human multiepoch EEG

Trial dependence of amplitude weights

MLE & human multiepoch EEG

Trial dependence of latency lags

Thank you for your attention…