Chaos in the brain Jan Kříž 5th Workshop on Quantum Chaos and Localisation Phenomena Warszawa 5th Workshop on Quantum Chaos and Localisation Phenomena WarszawaMay 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 EEG measures 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 ( )) 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), M. Schnabel, M. Kaschube, S. Lowel and F. Wolf, Eur. Phys. J. Special Topics, 145 (2007), 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),

Example 2: Directional selectivity & phase N. P. Issa, C. Trepel and M. P. Stryker, Journal of Neuroscience, 20 (2000),

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 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:Random matrix theory: T. Guhr, A. Müller-Groeling, H. A. Weidenmüller, Physics Reports 299 (1998), Maximum likelihood estimation:Maximum likelihood estimation: S.T. Merkel, C.A. Riofrío, S.T. Flammia, I.H.Deutsch, Phys. Rev. A 81 (2010), ArtNo (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 (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 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 x l (t j ) … EEG channel l at time t j N 1, N 2 chosen such that for Δ=150 ms - Experiment: clinical19 channel EEG device 15 – 20 minutes per measurements 90 volunteers measured without and with visual stimulation -ensemble of 7000 matrices per one measure

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

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 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“ low „SNR“ … noise (everything what we are not interested in including background activity of neurons)

Commonly used methods: Filtering + averaging, PCA MAXIMUM LIKELIHOOD ESTIMATION 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 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

[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 – XjXj =S+W j S=HθC T 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 – XjXj =k j S+W j X j =k j H θ C T R x j +W j

EEG & quantum mechanics IV … shift operator in matrix quantum mechanics: A. K. Kwasniewski, W. Bajguz and I. Jaroszewski, Adv. Appl. Clifford Algebras 8 (1998),

Experiment: Experiment: Pattern reversal MLE & human multiepoch EEG

Our 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…