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Chaos in the brain

Chaos in the brain. University of Hradec Králové , Doppler Institute for mathematical physics and applied mathematics Czech Republic. Jan Kříž. 5th Workshop on Quantum Chaos and Localisation Phenomena Warszawa May 22, 2011.

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Chaos in the brain

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  1. Chaos in thebrain University of Hradec Králové, Doppler Institute formathematical physics and appliedmathematics Czech Republic Jan Kříž 5th Workshop on Quantum Chaos and Localisation PhenomenaWarszawaMay 22, 2011

  2. What has thehumanbrain in commonwithquantummechanics?

  3. Human EEG measureselectricpotentials on thescalp (generated by neuronalactivity in thebrain) „Theanalysisof EEG has a longhistory.Beingused as a diagnostictool for 80 yearsitstillresists to be a subjectofstrict and objectiveanalysis.“

  4. QuantumMechanics Richard P. Feynman (1918 -1988)) I can safely say that nobody understands quantum mechanics

  5. EEG & quantum mechanics I • EEG signal = interference of electric signals produced by activity of huge number of neurons Superposition principle F. Wolf andT. Geisel.Nature, 395(1998), 73-74. M. Schnabel, M. Kaschube, S. Lowel and F. Wolf, Eur. Phys. J. SpecialTopics, 145 (2007), 137-157. Structuresemerging in thevisualcortexare described by random Gaussian fields (known from quantum chaotic systems)

  6. Example 1: Ocular dominance & nodal domains P. A. Anderson, J. Olavarria and R. C. Van Sluyters, Journal of Neuroscience, 8 (1988), 2183-2200.

  7. Example 2: Directional selectivity& phase N. P. Issa, C. Trepel and M. P. Stryker, Journal of Neuroscience, 20 (2000), 8504-8514.

  8. 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 frequencyspectrum and its time evolution Heisenberg uncertainty relations … S. Krishnan, Conference “Biosignal 2008”, Brno, Czech Republic, Opening Ceremony Keynote Lecture.

  9. EEG (biomedical signals) & quantum mechanics III • we use mathematical (statistical) tools known from quantum mechanics (chaos): • Randommatrix theory: • T. Guhr, A. Müller-Groeling, H. A. Weidenmüller, PhysicsReports299 (1998), 189-425. • Maximum likelihoodestimation: • S.T. Merkel, C.A. Riofrío, S.T. Flammia, I.H.Deutsch, Phys. Rev. A 81 (2010), ArtNo. 032126 • (implementationof QSR to quantumkicked 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 & chaoticscattering in overlappingresonators)

  10. HumanEEG & 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

  11. HumanEEG & Random matrix theory • xl(tj) … EEG channel l at time tj • N1, N2chosen such that for Δ=150 ms • Experiment: clinical19 channel EEG device • 15 – 20 minutes per measurements • 90 volunteers • measuredwithoutandwithvisualstimulation • ensemble of 7000 matrices per one measure

  12. HumanEEG & Random matrix theory Eigenvaluedensityfunction (log-log scale) Smalleigenvalues: subjectdependent Largeeigenvalues: subj. independent tailofthesameform as RandomLévymatrics Z. Burda, J. Jurkiewicz, M.A.Nowak, G. Papp, I. Zahed, Phys. Rev. E 65 (2002), ArtNo 021106 .

  13. HumanEEG & Random matrix theory Levelspacingdistribution (comparedwithWignerformulafor GOE) □ ... visuallystimulated + … no stimulation

  14. HumanEEG & Random matrix theory Number variance (comparedwithpredictionfor GOE) □ ... visuallystimulated + … no stimulation

  15. HumanEEG & Random matrix theory Summary • Levelspacingdistribution: verygoodagreementwiththe RMT predictions => universal behaviour • Number variance: sensitive when the subject is visually stimulated • Itisreasonable to assumethatalsosomepathologicalprocessescan influence thenumber variance

  16. Evoked response potentials - responsesto external stimulus (auditory, visual, ...)- sensoryand cognitiveprocessing in thebrain low „SNR“… noise (everythingwhatwe are not interested in including background activityofneurons)

  17. Evoked response potentials Commonlyusedmethods: Filtering + averaging, PCA Ourmethod: MAXIMUM LIKELIHOOD ESTIMATION • standard toolofstatisticalestimationtheory • by R. A. Fisher • datingback to 1920’s Corner stone: mathematical model

  18. MLE & human multiepochEEG Basic concept of MLE (R.A. Fisher in 1920’s) • assumepdffofrandomvectorydepending on a parameter set w, i.e. f(y|w) • itdeterminesthe probability ofobservingthe data vectory (in dependence on theparametersw) • however, we are facedwith inverse problem: wehavegiven data vector and we do not knowparameters • MLE: giventheobserved data (and a model ofinterest = set ofpossiblepdfs), findthepdf, thatis most likely to producethegiven data.

  19. MLE & human multiepochEEG [1] Baryshnikov, B.V., Van Veen, B.D.,Wakai R.T., IEEETrans. 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 oftemporal basis vectors, knownfrequency band isused to constructC H … unknown matrix ofspatial basis vectors θ … unknown matrix ofcoefficients

  20. MLE & humanmultiepochEEG [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=kjH θ CTRxj+Wj Xj =kjS +Wj

  21. EEG & quantum mechanics IV … shift operator in matrix quantum mechanics: A. K.Kwasniewski, W. Bajguz and I. Jaroszewski,Adv. Appl. CliffordAlgebras8 (1998), 417-432.

  22. MLE & humanmultiepochEEG Experiment: Pattern reversal

  23. MLE & humanmultiepochEEG Our MLE method Baryshnikov et al MLE method Averaging method

  24. MLE & humanmultiepochEEG Trial dependence of amplitude weights

  25. MLE & humanmultiepochEEG Trial dependence of latency lags

  26. Thank you for your attention…

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