Chaos in the brain

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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 thebrainUniversity of Hradec Králové,Doppler Institute formathematical physics and appliedmathematicsCzech RepublicJan Kříž5th Workshop on Quantum Chaos and Localisation PhenomenaWarszawaMay 22, 2011What has thehumanbrain in commonwithquantummechanics?Human EEGmeasureselectricpotentials on thescalp (generated by neuronalactivity in thebrain)„Theanalysisof EEG has a longhistory.Beingusedas a diagnostictoolfor 80 yearsitstillresiststo be a subjectofstrictand objectiveanalysis.“QuantumMechanicsRichard P. Feynman (1918 -1988))I can safely say that nobody understands quantum mechanicsEEG & quantum mechanics I
  • EEG signal = interference of electric signals produced by activity of huge number of neurons
  • Superposition principleF. 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)Example 1: Ocular dominance & nodal domainsP. 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 frequencyspectrum 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):
  • 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)
  • HumanEEG & Random matrix theoryP. 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
  • 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
  • HumanEEG & Random matrix theoryEigenvaluedensityfunction (log-log scale)Smalleigenvalues:subjectdependentLargeeigenvalues:subj. independent tailofthesameform as RandomLévymatricsZ. Burda, J. Jurkiewicz, M.A.Nowak, G. Papp, I. Zahed,Phys. Rev. E 65 (2002), ArtNo 021106 .HumanEEG & Random matrix theoryLevelspacingdistribution (comparedwithWignerformulafor GOE)□ ... visuallystimulated+ … no stimulationHumanEEG & Random matrix theoryNumber variance (comparedwithpredictionfor GOE)□ ... visuallystimulated+ … no stimulationHumanEEG & Random matrix theorySummary
  • Levelspacingdistribution: verygoodagreementwiththe RMT predictions => universal behaviour
  • Number variance: sensitive when the subject is visually stimulated
  • Itisreasonable to assumethatalsosomepathologicalprocessescan influence thenumber variance
  • Evoked response potentials- responsesto external stimulus (auditory, visual, ...)- sensoryand cognitiveprocessing in thebrainlow „SNR“… noise (everythingwhatwe are not interested in including background activityofneurons)Evoked response potentialsCommonlyusedmethods: Filtering + averaging, PCAOurmethod: MAXIMUM LIKELIHOOD ESTIMATION
  • standard toolofstatisticalestimationtheory
  • by R. A. Fisher
  • datingback to 1920’s
  • Corner stone:mathematical modelMLE & human multiepochEEGBasic 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.
  • 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+WjS=HθCTC … known matrix oftemporal basis vectors, knownfrequency band isused to constructCH … unknown matrix ofspatial basis vectorsθ … unknown matrix ofcoefficientsMLE & 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+WjXj=kjS+WjEEG & quantum mechanics IV… shift operator in matrix quantum mechanics:A. K.Kwasniewski, W. Bajguz and I. Jaroszewski,Adv. Appl. CliffordAlgebras8 (1998), 417-432.MLE & humanmultiepochEEGExperiment: Pattern reversalMLE & humanmultiepochEEGOur MLE methodBaryshnikov et al MLE methodAveraging methodMLE & humanmultiepochEEGTrial dependence of amplitude weightsMLE & humanmultiepochEEGTrial dependence of latency lagsThank you for your attention…
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