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Chaos in the brain
Jan Kříž
University of Hradec Králové,
Doppler Institute for mathematical physics
and applied mathematics
Czech Republic
5th Workshop on Quantum Chaos and Localisation Phenomena
Warszawa
May 22, 2011
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.“
Quantum Mechanics
Richard P. Feynman (1918 -1988))
I can safely say that
nobody understands
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
t f   const.
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)
Cl ,m (T ) 
… EEG channel l at time tj
N2
 x (t ) x
j  N1
l
j
m
(t j )
N1, N2 chosen such that t j  T , T    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 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)
Evoked response potentials
Commonly used methods: Filtering + averaging,
PCA
Our method: MAXIMUM LIKELIHOOD ESTIMATION
- standard tool of statistical estimation theory
- by R. A. Fisher
- dating back to 1920’s
Corner stone:
mathematical model
MLE & human multiepoch EEG
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.
Xj =S +W
Biomed.
Eng.
j 51 ( 2004), p. 2123 – 2128.
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,XM.I.,
Heethaar,
R.M.,
IEEE Trans.
TR
xj+W
=k
H
θ
C
j ( 2004),
j
j
Biomed. Eng. 51
p. 2123 – 2128.
0
X
 j
0
R 

0
1

1=k0S+W
 0
j
j
0 1  0
   

0 0  1
0 0  0 
EEG & quantum mechanics IV
0

0
R 

0
1

1 0  0

0 1  0
      exp iPˆ

0 0  1
0 0  0 
 
… shift operator in matrix quantum mechanics:
R q  q 1
A. K. Kwasniewski, W. Bajguz and I. Jaroszewski, Adv.
Appl. Clifford Algebras 8 (1998), 417-432.
MLE & human multiepoch EEG
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…