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Symbolic Dynamics of Neurophysiological Data
29. June 2007
PASCAL Workshop Berlin
Peter beim Graben [email protected]
School of Psychology and
Clinical Language Sciences
University of Reading
Content
symbolization techniques in neuroscience ERPs encodings complexity measures applications
CCCCOOCOCOCCCCCCOOCCOCOCCOCOOOCCCC
threshold current
Ion Channels I
(Kandel, Schwartz & Jessel 1991)
open probability at time t :
Trial 1: CCCOOOCOCOCCCCCCOOCCOCOCCOCOOOCCCCCTrial 2: CCCOOCCCCCCOOCCOCOCCOCOOOCOOCOCOCCCTrial 3: CCCOOOCOOOCOCOCOCOCOCOCOCOOOOCCOCCCTrial 4: CCCCCOOCOCOCCCCCCOCOCOCOCOCOOCOCCCC
PO(4) = 3 / 4
cylinder set
Ion Channels II
Markov-processes
stochastic resonance: Goychuk & Hänggi (2000)
α = α(Vm), β = β(Vm)
Vm = V0 sin (ω t)
Action Potentials I
tria
ls
(Grün 1996)spike rate of neuron i at time t : pi(t)
cylinder sets
Action Potentials II
algorithmic complexity: Rapp (1994)
spike train
1
2inter spike intervals
median
0 1 0 1 0 1 0 1
C (01010101) = (1 + ld 4 + 2) / 8 = 0.625
Event-Related Potentials ISetup
Event-Related Potentials II
(Lehmann 1971)
positive and negative maximal potentials
negative (minimum) positive (maximum)
Event-Related Potentials III
(Callaway & Halliday 1973)polarity histograms
threshold for coarse-graining:mean (baseline)
The Dynamical Approach
Başar (1980, 1983)beim Graben et al. (2000)
experimental manipulations are control parameters
ERP time series are images of trajectories of a dynamical system exploring the neural phase space under the EEG observable
ensembles of ERP epochs correspond to ensembles of trajectories starting from randomly distributed initial conditions
ERPs are order parameters
control condition
critical condition
“bottleneck”
Başar (1980, 1983)
Phase Space Portrait
control condition
critical condition
“bottleneck”
Coarse-Graining
cell 1
cell 2
cell 3
cell 0
cell 4
Symbolic Dynamics of Noisy Data
cell 1
noisy trajectory
1111111111111111111
noisy trajectory
1001010010100010100101
symbolic denoising
cell 1
cell 0
noise induced border crossings
Cylinder Sets of ERP
condition X epoch a 1 1 0 0 0 epoch b 0 1 0 0 1 epoch c 0 0 1 0 0 epoch d 1 1 0 0 1 time
t1
t2
t3
t4
t5
[10]t2 = {a, b, d}
cylinder set := set of all sequences agreeing in some affix.
Measures of Complexity
word statistics
Shannon entropy
Rényi entropies
redundancy: order parameter
Entropy
measure for disorder and unpredictability
0 1
prob
0,5
1,0
uniform distribution
max. Entropy: H = 1
1
prob
0,5
1,0
Dirac distribution
min. Entropy: H = 0
Signal-to-Noise Ratios
averaged ERP: ERP symbolization:
beim Graben (2001)
One-Threshold Encodings
ERP epoch
1100101010110111011001
• baseline (mean)*
• distribution (median)**
• half waves***
* Callaway & Halliday (1973)** Rapp (1994), Pompe (1998), beim Graben et al. (2000)*** Lehmann (1971), beim Graben (2001) beim Graben & Frisch (2004)
Two-Threshold Encoding
noisy signal
upper encoding threshold
lower encoding threshold
noise induced threshold crossings
beim Graben & Kurths (2003)
Symbolic Resonance Analysis
beim Graben & Kurths (2003)Frisch & beim Graben (2005)
threshold Θ1
threshold Θ2
threshold Θ3
3-symbol encodingBessel function + noise
0101211101212121101012010
3-symbol encoding word statistics
σ = 0.5, Θ = 0.5
Symbolic Resonance Analysis
(1+1)-dim 3-Pottsspin lattice magnetizations
spin flip transform
time
trial
Mean-Field Transformation
(1+1)-dim 3-Pottsspin lattice magnetizations
spin flip transform
time
trial
Mean-Field Transformation
(1+1)-dim 3-Pottsspin lattice magnetizations
spin flip transform
time
trial
Mean-Field Transformation
(1+1)-dim 3-Pottsspin lattice magnetizations
spin flip transform
time
trial
Mean-Field Transformation
(1+1)-dim 3-Pottsspin lattice magnetizations
spin flip transform
time
trial
Mean-Field Transformation
(1+1)-dim 3-Pottsspin lattice magnetizations
spin flip transform
time
trial
Mean-Field Transformation
(1+1)-dim 3-Pottsspin lattice magnetizations
spin flip transform
time
trial
Mean-Field Transformation
(1+1)-dim 3-Pottsspin lattice magnetizations
spin flip transform
time
trial
Mean-Field Transformation
(1+1)-dim 3-Pottsspin lattice magnetizations
spin flip transform
kind of “Reversi”time
trial
Mean-Field Transformation
3-symbols word statistics 2-symbols word statistics
Mean-Field Transformation
3-symbol encoding 2-symbol encoding
Mean-Field Transformation
Signal Dissociation Isimulated ERPs from different stochastic processes 1 vs 2.
Signal Dissociation IItime-threshold analysis
cylinder entropy differences χ2 error probabilities
beim Graben et al (in prep.)
Applications
Oddball Experiment
rare (10%)
frequent (90%)
P300
Baseline Encoding I
prestimulus average -200 to 0 ms.
Baseline Encoding II
word statistics (polarity histogram)
rare, `1´
rare, `0´
frequent, `1´
frequent, `0´
Baseline Encoding III
rarefrequent
sliding 1-word Shannon entropy
P300
Median Encoding I
median of epochs uniform distribution of symbols
Median Encoding II
word statistics
rare, `1´
rare, `0´
frequent, `1´
frequent, `0´
Median Encoding III
rarefrequent
sliding 1-word Shannon entropy
P300
Symbolic Resonance Analysis
Hans weiß, welchen Betrag der Bläser dem Geiger borgte.* Hans weiß, welchen Betrag der Bläser dem Geiger half.* Hans weiß, welchen Betrag der Bläser dem Geiger verbrauchte.
ERP averages resonance of the SNR
N400
dissociation of the N400
beim Graben et al (2005)
Symbolic Resonance Analysis
John knows which amount the trumpeter lend the violonist.* John knows which amount the trumpeter helped the violonist.* John knows which amount the trumpeter consumed the violonist.
ERP averages resonance of the SNR
N400
dissociation of the N400
beim Graben et al (2005)
Time-Threshold-Analysis
instantaneous word statistics
differences ofcylinder entropies:accusative - dative
voltage averages
Negative Polarity Processing
No man who had a beard was ever happy.
A man who had a beard was ever happy.
A man who had no beard was ever happy. Drenhaus et al. (2006)
N400
P600
N400
P6001
P6002
SRA word statistics
optimal discrimination threshold 7.7 μV
Acknowledgements
Douglas Saddy, Leticia Pablos Robles, Jürgen Kurths, Stefan Frisch, Heiner Drenhaus, Anja Meinke, Eva Brehm, Andrew Fink, Matthias Schlesewsky