Symbolic Dynamics of Neurophysiological Data 29. June 2007 PASCAL Workshop Berlin Peter beim Graben...

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

mailto:[email protected]

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


(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


3-symbols word statistics 2-symbols word statistics


3-symbol encoding 2-symbol encoding


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


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)


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

Symbolic Dynamics of Neurophysiological Data 29. June 2007 PASCAL Workshop Berlin Peter beim Graben...

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