Analyzing Brain Signals by Combinatorial Optimization

Analyzing Brain Signals

by Combinatorial Optimization

Justin DauwelsLIDS, MIT

Amari Research Unit, Brain Science Institute, RIKEN

September 25, 2008

Topics• Mathematical problem Similarity of Multiple Point Processes

• Motivation/Application Diagnosis of Alzheimer’s disease from EEG signals

CollaboratorsFrançois Vialatte*, Theophane Weber+, and Andrzej Cichocki* (*RIKEN, +MIT)

Financial Support

ABSP

DSM-IVmemory : amnesia, executive functions impairment (c'est-à-dire d'organisation et de réalisation d'une tâche complexe, comme par exemple remplir sa feuille de déclaration d'impôts), langage impairment (aphasie amnésique), apraxy (incapacite de réalisation de gestes complexes : par exemple utiliser la machine à laver), agnosia (troubles de reconnaissance) : par exemple de panneaux routiers, puis de visages etc.

Alzheimer's disease

• Mild (early stage)- becomes less energetic or spontaneous- noticeable cognitive deficits- still independent (able to compensate)

• Moderate (middle stage)- Mental abilities decline- personality changes- become dependent on caregivers

• Severe (late stage)- complete deterioration of the personality- loss of control over bodily functions- total dependence on caregivers

Apathy

Memory(forgettingrelatives)

Evolution of the disease (stages)One disease, many symptoms

Video sources: Alzheimer society

• 2 to 5 years before- mild cognitive impairment (MCI)- 6 to 25 % progress to Alzheimer‘s

memory, language, executive functions, apraxia, apathy, agnosia, etc…

• 2% to 5% of people over 65 years old• up to 20% of people over 80 Jeong 2004 (Nature)

EEG data

GOAL: Diagnosis of MCI based on EEG

• EEG is relatively simple and inexpensive technology• Early diagnosis: medication more effective, more time to prepare future care of patient, etc.

ABSP

DSM-IVmemory : amnesia, executive functions impairment (c'est-à-dire d'organisation et de réalisation d'une tâche complexe, comme par exemple remplir sa feuille de déclaration d'impôts), langage impairment (aphasie amnésique), apraxy (incapacite de réalisation de gestes complexes : par exemple utiliser la machine à laver), agnosia (troubles de reconnaissance) : par exemple de panneaux routiers, puis de visages etc.

Overview

Alzheimer’s Disease (AD) decrease in EEG synchrony Similarity of Point Processes

Two 1-D point processesTwo multi-D point processesMultiple multi-D point processes

Numerical Results Conclusion

Alzheimer's diseaseInside glimpse: abnormal EEG

• AD vs. MCI (Hogan et al. 203; Jiang et al., 2005)• AD vs. Control (Hermann, Demilrap, 2005, Yagyu et al. 1997; Stam et al., 2002; Babiloni et al. 2006)• MCI vs. mildAD (Babiloni et al., 2006).

Decrease of synchrony

Images: www.cerebromente.org.br

EEG system: inexpensive, mobile, useful for screening

Spontaneous (scalp) EEG

Fourier power

f (Hz)

t (sec)

ampl

itude

Fourier |X(f)|2

EEG x(t)

Time-frequency |X(t,f)|2(wavelet transform)

Time-frequency patterns(“bumps”)

Sparse representation: bump model

Bumps

Sparse representation

F. Vialatte et al. “A machine learning approach to the analysis of time-frequency maps and its application to neural dynamics”, Neural Networks (2007).

104- 105 coefficients

about 102 parameters

t (sec)

f(Hz)

f(Hz)

t (sec)

f(Hz)

t (sec)

Assumptions:

1. time-frequency map is suitable representation

2. oscillatory bursts (“bumps”) convey key information

Similarity of bump models

How “similar” are n ≥ 2 bump models?

Similarity of multiple multi-dimensional point processes

with and

“point” / ”event”

Overview




Two one-dimensional point processes

tx

x’

0

0 t

Generative model

v0 T0

T0

T0

0

0 T0

x

x’

0

0 T0

-δt /2

δt /2

non-coincident

non-coincident

x

x

Stochastic event synchrony (SES): delay δt , jitter st , non-coincidence ρ

Generative model

v0 T0

T0

T0

0

0 T0

x

x’

0

0 T0

-δt /2

δt /2

geometric prior for lenght

events i.u.d. in [0,T0]

Gaussian offsets withmean -δt /2 and variance st /2

Gaussian offsets withmean δt /2 and variance st /2

i.i.d. deletions with prob pd

i.i.d. deletions with prob pd

non-coincident

non-coincident

x

x

Marginalizing over v:

Generative model (2)

Cost function

unit cost of non-coincident event

unit cost of coincident pair

Model

Probabilistic inference

DYNAMIC PROGRAMMINGPARAMETER ESTIMATION

PROBLEM: Given 2 point processes x and x’, compute ρ and θ = δt , st

APPROACH: (j*, j’*,θ*) = argmaxj,j’,θ log p(x, x’, j, j’,θ)

SOLUTION: Coordinate descent

(j(i+1) , j’(i+1) ) = argmaxj,j’ log p(x, x’, j , j’ , θ(i)) θ(i+1) = argmaxx log p(x, x’, j(i+1) , j’(i+1) , θ)

0 x1 x2 x3 x4 x5 x6

0

x’1

x’2

x’3

x’4

x’5

x’6

xk non-coincident x’k’ non-coincident (xk x’k’ ) coincident pair

Application: spike trains

High reliabilityLarge timing dispersion

Low reliabilitySmall timing dispersion

jitter st = (15ms)2, non-coincidence ρ = 3% jitter st = (3ms)2, non-coincidence ρ = 27%

Overview




Similarity of two bump models...

... by matching bumps• Bumps in one model, but NOT in other → fraction of “non-coincident” bumps ρ

• Bumps in both models, but with offset → Average time offset δt (delay) → Timing jitter with variance st

→ Average frequency offset δf → Frequency jitter with variance sf

PROBLEM: Given two bump models, compute (ρ, δt, st, δf, sf )

Stochastic Event Synchrony (SES) = (ρ, δt, st, δf, sf )

Generative modelGenerate bump model (hidden)

• geometric prior for number of bumps

• bumps are uniformly distributed in rectangle

• amplitude, width (in t and f) all i.i.d.

Generate two “noisy” observations

• offset between hidden and observed bump = Gaussian random vector with mean ( ±δt /2, ±δf /2) covariance diag(st/2, sf /2)


• “deletion” with probability pd

yhidden

y y’

( -δt /2, -δf /2)

( δt /2, δf /2)

• Binary variables ckk’ : ckk’ = 1 if k and k’ are observations of same hidden bump, else ckk’ = 0• Constraints: sums Σk’ ckk’ and Σk ckk’ are binary (“matching constraints”)


MATCHINGPOINT ESTIMATION


APPROACH: (c*,θ*) = argmaxc,θ log p(y, y’, c, θ)

θ


c(i+1) = argmaxc log p(y, y’, c, θ(i) ) θ(i+1) = argmaxx log p(y, y’, c(i+1) ,θ )

MATCHING: c(i+1) = argmaxc log p(y, y’, c, θ(i) )

ALGORITHMS

• Polynomial-time algorithms gives optimal solution(s) (Edmond-Karp and Auction algorithm)• Linear programming relaxation: gives optimal solution if unique [Sanghavi (2007)]• Max-product algorithm gives optimal solution if unique [Bayati et al. (2005), Sanghavi (2007)]

EQUIVALENT to (imperfect) bipartite max-weight matching problem

c(i+1) = argmaxc log p(y, y’, c, θ(i) ) = argmaxc Σkk’ wkk’(i) ckk’

s.t. Σk’ ckk’ ≤ 1 and Σk ckk’ ≤ 1 and ckk’ 2 {0,1}

Probabilistic inference (2)

not necessarily perfectfind heaviest set of disjoint edges

Max-product algorithmMATCHING: c(i+1) = argmaxc log p(y, y’, c, θ(i) )

μ↑μ↑

μ↓ μ↓

• At convergence, compute marginals p(ckk’) = μ↓(ckk’) μ↓(ckk’) μ↑(ckk’)• Decisions: c*kk’ = argmaxckk’

p(ckk’) (optimal if solution unique)

Summary

MATCHING → max-productESTIMATION → closed-form



θ



Average synchrony

3. SES for each pair of models4. Average the SES parameters

1. Group electrodes in regions2. Bump model for each region

Overview




Beyond pairwise interactions...Pairwise similarity Multi-variate similarity

Similarity of multiple bump modelsy1 y2 y3 y4 y5

y1 y2 y3 y4 y5

Constraint: in each cluster at most one bump from each signal

Models similar if• few deletions/large clusters• little jitter

pc (i) = p(cluster size = i |y) (i = 1,2,…,M)

Parameters: θ = δt,m , δf,m , st,m , sf,m, pc


• geometric prior for number n of bumps



Generate M “noisy” observations

• offset between hidden and observed bump = Gaussian random vector with mean ( δt,m /2, δf,m /2) covariance diag(st,m/2, sf,m /2)



yhidden

y1 y2 y3 y4 y5



Exemplar-based formulationyhidden

y1 y2 y3 y4 y5

• Exemplars = identical copies of hidden bumps = cluster “center”• Other bumps in cluster = non-identical copies of exemplars

• Is event an exemplar?• If not, which exemplar is it associated with?• Several constraints

Integer program

Exemplar-based formulation: IPBinary Variables

Integer Program: LINEAR objective function/constraints

Equivalent to k-dim matching: for k = 2: in P but for k > 2: NP-hard!


CLUSTERING (Integer Program)ESTIMATION OF PARAMETERS

PROBLEM: Given M bump models, compute θ = δt,m , δf,m , st,m , sf,m, pc

APPROACH: (b*,θ*) = argmaxb,θ log p(y, y’, b, θ)


b(i+1) = argmaxc log p(y, y’, b, θ(i) ) θ(i+1) = argmaxx log p(y, y’, b(i+1) ,θ )

Integer programming methods (e.g., LP relaxation)• IP with 10.000 variables solved in about 1s• CPLEX: commercial toolbox for solving IPs (combines several algorithms)

NOTE: Max-product algorithm SUBOPTIMAL sometimes converged to “bad” solutions (how to fix??)

SummarySimilarity of multiple multi-dimensional point processes

Step 1: TWO ONE-dimensional point processes

Step 2: TWO MULTI-dimensional point processes

Step 3: MULTIPLE MULTI-dimensional point processes

Dynamic programming

Max-product/LP relaxation/Edmund-Karp

Integer Programming

Overview




EEG Data

EEG data provided by Prof. T. Musha

• EEG of 22 Mild Cognitive Impairment (MCI) patients and 38 age-matched control subjects (CTR) recorded while in rest with closed eyes → spontaneous EEG

• All 22 MCI patients suffered from Alzheimer’s disease (AD) later on

• Electrodes located on 21 sites according to 10-20 international system

• Electrodes grouped into 5 zones (reduces number of pairs) 1 bump model per zone

• Band pass filtered between 4 and 30 Hz

Sensitivity (average synchrony)

Granger

Info. Theor.

State Space

Phase

SES

Corr/Coh

Mann-Whitney test: small p value suggests large difference in statistics of both groups

Significant differences for ffDTF and SES (more unmatched bumps, but same amount of jitter)

Classification (bi-SES)

• Clear separation, but not yet useful as diagnostic tool• Additional indicators needed (fMRI, MEG, DTI, ...)• Can be used for screening population (inexpensive, simple, fast)

ffDTF

± 85% correctly classified

Overview




Conclusions

Measure for similarity of point processes

Key idea: matching of events

Application: EEG synchrony of MCI patients

About 85% correctly classified perhaps useful for screening a large population

Future work: Combination with other modalities (MEG, fMRI,...) Alternative inference techniques (variations on max-product, Monte-Carlo) More sophisticated models (e.g., interaction between events)

Analyzing Brain Signals

by Combinatorial Optimization

Justin DauwelsLIDS, MIT

Amari Research Unit, Brain Science Institute, RIKEN

September 25, 2008

References + softwareReferences

Quantifying Statistical Interdependence by Message Passing on Graphs: Algorithms and Application to Neural Signals, Neural Computation (under revision)

A Comparative Study of Synchrony Measures for the Early Diagnosis of Alzheimer's Disease Based on EEG, NeuroImage (under revision)

Measuring Neural Synchrony by Message Passing, NIPS 2007

Quantifying the Similarity of Multiple Multi-Dimensional Point Processes by Integer Programming with Application to Early Diagnosis of Alzheimer's Disease from EEG, EMBC 2008 (submitted)

Software

MATLAB implementation of the synchrony measures

Estimation

Deltas: average offset Sigmas: var of offset

...where

Simple closed form expressions

artificial observations (conjugate prior)

Large-scale synchrony

Apparently, all brain regions affected...

Alzheimer's disease

1980 1990 2000 2010 2020 2030 2040 20500

2

4

6

8

10

12

14

Outside glimpse: the future (prevalence)

USA (Hebert et al. 2003)

2000 2030 20500

20

40

60

80

100

120

Developped countriesDevelopping countries

World (Wimo et al. 2003)

Mill

ion

of s

uffe

rers

Mill

ion

of s

uffe

rers

• 2% to 5% of people over 65 years old

• Up to 20% of people over 80

Jeong 2004 (Nature)

Ongoing and future workApplications

alternative inference techniques (e.g., MCMC, linear programming) time dependent (Gaussian processes) multivariate (T.Weber)

Fluctuations of EEG synchrony Caused by auditory stimuli and music (T. Rutkowski) Caused by visual stimuli (F. Vialatte) Yoga professionals (F. Vialatte) Professional shogi players (RIKEN & Fujitsu) Brain-Computer Interfaces (T. Rutkowski)

Spike data from interacting monkeys (N. Fujii) Calcium propagation in gliacells (N. Nakata) Neural growth (Y. Tsukada & Y. Sakumura) ...

Algorithms

Fitting bump models

Signal

Bump

Initialisation After adaptationAdaptation

gradient method

F. Vialatte et al. “A machine learning approach to the analysis of time-frequency maps and its application to neural dynamics”, Neural Networks (2007).

Boxplots

SURPRISE!No increase in jitter, but significantly less matched activity!

Physiological interpretation• neural assemblies more localized?• harder to establish large-scale synchrony?


• geometric prior for number n of bumps p(n) = (1- λ S) (λ S)-n







yhidden

y y’

Easily extendable to more than 2 observations…

( -δt /2, -δf /2)

( δt /2, δf /2)



PROBLEM: Given two bump models, compute (ρspur, δt, st, δf, sf )


θ



Alzheimer's diseaseInside glimpse: abnormal EEG

• AD vs. MCI (Hogan et al. 203; Jiang et al., 2005)• AD vs. Control (Hermann, Demilrap, 2005, Yagyu et al. 1997; Stam et al., 2002; Babiloni et al. 2006)• MCI vs. mildAD (Babiloni et al., 2006).

Decrease of synchrony

Brain “slow-down”slow rhythms (0.5-8 Hz) fast rhythms (8-30 Hz)

(Babiloni et al., 2004; Besthorn et al., 1997; Jelic et al. 1996, Jeong 2004; Dierks et al., 1993).

Images: www.cerebromente.org.br

EEG system: inexpensive, mobile, useful for screening

focus of this project

Comparing EEG signal rhythms ?

PROBLEM I:

Signals of 3 seconds sampled at 100 Hz ( 300 samples)Time-frequency representation of one signal = about 25 000 coefficients

2 signals

Numerous neighboring pixels

Comparing EEG signal rhythms ?(2)

One pixel

PROBLEM II:

Shifts in time-frequency!

Strong (anti-) correlations „families“ of sync measures

Correlations









yhidden

y1 y2 y3 y4 y5





Average cluster size

Classification (multi-SES)

Average cluster size

Average bump freq

Average bump width

ffDTF

Similarity of bump models...

How “similar” or “synchronous” are two bump models?

Signatures of local synchronyf (Hz)

t (sec)

Time-frequency patterns(“bumps”)

EEG stems from thousands of neuronsbump if neurons are phase-locked= local synchrony

Alzheimer's diseaseInside glimpse: brain atrophy

Video source: P. Thompson, J.Neuroscience, 2003

Images: Jannis Productions.(R. Fredenburg; S. Jannis)

amyloid plaques andneurofibrillary tangles

Video source: Alzheimer society

POINT ESTIMATION: θ(i+1) = argmaxx log p(y, y’, c(i+1) ,θ )

Uniform prior p(θ): δt, δf = average offset, st, sf = variance of offset Conjugate prior p(θ): still closed-form expressionOther kind of prior p(θ): numerical optimization (gradient method)


MATCHING: c(i+1) = argmaxc log p(y, y’, c, θ(i) )

ALGORITHMS

• Polynomial-time algorithms gives optimal solution(s) (Edmond-Karp and Auction algorithm)• Linear programming relaxation: extreme points of LP polytope are integral• Max-product algorithm gives optimal solution if unique [Bayati et al. (2005), Sanghavi (2007)]

EQUIVALENT to (imperfect) bipartite max-weight matching problem

c(i+1) = argmaxc log p(y, y’, c, θ(i) ) = argmaxc Σkk’ wkk’(i) ckk’

s.t. Σk’ ckk’ ≤ 1 and Σk ckk’ ≤ 1 and ckk’ 2 {0,1}


not necessarily perfectfind heaviest set of disjoint edges

p(y, y’, c, θ) / I(c) pθ(θ) Πkk’ (N(t k’ – tk ; δt ,st,kk’) N(f k’ – fk ; δf ,sf, kk’) β-2)ckk’


Generative model


μ↑μ↑

μ↓ μ↓

Conditioning on θ

Max-product algorithm (2)• Iteratively compute messages

• At convergence, compute marginals p(ckk’) = μ↓(ckk’) μ↓(ckk’) μ↑(ckk’)• Decisions: c*kk’ = argmaxckk’

p(ckk’)

Algorithm

MATCHING → max-productESTIMATION → closed-form



θ











yhidden

y y’

Easily extendable to more than 2 observations…

( -δt /2, -δf /2)

( δt /2, δf /2)

Generative model (2)

• Binary variables ckk’

ckk’ = 1 if k and k’ are observations of same hidden bump, else ckk’ = 0 (e.g., cii’ = 1 cij’ = 0)

• Constraints: bk = Σk’ ckk’ and bk’ = Σk ckk’ are binary (“matching constraints”)

• Generative Model p(y, y’, yhidden , c, δt , δf , st , sf ) (symmetric in y and y’)

• Eliminate yhidden → offset is Gaussian RV with mean = ( δt , δf ) and covariance diag (st , sf)

• Probabilistic Inference:(c*,θ*) = argmaxc,θ log p(y, y’, c, θ)

y y’

( -δt /2, -δf /2)

( δt /2, δf /2)

i

i’ j’

p(y, y’, c, θ) = ∫ p(y, y’, yhidden , c, θ) dyhidden

θ

• Bumps in one model, but NOT in other → fraction of “spurious” bumps ρspur

• Bumps in both models, but with offset → Average time offset δt (delay) → Timing jitter with variance st

→ Average frequency offset δf → Frequency jitter with variance sf


APPROACH: (c*,θ*) = argmaxc,θ log p(y, y’, c, θ)θ

Summary

Objective function

• Logarithm of model: log p(y, y’, c, θ) = Σkk’ wkk’ ckk’ + log I(c) + log pθ(θ) + γ

wkk’ = -(1/st (t k’ – tk – δt)2 + 1/sf (f k’ – fk– δf)2 ) - 2 log β

β = pd (λ/V)1/2

Euclidean distance between bump centers

• Large wkk’ if : a) bumps are close b) small pd c) few bumps per volume element

• No need to specify pd , λ, and V, they only appear through β = knob to control # matches

y y’

( -δt /2, -δf /2)

( δt /2, δf /2)

i

i’ j’

Distance measures

wkk’ = 1/st,kk’ (t k’ – tk – δt)2 + 1/sf,kk’ (f k’ – fk– δf)2 + 2 log β

st,kk’ = (Δtk + Δt’k) st sf,kk’ = (Δfk + Δf’k) sf

Scaling

Non-Euclidean

p(y, y’, c, θ) / I(c) pθ(θ) Πkk’ (N(t k’ – tk ; δt ,st,kk’) N(f k’ – fk ; δf ,sf, kk’) β-2)ckk’

Generative model

Expect bumps to appear at about same frequency, but delayed Frequency shift requires non-linear transformation, less likely than delay Conjugate priors for st and sf (scaled inverse chi-squared):

Improper prior for δt and δt : p(δt) = 1 = p(δf)

Prior for parameters

CTR

MCI

Preliminary results for multi-variate modellinear comb of pc





θ



X

Y

Minx2 X, y2Y d(x,y)









(other prior pc0 for cluster size)

yhidden

y1 y2 y3 y4 y5



(Hebb 1949, Fuster 1997)

Stimuli Consolidation Stimulus

Voice Face Voice

Role of local synchrony

Assembly activation Hebbian consolidationAssembly recall


CLUSTERING (IP or MP)POINT ESTIMATION

PROBLEM: Given M bump models, compute θ = δt,m , δf,m , st,m , sf,m, pc




Integer program• Max-product algorithm (MP) on sparse graph• Integer programming methods (e.g., LP relaxation)

Fourier transform

High frequency

Low frequency

Frequency

1 23

2

13

Windowed Fourier transform

* =Fourier basis functions Window

function windowed basis functions

WindowedFourierTransform

t

f

Overview

Alzheimer’s Disease (AD): decrease in EEG synchrony Synchrony measure in time-frequency domain

Pairs of EEG signalsCollections of EEG signals


Average synchrony

3. SES for each pair of models4. Average the SES parameters

1. Group electrodes in regions2. Bump model for each region

Beyond pairwise interactions...Pairwise similarity Multi-variate similarity

Similarity measures• Correlation and coherence• Granger causality (linear system): DTF, ffDTF, dDTF, PDC, PC, ...

• Phase Synchrony: compare instantaneous phases (wavelet/Hilbert transform)

• State space based measures sync likelihood, S-estimator, S-H-N-indices, ...

• Information-theoretic measures KL divergence, Jensen-Shannon divergence, ...

No Phase Locking Phase Locking

TIME FREQUENCY

Analyzing Brain Signals by Combinatorial Optimization

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