Analyzing Brain Signals by Combinatorial Optimization

Analyzing Brain Signals

by Combinatorial Optimization

Justin DauwelsLIDS, MIT

Amari Research Unit, Brain Science Institute, RIKEN

December 1, 2008

Quantifying statistical interdependence of point processes

Application to spike data and EEG

Topics• Mathematical problem Similarity of Multiple Point Processes

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

• Along the way… Spike synchrony

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

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

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 1-dim point processesTwo multi-dim point processesMultiple multi-dim point processes


Two one-dimensional point processes

tx

x’

0

0 t

How synchronous/similar?

Classical methods for continuous time series faile.g., cross-correlation

Two aspects of synchrony

Analogy: waiting for a train

• Train may not arrive (e.g., mechanical problem) = Event reliability

• Train may or may not be on time = Timing precision

Two 1-dim point processes

Review of Spike Synchrony Measures Surrogate Spike Data Spike Trains from Morris-Lecar Neuron Conclusion

Spike Synchrony Measures

Von Rossum distance (mixed) Schreiber et al similarity measure (mixed) Hunter-Milton similarity measure (mixed) Victor-Purpura distance metric (event reliability) Event synchronization (mixed) Stochastic event synchrony

(timing precision and event reliability)

Van Rossum distance measure• Spikes convolved with exponential or Gaussian

function

→ spike trains converted into time series s(t) and s’(t)

• Squared distance between s(t) and s’(t)

• If x = x’, we have DR = 0

• Time constant τR

x

x’

0

0 τR

van Rossum M.C.W., 2001. A novel spike distance. Neural Computation 13, 751–63.

Schreiber et al. similarity measure

• Spikes convolved with exponential or Gaussian function

→ spike trains converted into time series s(t) and s’(t)

• Correlation between s(t) and s’(t)

• If x = x’, we have SS = 1

• Time constant τS

Schreiber S., Fellous J.M., Whitmer J.H., Tiesinga P.H.E., and Sejnowski T.J., 2003. A new correlation-based measure of spike timing reliability. Neurocomputing 52, 925–931.

x

x’

0

0

Victor-Purpura distance measure• Minimal cost DV of transforming x into x'• Basic operations

• event insertion/deletion: cost = 1• event movement: cost proportional to distance (constant CV)

• If x = x’, we have DV = 0

• Time constant τV = 1/CV

DELETION

INSERTION

Victor J. D. and Purpura K. P., 1997. Metric-space analysis of spike trains: theory, algorithms, and application. Network: Comput. Neural Systems 8(17), 127–164.

Stochastic Event Synchrony

• x and x’ synchronous if identical apart from• delay• little timing jitter• few deletions/insertions

• based on generative statistical model

x

x’

0

0

v0

Dauwels J., Vialatte F., Rutkowski T., and Cichocki A., 2007. Measuring neural synchrony by message passing, NIPS 20, in press.

Stochastic Event Synchrony

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 ρDauwels J., Vialatte F., Rutkowski T., and Cichocki A., 2007. Measuring neural synchrony by message passing, NIPS 20, in press.

Marginalizing over v:

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

xStochastic Event Synchrony


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

Spike Synchrony Measures

Von Rossum distance (mixed) Schreiber et al similarity measure (mixed) Hunter-Milton similarity measure (mixed) Victor-Purpura distance metric (event reliability) Event synchronization (mixed) Stochastic event synchrony

(timing precision and event reliability)



Surrogate Data

• pd = 0, 0.1, …, 0.4 (deletion probability)• δt = 0, 25, and 50 ms (delay)• σt = 10, 30, and 50 ms (timing jitter)• length of hidden sequence = 40/(1-pd)• expected length of x and x’ = 40

E{S} computed over 10’000 pairs

Surrogate Data: Results

• E{DR} increases with pd and σt

→ DR cannot distinguish timing dispersion from event reliability(likewise all measures except SES and DV)

• E{DV} increases with pd, practically independent of σt

→ DV measure for event reliability

• ONLY curves for δt = 0ms, measures strongly depend on lag

Victor Purpura measure DVVan Rossum measure DRsimilar for SS ,SH ,SQ

δt =0

Surrogate Data: Results for SES

• E{σt} increases with σt, practically independent of pd

→ σt measure for timing dispersion

• E{ρ} increases with pd, practically independent of σt

→ ρ measure for event reliability

• Curves for δt = 0, 25, and 50 ms practically coincident



Morris-Lecar Neurons• Simple neuron model• Exhibits behavior of Type I & II neurons (saddle-node/Hopf bifurc.)• Input current: baseline + sinusoid + Gaussian noise

• Membrane potentialType I Type II

5 trials

Spiking threshold

High reliabilityLarge timing dispersion

Low reliabilitySmall timing dispersion

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

Type I Type II

Morris-Lecar Neurons (2)

50 trials

Morris-Lecar Neurons: Results

• Small τ: Type II has larger similarity than type I (dispersion in Type I)• Large τ: Type I has larger similarity than type II (drop-outs in Type II)

• Observation:Similarity depends on time constant τ → similarity FUNCTION S(τ)SES AUTOMATICALLY selects st



Conclusion

• Similarity of pairs of spike trains: timing precision and reliability

• Comparison of various spike synchrony measures

• Most measures not able to separate the two aspect of synchrony• Exception: Victor-Purpura and Stochastic Event Synchrony

• Victor-Purpura: event reliability• SES: both timing precision and event reliability

• Most measures depend on time constant, to be chosen by user• Exception: Event Synchronization and SES

• Most measures sensitive to lags between the two spike trains• Exception: SES

• Future work: application to neurophysiological recordings

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 model

Generate 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)


Summary

MATCHING → max-productESTIMATION → closed-form


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 )


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

Dauwels J., Vialatte F., Weber T. and Cichocki. Analyzing Brain Signals by Combinatorial Optimization, Allerton 2008.

Generative model


• 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

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

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



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)


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

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

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

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

Correlations

Overview





Conclusions Measure for similarity of point processes

Key idea: matching of events

Applications Spiking synchrony (surrogate data/Morris Lecar neuron) EEG synchrony of MCI patients

SES allows to distinguish event reliability from timing precision

About 85-90% correctly classified MCI vs. healthy subjects perhaps useful for screening a large population

Future work: Combination with other modalities (MEG, fMRI, ...) Integration of biophysical models Alternative inference techniques (variations on max-product, Monte-Carlo)

Analyzing Brain Signals

by Combinatorial Optimization

Justin DauwelsLIDS, MIT

Amari Research Unit, Brain Science Institute, RIKEN

December 1, 2008

Quantifying statistical interdependence of point processes

Application to spike data and EEG

References + softwareReferences

Quantifying Statistical Interdependence by Message Passing on GraphsPART I: One-Dimensional Point Processes, Neural Computation (under revision)

Quantifying Statistical Interdependence by Message Passing on GraphsPART II: Multi-Dimensional Point Processes, Neural Computation (under revision)

Quantifying Statistical Interdependence by Message Passing on GraphsPART III: Multivariate Approach, Neural Computation (in preparation)

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

On the Early Diagnosis of Alzheimer's Disease Based on EEG, Current Alzheimer’s Research (in preparation, invited review)

Measuring Neural Synchrony by Message Passing, NIPS 2007

Analyzing Brain Signals by Combinatorial Optimization, Allerton 2008

SoftwareMATLAB implementation of the synchrony measuresMATLAB Toolbox for bump modelling

.

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

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 countries

Developping countries

World (Wimo et al. 2003)

Mil

lio

n o

f su

ffer

ers

Mil

lio

n o

f su

ffer

ers

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

• Up to 20% of people over 80

Jeong 2004 (Nature)

Ongoing and future work

Applications

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?

Generative modelGenerate bump model (hidden)

• 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)


MATCHINGPOINT ESTIMATION

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!

Generative model









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’

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

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



Generative modelGenerate bump model (hidden)








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)

Generative model









(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

1

3

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

Generative model (2)

Cost function

unit cost of non-coincident event

unit cost of coincident pair

Model

Surrogate Data: Results (2)

• SS depends on δt

• likewise other S except SES





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


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)

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!

Documents

Analyzing Brain Signals by Combinatorial Optimization