Transcript Human brain dynamics and synchrony measures, applications

Analyzing Brain Signals
by Combinatorial Optimization
Justin Dauwels
LIDS, 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
Collaborators
François Vialatte*, Theophane Weber+, and Andrzej Cichocki*
Financial Support
(*RIKEN, +MIT)
Alzheimer's disease
One disease,
many symptoms
memory, language, executive functions,
apraxia, apathy, agnosia, etc…
Memory
(forgetting
relatives)
Apathy
Video sources: Alzheimer society
• 2% to 5% of people over 65 years old
• up to 20% of people over 80
Evolution of the disease (stages)
•
-
2 to 5 years before
mild cognitive impairment (MCI)
6 to 25 % progress to Alzheimer‘s
•
-
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
EEG data
Jeong 2004 (Nature)
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.
Overview

Alzheimer’s Disease (AD)
decrease in EEG synchrony

Similarity of Point Processes
 Two
1-D point processes
 Two multi-D point processes
 Multiple multi-D point processes
Numerical Results
 Conclusion

Alzheimer's disease
Inside glimpse: abnormal EEG
EEG system: inexpensive, mobile, useful for screening
Decrease of synchrony
•
•
•
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).
Images: www.cerebromente.org.br
Spontaneous (scalp) EEG
Time-frequency |X(t,f)|2
(wavelet transform)
f (Hz)
Time-frequency patterns
(“bumps”)
Fourier |X(f)|2
Fourier power
amplitude
t (sec)
EEG x(t)
Sparse representation: bump model
f(Hz)
f(Hz)
Bumps
Sparse
representation
104-
105
coefficients
t (sec)
t (sec)
f(Hz)
t (sec)
Assumptions:
about 102 parameters
1. time-frequency map is suitable representation
2. oscillatory bursts (“bumps”) convey key information
F. Vialatte et al. “A machine learning approach to the analysis of time-frequency maps and its application to neural dynamics”, Neural Networks (2007).
Similarity of bump models
How “similar” are n ≥ 2 bump models?
Similarity of multiple multi-dimensional point processes
with
and
“point” / ”event”
Overview

Alzheimer’s Disease (AD)
decrease in EEG synchrony

Similarity of Point Processes
 Two
1-D point processes
 Two multi-D point processes
 Multiple multi-D point processes
Numerical Results
 Conclusion

Two one-dimensional point processes
x
0
t
0
t
x’
Generative model
non-coincident
x
x
0
T0
0
T0
v
0
0
T0
-δt /2
T0
δt /2
x
x’
0
non-coincident
T0
Stochastic event synchrony (SES): delay δt , jitter st , non-coincidence ρ
Generative model
non-coincident
x
x
i.i.d. deletions with prob pd
0
T0
Gaussian offsets with
mean -δt /2 and variance st /2
0
T0
geometric prior for lenght
v
0
T0
-δt /2
events i.u.d. in [0,T0]
Gaussian offsets with
mean δt /2 and variance st /2
0
T0
δt /2
x
x’
0
non-coincident
Marginalizing over v:
i.i.d. deletions with prob pd
T0
Generative model (2)
Model
Cost function
unit cost
of non-coincident event
unit cost
of coincident pair
Probabilistic inference
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)
DYNAMIC PROGRAMMING
PARAMETER ESTIMATION
x’6
x’5
x’4
x’3
x’2
x’1
xk non-coincident
0
0 x1 x2 x3 x4 x5 x6
x’k’ non-coincident
(xk x’k’ ) coincident pair
Application: spike trains
High reliability
Large timing dispersion
jitter st = (15ms)2, non-coincidence ρ = 3%
Low reliability
Small timing dispersion
jitter st = (3ms)2, non-coincidence ρ = 27%
Overview

Alzheimer’s Disease (AD)
decrease in EEG synchrony

Similarity of Point Processes
 Two
1-D point processes
 Two multi-D point processes
 Multiple multi-D point processes
Numerical Results
 Conclusion

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
Stochastic Event Synchrony (SES)
= (ρ, δt, st, δf, sf )
PROBLEM: Given two bump models, compute (ρ, δt, st, δf, sf )
Generative model
yhidden
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
y y’
( -δt /2, -δf /2)
( δt /2, δf /2)
• offset between hidden and observed bump
= Gaussian random vector with
mean ( ±δt /2, ±δf /2)
covariance diag(st/2, sf /2)
• amplitude, width (in t and f) all i.i.d.
• “deletion” with probability pd
• 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”)
Probabilistic inference
PROBLEM: Given two bump models, compute (ρ, δt, st, δf, sf )
θ
APPROACH:
(c*,θ*) = argmaxc,θ log p(y, y’, c, θ)
SOLUTION: Coordinate descent
c(i+1) = argmaxc log p(y, y’, c, θ(i) )
θ(i+1) = argmaxx log p(y, y’, c(i+1) ,θ )
MATCHING
POINT ESTIMATION
Probabilistic inference (2)
MATCHING: c(i+1) = argmaxc log p(y, y’, c, θ(i) )
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}
find heaviest set of disjoint edges
not necessarily perfect
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)]
Max-product algorithm
MATCHING: 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
PROBLEM: Given two bump models, compute (ρ, δt, st, δf, sf )
θ
APPROACH:
(c*,θ*) = argmaxc,θ log p(y, y’, c, θ)
SOLUTION: Coordinate descent
c(i+1) = argmaxc log p(y, y’, c, θ(i) )
θ(i+1) = argmaxx log p(y, y’, c(i+1) ,θ )
MATCHING → max-product
ESTIMATION → closed-form
Average synchrony
1. Group electrodes in regions
2. Bump model for each region
3. SES for each pair of models
4. Average the SES parameters
Overview

Alzheimer’s Disease (AD)
decrease in EEG synchrony

Similarity of Point Processes
 Two
1-D point processes
 Two multi-D point processes
 Multiple multi-D point processes
Numerical Results
 Conclusion

Beyond pairwise interactions...
Pairwise similarity
Multi-variate similarity
Similarity of multiple bump models
y1 y2 y3 y4 y5
Models similar if
• few deletions/large clusters
• little jitter
y1 y2 y3 y4 y5
Constraint: in each cluster at most
one bump from each signal
pc (i) = p(cluster size = i |y)
(i = 1,2,…,M)
Parameters: θ = δt,m , δf,m , st,m , sf,m, pc
Generative model
yhidden
Generate bump model (hidden)
• geometric prior for number n of bumps
• bumps are uniformly distributed in rectangle
• amplitude, width (in t and f) all i.i.d.
y1 y2 y3 y4 y5
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)
• amplitude, width (in t and f) all i.i.d.
pc (i) = p(cluster size = i |y)
(i = 1,2,…,M)
Parameters: θ = δt,m , δf,m , st,m , sf,m, pc
• “deletion” with probability pd
Exemplar-based formulation
yhidden
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: IP
Binary Variables
Integer Program: LINEAR objective function/constraints
Equivalent to k-dim matching: for k = 2: in P but for k > 2: NP-hard!
Probabilistic inference
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)
SOLUTION: Coordinate descent
b(i+1) = argmaxc log p(y, y’, b, θ(i) )
θ(i+1) = argmaxx log p(y, y’, b(i+1) ,θ )
CLUSTERING (Integer Program)
ESTIMATION OF PARAMETERS
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??)
Summary
Similarity of multiple multi-dimensional point processes
Step 1: TWO ONE-dimensional point processes
Dynamic programming
Step 2: TWO MULTI-dimensional point processes
Max-product/LP relaxation/Edmund-Karp
Step 3: MULTIPLE MULTI-dimensional point processes
Integer Programming
Overview

Alzheimer’s Disease (AD)
decrease in EEG synchrony

Similarity of Point Processes
 Two
1-D point processes
 Two multi-D point processes
 Multiple multi-D point processes
Numerical Results
 Conclusion

EEG Data
• 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
EEG data provided by Prof. T. Musha
Sensitivity (average synchrony)
Corr/Coh
Granger
Info. Theor.
State Space
Phase
SES
Significant differences for ffDTF and SES (more unmatched bumps, but same amount of jitter)
Mann-Whitney test: small p value suggests large difference in statistics of both groups
Classification (bi-SES)
± 85% correctly classified
ffDTF
•
•
•
Clear separation, but not yet useful as diagnostic tool
Additional indicators needed (fMRI, MEG, DTI, ...)
Can be used for screening population (inexpensive, simple, fast)
Overview

Alzheimer’s Disease (AD)
decrease in EEG synchrony

Similarity of Point Processes
 Two
1-D point processes
 Two multi-D point processes
 Multiple multi-D point processes
Numerical Results
 Conclusion

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 Dauwels
LIDS, MIT
Amari Research Unit, Brain Science Institute, RIKEN
September 25, 2008
References + software
References
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
Simple closed form expressions
Deltas: average offset
...where
Sigmas: var of offset
artificial observations (conjugate prior)
Large-scale synchrony
Apparently, all brain regions affected...
Alzheimer's disease
Outside glimpse: the future (prevalence)
Million of sufferers
USA (Hebert et al. 2003)
14
12
• 2% to 5% of people
over 65 years old
• Up to 20% of people
over 80
10
8
6
4
Jeong 2004 (Nature)
2
0
Million of sufferers
1980
1990
2000
2010
2020
2030
2040
2050
World (Wimo et al. 2003)
120
100
80
60
Developped
countries
Developping countries
40
20
0
Ongoing and future work
Applications

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
alternative inference techniques (e.g., MCMC, linear programming)
 time dependent (Gaussian processes)
 multivariate (T.Weber)

Fitting bump models

Initialisation
Adaptation
After adaptation
Signal
gradient method
Bump
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 model
yhidden
Generate bump model (hidden)
• geometric prior for number n of bumps
p(n) = (1- λ S) (λ S)-n
• bumps are uniformly distributed in rectangle
• amplitude, width (in t and f) all i.i.d.
Generate two “noisy” observations
y y’
( -δt /2, -δf /2)
( δt /2, δf /2)
• offset between hidden and observed bump
= Gaussian random vector with
mean ( ±δt /2, ±δf /2)
covariance diag(st/2, sf /2)
• amplitude, width (in t and f) all i.i.d.
• “deletion” with probability pd
Easily extendable to more than 2 observations…
Probabilistic inference
PROBLEM: Given two bump models, compute (ρspur, δt, st, δf, sf )
θ
APPROACH:
(c*,θ*) = argmaxc,θ log p(y, y’, c, θ)
SOLUTION: Coordinate descent
c(i+1) = argmaxc log p(y, y’, c, θ(i) )
θ(i+1) = argmaxx log p(y, y’, c(i+1) ,θ )
MATCHING
POINT ESTIMATION
Alzheimer's disease
Inside glimpse: abnormal EEG
EEG system: inexpensive, mobile, useful for screening
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).
Decrease of synchrony
•
•
•
focus of this project
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).
Images: www.cerebromente.org.br
Comparing EEG signal rhythms ?
2 signals
PROBLEM I:
Signals of 3 seconds sampled at 100 Hz ( 300 samples)
Time-frequency representation of one signal = about 25 000 coefficients
Comparing EEG signal rhythms ?(2)
One pixel
Numerous
neighboring pixels
PROBLEM II:
Shifts in time-frequency!
Correlations
Strong (anti-) correlations
„families“ of sync measures
Generative model
yhidden
Generate bump model (hidden)
• geometric prior for number n of bumps
p(n) = (1- λ S) (λ S)-n
• bumps are uniformly distributed in rectangle
• amplitude, width (in t and f) all i.i.d.
y1 y2 y3 y4 y5
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)
• amplitude, width (in t and f) all i.i.d.
pc (i) = p(cluster size = i |y)
(i = 1,2,…,M)
Parameters: θ = δt,m , δf,m , st,m , sf,m, pc
• “deletion” with probability pd
Classification (multi-SES)
Average bump freq
± 85% correctly classified
Average cluster size
Average bump width
Average cluster size
± 90% correctly classified
ffDTF
Similarity of bump models...
How “similar” or “synchronous” are two bump models?
Signatures of local synchrony
f (Hz)
Time-frequency patterns
(“bumps”)
EEG stems from thousands of neurons
bump if neurons are phase-locked
= local synchrony
t (sec)
Alzheimer's disease
Inside glimpse: brain atrophy
amyloid plaques and
neurofibrillary tangles
Video source:
Alzheimer society
Images: Jannis Productions.
(R. Fredenburg; S. Jannis)
Video source: P. Thompson, J.Neuroscience, 2003
Probabilistic inference
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 expression
Other kind of prior p(θ): numerical optimization (gradient method)
Probabilistic inference
MATCHING: c(i+1) = argmaxc log p(y, y’, c, θ(i) )
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}
find heaviest set of disjoint edges
not necessarily perfect
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)]
Max-product algorithm
MATCHING: c(i+1) = argmaxc log p(y, y’, c, θ(i) )
Generative model
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 algorithm
MATCHING: c(i+1) = argmaxc log p(y, y’, c, θ(i) )
Conditioning on θ
μ↓ μ↑ μ↓
μ↑
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
PROBLEM: Given two bump models, compute (ρspur, δt, st, δf, sf )
θ
APPROACH:
(c*,θ*) = argmaxc,θ log p(y, y’, c, θ)
SOLUTION: Coordinate descent
c(i+1) = argmaxc log p(y, y’, c, θ(i) )
θ(i+1) = argmaxx log p(y, y’, c(i+1) ,θ )
MATCHING → max-product
ESTIMATION → closed-form
Generative model
yhidden
Generate bump model (hidden)
• geometric prior for number n of bumps
p(n) = (1- λ S) (λ S)-n
• bumps are uniformly distributed in rectangle
• amplitude, width (in t and f) all i.i.d.
Generate two “noisy” observations
y y’
( -δt /2, -δf /2)
( δt /2, δf /2)
• offset between hidden and observed bump
= Gaussian random vector with
mean ( ±δt /2, ±δf /2)
covariance diag(st/2, sf /2)
• amplitude, width (in t and f) all i.i.d.
• “deletion” with probability pd
Easily extendable to more than 2 observations…
Generative model (2)
y y’
i
i’
( -δt /2, -δf /2)
j’
( δt /2, δf /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 )
θ
(symmetric in y and y’)
• Eliminate yhidden → offset is Gaussian RV with mean = ( δt , δf ) and covariance diag (st , sf)
p(y, y’, c, θ) = ∫ p(y, y’, yhidden , c, θ) dyhidden
• Probabilistic Inference: (c*,θ*) = argmaxc,θ log p(y, y’, c, θ)
Summary
• 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
PROBLEM: Given two bump models, compute (ρspur, δt, st, δf, sf )
θ
APPROACH:
(c*,θ*) = argmaxc,θ log p(y, y’, c, θ)
Objective function
y y’
i
( -δt /2, -δf /2)
i’
j’
( δt /2, δf /2)
• 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
Euclidean distance between bump centers
β = pd (λ/V)1/2
• 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
Distance measures
Scaling
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
Non-Euclidean
sf,kk’ = (Δfk + Δf’k) sf
Generative model
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’
Prior for parameters

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)
Preliminary results for multi-variate model
linear comb of pc
CTR
MCI
Probabilistic inference
PROBLEM: Given two bump models, compute (ρspur, δt, st, δf, sf )
θ
(c*,θ*) = argmaxc,θ log p(y, y’, c, θ)
APPROACH:
SOLUTION: Coordinate descent
c(i+1) = argmaxc log p(y, y’, c, θ(i) )
θ(i+1) = argmaxx log p(y, y’, c(i+1) ,θ )
MATCHING
POINT ESTIMATION
Minx2 X, y2Y d(x,y)
X
Y
Generative model
yhidden
Generate bump model (hidden)
• geometric prior for number n of bumps
p(n) = (1- λ S) (λ S)-n
• bumps are uniformly distributed in rectangle
• amplitude, width (in t and f) all i.i.d.
y1 y2 y3 y4 y5
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)
• amplitude, width (in t and f) all i.i.d.
pc (i) = p(cluster size = i |y)
(i = 1,2,…,M)
Parameters: θ = δt,m , δf,m , st,m , sf,m, pc
• “deletion” with probability pd
(other prior pc0 for cluster size)
Role of local synchrony
Stimuli
Consolidation
Assembly
activation
Assembly
recall
Voice Face
Stimulus
Hebbian consolidation
Voice
(Hebb 1949, Fuster 1997)
Probabilistic inference
PROBLEM: Given M bump models, compute θ = δt,m , δf,m , st,m , sf,m, pc
APPROACH:
(c*,θ*) = argmaxc,θ log p(y, y’, c, θ)
SOLUTION: Coordinate descent
c(i+1) = argmaxc log p(y, y’, c, θ(i) )
θ(i+1) = argmaxx log p(y, y’, c(i+1) ,θ )
CLUSTERING (IP or MP)
POINT ESTIMATION
Integer program
• Max-product algorithm (MP) on sparse graph
• Integer programming methods (e.g., LP relaxation)
Fourier transform
2
1
2
3
3
1
Frequency
High frequency
Low frequency
Windowed Fourier transform
Fourier basis functions
*
=
Window
function
windowed basis
functions
f
Windowed
Fourier
Transform
t
Overview
Alzheimer’s Disease (AD):
decrease in EEG synchrony
 Synchrony measure in time-frequency domain

 Pairs
of EEG signals
 Collections of EEG signals
Numerical Results
 Conclusion

Average synchrony
1. Group electrodes in regions
2. Bump model for each region
3. SES for each pair of models
4. Average the SES parameters
Beyond pairwise interactions...
Pairwise similarity
Multi-variate similarity
Similarity measures
•
•
Correlation and coherence
Granger causality (linear system): DTF, ffDTF, dDTF, PDC, PC, ...
TIME
•
Phase Synchrony: compare instantaneous phases (wavelet/Hilbert transform)
No Phase Locking
•
State space based measures
sync likelihood, S-estimator, S-H-N-indices, ...
•
FREQUENCY
Information-theoretic measures
KL divergence, Jensen-Shannon divergence, ...
Phase Locking