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## Analyzing Brain Signals by Combinatorial Optimization

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**Quantifying statistical interdependence of point processes**Application to spike data and EEG Analyzing Brain Signalsby Combinatorial Optimization Justin Dauwels LIDS, MIT Amari Research Unit, Brain Science Institute, RIKEN December 1, 2008**Topics**• Mathematicalproblem • Similarity of Multiple Point Processes • Motivation/Application • Earlydiagnosisof Alzheimer’sdiseasefrom EEG signals • Along the way… • Spike synchrony Collaborators François Vialatte*, Theophane Weber+, and AndrzejCichocki* (*RIKEN, +MIT) Financial Support**Alzheimer's disease**One disease, many symptoms Evolution of the disease (stages) EEG data • 2 to 5 years before • mild cognitive impairment (MCI) • 6 to 25 % progress to Alzheimer‘s memory, language, executive functions, apraxia, apathy, agnosia, etc… • Mild (early stage) • becomes less energetic or spontaneous • noticeable cognitive deficits • still independent (able to compensate) Memory (forgetting relatives) • Moderate (middle stage) • Mental abilities decline • personality changes • become dependent on caregivers Apathy • Severe (late stage) • complete deterioration of the personality • loss of control over bodily functions • total dependence on caregivers Video sources: Alzheimer society • 2% to 5% of people over 65 yearsold • up to 20% of people over 80 • Jeong 2004 (Nature) GOAL: Diagnosis of MCI based on EEG • EEG isrelativelysimple and inexpensivetechnology • Earlydiagnosis: 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 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). focus of this project 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 t (sec) amplitude EEG x(t)**Sparse representation: bump model**f(Hz) f(Hz) Bumps Sparse representation t (sec) f(Hz) t (sec) 104- 105 coefficients t (sec) • Assumptions: • time-frequency map is suitable representation • oscillatory bursts (“bumps”) convey key information about 102 parameters 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 multiplemulti-dimensional point processes with and “point” / ”event”**Overview**• Alzheimer’s Disease (AD) decrease in EEG synchrony • Similarity of Point Processes • Two 1-dim point processes • Two multi-dim point processes • Multiple multi-dim point processes • Numerical Results • Conclusion**Two one-dimensional point processes**x t 0 x’ 0 t How synchronous/similar? Classical methods for continuous time series fail e.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 0 τR x’ 0 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.**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 x DELETION 0 x’ 0 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 generativestatistical model x 0 v 0 x’ 0 Dauwels J., Vialatte F., Rutkowski T., and Cichocki A., 2007. Measuring neural synchrony by message passing, NIPS 20, in press.**Stochastic Event Synchrony**non-coincident x x T0 0 0 T0 v 0 T0 -δt /2 T0 x δt /2 0 x’ non-coincident 0 T0 Stochastic event synchrony (SES): delayδt, jitterst, non-coincidenceρ Dauwels J., Vialatte F., Rutkowski T., and Cichocki A., 2007. Measuring neural synchrony by message passing, NIPS 20, in press.**Stochastic Event Synchrony**non-coincident x x T0 i.i.d. deletions with probpd 0 Gaussian offsets with mean -δt /2 and variance st/2 0 T0 v geometric prior for lenght 0 T0 -δt /2 events i.u.d. in [0,T0] Gaussian offsets with mean δt /2 and variance st/2 T0 x δt /2 0 x’ i.i.d. deletions with probpd non-coincident 0 T0 Marginalizing over v: Dauwels J., Vialatte F., Rutkowski T., and Cichocki A., 2007. Measuring neural synchrony by message passing, NIPS 20, in press.**Probabilistic inference**PROBLEM: Given 2point processes x and x’, compute ρandθ = 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 x’k’non-coincident xknon-coincident (xkx’k’ ) coincident pair 0 0 x1 x2 x3 x4 x5 x6**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)**Two 1-dim point processes**• Review of Spike Synchrony Measures • Surrogate Spike Data • Spike Trains from Morris-Lecar Neuron • Conclusion**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**δt =0 Van Rossummeasure DR similar for SS ,SH ,SQ Victor Purpura measure DV • E{DR}increases with pd and σt • → DRcannot distinguish timing dispersion from event reliability • (likewise all measures except SES and DV) • E{DV}increases with pd,practically independent of σt • → DVmeasure for event reliability • ONLY curves for δt = 0ms, measures strongly depend on lag**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**Two 1-dim point processes**• Review of Spike Synchrony Measures • Surrogate Spike Data • Spike Trains from Morris-Lecar Neuron • Conclusion**Morris-Lecar Neurons**• Simple neuron model • Exhibits behavior of Type I & II neurons (saddle-node/Hopf bifurc.) • Input current: baseline + sinusoid + Gaussian noise • Membrane potential Spiking threshold Type II Type I 5 trials**Morris-Lecar Neurons (2)**Type II Type I 50 trials Low reliability Small timing dispersion High reliability Large timing dispersion jitterst= (3ms)2, non-coincidenceρ = 27% jitterst = (15ms)2, non-coincidenceρ = 3%**Morris-Lecar Neurons: Results**• Smallτ: Type II has larger similarity than type I (dispersion in Type I) • Largeτ: 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 AUTOMATICALLYselects st**Two 1-dim point processes**• Review of Spike Synchrony Measures • Surrogate Spike Data • Spike Trains from Morris-Lecar Neuron • Conclusion**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**• Alzheimer’s Disease (AD) decrease in EEG synchrony • Similarity of Point Processes • Two 1-dim point processes • Two multi-dim point processes • Multiple multi-dim point processes • Numerical Results • Conclusion**... 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 • → Frequencyjitter 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 • 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 y y’ ( -δt /2, -δf/2) ( δt /2, δf/2) Dauwels J., Vialatte F., Rutkowski T., and Cichocki A., 2007. Measuring neural synchrony by message passing, NIPS 20, in press.**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 Dauwels J., Vialatte F., Rutkowski T., and Cichocki A., 2007. Measuring neural synchrony by message passing, NIPS 20, in press.**Average synchrony**3. SES for each pair of models 4. Average the SES parameters • Group electrodes in regions • Bump model for each region**Overview**• Alzheimer’s Disease (AD) decrease in EEG synchrony • Similarity of Point Processes • Two 1-dim point processes • Two multi-dim point processes • Multiple multi-dim point processes • Numerical Results • Conclusion**Beyond pairwise interactions**Multi-variate similarity Pairwise similarity**Similarity of multiple bump models**y2 y1 y3 y4 y5 • Models similar if • few deletions/large clusters • little jitter y2 y1 y3 y4 y5 Constraint: in each cluster at most one bump from each signal Dauwels J., Vialatte F., Weber T. and Cichocki. Analyzing Brain Signals by Combinatorial Optimization, Allerton 2008.**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. y2 y1 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. • “deletion” with probability pd pc(i) = p(cluster size = i |y) (i = 1,2,…,M) Parameters: θ = δt,m, δf,m , st,m, sf,m,pc Dauwels J., Vialatte F., Weber T. and Cichocki. Analyzing Brain Signals by Combinatorial Optimization, Allerton 2008.**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) Dauwels J., Vialatte F., Weber T. and Cichocki. Analyzing Brain Signals by Combinatorial Optimization, Allerton 2008.**Overview**• Alzheimer’s Disease (AD) decrease in EEG synchrony • Similarity of Point Processes • Two 1-dim point processes • Two multi-dim point processes • Multiple multi-dim 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**Similarity measures**• Correlation and coherence • Granger causality (linear system): DTF, ffDTF, dDTF, PDC, PC, ... • Phase Synchrony: compareinstantaneous phases (wavelet/Hilbert transform) • State space based measures • sync likelihood, S-estimator, S-H-N-indices, ... • Information-theoretic measures • KL divergence, Jensen-Shannon divergence, ... FREQUENCY TIME No Phase Locking Phase Locking**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 • Clearseparation, but not yet useful as diagnostic tool • Additionalindicators needed (fMRI, MEG, DTI, ...) • Can be used for screening population (inexpensive, simple, fast)**Correlations**Strong (anti-) correlations „families“ of sync measures**Overview**• Alzheimer’s Disease (AD) decrease in EEG synchrony • Similarity of Point Processes • Two 1-dim point processes • Two multi-dim point processes • Multiple multi-dim point processes • Numerical Results • Conclusion**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)