Tools to monitor brain state

Alain de Cheveigné, CNRS / ENS / UCL

overview

• Two motivations

- importance of brain state

- data mining

• Algorithms

- segmentation

- clustering

a definition of state

"something that is true at some time and not at another"

- statistical distribution of values

- validity of a predictive model

- parameters of a predictive model

importance of brain state

importance of brain state

essential to have tools to monitor/characterize brain state

brain data miningbrain data mining

brain data miningbrain data mining

€

yk = w jkj∑ x j

lots of methods: PCA, ICA, beamforming, CSD, DSS, CSP, etc.

component analysis exploits correlation structure to improve SNR

brain data miningbrain data miningcomponent analysis can be extremely powerful:

simulated data: 10 channels, 1 target, 9 noise sources, random mix matrix, SNR=10-8

sources

sensors

noise

target

works if 9 noise sources, fails miserably if 10: dimensionality of noise subspace is critical

result of component analysis(DSS algorithm)

brain data miningbrain data mining

Dimensionality = (roughly) number of independent noise sources within data

If dim(noise) < n(channels) then there exists a projection of the data (= weighted sum of the channels) such that: (a) all noise sources are canceled,

(b) target activity is not (unless we're unlucky)

The aim of component analysis (ICA, beamforming, DSS, etc.) is to find such useful projections.

If dim(noise)=n(channels) they cannot succeed. We need: dim(noise) < n(channels)

brain data miningbrain data mining

Hypothesis:

There exists a partition of the time axis into subsets An such that the data are of rank < n(channels) over each subset.

€

A = Un Andim(An ) < J (number of channels)

Our task:

Find this partition:

--> related to manifold learning

brain data miningsignal state descriptors

Standard statistics:- mean- variance- covariance- autocorrelation (including multichannel)

€

mA (x) = (1/nA ) x(t)t∈A

∑

vA (x) = mA (x 2) −mA (x)2

CA (X) = (1/nA ) x j (t)t∈A∑ xk (t)[ ]

rA ,τ (x) = (1/n) x(t)t∈A

∑ x(t −τ )

brain data miningalgorithms

Two approaches: - segmentation - clustering

brain data miningsegmentation

find step in mean

find step in mean

algorithm 1

segmentation

find step in variance

algorithm 1 applied to xt2

segmentation

multichannel case: step in variance

data: 10 channels, 2-fold amplitude increase

sum of V statistics over channels: algorithm 2

segmentation

multichannel case: step in variance

data: 10 channels, 2-fold amplitude increase/decrease

sum of V statistics over channels: algorithm 2

segmentation

brain data miningalgorithms

multichannel case: step in covariance

data: 10 channels, 5 sources active in first half (rank=5), 5 sources active in second half (rank=5),rank of full data=10

algorithm 2 applied to xj(t) xj'(t)

None of these algorithms addresses our initial task:

Find:

€

A = Un Andim(An ) < J (number of channels)

segmentation

Segmentation by joint diagonalization (algorithm 3):

Rationale: - assume data X of rank J=n(channels) over entire segment A = A1 U A2, and of rank < J over both A1 and A2

- there exists a projection of data that is zero over A1 and non-zero over A2

- there exists a projection of data that is zero over A2 and non-zero over A1

- both can be found by joint diagonalization of covariance matrices of X over A1 and A:

- the first channel of Y=XP is zero over A1 and last channel zero over A2

segmentation

Segmentation by joint diagonalization (algorithm 3):

Algorithm: (a) choose initial arbitrary segmentation A = A1 U A2

(b) diagonalize covariance matrices of A and A1 (c) apply transform Y=XP (d) apply algorithm 2 to first and last columns of X new partition (e) go to (b) until no change in partition (or max iterations)

segmentation

multichannel case: step in covariance

data: 10 channels, 5 sources active in first half (rank=5), 5 sources active in second half (rank=5),rank of full data=10

algorithm 3

segmentation

clustering

- similar algorithms, similar results (on these example data)- segmentation or clustering? depends on data, depends on question

examples

monkey ECoG (NeuroTycho data)

injection of anaesthetic

examples

examples