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Some Mathematical Ideas for Attacking the Brain Computer Interface Problem

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Some Mathematical Ideas for Attacking the Brain Computer Interface Problem

Michael Kirby

Department of Mathematics

Department of Computer Science

Colorado State University

Department of Mathematics

- The Brain Computer Interface (BCI) Challenge
- Signal fraction analysis
- Takens’ theorem and classification on manifolds
- Nonlinear signal fraction analysis
- Conclusions and future work

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- Chuck Anderson (PI), Computer Science, Colorado State
- Michael Kirby (Co-PI), Mathematics, Colorado State
- James Knight, Ph.D. Student, Colorado State
- Tim O’Connor, Ph.D. Student, Colorado State
- Ellen Curran, Medical Ethics and Jurisprudence, Dept. of Law, Keele University, Staffordshire, UK
- Doug Hundley, Consultant, Department of Mathematics, Whitman
- Pattie Davies, Occupational Therapy Department, Colorado State
- Bill Gavin, Dept. of Speech, Language and Hearing Sciences, University of Colorado

“Geometric Pattern Analysis and Mental Task Design

for a Brain-Computer Interface”

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- Development Status: 1 - Planning
- Environment: Other Environment
- Intended Audience: Science/Research
- License: GNU General Public License (GPL)
- Natural Language: English
- Operating System: Linux, SunOS/Solaris
- Topic: Artificial Intelligence, Human Machine Interfaces, Information Analysis, Mathematics, Medical Science Apps.

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Department of Mathematics

Department of Mathematics

- Computers obey brain waves of paralyzed, Associated Press, appearing in MSNBC News, April 6, 2005
- Brainwaves Control Video Games, BBC March 2004
- Brainwave cap controls computer, BBC December 2004
- Brain Could Guide Artificial Limbs
- Patients put on thinking caps, Wired News, January 2005
- Monkey thoughts control computer, March 2002

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- Amyotrophic Lateral Sclerosis (ALS) , or “Locked-In Syndrome”, is an extreme neurological disorder and many patients opt against life support.
- Most commonly, the disease strikes people between the ages of 40 and 70, and as many as 30,000 Americans have the disease at any given time. (ALS Association website).
- Genetic factors appear to only account for 10 percent of all ALS cases. ALS can strike anyone, anytime.
- There are no effective treatments and no cure.
- Brain activity appears to remain vigorous while muscle control atrophies degeneritively and completely.

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The following information is from a news release sent out by the Department of Veteran Affairs on December 10, 2001. (ALS Association Web posting.)

“According to a news release on December 10, 2001 from the Department of Veteran Affairs, researchers conducting a large epidemiological study supported by both the Department of Veterans Affairs and the Department of Defense have found preliminary evidence that veterans who served in Desert Shield-Desert Storm are nearly twice as likely as their non-deploying counterparts to develop amyotrophic lateral sclerosis.”

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A means for communication between person and machine via measurements associated with cerebral activity, e.g., EEG, fMRI, MEG.

We assume that no muscle motion is employed such as eye twitching or finger movement.

Department of Mathematics

Department of Mathematics

- Duboi-Reymond (1848) reported the presence of electrical signals
- Caton (1875) measured “feeble” currents on the scalp
- Berger (1929) measured electrical signals with EEG
- 1930-50s EEG used in psychiatric and neurological sciences relying on visual inspection of EEG patterns
- 1960s-70s witness emergence of Quantitative EEG and confirmation of hemispheric specialization, e.g., left brain verbal and right brain spatial.
- 1980s+ observation of biofeedback

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- Delta waves [0,4] Hz associated with sleep. Also empathy.
- Theta waves [4, 7.5] associated with reverie, daydreaming, meditation, creative ideas
- Alpha waves [7.5,12] prevalent when alert and eyes closed. Associated with relaxed positive feelings.
- Beta waves 12Hz+ associated with active state, eyes open.

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- Electrical activity generated by complex system of billions of neurons
- Brain is a “gelatinous mass” suspended in a conducting fluid
- Difficult to “register” electrode location
- Artifacts from motion, eyeblinks, swallows, heartbeat, sweating…
- Food, age, time of day, fatigue, motivation of subject

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- Many EEG studies have reported reproducible changes in brain dynamics that are task dependent!
- People are able to control their brainwaves via biofeedback!

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Patients may “correct” their waveforms to achieve a normal state.

- Kamiya demonstrated the controllability of alpha waves in 1962.
- Communication in morse code by turning alpha waves on and off.
- Stress management and sleep therapy.
- Move a pac-man by stimulating alpha and beta waves.
Note that artifacts are a serious problem for real-time biofeedback applications.

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- Current biofeedback training requires 10 weeks to move a cursor.
- Typing requires 5 minutes/letter with 90% accuracy.
- Although there has been some mathematical work the field has been dominated by experiment and heuristics.
- Suggestions by clinical EEG experts that understanding EEG problem will have a strong mathematical component.
- Tremendous potential for results.

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- Resting task
- Imagined letter writing
- Mental multiplication
- Visualized counting
- Geometric object rotation
Keirn and Aunon, “A new mode of communication between man and his surroundings”, IEEE Transactions on Biomedical Engineering, 37(12):1209-1214, December 1990

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Frontal Lobes

Personality, emotions, problem solving.

Parietal lobes

Cognition, spatial relationships and mathematical abilities, nonverbal memory.

Occipital lobes

Vision, color, shape and movement.

Temporal lobes

Speech and auditory processing, language comprehension, long-term memory.

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Department of Mathematics

Given a data set

The Q fraction of a basis vector is defined as where

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Determine such that D() is a maximum.

Solution via the GSVD equation

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Department of Mathematics

SVD filter

Original Signal

Signal fraction filter

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SVD basis

GSVD basis

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SVD reconstruction

GSVD reconstruction

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Unknown (tall) m £ n signal matrix S

Unknown mixing n £ n matrix A

Observed m £ n data matrix X

Task: recover A and S from X alone.

In general it is not possible to solve this problem.

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Theorem: The solution to the signal fraction analysis optimization problem solves the signal separation problem X = SA given

1) is observed

2)

3)

In particular,

Where is the solution to the GSVD problem for signal fraction analysis.

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Original signals (unknown)

Mixed signals (observed)

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FastICA separation

Signal fraction separation

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Department of Mathematics

Given the separated signals = X we may filter the ith column of by setting

Where Id’ is the identity matrix with the ith row set to zero. The filtered version of the data is now

Where recall the original data is

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Department of Mathematics

Department of Mathematics

Department of Mathematics

- Can identity sources of noise such as respirators, eyeblinks, cranial heartbeat, line noise etc…
- Filtering works over short periods of the signal, i.e., can remove artifacts from a time series of length 500.
- Can use generalizations of the signal to noise ratio to separate quantities of interest.
- Simple and fast to compute.

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- Insert slide from Istec meeting

manifold: H(x) = 0

dist(A,B) large but H(A)=H(B)=0

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Assume a system is described by the dynamical equations

and that the solutions reside on an attracting set, e.g., a manifold. What can be said about the full system if it is only possible to observe part of the system? In the extreme, imagine we can only observe a scalar value

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We may embed the scalar observable into a higher dimensional state space via the construction

So now it is clear that

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Given a continuous time dynamical system with solution on a compact invariant smooth manifold M of dimension d, a continuous measurement function h(x(t)) can be time-delay embedded in to dimension 2d+1 such that there is a diffeomorphism between the embedded attractor and the actual (unobserved) solution set.

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Given a data point (x,y,z) we know which lobe by the sgn of x. But what if we only observe the z value? The lobe can be classified using Taken’s theorem and Time delay embedding.

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Department of Mathematics

Random data

Classification rate

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How many electrodes are needed? 6, 16, 32, 128, 256, 512? We should be able to answer this question by means of evaluating an objective function.

Through attractor reconstruction, time delay embedding techniques may practically enhance the resolution of skull caps leading to significant savings in time and equipment.

Colleagues working on EEG studies in children are very enthusiastic about this!

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Veronese embeddings of the data:

Degree 1: (x,y)

Degree 2: (x2, xy, y2)

Degree 3: (x3, x2y, xy2, y3)

Degree 1: (x,y,z)

Degree 2: (x^2, xy, xz, y^2, yz, z^2)

Degree 3: (x^3, x^2y, x^2z, xy^2, xz^2, xyz, y^3, y^2z, yz^2, z^3)

Such embeddings are behind one variant of kernel SVD.

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Numerical Experiments:

KSVD (KPCA) degree = 1, 2, 3, 4

KSFA degree = 1, 2, 3, 4

Objective: compare mode classification rates using knn for k = 1,…, 10.

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Department of Mathematics

Department of Mathematics

Department of Mathematics

Department of Mathematics

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- Present a geometric subspace approach for signal separation, artifact removal and classification.
- Provided evidence that brain dynamics might reside on an attractor and that time-delay embedding enhances classification rates.
- Illustrated a nonlinear extension to signal fraction analysis and compared with similar extension to svd.
- These ideas are presented in the context of EEG signals but are quite general and can be applied to images.

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