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Some Mathematical Ideas for Attacking the Brain Computer Interface Problem

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  1. 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

  2. Overview • The Brain Computer Interface (BCI) Challenge • Signal fraction analysis • Takens’ theorem and classification on manifolds • Nonlinear signal fraction analysis • Conclusions and future work Department of Mathematics

  3. NSF BCI Group • 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” Department of Mathematics

  4. SourceForge https://sourceforge.net/projects/csueeg/ • 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. Department of Mathematics

  5. Chuck Anderson Department of Mathematics

  6. Pattie Davies Department of Mathematics

  7. BCI Headlines in the News • 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 Department of Mathematics

  8. Lou Gehrig’s Disease (ALS) • 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. Department of Mathematics

  9. Gulf War Veterans and ALS 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.”  Department of Mathematics

  10. 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. The Brain Computer Interface (BCI) Department of Mathematics

  11. Low-Cost EEG Department of Mathematics

  12. History of EEG • 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 Department of Mathematics

  13. Characteristics of Brainwaves • 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. Department of Mathematics

  14. Reasons Why EEG Should Not Work for BCI • 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 Department of Mathematics

  15. Why EEG Can Work for BCI • Many EEG studies have reported reproducible changes in brain dynamics that are task dependent! • People are able to control their brainwaves via biofeedback! Department of Mathematics

  16. Biofeedback 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. Department of Mathematics

  17. Motivation for Our Work • 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. Department of Mathematics

  18. EEG Data Set: Mental Tasks • 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 Department of Mathematics

  19. Lobes of the Brain 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. Department of Mathematics

  20. Electrode Placementand Sample Data Department of Mathematics

  21. Geometric Filtering of Noisy Time-Series Given a data set The Q fraction of a basis vector is defined as where Department of Mathematics

  22. Signal Fraction Optimization Determine  such that D() is a maximum. Solution via the GSVD equation Department of Mathematics

  23. Department of Mathematics

  24. SVD filter Original Signal Signal fraction filter Department of Mathematics

  25. SVD basis GSVD basis Department of Mathematics

  26. SVD reconstruction GSVD reconstruction Department of Mathematics

  27. Blind Signal Separation 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. Department of Mathematics

  28. Signal Fraction Analysis Separation 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. Department of Mathematics

  29. Original signals (unknown) Mixed signals (observed) Department of Mathematics

  30. FastICA separation Signal fraction separation Department of Mathematics

  31. Department of Mathematics

  32. Artifact Removal 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 Department of Mathematics

  33. Signal Fraction Filters Department of Mathematics

  34. Constructing Signal Fraction Filter Department of Mathematics

  35. Department of Mathematics

  36. Benefits of Signal Fraction Analysis • 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. Department of Mathematics

  37. Classification on Manifolds • Insert slide from Istec meeting manifold: H(x) = 0 dist(A,B) large but H(A)=H(B)=0 Department of Mathematics

  38. Dynamical Systems Perspective 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 Department of Mathematics

  39. Time Delay Embedding We may embed the scalar observable into a higher dimensional state space via the construction So now it is clear that Department of Mathematics

  40. Taken’s Theorem (simplified) 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. Department of Mathematics

  41. The Lorenz Attractor 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. Department of Mathematics

  42. Do EEG data lie on an attractor? Department of Mathematics

  43. Elephants in the Clouds? Random data Classification rate Department of Mathematics

  44. Super Resolution Skull Caps 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! Department of Mathematics

  45. Manifolds and Nonlinear Methods(work with Fatemeh Emdad) 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. Department of Mathematics

  46. Kernel SVD versus Kernel SFA 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. Department of Mathematics

  47. KSFA, KPCA degree 1 Department of Mathematics

  48. KSFA, KPCA degree 2 Department of Mathematics

  49. KSFA, KPCA degree 3 Department of Mathematics

  50. KSFA, KPCA degree 4 Department of Mathematics