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Some Mathematical Ideas for Attacking the Brain Computer Interface Problem

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Title: Some Mathematical Ideas for Attacking the Brain Computer Interface Problem


1
Some Mathematical Ideas for Attacking the Brain
Computer Interface Problem
  • Michael Kirby
  • Department of Mathematics
  • Department of Computer Science
  • Colorado State University

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

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 OConnor, 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
4
SourceForge https//sourceforge.net/projects/csue
eg/
  • 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.

5
Chuck Anderson
6
Pattie Davies
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

8
Lou Gehrigs 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.

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. 

10
The Brain Computer Interface (BCI)
  • 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.

11
Low-Cost EEG
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

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.

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

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!

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.

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.

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

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.

20
Electrode Placement and Sample Data
21
Geometric Filtering of Noisy Time-Series
  • Given a data set
  • The Q fraction of a basis vector is defined
    as where

22
Signal Fraction Optimization
  • Determine ? such that D(?) is a maximum.
  • Solution via the GSVD equation

23
(No Transcript)
24
SVD filter
Original Signal
Signal fraction filter
25
SVD basis
GSVD basis
26
SVD reconstruction
GSVD reconstruction
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.

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.

29
Original signals (unknown)
Mixed signals (observed)
30
FastICA separation
Signal fraction separation
31
(No Transcript)
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

33
Signal Fraction Filters
34
Constructing Signal Fraction Filter
35
(No Transcript)
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.

37
Classification on Manifolds
  • Insert slide from Istec meeting

manifold H(x) 0
dist(A,B) large but H(A)H(B)0
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

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

40
Takens 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 2d1 such that there is a
    diffeomorphism between the embedded attractor and
    the actual (unobserved) solution set.

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 Takens
theorem and Time delay embedding.
42
Do EEG data lie on an attractor?
43
Elephants in the Clouds?
Random data
Classification rate
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!

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 (x2, xy, xz, y2, yz, z2)
  • Degree 3 (x3, x2y, x2z, xy2, xz2, xyz, y3,
    y2z, yz2, z3)
  • Such embeddings are behind one variant of kernel
    SVD.

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.

47
KSFA, KPCA degree 1
48
KSFA, KPCA degree 2
49
KSFA, KPCA degree 3
50
KSFA, KPCA degree 4
51
Relative Performance
52
Conclusions and Future Work
  • 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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