some mathematical ideas for attacking the brain computer interface problem l.
Skip this Video
Loading SlideShow in 5 Seconds..
Some Mathematical Ideas for Attacking the Brain Computer Interface Problem PowerPoint Presentation
Download Presentation
Some Mathematical Ideas for Attacking the Brain Computer Interface Problem

Loading in 2 Seconds...

play fullscreen
1 / 52

Some Mathematical Ideas for Attacking the Brain Computer Interface Problem - PowerPoint PPT Presentation

  • Uploaded on

Some Mathematical Ideas for Attacking the Brain Computer Interface Problem Michael Kirby Department of Mathematics Department of Computer Science Colorado State University Overview The Brain Computer Interface (BCI) Challenge Signal fraction analysis

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
Download Presentation

Some Mathematical Ideas for Attacking the Brain Computer Interface Problem

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
some mathematical ideas for attacking the brain computer interface problem

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

Department of Mathematics

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

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

chuck anderson
Chuck Anderson

Department of Mathematics

pattie davies
Pattie Davies

Department of Mathematics

bci headlines in the news
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

lou gehrig s disease als
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

gulf war veterans and als
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

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.

The Brain Computer Interface (BCI)

Department of Mathematics

low cost eeg
Low-Cost EEG

Department of Mathematics

history of eeg
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

characteristics of brainwaves
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

reasons why eeg should not work for bci
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

why eeg can work for bci
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


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

motivation for our work
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

eeg data set mental tasks
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

lobes of the brain
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

electrode placement and sample data
Electrode Placementand Sample Data

Department of Mathematics

geometric filtering of noisy time series
Geometric Filtering of Noisy Time-Series

Given a data set

The Q fraction of a basis vector is defined as where

Department of Mathematics

signal fraction optimization
Signal Fraction Optimization

Determine  such that D() is a maximum.

Solution via the GSVD equation

Department of Mathematics


SVD filter

Original Signal

Signal fraction filter

Department of Mathematics


SVD basis

GSVD basis

Department of Mathematics


SVD reconstruction

GSVD reconstruction

Department of Mathematics

blind signal separation
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

signal fraction analysis separation
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



In particular,

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

Department of Mathematics


Original signals (unknown)

Mixed signals (observed)

Department of Mathematics


FastICA separation

Signal fraction separation

Department of Mathematics

artifact removal
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

signal fraction filters
Signal Fraction Filters

Department of Mathematics

constructing signal fraction filter
Constructing Signal Fraction Filter

Department of Mathematics

benefits of signal fraction analysis
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

classification on manifolds
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

dynamical systems perspective
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

time delay embedding
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

taken s theorem simplified
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

the lorenz attractor
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

do eeg data lie on an attractor
Do EEG data lie on an attractor?

Department of Mathematics

elephants in the clouds
Elephants in the Clouds?

Random data

Classification rate

Department of Mathematics

super resolution skull caps
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

manifolds and nonlinear methods work with fatemeh emdad
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

kernel svd versus kernel sfa
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

ksfa kpca degree 1
KSFA, KPCA degree 1

Department of Mathematics

ksfa kpca degree 2
KSFA, KPCA degree 2

Department of Mathematics

ksfa kpca degree 3
KSFA, KPCA degree 3

Department of Mathematics

ksfa kpca degree 4
KSFA, KPCA degree 4

Department of Mathematics

relative performance
Relative Performance

Department of Mathematics

conclusions and future work
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.

Department of Mathematics