Loading...

PPT – Some Mathematical Ideas for Attacking the Brain Computer Interface Problem PowerPoint presentation | free to view - id: 798f6-ZDc1Z

The Adobe Flash plugin is needed to view this content

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
- Takens theorem and classification on manifolds
- Nonlinear signal fraction analysis
- Conclusions and future work

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

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.

Chuck Anderson

Pattie Davies

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

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.

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.

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.

Low-Cost 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

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.

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

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!

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.

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.

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

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.

Electrode Placement and Sample Data

Geometric Filtering of Noisy Time-Series

- Given a data set
- The Q fraction of a basis vector is defined

as where

Signal Fraction Optimization

- Determine ? such that D(?) is a maximum.
- Solution via the GSVD equation

(No Transcript)

SVD filter

Original Signal

Signal fraction filter

SVD basis

GSVD basis

SVD reconstruction

GSVD reconstruction

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.

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.

Original signals (unknown)

Mixed signals (observed)

FastICA separation

Signal fraction separation

(No Transcript)

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

Signal Fraction Filters

Constructing Signal Fraction Filter

(No Transcript)

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.

Classification on Manifolds

- Insert slide from Istec meeting

manifold H(x) 0

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

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

Time Delay Embedding

- We may embed the scalar observable into a higher

dimensional state space via the construction - So now it is clear that

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.

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.

Do EEG data lie on an attractor?

Elephants in the Clouds?

Random data

Classification rate

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!

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.

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.

KSFA, KPCA degree 1

KSFA, KPCA degree 2

KSFA, KPCA degree 3

KSFA, KPCA degree 4

Relative Performance

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.