Inferring Hand Motion Inferring Hand Motion from Multi-Cell from Multi-Cell

Recordings in Motor Recordings in Motor Cortex using a Kalman Cortex using a Kalman

FilterFilterWei Wu*, Michael Black†, Yun Gao*, Elie Bienenstock*§,

Mijail Serruya§, Ammar Shaikhouni§, Carlos Vargas§†,John Donoghue§

*Applied Mathematics †Computer Science §Neuroscience

Brown University

OutlineOutline

• Introduction

• Kalman Filter Model and its Algorithm

• Experimental Result

• Analysis• Optimal Time Lag

• Comparison with Linear Filter

• Conclusion

off-line data processing

GoalsGoals

neural signals

neural reconstruction

mathematicalalgorithm

hand position

KalmanFilter

Firingrates

observations

inference/decoding

on-line direct neural control

GoalsGoals

neural reconstruction

visual feedback

KalmanFilter

Firingrates

observations

inference/decoding

Related WorkRelated Work

• Georgopoulos et al. (1986)

• Taylor et al. (2002)

• Warland et al. (1997) Linear filter, ANN

• Wessberg et al.(2000) Linear filter, ANN

• Brown et al. (1998) Kalman filter

• Serruya et al.(2002) Linear filter

• Gao et al. (2002) Particle filter

Population Vector

spike wave form

Multi-electrode Array ImplantMulti-electrode Array Implantfor Spike Timing Recordingsfor Spike Timing Recordings

1 ms

80µV

Utah Array (4x4 mm)100 electrodes, 400m separation

Target Tracking TaskTarget Tracking Task

Motions: fast, unconstrained

Data (training 3.5 min, testing 1 min):• Position (Velocity, Acceleration)• Firing rate (42 cells, non- overlapping 70ms bins)

• has a sound probabilistic framework

• makes explicit assumptions about the data and noise

• indicates the uncertainty of the estimate

• requires a small amount of “training” data

• provides on-line estimation of hand position with short delay(within 200ms)

• has more accurate estimation than the standard linear filter does

Mathematical ModelMathematical Model

42 X 42 matrix

,2,1,0

) ,0(

k

k QNq

42 X 6 matrix

k

k

k

k

k

k

y

x

y

x

a

a

v

v

y

x

systemstatevector

42

2

1

k

k

k

z

z

z

firingratevector

6 X 6 matrix

Kalman Filter ModelKalman Filter Model

Measurement Equation:

6 X 6 matrixSystem Equation:

kkk wxAx 1

,2,1,0

),0(

k

k WNw

kkk qxHz

2||||argmin

kkk

H

xHzH

2

1 ||||argmin k

kkA

xAxA

)}({ 1 kkk xAxW cov

)}({ kkk xHzQ cov

System Encoding by System Encoding by Training DataTraining Data

Centralize the training data, such that

0})({Exp ,0})({Exp kk xz

11 and ˆ of estimate Initial k-k Px

Time UpdateMeasurement Update

Welch and Bishop 2002

Kalman Filter AlgorithmKalman Filter Algorithm

Prior estimateError covariance

Posterior estimate

Kalman gainError covariance

WAAPP

xAx

Tkk

kk

1

1ˆˆ

1)(

)(

)ˆ(ˆˆ

QHHPHPK

PHKIP

xHzKxx

Tk

Tkk

kkk

kkkkk

Reconstruction on Test Reconstruction on Test DataData

kkjk qxHz

• Uniform: lag j time steps (1 time step = 70ms)

Optimal LagOptimal Lag

• Non-uniform: lag time steps),,,( 4221 jjj

4,3,2,1,0j

Changing it in two ways:

Measurement Equation

kkk qxHz

Methods CC MSE (x , y)

Kalman(0ms lag) (0.77, 0.91) 6.96Kalman(70ms lag) (0.79, 0.93) 6.67Kalman(140ms lag) (0.81, 0.93) 6.09Kalman(210ms lag) (0.81, 0.89) 6.98Kalman(280ms lag) (0.76, 0.82) 8.91

Kalman(non-uniform) (0.82, 0.93) 5.24

Optimal Lag on Test DataOptimal Lag on Test Data

)( 2cm

Linear FilterLinear Filter

ax kk Zf

hand position vector of firing rates for 42 cells over 20 bins (1.4sec)

learned “filter”

Simple regression model,fast decoding, reasonable reconstruction

No explicitly probabilistic model,No uncertainty estimation,slow encoding

constant offset

Linear ReconstructionLinear Reconstruction

Methods CC MSE (x , y)

Kalman(140ms lag) (0.81, 0.93) 6.09Linear filter (0.76, 0.92) 8.30

)( 2cm

ConclusionConclusion

Kalman Filter:

• has sound probabilistic framework, explicit assumptions, and uncertainty in estimation

• is more accurate than linear filter in estimation

• provides efficient filtering algorithm

• shows better reconstruction with time lag analysis

Future WorkFuture Work

• Exploring Poisson model for spiking activity instead of Gaussian

• Exploring the non-linear system model

• Further comparison with population vector methods (Taylor et al, 2002) and particle filtering techniques (Gao et al, 2002)

• on-line experiment of direct neural control using the Kalman filter

ThanksThanksDavid Mumford Applied MathematicsJuliana Dushanova NeuroscienceLauren Lennox NeuroscienceMatthew Fellows NeuroscienceLiam Paninski NYU Neuroscience and MathematicsNicholas Hatsopoulos U. Chicago Comp. Neuroscience

Support:National Science Foundation Keck Foundation National Institutes of Health

1

1

)(

)()(

)ˆ(ˆˆ

QHHPHPK

HPQHHPHPPPHKIP

xHzKxx

Tk

Tkk

kT

kT

kkkkk

kkkkk

Firing rate gives better estimation

Linear filters built on-line

Mijail Serruya

target Neural control

• (off-line) reconstruct monkey’s hand trajectory from its neural activity

• (on-line) control cursor movement from monkey’s neural activity

• (ultimate) provide control of prosthetic devices for severely disabled humans

GoalsGoals