inferring hand motion from multi-cell recordings in motor cortex using a kalman filter wei wu*,...
TRANSCRIPT
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