Statistical modeling and analysis of neural data (NEU 560), Fall - - PowerPoint PPT Presentation

Jonathan Pillow Princeton University

Statistical modeling and analysis of neural data (NEU 560), Fall 2020 Lecture 7: least squares regression

Least-squares regression

(see whiteboard & notes)

Kaufman, Churchland, Ryu, & Shenoy Nature Neuroscience 2014

NEU 560, Lecture 7 part 2 (PCA and regression applications) Jonathan Pillow

Cortical activity in the null space: permitting preparation without movement

Motivation:

• how can we plan a course of action, while still

waiting for the right moment to act?

• preparatory activity occurs in motor cortex

prior to a movement; why doesn’t it cause movement? (sub-threshold? gating?)

no

• new proposed mechanism: linear algebra!

Fig 1

10 cm 1 a.u. 110 spikes per s 200 ms

b a

Vertical target position Vertical cursor position Central spot Firing rate of one PMd neuron Deltoid EMG Target Go Move

Methods:

• multi-electrode recordings:
• dorsal premotor cortex (PMd)
• primary motor cortex (M1)
• behavior: monkey cued about upcoming

movement

• preparatory activity: predicts aspects of

movement (reaction time, variability, etc)

task and typical data

M

WN

• muscles

time

1 T 1 m

=

1 m 1 n

neuron neurons …

1 n

neuron- muscle weights time

1 T

Model: regression!

• basic idea: neural activity patterns orthogonal to

the row space of W won’t affect the muscles

Fig 2

Firing rate neuron 1 Firing rate neuron 2 Preparation Baseline Reach right Go cue FR neuron 1 FR neuron 2 Output-potent projection Output-null projection Time T G Time T G Time T G Time T G Reach left

toy example: muscle force proportional to sum

• f two neural inputs

(If you understand this, you understand the entire paper)

n u l l s p a c e M = N1 + N2

1 2

rate (sp/s) neuron 1

1 2

rate (sp/s) neuron 2

500 1000

time (ms)

2 4

sum (1+2)

1 2

neuron 1 (sp/s)

1 2

neuron 2 (sp/s) state space view

1 2

rate (sp/s) neuron 1

1 2

rate (sp/s) neuron 2

500 1000

time (ms)

2 4

sum (1+2)

1 2

neuron 1 (sp/s)

1 2

neuron 2 (sp/s) state space view

right reach

W

}

my version

go cue

1 2

rate (sp/s) neuron 1

1 2

rate (sp/s) neuron 2

500 1000

time (ms)

2 4

sum (1+2)

1 2

neuron 1 (sp/s)

1 2

neuron 2 (sp/s) state space view

1 2

rate (sp/s) neuron 1

1 2

rate (sp/s) neuron 2

500 1000

time (ms)

2 4

sum (1+2)

1 2

neuron 1 (sp/s)

1 2

neuron 2 (sp/s) state space view

p

• t

e n t s p a c e

right reach

W

}

my version

go cue

n u l l s p a c e

1 2

rate (sp/s) neuron 1

1 2

rate (sp/s) neuron 2

500 1000

time (ms)

2 4

sum (1+2)

1 2

neuron 1 (sp/s)

1 2

neuron 2 (sp/s) state space view

n u l l s p a c e p

• t

e n t s p a c e W

}

left reach right reach my version

go cue

Fig 3:

115 Firing rate –400 Targ 400 –200 Move 600 –400 Targ 400 –200 Move 600 –400 Targ 400 –200 Move 600 85 Firing rate 95 Firing rate

+ c × =

Prep tuning / move tuning: 25% Prep tuning / move tuning: 150% Prep tuning / move tuning: 16%

a

illustrative pair: neuron 1 neuron 2 neuron 1 + neuron 2

s

Movement Preparation Go cue –0.5 0.5 Projection onto dim1 –0.5 0.5 Projection onto dim1 –0.5 0.5 Projection onto dim2 –0.5 0.5 Projection onto dim2 Monkey J, array Monkey N, array 115 Firing rate –400 Targ 400 –200 Move 600 –400 Targ 400 –200 Move 600 –400 Targ 400 –200 Move 600 85 Firing rate 95 Firing rate

+ c × =

Prep tuning / move tuning: 25% Prep tuning / move tuning: 150% Prep tuning / move tuning: 16%

a b

illustrative pair: population analysis (axes from PCA): neuron 1 neuron 2 neuron 1 + neuron 2

Approach: estimate output-potent (and output-null) dimensions from movement period activity only

• via principal components regression (PCR)
• then look at row space of W^T

(each column of W has weights for a single muscle) 6PCs for N 3PCs for M ⟹ W is 6 x 3 ⟹ 3D “potent” and 3D null space

=

• • • • • •
• • • • • •
• • • • • •

ˆ W = arg min

W ||M − WN||2

M W N

e s f e ,

• b

c

Output-potent

d

−400 Targ 400 −300 Move 600 −1 1 Projection (a.u.) From data set JA

fig 4:

• utput-potent dimension
• a

Output-null Targ 400 −300 Move −400 600 −1 1 Projection (a.u.) Test epoch Regression epoch

“output-null” dimension key panel! prep move prep move

fig 4:

, . t

• a

.

• J

N J Array N Array 1 Fraction of preparatory tuning 3.0× 8.2× 2.8× 5.6×

c

Output-potent Output- null Data set NA –400 Targ 400 –300 Move 0.32 Tuning

* * * *

Output- potent Output- null

d

tuning ratio: looking across all null and ‘potent’ directions: prep move

Accords nicely with observation that preparatory tuning often uncorrelated with peri-movement tuning

“Trial-averaged data were used except where noted: the primary goal of these analyses was to explain how there can be preparatory tuning without movement, not to explain trial-by-trial variability.”

caveat: trial-averaged activity only!

summary

• null spaces: simple reason preparatory neural

activity fails to generate movement

(i.e., muscles add it up in a way that cancels out)

• preparatory PMd activity also lies in null space
• f weights driving M1 from PMd

new technique:

• principal components regression (PCR) - first

project data onto top k PCs, then do regression.