Statistical modeling and analysis of neural data (NEU 560), Fall 2020 Jonathan Pillow Princeton University Lecture 7: least squares regression
Least-squares regression (see whiteboard & notes)
Cortical activity in the null space: permitting preparation without movement Kaufman, Churchland, Ryu, & Shenoy Nature Neuroscience 2014 NEU 560, Lecture 7 part 2 (PCA and regression applications) Jonathan Pillow
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!
• multi-electrode recordings: Methods: - dorsal premotor cortex (PMd) - primary motor cortex (M1) • behavior: monkey cued about upcoming movement • preparatory activity: predicts aspects of movement (reaction time, variability, etc) a b 10 Vertical target position cm Vertical cursor position Central spot Fig 1 0 1 a.u. Deltoid EMG task and 0 typical data Firing rate of one 110 spikes per s PMd neuron 0 Target Go Move 200 ms
� Model: regression! M WN time neuron time 1 T 1 n 1 T 1 1 1 neuron- muscles = muscle neurons weights m m … n • basic idea: neural activity patterns orthogonal to the row space of W won’t affect the muscles
Fig 2 toy example : muscle force proportional to sum of two neural inputs M = N 1 + N 2 Output-null projection FR neuron 2 Output-potent projection T G Time T G Time T G Time Firing rate neuron 2 Go cue Preparation Baseline n Reach right u l l s p a c Reach left e FR neuron 1 Firing rate neuron 1 (If you understand this, you T G Time understand the entire paper)
my version state space view state space view neuron 1 neuron 1 2 2 rate (sp/s) rate (sp/s) 2 2 right reach 1 1 go cue neuron 2 (sp/s) neuron 2 (sp/s) 0 0 1 1 neuron 2 neuron 2 2 2 rate (sp/s) rate (sp/s) 1 1 0 0 0 0 0 0 1 1 2 2 neuron 1 (sp/s) sum (1+2) neuron 1 (sp/s) sum (1+2) 4 4 2 2 0 0 } 0 0 500 500 1000 1000 W time (ms) time (ms)
t n my version e t e o c p a p s state space view state space view neuron 1 neuron 1 2 2 rate (sp/s) rate (sp/s) 2 2 right reach 1 1 go cue neuron 2 (sp/s) neuron 2 (sp/s) 0 0 1 1 neuron 2 neuron 2 2 2 rate (sp/s) rate (sp/s) 1 1 0 0 0 0 0 0 1 1 2 2 n neuron 1 (sp/s) u sum (1+2) neuron 1 (sp/s) sum (1+2) l l s 4 4 p a c e 2 2 0 0 } 0 0 500 500 1000 1000 W time (ms) time (ms)
t n my version e t e o c p a p s state space view neuron 1 2 rate (sp/s) 2 right reach 1 go cue neuron 2 (sp/s) 0 1 neuron 2 2 rate (sp/s) 1 0 left reach 0 0 1 2 n neuron 1 (sp/s) u sum (1+2) l l s 4 p a c e 2 0 } 0 500 1000 W time (ms)
neuron 1 Fig 3: neuron 1 neuron 2 + neuron 2 a Prep tuning / Prep tuning / Prep tuning / 85 115 95 move tuning: move tuning: move tuning: illustrative 25% 150% 16% Firing rate Firing rate Firing rate pair: + c × = 0 0 0 –400 Targ 400 –200 Move 600 –400 Targ 400 –200 Move 600 –400 Targ 400 –200 Move 600
neuron 1 s neuron 1 neuron 2 + neuron 2 a Prep tuning / Prep tuning / Prep tuning / 85 115 95 move tuning: move tuning: move tuning: illustrative 25% 150% 16% Firing rate Firing rate Firing rate pair: + c × = 0 0 0 –400 Targ 400 –200 Move 600 –400 Targ 400 –200 Move 600 –400 Targ 400 –200 Move 600 b Monkey J, array Monkey N, array 0.5 0.5 population Projection onto dim 2 Projection onto dim 2 analysis 0 0 (axes from PCA): –0.5 Movement –0.5 Preparation Go cue –0.5 0 0.5 –0.5 0 0.5 Projection onto dim 1 Projection onto dim 1
<latexit sha1_base64="2g1Cn1+E1VgH4Ux7bm5Y6gkTdk=">AB7XicbVBNS8NAEJ3Ur1q/qh69LBbBiyURQb0VvXhRKhTaEPZbDft0s1u2N0IJfRHePGg4tX/481/47bNQVsfDzem2FmXpRypo3rfjulpeWV1bXyemVjc2t7p7q796hlpgj1ieRStSKsKWeC+oYZTlupojiJOA2i4fXED56o0kyKBzNKaZjgvmAxI9hYKbhFJ+gOBd1qza27U6BF4hWkBgWa3epXpydJlBhCMdatz03NWGOlWGE03Glk2maYjLEfdq2VOCE6jCfnjtGR1bpoVgqW8Kgqfp7IseJ1qMksp0JNgM9703E/7x2ZuKLMGcizQwVZLYozjgyEk1+Rz2mKDF8ZAkmitlbERlghYmxCVsCN78y4vEP61f1t37s1rjqkijDAdwCMfgwTk04Aa4AOBITzDK7w5qfPivDsfs9aSU8zswx84nz/Dro4S</latexit> Approach : estimate output-potent (and output-null) dimensions from movement period activity only ˆ W || M − WN || 2 W = arg min - via principal components regression (PCR) - then look at row space of W^T (each column of W has weights for a single muscle) W M N • • • • • • 6PCs for N • • • • • • • • • • • • 3PCs for M = ⟹ W is 6 x 3 ⟹ 3D “potent” and 3D null space
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � “output-null” dimension fig 4: move a Output-null 1 prep Projection (a.u.) 0 Test epoch Regression epoch − 1 − 400 Targ 400 − 300 Move 600 output-potent dimension move b 1 Output-potent prep Projection (a.u.) 0 e s From f data set JA − 1 e − 400 Targ 400 − 300 Move 600 key panel! c d ,
� � � � � � � � � � � � � � � � fig 4: looking across all null and ‘potent’ directions: prep move tuning ratio: c d , 3.0 × 8.2 × 2.8 × 5.6 × 1 0.32 Fraction of preparatory tuning * * * * Output- . null Output- null t Tuning Output- Output-potent potent Data set NA 0 0 a J N J Array N Array –400 Targ 400 –300 Move .
Accords nicely with observation that preparatory tuning often uncorrelated with peri-movement tuning caveat: trial-averaged activity only! “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.”
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 of weights driving M1 from PMd new technique: • principal components regression (PCR) - first project data onto top k PCs, then do regression.
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