Correlations and neural information coding Shlens et al. 09
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1 Correlations and neural information coding Shlens et al. 09 Joel Zylberberg
![The neural code is not one-to-one ρ = 0.52 ρ = 0.80 d 3s [Max Turner, UW] b # of trials trial 1!! trial 2!](/docs-images/74/70257402/images/2-0.jpg "!.!.!. ρ = 0.52 10 20 30 ρ = 0.")
2 The neural code is not one-to-one ρ = 0.52 ρ = 0.80 d 3s [Max Turner, UW] b # of trials trial 1!! trial 2!!.!.!. ρ = ρ = 0.80 # of spikes Noise (trial-to-trial variability): many activity patterns per stimulus
3 Correlations in Retina DS Cell Pairs A Noise is correlated between cells B Cell 2 trial 1 Cell 1 trial 2.!.!. 3 s
4 Noise correlation ρ 0 ubiquitous in vivo: Responses vary from trial-to-trial Variability typically correlated Retina: Mastronarde LGN: Alonso et al 1996 V1: Kohn and Smith 2005 see also Ecker et al 2010 and Hansen & Dragoi 2012 IT: Gawne & Richmond 1993 PF: Constantinidis & Goldman-Rakic 2002 Parietal Cortex: Lee et al 1998 A1: decharms & Merzenich 1996 How do noise correlations relate SI: Romo et al 2003 Pre-motor cortex: Vaadia et al 1995 to (coding) function?
5 Correlations & information coding Geometrical picture: sign rule Strong vs. weak correlations Untuned neurons Information limiting correlations Retina Visual Cortex
6 Correlations & information coding Geometrical picture: sign rule Strong vs. weak correlations Untuned neurons Information limiting correlations Retina Visual Cortex
7 Basic Geometry: signal orthogonal to noise is benign (parallel is bad) Neural Response Space s = 1 s = 2 s = 1 s = 2 s = 3 Higher Info s = 3 Lower Info [Averbeck et al., 2006; Abbott & Dayan, 1999; Shamir, 2014; other work; Hu, Zylberberg, and Shea-Brown, PLoS Computational Biology 2014]
8 Correlations hinder coding Correlations improve coding Get this sign rule by considering pairs of cells CAUTION WARRANTED WITH THE SIGN RULE [Averbeck, Latham, & Pouget 2006]
9 Correlations & information coding Geometrical picture: sign rule Strong vs. weak correlations Untuned neurons Information limiting correlations Retina Visual Cortex
10 Weak correlations follow the sign rule (same sign of signal & noise correlations -> bad coding and vice versa) Proof: Symbolically calculate gradient of IFisher or EOLE at zero noise correlations.
11 Information is a concave function of noise correlations Strong correlations are very different from weak ones [Hu, Zylberberg, and Shea-Brown, PLoS Computational Biology 2014]
12 Correlation & covariance matrices are positive semi definite (restricts possible correlations) Consider 3 cells 1 ρ12 ρ13 Let ρ13 = ρ12 = ρ23 Then, by corollary, ρ23 = 1 Cell pairs overlap; noise correlations must be logically consistent over the population Restrictions on possible matrices (get a zero eigenvalue at boundaries)
13 Fisher info & Eigenvalues of Covariance Matrix A Neural Response Space Conditional Response Distribution I(s) = ~ f 0 (s) 2 X i cos i 2 i θ1 v1 θ2 v2 Signal Direction f (s) A zero eigenvalue (almost) always means infinite Fisher info I(s) ~ f(s) [ (~r s)] ~ f(s): mean resp. of cells to stim. s Σ(r s): conditional covariance matrix
14 Information is a concave function of noise correlations Info boundaries due to smallest eigenvalue > 0 Boundaries defined by logical consistency of correlations (PSD requirement) [Hu, Zylberberg, and Shea-Brown, PLoS Computational Biology 2014]
15 Strong correlations squish response distributions [da Silveira and Berry, PLoS Computational Biology 2014]
16 Can get very wrong inferences if some cells are ignored projections) s
17 (Interim Summary) Weak correlations obey the sign rule Strong correlations are almost always good (not necessarily follow sign rule): concavity Important to sample many cells
18 Correlations & information coding Geometrical picture: sign rule Strong vs. weak correlations Untuned neurons Information limiting correlations Retina Visual Cortex
19 Strong correlations are (almost) always good for info coding Info boundaries due to smallest eigenvalue > 0 [Hu, Zylberberg, and Shea-Brown, PLoS Computational Biology 2014]
20 [2014] (s) = 0 + ~ f 0 ~ f 0T Σ ο : some non-info-limiting covariance matrix ε: small scalar
21 Differential correlations (s) = 0 + ~ f 0 ~ f 0T A Neural Response Space Conditional Response Distribution Σ ο : some non-info-limiting covariance matrix ε: small scalar Large eigenvalue for θ=0 θ1 v1 θ2 v2 Signal Direction f (s) s = 1 s = 2 s = 3 Lower Info
22 Differential correlations (noise mimics changes in stimulus) (s) = 0 (s) = 0 + ~ f 0 ~ f 0T
23 (Interim Summary) Strong correlations are almost always good Exception: Differential correlations
24 Correlations & information coding Geometrical picture: sign rule Strong vs. weak correlations Untuned neurons Information limiting correlations Retina Visual Cortex
25 Do untuned neurons matter? (they are typically ignored) 30 Mean Firing Rate (Hz) Stimulus Angle ( o )
26 1. When untuned cells correlated with tuned ones noise correlation ρ Full Population Cell ID Fisher Information (rad -2 ) Tuned Subset(70%) Cell ID Population Size (# Neurons) Cell 1 (Untuned) Firing Rate... Cell N Firing Rate Cell 2 Firing Rate Cell 3 Firing Rate Stim. 1 Stim. 2 Stim. 3
27 2. When untuned cells independent of tuned ones Cell ID noise correlation ρ Fisher Information (rad -2 ) Full Population Tuned Subset (70%) Cell ID Population Size (# Neurons) Cell 1 (Untuned) Firing Rate... Cell N Firing Rate Cell 2 Firing Rate Cell 3 Firing Rate Stim. 1 Stim. 2 Stim. 3
28 noise correlation ρ Full Population Cell ID Fisher Information (rad -2 ) Tuned Subset(70%) Cell ID Population Size (# Neurons) Cell ID Fisher Information (rad -2 ) Full Population Tuned Subset (70%) Cell ID Population Size (# Neurons) When tuned & untuned neurons are correlated, untuned neurons contribute substantially to the neural code
29 Summary Weak correlations follow sign rule Strong correlations are almost always good (not necessarily follow sign rule) Exception: differential correlations always bad If correlations, untuned neurons contribute to neural code
30 [Annual Review of Neuroscience, 2016] [Nature Reviews Neuroscience, 2006]
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