NEURAL RELATIVITY PRINCIPLE
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1 NEURAL RELATIVITY PRINCIPLE Alex Koulakov, Cold Spring Harbor Laboratory 1
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4 Odorants are represented by patterns of glomerular activation butyraldehyde hexanone Odorant-Evoked SpH Signals in the Mouse Olfactory Bulb Bozza et.al., Neuron,
5 Odorants are represented by patterns of glomerular activation Glomeruli in the olfactory bulb (OB)
6 Odorant are represented by patterns of glomerular activation C1 C2 C3 How can odor identity be constant?
7 Solution primacy coding Template representing an odorant includes a very small number of glomeruli of highest affinity to the odorant (activated at the smallest concentration) C1 C2 C3 p (= 2) is an important parameter of the code 7
8 Different smells are represented by their primacy glomeruli
9 Temporal codes 9
10 Temporal codes in the olfactory bulb Cury and Uchida (2010) Shusterman, Smear, Koulakov, and Rinberg (2011) 10
11 Odor identity is represented in the sequence of sharp events generated by MCs Odorant 1 Cell number: Odorant 2 Time 11
12 Primacy coding p=2 First set of p active mitral cells determines odorant identity Prediction: odor identity is determined early in the sniff cycle
13 Smell A vs smell B 0.3nM 13 Chris Wilson (NYU) Dima Rinberg (NYU)
14 Primacy coding: Shazam service for the brain? p=2 14
15 evolution of olfactory receptors 15
16 2D odorspace Odor B binding affinity (k B ) R1 R2 Odor A binding affinity (k A ) 16
17 2D odorspace C k A A ~ 1 Odor B binding affinity (k B ) Inactive Active Odor A present Odor A binding affinity (k A ) 17
18 2D odorspace C k A A ~ 1 Odor B binding affinity (k B ) Inactive 2 p=2 O A representation 1 Active Odor A present Odor A binding affinity (k A ) 18
19 2D odorspace Odor B present Odor B binding affinity (k B ) O B representation Active Inactive C k B B ~ 1 Odor A binding affinity (k A ) 19
20 2D odorspace Odors A+B present Odor B binding affinity (k B ) O A + O B representation Odor A binding affinity (k A ) 20
21 p=2 primacy model Odor B binding affinity (k B ) Odor A binding affinity (k A ) 21
22 Individual odor identities are represented by simplexes p=2 p=3 p=
23 D=3, p=3 primacy hull 23
24 D=6, p=7 primacy hull 24
25 Prediction: Higher-order correlations in the response matrix D=6, p=7 Primacy hull Primacy Random Number of neighbors Overlap in odorant representation Primacy Random 25
26 Primacy structure in the response matrix Rˆor Sˆor Receptors or glomeruli Receptors or glomeruli Odorants Nearest neighbors p =
27 Fly data: Hallem and Carlson (2006) Rˆor Receptors Sˆor Receptors Odorants 27
28 Number of nearest neighbors, p=5 Primacy Random Number of neighbors pval Hamza Giaffar Overlap in odorant representation Primacy Random 28
29 Number of nearest neighbors, p=6 Primacy Random Number of neighbors pval Hamza Giaffar Overlap in odorant representation Primacy Random 29
30 Conclusions (OR evolution model) - Primacy model predicts that OR affinities occupy a narrow shell called primacy hull. - Topmost p responses of olfactory receptors share substantial overlaps - Increased overlaps are present in fly data (p=5) - These findings suggest substantial higher-order correlations in OR responses 30
31 Network models 31
32 Expand, sparsen, and decorrelate Olfactory bulb Antennal lobe Piriform cortex Mushroom body u 1 i u 1 i Vertices Simplexes (p=3) u 1 i 32
33 Connectivity between glomeruli and simplexes is given by the simplectic matrix Vertices (receptors/glomeruli) Ŝ Simplexes (cortical cells) 33
34 Dual networks 34
35 How does decoding work? Stimulus Glomerular responses x y Kx Cortical representation?? x x min ykx x α=1: Compressed sensing 35
36 Primacy (relativity) principle Stimulus Glomerular responses x y Kx Cortical representation?? x P P x min n x S y S y Pn n Pn n n Non-parametric conditions: Intensity invariance 36
37 Non-parametric problem: x min, x n S y S y Pn n Pn n n Fix the scale: x min, x n S y S y Pn n Pn n n Single condition: x min, us( Ssn yn ) 0 x n u s 1, s 1, s P P 37
38 Primary problem is difficult to implement: x min, u ( S y ) 0 x s sn n n y K x n nm m m u s 1, s 1, s P P Lagrangian (Karush-Kuhn-Tucker): L( x, ) x u ( S y ) 0 s s s sn n n n 38
39 Duality: Lagrange coefficients become variables Dual cost-function: ( ) min L( x, ) L( x, ) x ( ) Dual problem: n max ( ) 0 x x( ) Primal problem: x min, x + a bunch of very complex inequalities 39
40 Dual problem can be easily solved by a neural network Primal problem: Dual problem: Network Lyapunov function: x x + a bunch of very complex inequalities ( ) n 0 L( ) ( ) n 0 1 L( ) uss susgsquqq Gˆ SKK ˆ ˆ ˆ T Sˆ 2 s sq T Feedforward inputs Feedback 40
41 Dual network x y Ŝ u ˆK Gˆ SKK ˆ ˆ ˆ T Sˆ T Vertices Simplexes 41
42 Non-negativity constraint, x n 0 x L( x, ) x u ( S y ) x s s sn n n n s n n u y Ŝ, 0 n n ˆK Gˆ SKK ˆ ˆ ˆ T Sˆ T ˆK ˆ ˆ SK Granule cells 42
43 Features of dual networks: Operate with Lagrange-Karush-Kuhn-Tucker multipliers rather than with original stimulus variables Rely on inequalities -> intensity-invariant encoding Implement inequalities easily ( n 0) Sparse activity vectors New set of inequalities -> new set of λ-s -> new cell type 43
44 Conclusions: Primacy model: small number of receptors activated first code for odorant identity More general idea neural relativity the implementation of invariant stimulus percepts in dual networks. 44
45 Thanks to: Collaborators: Olfaction Dima Rinberg (NYU) Roman Shusterman (Haifa) Florin Albeanu (CSHL) Steve Shea (CSHL) Venki Murthy (Harvard) John Lisman (Braindeis) Lab Members Brian Kolterman Yi Wei Daniel Kepple Hamza Giaffar Animation Dancing-lemon-studio.com Funding: 45
46 Smell A vs smell B Behavioral performance 46 Chris Wilson (NYU) Dima Rinberg (NYU)
47 Mouse+human OR Isomap algorithm OR OR distance matrix Isomap algorithm Tenenbaum, de Silva and Langford (2000) 47
48 Mouse+human OR Isomap space Included variance Included Dimensions Can primacy model provide insight into the emergence of lowdimensional olfactory manifolds? 48
49 Dimension of olfactory space, D?? Primacy model (p~d) 49
50 Primacy and connectivity 50
51 Fly olfactory system Antennal lobe PNs Mushroom body KCs Vertices Simplexes (p=2) 51
52 Connectivity between glomeruli (PN, vertices) and KC (simplexes) looks random Vertices (glomeruli) Simplexes (kenyon cells) 52
53 Higher-order correlations in connectivity Primacy Random Count Similarity in KC connectivity D=6, p=7 primacy Random 53
54 Conclusions: Primacy model: small number of receptors activated first code for odorant identity Primacy code favors the representation of low-dimensional olfactory manifolds The network architecture based on sparse connectivity between AL and MB is ideal for implementing primacy coding 54
55 Thanks to: Collaborators: Olfaction Dima Rinberg (NYU) Roman Shusterman (Haifa) Florin Albeanu (CSHL) Steve Shea (CSHL) Venky Murthy (Harvard) Andreas Schaefer (MPI) John Lisman (Braindeis) Ivan Iossifov (CSHL) Lab Members Brian Kolterman Yi Wei Armen Enikolopov Toma Marinov Daniel Kepple Animation Dancing-lemon-studio.com Funding: 55
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