Phenomenological Models of Neurons!! Lecture 5!
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1 Phenomenological Models of Neurons!! Lecture 5! 1!
2 Some Linear Algebra First!! Notes from Eero Simoncelli 2!
3 Vector Addition! Notes from Eero Simoncelli 3!
4 Scalar Multiplication of a Vector! 4!
5 Vector Norm! 5!
6 Unit Vector! 6!
7 Inner Product of Vectors (Dot Product)! Note cos θ is a measure of similarity of two vectors 7!
8 Outer Product of Vectors! 8!
9 Linear Projection! 9!
10 Linear Projection! 10!
11 Linear Projection! 11!
12 Linear Combinations! 12!
13 Vector Space! 13!
14 Basis Vectors! 14!
15 Projection using Basis Vectors! 15!
16 Projection using Basis Vectors! 16!
17 Projection using Basis Vectors! 17!
18 Neural encoding problem! Notes from John Pillow 18!
19 Neural encoding problem! Notes from John Pillow 19!
20 Naïve Approach: A Huge Look-up Table! Notes from John Pillow 20!
21 Naïve Approach: A Huge Look-up Table! Notes from John Pillow 21!
22 Classical Approach! Notes from John Pillow 22!
23 Classical Approach: Receptive Fields! Hubel and Weisel, !
24 Classical Approach: Receptive Fields! Notes Georgopolous, from John 1982 Pillow 24!
25 Classical Approach: Receptive Fields! does not take time into account Notes from John Pillow 25!
26 Modern Approach! Notes from John Pillow 26!
27 Linear Models! 27!
28 Linear Models! 28!
29 Linear Models! 29!
30 Linear Models! You know Y (Whether neuron Fired or not) You know X (Stimulus given) Find k 30!
31 Finding Maxima and Minima! g Matlab demo! 31!
32 Vector/Matrix Calculus! lecture notes from Dr. Xia Hong 32!
33 Vector/Matrix Calculus! lecture notes from Dr. Xia Hong 33!
34 Vector/Matrix Calculus! If your notation is such that a is row vector (some text use this notation) "!g!w =!(aw) $!w = $ $ # a 1! a m % ' ' = a T ' & 34! lecture notes from Dr. Xia Hong
35 Vector/Matrix Calculus! 35! lecture notes from Dr. Xia Hong
36 Vector/Matrix Calculus! 36! lecture notes from Dr. Xia Hong
37 Vector/Matrix Calculus! 37! lecture notes from Dr. Xia Hong
38 Linear Models! You know Y (Whether neuron Fired or not) You know X (Stimulus given) Find k 38!
39 Lease Squares Estimate! S n!d # " k = R d!1 n!1 Find k # to minimize mean $ squared $ error E = (S " k # $ R) 2 39!
40 Lease Squares Estimate! S n!d " k = R d!1 n!1 Find k " to minimize mean # squared # error E = (S k " # R) 2 $E % $ (S " 2 k # R) $k " $k " = 0 40!
41 Lease Squares Estimate! S n!d " k = R d!1 n!1 Find k " to minimize mean # squared # error E = (S k " # R) 2 $E % $ (S " 2 k # R) $k " $k " = 0 % $ & (S " k # R) T (S " ) k # R) $k " ' ( * + = 0 (AB)T = B T A T % $ & " $k " (kt S T # R T )(S " ) ( k # R) + = 0 ' * % $ & " $k " (kt S T S " " k # k T S T R # R T S " ) ( k + R T R+ = 0 ' * 41!
42 Lease Squares Estimate! S n!d " k = R d!1 n!1 Find k " to minimize mean # squared # error E = (S k " # R) 2 $E % $ (S " 2 k # R) $k " $k " % $ & " ( $k " ' (kt % $ & " $k " ( ' (kt = 0 S T S " " T k # k 1!d S T S " k # 2 R T n!1 S T d!n R # R T n!1 n!1 S n!d k d!1 S n!d " ) + R T R+ = 0 * k d!1 " ) + R T R+ = 0 * 42!
43 Lease Squares Estimate! S n!d " k = R d!1 n!1 Find k " to minimize mean # squared # error E = (S k " # R) 2 $E % $ (S " 2 k # R) $k " $k " = 0 % $ & " $k " (kt S T S " ( k # 2 R T ' n!1 S n!d " ) + R T R+ = 0 * k d!1 "! d T x A! x = A T! x + A! % $ x ' # $ d! x &' % S T S " k + (S T S) T " k # 2S T R) = 0 % 2S T S " k = 2S T R % " k = (S!# T S) " #1 S $# T R Pseudo#inverse " d A! x % = A T $ ' # $ d! x &' A has row vectors 43!
44 Over determined systems! from Wikipedia 44!
45 Pseudo-inverse intuition! 45!
46 Linear Model! 46!
47 Covariance! g A measure of how two variables vary together! 47!
48 Linear Model! Notes from John Pillow 48!
49 An Example: Homework problem! 16 Stimulus: Olfactometer Valve Turning On Response: Sensory Neuron Firing Trials Time (ms) x !
50 Populate Matrices S and R! 2 sec stimulus history 16 Stimulus: Olfactometer Valve Turning On ms time bins Response: Sensory Neuron Firing Trials Time (ms) x !
51 Populate Matrices S and R! S R (# spikes) 51!
52 Best Linear Filter Model! 2 Weighting of the Stimulus " # k = (S!# T S) " $1 # S $ T R Pseudo$inverse Lag (Time in Seconds) Lag (Time in Seconds) 2 52!
53 Predicted Response Vs Actual Response! 9 8 Actual Response in different time bins Predicted Response !
54 2D Flickering Bars! 54!
55 Spike Triggered Average! 55!
56 Spike Triggered Average! 56!
57 Spike Triggered Average! 57!
58 Spike Triggered Average! 58!
59 Spike Triggered Average! 59!
60 Spike Triggered Average! 60!
61 Spike Triggered Average! 61!
62 Spike Triggered Average! 62!
63 Spike Triggered Average! 63!
64 Spike Triggered Average! 64!
65 Spike Triggered Average! 65!
66 Spike Triggered Average! 66!
67 Spike Triggered Average! 67!
68 Spike Triggered Average! 68!
69 Back to our homework! Spikes No Spikes 69!
70 Spike Triggered Average! Weighting of the Stimulus Lag (Time in Seconds) 2 70!
71 Spike Triggered Average! P(Stimulus) P(Stim, Spikes) Projection along STA axis 71!
72 Polynomial Model! 72!
73 Polynomial Model! 73!
74 For Homework Problem:! Spike Triggered Average Spike Triggered Covariance 74!
75 Linear-Nonlinear-Poisson Cascade Model! 75!
76 Linear-Nonlinear-Poisson Cascade Model! Notes from John Pillow 76!
77 Linear-Nonlinear-Poisson Cascade Model! Notes from John Pillow 77!
78 Linear-Nonlinear-Poisson Cascade Model! 78!
79 Linear-Nonlinear-Poisson Cascade Model! 79!
80 When does STA fail?! 80!
81 Suppressive interactions! 81!
82 Other Modifications! 82!
83 Multi-neuron GLM! 83!
84 Multi-neuron GLM! 84! JW Pillow et al. Nature 000, 1-5 (2008) doi: /nature07140!
85 85!
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