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Adaptive Cooperative Systems Chapter 8 Synaptic Plasticity 8.11 ~ 8.13 Summary by Byoung-Hee Kim Biointelligence Lab School of Computer Sci. & Eng. Seoul.

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Presentation on theme: "Adaptive Cooperative Systems Chapter 8 Synaptic Plasticity 8.11 ~ 8.13 Summary by Byoung-Hee Kim Biointelligence Lab School of Computer Sci. & Eng. Seoul."— Presentation transcript:

1 Adaptive Cooperative Systems Chapter 8 Synaptic Plasticity 8.11 ~ 8.13 Summary by Byoung-Hee Kim Biointelligence Lab School of Computer Sci. & Eng. Seoul National University

2 (C) 2009 SNU CSE Biointelligence Lab Contents 8.11 Principal component neurons  Introductory remarks  Principal components and constrained optimization  Hebbian learning and synaptic constraints  Oja’s solution / Linsker’s model 8.12 Synaptic and phenomenological spin models  Phenomenological spin models  Synaptic models in the common input approximation 8.13 Objective function formulation of BCM theory  Projection pursuit  Objective function formulation of BCM theory 2

3 Goals and Contents Goal: the information-processing functions of model neurons in the visual system Contents  Principal component neurons  Special class of synaptic modification models  Relation to phenomenological spin models  Objective function formulation of BCM theory (C) 2009 SNU CSE Biointelligence Lab3

4 Introductory Remarks Images are highly organized spatial structures – some common statistical properties Development of the visual system is influenced by the statistical properties of the images  knowledge of the statistical properties of natural scenes ~ understanding the behavior of cells in the visual system (C) 2009 SNU CSE Biointelligence Lab4

5 Scale Invariance in Natural Images Studies of image statistics reveal non-preferrence of angular scale  Decimation procedure with the grey-valued pixels of the image assuming the role of the spins  p.d.f. of image constrasts and image gradients are unchanged  (Field 1987), (Ruderman and Bialek 1994), (Ruderman 1994) Representing the scale invariance through the covariance matrix  Gives a constraint on the form of the covariance matrix  Starting point for the PCA  (Hancock, et al. 1992), (Liu and Shouval 1995), (Liu and Shouval 1996) (C) 2009 SNU CSE Biointelligence Lab5

6 Principal Components We are rotating the coordinate system in order to find projections with desirable statistical properties Projections: maximally preserve information content while compressing the data into a few leading components (C) 2009 SNU CSE Biointelligence Lab6 Variance of the data projected onto the axis is maximal

7 Principal Components and Constrained Optimization (C) 2009 SNU CSE Biointelligence Lab7 n-component random vector correlation matrix If =0, then the covariance matrix Introducing a fixed vector that satisfies the normalization condition use this to help us find interesting projections Variance after operation: Optimization problem: find the vector a that satisfies the normalization condition, and maximizes the variance The variance is equal to the eigenvalue The maximum variance is given by the largest root

8 Hebbian Learning and Synaptic Constraints (C) 2009 SNU CSE Biointelligence Lab8 The simplest form Hebb’s rule for synaptic modification [Problem] Unstable. The synaptic weights would undergo unbounded growth c: output activity m: synaptic weight vector d: input activity vector On reaching a fixed point m is an eigenvector of the input correlation matrix with eigenvalue equal to zero

9 Solutions for the Unbounded Growth Problem Oja’s solution Linsker’s model (C) 2009 SNU CSE Biointelligence Lab9 On reaching a fixed point Results in a synaptic vector m for which the projection of the input activity has a maximum variance The synaptic system may be characterized as performing a principal component analysis of the input data constraint on the total synaptic strength Clipping - The sum of the synaptic weights are kept constant - each synaptic weight lies within a set range E Q : the variance in the input activity E k : constraint

10 Properties of the Linsker’s Model Stability corresponds to a global near minimum of the energy function Equivalent to the maximum in the input variance subject to the constraint Dynamics of the model system  In different regimes for the parameters k 1 and k 2, different receptive field structures dominate  As k 1 and k 2 are varied, particular eigenvectors other than the principal one gain in relative importance (C) 2009 SNU CSE Biointelligence Lab10

11 Synaptic and Phenomenological Spin Models Theory on synaptic modification  Model to explain the emergence of these highly ordered repeating structures (C) 2009 SNU CSE Biointelligence Lab11 Phenomena  Cells in the primate visual cortex self-organize onto ocular dominance columns and iso-orientation patches  The patterns observed experimentally are highly ordered

12 Phenomenological Spin Models 2D Ising lattice of eye-specificity encoding spints (Cowan and Friedman 1991)  Coupling strengths  If we take with, this type of coupling generates a short-range attraction plus a long-range repulsion between terminals from the same eye Hamiltonian for iso-orientation (C) 2009 SNU CSE Biointelligence Lab12

13 (C) 2009 SNU CSE Biointelligence Lab13

14 Synaptic Models in the Common Input Approximation Consider an LGN-cortico-cortico network with modifiable geniculocortico synapses and fixed cortico-cortico-connections Design of an energy function s.t. the fixed point of the network correspond to the minima of the energy function The common input model by Shouval and Cooper  hamiltonian in this model: (C) 2009 SNU CSE Biointelligence Lab14 general form correlational hamiltonian

15 Information-processing Activities by Common Input Neurons For exclusive excitatory connections  symmetry breaking does not occur  all receptive fields have the same orientation selectivity Inhibition  affects both the organization and structure of the receptive fields  If there is sufficient inhibition, the network will develop orientation selective receptive fields The cortical cells self-organize into iso-orientation patches with pinwheel singularities (C) 2009 SNU CSE Biointelligence Lab15

16 Objective Function Formulation of BCM Theory - Intro Distinguishment between information preservation (variance maximization) and classification (multimodality) (C) 2009 SNU CSE Biointelligence Lab16

17 Projection Pursuit Projection pursuit  a method for finding the most interesting low-dimensional features of high-dimensional data sets  The objective is to find orthogonal projections that reveal interesting structure in the data  PCA is a particular case with the proportion of total variance as the index of interestingness  Why is it needed? High-dimensional spaces are inherently sparse, or “curse of dimensionality” For classification purpose  Interesting projection is one that departs from normalcy (C) 2009 SNU CSE Biointelligence Lab17

18 Objective Function Formulation of BCM Theory (1/3) In the objective (energy) function formulation of BCM theory, a feature is associated with each projection direction A one-dimensional projection may be interpreted as a single feature extraction Goal: to find an objective (loss) function whose minimization produces a one-dimensional projection that is far from normal (C) 2009 SNU CSE Biointelligence Lab18

19 Objective Function Formulation of BCM Theory (2/3) (C) 2009 SNU CSE Biointelligence Lab19 Redefining the threshold function Synaptic modification functions How? Introduce a loss function With some assumptions

20 Objective Function Formulation of BCM Theory (3/3) (C) 2009 SNU CSE Biointelligence Lab20 The risk, or expected value of the loss, which is continously differentiable We are able to minimize the risk by means of gradient descent w.r.t. m i Slightly modified, deterministic version of the stochastic BCM modification equation A BCM neuron is extracting third-order statistical correlates of the data This would be a natural extension of principal component processing in the retina

21 Take-Home Message (Tomasso Poggio, NIPS 2007 tutorial) (C) 2009 SNU CSE Biointelligence Lab21


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