1 / 41 Inference and Computation with Population Codes 13 November 2012 Inference and Computation with Population Codes Alexandre Pouget, Peter Dayan, and Richard S. Zemel Annual review of neuroscience 2003 Presenter : Sangwook Hahn, Jisu Kim

2 / 41 Inference and Computation with Population Codes 13 November 2012 Outline 1.Introduction 2.The Standard Model ( First Part ) 1.Coding and Decoding 2.Computation with Population Codes 3.Discussion of Standard Model 3.Encoding Probability Distributions ( Second Part ) 1.Motivation 2.Psychophysical Evidence 3.Encoding and Decoding Probability Distributions 4.Examples in Neurophysiology 5.Computations Using Probabilistic Population Codes

3 / 41 Inference and Computation with Population Codes 13 November 2012 Introduction  Single aspects of the world –(induce)> activity in multiple neurons  For example –1. Air current is occurred by predator of cricket –2. Determine the direction of an air current –3. Evade with other direction from predicted predator’s move air current

4 / 41 Inference and Computation with Population Codes 13 November 2012 Introduction  Analyze the example at the view of neural activity –1. Air current is occurred by predator of cricket –2. Determine the direction of an air current ( i. population of neurons encode information about single variable ii. information decoded from population activity ) –3. Evade with other direction from predicted predator’s move air current

5 / 41 Inference and Computation with Population Codes 13 November 2012 Guiding Questions (At First Part)  Q1: How do populations of neurons encode information about single variables? How this information can be decoded from the population activity? How do neural populations realize function approximation?  Q2: How population codes support nonlinear computations over the information they represent?

6 / 41 Inference and Computation with Population Codes 13 November 2012 The Standard Model – Coding  Cricket cercal system has hair cells (a) as primary sensory neurons  Normalized mean firing rates of 4 low-velocity interneurons  s is the direction of an air current (induced by predator)

7 / 41 Inference and Computation with Population Codes 13 November 2012 The Standard Model – Encoding Model  Mean activity of cell a depends on s – : maximum firing rate – : preferred direction of cell a  Natural way of describing tuning curves –proportional to the threshold projection of v onto

8 / 41 Inference and Computation with Population Codes 13 November 2012 The Standard Model – Decoding  3 methods to decode homogeneous population codes –1. Population vector approach –2. Maximum likelihood decoding –3. Bayesian estimator  Population vector approach ( sum ) – : population vector – : preferred direction – : actual rates from the mean rates – : approximation of wind direction (r is noisy rates)

9 / 41 Inference and Computation with Population Codes 13 November 2012 The Standard Model – Decoding  Main problem of population vector method –It is not sensitive to the noise process that generates –However, it works quite well –Estimation of wind direction to within a few degrees is possible only with 4 noisy neurons

10 / 41 Inference and Computation with Population Codes 13 November 2012 The Standard Model – Decoding  Maximum likelihood decoding –This estimator starts from the full probabilistic encoding model by taking into account the noise corrupting neurons activities –A –If is high -> those s values are likely to the observed activities –If is low -> those s values are unlikely to the observed activities rms = root mean square deviation

11 / 41 Inference and Computation with Population Codes 13 November 2012 The Standard Model – Decoding  Bayesian estimators –Combine likelihood P[r|s] with any prior information about stimulus s to produce a posterior distribution P[s|r] : –If prior distribution P[s] is flat, there is no specific prior information of s and this is renormalization version of likelihood –Bayesian estimator does a little better than maximum likelihood and population vector

12 / 41 Inference and Computation with Population Codes 13 November 2012 The Standard Model – Decoding  In homogenous population –Bayesian & Maximum likelihood decoding >>> population vector –‘the greater the number of cells is, the greater the accuracy is’ since more cells can provide more information about stimulus

13 / 41 Inference and Computation with Population Codes 13 November 2012 Computation with Population Code  Discrimination –If there are and where is a small angle, we can use Bayesian poesterior (P[s|r]) in order to discriminate those –It is also possible to perform discrimination based directly on activities by computing a linear : – : usually 0 for a homogeneous population code – : Relative weight

14 / 41 Inference and Computation with Population Codes 13 November 2012 Computation with Population Code  Noise Removal –Maximum likelihood estimator is unclear about its neurobiological relevance. 1. finding a single scalar value seems unreasonable because population codes seem to be used throughout the brain 2. while finding maximum likelihood value is difficult in general –Solution : utilizing recurrent connection within population to make it behave like an autoassociative memory Autoassociative memories use nonlinear recurrent interactions to find the stored pattern that most closely matches a noisy input

15 / 41 Inference and Computation with Population Codes 13 November 2012 Computation with Population Code  Basis Function Computations –Function approximation compute the output of functions for the case of multiple stimulus dimensions. –For example, –s h : head-centered direction to a target s r : eye-centered direction s e : position of eyes in the head

16 / 41 Inference and Computation with Population Codes 13 November 2012 Computation with Population Code  Basis Function Computations

17 / 41 Inference and Computation with Population Codes 13 November 2012 Computation with Population Code  Basis Function Computations –linear solution for homogeneous population codes (mapping from one population code to another, ignoring noise )

18 / 41 Inference and Computation with Population Codes 13 November 2012 Guiding Questions (At First Part)  Q1: How do populations of neurons encode information about single variables? -> p.6~7 How this information can be decoded from the population activity? -> p.8~12 How do neural populations realize function approximation? -> p.13~14  Q2: How population codes support nonlinear computations over the information they represent? -> p.15~17

19 / 41 Inference and Computation with Population Codes 13 November 2012 Encoding Probability Distributions

20 / 41 Inference and Computation with Population Codes 13 November 2012 Motivation  The standard model has two main restrictions :  We only consider uncertainty coming from noisy neural activities. (internal noise) : Uncertainty is inherent, independent of internal noise.  We do not consider anything other than estimating the single value. : Utilizing the full information contained in the posterior is crucial.

21 / 41 Inference and Computation with Population Codes 13 November 2012 Motivation  “ill-posed problems” : images do not contain enough information.  The aperture problem. : Images does not unambiguously specify the motion of the object.  Solution - probabilistic approach. : perception is conceived as statistical inference giving rise to probability distributions over the values.

22 / 41 Inference and Computation with Population Codes 13 November 2012 Motivation

23 / 41 Inference and Computation with Population Codes 13 November 2012 Psychophysical Evidence

24 / 41 Inference and Computation with Population Codes 13 November 2012 Psychophysical Evidence

25 / 41 Inference and Computation with Population Codes 13 November 2012 Encoding and Decoding Probability Distributions

26 / 41 Inference and Computation with Population Codes 13 November 2012 Encoding and Decoding Probability Distributions

27 / 41 Inference and Computation with Population Codes 13 November 2012 Encoding and Decoding Probability Distributions

28 / 41 Inference and Computation with Population Codes 13 November 2012 Encoding and Decoding Probability Distributions  Convolution encoding :  Can deal with non-Gaussian distributions that cannot be characterized by a few parameters, such as their means and variances.  Represent the distribution using a convolution code, obtained by convolving the distribution with a particular set of kernel functions.

29 / 41 Inference and Computation with Population Codes 13 November 2012 Encoding and Decoding Probability Distributions

30 / 41 Inference and Computation with Population Codes 13 November 2012 Encoding and Decoding Probability Distributions  Use large neuronal population of neurons to encode any function by devoting each neuron to the encoding of one particular coefficient.  The activity of neuron a is computed by taking the inner product between a kernel function assigned to that neuron and the function being encoded.

31 / 41 Inference and Computation with Population Codes 13 November 2012 Encoding and Decoding Probability Distributions

32 / 41 Inference and Computation with Population Codes 13 November 2012 Encoding and Decoding Probability Distributions

33 / 41 Inference and Computation with Population Codes 13 November 2012 Encoding and Decoding Probability Distributions

34 / 41 Inference and Computation with Population Codes 13 November 2012 Examples in Neurophysiology  Uncertainty in 2-AFC (2-alternative forced choice) : examples offer preliminary evidence that neurons represent probability distributions, or related quantities, such as log likelihood ratios.  There are also experiments supporting gain encoding, convolution codes, and DDPC, respectively.

35 / 41 Inference and Computation with Population Codes 13 November 2012 Computations Using Probabilistic Population Codes

36 / 41 Inference and Computation with Population Codes 13 November 2012 Computations Using Probabilistic Population Codes  If we use convolution code for all distributions –multiply all the population codes together term by term –requires neurons that can multiply or sum : achievable neural operation  If the probability distributions are encoded using the position and gain of population codes –Solution : Deneve et al. (2001) –Some limitations –Performs a Bayesian inference using noisy population codes

37 / 41 Inference and Computation with Population Codes 13 November 2012 Computations Using Probabilistic Population Codes

38 / 41 Inference and Computation with Population Codes 13 November 2012 Guiding Questions(At Second Part)  Q3: How may neural populations offer a rich representation of such things as uncertainty in the aspects of the stimuli they represent?  # 21 ~ # 24  Probabilistic approach : perception is conceived as statistical inference giving rise to probability distributions over the values.  Hence stimuli of neural populations represents probability distributions, which gives information of uncertainty.

39 / 41 Inference and Computation with Population Codes 13 November 2012 Guiding Questions(At Second Part)  Q4: How can populations of neurons represent probability distributions? How can they perform Bayesian probabilistic inference?  #25 ~ #31 (for first), #37 ~ #39 (for second)  Several schemes have been proposed for encoding probability distributions in populations of neurons : Log-likelihood method, Gain encoding for Gaussian distributions, Convolution encoding.  Bayesian probabilistic inference can be done by multiply all the population codes (convolution encoding), or using noisy population codes (gain encoding)

40 / 41 Inference and Computation with Population Codes 13 November 2012 Guiding Questions(At Second Part)  Q5: How multiple aspects of the world are represented in single populations? What computational advantages (or disadvantages) such schemes have?  # 25 ~ # 28 (first)  Log-likelihood : likelihood Gain encoding : mean and standard deviation Convolution encoding : probability distribution

41 / 41 Inference and Computation with Population Codes 13 November 2012 Guiding Questions(At Second Part)  Q5: How multiple aspects of the world are represented in single populations? What computational advantages (or disadvantages) such schemes have?  # 25 ~ # 28 (second)  Log-likelihood : decoding is simple, but some distribution limitation Gain encoding : strong distribution limitation. Convolution encoding : can work for complicated distribution.