1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 16 / 41 Inference and Computation with Population Codes 13 November 2012 Computation with Population Code  Basis Function Computations

17. 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. 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. 19 / 41 Inference and Computation with Population Codes 13 November 2012 Encoding Probability Distributions

20. 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. 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. 22 / 41 Inference and Computation with Population Codes 13 November 2012 Motivation

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

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

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

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

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

28. 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. 29 / 41 Inference and Computation with Population Codes 13 November 2012 Encoding and Decoding Probability Distributions

30. 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. 31 / 41 Inference and Computation with Population Codes 13 November 2012 Encoding and Decoding Probability Distributions

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

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

34. 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. 35 / 41 Inference and Computation with Population Codes 13 November 2012 Computations Using Probabilistic Population Codes

36. 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. 37 / 41 Inference and Computation with Population Codes 13 November 2012 Computations Using Probabilistic Population Codes

38. 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. 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. 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 / 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.