1. Projects: 1.Predictive coding in balanced spiking networks (Erwan Ledoux). 2.Using Canonical Correlation Analysis (CCA) to analyse neural data (David Schulz). 3.Encoding and Decoding in the Auditory System (Izzett Burak Yildiz). 4.Quadratic programming of tuning curves: a theory for tuning curve shape (Ralph Bourdoukan). 5.The Bayesian synapse: A theory for synaptic short term plasticity (Sophie Deneve).

2. Projects: 1.Choose a project. Send email to sophie.deneve@ens.fr sophie.deneve@ens.fr 2.Once project assigned, take appointment with advisor ASAP (before April 17). 3.Plan another meeting with advisor (mid- May). 4.Prepare Oral presentation (June 5). Pedagogy, context, clarity, results not so important.

3. The efficient coding hypothesis Predicting sensory receptive fields

4. Schematics of the visual system

5. The retina

6. Center-surround RFs

7. Hubel and Wiesel

8. V1 orientation selective cell

9. Hubel and Wiesel model

10. How are receptive fields measured?

14. It is a linear regression problem

17. Solution:

18. Receptive fields of V1 simple cells

19. Optimal sensory coding?

20. The notion of surprise

21. The entropy of a distribution

22. Minimal and maximal entropy

23. Maximizing information transfer Conditional entropy H(Y|X): Surprise about Y when one knows X Or more shortly: With:

24. Maximizing information transfer Conditional entropy H(Y|X): Surprise about Y when one knows X Mutual information between X and Y:

25. Maximizing information Mutual information between x and y: Maximize …or… Minimize Interesting! Boring! Unreliable! Precise!

26. Sensory system as information channel

27. Maximizing information transfer Mutual information between x and r: Fixed (no noise) Maximize Generative models Analysis models

28. Maximizing information transfer

29. Distribution of responses

30. Entropy maximization

31. Infomax activation function

32. An example in the fly

33. But: neurons cannot have any activation function!

34. Information maximization

37. Two neurons

38. Each neuron maximizing its own entropy

39. Entropy of a 2D distribution

40. Two neurons

41. Entropy maximization = Independent component analysis

42. Entropy maximization, 2 neurons

43. Independent component analysis, N neurons

44. Application: visual processing

45. Transformation of the visual input

46. Entropy maximization

48. Weights learnt by ICA (image patch)

49. The distribution of natural images

50. Geometric interpretation of ICA

51. First stages of visual processing

52. The efficient coding hypothesis

53. Limitations of ICA Works only once… Great!

54. Limitations of ICA Works only once… … and then what? Great!

56. Limitations of ICA Complete basis. Number of features = Number of pixels

57. Limitations of ICA Bottleneck Number optic nerve fibers << Number of retinal receptors

58. Maximizing information transfer Mutual information between x and r: Fixed Minimize Reconstruction error Fixed (no noise) Maximize Generative models Analysis models

59. Maximizing information Mutual information between x and y: FixedMinimize Unreliable! Precise!

60. Maximizing information Mutual information between x and y: FixedMinimize Unreliable! Precise! must predict the sensory input as well as possible

61. Generative model Generate Independent, prior

62. Generative model Generate Independent, prior

63. Generative model Generate Independent, prior Find the dictionary of features,, minimizing

64. The Gaussian Distribution Minimize mean squared error

65. Generative model, recognition model Generate Recognize Independent, prior Minimize entropy Minimize expected reconstruction error

66. Separate the problem in two: Given current sensory input, and dictionary estimate the hidden state Given the current state estimates and sensory input update the to minimize reconstruction error. Repeat until convergence. Start with some random dictionary

67. How to estimate r= h? Generate Recognize Maximum a-posteriori (MAP)

68. How to estimate r= h? Generate Recognize Bayes rule:

69. Reconstruction error and MAP Normal distribution Variance of pixel noise Minus log posterior equivalent to reconstruction error with cost: Prior Cost

70. Minimize reconstruction error Reconstructed sensory input Neural responses Dictionary of features Reconstruction error Penalty or cost

71. Generate Recognize How to estimate r= h? Maximize log posterior probability:

72. Generate Recognize How to update the dictionary Minimize mean-squared error:

73. Generative model, recognition model Generate Recognize 1. Find 2. Update to minimize MSE most probable hidden states

74. What prior to use? Sparse coding Cost = number of neurons with non-zero responses Good!Bad! Many cortical neurons are near-silent…

75. Sparse responses of an edge detector … … Sparse prior:

76. Elementary features found by sparse coding

77. Limitation of the sparse coding approach applied to sensory RFs Generate Recognize “Predictive fields” “Receptive fields” Different!

78. Receptive fields depend on stimulus type

79. Carandini et al, JNeurosci 2005

80. f t Responses to natural scene are poorly predicted by the RF. STRF: Machens CK, Wehr MS, Zador AM. J Neurosci. 2004