1. Chapter 2

2. Outline Linear filters Visual system (retina, LGN, V1)
Spatial receptive fields V1
LGN, retina
Temporal receptive fields in V1
Direction selectivity

3. Linear filter model
Given s(t) and r(t), what is D?

5. White noise stimulus
White noise is used to as a stimulus to measure the spike triggered average response as in the electric fish experiment.

6. Fourier transform
In practice, only approximate white noise signals can be generated with a flat spectrum up to a cut-off frequency.

7. H1 neuron in visual system of blowfly
A: Stimulus is velocity profile;
B: response of H1 neuron of the fly visual system;
C: rest(t) using the linear kernel D(t) (solid line) and actual neural rate r(t) agree when rates vary slowly.
D(t) is constructed using white noise

8. Deviation from linearity

10. Early visual system: Retina
5 types of cells:
Rods and cones: photo-transduction into electrical signal
Lateral interaction of Bipolar cells through Horizontal cells. No action potentials for local computation
Action potentials in retinal ganglion cells coupled by Amacrine cells. Note
G_1 off response
G_2 on response
A: Anatomy of retina of dog (Cajal 1911)
B: Recording in mud puppy (amphibian). Stimulus is flash of light (1 sec)

11. Pathway from retina via LGN to V1
Lateral geniculate nucleus (LGN) cells receive input from Retinal ganglion cells from both eyes.
Both LGNs represent both eyes but different parts of the world
Neurons in retina, LGN and visual cortex have receptive fields:
Neurons fire only in response to higher/lower illumination within receptive field
Neural response depends (indirectly) on illumination outside receptive field

12. Simple and complex cells
Cells in retina, LGN, V1 are simple or complex
Simple cells:
Model as linear filter
Complex cells
Show invariance to spatial position within the receptive field
Poorly described by linear model

13. Retinotopic map
Neighboring image points are mapped onto neighboring neurons in V1
Visual world is centered on fixation point.
The left/right visual world maps to the right/left V1
Distance on the display (eccentricity) is measured in degrees by dividing by distance to the eye
Distances on the display screen (eccentricity) are measured in degrees by dividing by distance to the eye

14. Retinotopic map
A: The pattern on the cortex was produced by imaging a radioactive analogue of glucose that was taken up by active neurons while a monkey viewed a visual image consisting of concentric circles and radial lines
$\lambda=12 mm,\epsilon_0=1^0$

15. Retinotopic map

16. Visual stimuli

17. Nyquist Frequency

18. Spatial receptive fields

19. V1 spatial receptive fields

20. Gabor functions

21. Response to grating

22. Temporal receptive fields
Space-time evolution of V1 cat receptive field
ON/OFF boundary changes to OFF/ON boundary over time.
Extrema locations do not change with time: separable kernel D(x,y,t)=Ds(x,y)Dt(t)
Cat V1

23. Temporal receptive fields

24. Space-time receptive fields

25. Space-time receptive fields

26. Space-time receptive fields

27. Direction selective cells

28. Complex cells
Early stage of invariant object recognition

29. Example of non-separable receptive fields LGN X cell

30. Example of non-separable receptive fields LGN X cell
Cat LGN X cell

31. Comparison model and data

32. Constructing V1 receptive fields
Oriented V1 spatial receptive fields can be constructed from LGN center surround neurons

34. Stochastic neural networks
The top two layers form an associative memory whose energy landscape models the low dimensional manifolds of the digits.
The energy valleys have names
2000 top-level neurons
10 label neurons
500 neurons
The model learns to generate combinations of labels and images.
To perform recognition we start with a neutral state of the label units and do an up-pass from the image followed by a few iterations of the top-level associative memory.
500 neurons
28 x 28 pixel image
Hinton

35. Samples generated by letting the associative memory run with one label clamped using Gibbs sampling
Hinton

36. Examples of correctly recognized handwritten digits that the neural network had never seen before
Hinton

37. How well does it discriminate on MNIST test set with no extra information about geometric distortions?
Generative model based on RBM’s %
Support Vector Machine (Decoste et. al.) %
Backprop with 1000 hiddens (Platt) ~1.6%
Backprop with >300 hiddens ~1.6%
K-Nearest Neighbor ~ 3.3%
See Le Cun et. al for more results
Its better than backprop and much more neurally plausible because the neurons only need to send one kind of signal, and the teacher can be another sensory input.
Hinton

38. Summary Linear filters Visual system (retina, LGN, V1) Visual stimuli
White noise stimulus for optimal estimation
Visual system (retina, LGN, V1)
Visual stimuli V1
Spatial receptive fields
Temporal receptive fields
Space-time receptive fields
Non-separable receptive fields, Direction selectivity
LGN and Retina
Non-separable ON center OFF surround cells
V1 direction selective simple cells as sum of LGN simple cells

39. Exercise 2.3 Is based on Kara, Reinagel, Reid (Neuron, 2000).
Simultaneous single unit recordings of retinal ganglion cells, LGN relay cells and simple cells from primary visual cortex
Spike count variability (Fano) less than Poisson, doubling from RGC to LGN and from LGN to cortex.
Data explained by Poisson with refractory period
Fig. 1,2,3