Chapter 2
Outline
• Linear filters
• Visual system (retina, LGN, V1)
• Spatial receptive fields
– V1
– LGN, retina
• Temporal receptive fields in V1
– Direction selectivity
Linear filter model
Given s(t) and r(t), what is D?
White noise stimulus
Fourier transform
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
Deviation from linearity
Early visual system: Retina
• 5 types of cells:
– Rods and cones: phototransduction 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
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
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
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
Retinotopic map
Retinotopic map
Visual stimuli
Nyquist Frequency
Spatial receptive fields
V1 spatial receptive fields
Gabor functions
Response to grating
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)
Temporal receptive fields
Space-time receptive fields
Space-time receptive fields
Space-time receptive fields
Direction selective cells
Complex cells
Example of non-separable receptive fields
LGN X cell
Example of non-separable receptive fields
LGN X cell
Comparison model and data
Constructing V1 receptive fields
• Oriented V1 spatial receptive fields can be constructed
from LGN center surround neurons
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
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
500 neurons
28 x 28
pixel
image
Hinton
Samples generated by letting the associative
memory run with one label clamped using
Gibbs sampling
Hinton
Examples of correctly recognized handwritten digits
that the neural network had never seen before
Hinton
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)
Backprop with 500 -->300 hiddens
K-Nearest Neighbor
See Le Cun et. al. 1998 for more results
1.25%
1.4%
~1.6%
~1.6%
~ 3.3%
• 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
Summary
• Linear filters
– 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
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