The Physics of the Brain

1
Receptive fields How they are constructed, how
their structure is deduced.
2
In general can sometimes approximate a system by
a linear approximation of the form These
are the first two terms in the more
general Volterra of Weiner expansions. The
simplest kernel is
an instantaneous kernel. Then
3
Receptive fields in the visual system
4
Retinal and LGN receptive fields
ON center OFF center
5
Retinal and LGN receptive fields
This DOG filter can be seen as a spatial
kernal. The cells response given an input I(r,t)
is
6
Retinal and LGN receptive fields
Using a linear filter
For an instantaneous temporal kernel
7
Retinal and LGN receptive fields
linear response of one neuron
linear response of another neuron at point a,c
neuron
Convolution!
8
Receptive fields in the visual system
9
Receptive fields in Visual Cortex
Orientation selective
Simple cell
Linear VC neuron
10
Simple cell
Gabor filter
11
Testing receptive fields with sinusoidal
gratings. LGN (show simulation examples)
Flat orientation tuning curve
12
Spatial frequency tuning
SF1
SF3
SF6
13
What is this?
Remember, a real neuron spikes
14
Visual cortex RFs (simple cells)
Tuning curve
15
(No Transcript)
16
Complex cells/ Energy model
F0 F?/2
Where L1 is the response to filter with phase 0
and L2 is the response for a filter with phase ?/2
17
Complex cells/ Energy model
L2
L1
L1
L2
18
Testing with gratings is related to the Fourier
transform
LGN spatial kernel Spectrum of kernel
19
Visual Cortex, FT of simple cell RF
20
Fourier Transform, review
A function f(t), can be decomposed as
Remember
How do we find ? This is
the Fourier transform.
and eq. 1 is the inverse Fourier
transform
21
This provides an inverse because
If f(t) is periodic f(t)f(tT) then can use a
Fourier series Where
22
or
When computed numerically, use
where and Using
this the discrete Fourier transform is
23
Examples
24
Convolution theorem In Fourier space
25
Spatial frequency tuning
SF1
SF3
SF6
26
Testing with gratings is related to the Fourier
transform
LGN spatial kernel Spectrum of kernel
27
Some more about kernels In general We
often assume a separable K of the form
28
Separable space time kernel (LGN cell)
29
Non separable space time kernel (VC cell)
30
COS(wt) inputs

In Fourier space If
then what is ?
31
and
Then
Therefore up
to a constant multiple and phase Or
in words, the kernel does not change the shape of
a pure cos (sin) input
32
Step function inputs
Therefore That is the kernel, in this case,
is a derivative of the rate
33

• Need to know
• Receptive fields in retina and LGN, center
  surround, DOG
• Simple and complex receptive field in VC, their
  properties and a feed forward way of constructing
  both.
• Fourier analysis and receptive field structure.
  Know how to carry out Fourier transforms for
  different simple functions.
• Reverse correlation coming next.

34
Reverse correlation methods
Temporal only
Define
White noise stimuli How can these be created in
practice?
35
Consider

36
Therefore, for white noise stimuli
Adding space That is the stimulus is
different in different points in space, and
changes in time. X is a vector in 2D space
X(x,y) and S is a function of this vector.
37
So, with space
38
Therefore, for white noise stimuli
Examples (with space)
Original spatial RF Extracted
spatial kernel (instantaneous-d function)
39
t1 t9
40
Homework 4 due March 28. Download the file
from web site. It includes a sequence of random
'white noise' inputs and the corresponding
outputs (res). Using reverse correlations induce
the spatio-temporal kernal. Is it separable? If
it is can you guess an exact form for the spatial
and temporal kernels? Extra credit (50)
Program a complex cell using an energy model with
shifted Gabor filters. Try to use reverse
correlation to find the spatio-temporal kernal of
this neuron. Explain in detail why you get these
results. Here the write-up counts at least as
much as the program. I will be away much of this
week, send question to Jeff !