Introduction to Neural Networks
U. Minn. Psy 5038

Representation of visual information

Primary visual cortex: anatomy, physiology, and functions

Goals

Provide an overview of a major brain subsystem to help anchor concepts in neural network theory.
Discuss issues of representation.
Functional requirements that determine the computations that networks must do.
Connect various parts and functions of the visual system with neural network ideas already studied, especially the (semi) linear neworks, as well as ones coming up in remaining lectures of the course.

Neural networks and the visual system

Retina-cortex pathway

Roughly ten million retinal measurements are sent to the brain each second, where they are processed by some billion cortical neurons.

Retinal-cortical pathway: light->retina->thalamus<->cortex
light->(receptors->bipolar cells->ganglion cells)->(lateral geniculate cells)->V1 cells in layer 4C
...also other neurons (e.g. in the retina such as horizontal and amacrine cells)
...also other pathways, e.g. to the superior colliculus

Summary approximation analogous to generic model neuron:

Despite several layers of synapses, many properties of ganglion cells, l.g.n. cells, and classes of V1 cells can be modeled using the generic model neuron, but with neural input replaced by light intensity. Where xi -> (Subscript[L, i]), and (Subscript[L, i])is the intensity of the light corresponding to location i. So firing rate is:

(Subscript[R, i])=σ((∑_(j=1)^n)(Subscript[w, ij])(Subscript[L, j]))

This corresponds to the steady-state lateral inhibition model we studied earlier, but has also been used to model cell responses in several areas of the mammalian visual system (e.g. "simple cells" of V1). We'll see below how more complex properties have been modeled using two layers of synaptic weights (e.g. "complex cells" of V1), corresponding to the non-linear multi-layer feedforward networks studied a couple of lectures ago.
We'll spend a considerable fraction of this lecture on the structure or spatial pattern of these weights.

There is an analogous model for time:

(Subscript[R, T])=σ((∑_(t=1)^(T-1))(Subscript[w, T,t])(Subscript[L, t]))

So what is new about the modeling? It's in the details--the pattern of weights, layers, and non-linearities. In vision research, a lot has been learned about the structure, extent, and possible development of the weights, and the forms of the squashing functions. We've also learned when the above generic model doesn't work. And also ideas about how to interpret the information processing functions of whole neural sub-populations modeled as non-linear feedforward networks in vision.

Pathways from eye-to-cortex

Schematic view

Retina

The primate retina has about 10^7 cones that send visual signals to the  optic nerve via about 10^6 ganglion cells.

The visual system in general responds well to contrasts over a middle range, but not as well to low or high spatial frequencies. The visual system is sometimes said to behave as a bandpass spatial filter. because it lets frequencies in a middle range of spatial frequencies pass through better than low or high frequencies. The lateral inhibition model we studied for the horseshoe crab is a bandpass spatial filter.

A center-surround spatial filter can be used to model both the low and high frequency fall-offs. One theory (the "single-channel") theory can account for this if the size and spacings of the excitatory and inhibitory regions of the receptive field weighting function,

are well-matched to frequency between the the low and high. Although the single-channel theory works here, it doesn't account for other results that we will get to later.

Ganglion cells do spatio-temporal filtering of incoming image intensities. They are band-pass filters meaning that low and high frequencies of image intensity get suppressed, and only spatial and temporal changes in a middle band of frequencies gets passed on. Why?
	->Lateral inhibition as redundancy reduction is one important idea that generalizes beyond early sensory coding.

Overall Retinal Functions: image sampling, contrast coding, gain control, spatio-temporal filtering (high pass--encoding change important), efficient coding

Functions of the Chiasm and Lateral geniculate nucleus (LGN)

Optic chiasm
The optic nerves from the  two eyes meet at the optic chiasm where about half of the fibers cross over and  the other half remain on the same side of the underside of the brain. Before  synapsing in the lateral geniculate nucleus, about 20% of these fibers that  now comprise the optic tract branch off to the superior colliculus--a structure  involved with eye movements. The rest of the optic tract fibers  synapse on cells in the lateral geniculate nucleus. Cells in the lateral  geniculate nucleus send their axons in a bundle called the optic radiation  to layer IV (one of six layers) of primary visual cortex.

The optic chiasm routes neuronal information so that information  from corresponding points on the left and right eyes can come together at  cortex for binocular vision, and in particular stereo vision.  Typically animals with frontal vision have nearly complete cross-over, and animals with strongly lateralized eyes (e.g. fish, rabbits) have little or no cross-over. There is a correlation between eye position and whether an animal is primarily prey or predator.
The nervous system has gone to considerable length  to bring information from the two eyes together early on. Although there seems to be little if any binocular interaction between neurons in the LGN, the arrangment of the optic chiasm is the first step towards the eventual construction of a topographic cortical map with information from both eyes.
In fact, there is a general principle that becomes even more apparent when  one looks at maps that pervade cortical organization: 
Neural computations that require bringing information together often require close physical proximity between between neurons to reduce connection lengths. 
This part of one solution to the "binding problem". There are other ways of bringing information together for a computation.
Later on we will see some of the consequences of self-organizing principles that serve to minimize wiring length when we study Kohonen networks and adaptive maps.
The neurons of lateral geniculate nucleus do more spatial and temporal band-pass filtering,  and the cells are characterized by fairly symmetrical center-surround  organization like the ganglion cells. They show even less response to uniform  illumination levels than ganglion cells. Despite the fact that neurons from the two  eyes exist within the same nucleus, no binocular neurons are found in LGN.  
We have to wait until cortex to see binocular neurons. Although the LGN is  often considered a relay station,  feedback from cortex suggests possible role of attention mechanisms (e.g. Crick, 1984 for  a speculative neural network theory of  LGN and reticular function). Sillito et al.  (1994) have found  "Feature-linked synchronization of thalamic relay cell firing induced by feedback from the visual cortex". Nature, 369, N. 9, 479-482. An estimated ratio of 10 fibers returning to LGN for every one going forward raises a major computational question.
See Sherman and Guillery (2002) for a discussion of thalamo-cortical anatomy and possible functions.
The superior colliculus has a key role is in the control of eye movements--a highly non-trivial  control problem requiring coordination of head and eye movements in the context of  a constantly changing environment.

Primary Visual Cortex

Cortex in general

Recall: four lobes: occipital, temporal, parietal, frontal. 
If flattened like a pancake, the gray matter would be about ~14 inches in diameter with, ~10^10 neurons. 2000 to 10000 synapses each.
Cortical modularity:
	e.g. ~30 visual areas based on primate studies (macaque), perhaps up to 12 with some degree of retintopic organization.
Largely inhibitory (GABA), typically quiet (because of high metabolic cost to firing, Lennie (2003)). Excitatory transmission (glutamate). (Lots of other transmitters).
Long range excitatory pyramidal cells (between areas). Within area excitatory and inhibitory connections.

Primary visual cortex: Large scale organization

Overview

primary visual cortex (striate cortex, V1 in primates, area 17 in cats)
		anatomical organization - topographic (i.e. retinotopic), later lecture on adapative maps
		functional cell types: simple, complex, end-stopped
		model of simple cells
			generic feedforward neural network models
			other than sigmoid non-linearities? Yes, shunting inhibition mentioned earlier.

Topographic map

The striate cortex is laid out as non-linear topographic map with 80%  of cortical   area devoted to about 20% of visual field, reflecting the higher  acuity of foveal vision. Because of the cross-over at the optic chiasm, the  left visual field (right retina) maps to right hemisphere. (see Engel et al., 1997 for fMRI measurments of human cortical magnification). Nearby locations in the retina map to nearby locations in primary visual cortex--thus, V1 is said to have a retinotopic map.
There are a number of interesting features of this map, where locations in retinal polar coordinates are approximately mapped to rectangular coordinates on V1. 
http://gandalf.psych.umn.edu/users/kersten/kersten-lab/demos/RetCorLogPolInt.mov
Lecture 21 describes the properties of this map in more detail. 
Why is visual information laid out topographically in many cortical areas? Are other features laid out topographically? In other areas?

Layers

Axons from the optic radiation synapse on layer IV neurons of the  primary visual cortex (area 17, striate cortex or V1).  Cortex is  anatomically structured in layers, numbered from I (superficial) to VI  (deep). Connectivity between areas is closely tied to layer structure.
Determining the network connectivity, and dynamics within the laminar structure of the cortex is a crucial step towards understanding whether there is a "common computational algorithm" that gets applied in other cortical areas.

Primary visual cortex: Neuron properties

V1 neurons respond well to contrast, both in terms of absolute sensitivity and contrast differences (Geisler and Albrecht, 1997; Boynton et al., 1999; Olman et al., 2004). Apart from the neurons the LGN fibers synapse on, and in contrast with  receptive field characteristics of earlier neurons, many cortical cells are tuned or selective to local stimulus attributes. Tuning varies along the dimensions of:
	• orientation (θ)
	• degree of binocularity 
	• motion direction
	• spatial frequency (f, or fx, fy)
	• spatial phase (ϕ)
Some of the major contributions to our understanding of visual cortex was due to the research of Hubel, D. H., & Wiesel, T. N. (see 1968 reference). Hubel and Wiesel won the Nobel prize for their work.

Columns and hypercolumns: orientation and degree of binocularity

The cells of the primary cortex  are organized into columns running roughly perpendicular to the surface in which  cells tend to have the same orientation preference and degree of binocularity (degree to which a a neuron is responsive to left vs. right eye input). A  "hypercolumn" is a group of columns spanning all orientations and one full range of eye-preferences. The size is on the order of cubic millimeters and includes about 100,000 cells. Different hypercolumns have different receptive field centers. In the cortex, we see for the first time binocular cells. 
Embedded in the cortical hypercolumns  are cytochrome oxidase blobs in which are found opponent color cells that seem to lack strong orientation selectivity  (Livingstone, M. S., & Hubel, D. H., 1984;  Livingstone, M. S., & Hubel, D. H., 1987).

see Kandel and Schwartz, http://www.amazon.com/exec/obidos/tg/detail/-/0838577016/qid=1066753949/sr=1-1/ref=sr_1_1/103-3512951-3346262?v=glance&s=books

If one views the cortex as essentially a 2-D sheet, it is reasonable to ask:
How can a high-dimensional feature space can be mapped into it and maintain a requirement that similar features (e.g. orientation, ocularity, spatial frequency, motion selectivity) get mapped to near-by locations. 
We return to this problem in the lecture on adaptive maps.

Simple cells

There are two main types of cells. The simple  cells are roughly linear except for  rectification, are spatially and temporally band-pass, and show spatial phase  sensitivity.

Note: The following illustrates no rectification, half-wave rectification, and full-wave rectification of a sinewave:

GraphicsGrid[{{Plot[Sin[x],{x,-12,12},Axes->False]
Plot[If[Sin[x]>0,Sin[x],0],{x,-12,12},Axes->False]
Plot[If[Sin[x]>0,Sin[x],-Sin[x]],{x,-12,12},Axes->False]}}]

A first-order approximation model for simple cell response firing rate (in impulses/sec) is:

Where Wij are the receptive field weights, and Lij the image intensity values at spatial location (i,j). An example would be a Gabor function (see discussion and figures below):

We can visualize the weights for particular values of the horizontal and vertical spatial frequencies fx, fy, and phase ϕ:

The point-wise non-linear function is a half-wave rectification operation, σ,  sets negative values to zero, and is linear for positive values:

You can see that the equation for R has the same form as the generic neuron model, except that the inputs are the physical stimulus values. The weights don't necessarily correspond to specific synapses, but are the net effect of a number of earlier layers.
And as we saw at the begining of the course, a more complete, but mainly descriptive, model is obtained by replacing the straight sloping line with one that saturates at high values. This model is steady state. To include time domain dependencies requires the introduction of a band-pass temporal tuning characteristics.

Complex cells

The second major class of neurons is the complex  cells. Like simple cells, complex cells are spatially and  temporally  band-pass, show orientation and motion direction selectivity, but are insensitive to  the phase of a stimulus such as a sine-wave grating. Rather than half-wave rectification, they show full-wave rectification. A model for complex cells would resemble the sum of the outputs of several  subunits positioned at several nearby spatial locations, relative to retinotopic space. Each subunit would resemble  simple cell with a linear spatial filter followed by a threshold non-linearity.
If true, the simplest neural-net like version of this model would correspond to two layers of weights, where the first set feed into simple cells, and the second set feed into complex cells. In actuality, complex cells may not be built out of simple cells, and as mentioned above, the generic connections model of simple cells collapses a number of neural layers to one effective layer. Another complication is that cells show a property called "response normalization" (see contrast normalization, below).
One way of obtaining the phase insensitivity would be to use subunits with cosine and sine phase receptive fields. We see below how a neural network can be built that can be used to detect edges--it  combines simple cell outputs into outputs similar to those of complex cells.
See Mechler, F., & Ringach, D. L. (2002) for a discussion of whether simple and complex cells make distinct classes.
The motion selectivity could be built in with appropriate inhibitory connections between subunits. Full-wave rectification could be built with subunit pairs that have excitatory and inhibitory receptive fields centers.

Contrast normalization

Both simple and complex cells show contrast normalization--a feature not included in the above simple model. For a discussion of steady-sate models of simple and complex cells see Heeger (1991) and Carandini et al. (1997)

(Subscript[R, i])=σ((∑_(j=1)^n)(Subscript[w, ij])(Subscript[L, j]))/(∑_(k∈(Subscript[N, i])))(Subscript[R, k]^2)

where (Subscript[N, i])is a neighborhood of neuron i

End-stopped cells

A third class of cells are the end-stopped  (or "hyper-complex") cells  that have an optimal orientation for a bar or edge stimulus, but fire most actively if the bar or edge terminates within the receptive field, rather than extending beyond it. It has been suggested that these cells act as "curvature" detectors. (Dobbins, A., Zucker, S. W., & Cynader, M. S., 1987).
These cells are also thought to be important for detecting occluding surfaces and the perception of illusory contours. (Heiter et al., 1992).
Whether or not these end-stopped cells should be considered a distinct functional class has been a matter of debate.

Functions of Primary Cortex

Functions of primary visual cortex

Emphasize local image measurements likely to be caused by surface properties, e.g. abrupt intensity changes due to object edges. But correlation between "edge detector neurons" and perceived image edges is weak.

Binocular vision and Stereo

What is stereo disparity? Demo with thumb and forefinger.

As mentioned above, primary cortex brings together information  from the two eyes in single neurons. This information is important for coordinated eye movements and stereo  vision. Although V1 cells are predominantly binocular, it was at first thought that disparity selectivity did not arise until V2 (Hubel and Wiesel, 1970). However, there is evidence for disparity selective cells in V1 and V2 (Poggio, G., F., & Poggio, T. ,1984). Disparity selectivity is a trivial task for single bar stimuli, but is a more challenging computational problem when there are many possibilities for incorrect matching of features between the two eyes.  Neurons have been found that seem to solve the problem of false matching (Poggio and Talbot, 1981).

One possible algorithm for stereo vision is discussed in: Poggio, T. (1984). Vision by Man and Machine. Scientific American, 250, 106-115. 
This algorithm is related to Hopfield networks that we will study later in this course.
Stereo vision has received a lot of attention in both computer and biological vision over the last 15 years. Later we will look at a neural network model of stereopsis.

Motion

motion measurements--local in space and time

The directional selectivity of cells in  striate cortex provide a form of early motion detection. This detection  is only local and  thus ambiguous. Cortical cells suffer from the "aperture problem", and further computation is  required to disambiguate object motion. Cortical cells are also selective for speed (Orban et al., 1983).   Both the motion selectivity and  binocularity suggest a general hypothesis for  cortical function: it  links information likely  to have a single environmental cause for  subsequent extra-striate processing.

Spatial filtering

Overview of spatio-temporal filtering functions

Spatial frequency filtering, temporal filtering- generic feedforward neural network models	(Lecture 3)
		Why spatial filtering?
			edge detection 
			cortical basis set and economic representations - self-organizing neural networks (Lecture 14)
		Why temporal filtering?
			transient detection
			economic representations
	Predictive coding? - A key theme in information processing by the brain

Beginning with the psychophysical results of Campbell and Robson  (1968), and continuing with studies of the spatial and temporal frequency  selectivity of simple and complex cells, there has emerged a detailed picture of how images may be processed in the visual cortex. Let us look at spatial frequency in detail with a view to understanding its computational function in vision.

The convolution model for neural networks

Spatial filtering can be realized as a linear feedforward network. This is how we modeled lateral inhibition. Point-wise non-linearities, and response normalization need to be added on to improve models' accounts of the data.
Recall our model of lateral inhibition. There are both ON-center and OFF- center types of ganglion cells, specified by the weights.  We noted that we could quantitatively model an array of ganglion cell responses as a matrix operation acting on an input image vector. 
We didn't make a deal of it at the time, but over small regions of space, neural image processing can be approximated as homogeneous. Thus we assumed "shift-invariance" (outputs are shifted linearly with shifts in inputs) which implies a particular structure and properties to the weight matrix--i.e. in 1D, the rows are identical except for progressive shifts as we go from one row to the next.Thus the rows the weight matrix are just shifted versions of each other. In other words, the basic properties of spatial filtering are reflected in the weight structure of a single neuron. The receptive fields for other neurons are basically the same, except for position. From a mathematical point of view, if space is treated as continuous, such homogeneous linear spatial filtering is called "convolution".

Let (Subscript[r, k,l]) be the response (in spikes/sec) of a ganglion cell at x-y location (k,l). The average response, to a first approximation, is determined by the weighted sum of the inputs, (Subscript[f, i,j]) at spatial location (i,j)

(Subscript[r, k,l])=(∑_(i,j))(Subscript[w, k,l; i,j])(Subscript[f, i,j])

If we assume spatial homogeneity, and thus shift-invariance:

(Subscript[r, k,l])=(∑_(i,j))(Subscript[w, k-i; l-j])(Subscript[f, i,j]). Or by suitable arrangement of rows and columns as matrix operation, r = W.g

In the continuous case,by the convolution integral:

r(x,y) = w*g = ∫∫w(x-x',y-y')f(x',y)dx'dy'

The fundamental computations in spatial filtering of input images are linear--image vectors are multiplied by a weight matrix. 
When it reached steady-state, the lateral inhibition network studied in Lecture 3 was essentially convolving the input with the weights. If you ever use a graphics package like Adobe Photoshop, you can easily convolve the image on the computer screen with any number of possible spatial filters (i.e. weight matrices). Mathematica has a built-in function ListConvolve[ ] that accepts as arguments an input vector and a "kernel". But we need to know how to specify the "kernel", i.e. the weights for the convolution.
So the questions are: 
	Can other neural systems be modeled as taking convolutions of their inputs? Yes. Simple cell responses are modeled as convolutions with a point-wise non-linearity.
	What is the structure of the effective weights for simple cell neurons in the visual cortex? Let's look at this more closely because it provides insight into issues of neural representation.

Basis set for representing visual information

Psychophysics and physiology
The results of masking, adaptation, and other psychophysical studies of spatial and orientation frequency selectivity in human vision are surprisingly consistent.
What is the form of the weights (Subscript[w, ij])=(Subscript[w, xy])?
A cortical basis set for images specifies the effective weights as a function of spatial position
	Both the psychophysical and neurophysiological data could be accounted for, in part,  by assuming the visual system  performed a quasi-Fourier analysis of the image. One possible model assumes that the visual system computes the coefficients (or spectrum) of an image with respect to the following basis set, called a Gabor set (Daugman, 1988):

The spectrum coefficients are represented by the firing rates of cells whose receptive field weights are represented by the above basis functions. In actuality, because as we saw earlier, simple cells behave more like linear filters followed by half-wave rectification, there should be two cells for each coefficient-- "on" and "off" cells). One difference between this basis set, and the Fourier basis set (i.e. the optical eigenfunctions) is that this set has a local spatial restriction because of the Gaussian envelope. A second difference, which has major implications for computation, is that the basis functions are, in general, not orthogonal. Graphs of these functions typically look like wave-packets:

Here we've plotted a one-dimensional slice (e,g, (Subscript[w, x])) through a sine, and cosine Gabor function. In two dimensions (with the standard deviation , and the x and y spatial frequencies equal to 1), we can visualize the receptive field weights as follows.

Visualizing the Gabor functions:

cgabor[x_,y_, fx_, fy_,s_] := 
Exp[-(x^2 +  y^2)/s^2] Cos[2 Pi(fx x + fy y)];
sgabor[x_,y_, fx_, fy_, s_] := 
Exp[-(x^2 +  y^2)/s^2] Sin[2 Pi(fx x + fy y)];

Relation of horizontal and vertical frequencies to orientation and oriented frequency

Make a list of the orientations, and center frequencies of the basis set

vtheta = Table[i1 Pi/4, {i1,4}];
vf = {.25, 1, 4, 8};

Plot the elementary basis functions with the width, s,  proportional to the reciprocal of spatial frequency. This maintains a constant bandwidth in octaves.

Table[DensityPlot[cgabor[x,y,vf[[i]] Cos[ vtheta[[j]] ], 
		vf[[i]] Sin[ vtheta[[j]] ],
		 1/vf[[i]] ], {x,-2,2}, {y,-2,2},
		PlotPoints->24, Mesh->False,ColorFunction -> "GrayTones",PlotRange->{-2,2}],
		{i, 2, 3}, {j, 1, 3}] // Short

We've discretized the above continuous specification of the basis set. This leaves several free parameters. Most models of detection and masking get by with about no more than 6 spatial frequencies, about 12 orientations (specified by the ratio of horizontal and vertical spatial frequencies), and two phases (cosine and sine) at each retinal location. A subset of neurons representing a particular spatial frequency bandwidth  makes up a spatial frequency channel. (Although there is neurophysiological evidence for pairs of V1 neurons having receptive fields with 90 deg phase shifted relative to each other, there is evidence against absolute phase--i.e. there is not a predominance of edge or bar type receptive fields. See Field and Tolhurst). One parameter still left unspecified is the standard deviation or spread of the Gaussian envelope. If large, this basis set approaches that of regular and  global Fourier analysis. The psychophysical data suggest that the standard deviation be such that the Gaussian envelope is about one cycle (at the 1/e point) of the sine wave. One cycle corresponds to about 1.5 octaves spatial frequency bandwidth.

Example: Visualizing receptive field weights as a basis set, with spatial scale decreasing in columns from left to right, and orientation changing from row to row:

Why a local Gabor-function representation of visual information?

Why  would the visual system have such a representation? We have several explanations. 
--One is that  encoding over multiple spatial scales is important for subsequent processing  that may involve edge detection, or stereoscopic matching, or specific kinds of recognition (e.g. low spatial frequency information is useful for providing "gist" information about a scene), and so forth.  
--A more abstract argument is similar to the famous Heisenberg uncertainty principle--in our case, if one wants to simultaneously represent spatial frequency and spatial location, there has to be a trade-off between these two.
The third explanation is in terms of economical encoding. This is a more general principle that we return to below and later in the context of self-organizing neural networks.
So-called "pyramid" schemes have been developed for computer vision, and have been applied in many vision areas, such as texture analysis and image compression.    (See Adelson, E. H., Simoncelli, E., & Hingorani, R., 1987; Simoncelli and Olshausen, 2001). Many of early attempts to understand visual cortical receptive fields in which the spatial filters have similar shapes across spatial scales,  were forerunners of modern wavelet theory (Simoncelli and Olshausen, 2001).

Edge Detection by Neural Networks

Because of the orientation selectivity of cortical cells, they have been interpreted as edge detectors. It is easy to see how a sine-phase Gabor function filter (1 cycle wide) would respond vigorously to an edge oriented with its receptive field. This type of receptive field behaves as a 2D smoothing operator followed by a first order directional derivative. There are various ways of "reverse engineering" the visual system to design edge detectors. Here is one that combines the outputs of two model simple cells.

Although one can build edge  detectors from oriented filters, simple cells cannot uniquely signal the  presence of an edge for several reasons. One is that their response is a  function of many different parameters. A low contrast bar at an optimal  orientation will produce the same response as a bar of higher contrast at a  non-optimal orientation.  There is a similar trade-off with other parameters  such as spatial frequency and temporal frequency. In order to make explicit the location of an edge from the responses of a population of cells, one would have to compute something like the "center-of-mass" over the population, where response rate takes the place of mass. Another problem is that edge detection has to take into account a range of spatial scales.  The cortical basis set does encompass a range of spatial scales, and in fact may be "self-similar" across these scales. See Koenderink (1990) for a theoretical discussion of "ideal" receptive field properties from the point of view of basis elements.

Motion Detection by Neural Networks - detecting orientation in space-time

Networks analogous to the Morrone-Burr detector (See Appendix below) have been proposed for detecting motion (Adelson and Bergen, 1985; Heeger et al, 1996; Simoncelli and Heeger, 1997). The intuition is that motion detection is detecting orientation in space-time.

Economical representations by neurons in primary cortex

We might expect something like Fourier analysis of the image to result in efficient coding because of  the close relationship between Fourier rotations and Principal Components Analysis (e.g. Appendix A, Andrews, 1983). Fourier coefficients for natural images tend to be uncorrelated. Some work has been completed toward a functional explanation for  the orientation and spatial frequency tuning properties of  cortical receptive fields based on the statistics of natural  images (Field, 1994), but the story is far from complete. Barlow has argued that a decorrelated representation of sensory information is important for efficient learning (Barlow, 1990).
There has been considerable work on the relationship between self-organizing models of visual cortex, and efficient coding of image information. For more on this, see:  Linsker, R. (1990), Barlow, H. B., & Foldiak, P. (1989), Olshausen and Field (1996), Simoncelli and Olshausen (2001). 
Olshausen and Field (1996) show that simple cell receptive field weights can emerge as a consequence of sparse coding. Imagine you don't know what the receptive field weights should be, and would like to write an algorithm to discover what the should be given simple principles. Olshausen and Field showed that if one enforces two constraints: 1) the image should be well-approximated by a weighted some of basis vectors ; 2) when representing natural images, the (squared) cooefficients (i.e. the projections of the image vector onto the basis vectors) should add up to a small number, then the receptive field weight structures emerge as a consequence.
We'll take a closer look at the topic of neural networks for efficient encoding in the next lecture (Lecture 14).

Extra-striate cortical visual areas

In lecture 1,  we saw that cells in the visual cortex send their visual information to an incredibly complex, and yet structured collection of extra-striate areas. Any hypothesized function of striate cortex must keep in mind what the information is to be used for. Two primary functions: object perception and recognition--"within object" processing, and spatial processing (between object, and view-object relations).

Spatial, action pathway 
	V1,MT,MST,LIP,...
Viewer-centered computations
"Where" (vs. "What")
("where" or "how" or "now")
Spatial computations,such as coordinate transformations for action

Object perception, recognition pathway ("what")
 	object perception, recognition pathway
 		V1, V2, V4, Posterior IT, Anterior IT, ... - generic feedforward neural network models PLUS feedback
 		Invariances required for recognition: 
 			photometric: illumination level, direction, shadows
 			geometrical: translation, size, orientation in depth
 			category-related: levels of abstraction
 		Binding problem:
 			grandmother cells, distributed codes, sparse codes - "binding by synchrony"

Just two cortical pathways? There are other proposals, such as for "tools", or for rapid coarse vs. slower fine processing.

Modeling large-scale neural systems & systems analysis

Much of the modeling of visual processing has been built on the tools that we've learned about. But there are many aspects of brain modeling that require additional tools and ways of thinking. 
Modeling information processing functions:

Feedback
		Information processing roles of feedback
		*Dynamical behavior
	Timing and sequences (e.g. speech, motor sequences)
	Dynamical issues for real-time control, visuo-motor control
	Large-scale architectures (e.g. Inter-area processing)
	*Handling uncertainty - Probabilistic models
Measuring and characterizing neural systems
	Linear and non-linear systems analysis, statistical and stochastic processes analysis (time series),...

Appendix

Morrone and Burr edge detector: combining sine and cosine phase filters

What kinds of computational function might complex cells serve?

Define the filters

cosinefilter[x_,sigma_,f_] := 
			Exp[-(x/sigma)^2] Cos[2 Pi f x]
sinefilter[x_,sigma_,f_] := 
			Exp[-(x/sigma)^2] Sin[2 Pi f x]

Plot[cosinefilter[x,.5,1],{x,-1,1}]

Define the input stimulus: an ideal edge

Plot[Sign[x],{x,-1,1},Axes->None]

Calculate the response of a bank of cosine filters to the edge

cr[x_] := 
NIntegrate[cosinefilter[(x - x1),.5,1] Sign[x1],{x1,-1.5,1.5}]

Plot[cr[x],{x,-1,1},PlotPoints->10]

Calculate the response of a bank of sine filters to the edge

sr[x_] := 
NIntegrate[sinefilter[(x - x1),.5,1] Sign[x1],{x1,-1.5,1.5}]

Plot[sr[x],{x,-1,1},PlotPoints->10]

Combine the outputs from the two banks by squaring and adding:

Plot[cr[x]^2 + sr[x]^2,{x,-1,1},PlotPoints->10]

The peak of the squared sum is at the location of the edge.
Note the basic structure of the neural network for edge detection: 
	1. A linear operation using weights from a cosine function, followed by a squaring operation (how does this squaring relate to the sigmoidal non-linearities we have been using?)
	2. A linear operation using weights from a sine function, followed by a squaring operation
	3. A linear sum of the outputs

Edges in the real world of natural images don't necessarily correspond to a sharp ideal edge, but can be blurry and noisy. Nevertheless they are perceived to be sharp. 
Morrone and Burr went on to show that one could do the same operation with different sizes of filters (i.e. different values of sigma), and each time the peak of the above operation for an ideal edge occurs at the edge transition. But even for blurry edges, the larger scale filters will still find a point in the transition region. Thus by adding up a whole set of neural outputs over a range of scales, one could detect an edge. Another way of viewing this network is one that detects phase coherence. Fourier theory shows that a step function can be built up of sine-waves of various frequencies whose zero crossings all line up with (say positive slope) at the edge transition.

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