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Receptive Fields: Simple Cells and Tuning Curves

Receptive Fields

The jumping spider makes use of pattern recognition to distinguish prey from mate. Their eyes allow them to detect specific features or templates, specifically, the bar-shaped feature similar to the legs of other spiders (Land, 1969). Frisby and Stone (2010) discuss the jumping spider because they provide an excellent paradigm for how template matching works through a series of steps.

–       First, the original image is projected on to the retina of the spider’s eye

–       Second, the spider’s eye focuses on a specific feature, in this case the leg

–       Then the leg’s image is cast onto the retina, and receptors project this information to the primary visual cortex (V1)

–       Neurons in V1 receive inhibitory and excitatory input from the receptors in the retina

–       The receptors not blocked by the leg have increased activity and send an excitatory light signal

–       The receptors blocked by the leg have decreased activity and send an inhibitory light signal

–       The neuron gathers the total input and if it exceeds its threshold, firing indicates that a bar is present

Template matching, whether in a spider or human, relies on encoding of a pattern whose shape directly matches the input pattern to be detected. Striate cortex cells in V1 are found in all mammals. These cells receive input from the retinal fibres, and each cell is responsible for a limited patch of the retina (receptive field). Accordingly, cell types in the striate cortex are classified according to their receptive fields (Hubel and Wiesel, 1981). Striate cells are broken down into two parts: simple cells and complex cells.

Templates can, however, be impractical because the amount needed would be exponential resulting in a binding problem (Frisby and Stone, 2010). For example, if we have 18 templates for orientation sensitive to 18 sizes, which are sensitive to 18 shades, you can image the ridiculous amount of templates you would need. This problem is known as a combinatorial explosion.

Simple Cells

Simple cells are named simple because they can be simply mapped into excitatory and inhibitory sub-regions. Simple cells are optimally excited by bar-shapes, which is why it makes sense that simple cells are also called slit- and line – detectors. Slit-detectors respond to a light bar on a dark surrounding, and line-detectors respond to the opposite (Hubel, 1988).  Light on dark or vice versa is important because simple cells respond best to patterns that generate luminance differences, in other words, edges. However, because simple cells are so sensitive to edges, the orientation of a bar is important. The optimal stimulus for a simple cell, to emphasize luminance differences, is one that provides maximum excitation and minimum inhibition (Frisby and Stone, 2010).  To provide maximum excitation and minimum inhibition, different orientations are dealt with by different cells (Hubel and Weisel, 1962). The angle to which each cell is tuned is determined by the pattern of its excitatory and inhibitory regions. ‘Slit’ and ‘edge’ simple cells exist for a full range of orientations, which is reflected by the brain’s wiring; the fibres going from the retina to cortex differ depending on which orientation they represent (Frisby and Stone, 2010).

Population Code

As discussed in the previous section, bars maximally excite simple cells; however, cells still respond even when they are not maximally excited. If part of a cells visual field is activated, a partial response will be initiated. As such, context is vital to making sense of our visual field. For example, a non-vertical stimulus stimulates the vertically oriented receptive field just as well as the vertical but faint edge. In order to distinguish between the two outputs, they must be considered in context of the activities of cells examining the same retinal patch.

Fortunately, having sensitive simple cells makes interpolation between neighboring orientation measurements possible. This “talk” or interpolation between cells is known as a “population code.” Even though there are only simple cells for 18-20 different preferred orientations, we manage discriminations of less than <0.26 degrees (Frisby and Stone, 2010). Communication between cells allows us to discriminate when orientations vary very slightly, in the grey area between defined orientations. Populations of cells that have same preferred value of particular stimulus, like orientation, are called a channel. Scientists are able to measure the preferred orientation of cells by recording the symmetric pattern of firing rates.

Unfortunately, a major consequence of having a limited number of cells tuned to a large number of orientations is that cells taking each measurement need to be “broadly tuned” for “coarse coading.” Cognitive psychologists wanted to know how many channels are necessary to resolve the ambiguity problem. The answer is technically two but then the cells would be so broadly tuned that you would not be establishing any type of context. In addition, the tuning curve would turn far to slow to interpret anything. Unless the input to the cell coincided with the flank of the curve, there would be very little difference between the cells outputs. Hence, the brain uses a large number of broadly tuned cells with tuning so that the most sensitive part of the tuning curve can always be representative of one orientation with the less sensitive parts being representative of slight deviations from the optimal orientation (Frisby and Stone, 2010). 

Tuning Curve

The overall relationship between orientation of input edge and the output of the cell is called the tuning curve (Frisby and Stone, 2010). Tuning curves are important because they allow you to pinpoint which cells are sensitive to what orientation. The flank mentioned in the section above is where the slope of the tuning curve is the greatest; it also represents the point of greatest change in firing rate. This peak in sensitivity is found half way from the top of the curve. The trough or top of the curve is the least sensitive part because the slop is equal to zero. Regan and Beverly (1985) proved that humans do have peaks and troughs in their orientation sensitivity.

Seeing Maps

seeing

All maps, whether they are used in the brain or not, represent a mathematical function (v = f(u)) transforming one points in space (the domain, u) to another (the codomain, v). For a map to be accurate, it must be continuous, without any breaks. Also, every pair of nearby points must correspond to two nearby points in the codomain. For example, take the cities Sheffield and Leeds. They represent two geographical points. For a map to be accurate, Sheffield and Leeds on a map must correspond proportionally in terms of distance, direction, etc. with Sheffield and Leeds in real life. The map of South Yorkshire would be the codomain and the real cities would be the domain. In the visual system, these points could be two retinal points (domain) accurately reflected onto the striatal cortex (codomain).  In addition, a map of the brain must specify direction. Direction here is not used in the common sense, direction refers to continuity between domain and codomain. For example, mapping from retina to the striate is actually discontinuous; however, mapping from each half of the retina to the striate cortex is continuous, giving rise to the retinotopic map. If you map against the specified direction, it is called inverse mapping. An example would be mapping from the cortex to the retina, which is discontinuous. This is because most nearby points in the striate cortex correspond to nearby points in the retina; however, if two striate points are located on different ocularity stripes, inverse mapping is discontinuous (Blasdel, 1992).

Scientists argue that the reason the striate cortex maintains the retinotopic map is because it economises the length of nerve fibres. Nerve fibres are necessary for inter-hypercolumn communication, and if neighbouring points were spread out, wiring would become chaotic.  Of course, mapping does not occur only in the visual system. The brain has a myriad of maps including ones for auditory, touch and motor output and many copies of these maps exist. All of these maps, like the retinotopic map must be continuous, suggesting spatial organisation is key to a healthy, functioning brain. Unfortunately, because there are so many features of the visual system that need to be represented in the map, singularities arise.

Singularities are jumps in continuity (Frisby and Stone, 2010), and they are the result of the packing problem. To put it simply, the brain wants to pack all features of the visual system into the brain; to maximise efficiency, the brain wants all similar variations of a feature in one place. In other words, the brain wants continuity. However, as was discussed with the binding problem, there are far too many variations and features of our visual system to account for them all in every possible detail. Hence, some continuity must be compromised (Hubel, 1981). The brain’s map of the visual system is continuous with respect to retinal position, but the map is discontinuous with respect to orientation. However, the brain still attempts to keep similar preferred orientations close together to maximise efficiency to the limited extent it can (Frisby and Stone, 2010).

Another way of thinking of singularities and the packing problem is in terms of parameters. Two points, as discussed above, define each retinal position otherwise known as position parameters (x, y). Correspondingly, each retinal point must correspond to a point on the cortex (x’, y’). Even though our cortex exists in three dimensions, there is still a finite amount of space to store information and parameters limit us to representing information in 2D. As orientation brings its own parameter (theta), the brain has to represent x, y and theta in 2D.  This cannot be done without introducing discontinuities in at least one of the parameters because we are limited to two parameters. As the cortex must maintain a smooth map of the retinal map, discontinuities must be introduced into the representation of orientation. Based on this information the packing problem can be redefined in terms of parameters. The packing problem arises from attempting to pack all three dimensions of a 3D parameter space into a 2D one. As a solution to the packing problem, the cortex treats two of the parameters with varying priority, in this case the domain and codomain. Low priority is given to orientation, hence singularities.

visual

Representation of point singularities in the Visual Cortex. Each color represents a different radial phase corresponding to an orientation column. Date 2 December 2011 Source Own work Author Rtang3

As singularities exist for orientation, the topological index was introduced to describe the number of singularities. Specifically, the index tells us how orientation varies as we move around the centre of a singularity (Frisby and Stone, 2010). This can be done by drawing a circle around a singularity and moving clockwise. If the underlying representation orientation changes clockwise, the singularity is positive; however, if the orientation changes anticlockwise, the singularity is negative. Singularities in the striate cortex rotate no more than 180 degrees, so the singularity variable is always between + – ½. Hypothetically, a pinwheel is a full rotation, with an index of +1. To date, now pinwheels have been found in our visual cortex. Tal and Schwartz (1997) found that for any neighbouring singularities, you could usually draw a smooth curve between them. The remaining cells form columns along the curve with the same orientation preference (iso-orientation). In addition, Tal and Schwartz confirmed that nearby singularities have the same topological indices with opposite signs.

In addition to orientation, ocularity is another parameter that has to be represented in the striate cortex. Ocularity refers to the extent in which cells respond to our eyes. The brain as ocularity stripes meaning columns in the striate alternate between monocular and binocular cells. The stripes suggest that the brain wants to ensure that pairs of L and R stripes process every part of the visual field. Unfortunately, adding another feature parameter only furthers the packing problem. Researchers have found that the brain maximised economy by ensuring that each iso-orientation domain in the orientation maps tends to cover a pair of L-R ocularity columns; in other words, each orientation is represented for each eye (Hubel and Wiesel, 1971). Furthermore, the brain needs to perceive lines of different widths. The brain has solved this problem by having cells tuned to the same orientation, sensitive to various widths of spatial frequencies. As with orientation, representation of spatial frequency is continuous except for some singularities.  Lastly, directionality is packed together with orientation. Except for sudden changes in direction (180 degrees), the direction map is continuous. It overlays the orientation map; however, it does not effect the continuity of orientation as orientations defines two possible directions (Frisby and Stone, 2010).

Fortunately, colour does not add to the packing problem! This is because colour is represented exclusively at the centre of orientation singularities. At the centre of singularities, cells have no preferred orientation, so colour does not add any parameter. To fully understand how orientation, ocularity and colour come together, the polymap was constructed to show an overlay of all the parameters. Based on observations from a polymap, in addition to the specific wiring of the brain, some scientists argue that the cortex is not really trying to solve the packing problem. All the maps try to do is to minimise the amount of wiring the brain needs to employ (Frisby and Stone, 2010). In fact, Swindale et al. 2000 found that the cortex does attempt to maximise coverage, and if any small changes were made to the current mapping system, wiring efficiency would be reduced.