Notes

A Section Through the Human Retina

Horizontal cells modulate the output of photoreceptors and play many roles in early visual processing contributing to contrast enhancement, color opponency, and the generation of center-surround receptive fields in cones. In the human retina there are approx. 1 million ganglion cells (per eye), which collect information about the visual world from bipolar and amacrine cells (retinal interneurons). There are many types of amacrine cells that are distinguished based on the way they respond to visual inputs. Then, the layer of photoreceptors responds to changes in membrane potential. Indeed, when the membrane changes potential, photoreceptors release neurotransmitters (All known NTs are present in the retina). Since they need nutrients, they are located closely to blood vessels, i.e., at the back of the eye in humans.

Rods & Cones

Phototransduction in Rods and Cones The retina has a very broad dynamic range, it functions under very varied illumination conditions.

• Rods: Vision in low light (e.g., night)
• Cones: Vision in stronger light (e.g., day). The graph below represents how changes the membrane potential to changes in light intensity shed on the photoreceptors.

Distribution of Rods and Cones: A View from the Side A different distribution of cones and rods is present on the retina. The cones work in day light and are capable of color vision, while the rods are saturated during day-time, but they provide uncolored night vision.

The figure above shows the distribution of rods and cones in the retina. We can notice that the fovea is rod-free and has a very high density of cones. The density of cones falls rapidly to a constant level at about 10-15 degrees from the fovea. Notice the blind spot which has no receptors. At about 15-20 degrees from the fovea, the density of the rods reaches a maximum. The color vision and resolution become quite poor in the periphery.

Response of a Cone to Light of Two Different Wavelengths In the picture below, light is flashed through one photoreceptor and changes in membrane potential are observed. What we can observe is that this cone responds differently to different wavelengths of photon light. Hence, the response of cones is wavelength-dependent. Indeed, there exist 3 types of cones:

• short-wavelength sensitive (blue cones)
• middle-wavelength sensitive (green cones)
• long-wavelength sensitive (red cones)

and only 1 type of rods. These different cones allow us to process different wavelengths and interpret them as colors. This concept is shown in the following figure, where the Principle of Univariance is explained.

Principle of Univariance

This principle states that one and the same visual receptor cell can be excited by different combinations of wavelength and intensity, so that the brain cannot know the color of a certain point of the retinal image. One individual photoreceptor type can therefore not differentiate between a change in wavelength and a change in intensity. Thus, the wavelength information can be extracted only by comparing the responses across different types of receptors. E.g., a rod could be equally excited by a low-intensity, well-tuned wavelength stimuli and by a high-intensity, badly-tuned wavelength stimuli. This also explains why rods (associated with vision in dark situation) tend to make us see everything in grey (but more or less bright) because cones are not active in such a situation, which instead allow us to disambiguate between the wavelength of the light stimuli.

There are cases of monochromats humans that have only one type of cones, which implies the loss of color perception. In such cases, humans can only perceive changes in light intensity. On the other side, there are women that are tetrachromats, which means they have 4 types of cones that allow the perception of "more" colors than a normal trichromats person.

Light Adaptation

• One of the main functions of the retina is light adaptation.
• "Cells get used to the light stimuli".
• The photoreceptors adapt very quickly to the average level of stimulation.
• It adjusts its sensitivity to the overall level of illumination.

Human Light and Dark Adaptation

The dark adaptation curve on the right shows the threshold (how much light) do you need to perceive something, i.e., how strong the light stimulus has to be in order for us to perceive it as a function of time when you walk in a dark environment. With the time passing we become more and more sensitive, which lowers the (log) threshold of the amount of light needed for us to see "something". The first dotted-line is the curve describing the adaptation of cones, followed by the adaptation of rods, which induce a significant decrease in threshold levels. The result is that the retina as a whole changes its sensitivity.

On the left we have the curve describing the minimal amount of light that is needed to see something. However, in this graph the horizontal axis defines the light intensity of the background instead of time. Thus, we can see that with a low-intensity of background light the threshold for perceiving a light is much lower than when the background has an "overwhelmingly" high-intensity level. Again, the dotted line describes the adaptation of cones, while the full-line describes the adaptation of rods.

The Jungfrau Viewed from Wengen We care for surface reflectance, not light intensity. Contrast is proportional to reflectance.

The intensity of light in different parts of the image is dependent on: reflectance*illuminance. So, assuming that the illuminance is always the same in an image, the intensity is then determined by the reflectance of the elements. The relevant parameter is the contrast, which tells us how much each part of the image differ in terms of intensity. In particular, the contrast remains constant to changes in illumination.

Cones Responses Adapt to Background Illumination This graph shows how individual cones respond to flashes of different light intensity presented in different background light conditions. It shows that cones become less and less sensitive the higher the light level in the background, which shows that the cone has adapted. Different parts of the retina can be adapted to different light intensities. Light adaptation is somewhat local in space.

The above effect can be explained by the local contrast, which changes across space.

Ganglion Cells Adapt to the Mean Light Intensity

In this graph, the response of an individual ganglion cell measured in terms of how many AP/s the cell fires under different background light conditions. When the background is dark the cell is quite responsive for low intensity stimuli and then adapts and reduces the firing rate for low-intensity stimuli.

Ganglion Cells have Center-Surround Receptive Fields

An Off-region strongly responds to reductions in light-intensity. The On-region responds to the opposite. In Center-Surround regions we can see that the two responses do not completely cancel each other.

Center-Surround Receptive Fields Enhance Edges Ganglion cells can be considered like filters, they do some spatial averaging of what is happening in the receptive field. The filter enhances the spatial locations associated with high contrast. Ganglion cells are good local contrast detectors. By changing the size of the receptive field (for example in the periphery RF are larger, while in the fovea are smaller), for example by reducing it as in the example to the right, the resolution is increased.

The Linear Model

A Model of the Ganglion Cell Receptive Field

Assumptions Implicit in the last 3 Slides:

• Receptive fields are difference of gaussians.
• Responses are a weighted average of the stimulus intensity, where the map of the weights is the receptive field.

Are these assumptions reasonable? The second assumption is true if and only if the cell is a linear system.

Linear systems L(x) obey:

• Homogeneity: L(a*x) = a L(x)
• Superposition (Additivity): L(x+y) = L(x) + L(y)

Homogeneity It means that if you have an input to the system x and you measure the response of the system to the input L(x), then this response satisfies: L(a*x) = a*L(x). The picture to the right presents this property by showing a plot of the retinal surface against the neural response, which evidences how the excitation elicited by an input scales linearly.

Superposition If you measure the response of the system to an input x: L(x) and consecutively to an input y: L(y). Then, if the inputs are presented together, the response of the system satisfies: L(x + y) = L(x) + L(y). The same discussion performed for the figure above applies here, which indeed shows the behaviour that the retinal surface should follow to satisfy the superposition property.

Linearity is often checked by using sinusoidal stimuli, because for a linear system:

• The responses to sinusoids are sinusoids.
• The dependence of response on stimulus frequency can be predicted from the shape of the receptive field.

So, if any of these two are false, the system is not linear.

Responses of a Linear System to Sinusoids Sinusoidal stimuli are a sequence of black/white bars that follows a sinusoidal distribution presented synchronously with different intensities.

A Sinusoid in 2-D: A Sinusoidal Grating

Predictions of the Linear Model with a "Difference of Gaussians" Receptive Field If you have a linear system, you can predict the particular response of each ganglion cell receptive field (center/surround/different of center and surround) to a specific spatial frequency, with the knowledge of its difference of gaussians.

Spatial frequency: within one degree of visual angle, how many sinusoidal cycles do we have?

Fitting the Model to the Data

In the figure we have measures of ganglion cells receptive fields responses to changes in spatial frequency, which were made on an on-center cell. They presented sinusoidal stimuli of different spatial frequency, from which we see that by increasing the spatial frequency we obtain a frequency which is optimal for the on-center and then drops again. The fits are good: the responses to sinusoids are predictable by a linear model with a "difference of gaussians" receptive field. Let's try another test of linearity. If it succeeds as well, we'll be happy with the model.

Spatial Frequencies

Making a Square Wave with Sinusoids

Fourier theorem tells us that the function at the bottom of the image (luminance as a function of space) can be approximated by summation of sinusoids. Square Waves in 2-D: by adding sine waves components we can obtain a square waves.

Responses of a Ganglion Cell to Edges

By making the assumption that the system is linear and that the receptive field is approximated by a "Mexican hat" we can see estimate the response of ganglion cells to edges. We can image an edge that is moving over the receptive field, we notice that the maximum response is obtained when the edge is bordering the on-center such that it covers the off-center and leaves the on-center fully illuminated. As soon as the edge covers the on-center, the response drops.

Chevreuil Illusion - Mach Bands If you look at the edges of the image to the right, you should notice that the contour to the edge appears brighter on the left than on the right. These slight increases/decreases are due to the shape (center/surround) of the ganglion cells receptive fields.

Sensitivity for Different Spatial Frequencies Spatial Frequency Tuning of a Ganglion Cell This represents again the responses to different spatial frequencies of the single ganglion cells receptive fields.

Spatial Frequency Sensitivity Curve of a Whole Brain Same kind of slide as before, but rather than being for a single cell, represents the overall spatial frequency sensitivity curve for a whole observer (person). To measure this kind of curve, you present on a screen a sinusoidal wave, then you make it flatter and flatter until it becomes undetectable (no more contrast respect to the background). However, if the frequency is increased over a certain frequency, the brain is incapable of detecting them and you will end up seeing it as a continuous stream rather than single sinusoidal waves.

Contrast Sensitivity Varies with Spatial Frequency On the x-axis the frequency of the sinusoidal waves increases, while on the y-axis the contrast decreases. The contrast sensitivity function represents what is in principle visible for an observer.

One Interpretation of the Contrast Sensitivity Curve

This picture shows spatial frequency against contrast sensitivity in a whole observer (Macaque monkey in this case). The curves without datapoints are the contrast sensitivity curves of a cortical cell/neuron, and what can be seen is that the cells are highly selective for spatial frequency. So, each individual cell sees only a small range of spatial frequency. Hence, the overall contrast sensitivity of an observer derives from the sum of many individual cells contrast sensitivities.