Speculations on Housecat Vision

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Watching Cats

When exploring the mathematics of vision, housecat eyes offer a richer opportunity for analysis than human eyes. Not only do cats have variable pupil shapes, they also use their eyelids to obscure their pupils.

I choose to speculate on the character of cat retinal diffraction patterns. In addition, I speculate on the character of neuron shapes and responses to determine what kind of value can be extracted from diffraction patterns.

Much information seems readily available without terribly sophisticated methods. Responses of neurons seem to be differential transients on biased equilibria and these are correlated by dendrites and convolved by axon terminal arbors.

If a gross assessment of housecat vision could be made it is this: For every pupil shape and lidding, and for every wavelength of visible light, the dendrites and axons form a sufficient transform to convert all possible diffraction patterns into a point-for-point map of the visual scene.

Gross Observations

I play hunting games with my houscat. She cooperates with me, sometimes playing predator, sometimes playing prey. It's all good fun for her, but gives me the opportunity to study her.

When she is hunting or knows I am hunting her, she opens her eyelids completely and dilates her pupils to a wide vertical oblong that approximates either an ellipse or the overlap of two circles. When she identifies me as prey or knows she has been chosen by me as prey, her pupils dilate to almost circular with still visible irises. At the moment of her own or my pounce her pupils dilate to full circular with irises dilated beyond the whites of her eyes for a fraction of a second.

When relaxed and wakeful, for instance when she is watching irrelevant scenes, she constricts her pupils to a narrow vertical oblong approaching a slit shape. When attending the scene, her pupils are still oblong. When ignoring the scene, her pupils narrow towards slit form. When I groom her, her pupils achieve full slit form. During these relaxed periods she sometimes partially closes her eyelids, partially obscuring her pupils with her upper eyelids.

When she sleeps, sometimes she is still a bit aware. Her eyelids cover her slit pupils from both above and below.

If I catch the attention of my sleeping cat, her eyelids open and her pupils rapidly dilate towards hunting/hunted shape. Sometimes, as if to fool me, she keeps her eyelids mostly closed but dilates her pupils.

So, to summarize, my cat's aperture has many reliable characteristics but has is no reliable shape. Her pupil has a reliable sequence of shape from slit to circular, but whatever shape it is has a secondary reliable sequence of lidding. This is nothing like what is typically understood about human apertures.

Typically, when vision is discussed, the seemingly circular aperture of the human eye pupil is used as an example. This circular aperture generates a well-known diffraction pattern called an Airy function with a central bright disk and limitless dim rings. The Airy disk radius of human vision extends the width of between 2 and 20 photoreceptors.

In my housecat, this analysis of vision mathematics is too blunt an instrument. Her slit pupil clearly ought to generate something like sinc(x).

The shape of her pupil is important, as will be discussed, but the when her eyelids partially obsure her pupils,

Appendix A: Testing Foundations

Many standard scientific teachings have statements that ought to be examined.

Flight of the cannonball

In physics, among the earliest problems given to students is to solve various equations for cannonball flight. The experimental conditions are given: "Let a cannonball be shot from a cannon at velocity V and angle theta. At what distance will it land?" Added to these are usually "ignore air drag", and to be correct one should add "assume uniform density per unit radius for both the cannonball and the earth". To be even more correct "assume there are no other sources of gravity" and "in the frame of reference of the earth" must also be added.

Ask almost anyone to give the mathematical name of the curve followed by the cannonball and almost universally they will say "parabola".

From the point of view of naval warfare in the 17th century, this is quite acceptable. But it is incorrect. The correct answer is "ellipse" and this has been known since Kepler since the cannonball is in orbit with the center of the earth as one of its foci. Physics itself is not in error, but the social traditions can lead to important errors.

Continuous or Discrete

First theorem of calculus (spatial):	$\frac{df(x)}{dx} = lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$
First theorem of calculus (temporal):	$\frac{df(t)}{dt} = lim_{\Delta t \to 0}\frac{f(t+\Delta t)-f(t)}{\Delta t}$

This formulation assumes that Δx and Δt approach zero. Let us test this assumption.

Fundamental to the theorem is the notion that Δx and Δt can be arbitrarily small. Max Planck (Planck units) determined that there is a minimum unit of space $1.616252 \times 10^{-35}meters$ and there is a minimum unit of time $5.39124 \times 10^{-44}seconds$ below which it is not possible to construe smaller division because of the uncertainty principle. An object may appear either here before or there after, but nowhere in between.

If this is true, the limit must always stop at these finite values and the approach to zero must terminate. Thus, if Planck units are valid, continuity cannot hold because both space and time form discrete meshes of finite increment.

Walther Nernst and Neuroscience

In 1920 Nernst received the Nobel prize in Chemistry. His Nobel speech/paper are a pleasure to read. He described how things distribute across a partition due to differential density pressure.

In neuroscience his work has been distilled down to the Nernst Eequation:

$V = \frac{RT}{zF} \times ln \frac{[C_o]}{[C_i]}$

where $[C i]$ is ion concentration inside and $[C o]$ is ion concentration outside a membrane.

This equation ought to cause alarm in the user. Under two conditions, this equation causes mathematical problems. If inside concentration drops to 0 (physically achievable) the fraction turns infinite. If outside concentration drops to 0 the natural log turns negative infinite.

This alarming equation is extended into the Goldman equation to account for the 3 dominant species of ion: Sodium, Potassium, and Chloride.

In thermodynamics, there are better formulations. Simply put, the probability of transfer across the partition boundary is the difference of probabilities of transfer in both directions. Both probabilities are controlled by exponential decay which has no mathematical pathology.

So, restated, the potential across a neuron membrane ought to be stated as the difference of two exponentials. A possible non-pathological foundation for this is the following naive equation:

$V \propto e^{[C_i] - [C_o]} - e^{[C_o] - [C_i]}$

Nernst wasn't wrong. But modern socially accepted adaptations need review.

Neuron voltage: membrane pumps or ordered water?

Appendix B: Traditional Math for Circular Pupils

Formula 1: $I=I_0*\left ( \frac{2 J_1(\rho)}{\rho}\right ) ^2$

Formula 2: $\rho=\frac{\pi q d}{\lambda R}$

where (in microns):

q is the radial distance from the optic axis on the focal plane (typically 1.0e0)
d is the aperture diameter (typically 3.0e3, smallest 1.0e3)
λ is the photon wavelength (typically 0.5e0)
R is the distance of focal plane to aperture(typically 1.0e4)

So for a 1mm pupil, a unit of x yields π * 1 * 1000 / ( 0.5 * 10000 ), or approximately 3/5. This means that the pupil diameter times 5/3 is the correct parameter for the wave function. Note that wave functions must be added before squaring for a proper intensity calculation. The splot command below shows this correct operation.

The following image results from these gnuplot commands:

square(x)=x*x
radius(x,y)=sqrt(x*x+y*y)
wave(mm,x,y)=2*besj1(5*mm*radius(x,y)/3)/(5*mm*radius(x,y)/3)
intensity(mm,x,y)=square(wave(mm,x,y))
set isosamples 21
splot square(wave(1,x-5,y+5)+wave(3,x,y)+wave(8,x+5,y-5))

In the plot, each unit of X and Y is approximately 1 micron, or the diameter of a photoreceptor in the foveal region of a human eye. Cursory examination of the above image shows that the diameter of the Airy disk for a human eye with a 1mm pupil is approximately 4 or 5 photoreceptors, with a 3mm pupil is 2 photoreceptors, and with an 8mm pupil is 1 or less photoreceptor.

This is important to note since the typical human pupil size is 3mm, and the blur radius of the Airy disk is greater than a single photoreceptor. Reading is done with a 1mm pupil and a blur radius of 2 or 3 photoreceptors.

Photoreceptor Transduction

Photoreceptors in all animals seek equilibrium. When light is absorbed, a chain reaction leads to chemical changes. The cell responds to these changes by seeking an intensity-appropriate equilibrium. A cell signal is a change in the neurotransmitter density in output synaptic junctions. The cell signals changes to equilibrium. Once in equilibrium, the cell does not signal. So, a photoreceptor in constant light does not signal.

Photoreceptors respond to light by changing voltage. At equilibrium, this voltage is poised on a biased curve. This curve results from the opposition of two exponential decays. A first approximation is the hyperbolic tangent:

Formula 3: $tanh(x)=\frac{e^{+x}-e^{-x}}{e^{+x}+e^{-x}}$

This transduction by opposed decay transforms the intensity function into a peak-widening, shoulder-narrowing function. The appearance of this is shown in this next image:

Initial Three Retina Layers

Photoreceptors make synapses with two kinds of cells, horizontal cells and bipolar cells. One type of horizontal cell averages the output of all photoreceptors. Horizontal cells also make synapses with bipolar cells. I have speculated that the bipolar cells measure the difference between photoreceptors and horizontal cells. If the light is going up on a photoreceptor and all its neighbors there is no difference measured by the bipolar cell. Only when the photoreceptor changes differ from average neighboring changes will the bipolar cell measure a difference.

Here is another gnuplot to show how these differences appear. In this image, the original diffraction pattern appears in red, the differences appear as green, and special functions were introduced to highlight the tips of the differences in magenta for the positive spikes and blue for the negative spikes. Note that these spikes are arranged very sharply constrained along circles around the center of the diffraction pattern. The deviation in radius of these spikes from the center is typically in the range of single-digit percents if not less.

square(x)=x*x
r(x,y)=sqrt(square(x)+square(y))
Wave(x,y)=besj1(r(x,y))/r(x,y)
Airy(x,y)=4*square(0.5*(Wave(x,y)+Wave(x+0.1,y+0.1)))
tAiry(a,x,y)=tanh(a*Airy(x,y))
avgtAiry(x,y,a,d)=(4*tAiry(a,x,y)+tAiry(a,x+d,y)+tAiry(a,x-d,y)+tAiry(a,x,y+d)+tAiry(a,x,y-d))/8
difftAiry(a,b,x,y)=tAiry(a,x,y)-avgtAiry(x,y,a,b)
threshold(a,b,c,x,y)=(difftAiry(a,b,x,y)>c)?1:((difftAiry(a,b,x,y)<-c)?-1:0)
set zrange[-1.1:+1.1]
set isosamples 30
splot Airy(x,y),threshold(2.5e2,1e-2,3e-4,x,y),-2+(r(x,y)<=3.79&&r(x,y)>=3.75),2-(r(x,y)<=3.55&&r(x,y)>=3.47)

So, there is an inner circular ring of positive spikes and an outer circular ring of negative spikes. These spikes occur exclusively where two photoreceptors best straddle a boundary between light and dark. These straddles identify locations of diffraction pattern edges with high precision. The width of the Airy disk diffraction pattern has no standing in the analysis. Only the boundary between light and dark where it is at its steepest.

Appendix C: Math for Cat Pupils

Sufficiency of Difference Image

In the analysis of human vision the speculation was that differencing yielded precision location information for the boundaries of diffraction patterns. It was also proposed that the bulk of the central feature plays no part.

Approaching Boundary Calculation

My housecat has a variable shape pupil. Calculating the diffraction pattern for each pupil shape and wavelength would take a great deal of effort and the result could only approximate the actual case. Here I propose a way to calculate just exactly what is needed at a very low computational complexity and cost.

The only information required to determine exactly where these boundaries are is the locations of the edge of the aperture. This is because the electric vector sum on the focal plane has relevant high spatial frequency contributions exclusively from the aperture edge. All other contributions average out.

The approach offered here is to pixelate the inner edge of the aperture itself and calculate the sum of electric vectors at the focal plane from all such pixels for a given wavelength. This should yield identical high precision boundary locations for the diffraction pattern.

So, from a 20mm circular pupil down to a 10mm by 0.5mm slit pupil I ought to be able to generate the diffraction patterns typically seen in my cat's eyes.

Initial Estimate

The solution for the circular pupil is already known. When my cat is ready to pounce, her pupil is 20mm across and so I can speculate that the diffraction pattern Airy disk is smaller than one photoreceptor. I consider this an extraordinarily clever way of maximizing photon absorption at photoreceptor sized discrimination.

Estimating is more difficult for her slit pupil. It is already known that a slit aperture yields a sinc(x) wave or

Formula 4: $I=I_0\left ( \frac{sin(x)}{x}\right )^2$

Note the resemblance to Formula 1. Instead of J0, a Bessel function, sine is used. These functions have substantially different frequency signatures. Also, sine operates primarily in the horizontal plane whereas J0 operates radially from the center. A way to view this is to subtract the local average from the point value. This is the same operation as described for bipolar cells differencing photoreceptors and horizontal cells. Here are gnuplot commands to illustrate this:

square(x)=x*x
wave(a,x)=2*besj1(a*x)/(a*x)
Airy(a,x)=square(wave(1.22*a,x))
sinc2(a,x)=square(sin(a*x)/(a*x))
Airy3(a,d,x)=(Airy(a,x-d)+Airy(a,x+d)+Airy(a,x))/3
sinc3(a,d,x)=(sinc2(a,x-d)+sinc2(a,x+d)+sinc2(a,x))/3
Airy4(a,d,x)=(Airy(a,x)-Airy3(a,d,x))/(Airy(a,x)+Airy3(a,d,x))
sinc4(a,d,x)=(sinc2(a,x)-sinc3(a,d,x))/(sinc2(a,x)+sinc3(a,d,x))
plot Airy4(1,.1,x),sinc4(1,.1,x)

Note how the first negative peak out from the center is synchronized, but the second and third negative peaks are relatively offset. The sinc function is understood to have peaks regluarly spaced. This illustrates that the Airy function does not have uniform inter-ring spacing. Instead the spacing becomes regular AFTER discounting the Airy disk radius.

To make this point more clearly, the same functions are given higher parameters. The Airy is turned upside down to bring its negative peaks close to the sinc negative peaks enabling visual examination of the frequency characteristics.

plot -1-Airy4(4,.4,x),1+sinc4(4,.4,x)

This leaves in question all the in-between oblong apertures. The general rule is that the narrower the aperture, the wider the diffraction pattern. So it could be said that the slit aperture has a very wide and short set of vertical stripes and the circular aperture has a very narrow and circular Airy disk/ring set, and that the in-between are vertically flattened and widened patterns reminiscent of Airy disk/rings. This will be tested with the calculation method described above.

Appendix Z: Notes

Light falls upon the retina in subtle patterns. Changes in light correlate with brain activity.

Photons

Photons start in a visible scene, propagate through the air

From a location in the visible scene, an event generates a photon. Sometimes this event is original such as a chemical bond forming. Other times it is the re-emission of an incident photon. From any given location, photons appear to project such that their electric vectors are synchronized.