Experimental data on the accuracy and frequency of saccades are incorporated into a model of the visual world and eye movements to determine the spatial distribution of visual objects on the retina. Visual scenes are represented as sequences of discrete small objects whose positions are initially uniformly distributed and then moved toward the center of the retina by eye movements. We then use this model to investigate whether the distribution of cones in the retina maximizes the information transferred about object position. Assuming for simplicity that a single cone is activated by the object, the rate of information transfer is maximized at the receptor stage if the probability that a target lies at a position on the retina is proportional to the local cone density. Although qualitatively it is easy to understand why the cone density is higher at the fovea, by linking the cone density with eye movements through information sampling theory, we provide an explanation for its quantitative variation across the retina. The human cone distribution and the object distribution in our model visual world are shown to have the same general form and are in close agreement between 5- and 30-deg eccentricity.

*D*(

*x*) at a retinal location

*x*is proportional to the probability density

*P*(

*x*) of attended objects at the same visual location (this is shown in ). In the absence of eye movements, it would be most efficient to have a uniform cone distribution (assuming that objects appear uniformly in the visual field). Instead, when an interesting object attracts our attention, we shift our gaze to fixate it at the center. Eye movements thus allow for a small high-resolution region, which is associated with the fovea. We expect fixation movements primarily to shape the arrangement of cones rather than rods, as rods become saturated and so contribute no information in daylight conditions. However, we also compare the distribution of ganglion cells, which transmit information from both rods and cones.

*P*(

*x*), the probability distribution of objects in visual space, from experimental data on the accuracy and frequency of saccades. As we focus on information about positions of objects, this approach is not applicable to the center of the fovea, the function of which is mainly to collect information about objects. We therefore ignore gaze-holding movements, which affect only the center of the fovea. Indeed, within the fovea, we don’t expect the arrangement of cones to match the target distribution determined by eye movements, because other factors become dominant. First of all, close to the foveola, the angular separation of cones approaches the optical limit on the eye resolution, setting a bound to the maximum useful cone density (Wandell, 1995). Second, the size of the peak density region is constrained by factors such as the proximity of capillaries to the cones. Finally, fovea and periphery serve different functions: The fovea is used mainly for object recognition and the periphery for object detection. The distribution of cones optimized for object detection is unlikely to be that for object recognition.

*P*(

*x*) with the cone distribution, we assess the extent to which information is maximized. Our model gives a quantitative link between two separate bodies of experimental data: cone density measurements and saccade psychophysics (Figure 1).

*x,ϕ*) to parameterize points in the visual hemisphere, where

*x*is the angular distance from the center of the eye and

*ϕ*the angle around the circle at

*x*. Clearly, the centering movements of the eye imply that

*P*(

*x,ϕ*) must decrease with eccentricity

*x*.

*P*(

*x,ϕ*) depends on one’s representation of the visual world. Here we adopt a very simple one that we can model easily from available experimental data. So we represent a subject’s visual life as a sequence of point object locations in the visual hemisphere. Note that this does not mean that there is only one object at a time in the scene, but that we pay attention to only one object at a time. Each object location is either a sample from an initial uniform distribution,

*P*

_{0}(

*x,ϕ*), or the result of one or more saccades to that sample point.

*P*(

*x,ϕ*) is the distribution of the object locations both before and after saccades (Figure 2).

*x*, the target’s eccentricity, and ignoring variations with the azimuth

*ϕ*. This makes sense because cone density variations and gaze shifts are much more strongly associated with changes in

*x*. In fact, in this work, we consider mainly saccades along the horizontal meridian,

*ϕ*= 0.

*x*of a target that lies initially at

*y*comes from a probability distribution

*K*(

*x,y*). We allow for the probability of making a saccade toward an object,

*α*

^{n}(

*y*), to depend both on the eccentricity of the target,

*y*, and the number of saccades previously made toward the target,

*n*. Thus the decision to switch attention to a new target can occur without making a saccade to the previous one detected at

*y*, with probability 1–

*α*

^{0}(

*y*), or after

*n*saccades have brought the previous target to eccentricity

*y*, with probability 1−

*α*

^{n}(

*y*). In either case, if attention is not switched to a new target, a saccade is made toward the currently attended target. The probabilities

*α*

^{n}(

*y*) are unknown, but they are closely related to the proportion of natural saccades made to a target at

*y*, denoted

*f*(

*y*), which can be determined from psychophysical experiments, as can

*K*(

*x,y*) (see the following sections).

*φ*, can be obtained from

*K*(

*x,y*),

*f*(

*y*) and

*P*

_{0}(

*x*) as where

*ω*is a free parameter representing the probability to saccade to an object. We derive this equation in , using the fact that our assumptions above define a stationary Markov process. An alternative (and equivalent) approach is to obtain

*P*(

*x*) numerically by simulating the saccadic process (see ).

*K*(

*x,y*) and

*f*(

*y*) from experimental data.

*y*from a subject’s fixation axis and ask her to saccade toward it, the probability that the post-saccade eccentricity lies between

*x*and

*x*+

*dx*is given by

*K*(

*x,y*)

*dx*. Thus

*K*(

*x,y*) can be thought of as the probability distribution for the error

*x*of a gaze shift as a function of initial target eccentricity

*y*.

*y*and the mean error

*μ*, namely

*μ*(

*y*) =

*a*(

*y*−

*y*

_{0}), with

*a*typically in the range 0.1 − 0.2 and

*y*

_{0}in the range 5° − 10° (Becker, 1991). Saccades to targets at

*y*>

*y*

_{0}usually undershoot the target, whereas saccades to targets at

*y*<

*y*

_{0}typically overshoot (Figure 1B). The scatter in the error of saccades also increases linearly with increasing eccentricity (van Opstal & van Gisbergen, 1989), so we take the standard deviation for saccade errors

*x*to be

*σ*(

*y*) =

*b*+

*cy*. A linear fit on data (from Frost & Poppel, 1976) (see Table 1) gives the following values for the parameters:

*a*= 0.15,

*y*

_{0}= 8.5°,

*b*= 0.5°,

*c*= 0.1.

*y*, we model

*K*(

*x,y*), the distribution of saccade errors, as a Gaussian over

*x*with mean

*μ*(

*y*) and standard deviation

*σ*(

*y*). where

*A*(

*y*) is the normalization factor that ensures

*∫dxK*(

*x,y*) =1, and the second term is the contribution from overshoots (

*μ*(

*y*) > 0 for most

*y*) under the assumption of no asymmetry between nasal and temporal fixations.

*μ*and

*σ*from data on eye-only saccades, studies on eye-head gaze shifts show that their accuracy is very much the same as that of eye-only saccades (Stahl, 2001). Therefore

*K*(

*x,y*) describes the redistribution of target eccentricities under a general saccadic gaze shift, involving both eye and head movements.

*a*and

*y*

_{0}are derived from data in which

*y*= 5° is the smallest initial target eccentricity. Below 5°, the simple linear relation between target eccentricity and saccade error ceases to hold. Indeed, saccades with small amplitudes can be extremely accurate (Kowler & Blaser, 1995). A point where the above expression for

*μ*manifestly breaks down is

*Y*with

*μ*(

*Y*) =

*a*(

*Y*−

*y*

_{0}) = −Y, because for

*y*<

*Y*, the mean error would become greater than the initial target eccentricity. For our values of

*a*and

*y*

_{0}, we get

*Ȳ*= 1.11°, so we take

*y*= 1.5° as the lower end for the range of validity of our model (the lower limits of our integrals are accordingly set to 1.5°). This is not a problem because, as discussed in the Introduction, we only expect saccade accuracy to determine the cone distribution at the larger eccentricities.

*f*(

*y*) is an experiment in which three subjects were asked to wander freely outdoors while their eye movements were recorded (Bahill, Adler, & Stark, 1975). They found the relative frequency of eye-only saccades to decay exponentially with amplitude, that is

*f*

_{E}(

*y*) ∝ exp(−

*y*/7.6°). (This means a mean saccade amplitude of 7.6°). This value has been confirmed in a more recent experiment (Andrews & Coppola, 1999), in which subjects with their heads fixed viewed a variety of scenes and performed different visual tasks. The mean size of the saccades recorded was very close to 7.6°.

*f*

_{E}(

*y*) differs from

*f*(

*y*) in two ways. First, saccades with amplitudes greater than 10° usually involve both eye and head movements, and their total amplitude is therefore larger. So we expect

*f*(

*y*) to take higher values at large eccentricities. Second, the argument of

*f*

_{E}is saccade amplitude, whereas that of

*f*is initial target eccentricity. We bridge over these differences one at a time: First, we find

*f*

_{G}, the frequency of eye-head saccades as a function of amplitude, from

*f*

_{E}and data on eye-head coordination; then we deconvolve

*f*

_{G}(

*y′*) with

*K*(

*y*−

*y′,y*) to obtain

*f*(

*y*). We assume that

*f*(

*y*) and

*f*

_{G}(

*y*) will have a similar functional form to

*f*

_{E}(

*y*), and so we model them as a constant term plus an exponential decay, namely .

*y*

_{E}and

*y*

_{H}, associated with a gaze-shift

*y*=

*y*

_{E}+

*y*

_{H}can be determined from experiments on combined eye-head visual fixations (Stahl, 1999, 2001). In these experiments, the total gaze shift and the eye movement relative to the head were measured as subjects moved their eyes and head freely to targets at various eccentricities. The head component was found to follow a piecewise linear fit as a function of the total gaze shift: If

*y*<

*B*then only the eyes move and

*y*

_{H}=

*0*, and if

*y*>

*B*, the average head component increases linearly with

*y*.

*B, D*, and

*m*vary considerably among the subjects, and even for left and right fixations by the same subject (for details, see Stahl, 1999, 2001). Because we need to combine these results with the average eye-only saccade frequency from a different experiment, we use the values that give the best fit to the combined data from five subjects. We get

*D*≈ 0,

*B*= 7.8°, and

*m*= 0.84. We also find that the standard deviation of

*y*

_{H}is

*s*= 7.6° for all

*y*>

*B*.

*y*

_{E}=

*y*−

*y*

_{H}, given a total gaze shift of size

*y*is then The normalization factor

*N*(

*y*) is obtained by requiring that ∫

_{0}

^{y}

*p*(

*y*

_{E}|

*y*)

*dy*

_{E}=1 for

*y*>

*B*. The frequency distribution of total gaze-shifts

*f*

_{G}(

*y*) and that of saccade amplitudes

*f*

_{E}(

*y*

_{E}) are then related by

*f*

_{G}, we could try to discretize the integral and then solve the resulting linear equations through

*f*

_{G}=

*p*

^{−1}

*f*

_{E}/

*dy*. The problem is that

*p*is not invertible: All the elements in the left-most columns of

*p*are zero, because for large gaze shifts the corresponding saccades are always accompanied by head movements. Instead we find the parameters in Equation 4, which give the best fit of the function ∫

_{0}

^{90}

*p*(

*y*

_{E}|

*y*)

*f*

_{G}(

*y*)

*dy*to

*f*

_{E}(

*y*)=exp(−

*y*/7.6°). This gives

*d*

_{G}= 0.017 and

*λ*

_{G}= 3.2°.

*y*−

*y*’ and dismiss the second term in Equation 2. This is needed to distinguish between undershoots and overshoots — which have a larger amplitude — to the same target. Equation 8 can be inverted to obtain a numerical solution for

*f*(

*y*). However, the resulting

*f*(

*y*) remains very close to

*f*

_{G}(

*y*), so that

*d*≈

*d*

_{G}and

*λ*≈

*λ*

_{G}. (The reason

*f*

_{G}is very close to

*f*is that at large

*y*, where saccadic errors are large,

*f*(

*y*) is almost constant.)

*d*and

*λ*were obtained by averaging the results of the eye-head coordination experiments for all five subjects. Because there is considerable variation between subjects, we repeated the computations using the individual subjects’ data for

*P*(

*y*

_{E}|

*y*). This gives values of

*λ*ranging from 2.0° to 7.6° and values of

*d*ranging from 0.004 to 0.09. Because these parameters were obtained using the average

*f*

_{E}(

*y*), for three different subjects in another experiment, this does not necessarily mean that the actual

*f*(

*y*) varies between individuals. Indeed the data that show large individual variation in eye-head coordination come from experiments in which all subject made saccades to targets with approximately the same distribution of positions (Stahl 1999, 2001). Thus, it is reasonable to assume, as we have done, that in a natural environment

*f*(

*y*) will not vary widely even if people use different eye and head movement strategies. However, in view of the wide variation in eye-head coordination strategies, our estimate of

*f*(

*y*) cannot be very reliable. To accurately determine

*f*(

*y*), and to check whether it really does vary between individuals or not, we would need to have data on eye-head coordination and on

*f*

_{E}(

*y*) for the same subjects (or ideally, to measure

*f*(

*y*) directly).

*K*

^{i}(

*x,y*) for each type and decomposing

*f*(

*y*) into the corresponding contributions,

*f*

^{i}(

*y*). We would then replace

*K*(

*x,y*)

*f*(

*y*) in Equation (1) by the sum

*μ*

_{i}

*K*

^{i}(

*x,y*)

*f*

^{i}(y). This would require a lot of experimental work on the accuracy and frequency of the various types of naturally occurring saccades. What we have done in effect is to assume that the accuracy of most natural saccades is well approximated by that of normal saccades.

*P*(

*x*) by using saccades to auditory targets. These are no doubt less frequent than visual saccades, but they cannot be neglected: We do often turn our heads looking for the source of a sound. In fact, only auditory saccades can possibly occur to targets at eccentricities outside the visual field. It therefore seems plausible that auditory saccades may have a significant effect on the target distribution.

*K*(

*x,y*) for auditory saccades by inserting the parameters

*a*= 0.32,

*x*

_{0}= 0,

*b*= 0, and

*c*= 0.23 in Equation (2) for auditory saccades.

*f*(

*x*) we used for visual saccades was based on an experiment in which

*f*

_{E}(

*y*) was measured for all saccades, visual and auditory. Because we do not know what fraction of natural saccades are auditory, we calculate

*P*(

*x*) in the two extreme cases, one where all saccades are visual and one where they are all auditory. The true

*P*(

*x*) will be somewhere between these extremes. This calculation is intended as a way of assessing how sensitive

*P*(

*x*) is to experimental uncertainty in

*K*(

*x,y*).

*P*(

*x*) was to compare it with the density distribution of cones on the retina

*D*(

*x*). For maximal information, the two should be proportional (a proof of this is given in ). We restrict the comparison to the cone density and gaze shifts along a horizontal axis through the fovea. According to our model where

*P*

_{0}= 1/(2π) is the initial uniform target distribution,

*f*(

*y*) is the normalized frequency of saccades to targets at eccentricity

*y*, and

*ω*is a free parameter which measures the fraction of targets that elicit saccades.

*D*(

*x*) we mean the number of cones per unit solid angle in the visual field found at eccentricity

*x*. We obtain it from standard cones/mm

^{2}density measurements using the curves for retinal projection given in Drasdo and Fowler (1974).

*P*to

*D*gives

*ω*= 0.09. In Figure 3, we have plotted

*D*(

*x*) and the

*P*(

*x*) corresponding to

*ω*= 0.09. The two distributions have the same general form. They both show a peak around the center of approximately equal width, and they are in close agreement between 5° and 30°. However, they behave differently within the fovea and at large eccentricities. For large

*x*,

*P*(

*x*) is constant, as a consequence of the assumption

*P*

_{0}= constant, whereas

*D*(

*x*) keeps decreasing with

*x*(this decrease does not show in Figure 1, because there we plotted

*D*(

*x*) in units of cones/mm

^{2}rather than cones/solid angle).

*P*(

*x*), because the RGCs determine the rate at which information can be transmitted from the retina. If we take

*D*(

*x*) to be the density of RGC (from Sjostrand, Olsson, Popovic, & Conradi, 1999), the best fit of

*P*to

*D*gives

*ω*= 0.13, and the results are similar to those for cones (see Figure 4).

*P*(

*x*) that is smaller at low eccentricities and larger at high eccentricities. The small departure from

*P*(

*x*) for visual saccades can be explained as follows. The distribution

*f*(

*y*) peaks at small eccentricities, so there are few saccades to targets at very large eccentricities, which is where the difference between auditory and visual saccades is most pronounced. In reality, only a fraction of saccades will be made to auditory targets, so the departure from

*P*(

*x*) will be even smaller. We can conclude that if our estimate of

*f*(

*y*) is accurate, auditory saccades have only a small influence on the cone distribution.

*f*(

*y*). In this case, the differences between the calculated probability distributions are relatively large, compared to the difference between

*P*(

*x*) and

*D*(

*x*) in Figure 3. These curves provide an estimate of how large is the uncertainty in

*P*(

*x*) due to the limited data available with which to calculate

*f*(

*y*).

*P*(

*x*) and

*D*(

*x*) are roughly proportional across a broad range of eccentricities, the two show important differences at very small and at large eccentricities.

*P*(

*x*). A first step forward in this case would be to do experiments to obtain a better estimate of

*f*(

*y*). Equally important would be to develop a model that allows for complex visual scenes and takes into account their effect on attention.

*x*Discrepancy

*P*(

*x*) goes like

*P*

_{0}(

*x*) at large eccentricities. Thus, our assumption that the initial target

*P*

_{0}is independent of eccentricity could be at the root of the large

*x*discrepancy between

*P*(

*x*) and

*D*(

*x*).

*P*

_{0}(

*x*) = constant makes sense if the positions of objects that we pay attention to are unrelated. However, when viewing complex scenes there may be correlations among the positions of some of the objects present (e.g., different targets for saccades may be part of the same large physical object). For this to induce a persistent decline of

*P*

_{0}(

*x*) at large eccentricities, there would have to be correlations at very large scales (e.g., to have

*P*

_{0}(60) >

*P*

_{0}(80), there should be significant correlations between objects 60° apart). Another important factor that is not yet included in our simple model is that as the cone density affects the resolution of the eye, it can itself influence the number of interesting objects that are detected, and so

*P*

_{0}(

*x*) might not be independent of the cone distribution.

*P*

_{0}(

*x*) as an independent function, according to Equation 13 in ,

*P*

_{0}(

*x*) is related to the frequency distribution of saccades

*f*(

*x*) and probabilities for making saccades

*α*

^{n}(

*x*). Indeed, for large

*x*, Equation 13 implies that

*f*(

*x*) is simply proportional to the product

*P*

_{0}(

*x*)

*α*

^{0}(

*x*). Because

*f*(

*x*) is a constant for large

*x*(Equation 3), our assumption that

*P*

_{0}(

*x*) is independent of

*x*at large eccentricities is equivalent to assuming that

*α*

^{0}(

*x*), the probability of making a primary saccade to an attended object, is a constant at large eccentricities. Because

*α*

^{0}drops out of Equation 1, the resulting probability distribution

*P*(

*x*) is independent of that constant. To get

*P*(

*x*) to fit the data

*D*(

*x*), we could instead assume that

*P*

_{0}(

*x*) is proportional to the cone density at large eccentricities. We would then have

*α*

^{0}(

*x*) ∝ 1/

*D*(

*x*), also at large eccentricities so that the resulting

*f*(

*x*) is constant there. Our hypothesis that the distribution of cones maximizes information then predicts the form of both the initial distribution of visual objects before saccades (

*P*

_{0}(

*x*)) and the probability of making at least one saccade to an attended object(

*α*

^{0}(

*x*)) at large

*x*. However, these predictions are sensitive to the form of

*f*(

*x*) at large

*x*, which we have estimated from very limited data. At smaller eccentricities,

*P*(

*x*) depends on all the functions

*α*

^{n}(

*x*), which is why we had to ignore them and use the experimentally observable

*f*(

*x*).

*x*departure between

*P*(

*x*) and

*D*(

*x*) may indicate that the information collected by the eye is not maximized there. For example, if visual attention declines with

*x*, there may be little to gain by fine-tuning the cone distribution to the target incidence curve there.

*P*(

*x*) relies on six parameters derived from available experimental results — four describing the accuracy and precision of saccades, and two describing their frequency. The first four seem well established after many experiments on the errors of normal saccades. However, a good deal of uncertainty clouds the two parameters in the saccade frequency curve. Perhaps the main improvement to our model would come from new experiments to re-evaluate

*f*(

*y*).

*f*(

*y*) could have a significant effect on our results. It would clearly be preferable to have data on both eye-movements and eye-head coordination for the same subjects, or even better, to be able to measure the frequency of gaze shifts involving both eyes and head directly. In other words, the uncertainty in

*f*(

*y*) can only be resolved by further experiments.

^{2}on the horizontal meridian and 16,000 cones/mm

^{2}on the vertical meridian (Curcio et al., 1990).

*f*(

*y*)); (c) and the initial probability

*P*

_{0}(

*x*) could be different. Of these, there is some evidence that vertical saccades are less accurate than horizontal saccades (Becker, 1991).

*P*(

*x*) is smaller at small values of

*x*. However, vertical saccades are still much more accurate than auditory saccades, which, as we saw, cause only a very small change in

*P*(

*x*). Thus the lower accuracy of vertical saccades cannot account for the substantial horizontal-vertical asymmetry of the cone distribution. Once more we are led to conclude that to exploit the predictive power of our model, we need more detailed information about the relative frequencies of vertical and horizontal gaze-shifts . This said, we also expect a contribution to the asymmetry from

*P*

_{0}(

*x*).

*P*

_{0}(

*x*) and

*f*(

*y*). For instance, animals that live in open environments often have horizontal streaks of high cone densities in their retinas. Cone distributions are already known for many species, for example, various monkeys, (Packer, Hendrickson, & Curcio, 1989; Wikler, Williams, & Rakic, 1990; Andrade, da Costa, & Hokoc, 2000), squirrels (Kryger, Galli-Resta, Jacobs, & Reese, 1998), and pigs (Chandler, Smith, Samuelson, & Mackay, 1999), but a systematic study of saccade accuracy and frequency across different species is still needed.

*f*(

*y*),

*P*(

*x*) is not sensitive to uncertainties in

*K*(

*x,y*). But a different

*K*(

*x,y*) implies a different

*f*(

*y*) [see Equation 11 and Equation 13]). Therefore, we could use the model to predict the accuracy of eye movements in different species: We expect those with a narrower peak to have more accurate eye movements.

*D*(

*x*) with the

*P*(

*x*) arising from data on eye movements through different models

*P*

_{0}(

*x*) of the visual world. The model that gives the best fit between

*P*(

*x*) and

*D*(

*x*) would most adequately describe the visual life of the species in question. Perhaps our simple model, which is at best partially adequate to describe human vision, would give a better fit between

*P*(

*x*) and

*D*(

*x*) in animals with a more basic visual system.

_{t},

*n*), where

_{t}*t*labels time step, u

_{t}=(

*x*,

_{t}*φ*) is the position of the current target in the visual hemisphere, and

_{t}*n*is the number of saccades made so far to that target. This chain is the outcome of a Markov process, because the probability of occurrence of different states at time

_{t}*t*depends only on the state at

*t*− 1, and not on previous states or on

*t*itself. If

*p*(u′, n′; u, n) represents the transition probability from “target at

*u*after

*n*saccades” to “target at u′ after

*n*′ saccades,” then clearly

*p*(u′, n′; u, n) = 0 unless

*n*′ =

*n*+ 1 or

*n*′ = 0, corresponding respectively to making a further saccade to the same target or switching attention to a new target.

*P*(u,

*n*) of the different states from

*P*

_{0}(

*u*) and the transition probabilities. Then we can compute

*P*(

*u*) as the sum

*P*(

*u*)=Σ

_{n}

*P*(u,

*n*). However, we first reduce the problem to a one-dimensional one, in which we consider only the effect of saccades on

*x*. This allows us to replace the two-dimensional probability

*P*(

*x,φ*), for which

*σP*(

*x,φ*)sin(

*x*)

*dx dφ*= 1, by a one-dimensional probability density

*Π*(

*x*), for which

*σΠ*(

*x*)

*dx*=1. If

*P*(

*x,φ*) is independent of

*φ*, we have

*Π*(

*x*)=2

*π*sin(

*x*)

*P*(

*x*). This is a good approximation because a saccade causes little change in

*φ*, and so

*P*(

*x,φ*) will only vary slowly with

*φ*. Once we have a model for

*Π*(

*x*), we can revert to the two-dimensional target distribution by dividing by sin(

*x*), as is needed to compare with the cone density distribution.

*t*and

*t*+ 1: Here

*Π*

_{0}(x) = sin(

*x*) is how the uniform distribution,

*P*

_{0}= 1/(2π), looks in the one-dimensional reduction. The function

*α*

^{n}(

*y*) is the probability that an object at eccentricity

*y*elicits a saccade, given that

*n*saccades have previously been made to that target. And

*K*(

*x, y*) is the probability distribution of saccade errors, as described previously.

*α*

^{n}(

*y*) decreases with

*y*. For instance, when shown two identical targets at different positions, we are more likely to saccade to the one at a lower eccentricity (Findlay, 1980). Another indication comes from recording saccade amplitudes for subjects in a natural environment: The frequency of saccades decreases as a function of amplitude (Bahill, Adler, & Stark, 1975). Indeed

*f*(

*y*) captures the joint effect of the

*α*′s on

*Π*, and so we will not need their explicit form. It would be very difficult to determine the

*α*′s from experiments. For one thing, there are many other variables affecting salience, such as target size and luminosity. Hence

*α*(

*y*) has to be interpreted as an average over all such other parameters. Over a long enough time, we would expect that objects of all sizes, luminosities, etc., will appear at all distances, and

*α*

^{n}(

*y*) can be interpreted as the proportion of all objects at eccentricity that elicited a saccade.

*t*.) For an ergodic Markov process with transition probabilities

*p*(

*s*;

*s*′), the relative frequency with which a state

*s*occurs satisfies

*Π*(

*s*) =

*Σ*(

_{s},P*s; s*′)

*Π*(

*s*′). In our case this means

*x*after

*n*saccade is the sum over

*y*that it was at

*y*after

*n*− 1 saccades, that a new saccade is made toward it, and that the error is

*x*. The second is the probability that the target at

*t*is no longer the target at

*t*+ 1 times the probability that a new interesting object is found at

*x*. Finally, the probability that the target lies at eccentricity

*x*is given by To compute this sum, we start by eliminating the

*α*’s in favor of

*f*(

*y*), the proportion of natural saccades made toward targets at eccentricity

*y*. The relative frequency of primary and secondary saccades to targets at eccentricity

*y*is given by In general

*f*(

^{n}*y*) ∝

*α*(

^{n}*y*)

*π*(

*y, n*), with the same proportionality factor for every

*n*and so

*f*(

*x*) = Σ

*f*(

^{n}*x*) satisfies

*f*(

*y*) to be normalized whereas the integral over

*y*of the right hand side is less than one because

*α*(

^{n}*y*) < 1. So

*ω*, a number between 0 and 1, represents the proportion of objects detected that elicit a saccade. We have no data from which to derive

*ω*, and so we will treat it as a free parameter of the model. Putting all this together, we get And then

*P*(

*x*) =

*Π*(

*x*)/(2

*π*sin(

*x*)) is as in Equation 1:

*P*(

*x*), which hold irrespective of the specific form of

*K*and

*f*. Provided that

*K*has a Taylor expansion,

*K*(

*x,y*) =

*K*(0,

*y*) + higher order terms in

*x*, with

*K*(0,

*y*) ≠ 0, for small

*x*we have where we have used the relation sin(

*x*) ≈

*x*+ … for small

*x*. So for small

*x*, we have

*P*(

*x*)∼

*a*+

*b*

*x*

^{−1}. At large

*x*, we expect

*K*(

*x, y*) ≈ 0 (because eye movements do bring the fovea close to the target), which implies that

*P*(

*x*) ≈ (1 −

*ω*)

*P*

_{0}at large

*x*.

*P*(

_{0}*y*), and simulated saccades as moving to a new position from the distribution

*K*(

*x, y*). Given any choice of the

*α*s, the simulation gives the resulting

^{n}*f*(

*y*) and

*ω*as well as

*P*(

*x*). This allows us to compare the simulated distribution to the probability distribution computed using Equation 1 (see Figure 7). Of course, this simulation is mathematically equivalent to a numerical integration of Equation 14, so it does not give us any new information, but it does provide a check of the correctness of the analysis leading to Equation 15.

*D*(

*x*) and

*P*(

*x*).

*ξ*to allow for the fact that the visual system already “knows” that it has just made a saccade.

*n*> 0 some of the information counted in

*H*is redundant: Knowing the target position after

_{n}*n*− 1 saccades and that a saccade will be made to the same target, we can predict the position after the new saccade, even if only imprecisely. Thus the additional information gained by detecting a target after

*n*saccades is only where

*I*

_{n−1}(

*x, ξ*′) is the information about the position

*x*after the

*n*′th saccade contained in the detection of the target by cone

*ξ*′ before the

*n*′th saccade. We can note that this cannot exceed the information about

*x*contained in the exact position of the target before the saccade, which we call

*x*′ [i.e.,

*I*(

*x, ξ*′) ≤

*I*(

*x, x′*)]. The equality would be achieved if

*ξ*gave enough information to determine

*x*′ with perfect precision. While this is clearly not the case for the retina, the precision with which eccentricities can be determined by the eye is much greater than the precision of saccades, which have standard deviations of a few degrees. Thus, it is a good approximation to take

*I*(

*x, ξ*′) =

*I*(

*x, x*′).

*π*≡

_{n}*σπ*(

*x,n*)

*dx*is the probability that

*n*saccades have been made to the target so far, and

*π*(

*x,n*) is the probability that the target is at position

*x*and

*n*saccades have been made. Our aim is to determine the function

*ξ*(

*x*) (or

*D*(

*x*))that maximizes

*I*, so if we use the approximation

*I*(

*x, ξ*′) =

*I*(

*x, x*′), we can ignore the terms

*I*(

*x, x*′), which do not depend on

*ξ*or

*D*(

*x*) at all:

*π*(

_{n}*ξ*). If we consider a small region between

*x*and

*x*+

*Δx*, the number of cones in that region is

*D*(

*x*)

*Δx*. The probability that a target is in that region, given that

*n*saccades have been made, is

*Δx π*(

*x,n*)/

*π*, and so the probability that a particular cone detects the target is Combining this with Equations 17 and 20, we obtain where “…” indicates terms that are independent of

_{n}*D*(

*x*).

*D*(

*x*) that gives the maximum value of this

*I*, with the constraint that the total number of cones is fixed, so

*σD*(

*x*)

*dx*=

*N*. The optimal

_{cones}*D*(

*x*) is the function for which the following functional derivative is 0: where

*λ*is a Lagrange multiplier. Thus the optimal distribution of cones is .

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