On the cortical mapping function – visual space, cortical space, and crowding

The retino-cortical visual pathway is retinotopically organized: Neighborhood relationships on the retina are preserved in the mapping to the cortex. Size relationships in that mapping are also highly regular: The size of a patch in the visual field that maps onto a cortical patch of fixed size, follows, along any radius and in a wide range, simply a linear function with retinal eccentricity. This is referred to as M-scaling. As a consequence, and under simplifying assumptions, the mapping of retinal to cortical location follows a logarithmic function along a radius, as was already shown by Fischer (1973) and Schwartz (1977, 1980). The M-scaling function has been determined for many visual tasks. It is standardly characterized by its foveal threshold value, together with the eccentricity where that value doubles, called E2. The cortical location function, on the other hand, is commonly specified by parameters that are separately determined from the empirical findings in neuroscience. Here, the psychophysical and neuroscience traditions are brought together by specifying the cortical equations in terms of the parameters customary in psychophysics. The equations go beyond those published in the past in being more explicit and ready for application, and they allow easy switching between M-scaling and cortical mapping. A new structural parameter, d2, is proposed to describe the cortical map, as a cortical counterpart to E2, and typical values for it are given. The resulting cortical-location function is then applied to data from a number of fMRI studies. One pitfall is discussed and spelt out as a set of equations, namely the common myth that a pure logarithmic function will give an adequate map: The popular omission of a constant term renders the equations ill defined in and around the retinotopic center. The correct equations are finally extended to describe the cortical map of Bouma’s law on visual crowding. The result contradicts recent suggestions that critical crowding distance corresponds to constant cortical distance.


Introduction
One of the most beautiful organizational principles of the human brain is that of topographical mapping. Whilst perhaps universal to the brain, its regularity is most apparent for the three primary senses mediated through the thalamus -sight, hearing, and touchi.e., in retinotopy, tonotopy, and somatotopy, respectively. For the visual domain with which we are concerned here, the regularity of topography is particularly striking and is at a level that lends itself to mathematical description by analytic functions. The seminal papers by Fischer (1973) and Schwartz (1977Schwartz ( , 1980 derive the complex logarithm as a suitable function for mapping the location in the visual field to the location of its projection's in (a flat-map of) the primary visual cortex, by which the visual field's polar-coordinate grid gets mapped onto a rectilinear cortical grid. The log function's image domain -the complex plane -is reinterpreted thereby as a two-dimensional real plane. 1 As Schwartz explains in the two papers 2 , the rationale for employing the log function in the radial direction is that its first derivative is an inverse linear function, as implicit in the cortical magnification concept for the visual field as proposed by Daniel & Whitteridge (1961) (see the next section for explanations of the concepts, or Strasburger, Rentschler, & Jüttner, 2011, Section 3, for review). Expressed more directly, the integral of an inverse linear function (as implied in the cortical magnification concept) is the logarithmic function. Intuitively, summing-up (integrating over) little steps on the cortical map where each step obeys cortical magnification will result in the log mapping. Schwartz's (1977Schwartz's ( , 1980 papers with the complex-log mapping have become rather popular in visual psychophysics and visual neurophysiology 3 . Van Essen, Newsome & Maunsell (1984), e.g., use it for explaining the topography of the macaque's primary visual cortex and write, "Along the axis corresponding to constant polar angle, magnification is inversely proportional to eccentricity, and hence distance is proportional to the logarithm of eccentricity (x  log E)" (p. 437). Klein & Levi (1987) derive, from the log rule that, if vernieracuity offsets are assumed to have a constant cortical representation (i.e. one that is independent of eccentricity), vernier offsets will depend linearly on eccentricity in the visual field. Horton & Hoyt (1991) use it to point out that the well-known inverse linear function for cortical magnification follows from a log-spaced cortical map. Engel et al. (1997, Fig. 9, Fig  12) and Larsson & Heeger (2006) use the (real-valued) log function implicitly when they use an exponential for the inverse location function (which corresponds to a log forward mapping). Duncan & Boynton (2003) fit their fMRI activity maps for the V1 topology using Schwartz's complex-log mapping. Providing an easy-to-apply closed-form mathematical basis for these mappings with explicit parameter equations by deriving what I will call the cortical location function will be the first major goal of the present paper. While Fischer's and Schwartz's papers present the mathematical relationships and give examples for their application, Klein & Levi (1987) go further and provide an empirical link between basic psychophysical data in the visual field, and the cortical map. For characterizing the psychophysical results they use a concept they had developed earlier (Levi, Klein, & Aitsebaomo, 1984;Levi, Klein, & Aitsebaomo, 1985): The slope of the normalized thresholds-vs.-eccentricity function can be quantified by a single number, called E 2 . In an x-y plot of, say, the vernier threshold shown across eccentricity, that value is the (negative) X-axis intercept, or, alternatively, the (positive) eccentricity value at which the foveal threshold value doubles (illustrated below in Figure 1B). To draw the link to the cortical logarithmic map, Klein & Levi (1987) show that relationships become simpler and more accurate if both psychophysical and cortical data are not treated as a function of eccentricity (E) itself but of a transformed eccentricity, E*, referred to as effective eccentricity. E* is defined as E* = E + E 2 . In the visual field, the linear cortical magnification function thereby turns into a simple proportionality. In the cortical map, distances are then proportional to the logarithm of effective eccentricity, x  log (E+E 2 ). The approach is verified by showing the empirical data both as thresholds and in cortical units (Klein & Levi, 1987, Fig. 5) 4 . However, the papers discussed so far have not yet fully exploited the tight mathematical link between psychophysics and the cortical map for its empirical use. The basic mathematical form of the mapping function  log (E) or log (E+E 2 )  is drawn upon and made use of, but further parameters are usually left free to vary and to be determined by data fitting. The derivations in the present paper thus take the log mapping approach one step further. Unlike in Schwartz (1980), Klein & Levi (1987), and quite a few other papers discussed below, the parameters for the logarithmic map are here obtained by mathematical derivation from those in the visual field. For the latter, i.e., for the psychophysical characterization, Levi and Klein's E 2 concept is again the basis. We thereby arrive at a set of fully explicit equations that allow converting the psychophysical description by E 2 to a description in the cortical map. These equations are the message of the paper. The empirical data for the cortical maps (from fMRI or single-cell analysis) are then, in a next step, used to verify the correctness of those parametrical equations. This approach represents a more principled one. It further places additional constraints on the describing functions, thus adding to their reliability.
Since such derivations have been attempted before and have led to erroneous results (e.g. in our own writing), or stopped short of exploring the implications, care is taken here to present the derivations step-by-step, considering at each step what it means. Key result equations are highlighted by surrounding boxes for easy spotting, i.e. those that should be of practical use in describing the cortical map, or, e.g., for obtaining improved estimates for the foveal cortical magnification factor.
Instead of the complex log, we here consider the simpler case of real-valued, 1D mapping, where eccentricity in the visual field, expressed in degrees of visual angle along a radius, is mapped onto the distance of its representation from the retinotopic center, expressed in millimeters. The resulting real-valued logarithmic function shall be called the cortical location function. This implies no loss of generality; that function is easily generalized to the 2D case by writing it as a vector function. Compared to the complex log, the real function has the added advantage of allowing separate parameters for the horizontal and vertical meridian, required to meet the horizontal-vertical anisotropy of the visual field. 5 Once these relationships for the cortical location function are established, they need to be verified by empirical data. We use data from the literature and own data for this. It turns out that not only do the fits work excellently, but the constraints imposed by the parametric equations can be used for the long-standing problem of improving estimates for the foveal cortical magnification factor (M 0 ). In one section, it is further argued that the simplified version (x  log E) that is occasionally used in the fMRI literature needs to be avoided and the full version with a constant term added in the log's argument needs to be employed (i.e., x  log (E + c)).

Goals of the Paper
Three goals are thus pursued in this paper. Firstly, relationships are derived that translate the nomenclature of psychophysics to that in cortical physiology. The approaches are closely linked, and results on the mapping functions can be translated back and forth. The key equations for the cortical location function will be eq. (10) and eq. (16) plus (17). The usefulness of these equations is shown in a subsequent section.
Secondly, it is explored how the cortical function looks like in the popular simplified case with omitted constant term (Figure 2 below). That section can be skipped if one is aware that this seemingly simpler method will lead to suboptimal and misleading results. It is argued there that this might have been a good solution at a time but is not now when we have detailed knowledge of the cortical mapping close to the retinotopic center. The linear M-scaling function shown below in Figure 1 is accurate down to very low eccentricities, which is not the case for the simplified location function. It makes little sense to continue working with equations that do not, and cannot, apply over the whole range.
Conversely, the usefulness of the derived improved equations is shown in a section that explores practical examples for the cortical mapping function, with data from the literature. The graphs look like those in Figure 2 but have realistic parameter values. Comprehensive and concise mathematical descriptions have been derived before (Schira, Tyler, et al., 2010); the purpose here is to do so with explicit cortical parameters using the nomenclature from psychophysics (i.e., using E 2 ), in an easily applicable way.
Thirdly, and finally, these concepts are applied to the cortical map for visual crowding. Crowding is the phenomenon of impaired recognition of a pattern in the presence of neighboring patterns. It is the prominent characteristic of peripheral vision yet is equally important in the fovea (Strasburger, 2020). Remarkably, unlike typical perceptual tasks like acuity where critical size scales with eccentricity, crowding is mostly independent of target size (Pelli, Palomares, & Majaj, 2004;Whitney & Levi, 2011). Instead of size, the critical distance between target and flankers scales with eccentricity. This characteristic has become to be known as Bouma's law. It follows the same linear eccentricity law as depicted in Figure  1 below; this time, however, it refers to distance between patterns instead of size of patterns. Consequently, the E 2 concept can be applied in the same way. The cortical location function derived in the first part can then be used to predict the cortical distances that correspond to the flanker distances in Bouma's law.
Guide to reading the paper. The key parts of the paper are Section 2 on the cortical location function and Section 3 on its extension to Bouma's law in the cortex. These are preceded by Section 1 on the involved concepts and history, which can be skipped if one is already familiar with that background.
Regarding the structuring of Section 2, it starts with the natural way of specifying the cortical location function which is relative to the retinotopic center (Section 2.1). However, that center's exact location is notoriously difficult to find, and so for the neuroscientist it is of interest to be able to use some other reference location instead. Ways to do that, and their limitations, are discussed in Section 2.2 and 2.3. The first of these (2.2) describes the way to go; the latter (2.3), however, is intended to show how not to do it. It can be skipped if that is agreed upon. Next is a section on practical examples (2.4), with three cases from the literature, further a method for estimating the foveal magnification factor (M 0 ), and finally the introduction of a new metric (d 2 ) for characterizing the cortical map that is the equivalent of E 2 in the cortex.
Section 3 for Bouma's law in the cortex can be read mostly independently of the preceding sections if the underlying location function from Section 2 is taken for granted.
In the derivations, care was taken to phrase the steps to be easy to follow. Yet a legitimate way of using the paper is to just take the highlighted final results as take-home message. These would be eq. (9), (10) or (13) for the cortical mapping of the visual field (i.e. the cortical location function), eq. (8a) for the new parameter d 2 , eq. (17) for M 0 , and eq. (32) -(34), or (38) -(40), for the mapping of Bouma's rule onto the cortex.

Background and concepts
Peripheral vision is unlike central vision as Ptolemy (90-168) already noted. Ibn al-Haytham (965-1040) was the first to study it quantitatively. Purkinje (1787-1869) determined the dimensions of the visual field with his sophisticated perimeter. Aubert and Foerster (1857) started modern quantitative research on the gradual variation across visual field eccentricities. Østerberg (1935) did meticulous measurements of retinal rod and cone receptor densities across the horizontal meridian (Strasburger et al, 2011, Fig. 4); these are still a part of modern textbooks on perception (see Wade, 1998, Strasburger & Wade, 2015a, 2015b, and Strasburger et al., 2011).
Yet we still lack a grip on what the nature of peripheral vision is. The goal here in the paper is to draw the attention to the highly systematic organization of neural input stage, by deriving equations that describe its retino-cortical architecture. But before we delve into the nittygritty of the equations in the main part, here is some background and the concepts involved are reviewed, to see the equations in perspective.
Peripheral vs. central vision: qualitative or quantitative difference? Whether the difference between central and peripheral vision is of a qualitative or a quantitative nature has long been, and still is, an issue of debate. Early perceptual scientists suggested a qualitative (along with a quantitative) difference: Al-Haytham, in the 11 th century, wrote that "form […] becomes more confused and obscure" in the periphery (Strasburger & Wade, 2015a). Porterfield (1696-1771) pointed out the obscurity of peripheral vision and called its apparent clearness a "vulgar error". Jurin's (1738, p. 150) observation that complexity of objects plays a role 6 suggested that more than a simple quantitative change is at play (and reminds of crowding). Similarly, Aubert & Foerster (1857, p. 30) describe the peripheral percept of several dots as "something black of indetermined form". Yet, perhaps due to the lack of alternative concepts (like Gestalt perception) or by a prevailing interest in the role of vision in astronomy, the underlying reasons for the differences were, in the past, invariably ascribed to a purely quantitative change of a basic property: spatial resolution. Trevarthen's (1968) two-process theory of focal, detail-oriented central vision, vs. ambient, spaceoriented peripheral vision, might seem a prominent example of a qualitative distinction. However, with its emphasis on separate higher cortical areas for the two roles (perhaps nowadays dorsal vs. ventral processing), it also does not speak to qualitative differences in the visual field's low-level representation.
On the quantitative side, concepts for the variations across the visual field only emerged in the 19 th century. Aubert and Foerster's (1857) characterization of the performance decline with retinal eccentricity as a linear increase of minimum resolvable size -sometimes referred to as the Aubert-Foerster law -is still the conceptual standard. It corresponds to what is now called M-scaling Virsu, Näsänen, & Osmoviita, 1987) or the change of local spatial scale (Watson, 1987). However, by the end of the 19 th century it became popular to use the inverse of minimum resolvable size instead, i.e. acuity, in an attempt to make the decline more graphic (e.g. Fick, 1898). And, since the inverse of a linear function's graph is close to a hyperbola, we arrive at the well-known hyperbola-like function of acuity vs. eccentricity seen in most textbooks, or in Østerberg's (1935) figure from which they are derived.
The hyperbola graph. Graphic as it may be, the familiar hyperbola graph implicit, for example, in Østerberg's (1935) receptor-density graph does not lend itself easily to a comparison of decline parameters. Weymouth (1958) therefore argued for returning to the original use of a non-inverted size by introducing the concept of the minimal angle of resolution (MAR). Not only as an acuity measure but also as a generalized size threshold. Based on published data, Weymouth summarized how the MAR and other spatial visual performance parameters depend on retinal eccentricity (MAR, vernier threshold, motion threshold in dark and light, Panum-area diameter and others, see Weymouth, 1958, e.g. Fig. 13). Importantly, Weymouth stressed the necessity of a non-zero, positive axis intercept for these functions. 7 This will be a major point here in the paper; it is related to the necessity of a constant term in the cortical-location function discussed below. The architecture of neural circuitry in the visual field thus appears to be such that processing units increase in size and distance from each other towards the periphery in retinal space. To Weymouth, these processing units were the span of connected receptor cells to individual retinal ganglion cells. Different slopes, Weymouth (1958) suggested, might arise from differing task difficulty, a view not shared by later authors, however.
6 "when we divide [a string of digits] so as to constitute several objects less compounded, we can more easily estimate the number of figures" (Jurin, 1738, p. 150). Jurin reports more examples that would count as qualitative differences; see Strasburger & Wade (2015a).
7 "If the threshold as a function of eccentricity were a straight line passing through the origin (this does not occur and would require an infinite foveal sensitivity) the threshold would be a constant percentage of the eccentricity. It is here claimed that these curves approximate a straight line, but with a finite and positive intercept; this would lead to a decreasing percentage, falling, at first, rapidly but changing more and more slowly in the periphery. The 'constant' percentage relation noted by Ogle is therefore a consequence of the straight line relationship here discussed and is secondary and less useful mathematically. Although Ogle must have observed this linear relationship, he does not seem to have developed its consequences as is here done." (Weymouth, 1958, p. 109); italics added.
Cortical magnification. The linear spatial concept was thus well established when, in the sixties and seventies, the cortex was taken into the picture and the role of cortical representation included in theories on visual field inhomogeneity. Daniel & Whitteridge (1961) and Cowey & Rolls (1974) introduced cortical magnification as a unifying concept which, for a given visual-field location, summarizes functional density along the retinocortical pathway into a single number, M. Linear M was defined as the diameter in the primary visual cortex onto which 1 deg of the visual field projects (alternatively, areal M was defined as the cortical area onto which 1 deg² projects). Enlarging peripherally presented stimuli by M was shown to counter visual-performance decline to a large degree for many visual tasks (reviewed, e.g., by Virsu et al., 1987) and was thus suggested as a general means of equalizing visual performance across the visual field ). Yet this socalled strong hypothesis was soon dismissed; an early critique was expressed by Westheimer (1982) on the grounds that vernier acuity thresholds cannot be explained with these concepts. 8 The relationship between the early visual architecture and psychophysical tasks is still a matter of debate and, with it, the question why different visual tasks show widely differing slopes of their eccentricity functions (see Figure 1). Yet the variation of the cortical magnification factor with eccentricity is largely agreed upon: M decreases with eccentricityfollowing approximately an hyperbola -and its inverse, M -1 , increases linearly. Klein & Levi (1987) point out that by replacing eccentricity by effective eccentricity, E* = E + E 2 , the dependency turns into proportionality (i.e., twice E* leads to twice M -1 ). The value of M, and its variation with eccentricity, can be determined anatomically or physiologically (Schwartz, 1980;Van Essen et al., 1984;Tolhurst & Ling, 1988;Horton & Hoyt, 1991, Slotnick, Klein, Carney, & Sutter, 2001, Duncan & Boynton, 2003Larsson & Heeger, 2006;Schira, Wade, & Tyler, 2007; see Figure 1, reproduced from Fig. 9 in Strasburger et al., 2011). Assuming that low-level tasks like measuring the MAR reflect cortical scaling, M can also be estimated psychophysically Virsu et al., 1987).  Oehler, 1985). B. An illustration of the E 2 concept. E 2 is the eccentricity where the foveal value doubles, 9 or, the eccentricity increment that leads to an increment by the foveal value. It is also the negative x-axis intercept.
The empirical data all fit the linear concept quite well, but some slight deviations are apparent in the considered range of about 40° eccentricity. These are asides here but should be mentioned. The linear equation for the eccentricity function was often "tweaked" a little to accommodate for these deviations: Rovamo, Virsu, & Näsänen (1978) added a small 3 rdorder term, Van Essen et al. (1984) and Tolhurst & Ling (1988) increased the exponent of the linear term slightly, from 1 to 1.1. Virsu & Hari (1996) took a rather different approach and used a sine function, based on geometrical considerations. Only a part of the sine's period (one-eighth) was used, though, so that the function is still close to linear in that range. The latter function is interesting because it is the only one that -because it is bounded -can be extended to larger eccentricities, i.e. 90° and even beyond that (note that the visual field extends beyond 90°; see Strasburger, 2020f for review).
Elliptical field. Another deviation from simple, uniform linearity is the fact that the visual field is not isotropic: Performance declines differently between radii (this is used by Greenwood, Danter, & Finnie, 2017, to disentangle retinal from cortical distance). Accordingly, iso-performance lines for the binocular field are approximately elliptical rather than circular outside the central visual field (e.g. Wertheim, 1894, Harvey & Pöppel, 1972Pöppel & Harvey, 1973, see their Fig. 6). At the transition from the isotropic to the anisotropic field (in the plateau region of Pöppel & Harvey, 1973), the scaling functions ( Figure 1A) not only have different slopes along the different meridians but also necessarily deviate from linearity. Correspondingly, early visual areas are also anisotropic (e.g. Horton & Hoyt, 1991). The effect of anisotropy on the cortical magnification factor is quantitatively treated by Schira et al. (2007Schira et al. ( , 2010; their M 0 estimate is the geometric mean of the isopolar and isoeccentric M estimates. In the equations presented below, the anisotropy can be 9 Note that it is incorrect to say that the foveal value doubles every E2 increment, as is found now and then in the literature (cf. Strasburger, 2020).
accommodated by letting the parameters depend on the radius in question. However, different parameters (slopes) along the radii in the log mapping are not sufficient to adequately account for the anisotropy, as Schira et al. (2007Schira et al. ( , 2010 have shown. For preserving area constancy across meridians, these authors thus extend the model by a shear function (using the hyperbolic secans; Schira et al., 2010, eq. 6 and Fig. 2) such that mappings differ between meridians, with deviations from linearity on the vertical meridian (and meridians close to that) at around 1° eccentricity (see Schira et al., 2010, Fig. 2). The derivations presented here, for simplicity, do not include these refinements and for the vertical meridian are thus only approximate.
The E 2 concept. For a quick comparison of eccentricity functions for psychophysical tasks, Levi et al. (1984, p. 794) introduced E 2 -a value which denotes the eccentricity at which the foveal threshold for the corresponding task doubles ( Figure 1B illustrates this). More generally, E 2 is the eccentricity increment at which the threshold increases by the foveal value. As a graphic aide, note that this value is also the distance from the origin of where the linear function crosses the eccentricity axis (i.e., E 2 is the negative abscissa intercept in Figure 1B). Eq.
(1) below states the equation for the cortical magnification factor's eccentricity function using the E 2 parameter. The function's slope is given by the fraction , so when these functions are normalized to the foveal value their slope is E 2 -1 . Parameter E 2 thus captures an important property of the functions in a single number. A summary of values was reported, e.g., by Levi et al. (1984), Levi et al. (1985), Klein & Levi (1987), or more recently by Strasburger et al. (2011, Tables 4-6). These reported E 2 values vary widely between different visual functions. They also vary considerably for functions that seem directly comparable to each other (Vernier: 0.62-0.8; M -1 estimate: 0.77-0.82; Landolt-C: 1.0-2.6; letter acuity: 2.3-3.3; gratings: 2.5-3.0). On the other hand, E 2 can also be surprisingly similar for tasks that seem entirely unrelated, like for example the E 2 of 1.22° for the perceived travel extent in the fine-grain movement illusion (up to several degrees, depending on eccentricity; it is elicited by two briefly flashed points separated by only a few minutes of arc (Foster, Thorson, McIlwain, & Biederman-Thorson, 1981). Note also the limitations of E 2 : since the empirical functions are never fully linear for example, the characterization by E 2 , by its definition, works best at small eccentricities.
The two centers. There is an important difference in difficulty between locating the fovea's center and the cortical retinotopic center. Whereas, for psychophysical tests, the measurement of the foveal value is particularly simple and reliable, the opposite is the case for the anatomical foveal counterpart, M 0 -1 . The latter is considered the most difficult to determine in the cortical map and is mostly extrapolated from peripheral values. The consequences of this include a different perspective between the two fields on research regarding the map. We will come back to that below.
Using the E 2 parameter, the inverse-linear scaling function can be concisely and elegantly stated as (1) M -1 in that equation, measured in °/mm, is the inverse cortical magnification factor as defined above; M 0 -1 is that value in the fovea's center. The left hand ratio in the equation, M -1 /M 0 -1 , is the ratio by which a peripherally seen stimulus needs to be size-scaled to occupy cortical space equal to a foveal stimulus. So the equation can equally well be written as where S is scaled size and S 0 is the size at the fovea's center. From eq. (1), M 0 -1 can be considered the size-scaling unit in the visual field, and E 2 the locational scaling unit (i.e. the unit in which scaled eccentricity is measured).
Cortical mapping: As mentioned above, Fischer (1973) and Schwartz (1977Schwartz ( , 1980 proposed the complex log function for mapping the visual field to the cortical area. The key property of interest for that mapping, however, is the behavior along a radius from the fovea in the visual field, which corresponds to the simpler real-valued log function instead of the complex logarithm. This, then, maps the eccentricity in the visual field to the distance from the retinotopic center on the cortical map. Neuroscience papers often prefer to show the inverse function (i.e. mirrored along the diagonal with the x and y axis interchanged, thus going "backwards" from cortical distance to eccentricity), which is the exponential function shown schematically in Figure 2. Schwartz (1980) has discussed two versions of the function that differ in whether there is a constant term added in the argument; the difference is illustrated in the graph. The version without the constant is often considered simpler and is thus often preferred (or the full version is ignored). An important point in the following will be that that simplicity is deceiving and can lead to wrong conclusions (and more complicated equations). The proposed term location function can refer to both the forward (log) and backward (exp) version, which are synonymous.  (1973) and Schwartz (1977Schwartz ( , 1980. A version with, and another without a constant term (parameter b in the equation) are shown. The constant term's omission was intended as a simplification for large eccentricities but is physically impossible for the foveal center. The graph shows E as a function of d, which is an exponential; Schwartz (1980) discussed mainly the inverse function, i.e. for cortical distance d as a function of eccentricity E, which is logarithmic. Symbols in the paper: To keep a better overview, symbols used in the paper are summarized in Table 1. Some of those are in standard use and some are newly introduced in the remainder. The ratio M -1 /M 0 -1 in eq. (1) is readily estimated in psychophysical experiments as the size of a stimulus relative to a foveal counterpart for achieving equal perceptual performance in a low-level task. However, in physiological experiments M is difficult to assess directly, even though it is a physiological concept. Instead, it is typically derived -indirectly -from the cortical-location function d = d(E) (Figure 2). The function links a cortical distance d in a retinotopic area to the corresponding distance in the visual field that it represents. More specifically, d is the distance (in mm) on the cortical surface between the representation of a visual-field point at eccentricity E, and the representation of the fovea center. Under the assumption of linearity of the cortical magnification function M -1 (E), this function is logarithmic (and its inverse E = E(d) is exponential as in Figure 2), as shown by Fischer (1973) and Schwartz (1977Schwartz ( , 1980. And, since E 2 allows a simple formulation of cortical magnification function in psychophysics, as e.g. in eq. 1, it will be useful to state that equation d = d(E) with those notations. This is the first goal of the paper. The location function allows a concise quantitative characterization of the early retinotopic maps (symbols used in the paper are summarized in Table 1).
To derive the cortical location function, notice first that, locally, the cortical distance of the respective representations d(E) and d(E+E) of two nearby points along a radius, at eccentricities E and E+E, is given by M(E)•E. This follows from M's definition and the fact that M refers to 1°. The cortical magnification factor M is thus the first derivative of d(E), i.e., ( Conversely, the location d on the cortical surface is the integral over M (starting at the fovea center):

Cortical location specified relative to a reference location
Implicit in the definition of d or d 2 is the knowledge about the location of the fovea center's cortical representation, i.e. of the retinotopic center. However, that locus has proven to be hard to determine precisely. Instead of the center, it has thus become customary to use some fixed eccentricity E ref as a reference. Engel et al. (1997, Fig . 9), for example, use E ref = 10°. Larsson & Heeger (2006, Fig. 5) To restate eq. (6) or (10) accordingly, i.e. with some reference eccentricity different from E ref = 0, we first apply eq. (10) to that reference: where d ref denotes the value of d at the chosen reference eccentricity, e.g. at 3° or 10°.
Solving that equation for d 2 and plugging the result into eq. (9) or (10), we arrive at Or, expressed to the base e, we have which represents the location function expressed relative to a reference eccentricity E ref and its equivalent in the cortical map, d ref .
(One could also derive eq. (13) directly from eq. (6).) Note that if, in that equation, E 2 is taken as the reference eccentricity for checking, it reduces to eq. (10) as expected. So, E 2 can be considered as a special case of a reference eccentricity. Note further that, unlike the location equations often used in the retinotopy literature (Van Essen et al., 1984, in the introduction; Duncan & Boynton, 2003;Larsson & Heeger, 2006), the equations are well defined in the fovea center: for d = 0, the eccentricity E is zero, as it should be.
What reference to choose is up to the experimenter. However, the fovea center itself cannot be used as a reference eccentricity -the equation is undefined for d ref = 0 (since the exponent is then infinite). Thus the desired independence of knowing the retinotopic center's location has not been achieved  that knowledge is still needed, since d, and d ref , in these equations are defined as the respective distances from that point.
Equations (12) and (13)   In the shifted system -i.e., with d instead of d as the independent variable -eq. (6) for example becomes The equation might be of limited practical use, however (like eq. (6) from which it was derived), since the parameters M 0 and E 2 in it are not independent; they are inversely related to each other as seen in eq. (8) or (8a) (or eq. 17). That interdependency is removed in eq. (9) or (10), (which work from the retinotopic center), or eq. (13) (which used a reference eccentricity). The latter (eq. 13), in the shifted system becomes That equation now has the advantage over eq. (15)  Equations (16) and (17) are crucial to determining the retinotopic map in early areas. They should work well for areas V1 to V4 as discussed below. The connection between the psychophysical and physiological/fMRI approaches in these equations allows cross-validating the empirically found parameters and thus leads to more reliable results. Duncan & Boynton (2003), for example, review the linear law and also determine the cortical location function empirically but do not draw the connection. Their's and others' approaches are discussed as practical examples in the section after next (Section 2.4).
2.3 Independence from the retinotopic center with the simplified function? Schwartz (1980) had offered a simplified location function (where the constant term is omitted) which works at sufficiently large eccentricities. Frequently that was the preferred one in later papers as being seemingly being more practical. The present section will show how this approach leads astray if pursued rigorously; the section can be skipped if that is understood, i.e. is not required for following the subsequent sections.
The simplified version of the location function E(d ) omits the constant term in eq. (6) and those that follow from it (i.e., the "-1" in eq. 6 to eq. 16). I.e., the equation is fit instead to the empirical data, with free parameters a and b. The distance variable in it is then d as before, i.e., the cortical distance, in mm from a reference, that represents some eccentricity E ref in the visual field. Engel et al. (1997, Fig . 9 (20) Now that we have parameters a and b we can insert those in the above equation and rearrange terms, by which we get or, expressed more conveniently to the base e, This is now the simplified cortical location function (the simplified analog to eq. 16), with parameters spelt out. One can easily verify that the equation holds true at the two defining points, i.e. at 1° and at the reference eccentricity. Note also that, as intended, knowing the retinotopic center's location in the cortex is not required since d is defined relative to a nonzero reference. Obviously, however, the equation fails increasingly with smaller eccentricities, for the simple reason that E cannot become zero in that equation. In other words, the fovea's center is never reached, even (paradoxically) when we are at the retinotopic center. Equation (18) where d, as before, is the distance from the retinotopic center. Naturally, by its definition, the equation behaves well at the two defining points (resulting in the values E ref , and 1°, respectively). However, in between these two points the function has the wrong curvature (see Fig. 4 in the next section, fat black line), and at the fovea center (i.e., at d = 0), the predicted eccentricity -instead of zero -takes on a meaningless non-zero value E 0 given by As seen in the equation, the value depends on the chosen reference eccentricity and its representation, and also the cortical representation of 1° eccentricity, all of which should not happen. So, the seeming simplicity of eq. (18) that we started out from leads astray in and around the fovea (which, after all, is of prime importance for vision). The next section illustrates the differences between the two sets of equations with data from the literature. For the reasons explained above, the retinotopic center is left undefined by Larsson & Heeger (2006), and a reference eccentricity of E ref = 3° is used instead. The fitted equation in the original graph in their paper is stated as E = exp (0.0577 (d +18.0)), which corresponds to eq. (18) with constants a = 0.0577, and b = -d 1° = 18.0. Its graph is shown in Figure 4 as the thick black line copied from the original graph. It is continued to the left as a dotted blue line to show the behavior toward the retinotopic center. At the value of -b, i.e. at a distance of d 1° = -18.0 mm from the 3° representation (as seen from eq. 19), the line crosses the 1° point. To the left of that point, i.e. towards the retinotopic center, the curve deviates markedly upward and so the retinotopic center (E = 0°) is never reached.  The pink and the green curve in Figure 4 are two examples for a fit of the equation with a constant term (i.e., for eq. 16). The pink curve uses E 2 = 0.6° and d ref = 38 mm, and the green curve E 2 = 1.0° and d ref = 35 mm. Note that smaller E 2 values go together with larger d ref values for a similar shape. Within the range of the data set, the two curves fit about equally well; the pink curve is slightly more curved (a smaller E 2 is accompanied by more curvature). Below about 1° eccentricity, i.e. around half way between the 3° point and the retinotopic center, the two curves deviate markedly from the original fit. They fit the data better there than the latter and, in particular, they reach a retinotopic center. Of the two, the pink curve (with E 2 = 0.6°) reaches the center at 38 mm from the 3° point, and the green curve at 35 mm.

Practical
The center cortical magnification factor, M 0 , for the two curves can be derived from eq. (17), giving a value of 35.4 mm/° and 25.3 mm/°, respectively. These two estimates differ substantially from one another -by a factor of 1.4 -even though there is only a 3-mm difference of the assumed location of the retinotopic center. This illustrates the large effect of the estimate for the center's location on the foveal magnification factor, M 0 . It also illustrates the importance of a good estimate for that location.
There is a graphic interpretation of the foveal magnification factor M 0 in these graphs. From eq. (6) one can derive that M 0 -1 is equal to the function's slope at the retinotopic center. Thus, if the function starts more steeply (as does the green curve compared to the pink one), M 0 -1 is higher and thus M 0 is smaller.
The figure also shows two additional curves (black and brown), depicting data from Duncan & Boynton (2003), which are discussed below. To better display the various curves' shapes, they are shown again in Figure 5 but without the data symbols. Figure 5 also includes an additional graph, depicting the exponential function E = exp(0.063(d + 36.54)) reported by Engel et al. (1997). In it, d is again the cortical distance in millimeters but this time from the 10° representation. E, as before, is the visual field eccentricity in degrees. For comparison with the other curves, the curve is shifted (by 19.1 mm) on the abscissa to show the distance from the 3° point. The curve runs closely with that of Larsson & Heeger (2006) and shares its difficulties. New Law (eq. 16), E2=0.6; dref=38 mm New Law (eq. 16), E2=1.0; dref=35 mm parameters from Duncan & Boynton (2003) parameters from Duncan & Boynton (2003), dref=15.45 Engel et al. (1997 Figure 5. Same as Figure 4 but without the data symbols, for better visibility of the curves. The additional dash-dotted curve next to that of Larsson & Heeger's depicts the equation by Engel et al. (1997). Duncan & Boynton (2003) In addition to the curves just discussed, Figure 4 and Figure 5 show a further E(d ) function that is based on the results of Duncan & Boynton (2003). That function obviously differs quite a bit from the others in the figure and it is thus worthwhile studying how Duncan & Boynton (2003) derived these values. The paper takes a somewhat different approach for estimating the retinotopic mapping parameters for V1 than the one discussed before.

The approach of
As a first step in Duncan & Boynton's paper, the locations of the lines of equal eccentricity are estimated for five eccentricities (1.5°, 3°, 6°, 9°, 12°) in the central visual field, using the equation w = k * log (z + a). The function looks similar to the ones discussed above, except that z is now a complex variable that mimics the visual field in the complex plane. On the horizontal half-meridian that is equivalent to eq. (6) in the present paper, i.e., to an E(d) function that includes a constant term (parameter a) and with the retinotopic center as the reference. At these locations, the authors then estimate the size of the projection of several 1°-patches of visual space (see their Fig. 3; this is where they differ in their methodology from other approaches). By definition, these sizes are the cortical magnification factors M i at the corresponding locations. Numerically, these sizes are then plotted vs. eccentricity in the paper's Fig. 4. Note that this is not readily apparent from the paper, since both the graph and the accompanying figure caption state something different. In particular the y-axis is labelled incorrectly (as is evident from the accompanying text). For clarity, therefore, Figure 6 here plots these data with a corrected label and on a linear y-axis. Duncan & Boynton (2003) Fig. 4, showing the cortical magnification factor's variation with eccentricity drawn on a linear y-axis and with a corrected y-axis label (M in mm/°). Note that the equation proposed earlier in the paper (p. 662), M = 9.81*E -0.83 , predicts an infinite foveal magnification factor (blue curve). In contrast, the inverse-linear fit M -1 = 0.065 E + 0.054 proposed later in the paper (p. 666) fits the data equally well in the measured range of 1.5° to 12° but predicts a reasonable foveal magnification factor of 18.5 mm/°. The E 2 value for the latter equation is E 2 = 0.83. The additional green curve shows an equation by Mareschal et al. (2010) (see next section). (B) The inverse of the same functions. Note the slight but important difference at 0° eccentricity, where the linear function is non-zero and its inverse is thus welldefined.
The authors next fit a power function to those data, stated as M = 9.81*E -0.83 for the cortical magnification factor (see Figure 6). There is a little more confusion, however, because it is said that, from such power functions, the foveal value can be derived by extrapolating the fit to the fovea (p. 666). That cannot be the case, however, since -by the definition of a power function (including those used in the paper) -there is no constant term. The function therefore goes to infinity towards the fovea center, as shown in Figure 6 (dashed line).
Furthermore, E 2 , which is said to be derived in this way in the paper, cannot be derived from a nonlinear function (because the E 2 concept requires a linear or inverse-linear function). The puzzle is resolved with a reanalysis of Duncan & Boynton's Fig. 4. It reveals how the foveal value and the connected parameter E 2 were, in fact, derived: as an inverse-linear function which fits the data equally well in the measured range of 1.5° -12° eccentricity ( Figure 6, continuous line). From that function, the foveal value and E 2 are readily derived. Indeed, the two values correspond to the values given in the paper.
The distance of the isoeccentricity lines from the retinotopic center is not specified in Duncan & Boynton (2003). We can derive that from eq. (17), though, because M 0 and E 2 are fixed: With the authors' parameters (M 0 = 18.5 mm/° and E 2 = 0.83), the scaling factor  in that equation comes out as  = 1.03 (from eq. 16). From that, d ref = d 1.5° = 15.87 mm. As a further check, we can also derive a direct estimate of d ref from their Fig. 3. For their subject ROD, for example, the 1.5° line is at a distance of d 1.5° = 15.45 mm on the horizontal meridian. That value is only very slightly smaller than the one derived above. For illustration, Figure 4 and Figure 5 in the previous section also contain a graph for that value (thin black line). Conversely, with d ref given, M 0 can be derived from eq. (17) (or eq. 26), which gives a slightly smaller value of M 0 = 18.0 mm/°. The two curves are hardly distinguishable; thus, as previously stated, d ref and M 0 interact, with different value-pairs resulting in similarly good fits.
In summary, the parameters in Duncan & Boynton's (2003) paper: M 0 = 18.5 mm/° and E 2 = 0.83, are supported by direct estimates of the size of 1°-projections. They are taken at locations estimated from a set of mapping templates, which themselves are derived from a realistic distance-vs.-eccentricity equation. The paper provides another good example how the linear concept for the magnification function can be brought together with the exponential (or logarithmic) location function. The estimate of M 0 comes out considerably lower than in more recent papers (e.g. Schira et al., 2009; see Figure 7 below). Possibly the direct estimation of M at small eccentricities is less reliable than the approach taken in those papers. Figure 6 shows an additional curve from a paper by Mareschal et al. (2010) on cortical distance, who base their cortical location function partly on the equation of Duncan & Boynton (2003). Mareschal et al. (2010) state their location function as

Mareschal, Morgan & Solomon (2010)
The upper part of the equation is that of Duncan & Boynton (pink curve) and is used below an eccentricity of 4°. The green continuous line shows Mareschal's log equation above 4°, and the dashed line shows how the log function would continue for values below 4°. Obviously, the latter is not meaningful and is undefined at zero eccentricity, which is why Mareschal et al. switched to the inverse-linear function (i.e. the pink curve) at that point. The problem at low eccentricity is apparent in Fig. 9 in their paper where the x-axis stops at ½ deg, so the anomaly is not fully seen. For their analysis, the switch of functions is not relevant since eccentricities other than 4° and 10° were not tested. The example is just added here as an illustration that the new equations derived here would have allowed for a single equation, with no need for a case distinction.

Toward the retinotopic center
As discussed above, predictions on the properties at the retinotopic center depend critically on determining its precise location and thus require data at small eccentricities. Schira, Tyler and coworkers have addressed that problem in a series of papers (Schira et al., 2007;Schira, Tyler, Breakspear, & Spehar, 2009;Schira et al., 2010) and provide detailed maps of the centers of the early visual areas, down to 0.075° eccentricity. They also develop parametric, closed analytical equations for the 2D maps. When considered for the horizontal 10 direction only, these equations correspond to those discussed above (eq. 1 and eq. 16/17) 11 . showing their V1 data (red curve), but redrawn on double-linear coordinates. As can be seen, the curve runs close to an hyperbola. Its inverse is shown in Figure 7C, which displays the familiar, close-to-linear behavior over a wide range, with a positive y-axis intercept that corresponds to the value at the fovea center, M 0 -1 . From the regression line, M 0 and E 2 are readily obtained and are E 2 = 0.21° and M 0 = 47.6 mm, respectively. Note that a rather large value of M 0 is obtained compared to previous reports. However, as can also be seen from the graph, if one disregards the most peripheral point, the centrally located values predict a somewhat shallower slope of the linear function with a thus slightly larger E 2 and smaller M 0 value: E 2 = 0.33° and M 0 = 34.8 mm. The latter values might be the more accurate predictors for V1's central point. In summary, the derived equations provide a direct link between the nomenclature used in psychophysics and that in neurophysiology on retinotopy. They were applied to data for V1 ( Fig. 2) but will work equally well for higher early visual areas, including V2, V3, and V4 (cf. Larsson & Heeger, 2006, Fig. 5;Schira et al., 2009, Fig. 7). M 0 is expected to be slightly different for the other areas (Schira et al., 2009, Fig . 7)) and so will likely be the other parameters.

d 2 -a structural parameter to describe the cortical map
As shown in Section 2.1 (eq. 9 or 10), a newly defined structural parameter d 2 can be used to describe the cortical location function very concisely. Parameter d 2 is the cortical representation of Levi and Klein's E 2 , i.e. the distance (in mm from the retinotopic center) of the eccentricity, E 2 , where the foveal value doubles. Eq. (8) can serve as a means to obtain an estimate for d 2 . Essentially, d 2 is the product of M 0 and E 2 with a scaling factor. Table 2 gives a summary of d 2 estimates thus derived. The value of d 2  8 mm with E 2 = 0.33°, based on Schira et al.'s (2009) (2). d 2 is the cortical representation of E 2 and characterizes the cortical location function in a single value.
*M 0 was not estimated in that paper; the mean of the preceding M 0 values was used for the calculation instead.

Crowding and Bouma's Law in the cortex
The preceding sections were about the cortical location function; in the final section the derived location function will be applied to an important property of cortical organization: visual crowding.Whereas in the preceding, cortical location was the target of interest, in this section we are concerned with cortical distances.
As reviewed in the introduction, MAR-like functions like acuity generally change in peripheral vision in that critical size scales with eccentricity, so deficits can (mostly) be compensated for by M-scaling (as, e.g. in . For crowding, in contrast, target size plays little role (Strasburger, Harvey, & Rentschler, 1991;Pelli et al., 2004;Whitney & Levi, 2011). Instead, the critical distance between target and flankers scales with eccentricity, though at a different rate than MAR. This scaling characteristic of crowding is known as Bouma's rule or Bouma's law (Bouma, 1970;Strasburger et al., 1991;Pelli et al., 2004;Pelli & Tillman, 2008;Strasburger, 2020). The corresponding distances in the primary cortical map are thus governed by differences of the cortical location function as derived here in Section 2. Crowding's critical distance (or indeed any distance, including acuity gap size) is thus, in a sense, a spatial derivative of location. Pattern recognition, at even slight eccentricities is, governed by the crowding phenomenon and is largely unrelated to visual acuity (or thus to cortical magnification) (Strasburger et al., 1991;Pelli et al., 2004;Pelli et al., 2007;Pelli & Tillman, 2008;Strasburger & Wade, 2015a). For understanding crowding it is paramount to look at its cortical basis, since we know since Flom, Weymouth, & Kahnemann (1963) that crowding is of cortical origin (as also emphasized by Pelli, 2008).
A question that arises naturally in that context then is how the cortical equivalent of critical crowding distance varies across the visual field. Klein & Levi (1987) were the first to consider the cortical distance for position thresholds (in a vernier task), and conclude that it is approximately constant. That conclusion was based on the observation that taking the first derivative of Schwartz's (1980) log mapping using the constancy assumption will result in the well-known inverse-linear cortical magnification function. Conversely, their empirically determined position thresholds, when mapped by an inverse-linear cortical magnification function (with an E 2 of 0.6), turned out mostly constant across a wide range of eccentricities (cf. their Fig. 5). Later, Duncan and Boynton (2003), after estimating M based on Schwartz's (1980) log mapping and applying that to obtain cortical distances (see Section 2.4.2), show that, for scaled vernier tasks and scaled gratings, the cortical equivalents are again mostly constant (above 1.5° eccentricity; 2003, Fig. 4). Motter & Simoni, 2007;Pelli, 2008;Mareschal, Morgan, & Solomon, 2010) 12 .
Elegant as it seems, however, it will be shown here that the constancy assumption is most likely incorrect as a general rule, and only true at sufficiently large eccentricities. If stated as a general rule, it rests on the same equating of linearity and proportionality -i. e. the omission of the constant term -that gave rise to those cortical location functions that miss the retinotopic center (discussed in Section 2.3). Based on the properties of the cortical location function derived in Section 2, it will turn out that the critical cortical crowding distance (CCCD) increases steeply within the fovea (where reading mostly takes place) and reaches an asymptote beyond perhaps 5° eccentricity, consistent with a constancy at sufficient eccentricity. Accordingly, Pelli (2008) warns against extrapolating the constancy toward the retinotopic center. Remarkably (and to my pleasant surprise), I found out only after I had completed the derivations, that the analytic equation exposed below nicely agrees with the data presented by Motter & Simoni (2007, Fig . 7). In that figure, reproduced here in Figure 8B, only the more peripheral data show the presumed constancy.
Let us turn to the equations. Bouma (1970) stated what is now known as Bouma's law for crowding: where space  is the free space between the patterns at the critical distance 13 and b is a proportionality factor. Bouma (1970) proposed an approximate value of b = 0.5 = 50%, which is now widely cited, but he also mentioned that other proportionality factors might work equally well. Indeed, Pelli et al. (2004) have shown that b can take quite different values, depending on the exact visual task. Yet even though this factor can be quite different between tasks, the implied linearity of eq. (27) holds up in most all the reviewed cases. The law is thus best stated as saying that free space for critical spacing is proportional to eccentricity, with the proportionality factor taking some value around 50% (or 40%; Pelli et al., 2004), depending on the task (Strasburger, 2020).
Today it has become customary to state flanker distance not as free space but as measured from the respective centers of the target and a flanker. The critical spacing then remains largely constant across sizes (Tripathy & Cavanagh, 2002, and others This equation no longer represents a proportionality yet is still linear in E. Importantly, however, going from Bouma's equation (eq. 27) to that in eq. (28) reflects adding the constant term in the argument that we talked about in the preceding sections. And formally, that equation (28) is analogous to M-scaling as in eq. (2). Analogously to Levi and Klein's E 2 we thus introduce a parameter 2 E for crowding, as the eccentricity where the foveal value of critical distance doubles. Denoting the foveal value of critical distance by 0  , we get, from eq. (28): Obviously, that equation is analogous to eq. (1) and (2) that we started out with; it describes how critical distance in crowding is linearly dependent on (but is not proportional to) eccentricity in the visual field. In this respect, it thus behaves like acuity and many other spatial visual performance measures, just with a different slope and axis intercept.
With the equations derived in the preceding sections, we can now derive the critical crowding distance in the cortical map, i.e. the cortical representation of critical distance in the visual field. Let us denote that distance by  (kappa). By definition, it is the difference between the map locations for the target and a flanker at the critical distance in the crowding task: The two cortical locations d f and d t are, in turn, obtained from the mapping function, which is given by inverting eq. (6) above: Note that we stated that equation previously (Strasburger &Malania, 2013, eq. 13, andStrasburger et al., 2011, eq. 28), but, alas, incorrectly: a factor was missing there.
To explore this function, its graph is shown in Figure 8A and we look at two special cases. In the retinotopic center, equation (32) predicts a critical distance 0  in the cortical map of With increasing eccentricity,  departs from that foveal value and increases (provided E 2 > Ê 2 ), depending on the ratio 2 2/ E E . Numerator and denominator are the E 2 values for the location function and the crowding function, respectively (eq. 1 vs. eq. 29). They are generally different, so that their ratio is not unity.
With sufficiently large eccentricity, the equation converges to The latter expression is identical to that for the foveal value in eq. (33) except that E 2 is now replaced by the corresponding value Ê 2 for crowding.  Levi et al., 1985, or Strasburger et al., 2011). M 0 = 29.1 mm was chosen to give a good fit with this E 2 in Fig. 2. Foveal critical distance was set to  0 = 0.1° fromSiderov, Waugh, & Bedell, 2013, 2014. An Ê 2 = 0.36° would obtain with this  0 and the value of 4° = 1.2° in Strasburger et al., 1991; it also serves as an example for being a clearly different value than E 2 for the cortical magnification factor, to see the influence of the 2 2/ E E ratio on the graph. Cortical critical distance  starts from the value given in eq. (33) for the fovea center (around 2 mm) and converges to the value in eq. (34). (B) Data for the cortical critical distance from Motter & Simoni (2007, Fig . 7), showing the qualitative similarity for the dependency.
Importantly, note that kappa varies substantially around the center, by around two-fold between the center and 5° eccentricity with realistic values of E 2 and Ê 2 . This, as said above, is at odds with the conjecture that the cortical critical crowding distance is a constant (Motter & Simoni, 2007;Pelli, 2008;Mareschal et al., 2010). Pelli (2008) presented a mathematical derivation for the constancy, very similar to the one presented above -based on Bouma's law and Schwartz' (1980) logarithmic mapping function. The discrepancy arises from the assumptions: Pelli used Bouma's law as proportionality, i.e., in its simplified form stated in eq. (27) (its graph passing through the origin). The simplification was done on the grounds that the error is small outside the retinotopic center and plays little role (and the paper appropriately warns that additional provisions must be made at small eccentricities). Schwartz's (1980) (simplified) mapping function was consequently also used in its simplified form (also leaving out the constant term), for the same reason. With these simplifications the critical distance in the cortex indeed turns out as simply being a constant.
As should be expected, at sufficiently high eccentricities  is close to constant in the derivations given above (Figure 8). These equations (eq. 32-34) can thus be seen as a generalization of Pelli's result that now also covers the (obviously important) case of central vision.
That said, an interesting (though unlikely) special case of eq. (32) is the one in which E 2 and

Outlook
Where does this leave us? The early cortical visual areas are very regularly organized. And, as apparent from the fMRI literature reviewed above and also earlier literature, the spatial maps of early visual areas appear to be pretty similar. Yet variations of visual performance across the visual field differ widely between visual tasks, as highlighted, e.g., by their respective, widely differing E 2 values. E 2 estimates for cortical magnification, on the other hand, appear to be quite similar. It is a puzzle how different spatial scalings in psychophysics can emerge from a largely uniform cortical architecture. It would seem, however, that there can be only one valid location function on any radius. The equivalence between psychophysical E 2 and the cortical location function in the preceding equations would then only hold for a single E 2 . That value is, probably, the one pertaining to certain low-level tasks, and likely to those tasks that are somehow connected to stimulus size. In contrast, Ê 2 for critical crowding distance is an example for a psychophysical descriptor that is not related to stimulus size (Pelli et al., 2004); it rather reflects location differences, as discussed in Section 4. The underlying cortical architecture that brings about different psychophysical E 2 values (like Ê 2 ) could be neural wiring differences, within or between early visual areas.
To go further, one of the basic messages of the cortical-magnification literature is the realization that by M-scaling stimulus sizes some, but not all, performance variations are equalized across the visual field (Virsu et al., 1987;Strasburger et al., 2011, Section 2.5). In parameter space, these other variables would be said to be orthogonal to target size. For pattern recognition, pattern contrast is such a variable (Strasburger, Rentschler, & Harvey, 1994;Strasburger & Rentschler, 1996), and pattern contrast needs to be scaled independently from pattern size to equalize performance. Temporal resolution is another example (Poggel, Calmanti, Treutwein, & Strasburger, 2012). Again, differing patterns of connectivity between retinal cell types, visual areas, and along different processing streams might underlie these performance differences. The aim of the present paper is just to point out that a common spatial location function underlies the early cortical architecture that can be described by a unified equation. This equation includes the fovea and the retinotopic center, and has parameters that are common in psychophysics and physiology.