Coordinate Systems in Vision Neurophysiology Marcelo Gattass1 and Ricardo Gattass2 1Tecgraf Institute, Department of Informatic, Pontificial Catolic University of Rio de Janeiro, PUC-Rio, Rio de Janeiro, RJ, 21941-902, Brazil 2Program of Neurobiology, Biophysics Institute Carlos Chagas Filho, Federal University of Rio de Janeiro, Rio de Janeiro, RJ, 21941-902, Brazil 1 Abstract Abstract: Algorithms have been developed to: 1. Compare visual field coor- dinate data, presented in different representation systems; 2. Determine the distance in degrees between any two points in the visual field; 3. Predict new coordinates of a given point in the visual field after the rotation of the head, around axes that pass through the nodal point of the eye. Formulas are proposed for the transformation of Polar coordinates into Zenithal Equatorial coordinates and vice versa; of Polar coordinates into Gnomic Equatorial of double merid- ians and vice versa; and projections of double meridians system into Zenithal Equatorial and vice versa. Using the transformation of polar coordinates into Cartesian coordinates, we can also propose algorithms for rotating the head or the visual field representation system around the dorsoventral, latero-lateral and anteroposterior axes, in the mediolateral, dorsoventral and clockwise directions, respectively. In addition, using the concept of the scalar product in linear alge- bra, we propose new algorithms for calculating the distance between two points and to determine the area of receptive fields in the visual field. 2 INTRODUCTION Coordinates of the visual field are challenged by the same problems that car- tographers have faced to represent points and areas on the earth's surface. The flat representation of the surface of a sphere has been the focus of proposals and attempts to accurately represent meridians and parallels on maps with minimal graphic distortion. The invention of a geographic coordinate system is generally credited to Eratosthenes of Cyrene, who composed his Geography at the Library of Alexandria in the 3rd century BC (Fraiser 1970). To establish the position 1 of a geographic location on a map, a map projection is used to convert geodetic coordinates to plane coordinates on a map. For the representations of the earth, cartographers have preferred the zenith equatorial representation because it has a flat representation that preserves the surface area with aesthetically acceptable distortions. Studies of the visual system require that the relationship between visual field and its representation in neural centers is accurately established. Topographic representation of the visual field is an essential feature of the organization of the visual system, at several stages of sensory processing. The methods most fre- quently used are based on the classical techniques developed for human perime- try, i.e., the use of campimeters or tangent screens. As a result, three types of representation of the visual field (sphere) are used in vision neurophysiology. The polar coordinate system, the zenithal equatorial system and the tangent representation. Perimetry using Landolt type campimeters enables the exploration of a large extent of the visual field without requiring realignment of the apparatus. When using this method, polar coordinate systems are commonly employed to define location in visual space. This method has been widely used in the study of the topography of visual projections in both cortical and sub-cortical structures in which displacement of the stimulus is manually controlled. Alternatively, the use of a hemisphere allows direct drawing of receptive field or scotomas during the mapping procedure. Different types of coordinates can be drawn on the surface of the sphere allowing the use of polar or equatorial zenithal coordinates which uses meridians and parallels. The use of the tangent screen, on the other hand, greatly facilitates the construction of automatic systems for controlling stimulus position and dis- placement; however, with the tangent screen it is not possible to explore a large extent of the visual field without re-aligning the animal or the screen-projector system (see Bishop, Kozak and Vakkur, 1962). For the study of the visual system of primates we used the following strategy: The nodal point of the eye contralateral to the recording sites was placed at the center of a 120 cm-diameter translucent plastic hemisphere. The cornea was covered with a contact lens selected by retinoscopy to focus the eye at 60cm. The locations of the fovea and the center of the optic disk were projected onto the hemisphere. The horizontal meridian was defined as a line passing through both of these points and the vertical meridian as an orthogonal line passing through the fovea (Figure 1). The representation of the visual field in different visual areas was described in reference to these meridians. The motivation for deducting the algorithms presented here was based on the work of visual projections in the primary visual cortex of the opossum. In that study, the authors utilized a Landolt' campimeter to map the opossum's re- ceptor fields. They positioned the animals head in the center of the campimeter oriented in the horizontal plane through a line that passed through the auditory canal and the roof of the mouth behind the upper canines. When they first analyzed the data, the map of the primary visual area showed a substantial invasion of the ipsilateral visual field and the vertical meridian did not coincide 2 Figure 1: Fundus photograph of the left eye of a capuchin monkey showing the horizontal (blue line) and vertical meridians (yellow dashed line). The horizontal meridian passes through the center of the optic disc (od) and fovea (f). with the edges of the primary visual area. This finding was different from those of all mammals (and primates) studied that always had the projection of the contralateral hemifield in the primary visual cortex and the representation of the vertical meridian always coincided with the outer border of V1 (see Daniel and Whitteridge, 1961). When evaluating the opossum's posture of the head in march, we found that the animal lowered the head at an angle of about 40 degrees, expanding the bilateral visual field (Figure 2). In this position, the ver- tical meridian changed its position and its projection coincided with the outer edge of the primary visual area. With this position the visual field projection in the primary visual area was restricted to the contralateral visual hemifield, similar to what was observed in all other mammal's species. For this reason, we deduct algorithms for the transformations of the coordinates with the changes in the position of the head and we applied it to the study published in 1978 (Sousa et al., 1978). In this paper we describe three types of coordinates and we deduct equations to transform different coordinates of the visual field. We propose a series of algorithms to transform points in the visual field with the movement of the head and to calculate the receptive field area. A preliminary version of these algorithms was presents in abstract form previously (Gattass and Gatttass, 1975). 3 Figure 2: Opossum posture during march. A Didelphis marsupilis aurita was forced to walk over a wood board in the dark. The photography was synchro- nized to examine the position of the head during the march. The angle of the head was about 40 degrees. 4 Figure 3: 3D representation of the visual axis. Left, representation of the visual field in polar coordinates. Right, 3D drawing of the animal´s head and the relation between the visual axis. 3 METHOD The transformation of the location of the projections of points in the visual field was shown in the 3D plane and single plane trigonometry was used to correlate the new location to movement of the eye and the original location. Utilizing the concept of the scalar product in linear algebra, we deduct numerous algorithms to transform locations in the visual field in different spherical coordinate system. Thus, each problem was represented in 3D space and the algorithms for the transformations were deducted using 3D trigonometry. We present the necessary equations for the practical use of 3D conformal transformations of coordination of the visual field. 4 RESULTS 4.1 DEDUCTION OF AN ALGORITHM In this section, we will present the steps used to deduct the the algorithm for the rotation of the head in the sagittal plane. Figure 4 show the first step of the deduction. First we declare the variables. In this case we will work with the following definitions: x = Rsin α sin β (1) y = Rcos β (2) z = Rsin β cos α (3) x α = arctan (4) z 5 Figure 4: Changes in visual axis position with the movement of the head. A- movement in the sagittal plane, B- movement in the horizontal plane, and C- movement in the frontal plane. y β = arccos (5) R Then, we determine the equations for the rotation of the animal's head clockwise or of the stimulation apparatus counterclockwise in the sagittal plane, along the latero-lateral axis (Figure 5). Under these conditions we will have: x0 = xcos θ + y sin θ (6) y0 = ycos θ − x sin θ (7) z0 = z (8) in this case, we have: x0 α0 = arctan (9) z0 y0 β0 = arccos (10) R0 replacing the values of x0, y0 and z0, we have: x cos θ + y sin θ α0 = arctan (11) z −x sin θ + y cos θ β0 = arccos (12) R replacing the values of x, y and z as a function of α and β , we will have equations presented previously as [1] and [2], sin β sin α cos θ + cos β sin θ α0 = arctan (13) sin β cos θ 6 β0 = arccos cos β cos θ − sin α sin β sin θ (14) where: α varies from 0 to 360 degrees, and β varies from 0 to 180 degrees.