We prove new results on the existence, non-existence, localization and multiplicity of non-trivial radial solutions of a system of elliptic boundary value problems on exterior domains subject to non-local, nonlinear, functional boundary conditions. Our approach relies on fixed point index theory. As a by-product of our theory we provide an answer to an open question posed by do Ó, Lorca, Sánchez and Ubilla. We include some examples with explicit nonlinearities in order to illustrate our theory.

Our first aim in this paper is to deal with the maximum principle for subfunctions in an arbitrary unbounded domain. As an application, we next give a result concerning the classical Phragmén–Lindelöf theorem for subfunctions in a cone. For a subfunction defined in a cone that is dominated on the boundary by a certain function, we finally generalize the Phragmén–Lindelöf type theorem by making a generalized harmonic majorant of it.

We prove new results on the existence, non-existence, localization and multiplicity of non-trivial solutions for perturbed Hammerstein integral equations. Our approach is topological and relies on the classical fixed-point index. Some of the criteria involve a comparison with the spectral radius of some related linear operators. We apply our results to some boundary-value problems with local and non-local boundary conditions of Neumann type. We illustrate in some examples the methodologies used.

This study defined the amplitude and phase characteristics of rod- and cone-isolated flicker electroretinograms (ERGs) and determined how these responses summate to generate the nonreceptor-specific ERG. Full-field ERGs were obtained from six normally sighted subjects (age 26 to 44 years) using a four-primary LED-based photostimulator and standard recording techniques. The four primaries were either modulated sinusoidally in phase to achieve simultaneous rod and cone activation (ERGR+C; nonreceptor-specific) or in different phases to achieve rod-isolated (ERGR) and cone-isolated (ERGC) responses by means of triple silent substitution. ERGs were measured at two mean luminance levels (2.4 and 24 cd/m2), two contrasts (20 and 40%), and four temporal frequencies (2–15 Hz). Fundamental amplitude and phase for each condition were derived by Fourier analysis. Response amplitude and phase depended on the stimulus conditions (frequency, mean luminance, and contrast), however, for all conditions: 1) response phase decreased monotonically as stimulus frequency increased; 2) response amplitude tended to decrease monotonically as stimulus frequency increased, with the exception of the 24 cd/m2, 40% contrast ERGR+C that was sharply V-shaped; 3) ERGR phase was delayed (32 to 210 deg) relative to the ERGC phase; 4) ERGR amplitude was typically equal to or lower than the ERGC amplitude, with the exception of the 2.4 cd/m2, 40% contrast condition; and 5) the pattern of ERGR+C responses could be accounted for by a vector summation model of the rod and cone pathway signals. The results show that the ERGR+C amplitude and phase can be predicted from ERGR and ERGC amplitude and phase. For conditions that elicit ERGR and ERGC responses that have approximately equal amplitude and opposite phase, there is strong destructive interference between the rod and cone responses that attenuates the ERGR+C. Conditions that elicit equal amplitude and opposite phase rod and cone responses may be particularly useful for evaluating rod–cone interactions.

Rod and cone photoreceptors possess ribbon synapses that assist in the transmission of graded light responses to second-order bipolar and horizontal cells of the vertebrate retina. Proper functioning of the synapse requires the juxtaposition of presynaptic release sites immediately adjacent to postsynaptic receptors. In this review, we focus on the synaptic, cytoskeletal, and extracellular matrix proteins that help to organize photoreceptor ribbon synapses in the outer plexiform layer. We examine the proteins that foster the clustering of release proteins, calcium channels, and synaptic vesicles in the presynaptic terminals of photoreceptors adjacent to their postsynaptic contacts. Although many proteins interact with one another in the presynaptic terminal and synaptic cleft, these protein–protein interactions do not create a static and immutable structure. Instead, photoreceptor ribbon synapses are remarkably dynamic, exhibiting structural changes on both rapid and slow time scales.

During the course of light adaptation, the amplitude and implicit time of the human cone ERG change systematically. In the present study, the effect of adapting field luminance on these ERG changes was assessed, and the hypothesis that light adaptation of the rod system is the primary determining factor was evaluated. Cone ERG responses, isolated through the use of 31.1-Hz flicker, were obtained from two visually normal subjects, initially under dark-adapted conditions and then repeatedly for 30 min following the onset of each of a series of ganzfeld adapting fields with luminances that ranged from –1.2 to 2.1 log cd/m. The increase in flicker ERG amplitude and decrease in implicit time during light adaptation were greatest at the highest adapting field luminances. Photopically equivalent achromatic and long-wavelength adapting fields induced comparable increases in flicker ERG amplitude, while scotopically equivalent adapting fields had considerably different effects. This latter finding demonstrates that the rod system is not a major determinant of the adaptation-induced increase in cone ERG amplitude.

The leading edge of the a–wave of the electroretinogram (ERG) was evaluated as a measure of human cone photoreceptor activity. The amplitude of the cone a–wave elicited by flashes of different energy was compared to the predictions of a class of models from in vitro studies of cone photoreceptors. These models successfully describe the leading edge of the a–wave. Thus, the human cone a–wave can be used to test hypotheses about normal and abnormal cone receptors. The ability of the human cone to adjust its sensitivity in the presence of steady adapting lights was assessed by recording cone a–waves to flashes on adapting fields up to 3.9 log td in intensity and by comparing these responses to quantitative models of adaptation. The first 10 ms of the cone's response is little affected by field intensities up to 2.9 log td. The 3.9 log td field reduced the response to weak flashes by about a factor of 2.5 (0.4 log unit). This relatively small reduction in sensitivity can be attributed to a combination of response compression, pigment bleaching, and an adaptation mechanism that changes the gain without changing the time course. We conclude that either the human cones show relatively little adaptation or that they have an adaptation mechanism that involves a time-course change. That is, as we are limited with the a–wave to the first 10 ms or so of the cone's response, we cannot rule out a gain mechanism linked to a time-course change.

The effects of cobalt ions on 502-nm rod- and 575-nm cone-driven components of the b-wave of the electroretinogram were studied in the isolated frog retina. Addition of 100–150 μM cobalt initially caused a suppression of rod-driven responses and an enhancement of cone-driven responses. In the continued presence of cobalt, however, the rod-driven responses gradually recovered and the cone-driven responses became suppressed. These concentrations of cobalt had no effect on the rod- and cone-driven mass receptor potentials which were isolated in the presence of 4 mM glutamate. At higher concentrations of cobalt (1 mM or greater), both rod- and cone-driven b-wave responses were eliminated and there was no recovery in the continued presence of cobalt. The results suggest that cobalt has markedly different, time-dependent effects on signal transmission from rods and cones to second-order cells.

The sensitivity of rod- and cone-driven responses was studied in the isolated frog retina during the period of rapid dark adaptation following a conditioning flash which bleached a negligible amount of visual pigment. Following a conditioning flash, cone-driven b-wave responses were first enhanced and then depressed. The time courses of the enhancement and subsequent depression of cone-drive responses varies greatly with the intensity and wavelength of the conditioning flash, but were identical when the conditioning' flashes were matched for equal excitation of 502 nm rods. These changes in cone-driven response sensitivity were correlated with the desensitization and recovery of rods following the conditioning flash. When signal transmission from rods to second-order cells was interrupted by the addition of L-glutamate, the conditioning flash did not produce the above-described enhancement and subsequent depression of long-wavelength receptor potential responses. The suppression of cone-driven response therefore appears to be due to a synaptically mediated influence from 502 nm rods which is maximal when the rods are in the dark-adapted state, with little or no contribution from 433 nm rods, and no involvement of the pigment epithelium.

The aim of the present study was to determine the number of cone classes in the Xenopus retina. We examined the dimensions and staining properties of cones, utilizing two monoclonal antibodies, COS-1 and OS-2, developed by Szel and Rohlich (1985). Living cones also were reacted with the plant lectins peanut agglutinin (PNA) and wheat germ agglutinin (WGA) and with a fluorescent stilbene dye, DIDS, which binds selectively to red-sensitive cones (Kleinschmidt, 1991; Kleinschmidt & Harosi, 1992a, b). Three cone populations were distinguished based on differences in size and staining properties. Eighty-eight percent of all cones were stained strongly by COS-1, PNA, and DIDS, but weakly by OS-2. The group of cones stained by COS-1 had the largest mean dimensions of outer segment length, width, and oil droplet diameter. COS-1 negative cones were divisible into two groups: a subclass of miniature cones (approximately 4% total cones) was stained strongly by OS-2, PNA, and DIDS. The balance, constituting approximately 9% total cones, were of intermediate size, were not stained by PNA and reacted weakly to OS-2 and DIDS. WGA stained all cones. Large, COS-1 + cones appear to be red-sensitive and belong to the class of anion-tunable cone pigments. We suggest that the intermediate size, COS-1 negative cones are blue-sensitive based on the finding that blue-sensitive chromatic horizontal cells connect to them preferentially (Witkovsky et al., work in progress). The remaining class of miniature cones may be UV-sensitive, since another amphibian, the salamander, has been shown to possess red-, blue-, and UV-sensitive cones (Perry & McNaughton, 1991).

The receptive-field profile of the cone in cat retina was computed. The computation was based on (1) the known anatomical circuit connecting cones via narrow-field bipolar cells to the on-beta ganglion cell; (2) the known physiological receptive-field profile of the on-beta (X) cell at the corresponding eccentricity; and (3) a model in which the beta receptive field arises by linear superposition of cone receptive fields. The computed cone receptive field has a center/surround organization with a center almost as broad as that of the beta cell center. The cone surround is comparably broad to that of the beta cell but somewhat lower in peak amplitude. The problems to which the center/surround receptive field are the solution, namely, signal compression and noise reduction, apparently must be solved before the first synapse of the visual pathway.

We study a class of second-order nonlinear differential equations on a finite interval with periodic boundary conditions. The nonlinearity in the equations can take negative values and may be unbounded from below. Criteria are established for the existence of non-trivial solutions, positive solutions and negative solutions of the problems under consideration. Applications of our results to related eigenvalue problems are also discussed. Examples are included to illustrate some of the results. Our analysis relies mainly on topological degree theory.

Retinal cones are depolarized in darkness, keeping voltage-gated Ca2+ channels open and sustaining exocytosis of synaptic vesicles. Light hyperpolarizes the membrane potential, closing Ca2+ channels and suppressing exocytosis. Here, we quantify the Ca2+ concentration in cone terminals, with Ca2+ indicator dyes. Two-photon ratiometric imaging of fura-2 shows that global Ca2+ averages ~360 nM in darkness and falls to ~190 nM in bright light. Depolarizing cones from their light to their dark membrane potential reveals hot spots of Ca2+ that co-label with a fluorescent probe for the synaptic ribbon protein ribeye, consistent with tight localization of Ca2+ channels near ribbons. Measurements with a low-affinity Ca2+ indicator show that the local Ca2+ concentration near the ribbon exceeds 4 μM in darkness. The high level of Ca2+ near the ribbon combined with previous estimates of the Ca2+ sensitivity of release leads to a predicted dark release rate that is much faster than observed, suggesting that the cone synapse operates in a maintained state of synaptic depression in darkness.

In this paper, existence criteria for multiple solutions of periodic boundary value problems for the first-order difference equation are established by using the Leggett–Williams multiple fixed point theorem and fixed point theorem of cone expansion and compression. Two examples are also given to illustrate the main results.

In the natural environment, color discriminations are made within a rich context of spatial and temporal variation. In classical laboratory methods for studying chromatic discrimination, there is typically a border between the test and adapting fields that introduces a spatial chromatic contrast signal. Typically, the roles of spatial and temporal contrast on chromatic discrimination are not assessed in the laboratory approach. In this study, S-cone discrimination was measured using stimulus paradigms that controlled the level of spatio-temporal S-cone contrast between the tests and adapting fields. The results indicate that S-cone discrimination of chromaticity differences between a pedestal and adapting surround is equivalent for stimuli containing spatial, temporal or spatial-and-temporal chromatic contrast between the test field and the surround. For a stimulus condition that did not contain spatial or temporal contrast, the visual system adapted to the pedestal instead of the surround. The data are interpreted in terms of a model consistent with primate koniocellular pathway physiology. The paradigms provide an approach for studying the effects of spatial and temporal contrast on discrimination in natural scenes.

To better understand the evolution of spatial and color vision, the number and spatial distributions of cones, rods, and optic nerve axon numbers were assessed in seven New World primates (Cebus apella, Saimiri ustius, Saguinus midas niger, Alouatta caraya, Aotus azarae, Calllithrix jacchus, and Callicebus moloch). The spatial distribution and number of rods and cones was determined from counts of retinal whole mounts. Optic axon number was determined from optic nerve sections by electron microscopy. These data were amassed with existing data on retinal cell number and distribution in Old World primates, and the scaling of relative densities and numbers with respect to retinal area, eye and brain sizes, and foveal specializations were evaluated. Regular scaling of all cell types was observed, with the exceptionally large, rod-enriched retina of the nocturnal owl monkey Aotus azarae, and the unusually high cone density of the fovea of the trichromatic howler monkey Alouatta caraya presenting interesting variations on this basic plan. Over all species, the lawful scaling of rods, cones, and retinal ganglion cell number is hypothesized to result from a conserved sequence of cell generation that defends retinal acuity and sensitivity over a large range of eye sizes.

During their complex life history, anguilliform eels go through a major metamorphosis when developing from a fresh water yellow eel into a deep-sea silver eel. In addition to major changes in body morphology, the visual system also adapts from a fresh water teleost duplex retina with rods and cones, to a specialized deep-sea retina containing only rods. The history of the rods is well documented with an initial switch from a porphyropsin to a rhodopsin (P5232 to P5011) and then a total change in gene expression with the down regulation of a “freshwater” opsin and its concomitant replacement by the expression of a typical “deep-sea” opsin (P5011 to P4821). Yellow eels possess only two spectral classes of single cones, one sensitive in the green presumably expressing an RH2 opsin gene and the second sensitive in the blue expressing an SWS2 opsin gene. In immature glass eels, entering into rivers from the sea, the cones contain mixtures of rhodopsins and porphyropsins, whereas the fully freshwater yellow eels have cone pigments that are almost pure porphyropsins with peak sensitivities at about 540–545 nm and 435–440 nm, respectively. However, during the early stages of metamorphosis, the pigments switch to rhodopsins with the maximum sensitivity of the “green”-sensitive cone shifting to about 525 nm, somewhat paralleling, but preceding the change in rods. During metamorphosis, the cones are almost completely lost.

Opsins, like many other G-protein-coupled receptors, sustain constitutive activity in the absence of ligand. In partially bleached rods and cones, opsin's activity closes cGMP-gated channels and produces a state of “pigment adaptation” with reduced sensitivity to light and accelerated flash response kinetics. The truncated retinal analogue, β-ionone, further desensitizes partially bleached green-sensitive salamander rods, but enables partially bleached red-sensitive cones to recover dark-adapted physiology. Structural differences between rod and cone opsins were proposed to explain the effect. Rods and cones, however, also contain different transducins, raising the possibility that G-protein type determines the photoreceptor-specific effects of β-ionone. To test the two hypotheses, we applied β-ionone to partially bleached blue-sensitive rods and cones of salamander, two cells that couple the same cone-like opsin to either rod or cone transducin, respectively. Immunocytochemistry confirmed that all salamander rods contain one form of transducin, whereas all cones contain another. β-Ionone enhanced pigment adaptation in blue-sensitive rods, but it also did so in blue- and UV-sensitive cones. Furthermore, all recombinant salamander rod and cone opsins, with the exception of the red-sensitive cone opsin, activated rod transducin upon the addition of β-ionone. Thus opsin structure determines the identity of β-ionone as an agonist or an inverse agonist and in that respect distinguishes the red-sensitive cone opsin from all others.