The inner regions of protoplanetary disks

To understand how planetary systems form in the dusty disks around pre-main-sequence stars a detailed knowledge of the structure and evolution of these disks is required. While this is reasonably well understood for the regions of the disk beyond about 1 AU, the structure of these disks inward of 1 AU remains a puzzle. This is partly because it is very difficult to spatially resolve these regions with current telescopes. But it is also because the physics of this region, where the disk becomes so hot that the dust starts to evaporate, is poorly understood. With infrared interferometry it has become possible in recent years to directly spatially resolve the inner AU of protoplanetary disks, albeit in a somewhat limited way. These observations have partly confirmed current models of these regions, but also posed new questions and puzzles. Moreover, it has turned out that the numerical modeling of these regions is extremely challenging. In this review we give a rough overview of the history and recent developments in this exciting field of astrophysics.Comment: 45 pages with 14 Figures. to appear in Annual Review of Astronomy and Astrophysics (2010, Vol. 48

Linear-nonlinear models of the red-green chromatic pathway

The mean hue of flickering waveforms comprising only the first two harmonics depends on their temporal alignment. We evaluate explanatory models of this hue-shift effect using previous data obtained using L- and M-cone-isolating stimuli together with chromatic sensitivity and hue discrimination data. The key questions concerned what type of nonlinearity produced the hue shifts, and where the nonlinearities lay with respect to the early band-pass and late low-pass temporal filters in the chromatic pathways. We developed two plausible models: (a) a slew-rate limited nonlinearity that follows both early and late filters, and (b) a half-wave rectifying nonlinearity-consistent with the splitting of the visual input into ON- and OFF-channels-that lies between the early and late filters followed by a compressive nonlinearity that lies after the late filter

Clinical vision and molecular loss: Integrating visual psychophysics with molecular genetics reveals key details of normal and abnormal visual processing

Over the past two decades we have developed techniques and models to investigate the ways in which known molecular defects affect visual performance. Because molecular defects in retinal signalling invariably alter the speed of visual processing, our strategy has been to measure the resulting changes in flicker sensitivity. Flicker measurements provide not only straightforward clinical assessments of visual performance but also reveal fundamental details about the functioning of both abnormal and normal visual systems. Here, we bring together our past measurements of patients with pathogenic variants in the GNAT2, RGS9, GUCA1A, RPE65, OPA1, KCNV2 and NR2E3 genes and analyse the results using a standard model of visual processing. The model treats flicker sensitivity as the result of the actions of a sequence of simple processing steps, one or more of which is altered by the genetic defect. Our analyses show that most defects slow down the visual response directly, but some speed it up. Crucially, however, other steps in the response sequence can make compensatory adjustments to offset the abnormality. For example, if the abnormal step slows down the visual response, another step is likely to speed up or attenuate the response to rebalance system performance. Such compensatory adjustments are probably made by steps in the sequence that usually adapt to changing light levels. Our techniques and modelling also allow us to tease apart stationary and progressive effects, and the localised molecular losses help us to unravel and characterise individual steps in the normal and abnormal processing sequences

A reinterpretation of critical flicker-frequency (CFF) data reveals key details about light adaptation and normal and abnormal visual processing

Our ability to see flicker has an upper frequency limit above which flicker is invisible, known as the “critical flicker frequency” (CFF), that typically grows with light intensity (I). The relation between CFF and I, the focus of nearly 200 years of research, is roughly logarithmic, i.e., CFF ∝ log(I)—a relation called the Ferry-Porter law. However, why this law should occur, and how it relates to the underlying physiology, have never been adequately explained. Over the past two decades we have measured CFF in normal observers and in patients with retinal gene defects. Here, we reanalyse and model our data and historical CFF data. Remarkably, CFF-versus-I functions measured under a wide range of conditions in patients and in normal observers all have broadly similar shapes when plotted in double-logarithmic coordinates, i.e., log (CFF)-versus-log(I). Thus, the entire dataset can be characterised by horizontal and vertical logarithmic shifts of a fixed-shape template. Shape invariance can be predicted by a simple model of visual processing built from a sequence of low-pass filters, subtractive feedforward stages and gain adjustment (Rider, Henning & Stockman, 2019). It depends primarily on the numbers of visual processing stages that approach their power-law region at a given intensity and a frequency-independent gain reduction at higher light levels. Counter-intuitively, the CFF-versus-I relation depends primarily on the gain of the visual response rather than its speed—a conclusion that changes our understanding and interpretation of human flicker perception. The Ferry-Porter “law” is merely an approximation of the shape-invariant template

Hue shifts produced by temporal asymmetries in chromatic signals depend on the alignment of the first and second harmonics

When M- or L-cone-isolating sawtooth waveforms flicker at frequencies between 4 and 13.3 Hz, there is a mean hue shift in the direction of the shallower sawtooth slope. Here, we investigate how this shift depends on the alignment of the first and second harmonics of sawtooth-like waveforms. Below 4 Hz, observers can follow hue variations caused by both harmonics, and reliably match reddish and greenish excursions. At higher frequencies, however, the hue variations appear as chromatic flicker superimposed on a steady light, the mean hue of which varies with second-harmonic alignment. Observers can match this mean hue against a variable-duty-cycle rectangular waveform and, separately, set the alignment at which the mean hue flips between reddish and greenish. The maximum hue shifts were approximately frequency independent and occurred when the peaks or troughs of the first and second harmonics roughly aligned at the visual input-consistent with the hue shift's being caused by an early instantaneous nonlinearity that saturates larger hue excursions. These predictions, however, ignore phase delays introduced within the chromatic pathway between its input and the nonlinearity that produces the hue shifts. If the nonlinearity follows the substantial filtering implied by the chromatic temporal contrast-sensitivity function, phase delays will alter the alignment of the first and second harmonics such that at the nonlinearity, the waveforms that produce the maximum hue shifts might well be those with the largest differences in rising and falling slopes-consistent with the hue shift's being caused by a central nonlinearity that limits the rate of hue change

Delayed cone-opponent signals in the luminance pathway

Cone signals in the luminance or achromatic pathway were investigated by measuring how the perceptual timing of M- or L-cone-detected flicker depended on temporal frequency and chromatic adaptation. Relative timings were measured, as a function of temporal frequency, by superimposing M- or L-cone-isolating flicker on "equichromatic" flicker (flicker of the same wavelength as the background) and asking observers to vary contrast and phase to cancel the perception of flicker. Measurements were made in four observers on up to 35 different backgrounds varying in wavelength and radiance. Observers showed substantial perceptual delays or advances of L- and M-cone flicker that varied systematically with cone class, background wavelength, and radiance. Delays were largest for M-cone-isolating flicker. Although complex, the results can be characterised by a surprisingly simple model in which the representations of L- and M-cone flicker are comprised not only of a fast copy of the flicker signal, but also of a slow copy that is delayed by roughly 30 ms and varies in strength and sign with both background wavelength and radiance. The delays, which are too large to be due to selective cone adaptation by the chromatic backgrounds, must arise postreceptorally. Clear evidence for the slow signals can also be found in physiological measurements of horizontal and magnocellular ganglion cells, thus placing the origin of the slow signals in the retina-most likely in an extended horizontal cell network. Luminance-equated stimuli chosen to isolate chromatic channels may inadvertently generate slow signals in the luminance channel

Hue shifts produced by temporal asymmetries in chromatic signals.

Observers viewed M- or L-cone-isolating stimuli and compared slowly-on and slowly-off sawtooth waveforms of the same mean chromaticity and luminance. Between 6 and 13 Hz, the mean hue of slowly-on L-cone and slowly-off M-cone sawtooth flicker appeared redder, and the mean hue of slowly-off L-cone and slowly-on M-cone sawtooth stimuli appeared greener-despite all the waveforms' having the same mean, near-yellow-appearing chromaticity. We measured the effect of the modulation depth and the slope of the sawtooth on the mean hue shifts as a function of temporal frequency. The results are complex but show that discriminability depended mainly on the second harmonic of the waveforms. We considered several models with combinations of linear and nonlinear stages. First, we considered models in which a nonlinear stage limits the rate of change of hue and restricts the steep slope of the sawtooth waveform more than its shallow slope, thus shifting the mean hue in the direction of the shallower slope (such a nonlinearity is also known as a slew-rate limit). Second, we considered saturation models in which the nonlinear stage compresses hue signals and thus shifts the mean of asymmetrical waveforms with or without differentiation before the nonlinearity. Overall, our modeling and results suggest that the hue shift occurs at some nonlinear mechanism in the chromatic pathway; and that, in terms of the Fourier components of the various waveforms, the effect of the nonlinearity depends crucially on the timing of the second harmonic relative to the first

Light adaptation controls visual sensitivity by adjusting the speed and gain of the response to light

The range of c. 1012 ambient light levels to which we can be exposed massively exceeds the <103 response range of neurons in the visual system, but we can see well in dim starlight and bright sunlight. This remarkable ability is achieved largely by a speeding up of the visual response as light levels increase, causing characteristic changes in our sensitivity to different rates of flicker. Here, we account for over 65 years of flicker-sensitivity measurements with an elegantly-simple, physiologically-relevant model built from first-order low-pass filters and subtractive inhibition. There are only two intensity-dependent model parameters: one adjusts the speed of the visual response by shortening the time constants of some of the filters in the direct cascade as well as those in the inhibitory stages; the other parameter adjusts the overall gain at higher light levels. After reviewing the physiological literature, we associate the variable gain and three of the variable-speed filters with biochemical processes in cone photoreceptors, and a further variable-speed filter with processes in ganglion cells. The variable-speed but fixed-strength subtractive inhibition is most likely associated with lateral connections in the retina. Additional fixed-speed filters may be more central. The model can explain the important characteristics of human flicker-sensitivity including the approximate dependences of low-frequency sensitivity on contrast (Weber’s law) and of high-frequency sensitivity on amplitude (“high-frequency linearity”), the exponential loss of high-frequency sensitivity with increasing frequency, and the logarithmic increase in temporal acuity with light level (Ferry-Porter law). In the time-domain, the model can account for several characteristics of flash sensitivity including changes in contrast sensitivity with light level (de Vries-Rose and Weber’s laws) and changes in temporal summation (Bloch’s law). The new model provides fundamental insights into the workings of the visual system and gives a simple account of many visual phenomena

The temporal characteristics of the early and late stages of L- and M-cone pathways that signal brightness

Flickering 560-nm light appears brighter and less saturated than steady light of the same average intensity. The changes in appearance are consistent with the visual signal's being distorted at some nonlinear site (or sites) within the visual pathway at which new temporal components, not part of the original waveform, are produced. By varying the input stimulus to manipulate these new temporal components--called distortion products--and measuring our observers' sensitivity in detecting them, we derived the temporal attenuation characteristics of the early (prenonlinearity) and late (post-nonlinearity) stages of the L- and M-cone pathway that signals brightness. We found that the early stage acts like a band-pass filter peaking at 10-15 Hz with sensitivity losses at both lower and higher frequencies, whereas the late stage acts like a two-stage low-pass filter with a corner frequency near 3 Hz. Although brightness is often associated with the fast achromatic or luminance pathway, these filter characteristics, and particularly those of the late filter, are consistent with comparable features of the L-M chromatic pathway that produce mainly chromatic distortion products (Petrova, Henning, & Stockman, 2013). A plausible site for the nonlinearity is after surround antagonism from horizontal cells. Modeling suggested the form of the nonlinearity to be initially expansive but possibly with a hard limit at the highest input levels

Modeling off-frequency binaural masking for short- and long-duration signals

Experimental binaural masking-pattern data are presented together with model simulations for 12- and 600-ms signals. The masker was a diotic 11-Hz wide noise centered on 500 Hz. The tonal signal was presented either diotically or dichotically (180° interaural phase difference) with frequencies ranging from 400 to 600 Hz. The results and the modeling agree with previous data and hypotheses; simulations with a binaural model sensitive to monaural modulation cues show that the effect of duration on off-frequency binaural masking-level differences is mainly a result of modulation cues which are only available in the monaural detection of long signals