**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.**

*or*M-cone isolating sawtooth flicker that went slowly-redder–rapidly-greener appeared redder, whereas the hue of sawtooth flicker that went slowly-greener–rapidly-redder appeared greener (Stockman et al., 2017b). Measurements of the effects of varying either the modulation of the waveforms or the rates of change of their rising and falling slopes suggested that the shifts in the mean hue depended mainly on the first and second harmonics of the flickering waveforms (Stockman et al., 2017b).

*slew-rate limited mechanism*that restricts the rate at which internal representations that determine hue can change. This restriction prevents the visual system from following the rapidly changing but not the slowly changing slope of low-frequency sawtooth waveforms. As a result, the mean output moves in the direction of the slowly changing slope.

^{1}Second, a

*compressive (or saturating) nonlinear mechanism*that instantaneously compresses hue signals—the larger the signal, the greater the compression—thus shifting the mean output of waveforms that are asymmetrical in extent, such as the rectangular waveforms shown as solid black lines in panels C and E of Figure 1. These waveforms should exhibit hue shifts in the direction opposite from the greater excursion from the mean.

^{1}, should occur with the slowly-off and slowly-on waveforms (panels B and D), which are asymmetrical in

*slope*, whereas the largest hue shift with the compressive nonlinearity alone should occur for the peaks-align and troughs-align waveforms (panels C and E), which are asymmetrical in

*extent*(see Stockman et al., 2017a, figures 12 and 13).

*input*to the visual system, the visual system

*as a whole*behaves more like a system that simply compresses hue signals rather than one that limits their rate of change.

*x*,

*y*color coordinates = 0.410, 0.514, luminance = 43.8 cd/m

^{2}), and could be L- or M-cone isolating or equiluminant, chromatic variations. Observers in the chromatic TCSF measurements viewed a circular 5.7° field and in the discrimination measurements compared different waveforms in two 5.7° semicircular fields separated by 0.6° and reported which half-field looked redder. For further details and for other procedures, please see the original publications.

_{10}chromatic sensitivity as a function of frequency (Hz, logarithmic axis). To obtain these data, the modulation of equiluminant red/green sinusoidal flicker was varied in a two-alternative forced-choice experiment to find the relation between modulation and detection performance. The reciprocal of the modulation corresponding to 75% correct detection was taken as the “threshold” sensitivity and the logarithm of the sensitivity is plotted as orange diamonds in both panels as a function of frequency. The chromatic TCSF is often taken as an indication of the overall attenuation characteristic of the chromatic pathway and represented by a single filter (e.g., Swanson, Ueno, Smith, & Pokorny, 1987). Although the dip in the TCSFs at 3 Hz is found in the data for both observers, and might suggest the involvement of multiple chromatic mechanisms at low frequencies rather than a single low-pass mechanism, the dip is not a consistent feature of other chromatic TCSF measurements (e.g., Kelly & van Norren, 1977; Petrova, Henning, & Stockman, 2013b; Varner, Jameson, & Hurvich, 1984). However, a similar dip is seen in figure 2 of de Lange (1958), and Cass, Clifford, Alais, and Spehar (2009) have identified low-pass and band-pass chromatic mechanisms in masking experiments that might give rise to such a dip.

*before*the nonlinear mechanism should be consistent with the second harmonic phase delays that are plotted twice as the symbols in Figure 3. Again, since the visible distortion product after the nonlinearity is assumed to be mainly a DC shift, the second harmonic phase delays

*after*the nonlinearity should be unimportant. In Figure 3, the symbols give the second harmonic phase delays (degrees) relative to the first harmonic in the two-component stimuli at the input to the visual system that produced the biggest mean hue shifts. Their fundamental frequencies ranged from 4 to 16 Hz (Stockman et al., 2017a). The label along the left ordinate of the figure corresponds to these measurements. The estimates are shown twice to illustrate that the greatest hue shifts occur at two different second harmonic phase delays for both L- and M-cone stimuli—one in the red direction and the other in the green direction. The data shown by the triangles for AS and inverted triangles for KR are based on the rectangular duty-cycle matches to mean hue made as a function of second harmonic phase (from figures 7–9 of Stockman et al., 2017a). There, the function relating the matching duty cycle of a rectangular waveform to the phase delay of the second harmonic was sinusoidal in form. From the fitted sinusoids, we extracted the second harmonic phases that produced the greatest mean hue shifts. The error bars are the standard errors of the phase in the sinusoidal fits.

*relative*phases of all the harmonics. Plotted as the second harmonic

*relative*to the first, as in Figure 3, a time delay would be a horizontal line at 0°.

*n*leaky integrator stages. However, a simple low-pass filter is an implausible approximation of chromatic filtering, since, for one thing, it ignores the prominent surround inhibition that occurs in the retina (e.g., Lee, Martin, & Grünert, 2010).

*n*-stage low-pass filter of the form given in Equations 1 and 2 with

*n*= 2:

*n*identical, cascaded, low-pass filters and

*n*identical low-pass filters has increased by

*G*, and the gain of the feedforward inhibition of the lead-lag filters is

_{e}*k*(if

*k*= 1 the lead-lag filter is a standard high-pass filter, while if

*k =*0, it is an all-pass filter). The lead-lag filter, which is mathematically equivalent to the filter that we called “divisive” in our earlier work (Petrova et al., 2013b; Petrova, Henning, & Stockman, 2013a; Stockman et al., 2014), is designed to capture the lateral interactions that contribute to the early filter as sketched in the left-hand panel of Figure 4. Considered alone, the early filter has three free parameters and the late filter has two but, since the two gains cannot be estimated independently, there are only four free parameters. To capture the overall gain, the gain of either the early or late filter can be set to unity, so the overall model has just four free parameters to be estimated from the data.

*relative*to the first,

*k*, and an overall gain factor,

*G*, to simultaneously optimize the fit to the amplitude and phase data for a model with the slew-rate limiting mechanism following the late filter.

*k*, and

*G*in the present case;

*n*was fixed at integer values) are varied to find the combination that produces the smallest mean square difference between the predictions of the model and the data. The value of

*n*was constrained to take on integer values in the final fits of each model. The fits were made to the phase delays measured in radians, because the phase delays in radians are comparable in magnitude to the log contrasts. The curve fitting procedure was the nonlinear regression implemented in SigmaPlot (Systat Software, San Jose, CA) based on the Marquardt-Levenberg algorithm (Levenberg, 1944; Marquardt, 1963) that minimizes the sum of the squared differences between the data and model predictions. Note that for nonlinear regression,

*R*

^{2}as a measure of goodness-of-fit is problematic (e.g., Kvalseth, 1985; Spiess & Neumeyer, 2010), so that as well as giving

*R*

^{2}values, we also give the standard error of the regression. Note that since the fits were in radians, the standard errors for the phase delay fits are also in radians.

*input*to the nonlinearity are either 0° or 180° as indicated by the horizontal gray lines in Figure 3 labeled “Slew phase.” These fits were again done simultaneously across the amplitude and phase data for all the observers using the same parameters with the exception that a different overall gain factor (

*k*= 0.78 ± 0.02.

*G*in log

_{10}units was 11.02 ± 0.15 for KR and 10.81 ± 0.15 for JA. The

*R*

^{2}value for the fit was 0.834 and the standard error of the regression was 0.202.

*discriminability*at, for example, 5 and 10 Hz, should depend on the sensitivity difference between the

*detection*of chromatic sinusoids at 10 and 20 Hz, and so on. In terms of shapes, this is equivalent to shifting the logarithmic chromatic TCSF functions (orange lines) by 0.3 log unit along the log frequency scale toward lower frequencies (or on a linear frequency plot halving the frequency), as shown by the red-green dashed lines. The shape of the shifted function agrees with the hue-discrimination data (red and green triangles), which supports the suggestion that the second harmonic transmitted through the late filter is the important factor (but see below).

*input*to the nonlinearity are either 90° or 270°. However, we could not find a pair of early and late filters that would account for both the amplitude and the phase data simultaneously, but we could find a combination in which the early filter alone accounts for the phase data while the early and late filter

*together*account for the amplitude data. This suggests that the important hue-shifting nonlinearity might lie between the early and late filters. In this section, we investigate a model with a nonlinearity between the early and late filters.

*relative*to the first,

*k*= 0.55 ± 0.02, and

*G,*in log

_{10}units, was 9.27 ± 0.11 for KR and 9.06 ± 0.11 for JA; the fits to the TCSFs are shown as orange lines in the panels of Figure 5. The fit is remarkably good and as good as the fits of the slew-rate limiting model in that the TCSF is well fit over the entire range of frequencies. The

*R*

^{2}value of 0.835 and the standard error of the regression was 0.201.

*n*= 1 in Equations 1 and 2. As well as changing its attenuation characteristics, this halves the phase delays caused by the late filter. The model fits to the phase data are shown by the red line in Figure 6. Again, the fits are good.

*prior*to the late filter. That distortion is a shift in the mean and apart from the gain of the late filter, which affects all frequencies equally, the attenuation characteristics of the late filter should not affect the hue shift. This means that hue discrimination as a function of frequency should follow the attenuation characteristics of the early filter alone. The early filter's attenuation hardly changes over the 4–13 Hz range (red lines in Figure 5) and is obviously dissimilar to the steeply low-pass shape of the hue-shift discrimination data (red and green triangles in Figure 5). Therefore, this model, with a single intermediate nonlinearity,

*cannot*predict the hue discrimination data. For similar arguments, see other descriptions of the linear–nonlinear–linear “sandwich” models (e.g., Marmarelis & Marmarelis, 1978; Petrova et al., 2013b; Victor, Shapley, & Knight, 1977).

_{ON}and M

_{OFF}receptive fields are combined in a unipolar “Red” channel, and the outputs of the M

_{ON}and L

_{OFF}receptive fields are combined in a unipolar “Green” channel before separately reaching identical late filters. Evidence in support of unipolar Red and Green mechanisms as opposed to bipolar red–green mechanisms is discussed by, for example, Eskew (2008), Sankeralli & Mullen (2001), and Wuerger, Atkinson & Cropper (2005). Our previous work suggested that the late filter acts as a two-stage low-pass filter that begins to attenuate chromatic flicker about 3 Hz (Petrova et al., 2013b; Stockman et al., 2014). However, the fits in the previous section suggest that the late filters might be single-stage, and that is how they are depicted in the blue-gray region of Figure 7. Lastly, compressive late or central nonlinearities in the red and green pathways are shown in the right-hand khaki region and represented by a saturating function.

*k*= 0.55 ± 0.02, and

*G =*9.27 ± 0.11 for KR and 9.06 ± 0.11 for JA. As already shown, this model fits the chromatic TCSFs and the phase data, but how can an intermediate nonlinearity made up of half-wave rectifiers and a late nonlinearity made up of a compressive input-output function account for the hue-shift discrimination TCSFs (red and green triangles, Figure 5)? To explain the final version of this model, we start with an illustrative example of the effects of the half-wave rectifier and the saturating nonlinearity on the two-component, troughs-align, L-cone–isolating waveform shown at A in Figure 8.

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*Vision Research*^{1}A related possibility is a mechanism made up of a differentiator followed by a symmetrical compressive (saturating) nonlinearity (Cavanagh & Anstis, 1986), but since this mechanism entails similar hue-shift predictions we just consider the slew-rate limited mechanism.

*f*, zero mean, and zero phase; that is,

*y*> 0 only between 0 and T/2 and

*y*< 0 only between T/2 and T. Solving Equations A1a and A1b for

*k*= 0, 1, 2, etc., gives the two complete Fourier series representations:

*φ*, of the second harmonic) they are the solutions for

*t*of: