purpose. Detection efficiency for flickering stimuli of constant duration decreases with increasing temporal frequency. Increasing frequency in this case also implies increasing number of flicker cycles. The current study was conducted to investigate whether this result could be due to the limited ability of the central detector to integrate flicker cycles.

methods. Flicker sensitivity was measured at 1 to 20 Hz in strong external temporal noise with increasing stimulus duration.

results. Sensitivity increased with stimulus duration in a nonsaturating manner up to the longest exposure times used, indicating probability summation. When expressed in terms of detection efficiency (η) as a function of number of cycles presented (*n*) all data could be modeled as a single decreasing function of the form η = 0.29*n* ^{−0.70}.

conclusions. The results show that the number of cycles, not time, is the determinant of probability summation of flicker. The results are consistent with the idea that the central detector is a suboptimal matched filter spanning less than one cycle.

^{ 1 }

^{ 2 }

^{ 3 }

^{ 4 }The signal-to-noise ratio at detection threshold can then be used to relate the performance measured directly to that of the “ideal observer” by determining detection efficiency.

^{ 5 }This approach has frequently been used to obtain the detection efficiency for spatial and spatiotemporal signals

^{ 1 }

^{ 6 }

^{ 7 }and more recently for purely temporal signals.

^{ 4 }

^{ 8 }

^{ 9 }

^{ 4 }

^{ 8 }used external white noise of various spectral densities to test a general model for flicker detection. The detector could be described as a suboptimal matched filter with detection efficiency that decreased with increasing temporal frequency. Because stimulus duration was constant in these experiments, the number of flicker cycles presented was greater the higher the temporal frequency, and the authors speculated that this could explain the decrease in efficiency. Failure to integrate the signal effectively over several cycles would result in a decrease in detection efficiency, analogous to what has been observed in spatial vision.

^{ 10 }

^{ 11 }

^{ 12 }

^{ 13 }

^{ 14 }

^{ 15 }Flicker sensitivity recorded without external noise has been found to increase with increasing exposure duration up to the longest durations tested. Log sensitivity as a function of log stimulus duration increases more or less linearly with slopes of approximately 0.19 to 0.25.

^{ 14 }

^{ 15 }According to Watson’s model

^{ 15 }for probability summation, this slope should be the reciprocal of the steepness parameter β of the psychometric or frequency-of-seeing (FOS) function. Indeed, FOS functions recorded with similar stimuli have roughly the predicted steepness, corresponding to a β of approximately 4 to 5.

^{ 15 }The only deviation from the slope of 0.19 to 0.25 was the steeper slope found at short exposure durations (100 ms), which may be due to the effects of the early modulation transfer functions (MTFs).

^{ 15 }

^{2}.

^{2}, corresponding to a scotopic luminance of 130 cd/m

^{2}. A summation device

^{ 16 }was used to combine the red, green, and blue outputs of the VGA board to obtain a monochrome signal of 256 intensity levels from a palette of 16,384 intensity levels. The amplitudes of the flickering signals were calibrated with a phototransistor (TIL81; Texas Instruments Inc., Dallas, TX). There was no attenuation in amplitude up to 30 Hz (for further details, see Ref.

^{ 4 }).

^{2}and 12.6 deg

^{2}, respectively. The stimulus field was surrounded by a circular equiluminous field limited to a diameter of 10° with a black cardboard mask. The flicker frequencies used were 1, 3, 10, 15, and 20 Hz. Flicker sensitivity was measured as a function of stimulus duration, which varied from 50 to 3000 ms, or from 1 to 60 cycles.

*E*) of flickering stimuli were calculated by numerical integration across time by adding up the products of the frame duration and frame contrast (

*c*) squared to remove its sign. Thus,

*c*(

*t*) = [

*L*(

*t*) −

*L*

_{0}]/

*L*

_{0};

*L*(

*t*) is the signal;

*L*

_{0}is the average luminance; and Δ

*t*is the duration of each frame (1/60 seconds). The root-mean-square (RMS) contrast is

*t*is the stimulus duration. For simple sinusoidal flicker, RMS contrast is equal to the Michelson contrast divided by √2. Flicker sensitivity (

*S*) was taken as the inverse of RMS contrast, i.e.

*S*= 1/

*c*

_{RMS}.

^{ 1 }as

*c*

_{ n }is the RMS contrast of noise and Δ

*t*is the duration of each frame, (i.e., 16.7 ms). In our experiments the spectral density of temporal noise was thus 15.0 × 10

^{−4}seconds at 1 Hz for subject LJ and 3.75 × 10

^{−4}seconds in all other conditions. The 60-Hz frame rate ensured that noise could be considered white at all temporal frequencies studied

^{ 17 }and even the highest flicker frequency of 20 Hz was within the 30-Hz noise band.

^{ 18 }The contrast step was constant at 0.1 log units throughout the algorithm. The threshold contrast at the probability of 84% correct was obtained as an arithmetic mean of eight contrast reversals.

*n*is the number of data points, η refers to data, and η

_{est}to predicted value. Logarithmic values were used for calculating the RMS error (ε), as data were plotted on a logarithmic scale. The value of

*k*is 1 for sensitivity and 0.5 for efficiency, because efficiency is based on contrast squared. If the average error between log η and log η

_{est}is Δη, then GoF = 100 [1 −

*k*abs(Δη)]. For example, if

*k*= 0.5 and Δη = ±0.30, then GoF = 0.85, which appears to be the lower limit for visually acceptable fit. The reason for using GoF instead of

*r*, the coefficient of determination, is that for fits with shallow slopes, both the explained variation and thus also the value of

*r*tend to be small, whereas GoF still gives reasonable values (for further details, see Ref.

^{ 9 }).

^{ 14 }

^{ 15 }The sensitivity at 1 Hz for LJ was lower than for AR, because a higher noise contrast was used (see the Methods section) to guarantee that external noise was dominant.

^{ 9 }The only difference that nearly reached statistical significance (

*P*< 0.05, Mann-Whitney test/Wilcoxon two-sample test; CoHort Software, Monterey, CA) between areas was at 10 Hz for subject AR.

^{ 15 }This supports Watson’s

^{ 15 }view that the initial steeper slope in his data was due to the early visual filters, the effect of which is not visible in our data measured in strong external noise.

^{ 4 }

^{ 8 }

*n*) = η

_{1}

*n*

^{−K }) where η

_{1}is efficiency at one cycle (

*n*= 1);

*K*= 1 − 2/β, which is the slope of linear decrease, and β is the steepness of the psychometric function (for the derivation of the equation on the basis of probability summation, see Appendix A). In Figure 3 the data from Figure 1 are thus combined in one frame and plotted in terms of detection efficiency, calculated according to equation 4 .

*n*) of cycles was similar at all temporal frequencies studied, forming a single cloud of data. The least-squares fit of equation 6 to the data in Figure 3 was found to be η(

*n*) = 0.29

*n*

^{−0.70}. The goodness of fit, calculated according to equation 5 , was 93%. The value of β was thus 6.7, and maximum efficiency at 1 cycle (η

_{1}), 0.29. The fact that the efficiency of detection is less than unity, even for single-cycle stimuli, means that temporal integration was incomplete for all our stimuli. The dotted line in Figure 3 shows the least-squares line fit to the raw data (i.e., individual thresholds measured for each data point) combined for all stimulus conditions and both subjects. The fit to raw data is very similar to the fit to median data (Fig. 3 , solid line).

*n*=

*tf*between the number of flicker cycles (

*n*) presented, time (

*t*) exposed and flicker frequency (

*f*) studied, equation 6 can be expressed as

*f*) at a constant exposure time (

*t*) in seconds also decreases with a slope of −0.70. This result is fairly close to our previous estimate of −0.58.

^{ 4 }

*n*) = 0.29

*n*

^{−0.70}describes well the linear decrease of log efficiency as a function of log number of cycles at all temporal frequencies and stimulus areas. The goodness of fit, calculated according to equation 5 , varied between 93% and 96%. When equation 6 was fitted separately to each frame in Figure 4 , the slope (and standard deviation) was found to be fairly constant. It was −0.64 (0.11), −0.83 (0.09), −0.70 (0.06), −0.66 (0.06), and −0.74 (0.05) at 1, 3, 10, 15, and 20 Hz, respectively.

*n*=

*tf*, sensitivity

*S*= 1/

*c*

_{RMS}can be described using two variables out of three (

*n*,

*t*,

*f*)

_{1}is efficiency at

*n*= 1,

*d*′ is the detectability index (1.4), and

*N*

_{t}is the spectral density of external temporal noise (3.75 × 10

^{−4}seconds for all other conditions except for LJ at 1 Hz, when it was 15.0 × 10

^{−4}seconds). The slope of increase of log(

*S*) as function of log(

*n*) or log(

*t*) was 1/β = 0.149 at all temporal frequencies, which is indicated by

*n*

^{1/β}in the middle part of equation 8 and

*t*

^{1/β}in the rightmost part of equation 8 .

^{ 4 }

^{ 8 }

^{ 9 }

^{ 14 }

^{ 15 }The nonsaturation of temporal integration suggests that the underlying process is probability summation over time. Our data suggest, however, that probability summation of cycles, not time, takes place. Probability summation implies that detection is limited by noise that is uncorrelated from one stimulus instant (e.g., stimulus cycle) to another, and thus ensures that within each instant there is some probability that threshold will be exceeded. With increasing exposure duration, the number of instants and thus the chances that the signal is detected increases. The slope of increase in sensitivity with exposure duration measured without external noise has been estimated

^{ 14 }

^{ 15 }to be approximately 0.19 to 0.25 at exposure durations longer than 100 ms. We found an approximate slope of 0.145 in strong external noise.

^{ 15 }experiments without noise, the sensitivity increase was steeper at the shortest stimulus durations than at longer durations, and Watson suggested that the deviation could be because, with wider stimulus bandwidth (as is inevitably the case for short durations) the gain of the modulation transfer functions was not constant. It is worth noting that any such nonlinearity connected with early gains that might act to steepen the slope would not be seen in our experiments using strong external noise.

^{ 15 }the result may be a weaker dependence of sensitivity on stimulus duration, as observed in the current study.

^{ 4 }

^{ 8 }

^{ 9 }The present experiments thus support the idea that the decrease in efficiency in the prior studies reflects the absence of any other means of temporal integration than probability summation across flicker cycles.

^{ 6 }

^{ 7 }such as noisy matched filter template, or inaccuracy of the template due to slight distortion of the flicker waveform, small error in flicker frequency, or small deviation in stimulus location in space or time, to mention a few. Additive neural noise can be neglected as insignificant in the present study, because strong external noise is the limiting factor for detection. Hence, neural noise cannot account for reduced efficiency.

^{ 19 }additive neural noise and decision process are preceded by a nonlinear transducer function and multiplicative noise that is a function of both signal and external noise, whereas in the model of Eckstein et al. (EAW)

^{ 20 }there is decisional uncertainty as a free parameter, but no transducer function, whereas multiplicative noise depends only on external noise. In principle, these models are capable of describing the current flicker data, but our model is simpler. However, in our data the contrasts were not extremely high, which may explain

^{ 19 }why distinction between additive and multiplicative noises was not necessary.

^{ 21 }

^{ 22 }

^{ 23 }that suggested constant time for linear temporal integration. In these studies, however, the experimental task was to detect a stationary spatial image with pulsed presentation in the presence of noise. It is not surprising therefore that temporal integration of these studies resembles that of spatial or spatiotemporal stimuli,

^{ 24 }

^{ 25 }which is different from the integration of purely temporal signals such as ours. In view of the general independence of the on and off systems,

^{ 26 }

^{ 27 }it seems likely that the integration intervals could be located at the fast luminance changes between maxima and minima of each flicker period.

^{ 28 }found that the situation in which temporal summation continues throughout the whole stimulus duration (giving a square-root dependence of sensitivity on stimulus duration) occurs only with very small stimulus fields. This type of temporal summation was limited or absent for large fields. In our experiments, we saw square root dependence for both large and small stimulus fields up to at least 0.5 seconds (Fig. 2) . We think this is because detection of both the small and the large field is limited by dominant purely temporal noise, and so there is no real difference in spatial integration between small and large fields. This can be explained easily in terms of the physical signal-to-noise ratio, which remains constant irrespective of the area for flickering stimuli when embedded in purely temporal noise with no spatial luminance variation. When noise is sufficiently strong to be the main determinant of the detection threshold, constant physical signal-to-noise ratio at all stimulus areas keeps the detection threshold constant.

^{ 15 }

^{ 29 }so that threshold is proportional to exposure duration raised to a small negative exponent.

^{ 30 }According to Tyler and Chen

^{ 31 }Weibull analysis is an accurate theory for the description of systems with Gaussian additive noise if 4 < β < 8, so that the additive noise distribution implied by Weibull analysis is approximately Gaussian, and the detection threshold sits at or above the maximum of the responses of all the monitored channels to additive noise.

*c*

_{ n }is the stimulus contrast of all

*n*cycles presented at detection threshold,

*c*

_{1}is the stimulus contrast of one cycle presented at detection threshold, and 1/β is the small exponent. Equation A1 can be readily transformed to

_{ n }= (

*d*′

^{2}

*N*

_{ t })/

*E*

_{th(n)}, where

*n*is the number of cycles presented,

*d*′ is the detectability index,

*N*

_{ t }is the spectral density of external noise, and

*E*

_{th(n)}is the energy threshold. Thus, η

_{ n }

*E*

_{th(n)}=

*d*′

^{2}

*N*

_{ t }. Similarly for

*n*= 1, η

_{1}

*E*

_{th(1)}=

*d*′

^{2}

*N*

_{ t }. They can be combined to η

_{ n }

*E*

_{th(n)}= η

_{1}

*E*

_{th(1)}, which can be written in a logarithmic form as

*E*

_{th(1)}=

*c*

_{1}

^{2}

*t*

_{1}and

*E*

_{th(n)}=

*c*

_{ n }

^{2}

*t*

_{ n }=

*c*

_{ n }

^{2}

*n t*

_{1}, where

*t*

_{1}is the exposure duration of one cycle and

*t*

_{ n }that of of

*n*cycles. The latter part of equation A3 (i.e., log

*E*

_{th(1)}− log

*E*

_{th(n)}) thus reduces to

_{ n }= η

_{1}

*n*

^{(1−2/β)}.

**Figure 1.**

**Figure 1.**

**Figure 2.**

**Figure 2.**

**Figure 3.**

**Figure 3.**

**Figure 4.**

**Figure 4.**