The pedestal effect is the improvement in the detectability of a sinusoidal grating in the presence of another grating of the same orientation, spatial frequency, and phase—usually called the *pedestal*. Recent evidence has demonstrated that the pedestal effect is differently modified by spectrally flat and notch-filtered noise: The pedestal effect is reduced in flat noise but virtually disappears in the presence of notched noise (G. B. Henning & F. A. Wichmann, 2007). Here we consider a network consisting of units whose contrast response functions resemble those of the cortical cells believed to underlie human pattern vision and demonstrate that, when the outputs of multiple units are combined by simple weighted summation—a heuristic decision rule that resembles optimal information combination and produces a contrast-dependent weighting profile—the network produces contrast-discrimination data consistent with psychophysical observations: The pedestal effect is present without noise, reduced in broadband noise, but almost disappears in notched noise. These findings follow naturally from the normalization model of simple cells in primary visual cortex, followed by response-based pooling, and suggest that in processing even low-contrast sinusoidal gratings, the visual system may combine information across neurons tuned to different spatial frequencies and orientations.

*detection*(no pedestal) in noise and find that the model correctly predicts how detection performance changes in spectrally flat and filtered noise. Finally, having estimated the likely number of contributing units, we demonstrate that a more realistic pool consisting of 250 units with correlated responses and tuned to a broad range of spatial frequencies robustly produces similar results.

*Invariant Response Descriptive Model*described in Albrecht et al. (2002), expanded to include an explicit selectivity parameter,

*Sel,*which varies between 0 (i.e., the unit is not sensitive to the signal) and 1 (i.e., the peak sensitivity of the unit's spatial-weighting function corresponds to the spatial frequency of the signal). The selectivity parameter is needed for the units' response functions to have the behavior demonstrated by cortical neurons (Albrecht et al., 2002; Geisler & Albrecht, 1997). The response functions, based on the Naka–Rushton equation, provide a good fit to the contrast response functions of striate cortex neurons to preferred (Sel = 1) and non-preferred (Sel < 1) stimuli (Albrecht et al., 2002; Geisler & Albrecht, 1997). Equation 1 shows the mean response of a unit,

*c,*expressed as a fraction of the unit's maximal firing rate:

*r*

_{0}is a spontaneous discharge rate. In the simulations,

*r*

_{0}is drawn from an exponential distribution with a mean value of 1.5% of the maximal firing rate (Olshausen & Field, 2005) [

*r*

_{0}∼ Exp(1.5)];

*r*

_{max}is the maximum firing rate, drawn from a normal distribution with mean 81.8 and standard deviation 12.2 [

*r*

_{max}∼

*N*(81.8, 12.2)];

*n*is the response exponent [

*n*∼

*N*(2.4, 0.18)];

*c*

_{50}is the semi-saturation contrast [

*c*

_{50}∼

*N*(0.387, 0.0351)]. The expressions in square brackets following the definition of the terms in equations, give, where appropriate, the form and parameters of the distribution from which values for the terms were randomly selected. The parameter distributions are based on neurophysiologically determined estimates (Albrecht et al., 2002). However, the exact parameter settings are not critical to any of the claims made in the paper. Nevertheless, we shall see that these distributions produce a remarkably good approximation of psychophysical data.

*R*

_{ u}(

*c*)) is the variance of a unit's response as a function of stimulus contrast. The scaling value of 1.5 is based on estimates provided in several papers on cortical cell response reliability (Albrecht et al., 2002; Geisler & Albrecht, 1997; Vogels et al., 1989). The particular value of the proportionality constant is not critical, but the fact that variance is proportional to mean activity is crucial.

*ω*

_{ u}(

*c*), is fully determined by its mean response, normalized, for convenience, by the sum of all the contributing weights and given by Equation 3:

*ω*

_{ u}(

*c*) is the weight of unit u at stimulus contrast

*c*in a network of

*N*units. In order to make the simulations tractable, the trial-to-trial variation in the weights was ignored; we used the mean responses in calculating the weights. This is a simplification. In a real nervous system, the means would not, of course, be available from a single unit and weights would necessarily be based on responses alone. In 1, we demonstrate that this simplification is immaterial with respect to the conclusions we draw.

*r*

_{uv}is the correlation coefficient between the

*u*th and the

*v*th unit. Correlation among units is, of course, notoriously difficult to determine. Nevertheless, it is an important—indeed a crucial—factor in some models of MT pooling (Shadlen, Britten, Newsome, & Movshon, 1996).

*d*′, is fully determined by the mean and variances of the pooled responses to the pedestal and to the pedestal-plus-signal (Green & Swets, 1966) and given by Equation 6:

*α,*controlling the effectiveness of noise in driving the normalization pool and reported that the values of

*α*resulting from the fits to 22 simple cells were equally spread (on logarithmic coordinates) between 0.1 and 10, indicating that the noise provided very strong inhibition for some cells but only weak inhibition for others.

*d*′ (Green & Swets, 1966). The ratio of the mean to the standard deviation for the pooled group is indicated by the green line, the same ratio for the most selective unit, by the purple line. The higher this ratio, the better the system is able to discriminate a low-contrast signal grating from a uniform field. At the very lowest stimulus contrasts, where the non-optimally tuned units mainly contribute noise, the purple line lies above the green line, indicating that performance of the most selective unit is slightly better than that of the pooled response. From a certain stimulus contrast on, however, the green line lies above the purple line, indicating that the pooled response outperforms the most selective unit. Moreover, as contrast increases, the difference between the two functions grows as a consequence of the proportionality rule and the changing weighting profile (Figure 4d). The pooled detectability function is thus more sharply accelerated and it has been suggested that this particular non-linearity underlies the pedestal effect (Nachmias, 1981; Nachmias & Sansbury, 1974; Smithson, Henning, MacLeod, & Stockman, 2009).

*p*< 10

^{−6}), and analysis of the

*β*-parameter—the parameter controlling the steepness of the cumulative Weibull function fitted to the data—reveals that the psychometric function is also steeper than for the notched noise condition (for each observer,

*p*< 0.05, for the combined data,

*p*< 10

^{−6}; Wichmann & Hill, 2001). These data are thus consistent with the notion that even detection of a sinusoidal grating may be based on pooled responses rather than on the most responsive channel.

*p*< 0.01) but steeper than in notched noise (

*p*< 0.05). Furthermore, as can be seen in Figure 7c, the signal contrast necessary to achieve 75% correct is higher in the white-noise condition than in both the no-noise condition (

*p*< 10

^{−6}) and the notched-noise condition (

*p*< 10

^{−6}). These data are thus consistent with the notion that noise may modify the neurons' contrast response functions, without, however, altering the pooling rules.

*d*′. Exactly one unit in the pool was optimally tuned to the signal. The selectivities of all other units were again randomly chosen from a Gaussian distribution centered at 0.50 with a standard deviation of 0.17 and clipped at 0 and 1. The average inter-unit correlation is coded by the color indicated by the color bar on the right of the figure. If the noise is uncorrelated [the highest (red) curve], addition of more units progressively improves detectability and, in the limit, would yield an errorless observer. On the other hand, weakly correlated noise (the other curves) shows that the addition of more units has very little effect on detectability once a certain critical number of units is reached. As the average inter-unit correlation increases (downward or more blue in Figure 8a), that critical number of units drops. For inter-neuron correlations between 0.1 and 0.2 (typical estimates from single cell recordings), the improvement with increasing numbers approaches its asymptotic level between 50 and 100 units (Zohary et al., 1994). Pool size is thus expected to be rather limited.

*d*′ =

*R*

_{ f}(

*c*) is the response of a linear filter as a function of signal contrast expressed as a fraction of the maximal response. As in Equation 2, the variance of the unit's response was proportional to its mean value.

*n*equals 2.4 and

*c*

_{50}equals 0.38. We performed simulations for a wide range of parameter values and noise levels and found that the results could be captured by resetting the parameters of the unit's response function as given by Equation B2 (fits to the mean response are shown in Figure B1).

*r*

_{Noise}is the average noise evoked response, and Δ

*n*and Δ

*c*describe the change of the response exponent and semisaturation contrast in noise. Based on our simulations, these three parameters were estimated analytically for each network unit as a function of external noise level (this is a free parameter in the model), the response exponent

*n,*and semisaturation contrast

*c*

_{50}(these are randomly selected parameters, as explained in Methods).

*α*introduced by Carandini et al. (1997) to capture non-specific suppression effects of broadband noise, as given by Equation B3.

*α*were drawn from an exponential distribution, appropriately scaled to approximate the estimates reported by Carandini et al. (1997). At the noise power chosen for the contrast-discrimination simulations in the paper, the response to noise alone was on average approximately four times higher than the spontaneous maintained discharge,

*r*

_{0}. This estimate is reasonably close to the roughly three-fold elevation found by Carandini et al. Further, at this noise level, Δ

*n*= 0.45 (standard deviation of 0.15 across units) and Δ

*c*= 0.11 (standard deviation of 0.03). Figure B2 illustrates the effects of broadband noise on the contrast response function for one unit, simulated with the normalization model—i.e., a narrowly tuned excitatory factor and a broadly tuned divisive inhibitory factor, both producing variable responses—and our approximation, making use of Equation B3.

*n*and Δ

*c*used to capture this modification were estimated for each unit based on the simulations of the simple model described above (see Equation B1). This is of course only an approximation but sufficient to capture the increase of Δ

*n*and Δ

*c*with stronger suppression. This can be seen in Figure B2, which illustrates the effects of notched noise on the contrast response function for one unit, simulated with the normalization model and our approximation, making use of Equation B4.

*n*= 0.43 (standard deviation of 0.3 across units) and Δ

*c*= 0.18 (standard deviation of 0.2). The average modification of the response exponent in notched noise thus closely resembles the results in white noise, while the noise-suppression is stronger. For both parameters, the standard deviation in notched noise is higher due to the effects of tuning. While this implementation captures the main effects of notched noise described above, it is a simplification and at best only an approximation. Nevertheless, this operationalization of notched noise effects proved to be sufficient to generate plausible contrast-discrimination data in noise.

*r*

_{max}. Finally, to estimate a unit's response variance, we used the ratio of the variance to the mean simulated in the simple model ( Equation B1) and multiplied this ratio with the mean responses deduced from Equations B3 and B4.

*Neural Computation*, 11, 91–101. [PubMed] [CrossRef] [PubMed]

*Nature Reviews, Neuroscience*, 7, 358–366. [PubMed] [CrossRef]

*Journal of the Optical Society of America A, Optics, Image Science, and Vision*, 19, 1267–1273. [PubMed] [CrossRef] [PubMed]

*The Journal of Physiology*, 203, 237–260. [PubMed] [Article] [CrossRef] [PubMed]

*The Journal of Physiology*, 197, 551–556. [PubMed] [Article] [CrossRef] [PubMed]

*PLoS Biology*, 2,

*Nature Neuroscience*, 9, 1412–1420. [PubMed] [Article] [CrossRef] [PubMed]

*Vision Research*, 45, 2943–2959. [PubMed] [CrossRef] [PubMed]

*Nature Neuroscience*, 2, 740–745. [PubMed] [CrossRef] [PubMed]

*Spatial vision*. Oxford: Oxford University Press.

*Neural Engineering: Computation, representation, and dynamics in neurobiological systems*. Cambridge: MIT Press.

*Neuron*, 54, 137–152. [PubMed] [Article] [CrossRef] [PubMed]

*Journal of the Optical Society of America A, Optics, Image Science, and Vision*, 11, 1710–1719. [PubMed] [CrossRef] [PubMed]

*Vision Research*, 46, 1585–1598. [PubMed] [CrossRef] [PubMed]

*Neuron*, 35, 759–771. [PubMed] [CrossRef] [PubMed]

*Visual Neuroscience*, 14, 897–919. [PubMed] [CrossRef] [PubMed]

*Journal of Vision*, 8, (15):17, 1–21, http://journalofvision.org/8/15/17/, doi:10.1167/8.15.17. [PubMed] [Article] [CrossRef]

*Journal of Vision*, 8, (9):4, 1–15, http://journalofvision.org/8/9/4/, doi:10.1167/8.9.4. [PubMed] [Article] [CrossRef]

*Visual pattern analyzers*. Oxford: Oxford University Press.

*Vision Research*, 11, 251–259. [PubMed] [CrossRef] [PubMed]

*Signal detection theory and psychophysics*. New York: John Wiley & Sons, Inc.

*Visual Neuroscience*, 9, 427–443. [PubMed] [CrossRef]

*Visual Neuroscience*, 9, 181–197. [PubMed] [CrossRef]

*Journal of the Optical Society of America A, Optics, Image Science, and Vision*, 19, 1259–1266. [PubMed] [CrossRef] [PubMed]

*Journal of Vision*, 7, (1):3, 1–15, http://journalofvision.org/7/1/3/, doi:10.1167/7.1.3. [PubMed] [Article] [CrossRef] [PubMed]

*Nature Neuroscience*, 9, 690–696. [PubMed] [CrossRef] [PubMed]

*Journal of the Optical Society of America*, 70, 1458–1471. [PubMed] [CrossRef] [PubMed]

*The Journal of Physiology*, 296, 27–47. [PubMed] [Article] [CrossRef] [PubMed]

*Journal of Neuroscience*, 27, 2338–2348. [PubMed] [CrossRef] [PubMed]

*Vision Research*, 21, 215–223. [PubMed] [CrossRef] [PubMed]

*Vision Research*, 14, 1039–1042. [PubMed] [CrossRef] [PubMed]

*Neural Computation*, 17, 1665–1699. [PubMed] [CrossRef] [PubMed]

*Visual Cognition*, 8, 531–547. [CrossRef]

*Journal of the Optical Society of America A, Optics and Image Science*, 2, 1508–1532. [PubMed] [CrossRef] [PubMed]

*Nature Reviews. Neuroscience*, 1, 125–132. [PubMed] [CrossRef] [PubMed]

*Nature*, 387, 281–284. [PubMed] [CrossRef] [PubMed]

*Journal of Vision*, 9, (5):21, 1–18, http://journalofvision.org/9/5/21/, doi:10.1167/9.5.21. [CrossRef] [PubMed]

*Experimental Brain Research*, 77, 432–436. [PubMed] [CrossRef] [PubMed]

*Some aspects of modelling human spatial vision: Contrast discrimination*. Oxford, UK: The University of Oxford.

*Perception & Psychophysics*, 63, 1314–1329. [PubMed] [Article] [CrossRef] [PubMed]

*Nature*, 370, 140–143. [PubMed] [CrossRef] [PubMed]