Most cortical visual neurons do not respond linearly with contrast. Generally, they show saturated responses to stimuli of high contrast, a feature often characterized by a divisive normalization function. This nonlinearity is generally thought to be useful in focusing the dynamic response range of the neuron on a particular region of contrast space, optimizing contrast gain. Some neurons not only saturate but also supersaturate; at high contrast, the response of the neuron decreases rather than plateaus. Under the contrast gain control theory, these cells would seem to reflect a nonoptimal normalization pool that provides excessive inhibition to the neurons. Since very few data on supersaturation are available, this article examines the frequency with which such neurons occur in macaque visual cortex by considering an extension of the Naka–Rushton equation with the capacity to represent nonmonotonic functions. The prevalence of gain-control theories for saturation has occluded an additional computational function for saturation, namely, in detecting the conjunction of certain features. A saturating nonlinearity is a critical part of the selective detection of compound stimuli over their components. In this role, the existence of saturating contrast response functions might be considered necessary rather than simply optimal.

^{−1}) and maintained during surgery with thiopental sodium. The monkey was intubated, the head placed in a stereotaxic frame, and a craniotomy made over the occipital cortex, centered on or near the lunate sulcus. Postsurgical anesthesia was maintained by continuous infusion of sufentanil citrate (4–12 μg·kg

^{−1}·hr

^{−1}) in physiological solution (Normosol-R, Abbott Laboratories, Illinois, USA) with added dextrose (2.5%). Muscular paralysis was then induced and maintained by continuous infusion of vecuronium bromide (100 mg·kg

^{−1}·hr

^{−1}). The monkey was respirated artificially to keep end-tidal CO

_{2}near 33 mm Hg. Electroencephalographic and electrocardiographic data were monitored continuously throughout surgery and the experiment. Rectal temperature was kept near 37°C using a heating blanket. Recordings were made using either tungsten-in-glass (Alan Ainsworth Electrodes, UK) or epoxy-coated tungsten electrodes (FHC Inc., Maine, USA) with an impedance of 1–5 MΩ. Spikes were isolated from single neurons using a combination of thresholding and template-matching techniques.

*f*

_{0}) or the amplitude of modulation of that response at the frequency of the stimulus drift rate (

*f*

_{1}), whichever was larger.

^{1}(Michaelis & Menten, 1913):

*R*is the output response of the neuron,

*B*is the baseline response of the neuron,

*R*

_{max}represents the maximum response of the neuron,

*c*

_{50}represents the contrast at which the response is halfway between baseline and maximum,

*n*is simply referred to as the exponent, and

*c*is the contrast presented to the neuron.

*s*is an additional parameter allowing the suppressive exponent to vary at a different rate to the excitatory exponent. The traditional Naka–Rushton equation is the specific form of this, with

*s*= 1. The characteristics of the function are similar to Equation 1 in that the exponents control the shape of the function,

*R*controls its amplitude, and

*c*controls its contrast scaling. Unfortunately, with

*s*≠ 1,

*R*

_{max}and

*c*

_{50}values do not relate simply to the intuitively appealing maximal response and half-maximum contrast of the function, respectively. The

*R*

_{max}value remains the point at which the curves asymptote, although for nonmonotonic forms, this is obviously less than the maximal response. The

*c*

_{50}value for nonmonotonic functions represents the contrast at the curve's “shoulder” rather than at its half-maximal response. Examples can be seen in Figure 1.

*i*is the index of this particular contrast level,

*e*is the expected response at this contrast level given the current model parameters,

*o*is the observed response, and

*σ*

^{2}is the trial-wise variance in responses at this contrast. Where there was zero variance in responses (arising from no response on any trial), the

*σ*

^{2}was set to 0.001 to prevent the error term from expanding to infinite. In fitting, the exponent terms of the models were bounded in the region 0.3–4.0.

*n*represents the number of data points being modeled,

*p*represents the number of parameters in the model, and RMS is the root-mean-square error of the model fit. The optimal model under this method is taken to be the one with the lowest AIC value (nearest to negative infinity). A related method is the Bayesian Information Criterion, which is identical except that the penalty term is log(

*n*)

*p*instead of 2

*p*. In this study, there were nearly always seven contrast levels, and because log(7) ≈ 2, the methods are equivalent.

*χ*

^{2}error term (Equation 3) and normalizes by the degrees of freedom for the model (Hoel, Sidney, & Stone, 1971):

*χ*

_{N}

^{2}. In practice, for the given data and models, this method typically inflicts a more stringent penalty on models with additional parameters.

*R*

_{max}is the maximum response of the neuron,

*R*

_{100}is the response at maximal contrast, and

*R*

_{0}is the baseline response. This index takes a value of 1 where the neuron's response rises monotonically and less than 1 for neurons demonstrating supersaturation. A neuron whose response at the maximal contrast falls fully back to baseline rate takes an MI of 0. In the current study, the MI was calculated for both the raw data and the model fit to the raw data.

*χ*

_{ N}

^{2}error term, which is lower for the Naka–Rushton equation (0.106) than for the modified version (0.117). The neurons shown in the remaining panels were all better fit by the modified Naka–Rushton model. In the supersaturating functions, it is clear that this is because of the inability of the Naka–Rushton function to provide a nonmonotonic response. However, the cell shown in Figure 2b shows a behavior that was also common, where the neuron did not show any sign of supersaturation but still showed a significantly improved fit with the modified model. This type of behavior can often only be shown for neurons with very consistent responses (note that the error bars in Figure 2b are almost obscured by the data points). The AIC error term, which is less stringent on models with additional parameters, was smaller for all of the six neurons shown in Figure 2.

*R*

_{50}represents the response to a grating of 50% contrast and

*R*

_{100}represents the response to a full-contrast grating. In both cases, the response is taken as either the amplitude of modulation at the temporal frequency of stimulation or the spike rate over and above the baseline rate of the neuron, whichever is greater. The index is bounded by zero and positive infinity. A CSI of 0 represents a linear neuron, for which a subsequent summing circuit would respond identically to a plaid and grating; a CSI of 1 indicates perfect saturation in the contrast range 50–100% such that the summing circuit's response to the plaid is 100% greater than the response to a grating. A CSI greater than 1 indicates a supersaturating cell, capable of providing inputs to a very selective summing circuit.

^{1}In fact, the equation given by Naka and Rushton did not include the exponent

*n*and was therefore

*identical*to that of Michaelis and Menton. Equation 1 is, however, used by most physiology laboratories and still typically referred to as the Naka–Rushton equation.

*Visual Neuroscience*, 2, 41–55. [PubMed] [CrossRef] [PubMed]

*Visual Neuroscience*, 6, 239–255. [PubMed] [CrossRef] [PubMed]

*Journal of Neurophysiology*, 88, 2530–2546. [PubMed] [Article] [CrossRef] [PubMed]

*Vision Research*, 43, 1983–2001. [PubMed] [CrossRef] [PubMed]

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

*Nature*, 420, 320–324. [PubMed] [CrossRef] [PubMed]

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

*Introduction to statistical theory*. Boston: Houghton Mifflin.

*Visual Neuroscience*, 22, 87–99. [PubMed] [CrossRef] [PubMed]

*Pflugers Archiv*, 401, 304–314. [PubMed] [CrossRef] [PubMed]

*Vision Research*, 13, 1255–1267. [PubMed] [CrossRef] [PubMed]

*Visual Neuroscience*, 15, 979–993. [PubMed] [CrossRef] [PubMed]

*Biochemische Zeitschrift*, 49, 333–369.

*Proceedings of the Royal Society of London Series B: Biological Sciences*, 216, 335–354. [PubMed] [CrossRef]

*Fitting models to biological data using linear and nonlinear regression: A practical guide to curve fitting*. New York: Oxford University Press.

*Journal of Physiology*, 185, 587–599. [PubMed] [Article] [CrossRef] [PubMed]

*Vision Research*, 39, 231–256. [PubMed] [CrossRef] [PubMed]

*Neuroscience*, 141, 15–18. [PubMed] [CrossRef] [PubMed]

*Science*, 292, 249–252. [PubMed] [CrossRef] [PubMed]

*Nature Neuroscience*, 4, 819–825. [PubMed] [Article] [CrossRef] [PubMed]

*Vision Research*, 30, 1–10. [PubMed] [CrossRef] [PubMed]

*Journal of Neuroscience*, 25, 4779–4792. [PubMed] [Article] [CrossRef] [PubMed]

*Journal of Neuroscience*, 24, 148–160. [PubMed] [Article] [CrossRef] [PubMed]

*Cerebral Cortex*, 8, 204–217. [PubMed] [Article] [CrossRef] [PubMed]

*Vision Research*, 40, 3145–3157. [PubMed] [CrossRef] [PubMed]

*Neural Computation*, 9, 305–318. [PubMed] [CrossRef] [PubMed]

*Science*, 289, 2347–2350. [PubMed] [CrossRef] [PubMed]

*Proceedings of the Royal Society of London Series B: Biological Sciences*, 202, 409–416. [CrossRef]

*Vision Research*, 25, 755–766. [PubMed] [CrossRef] [PubMed]

*Journal of Neuroscience*, 25, 11666–11675. [PubMed] [Article] [CrossRef] [PubMed]