J. W. Peirce (2007, p. 1) has proposed that saturating contrast-response functions in V1 and V2 may form “a critical part of the selective detection of compound stimuli over their components” and that supersaturating (non-monotonic) functions allow even greater conjunction selectivity. Here, we argue that saturating and supersaturating contrast-response functions cannot be exploited by conjunction detectors in the way that Peirce proposes. First, the advantage of these functions only applies to conjunctions with components of lower contrast than the equivalent non-conjunction stimulus, e.g., plaids (conjunctions) vs. gratings (non-conjunctions); most types of conjunction do not have this property. Second, in many experiments, conjunction and non-conjunction components have identical contrast, sampling the contrast-response function at a single point, so the function's shape is irrelevant. Third, Peirce considered only maximum-contrast stimuli, whereas contrasts in natural scenes are low, corresponding to a contrast-response function's expansive region; we show that, for naturally occurring contrasts, Peirce's plaid detector would generally respond more *weakly* to plaids than to gratings. We also reassess Peirce's claim that supersaturating contrast-response functions are suboptimal for contrast coding; we argue that supersaturation improves contrast coding, and that the multiplicity of supersaturation levels reflects varying trade-offs between contrast coding and coding of other features.

*r*(

*c*), where

*r*is the contrast-response function, and

*c*is the contrast of the stimulus component to which the neuron is tuned. The outputs of the two input neurons are then summed to give the output of the plaid detector. For a maximum-contrast plaid, the contrast,

*c*, of each component is 0.5, while, for a maximum-contrast grating, the contrast of the single component is 1 (see 1 for an explanation of this). The output of the plaid detector in response to the plaid is therefore

*r*(0.5) +

*r*(0.5) = 2

*r*(0.5), while the output in response to the grating is

*r*(1) +

*r*(0) =

*r*(1), assuming zero response to zero contrast.

*r*(0.5) =

*r*(1), so the plaid response equals the grating response (CSI = 0). Most of the V1 and V2 neurons that Peirce (2007) examined had contrast-response functions that either saturated (i.e., started to level off at high contrasts) or supersaturated (i.e., the response initially increased with increasing contrast but then reached a peak and declined with further increases in contrast). These neurons tended to give 2

*r*(0.5) >

*r*(1), so the plaid response was greater than the grating response (CSI > 0). On this basis, Peirce concluded that their contrast-response functions facilitate conjunction detection.

*c*. If we assume that each input neuron responds to only one of the stimulus components, then the response of the conjunction detector to a single component presented in isolation is

*r*(

*c*), and the response to the conjunction stimulus is

*r*(

*c*) +

*r*(

*c*) = 2

*r*(

*c*). The ratio of conjunction response to isolated‐component response is 2, whatever form the contrast-response function takes: Since all the components have the same contrast in both conjunction and non-conjunction stimuli, they all sample the contrast-response function at the same point, so its shape is irrelevant to detection of the conjunction in Peirce's model.

*weaker*response to plaids than to gratings of the same Michelson contrast.

*c*

_{M}, divided by the response to a grating of Michelson contrast

*c*

_{M}, minus 1:

*r*(0) = 0. In the generalized CSI, it is important to allow for the possibility that

*r*(0) ≠ 0, so that the estimated ratio of plaid response to grating response remains accurate at low contrasts for neurons that have a non-zero response to zero contrast.

*r*), we fitted Peirce's (2007) modified Naka–Rushton function to the data given in Peirce's Figure 2, which shows responses to a range of contrasts for six neurons with different levels of saturation, from weak saturation to extreme supersaturation. Given that these neurons were selected by Peirce, they were unlikely to show any selection bias in our favor.

*c*is the contrast of the stimulus component to which the cell is tuned:

*B*is the baseline firing rate at contrast

*c*= 0;

*A*controls the amplitude;

*q*controls the steepness of the initial rise in response;

*c*

_{50}controls the position of the response function along a log contrast axis; and

*s*controls the level of supersaturation. With 0 <

*s*< 1, the neuron never completely saturates: The function asymptotes toward

*r*=

*Ac*

^{ q(1−s)}+

*B*for large

*c*. With

*s*= 1, there is saturation but no supersaturation: The function reduces to the standard Naka–Rushton function, which rises with increasing contrast toward an asymptote of

*r*=

*A*+

*B*; in this case,

*c*

_{50}is the

*semi-saturation contrast*, the contrast at which the response is halfway between

*A*and

*B*. With

*s*> 1, the function supersaturates, rising to a peak at a contrast of

*r*=

*B*. Although, strictly speaking,

*c*

_{50}in Equation 3 is the “semi-saturation contrast” only when

*s*= 1, we will use this phrase more loosely to refer to this parameter whatever the value of

*s*.

*B*was constrained to be non-negative. The resulting contrast-response functions,

*r*, were substituted into Equation 2 to calculate the CSI for each neuron across the whole range of contrasts, 0 to 1.

*c*. In each case, the CSI is positive for high contrasts but negative for low contrasts. The critical contrast,

*c*

_{0}, at which the CSI crosses zero, is given by

*c*

_{0}values for the neurons (given in Table 1 and indicated by the red vertical lines in Figure 1) range from 0.0980 to 0.623, with a mean of 0.336. Whether we sample our contrasts from a natural distribution that peaks at zero (Brady & Field, 2000; Tadmor & Tolhurst, 2000) or close to 0.1 (Clatworthy et al., 2003), this set of contrast-response functions will usually give negative CSI values, indicating

*weaker*responses to plaids than to gratings of the same Michelson contrast for Peirce's (2007) plaid detector.

*c*

_{I}, corresponding to its point of inflection (i.e., point of steepest positive gradient) was close to the mean contrast of natural images (Ohzawa, Sclar, & Freeman, 1985), and this would mean that it saturated at lower contrasts than in the laboratory. However, gain control would also reduce

*c*

_{0}(the critical contrast at which the CSI switches from positive to negative) by the same factor; this is because the CSI function is calculated from the contrast-response function and inherits any horizontal compression applied to the latter. Thus, the ratio

*c*

_{0}/

*c*

_{I}would be unchanged by divisive gain control.

*c*

_{0}is higher than

*c*

_{I}. This means that, even if the contrast-response functions of these neurons did compress horizontally so that their points of inflection occurred at the prevailing mean contrast level,

*c*

_{0}would still be higher than the mean contrast, so most contrasts would still fall below

*c*

_{0}, giving a negative CSI for most natural stimuli. So, with or without contrast gain control, it is clear that, for naturally occurring contrast levels, Peirce's (2007) plaid detector would give a

*weaker*response to plaids than to gratings of the same Michelson contrast most of the time.

Neuron | Parameters of the contrast-response function | c _{0} | c _{I,linear} | c _{I,log} | ||||
---|---|---|---|---|---|---|---|---|

A | B | c _{50} | q | s | ||||

a | 33.0 | 1.66 | 0.363 | 2.23 | 0.93 | 0.623 | 0.248 | 0.430 |

b | 82.2 | 4.50 | 0.150 | 3.75 | 0.862 | 0.379 | 0.147 | 0.195 |

c | 19.4 | 4.21 | 0.234 | 3.11 | 1.06 | 0.400 | 0.182 | 0.219 |

d | 27.6 | 2.00 × 10^{−5} | 0.0639 | 2.33 | 1.04 | 0.0980 | 0.0420 | 0.0599 |

e | 16.0 | 9.61 × 10^{−5} | 0.235 | 3.81 | 1.12 | 0.397 | 0.193 | 0.214 |

f | 1.26 | 2.53 | 0.306 | 1.05 | 3.32 | 0.119 | 0.0586 | 0.140 |

*weaker*, not stronger, response to plaids than to gratings of the same contrast when they are presented at naturally occurring contrast levels. Furthermore, as noted by Peirce (2007), the CSI does not take account of cross-orientation suppression (Bonds, 1989; Freeman, Durand, Kiper, & Carandini, 2002; Morrone, Burr, & Maffei, 1982; Priebe & Ferster, 2006) and, therefore, greatly overestimates the response to the plaid; inclusion of the effect of cross-orientation suppression would extend even further the range of contrasts over which the plaid response was lower than the grating response.

*r*, is close to being proportional to

*c*

^{ q }when

*c*is small compared with

*c*

_{50}, and the distribution of fitted

*q*values in V1 peaks at around 2 using the standard Naka–Rushton function (Albrecht & Hamilton, 1982; Geisler & Albrecht, 1997; Sclar, Maunsell, & Lennie, 1990). If this is the approach that the brain uses to perform multiplication for conjunction detection, then the shape of the contrast-response function does play a critical role, but it is the

*expansive*portion of the function that is critical, not the saturating portion.

*a*−

*b*)

^{2}term in Equation 7 cannot be evaluated by a single neuron when

*a*is free to vary above or below

*b*. This problem can be easily dealt with by splitting the (

*a*−

*b*)

^{2}term into two half-squared terms, ⌊

*a*−

*b*⌋

^{2}and ⌊

*b*−

*a*⌋

^{2}, where ⌊

*x*⌋ = max(

*x,*0), ⌊

*a*−

*b*⌋

^{2}carries the (

*a*−

*b*)

^{2}signal when

*a*>

*b*, and ⌊

*b*−

*a*⌋

^{2}carries this signal when

*a*<

*b*. Assuming

*a*and

*b*to be positive, excitatory, inputs, the Babylonian trick then becomes

*a*and

*b*are the outputs of an initial layer of neurons, each tuned to one of the components of the conjunction, then the computation of Equation 8 would require two additional stages of neuronal processing: Layer 2 computes half-squared linear functions of the inputs (⌊

*a*+

*b*⌋

^{2}, ⌊

*a*−

*b*⌋

^{2}, and ⌊

*b*−

*a*⌋

^{2}), and layer 3 computes a linear function of layer 2's responses. This circuit produces a strong response when both stimulus components are present at sufficient contrast, and zero response when at least one component is missing; and this will still be true even if we introduce physiologically plausible non-linearities to the outputs of layers 1 and 3 (although in that case, the output will no longer be strictly proportional to

*a*×

*b*).

*A*and

*B*, which make excitatory synapses onto an output neuron,

*C*.

*C*has a threshold such that a spike from one of the input neurons is insufficient to trigger a spike in the output, but, if both input neurons produce a spike within a sufficiently short time period, their excitatory postsynaptic potentials sum to produce an above-threshold excitation, triggering a spike in the output neuron. Assuming that

*A*and

*B*fire independently, if the probability that input neuron

*A*spikes within a short time interval is

*P*(

*A*), and the corresponding probability for neuron

*B*is

*P*(

*B*), then the probability that they both spike within the same brief interval (thereby triggering a spike in the output neuron) is

*P*(

*A*)

*P*(

*B*). Since the probability of getting a spike within a short time interval is proportional to the firing rate (Dayan & Abbott, 2001, p. 10), the output firing rate is proportional to the product of the two input firing rates. Using the same basic principle (probability multiplication), Solomon, Chubb, John, and Morgan (2005) demonstrated that multiplicative behavior could occur without explicit multiplication in the kind of psychophysical motion task that van Santen and Sperling (1984) had offered as evidence of multiplication in motion processing.

*N*-methyl-

*D*-aspartate (NMDA). Synaptic input to an NMDA receptor has little effect when the postsynaptic cell is hyperpolarized, but has an increasingly large effect on membrane conductance as the cell becomes more depolarized, which leads to a multiplicative, rather than additive, effect of NMDA receptor activation on membrane potential (Dingledine, 1983). This non-linearity occurs because of a voltage-dependent block of NMDA receptors by Mg

^{2+}ions, which diminishes as the cell becomes depolarized (Mayer, Westbrook, & Guthrie, 1984; Nowak, Bregestovski, Ascher, Herbet, & Prochiantz, 1984).

*mutual information*between the stimulus contrast and the neural response, i.e., the amount of information that the neural response tells us about the stimulus contrast. Mutual information in this case is the reduction in

*entropy*(i.e., unpredictability) of the stimulus contrast that we achieve by receiving the neural response. Mutual information can be expressed as the entropy of the response minus the entropy of the noise so, assuming a constant noise entropy, we maximize the mutual information by maximizing the response entropy. For a single neuron, this is achieved by having a flat distribution of responses, so that the whole dynamic range of the neuron is used with equal probability (any deviation from a flat response distribution makes the neuron's response more predictable, so it has a lower entropy). The contrast-response function that flattens the response distribution has the shape of the cumulative probability distribution of contrasts in the environment, which is a saturating function (Laughlin, 1981). Several types of cell have been found to have this property [large monopolar cells in the compound eye of the fly (Laughlin, 1981), X and Y cells in retina and lateral geniculate nucleus (LGN) of the cat (Tadmor & Tolhurst, 2000), and M cells in macaque LGN (Tadmor & Tolhurst, 2000)].

*q*= 2; the maximum response and semi-saturation contrast were determined by parameters of the cost terms. Similar functions, with steeper or shallower slopes, were obtained by altering the exponent on one of the neural cost terms.

*B*), exponent

*q*= 3, and supersaturation parameter

*s*= 2. This value of

*s*is a special case, giving a contrast-response function that is even‐symmetric about

*c*

_{50}on a log contrast axis, with a similar shape to a Gaussian. The neurons in this panel differ only in

*c*

_{50}, which shifts the contrast-response function left or right on a log contrast axis, but otherwise leaves it unchanged.

*s*as high as 2. We include this example only to demonstrate how much better contrast coding would be if the visual system

*were*like this, as a counterargument to Peirce's claim that supersaturation is suboptimal for contrast coding. More typical values of

*s*are around 1.05 or 1.1, like neurons c, d, and e in Figure 1 and Table 1. Figures 3 and 4 show populations of neurons identical to Figure 2, except with

*s*= 1.1 in Figure 3 and

*s*= 1.05 in Figure 4. We now show how to evaluate all of these coding schemes formally.

*Fisher information*, which sets an upper bound on the accuracy with which a signal can be decoded. The Fisher information,

*I*

_{F}, for contrast

*c*is given by

*p*(r∣

*c*) is the probability of population response r, given contrast

*c*. The second derivative of the log likelihood in Equation 9 is a measure of curvature of the probability distribution, which in turn reflects how narrowly the probability is distributed across contrast. The integral in Equation 9 calculates the expected average curvature over many trials in response to contrast

*c*. For a neural population with a sufficiently high spike count, the Fisher information is very close to the reciprocal of the variance of the contrast estimate when decoding the population response using maximum-likelihood estimation (Dayan & Abbott, 2001, p. 109).

*I*

_{F}(

*c*) =

*Tr*′(

*c*)

^{2}/

*r*(

*c*), where

*T*is the trial duration,

*r*is the contrast-response function (measured in spikes per unit time), and

*r*′ is its first derivative with respect to contrast (Dayan & Abbott, 2001, Chapter 3). For a population of independent, Poisson-spiking neurons, the Fisher information is found by summing the Fisher information across the population:

*r*′, of the contrast-response function to be steep at the test contrast. In the upper panel of Figure 2, each neuron peaking to the right of the test contrast has a corresponding neuron that peaks the same distance from the test contrast but to the left. The pair have the same mean response and gradient magnitude as each other at the test contrast, so they make equal contributions to the sum in Equation 10. The move from supersaturation (upper panel) to saturation (lower panel) eliminates the contribution from the neurons that peak to left of the test contrast, and this halves the Fisher information.

*A*, is replaced with

*r*

_{max}, the peak response difference from baseline:

*r*is

*r*

_{max}+

*B*. It is not appropriate to use Equation 11 when

*s*< 1 (i.e., non-saturating functions) because, in this case, there is no peak response (other than the one imposed by the physical impossibility of exceeding a Michelson contrast of 1). In Equation 11 the contrast,

*c*, is in linear units. The near miss to Weber's law for suprathreshold contrast discrimination (Bird, Henning, & Wichmann, 2002; Nachmias & Sansbury, 1974; Swift & Smith, 1983) and the fact that cortical contrast adaptation is divisive (Ohzawa et al., 1985) suggest that the internal representation of contrast is closer to a log than linear scale, so we now express Equation 11 in terms of

*u*= log

_{ b }(

*c*):

*b*is the base of the logarithm, and

*z*= log

_{ b }(

*c*

_{50}). The first derivative of

*r*is given by

*u*, is given by

*I*

_{F}(

*u*) =

*T*(

*dr*/

*du*)

^{2}/

*r*, which, for a baseline,

*B*, of zero, simplifies to

*z,*so the population Fisher information at test contrast

*u*is found by summing Equation 14 across all the

*z*-values in the population. This gives the Fisher information for the supersaturating coding schemes given in the top panels of Figures 2, 3, and 4. For the saturating schemes in the lower panels, the procedure is the same, except that, for each neuron, we set the Fisher information to zero for any contrast above the peak (given by Equation 4), where the gradient of the contrast-response function is zero.

*b*= 10. For simplicity, we assumed that the log semi-saturation contrasts,

*z,*were uniformly distributed across the log contrast axis, from −1.95 to −0.05 in log contrast steps of 0.1. This range of semi-saturation contrasts (i.e., contrasts of around 0.01 to 1 in linear units) approximately reflects the range found physiologically (Chirimuuta et al., 2003; Clatworthy et al., 2003; Sclar et al., 1990), although monkeys do seem to have a separate population of neurons with semi-saturation contrasts clustered around a value greater than 1 (Clatworthy et al., 2003).

*s*= 1.05, 1.1, and 2) and compare these with the values obtained from the corresponding saturating schemes; the right, middle, and left columns of panels in Figure 5 correspond to the population coding schemes shown in Figures 2, 3, and 4, respectively. The unrealistically strong supersaturation scheme (

*s*= 2) is highly beneficial in terms of both decoding accuracy (indicated by Fisher information) and energy usage (indicated by the population spike rate). For the more realistic levels of supersaturation (

*s*= 1.05 or 1.1), the improvement in contrast decoding accuracy is very slight, but the supersaturating schemes show significant savings in energy usage for moderate and high contrasts.

*z,*described above. However, our conclusions do not depend strongly on this assumption or assumptions about the level of supersaturation. Adding a supersaturating region (i.e., a region of gradient less than zero) to the contrast-response function of any saturating neuron will improve the contrast decoding accuracy for contrasts falling within that region, while reducing metabolic costs. In the special case of

*s*= 2, the downward slope of the contrast-response function is an exact mirror image of the upward slope, and supersaturation improves contrast coding accuracy to the same extent as adding another saturating neuron, while consuming far less energy.

*all*supersaturate strongly, like in the upper panel of Figure 2?” The answer is probably that, as noted by Chirimuuta et al. (2003, p. 1259) and Geisler and Albrecht (1995), neurons do not just code for contrast; they also code for other stimulus attributes, such as orientation and spatial frequency, and the requirements of these different roles may come into conflict. If all neurons supersaturated strongly with contrast, then, for any stimulus contrast, only a small proportion of neurons could respond strongly enough to carry an accurate code for the other stimulus attributes, leading to poor performance on estimation of those attributes. With a saturating (but not supersaturating) scheme, most neurons respond close to maximum over much of the contrast range in response to their preferred stimulus orientation, spatial frequency, etc., and this would improve the accuracy with which those other attributes were coded (Geisler & Albrecht, 1995). According to this view, the multiplicity of different supersaturation levels is the result of varying trade-offs between accurate, energy-efficient, contrast coding, and accurate coding of everything else. This conclusion is consistent with the finding that strongly supersaturating neurons are much more prevalent in the auditory than the visual cortex (Brugge & Merzenich, 1973; Phillips, Semple, Calford, & Kitzes, 1994). The auditory signal has lower dimensionality than the visual signal, so coding of signal level has fewer competing stimulus attributes to trade off against; we would, therefore, expect the auditory cortex to have a greater tendency toward supersaturation, which benefits only coding of signal level.

*c*

_{M}, of a stimulus is defined as follows:

*L*

_{max}and

*L*

_{min}are the maximum and minimum stimulus luminance, respectively. Michelson contrast varies between 0 (when

*L*

_{min}=

*L*

_{max}) and 1 (when

*L*

_{min}= 0). The general formula for a plaid is

*L*(

*x, y*) is the luminance at position (

*x, y*);

*L*

_{0}is the background luminance;

*c*

_{1}and

*c*

_{2}are the contrasts of the two components of the plaid and specify the amplitude of each component as a proportion of the background luminance;

*g*

_{1}and

*g*

_{2}are (usually) sinusoidal functions of space, e.g., sin(2

*πfx*), for a vertical component with spatial frequency

*f,*and sin(2

*πfy*) for a horizontal component. If one component has zero contrast, e.g.,

*c*

_{2}= 0 and

*c*

_{1}=

*c*, then Equation A2 reduces to the equation for a grating:

*g*

_{1}and

*g*

_{2}vary between 1 and −1, so, for the grating in Equation A3, we have

*L*

_{max}=

*L*

_{0}(1 +

*c*) and

*L*

_{min}=

*L*

_{0}(1 −

*c*); substituting these values into Equation A1 gives

*c*

_{M}=

*c*, so the Michelson contrast of the stimulus is given by

*c*in Equation A3. For a plaid with equal-contrast components (

*c*

_{1}=

*c*

_{2}=

*c*), Equation A2 gives

*L*

_{max}=

*L*

_{0}(1 + 2

*c*) and

*L*

_{min}=

*L*

_{0}(1 − 2

*c*), giving

*c*

_{M}= 2

*c*, so the Michelson contrast of the plaid stimulus is twice the contrast of each component.

*P*(

*c*), of contrast

*c*is given by

*P*(

*c*) ∝ exp(−

*c*/

*λ*); but using equal-width bins on a log contrast axis, Clatworthy et al. (2003) found that the contrast distribution peaks close to a contrast of 0.1. This difference can be explained by the fact that switching from equal-width bins on a linear contrast axis to equal-width bins on a log contrast axis is equivalent to multiplying each value of the distribution by the contrast,

*c*, as proved below.

*u*, are given by

*u*= ln(

*c*). Then,

*c*= exp(

*u*). If the width of a bin in log units is

*δu,*and the width of the corresponding bin in linear units is

*δc*, then the ratio of bin widths (linear/log) as

*δu*→ 0 is given by

*c*. So, to calculate the shape of the probability distribution for equal-width bins on log axes, we need to take the probability distribution for equal-width bins on linear axes and multiply each value by the variable bin width, i.e., by

*c,*giving

*P*(

*c*) ∝

*c*exp(−

*c*/

*λ*). This peaks at

*c*=

*λ,*and setting

*λ*= 0.1 gives a distribution very similar to that of Clatworthy et al.

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