**The purpose of this article is to provide mathematical insights into the results of some Monte Carlo simulations published by Tolhurst and colleagues (Clatworthy, Chirimuuta, Lauritzen, & Tolhurst, 2003; Chirimuuta & Tolhurst, 2005a). In these simulations, the contrast of a visual stimulus was encoded by a model spiking neuron or a set of such neurons. The mean spike count of each neuron was given by a sigmoidal function of contrast, the Naka-Rushton function. The actual number of spikes generated on each trial was determined by a doubly stochastic Poisson process. The spike counts were decoded using a Bayesian decoder to give an estimate of the stimulus contrast. Tolhurst and colleagues used the estimated contrast values to assess the model's performance in a number of ways, and they uncovered several relationships between properties of the neurons and characteristics of performance. Although this work made a substantial contribution to our understanding of the links between physiology and perceptual performance, the Monte Carlo simulations provided little insight into why the obtained patterns of results arose or how general they are. We overcame these problems by deriving equations that predict the model's performance. We derived an approximation of the model's decoding precision using Fisher information. We also analyzed the model's contrast detection performance and discovered a previously unknown theoretical connection between the Naka-Rushton contrast-response function and the Weibull psychometric function. Our equations give many insights into the theoretical relationships between physiology and perceptual performance reported by Tolhurst and colleagues, explaining how they arise and how they generalize across the neuronal parameter space.**

^{1}using corresponding lowercase letters.

*X*and

*x*represent the stimulus level,

*R*and

*r*(

*x*) represent the mean spike count,

*N*and

*n*represent the actual spike count, and

**N**and

**n**are vectors holding the spike counts of all the neurons in the population. These variables are explained in more detail in our companion article (see the first paragraph of the section titled “The sensory coding model” in May & Solomon, 2015).

*r*(

*x*), specifies the mean spike count for stimulus

*x*. For visual stimulus contrast, the tuning function is called the contrast-response function. Tolhurst and colleagues modeled this function using the Naka-Rushton function (Naka & Rushton, 1966; Albrecht & Hamilton, 1982). For contrast,

*c*, in linear (e.g., Michelson) units, the Naka-Rushton function has the following form: If we measure contrast in log units,

*x*= log

*, then the Naka-Rushton is given by where See our companion article (May & Solomon, 2015) for a description of all the parameters and for plots of the Naka-Rushton function. On the log contrast scale, the gradient of the Naka-Rushton function peaks at a log contrast of*

_{b}c*x*=

*z*. The term

*c*

_{1/2}, called the semisaturation contrast, is the contrast for which the mean response exceeds

*r*

_{0}by

*r*

_{max}/2. Many authors use

*c*

_{50}to represent this contrast, but we use the less common term

*c*

_{1/2}(except when quoting other authors) because this form is easier to extend to other fractions of

*r*

_{max}, such as

*c*

_{1/3}(the contrast for which the mean response exceeds

*r*

_{0}by

*r*

_{max}/3, which we show to be the contrast that is coded most accurately by a model neuron with a Naka-Rushton contrast-response function with

*r*

_{0}= 0).

*contrast*without specifying the units, we mean log Michelson contrast. To be compatible with Clatworthy et al. (2003) and Chirimuuta et al. (2003), we always used log to base 10 in our modeling (i.e.,

*x*= log

_{10}

*c*and

*z*= log

_{10}

*c*

_{1/2}); however, our equations are derived for the general case of any base,

*b*.

*P*to indicate the type of stochastic process being used.

*r*(

*x*), always giving a Fano factor (ratio of variance to mean) of 1. In the visual cortex, the Fano factor is usually greater than 1 (Dean, 1981b; Tolhurst, Movshon, & Thompson, 1981; Tolhurst, Movshon, & Dean, 1983; Bradley, Skottun, Ohzawa, Sclar, & Freeman, 1987; Skottun, Bradley, Sclar, Ohzawa, & Freeman, 1987; Scobey & Gabor, 1989; Vogels, Spileers, & Orban, 1989; Snowden, Treue, & Andersen, 1992; Britten, Shadlen, Newsome, & Movshon, 1993; Softky & Koch, 1993; Swindale & Mitchell, 1994; Geisler & Albrecht, 1997; Bair & O'Keefe, 1998; Buracas, Zador, DeWeese, & Albright, 1998; McAdams & Maunsell, 1999; Oram, Weiner, Lestienne, & Richmond, 1999; Durant, Clifford, Crowder, Price, & Ibbotson, 2007). To get a Fano factor greater than 1, Tolhurst and colleagues (Chirimuuta et al., 2003; Clatworthy et al., 2003; Chirimuuta & Tolhurst, 2005a, 2005b) used a doubly stochastic Poisson process, which we refer to as the Tolhurst process. This process is a Poisson process in which the mean is itself a random variable sampled from a Poisson process with mean

*r*(

*x*): For this process, the mean spike count is

*r*(

*x*) and the variance is 2

*r*(

*x*), giving a Fano factor of 2. The infinite series in Equation 5 is difficult to handle, so in Supplementary Appendix B we derive a finite series expansion of the Tolhurst process that is more useful.

*F*, is set as a parameter and can take any value greater than or equal to 1. The Consul-Jain distribution takes the following form: where Like the Poisson and Tolhurst processes, the Consul-Jain process generates only nonnegative numbers of spikes, and the spike count variance is proportional to the mean. When

*F*= 1, the Consul-Jain process reduces to the ordinary Poisson process, given in Equation 4. However, the Consul-Jain process with

*F*= 2 is not identical to the Tolhurst process, even though the Fano factor is the same.

*x*that maximizes the likelihood,

*P*(

**N**=

**n**|

*X*=

*x*). In Tolhurst and colleagues' model, the neurons were statistically independent; in addition, if we use May and Solomon's (2015) parameterization of the Goris process, the neurons are implicitly decorrelated if the decoder knows the gain signal. For statistically independent neurons, the population likelihood is then given by the product of the likelihoods of the individual neurons:

*K*is the number of neurons.

*N*is a random variable representing the spike count of neuron

_{j}*j*, and

*n*is its value.

_{j}*R*is a random variable representing the mean spike count for neuron

_{j}*j*, and

*r*(

_{j}*x*) is its value. The second equality (Equation 9) arises because each

*r*(

_{j}*x*) is a deterministic function of the stimulus value,

*x*. To evaluate the probabilities in Equation 9, we use the appropriate expression, depending on which spiking process we are using. For the Consul-Jain process, we use Equation 6. For the Goris process, assuming the decoder knows the gain value, we use the gain-modulated Poisson distribution (May & Solomon, 2015; Equation 10). All the modeling in this article used the Tolhurst process, which is defined in Equation 5.

*T*is the number of trials (= 10,000),

*x̂*is the estimated log contrast estimate (note that we use an uppercase

*x̂*to represent the random variable and a lowercase

*x̂*to represent its value on a particular trial), and the denominator is the sum over all trials. For the models that we consider in this study, the log contrast estimate is largely unbiased (except at very low performance levels), so we have mean[

*x̂*] ≈

*x*. In this case, the accuracy score in Equation 10 is essentially the same as the precision. For consistency with Tolhurst and colleagues, we used their measure of decoding accuracy (Equation 10) when analyzing our Monte Carlo simulations. However, we refer to it as precision (except when directly quoting Tolhurst and colleagues) because our analytical approximations of it are formally measures of precision and because the term

*accuracy*is often used to mean the inverse of bias (e.g., Smith, 1999, chapter 2). We found that, except in degenerate conditions where the model's performance level was very low, it made a negligible difference whether we plotted Tolhursts's accuracy score or true precision. Note that in the Monte Carlo simulations in our companion article (May & Solomon, 2015), we calculated true decoding precision (i.e., reciprocal of the variance of the estimated stimulus value), not the accuracy score defined in Equation 10.

*τ*

_{Tolhurst}(

*x*) and

*τ*

_{C-J}(

*x*). As long as the mean spike count of the most informative neurons is not too low, both

*τ*

_{Tolhurst}(

*x*) and

*τ*

_{C-J}(

*x*) are very close to the following general approximation of the decoding precision,

*τ̃*(

*x*): where

*x*) is the first derivative of neuron

*j*'s tuning function with respect to

*x*. To parameterize

*τ̃*(

*x*) so that it approximates

*τ*

_{Tolhurst}(

*x*) or

*τ*

_{C-J}(

*x*),

*v*should be equal to the Fano factor; the Fano factor is fixed at 2 for the Tolhurst process and can take any value greater than or equal to 1 for the Consul-Jain process. The tilde (˜) above

*τ*indicates that this estimate of decoding precision is based on an approximation of the Fisher information that is not always accurate. With a Fano factor of 1, the Consul-Jain process is the ordinary Poisson, in which case

*τ̃*(

*x*) with

*v*= 1 is exactly equal to the Fisher information (see Dayan & Abbott, 2001, chapter 3). For the Tolhurst process, and the Consul-Jain process with

*F*≠ 1,

*τ̃*(

*x*) with

*v*equal to the Fano factor is an approximation of the Fisher information. For the Goris process, May and Solomon (2015) showed that an appropriate estimate of the decoding precision is given exactly by Equation 11 with

*v*= 1/(1 −

*v*is not the Fano factor—the Fano factor for Goris et al.'s (2014) spiking process is variable and depends on the mean spike count. When the gain is known by the decoder, the Fisher information of the Goris process varies from trial to trial due to the fluctuating gain, and

*τ̃*(

*x*) with

*v*= 1/(1 −

*r*(

_{j}*x*) and

*x*) in Equation 11 using the Naka-Rushton function (Equation 2), then we have Tolhurst and colleagues always used

*r*

_{0}= 0 in their modeling. Since we are focusing on their modeling results, we consider only the case of

*r*

_{0}= 0 in this article. In Equation 12, each neuronal parameter can vary from neuron to neuron, so, strictly speaking, each parameter should be indexed by the neuron number,

*j*, but we omit these indices to reduce notational clutter.

*K*= 1 in the case of a single neuron.

- “Increasing
*R*_{max}increases the contrast identification accuracy of single neurons at all contrasts, most obviously the peak accuracy, without changing the contrast at which accuracy is a maximum” (Clatworthy et al., 2003, p. 1991; note that they use*R*_{max}where we use*r*_{max}). This can be summarized by saying that there is an approximately multiplicative effect of*r*_{max}on decoding precision. - “The position of the accuracy peak along the contrast axis is consistently close to but, interestingly, slightly below the neuron's
*c*_{50}” (Clatworthy et al., 2003, p. 1989). - “The relationship between the maximum accuracy and
*q*is a steep straight line on log-log coordinates” (Clatworthy et al., 2003, p. 1989). - “To change the maximum accuracy . . . requires only a change in the product of
*R*_{max}and number of neurons, i.e., the total number of action potentials generated on average . . . ; for a given accuracy, there is a simple trade-off between the number of neurons and the response amplitude of individual neurons” (Clatworthy et al., 2003, p. 1990).

*τ̃*(

*x*), given in Equation 13.

*τ̃*(

*x*) was sufficiently close to the true decoding precision to make this approach valid. This is important because the Fisher information, on which

*τ̃*(

*x*) is based, is only an upper bound on the decoding precision and can far exceed the true decoding precision for small population sizes (Xie, 2002). In one of their investigations (shown in their figure 5A), Clatworthy et al. (2003) examined the effect of

*r*

_{max}on decoding precision for a single Tolhurst-spiking neuron with Naka-Rushton exponent (

*q*) equal to 2,

*c*

_{1/2}= 0.1, and

*r*

_{0}= 0.

*r*

_{max}took values of 5, 20, 50, 100, or 180 spikes. We replicated their methods (see Supplementary Appendix G for details) and obtained contrast decoding precision scores (calculated using Equation 10) that were essentially identical to those that we read off from their figure 5A. The small differences were almost certainly attributable to the stochastic nature of the simulations and small inaccuracies in our transcription of the data from Clatworthy et al.'s figure. The symbols in Figure 1 show these precision scores. The thin, colored curves show the true Fisher information, calculated numerically (for method, see Supplementary Appendix H). The thick, black curves show

*τ̃*(

*x*) with

*v*= 2. The two panels in Figure 1 are identical except that Figure 1A has a linear vertical axis and Figure 1B has a logarithmic one. The linear vertical axis facilitates comparison with Clatworthy et al.'s figure 5A, which used linear vertical axes, and the logarithmic vertical axis gives a clearer picture of the data for low

*r*

_{max}values, which are flattened out on the linear axis. The logarithmic vertical axis tends to exaggerate the deviations of the decoding precision from Fisher information because the precision scale is expanded for the worst-matching conditions (those with low precision) relative to the best-matching conditions.

**Figure 1**

**Figure 1**

*τ̃*(

*x*) is very close to the true Fisher information at the peaks (compare the thick, black lines against the thin, colored lines). This is important because the five observations that we analyze in this section are about the peaks, and our justification for using

*τ̃*(

*x*) to predict decoding precision is that it is approximately equal to the Fisher information.

*r*

_{max}of 50 spikes or more, the horizontal and vertical position of the peak of precision coincides closely with the peak of

*τ̃*(

*x*) (compare symbols against solid curves). This allows us to explain many of Clatworthy et al.'s observations about the precision peak by deriving equations that explain corresponding findings for

*τ̃*(

*x*). For

*r*

_{max}values substantially lower than 50 spikes, the decoding precision does not match

*τ̃*(

*x*) at all well, even at the peak, but Clatworthy et al.'s observations do not apply here either. In the following five subsections, we explain each of the five observations listed previously using the approximation of the decoding precision,

*τ̃*(

*x*), given by Equation 13.

*r*

_{max}

*τ̃*(

*x*) is proportional to

*r*

_{max}, so increasing

*r*

_{max}increases the precision by the same multiplicative factor for each contrast level; the position of the peak across contrast is unchanged, and the largest absolute change is for the peak precision. This multiplicative change gives rise to the vertical shifts seen for the black curves on the logarithmic vertical axis in Figure 1B.

*τ̃*(

*x*) along the contrast axis can be found by differentiating

*τ̃*(

*x*) with respect to

*x*, setting the derivative to zero, and solving for

*x*. From Equation 13, we have Setting this to zero gives Using Equation 15 to substitute for

*x*in Equation 2 gives (for

*r*

_{0}= 0) So, regardless of the values of

*v*,

*r*

_{max},

*q*, or

*c*

_{1/2},

*τ̃*(

*x*) for a single neuron will always peak at the contrast for which the mean response is

*r*

_{max}/3. We introduce the term

*c*

_{1/3}for the Michelson contrast that gives rise to a mean response of

*r*

_{max}/3, to be consistent with the term

*c*

_{1/2}for the semisaturation contrast. The value of log

_{10}(

*c*

_{1/3}) is indicated by the black vertical line in each panel of Figure 1, and it passes through the peak of each thick, black curve.

*τ̃*(

*x*) occurs at a log contrast given by Equation 15. Using this equation to substitute for

*x*in Equation 13, we find that the peak value of

*τ̃*(

*x*) is given by which is independent of

*c*

_{1/2}.

*q*is a steep straight line on log-log coordinates

*τ̃*(

*x*)] is proportional to

*q*

^{2}, giving a straight line on log-log coordinates. Figure 2 plots max[

*τ̃*(

*x*)] as a function of

*q*for a single Tolhurst-spiking neuron with

*r*

_{max}= 50 spikes and shows that max[

*τ̃*(

*x*)] is very close to the true peak of precision for this model neuron obtained using Clatworthy et al.'s Monte Carlo methods.

**Figure 2**

**Figure 2**

**Figure 3**

**Figure 3**

*r*

_{max}and number of neurons

*r*

_{max}and the number of neurons,

*K*, trade off so that, regardless of the individual values of

*r*

_{max}and

*K*, the decoding precision is a function of their product. In fact, unless

*r*

_{max}×

*K*is very low, the decoding accuracy is close to being proportional to

*r*

_{max}×

*K*. This observation is easily explained by Equation 13, which shows that

*τ̃*(

*x*) is proportional to

*r*

_{max}×

*K*.

*q*= 2 and

*c*

_{1/2}= 0.1. The near proportionality between decoding accuracy and

*r*

_{max}×

*K*is indicated by the fact that most of the points lie on a straight line of gradient 1 on these log-log axes. The straight line in our Figure 3 is a plot of max[

*τ̃*(

*x*)], calculated according to Equation 18. It clearly provides a very good match to the decoding precision data for

*r*

_{max}×

*K*≥ 50.

*τ̃*(

*x*) gives an exact linear trade-off between

*r*

_{max}and

*K*, the true Fisher information for the Tolhurst process does not. As

*r*

_{max}decreases,

*τ̃*(

*x*) tends to underestimate the true Fisher information.

*τ*

_{Tolhurst}(

*x*), derived in Supplementary Appendix E: where

*H*

_{Tolhurst}(

*x*) is plotted in Figure 4. As the spike rate of each individual neuron increases,

*H*

_{Tolhurst}(

*x*) for that neuron approaches 1/2, and so

*τ*

_{Tolhurst}(

*x*) approaches

*τ̃*(

*x*) (Equation 11) with

*v*= 2. However, as the spike rate decreases,

*H*

_{Tolhurst}(

*x*) increases, and so

*τ*

_{Tolhurst}(

*x*) exceeds

*τ̃*(

*x*) and better reflects the true decoding precision.

**Figure 4**

**Figure 4**

**Figure 5**

**Figure 5**

*r*

_{0}= 0, we can expand Equation 19 in a similar way to Equation 12: Unlike

*τ̃*(

*x*),

*τ*

_{Tolhurst}(

*x*) is specific to the Tolhurst spiking process rather than being a general approximation of the decoding precision that applies to several different spiking processes.

*τ*

_{Tolhurst}(

*x*). As

*r*

_{max}decreases,

*τ*

_{Tolhurst}(

*x*) and

*τ̃*(

*x*) start to diverge by a factor that approaches (1 – 1/

*e*)/0.5 = 1.264—that is, the ratio of the maximum to minimum values of

*H*

_{Tolhurst}(

*x*).

*r*

_{max}and

*K*applies to the Tolhurst process but not to the Poisson or Goris processes: for these,

*τ̃*(

*x*) is derived from the exact expression for the Fisher information, so the trade-off that we derived for

*τ̃*(

*x*) applies to these processes exactly.

*r*

_{max}(or, for populations,

*r*

_{max}×

*K*) is substantially less than 50 spikes, the peak of precision does not coincide closely with that of

*τ̃*(

*x*) or even the true Fisher information, so the explanations of Clatworthy et al.'s findings given above are no longer valid. However, most of these findings do not apply for these low

*r*

_{max}values either. The failure of observations 1 and 2 at low spike rates is clear from Figure 1, the failure of observation 5 at low spike rates is shown in Figure 3, and the failure of observation 4 at low spike rates is demonstrated in Figure 6.

**Figure 6**

**Figure 6**

*r*

_{max}for a 200-ms stimulus is only around 5.7 spikes for V1 neurons (Geisler & Albrecht, 1997), suggesting that, with single-neuron models, the spike count has to be implausibly high for the decoding precision to be well approximated by the Fisher information. However, as shown by Clatworthy et al. (2003), Chirimuuta and Tolhurst (2005a), and our Figure 3,

*r*

_{max}can be approximately traded off against the number of neurons so that what is important is the average total spike count of the population rather than that of the individual neurons. With a population code, it is possible to achieve a high total population spike count while keeping the spike counts for the individual neurons at a plausible level. This makes the Fisher information more relevant to understanding population-coding models than coding schemes based on a single neuron. So far we have considered only populations of identical neurons. We now turn to populations of differently tuned neurons.

*r*

_{max}= 10,

*r*

_{0}= 0, and

*q*= 2. One population had log

_{10}(

*c*

_{1/2}) values uniformly distributed between −3 and 0.1, and the other two had

*c*

_{1/2}values distributed according to the recorded values in either cat or monkey populations, found by arranging the neurons in ascending order of

*c*

_{1/2}and then sampling the population at equal percentile intervals. We did not have access to the exact sets of cat or monkey

*c*

_{1/2}values that they used, but we estimated them by fitting functions to the cat or monkey

*c*

_{1/2}distributions given in Clatworthy et al.'s figure 6 and then sampling these distributions in equal percentile steps (see Supplementary Appendix I for details). The advantage of our method is that it can easily be extended to neuronal populations of any size. Having set up the populations of neurons, we then calculated decoding precision scores using Clatworthy et al.'s Monte Carlo methods (see Supplementary Appendix G for details). Figure 7 shows that our decoding precision scores are very similar to those of Clatworthy et al., confirming that our method of generating the sets of

*c*

_{1/2}values is a good approximation to that of Clatworthy et al. The colored curves in Figure 7 show

*τ̃*(

*x*), calculated using Equation 12 with

*v*= 2. Since

*r*

_{max}= 10 in these simulations,

*H*

_{Tolhurst}(

*x*) ≈ 0.5 for the most informative neurons, so

*τ*

_{Tolhurst}(

*x*) ≈

*τ̃*(

*x*); there was no advantage in using the closer but more complex approximation. Figure 7 shows that, even for these small populations of neurons, with

*r*

_{max}of only 10 spikes, the decoding precision from the Monte Carlo simulations is very close to

*τ̃*(

*x*) over a wide range of contrasts.

**Figure 7**

**Figure 7**

*τ̃*(

*x*) is a good approximation of the decoding precision for the Tolhurst process as long as

*r*

_{max}is not too low. For low values of

*r*

_{max},

*τ̃*(

*x*) tends to underestimate the true Fisher information. As noted earlier, this means that there is not an exact trade-off between

*r*

_{max}and the number of neurons,

*K*. For low values of

*r*

_{max}, it is better to use

*τ*

_{Tolhurst}(

*x*) to accurately predict decoding precision. This is demonstrated in Figure 8.

**Figure 8**

**Figure 8**

*α*is the threshold—that is, the target contrast that gives

*P*(correct) = 1 – 0.5/e = 0.816…—and

*β*controls the function's shape on linear axes or slope on log axes (see May & Solomon, 2013, for an in-depth analysis of the Weibull function). In psychophysical contrast detection experiments with human observers,

*β*usually takes a value of about 3 (Foley & Legge, 1981; Nachmias, 1981; Mayer & Tyler, 1986; Meese, Georgeson, & Baker, 2006; Wallis, Baker, Meese, & Georgeson, 2013).

*r*

_{0}= 0 and

*q*= 2, Weibull

*β*for detection varied from 1.75 to 1.99 as the number of neurons increased from 1 to 23, but the model's

*β*never reached the normal human level of around 3. Chirimuuta and Tolhurst then introduced a threshold to the Naka-Rushton function by subtracting a constant value from the output and setting negative values to zero. With a threshold on the Naka-Rushton function,

*β*ranged from 2.25 to 4.20, providing a better match to psychophysically obtained values. Chirimuuta and Tolhurst assumed that their failure to obtain sufficiently high Weibull

*β*values with the standard Naka-Rushton function had been caused by the lack of a threshold, and they suggested that the standard, unthresholded Naka-Rushton function “may be crucially inadequate” as a model of the neuronal contrast-response function (Chirimuuta & Tolhurst, 2005a, p. 2956). However, we now show that, when

*r*

_{0}= 0, the model's psychometric function is close to a Weibull function with

*β*=

*q*. Thus, the real reason that Chirimuuta and Tolhurst always obtained a

*β*of about 2 with the standard Naka-Rushton function is that they always kept

*q*at 2 in these simulations. We show that one can obtain any Weibull

*β*by setting

*q*close to the required

*β*value.

*r*

_{0}= 0, the model's psychometric function for 2AFC contrast detection is close to a Weibull function with

*β*=

*q*

*r*

_{0}= 0, there is zero response to zero contrast: The nontarget stimulus can never elicit a single spike. If the target elicits at least one spike, the model will respond correctly. If the target fails to elicit any spikes, the model has to guess and will be correct half the time on a 2AFC task. In summary, the model will be correct on all 2AFC trials except half of those on which the target failed to elicit any spikes. This statement can be formalized as follows: where

*P*(no spikes) is the probability of getting no spikes in response to the target. We can already see that the model's psychometric function has a similar form to the Weibull function:

*r*(·), as a function of Michelson contrast (rather than log contrast) in Equation 24. For a population of

*K*statistically independent neurons, where

*r*(

_{j}*c*) is the contrast-response function of neuron

*j*. Using Equation 1 to expand

*r*(

_{j}*c*), we have where (

*r*

_{max})

*and (*

_{j}*c*

_{1/2})

*are the*

_{j}*r*

_{max}and

*c*

_{1/2}parameters, respectively, of neuron

*j*. Using Equation 26 to substitute for

*P*(no spikes) in Equation 23, we get an exactly correct expression for the model's psychometric function for contrast detection: If all the neurons in the population being monitored by the observer have the same contrast-response function, then Equation 27 reduces to In this case, Equation 28 shows that there is an exact linear trade-off between

*r*

_{max}and

*K*in the psychometric function for 2AFC detection rather than the approximate trade-off that we get with the Fisher information.

*r*

_{max}is sufficiently high, the contrast detection threshold will be somewhat lower than the lowest

*c*

_{1/2}in the population. Thus, at threshold (i.e., around the middle of the psychometric function), the denominators of Equations 26 to 28 become dominated by the

*c*term will make little difference. Dropping

^{q}*c*from the denominator of Equation 27, we get where

^{q}*α*is a constant, given by Relation 29 has the form of a Weibull function with

*β*=

*q*. The near equality in Relation 29 approaches equality as

*K*or

*r*

_{max}increase. □

*β*for all detectors. Nachmias (1981) called this assumption of identical

*β*for each detector the “homogeneity assumption,” and it is largely equivalent to the implicit assumption in the above proof that all neurons have the same

*q*. In the Discussion we expand on the links between our analysis and that of Quick (1974).

*P*(correct) = 1, with increasing target contrast. For most instantiations of the model, this is very close to the truth. However, when

*r*

_{max}and

*K*are both very low, the model's asymptotic performance is far below 1. This is because as

*c*increases, the denominator of Equation 26 becomes more and more dominated by the

*c*term, and the

^{q}*c*

_{1/2}

^{q}term makes less and less difference, so For very low

*r*

_{max}and

*K*, this value can be substantially above zero so that, even for infinite contrast, the model has to guess on a significant proportion of 2AFC trials.

*P*(correct) = ( 1 –

*λ*) as

*c*→ ∞. When

*λ*= 0, Equation 32 reduces to Equation 22. We now derive an expression for the model's “lapse rate” parameter. (Strictly speaking, the model never lapses; a low asymptotic performance level arises from a low spike rate rather than a finger error or a failure to look at the stimuli on some 2AFC trials.)

*P*(no spikes) in Equation 23, we obtain the asymptotic value of

*P*(correct) given by Since this asymptotic value of

*P*(correct) is (1 –

*λ*), we have This expression for

*λ*quickly approaches zero as

*r*

_{max}or

*K*increase.

*α*,

*β*, and

*λ*are plotted as symbols in Figures 9, 10, and 11, respectively. The solid lines plot the corresponding analytical expressions (

*α*given by Equation 30,

*β*given by

*q*, and

*λ*given by Equation 34).

**Figure 9**

**Figure 9**

**Figure 10**

**Figure 10**

**Figure 11**

**Figure 11**

*λ*values extremely well in Figure 11. Our analytical expressions for

*α*and

*β*are approximations that become increasingly accurate as

*K*or

*r*

_{max}increase.

*F*= 1). Equation E.17 of Supplementary Appendix E states that, for a single Consul-Jain-spiking neuron, where

*F*is the Fano factor. If we follow a series of mathematical steps analogous to Equations 23 to 29 above but use Equation 35 instead of Equation 24 to express the probability of a single neuron not spiking, we obtain an approximation of the psychometric function with the same form as Relation 29 but with where

*F*is the Fano factor or neuron

_{j}*j*. Similarly, if we follow a series of steps analogous to those in Equations 31 to 34 but use Equation 35 instead of Equation 24 to express the probability of a single neuron not spiking, we obtain the following expression for the “lapse rate” parameter:

*τ̃*(

*x*), which can be adjusted to apply to a variety of different spiking processes by setting the value of a single scalar parameter,

*v*. For the Tolhurst and Consul-Jain processes,

*v*should be set to the Fano factor. For the Goris process,

*v*should be set to 1/1(1 −

*τ̃*(

*x*) is an estimate of the Fisher information. For the Goris process, the Fisher information varies across trials, and

*τ̃*(

*x*) is its modal value.

*τ̃*(

*x*) revealed some surprisingly simple relationships between decoding precision and the neuronal parameters, and explained the five observations of Clatworthy et al. (2003) that we investigated. For example, Equation 18 shows that, for a population of identical, statistically independent neurons, the height of the peak of decoding precision is approximately proportional to

*r*

_{max}×

*K*×

*q*

^{2}and is independent of

*c*

_{1/2}. The expression for

*τ̃*(

*x*) also revealed that, to a good approximation, the contrast most accurately encoded by the neuron is that for which the mean response is

*r*

_{max}/3; we call this contrast

*c*

_{1/3}. Figure 7 shows that

*τ̃*(

*x*) matches the decoding precision very closely for a population of only 18 Tolhurst-spiking neurons with

*r*

_{max}of only 10 spikes.

*τ̃*(

*x*) is an exact expression for the modal Fisher information for the Goris process and gives the exact expression for the Fisher information for the Poisson process, which is the Consul-Jain process with a Fano factor of 1. However, for the Tolhurst process and the Consul-Jain process with

*F*> 1,

*τ̃*(

*x*) is an approximation of the Fisher information and is less accurate when the mean spike rate of the most informative neurons is very low. In Supplementary Appendix E, we derived expressions,

*τ*

_{Tolhurst}(

*x*) and

*τ*

_{C-J}(

*x*), that are close approximations of the Fisher information of the Tolhurst and Consul-Jain processes across all parameter values. These expressions reveal more complicated relationships between decoding precision and the neuronal parameters that hold when the spike rate is very low. Figure 8 shows the superiority of

*τ*

_{Tolhurst}(

*x*) over

*τ̃*(

*x*) at very low spike rates.

*β*=

*q*as

*r*

_{max}or

*K*increase. We thus refuted Chirimuuta and Tolhurst's conclusion that the standard Naka-Rushton function is unable to give Weibull

*β*values that are high enough to match those of human observers. To obtain

*β*values of around 3, typical of human observers, we need a Naka-Rushton exponent of around 3. Such levels are not atypical, and indeed Table 1 shows that the mean

*q*over 628 cells from three physiological studies of V1 is 2.9.

**Table 1**

*q*, at a physiologically plausible level of 2. However, this turned out to be a fatal decision because it prevented them from ever finding a parameterization of the model that gave a sufficiently high Weibull

*β*with the standard Naka-Rushton tuning function. Using the analytical approach, Relation 29 makes it immediately clear that the Naka-Rushton exponent,

*q*, is the key parameter for controlling

*β*, and that the model's psychometric function approximates a Weibull function with

*β*actually equal to

*q*.

*r*

_{0}= 0. With nonzero

*r*

_{0}, the analytical form of the psychometric function is different (we have analyzed this more general case and will present it in another article). It is implausible that, in human vision, contrast detection is mediated entirely by neurons with zero spontaneous firing rate. The assumption that there is no neuronal response to zero contrast is often called the high-threshold assumption.

^{2}Under the conventional assumption of additive, stimulus-independent noise, the lack of response to zero contrast implies that there is a threshold on the output of the sensory units that lies enough standard deviations above the mean of the noise for there to be a negligible probability of a response to zero contrast. Because Chirimuuta and Tolhurst's model has no sensory response to zero contrast, it is formally equivalent to a high-threshold model even though (with the standard Naka-Rushton function) it does not actually contain a sensory threshold. In high-threshold theory (and in Chirimuuta and Tolhurst's model), detection errors are always unlucky guesses on 2AFC trials that failed to elicit a response, whereas there is plenty of psychophysical evidence that incorrect responses are caused at least partly by hallucinations due to noise in the nontarget interval rather than entirely by unlucky guesses (Tanner & Swets, 1954; Swets, Tanner, & Birdsall, 1961; Nachmias, 1981; Solomon, 2007; Laming, 2013).

*r*

_{0}= 0 as a model of contrast detection, we presented our analysis of it for three reasons.

- It allows us to fully understand why Chirimuuta and Tolhurst's (2005a) contrast detection simulations resulted in a fitted Weibull
*β*that approached 2 with increasing number of neurons. This in turn allows us to refute their conclusion that the Naka-Rushton function requires a threshold to make it a plausible model of the neuronal contrast-response function. - Neurons with zero or negligible spontaneous firing rates do exist (e.g., see Dean, 1981a, figure 2), so it is not inconceivable that there are some organisms or experimental situations to which our analysis applies.
- It allows us to address Tyler and Chen's (2000) claim that high-threshold analysis of probability summation is “fundamentally flawed.” This idea is explored in the next subsection.

*r*

_{0}= 0, each neuron acts as a detector; the observer detects the target if at least one neuron responds during the target presentation, and has to guess the correct answer otherwise. The more neurons the observer is monitoring, the greater the chance that at least one neuron will respond. The term

*probability summation*refers to this increase in detection probability due to an increase in the number of detectors. The psychometric function in this case gives the probability that at least one neuron responds during the target presentation, or the observer guesses correctly if none respond.

*β*, then the observer's psychometric function will be a Weibull function with that

*β*-value and with the detection threshold parameter,

*α*, given by where

*α*is the detection threshold parameter of detector

_{j}*j*. For a very clear derivation of Equation 38, see Nachmias (1981), but note that his equation 4 has a typographical error: It is missing the minus sign on the exponent,

*β*. If all the detectors are identical (but statistically independent), then they all have the same

*α*, and Equation 38 reduces to

_{j}*β*parameter controls how much of a reduction in detection threshold we achieve by increasing the number of detectors,

*K*. If

*β*is low, then the detection threshold,

*α*, decreases quickly as

*K*increases; if

*β*is high, then the detection threshold decreases more slowly as

*K*increases.

*β*=

*q*and Thus, there is a near equivalence between Quick's model and that of Chirimuuta and Tolhurst. If all the neurons in Chirimuuta and Tolhurst's model have the same contrast-response function, then the model's detection threshold approximation given by Equation 30 reduces to Therefore, since

*β*≈

*q*in Chirimuuta and Tolhurst's model, their model shows approximately the same probability summation effects as Quick's model, with detection threshold proportional to

*K*

^{−1/}

*. For the modeling in Figure 9, all the neurons were identical, so Equation 41 is equivalent to Equation 30 (which was used to generate the solid lines in Figure 9), and it is clear that this equation does accurately predict the detection threshold of Chirimuuta and Tolhurst's model, particularly for the higher values of*

^{β}*r*

_{max}or

*K*.

- The observer monitors a set of channels.
- Within each channel is a continuous signal that increases linearly with the stimulus strength.
- Noise is added to this signal.
- The noise might be additive (standard deviation independent of the signal level) or multiplicative (standard deviation proportional to a power function of the signal level), or the noise could be the sum of an additive and a multiplicative component.
- A sensory threshold is applied to the noisy internal signal in each channel so that a stimulus is detected if and only if the noisy signal falls above threshold in at least one of the channels.

^{3}They showed that, for most Weibull

*β*values, the noise distribution deviated markedly from a Gaussian. They found this unacceptable because, according to the central limit theorem, the sum of a large number of non-Gaussian-distributed random variables is asymptotically Gaussian distributed. Therefore, if there are many different sources of noise from the stimulus to the neuronal decision mechanism, the noise is likely to be Gaussian at the decision mechanism. They then showed that high-threshold probability summation fails for additive Gaussian noise. They showed that, if the sensory threshold were low enough to be exceeded by the noisy signal in one channel 75% of the time, then the noisy signal would exceed the sensory threshold in at least one of 100 channels almost all the time, even when the stimulus intensity was reduced to zero. Thus, if the stimulus area or number of components increased so that the observer was monitoring many more channels, the observer would almost always be in a detect state, even when the stimulus was absent. The signal would have to be reduced to a physically unachievable negative contrast for the observer to be in a detect state 75% of the time. This problem does not occur with the noise distribution implied by the Weibull function because the Weibull probability density function falls to zero at the sensory threshold (see Supplementary Appendix K, especially Figure K.1); therefore, when the stimulus intensity is zero, no detector's noisy signal will exceed its sensory threshold. However, as already noted, Tyler and Chen (2000) rejected that distribution because of its “bizarre” non-Gaussian nature when

*β*is not close to 4 (p. 3127). They therefore concluded that probability summation with high-threshold theory is fundamentally flawed.

*K*

^{−1/}

*, as in standard high-threshold Weibull analysis. The contrast threshold for detection never reaches zero regardless of how many neurons the observer is monitoring, but we do not have to rely on bizarre model characteristics to achieve this. Aside from the zero spontaneous firing rate, the neurons in Chirimuuta and Tolhurst's model have contrast-response functions and noise distributions that are physiologically plausible to a reasonable extent. One could ask what the equivalent “continuous signal plus additive noise” model is. Since Chirimuuta and Tolhurst's model produces a psychometric function that closely approximates a Weibull function, the equivalent “continuous signal plus additive noise” model is closely approximated by the one derived by Tyler and Chen (and outlined in Supplementary Appendix K), with the “bizarre” noise distributions. However, this bizarreness comes from forcing Chirimuuta and Tolhurst's model into the Procrustean bed of “continuous signal plus additive noise” rather than being an implausible characteristic of the model itself.*

^{β}*τ̃*(

*x*), which approximates the Fisher information for a population of neurons with Tolhurst's spiking process as long as the mean spike count of the most informative neurons is around five spikes or more.

*τ̃*(

*x*) is also an estimate of the Fisher information for the Consul-Jain spiking process. Furthermore, it gives the exact Fisher information for the Poisson process and the exact modal value of the Fisher information for the Goris process when the decoder has access to the gain signal in Goris et al.'s (2014) model of neuronal spiking. Our expression for

*τ̃*(

*x*) revealed simple relationships between the properties of the neurons and the decoding precision that hold to a good approximation when the mean count rate is sufficiently high. We used this expression to explain five relationships between decoding precision and the neuronal parameter values that Clatworthy et al. (2003) observed from their Monte Carlo simulations.

*K*or

*r*

_{max}increase, the psychometric function asymptotically approaches a Weibull function with

*β*=

*q*. Our work therefore reveals a previously unknown theoretical connection between two of the most widely used functions in vision science: the Weibull psychometric function and the Naka-Rushton contrast-response function. This relationship explained why Chirimuuta and Tolhurst always obtained a Weibull

*β*of about 2 in their modeling (they always had

*q*= 2 in their assessments of the model's Weibull

*β*) and allowed us to refute their conclusion that it is necessary to have a threshold on the Naka-Rushton function to achieve Weibull

*β*values that match those found with human observers. Their threshold on the Naka-Rushton function had a similar effect to increasing

*q*, as it made the spike rate increase more abruptly with increasing contrast.

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^{1}In this paper, we use the word

*trial*in two ways. First, we use it in the way a physiologist would, to mean a stimulus presentation. Second, we use it to mean a trial on a two-alternative forced-choice (2AFC) psychophysical experiment, in which the observer is presented with two stimuli and has to make a response. To distinguish these two meanings, we always refer to the latter type of trial as a 2AFC trial.

^{2}In this discussion, we use the word

*threshold*in two ways: (a) to refer to an internal threshold on the sensory signal, and (b) to refer to the stimulus contrast corresponding to a particular level of detection performance. We have tried to make the meaning clear in each case by using the term

*sensory threshold*for the former case and the term

*detection threshold*for the latter.

^{3}As pointed out by Mortensen (2002), Tyler and Chen's (2000) published equation for the probability density function (PDF) of the noise (Tyler and Chen's equation 4b) contains errors. However, Tyler and Chen's plots of the noise PDFs (shown in their figure 2b) are correct, so we assume that the errors in Tyler and Chen's equation 4b are typographical errors rather than fundamental problems with their analysis. To clarify matters, we present in Supplementary Appendix K a derivation of the PDF that is based on Mortensen's derivation but is hopefully easier to follow than Mortensen's derivation or that of Tyler and Chen.