Near-Miss to Weber’s Law

Empirical tests of Weber’s law in the fields of acoustic [9], [18] and visual [11] perception have revealed that for very high stimulus magnitudes observers perform better than predicted. (Fechner [4] pointed out that the Weber fraction remains constant only for a limited range of stimulus magnitudes.) Discrimination performances in these studies are better fitted by an expression that allows sensitivity to grow as a power function of stimulus magnitude [9]–[12]:(3)where K(π) and β(π) are real valued parameters that may depend on the value of π, and ξ_π(a) gives the magnitude of a stimulus that is judged greater than a with probability π. If the value of β(π) is one, then Weber’s law is satisfied. Equation 3 has been demonstrated to hold over a wide range of magnitudes and because the exponent β is typically estimated around 0.9, equation 3 is referred to as the near-miss to Weber’s law [9]–[12], [18].

Here we consider a slightly different formulation of the near-miss to Weber’s Law and define the near-miss relative intensity of two stimuli with magnitudes x and a with x>a as(4)

The parameter β determines how strong the magnitude effect is with respect to the distance effect and if it equals 1, then near-miss relative intensity reduces to relative intensity. Thus, we consider Weber’s law to be satisfied when the parameter β is estimated to be one and invoke the near-miss to Weber’s law when β significantly differs from one.

Knowledge of gustatory information processing [7], [19], [20] is important for understanding the formation of economical food preferences [21]–[24] and may have implications for our understanding of the co-evolution of nectar rewards and animals’ discrimination abilities. Diverse groups of nectar-feeding animals such as bees [24]–[26], birds [27]–[29] and bats [23], [30], [31] show a general pattern of preference for sweeter sugar solutions and more precise discrimination at low concentrations. (Discrimination of nectar volume rewards also follows this pattern, so that choice proportions in 2AFC tests can be fitted against the relative intensity of the stimuli [22], [32], [33].) Although the results were consistent with Weber’s law, the law was not rigorously tested in these studies. Here, we present a series of 2AFC tests with nectar-feeding bats (Glossophaga soricina) designed to test directly whether concentration pairs with the same relative intensity result in equal discrimination probabilities (as predicted by Weber’s law). We also tested whether near-miss relative intensity is a better predictor of discrimination performance than relative intensity. We used the method of constant stimuli with a standard feeder giving rewards with 20% weight/weight sugar concentration and a test (referent) feeder, whose concentration was systematically varied. This allowed us to construct individual psychometric functions for the discrimination of sugar concentration. Finally, we reviewed previously published data on bats, hummingbirds, honeybees, and bumblebees in order to test whether Weber’s law or the near-miss to Weber’s law provide a better fit to sugar concentration discrimination performance.

Ethics Statement

Treatment of the experimental animals complied with the national laws on animal care and experimentation, under license of Veterinäramt Bielefeld. A specific ethical approval was not required due to the observational nature of the study that caused no suffering, damage, or pain to the animals.

Subjects

Experiments were carried out with five female and one male Pallas’s long-tongued bats (Glossophaga soricina) from the same colony at Bielefeld University. The climatic conditions in the housing room (ca. 2.2×3.4×3.7 m) were ca. 25°C and ca. 60% humidity. The colony received ad libitum 20% honey water, honey water mixed with supplementary nutrients: Nektar Plus (Nekton®, Günter Enderle, Pforzheim, Germany) and Alete2Folgemilch (Nestle), Multi-Mulgat® (BioWeyxin, Veyx-Pharma GmbH, Schwarzenborn, Germany) and bee-collected flower pollen. Once a month they were also provided with live flies (Musca domestica). The six experimental individuals were adult, older than one year of age. Bats were marked with unique Radio Frequency Identification tags (RFID: 12×2.1 mm, 124 kHz, Sokymat) using self-made silicon collars (total collar and RFID weight = 0.20 g, less than 2.5% of the body mass of the smallest bat). After the experiment, the animals were returned to the colony. Light conditions during the experiments were 12∶12 LD and all experiments were conducted during the dark phase.

Cage and Feeder System

During the experiments each bat was placed individually inside one of three adjacent cages (0.7×1.5×2.2 m) [34], inside a 3.4×3.7×3.8 m room. Each cage contained two computer-controlled feeders on the back wall and a hanging roost. Visits to the feeders were automatically detected by infrared sensors. Upon detecting a visitor, feeders delivered a fixed amount of 20 µL sugar water (hereafter ‘nectar’) as a reward that the bats removed by licking while briefly hovering in front of a feeder. Nectar reward delivery was controlled by two syringe pumps using two gas-tight Hamilton glass syringes (Series 1025). Feeders were connected to the pumps via two identical systems of pinch valves and tubes (Figure 1). Access to each feeder could be blocked automatically by moving a swivel arm with a plastic guard in front of the feeder opening. Details of the experimental apparatus are given in [34].

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Figure 1. Schematic overview of the cage and feeder system.

Two feeders (circles) were placed inside each cage (boxes with dashed outlines). Every feeder was connected via tubes (continuous lines) to two nectar pumping systems. One pump (black P) was connected to a 20% sugar solution reservoir (black N) and the other pump (gray P) was connected to the reservoir with the test concentration (gray N), which differed with test condition (Table 1). The flow of nectar was controlled with the pumps and pinch valves (black rectangles).

https://doi.org/10.1371/journal.pone.0074144.g001

Nectar consisted of equal parts of fructose, glucose, and sucrose dissolved in water, with a hexose to sucrose ratio similar to that in natural nectars of glossophagine-pollinated plants [35]. During a particular night one feeder in each cage received nectar from one pumping system, and the other feeder from the other system (Figure 1). System 1 was always filled with 20% w/w concentration and the concentration in System 2 varied throughout the experiment. Thus, during a single night the concentration offered at each feeder was fixed and did not change. In order to prevent bacterial and fungal growth inside the tubing systems, they were rinsed regularly with 70% ethanol and, or in addition to, water.

Experimental Schedule

Experiments were conducted consecutively with two groups of three animals each and each group was subjected to a series of 2AFC tests. The first group of three bats participated in calibration tests of the cage system for six nights before actual testing began (Table S1). The three bats of the second group started with the experiment on the day of the transfer to the cages.

The two feeders in every cage gave different sugar concentration rewards during each experimental session. Every session lasted 12 hours and consisted of two phases: a forced alternation phase, and a choice phase. The alternation phase lasted for 100 visits (50 samplings per feeder) and ensured that the bats experienced both nectar concentrations equally. Strict alternation was achieved by blocking a feeder opening with the plastic guard after every visit. During the choice phase the plastic guards were automatically removed from both feeders until the end of the session so that bats could access both feeders freely. During the inter-session interval (ISI = 12 h) the lights were on and all feeders were inaccessible.

Of the two feeders in each cage one (standard feeder) always gave rewards with 20% w/w sugar concentration (632 mmol L⁻¹ sucrose equivalents, [36]). The nectar concentration of the other feeder (test feeder) was systematically changed (Table 1) and ranged from 8 to 50% weight/weight (226 to 1796 mmol L⁻¹ sucrose equivalents). We avoided concentrations higher than 50%, because for sugar concentrations above ∼52% the increase in viscosity is expected to cause a reduction in net energy gain [37], [38].The test concentrations were chosen to be symmetrical around the standard concentration of 20% with respect to their relative intensity value. The sequence of test concentrations within both series was random. However, every condition was presented twice on consecutive nights on which the feeder positions for the test and standard concentration were exchanged (Figure 1, black and gray feeders), as a control for positional biases. Since the cages were supplied with nectar from the same two pumping systems, the sequence of test conditions was equal for bats within the same group. In each cage, the choice of position for the test feeder on the first presentation of a particular condition was pseudorandom, with an equal number of left and right starting positions for the test concentration.

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Table 1. Sequence of experimental conditions in the two subject groups.

https://doi.org/10.1371/journal.pone.0074144.t001

Data Analysis

Analysis was limited to the first one hundred visits of the choice phase, in order to analyze choice after an equal number of samplings at both feeders. For each bat and each condition we calculated the relative intensity and discrimination performance. The relative intensity was calculated as the absolute difference between the two sugar concentrations, divided by the mean concentration (see Equation 2). As explained in the introduction, this measure of intensity is analogous to the Weber ratio of Δ_π (a)/a and captures the expectations that discrimination performance should increase with the difference (distance effect) and decrease with the mean magnitude of the two options (magnitude effect). Discrimination performance was calculated over the two presentations of the same condition as the number of visits to the higher sugar concentration feeder divided by the total number of visits (N = 200). Data deposited in the Dryad Digital Repository: http://dx.doi.org/10.5061/dryad.0838c [39].

Psychometric Analysis

The data sets of each animal were separated into two subsets: the HIGH set contained the comparisons with concentrations larger than or equal to the 20% standard (Table 1) and the LOW set contained the comparisons with concentrations smaller than or equal to the 20% standard (Table 1). Psychometric analyses were performed on the two data sets from each individual and Weibull psychometric functions were fitted using the Bayesian algorithm proposed by Kuss et al. [15] using R 2.10.1 [40]. As prior function for the lapse rate we chose a beta distribution (2;10). For the threshold we chose a normally distributed prior with a mean of 1 and a standard deviation of 0.5, and for the slope a log-normal prior with a mean of 2 and a standard deviation of 1. We performed 5000 Markov Chain Monte Carlo (MCMC) sampling runs with a leapfrog step size of 100. From the individual psychometric functions obtained using this method, we calculated the mean threshold, slope, and lapse rate. We then used paired t tests to compare the three parameters of the psychometric functions obtained for the HIGH and LOW data sets. The prediction based on Weber’s law was that there would be no differences between the parameters from the two sets.

Weber’s Law vs. Near-Miss to Weber’s Law

In order to test whether the near-miss to Weber’s law provides a better fit to observed data than Weber’s law we individually fitted psychometric functions as Weibull sigmoid curves using the following equation [15], [23], [24]:(5)where x is the larger and a is the smaller of the sugar concentrations of the test and standard feeders, m is the threshold, s is the slope at the threshold, π_l is the lapse rate, and β is the exponent from Equation 4. In all models x and a were the independent variables, discrimination performance was the dependent variable, and m, s and π_l were estimated parameters. Using Akaike Information Criterion (AIC) scores we compared non-linear models in which the parameter β was either fixed at one (reducing near-miss relative intensity to relative intensity) or estimated within the model. We used the non-linear least-squares function nls in R 2.15.0 [41].

Reanalysis of Previously Published Data Sets

We reanalyzed data from sugar discrimination experiments with nectar-feeding bats, hummingbirds, bumblebees, and honeybees. When analyzing previously published work by other authors, we extracted numerical values from the published scatterplot figures using EasyNData [42] and converted the sugar concentration in percentage weight/weight units. In order to test whether the near-miss to Weber’s law provides a better fit to observed data than Weber’s law we analyzed the transformed data using the procedure described in the previous section. If the 95% confidence intervals for β in two different groups both spanned a convenient round number, this number was taken to be the β value of both groups. This was done because it is not otherwise possible to compare threshold and slope parameters for psychometric functions based on near-miss relative intensities with different β values. [Comparison of lapse rates can be done regardless of differences in β values.] Psychometric analyses on the different data sets were performed with Kuss’ algorithm [15] using as independent variables the near-miss relative intensity values calculated with the rounded β values.

Nectar-feeding Bats

We used data from two-alternative free-choice experiments on a population of wild electronically tagged G. commissarisi bats foraging at a 6×4 array of computer-automated artificial flowers that recorded individual choices [23]. Two reward types were presented simultaneously, with half of the feeders delivering one reward type and the other half the second type. The sugar concentrations were in the range of 5 to 50% w/w. Asymptotic relative visitation rates to the higher concentration feeders were measured in 23 individuals. Here we analyzed the pooled data from all 23 bats. It has been demonstrated that this type of data pooling only causes an underestimation of the slope parameter, but does not affect the threshold and lapse rate [24]. The same pooling was done in the remaining analyzed studies as well, even if individual data were available, for better comparison between data sets.

Hummingbird Data

Reanalyzed data were from a study on the concentration preferences in different hummingbird species [43]. The food intake from two adjacent feeders was monitored at half-hour intervals in nine individuals from five different species. The positions of the low and high concentration feeders were exchanged every half hour, for a total of 6 to 12 half-hour intervals. The sucrose concentrations were in the range of 0.15 to 1.10 M (5 to 33% w/w), with fixed differences of either 0.05 M (1.7% w/w), 0.10 M (3.4% w/w), or 0.20 M (6.7% w/w). Discrimination performance data were extracted from Figure 1 in [43].

Bumblebee Data

We used data from two-alternative free choice experiments on two B. impatiens colonies containing some electronically tagged bumblebees foraging at an array of computer-automated artificial flowers [24]. In these experiments 10 blue and 10 yellow feeders were used, in a staggered checker-board formation, on a 5×2 array. Rewards were delivered with a probability of about 50% and the sucrose concentrations were in the range of 15 to 50% w/w. Discrimination performance was measured as the asymptotic relative visitation rates to the higher concentration feeders. Data were pooled over three marked individuals and an unknown number of unmarked individuals and analyzed together.

Honeybee Data

Reanalyzed data were from a study on the concentration preferences in the Italian honeybee (Apis mellifera ligustica) [44]. In these experiments 18 blue and 18 white feeders were used, randomly distributed within a 6×6 square array and the concentrations of the two feeder colors were systematically varied. There were 27 different concentration pairings (7 experiments×4 treatments minus 1 treatment from the first experiment) for which relative visitation rates to the higher concentration feeders for different sets of 3–4 bees over 40 visits per bee per treatment were measured. The mean sucrose concentrations in the seven experiments were from 0.25 to 1.75 M (8.3 to 49% w/w), with differences between the two feeder colors of either 0 M (0% w/w), 0.2 M (6.7% w/w), 0.4 M (13.0% w/w), or 0.6 M (19.1% w/w). Discrimination performance data were extracted from Figures III though IX, Chapter 4 in [44].

Review of Sugar Discrimination in Different Nectar-feeding Animals

In all of the analyzed data sets the estimates for the exponent β from Equation 5 statistically differed from one (Figure 5; Table 3). In the vertebrate group of nectar-feeding animals β was estimated at 2.39 in G. soricina, 1.43 in G. commissarisi, and 2.09 in hummingbirds (Table 3). The 95% confidence intervals for these estimates spanned 2.0 in G. soricina and in hummingbirds, but neither of these intervals overlapped with the confidence interval estimated in G. commissarisi (Table 3). Thus, for psychometric analyses, β was set at 1.4 in G. commissarisi and at 2.0 in G. soricina and in hummingbirds. In the bees the estimate for β was 0.29 in A. mellifera ligustica and -0.04 in B. impatiens (Figure 5; Table 3). The 95% confidence interval for β in A. mellifera ligustica did not span zero, but that of B. impatiens did. However, since both confidence intervals overlapped and spanned 0.3, in further psychometric analyses we set β at 0.3 in A. mellifera ligustica and in B. impatiens.

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Figure 5. Psychometric curves based on near-miss relative intensities for different nectarivorous species.

The abscissa gives the near-miss relative intensities as defined in equation (4) and scaled with different scaling factors. Symbols give average discrimination performances measured in different experiments with different species. Lines give the fitted psychometric functions. (A) Psychometric functions for hummingbirds (thin line, empty circles) and G. soricina (thick line, black circles). The exponent β in equation (4) was fixed at 2 and the scaling factor was 10. G.s. – Glossophaga soricina, this study; Troch. – different hummingbird (Trochilidae) species, [43]. (B) Psychometric function for G. commissarisi (thick line, black circles). The exponent β was fixed at 1.4 and the scaling factor was 1. G.c. – Glossophaga commissarisi, [23]; (C) Psychometric functions for bumblebees (thick line, black circles) and honeybees (thin line, empty circles). The exponent β was fixed at 0.3 and the scaling factor was 0.1. B.i. – Bombus impatiens, [24]; A.m.l. – Apis mellifera ligustica, [44].

https://doi.org/10.1371/journal.pone.0074144.g005

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Table 3. Model comparison between near-miss to Weber’s law and Weber’s law in different nectar-feeding animals.

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The estimates for the threshold m, slope s, and lapse rate π_l for the psychometric functions of the different groups of animals were as follows: G. soricina: m = 0.12, s = 4.4, π_l = 0.11; G. commissarisi: m = 0.14, s = 9.7, π_l = 0.06; Trochilidae: m = 0.14, s = 4.3, π_l = 0.19; B. impatiens: m = 0.23, s = 4.1, π_l = 0.25; A. mellifera ligustica: m = 0.35, s = 1.9, π_l = 0.07. The values for the lapse rates were in the range of 0.06–0.25 and were, as expected, somewhat higher than the typical lapse rates in human studies (0.0–0.10; [15]). The psychometric functions are shown in Figure 5.