Many animals process visual information to inform decisions that result in fitness benefits to various species. Birds, for example, may first use their vision for locating a target of interest, and then for confirming correct identification (Troscianko et al., 2009). Whilst recognition of shape and texture is likely driven by achromatic processing in birds, discriminating the spectral component of a colour signal describing the quality of a stimulus is mainly driven by colour vision (Osorio et al., 1999).

Model bird species for studying evolution such as pigeons (Darwin, 1859, 1868) are known to be omnivorous, feeding on a wide variety of fruits, seeds, insects, flowers (Murton and Westwood, 1966; Crome, 1975; Innis, 1989; Baptista et al., 2009), and sometimes acting as pollinators when opportunistically feeding on nectar (Dalsgaard et al., 2016). In all these instances birds will likely use their colour vision to process information allowing for the initial detection of targets, and to subsequently discriminate a preferred option from sub-optimal alternatives simultaneously available.

When colour differences between stimuli are large, take for example preferred and opposing teams in a sport match dressed in yellow or blue, discrimination is both rapid and accurate. However, if respective teams were garbed in similar colours such as green and turquoise, both accuracy and speed of discrimination would likely be impaired leading to a less accurate and slower response (Chittka and Osorio, 2007). Thus, well-beyond any theoretical discrimination limit to colour vision imposed by physiological aspects, colour similarity affects performance as it has been demonstrated on important pollinators like bees (Dyer and Chittka, 2004) and hoverflies (Hannah et al., 2019). Indeed, classic colour discrimination experiments with humans (MacAdam, 1942) show that accuracy diminishes with similarity up to a point where target and distracter become indistinguishable from each other and the observer chooses randomly between them; by the same token, increasing colour differences facilitate detection of targets concealed by background matching in real scenarios (Niu et al., 2020; Dyer and Garcia, 2021). Olsson et al. (2015) showed that accuracy of discrimination by chickens also increases with colour dissimilarity from chance level, to a range between 80 and 100% in a non-linear but continuous fashion. Such psychophysics evidence is likely to be mediated by the probabilistic way in which neurons can code and respond to the salience of different colour signals (Komatsu and Ideura, 1993).

The ability to predict the likely outcome of a colour discrimination event from physiological and physical properties of both observer and stimuli is fundamental for plant-animal interaction studies. For example, one could measure the reflectance spectra from flowers of different species and use a model for formulating testable hypothesis on the effect of colour signalling as means to establish relationships between plant and animal which are evolutionary meaningful (Pauw, 2019). Whilst such models are now available for hymenopteran pollinators such as honeybees and bumblebees (Garcia et al., 2017), and have been applied to mapping plant-pollinator interactions (Shrestha et al., 2019; Garcia et al., 2020), currently there is no model which allows for an accurate and unbiased prediction of animal responses considering (dis)similar colours as those likely to be encountered by a bird observer. The development of such models likely serves to improve our understanding of the processing for visual information produced by stimuli like fruits (Schaefer et al., 2008), egg shells (Hanley et al., 2019), or mating partners and nesting choices (Endler and Day, 2006).

Most models currently employed to understand colour discrimination by birds are based on purely physiological data at photoreceptor level, and specifically ignore the effects of neural processing by the brain of colour information (Avilés, 2020). This position of physiologically mediated colour discrimination does not fit evidence from primates showing that processing of such signals is a multistage process involving different brain regions with various degrees of specialisation (Solomon and Lennie, 2007). Furthermore there is strong evidence in invertebrates that colour processing is mediated a high levels of the insect brain (Paulk et al., 2009; Mota et al., 2013; Lin et al., 2016), strongly suggesting that colour information processing requires higher level structures in a wide range of animals.

Considering the current use of colour models for birds, the receptor noise (RN) model (Vorobyev and Osorio, 1998) has been proposed to be a solution for predicting the minimum colour difference of two stimuli required by a bird observer to discriminate between them, the so called just noticeable difference (JND). Interestingly, and in spite of its wide implementation for avian studies, the RN model assumptions are built on noise receptor data and behavioural responses from an invertebrate model: the European honeybee (Apis mellifera; Vorobyev et al., 2001). In the RN model, the difference between two coloured stimuli is not expressed in a colour space but is equated with a perceived difference expressed in JND (Pike, 2012). Whilst this model is proposed to be useful for both invertebrate and vertebrate observers, behavioural validation of the RN model predictions with avian observers has provided mixed results (Lind and Kelber, 2009).

Currently, it is accepted that validation of RN's predictions is subject to the appropriate choice of parameters, particularly, the noise level within photoreceptors (Olsson et al., 2015; Avilés, 2020). Regrettably, these measurements remain outstanding for any bird species, and approximations for the real values for these parameters based on purely theoretical assumptions are used instead (Olsson et al., 2018). Even for the two key model species for which photoreceptor noise values are available, the honeybee (Vorobyev et al., 2001) and the bumblebee (Skorupski et al., 2007), RN fails to predict discrimination accuracy observed in behavioural experiments when considering experimentally measured noise values (Vorobyev et al., 2001; Avarguès-Weber et al., 2010; Garcia et al., 2017, 2018).

An alternative formulation for predicting colour discrimination in animals was proposed by von Helversen (1972a). His approach aims to formulate a monotonic function linking a physical measure of colour dissimilarity, as for example the distance of two stimuli in a colour space, to a measure of the accuracy achieved by an observer when discriminating between these stimuli. This monotonic function approach proposed by von Helversen (1972a) contrasts starkly with the position of producing a single metric predicting perceptual difference proposed by the RN model.

A monotonic function describing the accuracy of colour discrimination task from a subject-independent measurement of (dis)similarity constitutes a representation of the psychophysical law (Norwich, 1987). More specifically, a colour discrimination function links an objective measurement of dissimilarity, distance in colour space (ΔC), with the subjective perception of colour dissimilarity experienced by an observer. Implementation of the psychophysical law for describing the relationship between objective and perceived differences in magnitude of stimuli such as heaviness, loudness, taste, and other stimuli (Stevens, 1957) suggests that this relationship is non-linear, and can described by either a logarithmic or power function depending on conditions (Norwich, 1987). Thus, a relationship informed by physchophysical data more fully represents the complete perceptual processing by the sensory system of the animal, which is the essential driver of observed behaviour.

Recently, such a psyschophysical approach has been proposed for predicting colour discrimination in hymenopteran pollinators based on non-linear, sigmoidal functions rather than JND magnitudes. These models predict changes in the probability of discrimination with colour distance based on the result of psychophysics experiments (Garcia et al., 2017), rather than just predicting if the considered colour distance is either below or above a single theoretical JND discrimination threshold. In other words, models based on monotonic functions help assessing the uncertainty around an animal choice based on colour difference rather than just predicting if the animal can perceive, or not, the dissimilarity between two stimuli. The specific shape a colour discrimination function is given by the interaction between physiological aspects of colour vision for a given observer and the processing of such signals by the animal's brain as predicted by von Helversen (1972a).

In the present manuscript we develop a new framework for a colour discrimination function for a violet sensitive bird based on behavioural responses of the pigeon Columba livia. Importantly, pigeons are generalist foragers (Baptista et al., 2009; Dalsgaard et al., 2016) and a behaviourally accessible avian species for which it has been possible to collect precise psychophysics data on colour discrimination tasks (Wright, 1972). Our proposed model predicts the sensitivity index of colour discrimination, a more comprehensive measure of accuracy based on signal detection theory (see section 2 below), using a monotonic function described by a simple algebraic expression. We enable this solution without making assumptions about currently unknown photoreceptor noise parameters of a bird observer, thus overcoming one of the principal limitations of current modelling efforts. The function's accuracy to predict observed discrimination behaviour by the pigeon is then compared to that of the RN model for the same set of stimuli.

In the current study we consider if psychometric functions may offer a holistic modelling solution for understanding VS-sensitive, avian decision making considering (dis)similar colours when compared to the noise-limited colour discrimination model (RN).

Predictions from RN depend on the precise value of its various parameters, and in particular, on the magnitude of noise coefficients (e_i) assigned to the different photoreceptors (Lind and Kelber, 2009). In spite of the wide use of the RN model for answering questions regarding avian visual ecology such as: discrimination of parasitic eggs by hosts (Hanley et al., 2019), camouflage by female birds (Cain et al., 2019), and perceived flower colour variation by pollinating species at population level (Whitney et al., 2020); to cite just a few recent examples, no noise measurement data currently exist for any avian species. In most cases, the unknown but essential noise values are derived from applying theoretical assumptions of signal detection theory to photoreceptor density data (Vorobyev and Osorio, 1998), although it still remains unclear how neural processing by the brain may affect these assumptions. This is important as birds and other animals are known to use various strategies to counteract limitations imposed to vision by the physiological properties of their visual system. For example, spatial pooling for increasing signal-to-noise ratio under dim illumination (Warrant, 1999) is likely to occur in some species of owls (Orlowski et al., 2012) and nocturnal parrots (Corfield et al., 2011; Iwaniuk et al., 2020). Likewise, it remains largely unknown how external factors such as temperature and light intensity may affect photoreceptor noise values, and how neural processing may compensate for such fluctuations. Indeed, experiments on frogs under laboratory controlled conditions (Aho et al., 1988) suggest that visual performance could be affected by temperature if receptor noise were the only or primary factor mediating colour discrimination. Such basic physiological limitations would likely apply to many animals unless neural corrections resolve how colour can be reliably perceived in naturally occurring visual conditions.

The colour discrimination sensitivity function (CDSF) for a violet-sensitive bird observer (Figure 4), takes as input a measurement of colour dissimilarity and returns the likely outcome of the discrimination process expressed as a sensitivity index (d′). Through this approach CDSF accounts for the “inner-conditions” driving animal behaviour as originally hypothesised by von Helversen (1972a). Our function thus represents an application of the psychophysics law where the relationship between objective and perceived colour difference is modelled by a power function (Equation 6) as experimentally obtained for modelling perceived changes in the magnitude of stimuli such as heaviness and loudness (Stevens, 1957).

An additional advantage of the CDSF function is that it allows for a precise definition of the just noticeable difference for colour discrimination by means of the Weber fraction. The Weber fraction describes the difference in stimulus magnitude that is just noticeable by an observer (Debats et al., 2012), quantitatively expressed as a derivative of the psychophysics law (Norwich, 1987). By differentiating Equation (6) (Supplementary Figure 1), and evaluating the resulting function for a range of colour differences, we can see how for bird vision large Weber fraction values are initially obtained for small colour differences and subsequently falling with increasing stimuli dissimilarity into a plateau region, as observed for other perceptual tasks such as taste and brightness (Norwich, 1987). This approach allows for a more precise definition of the just noticeable colour differences perceivable by a bird observer, representing an improvement over the extrapolation of the behavioural and physiological results obtained from observations on the insect model species used for validating the RN assumptions (Vorobyev et al., 2001). Indeed, previous behavioural validation of RN assumptions for bird colour discrimination have provided inconclusive results on the predictive accuracy of this model in the absence of measured noise data. For example, in their experiment with domestic chicks, Olsson et al. (2015) found that a Weber fraction ν = 0.06 provided a good fit for observed behavioural data on a two option colour discrimination task under bright light conditions. This value is 40% smaller than the ν = 0.1 sometimes suggested as noise parameter for modelling bird vision (Vorobyev and Osorio, 1998; Lind et al., 2014), evidencing the susceptibility of RN predictions to errors when assumptions of their parameters are made (Lind and Kelber, 2009; Bitton et al., 2017).

In spite of its algebraic simplicity (Equation 1), the CDSF is flexible enough as to accommodate for potential changes in sensitivity produced by stimuli of different hue (Figure 5). Indeed, the statistical significance of the random term α_i in Equation (6) suggests that the magnitude of colour dissimilarity required to perceive two loci as being different changes with the spectral position of the transmittance peak of the stimuli (Figure 5); in other words, birds likely discriminate some colours better than others. Asymmetries in the ability to discriminate colour stimuli depending on hue have also been reported for human observers (Wyszecki and Stiles, 1982), macaque monkeys (Komatsu and Ideura, 1993), honeybees (von Helversen, 1972b; Dyer and Neumeyer, 2005), bumblebees (Dyer et al., 2008), stingless bees (Spaethe et al., 2014), Drosophila flies (de Salomon and Spatz, 1983), hoverflies (Hannah et al., 2019), and domestic chicks (Gallus gallus) following associative training (Olsson et al., 2015). Moreover, by using two coefficients the colour discrimination sensitivity function can take different shapes (Figure 3) potentially describing the effect of other cognitive-like processes affecting colour discrimination such as memory and conditioning as reported for some hymenopteran species (Dyer and Chittka, 2004; Giurfa, 2004; Avarguès-Weber et al., 2010; Dyer et al., 2011; Garcia et al., 2020).

Our data shows that modelling colour discrimination by means of the purely physiologically informed RN model (Figure 6A) leads to a greater RMSE, and thus poorer predictive power, than the CDSF either including or excluding the wavelength-specific random terms (Figure 6). For the empirical colour discrimination data for pigeons, RN predictions for the typically assumed ν = 0.05 value for modelling photoereceptor noise in birds, we obtained an RMSE error 95 % higher than the one resulting from implementing the discrimination function only including fixed terms (Figure 6C). This result is consistent with previous studies reporting that RN is not well suited for modelling perceptual colour distances of 1–3 jnd which are beyond the discrimination threshold (Bitton et al., 2017; Marshall, 2018; Olsson et al., 2018). Whilst an improvement of the RN model predictions was obtained when using a ν = 0.1, RMSE for this value was still 28% higher than the mean RMSE value for the CDSF. The CFSD model still provides a better estimate of colour discrimination by birds than the RN alternative when considering the 75–95% discrimination threshold range for which the RN was originally calibrated using an insect model (Vorobyev et al., 2001).

The CDFS function accurately predicts bird discrimination for colour differences between 0.25 < ΔC ≤ 0.5 THu for which pigeons show a sensitivity d′ > 2 (Figures 4, 5). Colour differences of this magnitude are large enough as to be discriminable with an accuracy of about 90 % with a false positive proportion of <10% (Figure 1) indicating that they are easily distinguishable by a violet sensitive bird. Such salient and robust colour signals are likely to be produced during plant pollinator interactions to attract animal pollination vectors like birds or bees (Lunau et al., 2011) as a solution to overcome the “colour noise” produced by natural variability of flower pigmentation that might confuse decision making (Dyer et al., 2012; Garcia et al., 2018; van der Kooi et al., 2019; Garcia et al., 2020).

Perceptual effects of colour differences ΔC > 6.0 are poorly predicted by the CDSF model. This result may be interpreted as the effect of processes such as categorisation when judging large colour differences as suggested by early experiments on this species (Wright and Cumming, 1971). Furthermore, a large colour difference between target and background increases salience of the signal attracting gaze and attention (Siuda-Krzywicka et al., 2019), which may have an effect on colour discrimination accuracy as suggested by experiments on honeybees (Giurfa, 2004; Avarguès-Weber et al., 2010). Another limitation of the CDSF model is that its predictions are currently based on pigeon responses to monochromatic stimuli. The intensity and purity of these signals may have an effect on the photoreceptor adaptation process potentially affecting discrimination accuracy when compared to responses obtainable from broad-band, colour stimuli as those typically produced by organic pigments. Nevertheless, the methodology we present can be used to re-calibrate the CDSF function to account for animal responses, which may even exist within subjects (Giurfa, 2004), to these type of stimuli once they become available. Indeed, such functions have already been derived for predicting the accuracy of colour discrimination by four hymenopteran species when observing broad-band stimuli (Garcia et al., 2017).

Pigeons have many feral and domestic breeds around the world and a demonstrated capacity to forage on a wide variety of foods including: flowers, insects, fruits, and seeds (Darwin, 1859, 1868; Murton and Westwood, 1966; Baptista et al., 2009). On the island of Cuba pigeons have been observed to feed from nectar rich flowers (Dalsgaard et al., 2016), and in Queensland, Australia, fruit pigeons were observed in a 5-year study to feed from 89 different species of plants and have diets including a variety of fruits and flowers (Innis, 1989). This suggests that pigeons require a good capacity to generalise information. Our findings that pigeons have a colour visual system that is more fully explained by a continuous discrimination function (Figures 4, 6) suggests that for other animals that have a requirement for foraging on a variety of colour stimuli it will be of value to consider this model of colour processing.

Psychometric functions linking objective measurements of dissimilarity, expressed as distance in a colour space, with perceived difference, measured as discrimination accuracy can easily accommodate a more holistic understanding of cognitive aspects of colour vision including memory, individual experience (Skorupski and Chittka, 2011), and effects of colour variability which cannot be accurately modelled by purely physiologically informed colour models (Garcia et al., 2020). This new generation of analytical tools open the door for testing interesting hypothesis about the perceptual effect of colour signalling such as flower display and plant—animal interactions with new, fresh eyes.

The datasets corresponding to posterior distributions of the CDSF coefficients, ΔS values for the various Weber fractions simulated and their specific values presented in this study can be found in Figshare data depository (doi: 10.6084/m9.figshare.13350926).

JG and AD designed the experiments. JG analysed the data and performed the statistical analyses. All authors wrote and edited the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.