Functional Ecology
Appendix S1 Description of bird, snake and crab vision models.
Bird vision model
Calculations were done based on a set of functions in Matlab R2009a (The MathWorks Inc,
USA). The model starts by calculating the cone quantum catches (qi) for each
photoreceptor class for froglets and ambient radiance spectra as
750
𝑞𝑖𝑏𝑖𝑟𝑑 =
∫ 𝑅( λ)𝐼(λ)𝐴ᵢ(λ)𝑑λ,
λ=300
here 𝐴𝑖 (𝜆) represents the absorptance spectrum for each of the four photoreceptor cone
classes of the bird integrated over 1 nm intervals from 300 to 750 nm. 𝑅(𝜆) and 𝐼(𝜆)
represent the reflectance spectrum of the froglet´s skin and the irradiance spectra measured
in the field respectively. Resulting photon quantum catches were standardized in order to
account for variation in light conditions, using the von Kries transformation adaptation
coefficient. This method assumes that photoreceptors adjust their sensitivity in proportion
to the background light environment
𝐾𝑖𝑏𝑖𝑟𝑑 =
1
,
750
∫𝜆=300 𝐴ᵢ(𝜆)𝐼(𝜆)𝑑𝜆
This procedure ensures that colour perception relies on colour constancy, whereby the
visual system removes variation in ambient light so that colours look similar under variable
light conditions (Cuthill 2006). Therefore, the adjusted quantum catch data for each
photoreceptor class was calculated as
𝑞𝑎 = 𝐾𝑖𝑏𝑖𝑟𝑑 𝑥 𝑞𝑖𝑏𝑖𝑟𝑑
Flores, E. E., Stevens, M, Moore, A. J., and Blount, J. D. Diet, development and the
optimisation of warning signals in post-metamorphic green and black poison frogs
Functional Ecology
Values of relative single cone quantum catches were then included in a Principal
Component Analysis (see Data analyses).
Following calculations of quantum catches, the model assumes that the signal of each cone
channel is proportional to the logarithm of the adjusted quantum catches; as such the
contrast between a pair of stimuli was calculated as the quotient of adjusted photon
quantum catches
∆𝑞𝑎 = 𝑙𝑜𝑔
[𝑞𝑎1 (𝑓𝑟𝑜𝑔 𝑠𝑝𝑒𝑐𝑡𝑟𝑎)]
[𝑞𝑎2 (𝑏𝑎𝑐𝑘𝑔𝑟𝑜𝑢𝑛𝑑 𝑠𝑝𝑒𝑐𝑡𝑟𝑎)]
The Vorobyev-Osorio colour discrimination model is based on evidence that colour
discrimination is determined by noise arising in the photoreceptors and is independent of
light intensity. Noise in each photoreceptor channel (𝑒𝑖 ) was calculated as
𝑒𝑖 =
𝜔𝑖
√𝑛𝑖
Where 𝜔𝑖 was taken as 0.05 and represents the Weber fraction of the most abundant cone
type (Siddiqi et al. 2004) and 𝑛𝑖 is the relative number of receptor types in the retina of the
blue tit (Hart et al. 2000) (𝑛𝐿 = 1.00, 𝑛𝑀 = 0.99, 𝑛𝑆 = 0.71, 𝑛𝑈𝑉 = 0.37). Colour
(chromatic) discrimination in the tetrachromatic visual model was calculated as JND values
using the following equation
𝐽𝑁𝐷𝑏𝑖𝑟𝑑 𝑐𝑜𝑙𝑜𝑟
(𝑒𝑈𝑉 𝑒𝑆 )2 (∆𝑞𝐿 − ∆𝑞𝑀 )2 + (𝑒𝑈𝑉 𝑒𝑀 )2 (∆𝑞𝐿 − ∆𝑞𝑆 )2 + (𝑒𝑈𝑉 𝑒𝐿 )2 (∆𝑞𝑀 − ∆𝑞𝑆 )2
√+(𝑒𝑆 𝑒𝑀 )2 (∆𝑞𝐿 − ∆𝑞𝑈𝑉 )2 + (𝑒𝑆 𝑒𝐿 )2 (∆𝑞𝑀 − ∆𝑞𝑈𝑉 )2 + (𝑒𝑀 𝑒𝐿 )2 (∆𝑞𝑆 − ∆𝑞𝑈𝑉 )2
=
(𝑒𝑈𝑉 𝑒𝑆 𝑒𝑀 )2 + (𝑒𝑈𝑉 𝑒𝑆 𝑒𝐿 )2 + (𝑒𝑈𝑉 𝑒𝑀 𝑒𝐿 )2 + (𝑒𝑆 𝑒𝑀 𝑒𝐿 )2
2
Flores, E. E., Stevens, M, Moore, A. J., and Blount, J. D. Diet, development and the
optimisation of warning signals in post-metamorphic green and black poison frogs
Functional Ecology
Since in our model, overall perceived luminance is considered to arise from stimulation of
double cone photoreceptors, luminance discrimination was evaluated as
∆𝑞𝐷
𝐽𝑁𝐷𝑏𝑖𝑟𝑑 𝑙𝑢𝑚𝑖𝑛𝑎𝑛𝑐𝑒 = (
)
𝑒𝐷
Overall variation in colour was based on values from single cone photon catch scores (and
a Principal Component Analysis of them) and variation in overall luminance was based on
double cone photon catch scores (ΔqD). Colour and luminance discrimination
(conspicuousness) was based on JND values.
Snake vision model
Calculations were done based on a set of functions in Matlab R2009a (The MathWorks Inc,
USA). The model starts by calculating the cone quantum catches (qi) for each
photoreceptor class for froglets and ambient radiance spectra as
750
𝑞𝑖𝑠𝑛𝑎𝑘𝑒 =
∫ 𝑅( λ)𝐼(λ)𝐴ᵢ(λ)𝑑λ,
λ=300
here 𝐴𝑖 (𝜆) represents the absorptance spectrum for each of the three photoreceptor cone
classes (i.e. UV, SW and LW) integrated over 1 nm intervals from 300 to 750 nm. 𝑅(𝜆) and
𝐼(𝜆) represent the reflectance spectrum of the froglet´s skin and the irradiance spectra
measured in the field respectively. Resulting photon quantum catches were standardized in
order to account for variation in light conditions, using the von Kries transformation
adaptation coefficient. This method assumes that photoreceptors adjust their sensitivity in
proportion to the background light environment
Flores, E. E., Stevens, M, Moore, A. J., and Blount, J. D. Diet, development and the
optimisation of warning signals in post-metamorphic green and black poison frogs
Functional Ecology
𝐾𝑖𝑠𝑛𝑎𝑘𝑒 =
1
,
750
∫𝜆=300 𝐴ᵢ(𝜆)𝐼(𝜆)𝑑𝜆
This procedure ensures that colour perception relies on colour constancy, whereby the
visual system removes variation in ambient light so that colours look similar under variable
light conditions (Cuthill 2006). Therefore, the adjusted photon quantum catch data for each
photoreceptor class was calculated as
𝑞𝑎 = 𝐾𝑖𝑠𝑛𝑎𝑘𝑒 𝑥 𝑞𝑖𝑠𝑛𝑎𝑘𝑒
Values of the relative single cone quantum catches were then included in a Principal
Component Analysis (see Data analyses).
Following calculations of quantum catches, the model assumes that the signal of each cone
channel is proportional to the logarithm of the adjusted quantum catches; as such the
contrast between a pair of stimuli was calculated as the quotient of adjusted quantum
catches
∆𝑞𝑎 = 𝑙𝑜𝑔
[𝑞𝑎1 (𝑓𝑟𝑜𝑔 𝑠𝑝𝑒𝑐𝑡𝑟𝑎)]
[𝑞𝑎2 (𝑏𝑎𝑐𝑘𝑔𝑟𝑜𝑢𝑛𝑑 𝑠𝑝𝑒𝑐𝑡𝑟𝑎)]
The Vorobyev-Osorio visual discrimination model is based on evidence that color
discrimination is determined by noise arising in the photoreceptors and is independent of
light intensity. Noise in each photoreceptor channel (𝑒𝑖 ) was calculated as
𝑒𝑖 =
𝜔𝑖
√𝑛𝑖
Where 𝜔𝑖 was taken as 0.05 according to Siddiqi et al. (2004) and represents the Weber
fraction of the most abundant cone type and 𝑛𝑖 is the relative number of receptor types in
the retina of the snake (data for Thamnophis sirtalis, Sillman et al. 1997) (𝑛𝐿 = 0.85,
Flores, E. E., Stevens, M, Moore, A. J., and Blount, J. D. Diet, development and the
optimisation of warning signals in post-metamorphic green and black poison frogs
Functional Ecology
𝑛𝑆 = 0.10, 𝑛𝑈𝑉 = 0.05). Colour (chromatic) discrimination in the trichromatic visual
model was calculated as JND values using the following equation
𝐽𝑁𝐷𝑠𝑛𝑎𝑘𝑒 𝑐𝑜𝑙𝑜𝑢𝑟
2
=√
(𝑒𝑈𝑉 )2 (∆𝑞𝐿 − ∆𝑞𝑆 )2 + (𝑒𝑆 )2 (∆𝑞𝐿 − ∆𝑞𝑈𝑉 )2 + (𝑒𝐿 )2 (∆𝑞𝑈𝑉 − ∆𝑞𝑆 )2
(𝑒𝑈𝑉 𝑒𝑆 )2 + (𝑒𝑈𝑉 𝑒𝐿 )2 + (𝑒𝑆 𝑒𝐿 )2
We considered overall perceived luminance in snakes as in other vertebrates to arise from
stimulation of LW cone photoreceptors with sensitivity in the long wavelength part of the
spectra; therefore luminance discrimination was evaluated as
∆𝑞𝐿
𝐽𝑁𝐷𝑠𝑛𝑎𝑘𝑒 𝑙𝑢𝑚𝑖𝑛𝑎𝑛𝑐𝑒 = (
)
𝑒𝐿
Overall variation in colour was based on values from single cone photon catch scores (and
a Principal Component Analysis of them) and variation in overall luminance was based on
LW cone photon catch scores (ΔqL). Colour and luminance discrimination
(conspicuousness) was based on JND values.
Crab vision model
Similar calculations were done based on a set of functions in Matlab R2009a (The
MathWorks Inc, USA). The model starts by calculating the cone quantum catches (qi) for
each photoreceptor class for froglets and ambient radiance spectra as
750
𝑞𝑖𝑐𝑟𝑎𝑏 =
∫ 𝑅( λ)𝐼(λ)𝐴ᵢ(λ)𝑑λ,
λ=300
Flores, E. E., Stevens, M, Moore, A. J., and Blount, J. D. Diet, development and the
optimisation of warning signals in post-metamorphic green and black poison frogs
Functional Ecology
here 𝐴𝑖 (𝜆) represents the absorptance spectrum for each of the two photoreceptor cone
classes (i.e. SW and LW) integrated over 1 nm intervals from 300 to 750 nm. Resulting
photon quantum catches were standardized in order to account for variation in light
conditions, using the von Kries transformation adaptation coefficient. This method assumes
that photoreceptors adjust their sensitivity in proportion to the background light
environment
𝐾𝑖𝑐𝑟𝑎𝑏 =
1
,
750
∫𝜆=300 𝐴ᵢ(𝜆)𝐼(𝜆)𝑑𝜆
This procedure ensures that colour perception relies on colour constancy, whereby the
visual system removes variation in ambient light so that colours look similar under variable
light conditions (Cuthill 2006). Therefore, the adjusted photon quantum catch data for each
photoreceptor class was calculated as
𝑞𝑎 = 𝐾𝑖𝑐𝑟𝑎𝑏 𝑥 𝑞𝑖𝑐𝑟𝑎𝑏
Values of the relative single cone quantum catches were then included in a Principal
Component Analysis (see Data analyses).
Following calculations of quantum catches, the model assumes that the signal of each cone
channel is proportional to the logarithm of the adjusted quantum catches; as such the
contrast between a pair of stimuli was calculated as the quotient of adjusted quantum
catches
∆𝑞𝑎 = 𝑙𝑜𝑔
[𝑞𝑎1 (𝑓𝑟𝑜𝑔 𝑠𝑝𝑒𝑐𝑡𝑟𝑎)]
[𝑞𝑎2 (𝑏𝑎𝑐𝑘𝑔𝑟𝑜𝑢𝑛𝑑 𝑠𝑝𝑒𝑐𝑡𝑟𝑎)]
Flores, E. E., Stevens, M, Moore, A. J., and Blount, J. D. Diet, development and the
optimisation of warning signals in post-metamorphic green and black poison frogs
Functional Ecology
The Vorobyev-Osorio visual discrimination model is based on evidence that colour
discrimination is determined by noise arising in the photoreceptors and is independent of
light intensity. Noise in each photoreceptor channel (𝑒𝑖 ) was calculated as
𝑒𝑖 =
𝜔𝑖
√𝑛𝑖
Where 𝜔𝑖 was taken as 0.12 based on physiological receptor noise data according to
Hempel De Ibarra et al. (2000) for honeybees Apis mellifera and 𝑛𝑖 is the relative number
of receptor types that for our purposes were set as (𝑛𝐿 = 0.5, 𝑛𝑆 = 0.5) following
(Cummings et al. 2008). Colour (chromatic) discrimination in the dichromatic visual model
was calculated as JND values using the following equation
2 (∆𝑞 − ∆𝑞 )2
𝑆
𝐿
𝐽𝑁𝐷𝑐𝑟𝑎𝑏 𝑐𝑜𝑙𝑜𝑢𝑟 = √
(𝑒𝑆 )2 + (𝑒𝐿 )2
We considered overall perceived luminance in crab to arise from stimulation of LW cone
photoreceptors with sensitivity in the long wavelength part of the spectra, luminance
discrimination was evaluated as
∆𝑞𝐿
𝐽𝑁𝐷𝑐𝑟𝑎𝑏 𝑙𝑢𝑚𝑖𝑛𝑎𝑛𝑐𝑒 = (
)
𝑒𝐿
Overall variation in colour was based on values from single cone photon catch scores (and
a Principal Component Analysis of them) and variation in overall luminance was based on
LW cone photon catch scores (ΔqL). Colour and luminance discrimination
(conspicuousness) was based on JND values.
Flores, E. E., Stevens, M, Moore, A. J., and Blount, J. D. Diet, development and the
optimisation of warning signals in post-metamorphic green and black poison frogs
Functional Ecology
References
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fiddler crab, Uca tangeri: effects of sex, viewer and background on conspicuousness.
Animal Behaviour, 75, 175–188.
Cuthill, I.C. (2006) Color perception. Bird coloration, vol. I: Mechanisms and
measurements (eds G.E. Hill & K.J. McGraw), pp. 3–40. Harvard University Press,
Cambridge.
Hart, N.S., Partridge, J.C., Cuthill, I.C. & Bennett, A.T.D. (2000) Visual pigments, oil
droplets, ocular media and cone photoreceptor distribution in two species of passerine
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Comparative Physiology A: Sensory, Neural, and Behavioral Physiology, 186, 375–
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and dim colours by honeybees. The Journal of Experimental Biology, 203, 3289–3298.
Siddiqi, A., Cronin, T.W., Loew, E.R., Vorobyev, M. & Summers, K. (2004) Interspecific
and intraspecific views of color signals in the strawberry poison frog Dendrobates
pumilio. Journal of Experimental Biology, 207, 2471–2485.
Sillman, A.J., Govardovskii, V.I., Rohlich, P., Southard, J.A. & Loew, E.R. (1997) The
photoreceptors and visual pigments of the garter snake (Thamnophis sirtalis): a
microspectrophotometric, scanning electron microscopic and immunocytochemical
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Flores, E. E., Stevens, M, Moore, A. J., and Blount, J. D. Diet, development and the
optimisation of warning signals in post-metamorphic green and black poison frogs