NEU 314: Mathematical Tools for Neuroscience
Lecture 3: September 15, 2022
Instructor: Sam Nastase
Princeton Neuroscience Institute
Trichromatic color vision
Maxwell’s color-matching experiment
Any “test” light can be matched by adjusting the intensities of any three other lights
Visible light
Visible light is just a thin slice of the electromagnetic spectrum: ~380 to ~750 nm
Visible light
Visible light is just a thin slice of the electromagnetic spectrum: ~380 to ~750 nm
Visible light
Visible light is just a thin slice of the electromagnetic spectrum: ~380 to ~750 nm
How many numbers would you need to write down to specify the color of a light source?
Just one? The wavelength?
620 nm?
Visible light
Visible light is just a thin slice of the electromagnetic spectrum: ~380 to ~750 nm
How many numbers would you need to write down to specify the color of a light source?
spectral properties
Visible light
Visible light is just a thin slice of the electromagnetic spectrum: ~380 to ~750 nm
How many numbers would you need to write down to specify the color of a light source?
spectral properties
Visible light
Visible light is just a thin slice of the electromagnetic spectrum: ~380 to ~750 nm
How many numbers would you need to write down to specify the color of a light source?
spectral properties
Visible light
Visible light is just a thin slice of the electromagnetic spectrum: ~380 to ~750 nm
How many numbers would you need to write down to specify the color of a light source?
spectral properties
Visible light
Visible light is just a thin slice of the electromagnetic spectrum: ~380 to ~750 nm
How many numbers would you need to write down to specify the color of a light source?
spectral properties
Visible light
Visible light is just a thin slice of the electromagnetic spectrum: ~380 to ~750 nm
How many numbers would you need to write down to specify the color of a light source?
spectral properties
Color vision
“Now, as it is almost impossible to conceive each sensitive point of the retina to contain an infinite number of particles, each capable of vibrating in perfect unison with every possible undulation, it is necessary to suppose the number limited, for instance, to the three principal colours, red, yellow, and blue, of which the undulates are related in magnitude.”
Young, 1805
Color vision
“Now, as it almost impossible to conceive each sensitive point of the retina to contain an infinite number of particles, each capable of vibrating in perfect unison with every possible undulation, it is necessary to suppose the number limited, for instance, to the three principal colours, red, yellow, and blue, of which the undulates are related in magnitude.”
Maxwell, 1864
Young, 1802
Helmholtz, 1850
How many measurements of this spectrum does the human eye take?
Visible light is just a thin slice of the electromagnetic spectrum: ~380 to ~750 nm
Color vision
How many measurements of this spectrum does the human eye take?
Visible light is just a thin slice of the electromagnetic spectrum: ~380 to ~750 nm
Only three! One for each cone (i.e. color-sensitive photoreceptor cell) class:
photoreceptor response
S cones: short wavelength (blue)
M cones: medium wavelength (green)
L cones: long wavelength (red)
Color vision
Visible light is just a thin slice of the electromagnetic spectrum: ~380 to ~750 nm
Human color vision relies entirely on the comparison of responses from three cone types!
photoreceptor response
S cones: short wavelength (blue)
M cones: medium wavelength (green)
L cones: long wavelength (red)
Color vision
Visible light is transduced by photoreceptor cells (rods and cones) in the retina
A brief tour of early vision in humans
Visible light is transduced by photoreceptor cells (rods and cones) in the retina
A brief tour of early vision in humans
Visible light is transduced by photoreceptor cells (rods and cones) in the retina
A brief tour of early vision in humans
Rods are sensitive in low-light conditions, are more prevalent in the periphery, and contain only one pigment type
Cones require bright light, are more prevalent in the fovea, and contain three different pigment types
Visible light is transduced by photoreceptor cells (rods and cones) in the retina
A brief tour of early vision in humans
Rods are sensitive in low-light conditions, are more prevalent in the periphery, and contain only one pigment type
Cones require bright light, are more prevalent in the fovea, and contain three different pigment types
Photoreceptors are tonically depolarized; when pigment proteins (opsins) absorb photons, the cell becomes hyperpolarized
Visible light is transduced by photoreceptor cells (rods and cones) in the retina
A brief tour of early vision in humans
Photoreceptors synapse on bipolar cells, which become depolarized or hyper- polarized depending on the receptor type
Two populations of bipolar cells: ON bipolar cells are excited by light; OFF bipolar cells are inhibited by light
Other cell types in the plexiform layers (e.g. horizontal cells) provide additional preprocessing; e.g. lateral inhibition
Visible light is transduced by photoreceptor cells (rods and cones) in the retina
A brief tour of early vision in humans
Photoreceptors synapse on bipolar cells, which become depolarized or hyper- polarized depending on the receptor type
Two populations of bipolar cells: ON bipolar cells are excited by light; OFF bipolar cells are inhibited by light
Other cell types in the plexiform layers (e.g. horizontal cells) provide additional preprocessing; e.g. lateral inhibition
This is where we start setting the stage for center-surround receptive fields and opponent processing!
Visible light is transduced by photoreceptor cells (rods and cones) in the retina
A brief tour of early vision in humans
Retinal ganglion cells (RGCs) receive input from bipolar cells and send action potentials to the brain via the optic nerve
A brief tour of early vision in humans
Livingstone & Hubel, 1988
Signals from the RGCs are sent to the lateral geniculate nucleus (LGN) in the thalamus, then on to primary visual cortex (V1)
The parvocellular pathway is color-selective, slow, high- resolution (small receptive fields)
The magnocellular pathway is color-blind, motion-sensitive, low- resolution (large receptive fields)
A brief tour of early vision in humans
Signals from the RGCs are sent to the lateral geniculate nucleus (LGN) in the thalamus, then on to primary visual cortex (V1)
The parvocellular pathway is color-selective, slow, high- resolution (small receptive fields)
The magnocellular pathway is color-blind, motion-sensitive, low- resolution (large receptive fields)
Visible light is just a thin slice of the electromagnetic spectrum: ~380 to ~750 nm
Human color vision relies entirely on the comparison of responses from three cone types!
photoreceptor response
S cones: short wavelength (blue)
M cones: medium wavelength (green)
L cones: long wavelength (red)
Color vision
Responses from a single cone type are ambiguous! (⊙_◎)
photoreceptor response
The problem of univariance
Absorption spectrum describes the response of a photoreceptor as a function of wavelength
Responses from a single cone type are ambiguous! (⊙_◎)
photoreceptor response
The problem of univariance
response ≈ .5
Absorption spectrum describes the response of a photoreceptor as a function of wavelength
Responses from a single cone type are ambiguous! (⊙_◎)
photoreceptor response
The problem of univariance
response ≈ .5
Absorption spectrum describes the response of a photoreceptor as a function of wavelength
Responses from a single cone type are ambiguous! (⊙_◎)
photoreceptor response
The problem of univariance
response ≈ .5
Photoreceptor has the same response to both of these wavelengths!
But the problem gets even worse…
Absorption spectrum describes the response of a photoreceptor as a function of wavelength
Responses from a single cone type are ambiguous! (⊙_◎)
photoreceptor response
The problem of univariance
response ≈ .5
The photoreceptor response scales with light intensity! Can’t differentiate weak intensity at peak absorption and high intensity at off-peak absorption
Absorption spectrum describes the response of a photoreceptor as a function of wavelength
×.5
×2
An infinite set of wavelength–intensity combinations can elicit the same response!
photoreceptor response
The problem of univariance
response ≈ .5
×.5
×2
Three-dimensional color space
We can compute 3D cone response vectors using matrix-vector multiplication
Three-dimensional color space
We can compute 3D cone response vectors using matrix-vector multiplication
illuminant spectrum
cone absorption spectra
cone responses
Maxwell’s color-matching experiment
Any “test” light (vector) can be matched by adjusting the intensities of any three other lights (basis vectors)
Metamers
Metamers are illuminants that are physically distinct but are perceptually indistinguishable
Metamers are a subspace of illuminants that have the same linear projection onto the three cone absorption spectra basis vectors!
Metamers
Metamers are illuminants that are physically distinct but are perceptually indistinguishable
illuminant spectrum
cone absorption spectrum
cone responses
Three-dimensional color space
Maxwell created the first color photograph using a composite of photographs with red, green, and blue filters
Maxwell, 1861
Three-dimensional color space
Monitors and digital images rely on only three illuminants to approximate any color
CRT monitor with three
(red, green, blue) phosphors
Color blindness
Dichromats typically only have two functional cone types
Protanopia: absence of L cones
Deuteranopia: absence of M cones
Tritanopia: absence of S cones
Color blindness
Dichromats typically only have two functional cone types
Protanopia: absence of L cones
Deuteranopia: absence of M cones
Tritanopia: absence of S cones
Color blindness
Dichromats typically only have two functional cone types
Protanopia: absence of L cones
Deuteranopia: absence of M cones
Tritanopia: absence of S cones
Color-blind folks are living in a two-dimensional subspace! (⌐▨_▨)
Color vision in other animals
There’s an incredible variety of color vision in the animal kingdom
—most mammals (dogs, cats, horses): dichromats
—old-world primates (including us): trichromats
—marine mammals: monochromats
—bees: trichromats (lack red “L” cone; ultraviolet instead)
—some birds, reptiles & amphibians: tetrachromats!
Color vision in other animals
There’s an incredible variety of color vision in the animal kingdom
Some tetrachromatic birds (e.g. zebra finches) use ultraviolet vision in foraging and display ultraviolet plumage during mate selection
Center-surround receptive fields
The receptive fields of retinal ganglion cells (and LGN cells) exhibit center-surround antagonism
Center-surround receptive fields
The receptive fields of retinal ganglion cells (and LGN cells) exhibit center-surround antagonism
Center-surround receptive fields
The receptive fields of retinal ganglion cells (and LGN cells) exhibit center-surround antagonism
Center-surround receptive fields
We can model this phenomenon using the dot product!
RGC
photoreceptors
Center-surround receptive fields
We can model this phenomenon using the dot product!
RGC
photoreceptors
Center-surround receptive fields
We can model this phenomenon using the dot product!
RGC
photoreceptors
Center-surround receptive fields
We can model this phenomenon using the dot product!
RGC
photoreceptors
Color constancy
The visual system discounts the illuminant; it cares more about the surface than the illuminant
Color constancy
The visual system discounts the illuminant; it cares more about the surface than the illuminant
Color constancy
But color constancy isn’t perfect! ｢(ﾟﾍﾟ)
dodecachromatic color vision?