1. Neurocomputational Overview & Principles of Chromatic Adaptation
Color is not an absolute physical property of electromagnetic radiation. Physical light
is defined purely by its radiant power distribution across wavelengths $\lambda \in [380, 780]\text{
nm}$. The subjective perceptual sensation of hue, saturation, and lightness is a biological fabrication
synthesized by the tri-receptor retinal cone mosaic and transformed through reciprocal opponent-color
pathways in the retina, lateral geniculate nucleus (LGN), and primary visual cortex (striate area V1).
"The Ghost Leaf" laboratory models the phototransduction limits of human vision. In the human fovea
centralis, high-acuity color photopic vision is mediated by three classes of cone photoreceptors
expressing distinct opsin G-protein-coupled receptors:
- L-Cones (Long-Wavelength Erythrolabe): Spectral absorption peak $\lambda_{\max}
\approx 564\text{ nm}$ (red-dominant spectrum).
- M-Cones (Medium-Wavelength Chlorolabe): Spectral absorption peak $\lambda_{\max}
\approx 534\text{ nm}$ (green-dominant spectrum).
- S-Cones (Short-Wavelength Cyanolabe): Spectral absorption peak $\lambda_{\max}
\approx 420\text{ nm}$ (blue-dominant spectrum).
Under steady-state illumination, light absorption drives the photoisomerization of the 11-cis retinal chromophore
bound within the photopsin GPCR complex into all-trans-retinal. This activates a G-protein-coupled signaling cascade
wherein the G-protein transducin ($\text{G}_{\text{t}}$) activates phosphodiesterase (PDE), catalyzing the hydrolysis
of cyclic guanosine monophosphate ($\text{cGMP}$). The reduction of cytoplasmic $\text{cGMP}$ concentrations induces the closure
of cyclic nucleotide-gated (CNG) cation channels, triggering hyperpolarization of the photoreceptor outer membrane
and a subsequent reduction in the tonic release of the neurotransmitter glutamate at the ribbon synapse.
When you maintain a steady, locked fixation on the high-contrast magenta silhouette, the localized foveal foveal
receptors receive intense stimulation exclusively across the L and S spectral channels while the M cones remain entirely
unstimulated in darkness. This sustained photon flux drives the rates of photoisomerization and subsequent bleaching
faster than the enzymatic visual cycle inside the adjacent retinal pigment epithelium (RPE) can regenerate 11-cis retinal
and reassemble functional photopsins. L and S photopsins become exhausted, while M photopsins retain a full 100%
biochemical reserve.
Upon instantaneous transition to a uniform, broadband white adaptation surface (which emits equal energy across the entire visible spectrum),
L, M, and S cones are subjected to identical photon densities. However, because the L and S cones are highly fatigued,
their receptor potentials are attenuated, producing minimal hyperpolarization and maintaining high glutamate release. Conversely, the rested
and hyper-sensitive M-cones hyperpolarize completely, inhibiting glutamate release. In the inner nuclear layers, this hyperpolarization
is processed by post-synaptic bipolar cells and parvocellular ganglion cells. The downstream visual pathway integrates the lopsided
response as an intense green stimulus, fabricating the appearance of a vivid emerald-green phantom leaf in the exact spatial geometry of the bleached foveal footprint.
3. Mathematical Foundations, Biophysical Kinetics & Architecture
The simulator continuously integrates a non-linear first-order differential model of photopigment
bleaching and metabolic regeneration based on the biophysical Rushton-Pugh kinetic equations. Let $S_i(t)
\in [0.08, 1.0]$ represent the fractional density of active, unbleached photopsin in receptor class $i
\in \{L, M, S\}$:
$$ \frac{dS_i}{dt} = -k_{\text{bleach}} \cdot I_i(\theta) \cdot S_i(t) + k_{\text{regen}} \cdot (1.0 - S_i(t)) $$
Where $I_i(\theta) \in [0, 1]$ represents the normalized spectral excitation intensity computed from
the stimulus color space, $k_{\text{bleach}} = \frac{1.85}{\tau}$ establishes the bleaching rate
scaled to the target exposure interval, and $k_{\text{regen}} \approx 0.055\text{ s}^{-1}$ models the
enzymatic resynthesis of 11-cis retinal by the RPE visual cycle (corresponding to a half-life of $t_{1/2}
\approx 12.6\text{ s}$).
Hering Retinal Ganglion Opponent Space Transformation
The visual cortex does not receive raw $L, M, S$ signals directly. Retinal ganglion cells and LGN
neurons convert photoreceptor outputs into three orthogonal physiological opponent channels:
$$
\begin{aligned}
\text{Luminance Channel } (A) &= L \cdot S_L + M \cdot S_M + 0.1 \cdot S \cdot S_S \\
\text{Red-Green Opponent } (RG) &= \frac{L \cdot S_L - M \cdot S_M}{L \cdot S_L + M \cdot S_M + \epsilon} \\
\text{Yellow-Blue Opponent } (YB) &= \frac{(L \cdot S_L + M \cdot S_M) - S \cdot S_S}{L \cdot S_L + M \cdot S_M + S \cdot S_S + \epsilon}
\end{aligned}
$$
During the blank afterimage phase with uniform white input $L = M = S = 1$, the depleted photopigments
($S_L \ll 1, S_M \approx 1, S_S \ll 1$) produce a strongly negative $RG$ value, compelling the striate
cortex to perceive vivid emerald green in the exact spatial geometry of the bleached macular footprint.
Procedural Foveal Mosaic & Fermat Spiral Packing
The embedded foveal mosaic displays 180 simulated cone photoreceptors arranged via a 2D Fermat phyllotaxis
spiral using the golden angle $\alpha \approx 137.5077^\circ$:
$$ r_n = c \sqrt{n}, \quad \theta_n = n \cdot \alpha $$
Cones are sampled against an analytical geometric boundary envelope representing the leaf margin:
$$ |x_n| < 0.58 \cdot \cos\left(\frac{y_n \cdot \pi}{1.8}\right) \quad \text{for } y_n \in [-0.85, 0.85] $$
Cone types are distributed according to physiological primate macular ratios ($\sim 60\%\text{ L}$,
$\sim 30\%\text{ M}$, $\sim 10\%\text{ S}$). Only photoreceptors falling geometrically inside the leaf
envelope undergo localized photopigment depletion.
Deterministic Audio Synthesis & Web Audio API
The simulation integrates an ambient Web Audio synthesizer designed with an exponential binaural carrier
drone ($160\text{ Hz}$ left / $162\text{ Hz}$ right creating a $2\text{ Hz}$ focus beat) that sweeps
upward to $220\text{ Hz}$ as photoreceptor fatigue intensifies. At the transition point, an exponential
decay chord ($660\text{ Hz} \to 440\text{ Hz}$) signals the visual shift, followed by a resonant harmonic
whose pitch reflects the active cortical opponent vector.