Module 6

Vision & Color

Telephoto eyes, retinal scanning, structural color, tetrachromatic vision, and courtship displays

Jumping spiders possess arguably the most sophisticated visual system of any arthropod. Their anterior median (AM) eyes function as miniature telephoto cameras with an f-number of 0.58 and angular resolution approaching 0.04° — rivaling some vertebrates. Combined with true tetrachromatic color vision, retinal scanning mechanisms, and the dazzling structural colors of peacock spiders, spider vision represents a masterclass in evolutionary optical engineering within extreme size constraints.

6.1 Jumping Spider Eye Architecture

All spiders have 8 eyes (with rare exceptions), but in jumping spiders (family Salticidae, >6,400 species), the arrangement is highly specialized. The eyes are organized into two functional groups:

Anterior Median (AM) eyes — the large, forward-facing "principal" eyes. These are telephoto cameras with the highest spatial acuity of any animal their size.
Anterior Lateral (AL) eyes — wide-angle motion detectors flanking the AM eyes, ~160° field of view.
Posterior Median (PM) eyes — small, sometimes vestigial in salticids.
Posterior Lateral (PL) eyes — rear-facing motion detectors, giving nearly 360° visual coverage.

The AM Telephoto Design

The AM eyes of Portia fimbriata and Hasarius adansoni achieve extraordinary spatial resolution through a unique telephoto optical design. Unlike typical simple eyes where the focal length roughly equals the eye diameter, the AM eyes use a deeply recessed retina in a long tube, creating an effective focal length \(f \approx 720\,\mu\text{m}\) with a corneal lens diameter of only\(D \approx 400\,\mu\text{m}\). This gives a focal ratio:

Focal Ratio of AM Eye

\[ f/\# = \frac{f}{D} = \frac{720\,\mu\text{m}}{400\,\mu\text{m}} \approx 1.8 \]

The effective f-number considering the Airy disk and retinal receptor spacing gives an operational performance equivalent to f/0.58 — among the fastest biological lenses known.

Diffraction-Limited Resolution

The angular resolution of the AM eye is ultimately limited by diffraction. For a circular aperture of diameter \(D\) at wavelength \(\lambda\), the Rayleigh criterion gives the minimum resolvable angle:

Rayleigh Criterion for AM Eyes

\[ \Delta\phi = 1.22 \frac{\lambda}{D} = 1.22 \times \frac{500\,\text{nm}}{400\,\mu\text{m}} = 1.22 \times \frac{5 \times 10^{-7}}{4 \times 10^{-4}} \approx 1.53 \times 10^{-3}\,\text{rad} \approx 0.088° \]

In practice, the inter-receptor spacing on the retina (\(\sim 1\,\mu\text{m}\)) combined with the long focal length gives an effective angular subtense per receptor of\(\Delta\phi_{\text{receptor}} = d/f = 1\,\mu\text{m}/720\,\mu\text{m} \approx 0.08° \approx 4.8\,\text{arcmin}\). Human foveal acuity is ~1 arcmin — so the jumping spider achieves roughly 1/5 of human resolution with an eye 100× smaller.

6.2 Eye Tube Scanning Mechanism

The AM eyes have fixed corneal lenses embedded in the carapace — they cannot rotate in their sockets like vertebrate eyes. Instead, the retina itself moves. Six muscles attach to each eye tube, allowing the bowtie-shaped retina to translate laterally, vertically, and rotate within the tube. This retinal scanning mechanism allows the spider to survey a visual field far wider than the retina's narrow 3–5° field of view.

Scanning Kinematics

Eye-tracking studies by Land (1969) and later by Jakob et al. (2018) revealed a characteristic scanning pattern with two modes:

Saccades — rapid repositioning movements (~200°/s) lasting 50–100 ms
Fixations — slow scanning at 1–5°/s, during which the retina systematically sweeps across a region of interest for 0.5–2 s

The scanning rate is approximately 1 Hz for complete scan cycles. The retina's bowtie shape (wide horizontally, narrow vertically) means each fixation-scan samples a horizontal strip. The effective spatial coverage can be derived:

Spatial Coverage per Scan

\[ \Omega_{\text{scan}} = \Delta\theta_{\text{retina}} \times v_{\text{scan}} \times t_{\text{fixation}} = 5° \times 3°/\text{s} \times 1.5\,\text{s} \approx 22.5\,\text{deg}^2 \]

where \(\Delta\theta_{\text{retina}} \approx 5°\) is the retinal width,\(v_{\text{scan}} \approx 3°/\text{s}\) is the scanning velocity, and\(t_{\text{fixation}} \approx 1.5\,\text{s}\) is the fixation duration. Multiple saccade-fixation cycles allow full coverage of objects in the ~28° AM field.

Information Processing Rate

The bowtie retina contains roughly 200×10 photoreceptors in its central region. At a scan rate of ~1 Hz and with the four-layer depth providing color information, the information throughput is:

\[ I = N_{\text{receptors}} \times N_{\text{layers}} \times f_{\text{scan}} \times \log_2(k) \approx 2000 \times 4 \times 1\,\text{Hz} \times 8\,\text{bits} = 64\,\text{kbit/s} \]

This is remarkably high for an arthropod and explains why jumping spiders can discriminate prey types, recognize conspecifics, and plan detour routes purely through vision.

Jumping Spider AM Eye Resolution & Scanning Coverage

Python

script.py121 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ========== AM Eye Optical Parameters ==========
wavelengths = np.linspace(350, 700, 200)  # nm
D = 400e-6  # corneal diameter (m)
f = 720e-6  # focal length (m)
d_receptor = 1.0e-6  # receptor spacing (m)

# Rayleigh criterion: angular resolution vs wavelength
delta_phi_rayleigh = 1.22 * wavelengths * 1e-9 / D  # radians
delta_phi_rayleigh_deg = np.degrees(delta_phi_rayleigh)
delta_phi_rayleigh_arcmin = delta_phi_rayleigh_deg * 60

# Receptor-limited resolution
delta_phi_receptor = d_receptor / f  # radians
delta_phi_receptor_arcmin = np.degrees(delta_phi_receptor) * 60

# Human comparison
human_acuity_arcmin = 1.0  # arcmin

fig, axes = plt.subplots(1, 3, figsize=(16, 5))

# Panel 1: Angular resolution vs wavelength
ax1 = axes[0]
ax1.plot(wavelengths, delta_phi_rayleigh_arcmin, color='#a78bfa', linewidth=2, label='Diffraction limit (Rayleigh)')
ax1.axhline(y=delta_phi_receptor_arcmin, color='#f472b6', linewidth=2, linestyle='--', label=f'Receptor limit ({delta_phi_receptor_arcmin:.1f} arcmin)')
ax1.axhline(y=human_acuity_arcmin, color='#fbbf24', linewidth=1.5, linestyle=':', label='Human foveal acuity')
ax1.fill_between(wavelengths, delta_phi_rayleigh_arcmin, delta_phi_receptor_arcmin,
                  where=delta_phi_rayleigh_arcmin < delta_phi_receptor_arcmin,
                  alpha=0.15, color='#a78bfa')
ax1.set_xlabel('Wavelength (nm)', fontsize=12, color='white')
ax1.set_ylabel('Angular Resolution (arcmin)', fontsize=12, color='white')
ax1.set_title('AM Eye Angular Resolution', fontsize=14, fontweight='bold', color='#c4b5fd')
ax1.legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#7c3aed', labelcolor='white')
ax1.set_facecolor('#0a0a1a')
ax1.tick_params(colors='white')
ax1.set_ylim(0, 12)
for spine in ax1.spines.values():
    spine.set_color('#7c3aed')

# Panel 2: Scanning coverage simulation
ax2 = axes[1]
np.random.seed(42)
n_saccades = 8
retina_width = 5.0  # degrees
retina_height = 1.5  # degrees
scan_speed = 3.0  # deg/s
fixation_time = 1.5  # s

saccade_positions = []
scan_patches = []
theta = 0
for i in range(n_saccades):
    cx = np.random.uniform(-12, 12)
    cy = np.random.uniform(-8, 8)
    saccade_positions.append((cx, cy))
    # During fixation, retina sweeps vertically
    scan_extent = scan_speed * fixation_time
    rect = plt.Rectangle((cx - retina_width/2, cy - scan_extent/2),
                           retina_width, scan_extent,
                           facecolor='#a78bfa', alpha=0.25, edgecolor='#c4b5fd', linewidth=1)
    ax2.add_patch(rect)
    # Retina at center position
    retina = plt.Rectangle((cx - retina_width/2, cy - retina_height/2),
                             retina_width, retina_height,
                             facecolor='#7c3aed', alpha=0.6, edgecolor='#e9d5ff', linewidth=1.5)
    ax2.add_patch(retina)

# Draw saccade paths
for i in range(len(saccade_positions) - 1):
    x0, y0 = saccade_positions[i]
    x1, y1 = saccade_positions[i + 1]
    ax2.annotate('', xy=(x1, y1), xytext=(x0, y0),
                arrowprops=dict(arrowstyle='->', color='#f472b6', lw=1.2))

# AM field boundary
am_field = plt.Circle((0, 0), 14, fill=False, edgecolor='#4c1d95', linewidth=2, linestyle='--')
ax2.add_patch(am_field)

ax2.set_xlim(-18, 18)
ax2.set_ylim(-18, 18)
ax2.set_aspect('equal')
ax2.set_xlabel('Azimuth (degrees)', fontsize=12, color='white')
ax2.set_ylabel('Elevation (degrees)', fontsize=12, color='white')
ax2.set_title('Saccade-Fixation Scanning Pattern', fontsize=14, fontweight='bold', color='#c4b5fd')
ax2.set_facecolor('#0a0a1a')
ax2.tick_params(colors='white')
for spine in ax2.spines.values():
    spine.set_color('#7c3aed')

# Panel 3: Comparison across species
ax3 = axes[2]
species = ['Human\nfovea', 'Eagle', 'Jumping\nspider AM', 'Honeybee', 'Fruit fly']
acuities = [1, 0.4, 4.8, 60, 300]  # arcmin
colors_bar = ['#fbbf24', '#f97316', '#a78bfa', '#22c55e', '#ef4444']

bars = ax3.barh(species, acuities, color=colors_bar, edgecolor='white', linewidth=0.5, height=0.6)
ax3.set_xlabel('Min. Resolvable Angle (arcmin)', fontsize=12, color='white')
ax3.set_title('Visual Acuity Comparison', fontsize=14, fontweight='bold', color='#c4b5fd')
ax3.set_xscale('log')
ax3.set_facecolor('#0a0a1a')
ax3.tick_params(colors='white')
for spine in ax3.spines.values():
    spine.set_color('#7c3aed')

for bar, val in zip(bars, acuities):
    ax3.text(val * 1.3, bar.get_y() + bar.get_height()/2,
             f'{val} arcmin', va='center', fontsize=10, color='white')

fig.patch.set_facecolor('#0a0a1a')
plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')
plt.close()
print("AM Eye Resolution Analysis Complete")
print(f"Diffraction limit at 500nm: {np.degrees(1.22 * 500e-9 / D) * 60:.1f} arcmin")
print(f"Receptor limit: {np.degrees(d_receptor / f) * 60:.1f} arcmin")
print(f"Human acuity: 1.0 arcmin")
print(f"Spider/Human ratio: {np.degrees(d_receptor / f) * 60 / 1.0:.1f}x coarser")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

6.3 UV Perception & Structural Color

Peacock spiders (genus Maratus, ~90 described species from Australia) produce some of the most spectacular color displays in the animal kingdom. Unlike pigment-based coloration, these colors arise from structural mechanisms — nanostructured cuticle elements that manipulate light through physical principles: diffraction gratings, multilayer thin-film interference, and 3D photonic crystals.

Thin-Film Interference

The simplest structural color mechanism involves thin-film interference from cuticular layers. When light hits a thin film of thickness \(d\) and refractive index \(n\), reflections from the top and bottom surfaces interfere constructively when:

Constructive Interference Condition

\[ 2 n d \cos\theta = m\lambda, \qquad m = 1, 2, 3, \ldots \]

where \(\theta\) is the angle of refraction within the film, \(\lambda\) is the vacuum wavelength, and \(m\) is the interference order. For chitin (\(n \approx 1.56\)) with \(d \approx 160\,\text{nm}\) at normal incidence:\(\lambda = 2nd/m = 2 \times 1.56 \times 160/1 \approx 499\,\text{nm}\) (green-blue).

Multilayer Reflectors in Maratus

In Maratus robinsoni and M. chrysomelas, Hsiung et al. (2017) found that the abdominal flap scales contain multilayer stacks of alternating high-\(n\) (chitin, 1.56) and low-\(n\) (air gaps, 1.0) layers. The reflectance of an \(N\)-layer stack at peak wavelength is:

Multilayer Reflectance

\[ R = \left(\frac{1 - (n_H/n_L)^{2N}}{1 + (n_H/n_L)^{2N}}\right)^2 \]

For \(N = 8\) bilayers: \(R = \left(\frac{1 - (1.56)^{16}}{1 + (1.56)^{16}}\right)^2 \approx 0.996\) — near-perfect reflection at the design wavelength, rivaling engineered dielectric mirrors.

3D Photonic Crystals

The rainbow scales of Maratus robinsoni are unique in the animal kingdom: they contain nanoscale airfoil-shaped structures that function as diffraction gratings combined with a 3D photonic crystal architecture. The grating equation gives the diffracted wavelength:

\[ d(\sin\theta_i + \sin\theta_d) = m\lambda \]

where \(d \approx 400\,\text{nm}\) is the grating period, and\(\theta_i, \theta_d\) are incident and diffracted angles. The nanoscale curvature of each airfoil further separates different wavelengths spatially, creating "super-black" regions (reflectance < 0.5%) adjacent to brilliant spectral colors — enhancing perceived chromatic contrast.

Multilayer Interference & Maratus Spectral Reflectance

Python

script.py158 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ========== Transfer Matrix Method for Multilayer Reflectance ==========
def transfer_matrix_reflectance(wavelengths, layers, theta_i=0):
    """
    Calculate reflectance of a multilayer thin film using transfer matrix method.
    layers: list of (n, d) tuples - refractive index and thickness (nm)
    """
    n_air = 1.0
    R = np.zeros_like(wavelengths, dtype=float)

for idx, lam in enumerate(wavelengths):
        # Start with identity matrix
        M = np.array([[1.0, 0.0], [0.0, 1.0]], dtype=complex)

cos_i = np.cos(np.radians(theta_i))

for n, d in layers:
            # Snell's law for this layer
            sin_t = n_air * np.sin(np.radians(theta_i)) / n
            if abs(sin_t) > 1:
                cos_t = 0
            else:
                cos_t = np.sqrt(1 - sin_t**2)

# Phase thickness
            delta = 2 * np.pi * n * d * cos_t / lam

# Layer matrix (TE polarization)
            eta = n * cos_t
            layer_M = np.array([
                [np.cos(delta), 1j * np.sin(delta) / eta],
                [1j * eta * np.sin(delta), np.cos(delta)]
            ])
            M = M @ layer_M

# Calculate reflectance
        eta_0 = n_air * cos_i
        eta_s = 1.56 * cos_i  # substrate (chitin)

r = (eta_0 * M[0,0] + eta_0 * eta_s * M[0,1] - M[1,0] - eta_s * M[1,1]) / \
            (eta_0 * M[0,0] + eta_0 * eta_s * M[0,1] + M[1,0] + eta_s * M[1,1])
        R[idx] = abs(r)**2

return R

wavelengths = np.linspace(300, 800, 500)

# ========== Different Maratus color patches ==========
fig, axes = plt.subplots(2, 2, figsize=(14, 10))

# Panel 1: Single thin film - varying thickness
ax1 = axes[0, 0]
thicknesses = [100, 130, 160, 200, 250]
for d in thicknesses:
    layers = [(1.56, d)]
    R = transfer_matrix_reflectance(wavelengths, layers)
    ax1.plot(wavelengths, R * 100, linewidth=1.8, label=f'd = {d} nm')

ax1.set_xlabel('Wavelength (nm)', fontsize=12, color='white')
ax1.set_ylabel('Reflectance (%)', fontsize=12, color='white')
ax1.set_title('Single Chitin Film: Varying Thickness', fontsize=14, fontweight='bold', color='#c4b5fd')
ax1.legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#7c3aed', labelcolor='white')
ax1.set_facecolor('#0a0a1a')
ax1.tick_params(colors='white')
for spine in ax1.spines.values():
    spine.set_color('#7c3aed')

# Panel 2: Multilayer stack - increasing N
ax2 = axes[0, 1]
n_bilayers_list = [1, 2, 4, 8, 12]
d_H, d_L = 80, 100  # quarter-wave at ~500nm

for N in n_bilayers_list:
    layers = []
    for _ in range(N):
        layers.append((1.56, d_H))  # chitin
        layers.append((1.0, d_L))   # air gap
    R = transfer_matrix_reflectance(wavelengths, layers)
    ax2.plot(wavelengths, R * 100, linewidth=1.8, label=f'N = {N} bilayers')

ax2.set_xlabel('Wavelength (nm)', fontsize=12, color='white')
ax2.set_ylabel('Reflectance (%)', fontsize=12, color='white')
ax2.set_title('Multilayer Stack: Increasing Bilayers', fontsize=14, fontweight='bold', color='#c4b5fd')
ax2.legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#7c3aed', labelcolor='white')
ax2.set_facecolor('#0a0a1a')
ax2.tick_params(colors='white')
for spine in ax2.spines.values():
    spine.set_color('#7c3aed')

# Panel 3: Angle-dependent reflectance (iridescence)
ax3 = axes[1, 0]
angles = [0, 15, 30, 45, 60]
layers_8 = []
for _ in range(8):
    layers_8.append((1.56, 80))
    layers_8.append((1.0, 100))

for theta in angles:
    R = transfer_matrix_reflectance(wavelengths, layers_8, theta_i=theta)
    ax3.plot(wavelengths, R * 100, linewidth=1.8, label=f'\u03B8 = {theta}\u00B0')

ax3.set_xlabel('Wavelength (nm)', fontsize=12, color='white')
ax3.set_ylabel('Reflectance (%)', fontsize=12, color='white')
ax3.set_title('Iridescence: Angle-Dependent Shift', fontsize=14, fontweight='bold', color='#c4b5fd')
ax3.legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#7c3aed', labelcolor='white')
ax3.set_facecolor('#0a0a1a')
ax3.tick_params(colors='white')
for spine in ax3.spines.values():
    spine.set_color('#7c3aed')

# Panel 4: Simulated Maratus spectral patches
ax4 = axes[1, 1]

# Blue patch: thin chitin layers
blue_layers = [(1.56, 60), (1.0, 80)] * 6
R_blue = transfer_matrix_reflectance(wavelengths, blue_layers)

# Green patch: medium chitin
green_layers = [(1.56, 80), (1.0, 100)] * 6
R_green = transfer_matrix_reflectance(wavelengths, green_layers)

# Red patch: thick chitin
red_layers = [(1.56, 110), (1.0, 130)] * 6
R_red = transfer_matrix_reflectance(wavelengths, red_layers)

# UV patch
uv_layers = [(1.56, 45), (1.0, 60)] * 6
R_uv = transfer_matrix_reflectance(wavelengths, uv_layers)

ax4.fill_between(wavelengths, R_uv * 100, alpha=0.3, color='#8b5cf6')
ax4.fill_between(wavelengths, R_blue * 100, alpha=0.3, color='#3b82f6')
ax4.fill_between(wavelengths, R_green * 100, alpha=0.3, color='#22c55e')
ax4.fill_between(wavelengths, R_red * 100, alpha=0.3, color='#ef4444')
ax4.plot(wavelengths, R_uv * 100, color='#8b5cf6', linewidth=2, label='UV patch')
ax4.plot(wavelengths, R_blue * 100, color='#3b82f6', linewidth=2, label='Blue patch')
ax4.plot(wavelengths, R_green * 100, color='#22c55e', linewidth=2, label='Green patch')
ax4.plot(wavelengths, R_red * 100, color='#ef4444', linewidth=2, label='Red patch')

ax4.set_xlabel('Wavelength (nm)', fontsize=12, color='white')
ax4.set_ylabel('Reflectance (%)', fontsize=12, color='white')
ax4.set_title('Simulated Maratus Color Patches', fontsize=14, fontweight='bold', color='#c4b5fd')
ax4.legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#7c3aed', labelcolor='white')
ax4.set_facecolor('#0a0a1a')
ax4.tick_params(colors='white')
for spine in ax4.spines.values():
    spine.set_color('#7c3aed')

fig.patch.set_facecolor('#0a0a1a')
plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')
plt.close()
print("Structural Color Analysis Complete")
print(f"Peak reflectance with 8 bilayers: >99%")
print(f"Blue shift at 60 deg: ~{int(500 * np.cos(np.radians(60)))} nm from 500 nm peak")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

6.4 Tetrachromatic Color Vision

Jumping spiders possess four classes of photoreceptor, each tuned to a different wavelength by virtue of distinct visual pigments and spectral filtering through the tiered retinal architecture. This constitutes true tetrachromacy — they can discriminate colors along four independent spectral axes, compared to three in most humans.

Photoreceptor Classes

UV Receptor

\(\lambda_{\max} \approx 380\,\text{nm}\) — Layer 4 (deepest)

Blue Receptor

\(\lambda_{\max} \approx 480\,\text{nm}\) — Layer 3

Green Receptor

\(\lambda_{\max} \approx 530\,\text{nm}\) — Layer 2

Red-shifted Receptor

\(\lambda_{\max} \approx 580\,\text{nm}\) — Layer 1 (front)

The spectral sensitivity of each receptor class follows a template based on the Govardovskii nomogram. For a receptor with peak sensitivity at \(\lambda_{\max}\), the normalized absorbance at wavelength \(\lambda\) is approximated by:

Govardovskii Visual Pigment Template

\[ S(\lambda) = \frac{1}{\exp\left[a_0\left(\frac{\lambda_{\max}}{\lambda} - 1\right)\right] + a_1 \cdot \exp\left[-a_2\left(\frac{\lambda_{\max}}{\lambda} - 1\right)^2\right] + 1} \]

with \(a_0 \approx 69.7\), \(a_1 \approx 0.29\), \(a_2 \approx 38.7\)for rhodopsin-based pigments. The tiered retinal architecture acts as a spectral filter: shorter wavelengths are absorbed by the front layers, progressively red-shifting the effective sensitivity of deeper layers.

Receptor Noise-Limited Color Discrimination

The ability of a tetrachromat to discriminate two colors depends on the signal-to-noise ratio in each receptor channel. The receptor noise-limited model(Vorobyev & Osorio 1998) predicts that two stimuli are discriminable when their chromatic distance \(\Delta S\) exceeds a threshold:

Chromatic Distance (Tetrachromat)

\[ (\Delta S)^2 = \frac{(\omega_1\omega_2)^2(\Delta f_4 - \Delta f_3)^2 + (\omega_1\omega_3)^2(\Delta f_4 - \Delta f_2)^2 + \ldots}{(\omega_1\omega_2\omega_3)^2 + (\omega_1\omega_2\omega_4)^2 + \ldots} \]

where \(\Delta f_i = \ln(q_i^A / q_i^B)\) is the log-ratio of quantum catches for stimuli A and B in receptor \(i\), and \(\omega_i\) is the Weber fraction (noise) of receptor \(i\). Colors with \(\Delta S > 1\) are discriminable.

The quantum catch \(q_i\) for receptor \(i\) viewing a stimulus with spectral reflectance \(R(\lambda)\) under illuminant \(I(\lambda)\) is:

\[ q_i = \int S_i(\lambda) \, R(\lambda) \, I(\lambda) \, d\lambda \]

6.5 Courtship Displays & Honest Signaling

Male jumping spiders perform some of the most elaborate courtship displays in the arthropod world.Maratus volans and its relatives raise their brilliantly colored abdominal flaps while performing choreographed dances, waving their third pair of legs, and producing substrate-borne vibratory signals. These multimodal displays are subject to intense sexual selection by highly visual, choosy females.

Honest Signaling Theory

The Zahavian handicap principle suggests that costly signals are inherently honest because only high-quality individuals can afford them. In peacock spiders, the UV-reflective nanostructures are metabolically expensive to produce and maintain. The signaling cost can be modeled as:

Costly Signal Model

\[ W(q, s) = B(s) - C(s, q) \]

where \(W\) is fitness, \(s\) is signal intensity (color saturation, dance vigor), \(q\) is male quality, \(B(s)\) is the mating benefit (increasing in \(s\)), and \(C(s, q)\) is the signaling cost (decreasing in \(q\) for the same \(s\)). The equilibrium signal intensity satisfies \(\partial B/\partial s = \partial C/\partial s\), and because\(\partial^2 C/\partial s \partial q < 0\), high-quality males produce more intense signals.

Multimodal Integration

Female jumping spiders integrate visual and vibratory information when evaluating males. Elias et al. (2012) showed that in Habronattus pyrrithrix, females require both visual and vibratory components for maximal receptivity. The probability of female acceptance follows:

\[ P(\text{accept}) = \sigma\!\left(\beta_0 + \beta_V V + \beta_A A + \beta_{VA} V \cdot A\right) \]

where \(\sigma\) is the logistic function, \(V\) is visual signal quality,\(A\) is vibratory signal quality, and \(\beta_{VA} > 0\) captures the synergistic interaction. The interaction term means that a high-quality visual display amplifies the effect of vibratory signals and vice versa — the backup signal hypothesis.

Tetrachromatic Color Discrimination & Courtship Signal Model

Python

script.py183 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ========== Photoreceptor Sensitivity Templates ==========
def govardovskii(lam, lam_max):
    """Simplified Govardovskii template for visual pigment absorbance."""
    x = lam_max / lam
    A = 69.7
    B = 28.0
    C = -14.9
    D = 0.674
    a = 0.8795 + 0.0459 * np.exp(-(lam_max - 300)**2 / 11940)
    S = 1.0 / (np.exp(A * (a - x)) + np.exp(B * (a - x)) + np.exp(C * (a - x)) + D)
    S = np.clip(S, 0, 1)
    return S / np.max(S)

wavelengths = np.linspace(300, 700, 400)

# Four receptor classes
lam_max = [380, 480, 530, 580]
labels = ['UV (380 nm)', 'Blue (480 nm)', 'Green (530 nm)', 'Red (580 nm)']
colors = ['#8b5cf6', '#3b82f6', '#22c55e', '#ef4444']

sensitivities = np.array([govardovskii(wavelengths, lm) for lm in lam_max])

fig, axes = plt.subplots(2, 2, figsize=(14, 10))

# Panel 1: Receptor sensitivities
ax1 = axes[0, 0]
for i in range(4):
    ax1.fill_between(wavelengths, sensitivities[i], alpha=0.2, color=colors[i])
    ax1.plot(wavelengths, sensitivities[i], color=colors[i], linewidth=2, label=labels[i])

ax1.set_xlabel('Wavelength (nm)', fontsize=12, color='white')
ax1.set_ylabel('Normalized Sensitivity', fontsize=12, color='white')
ax1.set_title('Jumping Spider Photoreceptor Sensitivities', fontsize=14, fontweight='bold', color='#c4b5fd')
ax1.legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#7c3aed', labelcolor='white')
ax1.set_facecolor('#0a0a1a')
ax1.tick_params(colors='white')
for spine in ax1.spines.values():
    spine.set_color('#7c3aed')

# Panel 2: Color space projection (UV-Blue vs Green-Red)
ax2 = axes[0, 1]

# Generate random color stimuli
np.random.seed(42)
n_stimuli = 200
stimuli_types = []
stim_colors_plot = []

for i in range(n_stimuli):
    # Random reflectance spectra (Gaussian bumps)
    center = np.random.uniform(350, 650)
    width = np.random.uniform(30, 80)
    reflectance = np.exp(-(wavelengths - center)**2 / (2 * width**2))

# D65 illuminant approximation
    illuminant = np.ones_like(wavelengths)

# Quantum catches
    q = np.array([np.trapezoid(sensitivities[j] * reflectance * illuminant, wavelengths) for j in range(4)])
    q = q / (q.sum() + 1e-10)

# Color opponent channels
    uv_blue = np.log(q[0] + 1e-10) - np.log(q[1] + 1e-10)
    green_red = np.log(q[2] + 1e-10) - np.log(q[3] + 1e-10)

stimuli_types.append((uv_blue, green_red))

# Color based on dominant wavelength
    if center < 420:
        stim_colors_plot.append('#8b5cf6')
    elif center < 500:
        stim_colors_plot.append('#3b82f6')
    elif center < 560:
        stim_colors_plot.append('#22c55e')
    else:
        stim_colors_plot.append('#ef4444')

uv_b = [s[0] for s in stimuli_types]
g_r = [s[1] for s in stimuli_types]

ax2.scatter(uv_b, g_r, c=stim_colors_plot, s=30, alpha=0.7, edgecolors='white', linewidth=0.3)
ax2.axhline(0, color='#4c1d95', linewidth=0.8, linestyle='--')
ax2.axvline(0, color='#4c1d95', linewidth=0.8, linestyle='--')
ax2.set_xlabel('UV-Blue Opponent Channel', fontsize=12, color='white')
ax2.set_ylabel('Green-Red Opponent Channel', fontsize=12, color='white')
ax2.set_title('Tetrachromatic Color Space', fontsize=14, fontweight='bold', color='#c4b5fd')
ax2.set_facecolor('#0a0a1a')
ax2.tick_params(colors='white')
for spine in ax2.spines.values():
    spine.set_color('#7c3aed')

# Panel 3: Chromatic distance (Delta S) for pairs
ax3 = axes[1, 0]

weber_fractions = [0.13, 0.07, 0.07, 0.10]  # UV noisier

# Generate pairs at different spectral separations
separations = np.linspace(5, 200, 50)
delta_S_values = []

for sep in separations:
    # Stimulus A: peak at 450 nm
    R_A = np.exp(-(wavelengths - 450)**2 / (2 * 40**2))
    # Stimulus B: peak shifted
    R_B = np.exp(-(wavelengths - (450 + sep))**2 / (2 * 40**2))

illuminant = np.ones_like(wavelengths)

q_A = np.array([np.trapezoid(sensitivities[j] * R_A * illuminant, wavelengths) for j in range(4)])
    q_B = np.array([np.trapezoid(sensitivities[j] * R_B * illuminant, wavelengths) for j in range(4)])

# Avoid log(0)
    q_A = np.maximum(q_A, 1e-10)
    q_B = np.maximum(q_B, 1e-10)

df = np.log(q_A) - np.log(q_B)
    w = np.array(weber_fractions)

# Simplified Delta S for tetrachromat
    num = 0
    den = 0
    for a in range(4):
        for b in range(a+1, 4):
            num += (w[a] * w[b] * (df[a] - df[b]))**2
            for c in range(4):
                if c != a and c != b:
                    den += (w[a] * w[b] * w[c])**2

delta_S = np.sqrt(num / (den + 1e-20)) if den > 0 else 0
    delta_S_values.append(delta_S)

ax3.plot(separations, delta_S_values, color='#a78bfa', linewidth=2.5)
ax3.axhline(y=1, color='#f472b6', linewidth=1.5, linestyle='--', label='Discrimination threshold')
ax3.fill_between(separations, 0, 1, alpha=0.1, color='#f472b6')
ax3.set_xlabel('Spectral Separation (nm)', fontsize=12, color='white')
ax3.set_ylabel('Chromatic Distance (\u0394S)', fontsize=12, color='white')
ax3.set_title('Color Discrimination Threshold', fontsize=14, fontweight='bold', color='#c4b5fd')
ax3.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#7c3aed', labelcolor='white')
ax3.set_facecolor('#0a0a1a')
ax3.tick_params(colors='white')
for spine in ax3.spines.values():
    spine.set_color('#7c3aed')

# Panel 4: Honest signaling - cost/benefit curves
ax4 = axes[1, 1]
signal = np.linspace(0, 10, 100)

# Benefit: saturating
benefit = 5 * signal / (signal + 2)

# Cost for different quality males
qualities = [0.3, 0.6, 1.0, 1.5, 2.0]
for q in qualities:
    cost = (signal**2) / (2 * q)
    fitness = benefit - cost
    ax4.plot(signal, fitness, linewidth=2, label=f'q = {q}')
    # Optimal signal
    opt_idx = np.argmax(fitness)
    ax4.plot(signal[opt_idx], fitness[opt_idx], 'o', markersize=8, color='white', zorder=5)

ax4.axhline(0, color='#4c1d95', linewidth=0.8, linestyle='--')
ax4.set_xlabel('Signal Intensity (display vigor)', fontsize=12, color='white')
ax4.set_ylabel('Net Fitness W(q, s)', fontsize=12, color='white')
ax4.set_title('Honest Signaling: Cost vs Quality', fontsize=14, fontweight='bold', color='#c4b5fd')
ax4.legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#7c3aed', labelcolor='white', title='Male quality', title_fontsize=10)
ax4.set_facecolor('#0a0a1a')
ax4.tick_params(colors='white')
for spine in ax4.spines.values():
    spine.set_color('#7c3aed')

fig.patch.set_facecolor('#0a0a1a')
plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')
plt.close()
print("Color Vision & Courtship Analysis Complete")
print(f"Min spectral separation for discrimination: ~{separations[np.argmax(np.array(delta_S_values) > 1)]:.0f} nm")
print(f"Optimal signal for high-quality male (q=2.0): s* = {signal[np.argmax(5*signal/(signal+2) - signal**2/4)]:.1f}")
print(f"Optimal signal for low-quality male (q=0.3): s* = {signal[np.argmax(5*signal/(signal+2) - signal**2/0.6)]:.1f}")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

References

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Jakob, E.M., Long, S.M., Harland, D.P., Jackson, R.R., Carey, A., Searles, M.E., Porter, A.H., Canavesi, C. & Rolland, J.P. (2018). Lateral eyes direct principal eyes as jumping spiders track objects. Current Biology, 28(18), R1092–R1093.
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M5. Sensory Systems M7. Reproduction & Development