Module 2: Vision & Sensory Biophysics

Effective Quantum Efficiency:

\[ \eta_{\text{eff}} = \eta \left(1 + R \cdot e^{-2\alpha d}\right) \]

where:

• \( \eta \) = base quantum efficiency of photoreceptors (~0.06 for rods)
• \( R \) = reflectance of the tapetum (~0.45 for cat tapetum cellulosum)
• \( \alpha \) = absorption coefficient of the retina (~0.02 \(\mu\)m\(^{-1}\))
• \( d \) = retinal thickness traversed by reflected light (~200 \(\mu\)m)

\[ P_1 = 1 - e^{-\alpha d} \]

If not absorbed on the first pass, the photon reaches the tapetum and is reflected with probability \( R \). On the return pass through the retina:

\[ P_2 = R \cdot e^{-\alpha d} \cdot (1 - e^{-\alpha d}) \]

Total absorption probability:

\[ P_{\text{total}} = P_1 + P_2 = (1 - e^{-\alpha d})(1 + R \cdot e^{-\alpha d}) \]

The enhancement factor relative to no tapetum:

\[ \frac{P_{\text{total}}}{P_1} = 1 + R \cdot e^{-\alpha d} \]

\[ \text{Enhancement} = 1 + 0.45 \cdot e^{-0.02 \times 200} = 1 + 0.45 \cdot e^{-4} = 1 + 0.45 \times 0.0183 = 1.008 \]

This seems small! But the calculation above assumes the entire retinal thickness acts as a uniform absorber. In reality, the outer segments of the photoreceptors (where absorption occurs) occupy only about 25–30 \(\mu\)m. With \( d_{\text{eff}} = 28 \) \(\mu\)m:

\[ \text{Enhancement} = 1 + 0.45 \cdot e^{-0.02 \times 28} = 1 + 0.45 \times 0.571 = 1.257 \]

Furthermore, the effective absorption coefficient for the outer segment alone is much higher (\( \alpha_{\text{os}} \approx 0.015 \)\(\mu\)m\(^{-1}\) at peak wavelength), and accounting for the rod waveguide properties and multiple scattering in the tapetum crystal lattice, the practical enhancement is 1.4–1.6, consistent with physiological measurements (Ollivier et al., 2004).

\[ S(\lambda) = \frac{1}{\exp\left[A_1\left(a_1 - \frac{\lambda_{\max}}{\lambda}\right)\right] + \exp\left[A_2\left(\frac{\lambda_{\max}}{\lambda} - a_2\right)\right] + 1} \]

with \( A_1 = 69.7 \), \( a_1 = 0.88 \), \( A_2 = 28.0 \), \( a_2 = 0.922 \)for the \( \alpha \)-band. The cat's rod-to-cone ratio of 25:1 (vs 20:1 in humans) further emphasizes scotopic (dim-light) sensitivity at the expense of color discrimination.

Round pupil:

\[ A = \pi r^2, \quad \frac{A_{\max}}{A_{\min}} = \left(\frac{r_{\max}}{r_{\min}}\right)^2 \approx \left(\frac{4}{1}\right)^2 = 16 \]

Slit pupil:

\[ A \approx w \cdot h, \quad \frac{A_{\max}}{A_{\min}} = \frac{w_{\max} \cdot h_{\max}}{w_{\min} \cdot h_{\min}} \]

The slit constricts primarily in the horizontal direction (\( w \)), going from ~14 mm to ~0.1 mm (140:1 ratio), while the vertical extent (\( h \)) changes only modestly. This two-dimensional constriction plus the ability to form an extremely narrow slit (limited by diffraction, not muscle mechanics) yields the ~135-fold range.

\[ \theta_{\text{circ}} = 1.22 \frac{\lambda}{D} \]

For a slit aperture of width \( w \), the diffraction pattern in the direction perpendicular to the slit is:

\[ \theta_{\text{slit}} = \frac{\lambda}{w} \]

When the cat's pupil is fully constricted (\( w \approx 0.1 \) mm) in bright light:

\[ \theta_{\text{horizontal}} = \frac{550 \times 10^{-6} \text{ mm}}{0.1 \text{ mm}} = 5.5 \times 10^{-3} \text{ rad} = 0.32° \]

\[ \theta_{\text{vertical}} = \frac{550 \times 10^{-6} \text{ mm}}{10 \text{ mm}} = 5.5 \times 10^{-5} \text{ rad} = 0.003° \]

The slit pupil creates anisotropic resolution: excellent vertical resolution (for detecting horizontal edges and movements) but poorer horizontal resolution in bright light. This is advantageous for a predator that needs to detect the horizontal motion of prey on the ground.

\[ n(r) = n_0 - \Delta n \cdot \left(\frac{r}{R}\right)^2 \]

where \( n_0 \approx 1.55 \) (center), \( \Delta n \approx 0.08 \), and \( R \)is the lens radius. Different radial zones bring different wavelengths to focus at the retina simultaneously, effectively correcting for chromatic aberration without the achromatizing lens doublet used in human-made optics.

\[ \frac{1}{f(\lambda)} = (n(\lambda) - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \]

For the eye, LCA shifts the focal point by:

\[ \Delta f = f \cdot \frac{\Delta n}{n - 1} = f \cdot \frac{n_{\text{blue}} - n_{\text{red}}}{n_{\text{mean}} - 1} \]

The multifocal lens solves this by arranging concentric zones such that the central zone focuses red/green light and the peripheral zone focuses blue light, both onto the retinal plane. When the slit pupil is constricted, only a thin strip of the lens is sampled, and the radial variation in focal length across that strip acts as an achromatizing element:

\[ P_{\text{eff}}(\lambda) = \int_0^{w/2} P(r, \lambda) \, T(r) \, dr \bigg/ \int_0^{w/2} T(r) \, dr \]

where \( P(r, \lambda) \) is the power at radius \( r \) for wavelength \( \lambda \)and \( T(r) \) is the slit transmission function. Malmström & Kröger (2006) demonstrated that this system achieves nearly diffraction-limited imaging across the entire visible spectrum.

\[ f_n = \frac{\beta_n^2}{2\pi L^2} \sqrt{\frac{EI}{\rho A}} \]

where \( \beta_n L \) are the eigenvalues of the cantilever boundary value problem:

• \( \beta_1 L = 1.875 \) (fundamental mode)
• \( \beta_2 L = 4.694 \) (2nd mode)
• \( \beta_3 L = 7.855 \) (3rd mode)

\[ EI \frac{\partial^4 w}{\partial x^4} + \rho A \frac{\partial^2 w}{\partial t^2} = 0 \]

Separation of variables: \( w(x,t) = W(x) \cdot T(t) \) gives:

\[ \frac{W''''}{W} = -\frac{\rho A}{EI} \frac{T''}{T} = \beta^4 \]

The spatial equation: \( W'''' - \beta^4 W = 0 \) has general solution:

\[ W(x) = C_1 \cosh(\beta x) + C_2 \sinh(\beta x) + C_3 \cos(\beta x) + C_4 \sin(\beta x) \]

Cantilever boundary conditions (fixed at \( x = 0 \), free at \( x = L \)):

• \( W(0) = 0 \), \( W'(0) = 0 \) (clamped end)
• \( W''(L) = 0 \), \( W'''(L) = 0 \) (free end: zero moment and shear)

Applying these yields the frequency equation:

\[ \cosh(\beta L) \cos(\beta L) + 1 = 0 \]

This transcendental equation has roots at \( \beta_n L = 1.875, 4.694, 7.855, \ldots \)The natural frequency is then \( \omega_n = \beta_n^2 \sqrt{EI/(\rho A)} \), giving the result above.

\[ M_{\text{base}} = F \cdot L \]

The bending strain at the follicle (outer fiber):

\[ \varepsilon_{\text{base}} = \frac{M_{\text{base}} \cdot r}{EI} = \frac{F \cdot L \cdot r}{E \cdot \frac{\pi r^4}{4}} = \frac{4FL}{\pi E r^3} \]

For a tip deflection \( \delta \), the force is \( F = 3EI\delta/L^3 \), so:

\[ \varepsilon_{\text{base}} = \frac{3r\delta}{L^2} \]

For a 5 cm whisker with base radius 100 \(\mu\)m deflected 1 mm at the tip:\( \varepsilon = 3 \times 10^{-4} \times 10^{-3} / (0.05)^2 = 1.2 \times 10^{-4} \), well above the mechanoreceptor detection threshold of ~\( 10^{-6} \).

\[ G = 10 \log_{10}\left(\frac{A_m}{A_t}\right) \text{ dB} \]

The cat pinna has mouth area \( A_m \approx 9 \) cm\(^2\) and ear canal area\( A_t \approx 0.2 \) cm\(^2\), giving a theoretical gain of\( G = 10 \log_{10}(45) = 16.5 \) dB. The actual measured gain is 10–20 dB depending on frequency and pinna orientation.

\[ H(f) = \frac{P_{\text{drum}}}{P_{\text{free}}} = \frac{1}{1 - \left(\frac{f}{f_0}\right)^2 + j\frac{f}{Q f_0}} \]

The first resonance of the cat ear canal (\( L \approx 2 \) cm, open-closed quarter-wave):

\[ f_0 = \frac{c}{4L} = \frac{343}{4 \times 0.02} = 4.3 \text{ kHz} \]

Higher harmonics at \( 3f_0 = 12.9 \) kHz, \( 5f_0 = 21.4 \) kHz, etc., extend amplification across the cat's sensitive range. The pinna's shape introduces direction-dependent spectral notches used for sound localization, particularly in the vertical plane.

Interaural Time Difference (ITD):

\[ \Delta t = \frac{d \sin\theta}{c} \]

For cat head width \( d \approx 6 \) cm: max ITD = \( 6/34300 \approx 175 \, \mu \)s. The cat's auditory system resolves timing differences as small as \( \sim 10 \, \mu \)s.

Interaural Level Difference (ILD):

\[ \Delta L = 20 \log_{10}\left(\frac{P_{\text{near}}}{P_{\text{far}}}\right) \]

Significant above ~2 kHz where \( \lambda < d \); ILD can reach 15–20 dB at high frequencies (> 10 kHz) due to head shadowing.