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.

Ear Canal as Quarter-Wave Resonator:

The ear canal acts as a tube closed at one end (tympanic membrane) and open at the other (pinna opening). Its resonant frequency is that of a quarter-wave resonator:

\[ f_{\text{res}} = \frac{c}{4L} \]

where \( c = 343 \) m/s is the speed of sound in air and \( L \) is the effective length of the ear canal. For the cat ear canal with \( L \approx 2.5 \) cm:

\[ f_{\text{res}} = \frac{343}{4 \times 0.025} = 3430 \text{ Hz} \approx 3.4 \text{ kHz} \]

This resonance provides passive amplification of ~10–15 dB near 3.4 kHz, with odd harmonics at \( 3f_{\text{res}} \approx 10.3 \) kHz and \( 5f_{\text{res}} \approx 17.2 \) kHz extending the gain across the cat's most sensitive hearing range. The full transfer function of the ear canal is:

\[ H(f) = \frac{1}{\cos\left(\frac{2\pi f L}{c}\right)} \]

with peaks at \( f = (2n-1) \cdot c/(4L) \) for \( n = 1, 2, 3, \ldots \)

Pinna Directional Gain:

For a conical pinna with opening half-angle \( \theta \), the directional gain is determined by the solid angle \( \Omega \) subtended:

\[ \Omega = 2\pi\left(1 - \cos\frac{\theta}{2}\right) \]

\[ G = 10 \cdot \log_{10}\left(\frac{4\pi}{\Omega}\right) \text{ dB} \]

For the cat pinna with an effective opening angle of \( \theta \approx 60° \):

\[ \Omega = 2\pi\left(1 - \cos 30°\right) = 2\pi(1 - 0.866) = 0.842 \text{ sr} \]

\[ G = 10 \cdot \log_{10}\left(\frac{4\pi}{0.842}\right) = 10 \cdot \log_{10}(14.9) \approx 11.7 \text{ dB} \]

In practice, the measured gain is approximately 8 dB due to imperfect conical geometry and diffraction effects. By narrowing the effective angle (orienting the ear toward a sound source), the cat can increase the gain further — a biological phased array antenna.

Interaural Time Difference (ITD) — effective below ~2 kHz:

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

where \( d \approx 4 \) cm is the effective interaural distance for cats and\( \theta \) is the azimuth angle of the source. Maximum ITD:\( \Delta t_{\max} = 0.04/343 \approx 117 \, \mu \)s. The cat's auditory brainstem resolves timing differences as small as ~10 \(\mu\)s.

Interaural Level Difference (ILD) — effective above ~3 kHz:

At frequencies where the wavelength is shorter than the head diameter (\( \lambda < d \)), the head casts an acoustic shadow. ILD can reach 15–20 dB at frequencies above 10 kHz. Cats excel at ILD-based localization because their independently mobile ears can be oriented to maximize the level difference between the two ears — effectively increasing the baseline \( d \) and amplifying the shadow effect.

Tonotopic Map — Position to Frequency:

The characteristic frequency at position \( x \) along the basilar membrane (measured from the base) follows a logarithmic mapping:

\[ x = L_{\text{BM}} \cdot \frac{\log(f / f_{\min})}{\log(f_{\max} / f_{\min})} \]

Inverting for the characteristic frequency at position \( x \):

\[ f(x) = f_{\min} \cdot \left(\frac{f_{\max}}{f_{\min}}\right)^{x / L_{\text{BM}}} \]

For the cat cochlea: \( L_{\text{BM}} \approx 25 \) mm, \( f_{\min} = 48 \) Hz,\( f_{\max} = 64{,}000 \) Hz. At the midpoint (\( x = 12.5 \) mm):

\[ f(12.5) = 48 \times \left(\frac{64000}{48}\right)^{0.5} = 48 \times \sqrt{1333} = 48 \times 36.5 \approx 1754 \text{ Hz} \]

The outer hair cells provide active amplification (the “cochlear amplifier”) of up to 40–60 dB through electromotility — voltage-driven length changes of the prestin motor protein embedded in their lateral cell membranes. This active process sharpens frequency tuning and extends the dynamic range of the ear to over 120 dB.