A pure tone at near the threshold of hearing deflects a hair bundle by rms

PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Electronics, Instrumentation & Communication · Bioelectricity, the senses and the physics of the body

Forces, Machines and Signals, Ch Hearing: from air pressure to a gated channel

Problem

A pure tone at

f_0 = 1\,\mathrm{kHz}

near the threshold of hearing deflects a hair bundle by

0.50\,\mathrm{nm}

rms. The bundle's total thermal jiggle,

\sqrt{k_BT/\kappa} = 2.07\,\mathrm{nm}

rms, is spread evenly over a band about as wide as

f_0

. Averaging the transduced current for a time

t

acts as a filter of bandwidth about

1/t

centred on

f_0

, so it admits a fraction about

1/(f_0t)

of the noise power. Assume that reliable detection needs a signal-to-noise ratio of about

3

. ((a)) Show that averaging for

t = 10\,\mathrm{ms}

admits the same fraction of the noise power as a filter of quality

Q = 10

, and find the rms noise left. ((b)) Find the averaging time a single hair cell needs to reach the detection criterion. ((c)) If the currents of

10

hair cells tuned near the same place, with independent noise, are pooled, the noise power falls by a further factor of

10

. Find the averaging time now needed.

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