A Finite-Difference Time Domain Cochlear Model performs the action of the inner ear of humans. The mechanical movement of the basilar membrane leads to activations of neurons. These activations lead to a cochleogram, at which time point a spike goes out at which point of the basilar membrane, and therefore at which frequency.

At the transition from the mechanical wave to spikes a synchronization of the phases of different frequencies happens. This is of importance for the perception of steady-state sounds, where the relative phase of partials play practically no role in perception.

Tuning curves of different frequencies on the basilar membrane of the model:

Spikogram of a harmonic sound with random phases, spikes over time and bark band

Synchronization of a lower partial to a higher one by systematically varying the phase of one partial with the phase of the other partial fixed

 

The cochlear is able to follow the double-slip motion of cello bowing by displaying the bifurcation in its interspike-intervals (ISI). The double-slip in the Helmholtz motion of cello or violin bowing results in a blurred sound, still with a clear pitch. The cochlear is able to follow the bifurcation quantitatively up to higher orders. This behaviour does not appear when calculating a Fourier or a Wavelet Transform of the sound. Although it is represented in these transformation too, it is coded in the amplitudes of higher frequencies rather than at the second partial, which bifurcates during the double splip.

Double-slip motion of cello bowing produces an interesting blurred sound. It is measured with this bowing machine at the University of Applied Sciences Hamburg. During one Helmholtz cycle a second tear-off the bow from the string appears. The time point of this tear-off changes over time

When tracking the peaks within this cycle it appears that the middle peak, the tear-off, is not perfectly in the middle of the cycle. Therefore two periodicities appear areound the second partial, one a bit smaller, one a bit larger than the periodicity of this second partial. Also the interval changes in time. this produces the blurred timbre of the sound.

Feeding the cello double-slilp sound into a cochlear model, at the frequency of the second partial interspike-intervals (ISI) appear (red and blue curves are bowing pressure and velocity).

Fitting the periodicities from the time series (red and blue) with the cochlear ISI the perfectly fit together. Therefore the double-slip motion is coded in the ISI.

 

Related publications

Bader, R. & Mores, R.: Cochlear detection of double-slip motion in cello bowing. arXiv:1804.05695v1 [q-bio.NC] 16 Apr 2018

Bader, R.: Phase synchronization in the cochlea at transition from mechanical waves to electrical spikes, Chaos 25, 103124,  2015.