1. TRAVELING WAVE LATENCY AND VELOCITY EFFECTS ON SOUND CODING
Petr Marsalek and Frank Jülicher s:
Max Planck Institute for the Physics of Complex Systems, Dresden, Germany
Faculty of Biomedical Engineering, Czech Technical University of Prague, Kladno, Czech Republic
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-10 -5 -2 2
x 10 -3
4.6 kHz
Traveling wave with envelope, high SPL -4
1.3 kHz
Distance to helicotrema, mm
370 Hz
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4.6 kHz
Traveling wave with envelope, low SPL
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Distance to helicotrema, mm
Introduction
The cochlear traveling wave is amplified by an active system. The passive properties of the basilar membrane stiffness introduce coarse frequency tuning already in the dead cochlea [1]. However, the cochlear amplifier is active and introduces sharpening of tuning, nonlinear gain and compression. The traveling wave therefore represents an active nonlinear wave phenomenon [2].
Figure 2: The traveling wave - low sound pressure
In low sound pressures around 40 dB SPL, only the small patches of cochlea resonate, therefore we show only parts of the cochlear partition distance from helicotrema in mm. The solution of the ODE is calculated from right to left. y-axis shows the displacement of the xbasilar membrane (BM) in nm, green is the amplitude and blue is the envelope of the BM excursions. Cochlear characteristic places for given three frequencies are shown first quarter of the BM length, 4.6 kHz, first half, 1.3 kHz and last quarter 370 Hz. The characteristic frequencies correspond to human hearing range.
Experimental measurements
Figure 1 shows experimental data compared to model outputs. Active properties are inferred from measurement of Brainstem Evoked Response to a narrow-band Auditory stimulation (BERA) [4]. The passive properties are the pioneer recordings on carefully prepared cochlea of human cadaver by G. von Bekesy [3]. Our model of human cochlea is based on generic set of parameters with only one free parameter (coefficient of nonlinear response in vicinity of resonance) [2]. Our aim is a qualitative description applicable to neural coding.
Nonlinear model
The basilar membrane oscillations are described by set of nonlinear equations, as shown in [2]. Pressure and amplitude are coupled together on a membrane with stiffness varying according to the Greenwood [1] function. This graded stiffness represents passive (linear) resonance properties. The active mechanism using generic Hopf oscillator is a standard physical description of nonlinear excitable properties of biological systems, including the active properties of hair cells within the organ of Corti. The active mechanism is responsible for gain in low sound pressure levels (SPL), Figure 2, and for compression in high SPL, Figure 3. The minimalistic set of parameters was extensively studied and enables a comparison of linear and nonlinear versions of the equations. The resulting nonlinear ordinary differential equation (ODE) is solved in backward direction from apex to base.
Figure 3: The traveling wave - high sound pressure
In higher sound pressures around 80 dB SPL, the compressive nonlinearity extends the dynamic range and the resonance profile spans greater parts of cochlear distance than for sounds of lower volume. y-axis shows basilar membrane displacement in length units, green is the amplitude and blue is the envelope of the BM displacements. Same cochlear characteristic places are shown, namely the first quarter, 4.6 kHz, the first half, 1.3 kHz and the last quarter 370 Hz of the human hearing range.
Results
We calculate numerically the velocity of the traveling wave, see example in Figure 4. We study both the active (live) case, Figure 6, and passive case (dead cochlea), Figure 5. We compare these velocities and latencies to experimental values in dead cochleas [3] and in normal hearing subjects [4], in subjects with hearing loss [5] and in mammals other than humans.
Figure 1: Experimental traveling wave velocities and cochlear latencies
Top panels show velocities and bottom panel show latencies. Left panels show active properties of alive cochlea. Right panels show passive properties of dead cochlea. x-axis shows cochlear distance in mm, direction from left-to-right is from oval window to helicotrema. Experimental data are green and blue with circles. Theoretical calculations are red and black and marked with crosses. Measurement of active properties is done by Brainstem Evoked Response to a narrow-band Auditory stimulation, abbreviated as BER(A). Passive properties are taken from original observations by G. von Bekesy.
Figure 6: (Active) Latencies and velocities for given frequencies
x-axis: cochlear distance in mm, red, +-marks, upper part - velocity, black, x-marks, lower part - latency, for the pure tones of three frequencies shown in the nonlinear wave example: 4.6 kHz, 1.3 kHz and 370 Hz.
Figure 4: Numerical calculation of latency and velocity
x-axis: cochlear distance in multiples of 3.5 mm, y-axis: BM amplitude. Circles show zero crossings and saw-tooth curve shows the corresponding phase. From the phase and known frequency, one cycle duration and therefore one cycle delay can be calculated. The cumulative latency is shown at the lower part in black and x-marks, calculated traveling wave velocity is shown in red and in
+-marks.
Figure 5: Passive velocity (top) and latency (bottom)
The latency is dependent on both sound pressure and frequency and is different for pure tones of given frequency, clicks and noises. This figure illustrates how we calculate the frequency independent latency. Briefly, this procedure is analogous to experimental procedures.
x-axis: cochlear distance in mm, y-axis: top velocity in m/s, bottom: latency in ms, red and black: velocities and latencies for three characteristic frequencies, 10 kHz, 1 kHz, 200 Hz, blue crosses: frequency independent.
Perspective and applications
The understanding of cochlear delays is of practical importance, because current cochlear implants reached such time precision that the delay between cochlear sites can be emulated. Cochlear delay can be related to azimuthal sound localization in binaural implantees [6].
Conclusions
The latencies of the traveling wave at different frequencies have consequences for sound encoding in the spike train of the auditory nerve. When hearing is restored by electrically stimulating the auditory nerve by cochlear implants, the traveling wave latency should be taken into account.
Acknowledgments: Supported by the Czech Ministry of Education (MSMT), Research Initiative no and by the Max Planck Society.
References:
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