A minimal physics-based model for musical perception

Abstract

Some people, entirely untrained in music, can listen to a song and replicate it on a piano with unnerving accuracy. What enables some to "hear" music so much better than others? Long-standing research confirms that part of the answer is undoubtedly neurological and can be improved with training. However, are there structural, physical, or engineering attributes of the human hearing mechanism apparatus (i.e., the hair cells of the internal ear) that render one human innately superior to another in terms of propensity to listen to music? In this work, we investigate a physics-based model of the electromechanics of the hair cells in the inner ear to understand why a person might be physiologically better poised to distinguish musical sounds. A key feature of the model is that we avoid a "black-box" systems-type approach. All parameters are well-defined physical quantities, including membrane thickness, bending modulus, electromechanical properties, and geometrical features, among others. Using the two-tone interference problem as a proxy for musical perception, our model allows us to establish the basis for exploring the effect of external factors such as medicine or environment. As an example of the insights we obtain, we conclude that the reduction in bending modulus of the cell membranes (which for instance may be caused by the usage of a certain class of analgesic drugs) or an increase in the flexoelectricity of the hair cell membrane can interfere with the perception of two-tone excitation.

Keywords: flexoelectricity; mechanics; music; soft matter.

Fig. 1.

Are there structural aspects of the hearing apparatus that render one human better poised than another to perceive music? Do external factors such as NSAID drugs play an important role in music perception and noise sensitivity?

Fig. 2.

Depicted is a schematic of a hair bundle with representative stereocilia. The active motion of stereocilia on the hair cells contributes to the sound amplification and several other compelling aspects of the human hearing mechanism.

Fig. 3.

Frequency response curve of two tones with different frequencies. When the two tones are well separated, the response curves do not have much overlap and the two tones are perfectly distinguishable. Adapted from ref. (1).

Fig. 4.

Schematic of stereocilia capturing its rotational motion and shape change.

Fig. 5.

The nonlinear compressibility aspect of the hearing mechanism. The ear amplifies faint sounds much more than the loud ones; therefore, the ear can perceive both high and low pitch to some extent without damaging its structure. The curves are interpolated based on the data points of the Fourier transform of the response. The amplification is calculated by dividing the response to the stimulation force, and the frequency response is normalized to the natural frequency of the system.

Fig. 6.

Suppression of the first tone by addition of the second tone where F₁ and F₂ stand for the amplitudes of the stimulus forces of the first and second tone. Note that the first tone is tuned to the natural frequency of the stereocilia (i.e., f₁ = f₀), and the frequency of the second tone is related to the natural frequency of the stereocilia by f₂ = 0.9f₁.

Fig. 7.

Sensitivity of the stereocilia to the flexoelectric constant. Response of the system to (A) the free oscillation, (B) forced oscillation involving two-tone excitation, where F₁ = 10 pm, F₂ = 0.5F₁, f₁ = f₀, and f₂ = 0.95f₁. Flexoelectricity is responsible for compensating for the damping due to the viscous fluid, and lack(addition) of it results in a faster (much slower or never-ending!) damping of the stimuli inside the ear.

Fig. 8.

Demonstration of the importance of the balance of flexoelectricity on inferring the second tone. The balance of flexoelectricity plays a crucial role in the perception of the two-tone excitation while dissipating for the lost energy due to friction. The system is excited with two tones where the first tone is fixed to the natural frequency of the system while the second tone frequency is f₂ = 0.95f₁.

Fig. 9.

The inference of the second tone is in the presence of the first tone for various frequencies of the second tone while the first tone is fixed at the natural frequency of the system. It can be inferred that there exists a threshold for the proximity of the two-tone frequencies where the hearing system can no longer distinguish between the two tones (here f₂ = 0.99f₁). The numerical values are F₂ = 0.5F₁.

Fig. 10.

The effect of the thickness of the stereocilia membrane on the two-tone interference and its influence on the amplitude of the first tone. The amplitudes of the response are normalized to the case where the second tone is absent. The 50% reduction in the membrane thickness can result in better distinguishment of two tones and less inference (this can also be concluded from the distance of normalized amplitude from 1). Also, it is assumed that f₂ = 0.98f₁. The numerical value for the membrane thickness is adopted from ref. (33).

Fig. 11.

The effect of NSAID drugs (e.g., ibuprofen) on the second tone inference. The maximum amplitude in the natural frequency of the system is normalized to the amplitude of the system on the one-tone stimulation. As it is shown, the decrease in the bending modulus by 50% which can be the result of consumption of NSAID drugs can disrupt the inference of the second tone where f₂ = 0.95f₁. Also, it is assumed that F₂ = 0.5F₁.