Exploring Nonlinear Oscillator Models for the Auditory Periphery

Transcription

1 Exploring Nonlinear Oscillator Models for the Auditory Periphery Andrew Binder Dr. Christopher Bergevin, Supervisor October 31, 2008

2 1 Introduction Sound-induced vibrations are converted into electrical signals via mechanotransducers in the hearing process. In vertebrates, acoustic signals stimulate the sensory cells in the inner ear to trigger the opening of transducer channels. Open transduction channels allow for the flow of ionic currents into the cell, thereby setting a basis to stimulate synapsed neurons. A short delay exists between the stimulation of the sensory cells, also known as hair cells in vertebrates, and a change in the rate of transduction current. This small delay suggests a direct coupling between the sensory cells and the transduction channels. A proposed mathematical model for the displacement of the sensory cells treats the system as a spring with a nonlinear stiffness attached to a gate that can be opened or closed [1]. The model is useful for approximating the behavior of sensory cells in response to a stimulus. The anatomy of the auditory periphery of invertebrates differs from vertebrates. Drosophila, a type of fly, have an antennal hearing organ to detect sound. When sound reaches the antenna, it drives the structure into motion. The displacement of the antenna drives a mechano-electrotransduction process. In turn, this displacement stimulates neurons that send an auditory stimulus signal to the brain. Vertebrate and invertebrate systems share many behavioral characteristics. Both systems demonstrate a nonlinear stiffness directly correlated to electrical responses, the ability to adapt to a constant stimulus, and adhere to a similar mathematical model for the displacement of sensory cells [2]. Inspired by the band pass filter behavior of sensory cells in the auditory periphery, we will approximate the displacement of the sensory cells as a second order harmonic oscillator with a nonlinear stiffness. We can then explore the properties of this nonlinear second order oscillator. The intermodulation distortion products that manifest from this system are of the most interest. Intermodulation distortion products are Figure 1: Examples of different degrees of nonlinearity. The shape is inspired from auditory physiology. 2

3 the responses of a nonlinear system resulting from multiple stimuli with different frequencies. Specifically, we want to examine the dependence of the magnitude and phase of the intermodulation distortion products on the amplitude and frequency of the stimulus, the tuning bandwidth of the filter, and the degree of nonlinearity in the second order oscillator. See Fig. 1 for examples of changes in the degree of nonlinearity. The goal is to build an understanding of the properties of the model and translate this insight back to the physiological data of both vertebrates and invertebrates. The model can then be modified in order to more clearly describe the underlying physiologies. A specific set of questions to guide this research are: Question 1: How does the magnitude and phase of the intermodulation distortion product with a specific frequency (2f 1 f 2 ) vary with respect to f 1 for a fixed ratio of f 2 f 1? Question 2: What qualitative changes result in the magnitude and phase of the intermodulation distortion products from varying the amplitude of the stimuli? Question 3: How do changes in the degree of nonlinearity affect the intermodulation distortion products? The model for the displacement of the sensory cells will be discussed in the context of the vertebrate system. 2 Vertebrate Hair Cells There are two types of hair cells in the cochlea, or inner ear: inner hair cells (IHCs) and outer hair cells (OHCs). The hair cells differ primarily in their purpose. IHCs are responsible for the transmission of the neural signals to the brain. OHCs are hypothesized to influence the stimulation of the IHCs. Despite their different functions, the hair cells have a similar structure. Rows of stereoecilia, comprising a hair bundle, protrude from the top surface of the cell. Each row is progressively taller than the last. Near the tips of the stereocilia are links that connect a stereocilia to a lower, adjacent one. One end of these links is connected to a transduction channel on the hairs. A sketch of an inner hair cell is shown in Fig. 2. The coupling between the two components allows the mechanically stimulated links to directly alter the state of a transduction channel. These transduction channels act like a gate on the cells that can be opened or closed and regulate ionic current across the membrane. The links behave like a spring in 3

4 Figure 2: Schematic of a hair cell and the spring link connected to a transducer channel gate on the the tips of the stereocilia. the model and have a spring constant. As the bundle of stereocilia is deflected, the springs can either be stretched or compressed. The state of the mechanically gated transduction channels can be altered by the force exerted by the spring. Since the system is proposed to behave like a spring, the driven oscillations may be considered in a first approximation to behave like a harmonic oscillator. 3 Hair Bundles as Harmonic Oscillators Hair cells can be modeled as damped, driven harmonic oscillators. Experimentally, hair cells have been shown to oscillate in response to a driving force and exhibit the effects of being damped [1]. In a working ear, sound waves provide a driving force and the viscous fluid surrounding the bundle acts as a damper on the bundle s motion. A damped, driven harmonic oscillator satisfies the nonhomogeneous second order linear differential equation dependent on the position (x): mẍ + bẋ + kx = F 0 e iωt (1) The equation depends on the mass of the hair bundle (m), the damping coefficient (b), bundle stiffness (k), driving force (F 0 ), driving frequency (ω), and time (t). The term describing the driving force is complex because it stores useful information about the magnitude and phase of the sinusoidal driving force. The analytical solution to the differential equation can be determined by finding the solution to the homogeneous equation and the steady state solution. The solution to the homogeneous equation is sometimes referred to as the transient solution because the amplitude of the oscillation 4

5 Figure 3: Model of the displacements of the system when ω D > ω 0 in the left graph and ω D = ω 0 in the right graph. dissipates with time. The steady state solution for the position (x) of the damped, driven harmonic oscillator is: x(t) = A D e i(ωt+φ) (2) The equation depends on the amplitude of the driving force (A D ), the driving frequency (ω), and the driving force phase shift (φ). There are three possible transient solutions to the problem. The appropriate solution to the differential equation is dependent on the size of the damping b in relation to the other parameters. As the oscillations in the ear do not converge directly to 0 and oscillate around the resting position [3], the underdamped transient solution is most appropriate. The underdamped transient solution is: Underdamped (b < 2 mk) : x(t) = A 1 e b 2m t e ±i(ω T t+φ T ) The equation depends on the amplitude (A 1 ), the transient phase shift (φ T ), and the transient frequency (ω T ). The general solution is the steady state solution plus the appropriate transient solution. As time approaches infinity, the general solution will become just the steady state solution. The analytic equation is useful for gaining an intuition as to how the hair bundles might behave with an acoustic driving force. When the driving frequency doesn t equal the natural frequency, the largest amplitudes will occur at the initial onset and eventually decrease to the steady state. When the driving frequency equals the natural frequency, the amplitude of the system increases until it reaches a steady state. Fig. 3 depicts a model for the displacements of the hair cell or fly antenna when the driving frequency is greater than the natural 5 (3)

6 Figure 4: Model of dependence of open state probability on displacement on the left and the transduction current of a hair cell on the right. frequency in the left graph and equal to the natural frequency in the right. A significant observation from experiments however, is that the stiffness of the hair bundle is nonlinear [1]. 4 Nonlinear Stiffness The configuration of the transduction channels is hypothesized to be the source of the nonlinear stiffness of the overall hair bundle. As the channels open, the gating springs relax. This relaxation results in a change in the stiffness of the hair bundle over the range of forces that alter the state of the channel. In a similar manner, closing the channels increases the stiffness. These changes in the stiffness imply a nonlinearity. The stiffness of the hair bundle (K B ) includes the stiffness of the stereociliary pivots, or base of the stereocilia (K S ), the stiffness contributed by the gating springs (K G ), and is dependent on the probability (p) of the channels being open [1]: K B = K S + K G Nz 2 p(1 p)/kt (4) The final term represents an increase in compliance or a decrease in stiffness of the bundle. This change depends on the number of channels (N), the gating force to open a single channel (z), Boltzmann s constant (k), and the absolute temperature (T ). The steady-state probability of the channel s being open has been suggested to be dependent upon the bundle s deflection (X), or displacement from its resting position as shown in Fig. 2 [1]: 6

7 1 p = (5) 1 + e z(x X 0 ) kt Fig. 4 shows a model of the probability of a channel being open at various displacements. This model is sigmoidal, positive, and approaches the limits of 0 and 1. When all of the channels are open, more ions should be able to flow into the cell leading to a higher current. When more channels are closed, the current should decrease. Fig. 4 provides a model of the transduction current dependent upon the bundle displacement. As expected, the current is greatest when the most channels are open and least when they are closed. Figure 5: The graph depicts the nonlinear stiffness of the hair bundle as a function of the hair bundle displacement [1]. Experimental data and the stiffness model equation (Eq. 4) as demonstrated in Fig. 5 show how the open state probability of the channels affects the stiffness. At large positive and negative displacements, the stiffness saturates. The smallest stiffness occurs when the probability for the channels to be open is approximately 50%. At this displacement, the transduction current model has its steepest slope. In order to create a better model for the hair cell system, the differential equation will now include the nonlinear stiffness. 5 Nonlinear Damped, Driven Oscillators The next step of the research process is to analyze the damped, driven oscillator model with the nonlinear stiffness both numerically and analytically. A numerical model of 7

8 the system with the nonlinear stiffness has already been created and an example of the nonlinear oscillator s behavior is shown in Fig. 6. An analytical solution will now be examined. First, a nonlinear oscillator model with a cubic nonlinearity for the stiffness will be studied in order to gain experience solving a simpler case. The solution will be used to provide a time waveform. A fast Fourier transform of the result can be used to determine the properties of the distortion product. This will indicate what could be expected from the auditory nonlinear second order differential equation. The process used to solve nonlinear differential equations will then be applied to the gating spring model. The solution should provide insight into the workings of the system and any unique situations that could arise due to the nonlinear stiffness. Previous research can provide an indicator of what is to be expected from the nonlinear model. Based on past work, the growth of the magnitudes of the distortion products for specific frequencies should be non-monotonic. A phase shift of the distortion products relative to the original frequency should have approximately a half cycle change. By varying the nonlinearity in the model, these properties should change. This information can be used to provide some insight into the physical system. Figure 6: A numerical approximation of a damped, driven oscillator using a fourth order Runge Kutta method on the left and its nonlinear stiffness on the left and its nonlinear stiffness on the right. 8

9 Appendix - Numerical Modeling of Hair Bundle Oscillations There are several methods to solving a differential equation numerically. The two methods used for this problem were Euler s method and the fourth order Runge- Kutta method. Euler s method models a solution by find the tangent line at a point and taking a small step along that line. Assuming the next point is still on the line, the slope is used to find the tangent line at that point and takes another step of the same size. This process continues for a number of steps. In equation form, this means that given the initial value problem for y: y (t) = f(t, y(t)), y(t 0 ) = y 0 (6) Using the derivative and an arbitrary step size h, the y n point can be determined by: y n+1 = y n + hf(t n, y n ) (7) The error of Euler s method can be determined by comparing the process to the Taylor Expansion of y a step h from t 0. Euler s method is the first two terms of the Taylor Expansion. The error of the method then, is the rest of the terms in the expansion: Error = 1 2 h2 y (t 0 ) + (8) The smaller the step size, the less error there will be in the approximation. A more accurate numerical method for approximating a solution is the fourth order Runge- Kutta method. The Runge-Kutta method works by taking the slope at various points along the interval giving greater weight to the slopes at the midpoint. This method provides the following equations: y n+1 = y n + h 6 (k 1 + 2k 2 + 2k 3 + k 4 ) (9) t n+1 = t n + h 9

10 The slopes k are equal to: k 1 = y (t n, y n ) k 2 = y (t n + h 2, y n + h 2 k 1) (10) k 3 = y (t n + h 2, y n + h 2 k 2) k 4 = y (t n + h, y n + hk 3 ) The Runge-Kutta method is more accurate than the Euler s method with an error on the scale of h 4. This method was used to create the model in Fig. 3 and Fig

11 References [1] Howard, J. and A. J. Hudspeth. (1988) Compliance of the Hair Bundle Associated with Gating of Mechanoelectrical Transduction Channels in the Bullfrog s Saccular Hair Cell. Neuron 1, [2] Albert, Jörg T., Björn Nadrowski, and Martin C. Göpfert. (2007) Mechanical Signatures of Transducer Gating in the Drosophila Ear. Current Biology 17, [3] Albert, Jörg T., Björn Nadrowski, and Martin C. Göpfert. (2007) Drosophila Mechanotransduction. Fly 1 (4), e1-e4 11