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Review Article
Models for response dynamics of vestibular end organs: a review
Jeong‑Yoon Choi1,2orcid
Research in Vestibular Science 2025;24(1):10-19.
DOI: https://doi.org/10.21790/rvs.2024.027
Published online: March 14, 2025

1Dizziness Center, Clinical Neuroscience Center, and Department of Neurology, Seoul National University Bundang Hospital, Seongnam, Korea

2Department of Neurology, Seoul National University College of Medicine, Seoul, Korea

Corresponding author: Jeong-Yoon Choi Department of Neurology, Seoul National University Bundang Hospital, 82 Gumi-ro 173 beon-gil, Bundang-gu, Seongnam 13620, Korea. E-mail: saideiju@gmail.com
• Received: December 19, 2024   • Revised: January 27, 2025   • Accepted: February 5, 2025

© 2025 The Korean Balance Society

This is an open access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

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  • The vestibular end organs, consisting of the semicircular canals and otoliths, sense physical variables related to head motion and generate neural signals according to their response dynamics. Therefore, understanding the dynamics is a crucial first step in interpreting vestibular responses under both normal and pathological conditions. This review addresses the response dynamics of vestibular organs using transfer functions. While the semicircular canals and otoliths are originally modeled as second-order overdamped systems, they can be approximated as first-order systems, when considering the range of head motion frequencies typically encountered during daily activities. This review also discusses the clinical implications of time constants in the transfer functions. In both the semicircular canals and otoliths, the dominant time constants are determined by their viscous-to-elastic properties. By analyzing changes in these properties under pathological conditions, the resulting alterations in system responses can be predicted. Such efforts may contribute to bridging the gap between mathematical modeling and clinical understanding.
  • Keywords: Vestibular system; Semicircular canals; Otoliths; Theoretical models; Biomechanical phenomena
The vestibular end organs, composed of semicircular canals and otoliths, are the primary sensors for head motion [1]. The neural signals generated in the vestibular end organs represent angular and linear accelerations associated with head motion [2] but are not identical to these physical variables [3]. Understanding the response dynamics of these peripheral organs is the first step in interpreting the diverse vestibular responses observed under normal and pathological conditions [2]. Additionally, this attempt must precede any explanation of how the central vestibular system compensates for imperfect sensory signals [2,3].
A fundamental approach for analyzing the response dynamics involves transforming time-domain differential equations into algebraic equations in the s-domain, representing the complex plane, using the Laplace transform [4]. Here, s is a complex number expressed as s=σ+jω, where σ is the real part, representing the growth or decay rate of the system in the time domain and jω is the imaginary part, indicating the oscillatory component of the system [4]. For the details, refer to Appendix 1. With the Laplace transform, the system behavior will be typically represented with a transfer function, which relates an input (e.g., head acceleration and velocity) to an output (e.g., cupula or otolith displacement or neural discharge). The Bode plot displays the gain (output/input) and phase (θ) of the system intuitively in the frequency domain [4]. In addition, the transfer function expressed with Laplace notation provides information about the system. In a stable system, the time constant helps determine the system’s corner frequency, influencing how the system attenuates or amplifies different frequency components [4]. In the frequency domain, time constants (tc) associated with zeros (in the numerator) and poles (in the denominator) of a transfer function establish characteristic corner frequencies, equal to the inverse of these time constants (f=1/tc). At these corner frequencies, the slope of the magnitude (gain) plot in dB, given by 20∙log10 (output/input) changes [4]. For a zero, above its corner frequency, the slope of the gain plot increases by 20 dB/decade and the phase of output leads the input. Conversely, for a pole, above its corner frequency, the slope of the gain plot decreases by 20 dB/decade and the phase of output lags the input. It is noteworthy that phase lead or lag does not mean that the output anticipates or lags the input. Instead, it means that the system output is proportional to the derivative or integral of the input. In the time domain, time constants characterize how quickly the system responds to or recovers from a stimulus [4]. This summary is illustrated in Fig. 1.
This review summarizes the response dynamics of peripheral vestibular end organs using transfer functions. In addition, by examining the meaning of time constants in both the frequency and time domains, as well as their relationships with the parameters that define them, this study aims to elucidate their clinical significance.
In response to head acceleration, the semicircular canals in the inner ear generate neural signals through cupula deflection, which is influenced by three factors: the acceleration of the cupula (proportional to head acceleration), viscous damping (mainly resulting from resistance due to endolymph viscosity), and the elastic restoring force (mostly arising from cupula elasticity) [5]. This relationship can be represented by:
(Eq. 1)
I⋅at=I⋅x¨t+c⋅x˙t+k⋅xt
Here, I represents the moment of inertia, analogous to mass in a translational motion equation; c denotes the viscosity-damping coefficient; k is the elastic restoring coefficient; a(t) is the head acceleration as a function of time; and x(t) is the displacement of the cupula over time. A single and double dot above x(t) represent the first and second derivatives with respect to time, indicating the velocity and acceleration of the cupula, respectively. The equation is then rewritten by dividing both sides by I:
(Eq. 2)
a=x¨+cIx˙+kIx
It can then be expressed as a transfer function using the Laplace transform [4], describing the relationship between the output (cupula displacement, x) and the input (acceleration, a), as shown below.
(Eq. 3)
A(s)=s2X(s)-sx(0)-x˙(0)+c/I(sX(s)-x(0))+k/IX(s)
Given the initial conditions x(0)=0 and x˙0=0 and moving X(s) to the left-hand side, the equation can be simplified as Eq. 4.
(Eq. 4)
XsAs=1s2+cIs+kI=1s+1TLs+1Ts
The Laplace term A(s) represents the given acceleration, while X(s) represents cupula displacement. The system exhibits a long-time constant (TL) and a short-time constant (Ts) [5]. Under the assumption of an overdamped system of the cupula [5], the long-time constant (TL) can be approximated as the ratio of the viscous coefficient to the elastic restoring coefficient (c/k), which has been reported to be about 5 seconds, while the short-time constant (Ts) can be approximated as the ratio of the inertial moment to the viscous coefficient (I/c), being estimated at about 0.003 seconds [5]. For details of these approximations, refer to the Appendix 1. Therefore, the cupula displacement in response to acceleration decreases above 0.03 Hz (≈1/2πTL) with a phase lag of 90°, and it becomes significantly less responsive beyond 53 Hz (≈1/2πTs) with an additional phage lag of 90° (Fig. 2A). The equation representing the relationship between cupula displacement and head velocity (v) is as follows:
(Eq. 5)
XsVs=ss2+k1s+k2=ss+1TLs+1Ts
From the equation, we can infer that the cupula response remains stable and accurate within the frequency range of 0.03 Hz and 53 Hz (Fig. 2B). Out of this range, the cupula displacement does not reliably correspond to head velocity and exhibits either a phase lead or lag [1]. Because natural head motion frequencies typically range from 0.5 to 10 Hz [6,7], the two transfer functions (Eq. 4 and Eq. 5) can omit the short-time constant [3]. Consequently, they can be simplified as follows:
(Eq. 6)
XsAs=1s+1TL=1s+0.2=55s+1 
(Eq. 7)
XsVs=ss+1TL=ss+0.2=5s5s+1
With the transfer function, we can now simulate the canal dynamics in response to various head motions. Fig. 2C presents the simulated canal response using the Simulink tool in MATLAB 2021b (MathWorks). The input is the angular head velocity. During transient or mid-range frequency continuous head motion, the vestibular response faithfully encodes the actual head velocity. However, during prolonged head motion, corresponding to low-frequency motion, the vestibular response decays over time, with a time constant of 5 seconds. This means that the initial response amplitude decreases to 37% every 5 seconds. Consequently, after 15 seconds and 25 seconds, the canal system generates only a minimal response, approximately 5% and 0.7% of the initial amplitude. Additionally, after rotation, the system generates a significant post-rotatory response.
The utricle and saccule detect the linear acceleration of the head within a non-inertial reference frame, where both inertial and gravitational acceleration act as inputs to the otolith organ [8-10]. External accelerations induce relative displacements among the components of the otolith system, including the otoconia, gel layers, and the surrounding endolymph fluid [10-12]. As a result, the transfer function describing otolith dynamics is inherently complex and often necessitates numerical or analytical simplifications. This review first examines the transfer functions derived from the works of Young and Meiry [13] (Eq. 8) and Hosman and Stassen [14] (Eq. 9).
(Eq. 8)
aΑs=1.5⋅s+ 113.2s+15.33⋅s+ 10.66
(Eq. 9)
AFRΑ=425⋅s+ 1s+10.5⋅s+ 10.016
In both transfer functions, there are two poles and one zero. For Eq. 8, the slope of the gain increases around 0.012 Hz, flattens near 0.03 Hz, and decreases above 0.24 Hz (Fig. 3A). Because the gain decreases tenfold (20 dB, 20⋅log10ouputinput) for every tenfold increase in frequency (e.g., from 0.1 Hz to 1 Hz or 1 Hz to 10 Hz), this equation suggests that linear acceleration during daily activities may not be faithfully transformed into neural signals. According to Eq. 9, the slope of the gain increases around 0.16 Hz, flattens near 0.32 Hz, and decreases above 10 Hz (Fig. 3B). Therefore, this model is more suitable than the Young and Meiry model for detecting linear acceleration related to head motion during daily activities.
McGrath [15], meanwhile, sought to simplify the otolith dynamics previously described using distributed parameter models. This review bypasses the detailed steps of simplifying the equations governing otolith dynamics and focuses solely on presenting the final form of the simplified equation, as shown in Eq. 10.
(Eq. 10)
δΑs=1−R1+12Rs2+M+12Rs+ε  
Here, R represents the density ratio of the endolymph to the otoconial layer (ρf/ρo), M denotes the viscosity ratio of the gel to the endolymph (μg/μf), and ε indicates the viscoelasticity of the gel layer, defined as G∙ρo∙b2/μf2, where G is the shear modulus of the gel layer and b is the thickness of the otoconial layer. Assuming the otolith system is overdamped, Eq. 10 can be further simplified into two first-order components.
(Eq. 11)
δΑs=1−R1+12Rs2+M+12Rs+ε∝ 1s+1TL⋅s+ 1Ts
Recalling that TL is the ratio of viscous damping coefficient to the elastic restoring coefficient, approximated by M+R/2/ε and Ts is the ratio of mass to the viscous damping coefficient, approximated by 1+R/2/M+R/2. Therefore, Eq. 11 can be rewritten as Eq. 12.
(Eq. 12)
δΑ=1−R1+12R⋅1s+ εM+12R⋅1s+ M+12R1+12R∝ 1s+1TL⋅s+ 1Ts
The specific values of TL and Ts were reported to range from 0.004 to 0.2 seconds and 0.02 to 0.4 milliseconds, respectively [15]. Therefore, the frequency associated with TL ranges from 0.7 to 40 Hz, and the frequency associated with Ts is at least above 400 Hz. The inverse value of TL indicates the corner frequency below which the system responds faithfully to the given acceleration, exhibiting no significant phase lag. Similarly, the inverse value of Ts represents the frequency above which shows a decrease in gain and an increase in phase lag. Between these two frequencies, the system exhibits a moderate decrease in gain and a modest phase lag. As suggested by Hosman and Stassen [14], the value of TL is set to 0.016 seconds here (the corresponding corner frequency is 10 Hz). Given the frequency range of normal head movements, the terms related to Ts can be ignored. Consequently, the Eq. 12 can be further simplified as:
(Eq. 13)
δΑ=1−R1+12R⋅1s+ εM+12R ∝ 1s+1TL
With the gain proportional, Eq. 12 can be expressed as:
(Eq. 14)
OotolithIotolith=62.5s+62.5
This suggests that the neural information in the otolith encodes head acceleration accurately for frequencies below 10 Hz. Beyond this range, the response progressively diminishes and is accompanied by an increasing phase lag. With the transfer function, the otolith dynamics in response to head motion can be simulated. Fig. 3C illustrates the simulated otolith responses to various types of head motion. As shown, during transient or mid-range frequency continuous head motion, the otolith-driven neural response faithfully encodes the actual head acceleration.
In both cupula and otolith dynamics, the time constants are critical in shaping the response dynamics. Specifically, the TL is a major role in the simplified model. It defines the corner frequencies at which the cupula in the semicircular canal effectively encodes velocity information for inputs above 0.03 Hz velocity input, while the otolith organs effectively encode acceleration for inputs below 10 Hz.
Recalling that TL and Ts are approximated by the ratio of viscosity coefficient (c) to elastic coefficient (k) and the ratio of mass (m) (or inertial moment, I) to the viscosity coefficient, respectively [5]. This indicates that any change in viscous or elastic properties—whether due to inner ear disorders or aging—can alter the time constants, thereby leading to changes in the response dynamics.
In the canal, viscosity is primarily influenced by the endolymph fluid, while elasticity is governed by the cupula [5]. An increase in endolymph viscosity—caused by elevated protein levels or dispersed (rather than aggregated) otoconia—or a decrease in cupula elasticity, resulting from factors such as apical detachment of the cupula from the roof or the disruption of key structural components like collagen, glycosaminoglycans, and elastin, leads to an increase in TL and a decrease in Tc.
In the case of otolith, the relationship would be more complicated. The static (or direct current, DC) gain and TL are determined as follows [15]:
Static gain=1−ρfρo1+12ρfρo,  TL=μgμf+ 12⋅ρfρoGμf
The decrease in otoconial density will decrease static gain of otolith as well as increase TL. This means that the gain for low-frequency acceleration (such as gravity) decreases, and the corner frequency (inverse of TL, approximately 62.5 radian/sec or 10 Hz) also decreases. As a result, high-frequency motion cannot be effectively encoded by the otolith.
The implications of these changes in clinical settings have not been actively explored or experimentally addressed. However, these changes clearly alter the neural signals transmitted to the central vestibular system, leading to a false sense of motion and necessitating new adaptive mechanisms. Therefore, further research on this topic is warranted.
This review examined the response dynamics of the vestibular end organs. By assuming an overdamped system and further simplifying the dynamics within the frequency range of natural head motion, the behavior of both the semicircular canals and otolith organs can be approximated as first-order transfer functions. Understanding these dynamics is crucial for interpreting vestibular responses, including eye movements [16,17], perception [18,19], and autonomic or postural reactions [20-23]. Further research is needed to investigate how pathological conditions modify these dynamics.

Funding/Support

This study was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (2020R1A2C4002281).

Conflicts of Interest

Jeong‑Yoon Choi is an Editorial Board member of Research in Vestibular Science and was not involved in the review process of this article. The authors declare no other conflicts of interest.

Availability of Data and Materials

The datasets are not publicly available but are available from the corresponding author upon reasonable request.

Fig. 1.
Transfer function and time constant. (A) This transfer function consists of one zero and one pole. The zero is the value of s that makes the numerator equal to zero, while the pole is the value of s that makes the denominator equal to zero. If τa is 1 second and τb is 10 seconds, the corner frequencies are 1 radian/sec and 0.1 radian/sec, respectively. For stimuli with frequencies above the corner frequency associated with the pole (indicated by the red dashed line) the slope of the gain plot decreases by 20 dB/decade, and the phase of the output lags behind the input. For stimuli with frequencies above the corner frequency associated with the zero (indicated by the blue dashed line), the slope of the gain plot stops decreasing, and the phase lag disappears. (B, C) The changes in the system output in response to pulse, ramp, and step stimuli in the time domain are analyzed with respect to changes in the time constant. The system is a one-zero low-pass filter with a time constant of 0.5 seconds (B) and 2 seconds (C). As τa increases from 0.5 to 2 seconds, the amplitude and latency of system decreases and increases.
rvs-2024-027f1.jpg
Fig. 2.
Response dynamics of the semicircular canals. (A, B) The transfer functions of cupula displacement relative to head acceleration (A) and velocity (B). The shaded area represents the frequency range of natural head motion (0.5–10 Hz, equivalent to 3.14–62.8 radian/sec). (C) The display of head velocity information in the time domain generated by the semicircular canals in response to pulse, ramp, and step velocity stimuli. The transfer function omits the first-order term related to short-time constant, which accounts for high-frequency head motion, outside of the natural head motion frequency.
rvs-2024-027f2.jpg
Fig. 3.
Response dynamics of the otoliths. (A, B) The transfer functions of otolith displacement relative to head acceleration, as proposed by Young and Meiry [13] (A) and Hosman and Stassen [14] (B). Both systems have one zero and two poles. The shaded area represents the frequency range of natural head motion (0.5–10 Hz, equivalent to 3.14–62.8 radian/sec). However, within this frequency range, the transfer function proposed by Hosman and Stassen better represents the otolith response to head acceleration in terms of both gain and phase. (C) The time-domain representation of head acceleration information generated by the otoliths in response to pulse, ramp, and step velocity stimuli. In this representation, the transfer function from Hosman and Stassen (B) is simplified by omitting the first-order terms in the numerator (τ1) and the long-time constant term (τ2) in the denominator. The gain term is adjusted to allow the otolith to accurately represent head acceleration.
rvs-2024-027f3.jpg
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Appendix 1.
Derivation of long and short time constants
rvs-2024-027-app1.pdf

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      Models for response dynamics of vestibular end organs: a review
      Image Image Image
      Fig. 1. Transfer function and time constant. (A) This transfer function consists of one zero and one pole. The zero is the value of s that makes the numerator equal to zero, while the pole is the value of s that makes the denominator equal to zero. If τa is 1 second and τb is 10 seconds, the corner frequencies are 1 radian/sec and 0.1 radian/sec, respectively. For stimuli with frequencies above the corner frequency associated with the pole (indicated by the red dashed line) the slope of the gain plot decreases by 20 dB/decade, and the phase of the output lags behind the input. For stimuli with frequencies above the corner frequency associated with the zero (indicated by the blue dashed line), the slope of the gain plot stops decreasing, and the phase lag disappears. (B, C) The changes in the system output in response to pulse, ramp, and step stimuli in the time domain are analyzed with respect to changes in the time constant. The system is a one-zero low-pass filter with a time constant of 0.5 seconds (B) and 2 seconds (C). As τa increases from 0.5 to 2 seconds, the amplitude and latency of system decreases and increases.
      Fig. 2. Response dynamics of the semicircular canals. (A, B) The transfer functions of cupula displacement relative to head acceleration (A) and velocity (B). The shaded area represents the frequency range of natural head motion (0.5–10 Hz, equivalent to 3.14–62.8 radian/sec). (C) The display of head velocity information in the time domain generated by the semicircular canals in response to pulse, ramp, and step velocity stimuli. The transfer function omits the first-order term related to short-time constant, which accounts for high-frequency head motion, outside of the natural head motion frequency.
      Fig. 3. Response dynamics of the otoliths. (A, B) The transfer functions of otolith displacement relative to head acceleration, as proposed by Young and Meiry [13] (A) and Hosman and Stassen [14] (B). Both systems have one zero and two poles. The shaded area represents the frequency range of natural head motion (0.5–10 Hz, equivalent to 3.14–62.8 radian/sec). However, within this frequency range, the transfer function proposed by Hosman and Stassen better represents the otolith response to head acceleration in terms of both gain and phase. (C) The time-domain representation of head acceleration information generated by the otoliths in response to pulse, ramp, and step velocity stimuli. In this representation, the transfer function from Hosman and Stassen (B) is simplified by omitting the first-order terms in the numerator (τ1) and the long-time constant term (τ2) in the denominator. The gain term is adjusted to allow the otolith to accurately represent head acceleration.

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      Models for response dynamics of vestibular end organs: a review
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