The velocity storage time constant: Balancing between accuracy and precision

Abstract

The velocity storage mechanism is often described in terms of the exponential decay in eye velocity in an upright subject who is accelerated to a constant velocity yaw rotation. The velocity storage time constant for this decay is roughly 10–30 s, which means that for low-frequency head rotations, eye velocity and perceptions have large errors. One may wonder if there would be benefits to having a longer time constant, which would improve accuracy. In this paper, simulations are used to highlight that improved accuracy may come at the cost of increased noise – i.e., reduced precision. Specifically, since the velocity storage mechanism extends the 5.7 s time constant of the semicircular canal, it must be performing an integration process over a certain frequency range. In fact, all mathematical models of velocity storage include an integration. This integration would also integrate neural noise. Thus, increasing the velocity storage time constant would lead to integration over a wider range of frequencies, resulting in more noise in the brain’s estimate of motion. Simulation results show this accuracy-precision tradeoff. Recent evidence is also reviewed supporting the hypothesis that the brain optimizes the velocity storage time constant to resolve this accuracy-precision tradeoff during aging and with variations in stimulus amplitude.

1. Introduction

The velocity storage mechanism (Raphan, Matsuo et al. 1977, Robinson 1977) is often described in terms of the dynamic response of eye movements (the vestibulo-ocular reflex; VOR) and perception (Bertolini, Ramat et al. 2011) in response to yaw rotations of an upright subject, notwithstanding its multifaceted behavior (Merfeld, Young et al. 1993). For example, when a subject is quickly accelerated to a constant-velocity rotation, the eye velocity initially has a similar magnitude, but opposite direction, of head velocity, but this eye velocity decays exponentially with a time constant of roughly 10–30 s (Dimitri, Wall et al. 2001).

During a presentation at the “Mathematical Modeling in Motor Neuroscience” meeting, the guest of honor (L Optican, June 7, 2018, Pavia, Italy) posed a question: Why shouldn’t this time constant be longer? In this paper, I will propose an answer to this question. Indeed, it seems that a longer time constant could provide behavioral benefits, since this would make the brain’s estimate of motion closer to the actual motion experienced. It would also reduce post-rotatory velocity storage effects that, for example, result in a perception of rotation in the opposite direction after rotation stops. These errors are certainly relevant to contemporary motion experiences that include sustained low-frequency vestibular stimulation, like airplanes and trains, but they could also apply to non-mechanized experiences like a canoe rocking on waves.

The answer to the question may lie in the role of neural noise in velocity storage (Laurens and Droulez 2007, Laurens and Angelaki 2011, Karmali and Merfeld 2012, Karmali, Whitman et al. 2018). All neural (and biological and physical) signals are altered by the presence of noise, which can be defined as random activity that does not encode useful information. In vestibular sensation, noise may be introduced by processes such as stimulus transduction, synaptic release, vestibular nerve depolarization and central processing (Faisal, Selen et al. 2008). During constant-velocity rotation, the semicircular canal afferent signal decays more quickly than behavioral measures, with a time constant of about 5.7 s (Fernandez and Goldberg 1971, Jones and Milsum 1971). To produce the longer behavioral time constant from the shorter peripheral time constant, the velocity storage mechanism requires a process which amplifies low-frequency content. In other words, the peripheral signal is integrated over a certain frequency range. All mathematical models of velocity storage (of which I am aware) implement this integration, including the early models proposed by Robinson (1977) and Raphan et al. (1977).

Importantly, the integration of the peripheral signal will also integrate noise (Karmali and Merfeld 2012). Thus, a longer time constant would cause integration over a broader range of frequencies and thus result in more noise being present in the brain’s estimate of motion, and also could result in the accumulation of noise which could cause drift over time (Merfeld, Clark et al. 2016). Noise in the brain’s estimate of motion is thought to be reflected in perceptual thresholds, VOR variability and postural performance (e.g., Benson, Hutt et al. 1989, Grabherr, Nicoucar et al. 2008, van der Kooij and Peterka 2011, Haburcakova, Lewis et al. 2012, Bermudez Rey, Clark et al. 2016, Karmali, Bermudez Rey et al. 2017, Nouri and Karmali 2018, Rosenberg, Galvan-Garza et al. 2018).

Thus, in selecting a velocity storage time constant, the brain may be resolving a tradeoff between being accurate and reducing noise – i.e., being precise. In this paper, simulations of simple models are used to investigate this tradeoff between accuracy and precision, and demonstrate the downsides of a longer time constant. While this trade-off is implicitly resolved in so-called optimal models that account for the statistics of sensory noise and motion (see the Discussion), in this paper it is illustrated explicitly using a simple transfer function model of velocity storage.

These analyses apply more broadly to our understanding of velocity storage. The velocity storage time constant varies with age (Dimitri, Wall et al. 2001), pathology (Dimitri, Wall et al. 2001, Cousins, Kaski et al. 2013, Priesol, Cao et al. 2015), stimulus velocity (Paige 1992) and species (e.g., Raphan, Matsuo et al. 1977, Robinson 1977, Collewijn, Winterson et al. 1980, Van Alphen, Stahl et al. 2001, Chen, Bockisch et al. 2014). Despite the importance of the velocity storage time constant in clinical diagnostics, a fundamental understanding of what causes these variations has been lacking. The evidence suggesting that these variations may be a response to variations in vestibular noise is summarized in the Discussion.

2. Methods

All simulations (Fig. 1) were implemented in Matlab 2017b using a time step of 1/2000 s. Peripheral responses were predicted by applying a first-order high-pass filter with the semicircular time constant of (Fernandez and Goldberg 1971, Jones and Milsum 1971) using the Matlab function “filter.” Noise was added to this peripheral response. A transfer function model of velocity storage was then applied to this noisy peripheral signal, with the form, where is the velocity storage time constant and is frequency. This is a simplified version of the commonly-used Observer model (Merfeld, Young et al. 1993, Karmali and Merfeld 2012, Clark, Newman et al. 2019), with scaling terms removed because only normalized responses are shown in this paper. The noisy peripheral signal was created by filtering Gaussian white noise generated using the Matlab function “randn” to concentrate noise below 2 Hz. This was done for illustration purposes to exaggerate the changes in noise that occur when low-frequency content is integrated. These analyses are intended to model abstracted activity across the population of neurons; we previously addressed how to model the convergence of individual elements that have higher individual noise in velocity storage (Karmali and Merfeld 2012). Cannon, Robinson et al. (1983) also showed that a realistic neural network implementation of an integrator is able to subtract background firing rate and variations in background rate that are correlated across the network, although they did not address variations that are not completely correlated across neurons (Lim, Karmali et al. 2017).

Figure 1:

Block diagram illustrating how noise is added to the peripheral signal arising from the semicircular canals, which is then processed by the velocity storage filter, which modifies the low-frequency dynamics. The output of velocity storage is known to affect perception and eye movements.

3. Results

The first row of Fig. 2 shows model predictions of peripheral responses. The left column shows responses to a step in head angular velocity from rest to a constant velocity (dashed line). The peripheral response decays exponentially with a time constant of 5.7 s. The effects of noise are also evident in the response, both higher-frequency noise that presents as thickness of the curve, and lower-frequency noise that presents as slow deviations from an exponential decay (e.g., at 35 s). Note that the response magnitude in these simulations is normalized to the amplitude of the input; this contrasts with experimental results and some model predictions that found that the vestibulo-ocular reflex in the dark has a gain of less than 1.

Figure 2:

First row shows noisy peripheral responses, while subsequent rows show central estimates. Grey dashed line: actual head angular velocity. Grey line: peripheral responses. Black lines: central estimates.

The second column shows the model predictions of peripheral responses for a sustained period with no head movement. Again, noise at a range of frequencies is evident; a long time range is used for this panel so that both low and high-frequency components of noise are visible. The third column shows the frequency response.

The second row of Fig. 2 shows model predictions of central estimates for a time constant of 23 s, which is typical for a 35 year old human (Dimitri, Wall et al. 2001). The first column shows the elongation of the central estimate (black) compared to the peripheral response (grey). Importantly, the central estimate remains closer to the actual velocity in comparison to the peripheral response – i.e., the estimate is more accurate. The second column shows that noise is slightly higher than in the peripheral response – i.e., the estimate is less precise. The third column shows the additional low-frequency bandwidth of the response.

The third and fourth rows of Fig. 2 show responses for time constants of 100 s and 1000 s, respectively. As the time constant increases, the central estimate becomes more accurate and less precise. The accumulation of noise is especially evident for a time constant of 1000 s.

4. Discussion

The key finding is that simulations confirm that there is a tradeoff between accuracy and precision that is determined by the velocity storage time constant. While one specific model of velocity storage was implemented, the results should extend to other models with similar dynamics – i.e., integration over a certain frequency range – which to my knowledge is a characteristic of all velocity storage models.

This result provides an intuitive description of the tradeoff that is implicitly implemented in optimal models of vestibular processing based on techniques like Kalman filters and particle filters (Borah, Young et al. 1988, Laurens and Droulez 2007, Karmali and Merfeld 2012, Laurens and Angelaki 2017, Karmali, Whitman et al. 2018). A few studies have found evidence to support the hypothesis that the brain optimizes the dynamics of vestibular processing based on the statistics of experienced motion and noise. For example, Laurens and Droulez (2007) modeled 3D vestibular estimation and found that the model predicted many experimental results; in this study the noise was selected by the modeler to produce the desired dynamics. We showed that increasing noise generally reduces the yaw-rotation velocity storage time constant, which is the correct response to maintain the precision-accuracy tradeoff (Karmali and Merfeld 2012). We extended that result to study the relationship between hair cell death and the velocity storage time constant during aging (Karmali, Whitman et al. 2018). When we assumed that the signal-to-noise ratio decreased in proportion to hair cell density, we found that a Kalman filter model predicted the reduction in velocity storage time constant with age across adulthood. Furthermore, we also recently found that VOR variability, which likely reflects vestibular noise, increases with yaw velocity magnitude (Nouri and Karmali 2018). Nouri and Karmali (2018) note that to maintain the precision-accuracy tradeoff, the velocity storage time constant should decrease with stimulus amplitude, which is consistent with experimental observations (Paige 1992). Finally, future work could investigate whether interspecies differences in noise and/or anatomy relate to interspecies differences in the velocity storage time constant (e.g., Raphan, Matsuo et al. 1977, Robinson 1977, Collewijn, Winterson et al. 1980, Van Alphen, Stahl et al. 2001, Chen, Bockisch et al. 2014)