Vestibular System

Prior to balance testing, participants complete a questionnaire regarding a history of dizziness and falls in the past 12 months. Balance testing consists of the modified Romberg Test of Standing Balance on Firm and Compliant Support Surfaces. This test examines the participant’s ability to stand unassisted under four test conditions that are designed to specifically test the sensory inputs that contribute to balance— the vestibular system, vision, and proprioception (Table 1; 7 (link)).
Balance testing was scored on a pass/fail basis. Test failure was defined as a subject 1) needing to open their eyes, 2) moving their arms or feet in order to achieve stability, or 3) beginning to fall or requiring operator intervention to maintain balance within a 30 second interval. Each subject who failed a Test Condition was eligible for one more attempt to pass. Because each successive Test Condition was more difficult than the condition preceding it, balance testing was concluded whenever a subject failed to pass a Test Condition (during the initial test or in the re-test). We focused on Test Condition 4 -- standing with eyes closed on a 16”×18”×3” foam pad – in which participants relied primarily on vestibular input. Of 5086 participants, 257 did not pass prior test conditions and did not participate in Test Condition 4. An additional 86 participants had missing data for Test condition 4, for a total of 343 excluded participants (6.7%). The time to failure (measured manually by stopwatch in seconds) was recorded in all subjects who participated in Test Condition 4; for subjects who passed Test Condition 4 the time to failure was set as 30 seconds (the maximum testing time). Further details of balance testing procedures are available (http://www.cdc.gov/nchs/data/nhanes/ba.pdf; 8).

The physical stimuli measured by the vestibular system are bidirectional. Therefore, vestibular responses are bidirectional. This characteristic fundamentally influences the application of detection theory to vestibular responses. Specifically, subjects can rotate to the right or left, translate up or down, or tilt forward or backward and sense these different directions of motion. In contrast, photons provided to a subject during a light detection task are unidirectional; perceptions of light opposite to those evoked by photons do not exist as common experience includes nothing “on the other side” of complete darkness. Similarly, the standard hearing test, which is a common clinical application of detection theory, is unidirectional.
Because large differences exist between psychophysical functions for unidirectional and bidirectional stimuli, a brief comparison is warranted. The log of the stimulus amplitude is typically used for unidirectional stimuli; the log is not typically used for bidirectional stimuli because the log of a negative number is imaginary. The theoretical psychophysical function for detection (Yes/No) of unidirectional stimuli tasks ranges between 0% yes for very small magnitudes and 100% yes for large magnitudes. In comparison, because standard detection paradigms for bidirectional stimuli require that all stimuli be either all positive or all negative, the theoretical psychophysical function for detection (Yes/No) of bidirectional stimuli ranges between 50% yes for very small magnitudes and 100% yes for large magnitudes. Furthermore, both unidirectional and bidirectional cumulative distribution functions have two free parameters, but very different, and even somewhat contradictory, terminologies are used. For a direction-recognition task using bidirectional stimuli, we refer to “bias” (e.g., vestibular bias) as the stimulus level that yields the percentage correct midway between the lower and upper bounds of the psychometric function, and we used “threshold”, which is linearly proportional to the standard deviation of the noise (c.f. Table 1), to refer to the width of the transition (e.g., the standard deviation of a Gaussian probability density function underlying the psychometric function). For unidirectional stimuli, the term “threshold” replaces bidirectional “bias”, and “slope” replaces bidirectional “threshold.”
Note that we are not claiming that vestibular responses are the only sensations that are bidirectional as there are bidirectional aspects of other modalities. We are simply noting that vestibular responses are bidirectional and that this fundamental characteristic impacts the application of detection theory. Specifically, vestibular bias—often simply defined as an offset from zero—arises, at least in part, due to the bidirectional nature of vestibular responses. For example, unequal contributions from the left and right labyrinths can lead to vestibular bias. A similar vestibular bias can arise centrally from asymmetric processing of peripheral information. The cause/source of a vestibular bias—whether peripheral or central—is not crucial to the following analysis as either can yield vestibular bias. Unfortunately, as described in more detail below, “bias” is used in the detection theory literature to refer to the fact that the detection criteria might not be the same for all subjects, which forms a basis for “criterion bias”. Criterion bias will be introduced mathematically later, but, briefly stated criterion bias simply represents the tendency for a subject to prefer one choice over another. To help distinguish these two independent effects, we will generally avoid the use of “bias” by itself and will instead refer specifically to “vestibular bias” or “criterion bias”.
We assume that any vestibular bias is constant (e.g., independent of stimulus amplitude and duration). The presence of vestibular bias means that we must distinguish the actual (“objective”) stimuli—known to the experimenter—from the sensed (“subjective”) stimuli—experienced by the subject. To do so, we use an example comparing two stimuli with specific values. Like some other examples, the values are arbitrary. (When values are not arbitrary, we will specifically state this.) Assume objective stimulus amplitudes of 0 and 6°/s and a subjective vestibular bias (μ) of −2°/s, where the vestibular bias simply means that, in the presence of null stimuli (zero amplitude), this subject will, on average, subjectively experience stimuli equivalent to −2°/s. (For example, the presence of a vestibular bias might manifest as a positive VOR bias, though we do not assume that the VOR bias and the subjective bias are necessarily one and the same.)
The probability density functions for the objective stimuli are shown as impulse functions (Fig. 1a), since the objective stimuli are presumed to have much less variability (“noise”) than the subjective experience (Fig. 1c). Specifically, it is presumed that the motion devices are well controlled and provide nearly the same stimuli each time. The vestibular bias is represented by a rightward shift of the subjective axes relative to the objective axes (Fig. 1b). Therefore, the mean subjective motions sensed are −2 and +4°/s, respectively. But the sensed stimuli will have physiologic noise that is assumed Gaussian, with a standard deviation (σ) of 2°/s chosen for this example. This noise includes all physiologic sources of variability (afferent noise, processing noise, etc.).
For all analyses included herein, the noise for small near-threshold stimuli is assumed constant and is assumed to sum with the signal. The distributions in Fig. 1c can be interpreted as indicating that for a given stimulus amplitude the subjective sensation (sensed signal) for a given trial will be randomly selected from this probability distribution. A sensed signal near the mean is most likely, but individual trials can yield sensed signals above or below the mean, with the prevalence proportional to the magnitude of the probability density function (PDF). The equation for a Gaussian PDF can be written as:
$$f(x)=\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{{(x-\mu )}^2}{2{\sigma }^2}}.$$
The cumulative distribution functions (CDFs) for these PDFs are shown Fig. 1d. The CDFs represent the percentage of times that the subject’s perception would be less than the value on the abscissa (x-axis) for the given mean stimulus. (An example follows three paragraphs below.) The CDF is the integral of the PDF:
$$\phi (x)=\underset{-\infty }{\overset{x}{\int }}f(x^{\prime })\text{d}{x}^{\prime }=\underset{-\infty }{\overset{x}{\int }}\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{{(x^{\prime }-\mu )}^2}{2{\sigma }^2}}\text{d}{x}^{\prime }.$$
This integral does not have a closed form solution, so it is solved using standard numerical methods that often involve a special function called the error function (Hildebrand 1976 ; Wikipedia 2010b ). Figure 1e and f show the PDFs and CDFs in objective coordinates, which is accomplished by simply reversing the shift due to vestibular bias from objective (Fig. 1a) to subjective coordinates (Fig. 1b).
The second column of Fig. 1 shows the same variables but the units have been changed. Specifically, we have normalized all values by the standard deviation of the original distribution. This process of normalizing by the standard deviation is sometimes called “standardizing” the variable as this normalization yields one as the standard deviation. For the rest of the paper, we will only use distributions with a standard deviation of one (implicitly assuming standardization). For simplicity, we further assume that this standard deviation of the noise is always constant and does not depend upon the stimulus but this assumption (like others) can be relaxed if required by experimental findings. For this distribution, this normalization yields objective stimuli of 0 and +3 and subjective stimuli means of −1 and +2. This normalization is equivalent to changing units and does not limit the generality of any findings.
The CDFs of Fig. 1j represent the percentage of times that the subject’s perception would be less than the value on the abscissa (x-axis) for the given mean stimulus. For example, for the dashed distribution, the subjective mean is +2, so 50% of the trials would be perceived as less than +2 (and 50% greater than +2), and 2.28% of the trials involving a mean subjective stimulus of +2 (objective stimulus of +3) would be perceived as being negative.
This value of 2.28% can be calculated using the cumulative distribution function in MATLAB Statistics Toolbox as cdf(‘norm’,0,2,1), where ‘norm’ indicates that the distribution is normal (Gaussian), zero represents the “decision boundary”, one is the standard deviation in standard deviation units, and two is the mean of the subjective distribution. Placing the decision boundary at zero represents that we asked subjects to indicate whether their perception was positive or negative. (Decision boundaries will be discussed in more detail later.) We will show similar MATLAB functions in the text to help directly illustrate the calculations. These can easily be mapped to any other program (e.g., Excel, etc.).
The Gaussian assumption is justified by the central limit theorem of statistics (Larsen and Marx 1986 ; Wikipedia 2010a ). More fundamentally, it is not essential that the distributions be Gaussian, though for bidirectional vestibular responses, the noise distribution will typically be symmetric (or at least nearly symmetric). If the distributions are not Gaussian, the approach outlined here is still valid, though calculations would need to be redone using an appropriate distribution. (For example, all of the standard z-scores and d′ calculations—to be discussed in the following paragraphs—assume Gaussian noise.) We will take a few paragraphs below to introduce some standard signal detection metrics but readers seeking details should refer to another source, like Macmillan and Creelman (2005) .