Postural impairments in unilateral and bilateral vestibulopathy

Chronic imbalance is a major complaint of patients suffering from bilateral vestibulopathy (BV) and is often reported by patients with chronic unilateral vestibulopathy (UV), leading to increased risk of falling. We used the Central SensoriMotor Integration (CSMI) test, which evaluates sensory integration, time delay, and motor activation contributions to standing balance control, to determine whether CSMI measures could distinguish between healthy control (HC), UV, and BV subjects and to characterize vestibular, proprioceptive, and visual contributions expressed as sensory weights. We also hypothesized that sensory weight values would be associated with the results of vestibular assessments (vestibulo ocular reflex tests and Dizziness Handicap Inventory scores). Twenty HCs, 15 UVs and 17 BVs performed three CSMI conditions evoking sway in response to pseudorandom (1) surface tilts with eyes open or, (2) surface tilts with eyes closed, and (3) visual surround tilts. Proprioceptive weights were identified in surface tilt conditions and visual weights were identified in the visual tilt condition. BVs relied significantly more on proprioception. There was no overlap in proprioceptive weights between BV and HC subjects and minimal overlap between UV and BV subjects in the eyes-closed surface-tilt condition. Additionally, visual sensory weights were greater in BVs and were similarly able to distinguish BV from HC and UV subjects. We found no significant correlations between sensory weights and the results of vestibular assessments. Sensory weights from CSMI testing could provide a useful measure for diagnosing and for objectively evaluating the effectiveness of rehabilitation efforts and future treatments designed to restore vestibular function such as hair cell regeneration and vestibular implants.


RMS sway calculation
RMS (root mean squared) sway measures were calculated by averaging the CoM or head sway angles across the last 11 cycles of the stimulus, subtracting the mean from the cycle-averaged sway, calculating the mean-squared value of this zero-meaned cycle-averaged sway, and then calculating the square-root of this mean-squared value.In equation form the mean response waveform is given by: where n is the sampling index, M is the number of stimulus cycles (nominally 12), Np is then number of samples per stimulus cycle, and r is the sampled response data (either CoM sway angle or head pitch angle).() is the mean response waveform that excluded the response data associated with the first stimulus cycle from the calculation of the mean.
Then the RMS value is calculated: Where ̅ is the mean value across the Np samples of ().

Remnant sway calculation
The RMS value of remnant sway gives a measure that quantifies the variability of the stimulus-evoked sway response that is not accounted for by the mean value of the stimulus-evoked sway.The remnant sway calculation is performed in the frequency domain by first calculating the discrete Fourier transforms (DFTs) of the last 11 cycles of the response waveforms using the Matlab 'fft' function and calculating the average of these DFTs: Where  [] is the one-sided DFT of the l th cycle of the response waveform,  ̂ is the mean DFT of the last M = 11 cycles of the response and k is the index of the frequency components of DFTs with k = 1 being the lowest frequency component of the DFT which has a value of ∆ which is equal to the inverse of the 20-s single-cycle duration of stimulus (i.e., ∆ = 0.05 Hz).Then the remnant power spectrum   is calculated which is based on a variance calculation given by the squared difference of the absolute value of individual cycle DFTs from the mean DFT: where Ksf is a factor that appropriately scales the power spectrum such that the area under the power spectrum is equal to the mean squared value of the signal.Specifically, Ksf is the inverse of the product of two times the time series sampling rate times the number of samples per stimulus cycle.Finally, the remnant RMS value is calculated by taking the square root of the summed value of the area under the remnant power spectrum: With kmax = 100 (corresponding to 5 Hz) being the highest frequency component index that was used in the summation.

Central Sensorimotor Integration (CSMI) analysis
The experimental frequency response function (FRF) provides a non-parametric, frequency-domain characterization of the dynamic properties of the balance control system.The experimental FRF was calculated by dividing the cycle-averaged DFT of the sway response by the cycle-averaged DFT of the stimulus: Where  ̂() is defined in equation S3 and  ̂() is the average stimulus DFT defined in a similar manner.
Smoothing was applied to Hexp by averaging adjacent frequency points to reduce the variance of Hexp and to provide Hexp measures that were approximately equally spaced on a logarithmic frequency scale.
Examples of Hexp calculations are shown in Figure 1C of the paper for a healthy control and bilateral vestibular loss subject.
The non-parametric experimental FRFs that characterized the dynamics of CoM responses to surface tilt or visual tilt stimuli were used to estimate the values of functionally relevant parameters of the balance control system by adjusting the parameters of a balance control model.
where 's' is the Laplace variable.Substituting S9 -S12 into S7 and S8 and setting  = 2, where j is the imaginary number √−1, allows for the calculation of the model H values as a function of the sinusoidal stimulus frequency, f.All transfer function equations assume that the sum of all sensory weights contributing to balance, in a given condition, sum to 1 meaning the value of a sensory weight represents the relative contribution of a sensory system to balance control.For example, in the eyes open surface stimulus condition proprioception, visual, and vestibular cues are the contributors to balance control.The curve fitting procedure will estimate the value of Wprop for this condition and then the vestibular plus visual contribution is given by Wvest + Wvis = 1 -Wprop.
The value of H, at any particular frequency, f, is a complex number that can be in terms of a 'magnitude function' |(2)| equal to the square root of the sum of squared values of the real and imaginary components of H, and 'phase function' ∠(2) equal to the arc tangent of the imaginary divided by the real components of H.The transfer function magnitude is also referred to as the system 'gain function' since it represents the body sway response magnitude normalized by the magnitude of the stimulus at each frequency value.
The free parameters were adjusted to optimally account for the experimental FRF, Hexp, derived from the experimental body sway responses to the pseudorandom stimulus (equation S6) using Matlab Optimization Toolbox function 'fmincon'.The free parameters include the sensory weight (W), motor activation 'stiffness' parameter (Kp) and 'damping' parameter (Kd), time delay (Td), and torque feedback (Kt).The body moment of inertia about the ankle joint, J, body mass, m, (excluding the feet), and body center-of-mass height above the ankle joints, h, are derived from direct measurement of body mass and based on anthropometric body measures [6] VS, stimulus under steady-state conditions when all transient responses that occur at stimulus initiation have decayed to negligible amounts.When all the dynamic elements of the model (which include the inverted pendulum body, B, the 'motor activation' component, MA, 'torque feedback', TF, and 'time delay', TD) are expressed in the Laplace domain, the equations relating CoM to SS (in both eyes open and closed conditions) and/or VS can be solved algebraically to define 'transfer functions', H, that express the dynamic relationship between the stimulus and the body sway response: The block diagram of the CSMI model in Figure1Bof the paper can be expressed as a differential equation that determines the body sway angle, CoM, relative to Earth vertical as a function of the support surface, SS, stimulus and/or the visual scene,

Table S1 : RMS values of stimulus-evoked CoM and head sway, remnant CoM and head sway for HC, UV, and BV groups and post-hoc comparisons between groups.
: SS/EO = surface stimulus eyes open, SS/EC = surface stimulus eyes closed, VS/EO = visual stimulus eyes open, SD = standard deviation.All sway measures have units of degrees.p-values indicate results of a One-Way Analysis of Variance (ANOVA)* or Kruskal-Wallis One-Way Analysis of Variance**.Post hoc analysis is either Dunn's (following Kruskal-Wallis) or Holm-Sidak (following ANOVA) method.In addition to the p value, we also calculated the Hedge's G value, which represents an effects size measure, measuring the difference between means relative to the standard deviation.Bolded outcomes indicate a significant group difference.EO = surface stimulus eyes open, SS/EC = surface stimulus eyes closed, VS/EO = visual stimulus eyes open, SD = standard deviation, s = seconds.p-values indicate results of a One-Way Analysis of Variance (ANOVA)* or Kruskal-Wallis One-Way Analysis of Variance**.Post hoc analysis is either Dunn's (following Kruskal-Wallis) or Holm-Sidak (following ANOVA) method.In addition to the p value, we also calculated the Hedge's G value, which represents an effects size measure, measuring the difference between means relative to the standard deviation.Bolded outcomes indicate a significant group difference.
, and g is the gravity constant.Examples of experimental FRFs and calculated FRFs derived from transfer function equations with optimally adjusted parameters are shown in Figure 1C in the paper.Note