SoS signals were delivered to perturb balance simultaneously in the roll and pitch directions during Condition (i), simultaneously in the RALP and LARP directions during Condition (ii), and in the roll, pitch, RALP and LARP directions simultaneously during Condition (iii). The roll and pitch signals were orthogonal to one another in both time and space. Similarly, the RALP and LARP signals were also orthogonal to one another in time and space. During Condition (iii), the roll, pitch, RALP and LARP stimuli were each orthogonal in time to all other stimuli, but roll and pitch stimuli were, by definition, not spatially orthogonal to the RALP and LARP stimuli.

Building on our earlier work that included two simultaneous SoS perturbations (38), we developed four distinct SoS balance perturbation trajectories that could be provided simultaneously during Condition (iii) as the stimuli for roll, pitch, RALP, and LARP. The trajectories used a fundamental frequency of 0.0055 Hz multiplied by four groups of interleaved multipliers ([11,59,109,179,241], [13,61,113,181,251], [15,71,131,193,263], and [17,67,127,191,257]). No multipliers were repeated; each multiplier, except 15, was a prime number. Neither of 15’s prime factors (3 or 5) was used; this avoids overlapping at higher harmonics, (e.g., 3rd harmonic associated with the multiplier 5). Each frequency component was orthogonal in time to all other frequency components. Each of these SoS trajectories was named to reflect the lowest multiplier (i.e., SoS₁₁, SoS₁₃, SoS₁₅, and SoS₁₇). When multiplied by the fundamental frequency, these integer sets yielded the following four frequency sets:

The four steady-state SoS displacement trajectories were generated by summing sinusoidal signals. For example, the following equation was used to generate the SoS₁₁ trajectory:

In this equation, t represents time, A 11 , i is the ith amplitude, f 11 , i is the ith frequency, and ∅ 11 , i is the ith phase value. These equations resulted in four SoS trajectories having interleaved, spectrally separated perturbation frequencies. Supplementary material shows the other three equations used to generate the rest of the trajectories (i.e., SoS₁₃, SoS₁₅, and SoS₁₇).

The phase values were determined via a systematic search using one-degree phase increments between 0 and 359° to minimize differences in the peak-to-peak amplitude of the four trajectories while keeping the velocity spectral magnitude constant across all frequencies (Table 1). The magnitude was scaled so that SoS₁₃ had a peak-to-peak amplitude of 1 deg. This yielded a 0.17°/s spectral velocity magnitude for each of the 5 frequency components. For the other trajectories, we maintained the magnitude of each spectral velocity component constant across all perturbation frequencies. This angular velocity magnitude was chosen based on Peterka (15) to make the task difficult while allowing participants to complete our test trials. This yielded peak-to-peak displacement amplitudes of 1.1051 deg. for the SoS₁₁ signal, 0.9592 deg. for the SoS₁₅ signal, and 0.8948 deg. for the SoS₁₇ signal. These four trajectories each had the exact same velocity magnitude of 0.17 deg./s at each of their five disturbance frequencies (Figure 2B).

We shifted the signals in time to begin motion at a displacement zero-crossing to minimize transients that otherwise might unduly perturb subjects at the beginning of the trial. Table 1 shows the original phase as well as the shifted phase values of SoS₁₁, SoS₁₃, SoS₁₅, and SoS₁₇ at each individual frequency.

We used a commercially available device (Virtualis MotionVR) to provide the platform tilt perturbations. The steady-state perturbation was designed to have a duration of 182.0444 s yielded by 16,384 (2¹⁴) points at the Virtualis platform’s sampling/refresh rate of 90 Hz. To allow the postural response to reach steady-state prior to the start of this steady-state cycle, 20s of stimuli identical to the last 20s of the steady state waveforms were added before the steady state waveform. We also added a 0.2 s ramp-up phase before the 20s steady state waveforms to minimize sudden motions that could potentially disturb our subjects or their responses. Since the steady-state cycle was repeated just once, the total perturbation duration was 202.2444 s for all trials. Supplementary Figure S1 shows example position and time trajectories. A custom Matlab script used to generate the waveforms is available on request.

We collected postural response data from 24 healthy participants. A custom script written in Matlab first generated the SoS time series and the resultant CSV files were imported into the Virtualis MotionVR Research software to deliver platform perturbations. No passive head movements were applied, which leads platform perturbations to stimulate a human postural control system that includes vestibular, proprioceptive, and visual sensors. Center of pressure (CoP) data were collected by two embedded force plates at a sampling rate of 90 Hz.

All of the participants completed 6 trials per day on each of two separate days—yielding 12 trials in total for each subject (Table 2) (i.e., no dropouts). The two test days were separated by more than 1 day but less than 18 days. See Supplementary Figure S2 for more details. All but one participant completed the testing within 9 days. We randomized the order of the 12 trials to mitigate order effects. A different combination of SoS perturbation signals (e.g., SoS₁₁ and SoS₁₃) was used for each trial. Participants are allowed up to three attempts if an incomplete trial occurs (e.g., touching the safety rail around the Virtualis or taking a step). No incomplete trials occurred in this cohort, so all participants experienced each trial once. Participants’ feet were bare, with the outside edge of feet separated by 28 cm to mimic the standard Sensory Organization Test (SOT) conditions that are widely used in clinics. Arms were crossed with hands on shoulders. Participants were asked to close their eyes. Participants wore a virtual reality headset (HTC VIVE) that showed a dark screen to eliminate visual cues if a participant opened their eyes. Prior to testing, participants were instructed to stand as still as possible without any extra motions to avoid active movements (e.g., at the head, knee, and hip) that are not induced by platform perturbations or verbal responses. The VR headset has an IMU sensor that records 3D head linear positions and 3D head angular positions. Each participant was asked to rest at least 1 min between trials to minimize fatigue.

The present study was approved by The Ohio State University Institutional Review Board. Prior to testing, all participants provided their written informed consent to participate in the study. Twenty-four healthy participants including 12 males and 12 females participated in the study. The age range of the participants was 21–65 yr (40.3 ± 13.1 years). To qualify as a healthy normal participant, strict criteria intended to eliminate conditions that might affect normal vestibular function or standing balance were assessed by a health questionnaire as a part of a central registry. These exclusion criteria included neurological disorders such as neurodegenerative disease, peripheral neuropathy, epilepsy, traumatic brain injury, or prior stroke; active vestibular disorders such as vestibular migraine, persistent postural-positional dizziness, Ménière disease, prior acute vestibular syndrome with peripheral vestibular loss, vestibular areflexia, or unresolved benign paroxysmal positional vertigo; orthopedic impairments affecting standing such as orthopedic injuries or impairments of the lower extremities; and active ongoing medical conditions (recent surgery, severe heart disease or pulmonary disease, and cancer).

We analyzed CoP data using Matlab R2024a. The first 20s of data for each trial were eliminated to remove the transient response at the onset of the perturbation. Each remaining CoP signal, having 16,384 data points representing 182.044 s was shifted to have zero mean and filtered using a zero-phase 2^nd order Butterworth low-pass filter with a cut-off frequency of 20 Hz.

To estimate RALP and LARP CoP sway, ML and AP CoP were rotated 45 degrees clockwise using the following equations.

A discrete Fourier transform (Matlab, fft.m) was applied to ML, AP, RALP, and LARP CoP signals and to roll, pitch, RALP and LARP perturbation signals. The complex numbers resulting from the discrete Fourier transform (DFT) were divided by the length of the data points (N = 16,384). The first half of the two-sided spectrum was removed, and the positive spectrum was multiplied by 2 to convert each two-sided amplitude spectrum to a single-sided amplitude spectrum. We used frequency response functions (FRF) to describe the sensitivity and phase of the CoP response relative to the tilt perturbation as a function of frequency. The FRF values are determined by dividing the complex DFT of the CoP by the complex DFT of the platform tilt angles ( So S 11 , So S 13 , So S 15 , and So S 17 ) at each of the five individual perturbation frequencies:

We calculated the sensitivity (units of mm/degree) of the CoP postural response by calculating the magnitude of these FRF values at each frequency. Sensitivity is a non-negative value, and phase estimated via the frequency response functions is limited to a range between −180° and 180°. Phase values beyond ± 180 ° were calculated using the unwrap function in Matlab.

Additionally, we calculated time domain metrics—the root mean square distance (RMSD) and the mean velocity of the ML, AP, RALP, and LARP CoP in each of the 12 conditions [(i)1–4, (ii)1–4, and (iii)1–4]. The ML, AP, RALP, and LARP RMSD were calculated by taking the root mean square distance of the zero-mean, low-pass filtered CoP signals. The ML, AP, RALP, and LARP mean velocity were determined by dividing the total distance by the duration of the trial.

For the time domain metrics – RMSD and mean velocity – we first performed eight sets of two-way repeated measures MANOVA to evaluate the effects of Condition (i, ii, iii) and Trial [e.g., (i)-1, (i)-2, (i)-3, and (i)-4] on the eight metrics that include RMSD and mean velocity in ML, AP, RALP, and LARP directions. Bonferroni-corrected pairwise comparisons were next performed to further examine differences across all 12 trials (3 conditions x 4 trials each). We then pooled the four trials within each condition and performed Bonferroni-corrected pairwise comparisons to evaluate condition-level differences.

For the spectral responses, Hotelling T-squared tests were used to determine whether spectral responses observed at perturbed frequencies were significantly different from zero. This helps demonstrate that those spectral responses are distinguishable from the remnant. We next performed four sets of four-way repeated measures MANOVA using real and imaginary components for ML, AP, RALP, and LARP frequency response functions (FRFs) that described the sensitivity and phase of CoP postural responses relative to the stimulus. For one-sample comparisons, we performed Hotelling T-squared tests using the real and imaginary parts of each FRF’s complex values. When comparing two different frequency sets, we calculated the 2-dimensional difference (i.e., the difference between real and the difference between imaginary components) between the two responses at each of the 5 perturbation frequencies. To address multiple comparisons, we applied both a commonly used Bonferroni correction to control the family-wise error rate by adjusting the significance level α from 0.05 to 0.05 divided by the number of comparisons, and the Benjamini-Hochberg (B-H) procedure to control the false discovery rate at q = 0.05, to mitigate the conservatism of the Bonferroni correction.

Repeated measures MANOVA revealed significant effects of Condition [e.g., (i), (ii), and (iii)] and Trial [e.g., (i)-1, (i)-2, (i)-3, and (i)-4] on Root mean square distance (RMSD) (Condition: p < 0.001, Trial: p < 0.001) and mean velocity (Condition: p < 0.001, Trial: p = 0.038). We also found a significant interaction effect of Condition and Trial on RMSD (p < 0.001) but not mean velocity (p = 0.62). Supplementary Tables S1, S2 show F-values and p-values that indicate the significant main effects and interaction effects of Condition and Trial on RMSD and Mean Velocity.

Further Bonferroni corrected pairwise comparisons found a significant difference between Condition (i) vs. (iii) (p < 0.001) and Condition (ii) vs. (iii) (p < 0.001) for both RMSD and mean velocity, which is shown in Figure 3. The asterisks delineate statistically significant differences (p < 0.05). The pairwise comparisons showed no significant differences among the four trials in mean velocity but indicated a significant difference between Trial 1 vs. 3 in ML RMSD and Trial 1 vs. 4 in LARP RMSD but not one of the other 22 trial comparisons of RMSD. The peak-to-peak amplitudes for Condition (iii) are larger than those for Condition (i) and (ii), so it was expected that Condition (iii) would evoke larger postural responses than Condition (i) or (ii) in all the directions. Peak-to-peak displacement was roughly 50% greater for Condition (iii) (range: 1.53° to 2.09°) than for Condition (i) and Condition (ii) (range 0.89° to 1.40°); peak-to-peak amplitudes for Condition (i) and Condition (ii) were about the same. Supplementary Table S3 provides the peak-to-peak amplitudes of the stimuli delivered during the four trials in the three conditions.

Supplementary Figure S4 provides box-and whisker plots that show RMSD and mean velocity of the CoP responses for all of the subjects and for all the trials for each condition. We pooled the four trials in each condition as the differences between trials were inconsistent. Figure 3 shows box- and whisker plots of the average RMSD (Figure 3A) and mean velocity (Figure 3B) of ML, AP, RALP, and LARP CoP responses across the four trials for each condition [e.g., (i)-1, (i)-2, (i)-3, and (i)-4]. Bonferroni corrected pairwise comparisons, performed using a paired t-test, confirmed that both RMSD and mean velocity were significantly larger for Condition (iii) when roll, pitch, RALP, and LARP perturbations were provided simultaneously versus Condition (i) when just roll/pitch stimuli were provided and Condition (ii) when just RALP/LARP stimuli were provided in separate trials (p < 0.0001). No significant differences were observed when comparing RMSD or mean velocity responses evoked by roll/pitch stimuli [Condition (i)] to those evoked by RALP/LARP stimuli [Condition (ii)) (p > 0.23].

Figure 4 shows the average ML, AP, RALP, and LARP CoP sway spectral response across the 24 subjects for the roll/pitch condition (i)-1 (i.e., when provided SoS₁₁ for roll and SoS₁₅ for pitch). Consistent with previous literature (38), ML CoP spectral peaks were observed at each of the five frequencies for the roll tilt stimuli (Figure 4A); ML sway at each of the roll perturbation frequencies were the predominant response components. Similarly, AP CoP spectral peaks were observed at each of the five frequencies for the pitch tilt stimuli (Figure 4B); AP sway at each of the pitch perturbation frequencies were the predominant response components. These spectral peaks were clearly distinguishable from one another in the frequency domain. RALP and LARP CoP spectral magnitudes each show 10 response components during this roll/pitch condition. This is a direct outcome of the fact that the RALP/LARP coordinates are rotated 45 degrees from the roll/pitch coordinates, so RALP spectral peaks were observed at each of the five frequencies for both roll tilt stimuli and pitch tilt stimuli (Figure 4C); similarly, LARP spectral peaks were observed at each of the five frequencies for both roll tilt stimuli and pitch tilt stimuli (Figure 4D).

Hotelling T-squared tests found that the spectral peaks observed at each perturbation frequency (i.e., ML for roll stimuli, AP for pitch stimuli, RALP for roll and pitch stimuli, and LARP for roll and pitch stimuli) were significantly different than zero (p < 0.0001). See Supplementary Table S4 for more details.

Figure 5 shows the average ML, AP, RALP, and LARP CoP sway spectral magnitude across the 24 subjects for RALP/LARP condition (ii)-1 (i.e., when provided SoS₁₃ for RALP and SoS₁₇ for LARP). RALP spectral peaks were observed at each of the five frequencies for the RALP tilt stimuli (Figure 5C); sway in the RALP plane at each of the RALP perturbation frequencies were the predominant response components (Figure 5C). LARP spectral peaks were observed at each of the five frequencies for the LARP tilt stimuli (Figure 5D); sway in the LARP plane at each of the LARP perturbation frequencies were the predominant response components (Figure 5D). The RALP and LARP spectral peaks were distinguishable from one another. Analogous to the RALP and LARP responses shown above (Figure 4) for roll and pitch stimuli, ML (Figure 5A) and AP (Figure 5B) CoP spectral magnitudes each show 10 response components—one for each of the RALP and LARP perturbation frequencies.

Although the primary focus of the study was CoP measurements, Supplementary Figure S5 shows spectral head RALP and LARP response components at RALP and LARP frequencies. RALP head response components were statistically larger than LARP head response components at all RALP frequencies (p < 0.0001); LARP head response components were statistically larger than RALP head response components at all LARP frequencies (p < 0.0001). This suggests that RALP/LARP platform perturbations evoke head motions in the perturbed directions.

Hotelling T-squared tests found that the spectral peaks observed at each perturbation frequency (i.e., ML for RALP and LARP stimuli, AP for RALP and LARP stimuli, RALP for RALP stimuli, and LARP for LARP stimuli) were significantly different than zero (p < 0.0001). See Supplementary Table S4 for more details.

Figure 6 shows the average ML, AP, RALP, and LARP CoP sway spectral magnitude across the 24 subjects for RALP/LARP condition (iii)-1 (i.e., when SoS₁₁ was provided for roll, SoS₁₅ was provided for pitch, SoS₁₃ was provided for RALP, and SoS₁₇ was provided for LARP). ML spectral peaks were observed at each of the roll, RALP, and LARP perturbation frequencies (Figure 6A); AP spectral peaks were observed at each of the pitch, RALP, and LARP perturbation frequencies (Figure 6B). Similarly, RALP spectral peaks were observed at each of the RALP, roll, and pitch perturbation frequencies (Figure 6C); LARP spectral peaks were observed at each of the LARP, roll, and pitch perturbation frequencies (Figure 6D). Each CoP spectral magnitude shows 15 response components at three sets of five perturbation frequencies in the directions described above. This demonstrates that spectral postural response components were observed at each perturbation frequency—except for responses that were spatially orthogonal to the stimuli (e.g., AP responses to roll stimuli or RALP responses to LARP stimuli)—when four SoS perturbations are provided in roll, pitch, RALP, and LARP directions simultaneously.

Hotelling T-squared tests found that the spectral peaks observed at each perturbation frequency (i.e., ML for roll, RALP, and LARP stimuli, AP for pitch, RALP, and LARP stimuli, RALP for RALP, roll, and pitch stimuli, and LARP for LARP, roll, and pitch stimuli) were significantly different than zero (p < 0.0001). See Supplementary Table S4 for more details.

Figures 7, 8 show the sensitivity (left, y-axis) and phase (right, y-axis) of the CoP spectral response relative to the perturbation stimuli in the ML, AP, RALP, and LARP planes. Consistent with earlier reports (15), (1) the sensitivity increased with increasing perturbation frequency for the low to mid-frequency range (0.06–0.70 Hz) and reached a plateau for the high-frequency range (0.70–1.42 Hz), and (2) phase lag increased with increasing perturbation frequency. The sensitivity of CoP RALP and LARP in response to roll stimuli is a little lower than the sensitivity in response to pitch stimuli at the low-frequency range (0.06–0.50 Hz) but a little higher at the mid to high-frequency range (0.50–1.42 Hz) (Figures 7C,D). The sensitivity of CoP ML and AP in response to RALP stimuli is slightly lower than the sensitivity in response to LARP stimuli in the low-frequency range (0.06–0.50 Hz) but slightly higher in the mid to high-frequency range (0.50–1.42 Hz) (Figures 8A,B). Supplementary Figure S6 shows the sensitivity and phase plots for Condition (iii).

The real and imaginary components of ML, AP, RALP, and LARP spectral responses were evaluated using a four-way repeated measures MANOVA to find a difference when the stimuli were provided in roll/pitch coordinates versus RALP/LARP coordinates. The four factors were (1) the three stimulus directions that were not orthogonal to the response direction (e.g., roll, RALP, and LARP tilts, but not pitch tilt, for the ML response), (2) the four sets of perturbation frequencies (i.e., SoS₁₁, SoS₁₃, SoS₁₅, and SoS₁₇), (3) the five frequencies for each set (i.e., f₁, f₂, f₃, f₄, and f₅), and (4) the three test conditions (i.e., roll/pitch only, RALP/LARP only, and simultaneous roll/pitch + RALP/LARP conditions). Mauchly’s test indicated the sphericity assumption was invalid. While some statistically significant differences were observed, a clear pattern of differences was not evident. See Supplementary Table S5 for details. That some differences were statistically significant, alongside violation of the sphericity assumption, encouraged additional statistical evaluation of these data using more refined analyses, Hotelling’s T-squared analyses with Bonferroni correction and the Benjamini-Hochberg (BH) procedure, which follow.

Earlier publications, as well as our data (Figures 7, 8), clearly establish that responses to balance perturbations vary systematically with frequency (11, 15, 38). Nevertheless, to help establish the sensitivity of these methods to small frequency changes and to tell if we can directly compare responses at nearby individual frequencies across our four different stimuli sets (SoS₁₁, SoS₁₃, SoS₁₅, SoS₁₇), we evaluated the impacts of small frequency changes on the postural responses utilizing the four SoS signals described in detail in the methods (Figure 2). Using formatting similar to Figure 9, Figure 10 shows the average frequency response functions of ML, AP, RALP and LARP CoP sway in the complex plane at the five perturbation frequencies for SoS₁₁, SoS₁₃, SoS₁₅, and SoS₁₇ when provided as: (i) roll/pitch only, (ii) RALP/LARP only, and (iii) roll/pitch/RALP/LARP. The data demonstrates distinct separations for the different sets of the five perturbation frequencies.

Analogous to earlier analyses, 240 comparisons were performed using Hotelling’s T-squared test to investigate how those four specific frequency sets (SoS₁₁, SoS ₁₃, SoS ₁₅, and SoS ₁₇) and their five frequencies (f₁ through f₅) affect ML, AP, RALP, and LARP CoP sway. After calculating the discrete Fourier transform, we took the vector difference between the real and imaginary components of the two spectra for each stimulation frequency for each participant and performed a Hotelling’s T-square test to compare CoP in response to (aa) SoS₁₁ and SoS₁₃, (bb) SoS₁₁ and SoS₁₅, (cc) SoS₁₁ and SoS₁₇, (dd) SoS₁₃ and SoS₁₅, (ee) SoS₁₃ and SoS₁₇, and (ff) SoS₁₅ and SoS₁₇. This yielded 60 comparisons (each of 6 SoS stimuli comparisons x each of 5 frequencies x each of 2 response directions) for (i) roll/pitch only, 60 comparisons for (ii) RALP/LARP only, and 120 comparisons for (iii) roll/pitch/RALP/LARP (each of 6 SoS stimuli comparisons x each of 5 frequencies x each of 4 response directions), which means 240 comparisons in total. Hotelling’s T-squared test found 168 significant differences. Bonferroni correction yielded 75 significant differences, which is 31.25% of the p-values, and the B-H procedure yielded 145 significant differences, which is 60.41% of the p-values. Table 5 shows the 10 lowest original p-values and the individual p-values’ BH critical values. Specifically, we found that small frequency shifts between (aa) SoS₁₁ and SoS₁₃, (bb) SoS₁₁ and SoS₁₅, (cc) SoS₁₁ and SoS₁₇, (dd) SoS₁₃ and SoS₁₅, (ee) SoS₁₃ and SoS₁₇, and (ff) SoS₁₅ and SoS₁₇ yielded significant differences in postural responses.

Stated briefly, when comparing responses at distinct frequencies from our four perturbation trajectories that were separated by 0.0110–0.1208 Hz (i.e., relatively small differences in perturbation frequencies), we found statistically significant differences for the majority of these comparisons. Since we already knew that postural responses vary with frequency, the finding that the majority of comparisons yielded significant differences helps establish the sensitivity of these Hotelling’s T-squared methods to detect even subtle differences associated with small changes in perturbation frequency, which bolsters the absence of significant differences reported above in the section entitled “Roll/Pitch stimuli and RALP/LARP stimuli evoke similar AP, ML, RALP, and LARP responses.”

Future studies with larger sample sizes could help verify these findings and improve the reliability and precision of effect size estimates.

Consistent with our earlier findings (38), we report that: (a) tilt perturbations in the roll plane yielded responses primarily in the ML direction (Figure 5A) with only small responses in the AP direction (Figure 5A), and (b) tilt perturbations in the pitch plane yielded responses primarily in the AP direction (Figure 5B) with only small responses in the ML direction (Figure 5B). As novel findings, we report that (c) tilt perturbations in the RALP plane yielded responses primarily in the RALP direction (Figure 6C; Supplementary Figure S5) with only small responses in the LARP direction (Figure 6C; Supplementary Figure S5), and (d) tilt perturbations in the LARP plane yielded responses primarily in the LARP direction (Figure 6D; Supplementary Figure S5) with only small responses in the RALP direction (Figure 6D; Supplementary Figure S5).

In Condition (i), where we provided simultaneous roll and pitch stimuli, we found that the largest response components were in the ML and AP directions (Figures 5A,B). When we look at these same data in coordinates that align with the RALP and LARP directions, we (by the mathematical relationship between these two different reference frames) found response components that reflect the projection of the roll and pitch responses onto the RALP and LARP directions (Figures 5C,D).

In Condition (ii), where we provided simultaneous RALP and LARP stimuli, we found that the largest response components were in the RALP and LARP directions (Figures 6C,D). Again, because of the mathematical nature of reference frames that are rotated with respect to one another, when we look at these same data in ML and AP coordinates, we found response components that reflect the projection of the predominant RALP and LARP responses onto the ML and AP directions (Figures 6A,B).

In Condition (iii), where we provided simultaneous stimuli in each of 4 planes (i.e., roll, pitch, RALP, and LARP)—each having 5 different stimulus frequencies, we found frequency response components at all 20 frequencies. Clear response components were found for: (a) roll, RALP, and LARP for each frequency of the roll stimuli, (b) pitch, RALP, and LARP for each frequency of the pitch stimuli, (c) roll, pitch, and RALP for each frequency of the RALP stimuli, and (d) roll, pitch, and LARP for each frequency of the LARP stimuli. In other words, clear response components were found at each stimulation frequency except that clear consistent response components were not found in the response direction that was spatially orthogonal to the stimulus direction.

The repeated measures MANOVA using real and imaginary components found that the ML and AP responses were not significantly different when the tilt stimuli were provided in roll/pitch coordinates vs. RALP/LARP coordinates [Pillai’s Trace, AP: p = 0.48, ML: p = 0.63]. Similarly, Hotelling’s T-square for further pairwise comparisons found no significant differences in the sensitivity of ML and AP spectral responses between roll and pitch tilt stimuli versus RALP and LARP tilt stimuli after the Bonferroni correction and the Benjamini-Hochberg (BH) procedure. This was true when the roll and pitch stimuli were provided in different trials (i.e., Condition (i)) than the RALP and LARP stimuli [i.e., Condition (ii)] and when the roll, pitch, RALP, and LARP stimuli were provided simultaneously in the same trial [i.e., Condition (iii)]. This suggests that we can compare ML and AP responses evoked by RALP and LARP stimuli to existing literature showing ML and AP responses evoked by roll and pitch stimuli. The central nervous system is known to integrate the signals from the three semicircular canals to estimate 3D angular velocity and 2D tilt (46, 47). While speculative, these multidimensional estimates that arise via sensorimotor integration may be used for postural control via ML and AP motor templates (34, 35).

Putting all of the above into context, this suggests that we can provide tilt stimuli in the RALP and LARP coordinates and:

For example, if a patient has a lesion in the right anterior semicircular canals, the patient will show a larger postural response to RALP tilt stimuli than to LARP tilt stimuli due to the limited contribution of the right anterior semicircular canals. We anticipate that 2D balance perturbations in roll/pitch coordinates are less likely to distinguish balance impairment due to asymmetric sensory deficits in the vertical semicircular canals (i.e., RALP and LARP) as all of the four vertical semicircular canals contribute to postural control at the same time when perturbations are provided in roll or pitch coordinates.

Furthermore, in the future, we intend to provide translation stimuli in the AP and ML directions while we simultaneously provide tilt stimuli in the RALP and LARP directions. Consistent with the approach shown herein, these four stimuli will all be orthogonal in time and spectrally separable. Nonetheless, providing the tilt stimuli in RALP/LARP directions and translation in the more traditional ML and AP directions adds spatial separation alongside the orthogonality in time and spectral separation mentioned earlier.

The plots of the sensitivity and phase of the CoP spectral responses (Figures 7, 8) are also qualitatively similar to earlier studies that provided pseudorandom pitch tilt stimuli with eyes closed (15) and roll tilt stimuli with eyes closed (20). Prior research generated support surface pitch tilts using a pseudorandom ternary sequence (PRTS) and quantified center of mass (CoM) AP body sway angles of 8 healthy subjects using a backboard (15). They found (1) that transfer function gain increased in the low-mid frequency range (0.0165–0.5 Hz) and decreased in the higher frequency range (0.5–2.48 Hz) and (2) that phase lag continuously increased with frequency (15). Another prior study estimated CoM ML body sway angles using a backboard and demonstrated that transfer function gain increased with frequency below 0.5 Hz and phase lag continuously increased with frequency in a bandwidth of 0.021–2.79 Hz in response to roll support surface PRTS tilts (20). When we estimate FRFs for the CoM by low-pass filtering CoP data (48) with a cut-off frequency of 0.47 Hz, our data (see Supplementary Figures S10–S12 for more details) demonstrate FRFs that mimic those from two earlier studies (15, 20).

We created four pseudorandom SoS trajectories that were designed to have frequency components that were close to each other (Table 1). Even though we were aware (see previous section) that frequency response functions varied with frequency (15), we hypothesized that the small frequency differences we designed would not yield significant response differences because the frequencies were near to one another (e.g., SoS₁₁ f₁ = 0.0604 Hz and SoS₁₇ f₁ = 0.0934 Hz) and because the response variability between participants would be too large to yield significant differences. However, the four-way repeated measures MANOVA showed a significant effect of the four pseudorandom SoS trajectories (i.e., SoS₁₁, SoS₁₃, SoS₁₅, and SoS₁₇) on the responses (p < 0.001). The Hotelling’s T-square for further pairwise comparisons also indicated that those four trajectories evoked significantly different responses (Table 5). This suggests to us we should not compare frequency response functions (FRFs) at different perturbation frequencies (e.g., f₁ of SoS₁₁ versus f₁ of SoS₁₃) as small changes in frequency matter for these FRFs. We were pleased by this unexpected finding because this (inadvertently) shows the power of the approach presented herein, where statistically significant differences in the postural response were found for nearby stimulation frequencies; this shows that response variability had to have been small enough to unveil the response differences evoked by small differences in the stimulus frequency. This would not have occurred if the methods did not yield clean spectral response components. The high precision in spectral CoP responses also implies complexity when comparing directly across studies unless selected perturbation frequencies are exactly identical. Modeling using transfer functions (15) will come into play to fit the experimental data to address this limitation.

Positive integers only were used to multiply the fundamental frequency to determine the frequency of all stimulus components. This means that complete cycles for each frequency are provided in each full period of stimulation. This makes the stimuli provided at each frequency orthogonal in time over the full steady-state period. Both theoretically and practically, this means that no stimulus frequency components interfere with any other stimulus frequency components. Theoretically, the same holds true for steady-state responses of any linear system, as is commonly assumed by many studies of the balance responses evoked by 1D balance perturbations (12, 20, 25–33).

We chose a smaller fundamental frequency (0.0055 Hz) than the one selected (0.044 Hz) in our previous study (38) to allow all perturbation frequencies to fit in our target bandwidth of 0.05–2.00 Hz. This bandwidth was selected because early literature showed that vestibular thresholds decreased with increasing frequency in the bandwidth (49). To define our frequency components, we multiplied the fundamental frequency of 0.0055 Hz by prime numbers with the single exception of 15; 15 was included, since its prime factors of 3 and 5 were not included as multipliers. This set of multipliers – like sets of prime number multipliers – reduces the chance of system nonlinearities yielding overlapping response frequency components.

We chose to begin with 11 as the lowest multiplier. This meant that a minimum of 11 cycles of stimuli were provided for each complete fundamental period of stimulation. This is above the minimum rule of 5 to 7 cycles often required for spectral analyses (15).

Choosing 11 as the lowest multiplier also allowed us to also include 13, 15, and 17 as the lowest integer multipliers, which leads to the fundamental frequency of the four trajectories used herein being separated only by a factor of 1.55. In other words, there was an increase of 55% from the lowest fundamental frequency of 0.060 Hz to the highest fundamental frequency of 0.093 Hz-meaning that these 4 stimuli all had similar—but separable—fundamental frequencies (see Figure 2; Table 1).

To accomplish an analogous goal, two pseudorandom ternary sequence (PRTS) stimuli with non-overlapping harmonics have been interleaved by doubling the fundamental frequency of the 2nd stimuli (e.g., a 100% increase) (22). Taking this approach for four stimuli—like those provided in our Condition (iii) would result in the fourth fundamental frequency (i.e., SoS₁₇ f₁) being 8 times greater than the first fundamental frequency (i.e., SoS₁₁ f₁) (i.e., a 700% increase). For 12 stimuli (e.g., 6 physical motion dimensions and 6 visual motion dimensions), which we may consider testing in the future, this approach of doubling the fundamental frequency for each motion dimension stimulated would mean that the highest fundamental frequency would be 2,048 times greater (i.e., 2^(12–1), which is more than 3 orders of magnitude greater) than the lowest fundamental frequency—almost certainly putting either the lowest or the highest fundamental frequency outside the physiologic range of interest.

We chose to perform our statistics using the real and imaginary Cartesian-coordinate response components instead of the more traditional polar coordinate response components of magnitude and phase. Standard coordinate transformations were used to convert back-and-forth from the standard Cartesian coordinates (i.e., real and imaginary components) and standard polar coordinates (i.e., magnitude and phase). The Cartesian coordinates offer the advantage that these two independent variables are each unbounded (i.e., range from minus ∞ to plus ∞). For comparison, magnitude is bounded at zero (i.e., is non-negative) and phase of the Fourier transform is bounded to fall in a range of −180 to 180°. As we discovered, these bounds alter statistical calculations even though the data points being compared are identical. While phase “unwrapping” can yield a greater phase range as the function shifts the angles to make the jump between consecutive angles less than 180° the unwrapping process is always subject to assumptions, which are avoided by simply using the real and imaginary components. Furthermore, the units on the real and imaginary variables are always the same, which means the weighting in all calculations—including, but not limited to, statistical calculations—is also naturally the same.

While the present study represents the potential of two-dimensional balance perturbations in canal coordinates (i.e., RALP and LARP), it does not demonstrate the contributions of the RALP and LARP semicircular canals to postural control. Future studies will need to show whether individuals with peripheral lesions in a specific vertical semicircular canal (i.e., impaired perception of RALP motion cues) show balance impairment in response to perturbations delivered in the same plane. That will demonstrate the potential of our pseudorandom balance perturbations provided in RALP/LARP coordinates to characterize how peripheral lesions impact postural control.

Our data sets did not allow us to determine the motor control strategies that led to the observed postural responses as we did not measure muscle activations using electromyography (EMG). Additional studies are required to determine whether the motor control system disaggregates the sensory signals provided in RALP/LARP coordinates into roll/pitch coordinates to sway in ML and AP directions or the motor control activates muscles in RALP/LARP coordinates when provided perturbations in RALP/LARP coordinates. We suspect that a mixture of different muscle synergies (34, 35, 50) (i.e., hip and ankle strategies (51–53)) are involved in responding to our two-dimensional balance perturbations delivered in RALP/LARP coordinates.

We identified some additional limitations of the study:

Finally, generally, we hypothesize that balance patient populations will show a larger postural response than healthy individuals. This would suggest that we should explore multidimensional balance assessments such as three-dimensional perturbations that add earth-vertical translations to the RALP and LARP tilt stimuli used for the present study. The earth-vertical perturbations might evoke a larger postural response in individuals with movement disorders than those with sensory deficits in the vertical semicircular canals as it would be hard for those with movement disorders to generate corrective torques using muscles to counteract gravitational torques to enhance their stability in response to the external perturbations.