We asked whether walking adaptation involves both forward model recalibration and stimulus-response mapping mechanisms learned in tandem (‘recalibration + mapping hypothesis’). We contrasted this with the alternative hypothesis that walking adaptation may involve only forward model recalibration mechanisms (‘recalibration only hypothesis’).

To test this, we devised the ‘Ramp Down’ paradigm depicted in Figure 1A (see Methods and Appendix 1). After adaptation, we gradually decreased the speed of the right belt every three strides. This enabled us to obtain reliable measurements of aftereffects across 21 predetermined speed configurations, spanning from the full split-belt perturbation to tied-belts.

We specifically measured aftereffects in step length asymmetry, the movement error that arises when the compensatory adjustments to step timing and position developed in adaptation (Δ motor output) fail to match the current treadmill speed difference (perturbation):(1)steplengthasymmetry=Δmotoroutput−perturbation=1stridelength∗Δsteplength\begin{document}$$\displaystyle {\rm step}\,{\rm length}\,{\rm asymmetry} = \Delta{{\rm motor}\, {\rm output}} - {\rm perturbation} = \frac{1}{\rm{stride\,length}} \ast \Delta \rm{step\,length}$$\end{document}(2)perturbation=meantimestridelength∗Δstepvelocity\begin{document}$$\displaystyle {\rm perturbation} = \frac{\rm{mean\,time}}{\rm{stride\,length}} \ast \Delta\rm{step\,velocity}$$\end{document}(3)Δmotoroutput=1stridelength∗Δstepposition−meanvelocitystridelength∗Δsteptime\begin{document}$$\displaystyle \Delta{\rm motor}\,{\rm output} = \frac{1}{\rm{stride\,length}} \ast \Delta{\rm step}\,{\rm position} - \frac{\rm{mean\,velocity}}{\rm{stride\,length}} * \Delta\rm{step\,time}$$\end{document}

where ‘Δ’ represents differences between right and left steps and ‘mean’ represents their average (see Methods and Box 1).

As depicted in Figure 1B, the recalibration-only hypothesis predicts that aftereffects will be present for all speed configurations in the Ramp Down. This is because the Δ motor output (blue) can only change gradually, through unlearning or forgetting of the forward model recalibration (Figure 1—figure supplement 1). This change is too slow to track the perturbation (red), which is ramped down to zero rapidly over ~80 seconds. Therefore, step length asymmetry (purple) becomes positive immediately in the Ramp Down.

In contrast, as depicted in Figure 1C, the recalibration + mapping hypothesis predicts that aftereffects will be present for some, but not all speed configurations in the Ramp Down. Specifically, there would be no aftereffects (~zero step length asymmetry) in the first portion of the Ramp Down, because the Δ motor output (blue) changes rapidly and matches the perturbation (red). This reflects the flexibility of the stimulus-response mapping (Figure 1—figure supplement 1). Aftereffects would emerge in the second portion of the task (positive step length asymmetry), reflecting forward model recalibration.

We include forgetting effects for the recalibration-only hypothesis to account for previous findings from Leech et al., 2018a; Box 2. The inset of Figure 1 (right column) illustrates how both hypotheses may account for near-zero step length asymmetry at the end of the speed match task. The Δ motor output decreases to match the perturbation either because the mapping mechanism flexibly scales down (recalibration + mapping hypothesis) or, alternatively, because of forgetting/unlearning (recalibration only hypothesis).

Figure 2A shows the paradigm used for Experiment 1, and the time course of step length asymmetry (group mean ± SE). The Ramp Down task corresponds to the manipulation illustrated in Figure 1 and is the focus of our analysis for Experiment 1. A similar ramp task was performed prior to adaptation as a control to see how participants responded to a gradually ramped perturbation at baseline (see Methods, ‘Ramp tasks’). Participants performed perceptual tests during the ramp tasks, which will be discussed in a later section (see ‘Perceptual test and results’). Patterns of step length asymmetry during other portions of the paradigm were consistent with a large body of previous work (e.g. Leech et al., 2018a; Leech et al., 2018b; Reisman et al., 2005; Roemmich et al., 2016; Rossi et al., 2019; Vazquez et al., 2015) and will not be discussed further.

Step length asymmetry during the baseline ramp and post-adaptation Ramp Down tasks are displayed in Figure 2B. For each speed configuration in the Ramp Down task, we statistically compared step length asymmetry to zero. We evaluated the emergence of aftereffects by examining when step length asymmetry became significantly positive. We found that step length asymmetry was not statistically different from zero during the first half of the Ramp Down task, for right belt speeds ranging from 1 m/s to 0.5 m/s faster than the left speed (these speed configurations are highlighted in green in Figure 2B right; all CI_LB [confidence intervals lower bounds] ≤ –0.001 and CI_UB [upper bounds] ≥ 0.009, Figure 2—figure supplement 1 and Supplementary file 1 - table 1). This result indicates that aftereffects do not emerge immediately in the Ramp Down task, a finding at odds with the recalibration-only prediction (Figure 2B ‘predictions’ inset, top). Step length asymmetry was instead significantly positive for the second half of the task, for right belt speeds ranging from 0.45 m/s to 0 m/s faster than the left speed (all CI_LB > 0.01). These results indicate that aftereffects emerge at a mid-point during the Ramp Down task and align with behavioral predictions from the recalibration + mapping hypothesis (Figure 2B ‘predictions’ inset, bottom).

As a control, we repeated the same analysis for the baseline ramp task (Figure 2B, left). We found that there was only one speed configuration for which step length asymmetry was not statistically different from zero (right speed 0.05 m/s slower than left; SL asym. = 0.006 [-0.017, 0.028], mean [CI]). For all other configurations, step length asymmetry was significantly positive (CI_LB ≥ 0.048 for right speed 0.1–0.15 m/s slower than left) or negative (CI_UB ≤ –0.019 for right speed 0–0.15 m/s faster than left). This is in stark contrast to the wide range of speed configurations with near-zero step length asymmetry observed in the Ramp Down period. Hence, the ability to walk symmetrically at different speed configurations is not innate but dependent on the adaptation process. We finally tested whether the pattern of motor variability during adaptation aligns with predictions for learning new stimulus-response maps. In contrast to recalibration, mapping mechanisms are predicted to be highly variable and erratic during early learning and stabilize as learning progresses (Tsay et al., 2024). Consistent with these predictions, the step length asymmetry residual variance (around a double exponential fit) decreased significantly between the start and end of adaptation (residual variance at start minus end of adaptation = 0.005 [0.004, 0.007], mean [CI]; Figure 2—figure supplement 2). These control analyses corroborate the hypothesis that the ‘no aftereffects’ region of the Ramp Down reflects the operation of a mapping mechanism.

To further test the competing hypotheses, we examined the perturbation and Δ motor output during the Ramp Down (Figure 3A). As the belt speed difference decreased, so did the perturbation component (red). In the first half of the task, the Δ motor output (blue) appeared to match the perturbation – consistent with the lack of step length asymmetry aftereffects (green shaded portion, matching Figure 2B). Additionally, in the second half of the task, the Δ motor output appeared larger than the perturbation – consistent with the positive step length asymmetry aftereffects observed in Figure 2B.

We formally contrasted predictions made by the competing hypotheses by developing mathematical models of the Δ motor output as a function of perturbation (Figure 3B–C). In the simplest framework, the Δ motor output behavior for the recalibration and mapping mechanisms can be formalized as follows:(4)ForwardModelRecalibration:u(k)=r\begin{document}$$\displaystyle {\rm Forward\,Model\,Recalibration}:u\left (k\right)=r$$\end{document}(5)Stimulus-ResponseMapping:u(k)= p(k)\begin{document}$$\displaystyle {\rm Stimulus{\text -}Response\,Mapping}:u\left (k\right)=\ p\left (k\right)$$\end{document}(6)Recalibration+Mapping:u(k)= {p(k)   ,    if p(k)≥r r         ,    otherwise  \begin{document}$$\displaystyle {\rm Recalibration + Mapping}\, :\, \textit{u}\left (k\right)=\mathrm{\ }\begin{cases} p\left (k\right)\ \ \ ,\ \ \ \ if\ p\left (k\right)\geq r\\\ r\ \ \ \ \ \ \ \ \ ,\ \ \ \ {\rm otherwise}\ \ \end{cases} $$\end{document}

where u is the modelled Δ motor output, k\begin{document}$k$\end{document} is the stride number, p\begin{document}$p$\end{document} is the perturbation, and r\begin{document}$r$\end{document} is a free parameter representing the portion of the Δ motor output related to recalibration. We used Equation 6 as our model for the recalibration + mapping hypothesis. This describes the scenario where participants modulate the Δ motor output to match the perturbation (u(k)= p(k))\begin{document}$(u(k)=\ p(k))$\end{document} for the first portion of the Ramp Down task. The Δ motor output remains constant during the second portion of the task (u(k)= r)\begin{document}$(u(k)=\ r)$\end{document} because participants are unable to reduce the Δ motor output to values smaller than ‘r\begin{document}$r$\end{document}’ (the amount achieved by recalibration). Parameters were estimated by fitting the model to individual participants’ Δ motor output data from the Ramp Down (see Methods).

The fitted Δ motor output is displayed in Figure 3B (mean ± SE of fits across participants; individual fits are shown in Figure 3—figure supplement 1). As expected, the recalibration + mapping fit captured the matching-then-divergent behavior of Δ motor output in response to the changing perturbation.

We used the dual state model of motor adaptation from Smith et al., 2006 as our model for the recalibration-only hypothesis (see Methods). The model has four parameters and can account for potential forgetting or unlearning of the Δ motor output that may occur during the Ramp Down. It can also account for the possibility that adaptation involves two recalibration mechanisms – that is, two distinct ‘fast’ and ‘slow’ mechanisms that both learn via a process of forward model recalibration, and both contribute to aftereffects, but that learn and forget at different rates. We chose this model because it is a well-established model that is widely used to capture the Δ motor output time course for traditional motor adaptation paradigms (i.e. those consisting of adaptation and post-adaptation phases). Yet, the performance of this model for a manipulation like the Ramp Down task performed here has not yet been tested. As such, if the recalibration + mapping model fit the Ramp Down data better than the dual state model, this would provide robust evidence for the presence of a stimulus-response mapping mechanism.

We show the Δ motor output fit by the recalibration only (dual state) model in Figure 3C (left panel, group mean ± SE; individual fits are shown in Figure 3—figure supplement 2). In contrast to the recalibration + mapping model, the dual state model was not able to capture the matching-then-divergent behavior of Δ motor output. The BIC statistic confirmed that the recalibration + mapping model fitted the data significantly better than the dual state (BIC difference = 8.422 [3.386, 13.778], mean [CI]).

We considered two additional prominent models for motor adaptation: optimal feedback control (Izawa and Shadmehr, 2011; Shadmehr and Krakauer, 2008; Todorov, 2004) and memory of errors (Herzfeld et al., 2014). Similar to the dual state, these models could not capture the matching-then-divergent behavior of Δ motor output and fitted the Ramp Down data significantly worse than the recalibration + mapping (Figure 3C; memory of errors minus recalibration + mapping BIC difference = 11.610 [6.068, 17.749], optimal feedback control minus recalibration + mapping BIC difference = 19.272 [14.216, 24.563], mean [CI]). In sum, the modeling analysis of the Δ motor output further supports the recalibration + mapping hypothesis.

The second goal of Experiment 1 was to evaluate the hypothesis that ‘perceptual realignment’ (a phenomenon leading to altered perception following adaptation) results from the operation of the same forward model recalibration mechanism involved in adaptation of the Δ motor output (Rossi et al., 2021a).

Previous work shows that perception realigns following gait adaptation: after adapting to a perturbation where the right treadmill belt is faster than the left, return to tied belts results in perception of the opposite asymmetry (i.e. right speed feels slower than left; Jensen et al., 1998; Vazquez et al., 2015). We therefore expected that people would perceive the belt speeds as equal at some point in the Ramp Down and eventually perceive the opposite asymmetry on tied belts. We measured this by asking participants to press a keyboard button at two separate occasions during the Ramp Down task: (1) when the belts first felt equal and (2) when they no longer felt equal (see Methods).

Figure 4A depicts the button presses for the Ramp Down perceptual test (top panel, group mean) overlayed onto the Ramp Down motor data (recalibration + mapping fit). Note that the task captures a range of belt speed configurations that participants perceive as ‘equal speeds’, with button presses (1) and (2) corresponding to the upper and lower bounds of this range. This is expected because perception is known to be noisy and may not be sensitive enough to discriminate between belt speed configurations that are too similar.

We quantified perceptual realignment using the established measure of point of subjective equality (PSE), defined as the belt speed difference perceived as ‘equal speeds’. A PSE of zero would indicate no perceptual realignment (accurate perception), and a PSE of 1 m/s (i.e. a magnitude equivalent to the difference between the belt speeds during adaptation) would indicate complete perceptual realignment such that the belt speeds feel equal during the adaptation phase.

We measured belt speed difference at the time of each button press. We computed PSE as the range of belt speed difference values between these two measurements:(7)PSEupperbound=[rightbeltspeed−leftbeltspeed]timeofbuttonpress1\begin{document}$$\displaystyle PSE_{\rm upper\, bound}= [right\,belt\,speed-left\,belt\,speed]_{\rm time\,of\,button\,press\,1}$$\end{document}(8)PSElowerbound=[rightbeltspeed−leftbeltspeed]timeofbuttonpress2\begin{document}$$\displaystyle PSE_{\rm lower\,bound}= [right\,belt\,speed-left\,belt\,speed]_{\rm time\,of\,button\,press\,2}$$\end{document}

We found that PSEupper bound\begin{document}$PSE_{\text{upper bound}}$\end{document} was 0.64 ± 0.03 m/s and PSElower bound\begin{document}$PSE_{\text{lower bound}}$\end{document} was 0.39 ± 0.05 m/s (Figure 4A, bottom panel, group mean ± SE). This was consistent with previous work (Leech et al., 2018a). This perceptual realignment was not present during baseline testing, as illustrated in Figure 4—figure supplement 1.

We aimed to evaluate the hypothesis that perceptual realignment arises from the forward model mechanism of motor adaptation and is unaffected by the stimulus-response mapping mechanism. This hypothesis predicts that the extent of perceptual realignment should be: (1) approximately equal to the extent of motor adaptation achieved by recalibration, and (2) less than the total extent of motor adaptation (which also includes mapping). We quantified (1) motor adaptation by recalibration as the fitted parameter ‘r’ from the recalibration + mapping model, and (2) total motor adaptation as the Δ motor output at adaptation plateau (Figure 4A, ‘r’ and ‘u_plateau’ bars in right panel, group mean ± SE). We then expressed these motor measures and perceptual realignment as ‘percent compensation for the perturbation’ (i.e. normalized to the perturbation magnitude in the respective units) so that they could be compared:(9)compensationperceptual=PSE1m/s\begin{document}$$\displaystyle compensation_{\rm perceptual} =\frac{PSE}{\rm 1{m}/{s}} $$\end{document}(10)compensationmotorrecalibration=rpplateau\begin{document}$$\displaystyle compensation_{\rm motor\,recalibration} =\frac{r}{{p}_{\rm plateau}} $$\end{document}(11)compensationmotortotal=uplateaupplateau\begin{document}$$\displaystyle compensation_{\rm motor\,total} =\frac{{u}_{\rm plateau}}{{p}_{\rm plateau}} $$\end{document}

where r\begin{document}$r$\end{document} is the fitted parameter from the recalibration + mapping model, pplateau\begin{document}$p_{\text{plateau}}$\end{document} is the mean perturbation over the last 30 strides of adaptation, and uplateau\begin{document}$u_{\text{plateau}}$\end{document} is the mean Δ motor output over the last 30 strides of adaptation (for Equation 9, note that 1 m/s is the belt speed difference in adaptation).

We show group-level compensation measures in Figure 4B. We found that compensationmotor recalibration\begin{document}$compensation_{\text{motor recalibration}}$\end{document} (56 ± 4%, group mean ± SE) fell within the compensationperceptual\begin{document}$compensation_{\text{perceptual}}$\end{document} range (39 ± 5% to 64 ± 3%): it was significantly smaller than the upper bound (difference = –8 [-14, –2]%) and significantly larger than the lower bound (difference = 18 [8, 28]%, mean [CI]). This supports our first prediction that perceptual realignment is comparable to the extent of motor adaptation achieved by recalibration. Furthermore, compensationperceptual\begin{document}$compensation_{\text{perceptual}}$\end{document} was significantly smaller than compensationmotor total\begin{document}$compensation_{\text{motor total}}$\end{document} (95 ± 2%, mean ± SE; difference = 31 [25, 38]% or 57 [48, 66]%, upper or lower perceptual bounds, mean [CI]). This supports our second prediction that perceptual realignment is less than the total extent of motor adaptation.

We show individual compensationmotor recalibration\begin{document}$compensation_{\text{motor recalibration}}$\end{document} and compensationperceptual\begin{document}$compensation_{\text{perceptual}}$\end{document} measures for each participant in Figure 4C–D. We evaluated Pearson’s correlation coefficients between these measures to test whether there is a direct relationship between perceptual realignment and the motor adaptation achieved by recalibration. We found that compensationmotor recalibration\begin{document}$compensation_{\text{motor recalibration}}$\end{document} was significantly correlated with the upper bound of compensationperceptual\begin{document}$compensation_{\text{perceptual}}$\end{document} – the value computed using the ‘speeds feel equal’ button press (r=0.64, p=0.002). This supports the hypothesized relationship between motor recalibration and changes to leg speed perception. Instead, compensationmotor recalibration\begin{document}$compensation_{\text{motor recalibration}}$\end{document} was not correlated with the lower bound of compensationperceptual\begin{document}$compensation_{\text{perceptual}}$\end{document} (r=0.30, p=0.195) – this value is computed using the ‘right feels slower than left’ button press, suggesting the true PSE may lay closer to the first button press. Together, our results support the hypothesis that perceptual realignment can be used as a proxy measure for the extent of motor adaptation achieved by forward model recalibration.

We asked whether the Ramp Down results could be explained by two recently developed frameworks designed to capture the relationship between perceptual and motor changes with reaching adaptation: the proprioceptive re-alignment model (PReMo; Tsay et al., 2022; Tsay et al., 2021) and the Perceptual Error Adaptation model (PEA; Zhang et al., 2024). To apply these models to walking adaptation, we translated reaching to walking adaptation variables using the conceptual equivalence outlined by Tsay et al., which extends the model to different perturbation types in reaching adaptation (from visual-proprioceptive to force-field perturbations – which are mechanical like split-belt; Appendix 2). Figure 5A–B shows the fitted Δ motor output and predicted perception of speed difference for PReMo and PEA. The models could not capture the matching-then-divergent behavior of Δ motor output, performing significantly worse than the recalibration + mapping model (PReMo minus recalibration + mapping BIC difference = 24.591 [16.483, 32.037], PEA minus recalibration + mapping BIC difference = 6.834 [1.779, 12.130], mean [CI]). Furthermore, they could not capture the perceptual realignment and instead predicted that the right leg would feel faster than the left throughout the entire Ramp Down.

While the mathematical formulations of the models failed to account for the observed patterns of motor recalibration and perceptual realignment, we suspected that their underlying principles remain valid for walking adaptation. We therefore developed a new model – the perceptuomotor recalibration + mapping model (PM-ReMap) – that preserved the main structure of PReMo while addressing its limitations, as identified through iterative simulations (Appendix 2). The final model retains PReMo’s core principle that both motor adaptation and perceptual realignment are driven by the altered perception of the Δ motor output, reflecting the Bayesian integration of predicted and actual Δ motor output further shifted by recalibration (see Methods). The PM-ReMap model also maintains the key operational properties of the mechanisms described by the recalibration + mapping model, but includes additional variables for forgetting or unlearning of recalibration during the Ramp Down.

As shown in Figure 5C, the PM-ReMap model captured the Δ motor output in the Ramp Down with performance comparable to that of the recalibration + mapping model (BIC difference = 2.381 [-0.739, 5.147], mean [CI]). It also captured perceptual realignment, predicting that some intermediate belt speed difference in the Ramp Down is perceived as ‘equal speeds’ (PSE^\begin{document}$\widehat{PSE}$\end{document}, Figure 5C). We compared this PSE^\begin{document}$\widehat{PSE}$\end{document} estimate to the actual PSE measured by the perceptual task, expressed as compensationperceptual\begin{document}$compensation_{\text{perceptual}}$\end{document} in normalized units as before. At group level, PSE^\begin{document}$\widehat{PSE} $\end{document} was comparable to the upper bound of compensationperceptual\begin{document}$compensation_{\text{perceptual}}$\end{document} (difference = –7 [–15, 1]%, mean [CI]), but significantly larger than the lower bound (difference = 19 [8, 31]%, mean [CI]). Furthermore, we found a significant correlation between individual participants’ PSE^\begin{document}$\widehat{PSE}$\end{document} and their upper bound of compensationperceptual\begin{document}$compensation_{\text{perceptual}}$\end{document} (r=0.63, p=0.003), but not their lower bound (r=0.30, p=0.203). Both sets of results are consistent with those observed for the recalibration + mapping model. We also confirmed that the significant correlation was driven by the model parameter ηp\begin{document}$\eta _{p}$\end{document}, which captures the extent of perceptual realignment at adaptation plateau (correlation of ηp\begin{document}$\eta _{p}$\end{document} with compensationperceptual\begin{document}$compensation_{\text{perceptual}}$\end{document} upper bound: r=0.57, p=0.009, lower bound: r=0.37, p=0.110). In sum, the PM-ReMap model accurately captures the observed patterns of motor recalibration and perceptual realignment, predicting that both reflect the same proportion of the perturbation size. As such, it extends the recalibration + mapping model by incorporating the ability to account for forgetting – typical of state space models – while still effectively capturing both recalibration and mapping mechanisms. However, the performance of the PM-ReMap model does not exceed that of the simpler recalibration + mapping model, suggesting that forgetting and unlearning do not have a substantial impact on the Ramp Down.

We performed six control experiments and reanalyzed previously-published data (Leech et al., 2018a) to replicate the findings of Experiment 1 across different paradigm conditions (Figure 6). This provided additional support for the recalibration + mapping hypothesis.

First, we checked that the motor and perceptual behaviors observed in the Ramp Down could be replicated using a different method of assessment. A ‘speed match’ task was given where participants use a joystick to increase the speed of the right belt (initially stationary) until they feel it matches the left belt (Figure 6B, ‘Ascend’; Leech et al., 2018a; Rossi et al., 2019; Statton et al., 2018; Vazquez et al., 2015). We tested this in a group that otherwise adapted with the same conditions as Experiment 1 (same duration, speeds, and schedule, ‘Medium Ascend’ in Figures 6–7). The ‘Ascend’ speed match approximates the second half of the Ramp Down (the portion when the right leg feels equal or slower than the left). Consistent with this, we observed step length asymmetry aftereffects early in the speed match task (initial SL asym. = 0.433 [0.271, 0.612], mean [CI]; Figure 7B, orange line and error bar to the right). Asymmetry decreased and was eventually near zero when the belt speeds felt equal (final SL asym. = 0.002 [-0.040, 0.045]; Figure 7B, orange error bar in inset).

We also used a ‘descend’ speed match task where participants decreased the speed of the right belt (initially fast as in adaptation) until they felt it matched the left (Figure 6B, ‘Descend’). This approximated the first half of the Ramp Down. We tested this in the ‘Medium Descend’ group that otherwise adapted as in Experiment 1. As expected, step length asymmetry was close to zero for the entire task (initial: –0.017 [-0.069, 0.039], final: 0.043 [-0.0004, 0.090], mean [CI]; Figure 7B, dark blue). For both speed match groups, the PSE was smaller than motor adaptation (compensationmotor total\begin{document}$compensation_{\text{motor total}}$\end{document} – compensationperceptual\begin{document}$compensation_{\text{perceptual}}$\end{document} = 48 [39, 57]% or 56 [42, 70]% for Medium Ascend or Descend, mean [CI]; Figure 7A). PSE was also comparable to that of Experiment 1 (compensationperceptual\begin{document}$compensation_{\text{perceptual}}$\end{document} difference between Ramp Down lower bound and Medium Ascend or Descend = –3 [–16, 10]% or 4 [-12, 18]%, mean [CI]).

Second, we replicated the findings of Experiment 1 in additional speed match experiments that varied in adaptation duration (3, 15, or 30 min), perturbation magnitude (1 m/s or 0.4 m/s speed difference), and schedule (abrupt or gradual). Across all conditions, compensationperceptual\begin{document}$compensation_{\text{perceptual}}$\end{document} was significantly smaller than compensationmotor total\begin{document}$compensation_{\text{motor total}}$\end{document} (Figure 7A; statistical results in Supplementary file 1, table 2). Moreover, all conditions exhibited the expected pattern of step length asymmetry in the first post-adaptation speed match task (Figure 7B; Supplementary file 1, table 3). Specifically, step length asymmetry was: (1) ~0 at the end of the task (when belt speeds felt equal) for all conditions, (2) significantly positive at the start of the task for Ascend conditions (Short, Medium, and Long Ascend), and (3) ~0 at the start of the task for Medium Descend and Small Abrupt Descend conditions. While step length asymmetry was initially negative for Short Descend and Small Gradual Descend conditions, this direction aligns with that expected from incomplete adaptation rather than aftereffects, thus supporting the recalibration + mapping hypothesis.

In sum, these control experiments corroborate the recalibration + mapping hypothesis by confirming that, across various conditions of perturbation duration, magnitude, and schedule, the step length asymmetry and PSE patterns are consistent with the presence of both mechanisms.

We examined how the relative contributions of recalibration and mapping evolve over time and are influenced by perturbation magnitude and schedule (Figure 8A). We compared Short, Medium, and Long Ascend (adapting for 3, 15, or 30 min) and observed progression from incomplete to complete adaptation: compensationmotor total\begin{document}$compensation_{\text{motor total}}$\end{document} significantly increased with each successive duration, reaching full adaptation only in Long Ascend (Supplementary file 1, table 4-5). Despite this progression, the relative contributions of recalibration and mapping did not change over time (the ratio compensationperceptual\begin{document}$compensation_{\text{perceptual}}$\end{document} / compensationmotor total\begin{document}$compensation_{\text{motor total}}$\end{document} did not differ between conditions; Figure 8A left and Supplementary file 1, table 6). We next compared Medium Descend and Small Abrupt (1 m/s or 0.4 m/s perturbation) and found that recalibration contributed significantly more for the smaller perturbation (larger compensationperceptual\begin{document}$compensation_{\text{perceptual}}$\end{document} / compensationmotor total\begin{document}$compensation_{\text{motor total}}$\end{document} in Small Abrupt than Medium Descend, Figure 8A middle and Supplementary file 1, table 6). Comparing Small Abrupt and Gradual (differing in adaptation schedule), recalibration appeared larger for Small Gradual, but this effect was not significant (Figure 8A right and Supplementary file 1, table 6). Finally, we confirmed no differences between the ‘ascend’ and ‘descend’ speed match tasks (Supplementary file 1, table 6).

We finally examined how motor aftereffects and perceptual realignment changed across the six iterations of the speed match task post-adaptation (Figure 8B; tasks are interleaved with varying intervals of tied-belt walking). We quantified motor aftereffects as the step length asymmetry expressed at the tied-belt configuration in each task (interpolated, see Methods), and perceptual realignment as the final speed difference, normalized to the respective adaptation perturbation like before. We focus on ‘Ascending’ groups because no tied-belt configuration is expected in ‘Descending’ tasks. At the end of adaptation, motor aftereffects were comparable to perceptual realignment in all groups (first task in Figure 8B; Supplementary file 1, table 7). However, motor aftereffects decayed faster with repeated iterations of the task, becoming significantly smaller than perceptual realignment after 1 or 2 min (significant difference in the second task at 1 min for Short and Long Ascend, and in the third task at 2 min for Medium Ascend, shaded in Figure 8B; Supplementary file 1, table 7). Motor aftereffects remained significantly smaller than perceptual realignment for the remainder of the post-adaptation phase in Medium and Long Ascend; however, the difference dissipated by 8 min for Short Ascend (Supplementary file 1, table 7).

In sum, we show that recalibration and mapping learn at similar rates during adaptation, resulting in comparable time courses, but their relative contributions are modulated by error size. Moreover, motor aftereffects and perceptual realignment decay differently with repeated post-adaptation tasks, potentially suggesting a differential effect on the mechanisms.

For all ramp tasks of Experiments 1 and 2, the speed of the right belt was changed by 0.05 m/s every 3 strides (defined below) during right leg swing. The baseline ramp of both experiments consisted of 7 increasing right speeds from 0.35 m/s to 0.65 m/s. The post-adaptation Ramp Down of Experiment 1 consisted of 21 decreasing right speeds from 1.5 m/s to 0.5 m/s. The post-adaptation Ramp Up & Down of Experiment 2 consisted of 41 total right speeds: 11 increasing speeds from 1.5 m/s to 2 m/s, followed by 30 decreasing speeds from 1.95 m/s to 0.5 m/s. The left belt speed was constant at 0.5 m/s for all ramp tasks.

In Experiment 1, a keyboard was placed on the treadmill handrail. In the post-adaptation Ramp Down task, participants were asked to press a button the first time they perceived the right belt to be (1) as fast as the left, and (2) faster than the left. In the baseline ramp, they pressed to report the right belt feeling (1) as fast as the left, and (2) slower than the left. This was explained prior to the experiment, and the above prompts were displayed during the tasks on the TV screen.

As described in Appendix 1, the Ramp Down task was specifically designed to measure the pattern of aftereffects in a way that ensured reliable and robust measurements with sufficient resolution across speeds and that minimized washout to prevent confounding the results. To balance time constraints with a measurement resolution adequate for capturing perceptual realignment, we used 0.05 m/s speed decrements, matching the perceptual sensitivity estimated from our re-analysis of the baseline data from Leech et al., 2018a. To obtain robust motor aftereffect measurements, we collected three strides at each speed condition, as averaging over three strides represents the minimum standard for consistent and reliable aftereffect estimates in split-belt adaptation (typically used in catch trials; Leech et al., 2018a; Rossi et al., 2019; Vazquez et al., 2015). To minimize unwanted washout by forgetting and/or unlearning, we did not pause the treadmill between adaptation and the post-adaptation ramp tasks and ensured the Ramp Down was relatively quick, lasting approximately 80 s on average. Of note, the Ramp Down design ensures that even in cases of partial forgetting, the emergence pattern of aftereffects remains consistent with the underlying hypotheses.

At the end of Experiment 2, participants answered this question on a computer: Did you deliberately change how you walked to account for how fast the belts were moving? If so, describe how. Note: deliberately means that you thought about and decided to move that way. The question refers to the entire central ~20 min walking block. To ensure participants remembered what phase they were asked about, before adaptation, we told participants that the following block will be called ‘central ~20 min walking block’.

The protocol and tasks for control experiments, shown in Figure 6, were based on previous work (Leech et al., 2018a; Rossi et al., 2019; Statton et al., 2018; Vazquez et al., 2015). The left belt was fixed at 0.4 m/s for Small Abrupt or Gradual and at 0.5 m/s otherwise. The right belt speed equaled the left during baseline, post-adaptation, and catch trial (administered two-thirds into adaptation to all but Small Gradual). Small Gradual adapted for 15 min, with the right belt at 0.4 m/s for the first 30 s, 0.8 m/s for the last 30 s, and increasing linearly in between. Small Abrupt adapted for 15 min with the right belt at 0.8 m/s. Short Ascend and Descend, Medium Ascend and Descend, and Long Ascend adapted for 3, 15, or 30 min with the right belt at 1.5 m/s.

Speed match tasks lasted 30 s with a time bar displayed on the TV screen. Participants controlled the right speed with a keypad with four buttons: (1) large increment varying between 50, 55, and 65 mm/s as in Vazquez et al., 2015, (2) large –50 mm/s decrement, (3) small 5 mm/s increment, (4) small –5 mm/s decrement. ‘Ascend’ tasks, beginning with the right belt stationary, were used in baseline for all experiments, and post-adaptation for Short Ascend, Medium Ascend, and Long Ascend (Figure 6). ‘Descend’ tasks, beginning at the adaptation speed, were used for post-adaptation of the other experiments.

We defined a stride as the period between two consecutive left heel strikes (LHS1toLHS2)\begin{document}$(LHS_{1}\,to\,LHS_{2})$\end{document}. We computed motor measures for each stride using Equations 1–3 (see Results), as in previous work (Finley et al., 2015; Sombric et al., 2017). The equation terms are defined as follows:Δsteplength=Rsteplength−Lsteplength\begin{document}$$\displaystyle \Delta \rm step\,length=R\,step\,length-L\,step\,length$$\end{document}stridelength=Rsteplength+Lsteplength\begin{document}$$\displaystyle \rm stride\,length=R\,step\,length + L\,step\,length$$\end{document}Δstep position=(R step positionRHS−L step positionLHS1)−(L step positionLHS2−R step positionRHS)\begin{document}$$\displaystyle \Delta \rm{step\ position} = \left( \rm{R\ step\ position}_{\rm{RHS}} - \rm{L\ step\ position}_{\rm{LHS1}} \right) - \left( \rm{L\ step\ position}_{\rm{LHS2}} - \rm{R\ step\ position}_{\rm{RHS}} \right) $$\end{document}Δsteptime=Rsteptime−Lsteptime\begin{document}$$\displaystyle \Delta \rm step\,time = R\,step\,time-L\,step\,time$$\end{document}mean time=Rsteptime+Lsteptime2\begin{document}$$\displaystyle \rm mean\ time =\frac{R\,step\,time + L\,step\,time}{2} $$\end{document}Δstepvelocity=Rstepvelocity−Lstepvelocity\begin{document}$$\displaystyle \Delta \rm step\,velocity = R\,step\,velocity-L\,step\,velocity$$\end{document}meanvelocity=Rstepvelocity+Lstepvelocity2\begin{document}$$\displaystyle \rm mean\,velocity=\frac{R\,step\,velocity + L\,step\,velocity}{2} $$\end{document}

Right (R)\begin{document}$(R)$\end{document} and left (L)\begin{document}$(L)$\end{document} step lengths are the anterior-posterior distance between the ankle markers of the two legs at right heel strike (RHS)\begin{document}$(RHS)$\end{document} and left heel strike (LHS2)\begin{document}$(LHS_{2})$\end{document} respectively. Step position is the anterior-posterior position of the ankle marker, relative to the average of the two hip markers, at heel strike of the same leg. Left and right step times are the times from LHS1\begin{document}$LHS_{1}$\end{document} to RHS\begin{document}$RHS$\end{document}, and from RHS\begin{document}$RHS$\end{document} to LHS2\begin{document}$LHS_{2}$\end{document}, respectively. Step velocity is the average anterior-posterior velocity of the ankle marker relative to the average of the two hip markers, over the duration of the step (left: LHS1\begin{document}$LHS_{1}$\end{document} to RHS\begin{document}$RHS$\end{document}, right: RHS\begin{document}$RHS$\end{document} to LHS2\begin{document}$LHS_{2}$\end{document}).

In Experiment 2, we also computed the measure of ‘strides to plateau’ using individual step length asymmetry data from the adaptation phase only. We first smoothed the data with a five-point moving average filter. We then calculated the number of strides until five consecutive strides fell within the ‘plateau range’, defined as the mean ± 1 SD of the last 30 strides of adaptation (Malone and Bastian, 2010; Rossi et al., 2019).

In both experiments, we assessed how adaptation influences variability by evaluating within-participant residual variance in step length asymmetry around a double exponential model fit during adaptation. The double exponential model equation is A1exp⁡-strideτ1+A2exp⁡-strideτ2\begin{document}$A_{1}\exp \left \{-\frac{stride}{\tau _{1}}\right \}+A_{2}\exp \left \{-\frac{stride}{\tau _{2}}\right \}$\end{document}, where stride is the stride number in adaptation, and A1\begin{document}$A_{1}$\end{document}, τ1\begin{document}$\tau _{1}$\end{document}, A2\begin{document}$A_{2}$\end{document}, τ2\begin{document}$\tau _{2}$\end{document}, are model parameters. We fitted this model to individual participants’ step length asymmetry data in adaptation, using the MATLAB lsqcurvefit function. Initial parameter values were set to [–0.177, 322.072, –0.216, 29.662] for Experiment 1 and [–0.177, 418.725, –0.203, 44.307] for Experiment 2, derived from fitting the model to group mean data (with initial parameters [−0.2, 300, –0.2, 30]). We computed residuals as the difference between individual step length asymmetry data and the corresponding double exponential model fit. We finally computed variability as the variance of these residuals over (1) the initial 30 strides of adaptation and (2) the final 30 strides of adaptation.

In the control experiments, we interpolated step length asymmetry as a function of speed in the task. We used locally weighted linear regression (MATLAB ‘smooth’, method = ‘lowess’, span = 20), with query speeds ranging from minimum to maximum speed at which a stride was taken (average across participants) in steps of 5 mm/s. The speed of a stride refers to the average speed for the duration of that stride; similarly, when multiple strides were taken at the same speed, the average step length asymmetry over those strides was used for our analysis.

We tested to see if there were separate clusters of participants in our dataset. For Experiment 1, the measure used for clustering was the number of strides in the Ramp Down with step length asymmetry above the baseline CI. For Experiment 2, it was the number of strides in the first portion of the Ramp Up & Down (first 60 strides, teal in Figure 9) with step length asymmetry below the baseline CI. For both experiments, ‘baseline CI’ was computed separately for each participant as the 95% CI of the mean of step length asymmetry data in the second baseline tied-belt block (after the baseline ramp task; see Statistical Analysis).

We used the MATLAB ‘dbscan’ density-based clustering algorithm (Ester et al., 1996) and automated the choice of parameters adapting previously published procedures (Naik Gaonkar and Sawant, 2013; Rahmah and Sitanggang, 2016) (Appendix 4). The resulting algorithm does not require any user input; it automatically assigns participants to clusters (whose number is not predefined), or labels them as outliers, based solely on each participant’s measure of interest. We also performed a silhouette analysis to assess the validity of the clusters identified in Experiment 2, using the MATLAB ‘silhouette’ function (Dalmaijer et al., 2022; Kaufman and Rousseeuw, 1990; Rousseeuw, 1987).

We developed and fitted our own recalibration + mapping model, presented in the Results section in Equation 6. The model has one parameter r\begin{document}$r$\end{document} representing the magnitude of forward model recalibration. We fitted r\begin{document}$r$\end{document} to individual participants’ Ramp Down data using initial parameter value =12*pplateau\begin{document}$=\frac{1}{2}*p_{\text{plateau}}$\end{document}, lower bound = 0, and upper bound =pplateau\begin{document}$=p_{\text{plateau}}$\end{document}, respectively corresponding to half, no, or total compensation of the adaptation perturbation by recalibration (pplateau\begin{document}$p_{\text{plateau}}$\end{document} is defined in Experiment 1 Results).

For the recalibration-only hypothesis, we fitted the dual state model defined as in Smith et al., 2006:Netinternalestimate:x(k)=xf(k)+xs(k)\begin{document}$$\displaystyle Net \,internal \,estimate:x\left (k\right)=x_{f}\left (k\right)+x_{s}\left (k\right)$$\end{document}Faststate:xf(k+1)=Af∗xf(k)+Bf∗(p(k)−x(k))\begin{document}$$\displaystyle Fast \,state:x_{f}\left (k+1\right)=A_{f}*x_{f}\left (k\right)+B_{f}* \left(p\left (k\right)-x\left (k\right)\right) $$\end{document}Slowstate:xs(k+1)=As∗xs(k)+Bs∗(p(k)−x(k))\begin{document}$$\displaystyle Slow \,state:x_{s}\left (k+1\right)=A_{s}*x_{s}\left (k\right)+B_{s}* \left (p\left (k\right)-x\left (k\right) \right)$$\end{document}Δmotoroutput:u(k)=x(k)\begin{document}$$\displaystyle \Delta \,motor \,output:u\left (k\right)=x\left (k\right)$$\end{document}

where k\begin{document}$k$\end{document} is the stride number, p\begin{document}$p$\end{document} is the perturbation, x\begin{document}$x$\end{document} is the net internal estimate of the perturbation, xf\begin{document}$x_{f}$\end{document} and xs\begin{document}$x_{s}$\end{document} are the fast and slow states contributing to the estimate. Participants generate Δ motor output u\begin{document}$u$\end{document} that equals the internal estimate of the perturbation x\begin{document}$x$\end{document} to minimize the predicted step length asymmetry. The model has four free parameters: Af\begin{document}$A_{f}$\end{document}, As\begin{document}$A_{s}$\end{document} are retention factors controlling forgetting rate, and Bf\begin{document}$B_{f}$\end{document}, Bs\begin{document}$B_{s}$\end{document} are error sensitivities controlling learning rate. We fitted the model parameters using initial values Af\begin{document}$A_{f}$\end{document}=0.92, As\begin{document}$A_{s}$\end{document}=0.99, Bf\begin{document}$B_{f}$\end{document}=0.1, and Bs\begin{document}$B_{s}$\end{document}=0.01 (as in Roemmich et al., 2016), and constraints 0<Af\begin{document}$A_{f}$\end{document}<As\begin{document}$A_{s}$\end{document}<1 and 0<Bs\begin{document}$B_{s}$\end{document}<Bf\begin{document}$B_{f}$\end{document}<1 (as defined by the model, Smith et al., 2006).

We implemented the optimal control theory model using the original equations provided by Todorov, 2005, and defined our state and observation systems to be consistent with previous implementations of optimal control theory models for motor adaptation (Izawa and Shadmehr, 2011). Given k\begin{document}$k$\end{document} = stride number, p\begin{document}$p$\end{document} = perturbation, s\begin{document}$s$\end{document} = step length asymmetry, and u\begin{document}$u$\end{document} = motor output, the equation provided in Box 1 (capturing the relationship between these standard parameters of walking adaptation) is rewritten as follows:s(k)=u(k)-p(k)\begin{document}$$\displaystyle s\left (k\right)=u\left (k\right)-p\left (k\right)$$\end{document}

Participants maintain an internal estimate of the perturbation, p^\begin{document}$\hat{p}$\end{document}, and update it similarly to other state space models:p^(k+1)=ap^(k)\begin{document}$$\displaystyle \hat{p}\left (k+1\right)=a\; \hat{p}\left(k\right)$$\end{document}

where 0≤a≤1\begin{document}$0\leq a\leq 1$\end{document} is a parameter capturing forgetting rate. They use this estimate to predict the step length asymmetry that will result from a motor command:s^(k+1)=u(k)−p^(k)\begin{document}$$\displaystyle \hat{s}\left (k+1\right)=u\left (k\right)-\hat{p}\left (k\right)$$\end{document}

According to optimal control theory, participants select motor commands u(k)\begin{document}$u\left (k\right)$\end{document} that minimize the cost associated with the motor task – including the base energetic cost of executing motor commands, plus additional cost associated with the observable state or performance of the motor system – such as accuracy, stability, state-dependent energy expenditure, etc. (Todorov, 2005). In walking adaptation, more asymmetric step lengths are linked with worse stability (Darter et al., 2018) and energy expenditure (Sánchez et al., 2019). Therefore, we set the observable state y\begin{document}$y$\end{document} to represent step length asymmetry.

Our system can be written in matrix form in the following way:Hiddenstate:x(k)=[p(k)s(k)]\begin{document}$$\displaystyle {\it Hidden\, state} : \boldsymbol{x}\left (k\right)=\begin{bmatrix}p\left (k\right) \\ s\left (k\right) \end{bmatrix} $$\end{document}Observablestate:y(k)=s(k).\begin{document}$$\displaystyle Observable \,state:y\left (k\right)=s\left (k\right).$$\end{document}Hiddenstateupdateequation:x(k+1)=Ax(k)+Bu(k)+εx(k)\begin{document}$$\displaystyle Hidden \,state \,update \,equation:\boldsymbol{x}\left (k+1\right)=A\; \boldsymbol{x}\left(k\right)+B\; u\left (k\right)+\boldsymbol{\varepsilon_{x}}\left (k\right)$$\end{document}Observablestateequation:y(k)=Hx(k)+εy(k)\begin{document}$$\displaystyle Observable \,state \,equation:y\left (k\right)=H\; \boldsymbol{x}\left (k\right) + \boldsymbol{\varepsilon }_{y}\left (k\right)$$\end{document}

The system dynamics and observation matrices are A=[a0−10],B=[01],H=[01]\begin{document}$A=\left [\begin{array}{cc}a & 0\\-1 & 0\end{array}\right ],\,B=\left [\begin{array}{c}0\\1\end{array}\right ],\,H=\left [\begin{array}{cc}0 & 1\end{array}\right ]$\end{document}.

The noise terms are εx∼N(0,Qx),Qx=diag(σp2,σs2),εy∼N(0,Qy),Qy=σy2\begin{document}$\boldsymbol{{\boldsymbol{\varepsilon_{x}}}}\sim N\left (0,Q_{x}\right),\,Q_{x}=diag\left (\sigma _{p}^{2},\sigma _{s}^{2}\right),\boldsymbol{\varepsilon }_{y}\sim N\left (0,Q_{y}\right),\,Q_{y}=\sigma _{y}^{2}$\end{document}.

a\begin{document}$a$\end{document}, σp2\begin{document}$\sigma _{p}^{2}$\end{document}, σs2\begin{document}$\sigma _{s}^{2}$\end{document}, and σy2\begin{document}$\sigma _{y}^{2}$\end{document} are parameters that vary across participants.Motoroutputgeneration:u(k)=−G(k)x^(k).\begin{document}$$\displaystyle Motor \,output \,generation:u\left (k\right)=-G\left (k\right)\; \hat{\boldsymbol x}\left (k\right).$$\end{document}Stateestimateupdate:x^(k+1)=Ax^(k)+Bu(k)+AK(k)(−p(k)+p^(k)).\begin{document}$$\displaystyle State \,estimate \,update:\hat{\boldsymbol x}\left (k+1\right)=A\; \hat{\boldsymbol x} \left (k\right)+B\; u\left (k\right)+A\; K\left (k\right) \left (-p\left (k\right)+\hat{p}\left (k\right)\right).$$\end{document}

where u\begin{document}$u$\end{document} is the motor output, p\begin{document}$p$\end{document} is the perturbation, p^\begin{document}$\hat{p} $\end{document} is the internal estimate of the perturbation, x^=[p^s^]\begin{document}$\boldsymbol{\hat{x}} = \left [\begin{array}{c}\hat{p} \\\hat{s} \end{array}\right ]$\end{document} is the internal estimate of the full state in vector form, G\begin{document}$G$\end{document} is the ‘feedback gain’ matrix computed to minimize the cost J\begin{document}$J$\end{document}, K\begin{document}$K$\end{document} is the ‘Kalman gain’ matrix defining the rate of learning from the sensory prediction error, and A\begin{document}$A$\end{document} and B\begin{document}$B$\end{document} are matrices defined as above. Note that we defined the sensory prediction error as the difference between the estimated and actual perturbation (−p(k)+p^(k))\begin{document}$\left(-p(k)+\hat{p}(k) \right)$\end{document} to be consistent with previous implementations of optimal control theory for the case of motor adaptation (Izawa and Shadmehr, 2011). Specifically, this is derived as follows:Sensorypredictionerror=y(k)−y^(k)=s(k)−s^(k)=(u(k)−p(k))−(u(k)−p^(k))=−p(k)+p^(k)\begin{document}$$\displaystyle {\rm Sensory\,prediction\,error}= y (k)-{\hat y}(k) =s(k)-\hat{s} (k)= \left(u(k)-p(k) \right) - \left(u(k)-\hat{p} (k) \right) =-p(k)+\hat{p} (k)$$\end{document}

We implemented the memory of errors model as defined by Herzfeld et al., 2014. Similar to the dual state, the model updates an internal estimate of the perturbation and uses it to produce a motor output equal to this perturbation. However, the learning rate is not fixed, but varies on each trial depending on the error and the history of errors:u(k+1)=au(k)+η(k,e(k))e(k)\begin{document}$$\displaystyle u\left (k+1\right)=a\,u\left (k\right)+\eta \left(k,e\left (k\right)\right) \,e\left (k\right)$$\end{document}e(k)=p(k)−u(k)\begin{document}$$\displaystyle e\left (k\right)=\,p\left (k\right)-u\left (k\right)$$\end{document}

where k\begin{document}$k$\end{document} is the stride number, u\begin{document}$u$\end{document} is the motor output, p\begin{document}$p$\end{document} is the perturbation, e\begin{document}$e$\end{document} is the sensory prediction error, a\begin{document}$a$\end{document} is a parameter capturing forgetting, and η\begin{document}$\eta $\end{document} is the learning rate. As mentioned, the learning rate depends on the error and history of errors. Specifically, it is computed on each trial as follows:η(k,e(k))=w(k)Tg(e(k))\begin{document}$$\displaystyle \eta \left (k,\,e\left (k\right)\right)=\boldsymbol{w}\left (k\right)^{T}\boldsymbol{g}\left( e\left (k\right) \right)$$\end{document}g(e(k))=exp⁡{−(e(k)−e˘)22σ2}\begin{document}$$\displaystyle \boldsymbol{g}\left( e\left (k\right) \right) =\exp \left\{\frac{-\left( e\left (k\right)-\breve{\boldsymbol{e}} \right) ^{2}}{2\sigma^2} \right \} $$\end{document}w(k+1)=w(k)+βsign{e(k−1)e(k)}g(e(k−1))g(e(k−1))Tg(e(k−1))\begin{document}$$\displaystyle \boldsymbol{w}\left (k+1\right)=\boldsymbol{w}\left (k\right)+\beta \,\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left \{e\left (k-1\right)e\left (k\right)\right \}\frac{\boldsymbol{g}\left( e\left (k-1\right)\right) }{\boldsymbol{g}\left(e\left (k-1\right)\right) ^{T}\,\boldsymbol{g}\left(e\left (k-1\right)\right) }$$\end{document}

Each g(e(k))\begin{document}$\boldsymbol{g}\left(e\left (k\right)\right)$\end{document} is a basis element with preferred error e˘\begin{document}$\breve {\boldsymbol e} $\end{document}, so that g(e(k))\begin{document}$\boldsymbol{g}\left (e\left (k\right)\right)$\end{document} has maximum value when e(k)=e˘\begin{document}$e\left (k\right)= {\boldsymbol{\breve{e}}}$\end{document}, and the value progressively decays, similar to a normal distribution, for ek\begin{document}$e\left (k\right)$\end{document} smaller or larger than e˘\begin{document}$\breve {\boldsymbol e} $\end{document}. σ2\begin{document}$\sigma ^{2}$\end{document} corresponds to the variance in a normal distribution, and here captures how sharply g(e(k))\begin{document}$\boldsymbol{g}\left(e\left (k\right)\right)$\end{document} decay as the difference between e(k)\begin{document}$e\left (k\right)$\end{document} and e˘\begin{document}$\breve {\boldsymbol e} $\end{document} increases.

wk\begin{document}$\boldsymbol{w}\left (k\right)$\end{document} is a vector that defines error sensitivity, and is updated each trial to increase sensitivity for errors that are experienced most consistently. Specifically, it is a vector that stores a scaling factor for each basis error center contained in the vector e˘\begin{document}$\breve {\boldsymbol e} $\end{document}: the learning rate is higher in response to errors associated with larger w\begin{document}$\boldsymbol{w}$\end{document} scaling factor. w\begin{document}$\boldsymbol{w}$\end{document} is updated such that if the error on one trial has the same sign as the error on the previous trial, the sensitivity to that error increases; otherwise, it decreases. The model updates the entire w\begin{document}$\boldsymbol{w}$\end{document} vector, rather than just the element corresponding to experienced error. The amount of the update is proportional to g(e(k−1))\begin{document}$\boldsymbol{g}\left(e\left (k-1\right)\right)$\end{document}, meaning that sensitivities to errors similar to that experienced on this trial are updated the most, while sensitivities to errors different from that experienced on this trial are updated the least. The update magnitude is additionally scaled by the learning rate β\begin{document}$\beta $\end{document}.

We set the initial value of w\begin{document}$\boldsymbol{w}$\end{document} as follows:wi=η0∑jg(0)j,forallwi elementsofw\begin{document}$$\displaystyle w_{i}=\frac{\eta _{0}}{{\sum _{j}}\,g\left (0\right)_{j}},\,\,\,\,\,for\,all\,w_{i}\ elements\,of\,\boldsymbol{w}$$\end{document}g(0)=exp⁡{−(0−e˘)22σ2}\begin{document}$$\displaystyle \boldsymbol{g}\left (0\right)=\exp \left \{\frac{-\left (0- {\boldsymbol{\breve{e} }}\right)^{2}}{2\sigma ^{2}}\right \}$$\end{document}

where ∑jg(0)j\begin{document}${\sum }_{j}\, g (0)_{j}$\end{document} is the sum of all elements of g(0)\begin{document}$\boldsymbol{g} (0)$\end{document}, and η0\begin{document}$\eta _{0}$\end{document} is a parameter representing the naive learning rate.

In our study, this model captures the scenario where the learning rate is largest for step length asymmetry values that have been experienced the most and most consistently. We fitted the model parameters, σ2\begin{document}$\sigma ^{2}$\end{document}, β\begin{document}$\beta $\end{document}, η0\begin{document}$\eta _{0}$\end{document}, and a\begin{document}$a$\end{document} using initial values [0.5, 0.001, 0.001, 0.9] and bounds [0, 0, 0, 0] (lower) and [10, 1, 1, 1] (upper).

We evaluated the PReMo model developed by Tsay et al., 2022; Tsay et al., 2021, using the following equivalencies with the walking adaptation variables (Appendix 2): xp\begin{document}$x_{p}$\end{document} = step length asymmetry, σp2,σu2\begin{document}$\sigma _{p}^{2},\,\sigma _{u}^{2}$\end{document} = proprioceptive and sensory prediction uncertainties, K\begin{document}$K$\end{document} = learning rate. For the model fits, we set the goal G\begin{document}$G$\end{document} = 0, as it represents explicit strategies (but alternative values are considered in Appendix 2).

The model operates by first integrating proprioceptive and predictive estimates for step length asymmetry (where prediction = goal):Integratedestimate:xpI(k)=σu2σu2+σp2xp(k)+σp2σu2+σp2G(k)\begin{document}$$\displaystyle Integrated\, estimate: x_{p}^{I}\left (k\right)=\frac{\mathrm{\sigma }_{u}^{2}}{\mathrm{\sigma }_{u}^{2}+\mathrm{\sigma }_{p}^{2}}\,x_{p}\left (k\right)+\frac{\mathrm{\sigma }_{p}^{2}}{\mathrm{\sigma }_{u}^{2}+\mathrm{\sigma }_{p}^{2}}\,G\left (k\right)$$\end{document}

In the original model, perception is further shifted by the visual-proprioceptive mismatch. However, this proprioceptive shift is zero in the absence of visual feedback, and step length asymmetry perception depends only on the integrated estimate:Proprioceptiveshift:βp(k)∝(visual information −xpI(k)) =0\begin{document}$$\displaystyle Proprioceptive\, shift:\mathrm{\beta }_{p}\left (k\right)\propto \left (visual\ information\ -x_{p}^{I}\left (k\right)\right)\ =0$$\end{document}Perceivedsteplengthasymmetry:xpper(k)=xpI(k)+βp(k)≡xpI(k)\begin{document}$$\displaystyle Perceived \,step \,length \,asymmetry:x_{p}^{per}\left (k\right)=x_{p}^{I}\left (k\right)+\mathrm{\beta }_{p}\left (k\right)\equiv x_{p}^{I}\left (k\right)$$\end{document}

Perceptual error drives implicit adaptation of step length asymmetry:Implicitadaptation:xp(k+1)=xp(k)+K(G(k)−xpper(k))\begin{document}$$\displaystyle Implicit \,adaptation:x_{p}\left (k+1\right)=x_{p}\left (k\right)+K \left( G\left (k\right)-x_{p}^{per}\left (k\right) \right)$$\end{document}

The original model does not contain a perturbation term, which is needed to fit the model to the Ramp Down data. To introduce this, we replace the implicit adaptation equation above with separate observation and hidden state update equations, where the hidden state xk\begin{document}$x\left (k\right)$\end{document} represents Δ motor output and p(k)\begin{document}$p\left (k\right)$\end{document} is the perturbation:Steplengthasymmetryobservation:xp(k)=x(k)−p(k)\begin{document}$$\displaystyle Step \,length \,asymmetry \,observation:x_{p}\left (k\right)=x\left (k\right)-p\left (k\right)$$\end{document}Hiddenstateupdate:x(k+1)=x(k)+K(G(k)−xpper(k))\begin{document}$$\displaystyle Hidden \,state \,update:x\left (k+1\right)=x\left (k\right)+K \left(G\left (k\right)-x_{p}^{per}\left (k\right) \right)$$\end{document}

While the original model does not explicitly define the perceived perturbation, we can infer this equation using the relationship between step length asymmetry, perturbation, and Δ motor output:pk=xk-xp(k)\begin{document}$$\displaystyle p\left (k\right)=x\left (k\right)-x_{p}\left (k\right)$$\end{document}pperk-pk=xperk-xk-xpperk-xpk\begin{document}$$\displaystyle \left \{p^{per}\left (k\right)-p\left (k\right)\right \}=\left \{x^{per}\left (k\right)-x\left (k\right)\right \}-\left \{x_{p}^{per}\left (k\right)-x_{p}\left (k\right)\right \}$$\end{document}

We assume no shift in the perception of Δ motor output as this is not defined by the model:Perceivedperturbation:pper(k)=p(k)−{xpper(k)−xp(k)}\begin{document}$$\displaystyle Perceived \,perturbation:p^{per}\left (k\right)=\,p\left (k\right)-\left \{x_{p}^{per}\left (k\right)-x_{p}\left (k\right)\right \}$$\end{document}

We used pper\begin{document}$p^{per}$\end{document} to obtain model predictions for the perception of belt speed difference, setting the prediction to ‘right leg feels faster’ when pper<0\begin{document}$p^{per}\lt 0$\end{document}, ‘feels equal’ when pper=0\begin{document}$p^{per}=0$\end{document}, and ‘feels slower’ when pper<0\begin{document}$p^{per}\lt0$\end{document}.

To limit redundancy during model fitting, we substitute individual uncertainty terms with the relative weight variable Wp=σu2σu2+σp2\begin{document}$W_{p}=\frac{\mathrm{\sigma }_{u}^{2}}{\mathrm{\sigma }_{u}^{2}+\mathrm{\sigma }_{p}^{2}}$\end{document}, where 1-Wp=σp2σu2+σp2\begin{document}$1-W_{p}=\frac{\mathrm{\sigma }_{p}^{2}}{\mathrm{\sigma }_{u}^{2}+\mathrm{\sigma }_{p}^{2}}$\end{document}. We then fitted the two model parameters K\begin{document}$K$\end{document} and Wp\begin{document}$W_{p}$\end{document} to individual data. The initial value of Wp\begin{document}$W_{p}$\end{document} was set to 13\begin{document}$\frac{1}{3}$\end{document} based on results from Zhang et al. showing that σp2≈2σu2\begin{document}$\sigma _{p}^{2}\approx 2\sigma _{u}^{2}$\end{document} (Zhang et al., 2024). The initial value of the learning rate K\begin{document}$K$\end{document} was set to 0.03; this value was selected to ensure that the time course of learning of xp\begin{document}$x_{p}$\end{document} would match that of recalibration observed in control experiments (learning of xp\begin{document}$x_{p}$\end{document} after 3 min reached 78% of the learning observed after 15 min; similarly, recalibration in the Short groups was 78% of that in the Medium groups – both averaged across Ascend and Descend group means). We set lower bounds = [0, 0] and upper bounds = [1, 1], and simulated the entire paradigm but computed error minimization on the Ramp Down task only as explained before.

We evaluated the PEA model developed by Zhang et al., which models perception as the Bayesian integration of any available sensory modalities as well as sensory predictions (Zhang et al., 2024). We set the following variable equivalencies: xp\begin{document}$x_{p}$\end{document} = step length asymmetry, σp2,σu2\begin{document}$\sigma _{p}^{2},\,\sigma _{u}^{2}$\end{document} = proprioceptive and sensory prediction uncertainties, A\begin{document}$A$\end{document} = retention rate, B\begin{document}$B$\end{document} = learning rate. We set target T\begin{document}$T$\end{document} = 0. As split-belt adaptation does not involve vision, we do not have the variables xv\begin{document}$x_{v}$\end{document} and σv2\begin{document}$\mathrm{\sigma }_{v}^{2}$\end{document} and simplify the model accordingly.

The perceived movement is the Bayesian integration of proprioceptive and predictive information:Perceivedmovement:x^p(k)=Wpxp(k)+(1−Wp)T(k)\begin{document}$$\displaystyle Perceived \,movement: \hat{x}_{p}\left (k\right)=W_{p}\; x_{p}\left (k\right)+(1-W_{p}) \; T\left(k\right)$$\end{document}withWp=1σp21σu2+1σp2\begin{document}$$\displaystyle with\, W_p=\, \frac{\frac{1}{\sigma^2_p}}{\frac{1}{\sigma^2_u}+\frac{1}{\sigma^2_p}}$$\end{document}

Motor adaptation is driven by the perceptual error – the discrepancy between perceived and target movement:Adaptation:xp(k+1)=A xp(k)+B (T(k)−x^p(k))\begin{document}$$\displaystyle Adaptation:x_{p}\left (k+1\right)=A\ x_{p}\left (k\right)+B\ \left(T\left(k\right)-\hat{x} _{p}\left (k\right) \right)$$\end{document}

Perceived and actual step length asymmetry are further combined in a Bayesian manner to produce the step length asymmetry report used for the perceptual task:Reportedsteplengthasymmetry:xreport(k)=WRxp(k)+(1−WR)x^p(k)\begin{document}$$\displaystyle Reported \,step \,length \,asymmetry: x_{report}\left (k\right)=W_{R}\; x_{p}\left (k\right)+(1-W_{R}) \; \hat{x}_{p}\left (k\right)$$\end{document}withWR=Wp1+Wp\begin{document}$$\displaystyle with \,W_{R} = \frac{W_{p}}{1 + W_{p}} $$\end{document}

We finally introduced variables for hidden state xk\begin{document}$x\left (k\right)$\end{document}, perturbation p(k)\begin{document}$p\left (k\right)$\end{document}, and reported perceived perturbation pper(k)\begin{document}$p^{per}\left (k\right)$\end{document} (used to predict speed difference perception) as we did for PReMo:Steplengthasymmetryobservation:xp(k)=x(k)−p(k)\begin{document}$$\displaystyle Step \,length \,asymmetry \,observation: x_{p}\left (k\right)=x\left (k\right)-p\left (k\right)$$\end{document}Hiddenstateupdate:x(k+1)=Ax(k)+B(T(k)−x^p(k))\begin{document}$$\displaystyle Hidden \,state \,update: x\left (k+1\right)=A\,x\left(k\right)+B\,\left(T\left (k\right)-{\hat x}_{p}\left (k\right) \right)$$\end{document}Reportedperceivedperturbation:pper(k)=p(k)−{xreport(k)−xp(k)}\begin{document}$$\displaystyle Reported \,perceived \,perturbation: p^{per}\left (k\right)=\,p\left (k\right)-\left \{\,x_{report}\left (k\right)-x_{p}\left (k\right)\right \}$$\end{document}

We fitted the model parameters A,B\begin{document}$A,\,B$\end{document}, and Wp\begin{document}$W_{p}$\end{document} to individual data using initial values = [0.97, 0.2, 1/3], selected approximately based on the average parameter values reported by Zhang et al. across experiments. We set lower bounds = [0, 0, 0], upper bounds = [1, 1, 1].

As described in Appendix 2, PM-ReMap was formulated as an extension of PReMo that addressed its key limitations. The first limitation was that PReMo assumes perceptual realignment arises from mismatches between sensory modalities, such as vision and proprioception. We addressed this by accounting for perceptual realignment driven by mismatches between predicted and actual sensory outcomes of motor commands. The second limitation was that PReMo assumes perceptual realignment changes immediately to reflect a proportion of the perturbation on each stride – causing it to return to zero in the Ramp Down. We addressed this by incorporating a learning rate that mediates gradual changes in perceptual realignment. The third limitation was that PReMo lacks an automatic mapping mechanism capable of driving immediate changes in motor output. While it includes a variable that may function as a mapping mechanism, this variable relies on an explicit strategy, adjusts motor output gradually through a learning rate, and lacks a defined equation to determine its value. We addressed this limitation by defining an equation for this variable consistent with an automatic mapping mechanism and removing the learning rate to enable immediate changes in motor output.

The final model equations are as follows:Actualsteplengthasymmetry:xv(k)=xp(k)−p(k)\begin{document}$$\displaystyle Actual \,step \,length \,asymmetry:x_{v}\left (k\right)=x_{p}\left (k\right)-p\left (k\right)$$\end{document}Integratedestimateformotoroutput:xpI(k)=σu2σu2+σp2xp(k)+σp2σu2+σp2G(k)\begin{document}$$\displaystyle Integrated \,estimate \,for \,motor \,output:x_{p}^{I}\left (k\right)=\frac{\mathrm{\sigma }_{u}^{2}}{\mathrm{\sigma }_{u}^{2}+\mathrm{\sigma }_{p}^{2}}\,x_{p}\left (k\right)+\frac{\mathrm{\sigma }_{p}^{2}}{\mathrm{\sigma }_{u}^{2}+\mathrm{\sigma }_{p}^{2}}\,G\left (k\right)$$\end{document}Integratedestimateforsteplengthasymmetry:xvI(k)=σu2σu2+σv2xv(k)+σv2σu2+σv2G(k)\begin{document}$$\displaystyle Integrated \,estimate \,for \,step \,length \,asymmetry:x_{v}^{I}\left (k\right)=\frac{\mathrm{\sigma }_{u}^{2}}{\mathrm{\sigma }_{u}^{2}+\mathrm{\sigma }_{v}^{2}}\,x_{v}\left (k\right)+\frac{\mathrm{\sigma }_{v}^{2}}{\mathrm{\sigma }_{u}^{2}+\mathrm{\sigma }_{v}^{2}}\,G\left (k\right)$$\end{document}Perceptualshiftformotoroutput:βp∗=ηp(xvI(k)−xpI(k))βp(k+1)=βp(k)+K(βp∗−βp(k))\begin{document}$$\displaystyle \begin{array}{ll}Perceptual \,shift \,for \,motor \,output:&\beta _{p}^{*}=\mathrm{\eta }_{p}\left( x_{v}^{I}\left (k\right)-x_{p}^{I}\left (k\right)\right) \\ &\beta_{p}\left(k+1\right)=\beta_{p}\left (k\right)+K\, \left( \beta _{p}^{*}-\beta _{p}\left (k\right) \right) \end{array}$$\end{document}Perceptualshiftforsteplengthasymmetry:βv∗=ηv(xpI(k)−xvI(k))βv(k+1)=βv(k)+K(βv∗−βv(k))\begin{document}$$\displaystyle \begin{array}{ll}Perceptual \,shift \,for \,step \,length \,asymmetry:&\beta _{v}^{*}=\mathrm{\eta }_{v}\left( x_{p}^{I}\left (k\right)-x_{v}^{I}\left (k\right) \right) \\ &\beta_{v}\left(k+1\right)=\beta_{v}\left (k\right)+K\, \left(\beta _{v}^{*}-\beta _{v}\left (k\right) \right) \end{array}$$\end{document}Perceivedmotoroutput:xpper(k)=xpI(k)+βp(k)\begin{document}$$\displaystyle Perceived \,motor \,output: x_{p}^{per}\left (k\right)=x_{p}^{I}\left (k\right)+\mathrm{\beta }_{p}\left (k\right)$$\end{document}Perceivedsteplengthasymmetry:xvper(k)=xvI(k)+βv(k)\begin{document}$$\displaystyle Perceived \,step \,length \,asymmetry:x_{v}^{per}\left (k\right)=x_{v}^{I}\left (k\right)+\mathrm{\beta }_{v}\left (k\right)$$\end{document}Perceivedperturbation:pper(k)=xpper(k)−xvper(k)\begin{document}$$\displaystyle Perceived \,perturbation:p^{per}\left (k\right)=\,x_{p}^{per}\left (k\right)-x_{v}^{per}\left (k\right)$$\end{document}Adaptationofmotoroutput:xp(k+1)=xp(k)+G(k)−xpper(k)\begin{document}$$\displaystyle Adaptation \,of \,motor \,output:x_{p}\left(k+1\right)=x_{p}\left(k\right)+G\left(k\right)-x_{p}^{per}\left (k\right)$$\end{document}Goalforimplicitadaptationsimulations:G(k)=0\begin{document}$$\displaystyle Goal\, for\, implicit \,adaptation \,simulations: G\left (k\right)=0$$\end{document}Goalformappingsimulations(RampDownonly):G∗=βp(k)Wp+p(k+1)G(k+1)={G∗ifG∗≥00otherwise\begin{document}$$\displaystyle Goal \,for \,mapping \,simulations \,(Ramp \,Down \,only): \quad & \, G^{*}=\frac{\beta _{p}\left (k\right)}{W_{p}}+p\left (k+1\right) \\ & \, G\left (k+1\right)=\begin{cases} G^{*} \quad if \quad G^{*}\geq 0 \\ 0 \quad otherwise \end{cases} $$\end{document}