Investigating the Effects of Cerebellar Transcranial Direct Current Stimulation on Post-Stroke Overground Gait Performance: a partial least- squares regression approach

Stroke often results in impaired gait, which can limit community ambulation and the quality of life. Recent works have shown the feasibility of transcranial Direct Current Stimulation (tDCS) as an adjuvant treatment to facilitate gait rehabilitation. Since the cerebellum plays an essential role in balance and movement coordination, which is crucial for independent overground ambulation, so, we investigated the effects of cerebellar tDCS (ctDCS) on the post-stroke overground gait performance in chronic stroke survivors. Fourteen chronic post-stroke male subjects were recruited based on convenience sampling at the collaborating hospitals where ten subjects finally participated in the ctDCS study. We evaluated the effects of two ctDCS montages with 2mA direct current, a) optimized configuration for dentate stimulation with 3.14cm2 disc anode at PO10h (10/5 EEG system) and 3.14cm2 disc cathode at PO9h (10/5 EEG system), and b) optimized configuration for leg lobules VII-IX stimulation with 3.14cm2 disc anode at Exx8 (electrodes defined by ROAST) and 3.14cm2 disc cathode at Exx7. We found ctDCS to be acceptable by all the exposed subjects. The ctDCS intervention had an effect on the 'Normalised Step length Affected side' (p=0.1) and 'Gait Stability Ratio' (p=0.0569), which was found using Wilcoxon signed-rank test at 10% significance level. Also, ctDCS montage specific effect was found using a two-sided Wilcoxon rank-sum test at a 5% significance level for 'Step Time Affected Leg' (p=0.0257) and '%Stance Time Unaffected Leg' (p=0.0376). Moreover, the changes in the quantitative gait parameters across both the montages were found to be correlated to the mean electric field strength in the lobules based on partial least squares regression analysis (R2 statistic = 0.6574) where the mean electric field strength at the cerebellar lobules, Vermis VIIIb, Ipsilesional IX, Vermis IX, Ipsilesional X, had the most loading. In conclusion, our feasibility study indicated the potential of a single session of ctDCS to contribute to the immediate improvement in the balance and gait performance in terms of gait-related indices and clinical gait measures.


Background
Stroke is a leading cause of disability across the globe with 80·1 million (74·1 to 86·3) prevalent cases globally and 116·4 million (111·4 to 121·4) disability-adjusted life-years in 2016 [1]. Gait impairments occur in more than 80% of stroke survivors [2], where gait impairments remain in 25% of all stroke survivors despite rehabilitation [3]. However, the recovery of independent walking requires considerable practice in stroke survivors [4]. Conventional gait training can play an important role [4] besides assistive robotic devices and brain-computer interfaces where a combination may be more effective than overground gait training alone [5]. During such gait training, neuroplasticity can be facilitated with adjuvant treatment with non-invasive brain stimulation (NIBS) techniques to the lower limb motor cortex of stroke survivors [6]. Here, moderate-quality evidence has been found on NIBS combined with various therapies to improve gait speed after stroke [7], [8]. Importantly, NIBS treatment effects were found signi cant only in the care of transcranial magnetic stimulation (TMS), while transcranial direct current stimulation (tDCS) did not show any signi cant therapeutic effects [8]. Here, it is important to note that the tDCS effects are governed by various parameters, including current intensity, electrode size, electrode placement, that may affect the e cacy of stimulation and the therapeutic outcomes [9]. Furthermore, the speci city of the tDCS is challenging, especially for the leg representations in the primary motor cortex at the interhemispheric ssure, as demonstrated by Foerster et al. in healthy humans based on computational modeling and neurophysiological testing [10]. Such speci city is particularly relevant for post-stroke patients with hemiplegia who may have an imbalance in the inter-hemispheric inhibition after stroke [11].
Transcranial direct current stimulation (tDCS) [12] involves passing constant weak direct current (generally of the order of 1-2 mA) through one's scalp via a pair of electrodes (anode and cathode) to stimulate speci c regions of one's brain. Research studies have shown that the application of tDCS on the leg area of the motor cortex can help to increase the motor evoked potential in the lower limb muscles [13]. Also, the application of tDCS at the primary and supplementary motor areas of the brain have been reported to affect the gait pattern of post-stroke patients [14], [15]. Speci cally, Tahtis et al. [15] showed that bi-cephalic tDCS, with anode placed over the ipsilesional lower limb primary motor cortex and the cathode placed over the contralesional leg motor cortex, can bring clinical improvement in the gait functionality of post-stroke patients. In the other study, Manji et al. [14] applied tDCS with anode placed in the front of 'Cz' (10/20 EEG montage) and the cathode placed at the inion improved gait speed (10-meter walk test [16]) and applicative walking ability (Timed-Up-and-Go [17]). Here, given the proximity of the representation of lower limbs in the primary motor cortex area, it is critical to ensure that the stimulation delivered using such montage was speci c to the targeted lower limb [18], [10]. Speci cally, the presumptive clinical applications of lower-limb tDCS need to take into account the unaffected contralesional hemisphere that produces inter-hemispheric inhibition where computational modeling can be used [18], [10].
Although the above-mentioned research studies focusing on the lower limb representation in the primary motor cortex have been promising, yet, it may also be important to investigate the role of cerebellar tDCS (ctDCS) [19] for post-stroke gait rehabilitation [20] since "the motor cortex retains what the cerebellum learns." Here, unlike the primary motor cortex stimulation that may increase the retention of newly learned visuomotor skills, ctDCS can facilitate motor adaptation during repetitive movements for gait rehabilitation [19]. Indeed, Zandvliet [20] showed that contra-lesional anodal cerebellar tDCS could improve standing balance performance; however, exploration of the dose is necessary. In this study, we aimed to investigate the lobular dosing of ctDCS based on two ctDCS montages, which were optimized using a computational pipeline [21]. Since the cerebellum plays an important role in balance and coordination, which are critical to gait recovery, as evidenced by TMS study [22], so it is postulated that ctDCS may also contribute towards improving the gait and balance of post-stroke patients. Here, the cerebellum can help in gait and posture by predicting the sensory output of motor commands and correcting the commands using a spatiotemporal interpretation of one's surroundings [23]. So, the function of the cerebellum is 'to build internal models that predict the sensory outcome of motor commands and correct motor commands through internal feedback [23]' based on the rehabilitation training. Prior works have shown that the cerebellum aids visually-guided limb movement and facilitates learning limb movement trajectories [24], [25], which is crucial for post-stroke gait rehabilitation.
Given the crucial role of the cerebellum in balance and gait, we aimed to investigate the effects of ctDCS on the post-stroke overground gait performance using regression analysis of the electric eld distribution with the behavioral outcome [26]. Speci cally, multivariate regression analysis can relate the changes in the balance and gait measures to the lobular electric eld distribution due to ctDCS along cerebellar functional gradients [27]. With this motivation, we conducted this study to investigate the effects of ctDCS of the dentate nuclei and the leg lobules VII-IX on balance and overground gait ability of chronic stroke survivors. Based on our prior works [26], we optimized the ctDCS electrode montage, where our goal was to target cerebellar regions related to dentate nuclei and the lower-limb representation (cerebellar lobules VII-IX). The dentate nucleus is involved in the planning, initiating, and modifying voluntary movements as well as cognition. For regression analysis, the lobular electric eld strength was proposed as a predictor of the changes in the gait parameters due to ctDCS intervention, which is based on our prior work on ctDCS effects on post-stroke standing balance [26]. So, the objectives of the current study were (i) evaluation of the acceptability of ctDCS by chronic stroke survivors, (ii) investigation of the effects of ctDCS on balance, clinical gait measures, as well as its regression analysis using the quantitative gait-related indices, (iii) investigation of the differential effects of the two ctDCS montages on balance, clinical gait measures and quantitative gait-related indices. In this study, we used our lowcost in-house built wearable gait characterizer [28] that can quantify gait performance in terms of Step Length [29], Walk Ratio [30], Gait Stability Ratio [31], and Symmetry Index [32]. The rest of the paper is organized as follows: Section II elaborates on the materials and methods used in our study, Section III presents the results, which is followed by the discussion and conclusion in Section IV.

Materials And Methods
Our wearable experimental setup comprised of (i) the Gait Quanti cation Shoe [28] (i.e., gait characterizer), and (ii) the wireless ctDCS cap, as shown in Fig. 1. The Gait Quanti cation Shoe characterized one's gait parameters during overground walking. The ctDCS intervention was evaluated with the Gait Quanti cation Shoe to investigate the effects of ctDCS on the gait of chronic post-stroke hemiplegic patients recruited by convenience sampling from collaborating hospitals.

i. Study Participants
The hemiplegic stroke survivors, who (i) were aged between 18 and 90 years, (ii) could walk independently for at least 10 meters, (iii) could provide informed and written consent, and (iv) could understand instruction from the experimenter were contacted. Twelve post-stroke male subjects (P1-P12, Mean (SD) = 46(± 13) years) were selected for the study (see Table 1) from fourteen volunteers. Here, volunteers who underwent any recent surgery or were in the acute phase of stroke were excluded from the study. Written informed consent was obtained from each subject, and the multi-center research protocol for this study was approved by the All India Institute of Medical Sciences, New Delhi, India Institutional

ii. Gait Quanti cation Shoe
In this study, we aimed to quantify one's gait in terms of gait-related indices by recording gait events using a pair of instrumented shoe [28]. Figure 1 shows the wearable device, namely the Gait Quanti cation Shoes (Gait Shoe henceforth) [28], that was used in this study to record one's gait events.
The Gait Shoe consisted of insoles instrumented with force-sensitive resistors (FSRs) that were placed below the greater toe, lateral heel and medial heel positions of each shoe to detect one's gait events, e.g., heel-strike, toe-off, etc. These gait events were used to compute different gait-related indices, e.g., Step Length [29], Walk Ratio [30], Gait Stability Ratio [31], and Symmetry Index [32], etc. The Gait Shoe was capable of transmitting the data wirelessly to a data logger computer for subsequent o ine analysis.

a. Computation of Step Length
Step Length is the distance between two successive contralateral heel-strikes during gait. We wanted to explore the implication of ctDCS on one's Step Length since this can be an essential indicator of the functional gait ability of hemiplegic post-stroke patients [29]. Here, we computed the average Step Length using the average Step Time (recorded by the Gait Shoe ) and the average Walking Speed. The Step Time was measured by the Gait Shoe from the time interval between two successive heel-strike events of contralateral legs. One's Walking Speed (during the overground walk) was computed from the time taken to walk through a pre-de ned distance. Subsequently, the Step Length was calculated using Eq. (1). Finally, the Normalized Step Length was computed using the individualized height information [28] (Eq. (2)).
Step Length = Step Time * Walking Speed …………………………………………………………………….. The Gait Stability Ratio (GSR) depends on one's Cadence (steps/sec) and Walking Speed (m/sec) [28]. The GSR is a good indicator of balance de cits in older adults [31]. The GSR takes into account the changes in one's Walking Speed that can in uence one's Step Length. Here, a decrease in GSR might indicate increased double support time during one's walk, thereby inferring an increase in dynamic stability. The GSR was computed using Eq. (3).
The Walk Ratio (WR) [30] can describe a relation between one's Step Length and Cadence during walking. Importantly, WR is invariant during different speeds, uneven surface conditions, but is affected by dual task-condition [33]. The Walk Ratio was computed using Eq. (4).

Walk Ratio = Cadence/ Step Length ……………………………………………………………………………(4) d. Computation of Symmetry Index
The Symmetry Index (SI) is a measure of the extent to which one makes symmetrical use of both legs during walking [32] -smaller the value of SI, the better is the gait symmetry. One of the distinctive characteristics of post-stroke gait is the impaired gait symmetry, particularly in hemiplegic patients [34].
Here, we computed the SI using the %stance phase (of a gait cycle) measured using the Gait Shoe while considering the %stance for each of the left (X L ) and right legs (X R ). The SI was calculated using Eq. (5).
SI= ((X L -X R ) / 0.5 * (X L +X R ) ) *100 …………………………………………………………………..……… (5) [Place Fig. 2 Here] iii. Optimization of the Electrode Montage (age-speci c computational modeling of ctDCS) We used age-speci c MRI templates that were obtained online at https://jerlab.sc.edu/projects/neurodevelopmental-mri-database/ with the permission of Dr. John Richards. The data comprised of average T1-weighted MRI for the head and brain and segmenting priors for gray matter (GM), white matter (WM), and cerebrospinal uid (CSF). For this, we chose the age group that matched the age of our subjects for this study. A Realistic volumetric Approach to Simulate Transcranial Electric Stimulation (ROAST) [35] was used to create a tetrahedral volume mesh of the head.
ROAST used SPM12("SPM -Statistical Parametric Mapping") to segment the head and brain. After segmentation, ve tissues were identi ed for the tetrahedral volume mesh, namely, Scalp, Skull, Cerebrospinal Fluid (CSF), Gray Matter (GM), and White Matter (WM). These different brain tissues for the volume mesh were modeled as different volume conductors for Finite Element Analysis (FEA) in the ROAST. Here, isotropic conductivity used for the different brain tissues [36] were (in S/m): Scalp = 0.465; Skull = 0.01; CSF = 1.654; GM = 0.276; WM = 0.126. For further details on the head modeling, please refer to our prior works [21], [26].
In this study, the Electric Field (EF) distribution was modeled for two different ctDCS montages for each subject's age-speci c head model created from MRI templates ( https://jerlab.sc.edu/projects/neurodevelopmental-mri-database/ ). The boundary condition was set as 2 mA injection current (Neumann boundary condition) with the following electrode con gurations from our prior work where we performed ctDCS optimization [26]: 1. a) Optimized con guration for dentate stimulation [26]: 3.14 cm 2 disc anode was PO10h (10/5 EEG system), and 3.14 cm 2 disc cathode was placed at PO9h (10/5 EEG system) for ctDCS with 2 mA direct current.
In all the simulations, the voxel size was considered as 1mm 3 . The contralesional anode and cathode injected the speci ed amount of current (source) in the volume conductor, i.e., the head model. Finite Element Analysis (FEA) was conducted on each head model to compute the ctDCS induced EF in the brain tissues. The electric eld was computed at all the voxels (voxel size 1mm 3 ) of the cerebellar lobules de ned by the cerebellar electric eld distribution was normalized using spatially unbiased atlas for the cerebellum and brainstem (SUIT) [38]. Subsequently, the cerebellar lobular electric eld distribution was found using SUIT [38], and T1-weighted images were tted to the SUIT template of the human cerebellum in SPM12 ("SPM -Statistical Parametric Mapping": https://www. l.ion.ucl.ac.uk/spm/software/spm12/).
The cerebellar mask was visually checked in MRIcron, and the non-linear deformation was then applied to each EF image. The volume of the cerebellar lobules, de ned by the SUIT atlas [38], was used for the extraction of the lobular EF distribution. Also, we customized SUIT codes to extract the left and the right dentate EF distribution.

iv. Experimental Setup and Data Analysis
Overground gait and balance evaluation were performed based on Timed-Up-and-Go (TUG) [17] and Berg Balance Score (BBS) [39] before and after the ctDCS intervention. Figure 1 shows the experimental setup for the clinical study in a low resource setting. The ctDCS setup consisted of a wireless STARSTIM 8 stimulator (Neuroelectrics, Spain). The study required a commitment of about 30 minutes from each participant.

a. Cerebellar tDCS intervention
Based on our prior work [26], 15 minutes of 2 mA bilateral ctDCS was delivered in a repeated measure single-blind crossover design using either of the two bipolar montages with a circular (1 cm radius) contralesional anode. The electrode locations were based on the ROAST toolbox [35], and "unambiguously illustrated (UI) 10/5 system" [37]; 1. PO9h -PO10h, and 2. Exx7 -Exx8. Overground quantitative and clinical gait evaluation was performed before and after the ctDCS intervention to compute a percent normalized change measures, (POST-PRE)x100/(POST + PRE).

b. Experimental setup for overground gait analysis
The experimental setup for the overground gait analysis consisted of (i) 10 m long straight pathway (overground) marked with start and end lines, (ii) data-logger computer, and (iii) a pair of Gait Shoes . In a repeated measure crossover design to compare the two ctDCS montages, we investigated the effects of ctDCS on one's gait characteristics during the overground walk. Once the participant arrived at the study hall, they were asked to sit and relax for about 5 minutes. Then, the experimenter explained to the participant what he was expected to do in the study as well as the risks. After informed consent, the baseline clinical measures, TUG, and BBS were recorded. Then, the experimenter helped the participant to wear the Gait Shoe [28]. Subsequently, the experimenter prepared the participant for ctDCS by placing the neoprene cap combined with a battery-driven wireless stimulator, Starstim 8 (Neuroelectrics, Spain), and the gel-based electrodes. The participants were informed that they could discontinue from the study in case of any discomfort.
Once the participant was ready to start the study, they were asked to walk on a 10 m long straight path (overground) marked with a start and stop lines at their self-selected comfortable speed, and the participant's overground Walking Speed (Speed OG ) was computed. After this, the participant was asked to sit and relax on a chair for about 5 min. Subsequently, ctDCS was administered in a repeated measure single-blind crossover design using one of the two ctDCS montages for 15 minutes at the rest condition with a dosage of 2 mA [26]. Following this, the participant repeated the 10 m overground walk, followed by an assessment of the clinical gait and balance measures (TUG and BBS). The gait performance of the post-stroke participants was quanti ed using Gait Shoe in terms of the gait-related indices, as described earlier. Therefore, the post-stroke participants performed two trials of the overground walk, pre, and post ctDCS intervention, at their self-selected walking speed while wearing the Gait Shoe , as illustrated in Fig. 2.
We also evaluated the acceptability of the ctDCS intervention in post-stroke subjects where we collected subjective feedback from the post-stroke participants prior to (Pre tDCS ), during (Active tDCS ), and post (Post tDCS ) application of ctDCS. c) Statistical analysis and the partial least squares regression Two-sided Wilcoxon rank-sum test was performed at the 5% signi cance level on the percent normalized change measures, for the null hypothesis that the two ctDCS montages led to the same percent normalized change in the quantitative gait parameters from the same continuous distributions with equal medians. Also, Wilcoxon signed-rank test was conducted to nd any change in the percent normalized change measures due to ctDCS intervention for the null hypothesis that data comes from a distribution whose median is zero at the 5% signi cance level. Multivariate regression analysis was conducted to relate the changes in the balance and gait measures to the lobular electric eld distribution due to ctDCS montages. Here, multicollinearity can occur when independent variables (predictors) are correlated. We have presented principal component regression analysis for multivariate linear regression of the lobular electric eld distribution as the predictor with the behavioral outcomes as the response variables [26]. The goal is to extract the relation between electric eld distribution and the behavioral effects of ctDCS where Partial Least Squares (PLS) is a promising multivariate statistical technique that can combine the information about the variances of both the predictors and the responses, while also considering the correlations among them [40]. In this study, we applied partial least squares regression approach to analyze the associations between the lobular electric eld distribution as the predictor with the gait outcome measures as the response variables. Although statistical inference is the strength of PLSR approach using computational cross-validation methods (e.g., jackknife, bootstrap) [40]; however, we will apply PLS as a correlation technique in this study. The matrix of correlations between the lobular electric eld distribution as the predictor with the gait outcome measures as the response variables is subjected to the singular value decomposition that results in the singular vectors called saliences. The lobular electric eld distribution as the predictor with the gait outcome measures as the response variables can be projected onto their respective saliences, which creates latent variables that are linear combinations of the original variables. Here, PLS searches for latent variables that express the largest amount of information common to both the lobular electric eld distribution as the predictor and the gait outcome measures as the response variables. This is a xed-effect model where the results can only be interpreted with respect to the current data sets from this study. In this study, PLS analysis was performed on the percent normalized change measures, of gait indices from the Gait shoe as the response variable, where the lobular electric eld distribution for both the montages across all the subjects (found after centering the data and then singular value decomposition) was the predictor.

i. Acceptability of ctDCS
Once the cap with the electrodes and the portable tDCS device was placed on the participant's head, we obtained Pre tDCS feedback from the participant to understand whether they were comfortable with the neoprene cap. Two subjects, P8 and P11, had challenges with the tting of the ctDCS cap and the gel electrodes on the scalp, so they left the study. The rest of the 10 participants expressed that they were comfortable wearing the ctDCS cap with gel electrodes. The Pre tDCS baseline feedback was followed by the feedback after the administration of the ctDCS, i.e., the Active tDCS . Except two subjects who left the study at the baseline (Pre tDCS ) stage, none of the ten participants expressed any discomfort with the ctDCS intervention (at Active tDCS ); and, they reported a minor tingling sensation on the scalp for the rst few seconds which was tolerable. After the application of ctDCS, the Post tDCS verbal feedback revealed that none of the participants had any adverse effects, such as a sensation of tissue burning, nausea, headache, etc. Also, no skin reddening (at the location of electrodes) of the scalp was noticed. From this, we concluded that post-stroke patients in this study could tolerate the ctDCS with 3.14 cm 2 disc gel electrodes at 2 mA direct current.
ii. Effects of ctDCS on Gait-related Indices measured using the Gait Shoe Effects of ctDCS on post-stroke overground gait were quanti ed using the Gait Shoe that measured gaitrelated indices, e.g., Step Length, Gait Stability Ratio, Walk Ratio, and Symmetry Index. Also, the mean lobular electric eld strengths for all the 10 participants using their age-speci c head model was found for both the ctDCS montages, dentate ctDCS and leg (lobules VII-IX) ctDCS, as shown in Fig. 3. Figure 3a shows that the leg ctDCS also affected the dentate nuclei besides the targeted lobules VII-IX at a comparable electric eld strength (0.1 V/m), while the Fig. 3b shows that the dentate ctDCS primarily affected the dentate nuclei at greater than 0.2 V/m electric eld strength. These results are based on computational modeling using subject's age-matched healthy MRI templates where four post-stroke subjects were left hemiplegic and the remaining six right hemiplegics, as shown in Table 1. Figure 4 shows the percent normalized change measures, in the gait parameters across the 10 participants due to the two ctDCS montages. The distribution shown in the violin plots in Fig. 4 was found to be mostly not normal. Therefore, non-parametric two-sided Wilcoxon rank sum test at 5% signi cance level was used to nd the difference in the effects between the two ctDCS montages where statistically signi cant effect was found for 'Step Time The effects of the two montages of the ctDCS were found similar across ten subjects for most gait parameters. Figure 5a shows a reasonable correlation between tted and observed responses using partial least squares (PLS) analysis with the mean lobular electric eld strength as the predictors, which was con rmed by the R 2 statistic = 0.6574. Residuals passed the Lilliefors test for two-sided goodness-oft for normality. Choosing the number of components in a PLS model is a critical step where greater than 60% of the variance in the response variables (percent normalized change in gait parameters) was explained by the rst ten components of the predictor variables (mean lobular electric eld strength), as shown in Fig. 5b.
Furthermore, the loadings of the latent variables on the response variables (percent normalized change in gait parameters) and the predictor variables (mean lobular electric eld strength) are shown in Figs. 6a and 6b, respectively. Here, we found that the cerebellar lobules, Vermis VIIIb, Ipsilesional IX, Vermis IX, Ipsilesional X, were related by the component 2 positively to the 'Step Time Affected Leg' (p = 0.0257) and '%Stance Time Unaffected Leg' (p = 0.0376), that showed signi cant effects between the two ctDCS montages using two-sided Wilcoxon rank-sum test. Also, the cerebellar lobules, Vermis VIIIb, Ipsilesional IX, Vermis IX, Ipsilesional X, were related by the component 2 negatively to the '%Swing Time Unaffected Leg' (p = 0.0539), '%Single Support Time Affected Leg' (p = 0.1212). Figure 6c shows that the cerebellar lobules, Vermis VIIIb, Ipsilesional IX, Vermis IX, Ipsilesional X, had the lowest difference in their mean lobular electric eld strength between the two ctDCS montages for all 10 subjects.

iii. Effects of ctDCS on the Balance and Gait related Clinical Measures
The results from the clinical assessment of the 10-meter walk test, TUG, and the BBS showed that there was a small improvement in the duration of the 10 m walk test and the TUG test (reduction was about 2.5% and 4%, respectively), which indicated an improved walking speed. Also, we found an improvement (about 5%) in the BBS score; however, the improvement in the TUG and BBS scores were similar across both the ctDCS montages.

Discussion:
In this study, we investigated the effects of two different ctDCS montages on overground gait parameters. Although the montages were computationally optimized for targeting the dentate nuclei and the leg representations in the cerebellum; however, both the ctDCS montages resulted in higher than 0.1V/m electric eld strength at the dentate nuclei, as shown in Fig. 3. In fact, we found it challenging to avoid affecting the dentate nucleus, the largest of the deep cerebellar nuclei when targeting leg lobules VII-IX. In this study, ctDCS intervention was found to affect the BBS, TUG, and the quantitative gait parameters, 'Normalised Step length Affected side' (p = 0.1) and 'Gait Stability Ratio' (p = 0.0569) based on the Wilcoxon signed rank test at 10% signi cance level. A decrease in 'Gait Stability Ratio' was found which indicates an increase in dynamic stability. Also, ctDCS montage speci c effects were found for the 'Step Time Affected Leg' (p = 0.0257) and '%Stance Time Unaffected Leg' (p = 0.0376) using two-sided Wilcoxon rank-sum test at 5% signi cance level, as shown in Fig. 4. These changes can be attributed to the ctDCS electric eld strength since individual changes in the quantitative gait parameters across both the montages were found to be correlated to the mean electric eld strength in the lobules based on partial least squares (PLS) regression analysis (see Fig. 5). The loadings of the latent variables found from PLS analysis (see Fig. 6 signi cance level. The ipsilesional cathode effects may be related to the cerebellar-brain inhibition, where prior work in healthy subjects showed robust effects of cathodal ctDCS after-effects on cerebellar-brain inhibition [41]. Such effects of cerebellar-brain inhibition on the contralesional leg motor cortex can be compared with the results from Tahtis et al. [15] where bi-cephalic tDCS with cathode placed over the contralesional leg motor cortex, i.e., postulated reduction of the excitability of the contralesional leg motor cortex, improved the gait functionality of post-stroke patients. Also, posterior vermis has been shown to be related to the performance on tandem walking [42], where lobule X was found essential in the vestibular system [43] while motor and somatosensory activation have been linked to the lobule VIIIb [44]. This is postulated to be related to the signi cant effect of ctDCS on the 'Gait Stability Ratio' which has been shown to be an indicator of balance during walking [31]. Moreover, step length variability during tandem walking has been related to the lesion in the lobules VIIIb and IX [25] that we did not investigate in this study; however, ctDCS effects were found for the 'Normalised Step length Affected side.' These posterior cerebellar lobules lobules VIIIb, IX, and X were affected by comparable mean electric eld strength across the two ctDCS montages, as shown in Fig. 6c, which is may have led to comparable balance and gait related behavioral outcomes across the two ctDCS montages. We found that PLS analysis can be an effective technique for regression modeling to understand the relation between the electric eld distribution and the behavioral effects where PLS results can also be generalized (i.e., to create a random effect model) using inferential analytical approach [40] when a larger dataset is available.
Clinical literature shows a crucial role of the cerebellum in coordinating voluntary movements (e.g., walking) and maintaining one's balance [45]. In this current study, the cerebellum was intact so that the ctDCS was performed to ameliorate de cits in the rest of the motor network in the cerebrum. Randomeffects modeling of the cumulative effect size by Oldrati and Schutter [46] showed that anodal and cathodal ctDCS were effective in changing motor-and cognitive-related behavioral performance in healthy volunteers. However, the polarity of ctDCS was not predictive of the direction of the behavioral changes in healthy volunteers [46]. Nevertheless, robust effects of the cathodal ctDCS after-effects have been found on cerebellar-brain inhibition [41], which may be related to the behavioral changes. However, the clinical applicability of ctDCS in improving the functional gait ability remains under-explored due to a diversity of ideas on cerebellar involvement in the movement and the inter-subject variability in the ctDCS effects [47]. We postulate based on cerebellar ataxia studies [48] that ctDCS may also be feasible in some cases of cerebellar damage. Here, besides neuroimaging guided subject-speci c head modeling, portable neuroimaging may be able to capture the immediate response of the ctDCS on the cerebrum [49] that may be used for individual dosing.
Our preliminary ndings in this study are encouraging; however, this study had certain limitations. Speci cally, one of the limitations was the small sample size of chronic post-stroke participants with heterogeneous conditions. Therefore, the low statistical power has a reduced chance of detecting a true effect [50], so this study can be considered as a feasibility study. Also, the convenience sampling in this study was biased since all of our participants were male hemiplegics, with four being left hemiplegic and the remaining six right hemiplegics. In the future, we plan to extend our study involving more post-stroke participants from both the genders. Although contralesional anode was used for bi-hemispheric ctDCS, the electrode locations were not optimized for individual cases. So, "one-size-ts-all" montage was used across the subjects that were optimized based on the subject's age-speci c head model from an MRI template that did not have any lesions. This limited any subject speci c inferences using the partial least squares analysis considering that the post-stroke participants had heterogeneous lesion conditions in the cerebrum. The cerebellum was intact in all the subjects in this study; however, heterogeneous lesion locations in the cerebrum need further investigation, especially in the non-responders, where multi-block or multi-table PLS can integrate one or more of these varieties in a common analysis [40]. In the future, we also plan to segregate the extended pool of post-stroke participants based on the portable neuroimaging of the ctDCS response [49] for such multi-table PLS analysis.

Conclusion:
Both the ctDCS montages targeted the dentate nuclei, and the "one-size-ts-all" ctDCS intervention was limited to a single session, which improved gait and balance performance with low statistical power, and no signi cant difference was found between the montages for most gait parameters. Nevertheless, the quantitative gait parameters across both the montages were found to be correlated to the mean lobular Block diagram for the experimental protocol a) Violin plot of the mean lobular electric eld strength (V/m) across 10 participants for the optimized con guration for leg lobules VII-IX stimulation. b) Violin plot of the mean lobular electric eld strength (V/m) across 10 participants for the optimized con guration for dentate stimulation. Violin plot allowed visualization of the distribution of the data and its probability density where the box plot (with median, interquartile range, upper adjacent value, lower adjacent value) is combined with the probability density placed on each side. Figure 4 a) Violin plot of the mean % change in the gait parameters across 10 participants due to ctDCS optimized for leg lobules VII-IX stimulation. b) Violin plot of the mean % change in the gait parameters across 10 participants due to ctDCS optimized for dentate stimulation. Violin plot allowed visualization of the distribution of the data and its probability density where the box plot (with median, interquartile range, upper adjacent value, lower adjacent value) is combined with the probability density placed on each side. a) The scatter shows a reasonable correlation between tted and observed responses after partial least squares (PLS) regression for all the response variables (percent normalized change in gait parameters) . b) Greater than 60% of the variance in the response variables (percent normalized change in gait parameters) was explained by the rst ten components of the predictor variables (mean lobular electric eld strength). between the dentate and the leg lobules VII-IX ctDCS montages. 'Contra' is contralesional and 'Ipsi' is ipsilesional.