# Postural Control in Children with Cerebellar Ataxia
## 2 Materials and Methods
### Data Processing  
During the 30 s quiet standing period, the statokinesigram, i.e., the trajectory of the CoP in the horizontal plane, was used to extract specific indexes of balance control. These indexes were: the area and the eccentricity of the ellipse containing 95% of CoP positions, the total length of CoP trajectory (CoP length), the average CoP velocity, and the peak-to-peak Mediolateral and Anteroposterior CoP displacements (ML and AP ranges, respectively) [31; 32]. In particular, the ellipse area (A) and eccentricity (_e_) were calculated according to the following formulae:  
\[\mathrm{A}=a*b*\pi \tag{1}\]  
\[e=\frac{\sqrt{\left|a^{2}-b^{2}\right|}}{a} \tag{2}\]  
in which \(a\) and \(b\) were the semi-major axis (i.e., half of the ellipse longest diameter) and the semi-minor axis (i.e., half of the shortest diameter), respectively.  
Gait initiation was subdivided into three phases [13]: the imbalance phase, in which CoP moves backward and towards the future swing foot; the unloading phase, in which CoP moves laterally towards the stance foot, and the first swing, in which CoP moves forwards along the stance foot, from toe-off to heel-strike of the swing foot. The temporal events delimiting each phase were determined by visual inspection of the CoP trajectory; in particular, the onset of the CoP backward shift represented the APA onset.  
For the _imbalance_ and _unloading_ phases, separately, we measured the phase duration, the length of the CoP trajectory, the maximum AP and ML shifts of both CoP and COM, and the distance from CoP to COM projection on the horizontal plane, at the end of the phase.  
The _first swing_ phase was evaluated by measuring the length of the first step, normalized to the lower limb length (_LL_), and its velocity (_v_), expressed in Froude number (Fr = \(\frac{v}{\sqrt{g+L}}\)_g_ being gravity acceleration [38]); also these measurements were obtained from kinematic data.  
For each subject, the kinematic and dynamometric variables were averaged over the three valid trials recorded. Data normality was evaluated by means of the Shapiro-Wilk test. Considering that data were not normally distributed, the differences among SlowP, NonP, and H groups were analyzed non-parametrically by using Kruskal-Wallis tests followed by Dunn post hoc, with Bonferroni adjustment. The level of significance was set to 0.05.  
The analysis of EMG recordings regarded the timing of muscle activation or inhibition, surrounding the APA onset at gait initiation. Raw EMG data were high-pass filtered (f\({}_{\mathrm{cut}}\) = 50 Hz) with a zero-phase shift 6 th-order elliptic filter, to remove movement artifacts, and then the signals were rectified. For each muscle, the traces of the recorded trials were time-aligned to the APA onset and averaged across trials. For each averaged EMG track, the period from 1 s to 0.5 s before the APA onset--where no EMG changes were observed--was assumed as reference. The signal was integrated (time constant = 11 ms), and the mean level in the reference period was subtracted. The onset of an excitatory or inhibitory EMG change was identified by a software algorithm, which searched the first time point in which the track fell above or below 2 SD of the reference signal (excitation or inhibition, respectively) and remained there for at least 50 ms. Whenever the criterion was met, the algorithm searched backward the point in which the signal started to deviate from the mean reference value [39]. No statistical analysis was performed on EMG timings because of the many cases in which no clear inhibitory or excitatory changes could be identified.