## 3 Materials and Method
### Metrics  
Within the purview of this investigation, an array of parameters underwent rigorous scrutiny, encompassing the following: maximum head angle deviation, effective range of head angle deviation (i.e., movement variability), and the Detrended Fluctuation Analysis (DFA) number. To commence, the angular deviation from the central position was meticulously quantified in degrees. The maximum magnitude among these calculated values was ascribed as the pinnacle of head angle deviation, while the arithmetic mean thereof was used to represent the average deviation of the head angle. Notably, the standard deviation within each registration affords insight into the amplitude of fluctuation characterizing the head angle deviation, thereby delineating its effective range. Furthermore, the DFA coefficient, elaborated upon in a subsequent section, was computed for each registration, adding an additional layer of complexity to the analytical framework.  
DFA NumberDrawing inspiration from the intricate fractal patterns arising from the juxtaposition of repetitive motifs, time series data can be scrutinized through a fractal lens. Within the framework of the DFA method, these fluctuations are conceptualized as a metric gauging randomness [46; 47]. By assessing the degree of randomness inherent in these fluctuations, the DFA method effectively quantifies the self-similarity embedded within the recorded time series. The fundamental underpinning of this method bears a resemblance to the root mean square error, distinguished by its superior resilience to the nonstationarity inherent in the examined processes [48]. Equation (4) furnishes the means to calculate the extent of fluctuations.  
\[F(n)=\sqrt{\frac{1}{N}\sum_{k=1}^{N}[y(k)-y_{n}(k)]^{2}} \tag{4}\]  
Here, the variable \(y(k)\) denotes the integral of the analyzed time series, spanning a length of \(N\), which is partitioned into sub-intervals of variable length n. The selection of 'n' is contingent upon the temporal span of the time series and the periodicity of embedded patterns, yielding diverse potential values. Concurrently, \(y_{n}(k)\) denotes the coefficients associated with the gradient of the least-squares error line, customarily fitted to each respective sub-interval. An exploration of the relationship between \(F(n)\) and '\(n\)' illuminates a notable exponential growth in the magnitude of fluctuations as the sub-interval length expands. Remarkably, this relationship exhibits near-linearity within the log-log framework. Specifically, \(F(n)\), which quantifies the extent of fluctuations, is effectively approximated by \(n^{\alpha}\). The parameter '\(\alpha\)', commonly referred to as the DFA number, represents the slope of the tangent line on the log-log graph correlating \(F(n)\) and '\(n\)'. This crucial parameter substantiates the extent of correlation among fluctuations. The interplay between \(\alpha\) and the stochastic nature of temporal variations across the successive phases is succinctly encapsulated within the confines of Equation (7).  
\[\alpha < 0.5\] Negative correlation (5) \[\alpha = 0.5\] Zero correlation (6) \[\alpha > 0.5\] Positive correlation-Completely random (7)  
The outcomes derived from this methodology serve a dual role, finding utility in the realm of biological elucidation of motor control dynamics [49]. Notably, a key advantage inherent in the application of this method lies in its capacity to provide substantiation for the underlying biological architecture of the central nervous system. For instance, the findings gleaned through this approach have been correlated with the intricate network of supra-spinal circuits implicated in motor control. Furthermore, the attained results furnish valuable insights into the operational mechanics of the central pattern generator responsible for orchestrating rhythmic motor movements.