# A Novel Criticality Analysis Method for Assessing Obesity Treatment Efficacy
## 2 Mathematical Model-Driven Criticality Analysis for Human Gait Assessment
As parameters are varied, the system exhibits complex phenomena, including bistability, limit cycles, spiralling trajectories, and chaos, particularly near critical transition points. Analysing the geometric patterns within the phase portrait provides valuable insights into the mechanisms underpinning self-organised criticality. Characteristics such as the number and stability of fixed points, oscillations, excitability, and susceptibility to perturbations can be deduced from phase space topology. Additionally, the fractal-like features within the phase portrait unveil the self-similar, scale-invariant nature of critical fluctuations. Moreover, this representation facilitates the quantification of nonlinear correlations that capture the intricacies of coupled dynamics. Therefore, phase space-based criticality analysis unveils the system's rich nonlinear behavior, phase transitions, and emergent complexity resulting from self-organised criticality. \[q_{f}=\frac{f}{f+\mu_{f}}, \tag{1}\]  
\[\sigma_{p}(q_{f})=f_{p}e^{(x_{p}q_{f})}, \tag{2}\]  
\[\sigma_{g}(q_{f})=f_{g}e^{(x_{g}q_{f})}, \tag{3}\]  
\[\frac{dm}{dt}=k_{g}\frac{fg}{K_{G}+f}-\frac{mp}{1+m}+r_{im}, \tag{4}\]  
\[\frac{df}{dt}=-k_{g}\frac{fg}{K_{G}+f}+\frac{mp}{1+m}-\frac{fp}{1+f}, \tag{5}\]  
\[\frac{dp}{dt}=\sigma_{p}(q_{f})\gamma\frac{f^{n}}{K_{R}^{n}+f^{n}}-k_{a}p^{2}, \tag{6}\]  
\[\frac{dg}{dt}=\sigma_{g}(q_{f})\beta\frac{f^{l}}{K_{S}^{l}+f^{l}}-k_{deg}\frac {gp}{K_{deg}+g}. \tag{7}\]Mathematically speaking, the CA model has several parameters, including \(\gamma=0.026\), \(\beta=0.00075\), \(K_{R}=4.5\), \(K_{S}=1\), \(K_{G}=0.1\), \(K_{deg}=1.1\), \(k_{g}=k_{deg}=0.05\), and \(k_{a}=\frac{k_{deg}}{k_{deg}}=0.0455\). Hill numbers \(n\) and \(l\) are also set to four. Bifurcation parameter \(r_{im}\) exhibits a wide range of dynamic behaviors, including stable periodic cycles, bistability, and chaos. This parameter remains constant for all oscillators within the chaotic domain. Additionally, an external input is applied as a perturbation to the \(r_{im}\) parameter as described in Equation (8). This parameter links different oscillators together by using a relative scale contribution from all other oscillators. RCC control parameters presented in Equations (1)-(3) (\(f_{p}=f_{g}=1\), \(x_{p}=x_{g}=-1\), and \(\mu_{f}=2\)) are kept constant throughout the experiment simulations in this paper, but can have different values that allow the local oscillator possibility to change its oscillatory orbits. \[r_{im}{}^{i}=\sum_{j=1,j\
eq i}^{n}w_{j}m_{j}+\varepsilon. \tag{8}\]  
The connectivity strength between various oscillators, represented by \(w_{j}\), can range from \(0.00011\), \(0.00012\), to \(0.00025\). External perturbations, represented by \(\varepsilon\), are uniformly distributed according to a Gaussian distribution and scaled within the domain of \([-1,1]\). These perturbations are observed over a range of evolution steps to explore the varying oscillatory cycles they produce. In this paper, a connectivity strength of \(w_{j}=0.0002\) was selected from the chaotic domain of the underlying oscillators to assess its effect on the dynamics while maintaining overall stability. The network of nonlinear models in this paper consists of 16 oscillators, each of which can adjust their local dynamics to adapt to external perturbations from their neighboring oscillators. The simulation of the entire model was carried out using EuNeuerone software (v2.3, 2013) and the Fehlberg-RK method as a fixed step integration for Ordinary Differential equations (ODEs). The total unweighted dynamics, represented by \(M\) and \(F\) in Equations (9) and (10), were measured as the net sum of the individual oscillators, allowing for observation by a remote observer who would otherwise be unable to detect the individual oscillators.