# Stochastic Disruption of Synchronization Patterns in Coupled Non-Identical Neurons
## 3 Stochastic Model
### Confidence Domain Method
This expansion marks the breakdown of the initial oscillatory regime and signals transitions to alternative oscillatory modes. Now, let us examine how the confidence domain method applies in the case of tristability. In Figure 6a, confidence ellipses are plotted around a point of cycle \(\mathcal{B}\) for two noise intensities: \(\varepsilon=0.0001\) and \(\varepsilon=0.0004\). For \(\varepsilon=0.0001\), the ellipse remains entirely within the solid region of the basin, preventing any noise-induced transitions. However, at \(\varepsilon=0.0004\), the ellipse extends into the fractal region of the basin, leading to the stochastic destruction of oscillations near \(\mathcal{B}\). A different scenario is observed near cycle \(\mathcal{G}\), where noise with \(\varepsilon=0.0004\) is insufficient to disrupt stochastic oscillations. As shown in Figure 6b, the confidence ellipse remains fully contained within the solid part of the \(\mathcal{G}\) basin, ensuring stability. The same holds for cycle \(\mathcal{O}\), as depicted in Figure 6c. Consequently, a higher noise intensity is required  
Figure 6: Illustration of the confidence domain method in the stochastic system (2) with \(\gamma=-1.75\), \(\Delta=0.002\), and \(\sigma=0.003\). Enlarged fragments of the phase plane of the deterministic system (1) show the solid and fractional parts of basins for cycle \(\mathcal{B}\) (light blue), cycle \(\mathcal{G}\) (light green), and cycle \(\mathcal{O}\) (white), along with confidence ellipses: (**a**) around the state \((2.3436,-1.0939)\) of cycle \(\mathcal{B}\) for \(\varepsilon=0.0001\) (blue) and \(\varepsilon=0.0004\) (red); (**b**) around the state \((0.1056,-1.1264)\) of cycle \(\mathcal{G}\) for \(\varepsilon=0.0004\); (**c**) around the state \((2.3088,2.2944)\) of cycle \(\mathcal{O}\) for \(\varepsilon=0.0004\); and (**d**) around the state \((2.3088,2.2944)\) of cycle \(\mathcal{O}\) for \(\varepsilon=0.05\) (blue) and \(\varepsilon=0.3\) (red). to destabilize cycle \(\mathcal{G}\). This effect is evident in Figure 6b, where the confidence ellipse for \(\varepsilon=0.004\) extends into the fractal region, signaling the onset of stochastic destruction. For cycle \(\mathcal{O}\), it remains stable even under stronger noise with \(\varepsilon=0.05\) (see blue confidence ellipse in Figure 6d). However, as shown in Figure 6d, when the noise intensity increases to \(\varepsilon=0.3\), the confidence ellipse extends into the fractal part of the basin, indicating the breakdown of noisy oscillations near \(\mathcal{O}\). Notably, the confidence ellipse method predicts that solutions starting from \(\mathcal{B}\) and \(\mathcal{G}\) transition to \(\mathcal{O}\) at \(\varepsilon=0.004\) (see Figure 6a,b). In contrast, the solutions originating from \(\mathcal{O}\) remain near \(\mathcal{O}\) even up to \(\varepsilon=0.05\) (see Figure 6d). This reveals a broad range of noise intensity, or an \"\(\varepsilon\)-window\", where all solutions of the stochastic system (2) reside near \(\mathcal{O}\). This finding is supported by the results of numerical simulations presented in Figure 5. Within this \(\varepsilon\)-window, regardless of the initial conditions, the \([0]\) synchronization mode persists. It is worth noting that the predictions made using the confidence ellipse method align well with the results of direct numerical simulations discussed in Section 3.1.