# Stochastic Disruption of Synchronization Patterns in Coupled Non-Identical Neurons
## 3 Stochastic Model
### Stochastic Deformations in the Tristability Case  
First, consider system (2) with \(\sigma=0.003\). As shown in the previous section, at this coupling strength, the deterministic system (1) exhibits the coexistence of three three-cycles (\(\mathcal{O}\), \(\mathcal{B}\), and \(\mathcal{G}\)), as illustrated in Figure 3a. Figure 4 demonstrates how stochastic perturbations in the coupling parameter influence the system dynamics. In Figure 4a-c, we plot the evolution of the difference between the \(x\)- and \(y\)-coordinates (\(x-y\)) of solutions corresponding to cycles \(\mathcal{O}\) (orange), \(\mathcal{G}\) (green), and \(\mathcal{B}\) (blue) under three different noise intensities. Figure 3: Attractors and their basins of attraction of system (1) with \(\gamma_{1}=-1.75\) and \(\Delta=0.002\) at coupling strengths: (**a**) \(\sigma=0.003\), tristability with coexisting 3-cycles \(\mathcal{B}\), \(\mathcal{G}\), and \(\mathcal{O}\) shown in blue, green, and orange; (**b**) \(\sigma=0.01\), bistability with coexisting 3-cycles \(\mathcal{G}\) and \(\mathcal{O}\) shown in green and orange; (**c**) \(\sigma=0.02\), monostability with the 3-cycle \(\mathcal{O}\) shown in orange; and (**d**) \(\sigma=0.05\), monostability with the chaotic attractor (orange) having positive Lyapunov exponent \(\Lambda=0.16\). For weak noise (\(\varepsilon=0.0004\)), stochastic solutions of system (2) that start at cycles \(\mathcal{O}\) or \(\mathcal{G}\) remain in these cycles, as shown in the upper two panels of Figure 4a. However, the trajectories that begin at cycle \(\mathcal{B}\) eventually switch to either cycle \(\mathcal{G}\) or \(\mathcal{O}\) after a transient period, as illustrated in the lower two panels. Thus, weak noise disrupts cycle \(\mathcal{B}\) while leaving the other cycles unaffected. As shown in Figure 4b, stronger noise (\(\varepsilon=0.004\)) not only eliminates cycle \(\mathcal{B}\) (lowest panel) but also destroys cycle \(\mathcal{G}\) (middle panel) while cycle \(\mathcal{O}\) remains unaffected (upper panel). Interestingly, solutions that start at \(\mathcal{G}\) (middle panel) exhibit mutual transitions between \(\mathcal{G}\) and \(\mathcal{B}\) (\(\mathcal{G}\leftrightarrow\mathcal{B}\)), eventually stabilizing near cycle \(\mathcal{O}\). Meanwhile, solutions originating from \(\mathcal{B}\) undergo sequential transitions \(\mathcal{B}\rightarrow\mathcal{G}\rightarrow\mathcal{O}\). Thus, regardless of the initial conditions, all transitions at this noise level ultimately lead to cycle \(\mathcal{O}\), making the system monostable. When the noise intensity increases to \(\varepsilon=0.1\), all three cycles lose stability. At this noise level, each cycle is disrupted by chaotic bursts, as shown in Figure 4c. The system undergoes the following stochastic transitions: \(\mathcal{O}\leftrightarrow\mathcal{CH}\), \(\mathcal{G}\rightarrow\mathcal{O}\leftrightarrow\mathcal{CH}\), and \(\mathcal{B}\rightarrow\mathcal{O}\leftrightarrow\mathcal{CH}\). Thus, regardless of the initial state, the stochastic system (2) transitions into an intermittent regime characterized by alternating small-amplitude stochastic oscillations near \(\mathcal{O}\) and large-amplitude chaotic oscillations \(\mathcal{CH}\). Random states of this chaotic regime are depicted in Figure 4d. Figure 5 illustrates the stochastic transformations of the \((x-y)\) variable as the noise intensity \(\varepsilon\) gradually increases, starting from deterministic cycles \(\mathcal{B}\) (Figure 5a), \(\mathcal{G}\) (Figure 5b), and \(\mathcal{O}\) (Figure 5c). These cycles are shown in the background in light colors. Colored dots  
Figure 4: Time series of the \((x-y)\) coordinate in the tristable system (2) with \(\gamma=-1.75\), \(\Delta=0.002\), and \(\sigma=0.003\), starting from cycles \(\mathcal{O}\) (orange), \(\mathcal{G}\) (green), and cycle \(\mathcal{B}\) (blue) under different noise intensities: (**a**) \(\varepsilon=0.0004\), (**b**) \(\varepsilon=0.004\), and (**c**) \(\varepsilon=0.1\).