# Stochastic Disruption of Synchronization Patterns in Coupled Non-Identical Neurons
## 3 Stochastic Model
### Stochastic Deformations in the Tristability Case
The arrows indicate the directions of transitions between coexisting 3-cycles. (**d**) Random states of system (2) with \(\varepsilon=0.1\). represent 100 successive random states of system (2) for solutions initiated at cycles \(\mathcal{B}\), \(\mathcal{G}\), and \(\mathcal{O}\), plotted after 1000 transient steps. The color of these dots matches the corresponding starting cycle but appears in a brighter shade. For weak noise, the random solutions remain close to their respective starting cycles. However, as the noise intensity increases, stochastic switches between coexisting states begin to appear, as shown in the time series in Figure 4a,b. These transitions are revealed in Figure 5 as dots between horizontal lines. In the case of \(\mathcal{B}\) (Figure 5a), these transitions disrupt the \([-]\) synchronization mode. As the noise intensity increases further, random states begin clustering around cycle \(\mathcal{O}\) (see the time series in Figure 4b). This indicates that both the \([-]\) and \([+]\) synchronization modes are destroyed, leaving only the \([0]\) synchronization mode intact. When the noise becomes sufficiently strong, solutions starting at \(\mathcal{B}\) transition directly into the intermittent regime \(\mathcal{O}\leftrightarrow\mathcal{CH}\), characterized by alternating small-amplitude stochastic oscillations near \(\mathcal{O}\) and large-amplitude chaotic oscillations near \(\mathcal{CH}\) (see the time series and random states in Figure 4c,d). Consequently, the \([0]\) synchronization mode is temporarily disrupted by random disturbances. Stochastic transformations of solutions starting from the three-cycle \(\mathcal{G}\) are shown in Figure 5b. For noise intensities up to \(\varepsilon=10^{-3}\), these solutions remain localized near the deterministic cycle. At \(\varepsilon=0.004\), stochastic transformations following the sequence \(\{\mathcal{G}\leftrightarrow\mathcal{B}\}\rightarrow\mathcal{O}\) are observed, as illustrated in the time series in Figure 4b (green trace). For stronger noise, the stochastic system transits to the regime \(\mathcal{G}\rightarrow\{\mathcal{O}\leftrightarrow\mathcal{CH}\}\) (see the time series shown in green in Figure 4c for \(\varepsilon=0.1\)). For solutions starting at the three-cycle \(\mathcal{O}\), only a single-stage stochastic transformation occurs, where noisy oscillations near \(\mathcal{O}\) transition into the intermittent mode \(\mathcal{O}\leftrightarrow\mathcal{CH}\) (see the orange time series in Figure 4c for \(\varepsilon=0.1\)). Figure 5: Stochastic transitions in the \((x-y)\) variable of solutions for the tristable system (2) with \(\gamma=-1.75,\Delta=0.002\), and \(\sigma=0.003\) as a function of noise intensity \(\varepsilon\), starting from (**a**) cycle \(\mathcal{B}\) (blue), (**b**) cycle \(\mathcal{G}\) (green), and (**c**) cycle \(\mathcal{O}\) (orange). The deterministic cycles are shown in the background by the corresponding light colors. To summarize the effect of noise in the tristability case, increasing noise intensity first destroys cycle \(\mathcal{B}\), followed by the destruction of cycle \(\mathcal{G}\), leaving cycle \(\mathcal{O}\) as the most resilient to noise.