# Stochastic Disruption of Synchronization Patterns in Coupled Non-Identical Neurons
## 4 Transients And Intermittent Synchronization
### Intermittent Synchronization  
We now examine how the synchronous behavior of the stochastic system (2) evolves in the excitability regime as the noise intensity increases. Figure 11 presents time series of the variable \(x\) and the synchronization indicator \(z\) for solutions of the stochastic system (2), initialized at the deterministic three-cycle \(\mathcal{O}\). Two noise intensities are considered: \(\varepsilon=0.05\) and \(\varepsilon=0.1\). The time series reveal two distinct phases: laminar and turbulent. During the laminar phase, the system maintains in-phase synchronization (\(z=1\)), with trajectories fluctuating slightly around the points of the periodic three-cycle. In contrast, the turbulent phase is characterized by burst-like oscillations and a breakdown of synchronization. It is important to note that the durations \(\tau\) of the laminar intervals are random and tend to decrease as the noise intensity increases (compare Figure 11a,b). Since the turbulent phase emerges due to the presence of noise, this phenomenon belongs to noise-induced intermittency. Similar behavior has been observed in previous studies on systems with randomly modulated parameters [39; 40; 41; 42; 43; 44]. Intermittency is typically characterized by the distribution of laminar phase durations, computed for fixed control parameters, and by analyzing the dependence of the mean laminar duration on a critical parameter. Figure 11: Time series of the \(x\)-coordinate (green) and the synchronization indicator \(z\) (blue) for solutions of the stochastic system (2), initialized at the deterministic three-cycle \(\mathcal{O}\) for noise intensities (a) \(\varepsilon=0.05\) and (b) \(\varepsilon=0.1\). Figure 10: Color map showing the convergence rate of solutions of the monostable deterministic system (1) with \(\gamma=-1.75\), \(\Delta=0.002\), and \(\sigma=0.015\) to the three-cycle \(\mathcal{O}\). The color indicates the number \(N\) of iterations required to achieve a convergence accuracy of \(0.001\). The red dashed line represents the confidence ellipse for the corresponding stochastic system (2) with noise intensity \(\varepsilon=0.1\). In the statistical analysis, the time series of \(10^{7}\) steps was used. Laminar phases were defined as sequences where \(z=1\), and both the mean duration \(\langle\tau\rangle\) and the probability \(P(\varepsilon)\) were computed based on this threshold. Figure 12a illustrates the dependence of the mean duration of the laminar phase \(\langle\tau\rangle\) on noise intensity \(\varepsilon\) for different values of the coupling strength \(\sigma\). For \(\sigma=0.012\), \(\sigma=0.015\), and \(\sigma=0.017\), the deterministic system (1) possesses a single in-phase three-cycle attractor, whereas for \(\sigma=0.02\), the system exhibits a single chaotic attractor. The sharp decline in \(\langle\tau\rangle\) for the three-cycle cases marks the onset of noise-induced excitation, characterized by the emergence of temporal bursts. In contrast, the horizontal plateau observed in the chaotic regime suggests that the mean interburst intervals remain unchanged as noise intensity increases. Figure 12b shows the probability of the stochastic system (2) being in the co-directed mode as a function of noise intensity \(\varepsilon\) for various coupling strengths \(\sigma\). For all considered values of \(\sigma\), including the chaotic regime (\(\sigma=0.02\)), the system predominantly remains in the co-directed mode, with probabilities \(p>0.5\). Figure 12c presents the dependence of the mean laminar duration \(\langle\tau\rangle\) on the deviation parameter \(d\) for different values of \(\varepsilon\). As expected, the farther \(\sigma\) is from the crisis bifurcation point \(\sigma_{*}\), the longer the mean duration of laminar (co-directed) behavior. Analysis of the results presented in Figure 12 allows us to conclude that intermittency observed in our study deviates from canonical scaling laws.