# Stochastic Disruption of Synchronization Patterns in Coupled Non-Identical Neurons
## 4 Transients And Intermittent Synchronization
### Basins of Short and Long Transients  
Although the monostable system (1) with \(\sigma=0.015\) ensures that all trajectories eventually converge to the three-cycle \(\mathcal{O}\), the rate of convergence strongly depends on the initial conditions. This behavior is illustrated in Figure 8(a), where initial points of solutions that take more than 200 iterations to converge to \(\mathcal{O}\) with an accuracy of 0.001 are marked in orange. These points define the long-transient basin (LTB), while the remaining region, corresponding to faster convergence, is marked in white and referred to as the short-transient basin (STB). Notably, the boundary between these basins exhibits a complex, fractal structure. In Figure 8(b), an enlarged fragment of the phase plane is shown near one of the points of three-cycle \(\mathcal{O}\), highlighting both the STB and LTB. The STB consists of a solid (white) region and a fractal part, represented by a mixture of white and orange areas. The selected three-cycle point lies within the solid part of the STB. We now examine how the transient behavior of solutions to the deterministic system (1) depends on the choice of the initial condition. Specifically, we consider two nearby points, \(A(2.32,2.28)\) and \(B(2.32,2.27)\). Point \(A\) lies within the solid part of the STB, whereas point \(B\) belongs to its fractal region. During transients, the system may alternate between synchronous (\(x=y\)) and asynchronous (\(x\
eq y\)) states. To quantify these states, we introduce the synchronization index defined as  
\[z_{t}=\text{sgn}[(x_{t+1}-x_{t})(y_{t+1}-y_{t})].\]  
Note that \(z_{t}=1\) when the neurons evolve synchronously (i.e., both variables increase or decrease together), and \(z_{t}=-1\) otherwise. Examples of synchronous and asynchronous transients are shown in Figures 8(c) and 8(d), respectively. Specifically, Figure 8(c) illustrates that the trajectory starting at point \(A\) converges to the three-cycle \(\mathcal{O}\) while maintaining in-phase synchronization (\(z_{t}=1\)). In contrast, Figure 8(d) shows that the trajectory from point \(B\) exhibits a large-amplitude burst with temporary desynchronization, characterized by the interval where \(z_{t}=-1\). The diversity of transient behaviors for various initial conditions \((x_{0},y_{0})\) near point \(B\), located in the fractal region, is illustrated in Figure 8(e). Notably, for \(y_{0}=2.2712\), the system exhibits three bursts before ultimately converging to the three-cycle \(\mathcal{O}\). The underlying reason for the noise-induced excitability described above and illustrated in Figure 8 lies in the presence of a fractal region associated with large-amplitude burst transient attractors in the deterministic system (1). For weak noise, random trajectories originating near the cycle \(\mathcal{O}\) remain close to its states within the solid part of the short-transient basin (STB), resulting in only minor stochastic fluctuations around the cycle. As the noise intensity increases, these trajectories may cross into the fractal region, triggering large-amplitude burst oscillations. This qualitative shift is anticipated by the positions of the confidence ellipses. In Figure 8(b), the ellipse for \(\varepsilon=0.02\) lies entirely within the solid part of the STB. However, the ellipse for \(\varepsilon=0.1\) partially overlaps the \"dangerous\" fractal zone. This overlap serves as an indicator of the onset of noise-induced excitation. The prediction aligns well with the results of direct numerical simulations (compare Figures 8 and 8(b)). It is important to note that the distinction between the solid and fractal regions in Figure 9a,b is based on a fixed threshold of \(N=200\) iterations. However, the convergence rate of solutions of system (1) to cycle \(\mathcal{O}\) depends on the specific initial condition. This dependence is illustrated in Figure 10, where the color map represents the number of iterations \(N\) required to reach the cycle with a convergence accuracy of 0.001.