# Stochastic Disruption of Synchronization Patterns in Coupled Non-Identical Neurons
## 2 Deterministic Model
### Dynamics of Coupled Neurons  
Now, consider the cooperative dynamics of the coupled system (1) with \(\gamma_{1}=-1.75\) and \(\gamma_{2}=-1.748\); i.e., the neurons are non-identical with a parameter mismatch of \(\Delta=0.002\). The influence of non-identity in coupled deterministic Rulkov neurons has been previously analyzed in detail in our recent paper [38] for \(\gamma=-1.75\), where we compared the dynamics for identical neurons (\(\Delta=0\)) and slightly non-identical ones (\(\Delta\in(0,0.002)\)) under varying coupling strength \(\sigma\). Our findings showed that although the overall qualitative behavior remains similar--i.e., transitions between ordered and chaotic regimes with increasing \(\sigma\)--the precise thresholds shift with increasing non-identity. Specifically, a larger \(\Delta\) tends to lower the critical value of \(\sigma\) at which the transition from regular to chaotic behavior occurs.  
Figure 1: Bifurcation diagram (blue) and Lyapunov exponent (red) of the isolated Rulkov system. The crisis bifurcation occurs at \(\gamma_{cr}=-1.74722\).  
While isolated neurons with these parameters exhibit three-cycles, the joint dynamics is governed by the coupling parameter \(\sigma\), as illustrated in Figure 2, which presents the bifurcation diagram of the \(x\) variable.  
At weak coupling strengths, system (1) exhibits tristability, with three coexisting three-cycles: \(\mathcal{B}\) (blue), \(\mathcal{G}\) (green), and \(\mathcal{O}\) (orange). These attractors and their basins of attraction are illustrated in Figure 3 for different values of the coupling parameter \(\sigma\). To compute the basins of attraction, we used a uniform grid of \(300\times 300\) initial conditions in the \((x,y)\) plane. Each point on the grid was integrated using system (1) for 1000 iterations. A trajectory was considered to belong to a given attractor if it converged to that attractor with a numerical tolerance of 0.001, based on the Euclidean distance between the final iterates and the attractor's orbit.  
Figure 3a illustrates the coexistence of three three-cycles at \(\sigma=0.003\). As the coupling increases, cycle \(\mathcal{B}\) is annihilated, reducing system (1) to a bistable state with two coexisting three-cycles, \(\mathcal{G}\) and \(\mathcal{O}\), at \(\sigma=0.01\) (Figure 3b). A further increase in \(\sigma\) eliminates cycle \(\mathcal{G}\), making the system monostable with a single remaining attractor, cycle \(\mathcal{O}\), at \(\sigma=0.015\) (Figure 3c). Finally, at the crisis bifurcation occurring at \(\sigma_{*}=0.019011\), cycle \(\mathcal{O}\) loses stability, causing the system to transition into chaos. The resulting chaotic attractor is shown in Figure 3d for \(\sigma=0.05\).  
The cycles \(\mathcal{B}\), \(\mathcal{G}\), and \(\mathcal{O}\) correspond to different synchronization patterns. Solutions originating from the basins of \(\mathcal{B}\) and \(\mathcal{G}\) exhibit synchronization in a one-step shift mode. In the basin of \(\mathcal{B}\), solutions stabilize to a three-cycle where the \(y\)-coordinate is one step ahead of the \(x\)-coordinate, which we denote as the \([-]\) synchronization mode. Conversely, in the basin of \(\mathcal{G}\), solutions stabilize to a three-cycle for which the \(y\)-coordinate lags one step behind the \(x\)-coordinate, referred to as the \([+]\) synchronization mode. Finally, solutions in the basin of \(\mathcal{O}\) stabilize to a three-cycle without any coordinate shift, denoted as the \([0]\) synchronization mode. Thus, as the coupling increases, the system first loses the \([-]\) synchronization mode, followed by the \([+]\) synchronization mode.  
Figure 2: Bifurcation diagram of system (1) with \(\gamma_{1}=-1.75\) and \(\Delta=0.002\). Here, the \(x\)-coordinates of the coexisting 3-cycles \(\mathcal{B}\), \(\mathcal{G}\), and \(\mathcal{O}\) are shown in blue, green, and orange, respectively. In the insert, we plot the enlarged fragment at small \(\sigma\).