# Stochastic Disruption of Synchronization Patterns in Coupled Non-Identical Neurons
## 3 Stochastic Model
### Confidence Domain Method  
The cycle breakdown process described above can be analyzed analytically using the confidence domain method [31], where the mutual arrangement of confidence domains and attractors' basins of attraction plays a key role. Below, we briefly present details of the confidence domain method. Consider a deterministic system with an exponentially stable three-cycle consisting of successive states \((\bar{x}_{1},\bar{y}_{1})\), \((\bar{x}_{2},\bar{y}_{2})\), and \((\bar{x}_{3},\bar{y}_{3})\). Under random disturbances, the stochastic sensitivity of these states is determined by the corresponding matrices \(W_{1}\), \(W_{2}\), and \(W_{3}\). These matrices satisfy the following algebraic equations (see mathematical details in [31]):  
\[W_{2}=F_{1}W_{1}F_{1}^{\top}+Q_{1},\quad W_{3}=F_{2}W_{2}F_{2}^{\top}+Q_{2}, \quad W_{1}=F_{3}W_{3}F_{3}^{\top}+Q_{3}.\]  
Here, \(F_{i}\) is the Jacobi matrix at the point \((\bar{x}_{i},\bar{y}_{i})\) of the three-cycle, while the matrices \(Q_{t}\) characterize the stochastic disturbances. For system (2), the matrix \(Q_{t}\) is given by  
\[Q_{t}=(g_{t}-\bar{x}_{t})^{2}\Bigg{[} \begin{array}{cc}1&-1\\ -1&1\end{array} \Bigg{]}.\]  
The stochastic sensitivity matrix \(W_{1}\) satisfies the matrix equation  
\[W_{1}=FW_{1}F^{\top}+Q,\]  
where \(F=F_{3}F_{2}F_{1}\), \(Q=Q_{3}+F_{3}Q_{2}F_{3}^{\top}+F_{3}F_{2}Q_{1}F_{2}^{\top}F_{3}^{\top}\). The other stochastic sensitivity matrices, \(W_{2}\) and \(W_{3}\), can be determined using the recurrence relations provided earlier. These stochastic sensitivity matrices are useful for approximating the dispersion of random states around the points of the deterministic three-cycle. Let \(\rho_{t}(x,y)\) be the stationary probability density distribution of stochastic solutions near the state \((\bar{x}_{t},\bar{y}_{t})\) of the deterministic cycle. Then, the Gaussian approximation of the function \(\rho_{t}(x,y)\) is written as  
\[\rho_{t}(x,y)\approx K\exp\Bigg{[}\frac{1}{2\varepsilon^{2}}\Bigg{(} \begin{array} []{c}x-\bar{x}_{t}\\ y-\bar{y}_{t}\end{array} \Bigg{)}^{\top}W_{t}^{-1}\Bigg{(} \begin{array}{c}x- \bar{x}_{t}\\ y-\bar{y}_{t}\end{array} \Bigg{)}\Bigg{]}. \tag{3}\]  
So, the stochastic sensitivity matrices \(W_{t}\) play a key role in describing the dispersion of random states near points of the deterministic cycle. The approximations (3) can be used to geometrically characterize the distribution of random states in the form of confidence ellipses. For each state \((\bar{x}_{t},\bar{y}_{t})\) of the three-cycle, the confidence ellipse is defined as  
\[\frac{\alpha_{1}^{2}}{\mu_{1,t}}+\frac{\alpha_{2}^{2}}{\mu_{2,t}}=-2\varepsilon ^{2}\ln(1-P),\qquad(t=1,2,3).\]  
Here, \(\alpha_{1}\) and \(\alpha_{2}\) are the coordinates of the ellipse in the basis of normalized eigenvectors \(u_{1,t}\) and \(u_{2,t}\) of the matrix \(W_{t}\), while \(\mu_{1,t}\) and \(\mu_{2,t}\) are the corresponding eigenvalues. The parameter \(P\) represents the fiducial probability. In Figure 6, we present magnified fragments of the phase plane of the deterministic system (1), highlighting the basins of attraction near specific states of the three-cycles. The basin points are color-coded: light blue for cycle \(\mathcal{B}\), light green for cycle \(\mathcal{G}\), and white for cycle \(\mathcal{O}\). A notable feature of these basins is the distinction between solid and fractal parts. The solid region surrounds the cycle points, while outside this area, the basins of all three three-cycles are fractally interwoven. In addition to these deterministic basins, Figure 6 also displays confidence ellipses for the stochastic system (2). As the noise intensity increases, these ellipses expand in size. As long as a confidence ellipse remains within the solid region of a basin, no transitions occur. However, once the ellipse extends beyond the solid part and overlaps with the fractal region, it begins to encompass points from the basins of other coexisting attractors.