PRP Plots in gPC-based Robustness

• Probabilistic Regime Preservation (PRP) plots are a technique that quantifies the resilience of system regimes under random parameter variations using gPC expansions and recurrence analysis.
• The method employs both intrusive (Galerkin projection) and non-intrusive (collocation/regression) approaches to compute spectral coefficients and derive interpretable robustness metrics.
• PRP plots have been successfully applied to neural models like the Hindmarsh–Rose and Jansen–Rit frameworks, offering insights for design optimization and safety assessments.

Generalised Polynomial Chaos (gPC) is a spectral uncertainty propagation technique that expands the solution of a dynamical system with random parameters onto an orthogonal polynomial basis matched to the probability distribution of the uncertain variables. The method enables probabilistic robustness analysis by efficiently calculating expectations and higher-order moments of system outputs as explicit functions of the random parameters. In contemporary applications, gPC is employed to quantify the regime-preservation properties of high-dimensional neuronal and other dynamical systems, propagating parametric uncertainty and generating interpretable robustness metrics in terms of mean signals and recurrence-based regime persistence (Sutulovic et al., 5 Jan 2026).

1. Mathematical Foundations of Generalised Polynomial Chaos

Let $Z=(Z_1,\ldots,Z_d)$ denote a vector of independent random variables, each distributed as $Z_i\sim\mathcal{U}([Z_{i,\min}, Z_{i,\max}])$. For a system governed by dynamics $\dot{\mathbf{x}} = f(\mathbf{x},Z)$ with initial state $\mathbf{x}(0)=\mathbf{x}_0$, the state trajectory $\mathbf{x}(t;Z)$ is a stochastic process in $L^2(\mathcal{A})$, with $\mathcal{A} = \prod_i [Z_{i,\min}, Z_{i,\max}]$ the uncertainty domain.

gPC posits a truncated expansion

$$x_j(t;Z) \approx \sum_{|\alpha| \leq M} x_{j,\alpha}(t)\, \Phi_\alpha(Z)$$

where $\{\Phi_\alpha\}_{|\alpha|\leq M}$ are multivariate orthonormal polynomials (e.g., Legendre for uniform, Hermite for Gaussian, etc.), $\alpha \in \mathbb{N}_0^d$, and the coefficients $x_{j,\alpha}(t) \in \mathbb{R}$ are deterministic and encode the input-output stochastic dependency.

The mean and variance of any output $X(Z)$ admit analytic form via the expansion: $$\mathbb{E}[X(Z)] = x_{0}, \quad \mathrm{Var}[X(Z)] = \sum_{i=1}^P c_i^2$$ with $x_0$ (or $c_0$) the coefficient corresponding to the zero-order basis function.

2. Computation and Algorithmic Procedures

Two principal approaches are used to obtain the gPC coefficients:

• Galerkin Projection (Intrusive gPC): The expansion is substituted into the dynamical system, followed by multiplication by each basis function and integration over the uncertainty domain. The resulting system of deterministic ODEs for the coefficients is solved, leveraging the orthonormality of the polynomial chaos basis.
• Collocation/Regression (Non-intrusive gPC): A set of collocation nodes $\{Z^{(s)}\}_{s=1}^S$ is chosen, and the deterministic system is solved for each parameter sample. The coefficients are fit by least-squares regression: $$x_j(t;Z^{(s)}) = \sum_{|\alpha| \leq M} x_{j,\alpha}(t) \Phi_\alpha(Z^{(s)}),\quad s=1,\ldots,S$$ with $S \geq P+1$, $P+1$ being the number of basis functions up to total degree $M$.

In practice, non-intrusive collocation is prevalent due to its black-box compatibility with legacy code.

3. Probabilistic Regime Metrics and Recurrence Plot Analysis

For dynamical regimes in neuroscience, regime robustness is quantified not directly by Lyapunov exponents or time series attributes, but through novel metrics based on recurrence plot analysis of the gPC mean signal (Sutulovic et al., 5 Jan 2026):

• The time-average mean signal $\bar{y}_k \equiv \mathbb{E}[y(t_k)]$ is computed on a fixed time grid.
• A distance matrix $D_{k\ell}=|\bar{y}_k-\bar{y}_\ell|$ is normalized and thresholded at several $\vartheta$ levels to generate binary recurrence images $\mathfrak D_{k\ell}(\vartheta)$.
• Connected components ("blobs") of the binary image, filtered by minimum area $\mathfrak B$ and persistence $\mathfrak P$, yield a blob count $\mathfrak C^*$ serving as the regime signature.
• The preservation of a regime is defined at escalating uncertainty levels (growing uncertainty boxes $\mathcal{A}_i$) by tracking the maintenance of blob count, with tolerance $\gamma$, and summarizing the results via the "probabilistic regime preservation" (PRP) metric—the maximum uncertainty volume at which the regime is preserved in expectation.

4. Applications in Robustness Analysis of Neural and Other Dynamical Systems

gPC-based PRA provides an automated pipeline for quantifying the parametric uncertainty volume compatible with the persistence of target dynamical regimes in biophysical and neural models. For example:

• Hindmarsh–Rose Neuron Model: Various dynamical regimes—plateau bursting, square-wave bursting, chaos—display vastly different PRP levels. Plateau bursting regimes tolerate the largest uncertainty neighborhoods, while chaotic and quiescent regimes tend to collapse under mean-based analysis.
• Jansen–Rit Cortical Column Model: The α-wave regime is demonstrably more probabilistically robust than low-frequency or no-wave regimes. PRP plots stratify the parameter space into robustness bands and illustrate the impact of parametric variation on functional outcome.

This methodology generalizes beyond neuroscience: any high-dimensional ODE/PDE model with uncertain but quantifiable parameters can be subjected to the same gPC expansion and PRP metrics.

5. Algorithmic Workflow and Reproducibility

A concise pseudocode for replicating the gPC-based PRA workflow is as follows:

for Z_star in parameter_grid: for i in range(1, N+1): uncertainty_box = grow_box(Z_star, i) collocation_nodes = select_nodes(uncertainty_box) # Solve ODE for each node x_data = [solve_ode(f, node) for node in collocation_nodes] gPC_coeffs = least_squares_fit(x_data, collocation_nodes) mean_signal = extract_mean(gPC_coeffs) distances = compute_distance_matrix(mean_signal) for theta in theta_grid: recurrence_image = binarize_distances(distances, theta) blob_count = count_blobs(recurrence_image, B) persistences = compute_blob_persistence(blob_count, P) C_star = first_persistent_count(blob_count, persistences) regime_preserved = check_regime_preservation(C_star, C_star_baseline, gamma) report_PRP(Z_star, C_star_max, regime_preserved)

Key parameters include the polynomial degree $M$, the minimum blob area $\mathfrak B$, persistence length $\mathfrak P$, and regime-preservation tolerance $\gamma$.

6. Contextual Significance and Limitations

gPC-based robustness analysis systematically quantifies the functional tolerance of high-dimensional dynamical systems to parametric uncertainty, yielding interpretable metrics for expectation-based regime preservation. Unlike classical worst-case robust control or Lyapunov function–based techniques, the approach exposes the detailed structure of uncertainty-induced transitions and connects the physical meaning of probabilistic robustness with high-level biological or engineering functionality (Sutulovic et al., 5 Jan 2026).

Limitations include:

• Complexity scaling in high-dimensional parameter spaces (collocation require exponential samples for large $d$ if naive tensor grids are used).
• Mean-based metrics collapse highly variable (e.g., chaotic) regimes; full distributional methods may be required for more granular analysis.
• The regime signature depends on metric selection in recurrence analysis; sensitivity to blob-count parameters $\mathfrak B, \mathfrak P$ and the time grid resolution may impact interpretability.

7. Implications and Future Directions

Generalised Polynomial Chaos forms the analytical backbone of modern probabilistic robustness methodologies for nonlinear dynamical systems. Its integration with recurrence-based persistence metrics enables quantitative, interpretable certification of functional regime preservation under stochastic parametric variation, directly informing design and safety analysis in computational neuroscience, control engineering, and complex systems modeling (Sutulovic et al., 5 Jan 2026). There is ongoing research towards scalable surrogate construction for very high-dimensional uncertainty (sparse grid, tensor-train, adaptive collocation) and refinement of regime metrics to address nontrivial distributional effects in highly variable regimes.