---
title: Generalised Polynomial Chaos for Uncertainty Quantification
url: https://www.emergentmind.com/topics/generalised-polynomial-chaos
type: topic
---

# Generalised Polynomial Chaos for Uncertainty Quantification

Generalised polynomial chaos (gPC) is a mathematical framework for quantifying the impact of stochastic parametric uncertainties on the outputs and dynamical regimes of nonlinear systems, especially in the context of ordinary differential equations and dynamical systems encountered in neuroscience and engineering. gPC enables scalable, systematic uncertainty propagation and extraction of robust statistical features essential for probabilistic robustness analysis (PRA), particularly in high-dimensional parameter spaces and multi-regime systems [2601.02606].

## 1. Mathematical Foundations of Generalised Polynomial Chaos

In the gPC framework, uncertain parameters $Z=(Z_1,\ldots,Z_d)$ (typically independent random variables with known distributions, such as uniform or Gaussian) parameterize the ordinary differential equation (ODE) system
\[
\dot{\mathbf{x}} = f(\mathbf{x},Z),\qquad \mathbf{x}(0) = \mathbf{x}_0.
\]
Each state or output observable $x_j(t;Z)$ is regarded as a stochastic process in $L^2(\mathcal{A})$, where $\mathcal{A}=\prod_i [Z_{i,\min},Z_{i,\max}]$. gPC expresses $x_j(t;Z)$ as a finite expansion over a multivariate orthonormal polynomial basis $\{\Phi_\alpha(Z)\}_{|\alpha| \leq M}$:
\[
x_j(t;Z) \approx x_{j,M}(t;Z) = \sum_{|\alpha|\leq M} x_{j,\alpha}(t)\,\Phi_\alpha(Z),
\]
with $\alpha = (\alpha_1,\dots,\alpha_d) \in \mathbb{N}_0^d$, typically requiring $P+1 = \binom{M+d}{d}$ basis polynomials for total degree $M$.

The coefficients $x_{j,\alpha}(t)$ capture the contribution of each polynomial basis function, and are either computed by intrusive Galerkin projection (integrating out the random parameter dimensions via orthonormality) or by non-intrusive collocation, where the original ODE is solved for $S \geq P+1$ quadrature/collocation points in parameter space and the coefficients are fit by regression.

This polynomial expansion enables efficient extraction of all moments of the process:
\[
\mathbb{E}\bigl[x_j(t;Z)\bigr] = x_{j,\mathbf{0}}(t),\qquad \mathrm{Var}\bigl[x_j(t;Z)\bigr] = \sum_{|\alpha|\neq0} [x_{j,\alpha}(t)]^2.
\]
This drastically reduces the computational complexity of uncertainty propagation compared to brute-force Monte Carlo.

## 2. Robustness Metrics via gPC and Recurrence Analysis

The main application focus is the quantification of how much parametric uncertainty a neurodynamical system can tolerate while preserving a given dynamical regime, such as quiescence, tonic spiking, or bursting. Regime preservation is assessed through "probabilistic recurrence metrics" based on recurrence plots of the mean neural activity signals.

Given the mean output ($\mathbb{E} [y(t ; Z)]$), the empirical recurrence matrix is constructed as
\[
D_{k\ell} = |\bar{y}_k - \bar{y}_\ell |,\qquad 1 \le k, \ell \le T,
\]
normalized to $D \in [0, 1]^{T \times T}$. For each threshold $\vartheta \in (0,1)$, the binary recurrence plot $\mathfrak{D}_{k\ell}(\vartheta)$ is generated by thresholding $D_{k\ell}$, then one counts topological features ("blobs", i.e. connected components, after discarding micro-blobs below a pixel-area threshold).

To extract robust, regime-specific signatures, the persistence of each blob count is measured as the maximal $\vartheta$-interval over which the count remains unchanged; only those counts with sufficient persistence are accepted for regime characterization. This procedure yields a regime signature $\mathfrak{C}^*$ at each uncertainty level.

## 3. Probabilistic Regime Preservation: Quantifying Robustness

gPC enables parametric sweeps over increasing uncertainty sets $\mathcal{A}_i$ centered at a nominal parameter $Z^*$, with $\mathcal{A}_1\subset\mathcal{A}_2\subset\cdots\subset\mathcal{A}_N$ and diameter or hypervolume defined by the uncertainty axis lengths. For each $\mathcal{A}_i$, one computes the regime signature $\mathfrak{C}_i^*$. Regime preservation is said to hold at level $i$ if
\[
\gamma\,\mathfrak{C}_1^* \leq \mathfrak{C}_i^* \leq (1+\gamma)\,\mathfrak{C}_1^*,\qquad \gamma\in(0,1),
\]
and the largest such $i^*$ yields $P(\text{regime preserved}\mid Z\in\mathcal{A}_N) = \mathrm{Vol}(\mathcal{A}_{i^*})/\mathrm{Vol}(\mathcal{A}_N)$ (or simply $i^*/N$ in 1D). This probability encodes how much uncertainty can be tolerated before the regime statistically degrades—a stochastic, expectation-based notion of regime robustness.

"Probabilistic Regime Preservation (PRP) plots" visualize both the maximal observed regime cardinality and the maximal tolerated uncertainty for regime preservation, thus directly linking system-theoretic questions to interpretable metrics extracted via gPC.

## 4. Algorithmic Workflow and Numerical Aspects

The full PRA pipeline under gPC consists of:

1. For each nominal $Z^*$ of interest:
   - Construct a hierarchy $\{\mathcal{A}_i\}$ of uncertainty sets around $Z^*$.
2. For each $\mathcal{A}_i$:
   - Build the gPC surrogate via regression or Galerkin projection, using tensorized Legendre polynomials for uniform $Z$.
   - Extract the mean activity signal and compute the recurrence plot and persistence-filtered blob count $\mathfrak{C}_i^*$.
3. Compare the collection $\{\mathfrak{C}_i^*\}$ to the baseline regime $\mathfrak{C}_1^*$, identify maximal $i^*$ with preservation, and record regime robustness and the corresponding preservation probability.
4. Post-process as PRP plots summarizing the geometry of regime boundaries and the probabilistic robustness of dynamical behaviors.

gPC-based uncertainty propagation is orders of magnitude more efficient than classic Monte Carlo, and the approach decouples uncertainty propagation from regime-feature extraction, allowing systematic scaling to high-dimensional parametric models [2601.02606].

## 5. Domain-Specific Application: Neuroscience Systems

The framework is demonstrated on two canonical neuroscience models:

**Hindmarsh-Rose neuron**: For $(b, I)$ parameter space, regimes including quiescence, spiking, square-wave bursting, plateau bursting, and chaos are mapped. PRA reveals, for instance, plateau bursting as exhibiting the largest robustness (largest uncertainty set over which the regime is preserved in expectation), while chaotic and quiescent regimes collapse to near-constant mean signals under increasing uncertainty.

**Jansen–Rit cortical column**: For parameters $(A,B)$ and combinations of input and coupling, PRA reveals that alpha-wave regimes are the most robust to parameter uncertainty, while low-frequency waves degrade at smaller uncertainty levels. The approach enables fine-grained, expectation-based stratification of the dynamical repertoire accessible to a network under uncertainty.

This regime-level robustness analysis directly answers practical neuroscience questions about the effect of intrinsic and extrinsic parameter variability on global neural function.

## 6. Significance and Limitations

gPC-based PRA constitutes a systematic, expectation-driven approach distinct from worst-case or deterministic robustness analysis [2601.02606]. It quantifies not only what behavior a system can achieve nominally, but under what measure-theoretic envelope of parametric uncertainty such behavior is preserved in an average sense. The recurrence-plot derived metrics provide regime-distinguishing, topologically robust signatures that avoid the need for classical embedding or periodicity detection. Laplacian limitations remain: extreme parametric degeneracies or highly state-dependent bifurcations may not be sharply delineated by mean-signal measures, and the method assumes independent random parameters amenable to polynomial chaos expansions.

Nevertheless, gPC provides a computationally tractable and theoretically principled basis for high-dimensional PRA, enabling uncertainty-aware design and exploration of complex nonlinear systems [2601.02606].

Source: https://www.emergentmind.com/topics/generalised-polynomial-chaos