Probabilistic robustness of dynamical systems via generalised Polynomial Chaos

Many natural systems are characterised by substantial uncertainty in model parameters, arising from biological variability, measurement limitations, or incomplete knowledge; still, they exhibit preservation of fundamental properties and qualitative behaviours. When desired behaviours are not guaranteed for all parameter values in a given set, probabilistic robustness methods can help identify the mechanisms responsible for their loss and the critical parameters for the system’s performance.
To quantify the preservation of dynamical behaviours affected by uncertain parameters with a known probability distribution, we present a probabilistic robustness analysis framework that achieves efficient uncertainty-quantification through generalised Polynomial Chaos (gPC) techniques, thus reducing the computational burden with respect to Monte Carlo approaches. After introducing the theoretical foundations of gPC expansions, we discuss how they can be employed to quantify the preservation of dynamical regimes in expectation. The methodology relies on efficiently computing mean system output through gPC surrogate models to then construct recurrence plots, which are used to detect loss of regime preservation via blob counts. We assess the probabilistic robustness of regimes for key models in neuroscience, the Hindmarsh-Rose and the Jansen-Rit model, showing how probabilistic robustness can be quantified systematically even in the presence of multiple uncertain parameters. The seminar highlights how probabilistic methods complement classical structural and robust approaches, enabling the analysis of systems whose behaviour depends critically on uncertain quantitative parameters.

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