gPC-based robustness analysis of neural systems through probabilistic recurrence metrics

TL;DR

The paper tackles how neural dynamical regimes endure parametric uncertainty by introducing probabilistic robustness analysis (PRA), which quantifies regime preservation in expectation rather than under all parameter realizations. It combines generalized polynomial chaos ($gPC$) surrogates to efficiently compute mean neural trajectories with recurrence-plot based persistence metrics, culminating in probabilistic regime preservation (PRP) plots that map regime robustness across parameter spaces. The authors demonstrate the framework on Hindmarsh-Rose (HR) single-neuron dynamics and Jansen-Rit (JR) cortical column models, revealing that plateau bursting is generally more robust than square-wave bursting and that chaotic regimes yield weak or indistinguishable mean patterns under averaging. The approach provides a versatile, uncertainty-aware toolkit for interpretable analysis of neural dynamics and can inform robust model validation and control strategies in neuroscience and related fields, with future work extending higher-order moments and broader applicability.

Abstract

Neuronal systems often preserve their characteristic functions and signalling patterns, also referred to as regimes, despite parametric uncertainties and variations. For neural models having uncertain parameters with a known probability distribution, probabilistic robustness analysis (PRA) allows us to understand and quantify under which uncertainty conditions a regime is preserved in expectation. We introduce a new computational framework for the efficient and systematic PRA of dynamical systems in neuroscience and we show its efficacy in analysing well-known neural models that exhibit multiple dynamical regimes: the Hindmarsh-Rose model for single neurons and the Jansen-Rit model for cortical columns. Given a model subject to parametric uncertainty, we employ generalised polynomial chaos to derive mean neural activity signals, which are then used to assess the amount of parametric uncertainty that the system can withstand while preserving the current regime, thereby quantifying the regime's robustness to such uncertainty. To assess persistence of regimes, we propose new metrics, which we apply to recurrence plots obtained from the mean neural activity signals. The overall result is a novel, general computational methodology that combines recurrence plot analysis and systematic persistence analysis to assess how much the uncertain model parameters can vary, with respect to their nominal value, while preserving the nominal regimes in expectation. We summarise the PRA results through probabilistic regime preservation (PRP) plots, which capture the effect of parametric uncertainties on the robustness of dynamical regimes in the considered models.

Figures (7)

• Figure 1: Workflow of our methodology for probabilistic robustness analysis (PRA) described in Section \ref{['sec:methodology']}. (a) Consider a dynamical system of interest, where the evolution of the state vector $\mathbf{x} \in \mathbb{R}^n$ depends on a vector $Z$ of $d$ uncertain parameters, independent and distributed according to uniform distributions within known bounding intervals. (b) Derive a surrogate model of the system with the gPC method. (c) Based on the surrogate model, efficiently compute the mean system output (blue) with respect to the parametric uncertainty $Z$; the corresponding deterministic output of the nominal, uncertainty-free system is shown (purple) for comparison. (d) To unravel key geometric patterns within the time series associated with the mean system output, construct the corresponding recurrence plot (left). Then, automatically extract persistent features (blob count) of recurrence plots corresponding to increasing uncertainty levels, so as to assess regime preservation: the regime is preserved for all the uncertainty levels that achieve a similar blob count, while it is no longer preserved once the blob count drops (right). (e) Capture the overall information about dynamical regimes and their robustness to parametric uncertainty in a probabilistic regime preservation (PRP) plot, where the colour encodes blob counts and the length of each side of the rectangles quantifies how much uncertainty the system can withstand in the corresponding parameter value while maintaining the same regime in expectation. (f) For comparison, in the same parameter space as the PRP plot, we present a bifurcation plot (image courtesy of Springer, see barrio2011parameter for additional details) that identifies the regimes resulting from different deterministic values of the parameters.
• Figure 2: For the HR model \ref{['eq:HR']}, deterministic realisations of $x_1(t)$ (top) and corresponding normalised recurrence plots in the time window $[600,1200]$ (bottom), with parameters $b \in [2.5,3.3]$ and $I \in [2.2,4.4]$ that yield different regimes: (A) $b=2.8$ and $I=4.2$, quiescence; (B) $b=3.1$ and $I=2.4$, tonic spiking; (C) $b=2.65$ and $I=2.4$, square-wave bursting; (D) $b=2.5$ and $I=4.0$, plateau bursting. In (C) and (D), the black horizontal line identifies a burst of spikes.
• Figure 3: Degradation of recurrence plot patterns associated with the HR square-wave bursting regime in Figure \ref{['fig:deterministic_RP']} (C) for increasing uncertainty levels $i=1,\ldots,N$, with $N=10$. Here $I=2.4$, while $b\sim\mathcal{U}([b^*,b^*+\Delta b_{i}])$ for the $i$-th uncertainty level, with $b^*=2.7$ (in red) and $\Delta b_i = \frac{i}{N} \Delta b_{\max}$ with $\Delta b_{\max}=0.2$ (i.e., $\Delta b_i=0.02i$ for $i=1,\ldots,10$). The nominal deterministic recurrence plot with $I=2.4$ and $b=b^*=2.7$ in shown in Supplementary Figure S1.
• Figure 4: Automated blob count for (a) the recurrence plot $D$ of the mean output $\mathbb{E}[x_1](t)$ for the HR model \ref{['eq:HR']}, with $b=2.5$ and $I\sim \mathcal{U}([3.6,3.8])$. (b)-(c)-(d) Matrix $D$ is visualised as a surface and different Boolean versions $\mathfrak{D}$ are obtained by processing with different thresholds: (b) $\vartheta=0.1$; (c) $\vartheta=0.5$; (d) $\vartheta=0.9$. (e) We discard blobs smaller than $\mathfrak{B}=150$ pixels and show the blob count $\mathfrak{C}$ as a function of $\vartheta$. (f) Persistence bar chart; among the blob counts with persistence larger than a user-defined minimum persistence $\mathfrak{P}=0.05$ (dashed red line), we select $\mathfrak{C}^*=24$ as it corresponds to lower thresholds; in fact, as shown in panel (e), $\mathfrak{C}=24$ is obtained for the thresholds highlighted in green, $\mathfrak{C}=0$ for those in orange.
• Figure 5: Blob counts with increasing uncertainty level for the recurrence plots of $\mathbb{E}[x_1]$ for the HR model \ref{['eq:HR']}, with $I=2.8$ and $b\sim \mathcal{U}([2.7,2.7+0.03 i])$, $i=1,\ldots,5$. A significant blob count change occurs for $i=3$, when the blob count drops below $50$% of the first blob count (dashed red line): the maximum tolerable uncertainty level is $i^*=2$.