Membrane Sensitivity and Deployment Fragility of Learnable Time Constants in Spiking Neural Networks

Spiking neural networks (SNNs) with learnable membrane time constants can improve temporal processing by adapting neuronal integration timescales, but they also turn the decay factor $\beta$ into an optimized dynamical parameter that is fragile under hardware-induced parameter mismatch. We develop a perturbation-based account of how time-constant variation affects trained SNNs. In controlled software simulations, this fragility is strongly task-dependent: Spiking Heidelberg Digits (SHD) and Spiking Speech Commands (SSC) settings lose $7\text{--}20$ percentage points of accuracy under $20$% coefficient-of-variation (CV) perturbations in $\beta$, whereas DVS-Gesture and CIFAR-10 lose less than $3$ pp. We show that, under a surrogate-linearized first-order analysis, the failure mode is governed by the membrane sensitivity $\Psi$. This quantity admits a closed-form online recursion requiring only one additional state per layer and no extra temporal storage. The analysis identifies time-averaged sensitivity energy as the controllable term in deployment fragility, leading to $\textbf{MARS}$ ($\textbf{M}$embrane-$\textbf{A}$ware $\textbf{R}$obustness through $\textbf{S}ensitivity$ Regularization), a training objective derived from perturbation analysis rather than heuristic noise augmentation. We prove a conservative worst-case output-perturbation bound and introduce a typical-case diagnostic explaining why sensitivity remains predictive when worst-case constants are vacuous. At $20$% CV perturbation, MARS reduces the accuracy drop in the sensitive settings to at most $0.5$ pp while preserving nominal accuracy, with minimal effect on DVS-Gesture and CIFAR-10.