Spatiotemporal First-Arrival Modeling and Parameter Estimation in Drift-Diffusion Molecular Channels

TL;DR

This work derives the first closed-form joint distribution for the first arrival time and position in drift-diffusion molecular channels, capturing the inherent spatiotemporal coupling. By formulating the Fisher information and CRLB for the joint FAT–FAP model, it shows that diffusivity estimation benefits linearly from spatial dimensionality and that lateral drift becomes observable via joint statistics. The authors introduce efficient MLEs for lateral drift and demonstrate a spatiotemporal modulation scheme, Drift Shift Keying, that enables reliable communication where timing-only receivers fail. Overall, the study reveals that joint spatiotemporal processing can substantially enhance sensing precision and communication capabilities in nanoscale MC systems.

Abstract

We derive a closed-form joint distribution of the first arrival time (FAT) and first arrival position (FAP) in drift-diffusion molecular communication (MC) channels. In contrast to prior studies that analyze FAT or FAP in isolation, our framework explicitly captures the spatiotemporal coupling inherent in multidimensional transport. Building on this derivation, we compute the Fisher information matrix (FIM) and demonstrate that estimation accuracy for diffusivity scales proportionally with the spatial dimension, enabling increased sensitivity in higher-dimensional environments. Furthermore, we show that lateral drift -- which is unobservable from timing data alone -- can be recovered via a closed-form Maximum Likelihood Estimator (MLE) with a simple physical interpretation. Leveraging this spatial degree of freedom, we propose Drift Shift Keying (DSK), proving that joint receivers can reliably detect signals that are undetectable to timing-only receivers due to identical marginal FAT distributions. These results highlight the significant potential of spatiotemporal processing for future nanoscale communication and sensing.

Figures (5)

• Figure 1: System model illustration. A molecule is released from the transmitter (TX) at $\mathbf{x}_0$ and propagates via Brownian motion with drift $\mathbf{v}$ until it hits the receiver (RX) hyperplane at $x^{(1)}=1$. The first arrival position (FAP) $\mathbf{X}_T$ and first arrival time (FAT) $T$ are recorded upon absorption.
• Figure 2: Performance of the derived lateral drift estimator. The simulation results (markers) tightly align with the theoretical CRLB (solid line), validating the efficiency of the proposed MLE in Eq. \ref{['eq:mle_drift']}.
• Figure 3: CRLB comparison for diffusivity estimation ($\sigma$). Joint FAT--FAP observation achieves a strictly lower CRLB than classical FAT-only methods across all sample sizes, indicating that spatial statistics contribute additional Fisher information not present in classical FAT-only models.
• Figure 4: Conditional joint distribution for drift velocity vector $(1,0)~\mu\text{m}/\text{s}$. The figure displays (a) Empirical results from particle-based simulations and (b) Analytical results. The empirical surface accurately reproduces the theoretical peak and decay profile for the longitudinal-only drift scenario.
• Figure 5: Conditional joint distribution for drift velocity vector $(1,-3)~\mu\text{m}/\text{s}$. (a) Empirical results. (b) Analytical results. The empirical surfaces match the analytical joint PDF not only in peak location but also in the skewed tail induced by the lateral drift component.