Corrected-Inverse-Gaussian First-Hitting-Time Modeling for Molecular Communication Under Time-Varying Drift

TL;DR

The paper addresses first-hitting-time modeling for molecular channels subject to time-varying drift, where stationary inverse-Gaussian models fail to capture phase and multi-pulse effects. It introduces a change-of-measure (Girsanov) framework that decomposes the FHT density into a macroscopic exponential core governed by the cumulative drift displacement $M(t)$ and a stochastic boundary-flux modulation via an Expected Positive Flux operator, yielding the explicit Corrected-Inverse-Gaussian (C-IG) density. The C-IG density, given by $f_{C-IG}(t)= \frac{F_{\text{smooth}}(t)}{\sqrt{2\pi\sigma^2 t^3}} \exp\left(- \frac{(\ell - x_0 - M(t))^2}{2\sigma^2 t}\right)$, is validated against high-precision Monte Carlo simulations for both pulsatile and abrupt-drift scenarios, demonstrating accurate phase modulation and backflow handling while preserving $\mathcal{O}(1)$ computational complexity. This physics-informed framework offers a practical tool for system-level analysis and receiver design in dynamic biological and molecular communication environments.

Abstract

This paper develops a tractable analytical channel model for first-hitting-time molecular communication systems under time-varying drift. While existing studies of nonstationary transport rely primarily on numerical solutions of advection--diffusion equations or parametric impulse-response fitting, they do not provide a closed-form description of trajectory-level arrival dynamics at absorbing boundaries. By adopting a change-of-measure formulation, we reveal a structural decomposition of the first-hitting-time density into a cumulative-drift displacement term and a stochastic boundary-flux modulation factor. This leads to an explicit analytical expression for the Corrected-Inverse-Gaussian (C-IG) density, extending the classical IG model to strongly nonstationary drift conditions while preserving constant-complexity evaluation. High-precision Monte Carlo simulations under both smooth pulsatile and abrupt switching drift profiles confirm that the proposed model accurately captures complex transport phenomena, including phase modulation, multi-pulse dispersion, and transient backflow. The resulting framework provides a physics-informed, computationally efficient channel model suitable for system-level analysis and receiver design in dynamic biological and molecular communication environments.

Figures (4)

• Figure 1: Schematic illustration of the 1D molecular communication system under pulsatile flow. Information molecules are released by the Tx nanomachine at $x_0$ and propagate through a fluid medium characterized by a constant diffusion coefficient $\sigma^2$ and a time-varying drift velocity $\mu(t)$. The channel impulse response is determined by the First-Hitting Time (FHT) $T$, the random instant when the stochastic trajectory $X_t$ first contacts the absorbing boundary at $x = \ell$.
• Figure 2: Methodological flowchart of the proposed framework. The approach leverages a change-of-measure formulation to decompose nonstationary transport into deterministic cumulative displacement and stochastic flux modulation, synthesized via the EPF correction into the final analytical C-IG model.
• Figure 3: First-hitting-time distributions under sinusoidal pulsatile drift. The proposed C-IG density captures phase shifts and amplitude modulation, whereas the classical IG model fails to represent nonstationary transport effects.
• Figure 4: First-hitting-time distributions under single-step drift switching with $T_{sw}=1.5$. The proposed C-IG density remains accurate under abrupt drift variation, while the classical IG model fails to align with the observed phase transition.

Theorems & Definitions (1)

• Remark : On Prefactor Stabilization