Molecular Timing Channels under Pulsatile Drift: A Corrected Inverse-Gaussian Approximation

Abstract

In a one-dimensional molecular timing channel with a perfectly absorbing receiver, constant positive drift yields an inverse-Gaussian (IG) first-hitting-time distribution, whereas pulsatile drift makes the arrival statistics depend on the molecular release phase. We propose a phase-normalized corrected inverse-Gaussian (C-IG) approximation for pulsatile drift with positive mean. The model combines an exponential term determined by cumulative drift with a Gaussian positive-part prefactor. A phase-dependent normalization factor ensures unit probability mass. For a known periodic environment, these factors can be tabulated offline, enabling subsequent pointwise density evaluation without solving a time-recursive integral equation. The model also recovers the classical IG law exactly under constant positive drift. Comparisons with a numerical Volterra solution and independent particle simulations assess both density and cumulative-distribution accuracy. In a representative pulsatile case, C-IG better captures the oscillatory density structure than constant-drift IG models, while an oracle IG fit achieves a smaller cumulative-distribution error. An amplitude-frequency sweep over release phases at a fixed Peclet number identifies the operating regimes in which C-IG meets prescribed accuracy criteria. Most tested settings below the flow-reversal threshold meet these criteria, whereas transient flow reversal substantially reduces accuracy and can lead to discrepancies in density peak counts. These results support C-IG as a computationally convenient model for phase-dependent arrival statistics within its numerically assessed range of validity.