I applied a dimension-reduction technique for stochastic ODEs to a stochastic PDE to quantify how noise affects wave speed.

In the stochastic FitzHugh–Nagumo equation (left), noise introduces not only a mean-zero wobble in the propagation speed but also a systematic drift—positive or negative depending on the noise’s correlation structure.

Using singular-perturbation theory, I derived a kinematic model and applied the Isochronal Phase Reduction method. This formulation allows computation of the average noise-induced drift completely deterministically, without sampling random realizations of the wave. The approach greatly improves efficiency when estimating whether stochastic fluctuations make the wave travel faster or slower than in the noise-free case.

Ion channels embedded in a cell membrane (such as in an axon) open and close in response to the local voltage potential, but their behavior is inherently stochastic. The collective, averaged activity of many neighboring channels can nevertheless produce the coherent propagation of an action potential, as shown in the simulated toy model on the left. In the figure, we simulate the upstroke of an action potential using a hybrid stochastic–deterministic system.

This work focuses on developing, simulating, and homogenizing such models to understand how microscopic channel variability shapes macroscopic voltage propagation.

Sound waves with long wavelengths can more easily penetrate walls or other barriers. In a similar way, a compression wave with a long wavelength traveling through a mass–spring lattice can overcome both the discreteness of the lattice and microscopic heterogeneities in its material properties. If the springs obey Hooke’s law, dispersion eventually causes the wave to vanish; however, if the springs are nonlinear, certain waves can propagate essentially undistorted for much longer times and distances.

My research focuses on the stochastic effects—and sometimes the surprising lack of them—on such nonlinear waves as they travel through heterogeneous lattices.