Interview with Paul Bressloff

Picture. Paul Bressloff and Sean Lawley

Who are you?

I am Paul C Bressloff, Professor of Mathematics at the University of Utah, USA. My collaborator is Sean D Lawley, a postdoctoral research fellow at the Department of Mathematics, University of Utah.

What prompted you to pursue this field of research?

I have a long-standing interest in applying the theory of stochastic processes to problems in cell biology. There are a growing number of such problems that involve the coupling between a piecewise deterministic dynamical system and a time-homogeneous discrete Markov chain, resulting in a so-called stochastic hybrid system. A classical example is the membrane voltage fluctuations of a single neuron due to the stochastic opening and closing of ion channels. Here the discrete states of the ion channels evolve according to a continuous-time Markov process with voltage-dependent transition rates and, in-between discrete jumps in the ion channel states, the membrane voltage evolves according to a deterministic equation that depends on the current state of the ion channels. In the limit that the number of ion channels goes to infinity, one can apply the law of large numbers and recover classical Hodgkin–Huxley type equations. However, finite-size effects can result in the noise-induced spontaneous firing of a neuron due to channel fluctuations. Stochastic hybrid systems also arise in motor-driven intracellular transport, gene networks, and the immune response of T-cells. One important generalization of a stochastic hybrid system is to take the underlying continuous process to evolve according to a Langévin (stochastic differential) equation rather than a deterministic equation. This could describe, for example, a macromolecule diffusing in a bounded domain, in which part, or all, of the boundary randomly switches between an open and a closed state.

What is this latest paper all about?

Our latest paper uses the theory of stochastic hybrid systems to analyze the diffusion equation on a finite interval with a randomly switching boundary condition at one end. It builds upon the thesis work of Sean, who studied this problem using the theory of random iterative systems. Here we establish a deep connection between the solution of the resulting piecewise deterministic partial differential equation (PDE) and the Brownian motion of individual particles. In particular, we show how the r-th moment of the solution to the PDE can be interpreted in terms of the probability that r non-interacting Brownian particles all exit at the same boundary; although the particles are non-interacting, statistical correlations arise due to the fact that they all move in the same randomly switching environment. Hence the stochastic diffusion equation describes two levels of randomness; Brownian motion at the individual particle level and a randomly switching environment at the multi-particle level.

What do you plan to do next?

An immediate extension of our work is to replace pure diffusion by Brownian motion in a potential well with a randomly switching boundary. This then raises the question of how the presence of the switching boundary modifies the classical Kramer's formulae for the rate of escape from the well. Another extension is to consider Brownian motion in a two-dimensional or three-dimensional domain with a randomly switching boundary.