Department of Mathematics
University of Utah
155 South 1400 East
Salt Lake City
Tel.: 801 585 1633
Fax.: 801 581 4148
Department of Mathematics
Tel.: 801 585 1633
There has been a resurgence of interest in non-equilibrium stochastic processes in biology, driven in part by the observation that the number of molecules involved in biochemical and gene networks are often small (low copy numbers). This means that deterministic mass-action kinetics tends to break down, and one needs to take into account the discrete, stochastic nature of biochemical reactions. Many switching processes involve the coupling between a piecewise deterministic dynamical system (continuous process) and a time-homogeneous Markov chain on some discrete space, resulting in a stochastic hybrid system, also known as piecewise deterministic Markov processes (PDMPs). The continuous process could represent the concentration of proteins synthesized by a gene, the membrane voltage of a neuron, the position of a molecular motor on a filament track, or the synaptic current into a population of neurons. The corresponding discrete process could represent the activation state of the gene, the state of voltage-gated ion channels, the velocity state of the motor, or the spiking activity of a neural population. Current research topics include the following
Diffusion in randomly switching environments: intracellular diffusion in domains with stochastically gated boundaries; intercellular diffusion via stochastic gap junctions; diffusion-limited biochemical reactions; age-structured switching processes.
Stochastic models of axonal and dendritic transport: PDE models of motor-based vesicular transport; synaptic democracy and vesicular transport; exclusion processes; aggregation models of intracellular transport; random intermittent search processes; protein transport and synaptic plasticity.
Stochastic hybrid systems : Analysis of noise-induced escape in metastable hybrid systems using large deviation theory and path-integrals; stochastic hybrid neural networks; phase reduction methods for hybrid neural oscillators.
A fundamental question in modern cell biology is how cellular and subcellular structures are formed and maintained given their particular molecular components. How are the different shapes, sizes, and functions of cellular organelles determined, and why are specific structures formed at particular locations and stages of the life cycle of a cell? In order to address these questions, it is necessary to consider the theory of self-organizing non-equilibrium systems. One major mechanism for self-organization within cells (and between cells) is the interplay between diffusion, active transport, and nonlinear chemical reactions. We are particularly interested in identifying and analyzing novel mechanisms for pattern formation that go beyond the standard Turing mechanism and diffusion-based mechanisms of protein gradient formation. Current research topics include the following
Cell polarization: Microtubule regulation in growth cone steering; symmetry unbreaking in fission yeast.
Intracellular pattern formation: Synaptogenesis in C elegans; Turing mechanism and active transport; intracellular waves and chemical signaling.
Cellular length control: axonal length sensing; axonal regeneration; intraflagellar transport in bacteria
Morphogenesis: cytoneme-mediated morphogen gradients; robustness
Bacterial quorum sensing: parallel signaling pathways in Vibrio; contraction mappings and mean field theory; coupled PDE-ODE systems; stochastic models
Our research in this area focuses on the spatio-temporal dynamics of continuum neural fields. Neural fields model the large-scale dynamics of spatially structured biological neural networks in terms of nonlinear integro-differential equations, whose associated integral kernels represent the spatial distribution of neuronal synaptic connections. They provide an important example of spatially extended excitable systems with nonlocal interactions, and exhibit a wide range of spatially coherent dynamics, including traveling waves, oscillations and Turing-like patterns. Current research topics include the following:
Neural field models of cortex: binocular rivalry waves; pattern formation, symmetric bifurcation theory and visual hallucinations; laminar neural fields.
Stochastic neural fields: stochastic traveling waves; variational methods and large deviations; homogenization of heterogeneous neural media
Recurrent network models of visual cortex: contextual image processing and the role of extrastriate feedback
Paul C. Bressloff Stochastic Processes in Cell Biology Interdisciplinary Applied Mathematics (Springer) August (2014)
Second edition is in preparation!
Paul C. Bressloff Waves in Neural Media: From Single Neurons to Neural Fields Lecture Notes on Mathematical Modeling in the Life Sciences (Springer) Published (2014)
Stephen Coombes and Paul C. Bressloff(eds.) Bursting: the Genesis of Rhythm in the Nervous System World Scientific (2005)
Dead Horse Point
Colorado River nr. Moab
Powder in Solitude