**Stochastic Processes in Molecular and Cell Biology, Statistical and Biological Physics, Mathematical Neuroscience**

A major goal of our research is to understand the fundamental biophysical mechanisms underlying cellular function in health and disease. This includes both single cells and multicellular systems. Our work draws upon a wide variety of methods in applied mathematics and theoretical physics including stochastic processes, statistical physics, nonlinear PDEs, and dynamical systems theory. In addition to specific applications, we are also developing new mathematical and numerical methods for analyzing complex and stochastic nonlinear systems. Current research topics include the following:

**I. Stochastic models of intracellular transport**

The efficient delivery of mRNA, proteins and other molecular products to
their correct location within a cell (intracellular transport) is of fundamental importance to
normal cellular function and development. The challenges of intracellular transport are particularly acute for neurons, which are amongst the largest and most complex cells in biology, in particular, with regards to the
efficient trafficking of newly synthesized proteins from the cell body or soma to distant locations on the axon and dendrites. In healthy cells, the regulation of mRNA and protein trafficking within a neuron provides an important mechanism for modifying the strength of synaptic connections between neurons, and synaptic plasticity is generally believed to be the cellular substrate of learning and memory. On the other hand, various types of dysfunction in protein trafficking appear to be a
major contributory factor to a number of neurodegenerative diseases associated with memory loss including Alzheimer's. Broadly speaking, there are two basic mechanisms for intracellular transport: passive diffusion within the cytosol or the surrounding plasma membrane of the cell, and active motor-driven transport along polymerized filaments such as microtubules and F-actin that comprise the cytoskeleton. Current research topics include the following:

**Spatially heterogeneous diffusion in cells**: Brownian motion with spatially-dependent switching conformational states; morphogen concentration gradients with switching diffusivities; homogenization theory for rapidly switching diffusions

**Diffusion in randomly switching environments**: intracellular diffusion in domains with stochastically gated boundaries; first passage time problems; intercellular diffusion via stochastic gap junctions; diffusion-limited biochemical reactions; age-structured switching processes; switching diffusivities.

**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 and extreme statistics.

**Stochastic models of receptor trafficking and synaptic plasticity **: Diffusion-trapping model of receptor trafficking; synaptic tenacity; receptor clustering

**Intracellular traveling waves **: CaMKII translocation waves; spike-diffuse-spike model of waves in active dendrites

**II. Self-organization in cell and developmental biology**

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

**Morphogenesis**: cytoneme-mediated morphogen gradients; robustness; effects of sticky boundaries; search-and-capture models; queuing models of morphogen bursts; diffusion-based morphogenesis and switching diffusivities

**Intracellular pattern formation**: Synaptogenesis in C elegans; Turing mechanism and hybrid reaction-transport models; stochastic pattern formation; intracellular waves and chemical signaling.

**Active phase separation and biological condensates**: Active suppression of Ostwald ripening; finite-size effects.

**Cell polarization**: Microtubule regulation in growth cone steering; symmetry unbreaking in fission yeast.

**Cellular length control**: axonal length sensing; axonal regeneration; intraflagellar transport in bacteria

**III. Stochastic neural and biochemical networks**

One of the major challenges in neuroscience is developing our understanding of how noise at the molecular and cellular levels affects dynamics and information processing at the macroscopic level of synaptically-coupled neuronal populations. Intrinsic noise sources include stochastic gene expression, the opening and closing of ion channels, membrane voltage fluctuations and synaptic noise. Extrinsic noise sources include environmental fluctuations and background synaptic inputs. A number of stochastic processes involve the coupling between continuous random variables and discrete random variables (stochastic hybrid systems). The continuous process could represent the concentration of proteins synthesized by a gene, the membrane voltage of a neuron, or population-level synaptic currents. The corresponding discrete process could represent the activation state of the gene, the conformational state of an ion channel, or population spiking activity. Current research topics include the following:

**Stochastic neural and biochemical oscillators **: Variational methods for analyzing stochastic limit cycle oscillators; variational methods for analyzing stochastic hybrid oscillators; noise-induced synchronization

**Stochastic hybrid systems**: Large deviation theory and path-integral methods for stochastic hybrid systems; Hamiltonian formulation; quasi-steady-state analysis; metastability

**Bacterial quorum sensing**: parallel signaling pathways in Vibrio; contraction mappings and mean field theory; coupled PDE-ODE systems; stochastic models

**Stochastic ion channels**: Stochastic Morris-Lecar model; spontaneous action potentials; dendritic NMDA spikes

**Hybrid chemical reaction networks**: Effects of a common noisy environment; robustness

**IV. Neural field theory and vision**

Our research in this area focuses on the spatio-temporal dynamics of continuum neural fields with particular applications to vision. 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:

**Stochastic neural fields**: attractor models of stimulus-dependent neural variability; variational methods and large deviations; stochastic traveling waves and bumps; homogenization of heterogeneous neural media

**Neural field theory**: laminar neural fields; neural fields on curved manifolds; neural fields on product spaces; pattern formation and symmetric bifurcation theory

**Vision**: contextual image processing and the role of extrastriate feedback; binocular rivalry waves; geometric visual hallucinations

**Paul C. Bressloff Stochastic Processes in Cell Biology** *Interdisciplinary Applied Mathematics (Springer)* August (2014)

**Second edition (two volumes) 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)