**News**

**New!** Congratulations to Dr. Sam Carroll, who successfully defended his thesis on Mar 6th 2018. Sam is staying on a a postdoctoral fellow at Utah.
**New!** Congratulations to Dr. Ethan Levien, who successfully defended his thesis on Feb 28th 2018. Ethan has been appointed as a postdoctoral fellow, SEAS Harvard
**New!** Congratulations to Dr. Heather Brooks, who successfully defended her thesis on Feb 23rd 2018. Heather has been appointed as a postdoctoral fellow, Department of Mathematics, UCLA
**New!** Congratulations to Dr. James MacLaurin, who has been appointed as an assistant professor, New Jersey Institute of Technology.
**New!** Congratulations to alumnus Dr. Jay Newby, who has been appointed as an assistant professor, University of Alberta.
PCB awarded a Distinguished Scholar and Creative Researcher Award, University of Utah (2017).
Dr. Bin Xu successfully defended her thesis on Mar 3rd 2017. Bin is currently a postdoctoral fellow in the Department of Applied Mathematics, Notre Dame.
Dr. Bhargav Karamched successfully defended his thesis on Feb 27th 2017. Bhargav is currently a postdoctoral fellow in the Department of Mathematics, University of Houston.
Dr. James Maclaurin joined our group as a 3-year postdoctoral fellow in July 2017. This position is jointly funded by the NSF and the Department of Mathematics.
PCB elected a Fellow of the Society for Industrial and Applied Mathematics (2016)
PCB appointed Associate Editor of the SIAM Journal of Applied Mathematics (2015)
PCB invited as a plenary speaker: SIAM Conference on Nonlinear waves and Coherent Structures (2014)
PCB published two books for Springer (2014): Stochastic Processes in Cell Biology, Waves in Neural Media

**Stochastic processes in molecular and cell biology**

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.

**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

**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

**Neural Field Theory and Vision**

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

**Lectures**

1. Equilibrium and non-equilibrium systems

2. Receptor-ligand binding and cooperativity

3. Stochastic ion channels

4. Random walks and Brownian motion

5. Langevin and Fokker-Planck equations

6. First passage time problems

7. Simple genetic circuits I

8. Simple genetic circuits II

9. Molecular noise in gene expression

10. System-size expansion and Gillespie

**Problem sheets**

Problem sheet I

Problem sheet II

Problem sheet III

Problem sheet IV

Problem sheet V

**Handouts**

I. Brief introduction to probability theory

II. Fourier and Laplace transforms

**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)