**News**

Congratulations to Dr. Sam Carroll, who successfully defended his thesis on Mar 6th 2018. Sam is staying on a a postdoctoral fellow at Utah.
Congratulations to Dr. Ethan Levien, who successfully defended his thesis on Feb 28th 2018. Ethan has been appointed as a postdoctoral fellow, SEAS Harvard
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
Congratulations to Dr. James MacLaurin, who has been appointed as an assistant professor, New Jersey Institute of Technology.
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

My research is at the multidisciplinary interface between applied mathematics, theoretical physics, neuroscience and cell biology. It draws upon a wide range of analytical tools and mathematical methods, including stochastic processes, partial differential equations, statistical physics, dynamical systems theory, pattern formation and nonlinear wave theory, and the theory of self-organizing systems. I am particularly interested in identifying the mathematical structures and biophysical principles underlying general aspects of cellular and multicellular function, and using the underlying theory to solve problems in cellular and systems neuroscience.

**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:

**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.

**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. 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

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

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

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

**III. 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

**IV. 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

**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

**Lecture notes**

11.Stochastic models of polymerization

12.Cell mitosis

13.Brownian ratchets

14.Flashing ratchet model of a processive molecular motor

15.Simple gene circuits I

16.Simple gene circuits II

**Problem sheets**

Problem sheet I

Problem sheet II

**Solutions**

Solutions I

**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)