**The Pandemic Years**

*Stochastic hybrid path integrals*

P. C. Bressloff. Path integrals for stochastic hybrid reaction-diffusion processes. *Submitted* (2021)

P. C. Bressloff. Construction of stochastic hybrid path integrals using operator methods. *J. Phys. A* **54** 185001 (2021)

P. C. Bressloff. Coherent spin states and stochastic hybrid path integrals. *J. Stat. Mech.* **043207** (2021)

*Stochastic search processes: narrow capture, resource accumulation, and stochastic resetting *

P. C. Bressloff and R. D. Schumm. The narrow capture problem with partially absorbing targets and stochastic resetting. *Submitted* (2021)

R. D. Schumm and P. C. Bressloff. Search processes with stochastic resetting and partially absorbing targets. *Submitted* (2021)

P. C. Bressloff. First-passage processes and the target-based accumulation of resources. *Phys. Rev. E* **103** 012101 (2021)

P. C. Bressloff. Asymptotic analysis of target fluxes in the three-dimensional narrow capture problem. *Multiscale Model. Simul.* **19** 612-632 (2021)

P. C. Bressloff. Asymptotic analysis of extended two-dimensional narrow capture problems. *Proc Roy. Soc. A* **477** 20200771 (2021)

P. C. Bressloff. Target competition for resources under multiple search-and-capture events with stochastic resetting. *Proc. Roy. Soc. A* **476** 20200475 (2020)

P. C. Bressloff. Queueing theory of search processes with stochastic resetting. *Phys. Rev. E* **102** 032109 (2020)

P. C. Bressloff. Search processes with stochastic resetting and multiple targets. *Phys. Rev. E* **102** 022115 (2020)

P. C. Bressloff. Modeling active cellular transport as a directed search process with stochastic resetting and delays. *J. Phys. A* **53** 355001 (2020)

P. C. Bressloff. Directed intermittent search with stochastic resetting. *J. Phys. A* **53** 105001 (2020)

*Stochastic processes with resetting*

P. C. Bressloff. Accumulation time of stochastic processes with resetting. *Submitted* (2021)

P. C. Bressloff. Drift-diffusion on a Cayley tree with stochastic resetting: the localization-delocalization transition. *Submitted* (2021)

P. C. Bressloff. Occupation time of a run-and-tumble particle with resetting. *Phys. Rev. E* **102** 042135 (2020)

P. C. Bressloff. Stochastic resetting and the mean-field dynamics of focal adhesions. *Phys. Rev. E* **102** 022134 (2020)

P. C. Bressloff. Switching diffusions and stochastic resetting. *J. Phys. A* **53** 275003 (2020)

*Active phase separation and biological condensates*

P. C. Bressloff. Two-dimensional Ostwald ripening in a concentration gradient. *J. Phys. A* **53** 365002 (2020)

P. C. Bressloff. Active suppression of Ostwald ripening: Beyond mean field theory. *Phys. Rev. E* **101** 042804 (2020)

* Cytoneme-based morphogenesis*

P. C. Bressloff. Directional search-and-capture model of cytoneme-based morphogenesis. *SIAM J. Appl. Math* In press (2021)

*Turing pattern formation with active transport*

P. C. Bressloff. Multi-spike solutions of a hybrid reaction-transport model. *Proc Roy. Soc. A* **477**
20200829 (2021)

H. Kim and P. C. Bressloff. Stochastic Turing pattern formation in a model with active and passive transport. *Bull. Math. Biol.* **82** 144 (2020)

*Stochastically-gated diffusion*

P. C. Bressloff. Diffusive search for a stochastically-gated target with resetting. *J. Phys. A* **53** 425001 (2020)

P. C. Bressloff. Stochastically-gated diffusion model of selective nuclear transport. *Phys. Rev. E* **101** 042404 (2020)

P. Murphy, P. C. Bressloff and S. D. Lawley. Interaction between switching diffusivities and cellular microstructure. *Multiscale Model. Simul.* **18** 572-588 (2020)

P. C. Bressloff, S. D. Lawley and P. Murphy. Effective permeability of gap junctions with age-structured switching *SIAM J. Appl. Math* **80** 312-337 (2020)

**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, asymptotic and perturbation methods, 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 passive and active 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. 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. An important feature of intracellular transport is that tends to be target-centric rather than searcher-centric. Active forms of transport also occur at the cellular level and multicellular levels, as exemplified by bacterial chemotaxis, and is part of the general theory of active matter.

**Stochastic search-and-capture processes**: stochastic resetting; conditional expectations and the strong Markov property; queueing theory and the target accumulation of resources; narrow capture problems; asymptotic methods for solving FPT problems; directed intermittent search processes; applications to motor-driven axonal transport and cytoneme-based morphogenesis

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

**Stochastically-gated diffusion**: Stochastically-gated diffusion and selective nuclear transport; intercellular diffusion via stochastic gap junctions; age-structured switching processes; intracellular diffusion in domains with stochastically gated boundaries; diffusion-limited biochemical reactions.

**Stochastic models of active transport**: run-and-tumble particles; PDE models of motor-based vesicular transport; quasi-steady-state approximations; synaptic democracy and vesicular transport; exclusion processes; aggregation models of intracellular transport; random intermittent search processes.

**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. 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.

**Active phase separation and biological condensates**: Active suppression of Ostwald ripening; Ostwald ripening in concentration gradients; asymptotic methods for 2D systems.

**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; multi-spike solutions; stochastic pattern formation; intracellular waves and chemical signaling.

**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.

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

**III. Stochastic hybrid systems**

One of the major challenges in biology is developing our understanding of how noise at the molecular and cellular levels affects dynamics and information processing at the macroscopic level. Sources of noise include stochastic gene expression, the opening and closing of ion channels, and environmental fluctuations. 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 calcium concentration. The corresponding discrete process could represent the activation state of the gene or the conformational state of an ion channel.

**Path integrals and large deviation theory**: Construction of hybrid path integrals; statistical field theory for hybrid systems; Hamiltonian formulation of large deviations; WKB analysis and metastability; Feynman-Kac formula for stochastic hybrid systems.

**Stochastic hybrid oscillators **: Variational methods for analyzing stochastic hybrid oscillators; noise-induced synchronization.

**Biochemical and neural hybrid systems**: Stochastic Morris-Lecar model; spontaneous action potentials; dendritic NMDA spikes; stochastic neural networks; gene networks with environmental noise.

**IV. Neural field theory and vision**

Our previous research in mathematical neuroscience focused 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.

**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)

**Paul C. Bressloff Stochastic and Nonequilibrium Processes in Cell Biology I: Molecular Processes** *Interdisciplinary Applied Mathematics (Springer)* Coming (2021)

**Paul C. Bressloff Stochastic and Nonequilibrium Processes in Cell Biology II: Cellular Processes** *Interdisciplinary Applied Mathematics (Springer)* Coming (2021)

**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)