## Research Highlights

### Dr. Michael Neilan

Dr. Neilan's research areas are computational and applied mathematics with an emphasis on numerical methods for partial differential equations (PDEs), in particular, finite element methods. Currently, with a main research focus on (i) computational methods for fully nonlinear second order PDEs and (ii) the construction of compatible mixed finite element pairs.

He is currently developing and studying practical and efficient finite element schemes for the Hamilton-Jacobi-Bellman equation; a fully nonlinear problem that naturally arises in the modeling of stochastic optimal control processes.

The main numerical difficulties of this problem stem from the non-variational structure of viscosity solution theory and the lack of regularity of the continuous solution. With graduate student Lauren Hennings, He has recently proposed and implemented a new simple finite element scheme that accurately captures weak solutions. He is currently developing a convergence theory of this discretization with a long -term goal of obtaining error estimates with explicit rates of convergence.

His second focus of research concerns mixed finite element discretizations for various PDE models. Recently with collaborators Johnny Guzman (Brown) and Rick Falk (Rutgers), They have developed the first family of stable finite element spaces for the Stokes and Navier-Stokes problems that produce exactly divergence-free velocity approximations with respect to simplicial partitions. The main idea to develop these spaces is to construct an exact discrete de Rham complex consisting of finite element spaces. Currently, with graduate student Duygu Sap, I am extending these results in the case of quadrilateral and hexahedral partitions.

## New Grants

**"The Mechanics of Neural Variability"**

PI: Dr. Brent Doiron,

Sponsor: National Science Foundation

Modern computers minimize noisy fluctuations in transistors and semiconductors in order to improve performance reliability. Put more simply, when two trials of an identical computation are performed (say adding 1+1) there are minimal differences between the currents that flow through a computer's electronic circuits on each trial. In stark contrast, brain dynamics show a sizable trial-to-trial variability of neural responses, and the biological currents flowing through neurons and synapses on distinct trials are often widely different. This makes it clear that the nervous system works under different principles than in silicon machines. Despite this long standing observation, there has only been considerable effort to characterize the trial averaged response of neurons, with the trial-to-trial variability of neural response remaining poorly understood.

The awarded grant to Prof. Doiron will build on past models from Prof. Doiron and colleagues that investigate the stochastic aspects of networks of neurons. New mathematical and computational challenges need to be overcome to gain deeper insight in neural variability. This will involve analysis that is considerably distinct from current techniques and will provide key tools for the field of stochastic biology. The awarded proposal aims to give theoretical insights into how internal neural architecture can both generate and propagate variability throughout a neural circuit. These will mark a significant advance within both theoretical and experimental communities, as well as provide the tools to generate new insights in sensory, motor, and cognitive neuroscience.

**"Generation and Control of Rhythmic Activity in Respiratory and Motor Networks"**

PI: Dr. Jonathan Rubin

Sponsor: National Science Foundation

The main point of the funded research will be to study how neural circuits, called central pattern generators, produce rhythms that drive repetitive behaviors, such as breathing and swimming, and how these rhythms are automatically controlled by internal feedback signals.

The interesting mathematical issues in this research include how to understand complicated activity patterns in systems where different processes evolve on multiple distinct time scales, how to predict activity patterns in networks with complex connection architectures, and how reweighting of inputs can allow a single rhythmic network to generate multiple distinct rhythms. The award will provide support for several graduate students (David Burstein, Abigail Snyder, and Yangyang Wang) who are working on aspects of these problems.

**“Partial Differential Equations in Conservation Laws and Applications”**

PI: Dr. Dehua Wang

Sponsor: National Science Foundation

This project is devoted to a mathematical study of some nonlinear partial differential equations in multi-dimensional conservation laws and related applications. In particular, the study focuses on the following topics from the theory of inviscid and viscous compressible flows and related applications:

• the mixed-type PDE problems for transonic flows past an obstacle,

• the mixed-type PDE problems for isometric embedding,

• the existence and regularity of global solutions to the compressible multi-dimensional Navier-Stokes equations, and

• the global solutions to the viscous flows of related applications in liquid crystals. The goals of the research are:

• developing novel analytic methods and efficient techniques for solving some important problems in multi-dimensional inviscid and viscous conservation laws and applications;

• exploring the qualitative behavior of flow motion;

• establishing new connections of the isometric embedding problem with elastodynamics; and

• gaining insights into other multi-dimensional problems of conservation laws and emerging applications.