# Faculty Research Talks - Spring 2022

Talks will be held at 2:30PM on every other Monday (M) at 2:30PM in CULM 611 unless otherwise noted. Please see more information below:

Talks will be held at 2:30PM on every other Monday (M) at 2:30PM in CULM 611 unless otherwise noted. Please see more information below:

Location: CULM 611

**Please note, this seminar will begin at 1:30 pm instead of our regularly scheduled time**

**Order-indeterminant maps derived from learning to keep a beat**

While many people say they have no rhythm, most humans when listening to music can easily discern and move to a beat. On the other hand, many of us are not so adept at actually generating and maintaining a constant beat over a period of time. Demonstrating a beat is a very complicated task. Among other things, it involves the ability of our brains to estimate time intervals and to make physical movements, for example hitting a drum, in coordination with the time estimates that we make. How our brain and body solve this problem is an open and active area of research. In this talk, I will discuss a mathematical model for a beat generator, which is defined here as a group of neurons that can learn to keep a constant beat represented by a periodic stimulus tone. The model leads to distinct mathematical questions involving order-indeterminate maps. These are maps in which the order of events is a priori not known, i.e. whether the next event is stimulus tone or beat generator spike.

Location: CULM 611

**Inverse problems in ocean acoustics**

In this presentation, we will provide a definition of an inverse problem and we will present the motivation behind the necessity for work on inverse problems in ocean acoustics. A historical perspective will be discussed that shows the success of existing methods along with the need for model improvement and acceleration of computations. Different approaches will be presented based on full-field inversion as well as inversion using distinct features of the acoustic field. Plans for future work will follow.

Location: CULM 611

**Instability and Breakup Dynamics of the Leapfrogging Vortex Quartet**

I discuss a well known system of ordinary differential equations modeling the interaction of point vortices in a two-dimensional inviscid fluid. I investigate the stability of a one-parameter family of periodic solutions of the four-vortex problem known as `leapfrogging' orbits. These solutions, which consist of two pairs of identical yet oppositely-signed vortices, were known to Gröbli (1877) and Love (1883), and can be parameterized by a dimensionless parameter α related to the geometry of the initial configuration. Simulations by Acheson (2000) and numerical Floquet analysis by Tophøj and Aref (2012) both indicate, to many digits, that the bifurcation occurs when α=1/ϕ^2, where ϕ is the golden ratio. These numerical studies indicated a sequence of behaviors that emerge as this parameter is further decreased, leading to the disintegration of the leapfrogging orbit into a pair of dipoles that escape to infinity along transverse rays.

This study has two objectives. The first is to rigorously explain the origin of this remarkable bifurcation value. The second is to understand the sequence of transitions in the phase space of the system that allows for the emergence of the various behaviors. While the first objective is essentially linear, finding the answer requires applying several tricks from the classical mechanics toolkit. The second objective is inherently nonlinear, and our approach involves both analysis and numerics. In particular, we make use of the recently developed technique of Lagrangian descriptors to visualize the phase space structures, including invariant manifolds. This work forms the dissertation research of my recently-graduated student Brandon Behring. I will describe an extension that I’ve been working on with NJIT undergraduate Noah Roselli and discuss some related open problems.

Location: CULM 611

**Resonant wave interactions and their oceanic applications**

Resonant interactions of weakly nonlinear waves have been considered one of main mechanisms for the long-term spectral evolution of ocean waves and have been studied extensively. In this talk, I first describe the resonant triad interaction of surface gravity-capillary waves on a homogeneous fluid layer, and present some relevant asymptotic models and their solutions. Then, the discussion will be extended to a two-layer system, where two different wave modes (surface and internal waves) can interact resonantly. Possible oceanic applications of the results will be also discussed.

Location: CULM 611

**Modeling single-neuron function: calcium-triggered neurotransmitter release and action potential propagation**

Understanding the function of a nervous system is not possible without deep knowledge of the electric activity of individual neuronal cells, as well as precise understanding of neurotransmitter release underlying inter-neuronal communication. This talk will focus on the general overview of the main aspects of neuronal excitability and neurotransmitter vesicle fusion, which rely on basic electrolyte ion flows across the cell membrane through membrane-spanning ion channel proteins. Neurotransmitter vesicle release in particular is triggered by the influx of calcium ions into the cell upon depolarization, followed by its binding to the calcium sensor proteins responsible for membrane fusion. Because of the small spatial and temporal scales involved in this process, mathematical and computational modeling is indispensable for a deep understanding of synaptic function and neuronal activity, and has been at the forefront of discovery in the field of neurophysiology for many decades. I will present the non-linear reaction-diffusion problems arising in the modeling of calcium-dependent synaptic transmission, as well as several active projects associated with gaining deeper understanding of this process. I will also describe a project dealing with a somewhat larger spatial scale, aimed at understanding the properties of action potential propagation in neuronal fibers of the mammalian auditory system.

*Updated: April 8, 2022*