Applied Math Colloquium - Spring 2019

Synchronization of neural activity in the brain is involved in a variety of brain functions including perception, cognition, memory, and motor behavior. Excessively strong, weak, or otherwise improperly organized patterns of synchronous oscillatory activity may contribute to the generation of symptoms of different neurological and psychiatric diseases. However, neuronal synchrony is frequently not perfect, but rather exhibits intermittent dynamics. The same synchrony strength may be achieved with markedly different temporal patterns of activity. I will discuss methods to describe these phenomena and will present the application of this analysis to the neurophysiological data in healthy brain, Parkinson’s disease, and drug addiction disorders. I will finally discuss potential cellular mechanisms and functional advantages of some of the observed temporal patterning of neural synchrony.

This presentation will describe acceleration of simulation methods for the Landau-Fokker-Planck equation, with a focus on a binary collision model that is solved using a Direct Simulation Monte Carlo (DSMC) method. Acceleration of this method is achieved by coupling the particle method to a continuum fluid description. Efficiency of the method is greatly increased by inclusion of particles with negative weights. This significantly complicates the simulation, and many difficulties have plagued earlier efforts to use negatively weighted particles. This talk will describe significant progress that has been made in overcoming those difficulties.

Computational methods that bridge multiple biological scales, including hybrid-dimensional numerical methods and subdivision-optimized geometric multigrid methods, will be presented in this talk. Using these techniques the talk will focus on a study of three-dimensional structure-function interplay at synaptic spines and neuronal dendrites, up to network level dynamics. The main focus will lie on the endoplasmic reticulum (ER), which, under certain structural conditions, gives rise to intracellular calcium waves. The ER forms a complex endomembrane network that reaches into the cellular compartments of a neuron, including dendritic spines. Combining a new 3D spine generator with 3D Ca2+ modeling, this talk addresses the relevance of ER positioning on spine-to-dendrite Ca2+ signaling. Simulations, which account for Ca2+ exchange on the plasma membrane and ER, show that spine ER needs to be present in distinct morphological conformations in order to overcome a barrier between the spine and dendritic shaft. RyR-carrying spine ER promotes spine-to-dendrite Ca2+ signals in a position-dependent manner and simulations indicate that RyR-carrying ER can initiate time-delayed Ca2+ reverberation, depending on the precise position of the spine ER. Upon spine growth, structural reorganization of the ER restores spine-to-dendrite Ca2+ communication, while maintaining aspects of Ca2+ homeostasis in the spine head. The presented work emphasizes the relevance of precise positioning of RyR-containing spine ER in regulating the strength and timing of spine Ca2+ signaling, which could play an important role in tuning spine-to-dendrite Ca2+ communication and homeostasis

We review some recent approaches to control large group of systems with sparse controls. At microscopic level we consider a number of multi-particle systems, used in applications to animal groups, opinion dynamics and other, while kinetic models are obtained as mean-field limit (Vlasov-Poisson type). Passing to the limit in the control strategies is particularly delicate and we will discuss some ways of overcoming the technical difficulty. Then we show how these approaches may be applied to control of traffic via autonomous and connected vehicles. Finally, a new class of equations, called Measure Differential Equations, are discussed as a way to encompass mean-field limits.

Understanding secondary droplet formation from fluid fragmentation is critical for industrial, environmental, and health processes including for predicting and controlling the transport of pathogen-bearing droplets created from contaminated fluids or surfaces. Despite the complexity and diversity of modes of unsteady fluid fragmentation into secondary droplets, universality across geometry and fluid systems emerges. We will discuss results from our recent joint experimental and theoretical investigations elucidating the role of unsteadiness in shaping a ubiquitous, yet neglected class of fluid fragmentation problems. In particular, we revisit fundamental assumptions of hydrodynamic instability and reveal how unsteadiness and multi-scale dynamics couple to select the sizes and speeds of secondary droplets generated. The implications for human health and food safety will be discussed.

Some brain disorders are hypothesized to have a dynamical origin; in particular, it has been been hypothesized that some symptoms of Parkinson's disease are due to pathologically synchronized neural activity in the motor control region of the brain. This talk will describe several different approaches for desynchronizing the activity of a group of neurons, including maximizing the Lyapunov exponent associated with their phase dynamics, optimal phase resetting, controlling the phase density, and controlling the population to have clustered dynamics. It is hoped that this work will ultimately lead to improved treatment of Parkinson's disease via targeted electrical stimulation.

Basic examples in phase transitions, such as the formation of crystals or the separation of phases, give rise to intricate random geometries with astonishing empirical universality. Physicists have invented several minimal models that attempt to capture the essence of these processes, one of which is diffusion limited aggregation (DLA). Despite an extensive physics literature, very little is known mathematically about DLA and its close relative, the Hastings-Levitov model for iterated conformal maps. I will describe a new model, that `interpolates' between these models and the Loewner theory for conformal maps. The main point of our model is its ease of description and analysis. In particular, it yields explicit stochastic PDE that are at least formally, the scaling limits of these processes. No background beyond basic familiarity with conformal maps will be presumed in this talk.

This is joint work with Vivian Olsiewski Healey (University of Chicago).

We model the joint evolution of a population density and the mean, and sometimes variance, of a quantitative trait (that is, a continuous random variable such as flowering time in plants) subject to selection toward an optimum value that varies in space. To do so, we study a family of deterministic models originating from the Kirkpatrick-Barton (1997) reaction-diffusion system. We use analysis and numerics to identify conditions under which the models predict range pinning due to an influx of locally maladapted individuals from the center of a species' range to its borders (“genetic swamping”) versus invasions represented as travelling waves. We highlight differences between the predictions of the Kirkpatrick-Barton model and those of related models incorporating features, such as non-Gaussian dispersal kernels and patchy habitat, that are often represented in nongenetic invasion models.

 

Resonance Between Surface and Internal Gravity Waves

The ocean is stratified into a lower layer of cold, salty and dense water and an upper layer of warmer, sweeter and lighter water due to the transport of fresh water from river deltas, estuaries and heavy rainfall over the ocean surface. In between these layers, internal gravity waves are generated by disturbances caused by tidal currents that flows over ocean floor bathymetry such as continental shelfs or ocean floor hills or mountains. In height, a wave may rise to hundreds of feet and in length, a wave train may propagate for hundreds of miles. Being ubiquitous, due the periodic motion of the moon and the tides, it is believed that internal waves play a significant role in transferring heat and nutrients between ocean layers. Satellite images from NASA's archive will be presented.

For a two-layer fluid, a closed Hamiltonian system of second-order nonlinear evolution equations was derived in Choi (2019). A numerical solution, employing a pseudo-spectral (PS) method, is presented. A triad of internal and surface gravity waves can resonate at second-order nonlinearity. The resonance may transfer energy from a internal wave to surface waves in which a monochromatic and uniform surface wave may become modulated and evolve into envelopes. Resonance conditions are presented with special emphasis on the group resonance which is admissible when the group velocity of the surface waves coincides with the phase speed of the internal wave. The PS model is compared to experiments and modulation theory of Lewis, Lake, and Ko (1974).

For the initial wave evolution the PS model affirms the experiments and theory. The modulation theory is revisited and the governing equations are solved as an initial value problem to suit the PS model. Simulations with higher nonlinearity and with realistic ocean parameters are performed and compared to the theory. The results will be presented along with a series of wave animations.

References

W. Choi. Nonlinear interaction between surface and internal waves. Part I: Nonlinear models and spectral formulation. Preprint, 2019.

J. E. Lewis, B. M. Lake, and D. R. S. Ko. On the interaction of internal waves and surface gravity waves. J. Fluid Mech., 63:773-800, 1974.