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## Analysis and its Applications Seminar

#### Modeling and Analysis of patterns in multi-constituent systems with long range interaction

### Abstract

Skin pigmentation, animal coats and block copolymers can be considered as multi-constituent inhibitory systems. Exquisitely structured patterns arise as orderly outcomes of the self-organization principle. Analytically, via the sharp interface model, patterns can be studied as nonlocal geometric variational problems. The free energy functional consists of an interface energy and a long range Coulomb-type interaction energy. The admissible class is a collection of Caccioppoli sets with fixed volumes. To overcome the difficulty that the admissible class is not a Hilbert space, we introduce internal variables. Solving the energy functional for stationary sets is recast as a variational problem on a Hilbert space. We prove the existence of a core-shell assembly and the existence of disc assemblies in ternary systems and also a triple-bubble-like stationary solution in a quaternary system. Numerically, via the diffuse interface model, one open question related to the polarity direction of double bubble assemblies is answered. Moreover, it is shown that the average size of bubbles in a single bubble assembly depends on the sum of the minority constituent volumes and the long range interaction coefficients. One further identifies the ranges for volume fractions and the long range interaction coefficients for double bubble assemblies.## Quantitative Biology Colloquium

#### Normalization Methods in Single-cell RNA Sequencing

### Abstract

Through gene sequencing experiments, researchers can analyze the genetic content of tumors or developing embryos and better understand the importance of particular genes during stages of development. Single-cell RNA-sequencing (scRNA-seq) provides a means to assess transcriptomic variations among individual cells, rather than over the tumor as a whole, giving an advantage over bulk sequencing methods that fail to detect subgroups and rare cell types. However, restrictions such as amplification bias, technical noise, and dropout events often limit the power of scRNA-seq results. To address these issues, various normalization methods have been developed that correct observed gene counts to account for existing noise and more accurately represent the true biological signal of interest. Eliminating technical noise and amplification error often involves the use of a set of exogenous genes injected into the cell in known quantities, referred to as “spike-in genes”. By statistically modeling the difference between observed gene counts and known gene counts, the resulting model can then apply to all other genes present in the cell, adjusting observed gene counts accordingly. I propose a novel scRNA-seq normalization method that normalizes between a data set’s groups while also using dropout imputation to adjust for missing values. I compare this method with existing normalization approaches, using real data sets to support my results.## Modeling and Computation Seminar

#### Inverse source problem in a forced network

### Abstract

We address the nonlinear inverse source problem of identifying a time- dependent source occurring in one node of a network governed by a wave equation. This is an important problem for applications like monitoring the electrical grid or testing a network of acoustic pipes. We prove that time records of the associated state taken at a strategic set of two nodes yield uniqueness of the two unknown elements: the source position and the emitted signal. Using graph theory, we discuss the number and location of the observation nodes. A non-iterative identification method that localizes the source node by solving a set of well posed linear systems is developped. Once the source node is localized, the emitted signal is identified using a deconvolution problem or a Fourier expansion. Numerical experiments on a five node graph confirm the feasability of the approach.## Brown Bag Seminar

#### Drive-Based Motivation for Coordination of Limit Cycle Behaviors

### Abstract

Navigation is often be considered a prototypical behavior for a robotic system. Constructing robots capable of complex behaviors requires developing a method for switching among possible behaviors, for example using a hybrid automaton. Recent work has developed an alternative approach using continuous dynamical systems that use an internal drive state to select the desired behavior. In one particular result, authors developed drive dynamics that result in trajectories where the robot repeatedly flows to one point attractor and then another, yielding a limit cycle where the robot patrols between two points. A further level of complexity arises when one seeks to create a system that switches between these basic limit cycles instead of the original point attractors. This work outlines the problem using the recently-developed drive-based dynamical framework. As a proof-of-concept we demonstrate the existence of an attracting set consisting of orbits that repeatedly flow between two canonical limit cycles (e.g., Hopf oscillators). This attracting set connecting limit cycle behaviors arises from a bifurcation depending on a family of parameters which is analogous to the bifurcation that produces the limit cycle connecting two point-attractor behaviors in previously-published system. Implications and next-steps in the case of arbitrary limit cycles are discussed.## Applied Math Colloquium

#### Resonance-based mechanisms of generation of oscillations in networks of non-oscillatory neurons

### Abstract

Several neuron types have been shown to exhibit (subthreshold) membrane potential resonance (MPR), defined as the occurrence of a peak in their voltage amplitude response to oscillatory input currents at a preferred (resonant) frequency. MPR has been investigated both experimentally and theoretically. However, whether MPR is simply an epiphenomenon or it plays a functional role for the generation of neuronal network oscillations, and how the latent time scales present in individual, non-oscillatory cells affect the properties of the oscillatory networks in which they are embedded are open questions. We address these issues by investigating a minimal network model consisting of (i) a non-oscillatory linear resonator (band-pass filter) with 2D dynamics, (ii) a passive cell (low-pass filter) with 1D linear dynamics, and (iii) nonlinear graded synaptic connections (excitatory or inhibitory) with instantaneous dynamics. We demonstrate that (i) the network oscillations crucially depend on the presence of MPR in the resonator, (ii) they are amplified by the network connectivity, (iii) they develop relaxation oscillations for high enough levels of mutual inhibition/excitation, and (iv) the network frequency monotonically depends on the resonator’s resonant frequency. We explain these phenomena using a reduced adapted version of the classical phase-plane analysis that helps uncovering the type of effective network nonlinearities that contribute to the generation of network oscillations. Our results have direct implications for network models of firing rate type and other biological oscillatory networks (e.g, biochemical, genetic).