Coarse-graining Hamiltonian systems using WSINDy.

Weak form equation learning and surrogate modeling has proven to be computationally efficient and robust to measurement noise in a wide range of applications including ODE, PDE, and SDE discovery, as well as in coarse-graining applications, such as homogenization and mean-field descriptions of interacting particle systems. In this work we extend this coarse-graining capability to the setting of Hamiltonian dynamics which possess approximate symmetries associated with timescale separation. A smooth -dependent Hamiltonian vector field possesses an approximate symmetry if the limiting vector field possesses an exact symmetry.

Weighted Composition Operators for Learning Nonlinear Dynamics.

Operator theoretic methods in dynamical system have been dominated by the use of Koopman operators and their continuous time counterparts, such as Koopman Generators and Liouville Operators. The advantage gained from their use primarily stems from the ability to extract subspaces and eigenfunctions within a space of observables that are invariant with respect to the Koopman operator over that space. When this occurs, a dynamic mode decomposition of the systems state provides a linear model for the dynamical system.

Direct Estimation of Parameters in ODE Models Using WENDy: Weak-Form Estimation of Nonlinear Dynamics.

We introduce the Weak-form Estimation of Nonlinear Dynamics (WENDy) method for estimating model parameters for non-linear systems of ODEs. Without relying on any numerical differential equation solvers, WENDy computes accurate estimates and is robust to large (biologically relevant) levels of measurement noise. For low dimensional systems with modest amounts of data, WENDy is competitive with conventional forward solver-based nonlinear least squares methods in terms of speed and accuracy.

Learning mean-field equations from particle data using WSINDy.

We develop a weak-form sparse identification method for interacting particle systems (IPS) with the primary goals of reducing computational complexity for large particle number and offering robustness to either intrinsic or extrinsic noise. In particular, we use concepts from mean-field theory of IPS in combination with the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy) to provide a fast and reliable system identification scheme for recovering the governing stochastic differential equations for an IPS when the number of particles per experiment is on the order of several thousands and the number of experiments is less than 100. This is in contrast to existing work showing that system identification for less than 100 and on the order of several thousand is feasible using strong-form methods.

Medial Prefrontal Cortex to Medial Septum Pathway Activation Improves Cognitive Flexibility in Rats.

Background: The medial prefrontal cortex (mPFC) is necessary for cognitive flexibility and projects to medial septum (MS). MS activation improves strategy switching, a common measure of cognitive flexibility, likely via its ability to regulate midbrain dopamine (DA) neuron population activity. We hypothesized that the mPFC to MS pathway (mPFC-MS) may be the mechanism by which the MS regulates strategy switching and DA neuron population activity.

Direct Estimation of Parameters in ODE Models Using WENDy: Weak-form Estimation of Nonlinear Dynamics.

We introduce the Weak-form Estimation of Nonlinear Dynamics (WENDy) method for estimating model parameters for non-linear systems of ODEs. Without relying on any numerical differential equation solvers, WENDy computes accurate estimates and is robust to large (biologically relevant) levels of measurement noise. For low dimensional systems with modest amounts of data, WENDy is competitive with conventional forward solver-based nonlinear least squares methods in terms of speed and accuracy.

ANALYTICAL SINGULAR VALUE DECOMPOSITION FOR A CLASS OF STOICHIOMETRY MATRICES.

We present the analytical singular value decomposition of the stoichiometry matrix for a spatially discrete reaction-diffusion system. The motivation for this work is to develop a matrix decomposition that can reveal hidden spatial flux patterns of chemical reactions. We consider a 1D domain with two subregions sharing a single common boundary.

Online Weak-form Sparse Identification of Partial Differential Equations.

This paper presents an online algorithm for identification of partial differential equations (PDEs) based on the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy). The algorithm is online in the sense that if performs the identification task by processing solution snapshots that arrive sequentially. The core of the method combines a weak-form discretization of candidate PDEs with an online proximal gradient descent approach to the sparse regression problem.

Medial septum activation improves strategy switching once strategies are well-learned via bidirectional regulation of dopamine neuron population activity.

Strategy switching is a form of cognitive flexibility that requires inhibiting a previously successful strategy and switching to a new strategy of a different categorical modality. It is dependent on dopamine (DA) receptor activation and release in ventral striatum and prefrontal cortex, two primary targets of ventral tegmental area (VTA) DA projections. Although the circuitry that underlies strategy switching early in learning has been studied, few studies have examined it after extended discrimination training.

Protocol for Analysis and Consolidation of TrackMate Outputs for Measuring Two-Dimensional Cell Motility using Nuclear Tracking.

Collective cellular migration plays a key role in many fundamental biological processes including development, wound healing, and cancer metastasis. To understand the regulation of cell motility, we must be able to measure it easily and consistently under different conditions. Here we describe a method for measuring and quantifying single-cell and bulk motility of HaCaT keratinocytes using a nuclear stain.

WEAK SINDY FOR PARTIAL DIFFERENTIAL EQUATIONS.

Sparse Identification of Nonlinear Dynamics (SINDy) is a method of system discovery that has been shown to successfully recover governing dynamical systems from data [6, 39]. Recently, several groups have independently discovered that the weak formulation provides orders of magnitude better robustness to noise. Here we extend our Weak SINDy (WSINDy) framework introduced in [28] to the setting of partial differential equations (PDEs).

Estimating the Impact of Statewide Policies to Reduce Spread of Severe Acute Respiratory Syndrome Coronavirus 2 in Real Time, Colorado, USA.

The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) pandemic necessitated rapid local public health response, but studies examining the impact of social distancing policies on SARS-CoV-2 transmission have struggled to capture regional-level dynamics. We developed a susceptible-exposed-infected-recovered transmission model, parameterized to Colorado, USA‒specific data, to estimate the impact of coronavirus disease‒related policy measures on mobility and SARS-CoV-2 transmission in real time. During March‒June 2020, we estimated unknown parameter values and generated scenario-based projections of future clinical care needs.

WEAK SINDy: GALERKIN-BASED DATA-DRIVEN MODEL SELECTION.

We present a novel weak formulation and discretization for discovering governing equations from noisy measurement data. This method of learning differential equations from data fits into a new class of algorithms that replace pointwise derivative approximations with linear transformations and variance reduction techniques. Compared to the standard SINDy algorithm presented in [S.

BOUNDEDNESS OF A CLASS OF SPATIALLY DISCRETE REACTION-DIFFUSION SYSTEMS.

Although the spatially discrete reaction-diffusion equation is often used to describe biological processes, the effect of diffusion in this framework is not fully understood. In the spatially continuous case, the incorporation of diffusion can cause blow-up with respect to the norm, and criteria exist to determine whether the system is bounded for all time. However, no equivalent criteria exist for the discrete reaction-diffusion system.

Competitive exclusion in a DAE model for microbial electrolysis cells.

Microbial electrolysis cells (MECs) are devices that employ electroactive bacteria to perform extracellular electron transfer, enabling hydrogen generation from biodegradable substrates. In our previous work, we developed and analyzed a differential-algebraic equation (DAE) model for MECs. The model resembles a chemostat or continuous stirred tank reactor (CSTR).

Life cycle of a cyanobacterial carboxysome.

Carboxysomes, prototypical bacterial microcompartments (BMCs) found in cyanobacteria, are large (~1 GDa) and essential protein complexes that enhance CO fixation. While carboxysome biogenesis has been elucidated, the activity dynamics, lifetime, and degradation of these structures have not been investigated, owing to the inability of tracking individual BMCs over time in vivo. We have developed a fluorescence-imaging platform to simultaneously measure carboxysome number, position, and activity over time in a growing cyanobacterial population, allowing individual carboxysomes to be clustered on the basis of activity and spatial dynamics.

Mechanical regulation of photosynthesis in cyanobacteria.

Photosynthetic organisms regulate their responses to many diverse stimuli in an effort to balance light harvesting with utilizable light energy for carbon fixation and growth (source-sink regulation). This balance is critical to prevent the formation of reactive oxygen species that can lead to cell death. However, investigating the molecular mechanisms that underlie the regulation of photosynthesis in cyanobacteria using ensemble-based measurements remains a challenge due to population heterogeneity.

Medial septum activation produces opposite effects on dopamine neuron activity in the ventral tegmental area and substantia nigra in MAM vs. normal rats.

The medial septum (MS) differentially impacts midbrain dopamine (DA) neuron activity via the ventral hippocampus, a region implicated in DA-related disorders. However, whether MS regulation of ventral tegmental area (VTA) and substantia nigra pars compacta (SNc) is disrupted in a developmental disruption model of schizophrenia is unknown. Male Sprague-Dawley rats were exposed at gestational day 17 to methylazoxymethanol (MAM) or saline.

Medial septum differentially regulates dopamine neuron activity in the rat ventral tegmental area and substantia nigra via distinct pathways.

The medial septum (MS) impacts hippocampal activity and the hippocampus, in turn, regulates midbrain dopamine (DA) neuron activity. However, it remains to be determined how MS activation impacts midbrain DA activity. This question was addressed by infusing NMDA (0.

A numerical framework for computing steady states of structured population models and their stability.

Structured population models are a class of general evolution equations which are widely used in the study of biological systems. Many theoretical methods are available for establishing existence and stability of steady states of general evolution equations. However, except for very special cases, finding an analytical form of stationary solutions for evolution equations is a challenging task.

A numerical framework for computing steady states of structured population models and their stability.

Structured population models are a class of general evolution equations which are widely used in the study of biological systems. Many theoretical methods are available for establishing existence and stability of steady states of general evolution equations. However, except for very special cases, finding an analytical form of stationary solutions for evolution equations is a challenging task.

Modeling keratinocyte wound healing dynamics: Cell-cell adhesion promotes sustained collective migration.

The in vitro migration of keratinocyte cell sheets displays behavioral and biochemical similarities to the in vivo wound healing response of keratinocytes in animal model systems. In both cases, ligand-dependent Epidermal Growth Factor Receptor (EGFR) activation is sufficient to elicit collective cell migration into the wound. Previous mathematical modeling studies of in vitro wound healing assays assume that physical connections between cells have a hindering effect on cell migration, but biological literature suggests a more complicated story.

An Inverse Problem for a Class of Conditional Probability Measure-Dependent Evolution Equations.

We investigate the inverse problem of identifying a conditional probability measure in measure-dependent evolution equations arising in size-structured population modeling. We formulate the inverse problem as a least squares problem for the probability measure estimation. Using the Prohorov metric framework, we prove existence and consistency of the least squares estimates and outline a discretization scheme for approximating a conditional probability measure.

Laplacian Dynamics with Synthesis and Degradation.

Analyzing qualitative behaviors of biochemical reactions using its associated network structure has proven useful in diverse branches of biology. As an extension of our previous work, we introduce a graph-based framework to calculate steady state solutions of biochemical reaction networks with synthesis and degradation. Our approach is based on a labeled directed graph G and the associated system of linear non-homogeneous differential equations with first-order degradation and zeroth-order synthesis.