SIAM Student Chapter Seminar

Forest ecosystems often experience sudden insect outbreaks in which populations remain low for long periods before rapidly exploding to damaging levels. These abrupt transitions cannot be explained by simple population growth models alone. In this talk, we introduce a dynamical systems model that incorporates both logistic growth and saturating predation to describe insect population dynamics. Using graphical and qualitative analysis, we show how the model can admit multiple equilibria corresponding to low and high population states. As environmental parameters change, these equilibria can appear or disappear through saddle-node bifurcations, leading to sudden shifts in population levels. We also discuss the phenomenon of hysteresis, which explains why outbreaks may persist even after environmental conditions improve. This example provides an accessible introduction to bifurcation theory and illustrates how dynamical systems can help explain sudden transitions in ecological systems.

Modern vehicle trajectory datasets, such as the I-24 MOTION, provide high-resolution microscopic information on traffic flow, but are often affected by missing data, measurement noise, and incomplete structural information. These limitations make it challenging to reconstruct reliable macroscopic fields, which are required to assess how well macroscopic traffic models reproduce observed traffic dynamics.   

In this talk, we present a modeling framework for reconstructing consistent macroscopic fields from imperfect microscopic trajectory observations. The approach combines (i) an acceleration-based minimum cost maximum bipartite matching procedure to identify physically feasible connections between trajectory fragments, and (ii) a probabilistic phantom-density model that distributes missing vehicle mass in regions with incomplete observations. This leads to a reliable reconstruction that preserves mass (i.e., avoids spurious gains or losses of mass), while accounting for the uncertainty induced by incomplete microscopic information. We present visualizations of naively reconstructed density fields and show how incomplete data can introduce spurious variation in mass and artificial discontinuities in the density fields. We then demonstrate that our proposed method reduces these spurious artifacts and produces smoother, physically consistent fields, enabling reliable validation of macroscopic traffic models. 

Dentate gyrus granule cells are among the most important neurons in the hippocampus, playing a key role in how the brain forms and separates memories. What makes these cells particularly interesting is how sensitive they are to calcium. Calcium is not just another ion here—it links electrical signals to internal cellular processes, shaping how the neuron responds, adapts, and stores information.

As the brain ages, these neurons do not stay the same. Their structure begins to change, dendrites can shrink, synaptic connections may become less dense, and the distribution of ion channels can shift. At the same time, the internal calcium machinery also evolves. Together, these changes alter how the neuron behaves. Instead of a single factor, neuronal excitability emerges from the interaction of those multiple factors—intracellular infrastructure, dendritic shrinkage, synaptic density, and channel distribution.

Understanding how all of these pieces work together is still a challenge, since many existing models focus on only one aspect at a time. More importantly, there is no way of experimentally measuring how these pieces work together. To address this gap, we built a multi-compartment neuron model using a realistic morphology. The model combines voltage dynamics with intracellular calcium processes. It also enables inhomogeneous behavior across the neuron by assigning region-specific properties and synaptic inputs.

The goal is to understand how aging-related changes work together to influence neuronal excitability and how their interactions may lead to hyperexcitability. This model is currently under development and serves as a framework for exploring these complex interactions, with the aim to answer how aging affects calcium signaling and how to possibly predict and address aging-related dysfunction.