Research Interests

My research is driven by a desire to understand the roles of stochasticity, structure, and evolution in shaping the dynamics of biological systems. I develop and analyze mathematical models, combining methods from probability, dynamical systems, and random graph theory to shed light on biological issues while generating new mathematical questions. In particular, I study stochastic processes on networks with applications in neuroscience (and recently epidemiology), stochastic models in genetics, and a variety of statistical applications. For more information, see my Research Summary and Publications.

Complexity Reduction for Stochastic Network Models in Neuroscience

The inclusion of stochastic effects within dynamical neural models is a topic of growing importance in mathematical neuroscience. A major source of noise in neural systems is random gating of ion channels. Such systems are typically represented as a Markov process on a graph. Recently, Schmandt and Galan (PRL, 2012) introduced the stochastic shielding approximation, a new approach to efficient, accurate simulation of Markov chain models. They show that neglecting a carefully chosen subset of noise sources within the Markov process leads to significantly faster simulations with practically no observable error. Inspired by their work and in collaboration with Peter Thomas (CWRU), I conducted a thorough mathematical analysis of the stochastic shielding approximation, both in toy models, in the Hodgkin-Huxley ion channel model, and in a broad class of random graph models. This method is based on replacing a high-dimensional stochastic process defined on a graph with a lower-dimensional process on the same graph, rather than replacing a complex network with a simpler one. We show that this form of model reduction can be represented as a mapping from the original process to an approximate process defined on a significantly smaller sample space (Schmidt and Thomas, JMN 2014). Our analysis results in a new, quantitative measure of the importance of individual edges within a Markov process on a graph. Our measure not only confirms the optimality of the stochastic shielding approximation, but also sheds new light on to the contributions of different ion channel transitions to the variability of neural systems.

In Schmidt, Galan, Thomas (PLoS Comp Biol 2018), we extended this work by exploring the robustness of the stochastic shielding phenomenon and the accuracy of the approximation under conditions of timescale separation and sparsity in the stationary distribution, via the edge importance measure described Schmidt and Thomas (JMN 2014). We show that typical edge importance hierarchy is robust to the introduction of timescale separation for a class of simple networks, but that it can break down for more complex systems with three or more distinct timescales, such as the nicotinic acetylcholine receptor, a well-studied ion channel that can exhibit bursting behavior. We also establish that the edge importance measure remains a valid tool for analysis for arbitrary networks regardless of multiple timescales. I have two projects in progress that apply these methods to other systems: calcium release in neural signaling and flows on river delta networks.

Contagion Dynamics on Networks and Adaptive Networks

I am collaborating with Biostatistics graduate student Brittany Lemmon (UC Davis) and Paul Hurtado (UNR) to study an epidemic spreading on a network that changes over time, known as an adaptive network. Classical contagion models, such as the Susceptible-Infected-Recovered (SIR) model, and other infectious disease models typically assume a well-mixed contact process. This may be unrealistic for infectious disease spread where the contact structure changes due to individuals' responses to the infectious disease. For instance, individuals showing symptoms might isolate themselves, or individuals that are aware of an ongoing epidemic in the population might reduce or change their contacts. Here we investigate contagion dynamics in an adaptive network context, meaning that the contact network is changing over time due to individuals responding to an infectious disease in the population. This is work in progress that was initiated for Brittany's Honors Thesis at UNR, and now we are extending and generalizing her results.

Dynamic network models are used to model infectious diseases as well as social contagions and the adoption of fads. My former Master's student Natasha Lang and I looked at threshold models on a network and analyzed how different parameters affect cascade behavior on the network, thinking specifically of the spread of a social contagion or infectious disease. This work is an extension of the original Watts model that includes various heterogeneous thresholds, and a variant that includes heterogeneous edge weights. We also relate some insights from these results to COVID-19 data in Nevada.

In addition, I wrote two book chapters: one on network structure and dynamics of biological systems geared for undergraduate (or beginning graduate) researchers and their advisors (Schmidt 2020), and another more general chapter on stochastic models in biology (Schmidt 2024).

Network Structure and Dynamics of Sleep-Wake Regulation

I am collaborating with Janet Best (OSU) and Peter Kramer (RPI) to model sleep-wake regulation in mammals. Sleep-wake behavior is governed by interacting networks of neurons in the brain that can be well-modeled as stochastic processes on random graphs. An important first step in modeling sleep-wake regulation is to understand how network architecture influences the dynamics of activity occurring on the network. Primarily using degree distribution as a marker of graph structure, we ask to what extent the graph structure is reflected in the dynamics of the process on the graph. Our findings suggest that memoryless processes, such as percolation-like processes and spiking of integrate-and-fire neurons, are more likely to reflect the degree distribution (Schmidt, Best, Blumberg 2011). We find that processes with memory, such as those in which nodes change their firing rate in response to inputs, can robustly produce power law and other heavy-tailed distributions of activity regardless of degree distribution. Sleep-wake regulation offers many avenues in which to extend this work and to better understand the interplay of genetic and biochemical mechanisms in shaping complex physiological dynamics. I am looking at how sleep-wake dynamics change during neural development alongside the developing neural network structure, and also how differences in gene regulation and function may influence sleep-wake patterns.

We recently developed a new method for estimating a power law region, including its lower and upper bounds, of the probability density in a set of data that can be modeled as a continuous random sample (Olmez et al, J Stat Comp and Sim 2021). Our method is based on adaptively penalizing the Kolmogorov-Smirnov method. This work was motivated by our current project on modeling sleep-wake regulation in a network context. We needed to first develop new methods to properly analyze sleep and wake bout distributions from the model, as wake bout distributions tend to show power law behavior only over an intermediate range (not a full tail). I am currently working with recent UNR graduate Adrian Samberg on an integrate-and-fire network version of this model that we plan to submit for publication soon.

Mathematical Modeling of Retinal Degeneration

I started a collaboration with five women at IPAM's summer 2019 Program for Women in Mathematical Biology to study retinal degeneration. Cell degeneration, including that resulting in retinal diseases, is linked to metabolic issues. In the retina, photoreceptor degeneration can result from an imbalance in lactate production and consumption as well as disturbances to glucose and pyruvate levels. We analyzed a novel mathematical model for the metabolic dynamics of a cone photoreceptor, which is the first model to account for energy generation from fatty acid oxidation of shed photoreceptor outer segments (Camacho et al, 2021). Multiple parameter bifurcation analysis shows that joint variations in key parameter ranges affect the cone’s metabolic vitality and its capability to adapt under glucose-deficient conditions. Time-varying global sensitivity analysis (Dobreva et al, 2022) is used to assess the sensitivity of model outputs of interest to changes and uncertainty in the parameters at specific times. The results reveal a critical temporal window where there is more flexibility for therapeutic interventions to rescue a cone cell from the detrimental consequences of glucose shortage.

The first paper cited above (Camacho et al, 2021) models the dynamics of photoreceptor metabolism in a single cone cell, and the second paper (Dobreva et al, 2022) looks at the implications of that model's dynamics for various disease states. Work in progress includes extending this single photoreceptor model into a network of three interacting neural cell types: photoreceptors (rods and cones), retinal pigment epithelium, and Mueller cells. The network dynamics in this case are very complicated and will likely benefit from the complexity reduction methods described above.

Stochastic Gene Regulation and Carcinogenesis

Gene regulatory networks dynamically orchestrate the level of expression for each gene in the genome by controlling how vigorously that gene will be expressed. Gene expression is inherently stochastic due to the small number of regulatory molecules typically present within a cell. Genetic feedback loops in cells break detailed balance and involve bimolecular reactions and, hence, exact solutions revealing the nature of the stochastic fluctuations in these loops are lacking. In collaboration with Ramon Grima (Univ. of Edinburgh) and Timothy Newman (Univ. of Dundee), I analyzed a stochastic model of a gene regulatory feedback loop using the chemical master equation for theoretical analysis. The network we consider breaks detailed balance and involves a single bimolecular reaction step. We provide an exact solution of the steady-state master equation for arbitrary values of the parameters, and present simplified solutions for a number of special cases. Our results emphasize the importance of stochastic effects in modeling gene regulation.

For my PhD research, I developed and analyzed models of gene regulatory sequence evolution which yield new insights into evolutionary genetics, cancer progression and related diseases. Specifically, I used probabilistic models to understand how population genetic factors influence this evolutionary process and shape the distribution of waiting times for the generation of new transcription factor binding sites in different species. In collaboration with Rick Durrett and Jason Schweinsberg, we generalized these results by considering the time it takes to acquire k such mutations in a population. The 2-step process applies to gene regulatory sequence evolution, and our findings show that regulatory sequences can evolve at rates faster than divergence times between species. This supports recent empirical studies that new regulatory sequences typically come from small modifications to existing sequences. Results from the general k-step model give estimates for the rate of cancer progression in colon cancer and other diseases resulting from a sequence of mutations occurring in a collection of cells.

In collaboration with undergraduate Honors student Dana Winterringer, we studied the influence of diet in colorectal cancer using continuous-time Markov chains to model the process of carcinogenesis. See Dana's honors thesis for details.

Mentoring Experience

See my People page for a list of current and past students at UNR.

Mentoring Experience Prior to UNR

In 2014, I directed a small group project on complex networks as part of MBI's Summer Undergraduate Program. The students learned about network topologies, random graph models, the small world property, clustering, and other properties of networks. They also wrote MATLAB and Mathematica code to explore various properties of three key families of random graphs.

I directed a research project for the 2009 MBI Summer Graduate Program at Ohio State University. We looked at the evolution of variance in mate choice, thinking specifically about fish and other organisms that mate in leks.

During the summer of 2007, I directed a project in mathematical population genetics for the Mathematics Department's Research Experience for Undergraduates (REU) at The University of Akron. This project looked at how blood type distributions in different human populations might be evolving over time. We used both a stochastic model and its corresponding mean-field model (system of ordinary differential equations) which looks at the average behavior of the system.


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