You are here

Invited Paper Session Abstracts - Modeling Biological Rhythms

Friday, August 3, 1:30 p.m. - 4:50 p.m., Plaza Ballroom E, Plaza Building

Periodic oscillations are a characteristic feature of many living systems. Cells, organs, and whole organisms often exhibit regular clock-like behavior. Examples include cell division, circadian rhythms, heartbeats, and brain waves. Researchers seek to understand how these oscillations are generated, how they interact with external cues, and how disruptions in biological rhythms may be associated with pathologies. One indication of the current level of interest in these topics is the awarding of the 2017 Nobel Prize in Physiology or Medicine to a group of researchers who investigated the molecular mechanisms controlling circadian rhythms.

Mathematical modeling has proven to be an invaluable tool for investigating biological rhythms. Drawing on the theory of dynamical systems, mathematical biologists have made important contributions to understanding the structure and behavior of biological oscillators. In addition, biological oscillators are a rich source of topics for classroom explorations and student research projects.

Speakers in this IPS will illustrate the breadth of biological questions and mathematical techniques that are used to study the rhythms of life. They will highlight recent advances and open questions.

David Brown, The Colorado College

Order Emerging from Chaos: The Mathematics of Firefly Synchronization

1:30 p.m. - 1:50 p.m.
Matthew Mizuhara, The College of New Jersey

Tourists flood Amphawa, Thailand every year to witness nature’s light show: trees lined with fireflies blinking in near perfect unison. Such collective behaviors arise spontaneously in many biological systems ranging from bird flocks to neuron synapses to bacterial swimmers. In this talk we will study the Kuramoto model, a system capturing synchronous dynamics of random, coupled oscillators. We will explore the role of network connectivity on synchronization, as well as a variety of coherent structures that can arise. This work is in collaboration with Hayato Chiba and Georgi Medvedev.

Optimizing Flexibility in the Collective Decisions of Honeybees

2:00 p.m. - 2:20 p.m.
Subekshya Bidari, University of Colorado

Honeybees make decisions as a group while searching for a new home site or foraging. The quality of each choice influences the rate at which scout bees recruit others via a waggle dance. In addition, decided bees can influence those with opposing opinions to change their minds via “stop-signals.” Most previous experimental studies have assumed bee swarms make decisions in static environments, but most natural environments are dynamic. In such cases, bees should adapt to new evidence as the environment constantly changes. One way of adapting is to abandon one’s current opinion and restart the evidence-accumulation and decision process (Seeley et al 2012). Incorporating such individual behavior into a dynamical model leads to a collective decision-making process that discounts previous evidence and weights newer information more strongly. We show that properly tuning this “forgetting” process can improve a swarm’s performance on a foraging task in a dynamic environment. Individual forgetfulness allows the group to change its mind, and move to a higher yielding foraging site. Our analysis explores parameter-dependent changes in the foraging yield using bifurcation theory and fast/slow analysis in a mean field version of the collective decision-making model.

Patterns of Collective Oscillations: Effects of Modularity and Time-Delay

2:30 p.m. - 2:50 p.m.
Per Sebastian Skardal, Trinity College

Synchronization in large ensembles of coupled oscillators is a nonlinear phenomenon of vital importance in applications ranging from cardiac pacemakers to circadian rhythms. The analytical treatment of such systems has a rich history and remains an active area of research. In this talk we will first explore some of these analytical methods, including a recent dimensionality reduction method discovered by Ott and Antonsen. Using this technique we will then analyze the dynamics of two different oscillator systems with important properties that arise in a wide variety of biological models: modular (i.e., community) structure, and time-delayed interactions. In modular systems, when a system is comprised of a small number of modules we uncover dynamical states corresponding to incoherence, global synchronization, as well as another state characterized by complex oscillations, and bistability between these different states. However, when the number of modules is large we uncover a hierarchical path to synchronization, where modules first synchronize within, then different modules synchronize with one another. In time-delayed systems, we characterize a transition between incoherence and global synchronization that becomes subcritical for sufficiently large time-delays, creating a hysteresis loop between these two states with a region of bistability.

Establishing a Theoretical Framework for Ultradian Forced Desynchrony Protocols

3:00 p.m. - 3:20 p.m.
Nora Stack, Colorado School of Mines

Humans have an average intrinsic circadian period of ~24.2 hours but are entrained to a 24 h day by environmental cues such as light, eating, and exercise. Our work is focused on optimal protocol design and data mining for circadian studies. Recently, we have focused on ultradian forced desynchrony (FD) protocols. These protocols are used to assess the intrinsic period of an individual using short light/dark (LD) cycles (e.g., 4 h LD cycles). These cycles are too short to entrain the circadian pacemaker. Therefore, they decouple the individual’s circadian and rest/activity cycles. These short LD cycle protocols have been used less widely compared to traditional FD protocols involving long (e.g., 28 h) LD protocols and an optimal design for ultradian FD protocols has not been established. It is cost prohibitive to optimize ultradian FD protocols experimentally. However, we used a theoretical approach to optimize protocol design and quantify the relative error associated with estimates made using ultradian FD protocols under different experimental conditions. Applying a mathematical model of the circadian pacemaker, we simulated the effects of varying light intensity, light/dark cycle duration, and phase onset for a range of intrinsic periods to determine optimal ultradian protocol design as well as an analysis of potential error present in current ultradian data sets. By investigating the properties of these protocols, we are able to recommend optimal features for protocol design and establish error bounds for existing data sets.

Multiple Time Scale Bursting Dynamics and Complex Bursting Patterns in Respiratory Neuron Models

3:30 p.m. - 3:50 p.m.
Yangyang Wang, The Ohio State University

Central pattern generators may exhibit behavior, including bursting, involving multiple distinct time scales. Our goal is to understand bursting dynamics in multiple-time-scale systems, motivated by respiratory central pattern generator neurons. We apply geometric singular perturbation theory to explain the mechanisms underlying some interesting forms of bursting dynamics involving multiple forms of activity within each cycle. We consider how many time scales are involved, obtain some non-intuitive results, and identify solution properties that truly require three time scales.

Quasicycles in the Stochastic Hybrid Morris-Lecar Neural Model

4:00 p.m. - 4:20 p.m.
Heather Zinn Brooks, University of Utah

Intrinsic noise arising from the stochastic opening and closing of voltage-gated ion channels has been shown experimentally and mathematically to have important effects on a neuron’s function. Study of classical neuron models with stochastic ion channels is becoming increasingly important, especially in understanding a cell’s ability to produce subthreshold oscillations and to respond to weak periodic stimuli. While it is known that stochastic models can produce oscillations (quasicycles) in parameter regimes where the corresponding deterministic model has only a stable fixed point, little analytical work has been done to explore these connections within the context of channel noise. Using a stochastic hybrid Morris-Lecar (ML) model, we combine a system-size expansion in K+ and a quasi-steady-state approximation in persistent Na+ in order to derive an effective Langevin equation that preserves the low-dimensional (planar) structure of the underlying deterministic ML model. By calculating the corresponding power spectrum, we determine analytically how noise significantly extends the parameter regime in which subthreshold oscillations occur. This work is joint with Paul Bressloff at University of Utah.

Investigation of Calcium Dynamics in Astrocytes via Bifurcation Analysis

4:30 p.m. - 4:50 p.m.
Greg Handy, University of Utah

Astrocytes are glial cells in the brain that make up 50% of brain volume, with each one wrapping around thousands of synapses. In the presence of neuronal activity, astrocytes exhibit calcium transients, hinting that these cells may be playing an active role in regulating brain activity. As a first step in understanding these calcium transients, we examine experimental data collected by our collaborators, in which the calcium responses in astrocytes are evoked artificially by brief stimulant applications. Surprisingly, even in this controlled experimental setup, calcium transients exhibit a vast range of amplitudes and durations, with some presenting multiple calcium peaks after one stimulus. In order to better understand this experimentally observed diversity of calcium transients, we develop an idealized differential equations (ODE) model and investigate the underlying structure of steady states and oscillatory attractors (bifurcation diagram). We use this analysis to propose a classification system for the types of calcium transients observed, and make experimentally testable predictions regarding the mechanisms responsible for the variability observed in our collaborators’ experimental data.