Research:

Many fundamental rhythmic activities such as breathing and swallowing are generated by neural networks referred to as central pattern generators (CPGs). To survive and reproduce, an animal must adjust to changes in its internal state and the external environment. Most animals control their adaptive motor behaviors thanks to a combination of CPGs and sensory feedback from the peripheral nervous system that can alter the ongoing intrinsic neural dynamics in CPGs. 

My primary research interests lie in the area of mathematical and computational neuroscience, focusing on understanding both the neural activities that generate rhythmic movements, modulation of neural networks, and the interplay between CPGs and sensory feedback, using a variety of mathematical ideas including 


Check out the selected projects listed below for more details! 

Impact of modulating intrinsic and synaptic properties on network synchrony in a respiratory network model


Neural networks are constantly modulated by numerous neuromodulators through altering properties of neurons, synapses and networks. Understanding action of neuromodulation on respiratory networks is critical to understanding neural control of breathing and disorders such as opiod-induced respiratory depression.  I have been working closely with experimental collaborators to develop a framework that integrates computational modeling  and experimental approaches to examine the modulatory actions that affect respiratory neural control.  

Mixed-mode oscillations in cortical theta oscillator model and in vitro (Pittman-Polletta et al 2021)

An important feature of most biological rhythms is that they evolve on different timescales. I'm particularly interested in modeling and analyzing multiple-timescale spiking activity patterns in neuronal systems. To this end, we have extended the standard fast-slow decomposition method/geometric singular perturbation theory to three or more timescale settings (e.g., Nan et al 2015, Wang and Rubin 2016). Applications of interest include uncovering mechanisms underlying non-standard respiratory output patterns such as the sigh, mixed-mode oscillations in cortical oscillators, and symmetry-breaking in cell cycle dynamics. 

Analysis of mixed-mode oscillations in a three-timescale system (Phan and Wang 2024)

Different types of activity patterns in an embryonic respiratory neuron model (Wang and Rubin 2020)

Breathing is a critical rhythmic behavior that continues from birth up until death. While much is known about respiratory neuronal dynamics after birth, less is understood regarding the mechanisms driving inspiratory network rhythms at embryonic stages. Through computational modeling and dynamical systems methods, we elucidate the intrinsic mechanisms underlying various types of prenatal bursting neuron behavior. Our work predicts how developmental changes in key ionic currents influence the bursting activity of these neurons. Our goal is to shed light on how intrinsic bursting properties and synaptic interactions evolve during prenatal development to support respiratory network rhythmogenesis. 

Isochrons for a limit cycle with sliding components  (Wang et al 2021a)


Motor systems show an overall robustness, but because they are highly nonlinear, understanding how they achieve robustness due to their different components such as sensory feedback and biomechanics is difficult.  Through collaboration with experimentalists and theoreticians, we aim to dissect motor robustness against external perturbations, a key but challenging question in motor control.  


As a concrete example, we focus on a neuromechanical model of rhythmic feeding behaviors of the marine mollusk, Aplysia californica (Wang et al 2022). This project has also motivated us to develop new methods for studying  motor robustness such as infinitesimal shape response curve and local timing response curve, as well as extend classic infinitesimal phase response curve to systems operating within hard limits (Wang et al 2021). 

Identify homeostasis motifs from a network based solely on its structure (Figures adapted from Wang et al 2021b)


Networks of coupled dynamical systems arise in many branches of science. In many examples, the network structure influences the dynamics and bifurcations that can be expected to occur generically. We use singularity theory and combinatorial matrix theory to study how the network structure influences the dynamics and bifurcations without knowing specific equations for each cell (Gandhi et al., 2020). 


Besides classifying bifurcations in networks, we are also interested in studying homeostasis, an important biological phenomenon whereby the output of a system (say, body temperature) is approximately constant despite changes of an input (say, ambient temperature). Our analysis allows us to develop an algorithm for determining all the subnetwork motifs that support homeostasis. Notably, our algorithm does not require performing any numerical simulations on model equations and hence is easy to implement in real applications. 

Sensitivity and uncertainty analysis using Latin hypercube sampling of parameter space and partial rank correlation coefficients for both proposed argasid tick models (Clifton et al., 2018) 

Unlike the well-studied hard-bodied (ixodid) ticks, soft-bodied (argasid) ticks exist in relative obscurity.  The life cycle of ixodid ticks, the vectors for diseases like Rocky Mountain spotted fever and Lyme disease, have been characterized quantitatively, yet no mathematical model of any soft-bodied tick exists. This void is not trivial; argasid ticks such as Ornithodoros moubata are vectors of devastating human and animal diseases including African swine fever (ASF) in domesticated pigs and African relapsing fever in humans. Therefore, it is imperative to quantitatively understand the complex life cycle of this tick.  We have proposed the first two mathematical models of the life cycle of O.~moubata to explore the dynamics and identify knowledge gaps of these poorly studied ticks. These models provide the basis for developing future models that include disease states to explore infection dynamics and possible management of ASF.