Research

My Google Scholar

Analysis of Neural Circuit Dynamics

One of the main research interests of our group is understanding the collective dynamics of neuronal networks coupled through gap junctions or synaptic connections. In particular, we develop mathematical models and dynamical systems methods to investigate how central pattern generators (CPGs), neural circuits responsible for producing rhythmic activity, are interconnected and coordinate to generate and control locomotion.

Stick Insect Locomotion

A wide range of vital behaviors, including locomotion, digestion, and respiration, emerge from multiphase rhythmic activity generated by closed-loop networks that integrate neural and biomechanical components. Our research aims to understand the principles by which (1) neural dynamics and coupling coordinate rhythmic stepping patterns in six-legged stick insects, and (2) neural drives and biomechanical feedback interact within a closed-loop network to control single-leg stepping.

For some examples, see the following papers:

The mesothoracic limb neural circuit.

Coupled FitzHugh-Nagumo Models

To better understand the modeling and analysis of CPGs as complex neural circuits, we study the dynamics of simpler neuronal models, such as FitzHugh-Nagumo (FHN), coupled via gap junctions or synaptic connections. We are interested in phenomena such as mixed-mode oscillations (MMOs), spike-adding mechanisms, phase locking, and other complex dynamical behaviors that play an important role in shaping and generating locomotor patterns.

For some examples, see the following papers:

The spike-adding diagram of a forced FHN system as the frequency and amplitude of the periodic input vary.

Modeling & Analyzing Epithelial Cells

Cells must counter osmotic water influx driven by impermeant intracellular molecules to maintain stable volume and proper function. In epithelial tissues, cell volume and ion homeostasis are regulated through the coordinated activity of ion pumps, channels, and transporters. Classical five-dimensional differential-algebraic pump–leak equations (PLEs) provide a mathematical framework for describing these processes in single cells.

Building on this framework, we develop mathematical models to investigate how passive transporters interact with active pumps to regulate cellular ion concentrations, membrane potential, and volume. We have also developed a two-compartment PLE model that couples an epithelial cell with a fluid-filled lumen, allowing us to study the coupled regulation of cell and lumen volumes in epithelial systems. Check these papers for more details:

A comprehensive mathematical framework that reveals how pump localization and membrane properties shape epithelial function and dysfunction.

Contraction Theory & Network Synchronization

A significant part of our research focuses on developing and applying techniques from contraction theory to study the stability and robustness of dynamical systems. We develop contraction-based approaches for a broad range of systems, including ordinary differential equations (ODEs), partial differential equations (PDEs), stochastic differential equations (SDEs), and discrete-time systems. For some examples, see the following papers:

We are particularly interested in understanding synchronization and cluster synchronization in complex networks, as well as their stability and robustness to perturbations and uncertainties. We develop mathematical techniques to characterize when synchronization emerges and persists in networks of coupled dynamical systems. For some examples, see the following papers:

Spatial uniformity of solutions to reaction–diffusion PDEs.

Bacterial Chemotaxis

E. coli chemotaxis across scales: from the intracellular signaling pathway and agent-based description of individual bacteria to a PDE approximation of the population-level dynamics.

Another direction of our research focuses on understanding how individual cellular dynamics and interactions with the environment give rise to collective behavior at the population level. In particular, we use PDEs to model the collective motion of E. coli populations in response to chemical signals in their environment. Starting from stochastic descriptions of individual bacterial motion, such as Fokker–Planck equations, we derive and analyze population-level advection–diffusion models that connect intracellular sensing and individual movement to macroscopic population dynamics. For some examples, see the following papers:

My Dissertation