Thursday 07 Apr 2016: Dynamics seminar - Mathematical exploration of the cardio-respiratory system
Alona Ben-Tal - Massey University
Studying the cardio-respiratory system is important from the point of view of physiology and medical care but is also interesting mathematically. I will give two recent examples related to the exploration of the cardio-respiratory system.
The first example will show how a new numerical method, based on an equation free approach, can help us simplify the dynamics of bursting neurons. The method maps between the variables of a bursting neural network (for which the equations are known) and the variables of a simplified model (for which the equations are in principle unknown). By moving back and forth between the variables of the detailed system and the variables of the simplified system using restriction and lifting operators, and simulating the detailed system for short periods of time, we can calculate the stationary solutions of the simplified model and their stabilities and create bifurcation diagrams for the simplified model. This work is motivated by a more general question regarding the modelling of complex biological systems which I will also discuss.
The second example concerns breathing in birds. Avian lungs are remarkably different from mammalian lungs in that air flows unidirectionally through rigid tubes where gas exchange occurs. It has been hypothesized that the unidirectional flow is due to aerodynamic valving, resulting from the complex anatomical structure of the avian lung and the fluid dynamics involved. To test this hypothesis, we have constructed a novel mathematical model that, unlike previous models, can produce unidirectional flow which is robust to changes in model parameters, breathing frequency and breathing amplitude. The model consists of two piecewise linear ordinary differential equations with lumped parameters and discontinuous, flow-dependent resistances that mimic the experimental observations. The model provides several new physiological insights into the lung mechanics of birds and also serves as a new example of a piecewise linear model that exhibits nonlinear phenomena.