Traveling waves in biology
Organizer: Faustino Sánchez Garduño, firstname.lastname@example.org
There is a vast number of phenomena in biology in which a key element or precursor to a developmental process seems to be the appearance of a travelling wave of chemical concentration, mechanical deformation, electrical signal and so on. For instance, chemical and mechanical waves which propagate on the surface of many vertebrate eggs, the analysis on insect dispersal, in interacting population models where spatial effects are important, the progressing wave of an epidemic, the movement of microorganisms moving into a food source, to name but a few. The purpose of this session is to exhibit the new developments in this area.
Elizabeth Cherry, Georgia Tech
Flavio Fenton, Georgia Tech
Luis López Ríos, IIMAS-UNAM
Ramón Plaza, IIMAS-UNAM
Stabilization effects of chemotaxis on bacterial aggregation patterns
Ramón Plaza, IIMAS-UNAM
We consider a chemotaxis-reaction-diffusion system that models the dynamics of colonies of Bacillus subtilis on thin agar plates. The system of equations reproduces the dense branching patterns observed experimentally in the semi-solid agar, low-nutrient regime. Numerical simulations show that, when the chemotactic sensitivity toward nutrients is increased, the morphology of the colony changes from a dense branched pattern to a uniform envelope that propagates outward. In this talk I will provide a quantitative argument that explains this change in morphology. This result is based on a traveling wave approach: energy estimates on the spectral equations for perturbations around the envelope front imply the suppression of colony branching as a result of the stabilizing effect of the increasing chemotactic signal. This is a collaboration with A. Butanda (CIMAT) and C. Malaga (UNAM).
Stability of traveling waves for reaction diffusion-degenerate equations
Luis Fernando López Rios, IIMAS-UNAM
We will present several results about the spectral stability of monotone traveling waves solutions for reaction-diffusion equations, where the reaction is of bistable type and the diffusion coefficient is density dependent and degenerate. In terms of continuous models of the spread of biological populations, this type of reaction describes the Allee effect, in which aggregation can improve the survival rate of the individuals. On the other hand, the type of diffusion we consider describes populations where the motility depends on the density itself, with no motility in regions where the population is very scarce. This is a joint work with Ramón Plaza and Francisco Leyva
Reconstructing cardiac electrical dynamics using data assimilation
Elizabeth Cherry, Georgia Tech, USA
The heart is an electro-mechanical system in which, under normal conditions, electrical waves propagate in a coordinated manner to initiate an efficient contraction. In pathologic states, known as cardiac arrhythmias, single and multiple rapidly rotating spiral and scroll waves of electrical activity can appear and generate complex spatiotemporal patterns of activation that inhibit contraction and can be lethal if untreated. Despite this understanding of the involvement of such reentrant electrical scroll waves, experimental limitations have hampered a detailed understanding of the specific mechanisms responsible for reentrant wave formation and breakup. To further this effort, we recently have begun to apply the technique of data assimilation, widely used in weather forecasting, to reconstruct time series in cardiac tissue. Here we use model-generated synthetic observations from a numerical experiment to evaluate the performance of the ensemble Kalman filter in reconstructing such time series for a discordant alternans state in one spatial dimension and for scroll waves in three dimensions. We show that our approach is able to recover time series of both observed and unobserved variables that match the truth. Where nearby observations are available, the error is reduced below the synthetic observation error, with a smaller reduction with increased distance from observations. Using both one- and three-dimensional cases, we provide a deeper analysis showing that data assimilation can provide high-quality estimates under a wide range of model error conditions, ranging from varying a single parameter value to using an entirely different model to generate the truth state. We illustrate how algorithmic settings including multiplicative and additive inflation as well as localization radius can be used to reduce error in the state estimates. Our findings demonstrate that state reconstruction for spatiotemporally complex cardiac electrical dynamics is possible and has the potential for successful application to real experimental data.
Simulation of Complex Electrical Waves of the Heart in real time using GPUs
Flavio Fenton, Georgia Tech, USA
The heart is an electro-mechanical system in which, under normal conditions, electrical waves propagate in a coordinated manner to initiate an efficient contraction. In pathologic states, single and multiple rapidly rotating spiral and scroll waves of electrical activity can appear and generate complex spatiotemporal patterns of activation that inhibit contraction and can be lethal if untreated.
In this talk we will show several dynamics of these waves experimentally and then describe how we use mathematical models to simulate and study these waves. Then we present a new formalism to solve these equations using graphic cards of laptops and cellphones in real time, solving more than 1.7 billion differential equations per second. This allows for the first time, high performance computer simulations of accurate cardiac models in anatomically accurate heart structures without the need of supercomputers, and a first step towards patient specific modeling.
15:30 - Elizabeth Cherry
16:00 - Flavio Fenton
16:30 - Luis López Ríos
17:00 - Ramón Plaza