Mathematical Biology 3380

Some references

• Synchronization A. Pikovsky et al, Cambridge, 2003
• Chemical Oscillations, Waves and Turbulence Y. Kuramoto, Dover reprint
• Cross and Hohenberg

Prerequisites

• Mathematics
• Some differential equations
• Some basic linear algebra (eigenvalues, etc)
• Some Fourier series
• Computing. You will be required to do many computer exercises. Free software XPPAUT is available that will allow you to do most of the exercises.

• Homework 60%
• Final Project 40%

• Probable topics (but, you are welcome to ask for others)
• General results on coupling and synchronization with many examples
• Discrete maps and ecology
• Huygens pendula
• Millenium bridge
• Genetic oscillators
• Pecora theory
• Theory of phase reduction
• Phase resettimng curves
• Development of maps for resetting
• Weak coupling theory and phase models
• Kuramoto theory
• Basic Kuramoto model
• Ott-Antonsen Ansatz
• Extensions to more general coupling
• Non-global approaches
• Spatial arrangements
• Synchrony
• Waves
• Spiral waves and rotors
• Phase reduction and PDEs
• Stochastic synchronization
• Some stochastic theory
• Phase reduction
• Correlation in vs correlation out

• Lecture 1
• Some videos (more will be added)
• Millenium Bridge
• Synchronized metronomes
• Fireflies from SE Asia
• Termites making synchronized threat noises

When they are disturbed, soldiers of the two termite species Pseudacanthotermes spiniger and P. militaris hit the substratum with their head, thereby producing sounds. High-speed video recordings allowed us to analyze the movement. The sound emissions were recorded and their temporal structure was analyzed. Artificial stimulation proved that head-banging acts as an alarm signal transmitted through the vibrations produced in the substratum. Perception of these vibrations induced a polyethic response. Workers reacted to head-banging by escaping. Minor soldiers reacted by escaping, becoming immobile, or head-banging, thereby indicating the existence of positive feedback in signal production. Differences in the time patterns of the drumming appeared between both species but could not be shown to play a role in species recognition. Link

• Homework 1, due in 2 weeks Below are som\ e matlab and XPP files
• Lectures 2-4 or so
• Lecture for Monday Jan 30
• Liapunov exponent calculations
• Link to chapter 8 of my text book. Phase models We will be studying this over the next several weeks.
• Homework 2 due Feb 15th
• Week of February 13-15: more phase models including chains of oscillators
• Probably will get into the Kuramoto theory as we turn more toward symmetry breaking
• Here are my notes on the Ott-Antonsen applied to the Kuramot model
• Here is a longish HW assignment and XPP code to do the big 400 neuron simulation and the 3 variable simplified version using OA

Part II Pattern formation

• You have already seen some examples of this in the analysis of synchronization onset in the Kuramoto model
• Example with coupled chain of oscillators
• End of March and most of April