Computer Simulation and Applications September 16
Computer Simulation and Applications September 19 Simple Systems, Complex Behavior
Fireflies: mysterious mass synchrony • Actually beetles , also called lightning bugs (for their conspicuous use of bioluminescence to attract mates or prey) • Thailand, with fantastic, out of this world firefly shows; enormous congregations of fireflies blinking on and off in unison , in displays that supposedly stretched for miles along the riverbanks. (Also occuring in Africa, and some more places ) • Accounts on this phenomenon by Western travelers to South East Asia go back as far as 300 years. • Mysterious form of mass synchrony. • In 1917 Philip Laurent wrote up an explanation in Science: “the apparent phenomenono was caused by the twisting or sudden lowering and raising of my eylids the insects had nothing to do with it” • In the years between 1915 and 1935, Science published 20 other articles on this phenomenon (some said it was just coincidence, others ascribed it to peculiar atmospheric conditions such as exceptional humidity, calm or darkness. Some (eg George Hudson) believed there must be a maestro that cues all the rest of the fireflies • George Hudson in 1918 in Science: “if it is desired to get a body of men to sing or play together in perfect rhythm they not only must have a leader but must be trained to follow the leader ... Do these insects inherit a sense of rhythm more perfect than our own?” He concluded there must a leader among the fireflies. • Hugh Smith lived in Thailand from 1923 to 1934 witnessed the phenemon countless times wrote in exasperation: “some of the published explanations are more remarkable than the phenomenon itself” – he confessed though that he could not offer an explanation. • Late 1960s people started to gain understanding • One clue was so obvious that nearly everyone missed it: synchronous fireflies not only flash unison – they flash in rhythm , at constant tempo (even when isolated from one another, they still keep a constant beat • That means each insect must have its own means of keeping time, some sort of internal clock. Hypothetical oscilator (still unidentified) anatomically but is presumed to a cluster of neurons somewhere in the firefly’s brain much like a natural pacemaker in our hearts. • The oscillator fires repetively, generating an electrical rhythm that travels to the firefly’s lantern and ultimately trigger s its periodic flash http://www.youtube.com/watch?v=IBgq-_NJCl0&feature=related movie of synchronizing fireflies (courtesy of Johannes)
Fireflies conceived as mindless oscillators • Second clue came from the work of the biologist John Buck Informal experiment: he captured a lot of fireflies near Bangkok and released them in his darkened hotel room. The insects flitted about nervously, then settled down all over the walls and ceiling, always spacing themselves 10 centimeters apart. At first the fireflies twinkled incoherently. Then pairs, then trios began to pulse in unison. Pockets of synchrony continued to emerge and grow, until as many as a dozen were blinking on and off in perfect concert This suggested that fireflies must be somehow adjusting their rhythms in response to the flashes of others. John Buck and his colleagues later in the lab started to test this hypothesis directly: they flashed an artificial light at the firefly (to mimick the flash of another) and measured its response. The firefly will shift the timing of its subsequent flashes in a consistent and predictable manner, the size and direction of the shift depend on when in the cycle the stimulus was received. For Some species the stimulus always advanced the firefly’’s rhythm, as if setting the clock ahead Other species delayed or advanced the clock depending on where they were in the cycle of the other • Two clues the flash rhythm was regulated by an internal, resettable oscillator. • These two clues suggested a sync mechanism for a congregation of fireflies: each is continually sending and receiving signals, shifting the rhythm of others and being shifted by them in turn. Out of cacaphony sync emerges spontaneously, fireflies organize themselves, no maestro. Synchronization occurs through mutual cuing in the same way an orchestra can keep perfect timing without a conductor Counterintuitive: insects don’t need to be intelligent The insects have all the ingredients: each firefly contains an oscillator, a little metronome, whose timing adjusts automatically in response to the flashes of others. That is all there is to it! Except for one thing : is such a scenario possible? • Can perfect synchrony emerge from a cacaphony of thousands of mindless metronomes ? S. Strogatz and R. Mirollo proved that this is possible. Moreover they proved that not only is it possible but under certain conditions it will always occur. • The tendency to synchrony is one of the most pervasive drives in the universe, for reason not understood yet, extending from atoms to animals, from people to planets.
Fireflies = simple model of self-organizing system • Examples: sperm swimming side by side en route to the egg beat their tails in unison, in a premordial display of synchronized swimming Eplipsy (in this case synchrony is most undesirable) caused by millions of brain cells discharging in pathological lockstep, causing rhythmic convulsions associated with seizures. Also lifeless can synchronize: amazing coherence of a laser beam comes from trillions of atoms pulsing in concert, all emitting photons f the same phase and frequency. Over the course of millenia, the incessant effects of the tides have locked the moon’s spin to its orbit; it now turns on its axis at precisely the same rate as it circles the earth (that is why we never see its dark side) • On the surface these phenomena seem unrelated (brain and lasers) but at a deeper level there is a connection , transcends the details of particular mechanism: mathematics • All examples are variations of the same math theme: self organization, the spontaneous emergence of order out of chaos. • We are beginning to unlock , by the study of simple MODELS of fireflies and other self-organizing systems, the secrets of this dazzling kind of order in the universe. • Ch. Peskin is the first person to pose the question about self-organization : Can perfect synchrony emerge from a cacaphony of thousands of mindless metronomes ? Fireflies they gave us insight into the inevitability of sync
Pacemaker of the Heart • Ch. Peskin also proposed a schematic (according to the dictum of Einstein: everything should be made as simply as possible, but not simpler) • model for how the pacemaker cells of the heart synchronize themselves • Pacemaker of the heart • marvel of evolutioin • most impressive oscillator ever created • a cluster of 10,000 cells called sinoatrial node • generates electrical rhythm that commands the rest of the heart to beat • has to be done reliably, minute after minute • three billion beats in a lifetime • unlike most cells in heart, the pacemaker cells oscillate automatically: isolated in petri dish, their voltage rises and falls in a regular rhythm • All of which raises the question: Why do we need so many cells, if one can do the job? • probably because a single leader is not robust design: a leader can malfunction or die • instead evolution ahs produced a more reliable, democratic system in which thousands of celss collectively set the pace • democracy has it own problems: somehow the cells have to coordinate their firings – conflicting signals and the heart becomes deranged Charles S. Peskin http://www.math.nyu.edu/faculty/peskin/ also site for the book: modeling and simulation in the life sciences
Cardiac pacemaker simulated as a net of simple oscillating circuits • This became Peskin’s question: How do these cells with no leader or outside instructions , manage to get in sync? • Notice the how similar this question is to the earlier one about fireflies both involve large populations of rhythmic individuals that fire off sudden pulses that jolt the rhythms ot others in their group, speeding them up or slowing them down according to specific rules. In both cases, synchronization seems to be inevitable. Challenge question: explain why is it inevitable • Peskin examined this question within a framework of a simplified model each pacemaker cell as an oscillating electrical circuit is equivalent to a capacitor in parallel with resistor • The whole pacemaker he idealized as enormous collection of these simple mathematical oscillators; for simplicity he • Assumed • all oscillators are identical – follow the same charging curve • each oscillator is strongly equally strongly to all others • oscillators effect one another when they fire • He worked on showing that synchrony emerges out of the cacaphony, but was only able to show this for a system with two oscillators. • This problem lay dormant for 15 years, eventually the mathematicians Strogatz and Mirello showed that synchrony will always emerge under certain conditions
• “En forgerant on devient forgeron” • “door te smeden wordt men smid” • ”by forgeing one becomes ´forgerer´= blacksmith ” • So lets continue! • In what way? Geometrically, numerically, and algebraically ...
Logistic Family of Maps f(x) = a x (1-x), continued • Proposed by Verhulst: try to derive this model or otherwise ask me
Recommend
More recommend
