Arbit Scratchpad

Ewww!! Yes that's true of some Darter , pupfish varieties and other fish species too ( Filial Cannibalism )! The dynamics of the population with adults cannibalizing on their offspring can be described by the Ricker map, given by the equation: Figure 1: Bifurcation Diagram (Click to enlarge) -Plot between R and eta . Chaos and bifurcations and period doubling observed.  We observe the 2 point limit cycle is observed at R value > 7.4 Figure 2: We observe a 4-point limit cycle for values of R > 12.5, see above figure The second iterate results in a 2-cycle periodic solution and then later this branches out and a 4-cycle periodic solution appears. Thus as R passes through a series of bifurcation values the character of the solution passes through a series of bifurcations, as period doubling of the periodic solutions. As R increases through successive bifurcations, every even periodic solution branches into a double-periodic solution and this happens wh...

What started as a measles outbreak among seven people who visited Disneyland in December has spread to more than 26, as an unvaccinated California woman apparently transmitted the virus through airports and the theme park, health officials said. (Source: Guardian ). The Ebola is another outbreak we saw a few months back... Taken together, the 26 cases would account for almost 12% of the expected measles cases for the entire year in the US. Photograph: John Heseltine/Corbis S - Susceptible , I - Infected , R - Recovered The SIR Model a three- compartment model , for disease spread can be used to explain such outbreaks. In this model each individual is either Susceptible to the disease, Infected, or has Recovered and is immune. Infected individuals infect susceptibles they meet with some rate and recover with some rate. The ratio of these rates determine if the disease will become an epidemic or not. The simulation for the SIR Model I had done recently as...

Global synchronization of oscillators is found abundantly in nature, emerging in fields from physics to biology. The Kuramoto model describes the synchronization behavior of a generalized system of interacting oscillators. A particularly beautiful example comes from certain species of fireflies , with stories of huge populations of fireflies all flashing in perfect unison, making long swaths of light flashing on and off in the darkness. It was not until the late 1960s that anyone understood what was really going on—that the rhythm was not being set by any single “conductor” firefly, but rather by the interactions among all of them. Somehow the oscillator in each firefly (presumably some patch of neurons in each firefly’s brain) corrects itself to flash in unison with all the others. With a large number of oscillators with different natural frequencies, the Kuramoto model predicts that, if they are allowed to interact strongly enough, they will all start oscillating at the same ...

Here is a presentation on Integrated Information Theory of Consciousness, that was given as part of the Complex Systems Seminar, at Chalmers University of Technology: Consciousness as Integrated Information from Shamit Bagchi More details here:  http://integratedinformationtheory.org/

A topographic map is the ordered projection of a sensory surface, like the retina or the skin, or an effector system, like the musculature, to one or more structures of the central nervous system. Topographic maps can be found in all sensory systems and in many motor systems.  “Topographically” mapped visual cortical areas,such as primary visual cortex map Cartesian X-­Y positions in visual space using an approximate log conformal mapping (From The Conscious Grid). Images of concentric circles of radii between  and  around the origin under the (conformal) map . Large circles are mapped into parallel lines with increasing real parts. (Courtesy: http://functions.wolfram.com/ElementaryFunctions/Log/visualizations/4/) The maps are not a strict isomorphic representation of XY space. Also, neurons in such maps do not stand for “pixels” of XY space. Kohonen Maps are relevant as a first step in understanding such topographic maps. More here on...

Kohonen Self Organization Map  To map a 2D triangle (set of vectors) to a 1-D vector: Input to the Kohonen Map: Simulation - Ordering and beginning of Convergence: Final output (below) - end of Convergence phase: From a project done in Complex Adaptive Systems course 'Neural Networks' at the Chalmers University of Technology, Gothenburg, Sweden, 2014.