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Figure (clik to enlarge) – Applying P(0)=0.6; r=4; N=100000; for(n=0;n<=N;n++) { P(n+1)=r*P(n)*(1-P(n)); } *Robert May* Population Dynamics equation [1974-76] (do check on Logistic maps) for several iterations (generations). After 780 iterations, *P* is attracted to 1 (max. population), and then suddenly, for the next generations the very same population is almost extinguish.

“** Not only in research, but also in the everyday world of politics and economics, we would all be better off if more people realised that simple non-linear systems do not necessarily possess simple dynamical properties**.” ~

*Robert M. May*, “Simple Mathematical models with very complicated Dynamics”,

*Nature*, Vol. 261, p.459, June 10, 1976.

(…) The fact that the simple and deterministic equation (1) can possess dynamical trajectories which look like some sort of random noise has disturbing practical implications. It means, for example, that apparently erratic fluctuations in the census data for an animal population need not necessarily betoken either the vagaries of an unpredictable environment or sampling errors: they may simply derive from a rigidly deterministic population growth relationship such as equation (1). This point is discussed more fully and carefully elsewhere [1]. Alternatively, it may be observed that in the chaotic regime arbitrarily close initial conditions can lead to trajectories which, after a sufficiently long time, diverge widely. This means that, even if we have a simple model in which all the parameters are determined exactly, long term prediction is nevertheless impossible. In a meteorological context, *Lorenz* [15] has called this general phenomenon the “** butterfly effect**“: even if the atmosphere could be described by a deterministic model in which all parameters were known, the fluttering of a butterfly’s wings could alter the initial conditions, and thus (in the chaotic regime) alter the long term prediction. Fluid turbulence provides a classic example where, as a parameter (the

*Reynolds*number) is tuned in a set of deterministic equations (the

*Navier-Stokes*equations), the motion can undergo an abrupt transition from some stable configuration (for example, laminar flow) into an apparently stochastic, chaotic regime. Various models, based on the

*Navier-Stokes*differential equations, have been proposed as mathematical metaphors for this process [15,40,41] . In a recent review of the theory of turbulence,

*Martin*[42] has observed that the one-dimensional difference equation (1) may be useful in this context. Compared with the earlier models [15,40,41] it has the disadvantage of being even more abstractly metaphorical, and the advantage of having a spectrum of dynamical behaviour which is more richly complicated yet more amenable to analytical investigation. A more down-to-earth application is possible in the use of equation (1) to fit data [1,2,3,38,39,43] on biological populations with discrete, non-overlapping generations, as is the case for many temperate zone arthropods. (…) in pp. 13-14,

*Robert M. May*, “Simple Mathematical models with very complicated Dynamics“,

*Nature*, Vol. 261, p.459, June 10, 1976 [PDF link].

Figure – Understanding the Brain as a Computational Network: significant neuronal motifs of size 3. Most over-represented colored motifs of size 3 in the C. elegans complex neuronal network. Green: sensory neuron; blue: motor neuron; red: interneuron. Arrows represent direction that the signal travels between the two cells. (from *Adami* et al. 2011 [Ref. below])

** Abstract**: […] Complex networks can often be decomposed into less complex sub-networks whose structures can give hints about the functional organization of the network as a whole. However, these structural motifs can only tell one part of the functional story because in this analysis each node and edge is treated on an equal footing. In real networks, two motifs that are topologically identical but whose nodes perform very different functions will play very different roles in the network. Here, we combine structural information derived from the topology of the neuronal network of the nematode C. elegans with information about the biological function of these nodes, thus coloring nodes by function. We discover that particular colorations of motifs are significantly more abundant in the worm brain than expected by chance, and have particular computational functions that emphasize the feed-forward structure of information processing in the network, while evading feedback loops. Interneurons are strongly over-represented among the common motifs, supporting the notion that these motifs process and transduce the information from the sensor neurons towards the muscles. Some of the most common motifs identified in the search for significant colored motifs play a crucial role in the system of neurons controlling the worm’s locomotion. The analysis of complex networks in terms of colored motifs combines two independent data sets to generate insight about these networks that cannot be obtained with either data set alone. The method is general and should allow a decomposition of any complex networks into its functional (rather than topological) motifs as long as both wiring and functional information is available. […] from Qian J, Hintze A, Adami C (2011) Colored Motifs Reveal Computational Building Blocks in the C. elegans Brain, PLoS ONE 6(3): e17013. doi:10.1371/journal.pone.0017013

[…] The role of evolution in producing these patterns is clear, said *Adami*. “Selection favors those motifs that impart high fitness to the organism, and suppresses those that work against the task at hand.” In this way, the efficient and highly functional motifs (such as the sensory neuron-interneuron-motor neuron motif) are very common in the nervous system, while those that would waste energy and give no benefit to, or even harm, the animal are not found in the network. “*Adami* and his team have used evolutionary computation to develop hypotheses about the evolution of neural circuits and find, for these nematode worms, that simplicity is the rule,” says *George Gilchrist*, program director in NSF’s Division of Environmental Biology (in the Directorate for Biological Sciences), which funds BEACON. “By including functional information about each node in the circuit, they have begun decoding the role of natural selection in shaping the architecture of neural circuits.” […] from Danielle J. Whittaker “*Understanding the Brain as a Computational Network*“, NSF, April 2011.

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