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This Concept Map, created with IHMC CmapTools, has information related to: n4. Boolean Networks, In computer simulations, the networks do adapt and improve and they evolve, not to the very edge of chaos, but in the ordered regime, not too far from the edge of chaos. It is though a position in to ordered regime near the transition to chaos affords the best mixture of stability and flexibility. ???? The exquisite power of self-organization in enormous Boolean networks may be the ultimate wellspring of dynamical order., If a single node is flipped from onto off,it leaves the system in the same basin of attraction. The system will return to the same state cycle from which it was perturbed. That is the essence of homeostatic stability. State cycle 3 above(111), is stable in this way. If the network is in this basin, flipping the activity of any node will have node in this way will return the system to the same state cycle. ???? Homeostatic stability does not always arise. State cycle 1 (000) is an isolated steady state and is unstable to the slightest perturbation. After a flip in the state of one node, the system is shoved into a different basin of attraction. It is unstable, and "can’t go home again"., Then the “rule tables” specifying whether each node will be active (1) or inactive (0) for each of the four possible signals would be: “AND” table “OR” table “OR” table 2 3 1 1 3 2 1 2 3 0 0 - 0 0 0 – 0 0 0 - 0 0 1 - 0 0 1 – 1 0 1 - 1 1 0 - 0 1 0 – 1 1 0 - 1 1 1 - 1 1 1 – 1 1 1 – 1 ???? [The output is positive if either or both of the inputs are positive. If both inputs are negative, then the output is negative.] OR gate Input 1 Input 2 Output 0 0 0 0 1 1 1 0 1 1 1 1, Systems capable of complex behavior have a decided survival advantage, and thus natural selection finds its role as the molder and shaper of the spontaneous order for free. ???? In the ordered regime, similar initial stages tend to become more similar, hence converging along their trajectories. This is another expression of homeostasis. Perturbations to nearby states tend to damp out., In the ordered regime, similar initial stages tend to become more similar, hence converging along their trajectories. This is another expression of homeostasis. Perturbations to nearby states tend to damp out. ???? You measure convergence or divergence along the trajectories of a network to determine its position on the order-chaos axis. In this measure, networks at the phase transition have the property that trajectories neither diverge nor converge., If different networks are built with increasing P biases, starting from the no-bias value of 0.5, (where half the nodes are 1’s ands half are 0’s) to a maximum of 1.0 (where all nodes either 1’s or 0’s), networks with P = 0.5 or slightly greater than 0.5, are chaotic, and networks with P near 1.0 are orderly. ???? When the P parameter is 1.0, the nodes in the network are of only two types. One responds with a 0 to any input pattern, and one responds with a 1 to any input pattern. So if you start the network in any state at all, the 0 nodes responds with a 0 and the 1 nodes respond with a 1. The network freezes into the corresponding pattern of 0’s and 1’s, and remains in that steady state forever., In addition, adjusting the P bias parameter from 0.5 to 1.0 also determines whether the network is in the ordered or chaotic regime. ???? There is a sharp phase transition when the network passes from the ordered regime to the chaotic regime. Just near this phase transition, just at the edge of chaos, the most complex behavior can occur; orderly enough to ensure stability, yet full of flexibility and surprise. This is what is meant by Complexity!, The main rules are simple to summarize. Two features of the way networks are constructed can control whether they are in an ordered regime, a chaotic regime, or a phase transition regime between these two, “on the edge of chaos”. ???? The second feature that controls the emergence of order or chaos is simple biases in the control rules themselves. Some control rules, the AND and OR of Boolean networks, tend to create orderly dynamics, others create chaos., The new and interesting hypothesis is that cells achieve both stability and flexibility by achieving a kind of poised state, balanced "on the edge of chaos". ???? Sparsely connected networks with K = 1 or K = 2, spontaneously exhibit powerful order. Networks with higher numbers of inputs per node, K = 4 or more, show chaotic behavior. Tuning the number of inputs per node – and thus the density of the web of connections among the nodes – from low to high, tunes the network from orderly to chaotic behavior., Now consider a network in which K = N, meaning that each node receives input from all other nodes, including from itself. Then there are an enormous number of possible states, and the length of the cycles is on tne order of 1030. The network would be changing for all time. ???? If a network is perturbed by flipping a node from on to off, or vice versa, the K=N networks exhibit an extreme version of the butterfly effect. Flip a node, and the system almost certainly falls under the sway of another attractor. Since there may be thousands of attractors, the tiny fluctuations will utterly change the future evolution of the system. K = N networks are massively chaotic. No order for free in this family., The second state cycle has two states, (001) and (010). The network oscillates between these two. No other states drain into this attractor. Launch the network with either of these two patterns, and it will remain in that cycle, blinking back and forth between the two states. ???? The third stare consists of the steady state (111). This attractor lies in a basin of attraction draining the four other states. Start the network in any one of these patterns and it will quickly flow to the steady state and freeze up., In these networks, (? K=2) nearby states (trajectories?) converge in state space, and the two similar initial patterns will likely lie in the same basin of attraction, driving the system to the same attractor. Thus these systems do not show sensitivity to initial conditions; they are not chaotic. The consequence in homeostasis. ???? Once such a network is perturbed, it will return to the same attractor with a very high probability., The simplest possible behavior would occur if the network fell immediately into a never ending state cycle consisting of a single pattern of 1’s ands 0’s. A system started in such a state never changes; it is said to be stuck in a cycle length of 1. ???? The first thing to appreciate about a Boolean network is that any network will settle down to a state cycle, but the number of states on such a recurrent pattern may be tiny, as small as a single state, or so hyper-astronomical that the numbers are meaningless., In computer simulations, the networks do adapt and improve and they evolve, not to the very edge of chaos, but in the ordered regime, not too far from the edge of chaos. It is though a position in to ordered regime near the transition to chaos affords the best mixture of stability and flexibility. ???? The hypothesis that complex systems may evolve to the edge of chaos or to the ordered regime near that poised edge, appears to account for a very large number of features of ontology, that magnificent, ordered dance of development from fertilized egg to adult., Any Boolean network has attractors, each draining some basin of attraction, but the state space of networks with thousands of kinds of molecular species are vast. The system might be located on any one of them. That is notorder. ???? After perturbation by mutation, catastrophic changes might occur, causing the death of the organism. Alternately, a mutation could cause the network to assume a new dynamical form. Some state cycles remain, but others are changed. New basins of attraction steer the network into different patterns., The first thing to appreciate about a Boolean network is that any network will settle down to a state cycle, but the number of states on such a recurrent pattern may be tiny, as small as a single state, or so hyper-astronomical that the numbers are meaningless. ???? More than one trajectory can flow from the same state cycle. Start a network with any of these different initial patterns, and after churning through a sequence of states it will settle into the same repeated state cycle, the same pattern of blinking onand offnodes., Kauffman – The reason that complex systems exist on, or in, the ordered regime near the edge of chaos is because evolution takes them there. ???? Systems capable of complex behavior have a decided survival advantage, and thus natural selection finds its role as the molder and shaper of the spontaneous order for free., More than one trajectory can flow from the same state cycle. Start a network with any of these different initial patterns, and after churning through a sequence of states it will settle into the same repeated state cycle, the same pattern of blinking onand offnodes. ???? A Boolean network may harbor many state cycles, each draining its own basin of attraction., A Boolean network is one in which the actors are joined with either an AND or an OR prefix. Consider a Boolean network with three nodes connected to each other, each receiving inputs from the other two. Each node can have one of two values, 1 or 0. There are four possible input patterns each node can receive from its neighbors. Both inputs could be OFF (00), or both could be ON (11), or one ON and the other OFF (10, or 01), for four separate states. ???? Then the “rule tables” specifying whether each node will be active (1) or inactive (0) for each of the four possible signals would be: “AND” table “OR” table “OR” table 2 3 1 1 3 2 1 2 3 0 0 - 0 0 0 – 0 0 0 - 0 0 1 - 0 0 1 – 1 0 1 - 1 1 0 - 0 1 0 – 1 1 0 - 1 1 1 - 1 1 1 – 1 1 1 – 1, Attractors are the ultimate source of homeostasis ensuring that a system is stable. ???? If a single node is flipped from onto off,it leaves the system in the same basin of attraction. The system will return to the same state cycle from which it was perturbed. That is the essence of homeostatic stability. State cycle 3 above(111), is stable in this way. If the network is in this basin, flipping the activity of any node will have node in this way will return the system to the same state cycle.