#Attractors
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Shannon, C.D., Weaver, W. (1949) The Mathematical Theory of Communication. University of Illinois Press, Urbana Sharkovskii, A.N. (1964): Coexistence of the cycles of a continuous mapping of the line to itself. Ukranian Mat. Z. 16:61-71 Shaw, R. (1981): Strange attractors, chaotic behavior and information flow. Z. Naturfursch. 36a:80-112 Shenker, S.J., Kadanoff, L.P. (1982): Critical behavior of a KAM surface. J. Stat. Phys. 27:631-645 Shlesinger, M.F. (1988) Fractal time in condensed matter. Ann. Rev. Phys. Chem. 39:269-290 Shlesinger, M.F. (1996) Random Processes. Encyclopedia of Applied Physics. 16:45-70 Shlesinger, M.F., Klafter, J., Wong, Y.M. (1982) Random walks with infinite spatial and temporal moments. J. Stat. Phys. 27:499-512 Shlesinger, M.F., Zaslavsky, G.M. (1996) Strange kinetics. Physics Today. Feb. pg. 33-39. Shlesinger, M.F., Zaslavsky, G.M., Frisch, U. (1995) Levy Flights and Related Topics in Physics. Springer-Verlag 287 HOUSE_OVERSIGHT_013787
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Romeiras, F.J., Bondeson, A., Ott, E., Antonsen, T.M., Grebogi, C. (19987) Quasioperiodically forced dyanmical systems with strange, nonchaotic attractors. Physica 26D:277-294. Roschke, J. (1992) Strange attractors, chaotic behavior and informational aspects of sleep EEG data. Neuropsychobiology 25:172-176 Roschke, J., Aldenhoff, J.B.(1992) A nonlinear approach to brain function: deterministic chaos and sleep EEG. Sleep 15:95-101 Roschke, J., Aldenhoff, J.B. (1993) Estimation of the dimensionality of sleep-EEG data in schizophrenics Eur Arch Psychiatry Clin Neurosci 242:191-196 Roschke, J., Fell, J., Mann, K. (1997) Non-linear dynamics of alpha and theta rhythm: correlation dimensions and Lyapunov exponents from healthy subject's spontaneous EEG Int J Psychophysiol 26:251-61 Réssler, O. (1976): An equation for continuous chaos. Phys. Lett. A57:397-398 Réssler, O.E. (1979) An equation for hyperchaos. Phys. Lett. 71A:155-157 Ruelle, D. (1978) What are measures describing turbulence. Prog. Theor. Phys. Supp. 64:339-345 Ruelle, D.(1979) Ergodic theory of differentiable dynamical systmes. Publ. Math. IHES 50:27-58 Ruelle, D. (1987) Chaotic Evolution and Strange Attractors. Cambridge University Press. Cambridge 283 HOUSE_OVERSIGHT_013783
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Grassberger, P. (1983) Generalized dimensions of strange attractors. Phys. Lett. A97:227-230. Grassberger, P, Procaccia, |. (1983) Measuring the strangeness of strange attractors. Physica 9D:189-208 Gregson, R.A.,Britton, L.A., Campbell, E.A., Gates, G.R. (1990) Comparisons of the nonlinear dynamics of electroencephalograms under various task loading conditions: a preliminary report. Biol Psychol 31:173-191 Grimmett, G. (1989) Percolation. Springer-Verlag. Guckenheimer, J., Holmes, P. (1983): Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer. N.Y. Guillemin, R.C., Brazeau, P., Briskin, A., Mandell, A.J. (1983) Evidence for synergetic dynamics in a mammalian pituitary cell perfusion system in (ed. E.Basar, H. Flohr, H. Haken, A.J. Mandell) Synergetics of Brain. Springer-Verlag. pp 155-162 Gupta, V., Suryanarayanan, S., Reddy, N.P. (1997) Fractal analysis of surface EMG signals from the biceps. Int J Med Inf 45:185-192 Guzzetti, S.,Signorini,M.G., Cogliati, C., Messetti,S., Porta, A., Cerutti, S., Malliani, A. (1996) Non-linear dynamics and chaotic indices in heart rate variability of normal subjects and heart-transplanted patients. Cardiovasc Res 31:441-446 Hadamard, J. (1898) Les surfaces a courbures opposees et leur lignes geodesic. J. Math. Pures. Appl. 4:27-73 263 HOUSE_OVERSIGHT_013763
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Briggs, K. (1990) An improved method for estimating Lyapounov exponents of chaotic time series. Phys. Lett. A151:27-32 Brodskii,V.(1998) The nature of the circahoralian (ultradian) intracellular rhythms. Similarity to fractals Izv Akad Nauk Ser Biol 3:316-29 Broomhead, D.S., King, G.P. (1986) Extracting qualitative dynamics from expeirmental data. Physica D20:217-236. Brown, R., Bryant, P., Abarbanel, H.D. (1991) Computing the Lyapounov spectrum of a dynamical system from an observed time series. Phys. Rev. A43:2787-2806 Bryant, P., Brown, R., Abarbanel, H.D. |. (1990) Lyapounov exponents from observed time series. Phys. Rev. Lett. 65:1523-1526 Bullard, W.P., Guthrie, P.B., Russo, P.V., Mandell, A.J. (1978) Regional and subcellular distribution and some factors in the regulation of reduced pterins in rat brain. J. Pharmacol. Exp. Therap. 206:4-20 Buzug, Th., Reimers, T., Pfister, G. (1990) Optimal reconstruction of strange- attractors from purely geometrical arguments. Europhys. Lett. 13:605-610 Callahan, J., Sashin, J.I. (1987) Models of affect-response and anorexia nervosa. Ann. N.Y. Acad. Sci. 504:241-259 Canavier, C.C., Clark, J.W., Byrne, J.H. (1990): Routes to chaos in a bursting neuron. Biophys J. 57:1245-1251 Carpenter, G.A. (1979): Bursting phenomena in excitable membranes. SIAM J. Appl. Math. 36:334-352 255 HOUSE_OVERSIGHT_013755
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made on their attractors when they were much more densely filled (Lathrop and Kostelich , 1989). Bowen’s “shadow lemma’ in support of a thin film of points over the skelton of unstable fixed points of attractors is the fundamental reason that short sample length time series can often discriminate between control and experimental conditions in brain research studies. Another recently implemented entropy, called “approximate entropy,” is exploiting the underlying unstable fixed point skeletal shadowing principle in expansive dynamical systems to find statistically significant differences between control and experimental results in reasonably short, physiologically realistic, sample lengths (Pincus, 1991; Pincus et al, 1991). This algorithm is somewhat derivative of those involved in the computation of the correlation dimension (see above). Instead of computing across a range (and taking the limits) of embedding dimensions, d, and sequential paired-vectorial distances, ¢, it empirically tailors and fixes them to compute a “logarithmic likelihood” that points remains close through incremental change in the time series. The “approximate entropy” is not easily relatable to either hr and hy. One is tempted to predict that this geometrically oriented algorithm might be fooled into a postive entropy diagnosis if applied to strange, nonchaotic dynamical systems with fractal dimension but no A+) -related mixing. Since sequence position is conserved in this computation, two simultaneously studied (“multiparameter”) systems can be examined for their mutual coherence as the “cross approximate entropy.” Among the interesting findings from applications of this index to neuroendocrine studies are an increase in approximate entropy in LH and FSH secretory patterns with age in both sexes, perhaps quantitatively heralding menopause (Pincus and Minkin, 1998) and decreased cross approximate entropy, a decrease in regulatory coupling between ACTH and cortisol secretion patterns in
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role of self-intersection on manifolds in this new source of irreversibility (with a resulting “arrow of time”) is developed in Mackey (1992). As noted, the skeleton which configures attractors is composed of unstable, “saddle” fixed points, each of which attract (iron down) the trajectory along one dimension and repel or spread it out along another. Systems fulfilling the criteria for a chaotic dynamical system have the property of a countably infinite number of unstable periodic orbits composed of these unstable fixed points. Depending upon parameters, the orbital points can pull up their tails to be discrete with respect to each other or spread along the unstable direction to connect smoothly with others along a curve such as a saddle cycle. Parametric control of the strengths and structures of the saddle point skeleton of typical attractors can be used to change both the rate of generation of novel symbols as well as recurrances to old ones in the symbolic dynamics generating a brain dynamical system’s lexagraphic products (Bowen, 1978; Alexeev and Jacobson, 1981)). Using a variety of techniques to algorithmically register “return times,” experimental condition-sensitive “saddle orbits” composing unstable periodic orbits have been demonstrated in geometric reconstructions of real data series generated by a 40+ component chemical reaction (Lathrop and Kostelich, 1989), in response to natural stimuli in the time dependent behavior of the crayfish caudal photoreceptor (Pei and Moss, 1996) and in the interburst interval sequences recorded in hippocampal slices of the rat (So et al, 1997; So et al, 1998). If the reader uses the software listed above to simulate the time evolution of one of these attractors of abstract or real systems , she will learn that a remarkably small number of points, a very short time sample, will outline, “shadow” (Bowen, 1978), the complete array of unstable fixed points before filling in the attractor. It is tempting to speculate about
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unstable fixed points, thus forming unstable periodic orbits. The latter, one of three diagnostic features of chaotic attractors (see above), can also be seen as resulting from the “coarse-grained” imprecision of real world neurobiological measurement such that two points that are brought close to attractive-repelling points are, within measurement error, recorded as having the same value. 240 HOUSE_OVERSIGHT_013740
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period doubling bifurcations accumulate (Grassberger, 1981). This is a dust-like region, which when endlessly dilated looks like the same dust. Some mathematicians call these objects “Lebesgue points” because even though at low magnifications when they look rather solid, they are not. Composed of points, they have topological measure zero (a line has measure one) and non-integer fractal dimension. These 4=0, D # Integer, period doubling accumulation points can be found in a wide variety of attractors, though in each case the parameter space in which they are located is so small (in point set topology also called “Lebesgue measure zero”) that they are very difficult to locate and therefore have little chance of being physiologically significant. This constrasts with a relatively new category of dynamical systems which promises to be important in studies of the nervous system. These are ones that are driven by two or more independent frequencies (called quasiperiodic driving). We found them to be relevant to brain stem, thalamocortical neurophysiology of perceptual processes and states of consciousness. They have the properties, A=0, Do and D, # integer and a characteristic scaling “spectral distribution function” (see below). They have been named “strange nonchaotic attractors” (Grebogi et al, 1984; Romeiras et al, 1987; Ding et al, 1989). In addition, the strange nonchaotic behavior of these quasiperiodically-driven, nonlinear oscillators has positive (>0) measure in parameter space and thus is of potential physiological significance. A good demonstration of a multiple frequency driven strange nonchaotic attractor can be found and manipulated in the software package of Nusse and Yorke (1991). The neurobiological substrate for this system is the brain stem neuronal modulatory driving of on- going thalamocortcal oscillatory brain waves (once called “recruitment waves” in the 7-14 Hz, 8 to a, day dreaming to quiet alert range) and as perturbed by multifrequency
February 7, 2026 at 11:22 AM
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higher EEG D2 values were frontal in schizophrenic patients and more central in controls (Elbert et al, 1992). The Dz computed on the EEG during Stage IV (“delta”) sleep was sensitive to acute sleep deprivation and recovery, but demonstrated compensation (Cerf et al, 1996). Non-alcholic children of alcoholic parents manifested lower values for D, in their EEG attractors than the children of a normal control group (Ehlers et al, 1995). Higher |.Q. correlated with EEG Dz in most leads in the resting state but not during a visual imagery task (Lutzenberger et al, 1992). These differences also correlated with individual differences in task performance in a perceptual pattern predictive task (Gregson et al, 1990) and with a working memory task load with regional differences most marked in the right fronto-temporal cortex (Sammer, 1996). Peripheral nerve stimulation in the earlobe and trapezius muscle induced increments in D2 in the EEG of specific brain regions (Heffernan, 1996). Memory for but not induced pain increased EEG Dz in chronic pain patients but not in normal controls (Lutzenberger et al, 1997). Using contingent reinforcement of brain wave modes by hypothalamic, but not cerebral hemispheric, stimulation reduced Dz in the EEG (Mogilevskii et al, 1998) resembling the changes accompanying defensive reflex conditioning in the rabbit between the early and late stages of the process (Efremova and Kulikov, 1997). Difficult to diagnose “periodic lateralized epileptiform discharge” syndromes have apparently yielded to D2 computations (Stam et al, 1998). In equally problematic “atypical seizure” syndromes in children, Dz computed on the autocovariance functions of 200 Hz digitized EEG records from multiple channels demonstrated characteristic changes (Yaylali et al, 1996). Unlike computing a reliable leading 2(+)on a point set of a time series reconstruction denoting the “sensitivity to initial conditions’ requirement for the diagnosis of chaos (and a potential for
February 7, 2026 at 11:21 AM
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(Edwards and Griffith, 1997). The role of central and autonomic innervation in cardiac interval dynamics has been explored using D2 in various ways. For examples, the transplanted heart rhythm in man has a lower D2 than that of the normal heart (Guzzetti et al, 1996) and general anesthesia and cholinergic (but not B-adrenergic) blockade decreased multisystem Dz in a series of multiparameter (respiration, mean blood pressure and heart rate) studies in piglets (Zwiener et al, 1996; Hoyer et al, 1998). The activities of single and aggregates of neurons are being described and differentiated by the Dz of their interevent interval attractors. Early and important studies related to both neuronal and field electrical activity indicated their promise (Rapp et al, 1985; Zimmerman and Rapp, 1991). The olefactory bulb demonstrated spatially uniform scaling dimensions that changed with event-related perturbation (Skinner et al, 1990). An iron-induced spiking focus in the rat hippocampus in vivo manifested the same decrease in D2 as it did in the kindled in vitro hippocampal slice (Koch et al, 1992). Dz also differentiated among characteristic single unit time series in norepinephrine, dopamine and serotonin neurons (Selz and Mandell, 1991) and among A8, A9 and A10 dopamine neurons (Selz and Mandell, 1992). Attractors reconstructed from single unit interspike intervals in the substantia nigra pars compacta and the auditory thalamus manifested discriminatable values for D2 in neurons recorded by the same electrode (Celletti and Villa, 1996) and changes in state manifested in patterns of subthreshold oscillations in single neurons in the inferioir olivary nucleus could be characterized using this index (Makarenko and Llinas, 1998). Dz reliably discriminated between states of arousal and between the multiparameter (eye movements, neck muscle tone, EEG stage) defined EEG stages of sleep (Bablyoyantz, 1986; Rapp et al, 1989; Ehlers et al, 1991) with non- REM having a lower D2 than
February 7, 2026 at 11:21 AM
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Orbital Divergence Characterizes Expansive Dynamics on_ Biological Attractors In the dynamical world of equilibria (fixed points in phase space) and periodic cycles (fixed points of a return map), a common concern involves their stability. What happens if an adventitious jiggle moves the orbit a little distance away from the fixed point? Would the wind wiggled suspension bridge start to flap with increasing amplitude or would it damp back down quickly to its stable state. A “Lyapounov functional,” L, is constructed which can be visualized like a smooth potential bowl around the fixed point such that any L stable solution that starts at its bottom tends to stay there or is asymptotically L stable if the solution converges to the fixed point at the bowl’s bottom as ¢— oo. If the point is not L stable, it is L unstable. The modern study of nonlinear systems have produced another kind of stability issue with a similar appellation yielding other direction specific indices, the Lyapounov characteristic exponents, 2 (Oseledec, 1968; Eckmann and Ruelle,1985; Ruelle, 1990; Ott et al, 1994). In this context, the instability is not one of perturbative escape from a fixed point, but of the average rate with which the (theoretically infinitesimal) distances among a handful of points representing a set of initial conditions (each a precision limited, hypothetical repetition of the same experiment), are being stretched apart by the expansive action of a strange attractor system. In three dimensions, one can envision a ball of initial conditions being elongated along the unstable direction and ironed down from both sides along the stable direction over time, transforming the ball into an ellipsoid and then into a (recurrent) curve. In simplest terms and thinking about a one dimensional scalar time series, the Lyapounov exponent reflects the multiplicative average (logarithmic addition) of the sequence of slopes of the series of straight lines connecting the points. An average
February 7, 2026 at 11:17 AM
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Although counter-intuitive when expressed in words, the trajectories that one sees in the graphics of chaotic attractors result from the actions along the “unstable” directions of the stretching distortion; the actions in the otherwise invisible stable directions “iron down” the points onto this unstable manifold (n dimensional abstract surface). As might be expected from this set of characteristic motions, the diagnostic triad of chaotic dynamical systems are: (1) Sensitivity to initial conditions—tiny distances between starting points are magnified and large distances between starting points are reduced under the stretching and folding actions of the system; (2) The presence of a theoretically infinite but countable number of unstable periodic orbits of theoretically all period lengths—points in phase space can be viewed an attractive-repellers, visited and left by the orbits recursively as the dynamics proceed; and (3) Indecomposability—the attractor is not separable into isolated regions and no points escape (see Devaney, 1989, for one of the clearest definitions). Of particular relevance to information encoding and transport by brain mechanisms, it is important to visualize that new information in the form of unstable periodic orbits is being created as well as destroyed by the dynamics. The logarithmic rate of formation of these new orbits is computed as the system’s topological entropy (see below). Assuming the real neurobiological system under study is behaving in these ways (and often much has to be done to help justify such a claim), the observables take the form of an irregular and/or episodic time series of amplitudes, as in repeated sample, neuroendocrine studies of plasma hormone levels (Veldhuis and Johnson, 1992) or a sequence of times between events as in neuronal interspike intervals (Katz, 1966; Perkel et al, 1967). These time or time-sequence series are generally studied from three relatively distinct yet complementary
February 7, 2026 at 11:14 AM
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as importantly, far apart initial values could be found close together in the limit set. This behavior was called “sensitivity to initial conditions” by David Ruelle (1978; Ruelle and Takens, 1971). It is noteworthy, however, that over a range of values of the parameters, the overall pattern of the orbits of the Lorenz attractor results in characteristic geometric pictures as well as invariant statistical descriptors. Qualitative and quantitative global similarities were gained while specific solutions were lost in these “strange attractors” of nonlinear systems. Analog computer simulation of a simpler set of equations inspired by nonlinear chemical reaction kinetics led to the discovery by Réssler (1976) of another early and generic strange attractor combining sensitivity to initial conditions and characteristic geometries and measures. It was the Russian mathematicians, ALN. Kolmogorov (1957), Sinai (1959) and V.|. Arnold (Arnold and Avez, 1968), the French mathematicians, Rene Thom (1972) and David Ruelle (1978) and the U.C. Berkeley mathematicians, Steve Smale (1967) and his student, Rufus Bowen (1975), and their associates who gathered together these and other related computational discoveries and embedded them in a qualitative theory of nonlinear differential equations, using a variety of formalisms, including point set and differential topology, geometry, analysis and ergodic (having an invariant statistical description) measure theory that formally established the fundamentals for research in nonlinear dynamical systems. Here a dynamical system refers generally and simply to the components and nonlinear processes (transformations) that move points (values) in discrete (“map”) or continuous (“differential equation”) time around in an appropriately defined space. The phrase, “nonlinear transformation” in this context does not imply easily solvable curved functions, such as those representing the sigmoid kinetic or threshold functions of enzymes and neuronal
February 7, 2026 at 11:10 AM
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related knowledge to the most abstract relationships. But they’re also weighted with values similar to neural net weights, and pass around quantities (importance values, discussed in Chapter 23 of Part 2) similar to neural net activations, allowing emergent attractor/assembly based knowledge representation similar to attractor neural nets. The concept of glocality lies at the heart of this combination, in a way that spans the pseudo- dichotomy: e Local knowledge is represented in abstract logical relationships stored in explicit logical form, and also in Hebbian-type associations between nodes and links. e Global knowledge is represented in large-scale patterns of node and link weights, which lead to large-scale patterns of network activity, which often take the form of attractors qualitatively similar to Hopfield net attractors. These attractors are called maps. The result of all this is that a concept like “cat” might be represented as a combination of: e A small number of logical relationships and strong associations, that constitute the “key” subnetwork for the “cat” concept. e A large network of weak associations, binding together various nodes and links of various types and various levels of abstraction, representing the “cat map”. HOUSE_OVERSIGHT_013185
February 7, 2026 at 9:17 AM