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“Ergodic” (Invariant) Measures Applied to n- Dimensional, Lag Embeddings of Expanding and Mixing, Biological Dynamical Systems Arnold J. Mandell M.D. Multi.

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Presentation on theme: "“Ergodic” (Invariant) Measures Applied to n- Dimensional, Lag Embeddings of Expanding and Mixing, Biological Dynamical Systems Arnold J. Mandell M.D. Multi."— Presentation transcript:

1 “Ergodic” (Invariant) Measures Applied to n- Dimensional, Lag Embeddings of Expanding and Mixing, Biological Dynamical Systems Arnold J. Mandell M.D. Multi Modal Imaging Laboratory, MMIL Department of Psychiatry, UCSD and Fetzer-Franklin Fund Ornstein theorem: Most/All suitably normalized measures made on chaotic dynamical systems are equivalent to their informational entropy, h (which has no single or definitive algorithmic computation). Frobenius-Perron theorem: Square matrices with non-negative entries have at at least one positive eigenvalue, λ ≥ 0 and log λ ≈ h. T, Pesin-Young-Mandell (conservation of) : For uniformly & non- uniformly hyperbolic systems, the topological entropy, h T, varies as the product of the capacity dimension, Dc & the leading Lyapounov exponent, Λ 1. hT = Dc Λ 1.

2 Bifurcation Entropy Fractal Leading Lyapounov Nonlinear Dynamics Chaos Google Book Title Word Count From 1750 to Note decline of relevant words beginning in the vicinity of Adoption of new science characteristically comes late to biology

3 Dynamical systems as qualitative nonlinear orbital behavior (in for example, phase portraits) “The qualitative theory of differential equations” see classic book by VI Arnold,(1983) global behavior Statistical mechanics’ central point is the dependence on microscopic variables. Distributions/moments versus. “Invariant Measure” “Ergodic theory of dynamical systems” Topological. metric and nonuniform entropy, the dimensions, the expansion exponents, graph theoretic exponents….see Ergodic Theory by Cornfield, Fommin &Sinai (1982)

4 Multiply periodic ? Chaotic? Irrational winding on the torus Lyapounov ≤ 0 Stretch, fold, shuffle. Points get out of order; “mixing”Lyapoun ov > 0 Generic membrane equations Mandell and Selz, 1990 Quasi-periodic and chaotic van der pol. Phase Portraits

5 In place of n-tori of stable periodic orbits, we may have: Homoclinic Orbits Which Join Unstable Fixed Points to Themselves Recurrence via unstable periodic orbits of increasing lengths and self- similar structure. “sizes” are hierarchical and scale with a power law. Intersection of stable and unstable manifolds of unstable fixed point A two dimensional section showing the unstable and stable manifolds of an orbital flow meeting at a “homoclinic” point. X “A small disc centered near a homoclinic point includes infinitely many periodic points of different periods” Poincare; Smale; Yorke; etc. WUWU WsWs

6 UNSTABLE PERIODIC ORBITS, UPOs, (HOMOCLINIC TO UNSTABLE FIXED POINTS) OF HIPPOCAMPAL NEURON INTERSPIKE INTERVALS (period one) So et al, 1998 (Steve Schiff’s group) Find periodic orbit (period one) Hippocampal neuron spikes Interspike intervals Phase portraits Of ISI; stable and unstable manifolds of unstable fixed points defining unstable periodic orbits Return embedding of ISI; colored points UPOs Stable and unstable manifolds support chaotic dynamical systems Aattractors.

7 Phase Portrait of Temperature Time Series Before a Catch X(t) X (t – 1) Effect of “catch and return” on fresh water pike fish temperature dynamics An example of phase space reconstruction and measures made on phase portrait of a physiological time series.

8 Phase Portrait of Temperature Time SeriesAfter a Catch X(t) X (t-1)

9 Spontaneous, bursting, intermittent patterns of brain stem neuronal discharge drive subcortical and neocortical membrane fluctuations: both at several time scales with power spectral power law scaling. Thanks to Carlson, Foote, Guillemin,

10 AG YM SP “…equating entropy increase as the spontaneous dispersal of energy, namely how much energy is spread out in a process, or how widely dispersed it becomes...” Leff, 1996,2007 Phase portraits of IMF 3,4,5 of C16-ssds(i) in AG, control, YM, intermediate state, and SP, typical medicated schizophrenic proband.

11 Modal Descriptions Frequency (power) Spectrum in Log-Log plot slope → α Broomhead-King autocovariance eigenfunctions, Ψ Morlet mother wavelet, wavelet transformation of Ψ, W(Ψ) Emperical Mode Decomposition →Intrinsic Mode Functions, EMD →IMF

12 The continuous power spectrum with complex singularities of a chaotic attractor (from “hydrodynamic” equation) Farmer, Crutchfield, Froehling, Packard and Shaw, “The original UCSC chaos kids” The “funnel” Positive Lyapounov exponent Continuous power spectrum Complex singularities Unstable periodic orbits Recursive homoclinic behavior near unstable fixed points Chaotic dynamical systems manifest a variety of power spectral exponents. Universality classes??

13 Log(frequency)--> log(amplitude 2 ) (i.e., power) A common manifestation of hierarchical, multiscale, self- similar, fractal statistical dynamics is “1/f α noise.” The system manifests correlations at many scales. This “signature” is common to many systems with strong (cooperative) interactions and many degrees of freedom (e.g. the brain’s electromagnetic systems). It may also accompany distribution functions with infinite second moments as in a Levy Process “Universal” scaling laws show up in a wide range of contexts:

14 f --α, Mean α = -1.67± 0.43; Median = -1.59; Var = 0.18; fs = 600/150 ComputHz Log-log Power Spectra of Three Minutes (108,000 points) of MEG Central C16 ssds(i) in Ten Normal Controls The Spectral Power law, α, approximates the Kolmogorov scaling of 5/3 (1)Use standard Fourier transform of time series; (2) log transform the frequency and power axes. (3) Compute the slope of the middle third of log-log plot; (4) –slope = α

15 To Study the Inverse of the Time-Dependent Wavelengths, Broomhead-king Decomposition Uses the Leading Eigenvectors of a Lagged Autocovariance Matrix, Composed with the Original Series to Generate Two or Three Leading B/K Eigenfunctions

16 To Observe the Dynamics in Time of Scaling System(s) We Apply the Wavelet Transformation in which a “Mother” Wavelet is Convolved with the Data as it is Translated Down the Series, “b”, at Various Dilations, “a”. Morlet mother wavelet, w, = w = sine wave x Gaussian

17 Intermittent, hierarchical scaling vortices, we call strudels, the German word for “whirlpools” or “eddies,” in the 3-5 to 20+ second time scales. Here, two strudels:

18 Four intermittent, hierarchical, scaling strudels, i(S), i = 4, are seen, defined by their near continuity beginning below the middle scales and, over shorter or longer times, reaching or exceeding the upper scaling bound of the wavelet graph. Both the incidence and durations are within the range reported for TUTIs (task unrelated thoughts and images): 5 to 20+ seconds. Note how much detail and texture of the time series would be lost reporting only their means, variance and higher moments Intermittent Vortices in the 3-5 to 20+ second BK eigenfunction time scales.

19 MEG, symmetric sensor difference series, ssds EEG, multi- electrode Local field potentials from Pyramidal cell layers II and III (self-referential network) Local field potentials from neocortical pyramidal cell network in taskless, resting monkey statistically resemble MEG ssds (Shew and Plenz, 2009)

20 Morlet wavelet decomposition of 14 second,C16 sensor difference sequence During brief “petite absence” seizure demonstrate spike, dome and Vertically coherent strudels. Fast spiking drives an expanding flow. >100 Hz ~3-4Hz ~2.0 Hz ~0.8 Hz ~ 14 seconds


22 Intermittent 3 to 8 Second Strudel “Absences” Persist Over the 2.3 min of Eyes-Closed Resting Record in Proband, YM Λ = 0.451; D C =1.61; h T =0.348 memv = (0.572) α = 2.79; X 4 =0.441;

23 Identify successive pairs of zero crossings, identify local extrema Connect max, min with cubic splines (upper and lower “envelopes”) Compute first mean, m 1, of the envelopes [ssds]-m 1 = h 1 ; h 2 = h 1 – m 2…. ….+ …residue(“sifting”) Inter-maxima distance is the local time scale Allows real time snap shots of nonstationary, “instantaneous” fluctuations growing in scale (wavelength) from left to right. h i I = 0.. n To empirically “unpeel” the hierarchical scales revealed in the log-log power spectra use Huang’s Empirical Mode Decomposition yielding an array of Intrinsic Mode Functions.

24 AG YM SP Hilbert-Huang Intrinsic Mode Functions, IMF 1,2,3,4,5 MEG, C16(ssds),16.66 sec Note Loss of modular amplitudes in proband SP’s intermediate time scales hnhn

25 Invariant Measure Theory Using “Blind Boys and the Elephant” Metrics

26 Battery of measures being applied to resting, eyes closed, asymmetric MEG sensor difference sequences: ssds: symmetric sensor difference sequences Note that axiosymmetric B fields would cancel Symbolic dynamics; Information Phase space geometry Modal analysis Probability distribution

27 Algorithmic Themes in Quantifying Global Fine Structure in Expanding & Mixing Dynamical Systems 1.The quantities sought are fractional exponents (logarithms) 2.These quantities may indicate hierarchical scaling relations, “self similarity” capacity, correlation, Hausdorff dimensions.Dc 3.Logarithmic relation between the measure (abscissa) and the measurement (ordinate), capacity, correlation, Hausdorff dimensions Dc. 4.From the growth rate of the trace of the exponentiated transition incidence matrix, Rate of appearance of new recursive orbits, topological entropy, hT 5.From the transition matrix, distribution of weights in normalized, exponentiated Markoff matrix,metric entropyhM 6.Using the phase space reconstruction, determination of the separation rate of recursively renewed “nearby initial conditions.”. Leading (positive) Lyapounov exponent, λ or Λ Partition (“generating”?), transition matrix, with or without symbol substitution, and symbolic dynamics

28 Sensitivity to initial conditions, Lyapounov Exponent (rate of Expansion) Slope Log measure (scales ) Log measu rement “box” capacity dim ension Complexity of manifold Growth rate of new recursive orbits…..topological entropy Measures: Λ, Dc, hT Partition→transition incidence matrix→growth rate of trace while exponentiating matrix

29 Leading Lyapounov exponent ; divergence, expansion, mixing Power spectral scaling exponent:log-log slope, global, scaling Topological entropy; rate of new “loop” formation Metric entropy: distribution of weights on loops Capacity dimension: complexity of manifold of support Non-uniformity ||top-met| difference Measureable entropy manifold volume top x Lya x dimension Unwinding number lags to asymptotic capacity dimension Skew distributional asymmetry Kurtosis peakedness and heavy tail Levy exponent rate of converrgence of tail of distribution Hurst exponent persistence vs antipersistent Multi-parameter quantitative-qualitative descriptions of dynamical systems.

30 Λ = log (rad B/rad B i ) h T = log(#B i ) d = lim (me)/log(e) Fixing entropy, h T, log (3) dimension, d, goes down as the Lyapounov increases Measurable entropy manifold volume, MEMV = [h T l D C ] Thereom : h T = l D C Pesin, Young, Manning, topological entropy = product of the leading lyapounov exponent and the capacity dimension; Premise: conservation of brain entropy Intuition about the relations between entropy, Lyapounov and dimension Fixing Λ, h T and d C go up. Λ increases d C increases h T increases

31 REDUCED VOLUME OF FRONTAL LOBE (F14 SSDS) MEASUREABLE ENTROPY MANIFOLD VOLUME, MEMV, IN PROBANDS Implicit function representation of the Pesin-Young Ansatz: ld= h T ( means of 10 SSDS, each of which were computed on 32,000 points) l Lyapounov exponent d capacity dimension Ten control subjects Ten medicated schizophrenic patients Topological Entropy h T memv = 3.13 log unitsmemv = 2.37 log units

32 Third new findings: aggregate measure relations and memv decreased in “abnormal” brain plasmas.

33 Phase Portraits and Recurrence Plots of the Three Leading Autocovariance Matrix Eigenfunctions

34 Data: symmetric sensor difference series, ssds, Four minutes 600/150 Hz Recurrence plot B/K phase Space New: MEG-ssds Measure Suite


36 Topological entropy, h T, capacity dimension, D C, and leading Lypounov exponent, Λ, and their Cartesian product, measurable entropy manifold volume, memv = Π[Λ D C h T ] on ssds(i) discriminate controls from probands. x ≡ L, y ≡ D C, Z ≡ h T

37 From Intermittency to Transitivity in Neuropsychobiological Flows AJ M, Am. J. Physiol. 245: R484-R494, 1983 Intermittency Transitivity ∂(Λ) > 0 ∂(α) < 0 ∂(h T ) > 0 ∂(h M ) > 0 ∂|h T – h M | < 0 ∂(D C ) > 0 ∂(σ 3 ) < 0 ∂(σ 4 ) < 0 ∂( memv ) > 0 Aggregate of relative changes in measures

38 A few (personal) references Mandell, AJ (1987) Dynamical complexity and pathological order in the cardiac monitring problem. Physica D 27: Mandell, AJ & Selz, KA(1993) Brain stem neuronal noise and neocortical resonance. J. Stat. Phys. 70: Mandell, AJ & Selz, KA(1997) Entropy conservation as hT = Λ*Dc. (1997) Chaos 7: Mandell, AJ & Shlesinger, MF(1990) Lost choices, parallelism and topological entropy decrements in neurobiological aging. AAAS Washington. Selz, KA & Mandell, AJ. (1991) Bernoulli partition equivalence of intermittent neuronal discharge patterns. Int. J. Bifurcation Chaos 1: Mandell, AJ (2013) Intermittent turbulent eddies in brain magnetic fields. Chaos, Solitons & Fractals 55: Robinson, S, Mandell, AJ & Coppola, R (2013) Spatiotemporal imaging of complexity.Frontiers in Comp. Neurosci. 6:1-14 (#101).

39 Data: symmetric sensor difference series, ssds, Four minutes 600/150 Hz Recurrence plot B/K phase Space New: MEG-ssds Measure Suite

40 Intermittent 3 to 8 Second Strudel “Absences” Persist Over the 2.3 min of Eyes-Closed Resting Record in Proband, YM Λ = 0.451; D C =1.61; h T =0.348 memv = (0.572) α = 2.79; X 4 =0.441;

41 Daydreaming, Thought Blocking and Strudels in the Task-free, Resting Consciousness of the Brain Plasma’s Magnetic Fields* Arnold J. Mandell a,b,d, Karen A. Selz b, John Aven a,c, Tom Holroyd a and Richard Coppola a a. NIMH Core MEG Facility, Building 10, NIMH, Bethesda, MD b. Cielo Institute, 486 Sunset Dr., Asheville, NC c. Fetzer-Franklin Fellow in Consciousness Studies at NIMH; d. Corresponding Author * Supported by the Fetzer-Franklin Trust, DARPA ( Microelectronics), and the Space and Naval Warfare Systems Center. “A plasma is lawfully and intrinsically multidisciplinary”

42 The plasma of consciousness includes observable and subjective elements ( 1) “Particle” lengths overlap; each particle effects many. Motion is intrinsically cooperative; (2) Important interactions in the bulk, not like dipole magnets (Stokes Theorem) at the surface. (3) Elemental oscillations mucn faster than collisions:→ EM instantaneous forces dominate gas and chemical kinetics. (4) Responds strongly to electromagnetic fields which can generate transient structures in the plasma. (5)Need not have specific shape or size (6) Overlapping fields: chemical, electromagnetic, psychological (6) Composed of ionized and neutral particles “balanced” (8) Small space charge (7) High density of charge carriers (ions, electrons, neutrals, hydrophobic charges). (8) Spontaneous currents and return currents., moving charges. (9) Persistent magnetic fields Consciousness :One of the Properties of the Body Temperature Brain Plasma slower and cooler

43 We study the mean field approximation of the global magnetic field component of the conscious brain plasma. Traveling charges in apical dendrites of the neocortical pyramidal cell networks (MU) are associated with extracellular return currents with varying impedance, capacitance, inductances which constitute electromagnetic “ephaptic” fields (LFP). These, in turn, “feed back” to modulate the thresholds and dynamics of the networks (Frolich, 2009) Electrical currents flowing within the apical dendrites of pyramidal cells generate the surrounding magnetic field apical dedrites Cahal Temporally covariant large volume recorded from single central pair of sensors. Ctx layers II and III

44 Some psychoanalytic characteristics of the conscious brain plasma: 1.All properties of the conscious brain plasma are deterministic; not random. 2.The plasma of consciousness is ceaselessly driven by energizing “drives”. 3.Psychic energy (entropy) is conserved. 4.The plasma of consciousness has levels of topographic scaling: unconscious, preconscious and conscious, Ucs, Pcs, Cs. 5.The plasma of consciousness has finite set of thematic dynamical quasi- specialized components: id, ego, superego (flavored energetics). These components are not necessarily aware of each other. 6.Inhibitory “defenses” modulate access between levels of plasma consciousness by primarily repression, more primitively denial, dissociation, conversion, and undoing, and if persistent and stereotyped, character formation, for example obsessive compulsive (doing and undoing), and hysterical personality (dissociation and display). 7.Normal failures of defenses create leaks between levels of the plasma of consciousness into the preconscious and conscious: dreaming, parapraxes, “”I was thinking about one thing and I said my mother” “free associations”, and day dreaming.

45 1.Using probe windows of five to twenty-five seconds (“beep”) TUTI buttons are pushed after daydreams one or more times in 60-70% of the windows during repetitious tasks and/or taskless resting conditions 2.TUTIs are increased with psychologically perturbing preconditions. 3.Interrupting TUTIs makes the next one occur more quickly (“pressure”) 4.TUTI deficiency has been reported with aging, Alzheimer's, mTBI, interictal epilepsies and schizophrenia. 5.Brain damage to some areas of the default network leads to “mental emptiness”, TUTI deficiency and reduction in spontaneous speech and thoughts. 6.Anterior cingulotomy (for OCD)---prominent in Columbia-Greystone Project ( ) leads to very vivid daydreams that are often confused with reality. 7.Increased demand via speed of signal processing or task difficulty decreases the frequency and extent of TUTIs Klinger, Antrobus, Singer, Giambra, Binder, Smallwood and others ( ). Psychological transients, eddies in the flow of consciousness in the brain’s plasma Task Unrelated Thoughts and Images, TUTIs. “…thoughts, images (and sounds) that intrude into a person’s Cs unintentionally (involuntarily)…and are unrelated to their activity, Giambra, 1995

46 The products of ceaseless Ucs activity intrude into Pcs and Cs as Task Unrelated Thoughts and Images, TUTIs, when internal entropies increase (alertness, arousal, fearfulness) or when ongoing attention demanding tasks are minimized Implicit model of Antrobus, Singer, Giambra, Binder and others, 1960 to 1999 Changes in total brain entropies “psychic energies” Purposeful thoughts, plans and actions governed by The Reality Principle, Freud, 1911 Involuntary thoughts and images, daydreaming: ↑ frequency/duration of TUTIs when quiet governed by the Pleasure/Displeasure Principle. Reciprocal partitions of available entropies (“psychic energies” ) Conservation of psychic entropies Psychic entropies not conserved.

47 Marcus Raichle’s 2001 default activation anatomy of a component of the conscious plasma of the brain fMRI regions light up when task-free and resting, Pleasure/Displeasure Principle; but are dark when purposefully thinking or doing, the Reality Principle. Averages of nine subjects: medial prefrontal, medial parietal, anterior and posterior cingulate, habenula, etc.; generally the medial brain.

48 Tesla = Cooper electron pairs tunnel across JJs at critical current; this process is perturbed by a change in magnetic flux density. To study the magnetic fields components of the plasma of consciousness

49 The CTF gradiometer records the magnetic fields of the concious brain plasma The ssds primary data range in amplitude from fT 10 3

50 SYMMETRIC SENSOR DIFFERENCE SEQUENCES, ssds L-R: C16, F14, T44, P57 1.Minimizes physical artifacts such as coughing/ blinking; 2.Reduce-covariances-including d,t,a,b,g modes; examining “similarity regime”(Novikov, 1991) 3.Global, scalar fields makes location less relevant. 4.Imposes a local gauge, [(0-ssds max) fT/Hz]. 5.Relative motion regularizes locally (in time) the globally nonstationary MEG signals 6.Difference metric (like velocity increment) is a common variable in turbulence dynamics and statistics. 7.Evokes intuitions and techniques of magnetic hydrodynamic, MHD, plasmas and fields 8.ssds minimizes central values and emphasizes outliers seconds of ssds 350 seconds of fMRI fluctuations Reichle, 2009 Why ssds 275-channel, superconducting quantum interference device (SQUID),radial gradiometer system from VSM MedTech Ltd., C16 Mean Field Approximation of the Brain Plasma’s Magnetic Field/t

51 Central pair difference reduces sway and other artifacts. A single central ssds pair “sees” or “ is responsive with” a large volume of the always conscious neocortical plasma Lighter area indicates neocortical volume to which the red C16 sensor pair’s ssds is similarly directionally changing with arbitrary threshold. ~ 80% left right ssds =left-right 16.7 seconds

52 f --α, Mean α = -1.67± 0.43 Median = Var = 0.18 fs = 600/150 Hz Kolmogorov 5/3/s First new observation: Log-log Power Spectra of 3 Minutes of Central C16 ssds(i) in 10 NIMH Controls


54 To Study the Inverse of the Time-Dependent Wavelengths, Broomhead-king Decomposition Uses the Leading Eigenvectors of a Lagged Autocovariance Matrix, Composed with the Original Series to Generate Two or Three Leading B/K Eigenfunctions

55 To Observe the Dynamics in Time of Scaling System(s) We Apply the Wavelet Transformation in which a “Mother” Wavelet is Convolved with the Data as it is Translated Down the Series, “b”, at Various Dilations, “a”. Morlet mother wavelet, w, = w = sine wave x Gaussian

56 Morlet wavelet transformation of the leading B/K eigenfunctions of 66.6’’ symmetric, ssds (top), and asymmetric, asds, sensor difference sequences yielding intermittent, scaling strudels C16-C16 C16-T44 C16-P57 T44 further from C16 then P57 yet more similar to C16.

57 1.Minimizes physical artifacts (e.g. coughs/blinks) 2.Reduces modal covariance including d, t, a, b, g 3.ssds as global, scalar fields reduce role of location and emphasize time. 4.Imposes a travelling local gauge, [0-ssds max) fT/Hz], 5.Locally normalizes the nonstationary signal. 6.Difference metric like velocity derivative in turbulence dynamics and statistics 7.Power spectra, Morlet wavelets of Broomhead/King leading eigenfunctions, and multiple measures of ssds-MEG resemble closely those of ssds-local field potentials of neocortical pyramidal cell layers II and III (monkey/Shew-Plenz). Symmetric Sensor Difference Sequences: ssds C16 left C16 right L – R ssds 15 seconds; 600 Hz with 150 cut off

58 The value off the ssds of a single pair, C16 (red), changes in > 0.30 correlation with large regions of the neocortex.

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