“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 . - PowerPoint PPT Presentation
“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. Pesin-Young-Mandell (conservation of) : For uniformly & non-uniformly hyperbolic systems, the topological entropy, hT , varies as the product of the capacity dimension, Dc & the leading Lyapounov exponent, Λ 1 . hT = Dc Λ1.
Transcript
“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, MMILDepartment 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.Pesin-Young-Mandell (conservation of) : For uniformly & non-
uniformly hyperbolic systems, the topological entropy, hT , varies as the product of the capacity dimension, Dc & the leading Lyapounov
exponent, Λ 1 . hT = Dc Λ1.
Bifurcation
Entropy
Fractal
Leading Lyapounov
Nonlinear Dynamics
Chaos
Google Book Title Word Count From 1750 to 2005.Note decline of relevant words beginning in the vicinity of 2000.Adoption of new science characteristically comes late to biology
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)
Multiply periodic?
Chaotic?
Irrational windingon the torusLyapounov ≤ 0
Stretch, fold, shuffle.Points get out of order; “mixing”Lyapounov > 0
Generic membraneequations
Mandell and Selz, 1990
Quasi-periodic andchaotic van der pol.
Phase Portraits
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 ofstable and unstablemanifolds of unstablefixed point
A two dimensional section showingthe unstable and stable manifoldsof 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.
Stable and unstablemanifolds supportchaoticdynamicalsystemsAattractors.
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.
Phase Portrait of Temperature Time
SeriesAfter a Catch
X(t)
X (t-1)
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,
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 IMF3,4,5 ofC16-ssds(i) in AG, control, YM,intermediate state, and SP, typical medicated schizophrenic proband.
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)
The continuous power spectrum with complex singularities of a chaotic attractor (from “hydrodynamic” equation)
Farmer, Crutchfield, Froehling, Packard and Shaw, 1978-89“The original UCSC chaos kids”
The “funnel”Positive Lyapounov exponentContinuous power spectrumComplex singularitiesUnstable periodic orbitsRecursive homoclinic behavior near unstable fixed points
Chaotic dynamical systemsmanifest a variety of powerspectral exponents.Universality classes??
Log(frequency)-->
log(
ampl
itude
2 )(i.
e., p
ower
)
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:
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
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
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
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:
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.
MEG, symmetric sensor difference series, ssds
EEG, multi-electrodeLocal fieldpotentials from Pyramidalcell layersII and III (self-referentialnetwork)
Local field potentials from neocortical pyramidal cell network in taskless, resting monkey statistically resemble MEG ssds (Shew and Plenz, 2009)
Morlet wavelet decomposition of 14 second,C16 sensor difference sequenceDuring 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
Intermittent 3 to 8 Second Strudel “Absences” Persist Over the 2.3 min of Eyes-Closed Resting Record in Proband, YM
• Identify successive pairs of zero crossings, identify local extrema
• Connect max, min with cubic splines (upper and lower “envelopes”)
• Compute first mean, m1, of the envelopes• [ssds]-m1 = h1; h2 = h1 – m2…. ….+ …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.
• hi 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.
AG
YM
SP
Hilbert-Huang Intrinsic Mode Functions, IMF1,2,3,4,5 MEG, C16(ssds),16.66 sec
Note Loss of modular amplitudes in proband SP’s intermediate time scales
hn
Invariant Measure Theory
Using “Blind Boys and the Elephant” Metrics
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
Modalanalysis
Probabilitydistribution
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
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
Sensitivity to initial conditions, LyapounovExponent (rate of Expansion)
Slope
Log measure (scales)
Log measurement
“box” capacity dimensionComplexity 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
Leading Lyapounov exponent; divergence, expansion, mixingPower spectral scaling exponent:log-log slope, global, scalingTopological entropy; rate of new “loop” formationMetric entropy: distribution of weights on loopsCapacity dimension: complexity of manifold of supportNon-uniformity ||top-met| differenceMeasureable entropy manifold volume top x Lya x dimensionUnwinding number lags to asymptotic capacity dimensionSkew distributional asymmetryKurtosis peakedness and heavy tail Levy exponent rate of converrgence of tail of distributionHurst exponent persistence vs antipersistent
Multi-parameter quantitative-qualitative descriptions of dynamical systems.
Fixing entropy, hT, log (3)dimension, d, goes down as the Lyapounov increases
Measurable entropy manifold volume, MEMV = [hT l DC ] Thereom: hT = lDC 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, hT and dC go up.
Λ increases dC increases hT increases
REDUCED VOLUME OF FRONTAL LOBE (F14 SSDS) MEASUREABLE ENTROPY MANIFOLD VOLUME, MEMV, IN PROBANDSImplicit function representation of the Pesin-Young Ansatz: ld=hT
(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
TopologicalEntropy hT
memv = 3.13 log units memv = 2.37 log units
Third new findings: aggregate measure relations and memv decreased in“abnormal” brain plasmas.
Phase Portraits and Recurrence Plots of the Three Leading Autocovariance Matrix Eigenfunctions
Daydreaming, Thought Blocking and Strudels in the Task-free, Resting Consciousness of the Brain Plasma’s
Magnetic Fields*Arnold J. Mandella,b,d, Karen A. Selzb, John Avena,c, Tom Holroyda and Richard Coppolaa
a. NIMH Core MEG Facility, Building 10, NIMH, Bethesda, MD b. Cielo Institute, 486 Sunset Dr., Asheville, NC 28804-3727
c. Fetzer-Franklin Fellow in Consciousness Studies at NIMH; d. Corresponding Author www.cieloinstitute.org
* Supported by the Fetzer-Franklin Trust, DARPA ( Microelectronics), and the Space and Naval Warfare Systems Center. “A plasma is lawfully and intrinsically multidisciplinary”
The plasma of consciousnessincludes 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
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).
103
Electrical currents flowing within the apical dendrites of pyramidal cells generate the surrounding magnetic field
apicaldedritesCahal
Temporally covariant large volume recorded from single central pair of sensors. Ctx layers II and III
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.
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
(1948-1956) 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 TUTIsKlinger, Antrobus, Singer, Giambra, Binder, Smallwood and others (1960-2000).
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
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 (“psychicenergies” )
Conservation ofpsychic entropies
Psychic entropiesnot conserved.
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.
0.00005 Tesla =
Cooper electron pairs tunnel across JJs at critical current; this process is perturbed by a change in magnetic fluxdensity.
To study the magnetic fieldscomponents of the plasmaof consciousness
The CTF gradiometer records the magnetic fields of the concious brain plasma
The ssds primary data range in amplitude from 50-4000 fT
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 signals6. 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.
16.6 seconds of ssds
350 seconds of fMRI fluctuationsReichle, 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
Central pairdifference reduces swayand other artifacts.
A single central ssds pair “sees” or “ is responsive with” a large volume of the always conscious neocortical plasma
Lighter areaindicates neocortical volume towhich the red C16 sensor pair’s ssds is similarly directionally changing witharbitrary threshold. ~ 80%
left
right
ssds =left-right
16.7 seconds
f--α, Mean α = -1.67± 0.43
Median = -1.59 Var = 0.18
fs = 600/150 HzKolmogorov 5/3/s
First new observation: Log-log Power Spectra of 3 Minutes of Central C16 ssds(i) in 10 NIMH Controls
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
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
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.
1. Minimizes physical artifacts (e.g. coughs/blinks)2. Reduces modal covariance including d, t, a, b, g3. 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 with150 cut off
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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