Complexity

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Nonlinear dynamics in biophysics Dresden June/July 2005 Bacterial colony The genome The universe Neural network Galaxy

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From Complexity to Perplexity Horgan Sci. Am June 1995 “I think the next century will be the century of complexity” Steven Hawking Can science achieve a unified theory of complex system? Even at the Santa Fe institute, some researchers have their doubts. function Functional (healthy) complexity Latent information

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Hubberman and Hogg, Physica D 1986, Gellmann “The Quark and the Jaguar” The commonly accepted naïve picture

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Lifting the Perplexity off the Complexity Eshel Ben-Jacob We do not propose “The world best definition of Complexity” Quantitative observables associated with the intuitive notion Inspired from the activity of Neural Networks Eyal Hulata et al Phys Rev lett. 2004 “Brain in a Nutshell” The Complexity-Regularity plane The Time-frequency domain Functional Complexity Recorded Brain Activity

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Segev et al Phy Rev lett 2000,2001,2002,2003 Baruchi et al, Complexity 2005 “Brains” in a Nutshell A lesson from Cultured Neural Networks Regulated spontaneous activity– the basic templates for computability?

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Formation of Synchronizes Bursting Events - SBEs

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I(n) I(n+1) Multi-time scales (large variations) in the Inter-Spike-Intervals (ISI) time Binary sequence (Bar-Code) representation Time bins of 200ms

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Itypical IAverage The statistical scaling characteristics Distribution of ISI # Long tail behavior Symmetric Le’vy distribution With δ = Itypical

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ISI # δ Slope -   Typical interval width The Le’vy distribution 1/ The sequence's plasticity 1/ The sequence's regularity Gaussian distributions -  = 2

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recorded shuffled Self-regulated temporal ordering recorded shuffled (Behavior of the second moment)

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Our requirements: Complexity-Regularity plane Regularity 0 1 Complexity Disorder Order δ=0 δ≠0 Longer tail Longer tail Recorded Shuffled

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(Based on clues from neural networks activity) Guiding ideas 1.Tiling the Time-frequency domain Using a cost function – for maximal information 2. Regularity – a measure of the uni-formity 3. Complexity – a measure of the vari-formity (Using sequences with Le’vy distribution as test functions)

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Time-frequency representations ∆ω ∆t Original signal Fourier transform The trivial examples The waves constraints (“Heisenberg uncertainty”) In general “Information cell” “Information cell” ∆ω ∆t

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The Challenge – Looking for “Best Tilling” Division of the Time-frequency domain to N “information cells” (for a sequence of N time bins) Looking to capture the maximum amount of information Higher time resolution vs. higher frequency resolution

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Stage 1: Using wavelet-packet-decomposition All possible divisions to equal cells – example for N=64 Maximum time resolution Maximum “frequency” resolution Low “energy” q High q

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Maximizing a global Shannon-like cost function Thiele & Villemous, A.C.H.A., 1996 Hulata et al Phys. Rev. Lett. 2004

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Thiele & Villemous, A.C.H.A., 1996

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Examples

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time frequency “energy” ∆t ∆ω “Information cell” q

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Physical intuition Assigning a spin - Sn for each rectangle “energy” ∆t ∆ω Sn ≡ ( ) ∆ω ∆t ( ) Nbin -1 ≤ Sn ≤ 1 qn

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The Regularity - R R ≡ <Sn> “energy” ∆t ∆ω Physics intuition: The total magnetization (The information cell) Sn

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Generation of artificial sequences Ordered side Disordered side δ δ = 0 ≠ 0  Increasing  Increasing A family for a given  Larger  Smaller  Regularity 0 1

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The regularity - R = <Sn> Disorder <Sn> 3 3 2 2 1 1 4 4 Order

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Structural Complexity 1. The Complexity of a “Word” -j Cj = ∑ ∑ I Sn - SmI n m(n) 2. The Sequence Complexity: C ≡ Variance ( Cj ) (Interpretation: Local and global variations) j

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Regularity Complexity Disorder Order 1 1 2 2 3 3 4 4

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The Regularity-Complexity Plane Man-made sequences with Le’vy distribution For different α (tail slope) Small α Large α  

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Different δ (the typical interval)

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Reflections on the naive picture

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recorded shuffled Back to experiments

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recorded shuffled Regularity-Complexity Plane Complexity Regularity 0 1 Recorded Shuffled Hulata et al Phys Rev Lett 2004 Testing the new observables

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Complexity Regularity 0 1 Recorded Shuffled Functional complexity Possible connection with information? Looking at networks development

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Bursts of SBEs , bursts of bursts of SBEs … Time cascades Hierarchical temporal organization “Words” complexity

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Characterization of network development (Pictures from joint studies with Pablo Blinder and Danny Baranes)

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Complexity Regularity Time (days) Network development Regularity Complexity

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Testing for Generality I Applying to cultured networks from the frontal ganglion (FG) of locust. Fuchs et al J. Complexity (2004)

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Generality

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From “Brains” in a Nutshell to the Human Brain ECoG recording With Leo Towle (latter in the meeting)

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From binary sequences to Continuous signals

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(inter ictal) (ictal) (onset) The challenge From binary sequences to Continuous signals

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Inter-Ictal Ictal (seizure) Low regularity Low complexity RM=0.12 SF=1.84

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Shuffling of continuous signals Recorded Shuffled (Mixing the phases) RM=0.67 SF=1.5 RM=0.12 SF=1.84

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Inter-Ictal RM=0.35 SF=1.62 Ictal RM=0.07 SF=1.69 Onset RM=0.12 SF=1.84 Regularity Complexity Inter-Ictal Ictal Onset

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Looking a head Complexity and Latent information Functional Complexity The principle of matched complexities

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Thank you