Slide1: Nonlinear dynamics in biophysics Dresden June/July 2005 Bacterial colony The genome The universe Neural network Galaxy


Slide2: 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


Slide3: Hubberman and Hogg, Physica D 1986, Gellmann “The Quark and the Jaguar” The commonly accepted naïve picture


Slide4: 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


Slide5: 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?


Slide6: Formation of Synchronizes Bursting Events - SBEs


Slide7: 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


Slide8: Itypical IAverage The statistical scaling characteristics Distribution of ISI # Long tail behavior Symmetric Le’vy distribution With δ = Itypical


Slide9: ISI # δ Slope -   Typical interval width The Le’vy distribution 1/ The sequence's plasticity 1/ The sequence's regularity Gaussian distributions -  = 2


Slide10: recorded shuffled Self-regulated temporal ordering recorded shuffled (Behavior of the second moment)


Slide11: Our requirements: Complexity-Regularity plane Regularity 0 1 Complexity Disorder Order δ=0 δ≠0 Longer tail Longer tail Recorded Shuffled


Slide12: (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)


Slide13: 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


Slide14: 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


Slide15: 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


Slide16: Maximizing a global Shannon-like cost function Thiele & Villemous, A.C.H.A., 1996 Hulata et al Phys. Rev. Lett. 2004


Slide17: Thiele & Villemous, A.C.H.A., 1996


Slide18: Examples


Slide19: time frequency “energy” ∆t ∆ω “Information cell” q


Slide20: Physical intuition Assigning a spin - Sn for each rectangle “energy” ∆t ∆ω Sn ≡ ( ) ∆ω ∆t ( ) Nbin -1 ≤ Sn ≤ 1 qn


Slide21: The Regularity - R R ≡ “energy” ∆t ∆ω Physics intuition: The total magnetization (The information cell) Sn


Slide22: Generation of artificial sequences Ordered side Disordered side δ δ = 0 ≠ 0  Increasing  Increasing A family for a given  Larger  Smaller  Regularity 0 1


Slide23: The regularity - R = Disorder 3 3 2 2 1 1 4 4 Order


Slide24: 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


Slide25: Regularity Complexity Disorder Order 1 1 2 2 3 3 4 4


Slide26: The Regularity-Complexity Plane Man-made sequences with Le’vy distribution For different α (tail slope) Small α Large α  


Slide27: Different δ (the typical interval)


Slide28: Reflections on the naive picture


Slide29: recorded shuffled Back to experiments


Slide30: recorded shuffled Regularity-Complexity Plane Complexity Regularity 0 1 Recorded Shuffled Hulata et al Phys Rev Lett 2004 Testing the new observables


Slide31: Complexity Regularity 0 1 Recorded Shuffled Functional complexity Possible connection with information? Looking at networks development


Slide32: Bursts of SBEs , bursts of bursts of SBEs … Time cascades Hierarchical temporal organization “Words” complexity


Slide33: Characterization of network development (Pictures from joint studies with Pablo Blinder and Danny Baranes)


Slide34: Complexity Regularity Time (days) Network development Regularity Complexity


Slide35: Testing for Generality I Applying to cultured networks from the frontal ganglion (FG) of locust. Fuchs et al J. Complexity (2004)


Slide36: Generality


Slide37: From “Brains” in a Nutshell to the Human Brain ECoG recording With Leo Towle (latter in the meeting)


Slide38: From binary sequences to Continuous signals


Slide39: (inter ictal) (ictal) (onset) The challenge From binary sequences to Continuous signals


Slide40: Inter-Ictal Ictal (seizure) Low regularity Low complexity RM=0.12 SF=1.84


Slide41: Shuffling of continuous signals Recorded Shuffled (Mixing the phases) RM=0.67 SF=1.5 RM=0.12 SF=1.84


Slide42: 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


Slide43: Looking a head Complexity and Latent information Functional Complexity The principle of matched complexities


Slide44: Thank you