Complex Network Dynamics:

Transcription

1 Complex Network Dynamcs: Theory & Applcaton Yur Mastrenko E-mal: Lecture 2

2 Network Scence Complex networks versus Dynamcal networks Oscllatory networks Ensembles of oscllators Structure Topology Dynamcs or versus or Actvty

3 We want to study Emergent network dynamcs In phlosophy, system theory, scence, and art, EMERGENCE s the way how complex systems and patterns arse out of a multplcty of relatvely smple nteractons. Wkpeda

4 Chmera stare n network of 100x100x100 oscllators n 3Dm cube. Two ncoherent rolls n coherent surroundng

5 What s a network? Network: a set of nodes joned by lnks oscllators connectons node, oscllator lnk, connecton A network wth N=8 nodes and K=10 lnks (Newman, 2005) - mathematcally, network s a graph - there s dynamcs n each node/oscllator - our task: to study collectve dynamcs of the network

6 We need dfferental equatons or terated maps

7 Network: Coupled oscllators Two types of dynamcal systems for networks: Dfferental equatons: dx dt f ( x ) couplng terms Dfference equatons (terated maps): x x 1 x 1 network x t1 f ( x t ) couplng terms M M f ( x) : R R - nonlnear functon

8 Nonlnear Dynamcs of Networks - regular (statonary, perodc,or quasperodc) - chaotc (strange attractors, chmera states)

9 THE ESSENCE OF CHAOS Process (dynamcs) determnstc fully determned by ntal state and system equatons Long-term behavor unpredctable butterfly effect CHAOS = senstve dependence on ntal condtons

10 CHAOS = BUTTERFLY EFFECT Henr Poncaré: (1880) It so happens that small dfferences n the ntal state of the system can lead to very large dfferences n ts fnal state. A small error n the former could then produce an enormous one n the latter. Predcton becomes mpossble, and the system appears to behave randomly. Ray Bradbury A Sound of Thunder (1952)

11 EXAMPLES OF CHAOTIC SYSTEMS the solar system (Poncare) the weather (Lorenz) turbulence n fluds populaton growth lots and lots of other systems HOT APPLICATIONS neuronal networks of the bran (dynamcal chaos) gene regulatory networks (spatal chaos)

12 WEATHER UNPREDICTIBILITY Edward Lorenz: (1963) Dffcultes n the weather forecast are not related to the complexty of the Earths clmate but to CHAOS n the global weather dynamcs (gven by nonlnear equatons).

13 LORENTZ ATTRACTOR (1963) butterfly effect a trajectory n the phase space The Lorenz attractor s generated by the system of 3 dfferental equatons dx/dt= -10x +10y dy/dt= 28x - y -xz dz/dt= -8/3z +xy.

14 Dynamcs of oscllators placed n a network: Coupled oscllators Dynamcal system: a system of one or more varables whch evolve n tme accordng to a gven rule Two types of dynamcal systems: Dfferental equatons: dx dt f ( x ) couplng terms Dfference equatons (terated maps): x x 1 x 1 network x t1 f ( x t ) couplng terms M M f ( x) : R R - nonlnear functon

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16 Networks of terated maps

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18 Networks of chaotc maps: Coherent states x t1 f ( x t ) 2P P j-p [ f ( x t j ) f ( x t )] f ( x) ax(1 x), a 3.8 (chaotc map) Parameter regons of coherence couplng strength σ C O H E R E N C E N = 100 oscllators Chaotc synchronzaton couplng radus r = P/N Space-temporal chaos couplng radus r = P/N

19 Network of Lorenz systems: Travelng waves ) 1,..., ( ) ( 2 ) ( 2 N cz y x z y y P xz y bx y x x P ay ax x j P P j j P P j a = 10, b = 28, c = 8/3 Lorenz attractor Chaotc synchronzaton Space-temporal chaos σ = 16, r = 0.1, N = 100 σ = 13.3, r = 0.1, N = 100 σ = 13.8, r = 0.05, N = 300

20 Network of Rössler systems: Chmera states ) ( 2 z j P P j x x P y x ) 1,..., ( N z x cz b z ay x y

21 Neuronal Networks Human bran: ~ neurons - how complex are neuronal networks n the bran? - are they locally or globally coupled? - strength of couplng between ndvdual neurons? - exctatory and nhbtory neurons, why so? How to get modellng? 21

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25 - bdrectonal connectons are more common than one can expected, f the network connectons be random - connecton strength dstrbuton dffers sgnfcantly from random and characterzed by a long tal - synaptc weghts are concentrated among few strong synaptc connectons Neuronal connectvty represents a skeleton of stronger connectons n the sea of weaker ones! 25

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27 The next UNIT to model (The Neocortcal - column ) (1mm 3 ) Sze of a pn head

28 Cortcal column and neuronal mcrocrcuts 28

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30 Why do we need to buld a model? I have all these data n the cortex - cell types, ther frng propertes, dendrtc exctablty, connectvty, synaptc dynamcs But I don t Understand t. I need to model t. Bert Sakmann Nobel Prze n Physology and Medcne, 1991 Unversty of Hedelberg 30

31 How can one model neuronal networks? Kss - detaled Kss reduced Rodn Brancus

32 Hodgkn-Huxley model (1952) lan Lloyd odgkn Andrew Feldng Huxley The H&H model; (1) Bophyscal, (2) Compact, (3) Predctve

33 Network of Hodgkn-Huxley neurons Synaptc current Couplng functon

34 Network of Hodgkn-Huxley neurons : Chmera state TIME

35 Chmera state can reflect workng memory n neuronal networks Bell-shaped perssted neural actvty Network archtecture Space-temporal network actvty n a bump state Renart, Song, Wang "Robust spatal workng memory (Neuron 2003)