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Complex Network Dynamics: Theory & Application Yuri Maistrenko E-mail: [email protected] Lecture 2 Network Science Dynamical networks Complex networks versus Oscillatory networks Ensembles of oscillators Structure or Topology versus Dynamics or Activity We want to study Emergent network dynamics “In philosophy, system theory, science, and art, EMERGENCE is the way how complex systems and patterns arise out of a multiplicity of relatively simple interactions.” Wikipedia Chimera stare in network of 100x100x100 oscillators in 3Dim cube. Two incoherent rolls in coherent surrounding What is a network? Network: a set of nodes joined by links oscillators connections node, oscillator link, connection A network with N=8 nodes and K=10 links (Newman, 2005) - mathematically, network is a graph - there is dynamics in each node/oscillator - our task: to study collective dynamics of the network We need differential equations or iterated maps Network: Coupled oscillators Two types of dynamical systems for networks: • Differential equations: xi dxi f ( xi ) coupling terms dt • Difference equations (iterated maps): xit 1 f ( xit ) coupling terms f ( x) : R M R M - nonlinear function xi 1 xi 1 network Nonlinear Dynamics of Networks - regular (stationary, periodic ,or quasiperiodic) - chaotic (strange attractors, chimera states) THE ESSENCE OF CHAOS • Process (dynamics) deterministic fully determined by initial state and system equations • Long-term behavior unpredictable butterfly effect CHAOS = sensitive dependence on initial conditions CHAOS = BUTTERFLY EFFECT Henri Poincaré: (1880) “ It so happens that small differences in the initial state of the system can lead to very large differences in its final state. A small error in the former could then produce an enormous one in the latter. Prediction becomes impossible, and the system appears to behave randomly.” Ray Bradbury “A Sound of Thunder “ (1952) EXAMPLES OF CHAOTIC SYSTEMS • • • • • the solar system (Poincare) the weather (Lorenz) turbulence in fluids population growth lots and lots of other systems… “HOT” APPLICATIONS • neuronal networks of the brain (dynamical chaos) • gene regulatory networks (spatial chaos) WEATHER UNPREDICTIBILITY Edward Lorenz: (1963) Difficulties in the weather forecast are not related to the complexity of the Earths’ climate but to CHAOS in the global weather dynamics (given by nonlinear equations). LORENTZ ATTRACTOR (1963) butterfly effect a trajectory in the phase space The Lorenz attractor is generated by the system of 3 differential equations dx/dt= -10x +10y dy/dt= 28x -y dz/dt= -8/3z +xy. -xz Dynamics of oscillators placed in a network: Coupled oscillators Dynamical system: a system of one or more variables which evolve in time according to a given rule Two types of dynamical systems: xi • Differential equations: dxi f ( xi ) coupling terms dt • Difference equations (iterated maps): xit 1 f ( xit ) coupling terms f ( x) : R M R M - nonlinear function xi 1 xi 1 network Networks of iterated maps Networks of chaotic maps: Coherent states t 1 i x f (x ) t i iP t t [ f ( x ) f ( x j i )] 2P ji-P f ( x) ax(1 x), a 3.8 (chaotic map) 𝑁 = 100 oscillators Parameter regions of coherence coupling strength 𝝈 Chaotic synchronization COHERENCE coupling radius 𝒓 = 𝑷/𝑵 coupling radius 𝒓 = 𝑷/𝑵 Space-temporal chaos Network of Lorenz systems: Traveling waves xi axi ayi i P 2P y i bxi yi xzi (x j i P 2P zi xi yi czi j xi ) i P (y j i P j Lorenz attractor yi ) ( i 1,..., N ) 𝑎 = 10, 𝑏 = 28, 𝑐 = 8/3 Chaotic synchronization 𝜎 = 16, 𝑟 = 0.1, 𝑁 = 100 𝜎 = 13.3 , 𝑟 = 0.1, 𝑁 = 100 Space-temporal chaos 𝜎 = 13.8 , 𝑟 = 0.05, 𝑁 = 300 Network of Rössler systems: Chimera states xi yi z i y i xi ayi zi b czi xi zi iP 2P (x j i P j xi ) (i 1,..., N ) Neuronal Networks Human brain: ~ 100 000 000 000 neurons - how complex are neuronal networks in the brain? - are they locally or globally coupled? - strength of coupling between individual neurons? - excitatory and inhibitory neurons, why so? How to get modelling? 21 22 24 - bidirectional connections are more common than one can expected, if the network connections be random - connection strength distribution differs significantly from random and characterized by a “long tail” - synaptic weights are concentrated among few strong synaptic connections Neuronal connectivity represents a skeleton of stronger connections in the sea of weaker ones! 25 26 The next UNIT to model (The Neocortical - column ) (1mm3 ) Size of a pin head Cortical column and neuronal microcircuits 28 29 Why do we need to build a model? “I have all these data in the cortex - cell types, their firing properties, dendritic excitability, connectivity, synaptic dynamics… But I don’t Understand it. I need to model it.” Bert Sakmann Nobel Prize in Physiology and Medicine, 1991 University of Heidelberg 30 How can one model neuronal networks? Kiss - detailed Rodin Kiss – reduced Brancusi Alan Lloyd Hodgkin Hodgkin-Huxley model (1952) Andrew Fielding Huxley The H&H model; (1) Biophysical, (2) Compact, (3) Predictive Network of Hodgkin-Huxley neurons Synaptic current Coupling function Network of Hodgkin-Huxley neurons : Chimera state TIME Chimera state can “reflect” working memory in neuronal networks Bell-shaped persisted neural activity Network architecture Space-temporal network activity in a bump state Renart, Song, Wang "Robust spatial working memory…”(Neuron 2003)