in neural models in networks of pendulum-like elements in networks of biological neurons and coupled damped pendulums J. Hizanidis 1, V. Kanas 2, A. Bezerianos 3, and T. Bountis 4 1 National Center for Scientific Research, Demokritos 2 Department of Electrical and Computer Engineering, University of Patras 3 Cognitive Engineering Lab, Singapore Institute for Neuroengineering, National University of Singapore 4 Department of Mathematics, University of Patras July 13 2013 1 / 35
in neural models in networks of pendulum-like elements Outline 1 Synchronization in complex networks The Kuramoto model 2 History Example of systems exhibiting chimera states 3 in neural models 4 in networks of pendulum-like elements 5 2 / 35
in neural models in networks of pendulum-like elements Synchronization in complex networks Synchronization in complex networks The Kuramoto model Emerging phenomenon in which an enormous system of oscillators spontaneously locks to a common frequency, despite the inevitable differences in the natural frequencies of the individual oscillators. Synchronization processes are ubiquitous in nature and play an important role in many contexts (biology, ecology, technology) networks of pacemaker cells in the heart synchronously flashing fireflies and crickets that chirp in unison arrays of lasers and microwave oscillators 3 / 35
in neural models in networks of pendulum-like elements The Kuramoto model Synchronization in complex networks The Kuramoto model 1967: Arthur T. Winfree considered biological oscillators as phase oscillators, neglecting the amplitude 1975: Yoshiki Kuramoto developed an analytical theory to explain synchronization of globally coupled phase oscillators θ i = ω i + K N N sin(θ j θ i ), j=1 i = 1... N K: coupling constant N: total numer of oscillators g(ω): natural frequency distribution, symmetric about Ω 4 / 35
in neural models in networks of pendulum-like elements Synchronization in complex networks The Kuramoto model The Kuramoto model describes a large population of coupled phase oscillators with distributed natural frequencies. If the coupling strength exceeds a certain threshold, the system exhibits a phase transition: some of the oscillators spontaneously synchronize, while others remain incoherent. Globally coupled oscillators settle in one of a few basic patters: Synchrony Travelling waves (1D), spiral waves (2D) Incoherence (no spatial structure) Spatiotemporal chaos 5 / 35
in neural models in networks of pendulum-like elements Kuramoto and Battogtokh (2002) Synchronization in complex networks The Kuramoto model Complex Ginzburg-Landau equation with weak coupling: φ(x, t) t = ω 1 0 G(x x ) sin[φ(x, t) φ(x, t) + α]dx φ(x, t): phase of the oscillator at position x and time t G(x x ) e κ x x nonlocal coupling 6 / 35
in neural models in networks of pendulum-like elements Synchronization in complex networks The Kuramoto model Coexistence of Coherence and Incoherence 7 / 35
in neural models in networks of pendulum-like elements Abrams and Strogatz (2004) Synchronization in complex networks The Kuramoto model θ(x, t) t π = ω G(x x ) sin[θ(x, t) θ(x, t) + α]dx π G(x) = 1 (1 + A cos x), 0 A 1 2π cosine kernel allows the model to be solved analytically: stability of chimera states dynamics and bifurcations 8 / 35
in neural models in networks of pendulum-like elements The chimera state History Example of systems exhibiting chimera states strange mode of syncrhonization: chimera state never been seen in in systems with local or global coupling nothing to do with partially locked/partially incoherent states that occur in populations of nonidentical oscillators 9 / 35
in neural models in networks of pendulum-like elements History Example of systems exhibiting chimera states 10 / 35
in neural models in networks of pendulum-like elements History Example of systems exhibiting chimera states Dichotomy between synchrony and disorder With local or global coupling, identical oscillators either synchronize or oscillate incoherently, but never do both simultaneously With nonlocal coupling, identical oscillators can split into two coexisting domains, synchronous and asynchronous Chimera state: counterintuitive phenomenon: coexistence of coherent and incoherent oscillations in populations of symmetrically coupled identical oscillators 11 / 35
in neural models in networks of pendulum-like elements Coupling schemes in networks History Example of systems exhibiting chimera states r = R/N coupling radius N: total number of elements in the network R: number of coupled nearest neighbors 12 / 35
in neural models in networks of pendulum-like elements in two subnetworks Abrams et al PRL, (2008) History Example of systems exhibiting chimera states dθ 1 i dt dθ 2 i dt j=1 j=1 = ω + µ sin(θj 1 θj 1 α) + ν sin(θj 2 θj 1 α) N N N N = ω + µ j=1 sin(θj 2 θj 2 α) + ν j=1 sin(θj 1 θj 2 α) N N N N 13 / 35
in neural models in networks of pendulum-like elements in chaotic systems Coupled chaotic logistic maps: History Example of systems exhibiting chimera states z i n+1 = f (z i n ) + σ 2R Coupled Rössler systems: i+r j=i R [f (z j n ) f (z i n )], f (z) = az(1 z) ẋ i = y i z i + σ 2R i+r j=i R (x j x i ) ẏ i = x i + ay i ż i = b + z i (x i c), i = 1... N 14 / 35
in neural models in networks of pendulum-like elements Experimental evidence of chimera states History Example of systems exhibiting chimera states 15 / 35
in neural models in networks of pendulum-like elements History Example of systems exhibiting chimera states in the context of neuroscience Unihemispheric sleep Many birds as well as dolphins sleep with one eye open, in the sense that one hemisphere of the brain is synchronous while the other is asynchronous. 16 / 35
in neural models in networks of pendulum-like elements History Example of systems exhibiting chimera states in the FitzHugh-Nagumo model Omelchencko et al. PRL 2013 ɛ u k = u k u3 k 3 v k v k = u k + a 17 / 35
in neural models in networks of pendulum-like elements Single neuron dynamics History Example of systems exhibiting chimera states The FitzHugh-Nagumo system reproduces neuron spiking behaviour and is a simplification of the Hodgkin-Huxley model (1952): Set of nonlinear ordinary differential equations that approximate ionic mechanisms underlying the initiation and propagation of action potentials We will use a more realistic system: the Hindmarhs-Rose model 18 / 35
in neural models in networks of pendulum-like elements History Example of systems exhibiting chimera states The Hindmarsh-Rose (HR) model (1984) ẋ = ax 3 + bx 2 + y z + J ẏ = c dx 2 y ż = r(s(x x 0 ) z) x: membrane potential y: activity of fast gated ion channels (Na + and K + ) z: activity of slow gated ion channels (Ca and Cl ) system parameters: s = 4, x 0 = 1.6, a = 1, b = 3, c = 1, d = 5, r = 0.001 19 / 35
in neural models in networks of pendulum-like elements Typical HR bursting patterns History Example of systems exhibiting chimera states external stimulus J = 2 20 / 35
in neural models in networks of pendulum-like elements Two-dimensional HR model (1982) ẋ = ax 3 + bx 2 + y + J ẏ = c dx 2 y z variable eliminated no ability to produce bursting modes no firing frequency adaptation 21 / 35
in neural models in networks of pendulum-like elements Network of 2D-HR oscillators ẋ k = y k xk 3 + 3x 2 + J + σ j=k+r x [b xx (x j x k ) + b xy (y j y k )] 2R j=k R ẏ k = 1 5xk 2 y k + σ j=k+r y [b yx (x j x k ) + b yy (y j y k )] 2R j=k R Coupling matrix (Omelchencko et al. 2013): ( ) ( buu b B = uv cosφ sinφ = sinφ cosφ b vu b vv ) 22 / 35
in neural models in networks of pendulum-like elements Snapshots at t = 3000, N = 1000, σ x = σ y = 0.1, J = 0 Mixed oscillatory states (MOS), wave-like patterns Chimera state for diagonal coupling Multi-chimera states for smaller R 23 / 35
in neural models in networks of pendulum-like elements Bistability in 2D-HR model (J = 0) MOS due to bistability Uncoupled neuron: LC coexists with 3 fixed points For J > 0 the x-nullcline is lowered and LC is the only stable attractor 24 / 35
in neural models in networks of pendulum-like elements Coupling in x variable only (σ y = 0, N = 1000, R = 350) Coupling in voltage variable is more realistic Chimera state with one incoherent domain appears as intermediate pattern between desych and synch as σ increases 25 / 35
in neural models in networks of pendulum-like elements Network of 3D-HR oscillators ẋ k = y k x 3 k + bx 2 k + J z + σ j=k+r x (x j x k ) 2R j=k R j=k R ẏ k = 1 5x 2 k y k + σ j=k+r y (y j y k ) 2R ż k = r(s(x k + 1.6) z k ) s = 4: firing frequency adaptation and burst production r = 0.01: spiking frequency, number of spikes per bursting b: transition between spiking and bursting 26 / 35
in neural models in networks of pendulum-like elements Rich bifurcation scenarios: Storace et al. (2008) prepare system in spiking regime (b = 3, J = 5) 27 / 35
in neural models in networks of pendulum-like elements Snapshots at t = 5000 (N = 1000, R = 350, σ x = σ y ) MOS: regularly spiking neurons synchronize, irregularly spiking neurons do not Complete syncrhonization for large σ No chimera states observed 28 / 35
in neural models in networks of pendulum-like elements Coupling in x variable only (σ y = 0, N = 1000, R = 350) Desynch alternates with full synch Chimera state associated with change in dynamics Mulistability and sensitive dependence on initial conditions 29 / 35
in neural models in networks of pendulum-like elements Chimera States in Mechanical Oscillator Networks Large spring coupling k: Synchronization Low k values: Antiphase synchronization Intermediate k values: Chimeras emerge naturally Martens et al. (2013) 30 / 35
in neural models in networks of pendulum-like elements Networks of pendulum-like elements m d 2 θi 1 dt 2 +ɛdθ1 i dt = ω 1 i + µ N N sin(θj 1 θi 1 α)+ ν N j=1 N sin(θj 2 θi 1 α) j=1 m d 2 θi 2 dt 2 +ɛdθ1 i dt = ω 2 i + µ N N sin(θj 2 θi 2 α)+ ν N j=1 N sin(θj 1 θi 2 α) j=1 ω i taken from Lorentzian distribution ɛ: damping parameter m: inertia term 31 / 35
in neural models in networks of pendulum-like elements µ = 0.6 strong coupling within each population ν = 0.4 weak coupling between populations 32 / 35
in neural models in networks of pendulum-like elements ɛ = 1, increase m 33 / 35
in neural models in networks of pendulum-like elements ɛ = 1, increase m 34 / 35
in neural models in networks of pendulum-like elements Discussion and future work Hindmarh-Rose model Consider two populations and look for chimera states Bifurcation diagram in (σ, R) plane Effect of inhomogeneities in single dynamics and network topology on chimera states Pendulum-like model Check stability of chimera state for decreasing damping Scan (ɛ, m) plane and identify dynamical behaviours Check role of phase lag α Analytic results? 35 / 35