Switching in Complex Networks of States: A New Paradigm for Natural Computation Marc Timme in collaboration with Fabio Schittler Neves - Network Dynamics Group - Network Dynamics Group Max Planck Institute for Dynamics & Self- -Organization Organization Max Planck Institute for Dynamics & Self Bernstein Center for for Computational Neuroscience, G , Gö öttingen ttingen Bernstein Center Computational Neuroscience Georg August University, G Georg August University, Gö öttingen ttingen
Biological and bio-inspired computation Biological Networks • Neural circuits (computation & learning) • „Tree“ of life (evolution) Bio-inspired networks • Autonomous robots • Natural computing devices Marc Timme - Network Dynamics Group, Max Planck Institute for Dynamics & Self-Organization
Towards Natural Computation Biological Biological Processes: Processes : • • are are nonlinear • • exploit exploit self-organized, emerging collective collective states states • based • based on learning on learning, , adaptation adaptation, , evolution evolution Technical Technical computing computing and behaving and behaving (robotic ( robotic) ) systems systems: : • • may may be be realized realized in a neuro-analogous way in a (bio-inspired development & possible explanation of biol. phenomena) • require understanding of collective nonlinear dynamics & self-organization How to build a natural computer?
Outline Model: Networks of symmetrically pulse- -coupled coupled oscillators Model: Networks of symmetrically pulse oscillators Phenomenon: : Periodic orbit (in the the sense of Milnor Milnor) ) … … Phenomenon Periodic orbit attractors (in sense of attractors … that are … that are unstable unstable Analytically Tractable Example: : Unstable modes Analytically Tractable Example Unstable modes Switching among among attractors attractors Switching System- -independence independence System Asymmetries: : Switching Asymmetries Switching of complex complex periodic orbits Selection of periodic orbits Selection Universal Computation Computation: : k- -winner winner takes all, binary binary & n Universal k takes all, & n- -ary ary logics logics N=5 versatility versatility; N=100 & ; N=100 & expon N=5 expon. . scaling scaling Robots: : phototaxis & obstacle obstacle avoidance Robots phototaxis & avoidance Marc Timme - Network Dynamics Group, Max Planck Institute for Dynamics & Self-Organization
Neural Model and Phase Description networks of neurons - oscillatory if driven by current • uncoupled neurons have increasing (concave) potential • spike sent at threshold • received after time delay ∝ • coupling strength original model: R.E. Mirollo, S.H. Strogatz; SIAM J. Appl. Math . 50:1645 (1990) model with delay: U. Ernst, K. Pawelzik, T. Geisel; Phys. Rev. Lett. 74:1570 (1995) Marc Timme - Network Dynamics Group, Max Planck Institute for Dynamics & Self-Organization
Neural Model and Phase Description Membrane potential dynamics dynamics Membrane potential : spike spike sending sending Pulse interactions interactions: Pulse and reset and reset time Received after delay Received after delay time U ( φ ) = ˜ ˙ ˜ ˜ ˜ V ( φ T ) V = f ( V ); V (0) = 0 , V ( T ) = 1 Marc Timme - Network Dynamics Group, Max Planck Institute for Dynamics & Self-Organization
All-to-all Connectivity: Partial Synchrony and Switching η = 10 − 3 deterministic: weak noise : units synchronize clusters decay into groups (clusters) ε ij = const > 0 τ > 0 attractor switching attractor switching U. Ernst et al., Phys. Rev. Lett. 74:1570 (1995) Switching persists for small noise strengths η = 10 − 22 Origin of switching dynamics? Marc Timme - Network Dynamics Group, Max Planck Institute for Dynamics & Self-Organization
Attracting and yet unstable? ( one single random perturbation = 10 -4 ) switching towards another attractor ( decay also occurs for very small pertubations = 10 -22 ) Marc Timme - Network Dynamics Group, Max Planck Institute for Dynamics & Self-Organization
New Kind of Invariant Set: Unstable Attractor Basin of attraction perturbations induce switching perturbations induce switching (2D section through state space) Marc Timme - Network Dynamics Group, Max Planck Institute for Dynamics & Self-Organization
Analysis Confirms: Unstable & Attracting Locally unstable although attracting (saddle periodic orbit with positive measure basin) new kind of (Milnor) attractor: unstable attractors First identification and analysis: M.T. et al.; Phys. Rev. Lett. 89:154105 (2002a) Large networks: M.T. et al.; Chaos 13:377 (2003) Rigorous results: P. Ashwin and M.T.; Nonlinearity 18:2053 (2005) Functional relevance of switching: P. Ashwin and M.T., Nature 436:36 (2005) Bifurcation: C. Kirst and M.T., Phys. Rev. E (R) (2008). Marc Timme - Network Dynamics Group, Max Planck Institute for Dynamics & Self-Organization
Cartoon of Heteroclinic Cycle in Symmetric Oscillator Systems saddle 1 saddle 1 heteroclinic heteroclinic connection connection saddle 3 saddle 3 three heteroclinic connections saddle 3 saddle 3
Breaking the Symmetry Periodic Orbit Close to Heteroclinic Cycle saddle 1 saddle 1 heteroclinic heteroclinic connection connection saddle 3 saddle 3 broken broken one one complex periodic orbit saddle 3 saddle 3
Full symmetry in a network of N oscillators N=5: only three parameters: I , ε , τ . V=(V1,V2,V3,V4,V5). (independent of N) N=5: cluster states of different symmetries: V=(a,a,a,b,b) V=(a,a,a,a,b) V=(a,a,b,b,c) 5!/(2!2!) = 30 saddle saddle states 5!/(2!2!) = 30 states
Saddle Instabilities and Heteroclinic Switching (c,a,a + ∆ ,b,b) (b,b,c,a,a) arbitrarily small perturbation induces arbitrarily small perturbation induces controlled switching
Two ways to switch: network of states V=(a,a,c,b,b); gray ‘a’ unstable, black ‘b’ and ‘c’ stable, (a+ ∆ ,a,b,b,c) (c,b,a,a,b) (a,a+ ∆ ,b,b,c) (b,c,a,a,b)
Symmetry breaking induces cyclic switching Symmetry breaking input currents: I 1 >I 2 >I 3 >I 4 >I 5 -- symmetric -- ------ symmetry broken ------ -- symmetric -- ------ symmetry broken ------ Cyclic switching along Cyclic switching along complex periodic orbit
Complex Network of Saddle States I I = (4,3,2,1,0) Noise + asymmetry asymmetry = (1,4,3,2,0) Noise + I = (4,3,2,1,0) I = (1,4,3,2,0)
Symmetry breaking classification (a + ∆ ,b,c,b,a) (c,a,b,a,b) time I ext a I 1 I 2 b c I 3 b I 4 a I 5 Marc Timme - Network Dynamics Group, Max Planck Institute for Dynamics & Self-Organization
Symmetry breaking classification (a + ∆ ,b,c,b,a) (c,a,b,a,b) time I ext c a + ∆ I 1 a I 2 b b c I 3 b a I 4 a b I 5 I 1 >I 5 Marc Timme - Network Dynamics Group, Max Planck Institute for Dynamics & Self-Organization
Symmetry breaking classification (c,a + ∆ ,b,a,b) (b,c,a,b,a) time I ext c a b I 1 a + ∆ c I 2 b b c a I 3 b a b I 4 a b a I 5 I 1 >I 5 I 2 >I 4 Marc Timme - Network Dynamics Group, Max Planck Institute for Dynamics & Self-Organization
Symmetry breaking classification (b,c,a + ∆ ,b,a) (a,b,c,a,b) time I ext c a b a I 1 a c b I 2 b b c a + ∆ c I 3 b a b a I 4 a b a b I 5 I 1 >I 5 I 2 >I 4 I 3 >I 5 Marc Timme - Network Dynamics Group, Max Planck Institute for Dynamics & Self-Organization
Symmetry breaking classification (a + ∆ ,b,c,a,b) (c,a,b,b,a) time I ext c a b a + ∆ c I 1 a c b a I 2 b b c a c b I 3 b a b a b I 4 a b a b a I 5 I 1 >I 5 I 2 >I 4 I 3 >I 5 I 1 >I 4 Marc Timme - Network Dynamics Group, Max Planck Institute for Dynamics & Self-Organization
Symmetry breaking classification (c,a + ∆ ,b,b,a) (b,c,a,a,b) time I ext c a b a c b I 1 a c b a + ∆ c I 2 b b c a c b a I 3 b a b a b a I 4 a b a b a b I 5 I 1 >I 5 I 2 >I 4 I 3 >I 5 I 1 >I 4 I 2 >I 5 Marc Timme - Network Dynamics Group, Max Planck Institute for Dynamics & Self-Organization
Symmetry breaking classification (b,c,a + ∆ ,a,b) (a,b,c,b,a) time I ext c a b a c b a I 1 a c b a c b I 2 b b c a c b a + ∆ c I 3 b a b a b a b I 4 a b a b a b a I 5 I 1 >I 5 I 2 >I 4 I 3 >I 5 I 1 >I 4 I 2 >I 5 I 3 >I 4 Marc Timme - Network Dynamics Group, Max Planck Institute for Dynamics & Self-Organization
Symmetry breaking classification (a + ∆ ,a,b,b,c) (c,b,a,a,b) time I ext c a b a c b a I 1 a c b a c b I 2 b b c a c b a c I 3 b a b a b a b I 4 a b a b a b a I 5 I 1 >I 5 I 2 >I 4 I 3 >I 5 I 1 >I 4 I 2 >I 5 I 3 >I 4 result: {I 1 ,I 2 ,I 3 }>{I 4 ,I 5 } Marc Timme - Network Dynamics Group, Max Planck Institute for Dynamics & Self-Organization
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