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Strange Nonchaotic Firings in the Quasiperiodically Forced Neural Oscillators Sang-Yoon Kim Department of Physics Kangwon National University • Dynamical Response of Biological Oscillators under External Stimulus External Stimulus Nonlinear Self-Sustained Biological Oscillator Response • Periodic Stimulation [Fext(t) = A cos2ft] Squid Giant Axon (Neural Oscillator), Aggregates of Embryonic Chick Heart Cells (Cardiac Oscillator) • Quasiperiodic Stimulation [Fext(t) = A1 cos2f1t + A2 cos2f2t] Forcing with Two Incommensurate Frequencies 1 Regular Response Under Periodic Stimulus [Fext = A cos 2ft] Periodically Forced Swing Dynamical Response: Regular [Periodic Phase-Locking (Entrainment) or Quasiperiodicity] and Chaotic Behaviors Order Chaos • Regular Response (Phase Locking or Quasiperiodicity) ¼-Phase Locking in the Periodically Forced Squid Giant Axon [K. Aihara, G. Matsumoto, and M. Ichikawa, Phys. Lett. A 111, 251 (1985)] 2 Chaotic Response Under Periodic Stimulation • Chaotic Firings in the Periodically Forced Squid Giant Axon [K. Aihara, G. Matsumoto, and M. Ichikawa, Phys. Lett. A 111, 251 (1985)] • Lorentz Attractor [Strange Chaotic Attractor] [Lorenz, J. Atmos. Sci. 20, 130 (1963)] Butterfly Effect [Small Cause Large Effect] Sensitive Dependence on Initial Conditions Infinite Complex of Surfaces: A Set with Zero Volume But Infinite Surface Area (D0 ~ 2.09) • Fractal Phase Space Structure [Complex Geometric Object with Fine Structure at Arbitrary Small Scales] Scale Invariance Self-Similarity 3 Fractal (Self-Similar Object at Arbitrary Small Scales) • Cantor Set Cantor Set [D0=ln2/ln3] • Koch Curve Von Koch Curve [D0=ln4/ln3] 4 Dynamical Response Under Quasiperiodic Stimulus [Fext = A1 cos 2f1t + A2 cos 2f2t] Quasiperiodic Forcing (with two incommensurate frequencies f1 and f2) f1 Nonlinear Self-Sustained Oscillator f2 Response • Appearance of Strange Nonchaotic Attractors (SNAs) Between Order and Chaos Order Smooth Torus SNA Chaos SNA Chaotic Attractor Properties of SNAs: (1) No Sensitivity to Initial Conditions (2) Fractal Phase Space Structure 5 Neural Systems • Neural Signal (Electric Spikes) Electrode signal (mV) Stimulus Sensory Spikes (Neurons) Brain Motor Spikes (Neurons) Response [Strength of a Stimulus > Threshold Generation and Transmission of Spikes by Neurons] • Neurons ~ 1011 (~100 billion) neurons in our brain [cf. No. of stars in the Milky Way ~ 400 billion] (Typical Size of a Neuron ~ 30m, Each neuron has 103 ~ 104 synaptic connections: Synaptic Coupling) Sum of the input signals at the Axon Hillock Sum > Threshold Generation of a Spike Synaptic Coupling Excitatory Synapse Exciting the Postsynaptic Neuron Inhibitory Synapse Inhibiting the Generation of Spikes of the Postsynaptic Neuron 6 Hodgkin-Huxley Model for the Squid Giant Axon • A Series of Five Papers: Published in 1952 (First four papers: experimental articles Conductance-Based Physiological Model: Suggested in the fifth article) • Nobel Prize (1963) Unveiling the Key Properties of the Ionic Conductances Underlying the Nerve Spikes Giant axon Brain 1st-level neuron Presynaptic (2nd level) 2nd-level neuron Stellate ganglion Stellate nerve Smaller axons 3rd-level neuron Postsynaptic (3rd level) (A) Stellate nerve with giant axon 1mm 1mm (B) Cross section (C) Squid giant axon = 800m diameter Mammalian axon = 2 m diameter 7 Hodgkin-Huxley (HH) Model [A.L. Hodgkin and A.F. Huxley, J. Physiol. 117, 500-544 (1952)] Modeling Based on the Experimental Results on the Giant Squid Axons Voltage-Dependent Conductances of the Na and K Channels 1963 Nobel Prize Resting dV I ion I ext I Na I K I L I ext , dt g Na m 3 hV ENa gK n 4 V EK gL V EL I ext dx x (V )( 1 x ) x (V )x , x m, h , n. dt [Ohm’s Law for each ion current: I = g(VE); g=g(V): voltage-dependent conductivity] Inactivated • Action Potential (Spikes) 50 ENa Conductance: Voltage-Dependent Activation potential 0 Na+ conductance –50 K+ conductance 40 20 0 EK Neuron: Excited Generation of Spikes Open channels per m2 of membrane V: Membrane Potential m: Activation Gate Variable of the Na+ Channel h: Inactivation Gate Variable of the Na+ Channel n: Activation Gate Variable of the K+ Channel (): Rate Constant Activated Membrane potential (mV) C 8 Generation of Action Potentials A Stimulus: t (1) All or None Response • Subthreshold case (Pulse) • Suprathreshold case A 2 μA / cm 2 ( 9.78μA / cm 2 ), t 100ms A 10μA / cm2 • Suprathreshold case A 20A/cm 2 No spike Increase in the stimulating intensity Increase in the frequency of spikes (2) Refractory Period During the refractory period, it is impossible to excite the cell no matter how great a simulating current is applied. 14ms 30ms 5ms A 10 A/cm 2 t 1ms 9 Intermittent Route to Chaos (Periodic Forcing) I ext I dc A1 sin( 2f1t ); A1 1A/cm 2 and f1 60Hz • Subthreshold Periodic Oscillation (Silent State) for Idc=2.5A/cm2 • Occurrence of Chaotic Spiking Intermittent Route to Chaos for Idc>Idc*(=3.058 824 A/cm2) I dc 3.5A/cm 2 10 Effect of Quasiperiodic Forcing on the Spiking Transition I ext I dc A1 sin( 2f1t ) A2 sin( 2f 2t ); f 2 / f1 ( 5 1) / 2, A1 1A/cm 2 and f1 60Hz [Phase of the 2nd driver: f2t = ( f1t: Phase of the 1st driver)] Appearance of Strange Nonchaotic Spikings Occurrence of Strange Nonchaotic Spiking State (SN) between the Silent State (S) and the Chaotic Spiking State (C) Properties of the Strange Nonchaotic State (SN) No Sensitivity to Initial Conditions < 0 Strange Geometry Fractal Silent State (Idc=1.7) Strange Nonchaotic Spiking State (Idc=2.85) Chaotic Spiking State (Idc=3.9) 11 Phase Sensitivity of an Attractor [A.S. Pikovsky and U. Feudel, Chaos 5, 253 (1995)] Smooth Torus SNA Idc=2.85 1=-0.247 < 0 =1.69 > 0 Idc=1.7 1=-1.679 < 0 • Sensitivity of an Attractor with respect to the Phase of the External Quasiperiodic Forcing Phase Sensitivity: Characterized by Differentiating V with respect to at a discrete time t=nP1 (P1=1/f1) |Sn|: bounded for all n Smooth Geometry |Sn|: unbounded (a dense set of singularities) Strange Geometry (Fractal) max | Sn ( x (0)) | Phase Sensitivity Function N min V Sn , n 1, 2, ... t nP1 { x ( 0 )} 0 n N Nonchaotic Attractor with <0 Smooth Torus: N: bounded (=0) SNA: N ~ N (: phase sensitivity exponent) > 0 Sensitive to the External Phase : Used to Measure the Degree of Strangeness 12 Characterization of the Silent and Spiking States Silent State (Idc=1.7) Idc*=1.9169 1=-1.679 Strange Nonchaotic Spiking State (Idc=2.85) 1=-0.247, =1.69 Strange Nonchaotic Spiking State (Black) Idc=3.572 Chaotic Spiking State (Idc=3.9) 1=0.165 As Idc Is Increased from Idc* (=1.9169), the Degree of Strangeness of the SNA Increases. * I dc I dc I dc 13 Mechanism for the Transition to Strange Nonchaotic Spiking Rational Approximation (RA) Investigation of the Transition to Strange Nonchaotic Spiking in a Sequence of Periodically Forced Systems with Rational Approximants k to the Inverse Golden Mean ( f 2 / f 1 ) ( 5 1) /2 k Fk 1 / Fk ; Fk 1 Fk Fk 1 , F0 0 and F1 1. Phase-Dependent Saddle-Node Bifurcation in the RA of level k=6 Idc=1.9A/cm2 Smooth Stable (black, corresponding to the silent state) and Unstable (gray) Tori Phase-Dependent Saddle-Node Bifurcation Idc=2.6A/cm2 Nonsmooth Spiking Attractor with Gaps, Filled with Intermittent Chaotic Attractors Average Lyapunov exponent <1> = -0.392 The nonsmooth spiking attractor is nonchaotic 14 Characterization of Spiking States Sequences and Histograms of Interspike Interval (ISI) Strange Nonchaotic Spiking State (Idc=2.85) Chaotic Spiking State (Idc=3.9) Increase in Idc ISI decreases. Aperiodic Sequences and Multimodal Histograms of ISI Aperiodic Complex Spikings Resulting from both SN and Chaotic Spiking States 15 Hindmarsh-Rose Model of Bursting Neurons • Bursting Bursting Phase Silent Phase Neural Activity: Alternation between Bursting and Silent Phases (bursting phase: consisting of rapid spikes) Representative Bursting Cells: (1) Thalamic Neurons: Relaying information from the sensor organs to cortex (2) Pancreatic -cells: Release of insulin controlling the level of glucose (sugar) in the blood • Hindmarsh-Rose (HR) Neuron [Abstract Polynomial Model of Bursting Neurons] [Refs.: J.L. Hindmarsh and R.M. Rose, Nature 296, 162 (1982). J.L. Hindmarsh and R.M. Rose, Proc. R. Soc. Lond. B 221, 87 (1984). J.L. Hindmarsh and R.M. Rose, Proc. R. Soc. Lond. B 225, 161 (1985).] dX Y aX 3 bX 2 Z I ext , dt dY c dX 2 Y , dt dZ r sX X 0 Z dt X: voltage-like variable Y: voltage recovery variable Z: slow adaptation variable (controlling the transition between the bursting and silent phases) (a=1, b=3, c=1, d=5, s=1, r=0.001, and X0=-1.6) 16 Intermittent Route to Chaotic Bursting (Periodic Forcing) I ext I dc A1 sin(2f 1t ); I dc 0.3 (constant bias) • Silent State for Idc=0.3 (1=-0.133) A1 0.5 and f 1 30Hz (weak periodic forcing) • Transition to a Chaotic Bursting Intermittent Route to Chaos for Idc=0.416 720 … via a Subcritical Hopf Bifurcation [Poincaré Map: Stroboscopic Sampling at Multiples of the External Period P1 (=1/f1)] • Chaotic Bursting State for Idc=0.5 (1=0.406) “Hedgehog-like” Chaotic Attractor 17 Effect of Quasiperiodic Forcing on the Bursting Transition I ext I dc A1 sin( 2f1t ) A2 sin( 2f 2t ); f 2 / f1 ( 5 1) / 2, A1 0.5 and f1 30Hz Direct Transition to Chaotic Bursting for Small A2 (<0.4) Transition to bursting state for Idc~0.3963 when A2=0.2 Silent State (Idc=0.39) Chaotic Bursting State (Idc=0.4, 1=0.154) Like the periodically-forced case, 1 seems to jump to a finite positive value near the transition point. Transition to Strange Nonchaotic Bursting for A2=0.5 Appearance of Strange Nonchaotic Bursting State between the Silent State and the Chaotic Bursting State Silent State Idc ~ 0.2236 Strange Nonchaotic Idc ~ 0.271 Chaotic Bursting State (Idc=0.29) Bursting State (Idc=0.24) (Idc=0.21) 1=-0.029 1=0.074 Properties of the Strange Nonchaotic Attractor (SNA) No Sensitivity to Initial Conditions < 0 Fractal with Strange Geometry 18 Characterization of Silent and Bursting States Lyapunov Exponents Poincaré Maps Quasiperiodic Silent State (Idc=0.21) 1=-0.206 Strange Nonchaotic Bursting State (Idc=0.24) 1=-0.029, =1.74 Chaotic Bursting State (Idc=0.29) 1=0.074 Black curve: Strange nonchaotic bursting • Sensitivity of an Attractor with Respect to the Phase of the External Quasiperiodic Forcing [Ref. A.S. Pikovsky and U. Feudel, Chaos 5, 253 (1995)] Phase Sensitivity: Characterized by Differentiating X with respect to at a discrete time t=nP1 (P1=1/f1) X Sn , n 1, 2, ... t nP1 Phase Sensitivity Function N min max | Sn ( x (0)) | { x ( 0 )} 0 n N Phase Sensitivity Exponents N ~ N |Sn|: bounded for all n Smooth Geometry |Sn|: unbounded (a dense set of singularities) Strange Geometry (Fractal) Nonchaotic Attractor with <0 Smooth Torus: N: bounded (=0) SNA: > 0 Sensitive to the External Phase : Used to Measure the Degree of Strangeness Increase in Idc from I*dc (=0.2236) Increase in the Degree of 19 Strangeness of the SNA Mechanism for the Transition to Strange Nonchaotic Bursting Rational Approximation (RA) Investigation of the Transition to Strange Nonchaotic Bursting in a Sequence of Periodically Forced Systems with Rational Approximants k to the Inverse Golden Mean ( f 2 / f 1 ) ( 5 1) /2 k Fk 1 / Fk ; Fk 1 Fk Fk 1 , F0 0 and F1 1. Smooth Torus Corresponding to the Silent State in the RA of level k=7 Idc=0.22 RA of the smooth torus consists of the stable F7(=13)-periodic orbits in the whole range of Transition to a Nonsmooth Bursting Attractor Idc=0.222 Appearance of F7 gaps filled with chaotic attractors Nonsmooth bursting attractor Idc=0.222 Transition to a nonsmooth bursting attractor via a subcritical perioddoubling bifurcation Idc=0.222 Average Lyapunov exponent <1> = -0.073 The nonsmooth bursting attractor is nonchaotic 20 Characterization of Bursting States Characterization of Bursting States ith interburst interval (IBI): Time interval between the ith and (i+1)th bursts (i = 1, 2, …) ith bursting length (BL): Time interval between the first and last spikes in the ith burst No. of spikes in the ith burst (n) Sequences and Histograms of IBI, BL, and n Strange Nonchaotic Bursting State (Idc=0.24) Chaotic Bursting State (Idc=0.29) Increase in Idc IBI decreases, and BL and n increase. Aperiodic Sequences and Multimodal Histograms of IBI, BL, and n Aperiodic Complex Burstings Resulting from both SN and Chaotic Bursting States 21 Summary Typical Occurrence of Strange Nonchaotic Spikings and Burstings in Quasiperiodically Forced Neurons Silence Strange Nonchaotic Firing Chaotic Firing Aperiodic Complex Firings: Resulting from Two Dynamically Different States with Strange Geometry (One is Chaotic and the Other One is Nonchaotic) Strange Nonchaotic Spiking Chaotic Spiking Strange Nonchaotic Bursting Chaotic Bursting 22