Entrainment and Chaos in the Hodgkin-Huxley Oscillator

Transcription

1 Entrainment and Chaos in the Hodgkin-Huxley Oscillator Kevin K. Lin klin Courant Institute, New York University Mostly Biomath p.1/42

2 Overview (1) Goal: Show that the Hodgkin-Huxley neuron model, driven by a periodic impulse train, can exhibit entrainment, transient chaos, and fully chaotic behavior. Mostly Biomath p.2/42

3 Overview (2) Why? 1. Suggested by general, rigorous perturbation theory of kicked oscillators (Qiudong Wang & Lai-Sang Young). 2. Hodgkin-Huxley is a paradigm for excitable biological systems where pulse-like inputs and outputs are natural. Mostly Biomath p.3/42

4 Overview (3) Results: 1. Entrainment and chaos are readily observable in the pulse-driven Hodgkin-Huxley system. 2. The pulse-driven Hodgkin-Huxley system prefers entrainment. 3. Strong expansion is caused by invariant structures. Mostly Biomath p.4/42

5 Outline Classical Hodgkin-Huxley neuron model Kicked nonlinear oscillators & Wang-Young theory Pulse-driven Hodgkin-Huxley neuron model Mostly Biomath p.5/42

6 Squid giant axon Mostly Biomath p.6/42

7 Schematic (rest state) References: Abbott and Dayan, Theoretical Neuroscience, MIT Press 2001 Cronin, Mathematical Aspects of Hodgkin-Huxley Neural Theory, Cambridge University Press 1987 Mostly Biomath p.7/42

8 Hodgkin-Huxley equations v = C 1 [ I ḡ K n 4 (v v K ) ḡ Na m 3 h(v v Na ) ḡ leak (v v leak ) ] ṁ = αm(v)(1 m) βm(v)m ṅ = αn(v)(1 n) βn(v)n ḣ = α h (v)(1 h) β h (v)h αm(v) = Ψ ( ) v+25 10, β m(v) = 4 exp (v/18), αn(v) = 0.1Ψ ( ) v+10 10, β n(v) = exp (v/80), 1 α h (v) = 7 exp (v/20), β h (v) = 1+exp( v ), v Ψ(v) = exp(v) 1. Mostly Biomath p.8/42

9 Equivalent circuit [ ] v = C 1 I ḡ K n 4 (v v K ) ḡ Na m 3 h(v v Na ) ḡ leak (v v leak ) Mostly Biomath p.9/42

10 Parameters In this study: All parameters take on original squid values except the injected current I This guarantees stable oscillations Mostly Biomath p.10/42

11 Parameters (cont d) v Na = 115 mv, ḡ Na = 120 mω 1 /cm 2, v K = +12 mv, ḡ K = 36 mω 1 /cm 2, v leak = mv, ḡ leak = 0.3 mω 1 /cm 2, C = 1 µf/cm 2, I = Mostly Biomath p.11/42

12 Parameters (cont d) Stable fixed point Unstable fixed point V Unstable cycle -80 Limit cycle I Mostly Biomath p.12/42

13 Dynamics without kicks v -4 n t v -2 Mostly Biomath p.13/42

14 Outline Classical Hodgkin-Huxley neuron model Kicked nonlinear oscillators & Wang-Young theory Pulse-driven Hodgkin-Huxley neuron model Mostly Biomath p.14/42

15 Kicked oscillators A stable, nonlinear oscillator is a flow with a limit cycle γ (period=t γ ) and basin of attraction U. A kick instantaneously changes the system s state: Mostly Biomath p.15/42

16 Examples of kicked oscillators Circadian rhythm, phase reset experiments (Winfree). Possible approach to disrupting synchronous firing of neuron (Tass). Mostly Biomath p.16/42

17 Simple Example 1.0 y x ṙ = (µ αr 2 )r+ 1 2 sin(3θ) δ(t nt) n Z θ = ω + βr 2 Mostly Biomath p.17/42

18 Effect of Kick-and-Flow on Phase Space t = 0 t = 0 (after kick) t = t = 2 t = 2 (after kick) t = 4 Mostly Biomath p.18/42

19 Discrete time map Define F T : R 4 R 4 by F T (x) = Φ T (K(x)), where K(x) represents kicks Φ T (x) = flow map T = period of kicks. Continuous time Discrete time Entrainment F T has sinks Chaos F T chaotic Mostly Biomath p.19/42

20 Reduction to 1-D Wang and Young start with F T and 1. Reduces from the map F T on R n to a circle map f T : lim F T+nT γ (x) n induces a map f T on γ S 1. We refer to f T as the singular limit or the phase resetting curve. 2. Analyze f T and infer properties of F T. Mostly Biomath p.20/42

21 Wang-Young Conditions If 1. Kicks do not send limit cycle to bad places, i.e. K(γ) does not go outside the basin of γ 2. Kicks are in the right directions (e.g. not along W ss (x)) to take advantage of shear Mostly Biomath p.21/42

22 Wang-Young Consequences Then for different kick amplitude A & kick period T the discrete-time system F T can have 1. Rotation-like behavior (small A) 2. Sinks and sources (for all A large enough) 3. Transient chaos / horseshoes (for large A & T) 4. Strange attractor & chaos (for large A & T 1) Mostly Biomath p.22/42

23 Wang-Young Theory (cont d) Notes: 1. The conditions are satisfied for the simple example. 2. For Hodgkin-Huxley there is not too much choice if we want to stay close to physical interpretation of model. Mostly Biomath p.23/42

24 Lyapunov exponents The largest Lyapunov exponent λ of F T is useful for distinguishing different scenarios numerically: Rotations λ = 0 Sinks λ < 0 Chaos λ > 0 Mostly Biomath p.24/42

25 Outline Classical Hodgkin-Huxley neuron model Kicked nonlinear oscillators & Wang-Young theory Pulse-driven Hodgkin-Huxley neuron model Mostly Biomath p.25/42

26 Hodgkin-Huxley equations v = C 1 [ I ḡ K n 4 (v v K ) ḡ Na m 3 h(v v Na ) ḡ leak (v v leak ) ] +A n Z δ (t nt) ṁ = αm(v)(1 m) βm(v)m ṅ = αn(v)(1 n) βn(v)n ḣ = α h (v)(1 h) β h (v)h Prior work: Winfree, Best on null space and degree-change. Mostly Biomath p.26/42

27 Entrainment A = 10, T = v t v t Mostly Biomath p.27/42

28 Entrainment (cont d) -2 A = 10, T = 81.0 v t v1(t)-v2(t) t Mostly Biomath p.28/42

29 Entrainment (cont d) Time-T map: F T = Φ T K 2 v1(n)-v2(n) Multiple of T (n) 2 Mostly Biomath p.29/42

30 Chaos v A = 10, T = t 2.0 Log10(dist) t Mostly Biomath p.30/42

31 λ (F T ) versus T Largest Lyapunov exponent of F_T T / T_gamma Mostly Biomath p.31/42

32 λ (F T ) versus A ROTATIONS 0.6 SINKS CHAOS UNKNOWN Drive amplitude Chaos: Prob( ˆλ > 3ǫ) Rotations: Prob( ˆλ < ǫ/3) Sinks: Prob( ˆλ < 3ǫ) Unknown: everything else Mostly Biomath p.32/42

33 Phase resetting curves ( f T ) A=5, T=101.5 A=10, T= A=20, T= Mostly Biomath p.33/42

34 Plateau and fixed points The first return map R ft to the interval [4, 10] (enclosing the plateau), for A = 10 and T = Mostly Biomath p.34/42

35 Plateau and fixed points Drive amplitude A Probability of sink near plateau 5 41% 10 58% 20 68% 30 76% Mostly Biomath p.35/42

36 Zooming into the kink Mostly Biomath p.36/42

37 Why the kink? h 0.34 n v m But Hodgkin-Huxley lives in R 4... Mostly Biomath p.37/42

38 Why the kink? (cont d) Approaching critical A crit : A=13.58 A= A= A= A=13.59 A= Mostly Biomath p.38/42

39 Why the plateau? Graph of f T for A = 10, around plateau f_t phase phase Black: log 10 f Blue: log 10 (Ess(K(γ(θ)), γ(θ))) Mostly Biomath p.39/42

40 Finding horseshoes Horseshoes can produce transient chaos: A = 10, T = Mostly Biomath p.40/42

41 Summary Can observe entrainment and chaos in the pulse-driven Hodgkin-Huxley neuron model. The pulse-driven Hodgkin-Huxley model prefers entrainment. This can be explained. Complex phase response can arise from kicks going near invariant structures. Mostly Biomath p.41/42

42 References 1. Eric N. Best, Null space in the Hodgkin-Huxley equations, Biophys. J. 27 (1979) 2. Kevin K. Lin, Entrainment and chaos in the Hodgkin-Huxley oscillator, in preparation 3. Qiudong Wang and Lai-Sang Young, Strange attractors in periodically-kicked limit cycles and Hopf bifurcations, Comm. Math. Phys. 240 (2003) 4. Arthur Winfree, The Geometry of Biological Time, 2nd Edition, Springer-Verlag (2000) Acknowledgements I am grateful to Lai-Sang Young for her help with this work, and to Eric Brown, Adi Rangan, Alex Barnett, and Toufic Suidan for critical comments. Many thanks to Charlie Peskin for the invitation. This work is supported by the National Science Foundation. Mostly Biomath p.42/42