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Entrainment and Chaos in theHodgkin-Huxley Oscillator
Kevin K. Lin
http://www.cims.nyu.edu/∼klin
Courant Institute, New York University
Mostly Biomath - 2005.4.5 – p.1/42
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.
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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.
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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.
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Outline
Classical Hodgkin-Huxley neuron model
Kicked nonlinear oscillators &
Wang-Young theory
Pulse-driven Hodgkin-Huxley neuron
model
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Squid giant axon
http://hermes.mbl.edu/publications/Loligo/squid
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Schematic (rest state)
References:
Abbott and Dayan, Theoretical Neuroscience, MIT Press 2001
Cronin, Mathematical Aspects of Hodgkin-Huxley Neural Theory, Cambridge
University Press 1987
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Hodgkin-Huxley equations
v = C−1[
I − gKn4(v − vK) − gNam3h(v − vNa) − gleak(v − vleak)]
m = αm(v)(1− m)−βm(v)m
n = αn(v)(1− n)−βn(v)n
h = αh(v)(1− h)−βh(v)h
αm(v) = Ψ(
v+2510
)
, βm(v) = 4 exp (v/18) ,
αn(v) = 0.1Ψ(
v+1010
)
, βn(v) = 0.125 exp (v/80) ,
αh(v) = 0.07 exp (v/20) , βh(v) = 11+exp( v+30
10 ),
Ψ(v) = vexp(v)−1
.
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Equivalent circuit
v = C−1[
I − gKn4(v − vK) − gNam3h(v − vNa) − gleak(v − vleak)]
http://www.syssim.ecs.soton.ac.uk/vhdl-ams/examples/hodhuxneu/hh2.htm
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Parameters
In this study:
All parameters take on original squid
values except the injected current I
This guarantees stable oscillations
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Parameters (cont’d)
vNa = −115 mV, gNa = 120 mΩ−1/cm2,
vK = +12 mV, gK = 36 mΩ−1/cm2,
vleak = −10.613 mV, gleak = 0.3 mΩ−1/cm2,
C = 1 µF/cm2, I = −14.2211827403
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Parameters (cont’d)
-100
-80
-60
-40
-20
0
20
6 7 8 9 10 11 12 13 14-I
V
Unstable fixed pointStable fixed point
Limit cycle
Unstable cycle
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Dynamicswithout kicks
40.030.020.010.00.0
t
0.0
-20.0
-40.0
-60.0
-80.0
v
0.0-20.0-40.0-60.0-80.0
v
0.7
0.65
0.6
0.55
0.5
0.45
n
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Outline
Classical Hodgkin-Huxley neuron model
Kicked nonlinear oscillators &
Wang-Young theory
Pulse-driven Hodgkin-Huxley neuron
model
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Kicked oscillators
A stable, nonlinear oscillator is a flow with
a limit cycle γ (period=Tγ) and basin of
attraction U.
A kick instantaneously changes the
system’s state:
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Examples of kicked oscillators
Circadian rhythm, phase reset
experiments (Winfree).
Possible approach to disrupting
synchronous firing of neuron (Tass).
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Simple Example
2.01.00.0-1.0
x
1.0
0.0
-1.0
y
r = (µ −αr2)r+1
2sin(3θ) · ∑
n∈Z
δ(t − nT)
θ = ω + βr2
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Effect of Kick-and-Flow on Phase Space
1.00.80.60.40.20.0-0.2-0.4-0.6-0.8
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
1.00.50.0-0.5-1.0
1.0
0.5
0.0
-0.5
-1.0
-1.51.00.50.0-0.5-1.0
1.0
0.5
0.0
-0.5
-1.0
t = 0 t = 0 (after kick) t = 1
1.00.50.0-0.5-1.0
1.0
0.5
0.0
-0.5
-1.0
1.00.0-1.0
1.0
0.5
0.0
-0.5
-1.0
-1.5
1.00.50.0-0.5-1.0
1.0
0.5
0.0
-0.5
-1.0
t = 2 t = 2 (after kick) t = 4
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Discrete time map
Define FT : R4 → R
4 by FT(x) = ΦT(K(x)), where
K(x) represents kicks
ΦT(x) = flow map
T = period of kicks.
Continuous time ⇔ Discrete time
Entrainment ⇔ FT has sinks
Chaos ⇔ FT chaotic
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Reduction to 1-D
Wang and Young start with FT and
1. Reduces from the map FT on Rn to a circle
map fT:
limn→∞
FT+nTγ(x)
induces a map fT on γ ∼ S1. We refer to fT
as the singular limit or the phase resetting
curve.
2. Analyze fT and infer properties of FT.
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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 Wss(x)) to take advantage of shear
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Wang-Young Consequences
Then for different kick amplitude A & kick
period T the discrete-time system FT 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)
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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.
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Lyapunov exponents
The largest Lyapunov exponent λ of FT is
useful for distinguishing different scenarios
numerically:
Rotations ⇔ λ = 0
Sinks ⇔ λ < 0
Chaos ⇔ λ > 0
Mostly Biomath - 2005.4.5 – p.24/42
Outline
Classical Hodgkin-Huxley neuron model
Kicked nonlinear oscillators &
Wang-Young theory
Pulse-driven Hodgkin-Huxley neuron
model
Mostly Biomath - 2005.4.5 – p.25/42
Hodgkin-Huxley equations
v = C−1[
I − gKn4(v − vK) − gNam3h(v − vNa) − gleak(v − vleak)]
+A ∑n∈Z δ (t − nT)
m = αm(v)(1− m)−βm(v)m
n = αn(v)(1− n)−βn(v)n
h = αh(v)(1− h)−βh(v)h
Prior work: Winfree, Best on “null space” and
degree-change.
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Entrainment
A = 10, T = 81.0
150.0100.050.00.0
t
0.0
-20.0
-40.0
-60.0
-80.0
v
1500.01000.0500.00.0
t
60.0
40.0
20.0
0.0
-20.0
-40.0
-60.0
v
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Entrainment (cont’d)
A = 10, T = 81.0
150.0100.050.00.0
t
0.0
-20.0
-40.0
-60.0
-80.0
v
1500.01000.0500.00.0
t
100.0
50.0
0.0
-50.0
-100.0
v1(t)-v2(t)
Mostly Biomath - 2005.4.5 – p.28/42
Entrainment (cont’d)
Time-T map: FT = ΦT K
20.010.00.0
Multiple of T (n)
20.0
0.0
-20.0
-40.0
v1(n)-v2(n)
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Chaos
A = 10, T = 80.8
6000.04000.02000.00.0
t
100.080.060.040.020.00.0
-20.0-40.0-60.0-80.0
v
6000.04000.02000.00.0
t
2.0
0.0
-2.0
-4.0
-6.0
Log10(dist)
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λ (FT) versusT
8.07.06.05.04.03.02.0
T / T_gamma
0.0
-2.0
-4.0
-6.0
-8.0
Largest Lyapunov exponent of F_T
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λ (FT) versusA
40.030.020.010.0
Drive amplitude
1.0
0.8
0.6
0.4
0.2
0.0
SINKS
CHAOS
ROTATIONS
UNKNOWN
Chaos: Prob(λ > 3ǫ) Sinks: Prob(λ < −3ǫ)
Rotations: Prob(∣
∣λ∣
∣ < ǫ/3) Unknown: everything else
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Phase resetting curves (fT)
12.510.07.55.02.50.0
20.0
15.0
10.0
5.0
0.0
A=5, T=101.5
12.510.07.55.02.50.0
15.0
10.0
5.0
0.0
A=10, T=80.8
12.510.07.55.02.50.0
15.0
10.0
5.0
0.0
A=20, T=101.5
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Plateau and fixed points
The first return map R fTto the interval [4, 10]
(enclosing the plateau), for A = 10 and
T = 17.6.
10.09.08.07.06.05.04.0
10.0
9.0
8.0
7.0
6.0
5.0
4.0
Mostly Biomath - 2005.4.5 – p.34/42
Plateau and fixed points
10.09.08.07.06.05.04.0
10.0
9.0
8.0
7.0
6.0
5.0
4.0
Drive amplitude A Probability of sink near plateau
5 41%
10 58%
20 68%
30 76%
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Zooming into the kink
1.00.80.60.40.20.0
16.0
14.0
12.0
10.0
8.0
6.0
4.0
2.0
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Why the kink?
-2.0-4.0-6.0-8.0-10.0-12.0-14.0
v
0.38
0.36
0.34
0.32
h
0.180.160.140.120.1
m
0.45
0.44
0.43
0.42
n
But Hodgkin-Huxley lives in R4...
Mostly Biomath - 2005.4.5 – p.37/42
Why the kink? (cont’d)
Approaching critical Acrit ≈ 13.5895...:
10.510.09.59.0
30.0
20.0
10.0
0.0
A=13.58
10.510.09.59.0
40.0
30.0
20.0
10.0
0.0
A=13.589
10.510.09.59.0
50.0
40.0
30.0
20.0
10.0
0.0
A=13.5895
10.510.09.59.0
40.0
30.0
20.0
10.0
0.0
A=13.5896
10.510.09.59.0
30.0
20.0
10.0
0.0
A=13.59
10.510.09.59.0
20.0
10.0
0.0
A=13.6
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Why the plateau?
Graph of fT for A = 10, around plateau.
9.08.07.06.05.0
phase
9.0
8.0
7.0
6.0
5.0
f_T
9.08.07.06.05.0
phase
0.0
-1.0
-2.0
-3.0
-4.0
-5.0
Black: log10 | f ′ | Blue: log10 |∠(Ess(K(γ(θ)), γ(θ)))|
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Finding horseshoes
Horseshoes can produce transient chaos:
A = 10, T = 81
0.70.60.50.40.3
12.0
10.0
8.0
6.0
4.0
2.0
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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.
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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.
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