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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 cos2ft]
Squid Giant Axon (Neural Oscillator),
Aggregates of Embryonic Chick Heart Cells (Cardiac Oscillator)
• Quasiperiodic Stimulation [Fext(t) = A1 cos2f1t + A2 cos2f2t]
Forcing with Two Incommensurate Frequencies
1
Regular Response Under Periodic Stimulus [Fext = A cos 2ft]
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 2f1t + A2 cos 2f2t]
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 ~ 30m, 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 = 800m 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 hV  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(VE);
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  20A/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( 2f1t ); A1  1A/cm 2 and f1  60Hz
• Subthreshold Periodic Oscillation (Silent State) for Idc=2.5A/cm2
• Occurrence of Chaotic Spiking
Intermittent Route to Chaos for Idc>Idc*(=3.058 824 A/cm2)
I dc  3.5A/cm 2
10
Effect of Quasiperiodic Forcing on the Spiking Transition
I ext  I dc  A1 sin( 2f1t )  A2 sin( 2f 2t );   f 2 / f1  ( 5 1) / 2, A1  1A/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=1.9A/cm2
Smooth Stable (black, corresponding to
the silent state) and Unstable (gray) Tori
Phase-Dependent
Saddle-Node Bifurcation
Idc=2.6A/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 sX  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(2f 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( 2f1t )  A2 sin( 2f 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