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Article
Biophysics
© The Author(s) 2011. This article is published with open access at Springerlink.com csb.scichina.com www.springer.com/scp
SPECIAL TOPICS:
January 2011 Vol.56 No.2: 151–157
doi: 10.1007/s11434-010-4281-2
Channel noise-induced phase transition of spiral wave in
networks of Hodgkin-Huxley neurons
MA Jun1,2*, WU Ying3, YING HePing4 & JIA Ya2
1
Department of Physics, Lanzhou University of Technology, Lanzhou 730050, China;
Department of Physics, Huazhong Normal University, Wuhan 430079, China;
School of Science, Xi’an University of Technology, Xi’an 710048, China;
Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310027, China
2
3
4
Received July 19, 2010; accepted September 21, 2010
The phase transition of spiral waves in networks of Hodgkin-Huxley neurons induced by channel noise is investigated in detail.
All neurons in the networks are coupled with small-world connections, and the results are compared with the case for regular
networks, in which all neurons are completely coupled with nearest-neighbor connections. A statistical variable is defined to study
the collective behavior and phase transition of the spiral wave due to the channel noise and topology of the network. The effect of
small-world connection networks is described by local regular networks and long-range connection with certain probability p. The
numerical results confirm that (1) a stable rotating spiral wave can be developed and maintain robust with low p, where the
breakup of the spiral wave and turbulence result from increasing the probability p to a certain threshold; (2) appropriate intensity
of the optimized channel noise can develop a spiral wave among turbulent states in small-world connection networks of H-H
neurons; and (3) regular connection networks are more robust to channel noise than small-world connection networks. A spiral
wave in a small-world network encounters instability more easily as the membrane temperature is increased to a certain high
threshold.
breakup, channel noise, factor of synchronization, probability of long-range connection
Citation: Ma J, Wu Y, Ying H P, et al. Channel noise-induced phase transition of spiral wave in networks of Hodgkin-Huxley neurons. Chinese Sci Bull, 2011, 56:
151−157, doi: 10.1007/s11434-010-4281-2
Collective electrical behaviors of neurons and oscillators in
networks often have spatiotemporal patterns [1–11]. A spi-
ral wave is one such spatial pattern and is often observed in
excitable and oscillatory media. In experimental studies,
most of the works used to study the chemical wave in the
Belousov-Zhabotinsky reaction [12], and many other theo-
retical and numerical works on spiral waves have been re-
ported [13–18]. The importance of studying spiral waves is
that it gives important clues as to how to remove spiral
waves in cardiac tissue and prevent ventricular fibrillation
[19] and allows a better understanding of the nonlinear dy-
namics from a spiral wave to turbulence. There is evidence
that a spiral wave in cardiac tissue is harmful, and thus,
*Corresponding author (email: hyperchaos@163.com)
many effective schemes have been proposed to eliminate
spiral waves in media. For example, the scheme of periodi-
cal forcing is proposed to eliminate the spiral waves and
turbulence by generating a target wave or travelling wave in
the media [20,21]. Transition from a spiral wave to other
states induced by a polarized field [22], deformation of me-
dia [23,24], and the synchronization of spiral waves [25]
have also been investigated extensively. In particular, the
noise-induced formation and development of spiral waves
in a reaction-diffusion system was discussed by Hou and
Xin in detail [26]. The dynamics of spiral waves and control
pattern selection in reaction-diffusion systems have been
studied extensively while few works have been reported on
the development and phase transition of the spiral wave in
the networks of neurons, and its role in signal communica-
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152 Ma J, et al. Chinese Sci Bull January (2011) Vol.56 No.2
tion in the networks is unknown. A neuronal system con-
sists of a large number of neurons [27] with complex con-
nections. Normal electric activity of neurons is critical for
signal communication among neurons. Stable rotating spiral
waves in rat neocortical slices visualized by voltage-
sensitive dye imaging were found in experiments [6,7]. As
noted in [6,7], spiral waves might serve as pacemakers for
the emergent population to generate periodic activity in a
nonoscillatory network without the need for individual cel-
lular pacemakers. It is interesting to simulate and investi-
gate the formation and breakup of spiral waves in networks
of neurons with different topologies. Perc et al. made great
progress in the formation of spiral waves in networks
[8–10,28,29]. He et al. [30] presented excellent results on
the formation of spiral waves in small-world networks of
FitzHugh-Nagumo neurons and confirmed that the destruc-
tive effect of an inhomogeneous medium on spiral waves
can be decreased or removed by appropriate small-world
connections. Some aspects of this topic remain unclear; for
example, the dynamics of spiral waves in small-world net-
works of neurons and the effects of channel noise and the
size of the network on a spiral wave. It is better to study
spiral wave dynamics in networks of Hodgkin-Huxley
(H-H) neurons than in networks of Hindmarsh-Rose (H-R)
neurons [4,5] because the H-R neuron model is a simplified
version of the realistic H-H model. Channel noise can
change the dynamics of H-H neurons greatly [31,32]. White
et al. [31] pointed out that the probabilistic gating of volt-
age-dependent ion channels is a source of electrical ‘chan-
nel noise’ in neurons. Schmid et al. [32] reported the ca-
pacitance fluctuations reducing channel noise in stochastic
H-H systems. Fox et al. [33] presented the autocorrelation
functions of channel noise to estimate the effect of channel
noise. The present work investigates the robustness and
breakup of a spiral wave in small-world networks of H-H
neurons in the presence of channel noise. A statistical vari-
able is defined to study the phase transition of spiral waves,
and the results are compared with those for regular net-
works. The external forcing current at all sites (neurons) is
set at zero, which makes each single H-H neuron quiescent.
It is found that appropriate channel noise actively develops
spiral waves and maintains its robustness so that signal
communication still can pass through these quiescent areas.
1 Mathematical model and discussion
The H-H neuron mode is more realistic than other presented
neuron models. Small-world networks of H-H neurons are
described as follows:
4
ij
3
ij ij
∑
KK Na Na
LL
d
()()
d
()( ), (1)
ij
m ijij
ijij ijkl klij
kl
V
C g n V
?
V g m h V
?
V
t
g V
?
VIDVV
ε
=−+−
+−++−
d
( )(1)() ( ),t
d
ij
m ijijm ij ijm
m
VmV m
t
αβξ=−−+
(2)
d
( )(1)() ( ),t
d
ij
t
h ij ijmijijh
h
VhV h
αβξ=−−+
(3)
d
()(1)() ( ),t
d
ij
m ij ijm ij ijn
n
VnV n
t
αβξ=−−+
(4)
0.1(
−
40) ( )
40)/10)
ij
V
V
−
,
1 exp( (
Φ
4 ( )exp( (T65)/18),
ij
−
m
m ij
VT
Φ
α
β
+
=
+
=+
(5)
( )T
V
,
1 exp( (
Φ
35)/10)
−
0.07 ( )exp( (T 65)/20),
h
ij
h ij
V
Φ
−
β
α
=
−+
=+
(6)
0.01(
1 exp( (
−
55) ( )
55)/10)
ij
V
T
,
0.125 ( )exp( (
Φ
65)/80),
ij
n
n ij
V
−
T
V
Φ
α
β
+
=
+
=−+
(7)
( 6.3 )/10
( )T3.
T
φ
−
=
℃℃ (8)
Here the variable Vi,j describes the membrane potential of
the neuron in site (i, j) and the subscripts (i, j) indicate the
site of the neuron. m, n and h are parameters for the gate
channel, and the capacitance of the membrane is Cm = 1
μF/cm2. D is the intensity of coupling, εklij describes the
connection state (on or off) between site (k, l) and site (i, j),
i and j are integers, εklij = 1 if site (k, l) is connected with
site (i, j) and εklij = 0 otherwise. Clearly, if the fraction of
randomly introduced shortcuts (i.e. rewired links) p (prob-
ability) equals zero, εklij = 0 only if site (k, l) is one of the
four nearest neighbors of site (i, j). The maximal conduc-
tance of potassium is
K
g ? = 36 mS/cm2, the maximal con-
ductance of sodium is
Na
g ?
= 120 mS/cm2, the conductance
of leakage current is
L
g ? = 0.3 mS/cm2 and the external in-
jection current Iij = 0. The reversal potential VK = –77 mV,
VNa = 50 mV and VL = –54.4 mV. ξm(t), ξh(t) and ξn(t) are
independent Gaussian white noise, and the statistical prop-
erties [33] of the channel noise are defined as follows:
ξ
ξξα β δ
δ
=−
K
( )
( )t
0;
t
′
( )2
D
(
t
)/[
′
( )]
( ).
′
m
mmmm
t
mm
m
t
ttN
αβ
<
<
>=
>=−+
(9)
Na
( )
( )t
0;
′
( )t2
D
()/[
′
()]
().
′
n
nnn
δ
n
t
nn
n
t
t
t
tN
ξ
ξξα β δαβ
<
<
>=
>=−+
=−
(10)
Na
( )
( )t
0;
′
( )t2
D
( )/[
′
( )]
( ).
′
h
hhh
δ
h
t
hh
h
t
t
t
tN
ξ
ξξα β δαβ
<
<
>=
>=−+
=−
(11)
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Ma J, et al. Chinese Sci Bull January (2011) Vol.56 No.2
153
Here, Dm, Dn and Dh describe the intensity of noise, function
δ(t–t′) = 1 at t = t′ and δ(t–t′) = 0 at t ≠ t′, and NNa and NK
are the total numbers of sodium and potassium channels
present in a given patch of the membrane, respectively. In
the case of homogeneous ion channel density, ρNa = 60
μm–2 and ρK = 18 μm–2, the total channel number is decided
by NNa = ρNa·s and NK = ρK·s, and s describes the mem-
brane patch. Using mean-field theory, a statistical variable
[5,34,35] is defined to study the collective behaviors and
statistical properties.
2
11
1
,
NN
i js
ji
FVV
N
==
= =<>
∑∑
(12)
22
22
2
11
.
1
()
NN
i ji j
ji
FF
R
VV
N
==
< > − <>
=
< > − <>
∑∑
(13)
Here R is a factor of synchronization, N2 is the number of
neurons and Vij is the membrane potential of the neuron. It
is necessary to define the statistical variable R to character-
ize the system’s normalized variation and thus synchroniza-
tion. R may not be suitable to characterize the synchroniza-
tion of the pattern of spiral waves, while it could be useful
in detecting the critical bifurcation parameter inducing
breakup or elimination of the spiral wave in networks of
neurons as previously mentioned. As previously mentioned
[34,35], the curve of the factor of synchronization vs. bifur-
cation parameter illustrates the phase transition of the spiral
wave through points of sudden change. In [34], the author
of the present work reported the additive Gaussian-colored
noise-induced breakup in a regular network of H-R neurons,
and multiplicative noise in the development of a spiral wave
in regular networks of neurons (H-R, H-H) has also been
investigated in detail [35]. Further numerical results have
confirmed that a spiral wave can develop in networks
(regular or small-world type) of neurons even if there is no
external forcing current. The following section presents a
numerical investigation of the robustness and phase of spiral
waves in the small-world networks of H-H neurons in the
presence of channel noise where there are no external forc-
ing currents acting on neurons.
2 Numerical results and discussion
The numerical studies have a time step h = 0.001, external
forcing current Iij = 0, 40000 neurons in a two-dimensional
array of 200 × 200 sites, and a no-flux boundary condition.
The small-world connection network can be described by
local regular networks (complete nearest-neighbor connec-
tions) and a long-range connection (shortcut) with a certain
probability p. First, the case of no channel noise is consid-
ered, and the snapshots of the membrane potentials of neu-
rons under different probabilities (p = 0.02, 0.03, 0.04 and
0.05) are plotted with a transient period of about 500 time
units.
The numerical results presented in Figure 1 show that a
stable rotating spiral wave can develop completely with
appropriate long-range probability, and no regular spiral
wave is generated when the long-range probability exceeds
a certain threshold. Note that the patterns in the figure are
transient snapshots at t = 500 time units, and the shape and
contour of a stable spiral wave often remain unchanged as
Figure 1 Spatiotemporal patterns developed within a transient period of about 500 time units for long-range probability p = 0.02 (a), 0.03 (b), 0.04 (c) and
0.05 (d). The snapshots are plotted in grayscale from black (about –80 mV) to white (about –40 mV) and the coupling coefficient D = 1.
Page 4
154 Ma J, et al. Chinese Sci Bull January (2011) Vol.56 No.2
new segments of the spiral wave emerge for broken waves.
The transient snapshots for any fixed duration show the
distribution of membrane potentials of neurons, and stable
spiral waves are maintained even though the membrane
potential of a neuron at a site in the network varies with
time. The corresponding factor of synchronization R is
given in Table 1.
It is found that a spiral wave can emerge and cover more
area of the network if a lower long-range connection prob-
ability is used, and a smaller factor of synchronization is
often employed. A smaller factor of synchronization also
indicates a shorter transient period required to develop a
spiral wave in a network. It is important to study the effect
of channel noise on the phase transition of a spiral wave.
Figure 2 illustrates the correlation of the synchronization
factor and the membrane patch, which describes the inten-
sity of channel noise, and the snapshots of membrane po-
tentials of neurons for different fixed membrane patches
(intensities of channel noise).
The results in Figure 2(b) confirm that the spiral wave
breaks up when the intensity of channel noise increases to a
certain threshold. A spiral wave emerges and covers a
greater area in the case of weak channel noise, as seen by
comparing the results in Figure 2(b) with those in Figure
1(a). The curve in Figure 2(a) shows that the factor of syn-
chronization decreases with increasing intensity of the
channel noise (smaller membrane being used). There are
two distinct peaks (s = 15 and 20) in the curve in Figure
2(a), and the development of the spiral wave under channel
noise close to the two peaks is investigated by checking the
growth rate of a spiral wave in the networks. Figure 2(c)
confirms that a longer transient period is required for a spi-
ral wave to emerge and cover a greater area of a network in
the case that the membrane patch (channel noise) corre-
sponds to the two peaks in the curve. A spiral wave can
emerge and cover the entire system with low long-range
connection probability [36]. It is interesting to check the
active role of channel noise in supporting a spiral wave in a
network of neurons. As illustrated in Figure 1(c), no regular
and distinct spiral wave occupies the networks with long-
range connection probability p = 0.04. Channel noise is se-
lected with different intensities to check the effect of chan-
nel noise on the formation of the spatiotemporal pattern.
The results in Figure 3 show that appropriate channel
noise can induce and develop a spiral wave in the networks
of neurons at a certain long-range connection probability
although the channel noise often induces breakup of the
spiral wave. Comparing the results presented in Figure 3(b)
with those presented in Figure 1(c), it is seen that the spiral
Table 1 Factors of synchronization under different long-range probability
Parameter
p
R
Value
0.04
0.192383
0.02
0.091359
0.03 0.05
0.182009
0.06
0.180047 0.246083
Figure 2 Calculated factor of synchronization vs. channel noise, de-
scribed by the membrane patch (a) and spatiotemporal patterns developed
within a transient period of about 2000 time units for s = 2 (b1), 3 (b2), 4
(b3), 15 (b4), 17 (b5), 20 (b6), 28 (b7), 30 (b8) and 36 (b9). The snapshots
of the development of a spiral wave are plotted for a transient period of
about 200 time units for s = 13 (c1), 15 (c2), 17 (c3) and 20 (c4). The
long-range probability is fixed at p = 0.02, the coupling intensity is D = 1
and the membrane temperature T = 6.3°C. The snapshots are plotted in
grayscale from black (about –80 mV) to white (about –40 mV).
wave covers a greater area when appropriate channel noise
is introduced into networks of neurons. Clearly, channel
noise can optimize the order of the spatiotemporal pattern in
a network, and the optimized intensity of channel noise is
close to the peak of the curve of the factor of synchronization
vs. membrane patch. As is well known, a high probability of
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Ma J, et al. Chinese Sci Bull January (2011) Vol.56 No.2
155
Figure 3 Calculated factor of synchronization vs. channel noise de-
scribed by the membrane patch (a) and a spatiotemporal pattern developed
within a transient period of about 2000 time units for s = 17 (b1), 23 (b2),
25 (b3) and 28 (b4) at fixed long-range probability p = 0.04, coupling
intensity D = 1, and membrane temperature T = 6.3°C. The snapshots are
plotted in grayscale from black (about –80 mV) to white (about –40 mV).
long-range connection and noise often destroy the order of
the spatiotemporal pattern and break up the spiral wave. An
ordered state can be generated when appropriate channel
noise is introduced into media with small-world connec-
tions. It is the channel noise that optimizes the order of
small-world networks, although it can also destroy the or-
der. The membrane temperature often has an important role
in determining the dynamics of neurons. Therefore, it is
interesting to study the collective behaviors of spiral waves
in networks with small-world connections. Figure 4 gives
the factors of synchronization at different membrane tem-
peratures and a fixed probability of a long-range connec-
tion.
The results in Figure 4 show that the factors of synchro-
nization decrease with increasing membrane temperature
and breakup of the spiral wave is induced in the small-world
networks of H-H neurons with fixed long-range probability
p = 0.02. It is the small-world effect that destroys the or-
dered state of networks, which differs from the case for
regular networks, in which a certain high membrane tem-
perature simply synchronizes all neurons with complete
nearest-neighbor couplings (the media become homogene-
ous at a certain membrane temperature). To make a distinct
Figure 4 Calculated factor of synchronization vs. membrane temperature
(a) and spatiotemporal patterns developed within a transient period of
about 2000 time units for T = –4°C (b1), 0°C (b2), 2°C (b3), 8°C (b4),
10°C (b5), 12°C (b6), 16°C (b7), 18°C (b8) and 20°C (b9) at fixed
long-range probability p = 0.02 and coupling intensity D = 1. The snap-
shots are plotted in grayscale from black (about –80 mV) to white (about
–40 mV).
comparison, the factors of synchronization for various
membrane patches in regular networks are calculated and
the results are shown in Figure 5.
The results in Figure 5 show that the factor of synchro-
nization changes slowly with the membrane patch size, and
there are no sudden changes in the curve of the synchroni-
zation factor vs. channel noise (membrane patch). This in-
dicates that no phase transition occurs as the membrane
patch size increases (decrease in the intensity of the channel
noise), and a stable rotating spiral wave finally emerges to
cover the network of neurons. On the other hand, breakup of
the spiral wave is induced by increasing the intensity of
channel noise (or decreasing the membrane patch size).
These statements are confirmed by the snapshots of mem-
brane potentials of neurons in the networks.
Comparing the results for the regular networks with
those for small-world networks of neurons, it is found that a
regular network actively supports the spiral wave and
maintains its robustness against channel noise while the
small-world network often induces the breakup of a spiral
wave when the long-range connection probability exceeds a
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156 Ma J, et al. Chinese Sci Bull January (2011) Vol.56 No.2
Figure 5 Calculated factor of synchronization vs. membrane patch
(channel noise) within a transient period of about 2000 time units for
membrane temperature T = 6.3°C (a) and T = 16.3°C (b). The inserted
figures are enlarged illustrations for the membrane path s ≥ 5.
certain threshold. Channel noise can play an active role in
developing a spiral wave in small-world networks of H-H
neurons only when an appropriate intensity is selected. To
date, most works have claimed that small-world connections
better describe the complex connections of neurons tha
regular networks, in which a neuron is only coupled with
the four nearest adjacent neurons. To our knowledge, a
regular connection supporting a spiral wave and long-range
connections in small-world networks often destroys the spi-
ral wave in homogeneous media. The reason could be that a
regular connection results in regular variation in the poten-
tials of the (five adjacent) neurons in the local domain ow-
ing to strong local coupling, and the long-range connection
with high probability simply prevents neurons from chang-
ing simultaneously.
3 Conclusions
In this work, the channel noise-induced formation and
changes in spiral waves in networks of H-H neurons were
investigated and some interesting results were found. A
statistical variable referred to as the factor of synchroniza-
tion was defined to measure the phase transition of the spi-
ral wave. The small-world networks are described by the
combination of local regular connection and long-range
connenction with certain probability p. Long-range connec-
tions with high probability often prevent the formation of a
spiral wave, and a generated spiral wave can cover a net-
work of neurons only when appropriate intensity of the
channel noise is selected. The corresponding curve of the
factor of synchronization vs. channel noise (membrane
patch) indicates coherent resonance-like behavior, odder
selection and optimization with the channel noise. Breakup
of the spiral wave in a small-world network occurs more
easily than that in regular networks of neurons as the mem-
brane temperature increases; that is, higher membrane tem-
perature can induce breakup of the spiral wave more easily
owing to the effect of small-world connections. The factor
of synchronization changes slowly as the membrane patch
size increases (or the intensity of channel noise decreases),
and the spiral wave maintains its robustness against certain
channel noise. As a result, selecting optimized channel
noise is helpful in developing a stable spiral wave in the
small-world networks of neurons through measuring and
detecting the critical factor of synchronization vs. channel
noise curve owing to its active role in propagating the elec-
trical signal in the quiescent domain.
This work was partially supported by the National Natural Science Foun-
dation of China (10747005 and 10972179) and the Natural Foundation of
Lanzhou University of Technology (Q200706).
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