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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 NaNa

LL

d

()()

d

()( ), (1)

ij

m ij ij

ijij ijklklij

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

Vm V m

t

αβξ=−−+

(2)

d

( )(1)()( ),t

d

ij

t

h ijijm ijijh

h

Vh V h

αβξ=−−+

(3)

d

( )(1)()( ),t

d

ij

m ij ijm ijijn

n

Vn V n

t

αβξ=−−+

(4)

0.1(

−

40) ( )

40)/10)

ij

V

V

−

,

1 exp( (

Φ

4 ( )exp( (T 65)/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

nij

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

ξξ α 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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