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我们研究了由不同信息传输延迟和网络拓扑结构引起的嘈杂的霍奇金-赫克斯利神经元合奏中螺旋波的空间动力学。 在相干共振的经典设置中,对噪声的强度进行了微调,以优化系统的响应。 在这里,我们使噪声强度保持恒定,而改变耦合神经元之间信息传输延迟的长度。 我们表明存在一个中间传输延迟,通过该延迟可以优化螺旋波的排序,从而表明在所检查的系统中存在延迟增强的空间动力学相干性。 此外,随着扩散相互作用拓扑向小世界类型变化,我们研究了这种现象的鲁棒性,并发现远处神经元之间的捷径链接阻碍了相干螺旋波的出现,而与传输延迟长度无关。 因此,提出的结果提供了见识,可以促进对现实神经元网络上信息传输延迟的理解。 (C)2008 Elsevier BV保留所有权利.Source @@@ Physics Letters A,372(35),pp 5681-5687,2008/8/25(ISI Web of Knowledge)来源:Physics Letters A,372(35) ,pp 5681-5687,2008年。
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Physics Letters A 372 (2008) 5681–5687
Contents lists available at ScienceDirect
Physics Letters A
www.elsevier.com/locate/pla
Delay-enhanced coherence of spiral waves in noisy Hodgkin–Huxley neuronal
networks
Qingyun Wang
a,b,∗
, Matjaž Perc
c
, Zhisheng Duan
a
, Guanrong Chen
a,d
a
State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China
b
School of Statistics and Mathematics, Inner Mongolia Finance and Economics College, Huhhot 010051, China
c
Department of Physics, Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia
d
Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China
article info abstract
Article history:
Received 16 April 2008
Received in revised form 24 June 2008
Accepted 2 July 2008
Available online 5 July 2008
Communicated by C.R. Doering
PACS:
05.45.-a
05.40.-a
89.75.Kd
Keywords:
Neuronal networks
Delay
Spiral wave
Coherence resonance
We study the spatial dynamics of spiral waves in noisy Hodgkin–Huxley neuronal ensembles evoked
by different information transmission delays and network topologies. In classical settings of coherence
resonance the intensity of noise is fine-tuned so as to optimize the system’s response. Here, we keep the
noise intensity constant, and instead, vary the length of information transmission delay amongst coupled
neurons. We show that there exists an intermediate transmission delay by which the spiral waves are
optimally ordered, hence indicating the existence of delay-enhanced coherence of spatial dynamics in
the examined system. Additionally, we examine the robustness of this phenomenon as the diffusive
interaction topology changes towards the small-world type, and discover that shortcut links amongst
distant neurons hinder the emergence of coherent spiral waves irrespective of transmission delay length.
Presented results thus provide insights that could facilitate the understanding of information transmission
delay on realistic neuronal networks.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
Spiral wave dynamics is the subject of ongoing and intense in-
vestigations in diverse fields of research ranging from physics and
chemistry to biology [1–3]. Especially within real neuronal net-
works spiral waves are common, and temporal variations of spi-
ral core numbers have been studied extensively [4,5]. To deepen
the understanding of mechanisms behind the generation of spiral
waves in neuronal systems, several theoretical as well as experi-
mental studies have been conducted, resulting in fascinating new
discoveries and insights. In particular, self-maintaining spiral waves
have been observed in a non-homogeneous neuronal network with
uniform initial conditions, whereas the breakup of outwardly ro-
tating spiral waves has been reported also in settings different
from the neuronal dynamics [6–8]. Moreover, the breakup of in-
wardly rotating spiral waves has been investigated in an oscillatory
FitzHugh–Nagumo system near a Hopf bifurcation [9], where it has
been shown that the breakup first occurred by regions that are far
*
Corresponding author at: State Key Laboratory for Turbulence and Complex Sys-
tems, Department of Mechanics and Aerospace Engineering, College of Engineering,
Peking University, Beijing 100871, China. Tel./fax: +86 10 62765037.
E-mail address: nmqingyun@163.com (Q. Wang).
away from the core area and then gradually penetrated the whole
medium as the diffusion coefficient ratio between the two compo-
nents of the oscillatory system increased. Two-dimensional lattices
of non-homogeneous cardiac cells have also been shown to gener-
ate spiral waves due to spatial discreteness and inhomogeneity of
the model [10]. Interestingly, it has been argued that the breakup
of spiral waves may be an important mechanism by cardiac fibril-
lation [11], as it was discovered that spatiotemporal patterns of
cardiac activity by human seizures have a consistent dynamical
evolution by the initialization, development as well as termination
of each seizure [12].
Intimately related to the above findings concerning spiral wave
formation in non-homogeneous two-dimensional excitable media
are studies reporting stochastic and coherence resonance phenom-
ena in dynamically similar excitable systems [13–24].Asnoiseis
an inseparable part of any real-life process, the understanding of
its potential impacts is of vital importance. To date, it has become
an established and well-accepted fact that biological neurons can
exploit the constructive role of noise to extract hindered informa-
tion and enhance weak stimuli via stochastic resonance [25–27].
Aside from these two rather main-stream phenomena, there exist
several related reports on noise-induced order either from chaotic
states [28], by means of variations in system size [29,30] and di-
0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.physleta.2008.07.005
5682 Q. Wang et al. / Physics Letters A 372 (2008) 5681–5687
versity [31], or via an enhancement of synchronization in coupled
systems [32–34]. Furthermore, order out of noise and the dynam-
ical properties of excitatory events in general have been studied
extensively also in non-identical ensembles of systems governed
by nonlinear dynamics, such as neurons [35], as well as in small-
world neuronal networks [36,37] and ensembles of bistable over-
damped oscillators [38]. Following initial advances on isolated dy-
namical systems, the phenomena of stochastic and coherence res-
onance have been generalized also to two-dimensional media. In
particular, spatiotemporal stochastic resonance has been reported
in [39], while spatial coherence resonance has been introduced
first near pattern forming instabilities [40] and subsequently also
in excitable media [41]. The characteristics of noise-induced pat-
terns have also been investigated in a spatially extended Hodgkin–
Huxley (HH) neuronal network in dependence on the coupling
strength [42], and it was found that the spatiotemporal dynamics
could be enhanced as the connections amongst neurons became
stronger. Aside from these examples, the body of recent literature
devoted to the study of noise and other stochastic influences on
the dynamics of spatially extended systems is huge, so that we
found it impossible to select or review here all relevant contri-
butions. The interested reader is pointed towards Ref. [43], while
some exemplary studies are also given in [44–50].
Recently, the body of literature devoted to studying effects of
noise on nonlinear, and in particular excitable dynamical systems
has been supplemented extensively by the addition of effects of
different delay lengths, either in terms of information transmis-
sion delay [51] or a delayed inhibitory feedback [52]. It has been
shown (see e.g. [51]) that the coherence of noise-induced pulses
may exhibit a double resonance outlay that emerges if the delay
length is optimally fine-tuned. Furthermore, the phenomenon of
stochastic resonance has also been investigated in related systems
incorporating time-delay feedback mechanisms [53,54].However,
since information transmission delays are inherent in intra- and
inter-neuronal communication because of finite propagation veloc-
ities governing the conduction of signals along neuritis as well as
delays in the synaptic transmission along chemical synapses [55],
it is thus important to understand the dynamics of coupled neu-
ronal ensembles when such temporal delays are not negligible, as
was comprehensively exemplified in [56]. Notably, it has been sug-
gested that time delays can facilitate neural synchronization and
lead to many interesting and even unexpected phenomena [57,58].
To elaborate on the latter statement and to extend the scope
of the subject, we presently study the dynamics of spiral waves in
noisy neuronal networks subject to information transmission delay.
We employ the realistic Hodgkin–Huxley (HH) [59] model of neu-
ronal dynamics and, in addition to different delay lengths and the
classic diffusive coupling, consider also small-world interactions
governing the dynamics of the ensemble. More precisely, we exam-
ine the impact of different transmission delay lengths on the spiral
wave dynamics. We find that, while short transmission delays do
enable the existence of spiral waves, their spatial order can be sub-
stantially improved if the delay length is fine-tuned. On the other
hand, long information transmission delays completely hinder the
emergence of spiral waves, thus clearly indicating a resonance-like
dependence of the spatial order of spiral waves on the trans-
mission delay length. In particular, we find that there exists an
intermediate information transmission delay length by which the
spiral waves are optimally ordered in space, hence reporting delay-
enhanced coherence of spatial dynamics in the examined system.
Importantly, this phenomenon can be observed best on diffu-
sive neuronal networks, whereas the introduction of shortcut links
amongst distant units, eventually constituting a small-world topol-
ogy, progressively hinders coherent spiral wave formation. By em-
ploying the circularly averaged spatial structure function [43],we
provide conclusive evidences for the delay-enhanced coherence of
spiral waves on diffusively coupled HH neuronal networks, as well
as for the small-world induced breakup of spiral waves and related
decoherence of spatial dynamics of excitatory events. Interestingly
though, the optimal information transmission delay is almost im-
mune to the transition from diffusive to small-world interactions
amongst coupled neurons, suggesting that the delay-enhanced co-
herence is a robust mechanism warranting an enhancement of
spatial order by the formation of spiral waves in noisy neuronal
environments. These findings can have important practical imple-
mentations by various tasks that fall into the domain of neuronal
networks (e.g. communication, information processing or compu-
tation), where noise and information transmission delays, as well
as small-world-like interconnectedness, appear to be universally
present.
The Letter is organized as follows. Section 2 is devoted to the
description of the HH model and employed networks, whereas Sec-
tion 3 evidences the phenomenon of delayed-enhanced coherence
of spiral waves if the diffusive topology is used. Effects of small-
world topology on the delayed-enhanced coherence are presented
in Section 4, while the last section summarizes the results.
2. Mathematical model and setup
The spatiotemporal dynamics of studied HH neuronal networks
is governed by the following differential equations [59]:
C
dV
i, j
dt
=−g
Na
m
3
i, j
h
i, j
(V
i, j
− V
Na
) − g
L
(V
i, j
− V
L
)
−
g
K
n
4
i
, j
(V
i, j
− V
K
) + I
+ D
k,l
ε
i, j,k,l
V
k,l
(t − τ ) − V
ij
+
σ ξ
i, j
(t), (1)
dm
i, j
dt
= α
m
i, j
(1 − m
i, j
) − β
m
i, j
m
i, j
, (2)
dh
i, j
dt
= α
h
i, j
(1 − h
i, j
) − β
h
i, j
h
i, j
, (3)
dn
i, j
dt
= α
n
i, j
(1 − n
i, j
) − β
n
i, j
n
i, j
, (4)
where V is the transmembrane voltage in mV, m is the acti-
vation and h the inactivation coefficient of sodium conductance,
n is the activation coefficient of potassium channels, while the
time
(t) units are seconds. For a more detailed descriptions of
all variables we refer the reader to the original work [59].The
sum in Eq. (1) runs over all lattice sites whereby
ε
i, j,k,l
= 1if
the site
(k,l) is coupled to (i, j) and ε
i, j,k,l
= 0 otherwise. When
ε
i, j,k,l
= 1onlyif(k, l) is one of the four nearest neighbors of
the focal site
(i, j) we obtain a diffusively coupled network of
HH neurons each having degree z
= 4, as depicted in Fig. 1(a).
The latter will be used throughout Section 3.However,ifacer-
tain fraction 0
< q 1 of links constituting the diffusively coupled
network is randomly rewired, as exemplified in Fig. 1(b), the re-
sulting network is of small-world type [60].Presentlyweemploy
the rewiring procedure described in [61] to preserve the degree
of each neuron (z
= 4), which enables us to focus explicitly on
the effect of network topology rather than possible effects origi-
nating from different numbers of inputs per neuron. Small-world
networks will be used in Section 4. Importantly, we generated each
interaction network at the beginning of a particular simulation and
kept it fixed the whole time; and moreover, if necessary below
results were averaged over 30 different realizations of the inter-
action network by each q. Quantities
σ and τ in Eq. (1) denote
the standard deviation of additive uncorrelated Gaussian noise
ξ
i, j
,
satisfying
ξ
i, j
(t)=0 and ξ
i, j
(t), ξ
m,n
(t
)=δ(t − t
)δ
i,m
δ
j,n
, and
the transmission delay, respectively. In order to introduce a noisy
background for the dynamics, we set
σ = 1.0 and do not vary this
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