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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

, Zhisheng Duan

, Guanrong Chen

a,d

State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China

School of Statistics and Mathematics, Inner Mongolia Finance and Economics College, Huhhot 010051, China

Department of Physics, Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia

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.

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-

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]:

i, j

=−g

i, j

− V

) − g

i, j

− V

)

−

, j

i, j

− V

) + I

+ D



k,l

i, j,k,l



k,l

(t − τ ) − V



σ ξ

i, j

(t), (1)

i, j

= α

i, j

(1 − m

i, j

) − β

i, j

, (2)

i, j

= α

i, j

(1 − h

i, j

) − β

i, j

, (3)

i, j

= α

i, j

(1 − n

i, j

) − β

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

i, j,k,l

= 1if

the site

(k,l) is coupled to (i, j) and ε

i, j,k,l

= 0 otherwise. When

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

i, j

satisfying

ξ

i, j

(t)=0 and ξ

i, j

(t), ξ

m,n



)=δ(t − t



)δ

i,m

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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