An optimally evolved connective ratio of neural networks that maximizes the occurrence of synchronized bursting behavior
 ChaoYi Dong^{1} and
 KwangHyun Cho^{2}Email author
DOI: 10.1186/17520509623
© Dong and Cho; licensee BioMed Central Ltd. 2012
Received: 14 December 2011
Accepted: 31 March 2012
Published: 31 March 2012
Abstract
Background
Synchronized bursting activity (SBA) is a remarkable dynamical behavior in both ex vivo and in vivo neural networks. Investigations of the underlying structural characteristics associated with SBA are crucial to understanding the systemlevel regulatory mechanism of neural network behaviors.
Results
In this study, artificial pulsed neural networks were established using spike response models to capture fundamental dynamics of large scale ex vivo cortical networks. Network simulations with synaptic parameter perturbations showed the following two findings. (i) In a network with an excitatory ratio (ER) of 8090%, its connective ratio (CR) was within a range of 1030% when the occurrence of SBA reached the highest expectation. This result was consistent with the experimental observation in ex vivo neuronal networks, which were reported to possess a matured inhibitory synaptic ratio of 1020% and a CR of 1030%. (ii) No SBA occurred when a network does not contain any allpositiveinteraction feedback loop (APFL) motif. In a neural network containing APFLs, the number of APFLs presented an optimal range corresponding to the maximal occurrence of SBA, which was very similar to the optimal CR.
Conclusions
In a neural network, the evolutionarily selected CR (1030%) optimizes the occurrence of SBA, and APFL serves a pivotal network motif required to maximize the occurrence of SBA.
Background
In the brain development, neurons are assembled together via numerous synapses to build up complicated neuronal networks performing specific behaviors, such as transient or sporadic activity, synchronized bursting activity (SBA), and hyperexcitable activity. One of the most prominent behaviors in cortical networks is the synchronized bursting spikes occurring in the brain development and maturation [1–3]. The behavior is not only found in ex vivo cultured cortical networks [4] but also in the brain regions of several in vivo animal models like visual cortex [5], hippocampus [6], and auditory neocortex [7]. In particular, under in vivo conditions, SBA is considered highly related to a variety of crucial biophysical functions, such as attentional selection [8–10], cognitive motor processes [11], visual pattern recognition [12], auditory object perception [13].
Although SBA is an unique phenomenon in neuronal networks, characteristics of the neural networks causing SBA remain unknown, in contrast to the study on the function significance of the SBA [14]. Presently, large random ex vivo cortical networks are more appropriate experimental model systems in the studies on the universal mechanisms governing the formation and conservation of neural network activities. Experiments using ex vivo cultured neural networks have demonstrated that the adjustment of synaptic connections is highly correlated with the development of neuronal network behavior such as the evolution of spontaneous electrical activity [15, 16]. In the matured phase of an ex vivo cultured neural network, each neuron builds up synaptic connections with 1030% of other neurons within the neural network [17, 18]. Another line of evidence has indicated that the electrical activity of neurons can directly affect the outgrowth of neurites, and such reconfiguration of neuronal networks in turn causes adaptive adjustment of the neuronal electrical activity [19]. This behaviordependent regulatory mechanism precisely drives and controls networks to grow, prune, and finally converge to a proper connective ratio (CR) (1030%).
According to the above described connectivity characteristics of ex vivo cultured neural networks, two interesting questions arise: why do such a matured neural network keep its CR within a fixed range (1030%); and what biological significance and associated implications does this fixed CR have? To answer these intriguing questions, we hypothesize that the CR is associated with the facilitation of synchronized bursting network behaviors, since synaptic connections are always found correlated with network behaviors in ex vivo experiments. Spike response models [20–23] were used to construct randomly connected artificial pulsed neural networks. The connective weights between two neurons were randomly selected, and the CR of the networks was increased progressively to mimic the process of development of cultured neural networks. The correlation between network behavior and structure was investigated using simulations. Subjecting the simulations to parameter perturbations revealed that, for a network with an excitatory ratio (ER) at 8090% (a realistic ratio for ex vivo networks), the CR of the network always lies in a range of 1030% when the occurrence of SBA reaches its highest expectation. This value is consistent with the matured CR of ex vivo neuronal networks with the inhibitory synaptic ratio at 1020% [24, 25]. This result reveals that the networks are evolved to form such a CR for optimizing the occurrence of SBA rather than randomly connected.
This study also explored the relationship between the occurrence of SBA and the composition of network motifs in the neural networks [26, 27]. We found that SBA can be found only in the networks containing an allpositiveinteraction feedback loop (APFL) [2]. For networks containing APFLs, the number of APFLs also demonstrates an optimal range corresponding to the maximized occurrence of SBA, close to the CR. Thus, we infer that the APFL may serve a crucial network motif underlying to maximize the occurrence of SBA.
For a pilot study in real neural networks, we have employed the neural network of nematode worm C. elegans[28, 29]. The nervous system of C. elegans consists of 302 neurons and the number of neurons is almost same for different individuals. Each neuron in C. elegans' nervous system has distinct properties in view of morphology, connectivity, and position, and therefore it can be labelled specifically. The neural network of C. elegans is highly clustered like regular lattices and also has small characteristic path lengths like random graphs. So, it is well represented by smallworld networks [30, 31]. We investigated the egglaying circuit of C. elegans including 11 neurons or neuron classes to examine our major claims [32, 33]. As a result, we found that the egglaying circuit has 17.3% CR and 10.5% ER which lie within the aforementioned evolved ranges. We also found that three twonode APFLs included in this circuit contribute to inducing a much higher level of SBAs in contrast to the randomly connected networks with the same number of network nodes.
Results
The optimal CR at the maximal occurrence of SBA
To unravel the biological significance of the CR of matured neural networks (1030%), we first investigated the relationships between CR, ER, and the occurrence of SBA. Spike response models (SRM) can be used to simulate random ex vivo cortical networks so that their fundamental dynamical properties can be modeled [34]. The detailed simulation protocols for the artificial pulsed neural networks constructed by SRM are introduced in the Methods section. The connective weights among neurons were randomly assigned in a certain range to obtain a result that was irrelevant to the specific value of connective weights. For each CR and ER, 1,000 randomly connected pulsed neural networks were constructed to generate various network behaviors which were then further classified into four major categories with respect to the proposed criteria in the section entitled "The typical behaviors of spike neural networks".
The optimal number of APFLs causing maximal occurrence of SBA
Figures 2 a1 and a2 show the correlation between two typical network behaviors, SBA and hyperexcitable activity (HEA), and the total number of 2, 3, and 4node APFL motifs in 12node pulsed neural networks (see Figure 2 a1, a2). When the number of APFL motifs increases, the mean of SBA occurrence initially increases, then reaches a peak (maximum SBA occurrence of 4.3, corresponding to four APFLs in the 12node networks), and finally returns to zero (after 37 APFLs in the 12node networks, see Figure 2 a1). However, the occurrence of HEA always increased with the increase in the number of APFL motifs (see Figure 2 a2). If the number of APFLs exceeded 37 in a 12node network, the occurrence of 2channel HEAs (see the section of 'Simulation protocols" for its definition) approached the maximum, ${C}_{2}^{12}$, which corresponds to the case in which all pairs of channels are hyperexcitable. Thus, the decreased the occurrence of SBA can be explained by the increase of HEA occurrence when APFLs are sufficiently enriched in a network. A similar result was obtained in 60node networks (see Figures 2 b1 and b2). The maximum 2channel SBA occurrence was 727, corresponding to 400 APFL motifs (2, 3, 4node) in the 60node networks. When the number of 2, 3, 4node APFL motifs exceeded 900, HEA fully dominated, and all other behaviors including SBA vanished from the 60node networks. Notably, the first points (mean = 0 and standard deviation = 0) in Figures 2 a1 and b1 imply that no SBA occurs when no APFL motif is present in the network. Therefore, the APFLs is necessary to trigger SBA in a pulsed neural network.
Figures 1 and 2 clearly indicate that SBA occurs significantly within an optimal range of CR and APFL number. In fact, the simulations demonstrate that the number of APFL motifs increases along with the increase of ER and CR in randomly generated synthetic networks (data not shown). Therefore, we infer that the primary factor inducing the maximal occurrence of SBA may be the formation of a suitable number of APFLs in neural networks.
The relationship between the number of 2, 3, or 4node APFLs and the occurrence of SBA
How is SBA inhibited when each type of APFL motif is absent from the pulsed neural networks? Figure 3 a2 shows the occurrence of SBA when one type or a combination of APFL motifs is not present in the 12node networks. With the exclusion of more types of APFL motifs, both the mean and standard deviation approached zero. For example, if we take the absence of 2, 3, 4, 5, and 6node APFLs into account, the occurrence of SBA is only 3.418 × 10^{5} ± 0.0101 (mean frequency and standard deviation). Thus, the loss of more types of APFL motif gradually inhibits the occurrence of SBA. In 60node networks, the observed trend slightly differs in that the exclusion of 2node APFLs completely prohibits the occurrence of SBA (see the first point in Figure 3 b2). This fact implies that 2node APFLs may function dominantly in the inhibition of SBA in largescale networks, compared with 3node and 4node APFLs.
A case study of the egglaying circuit of C. elegans
We carried out simulations over two network groups for 20,000 times with different synaptic weight perturbations. One group of networks are randomly connected with any CR between 0 and 1, and the other group of networks have the same topological structure as shown in Figure 4. Twosample ttest was carried out for these two groups to examine whether the real biological neural network induce higher SBAs compared to the randomly connected neural networks. Details are as follows: Let vector x denote the SBAs of 20,000 randomly connected networks and vector y represent the SBAs of the egglaying neural network of C. elegans with the 20,000 synaptic weight perturbations. Assuming that the variances of x and y are unknown, it becomes the BehrensFisher problem [35]. The ttest statistic is $T=\frac{\stackrel{\u0304}{x}\mathit{\u0233}}{\sqrt{{s}_{x}^{2}/{n}_{x}+{s}_{y}^{2}/{n}_{y}}}$ ~t(n_{ x } + n_{ y }  2) where s_{ x } = 4.3695 and s_{ y } = 6.9421 are the standard deviations of x and y; $\stackrel{\u0304}{x}=1.0131$ and ȳ = 4.9804 are the means of x and y; n_{ x } and n_{ y } are the numbers of data of x and y. The pvalue for the null hypothesis ${H}_{0}:\stackrel{\u0304}{x}\ge \mathit{\u0233}$ is less than 0.0001 which is much less than the significance level α = 0.05. Therefore, the null hypothesis H_{0} should be rejected and the alternative is accepted.
Discussion
The present study unraveled the direction of neural network development to facilitate a relatively high level of SBA. Thus, the CR range of a mature cultured neural network may represent a delicate design and not the result of random selection. In addition, such biological interpretation of the optimal CR may be further applicable to in vivo situations, since the distribution of cell types in ex vivo networks is often similar to those found in vivo[25, 36]. Some evidence indicates that neural networks first develop toward certain connective structures and then form specific functions by adjusting their synaptic efficacies according to the external stimulus [37]. Our simulations suggest that neurons may connect with each other at a 1030% CR to achieve the highest possibility of SBA occurrence in the early stage, and then, based on such an optimal CR range, the constructed networks further recruit and control SBA by chemically adjusting their synaptic efficacies.
We showed that our main results are quite robust to variations of network scales, network topological properties, and simulation parameters. We carried out simulations (see Simulation protocols) for a variety of neural networks with 10300 nodes and found that the mean value of all the optimal CRs is 13% (with a standard deviation 0.0181) which lies within the evolutionarily selected range of CR (1030%). In addition, note that the networks used for simulations in the early part (1,000 networks were constructed for each CR and ER) were based on random connections and therefore various possible topological structures were already taken into account. So, we confirmed that our results hold regardless of particular connective forms. We have also investigated the possible influences by perturbation of parameters {τ_{ s }, τ_{ m } , Δ_{ ax }, τ} (over a range from 95% to 250% perturbations of the original simulation parameters). For instance, the mean of all the optimal CRs for 10node networks with the parameter perturbations was 18.67% (with a standard deviation 0.0183). In this way, we could also confirm that our results, the evolutionarily selected range of CR of 1030%, still hold against the parameter perturbations.
Conclusions
In this study, we investigated the underlying cause of the evolutionarily selected CRs of neural networks. Artificial pulsed neural network simulation has shown that an optimal CR range (1030%) maximizes the occurrence of synchronized bursting behaviors (when ER = 0.9), which is consistent with previous ex vivo experimental observation, in which the CRs of cultured cortical networks consistently lie in a range of 1030% with an ER of 8090%.
Employing timeseries data from multielectrode array experiments, we identified some APFL motifs in cultured cortical networks of E18 SpragueDawley rats [2]. To further unravel the crucial role of the APFL motifs, we investigated the relationship between specific network structures (network motifs) and network behaviors in artificial pulsed neural networks. This study can readily be used to capture the fundamental dynamical characteristics of cultured neural networks. We found that the existence of APFL motifs is a necessary condition of SBA, not only for smallscale networks but also for largescale networks. To recruit a high level of SBA, networks must have an optimal number of APFL motifs. Therefore, we infer that the formation of the appropriate number of APFLs is related to the maximal occurrence of SBA, whereas the optimal CR is only a necessary condition to achieve the required APFLs.
Furthermore, we investigated the distribution of each type of APFL motif (2, 3, or 4node) at different SBA levels. In both 12node and 60node networks, the 2node APFL motif dominated among APFL motifs at high SBA levels. More importantly, the contribution of each type of APFL motif to SBA was demonstrated by comparing the inhibitory effect of each APFL motif against SBA. For largescale networks, the exclusion of 2node APFLs almost fully prohibits the occurrence of SBA, implying that compared to other APFL motifs, 2node APFL motifs may be crucial for neural networks to produce SBA.
Methods
Definitions of network motifs and feedback loops
A network motif is defined as an enriched subnetwork pattern in complex networks that occurs more frequently than in randomized networks [38–41]. Here, this concept was extended to a more general definition. A motif refers to any subnetwork with a particular structure. To relate the structure of various distinct network motifs and their dynamic behaviors, a range of different network structures can be considered, and their correlations to specific dynamic behaviors investigated. This paper focused on synchronized bursting activities.
Pulsed neural networks
To infer the relationship between a type of feedback motif and its network behaviors, typical network motifs were constructed based on pulsed neural networks, and their network responses to randomly assigned initial states and simulation parameters were observed. The pulsed neural networks, also called the third generation of artificial neural networks, are based on spiking neurons, or "integrateandfire" neurons [22, 23]. These neurons utilize recent insights from neurophysiology, specifically the use of temporal coding to pass information between neurons [11, 42, 43], which closely mimic realistic communication between neurons. Therefore, pulsed neural networks are commonly applied to study the properties of neural networks.
where τ_{ s }and τ_{ m }are time constants describing axonal transmission dynamics and membrane dynamics, respectively, and Δ_{ ax }is axonal transmission delay. H(t  Δ_{ ax }) is the Heaviside step function which vanishes for t ≤ Δ_{ ax }, and set t > Δ_{ ax }equal to 1.
Networks of different sizes can exhibit similar network behaviors (or dynamics) if their neurons are supplied with the same average inputs [22]. To make networks of two sizes comparable with respect to the same average input of each neuron, the networks must have weight scopes scaled by the number of network nodes. For example, take an n_{1}node network as a nominal case with an allowed weight scope of [W_{max}, W_{max}]. Then the weight scope of an n_{2}node network should be assigned as $\frac{\left[{W}_{max},{W}_{max}\right]}{\left({n}_{2}1\right)/\left({n}_{1}1\right)}$. In practice, the smaller network (n_{1} = 12) was set as the nominal network. Thus, the weight scope of the other network (n_{2} = 60) was scaled by the factor (60  1)/(12  1) ≈ 5.
The typical behaviors of spike neural networks
where $S{S}_{xx}=\sum _{k=1}^{n}{x}^{2}\frac{{\left(\sum _{k=1}^{n}x\right)}^{2}}{n}$, $S{S}_{yy}=\sum _{k=1}^{n}{y}^{2}\frac{{\left(\sum _{k=1}^{n}y\right)}^{2}}{n}$, and $S{S}_{xy}=\sum _{k=1}^{n}xy\frac{\left(\sum _{k=1}^{n}x\right)\left(\sum _{k=1}^{n}y\right)}{n}$.
Simulation protocols
Simulations were carried out using SRMs for the randomly connected networks where all neurons were assumed to have an identical parameter set {τ_{ s }, τ_{ m } , Δ_{ ax }, τ, δ}, and the synaptic efficacies w_{ ij } were randomly chosen from a uniform distribution [1, 1] (see the section of 'Pulsed neural networks' for further details on SRM). For a randomly generated network with n neurons and m synaptic connections (suppose that m_{ e } denotes excitatory connections and mm_{ e } represents inhibitory connections), the CR is defined as the percentage of the number of existing synaptic connections of the network divided by that of the fully connected network with node number n, i.e., $\frac{m}{n\left(n1\right)}$. Here, only a single connection between two different neurons is allowed for simplification. Hence, the possibility of selfconnection of one neuron and multiple connections between any pair of neurons are excluded. The ER is referred to as a quotient of excitatory synaptic number over the total number of synaptic connections, i.e., $\frac{{m}_{e}}{m}$.
To investigate the relationships among CR, ER, and the occurrence of SBA, simulations with both synaptic efficacies and network structures randomly perturbed were carried out using different combinations of CR and ER. Classification of four typical network behaviors (TRA, SBA, ASBA, and HEA) can be found in the section entitled "The typical behaviors of spike neural networks". For each ratio pair (CR, ER), 1,000 randomly connected artificial neural networks were constructed, and simulations based on these networks were carried out. For each constructed network (corresponding to one simulation), the total number of APFL motifs (2, 3, and 4node) and the occurrences of kchannel SBA and HEA (2 ≤ k ≤ n and k ∈ ℤ) were investigated to show the correlations between APFL number and such typical behaviors. Our simulations demonstrated that the change of k or inclusion of more (greater than 4) node motifs had no effect on evaluating the dynamical characteristics of networks. In practice, only the 2channel SBA and 2channel HEA were evaluated and used to measure the levels of synchronization and hyperexcitation in networks. The correlation between the occurrence of 2channel SBA or 2channel HEA and network topological characteristics was obtained from the statistics of the occurrence of such behaviors with various CRs, ERs, and randomly assigned synaptic efficacies.
Abbreviations
 APFL:

Allpositiveinteraction feedback loop
 TRA:

Transient response activities
 SBA:

Synchronized bursting activities
 ASBA:

Asynchronous bursting activities
 HEA:

Hyperexcitable activities
 CR:

Connective ratio
 ER:

Excitatory ratio
 SI:

Synchrony index
 SRM:

Spike response model.
Declarations
Acknowledgements
We thank Dongkwan Shin and Chaoxuan Dong for their critical reading of this paper. This work was supported by the National Research Foundation of Korea (NRF) grants funded by the Korea Government, the Ministry of Education, Science & Technology (MEST) (20090086964, 20100017662, and 20110006314). It was also supported by the WCU (World Class University) program (R322008000102180) through the NRF funded by MEST.
Authors’ Affiliations
References
 Aradi I, Maccaferri G: Cell typespecific synaptic dynamics of synchronized bursting in the juvenile CA3 rat hippocampus. J Neurosci 2004, 24: 9681. 10.1523/JNEUROSCI.280004.2004View ArticleGoogle Scholar
 Dong CY, Lim J, Nam Y, Cho KH: Systematic analysis of synchronized oscillatory neuronal networks reveals an enrichment for coupled direct and indirect feedback motifs. Bioinformatics 2009, 25: 16801685. 10.1093/bioinformatics/btp271View ArticleGoogle Scholar
 Klemm K, Bornholdt S: Topology of biological networks and reliability of information processing. Proc Natl Acad Sci 2005, 102: 1841418419. 10.1073/pnas.0509132102View ArticleGoogle Scholar
 Kamioka H, Maeda E, Jimbo Y, Robinson HPC, Kawana A: Spontaneous periodic synchronized bursting during formation of mature patterns of connections in cortical cultures. Neurosci Lett 1996, 206: 109112. 10.1016/S03043940(96)124484View ArticleGoogle Scholar
 Gray CM, Singer W: Stimulusspecific neuronal oscillations in orientation columns of cat visual cortex. Proc Natl Acad Sci 1989, 86: 16981702. 10.1073/pnas.86.5.1698View ArticleGoogle Scholar
 Bragin A, Jando G, Nadasdy Z, Hetke J, Wise K, Buzsaki G: Gamma (40100 Hz) oscillation in the hippocampus of the behaving rat. J Neurosci 1995, 15: 4760.Google Scholar
 Traub R, Contreras D, Cunningham M, Murray H, LeBeau F, Roopun A, Bibbig A, Wilent W, Higley M, Whittington M: Singlecolumn thalamocortical network model exhibiting gamma oscillations, sleep spindles, and epileptogenic bursts. J Neurophysiol 2005, 93: 21942232.View ArticleGoogle Scholar
 Womelsdorf T, Fries P: The role of neuronal synchronization in selective attention. Curr Opin Neurobiol 2007, 17: 154160. 10.1016/j.conb.2007.02.002View ArticleGoogle Scholar
 Siegel M, Donner TH, Oostenveld R, Fries P, Engel AK: Neuronal synchronization along the dorsal visual pathway reflects the focus of spatial attention. Neuron 2008, 60: 709719. 10.1016/j.neuron.2008.09.010View ArticleGoogle Scholar
 Engel AK, Fries P, Singer W: Dynamic predictions: oscillations and synchrony in topdown processing. Nat Rev Neurosci 2001, 2: 704716. 10.1038/35094565View ArticleGoogle Scholar
 Riehle A, Grun S, Diesmann M, Aertsen A: Spike synchronization and rate modulation differentially involved in motor cortical function. Science 1950, 1997: 278.Google Scholar
 Gray CM, Konig P, Engel AK, Singer W: Oscillatory responses in cat visual cortex exhibit intercolumnar synchronization which reflects global stimulus properties. Nature 1989, 338: 334337. 10.1038/338334a0View ArticleGoogle Scholar
 deCharms RC, Merzenich MM: Primary cortical representation of sounds by the coordination of actionpotential timing. Nature 1996, 381: 610613. 10.1038/381610a0View ArticleGoogle Scholar
 Kim JR, Shin D, Jung SH, HeslopHarrison P, Cho KH: A design principle underlying the synchronization of oscillations in cellular systems. J Cell Sci 2010, 123: 537. 10.1242/jcs.060061View ArticleGoogle Scholar
 Muramoto K, Ichikawa M, Kawahara M, Kobayashi K, Kuroda Y: Frequency of synchronous oscillations of neuronal activity increases during development and is correlated to the number of synapses in cultured cortical neuron networks. Neurosci Lett 1993, 163: 163165. 10.1016/03043940(93)90372RView ArticleGoogle Scholar
 Nakanishi K, Nakanishi M, Kukita F: Dual intracellular recording of neocortical neurons in a neuronglia coculture system. Brain Res Protoc 1999, 4: 105114. 10.1016/S1385299X(99)000033View ArticleGoogle Scholar
 Jimbo Y, Tateno T, Robinson HPC: Simultaneous induction of pathwayspecific Potentiation and depression in networks of cortical neurons. Biophys J 1999, 76: 670678. 10.1016/S00063495(99)772346View ArticleGoogle Scholar
 Marom S, Shahaf G: Development, learning and memory in large random networks of cortical neurons: lessons beyond anatomy. Q Rev Biophys 2002, 35: 6387.View ArticleGoogle Scholar
 Fields RD, Nelson PG: Activitydependent development of the vertebrate nervous system. Int Rev Neurobiol 1992, 34: 133214.View ArticleGoogle Scholar
 Segev R, Shapira Y, Benveniste M, BenJacob E: Observations and modeling of synchronized bursting in twodimensional neural networks. Phys Rev E 2001, 64: 11920.View ArticleGoogle Scholar
 Izhikevich EM, Gally JA, Edelman GM: Spiketiming dynamics of neuronal groups. Cereb Cortex 2004, 14: 933. 10.1093/cercor/bhh053View ArticleGoogle Scholar
 Maass W, Bishop CM: Pulsed Neural Networks. Cambridge: MIT Press; 1999.Google Scholar
 Maass W: Networks of spiking neurons: the third generation of neural network models. Neural Netw 1997, 10: 16591671. 10.1016/S08936080(97)000117View ArticleGoogle Scholar
 Eckenstein F, Thoenen H: Cholinergic neurons in the rat cerebral cortex demonstrated by immunohistochemical localization of choline acetyltransferase. Neurosci Lett 1983, 36: 211. 10.1016/03043940(83)900022View ArticleGoogle Scholar
 Huettner JE, Baughman RW: Primary culture of identified neurons from the visual cortex of postnatal rats. J Neurosci 1986, 6: 30443060.Google Scholar
 Lodato I, Boccaletti S, Latora V: Synchronization properties of network motifs. EPL (Europhysics Letters) 2007, 78: 28001. 10.1209/02955075/78/28001View ArticleGoogle Scholar
 DHuys O, Vicente R, Erneux T, Danckaert J, Fischer I: Synchronization properties of network motifs: Influence of coupling delay and symmetry. Chaos 2008, 18: 037116. 10.1063/1.2953582View ArticleGoogle Scholar
 Chatterjee N, Sinha S: Understanding the mind of a worm: hierarchical network structure underlying nervous system function in C. elegans. Prog Brain Res 2007, 168: 145153.View ArticleGoogle Scholar
 Hobert O: Specification of the nervous system. WormBook: the online review of C elegans biology 2005, 12: 1.Google Scholar
 Watts DJ, Strogatz SH: Collective dynamics of'smallworld' networks. Nature(London) 1998, 393: 440442. 10.1038/30918View ArticleGoogle Scholar
 Varshney LR, Chen BL, Paniagua E, Hall DH, Chklovskii DB: Structural Properties of the Caenorhabditis elegans Neuronal Network. PLoS Comput Biol 2011, 7: e1001066. 10.1371/journal.pcbi.1001066View ArticleGoogle Scholar
 Ambros V, Horvitz HR: Heterochronic mutants of the nematode Caenorhabditis elegans. Science 1984, 226: 409. 10.1126/science.6494891View ArticleGoogle Scholar
 Trent C, Tsung N, Horvitz HR: Egglaying defective mutants of the nematode Caenorhabditis elegans. Genetics 1983, 104: 619.Google Scholar
 Gerstner W, Kistler W: Spiking Neuron Models: An Introduction. NY: Cambridge University Press New York; 2002.View ArticleGoogle Scholar
 Kim SH, Cohen AS: On the BehrensFisher problem: a review. J Educ and Behav Stat 1998, 23: 356377.View ArticleGoogle Scholar
 Nakanishi K, Kukita F: Intracellular [Cl] modulates synchronous electrical activity in rat neocortical neurons in culture by way of GABAergic inputs. Brain Res 2000, 863: 192204. 10.1016/S00068993(00)021521View ArticleGoogle Scholar
 Hagan MT, Demuth HB, Beale MH: Neural network design. MA: PWS Boston; 1996.Google Scholar
 Kim JR, Yoon Y, Cho KH: Coupled feedback loops form dynamic motifs of cellular networks. Biophys J 2008, 94: 359365. 10.1529/biophysj.107.105106View ArticleGoogle Scholar
 Kim TH, Kim J, HeslopHarrison P, Cho KH: Evolutionary design principles and functional characteristics based on kingdomspecific network motifs. Bioinformatics 2010, 27: 245.View ArticleGoogle Scholar
 Kwon YK, Cho KH: Quantitative analysis of robustness and fragility in biological networks based on feedback dynamics. Bioinformatics 2008, 24: 987994. 10.1093/bioinformatics/btn060View ArticleGoogle Scholar
 Milo R, ShenOrr S, Itzkovitz S, Kashtan N, Chklovskii D, Alon U: Network motifs: simple building blocks of complex networks. Science 2002, 298: 824827. 10.1126/science.298.5594.824View ArticleGoogle Scholar
 Buracas GT, Zador AM, DeWeese MR, Albright TD: Efficient discrimination of temporal patterns by motionsensitive neurons in primate visual cortex. Neuron 1998, 20: 959969. 10.1016/S08966273(00)804778View ArticleGoogle Scholar
 Thorpe S, Fize D, Marlot C: Speed of processing in the human visual system. Nature 1996, 381: 520522. 10.1038/381520a0View ArticleGoogle Scholar
 Mehring C, Hehl U, Kubo M, Diesmann M, Aertsen A: Activity dynamics and propagation of synchronous spiking in locally connected random networks. Biol Cybern 2003, 88: 395. 10.1007/s0042200203844View ArticleGoogle Scholar
 Brunel N: Dynamics of networks of randomly connected excitatory and inhibitory spiking neurons. J Physiol Paris 2000, 94: 445. 10.1016/S09284257(00)010846View ArticleGoogle Scholar
 Silberberg G, Bethge M, Markram H, Pawelzik K, Tsodyks M: Dynamics of population rate codes in ensembles of neocortical neurons. J Neurophysiol 2004, 91: 704.View ArticleGoogle Scholar
 Bonneau R: Learning biological networks: from modules to dynamics. Nat Chem Biol 2008, 4: 658664. 10.1038/nchembio.122View ArticleGoogle Scholar
 Gerstein GL, Kiang NYS: An approach to the quantitative analysis of electrophysiological data from single neurons. Biophys J 1960, 1: 1528. 10.1016/S00063495(60)868725View ArticleGoogle Scholar
 Wylie DR, De Zeeuw CI, Simpson JI: Temporal relations of the complex spike activity of Purkinje cell pairs in the vestibulocerebellum of rabbits. J Neurosci 1995, 15: 2875.Google Scholar
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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.