Department of Automatic Control, Inner Mongolia University of Technology, Huhhot 010080, People's Republic of China

Department of Bio and Brain Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 305-701, Republic of Korea

Abstract

Background

Synchronized bursting activity (SBA) is a remarkable dynamical behavior in both

Results

In this study, artificial pulsed neural networks were established using spike response models to capture fundamental dynamics of large scale

Conclusions

In a neural network, the evolutionarily selected CR (10-30%) 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 hyper-excitable activity. One of the most prominent behaviors in cortical networks is the synchronized bursting spikes occurring in the brain development and maturation

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

According to the above described connectivity characteristics of

This study also explored the relationship between the occurrence of SBA and the composition of network motifs in the neural networks

For a pilot study in real neural networks, we have employed the neural network of nematode worm

Results

The optimal CR at the maximal occurrence of SBA

To unravel the biological significance of the CR of matured neural networks (10-30%), we first investigated the relationships between CR, ER, and the occurrence of SBA. Spike response models (SRM) can be used to simulate random

The SBA properties of networks were investigated at two different scales: small networks with 12 nodes and large networks with 60 nodes. We recorded and calculated the expectation and standard deviation of SBA occurrence over the 1,000 networks (with a stereotyped CR and ER). Figure

The relationship between the occurrence of SBA and CR (or ER) for 12-node networks and 60-node networks

**The relationship between the occurrence of SBA and CR (or ER) for 12-node networks and 60-node networks**. **(a-1), (a-2)** The relationships among CR, ER, and the occurrence of SBA for 12-node networks from two different perspectives. The occurrence of SBA in 1,000 randomly connected networks was evaluated on each point of a lattice composed of CRs (0:0.05:1) and ERs (0:0.1:1). The surface points represent the mean value of the occurrence of SBA, while the upper and lower red bars show their standard deviations over 1,000 simulations. **(a-3) **The relationship between CR and the occurrence of SBA for 12-node networks when ER equals 0.9, which is consistent with the actual biological level. **(b-1), (b-2) **The relationship among CR, ER, and the occurrence of SBA for 60-node networks from two different perspectives. The occurrence of SBA in 1,000 randomly connected networks was evaluated on the same lattice as (a). The surface points represent the mean value of the occurrence of SBA, while the upper and lower red bars show their standard deviations over 1,000 simulations. **(b-3) **The relationship between CR and the occurrence of SBA for 60-node networks when ER equals 0.9 (the biological level).

The optimal number of APFLs causing maximal occurrence of SBA

The existence of APFLs was shown to be a prerequisite for inducing SBA for 2, 3, and 4-node pulsed neural networks

The relationship between the total number of 2, 3, and 4-node APFL motifs and the occurrence of SBA or HEA for the 12-node networks and 60-node networks

**The relationship between the total number of 2, 3, and 4-node APFL motifs and the occurrence of SBA or HEA for the 12-node networks and 60-node networks**. **(a-1)** The relationship between the total number of 2, 3, and 4-node APFL motifs and the occurrence of SBA for the 12-node pulsed neural networks. **(a-2)** The relationship between the total number of 2, 3, and 4-node APFL motifs and the occurrence of 2-channel HEA for the 12-node networks. **(b-1)** The relationship between the total number of 2, 3, and 4-node APFL motifs and the occurrence of SBA for the 60-node pulsed neural networks. **(b-2)** The relationship between the total number of 2, 3, 4-node APFL motifs and the occurrence of 2-channel HEA for the 60-node networks.

Figures

Figures

The relationship between the number of 2, 3, or 4-node APFLs and the occurrence of SBA

We also investigated the relationship between the distribution of each type of APFL and the level of SBA. In the 12-node neural networks, we found that 2-node APFLs are significantly enriched compared to 3-node or 4-node APFLs for all levels of SBA (Figure

The influence of 2, 3, or 4-node APFL motifs on inducing SBA

**The influence of 2, 3, or 4-node APFL motifs on inducing SBA. (a-1) **The number of 2, 3, or 4-node APFL motifs with respect to the level of SBA (12-node networks). **(a-2) **The occurrence of SBA along with the absence of particular APFL motifs or combinations of motifs in 12-node networks. **(b-1) **The number of 2, 3, or 4-node APFL motifs with respect to the level of SBA (60-node networks). **(b-2) **The occurrence of SBA along with the absence of particular APFL motifs or their combinations in 60-node networks.

How is SBA inhibited when each type of APFL motif is absent from the pulsed neural networks? Figure ^{-5 }± 0.0101 (mean frequency and standard deviation). Thus, the loss of more types of APFL motif gradually inhibits the occurrence of SBA. In 60-node networks, the observed trend slightly differs in that the exclusion of 2-node APFLs completely prohibits the occurrence of SBA (see the first point in Figure

A case study of the egg-laying circuit of

We investigated the egg-laying circuit of

The egg-laying circuit of

**The egg-laying circuit of C. elegans**. Each node represents a neuron or a neuron class. The arrows represent excitatory synaptic connections and the line bars denotes inhibitory synaptic connections. In this network diagram, all outer connections are excluded for simplicity.

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 _{x }
_{y }
_{x }
_{y }
_{x }
_{y }
_{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

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 10-300 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 (10-30%). 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 {_{s}, τ_{m}
_{
ax
},

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 (10-30%) maximizes the occurrence of synchronized bursting behaviors (when ER = 0.9), which is consistent with previous

Employing time-series data from multi-electrode array experiments, we identified some APFL motifs in cultured cortical networks of E18 Sprague-Dawley rats

Furthermore, we investigated the distribution of each type of APFL motif (2, 3, or 4-node) at different SBA levels. In both 12-node and 60-node networks, the 2-node 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 large-scale networks, the exclusion of 2-node APFLs almost fully prohibits the occurrence of SBA, implying that compared to other APFL motifs, 2-node 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 sub-network pattern in complex networks that occurs more frequently than in randomized networks

A feedback loop (FBL) is defined as a network motif composed of network nodes (neurons) and closed directed paths (synapses). The example network shown in Figure ^{-}
^{
1
}), where

Illustration of feedback loops and APFLs

**Illustration of feedback loops and APFLs. (a) **An example network composed of five different feedback loops. An interaction between nodes is represented by arrows to denote excitatory regulation and blunt lines for inhibitory regulation. The sub-networks colored in red denote the two APFL motifs. **(b) **Five feedback loops (FBL) contained in the network shown in (a).

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 "integrate-and-fire" neurons

For a spiking neuron _{i}
_{i }

where _{i}
_{
i
}. In the biological context, Ψ_{i }

A pre-synaptic spike at time _{i }
_{ij}
_{ij }
_{ij }
_{ij }
_{i }

The models described by (1)-(3) are referred to as SRMs

where the kernel function _{i}
_{
ij
}, and membrane dynamics _{i}

The functions describing the dynamics of a spike neuron

**The functions describing the dynamics of a spike neuron. (a)** The kernel ε_{ij}(t) describing the response of _{i}_{ax }= 50 _{s }= 3.5 _{m }= 8 **(b) **The function _{i }**(c) **The kernel _{i}_{m }in (a). **(d) **The membrane voltage _{i }_{i}

where τ_{
s
}and τ_{
m
}are time constants describing axonal transmission dynamics and membrane dynamics, respectively, and Δ_{
ax
}is axonal transmission delay. _{
ax
}) is the Heaviside step function which vanishes for _{
ax
}, and set _{
ax
}equal to 1.

One typical membrane voltage reset function is

where _{refractory}

Networks of different sizes can exhibit similar network behaviors (or dynamics) if their neurons are supplied with the same average inputs _{1}-node network as a nominal case with an allowed weight scope of [-_{max}, _{max}]. Then the weight scope of an _{2}-node network should be assigned as _{1 }= 12) was set as the nominal network. Thus, the weight scope of the other network (_{2 }= 60) was scaled by the factor (60 - 1)/(12 - 1) ≈ 5.

The typical behaviors of spike neural networks

Four typical spontaneous network behaviors appeared during the simulations: transient response activity (TRA), SBA, asynchronized bursting activities (ASBA), and HEA

where

By its definition, _{refractory}

Four possible network behaviors for a 2-node PFL motif with various synaptic efficacies

**Four possible network behaviors for a 2-node PFL motif with various synaptic efficacies**. The initial state of the network is taken as (1,1). **(a) **The scheme of a 2-node PFL motif. **(b) **Transient response activity. **(c) **Synchronized bursting activity. **(d) **Asynchronized bursting activity. **(e) **Hyper-excitable activity.

Simulation protocols

Simulations were carried out using SRMs for the randomly connected networks where all neurons were assumed to have an identical parameter set {_{s}, τ_{m}
_{
ax
}, _{ij }
_{e }
_{e }

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 4-node) and the occurrences of

Abbreviations

APFL: All-positive-interaction feedback loop; TRA: Transient response activities; SBA: Synchronized bursting activities; ASBA: Asynchronous bursting activities; HEA: Hyper-excitable activities; CR: Connective ratio; ER: Excitatory ratio; SI: Synchrony index; SRM: Spike response model.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

KHC designed the study, CYD performed the simulations, CYD and KHC analyzed the data, and CYD and KHC wrote the paper. All authors read and approved the final manuscript.

Acknowledgements

We thank Dongkwan Shin and Chaoxuan Dong for their critical reading of this paper. This work was supported by the National Re-search Foundation of Korea (NRF) grants funded by the Korea Government, the Ministry of Education, Science & Technology (MEST) (2009-0086964, 2010-0017662, and 2011-0006314). It was also supported by the WCU (World Class University) program (R32-2008-000-10218-0) through the NRF funded by MEST.