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 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 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 1 shows the relationships among CR, ER, and the occurrence of SBA in the 12-node networks (Figure 1, a-1, a-2, a-3) and the 60-node networks (Figure 1, b-1, b-2, b-3). The mean and standard deviation of the occurrence of SBA were calculated over the 1,000 spike neural networks in each possible combination of CR and ER. The interesting profiles concerning the relationship between CR and the occurrence of SBA are found when the ER was equal to 0.9, which coincides with the experimental observation in cultured neural networks [25]. Figures 1 a-3 and b-3 show an optimal CR of 15% in the 12-node networks and 10% in the 60-node networks, respectively. This relation between CR and SBA suggests that the direction of evolutionary selection of CRs (not only in artificial pulse neural networks but also in in vitro neural networks) is to maximize the possibility for synchronized bursting behavior by networks.

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 [2]. SRM simulations (2, 3, and 4-node small networks) indicated that only those networks containing APFLs could produce SBA. Other networks containing only negative feedback loops or double-negative positive feedback loops (referred to the section entitled "Definition of network motifs and feedback loops" for its definition) and those without any feedback loop, could not generate SBA irrespective of synaptic efficacy. We were intrigued by these observations and further investigated the relationship between APFLs and the occurrence of SBA in larger-scale networks. Figure 2 provides detailed descriptions of this relationship in both 12-node and 60-node networks. Statistical tests on these relationships were carried out for all 1,000 simulations on the same lattices, ERs (0:0.05:1) × CRs (0:0.1:1), shown in Figure 1 a-1, a-2 and Figure 1 b-1, b-2.

Figures 2 a-1 and a-2 show the correlation between two typical network behaviors, SBA and hyper-excitable activity (HEA), and the total number of 2, 3, and 4-node APFL motifs in 12-node pulsed neural networks (see Figure 2 a-1, a-2). 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 12-node networks), and finally returns to zero (after 37 APFLs in the 12-node networks, see Figure 2 a-1). However, the occurrence of HEA always increased with the increase in the number of APFL motifs (see Figure 2 a-2). If the number of APFLs exceeded 37 in a 12-node network, the occurrence of 2-channel 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 hyper-excitable. 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 60-node networks (see Figures 2 b-1 and b-2). The maximum 2-channel SBA occurrence was 727, corresponding to 400 APFL motifs (2, 3, 4-node) in the 60-node networks. When the number of 2, 3, 4-node APFL motifs exceeded 900, HEA fully dominated, and all other behaviors including SBA vanished from the 60-node networks. Notably, the first points (mean = 0 and standard deviation = 0) in Figures 2 a-1 and b-1 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 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 3 a-1). Figure 3 b-1 shows the relationship between the number of 2, 3, or 4-node APFLs and the occurrence of SBA for 60-node pulsed neural networks. The number of 2-node APFLs significantly exceeds the number of 3-node or 4-node APFLs only when the occurrence of SBA exceeds a high level (1,200 events). Therefore, the number of 2-node APFLs dominates the total number of APFLs subject to a high level of SBA occurrence in both small- and large-scale networks.

How is SBA inhibited when each type of APFL motif is absent from the pulsed neural networks? Figure 3 a-2 shows the occurrence of SBA when one type or a combination of APFL motifs is not present in the 12-node 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 6-node 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 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 3 b-2). This fact implies that 2-node APFLs may function dominantly in the inhibition of SBA in large-scale networks, compared with 3-node and 4-node APFLs.

A case study of the egg-laying circuit of C. elegans

We investigated the egg-laying circuit of C. elegans composed of 11 neurons or neuron classes for a pilot study of real neural networks [32, 33]. We identified that this circuit contains three two-node APFLs as indicated by the bi-directional red arrows (Figure 4). In addition, we found that the CR of this circuit is 17.3% and the ER of it is 10.5%.

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. Two-sample t-test 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 egg-laying 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 Behrens-Fisher problem [35]. The t-test statistic is $T=\frac{\bar{x}-\mathit{ȳ}}{\sqrt{s_x^2/n_x+s_y^2/n_y}}$ ~t(n _x + n _y - 2) where s _x = 4.3695 and s _y = 6.9421 are the standard deviations of x and y; $\bar{x}=1.0131$ 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 p-value for the null hypothesis $H_0:\bar{x}\ge \mathit{ȳ}$ is less than 0.0001 which is much less than the significance level α = 0.05. Therefore, the null hypothesis H₀ should be rejected and the alternative is accepted.

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 [38–41]. Here, this concept was extended to a more general definition. A motif refers to any sub-network 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.

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 5a contains five FBLs (Figure 5b). FBL1, FBL2 and FBL3 are positive feedback loops (PFLs); FBL4 and FBL5 are NFLs. The sign of a feedback loop is determined by (^{- 1} ), where q is the total number of inhibitory interactions contained in the loop. Moreover, if a positive feedback loop contains only positive interactions like FBL1 and FBL2, it is referred to as an all-positive-interaction feedback loop (APFL).

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 [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 the kernel function e(s) mimics the dynamic from the local noise current stimulation to the membrane voltage of neuron i. As usual, several typical mathematical formulations were adopted (illustrated in Figure 6) to describe refractoriness Ψ _i , post-synaptic potential ε _ij , and membrane dynamics e _i . For instance, let

${\epsilon }_{ij}\left(t\right)=\frac{1}{1-\left({\tau }_s/{\tau }_m\right)}\left[exp\left(-\frac{t-{\Delta }_{ax}}{{\tau }_m}\right)-exp\left(-\frac{t-{\Delta }_{ax}}{{\tau }_s}\right)\right]H\left(t-{\Delta }_{ax}\right)$

(5)

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 n₁-node network as a nominal case with an allowed weight scope of [-W_max, W_max]. Then the weight scope of an n₂-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 (n₁ = 12) was set as the nominal network. Thus, the weight scope of the other network (n₂ = 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 $S{S}_{xy}=\sum _{k=1}^{n}xy-\frac{\left(\sum _{k=1}^{n}x\right)\left(\sum _{k=1}^{n}y\right)}{n}$.

By its definition, r is a value in the range of [0, 1]. If r exceeds a threshold (0.7 was used in our simulations), the two spike trains are considered synchronized with each other. Otherwise, they are considered asynchronized. In the case of HEA, all nodes permanently fire with an interval of absolute refractory period T _refractory . Notably, hyper-excited spike trains also have a high SI, which is quite similar to SBA. However, they fire as closely as possible without any perceivable intermittence during their final steady state. Figure 7 illustrates these four typical activities using a simple 2-node PFL motif. For a larger-scale network with n nodes, k-channel SBAs (2 ≤ k ≤ n and k ∈ ℤ) can be calculated by the same criteria. For simplicity, only 2-channel SBA was selected in this study to represent the level of k-channel SBAs, considering that measurements greater than 2-channel do not change the final results and major conclusions of the study. Similarly, 2-channel HEA was selected in to represent the level of k-channel HEAs either.

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 m-m _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., $\frac{m}{n\left(n-1\right)}$. Here, only a single connection between two different neurons is allowed for simplification. Hence, the possibility of self-connection 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 4-node) and the occurrences of k-channel 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 2-channel SBA and 2-channel HEA were evaluated and used to measure the levels of synchronization and hyper-excitation in networks. The correlation between the occurrence of 2-channel SBA or 2-channel 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.