# Macroscopic Cluster Organizations Change the Complexity of Neural Activity

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## Abstract

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## 1. Introduction

- Construct neuron groups consisting of spiking neurons that have weighted connections to randomly selected neurons in the same neuron group (intraconnections). We assumed that a neuron group and the intraconnections inside it correspond to a brain region and the intraconnections in the regions, respectively.
- Determine the initial macroscopic network structure of neuron groups (fundamental network) based on the WS model. Then, if an edge exists between two neuron groups in the fundamental network, construct synaptic connections from the neurons in the group to the neurons in the other group (interconnections). We assumed that the average of interconnections between neuron groups correspond to the long-ranged interconnectivity between brain regions (synaptic network, Figure 1b).
- Apply a plasticity rule to synaptic weights and self-organize the network (Figure 1c). If an edge does not exist between two neuron groups in the fundamental network, the weights between them remain zero to keep the given small-world structure.
- Analyze the complexity of the activity and structural properties of the self-organized synaptic network using MSE and complex network theory to show their relationship (Figure 1d).
- Investigate the frequency characteristics, firing rate, and intraconnections in each neuron group to explore the possible mechanisms of decrease in the complexity of neural activity.

## 2. Materials and Methods

#### 2.1. Spiking Neural Network Model

#### 2.1.1. Neuron Model

#### 2.1.2. Construction of Fundamental Network Using the Watts and Strogatz Model

- Construct a lattice (regular) network, where each neuron group is connected to k neighboring neuron groups (Figure 2a). the lattice network has numerous clusters and a long path length.
- Randomly rewire each edge according to a rewiring probability, ${p}_{WS}$. This creates a shortcut between neuron groups, as shown by the red line in Figure 2b. The network structure becomes random, and the number of clusters and path length decrease as ${p}_{WS}$ increases (Figure 2c). The typical value of ${p}_{WS}$ used to construct the small-world network is between $0.01$ and $0.1$.

#### 2.2. Parameters and Simulation Setting

#### 2.3. Analysis Method for Neural Activity and Network Structure

#### 2.3.1. Analysis Method for Complexity of Neural Activity

#### 2.3.2. Analysis Method for Neural Activation in Neuron Groups

#### 2.3.3. Complex Network Analyses of Structural Properties

- Clustering coefficient: The proportion of connections with the shape of a closed triplet over all possible combinations of triplets formed by three nodes in a network. This is defined as follows:$${C}_{i}=\frac{\mathrm{number}\phantom{\rule{0.166667em}{0ex}}\mathrm{of}\phantom{\rule{0.166667em}{0ex}}\mathrm{closed}\phantom{\rule{0.166667em}{0ex}}\mathrm{triangles}}{\mathrm{number}\phantom{\rule{0.166667em}{0ex}}\mathrm{of}\phantom{\rule{0.166667em}{0ex}}\mathrm{possible}\phantom{\rule{0.166667em}{0ex}}\mathrm{triangles}}.$$
- Average shortest path length: the shortest path length is defined as the minimum number of steps required to pass from one node to another node in a network [26].
- Degree centrality: it refers to the number of connections of a node [27].

## 3. Results

- The analysis of MSE showed that each neuron group had different levels of the complexity of neural activity and the average complexity of all neuron groups decreased if the fundamental network had small ${p}_{WS}$ (Section 3.1).
- The analysis of neural activity in neuron groups with different values of MSE showed that a neuron group with low complexity exhibited increased signal amplitude in two frequency bands (20–40 and 40–60 Hz) of neural activity (Section 3.2).
- The complex network analyses for each neuron group showed that the local over-connectivity (the clustering coefficient and degree centrality were high) and complexity of a neuron group had a negative relationship (Section 3.3).

#### 3.1. Relationship between MSE and WS Model

#### 3.2. Neural Activities in Neuron Groups with Different Levels of Complexity

#### 3.3. Relationship between Neural Activity and Structural Properties

## 4. Discussion

#### 4.1. Hypothetical Mechanism of Low Complexity because of Local Over-Connectivity

- The firing rates of neurons in a neuron group with local over-connectivity increases because of excessive input from the connected neuron groups (see Figure 7). As a result, to maintain a certain amount of the activities of neurons, strengthened intraconnections from excitatory to inhibitory neurons and weakened intraconnections between excitatory neurons are self-organized by STDP (see Figure 8).
- The increase in the strength of intraconnections to inhibitory neurons induces the oscillation of excitatory neurons, which increases the intensity of the several specific frequency components of neural activity (see Figure 5).
- The signals of specific frequency components become robust. As a result, complexity decreases (Figure 6).

#### 4.2. Relationship with Studies on ASD

#### 4.3. Limitations and Future Work

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**Multiscale entropy (MSE)-based complexity curves of each neuron group in a synaptic network. (

**a**) lattice network (${p}_{\mathrm{WS}}=0.0$); (

**b**) a small-world network (${p}_{\mathrm{WS}}=0.1$); (

**c**) a random network (${p}_{\mathrm{WS}}=1.0$). The y-axis indicates sample entropy, and the x-axis indicates scale factor $\u03f5$.

**Figure A2.**The differences of peak amplitude of spontaneous neural activity in some frequency bands between neuron groups with low and high complexity when ${p}_{WS}=0.0$. We used 10 neuron groups with high and low complexity in each simulation as comparison data. The number on above each violin plot denotes the average value for ten simulations. Wilcoxon signed-rank test was used for statistical test. (

**a**) amplitude in the 20–40 Hz band (Wilcoxon signed-rank test, statistic = 6.0, p-value = 4.6706 × ${10}^{-18}$); (

**b**) amplitude in the 40–60 Hz band (Wilcoxon signed-rank test, statistic = 11.0, p-value = 5.4302 × ${10}^{-18}$).

**Figure A3.**Relationship between the connectivity structure and the complexity of neural activity. Each marker corresponds to a neuron group in the network, and its color indicates the summation of the sample entropy for all 80 scale factors. The x-axis indicates the degree centrality, and the y-axis indicates the clustering coefficient.

**Figure A4.**Relationship between the peak frequency and the complexity of neural activity. (

**a**) relationship in the 0–20 Hz band; (

**b**) relationship in the 20–40 Hz band; (

**c**) relationship in the 40–60 Hz band. Each marker corresponds to a neuron group in the network, and its color indicates the summation of the sample entropy for all 80 scale factors. The x-axis indicates the peak frequency of the neural activity, and the y-axis indicates the amplitude. Figure shows that the amplitude in increases as the complexity of neural activity decreases in 20–40 Hz and 40–60 Hz bands. Furthermore, the peak frequency increases as the complexity of neural activity decreases in the 40–60 Hz band. However, amplitude decreases as the complexity of neural activity decreases in the 0–20 Hz band. This result indicates that the robust frequency components in neural activity of the neuron groups with low complexity shifted to the high frequency bands (20–40 Hz and 40–60Hz) from the low frequency bands (0–20 Hz).

**Figure A5.**Relationship between the connectivity structure and the firing rate of excitatory and inhibitory neurons. Each marker corresponds to a neuron group in the network, and its color indicates the average firing rate of excitatory and inhibitory neurons. The x-axis indicates the degree centrality, and the y-axis indicates the clustering coefficient. (

**a**) relationship between structural properties and firing rate of excitatory neurons; (

**b**) relationship between structural properties and firing rate of inhibitory neurons.

**Figure A6.**Relationship among the weight of intraconnection, structural properties, and complexity of neural activity. (

**a**) relationship among the weight of intraconnection from excitatory to excitatory neuron, clustering coefficient based on the interconnection, and complexity; (

**b**) relationship among the weight of intraconnection from excitatory to inhibitory neuron, clustering coefficient based on the interconnection, and complexity; (

**c**) relationship among the weight of intraconnection from excitatory to excitatory neuron, degree centrality based on the interconnection, and complexity; (

**d**) relationship among the weight of intraconnection from excitatory to inhibitory neuron, degree centrality based on the interconnection, and complexity. Each marker corresponds to a neuron group in the network, and its color indicates the summation of the sample entropy for all 80 scale factors. The x-axis indicates the average of weight of interconnection, and the y-axis the structural properties of interconnection.

**Figure A7.**Relationship between shortest path length and complexity for each neuron group. The x-axis is the shortest path length, and the y-axis is the summation of the sample entropy for all 80 scale factors.

## Appendix B. Complexity of Neural Activity without STDP

**Figure A8.**Relationship between the connectivity structure without STDP and the complexity of neural activity with the tonic input. Duration for tonic input was set as 100 s. Here, we used the same initial weights of the synaptic networks in Figure A3 and fixed the weights during the tonic input. Each marker corresponds to a neuron group in the network, and its color indicates the summation of the sample entropy for all 80 scale factors. The x-axis indicates the degree centrality, and the y-axis indicates the clustering coefficient.

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**Figure 1.**Hypothesis and assumptions about relationships among the fundamental network, synaptic network and the complexity of neural activity in this study. The black node and empty black circle represent a neuron group and a neuron in the group, respectively. The black and green lines indicate an intraconnection between neurons in a neuron group and an interconnection between neurons in different neuron groups, respectively. The red and blue dots show neuron groups with and without high clustering coefficient and high shortest path length, respectively.

**Figure 2.**Overview of the spiking neural network model. A network is created using 100 neuron groups with macroscopic connections between neuron groups based on the Watts and Strogatz model [21]. The black nodes and green edge represent the neuron groups and macroscopic connections, respectively. (

**a**) a lattice network where each node connected with neighboring nodes has local over-connectivity. All connections are rewired with rewiring probability ${p}_{WS}$, and larger ${p}_{WS}$ yields more random network; (

**b**) a small-world network with a large number of clusters and shorter path length compared with other networks; (

**c**) a random network where nodes are completely randomly connected to each other; (

**d**) each neuron group contains 800 excitatory (red circles) and 200 inhibitory (blue circles) spiking neurons, and each neuron has intra- (black arrow) and inter-connections (green arrow).

**Figure 3.**Time schedule for simulation. Each colored area indicates the period for which the event occurred. Neural activities during 1110 s to 1200 s were analyzed to determine the relationship among the structural properties of the synaptic network and neural activity.

**Figure 4.**Relationship between sample entropy and ${p}_{\mathrm{WS}}$ of the WS model. The x-axis in all graphs represents ${p}_{\mathrm{WS}}$. (

**a**–

**f**) average sample entropy of all neuron groups with ten independent simulations at scale factors ($\u03f5$ in Equation (1)) of multiscale entropy (MSE) at 1, 10, 20, 40, 60, and 80. The error bars indicate the standard deviation.

**Figure 5.**Amplitude of each frequency spectrum sampled from the local averaged potentials of the 10 neuron groups with low complexity (blue) and high complexity (red) in a network. The peak envelopes is used to plot the curve in the figure. Color curves and color-shaded areas represent average and standard deviation values for ten simulations, respectively. (

**a**) the lattice network (${p}_{\mathrm{WS}}=0.0$) during self-organization by spike-timing-dependent plasticity (STDP) (0–100 s); (

**b**) the random network (${p}_{\mathrm{WS}}=1.0$) during self-organization by STDP (0–100 s); (

**c**) the lattice network (${p}_{\mathrm{WS}}=0.0$) after self-organization by STDP (1100–1200 s); (

**d**) the random network (${p}_{\mathrm{WS}}=1.0$) after self-organization by STDP (1100–1200 s).

**Figure 6.**Relationship between the connectivity structure and complexity of neural activity. Each marker corresponds to a neuron group in the network, and its color indicates the summation of the sample entropy for all 80 scale factors. The x-axis indicates the degree centrality, and the y-axis indicates the clustering coefficient.

**Figure 7.**Relationship between the connectivity structure and firing rate of excitatory and inhibitory neurons. Each marker corresponds to a neuron group in the network, and its color indicates the average firing rate of excitatory and inhibitory neurons. The x-axis indicates the degree centrality, and the y-axis indicates the clustering coefficient. (

**a**) relationship between structural properties and firing rate of excitatory neurons; (

**b**) relationship between structural properties and firing rate of inhibitory neurons.

**Figure 8.**Relationship among the weight of intraconnection, structural properties of the synaptic network, and complexity of neural activity. (

**a**) relationship among the average weight of intraconnections from excitatory neurons to excitatory neurons, clustering coefficient based on the interconnection, and complexity; (

**b**) relationship among the average weight of intraconnections from excitatory neurons to inhibitory neurons, clustering coefficient based on the interconnection, and complexity; (

**c**) relationship among the average weight of intraconnections from excitatory neurons to excitatory neurons, degree centrality based on the interconnection, and complexity; (

**d**) relationship among the average weight of intraconnections from excitatory neurons to inhibitory neurons, degree centrality based on the interconnection, and complexity. Each marker corresponds to a neuron group in the network, and its color indicates the summation of the sample entropy for all 80 scale factors. The x-axis indicates the average weight of interconnection, and the y-axis the structural properties of interconnection.

Parameters | Values | Descriptions | Notes |
---|---|---|---|

${D}_{\mathrm{intra},\mathrm{exc}}$ | [0, 20] | Transfer delay of excitatory synapse in neuron group | (uniform dist., ms) |

${D}_{\mathrm{intra},\mathrm{inh}}$ | 1 | Transfer delay of inhibitory synapse in neuron group | (ms) |

${w}_{\mathrm{init},\mathrm{exc}}$ | 6.0 | Initial weight of excitatory synapse | - |

${w}_{\mathrm{init},\mathrm{inh}}$ | −5.0 | Initial weight of inhibitory synapse | - |

${w}_{\mathrm{upper}}$ | 10.0 | Maximum value of weight | - |

${N}_{\mathrm{E}}$ | 800 | Number of excitatory neurons in a neuron group | - |

${N}_{\mathrm{I}}$ | 200 | Number of inhibitory neurons in a neuron group | - |

N | 1000 | Number of neurons in a neuron group | $={N}_{E}+{N}_{I}$ |

${C}_{\mathrm{intra}}$ | 100 | Number of intraconnections of a neuron | - |

${D}_{\mathrm{inter},\mathrm{exc}}$ | [10, 30] | Transfer delay of excitatory synapse between neuron groups | (uniform dist., ms) |

${N}_{\mathrm{group}}$ | 100 | Number of neuron groups | - |

k | 6 | Number of edges for each neuron group | - |

${C}_{\mathrm{inter}}$ | 3 | Number of interconnections of a excitatory neuron | - |

${p}_{\mathrm{WS}}$ | [0.0, 1.0] | Rewiring probability | - |

${t}_{\mathrm{step}}$ | 1 | Time step | (ms) |

${T}_{\mathrm{total}}$ | 1200 | Total simulation time | (s) |

${T}_{\mathrm{tonic}}$ | 1100 | Time length of tonic input | (s) |

${T}_{\mathrm{STDP}}$ | 1000 | Time length of self-organization through STDP | (s) |

${N}_{\mathrm{sim}}$ | 10 | Number of independent simulations | - |

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## Share and Cite

**MDPI and ACS Style**

Park, J.; Ichinose, K.; Kawai, Y.; Suzuki, J.; Asada, M.; Mori, H.
Macroscopic Cluster Organizations Change the Complexity of Neural Activity. *Entropy* **2019**, *21*, 214.
https://doi.org/10.3390/e21020214

**AMA Style**

Park J, Ichinose K, Kawai Y, Suzuki J, Asada M, Mori H.
Macroscopic Cluster Organizations Change the Complexity of Neural Activity. *Entropy*. 2019; 21(2):214.
https://doi.org/10.3390/e21020214

**Chicago/Turabian Style**

Park, Jihoon, Koki Ichinose, Yuji Kawai, Junichi Suzuki, Minoru Asada, and Hiroki Mori.
2019. "Macroscopic Cluster Organizations Change the Complexity of Neural Activity" *Entropy* 21, no. 2: 214.
https://doi.org/10.3390/e21020214