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Edited by: Ralph G. Andrzejak, Universitat Pompeu Fabra, Spain

Reviewed by: Roberto Barrio, University of Zaragoza, Spain; Syamal Kumar Dana, Jadavpur University, India

This article was submitted to Dynamical Systems, a section of the journal Frontiers in Applied Mathematics and Statistics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We investigate synchronization patterns and chimera-like states in the modular multilayer topology of the connectome of

Synchronization phenomena are widely studied across fields, from classical mechanics [

Chimera states were first reported in rings of non-locally and symmetrically coupled identical phase oscillators [

Previous works on the effect of nontrivial topologies on chimera states have involved scale-free and random networks [

The nematode

It has been found that some of

In our multilayer approach to modeling the neuronal connectivity of

Layers of the

In order to investigate the synchronization patterns of the neuronal network at hand, we will first investigate the dynamics based on a previous approach from Hizanidis et al. [

Before investigating the

Topology of the artificial

In _{1} to γ_{6}, the global level of synchronization γ of the whole network, the chimera-like index χ_{γ} as well as the metastability index λ_{γ} are shown for a large range of electrical and chemical couplings. Note that the global level of synchronization γ is computed from the pairwise Euclidean distances of the three dimensional coordinates defined by the dynamical parameters _{γ} and λ_{γ} is independent of the community structure.

Synchronization parameter scans of the designed network. Changes in the community dynamical properties and global dynamical properties as the electric and chemical coupling strengths vary. _{1} to γ_{6}. _{γ}. _{γ}. System parameters: _{0} = −1.6, _{ext} = 3.25, _{syn} = 2 and _{wl} = 0.

It can be seen that for an electrical coupling strength of _{el} = 0.4 and no chemical coupling (_{ch} = 0) the chimera-like index based on Euclidean distances attains a value of χ_{γ} ≈ 0.25. Furthermore, the metastability index for these coupling strengths λ_{γ} ≈ 0.12 is only half as large as the chimera-like index. This serves as justification why we may call the state a “chimera-like” state, in contrast to a “metastable” state where the metastability index prevails.

_{el} = 0.4 and _{ch} = 0, that is, that communities are not connected. It can be seen that especially the nodes in the larger communities 2 and 4 are much more synchronized than the nodes in the small communities. The chimera-like index is still large in a close neighborhood of this point (cf. _{4} ≈ 0.95.

Space-time plots of the designed network. _{el} = 1.76, _{ch} = 0. Keeping the chemical coupling at _{ch} = 0 and decreasing the electrical coupling to _{el} = 0.4 leads to the timeseries in panel _{m} and

The actual size of the community affects the order parameter only in an indirect way. What seems to be more important is the higher mean degree of nodes in the larger communities. In

In order to investigate the synchronization of the

The communities are first evaluated using a Multilayer-Louvain community detection algorithm (see Methods 5.4), which yields 8 communities instead of 6 as in the aggregated case discussed in the previous section (cf.

Communities of the Multilayer-Louvain

The unclear separation of edge types in the partition shows its effects when analyzing the dynamics of the system using the Hindmarsh-Rose equations (see Methods 5.2). _{γ} and the metastability index λ_{γ} for two different wireless coupling strengths. For the level of synchronization γ_{1} to γ_{8} within the individual communities (see

Synchronization parameter scans of the Multilayer-Louvain network. Changes in the global dynamical properties as the electric and chemical coupling strengths vary. _{wl} = 0) for different electrical and chemical couplings: the global level of synchronization of the whole network γ, the chimera-like index χ_{γ} and the metastability index λ_{γ}. _{γ}, and λ_{γ} for _{wl} = 0.2. The system parameters are the same as in _{wl}.

Even though the parameters used to identify chimera-like states (γ, χ_{γ}, and λ_{γ}) are significantly lower than in the model by design, the timeseries of the neuron membrane potential _{wl}, _{el} and _{ch}. The corresponding coupling strengths and synchronization parameters are noted in _{wl} = 0.0, and _{wl} = 0.2. One observes that for high electrical coupling strengths (

Space-time plots of the Multilayer-Louvain network. The system parameters are the same as in _{wl}. _{el} and _{ch} with _{wl} = 0. _{wl} = 0.2. The values of all coupling strengths are summarized in

Parameter sets used in

_{wl} |
_{el} |
_{ch} |
_{γ} |
_{γ} |
||
---|---|---|---|---|---|---|

7 (A) | 0.00 | 1.80 | 0.05 | 0.35 | 0.05 | 0.07 |

7 (B) | 0.00 | 0.50 | 0.20 | 0.15 | 0.02 | 0.02 |

7 (C) | 0.00 | 0.10 | 0.25 | 0.13 | 0.02 | 0.02 |

7 (D) | 0.20 | 1.80 | 0.05 | 0.22 | 0.04 | 0.03 |

7 (E) | 0.20 | 0.50 | 0.20 | 0.11 | 0.02 | 0.01 |

7 (F) | 0.20 | 0.10 | 0.25 | 0.10 | 0.01 | 0.01 |

Using the Multilayer-Louvain community detection approach to partition the network, one observes a system expressing different synchronization patterns depending on the interplay of the coupling strengths of the three layers. However, even though these patterns are visible in the evolution of the

In the previous section it was shown that the Multilayer-Louvain partition already leads to significant differences in the synchronization behavior of the distinct communities. Yet, as this barely becomes apparent in the chimera-like index, we investigated an alternative approach of finding communities, based on dynamical correlations between the time series of the _{m}, χ_{γ}, and λ_{γ}, see Methods 5.3) do intrinsically depend on the particular partition.

Communities of the dynamical correlation-based

Regarding the dynamical properties, the dynamical correlation-based partition leads to qualitatively similar results as the Multilayer-Louvain partition. Compare _{γ} are still obtained for high electrical couplings and small chemical couplings. Increasing the wireless coupling as in _{γ}, similar to what has been observed previously in the dynamical analysis of the Multilayer-Louvain partition. However, the value of the highest chimera-like index (χ_{γ} ≈ 0.14, obtained at _{el} = 1.96 and _{ch} = 0.04) is significantly higher for the correlation-based partition. Moreover, it is also higher than the corresponding metastability index (λ_{γ} ≈ 0.07). Therefore, we may indeed call the state “chimera-like" [

Synchronization parameter scans of the dynamical correlation-based network. Changes in the global dynamical properties as the electric and chemical coupling strengths vary. _{wl} = 0) for different electrical and chemical couplings: the global level of synchronization of the whole network γ, the chimera-like index χ_{γ} and the metastability index λ_{γ}. _{γ}, and λ_{γ} for _{wl} = 0.2. The system parameters are the same as in _{wl}. Refer to _{i}.

The reason for the high chimera-like index can be observed in the space time plots of the _{3} ≈ 0.40 and γ_{6} ≈ 0.67) are higher than for the other communities, which raises the chimera-like index. The reason for this high level of synchronization is the strong intra-community coupling of the two large communities in the electrical layer, since the electrical coupling strength is the very high in parameter set (_{el} = 1.80).

Space-time plots of the dynamical correlation-based network. The system parameters are the same as in _{wl}. _{el} and _{ch} with _{wl} = 0. _{wl} = 0.2. The values of all coupling strengths are summarized in

For time series with small electrical coupling (see

In the case of intermediate electrical, chemical coupling and no wireless coupling (see

For a full review of the different coupling strengths of to the system that lead to the time series in _{γ}, and λ_{γ}, please refer to

Parameter sets used in

_{wl} |
_{el} |
_{ch} |
_{γ} |
_{γ} |
||
---|---|---|---|---|---|---|

10 (A) | 0.00 | 1.80 | 0.05 | 0.35 | 0.11 | 0.06 |

10 (B) | 0.00 | 0.50 | 0.20 | 0.15 | 0.03 | 0.03 |

10 (C) | 0.00 | 0.10 | 0.25 | 0.13 | 0.03 | 0.02 |

10 (D) | 0.20 | 1.80 | 0.05 | 0.22 | 0.07 | 0.03 |

10 (E) | 0.20 | 0.50 | 0.20 | 0.11 | 0.02 | 0.01 |

10 (F) | 0.20 | 0.10 | 0.25 | 0.10 | 0.02 | 0.02 |

Interesting synchronization patterns were found using different modeling approaches in the multilayer network of

Following the approach of Hizanidis et al. [

Moving toward a more biologically-inspired modeling (section 3), these synchronization states are more difficult to observe. Since the edge types are not clearly separated anymore and it is therefore impossible to tune intra- and intercommunity coupling separately, the nodes within one community cannot synchronize as easily. This is especially the case when partitioning the network with the Multilayer-Louvain algorithm: the synchronization patterns are visible in the time series (see

We also discussed an alternative way to identify correlated clusters in the network, namely to sort nodes in communities according to the Pearson correlation matrix of the

Neuron functions of nodes in the dynamical correlation-based partition.

Interneuron | 10 | 7 | 32 | 11 | 5 | 16 | 8 | 0 | 89 |

Motor | 7 | 2 | 32 | 7 | 3 | 49 | 6 | 2 | 108 |

Sensory | 8 | 11 | 27 | 15 | 7 | 10 | 0 | 4 | 82 |

All | 25 | 20 | 91 | 33 | 15 | 75 | 14 | 6 | 279 |

In this context, a question could be raised regarding the multilayer nature of the studied network. Since the three layers do not share the same number of neurons (only 253 of the 279 neurons are connected by electrical gap junctions), a certain group of nodes are prone to remain desynchronized for certain combinations of _{el}, _{ch} and _{wl}. However, the strong biological interplay between synapse types is crucial to the understanding of the neuronal network as an entity [

Keep in mind that the studied three-layer network contains information only about the electrical, chemical and monoamine connections. Another layer could be added for the neuropeptide wireless network, which was not included since many neuropeptide receptors, as well as ligands for many neuropeptide receptors are unknown. Also, the distance over which neuropeptide signaling can occur is uncharacterized for many of them.

Furthermore, concerning the synchronization metric based on Euclidean distances (see section 5.3), the threshold parameter which defines the limit between synchronized and desynchronized nodes has been set to δ = 0.01 as in reference [

This work presents an approach for analyzing the complex biological network of

The gap junction and chemical synapse networks of a hermaphrodite

The electrical sub-network consists of 253 neurons and 890 synapses or gap junctions from 517 unique neuron pairs (including 3 self-connections). A total of 352 out of 517 neuron pairs have only one synapse between them, while the other 165 pairs show multiple parallel connections, with a maximum value of 23. This means that the respective symmetric (undirected) adjacency matrix has weights varying from 1 to 23 for connected neurons.

The chemical sub-network contains 253 source and 268 target neurons, the union of both sets is composed of 279 neurons, which is the total number of nodes of the modeled

The wireless sub-network of this study is restricted to the monoamine network in Bentley et al. [

The functional classification of the neurons in three categories (sensory, motor and interneurons) has also been obtained from [

For details relevant to our study on the individual neurons and their characteristics, please refer to the

We consider a network of neurons locally characterized by Hindmarsh-Rosedynamics [

where _{i} is the membrane potential of the _{i} is associated with the fast current, either Na^{+} or K^{+}, and _{i} with the slow current, for example Ca^{2+}. The parameters of Equation (1) are chosen such that _{0} = −1.6, and _{ext} = 3.25, for which the system exhibits a multi-scale chaotic behavior characterized as spike bursting [

The connectivity structure of the electrical synapses is described in terms of the Laplacian matrix _{ij} = 1 if _{ij} = 0 otherwise. ^{el} is the symmetric adjacency matrix whose elements are _{el} and its functionality is governed by the linear function

The connectivity structure of the chemical synapses is described by the adjacency matrix ^{ch}, whose elements are _{ch}. For the chosen set of parameters, |_{i}| < 2 and thus (_{i}−_{syn}) is always negative. Therefore, the chemical coupling is excitatory if _{syn} = 2. The other parameters are θ_{syn} = −0.25 and λ_{syn} = 10, following references [

The wireless connectivity structure is described by the adjacency matrix ^{wl}. It is also considered nonlinear; however, much slower than the chemical synaptic coupling [_{syn} with _{syn} = 2 and θ_{syn} = −0.25, as for the chemical coupling. Furthermore, the wireless coupling is considered as an additional disrupting signal to the synchronization of the network. It is therefore treated like excitatory chemical synapses.

We adapt an approach based on Kemeth et al. [_{1}, _{2}, …, _{N}} at every time step _{i} = (_{i}, _{i}, _{i}). For all

Two nodes _{i}(_{j}(_{i}(_{j}(_{max} is a threshold value. The value _{max} is the maximum possible Euclidean distance between a pair of nodes:

where _{max} = (_{max}, _{max}, _{max}) and _{min} = (_{min}, _{min}, _{min}) are vectors containing the maximum and minimum values of the dynamical variables for all time steps

Based on the set of Euclidean distances, we can measure the amount of spatially coherent nodes at each time step

The fraction between the number of distances within the range of the threshold value and the possible number of distances then results in the amount of synchronized node pairs. Note that the number of node pairs grows at a rate of ^{2}. It is therefore necessary to take the square root of this value, in order to make it comparable across network sizes. We call the resulting value “level of synchronization:”

If γ(_{m}(

at time _{m}(_{M} denotes the average level of synchronization at time

Similarly we can compute the metastability index:

of community _{m}(_{T} denotes the temporal mean of γ_{m}(

The subscript γ is utilized to emphasize that these parameters differ from the parameters in Shanahan [

In order to compare community partitions with different numbers of communities, it is important to know that the ranges of χ_{γ} and λ_{γ} depend on the number of communities

since the deviation from the mean is 0.5 for every community. The same considerations lead to a maximum metastability index of:

which is approximately 0.25, since the total number of time steps

and,

The communities discussed in section 3 are computed based on a multiplex Louvain community algorithm [

where _{ij} is the graph's _{i}, _{j}) = 1 if nodes _{i}, _{j}) = 0 otherwise. Therefore, the term

For multilayer networks, the modularity as defined in Equation (14) is not well suited as it does not differentiate if nodes are connected by different layers. In order to extend the modularity to multilayer applications, consider a network with _{multilayer} for a multilayer network with

where _{l} in Equation (14). Note that in the case of the considered

Community partitions of the biological

We present a heuristic approach to finding meaningful communities based on the dynamics of the system. While previous approaches aimed to find a community structure based on the topology, we propose an algorithm which partitions the network based on the nodes' correlations of the

In the hereby created matrix, every entry represents the correlation value of two time series of the two respective nodes (cf. the matrix in

Schematic description of the dynamical correlation-based community detection algorithm. In the sorted matrices red lines represent the borders between two communities.

In order to find a community partition in the correlation matrix, we employ the stochastic block model approach from the

For every parameter set (i.e. every combination of the three different coupling strengths _{el}, _{ch} and _{wl}) we obtain one correlation matrix, on which we apply the _{m}, since one node in a small community _{m}. Therefore, very small communities (especially communities consisting of only two or three nodes) can have a significantly stronger influence on the chimera-like and the metastability index than large communities. This is why we only consider community partitions with at least six nodes per community.

Another constraint applied to the partitions is a lower boundary for the level of synchronization in at least one community. The constraint is needed, because nodes do not synchronize as easily in the system based on the connectivity data (see section 3). However, a highly synchronized community is crucial to finding chimera-like states. The threshold value used to filter out partitions containing low-synchronized communities was set to γ_{thr} = 0.30. This is a reasonable compromise between reaching a high level of synchronization in at least one community and still keeping a relatively high number of partitions.

The algorithm finds 582 partitions that satisfy the constraints out of an initial set of 50 · 15 · 7 = 5250 possible partitions (_{el} ∈ [0.04, 0.08, …, 2.00], _{ch} ∈ [0.02, 0.04, …, 0.30] and _{wl} ∈ [0.00, 0.05, …, 0.30]). Subsequently, we iterate over all pairs of nodes (

As a final step, this histogram is sorted using the _{el}, _{ch}, and _{wl}. The resulting sorted histogram is shown in

Please note that the proposed algorithm is only

AP, LM, and JR had the lead analyzing the network data and performing the numerical simulations. All authors designed the study, analyzed the results, and wrote the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors would like to thank P. Orlowski for technical support.

The Supplementary Material for this article can be found online at: