3.1. Two Motif Models
By constructing the motif model of two neurons based on dynamic synapse, the synchronization phenomenon and effective link in the network are studied. The neuron node is driven by external stimulus and its amplitude intensity is
. The signal frequency is
. The discharge sequence between two neurons in different coupling connection states and the statistical distribution of two neurons in the connection state are shown in
Figure 2 and
Figure 3, respectively.
Figure 2 shows the discharge sequence between two neurons with different coupling connections.
Figure 2a shows that, in the unidirectional coupling state, such as the structure of
Figure 1(a-1), discharge sequence between neurons demonstrates a lack of consistency, and the synchronization activity between neuron
and neuron
is weak. This is because there is only one-way action between neurons and there is no feedback link. When the neurons are in the state of two-way coupling connection (i.e., feedback link exists), as shown in
Figure 1(a-2) structure and the discharge sequence shown in
Figure 2b, it can be seen that the discharge sequences between neurons are highly consistent and the pulse sequences coincide at a fixed time. This phenomenon is called isochronous synchronization.
Figure 3 is a box-line plot that quantitatively portrays the values of the
taken under different coupling connections. The upper and lower limits of the bins each indicate the upper and lower quartiles, the entire box indicates the dispersion of the
and the line in the middle indicates the median of the dataset. Through running the experimental data 100 times,
Figure 3 quantitatively describe the values under different coupling connections. For the dual coupled motif model, the
exponent is higher, i.e., it shows that it is more sensitive to the phase information carried by the input information. For the single coupling model, the
value is around 0.28. The smaller
value indicates that the phase carried by the input information has less impact on it under this structure, i.e., the synchronization of the nodes is less affected when the stimulus information changes the carried phase, which, in turn, cannot regulate the effective links in the network.
In the brain network, the
is affected not only by the phase shift of the stimulus, but also by the time delay. Therefore, this study considers the transmission delay between transmission paths and describes the effect of delay on the
. The simulation is shown in
Figure 4.
Figure 4 represents the
effect under the time delay scale of
. From
Figure 4, it is seen that the
value under single coupling fluctuates around 0.43 and does not change significantly with the increase in time delay, indicating that the phase information carried by the stimulus phase has less ability to regulate under the single coupling structure under the influence of time delay, and its coherence value
is more stable. In bidirectional coupling, the
value has a strong fluctuation between
. With the increase in time delay, the
value appears to show a trend of in-phase and inverse-to-phase changes, which shows that the synchronization of neural nodes can be regulated in-phase or inverse-to by means of time delay [
17,
18,
19]. Therefore, exploring the effect of time delay on
between coupled nodes is important to investigating the relationship between stimulus information phase shift and the selection mechanism of network paths.
3.2. Three Motif Models
In the previous section, the synchronization status of different structural links under the two motif models is analyzed, and it is concluded that the connection structure and transmission delay have influence on the synchronization characteristics. Because the structure of the three motif models is changeable, but their essence is the extension of the two motif models, the analysis of the three motif models in
Figure 1b mainly considers the synchronization state between the links of the motif model and the effective links when the structural links are missing. Therefore, we next focus on the motif model of
Figure 1(b-2,b-3) in order to analyze the synchronization state and effective link under this link.
Figure 1(b-2) structure is a connection model in which neuron
is inserted between doubly coupled neurons
and
. From the physical structure neuron
is directly regulated by neuron
and indirectly regulated by neuron
. Its discharge sequence and the
distribution are shown in
Figure 5 and
Figure 6.
Figure 5 is a discharge sequence for three motif models. Its analysis method is similar to that of two motif models. However, in
Figure 5, the discharge sequence diagrams of neurons with structural links (neuron
and neuron
) and without structural links (neuron
and neuron
) are simulated, respectively. It can be seen from
Figure 5 that there is a certain difference in the pulse sequence between neuron
and neuron
, but their discharge activities are regular and consistent. Although there is no structural link between neuron
and neuron
, there is a phase synchronization state in the discharge sequence, which is consistent with the theoretical analysis. This shows that the synchronization state between neurons is an important means of neural information transmission. Therefore, it can be shown from
Figure 5 that there is an effective transmission between neuron
and neuron
in the process of information transmission; synchronization characteristics can build dynamic effective links between neurons and these effective links do not necessarily correspond to structural links one by one, which may be the main reason for flexible path switching in brain networks.
Figure 6 is a boxplot of the three Motif models. It also conducted 100 simulation experiments, and the statistical analysis concluded that the
values between neurons
are relatively stable. The
value is maintained between
. Under this structure, the phase information of the stimulus signal has a certain promoting effect on the synchronous state, thus resulting in a “new” connected pathway
. The simulation results show that there is an effective pathway between the neuronal nodes, that the synchronization between the neural nodes can be regulated by the phase offset of the stimulus signal and that the degree of regulation depends on the size of the
value.
Under the model structure of
Figure 1(b-2), it also considers the effect of time delay on
. The simulation results are shown in
Figure 7. The value range of delay is
. Through the analysis of the simulation results, it is found that, compared with the two neuron structure links, the
of
Figure 1(b-2) model is lower. It is maintained between [0.28, 0.36] and there is a weak fluctuation phenomenon with the increase in time delay, indicating that the time delay also has a certain effect under this structure. In addition, the reason for the small change trend of
is that the increase in the number of central nodes affects the transmission of nerve impulses in the whole link, and the signal is annihilated in the bottom noise of the nervous system in the process of transmission due to the increase in the number of nodes. Therefore, the future multi-node in-depth study will consider the impact of the number of central nodes on
.
In order to more comprehensively study the synchronization and effective link between the three motif models, we next simulate and analyze the motif model of
Figure 1(b-3).
Figure 1(b-3) is based on
Figure 1(b-2); a structural link is added between neuron
and
to form a double coupling between neuron
and
(with feedback link). The discharge sequence and
statistical diagrams of the structure of
Figure 1b-3 are shown in
Figure 8 and
Figure 9, respectively.
Figure 8 and
Figure 9 show discharge sequences and
statistical diagrams of the motif model of
Figure 1(b-3), respectively. According to the consistency of the discharge sequence between neurons in
Figure 8, no matter whether there is a structural link between neurons—for example, there is a structural link between neurons
and
, and there is no structural link between neurons
and
—neurons achieve isochronous synchronization in the firing process; that is, an effective link is formed between any two neuron nodes. In addition, it can be seen that the
data distribution in the box diagram of
Figure 9 is more stable than that of
Figure 6, but the fluctuation of the
value of neuron
is more obvious from the box diagram, indicating that it is more easily regulated by the phase information of stimulus signal. Therefore, it is found that the synchronization between neural nodes is an important index to form an effective link, and the synchronization between neurons is regulated by many factors.
The relationship between the
and the transmission delay of motif model 1(b-3) is shown in
Figure 10. It can be seen that the value of
changes obviously with the increase in delay, and that its value fluctuates greatly in the range of
. This indicates that the stimulus node is more vulnerable to in-phase or out-of-phase stimulation; that is, it is more sensitive to the phase information of external stimulus signal. In addition, it can be seen that when the delay is in the range of
, the
value changes periodically with the change of delay, which reflects that the information transmission between nodes is related to phase locking. In this structure,
will dynamically adjust the intensity of synchronization with the size of the time delay, so that the synchronization between nodes has a dynamic change, and then affect the connection state of effective links in the network. This indicates that the brain network can reorganize nodes in different time dimensions with synchronous state, and then achieve a variety of effective links to facilitate task efficiency.
3.3. Regulation of Path Selection by Different Phases of Stimulus Signal
For the study of the motif model, it can be found that in the motif model with two or three nodes connected to each other, the synchronization between nodes shows a complex dependence on the phase offset and transmission delay of the stimulus signal. Therefore, in order to extend the analysis to the more complex network model, we next construct a network model with 20 delay nodes to simulate the brain, and each network node is a neuron group of a small-world network with 100 neuron nodes. In addition, there is a double coupling state between the nodes.
Figure 11 shows the schematic diagram of the relationship between the
values of some nodes and the central nodes. It can be seen that the
values of the stimulus node pairs
are different between the paths passing through one central node and those passing through two nodes, which indicates that there is a certain relationship between the path choice between the stimulus information pairs and the number of central nodes. Therefore, the box diagram of
Figure 12 is used to describe the influence of the number of central nodes on the
index between node pairs. As can be observed from the boxplot, when and only when there is one pivot node, its
value is distributed in
. As the number of pivot nodes increases, the value of
and the distribution interval increases; when the number of pivot nodes is four, the value of
is distributed between
, indicating that the data of
fluctuates more, but when the number of pivot nodes is five or the number of pivot nodes is greater than five, the value of
decreases and the fluctuation range becomes smaller. From the perspective of the significance of
, when the number of hub nodes increases to a certain number, the
values between nodes in the network can be regulated by the phase shift of the stimulus information. This shows that the path selection strategy in the network is affected not only by the phase of stimulus information, but also by the number of central nodes. Therefore, in the complex link structure composed of twenty nodes, limiting the maximum number of central nodes to five, which means that the path of up to six nodes is mainly searched in the path search, and 95% confidence interval is used for evaluation.
In addition to the number of central nodes, the synaptic strength
of node pairs also plays an important role in synchronization characteristics. By obtaining the
values at different synaptic strengths
, it can be found that there is a close relationship between the effective link between stimulated nodes and the synaptic strength. By fitting the data, there is a high degree of coherence between the
values
at different strengths and different nodes. In addition, it can be seen by the simulation results in
Figure 13 that as the synaptic strength
increases,
value becomes smaller and smaller, which indicates that when the coupling strength between the nodes gradually increases, resulting in a strong structural link between the nodes, it is more difficult to regulate the synchronization between the nodes using the phase information of the stimulus information. This indicates that there should be a general weak coupling phenomenon between the nodes of the brain network.
In order to evaluate the path transmission capability between a pair of excited nodes in the network, the model gives the proposed metric of the degree of activation of an effective path and the degree of interaction between pairs of information transmission nodes at a given stimulus phase offset
, given the physical link determination. It is assumed that
is denoted as the set of all paths between node pairs
, thus setting the maximum activation state (pathway activation,
) of a path through
pivot nodes at a stimulus phase offset of
.
denotes the coherence of a path between a pair of stimulus node pairs
after passing through different pivot nodes, which measures the transmission capability of the whole path or the degree of effective path activation and is defined as Equation (15):
The mechanism of information transmission between network nodes must have a success or failure state. In Equation (15), is used to measure the success rate of information transmission between node pairs, where denotes the coherence value of the phase deviation of the stimulus information passed between node pairs , and to retain the link selection characteristics that the short path is better than the long path in the study process.
In addition, to determine the relationship between the information transmission path between a particular node pair
and the offset phase
of the stimulus information, here, we give the preference selection index (pathway-phase-selectivity,
) of a specific pathway when the stimulus information carries different phase information, as defined in Equation (16):
where
is determined by the activation state index
of a path between node pairs. The
is used to measure the activation capability and transmission capability of a path given any path
between node pairs
in the network with different phase information, i.e.,
.
In addition, the selection strategy of the path with the largest transmission capacity between nodes is measured by using the phase information of the excitation signal, giving the quantitative index of the path selection mechanism (pathway-switching-selectivity,
), which is expressed in the form of (17):
where the
metric measures the stimulus phase difference at
and selects the path with optimal transmission capability in the path set
between node pairs
.
represents the path set between a pair of stimulus nodes
and
. It is assumed that
represents the path with the strongest activation index in the path concentration
, and
represents the path of the second strongest activation index
in the path concentration. In addition, when
, it means that the
path is more active in phase under the
condition, and when
, it means that the
path is more active.
In the process of analyzing the path activation index , all the paths between the five delay nodes are selected for evaluation, i.e., . The phase information carried by the stimulus information is used to modulate these node pairs, and then values under different phase offsets are calculated. Due to the existence of obvious or obscure phase relations, the simulation results are presented in polar coordinates. The radius in the figure indicates the magnitude of the value and the angle indicates the offset between different stimulus phases. The analysis is also performed for any of the pathways in for a given node, and the path selectivity index is given to explore the phase dependence properties under a specific pathway.
The blue curve in
Figure 14 indicates the path with the strongest
between two node pairs, and the orange color indicates the second strongest path. For
Figure 14a, the results show that both the strongest path
value and the second strongest path
value between a pair of nodes
are relatively stable and do not change significantly with the phase shift of the stimulus information.
Figure 14b indicates that the strongest path
value between node pair
changes with the stimulus phase offset, indicating that the phase information carried by the stimulus information will have some influence on the
value; that is, the phase information can be used to modulate the network and thus find the optimal path.
Figure 14c shows that the strongest path and the second strongest path
values between the nodes to
change almost together, i.e., it shows that the
values will be modulated by the role of phase information, and the selection of the path by the information flow can be either of the two paths, indicating that both paths can characterize the information.
Figure 14d shows that the phase information carried by the stimulus information in the node pair
has obvious modulation on the
values of the strongest path and the second strongest path (blue in the figure indicates
, orange indicates
), which mainly shows that the strongest path and the second strongest path have a preferential choice of phase between them under different phase offsets, i.e., under different phase offsets of the stimulus information.
When studying the path activation index
,
Figure 14 analyzes the relationship between the strongest path and the second strong path with respect to the stimulus phase in the node
path set
. Based on this, it adopts the
to analyze any pathway in the path set
, and then explore the phase dependence of the specific pathway. Through the statistics of the data results, the
values of all node pairs in the network composed of five nodes are drawn, and the phase selectivity histogram is shown in
Figure 15.
Figure 15 analyzes all paths between node pairs
. Statistical analysis of the data results is performed to plot the values of all node pairs between
. The results displayed are consistent with the conclusions of the theoretical analysis; that is, the larger value of
indicates that the path activation index is unstable and vulnerable to the modulation and influence of the phase information of the stimulus information.
Figure 15 shows that the value of
is between 0 and 1, and the larger the value of
, the more the path between nodes are affected by the phase shift of the stimulus information. After counting the path values between 180 node pairs,
,
,
and
correspond to the subplots in
Figure 14. It can be seen that when the change of
value is small, the value of
is closer to 0, and when the change of
value is larger, the value of
converges to 1. Through the statistical analysis, there exists a large number of paths between the network node pairs that all have the
effect, i.e., in a specific path state, they will have a certain preference depending on the phase information of the stimulus information, so this simulation shows that the phase information carried by the stimulus information can be used to determine the path based on the phase preference.
Theoretical analysis shows that in the information transmission, there should be nodes in the network to control the phase offset of the stimulated information so that they can preferentially choose a path as the main path of the network information flow transmission. i.e., as the optimal path to represent the information.
Therefore, this study gives the conditions under which the information flow will choose the path with the optimal information representation path (
) under the condition that the phase offset of the stimulus information is determined:
where
denotes the strategy for selecting the optimal path between node pairs when the stimulus information phase
is perturbing the network. Its physical meaning indicates that in the path set of a pair of network node pairs, there must exist one or more optimal paths to enable them to have optimal transmission capability and more accurate characterization capability.
By analyzing the
values of the strongest path
and the sub-strong path
, it can be found that a large number of paths between the nodes have phase preference. Therefore, we next count the phase preference of the strongest path
and the sub-strong path
. The statistical histogram is shown in polar coordinates. The angle represents the phase shift of the stimulus information and the radius represents the frequency of path selection. The result is shown in
Figure 16.
Figure 16a shows the phase preference exhibited by the strongest path between a pair of nodes with phase shift,
, which is significantly different from the uniform distribution. It indicates that there is a significant peak at
, showing that the strongest path has a strong dependence on the phase information.
Figure 16b shows the preference exhibited by the phase shift under the next strongest path, which is
with some differences from the uniform distribution. Therefore, it can be found that the path with the strongest path activation index
has a similar preference to the coherence
between a pair of stimulated nodes, which indicates that using the phase information of the stimulus signal as a path selection switch for the information flow in the network is a feasible way. In addition, the present findings show that the phase offset of the stimulus information selects the path with the maximum transmission capacity (path activation index
is maximum) when choosing the path, finding the optimal information representation loop. In other words, in brain-like intelligent networks, the functional loop can be switched by modulating the phase information carried by the stimulus information to achieve different functional tasks.
The phase preference between the strongest path and the second strongest path between any pair of nodes is analyzed in
Figure 16. On this basis, we continue to count the
values of 180 node pairs (indifference filtering) under the disturbance of stimulus information with different phase differences in order to obtain the normalized path selection histogram between the strongest path and the second strongest path. By normalizing the square root of the
value, the simulation results in
Figure 17 are obtained.
Figure 17 measures the modulation effect of phase information on the node-to-node paths by calculating the standard deviation of
.
,
,
and
in
Figure 17 correspond to the subplots in
Figure 14; when the value of
is positive, it means that the path switching index is low and the strongest path
is more active, indicating that there is no need to choose other paths; when the value of
is negative, it means that the path switching index is high, and the optimal path can be switched by stimulus phase modulation.
In this study, through the analysis of all node pairs in a network of twenty nodes, it can be found that a significant portion of the paths between node pairs (about 42%) can be switched between the strongest path and the second strongest path by phase shifting of stimulus information. The simulation results indicate that such switching is one of the means of achieving structural robustness in the network and that a similar mechanism can be used in brain-like networks to accomplish the switching of functional loops and thus achieve optimal representation of information.