A model of time-dependent structural plasticity for the synchronization of neuron networks is presented. It is known that synchronized oscillations reproduce structured communities, and this synchronization is transient since it can be enhanced or suppressed, and the proposed model reproduces this characteristic. The evolutionary behavior of the couplings is comparable to those of a network of biological neurons. In the structural network, the physical connections of axons and dendrites between neurons are modeled, and the evolution in the connections depends on the neurons’ potential. Moreover, it is shown that the coupling force’s function behaves as an adaptive controller that leads the neurons in the network to synchronization. The change in the node’s degree shows that the network exhibits time-dependent structural plasticity, achieved through the evolutionary or adaptive change of the coupling force between the nodes. The coupling force function is based on the computed magnitude of the membrane potential deviations with its neighbors and a threshold that determines the neuron’s connections. These rule the functional network structure along the time.

The human brain comprises approximately

Synchronization is a phenomenon that occurs when units of a set interact dynamically and adjust some of their properties to arrive at simultaneity in time. It is intrinsic, from the highest levels of organization: the world economy, the stock market, and ecological systems [

From the moment when oscillatory responses were discovered in the visual cortex of cats and between areas of the human brain [

Models of neural networks have been proposed to elucidate the interaction’s laws between regions of the brain’s functional networks, using synchronization theory. Such models have included characteristics and topological properties of the brain network, and they have shown that they could reach synchronization. For example, Kuramoto’s generalized model has helped to describe how frequencies and synaptic plasticity affect the synchronization of the neural network in a more realistic way [

In the study of the brain, one of the goals is to find out how the structural network, relatively fixed, produces the functional network’s evolutive patterns at the same time reaches synchronization [

In this work, we propose a model that reproduces the evolutionary dynamics of a functional network of neurons from a fixed structural network. It is essential to mention that there are several possible terms within the category of temporal networks to name these models [

A system composed by

In the model,

Each node dynamics is represented by a neuron dynamics using the Hodgkin-Huxley model [

The first state variable is the membrane potential

The network synchronization of neurons means synchronization in the membrane potential. Therefore, the

The concept of evolution in a network leads us to think that it is necessary to change the topology derived from the nodes’ dynamic interaction. In other words, it means that the network matrix

In this work, a vector and a matrix used in coupled maps are proposed, which models coupled dynamic systems whose couplings change depending on the system elements’ state variables and their interactions [

The vector

The structural network characterizes the physical wiring. In other words, it is the network that describes the configuration of the axonal and dendritic connections between neurons. Then, in our model, this network is specified as a set of spatial regions

The construction process of the structural network, using the domains

The proposed model of the functional evolutionary network of neurons, generated under a fixed structural network, is:

The function

The objective of a controller is to force the error system to converge to zero and to be able to obtain outputs similar to the input reference. If the complex network’s model (

Assuming

Up to this point, the system is in open loop. Now, if the

where

If the right-hand side of the first equation in (

In order for a network with a fixed configuration to reach synchronization, it is enough to show that the error system converges to zero, as was reported in [

The value for the parameter

The elements

To analyze the synchronization level of the connected neurons, a measure of the synchronization is desirable. A synchronization measure of the synchronized neurons in a network or a specific community can be defined as the logarithm of the standard deviation of the membrane potential over the network (or community) and the average of the network’s potentials (or community’s potentials). Consider the average of the states (membrane potential) in the time interval

Now, the standard deviation is calculated as

If the synchronization index

One of the fundamental behaviors produced by model (

In this way, once the communities are detected, the index (

The numerical simulation of the model was carried out with the values of the parameters given in Table

Parameters used in the numerical simulation of the neural network.

Parameter | Value | Parameter | Value |
---|---|---|---|

120 | 55 | ||

36 | -72 | ||

0.3 | -49.4 | ||

1 | 18.82 | ||

100 | |||

64 | 2 | ||

25 |

Network synchronization. The

The structural network used in the simulation was a lattice network. All the neurons’ dendrites and axons were connected. Evolution exists when there are different configurations of the network over time, and the coupling function depends on some state of the system. Therefore, for the proposed model (

The evolution of the functional network obtained in the simulation is shown in Figure

Evolution of the functional network. The

It is essential to highlight that although the network’s topology is evolving concerning the state, it reaches a synchronization in the neurons’ membrane potentials, as Figure

Figure

The maximal eigenvalue

In Figure

Mean value of the neuron potential (

Moreover, the coupling evolutionary model and the synchronization criteria generate communities of neurons with similar synchronous behavior. The algorithm reported in [

Community detection in the neural network. The

Suppose a group of neurons in a network is very connected, in that case, the neurons have more probabilities of exhibiting similar behaviors because the connections work as controllers that force the neurons to reach synchronization. To illustrate by means of simulation this previous idea in simulation, based on identifying the neurons that belong to each community formed over time, from 400 to 3000, each neuron probability membership to each community formed is computed. Neurons that belong with more than

Index’s values

The result shows the evolutionary synchronization of a network of neurons. Evolution is understood as the change in the functional network structure in terms of the connected neurons’ potentials, those neurons whose membrane potential is close. Then, the proposed model generates a class of evolutionary patterns in the functional network of neurons. This evolutionary behavior represents the attenuation or increment of the electrical connection between neurons. Similar behavior has been observed experimentally in living beings. The coupling between neurons can be seen as an adaptive controller that forces the network to converge to practical synchronization between subgroups of neurons, even as the couplings matrix evolves. The coupling matrix, being dependent on the membrane potential through a function that reproduces social behaviors, generates changes in the topology, which is purely defined by the connections between neurons at a particular time. Furthermore, the affinity between the potentials of neurons with synchronous behavior sets the guideline for such connections. Finally, there was a finding that different subgroups of neurons with different behaviors can be generated in the same network. This phenomenon can be understood as executing different tasks performed by the same network of neurons, where each task can be seen as a particular synchronous behavior. Even though it is a simplified model of the human connectome, the results in this work can be extended to larger dimensions. Each periodic region proposed in the model representing the neuron’s space can be whichever topological manifold. Thus, there exist manifolds that correctly model the neuron’s space of whichever structural network.

All data supporting the results can be found in the manuscript.

The author(s) declare(s) that they have no conflicts of interest.