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Complexity is the undeniable part of the natural systems providing them with unique and wonderful capabilities. Memristor is known to be a fundamental block to generate complex behaviors. It also is reported to be able to emulate synaptic long-term plasticity as well as short-term plasticity. Synaptic plasticity is one of the important foundations of learning and memory as the high-order functional properties of the brain. In this study, it is shown that memristive neuronal network can represent plasticity phenomena observed in biological cortical synapses. A network of neuronal units as a two-dimensional excitable tissue is designed with 3-neuron Hopfield neuronal model for the local dynamics of each unit. The results show that the lattice supports spatiotemporal pattern formation without supervision. It is found that memristor-type coupling is more noticeable against resistor-type coupling, while determining the excitable tissue switch over different complex behaviors. The stability of the resulting spatiotemporal patterns against noise is studied as well. Finally, the bifurcation analysis is carried out for variation of memristor effect. Our study reveals that the spatiotemporal electrical activity of the tissue concurs with the bifurcation analysis. It is shown that the memristor coupling intensities, by which the system undergoes periodic behavior, prevent the tissue from holding wave propagation. Besides, the chaotic behavior in bifurcation diagram corresponds to turbulent spatiotemporal behavior of the tissue. Moreover, we found that the excitable media are very sensitive to noise impact when the neurons are set close to their bifurcation point, so that the respective spatiotemporal pattern is not stable.

The brain is composed of an extremely large number of neurons [

Many studies prove that biological behaviors are the outcome of collective activities of the neurons in neural network [

The point at which neurons are able to communicate with each other is called synapse [

Real cortical tissue has a laminar structure [

In order to study the factors affecting wave propagation, it is interesting to figure out what a memristor-type synaptic connection exactly does, not only for one limited agent but also for large number of neuronal units and how much it affects the spatial distribution of the cell membrane potential and leads to wave propagation via the complex demonstrations. Actually, the answer of these questions may also reflect the influence of memory and learning process in a neuronal network through the emerged patterns. In other words, we examine different plasticity levels for the synaptic connections by means of different memristor contributions. On the other hand, by noticing differential equation models, which are used in this study, the initial states of the variables of a system refer to the result of their past dynamics. Therefore, we choose a different initial condition for a local area of the network indicating the different input sensory signals that have been applied to that specific area in the past. After that, we investigate the effect of memristor-type synapse against resistor-type synapse on the pattern formation in the network. Plus, we also expand our computations to noise considerations in some separated snapshots, because noise plays an important role in dynamical response of oscillatory systems.

The results show that different spatiotemporal patterns take place in the excitable tissue without supervision. As is clear through the snapshots, the overall pattern is mostly determined by memristor-type coupling. In accordance with some reports on the role of synaptic plasticity in some important high-order cortical activities, our results confirm that synaptic plasticity makes the tissue capable of representing different complex demonstrations. In fact, the increase and decrease of the memristor effect greatly changes the ultimate appearance of the tissue, which, in turn, actually resulted from the pattern of electrical activity of each neuron interacting with the neighbor neurons in the whole tissue. Moreover, the resulting patterns are found to be robust against noise for all the cases except for

The rest of the paper is organized as follows.

In the next section, the mathematical model is introduced with description of its variables and parameters. After that, in the third section, our numerical method is explained and the results are displayed. The computational analysis for variation of

There are a number of mathematical neural models capable of representing complex dynamic behaviors. These models introduced for a large number of neurons have properties that benefit investigations on biological neuronal network. Usually, in these models, it is assumed that the presynaptic firing rate determines the synaptic input current [

Neurons have a selective response to a compact range of parameters. In our study, the idea is to provide a compact range of connections and interactions in a neuronal network. For this purpose, we designed a square array of neuronal units with nearest neighbor connections. Each unit has a topology with hyperbolic-type memristor-based connection. A hyperbolic-type Hopfield neural network is considered for each agent. In this 3-neuron Hopfield neural network, one of the connection weights is defined as a memristive-type weight. The Hopfield equation for the

The differential equations describing the desired memristor-type neuronal unit can be expressed as follows:

In our study, we design a square array network consisting of

Considering (

It seems that neurons need to have an appropriate level of memristor effect to be capable of responding to the received stimulus in a desired pattern leading to wave propagation. Otherwise, the generated circular wave (in the central area of the network) will not be developed. Even though a propagated wave can travel a further distance by strengthening the resistive coupling intensity between the agents (which is adjustable by parameter

Firstly, Figure

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

For the next step, we provide the tissue a nonzero memory effect with

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

After that, on the way of increasing memory properties, leading to more synaptic plasticity, we set

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

Furthermore, we examine a slightly further increase in the level of memory effects by applying

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

Considering all the above, we set a higher level of memristor-type coupling by

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

As is visible in the displayed results in Figures

For further investigation, we put the memory influence of the excitable tissue in a higher level by setting

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

In this part, we expand our computational analysis to investigate stability of the emerged patterns under the noise effect. For this purpose, the Gaussian white noise

The results show that the spatial pattern totality is not influenced by the noise effect for

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network for

Knowing that the dynamic behavior of a system can be revealed from its bifurcation analysis, in this subsection, the bifurcation analysis is carried out by setting different memristor-type couplings for a system of two coupled neuronal units. In this way, a qualitative analysis of the role of parameter

Bifurcation diagram of two coupled neurons for different memristor-type coupling intensities (

In Figure

In this paper, it was reported in Section

Bifurcation diagram of two coupled neurons for different memristor-type coupling intensities (

In this study, the synaptic plasticity by means of memristor was investigated and the potential spatiotemporal patterns were detected. We showed that the memristive neuronal network is capable of representing plasticity phenomenon observed in biological cortical synapses. Excitable media were modeled by a network of

Finally, for further study, we sought to discover whether it is possible to find a meaningful relationship between the qualitative properties of the coupled neurons and the spatiotemporal demonstrations from a two-dimensional lattice. Therefore, first the bifurcation analysis was carried out for different intensities of memristor-type coupling to see the possible mutual influence of the coupled neurons under memristive effects, and then the results were compared to our two-dimensional analysis. Our study revealed that the spatiotemporal patterns of electrical activity of the tissue concur with the bifurcation analysis. It was shown that the memristor coupling intensities, by which the system undergoes periodic behavior, prevent the tissue from holding wave propagation. In addition, the chaotic-like behavior in bifurcation diagram concurs with the turbulent spatiotemporal electrical activity of the tissue. Moreover, we showed that the excitable media is very sensitive to noise impact when the neurons are set close to their bifurcation point, so that the spatiotemporal pattern is not stable.

The authors declare no conflicts of interest.

This paper is partially supported by the National Natural Science Foundation of China under Grant 61561023, the Key Project of Youth Science Fund of Jiangxi China under Grant 20133ACB21009, the Project of Science and Technology Fund of Jiangxi Education Department of China under Grant GJJ160429, and the Project of Jiangxi E-Commerce High Level Engineering Technology Research Centre. Sajad Jafari was supported by Iran National Science Foundation (no. 96000815).