Complexity

Volume 2018, Article ID 6427870, 20 pages

https://doi.org/10.1155/2018/6427870

## Investigation of Cortical Signal Propagation and the Resulting Spatiotemporal Patterns in Memristor-Based Neuronal Network

^{1}School of Information Technology, Jiangxi University of Finance and Economics, Nanchang 330013, China^{2}Jiangxi E-Commerce High Level Engineering Technology Research Centre, Jiangxi University of Finance and Economics, Nanchang 330013, China^{3}Biomedical Engineering Department, Amirkabir University of Technology, Tehran 15875-4413, Iran^{4}Neuroscience Research Center, Baqiyatallah University of Medical Sciences, Tehran, Iran

Correspondence should be addressed to Boshra Hatef; moc.liamg@fetaharhsob

Received 27 December 2017; Revised 8 March 2018; Accepted 28 March 2018; Published 27 June 2018

Academic Editor: Daniela Paolotti

Copyright © 2018 Ke Ding et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

The brain is composed of an extremely large number of neurons [1], as the basic and also complex adaptive blocks of the brain system [2, 3]. Neuronal information transference is possible via propagation of the electrical and chemical signals in neuronal network [4, 5]. Indeed, fluctuation of the membrane potential of the neurons has a specific pattern in both time domain and space domain within the information processing [6]. These fluctuations actually bring a functional coherence and interplay between different parts, so that the related controlling behaviors are possible [7]. It is confirmed that emergence of a particular spatiotemporal pattern is in direct relationship with the intrinsic properties of the system [6, 8, 9]. Metabolically, generation and transference of the information and the related signals are costly [4, 10]. However, the way that the brain is wired has made it a nature-made computer with high level of efficiency in computation and cognition [4]. Although the human brain is not very quick at handling complex calculations, it beats a traditional computer system when it comes to energy efficiency [11]. Therefore, many efforts have been made to build up a hardware structure and a software design or even employ a mathematical model to study the brain system [12, 13], regarding some functional properties or physical effects [14]. There can be considered some factors responsible for energy efficiency of neuronal network including mechanisms of action potential propagation, synaptic neurotransmitters, and other factors that are studied in more detail in [4]. In this regard, it is interesting to investigate the emerged spatiotemporal patterns accompanied by the complex dynamics [8]. The recent studies on neural network and emulator circuit implementations [13–15] have led to the fourth circuit element called memristor [16] in addition to resistor, capacitor, and inductor [15, 17, 18]. Memristor (contraction of memory and resistor) is a two-terminal circuit element with nonlinear characteristics and high performance [19]. It has attracted much attention in biological models, adaptive filters, or integrated circuits [20] due to multistability phenomenon in coexisting attractors with chaotic demonstrations [21]. For the application of biological models, it can mimic how neurons in a network change their behavior when they are activated. Actually, its respond at each moment does not rely only on the signals it is receiving at that exact moment but is influenced by its own recent activity, too.

Many studies prove that biological behaviors are the outcome of collective activities of the neurons in neural network [22]. Indeed, the brain activities are not determined by each individual neuron, separately, but the intrinsic coherence coming from collective activities and patterns of maintenance or destruction of the local synchronization between the agents [23, 24]. As a result, it is not only about the components and their fluctuation pattern, individually, but also about the connections among them. In other words, neuronal information is carried by evolutionary spatiotemporal patterns indicating a powerful, efficient, and purposeful functional connectivity. Besides, the connection strength determines the specific spatial patterns in the network, as well [24]. Actually, what seems to have the most significant and delicate influence over the ultimate performance of the system is the interactions between the components, both quantitatively and qualitatively. Thus, we need to know the spatiotemporal distribution of the brain cells membrane potential and the pattern of propagated waves in the two-dimensional space focusing on the connections [8]. However, obviously, association of a large number of unites within nonlinear connections and interactions makes it so hard to grasp some of the concepts and deal with the related topics. Therefore, some simplifications with acceptable range of reductions are needed [25]. Regardless of these reductions in what is actually happening in the real neuronal network, this procedure can help us understand the related complicated occurrence, to some extent. In this study, the main idea is to notice the neurons at the level of population and consider a group of correlated neurons within more realistic interconnections and communication tools.

The point at which neurons are able to communicate with each other is called synapse [3], which bridges the neurons in the neuronal network [26]. In fact, transmission of the electrical signal in neuronal system can take place through synapses. It is also found that learning and memory are the two significant brain abilities attributed to synapses and their functional properties [3]. From a perspective, learning and memory are interrelated with each other. Memory is the internal mental recorded information, while learning is the ability of modifying the information stored in the memory based on the new inputs. More precisely, it also can be said that learning is the first step of memory since the sensory system sends information to the brain. Synaptic plasticity is postulated to be one of the important foundations of learning and memory [27]. Furthermore, plasticity is reported to be responsible for certain abilities like rapid response to threat stimuli and localization of the sound source [28]. The invention of memristor has made it possible to realize some complex activities which were impeded by lack of an appropriate device to model synaptic plasticity. The focus of this study is to demonstrate the capability of memristive neuronal network to represent some complex behaviors and large-scale plasticity which is also well described via experimental observations in the prefrontal cortex [29], visual cortex [30], and neocortex [31].

Real cortical tissue has a laminar structure [32]. Indeed, neurons of cerebral cortex are arranged in characteristic layers [3]. Primarily, presence or absence of neuronal cell bodies specifies the layers of cerebral cortex. This laminar structure of the cortex plays a significant role in organizing the inputs and outputs of the brain [3]. In fact, different inputs need to be processed in different ways while the outputs arise from different cortical regions. Accordingly, the laminar structure of cerebral cortex helps providing required circumstances. Considering distribution of the cortical electrical activities, related spatiotemporal patterns arise from the interface between the levels of activities of neurons in the surfaces. With given explanations, in this study, we simplify the case to a two-dimensional network of neuronal models, expressing an excitable cortical tissue to investigate the resulting pattern of the wave propagation in the surface.

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 , in which some concentric circular patterns are formed in the two-dimensional space. For further study, we sought to realize 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, the bifurcation analysis is carried out for different intensities of memristor-type coupling. It is revealed that the spatiotemporal electrical activity of the tissue concurs with the bifurcation analysis. We show the memristor coupling intensities by which the system undergoes periodic behavior and prevents the tissue from holding wave propagation. In addition, the chaotic-like behavior in bifurcation diagram corresponds to turbulent spatiotemporal behavior of the tissue. Moreover, it is found that the excitable media is 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 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 and can be found in Section 3.1. Sections 3.2 and 3.3 include the noise and the bifurcation analysis, respectively. Finally, the fourth section concludes our study.

#### 2. Model and Description

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 [33]. Hopfield neural model is defined as a graded response model [34]. This model has been successful in representing different dynamical behaviors including chaotic behaviors [35, 36] having to do with nonlinear demonstrations of the brain performance.

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 -th neuron is described as follows:where variable denotes the voltage across the capacitor , stands for membrane resistance between the inside and outside of the neuron, is an input bias current, is the neuron activation function for voltage input from the -th neuron, and is synaptic weight that illustrates the strength of connections between -th and -th neurons. In our work, the proposed Hopfield network is achieved by replacing resistive connection with hyperbolic-type memristor, which is discussed in detail in [35]. The set of parameters are , , . The weight matrix is considered aswhere is the synaptic weight connecting the first and the third neurons with the proportion of . The parameter is a constant indicating the strength of hyperbolic-type memristor-type coupling.

The differential equations describing the desired memristor-type neuronal unit can be expressed as follows:After that, we develop the case to a large array network of neuronal units within coupling intensities. Therefore, the equations for the square array network are represented as follows:where the subscript denotes the position of each neuronal unit in the two-dimensional square array network. is the resistor-type coupling intensity. and are constant. The reader will pay attention that the parameter denotes the resistor-type coupling strength in this study while parameter shows the memristor-type coupling strength.

#### 3. Numerical Results and Discussions

In our study, we design a square array network consisting of neuronal units. Our numerical results are calculated by Runge-Kutta 4th-order method with the time step under no flux boundary. The initial states of the network except the small central area, which is observable in the following images, are set as . In addition, the initial condition of the central local area is .

##### 3.1. Computational Analysis for Variation of and

Considering (2), there are two types of coupling between neurons in the whole network, namely, the memristor-type coupling and the resistor-type coupling. We pick different levels of memristor-type coupling by adjusting parameter , and then we consider two levels of resistor-type coupling intensity by adjusting parameter , in each case of adjustments.

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 ), still the lack of wave propagation remains and the ultimate general pattern does not change. Moreover, the excitable media are able to switch over different spatial behaviors by varying the memristive coupling strength. Hence, from this point of view, memristor-type coupling is more noticeable against resistor-type coupling. To put it more clearly, the distribution of membrane potential in the two-dimensional excitable media is shown in colored levels. In some cases, the central local area is maintained for a few seconds or continues to grow under a particular pattern, while in others the continuity of the central part is very short.

Firstly, Figure 1 shows the result in two snapshots when there is no effect of memristor in the network . Moreover, the neuronal units are connected to each other with the coupling intensity of . As it is observable, there is no propagation in this case and the generated signals find no path to travel the tissue. In fact, the generated wave front dies right at the beginning of its existence. Besides, there is no sign of propagation even when we choose a higher resistor-type coupling intensity by under no memory effect by (Figure 2). Here the only difference that can be seen between the results in Figures 1 and 2 is in the increased radius of the initiated wave front in Figure 2 in comparison with Figure 1. In this case, the central concentrated energy does not flow to the rest of the neurons and vanishes right at the beginning of its existence.