Dynamical robustness and firing modes in multilayer memristive neural networks of nonidentical neurons
Introduction
Networks are ubiquitous around us. They can be tangible networks, such as transportation networks, electric grids and neural networks. Or they can be abstract networks, such as the actor’s collaboration networks, acquaintances of networks and citation networks from Science Citation Index Database. Most of the interest involved with network is to understand the mechanism of many networked systems composed of dynamical units. The nature of collective dynamical behaviors of coupled oscillators networks are critical to understand the proper functioning of systems in biology, physics, chemistry, neuroscience and engineering [1], [2], [3], [4], [5]. Depending on individual elements, coupling schemes and network topologies, various collective behaviors can be observed in networked systems composed of a large ensemble of units, such as synchronization [6], [7], [8], oscillation quenching [9], [10], etc. Among various collective dynamical behaviors occurring in coupled oscillators networks, aging transition is such a phenomenon that the weakening macroscopic oscillation of coupled oscillators network composed of active (or self-sustained) oscillators and inactive (or non-self-oscillatory) oscillators collapses at a critical ratio [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. Another exploration of aging transition is concerned with network’s dynamical robustness, which refers to the ability of a network to confront damage or deterioration of some elements and maintain its dynamical behaviors. The critical ratio is provided as a measure of the network’s dynamical robustness; that is, a high critical ratio reveals strong dynamical robustness of coupled oscillators network [22], [23], [24], [25], [26].
The traditional network models which all the network’s links are treated on an equivalent footing may not fully capture the details and even to accurately describe some collective behaviors occurring in real-world networks. The multilayer networks, which include multiple subsystems and inter-layer dynamical interactions are take into account, are presented as a proper framework where correct descriptions of collective behaviors occurring in real-world networks can be made. For example, the information or rumor spreading between social groups (or layers) of users on social networks [27], the problem of passenger rescheduling between commercial airlines (or layers) when some flights (or links) of Air Transportation Network are removed [28], the description of C. elegans neural network which consists of 281 neurons and two layers [29]. Lately, some of intensive researches on dynamical robustness of networks has been switched to multilayer networks. In [30], authors explored dynamical robustness of multilayer oscillator networks and shown that the changes of dynamical robustness based on interlayer coupling schemes. Kundu et al. [31] found that chemical synapses through interlayer connections enhance rhythmicity in multiplex neuronal networks.
The idea of memristor was already originated by Chua in 1971 [32]. But until 2008 a two-terminal electrical device which behaves like a memristor was presented [33]. As the fourth fundamental passive circuit element, memristor used to describe memory effect has attracted extensive research, which with an emphasis on how electromagnetic induction affect the dynamics of neuronal systems [34], [35], [36], [37]. Memristors can efficiently emulate the dynamical behavior of biological synapses in neuromorphic systems, which helps to understand the information process and regulation mechanism of neural system. In [38], authors investigated two neurons coupled through memristor and shown that memristor coupling facilitated synchronous behavior. Various dynamical behaviors could be observed in neurons systems whose elements were coupled through cubic flux-controlled, quadratic flux-controlled, or exponential flux-controlled memristors [39]. The chimera state and synchronous behavior in multilayer neural networks with memristive synapses between layers were examined [40]. However, previous studies on dynamical robustness do not consider electromagnetic induction in the neural networks. Thus, we know little about the effects of the electromagnetic induction on the dynamical robustness of neural networks.
Motivated by these key facts, here we construct a multilayer memristive neural network composed of mixed populations with active and inactive neurons. The neurons are coupled with electrical synapses within each layers whereas interactions between neurons in different layers are through memristors so that we explore the effect of electromagnetic induction on the network's dynamical robustness and firing modes on the road to aging transition.
The structure of this paper is as follows. In Section 2, a multilayer memristive neural network composed of active and inactive neurons is presented. Section 3 is devoted to the theoretical analysis of multilayer memristive networked systems and memristive synapse coupling effects on dynamical robustness and firing modes. The experimental results and discussion is carried out on Multisim in Section 4. The concluding remarks on the obtained results are offered in Section 5.
Section snippets
Model description and preliminaries
The neural model proposed by Hindmash and Rose [41] can be describe by the following equations:where , , represent the membrane potential, slow current for recovery variable, and adaption current, respectively. is the forcing current, which is closely related to abundant dynamical behavior of HR neuron, such as bursting, spiking and resting. The parameters could been taken as , , , , and .
In this paper, a multilayer
Analysis and results
As mentioned in Section 2, the HR neuron could exhibit rich dynamical behaviors by changing the forcing current. An isolated HR neuron exhibits bursting state if and exhibits resting state if . Hence, the neurons in multilayer memristive neural network (2) with possess the self-oscillatory and are tagged with active neurons, whereas neurons with lose the self-oscillatory and are tagged with inactive neurons, where , , , and
Electric circuit implementation and discussion
In this section, an analog circuit of the multilayer memristive neural network (2) in which each layer consists of 2 neurons is built on Multisim, and the circuit realization is shown in Fig. 8.
Operational amplifiers in use are TL082 type ones and their power supplies are V. The operational amplifiers and are summator and inverting amplifier, respectively. They both are used to realize the electronic synapse coupling within layer-1 characterized by
Conclusions
In conclusion, the effects of electromagnetic induction on dynamical robustness and firing modes is investigated by constructing an multilayer memristive neural network, in which either cubic order flux-controlled or quadratic flux-controlled memristors are used as synapses to connect the neurons between two layers. In the case of memristive synapse coupling by cubic order flux-controlled memristors, it is found that the increase of electromagnetic induction parameter and parameter of memristor
CRediT authorship contribution statement
Yuanyuan Liu: Conceptualization, Investigation, Methodology, Visualization, Writing – original draft. Zhongkui Sun: Writing – review & editing, Supervision. Xiaoli Yang: Supervision. Wei Xu: Supervision.
Acknowledgements
We thank Prof. Jun Ma and Mr. Zhao Yao for help with electric circuit implementation. And we thank the anonymous reviewers for valuable comments that helped improve the paper. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11772254, 11972288) and by Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University.
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2022, AEU - International Journal of Electronics and CommunicationsCitation Excerpt :The study of chaos theory originated in atmospheric science [1] and many chaotic systems have been constructed to mimic the chaos phenomena that commonly appear in natural and artificial scenarios [2–4]. Memristor, possessing the special nonlinearity with inner states [5], has been validated to be a great candidate for building memristive chaotic circuits with hidden chaotic or hyperchaotic oscillations [6–8], memristive neural networks with chaotic bursting and spiking [9,10], and memristive chaotic systems with (extreme) multi-stability [11–13]. To date, the memristor-based applications and advancements have been integrated with many disciplines in recent decades [14–16], since the unique properties of non-volatile, nano-size, local-activity, and so on.