Influences of electromagnetic radiation distribution on chaotic dynamics of a neural network☆
Introduction
The working mechanisms of a nervous system, including consciousness, thinking and memory are thought as the last frontier of the biological sciences. A nervous system contains a huge number of neurons which are the basic physical and functional unit of a neural network. Numerous experiments have proved that nonlinear dynamical behaviors exist in brain, and these behaviors are highly related to the underlying mechanism of normal function of nervous system [1], [2]. As we all known, the famous Hopfield neural network is a simplified nervous system, which provides a reliable model to emulate the intricate dynamics of the brain [3]. By now, the dynamical analysis has been extensively discussed on Hopfield neural network [4]. The investigation on nonlinear dynamics in Hopfield neural network model makes it possible to imitate some behaviors of biological nervous system. Such research provides the theoretical basis and guidance for practical physiological experiments, which is extremely significant for the development of brain science and medicine.
It has been confirmed that several factors including network structure [5], [6], time delay [7], noise [8] and electric field [9] can change the dynamics of a neural network. Particularly, electromagnetic radiation is one of the factor that cannot be ignored, which is almost full of the human living environment due to the widely application of various electromagnetic communication instruments. In recent years, the study of the effects of electromagnetic radiation on neuronal dynamics has opened up a new direction for neurodynamics. Li et al. [10] first explored the influence of electromagnetic radiation on single neuron and neuronal network through introducing a new external membrane current variable in the Hodgkin–Huxley neuron model. The numerical and experimental results show that electromagnetic radiation can inhibit the electrical activities of an isolate neuron but also regulate the collective electrical activities of the neurons network. Lv and Ma [11] believed that because the membrane potential and ion concentration of neurons are constantly changing, according to the law of electromagnetic induction, the effects of external magnetic field on a neuron can be equivalent to the magnetic flux flowing through the cell membrane. Based on the above theory, they established a new Hindmarsh–Rose neuron model under electromagnetic radiation and used it to research the effects of electromagnetic radiation on the electrical activities of neurons. And some other investigators [12], [13], [14], [15], [16], [17] found that the electrical activity patterns of neurons can be controlled through adjusting the external magnetic field parameters, which includes spiking state, bursting state, periodic state and even chaotic state. In addition, the influences of electromagnetic radiation on the synchronous dynamics of neurons have been also investigated. For example, Ma et al. [18] detected that appropriate electromagnetic radiation can effectively generate intermittent synchronization while excessive electromagnetic radiation will induce disorder of neuronal system. Ren et al. [19] proved that electromagnetic radiation can increase the synchronization degree of negative feedback coupled neurons and also contributes to the discharge of positive feedback couple neurons. And electromagnetic radiation has a stable effect on the phase synchronization of neurons, which is discovered in Ref. [20].
Chaos is a significant dynamical phenomenon that is ubiquitous in nonlinear systems in nature [21], [22]. According to the complexity of chaos, it can be divided into chaos, transient chaos and hyperchaos [23], [24], [25]. It has been affirmed experimentally that chaos exists in nervous system and various neurological diseases including migraines [26], epilepsy [27], and Parkinson diseases [28]. Undoubtedly, research on the chaotic dynamics of the nervous system is helpful to understand the generation mechanism of neuronal disease. As a result, chaotic behaviors of the neural system have been widely studied by using the Hopfield neural network which is a common model for simulating the biological neural network. Multiple chaotic phenomena have been found in the Hopfield neural network, such as chaos [29], [30], transient chaos [31], hyperchaos [32], [33], and hidden attractors [34], [35]. However, the chaotic dynamics of Hopfield neural network under electromagnetic radiation is rarely investigated. In Ref. [36], Hu et al. researched the chaotic dynamics of a small Hopfield neural network consisting of three neurons which one of neuron under electromagnetic radiation. Their simulation results show that the periodic and chaotic motions in the neural network can appear intermittently via altering the electromagnetic radiation parameters, that is to say, the chaotic behaviors of the neural network can be controlled by external electromagnetic radiation intensity. According to the above research results, it can be inferred that the purpose of treating certain neurological diseases can be achieved by adding electromagnetic radiation on neurons to change the dynamics behaviors of the nervous system. Indeed, some biological experiments have demonstrated that suitable electromagnetic radiation stimulation is beneficial for the recovery of mental illness [37], [38]. Nevertheless, it is noted from the above reviews that all researchers only consider one case that one neuron in a neural network is stimulated by electromagnetic radiation. Actually, there are multiple neurons affected by electromagnetic radiation in a nervous system under electromagnetic radiation. It is well know that the electrical activities of a neuron can be changed due to the effects of electromagnetic radiation. In other words, the dynamical behaviors of the neural network become more complicated when multiple neurons are subjected to magnetic filed. Therefore, the difference about the number of neurons affected by electromagnetic radiation in a neural system should be taken into account. Such difference is valuable for better understanding chaotic dynamics of the real nervous system in the brain.
In this paper, we investigate the dynamic effects of electromagnetic radiation on multiple neurons in a neural network, which has not been reported in the previous works. First a novel mathematical model is proposed to describe the influences of electromagnetic radiation distribution on neural network with n neurons. In the proposed model, the effect of external magnetic field on a neuron is replaced by the magnetic flux across the membrane of the neuron, the coupling relationship between the magnetic flux and the membrane potential is described by a memristor [39], [40], and the variation and leakage of magnetic flux are also considered. Based on the novel neural network model, we analyze and discuss the boundedness, symmetry and equilibrium stability of the nervous system under electromagnetic radiation. Then, one novel Hopfield neural network with three neurons is realized using the new model, and the dynamics effects of the different number of neurons stimulated by electromagnetic radiation in the neural system are orderly revealed through employing phase portrait, bifurcation diagram, Lyapunov exponents and Poincar maps. Finally, the numerical analysis results are verified by hardware experiments. The results obtained in this paper mainly include the following points: (i) the effects of electromagnetic radiation distribution on neural network consisting of n neurons is modeled via using a set of new state equations, which contributes to simulate the study of biological neurons and neural network; (ii) the basic nonlinear dynamics of the neural network is analyzed, and the analysis results prove that the neural network with n neurons exposed to electromagnetic radiation is a bounded, symmetric and unstable nonlinear system within a certain range; (iii) the dynamic state of the neural network is very sensitive to electromagnetic radiation, that is to say, various dynamical phenomena including periodicity, quasi-periodicity, and kinds of chaos can be observed through adjusting the parameters of electromagnetic radiation; (iv) the complexity of the chaotic dynamics of the neural system is associated with the distribution of electromagnetic radiation, that is, with the increasing of the number of neurons stimulated by electromagnetic radiation, the state of neural network gradually transforms from an initially stable periodic to chaos, transient chaos and complex hyperchaos; (v) a novel hardware circuit of the neural network is provided by using active and passive electronic components, and the results captured from hardware experiment are basically consistent with the software simulation results.
The content of the paper is typeset as follows. Section 2 presents a new neural network model with n neurons under electromagnetic radiation and further analyzes its basic dynamic characteristics; Section 3 focuses on the investigation and discussion of the effects of the electromagnetic radiation on multiple different neurons in a neural network; Hardware experiments of the neural network are implemented in Section 4. Section 5 summarizes the full text.
Section snippets
Mathematical model description
Hopfield neural network model not only enjoys a similar network structure to the real biological neural system, but also can generate some dynamical phenomena similar to the brain chaos, which is a simplified biological nervous system model. The expression of this model consisted of n neurons is [3]:
It should be noted that xi, IBi are column matrices, and Ci, Ri are diagonal matrices. Among them, xi and IBi are the membrane voltage and bias current of the
Dynamics analysis of the neural network under different electromagnetic radiation distribution
For a neural network, the distribution of electromagnetic radiation is usually uneven, that is, the number of neurons exposed to electromagnetic radiation in the neural network is uncertain. In order to study the effects of the distribution of electromagnetic radiation on the dynamics of the neural network, we present a neural network including three neurons using the new neural network model described above, and analyze the chaotic dynamics of the neural network with different number of
Circuit implemention and hardware measurements
In this section, the circuit of the neural network model (17) is realized by using discrete electronic components, and the results of the above theoretical analysis and numerical simulation are proved through hardware experiments. Before implementing the complete neural network circuit, we first built two basic nonlinear circuit units, one is a hyperbolic tangent function circuit [39], and other is a negative absolute value circuit [43], as shown in Fig. 18. The bipolar transistors S8050,
Conclusions
In this work, we present a mathematical model to simulate the effects of electromagnetic radiation distribution on the neural network based on n neurons. The theoretical analysis results show that the neural system subjected to electromagnetic radiation has boundedness, symmetry, and zero equilibrium point instability under certain contions. When n is equal to 3, the chaotic dynamics of the neural network is investigated in detail by using magnetic field to stimulate different number of
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