Spatiotemporal pattern in a neural network with non-smooth memristor

: Considering complicated dynamics of non-smooth memductance function, an improved Hindmarsh-Rose neuron model is introduced by coupling with non-smooth memristor and dynamics of the improved model are discussed. Simulation results suggest that dynamics of the proposed neuron model depends on the external stimuli but not on the initial value for the magnetic flux. Furthermore, a network composed of the improved Hindmarsh-Rose neuron is addressed via single channel coupling method and spatiotemporal patterns of the network are investigated via numerical simulations with no-flux boundary condition. Firstly, development of spiral wave are discussed for different coupling strengths, different external stimuli and various initial value for the magnetic flux. Results suggest that spiral wave can be developed for coupling strength 0 when the nodes are provided with period-1 dynamics, especially, double-arm spiral wave appear for can make spiral wave collapse and the network demonstrates chaotic state. Alternation of initial value for the magnetic flux hardly has effect on the developed spiral wave. Secondly, formation of target wave are studied for different coupling strengths, different sizes of center area with parameter diversity and various initial value for the magnetic flux. It can be obtained that, for certain size of center area with parameter diversity, target wave can be formed for coupling strength 0  , while for too small size of center area with parameter diversity, target wave can hardly be formed. Change of initial value for the magnetic flux has no effect on the formation of target wave. Research results reveal the spatiotemporal patterns of neuron network to some extent and may provide some suggestions for exploring some disease of neural system.


Introduction
Collective behaviors of neural network received considerable attention because of its application in many fields. It was found that various dynamics could be observed in physical, chemical, and biological systems, from which it can be believed that numerical investigation could be feasible to explore some main properties of nonlinear dynamical systems.
As one sort of collective behavior, synchronization of neural network was investigated via various aspects and many results have been obtained. Synchronization transitions induced by information transmission delay and coupling strength in scale-free neuronal networks with different average degrees and scaling exponents were revealed and it can be known that delay plays a more subtle role than coupling strength [1]. Synaptic effect on the excitement and synchronization of synaptic coupled neuronal networks are discussed by a modified Oja learning rule and it was found that, synaptic learning can suppress the over-excitement and is helpful for realizing synchronization [2]. Synchronization transitions of bursting oscillations dependent on the information transmission delay over scale-free neuronal networks with attractive and repulsive coupling was explored and it was shown that the change of delay can promote or impair spatiotemporal synchrony [3]. Synchronization of a modified bursting Hodgkin-Huxley neuronal model is studied in various conditions and it can be obtained that various synchronizations can be achieved via different kinds of synapses [4]. Finite-time projective synchronization of memristor-based neural networks with leakage and time-varying delays was concerned and two types of projective synchronizations of were reached with several stability conditions being presented [5]. Inhibitory synchronization inspired by firing rate in E/I neuronal networks was discussed and it was discovered that firing rate contrast enhancement may play an important role for inhibitory synchronization [6].
Simultaneously, as two typical regular phenomena of spatiotemporal pattern, spiral wave and target wave attracted much attention in neuroscience. Spiral wave, which can be formed when system is far away from stable state, can be found in many systems. Existing results indicated that spiral wave may be induced by parameter heterogeneity [7], broken target wave [8], autapses in the media [9], non-smooth memristor was hardly reported, the complex behavior for both single neuron and neural network with non-smooth memristor is worthy of in-depth study and discussion.
To further explore the spatiotemporal pattern in neural network under different electromagnetic induction, spatiotemporal pattern in a network composed of Hindmarsh-Rose neurons with nonsmooth memristor is to be investigated. Other parts of this paper are arranged as follows. Hindmarsh-Rose neuron with non-smooth memristor along with its dynamical behavior is depicted in Section 2. In Section 3, a neural network is constituted by the revised Hindmarsh-Rose neuron model and spatiotemporal pattern of the network is discussed under different conditions. Section 4 gives some conclusions.

Hindmarsh-Rose neuron coupled by non-smooth memristor
In neural system, neurons receive or transmit signals via synapses. Therefore, synaptic property has much effect on the electrical activity of neuron or neural network. As a matter of fact, information transmission between neurons often appears discontinuous. To understand and master the dynamics in this case, a kind of non-smooth memristor is used to simulate the synapse of Hindmarsh-Rose neuron and a revised Hindmarsh-Rose neuron model is proposed in this section. A non-smooth memristive [19] is considered as which is non-ideal voltage controlled memristor with absolute value nonlinearity, where  and  are two positive memristor parameters,  is internal state variable of voltage-controlled memristive. And then, Hindmarsh-Rose neuron with non-smooth memristor can be depicted as   3 ) 1.56 where x represents membrane potential of neuron, y means the exchange of ions in neuron membrane, z denotes adaption current and describes the magnetic flux across membrane. 3

 
w is the memory conductance of memristor used to describe the coupling between magnetic flux and membrane potential of neuron. 2 kw  is defined as the effect of self-inductance with 2 k being the gain dependent on the media. a , b , c , d , r , S , 1 k , 2 k ,  and  are the parameters which govern the dynamics of the neural system. ext I is external forcing current. When system parameters are chosen as    x  ) in system (2) with bifurcation parameter

Spatiotemporal pattern of Hindmarsh-Rose neural network with non-smooth memristor
To discuss the collective behavior of neural network composed of Hindmarsh-Rose neuron with non-smooth memristor, the network with nearest-neighbor coupling via single channel is presented as 720
where D is coupling coefficient between adjacent neurons, a , b , c , d , r , S , ext I , 1 k , 2 k ,  and  are defined the same as in system (2). It is supposed that the parameter values of neurons in network (3) are all the same. Therefore, all the nodes in the network are identical. For simplicity, it is assumed that network (3) is composed of 200 × 200 neurons, which are evenly distributed in a twodimensional lattice. And then spatiotemporal pattern of the addressed neural network is to be explored. , double-arm spiral wave can be observed ( Figure 6). That is to say, by selecting suitable parameters, spiral wave can be developed in neural network (3) when the nodes exhibit period-1 bursting.

Spiral wave and its collapse
, numerical simulations indicate that initial value for the magnetic flux hardly has effect on the dynamics of system (2). Given examples, for initial value

Target wave and its collapse
In this section, choose  Factually, when the center area with parameter diversity is too small, the local heterogeneity of parameter can hardly induce target wave in neural network (3). To illustrate this phenomenon, center area is chosen as 3 × 3 nodes, the dynamical behaviors of network (3) are simulated with no-flux boundary condition and initial value (−1.31742, −7.67799, 1.1302, 1.302) (see Figure 16), from which it is obvious to know that, when the size of center area is two small, target wave can hardly be formed even for large coupling strength. The reason may be that the energy change caused by parameter heterogeneity is too small to induce continuous diffusion outward.
To further explore the effect of size of center area with local parameter heterogeneity on the formation of target wave, the center area with local heterogeneity of parameter is chosen as 11 × 11, 15 × 15, respectively, with 0.9 a  for center nodes while 1.0 a  for other nodes. Dynamical behaviors of network (3) are calculated with no-flux boundary condition and initial value (−1.31742, −7.67799, 1.1302, 1.302) for different coupling intensity and given in Figures 17 and 18, which demonstrate that, stable target wave can be induced by parameter heterogeneity for coupling intensity 0 1 D  . In addition, it can be known that, the larger the coupling intensity is, target wave can be formed easily.      Figures 19 and 20, which indicate that, local heterogeneity of parameter can delay the development of spiral wave temporarily. But spiral wave finally comes into formation. Namely, spiral wave can be formed in the competition of target wave and spiral wave. Larger coupling intensity is helpful for inducing spiral wave. Then, take =1.5 ext I , 2.5, 2.8, spatiotemporal patterns of network (3) appear chaotic (Figures 21 and 22) over time. From Figures 19-22, it is easy to know that, when the nodes in centre area appear period-1 dynamics, parameter heterogeneity can induce stable spiral wave, but when the nodes in centre area are in multiple-period or chaotic state, parameter heterogeneity cannot induce spiral wave.  Thirdly, by numerical simulation with no-flux boundary condition, we can draw a conclusion that initial value for the magnetic flux has no effect on the developed target wave. To illustrate this result, choose initial value 0 0 ( 1.31742, 7.67799,1.1302, )    x w with 0 w =0, 10, and center area 11 × 11 nodes, spatiotemporal pattern of network (3) are given in Figure 23. Compared Figure 23 with Figure 14, we can conclude that the development of target wave could not be affected by the change of initial value for the magnetic flux.

Conclusions
Considering the electromagnetic environment which neural system is in, Hindmarsh-Rose neuron model with non-smooth memristor is presented as well as its dynamics being depicted and a neural network is constituted based on it. Then, spatiotemporal patterns of the proposed network are investigated. Some results are acquired.
Firstly, stable spiral wave can be developed for coupling intensity 0 1 D  when the nodes in the network is provided with period-1 bursting. The dynamics of nodes changing into multiple-period or chaos can make spiral wave collapse and hardly can be formed. Secondly, if the size of center area with local heterogeneity of parameter is large enough, when the nodes in centre area are in period-1 bursting state while other nodes converge to equilibrium point, local heterogeneity of parameter a can induce stable target wave in the addressed neural network. The change of external forcing current can also make target wave collapse. Thirdly, when the nodes in centre area are in period-2 bursting state while other nodes are in period-1 state, by setting parameter with local heterogeneity, an interesting phenomenon can be observed that spiral wave can form in the competition of target wave and spiral wave. But the increasing of external forcing current can destroy the development of spiral wave even for large coupling intensity. Additionally, whether the dynamics of the proposed neuron model or the spatiotemporal patterns of the addressed neuron network does not depend on the initial value for the magnetic flux.
The findings can provide additional insights into the pattern dynamics of networks, which is relevant to many physical, chemical, and biological systems.