Synchronous behavior can be responsible for the function or dysfunction of a neural network. To employ a memristor with threshold memductance as a bidirectional synapse, a memristive synapse-connected Chay twin-neuron network is constructed. This paper numerically presents the synchronous behavior for four representative firing activities in the memristive twin-neuron network by utilizing time-domain waveforms, synchronized transition states (STSs), and mean synchronization errors (MSEs). Indeed, the synchronous behaviors are truly related to the coupling strength and initial condition of the memristor. Besides, utilizing the powerful XC7Z020 FPGA, a digitally circuit-implemented electro-neuron and the memristive synapse-connected Chay twin-neuron network are made. Thereafter, the four representative firing activities and their STSs are experimentally captured to further confirm the numerical simulations.

Biological system propagates and handles neural signal via abundant collective behaviors [

Based on the law of electromagnetic induction theory, Lv et al. have come up with the idea that the memristor can be employed as a connected synapse to describe transmembrane current generated by the membrane potentials’ difference between two coupled neurons [

Three-dimensional (3D) Chay neuron model is a more physiologically excitable three-variable neuron model with simple mathematical form and rich essential features of electrical activities [^{+} ions and the inward one by Na^{+} and Ca^{2+} ions play key role in living biological activities. Thus, beyond all these investigations, by selecting four sets of the maximal conductances for the two ions currents, four representative firing activities are revealed. Besides, the synchronization for these four representative firing activities in coupled Chay neuron models are barely disclosed, which hinders the process of unveiling living biological activities.

The theoretical explorations and numerical simulations are classical methods to investigate neural dynamics, but hardware implementations and experimental measurements have become increasingly promising for diverse neuron-based engineering applications [

The arrangement of this paper is well designed as follows. The representative firing activities in a 3D Chay neuron model are briefly reviewed by employing numerical plots of phase trajectories and time-domain waveforms. Besides, a memristive synapse-connected Chay twin-neuron network is built in Section

The 3D Chay neuron model is an effective candidate in the numerical simulation for electrical activities in single biological neuron [_{∞} = _{y}_{y} _{y}) is unified to the expressions of _{∞}, _{∞}, and _{∞}, in which _{m}, _{m}, _{h}, _{h}, _{n}, _{n}, and _{n} are

Parameters’ values and significations of the 3D Chay neuron model.

Parameters | Significations | Values |
---|---|---|

_{I} | Reversal potentials for mixed Na^{+}-Ca^{2+} ions | 100 mV |

_{K} | Reversal potentials for K^{+} ions | –75 mV |

_{L} | Reversal potentials for leakage ions | –40 mV |

_{C} | Reversal potentials for Ca^{2+} ions | 100 mV |

Maximal conductance of mixed Na^{+}-Ca^{2+} channel | 1925 mS/cm^{2} | |

Maximal conductance of K^{+} channel | 1700 mS/cm^{2} | |

Maximal conductance of Ca^{2+}-sensitive K^{+} channel | 12 mS/cm^{2} | |

Maximal conductance of leakage channel | 7 mS/cm^{2} | |

_{n} | Time kinetic constant of the fast variable | 230 ms |

_{C} | Rate constant for the efflux of intracellular Ca^{2+} ions | 3.3/18 ms^{−1} |

Time constant determines the changing rate | 0.27 ms |

On account of the fast-slow effect, the bursting and spiking behaviors are two representative firing activities generated in this neuron model [^{−3} s, fixing the initial conditions (0.1 mV, 0.1, 0.1 nmol/L), and employing the model parameters in Table ^{−3} s and 10^{−4} s, respectively.

The firing activities related to the reversal potentials [^{+}-Ca^{2+} channel and maximal conductance

Trajectories in the ^{2} and ^{2}; (b) chaotic bursting behavior for ^{2} and ^{2}; (c) chaotic spiking behavior for ^{2} and ^{2}; (d) periodic bursting behavior with 5 spikes per burst for ^{2} and ^{2}.

Synchronous behavior for the memristive twin-neuron network with various _{0}, where the lefts are time-domain waveforms and the rights are STSs. (a) _{0} = –2 mWb; (b) _{0} = –2 mWb; (c) _{0} = –2 mWb; (d) _{0} = 1 mWb.

Support two identical neurons are bidirectionally coupled by memristor synapse to represent the electromagnetic induction effect induced with the membrane potential differences between them. Herein, the memristor with threshold memductance _{1} and _{2} are the membrane potential, _{1} and _{2} are probability of the voltage-sensitive K^{+} channel for the two identical Chay neuron models, _{1} and _{2} are the intracellular concentration of calcium ions for them, respectively, and _{i∞} = _{iy}_{iy} _{iy}) can be unified to the explications of _{i∞}, _{i∞}, and _{i∞}, in which _{im}, _{im}, _{ih}, _{ih}, _{in}, _{in}, and _{in} are

Thus, the synchronous behaviors for the memristive synapse-connected Chay twin-neuron network can be disclosed by system (

The exploration of synchronous behaviors for the Chay twin-neuron network is performed by MATLAB numerical simulations. Herein, we mainly focus on the two coupled neurons with only difference initial membrane potentials between them. Thus, the Chay twin-neuron network is triggered by initial conditions (0.1 mV, 0, 0.1 nmol/L, 1 mV, 0, 0.1 nmol/L, _{0}) without loss of the generality, within which the memristor initial condition _{0} is tunable.

The electrical activity is chaotic spiking under the model parameters in Table _{1}-_{2} plane, which means that the two neurons are in sync. Otherwise, the relatively large errors between the two membrane potentials indicate the two neurons out of synchronization. When _{0} = –2 mWb, the difference between the two membrane potentials become smaller with _{0} as –2 mWb and 1 mWb, the difference between the two membrane potentials become larger, as shown in Figures _{0}, the synchronization can be realized and the difference between the two membrane potentials is becoming smaller in the memristive synapse-connected Chay twin-neuron network. The mechanism for this process is that the coupling memristor exchanges the magnetic flux. Thus, the induced current is the carrier to drive the two Chay neurons in sync. Otherwise, the two connected Chay neurons are loss synchronization under tiny electromagnetic induction outputs.

To fully explore the synchronous behavior associated to the coupling strength _{0} of the memristor synapse simultaneously, mean synchronization error (MSE) _{i} (_{i} (_{i} (

Herein, the time sequence interval (600 s and 700 s) and the time step 0.01 s are selected. Thus, the number of total samples is _{0}, as well as different _{0} − _{0}, the normalized MSEs _{0}.

Normalized MSEs of the memristive synapse-connected Chay twin-neuron network in the _{0} − ^{2} and ^{2}; (b) ^{2} and ^{2}; (c) ^{2} and ^{2}; (d) ^{2} and ^{2}.

The normalized MSEs is given in Figure

It is more complex to physically realize the memristive synapse-connected Chay twin-neuron network than a single 3D Chay neuron model by FPGA. Thus, only the memristive synapse-connected Chay twin-neuron model by the digital electronic platform is demonstrated with representativeness. For this aim, fourth-order Runge–Kutta algorithm is utilized to obtain the discrete-time form for model (_{q} (

To achieve the discrete-time memristive synapse-connected Chay twin-neuron model (_{1} = _{4} = 1 and _{2} = _{4} = 2 are set up. The parameters _{0}, which impact the firing activities and synchronous behaviors, are changed by the software program to capture the four typical firing activities, time-domain waveforms, and STSs corresponding to the numerical simulations.

The hierarchical structure of the Verilog HDL program is illustrated in Figure _{i}, _{i}, _{i}, and _{V1}, _{n1}, _{C1} _{V2}, _{n2}, _{C2}, and _{φ}). The intermediate variables _{1∞}, _{1∞}, _{1∞}, _{2∞}, _{2∞}, and _{2∞} have to be calculated before the calculations of _{V}, _{n}, and _{C}; thus, a sublayer for intermediate variables is hired in this layer. In the iteration layer, the functions _{V1}, _{n1}, _{C1} _{V2}, _{n2}, _{C2}, and _{φ} are transmitted by time multiplex way to compute the intermediate vectors _{1}, _{2}, _{3}, and _{4}) in (

Hierarchical structure of the Verilog HDL program for the memristive synapse-connected Chay twin-neuron model.

RTL schematics of the discrete-time memristive synapse-connected Chay twin-neuron model executed on XC7Z020 FPGA.

Since the outputs of a FPGA are digital, they are fed to a two-channel 14 bit D/A converter (AD9767) combined with the peripheral circuit to convert the digital outputs into analog ones. Herein, an oscilloscope Agilent DSO-X 3012A is employed to display the output analog signals. Note that, all variables are in the single-precision floating-type during the computation process, so they must be converted into integers and enlarged to the range of [−8192, 8191] to take full use of 14 bits of the DAC digital input ports. The time-domain waveforms of _{1} and _{2} and synchronous transition states in the _{1} − _{2} plane for different coupling strength _{0} in the Chay bi-neuron network are captured and displayed in Figure _{1} and _{2} on the oscilloscope are differed from that of MATLAB simulations in Figure

Hardware measured time-domain waveforms and synchronous transition states with different coupling strength

Besides, a FPGA-based digital hardware electronic neuron is simply realized to confirm the four representative firing activities in the 3D Chay neuron model. The trajectories in the _{1} − _{2} phase plane and time sequences of _{1} and _{2} are captured and displayed in Figure

Experimentally captured trajectories in the C-V phase plane and time-domain waveforms. (a) Periodic bursting behavior for ^{2} and ^{2}; (b) chaotic bursting behavior for ^{2} and ^{2}; (c) chaotic spiking behavior for ^{2}and ^{2}; (d) periodic bursting behavior with burster-5 for ^{2} and ^{2}.

Four kinds of representative firing activities classified on the dependence of two maximal conductances in a 3D Chay neuron model are briefly reviewed. Then, a memristive synapse-connected Chay twin-neuron network is built, upon which synchronous behaviors are explored by utilizing time-domain waveforms, STSs, and MSEs. The numerical simulations demonstrated the success and effectiveness of employing the memristor synapse to achieve synchronization. It is found that, associating with the large coupling strength and more negative initial condition of the memristor, synchronous behaviors are achieved. An effective approach to implement the electronic neuron and the Chay twin-neuron network via FPGA are employed, from which the four kinds of representative firing activities of chaotic and periodic bursting/spiking behaviors, as well as the STSs are experimentally captured to confirm the correctness of the numerical ones. Synchronous behavior disclosed in neuronal network can well reveal the benefit for understanding the dynamical intricacy in the biological neurons and reflect the feasibility of diverse neuron-based applications.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China under Grant nos. 61801054 and 51777016, Natural Science Foundations of Jiangsu Province, China, under Grant nos. BK20160282 and BK20191451, and Postgraduate Research and Practice Innovation Program of Jiangsu Province, China, under Grant nos. KYCX19_1768 and KYCX20_2550.

^{2+}concentration with Chay neuron model