Periodic Spiking Research Articles

Random-Coupled Neural Network

Improving the efficiency of current neural networks and modeling them on biological neural systems have become prominent research directions in recent years. The pulse-coupled neural network (PCNN) is widely used to mimic the computational characteristics of the human brain in computer vision and neural network fields. However, PCNN faces limitations such as limited neural connections, high computational costs, and a lack of stochastic properties. This study proposes a random-coupled neural network (RCNN) to address these limitations. RCNN employs a stochastic inactivation process, selectively inactivating neural connections using a random inactivation weight matrix. This method reduces the computational burden and allows for extensive neural connections. RCNN encodes constant stimuli as periodic spike trains and periodic stimuli as chaotic spike trains, reflecting the information encoding characteristics of biological neural systems. Our experiments applied RCNN to image segmentation and fusion tasks, demonstrating its robustness, efficiency, and high noise resistance. Results indicate that RCNN surpasses traditional methods in performance across these applications.

A fractional-order improved FitzHugh-Nagumo neuron model

Abstract In this paper, we propose a fractional-order improved FitzHugh-Nagumo (FHN) neuron model in terms of a generalised Caputo fractional derivative. Following the existence of a unique solution for the proposed model, we derive the numerical solution using a recently proposed L1 predictor-corrector method. The given method is based on the L1-type discretization algorithm and the spline interpolation scheme. We perform the error and stability analyses for the given method. We perform graphical simulations demonstrating that the proposed FHN neuron model generates rich electrical activities of periodic spiking patterns, chaotic patterns, and quasi-periodic patterns. The motivation behind proposing a fractional-order improved FHN neuron model is that such a system can provide a more nuanced description of the process with better understanding and simulation of the neuronal responses by incorporating memory effects and non-local dynamics, which are inherent to many biological systems.

A Rectified Linear Unit-Based Memristor-Enhanced Morris–Lecar Neuron Model

This paper introduces a modified Morris–Lecar neuron model that incorporates a memristor with a ReLU-based activation function. The impact of the memristor on the dynamics of the ML neuron model is analyzed using bifurcation diagrams and Lyapunov exponents. The findings reveal chaotic behavior within specific parameter ranges, while increased magnetic strength tends to maintain periodic dynamics. The emergence of various firing patterns, including periodic and chaotic spiking as well as square-wave and triangle-wave bursting is also evident. The modified model also demonstrates multistability across certain parameter ranges. Additionally, the dynamics of a network of these modified models are explored. This study shows that synchronization depends on the strength of the magnetic flux, with synchronization occurring at lower coupling strengths as the magnetic flux increases. The network patterns also reveal the formation of different chimera states, such as traveling and non-stationary chimera states.

Periodic spiking pulse formation in broadband nonlinear optoelectronic oscillator.

An approach to generating periodic spiking pulses in a broadband opto-electronic oscillator (OEO) without external injection is proposed and demonstrated. Through biasing the electro-optic Mach-Zehnder modulator (MZM) in the OEO cavity at its nonlinear working point, spiking pulses are excited by the self-generated chaos under the combined action of the loop filter response time and the nonlinear excitation effect. An interaction force is introduced between pulses. As a result, the pulse number in a single roundtrip duration and the temporal distribution of the generated spiking pulses can be tuned by adjusting the loop gain and the direct-current (DC) bias voltage of the MZM. Both numerical simulation and experiment are carried out to demonstrate the spiking pulse generation mechanism, where ultra-short spiking pulse trains with even and uneven temporal distribution are generated.

Potentiostatic oscillations of current and surface structure in a 3D cellular automaton model of metal passivation

Some recently obtained data from simulations in our 3D cellular automaton model on metal passivation are presented and discussed. The model is a non-equilibrium heterogeneous Ising-like model with moving boundary due to metal depletion by corrosion. The mobile particles representing metal oxide are produced in corrosion events. They bind to the metal surface and to each other with short range interaction. They diffuse by random walk with asymmetric exclusion and disappear in the bulk solution.The results of this paper are obtained in potentiostatic conditions. In our model, this amounts to a fixed bare metal corrosion rate. The results include current transients and surface snapshots at a number of moments of time. We define several other dynamic variables. Their transients allow us to construct phase portraits, to identify the bare metal surface area and the oxide layer thickness as the essential variables for the oscillations. The first of those is equivalent to current density by model construction. The second plays a peculiar role in periodic current spiking, as discussed in the text.

Biphasic action potentials in an individual cellular neural network cell

Hardware circuit that can effectively simulate biological neurons is an important basis for neuromorphic computation. Cellular neural network (CNN) cell is the basic information processor of a CNN, which acts like a neuron in the brain and has good circuit realizability. An individual memristive CNN cell is constructed by using a memristor instead of a linear resistor for imitating the ion channel time-varying conductance, in which abundant biphasic chaotic and periodic spiking activities are uncovered. This provides a new way to simulate biological neurons at the level of analog circuits. This paper first deduces the mathematical model of the memristive CNN cell, analyzes the equilibrium stability and then explores its dynamical behaviors based on numerical simulation. The results display that the different spiking activities can be effectively regulated by the system parameters and excitation parameters. Furthermore, the analog circuit of the memristive CNN cell is designed and the PSpice-based circuit simulations are performed to verify the correctness of the numerical simulations.

Emergence of spiral and antispiral patterns and its CGLE analysis in leech-heart interneuron model with electromagnetic induction

Neurons can exhibit various rhythmic activities such as bursting, spiking, and quiescent states when exposed to external input current stimulus. In this paper, a model of medicinal leech's heart (LH) interneuron is considered to describe the dynamics of neurons with a varied range of electrical activities. The crucial insights into the model's dynamics are explored in three different parameter regimes: phasic spiking, regular spiking, and bursting, based upon the codimension-one bifurcation of the model by considering VK2shift as a bifurcation parameter. The spatiotemporal dynamics of the model are explored by allowing 1D and 2D diffusion in the membrane voltage. The 1D diffusive system produces irregular bursting dynamics for the intermediate value of diffusion coefficients, whereas, at higher values, it shows synchronized oscillations. In the presence of 2D diffusion, the emergence of different types of spiral patterns is observed in the system. Furthermore, the system is extended by incorporating electromagnetic induction in the membrane voltage to explore the effect of induction on the various dynamics of neural model. By varying its intensities, the membrane voltage in the extended model produces a variety of discharge modes, such as periodic spiking, fast-spiking, resting, and spike-adding phenomena. In addition, the emergence of anti-spiral patterns in the extended model near subcritical Hopf bifurcation is analytically verified using the complex Ginzburg-Landau equation (CGLE). These findings demonstrate that the firing patterns vary based on the control parameters, and these variations contribute to our understanding of how the brain system transmits and processes the signals.

Periodic and chaotic spiking behaviors in a simplified memristive Hodgkin-Huxley circuit

The famous Hodgkin-Huxley circuit contains two time-varying resistors to describe the electrophysiological characteristics of sodium and potassium ion channels. But the time-varying resistors are expressed by mixed exponential equations, which are unavailable to directly implement in analog circuit and hinders hardware application of the Hodgkin-Huxley circuit. To hit this issue, a simplified memristive Hodgkin-Huxley (m-HH) circuit is proposed, which only involves two locally active memristors (LAMs) to depict the sodium and potassium ion channels, a capacitor to describe the neuron membrane, a DC current to represent external stimulus, and two DC voltages to express the reversal potentials. MATLAB-based numerical simulations and analog circuit-based hardware measurements display that the simplified m-HH circuit can generate memristor parameter- and DC current-related periodic spiking behaviors with different periodicities and chaotic spiking behavior. This delights that electrophysiological characteristics of ion channels and external stimulus can be employed for regulating these periodic and chaotic spiking behaviors. It is interesting that the simplified m-HH circuit can generate frequency self-adaptation and coexisting firing patterns. The inter-spike interval bifurcation diagram shows that the frequency of the periodic spiking behavior increases with the increase of externally applied DC current. Besides, the hybrid parameter bifurcation diagram displays that the simplified m-HH circuit can generate memristor initial state-related coexisting firing patterns of period-6 with chaos, period-3 with chaos, and period-3 with period-8. These offer us the unique insight into explaining some biological firing patterns. The most significance is that the analog hardware circuit is feasible in reproducing these periodic and chaotic spiking behaviors and benefits for developing spiking-based neuromorphic hardware.

A New Compact Model for Third-Order Memristive Neuron With Box-Shaped Hysteresis and Dynamics Analysis

This paper proposes a new compact model and presents circuit-theoretical analysis for a third-order memristive neuromorphic element fabricated by S. Kumar et al. The proposed model mainly consists of a box-shaped resistor model and a simplified piecewise-linear memristor model. Since the dynamic behavior of the box-shaped hysteresis in the quasi-static current-voltage curve is mostly unexplored, we firstly extract the box-shaped resistor model and construct its oscillators. The coordinate system shifting method and dynamic route analysis method are used to reveal the operating mechanism of box-shaped resistor-based oscillators. Both the theoretical analysis and simulation verification indicate that the box-shaped hysteresis characteristic facilitates the generation of neuromorphic action potentials. The proposed new compact model not only captures quasi-static characteristics, but also includes dynamic behaviors, such as action potential, periodic spiking, and periodic bursting. The influences of the model parameters are further investigated to reveal the mechanism of the neuromorphic behaviors in the box-shaped hysteresis and positive differential resistance regions. Simulation results manifest the feasibility of the proposed model and the correctness of the presented analysis methods, which pave the way to the optimized design of memristive devices and the research of neuromorphic dynamics.

Spiking ruby revisited: self-induced periodic spiking oscillations leading to chaotic state in a Cr:Al2O3 laser with cw 532-nm pumping

This paper reexamines a 60-year-old mystery of spiking behavior in ruby lasers with a cw 532-nm pump, paying special attention to mode matching between the pump and lasing beam within a ruby crystal placed in a semi-confocal laser cavity. Periodic spiking oscillations were observed in a limited pump power regime, where spikes obeying the generic asymmetric hyperbolic function appeared at a repetition rate around 50 kHz and with a 130-150 ns width and 0.1-0.6 µJ energy depending on the pump power. The physics of the spiking behavior based on Kleinman’s mechanical approach and a plausible interpretation for the periodic spiking oscillation in terms of self-induced mode matching between the pump and laser beams through the self-induced Kerr-lens effect are addressed. The statistical nature inherent to spiking and the associated self-organized critical behavior in the quasi-periodic spiking oscillations as well as chaotic states occurring outside the periodic spiking regime are clarified from a nonlinear dynamics point of view and intriguing statistical properties inherent to chaotic oscillations in usual solid-state lasers subjected to external modulations are shown to be present in the self-induced instabilities of the ruby laser

A magnetic-field-driven neuristor for spiking neural networks

Artificial intelligence has been widely deployed in many fields with remarkable success. Among various artificial neural network structures in artificial intelligence, the spiking neural network, as the next-generation artificial neural network, closely mimics the natural neural networks. It contains the all-or-nothing and diverse periodic spiking, which is an analogy to the behavior of natural neurons. Artificial devices that perform the function of neurons are called neuristors. Most existing neuristors are driven by electrical signals, which suffer the problem of impedance mismatch between input and output neuristors. By exploiting the S-shape negative differential resistances element that is sensitive to the external magnetic field, we constructed a magnetic-field-driven neuristor. Magnetic fields can stimulate all-or nothing spiking, and its shape and frequency can be modulated through capacitances in the circuit. As magnetic fields serve as the information carrier, the cascading of our neuristors can get rid of the electrical impedance mismatch, promising a scalable hardware platform for spiking neural networks.

Firing pattern in a memristive Hodgkin–Huxley circuit: Numerical simulation and analog circuit validation

Spiking firing patterns and their hardware implementation can assist us in exploring spike-based applications. The classical Hodgkin–Huxley circuit can generate spiking firing patterns, but it is hard to implement on analog platform because of the involvement of complex exponential nonlinearities for characterizing the sodium and potassium ion channels. To solve this issue, this paper proposes a second-order local active memristor (LAM) and a first-order LAM to respectively characterize the sodium and potassium ion channels in the Hodgkin–Huxley circuit, thereby obtains a memristive Hodgkin–Huxley circuit. Numerical simulations demonstrate that the memristive Hodgkin–Huxley circuit can generate abundant spiking firing patterns, i.e. periodic and chaotic spiking firing patterns. These abundant firing patterns can be effectively regulated by memristor- and stimulus-related parameters. Moreover, a PCB-based analog circuit employing off-the-shelf discrete circuit components is manually constructed and hardware experiments are executed. Experimental results captured from hardware measurements satisfactorily verify the numerically simulated spiking firing patterns and effectively exhibit the availability of the memristive Hodgkin–Huxley circuit in generating spiking firing patterns.

Hidden firing patterns and memristor initial condition-offset boosting behavior in a memristive Hindmarsh-Rose neuron model

Electromagnetic induction can effectively induce abundant firing patterns in neurons. In modeling a neuron model with the electromagnetic induction effect, an electromagnetic induction current is frequently added to the state equation of membrane potential. To more properly reflect the non-uniform distribution of the ions inside and outside the neuron membrane, an ideal flux-controlled memristor with sinusoidal memductance function and non-linearly modulated input is raised to depict an electromagnetic induction effect on a Hindmarsh–Rose neuron model, and thereby, a three-dimensional (3D) memristive Hindmarsh–Rose (mHR) neuron model is built in this paper. The proposed mHR neuron model possesses no equilibrium point since the involvement of the ideal flux-controlled memristor, which induces the generation of hidden dynamics. Numerical results declare that the mHR neuron model can generate abundant hidden dynamics, i.e., periodic spiking, chaotic spiking, period-doubling bifurcation route, tangent bifurcation, and chaos crisis. These hidden dynamics are much related to the memristor coupling strength and externally applied stimulus. Afterward, the memristor initial condition-offset boosting behavior is revealed. This can trigger the generation of infinite multiple coexisting firing patterns along the memristor variable coordinate. These coexisting firing patterns have identical attractor topology but different locations in the phase plane. Finally, an analog circuit is designed for implementing the mHR neuron model, and PSIM-based circuit simulation is executed. The circuit-simulated results perfectly verify the generation of hidden infinite multiple coexisting initial condition-offset boosting firing patterns in the proposed mHR neuron model.

A Scalable Neuristor Based on a Half-Wave Memristor Emulator

The neuristor based on memristors can be used to mimic synapse and neurons of biological neural systems, and it is the key unit of spiking neural networks. However, the resistance states of realistic memristors are nonvolatile, which is not conducive to mimicking the forgetting function of the brain. Given that the resistance states of memristor emulators are volatile after power down, this paper exhibits a scalable neuristor built with a half-wave memristor emulator. The proposed neuristor demonstrates four critical features for action-potential-based computing: the all-or-nothing spiking of an action potential, threshold-driven spiking, diverse periodic spiking and symmetric anti-Hebbian learning rule of spike-timing-dependent plasticity. Particularly, there are no complex shape and duration constraints on pre- and post-spikes for implementing the symmetric anti-Hebbian learning rule.

Simplification of Chua corsage memristor and hardware implementation of its neuron circuit

<sec>The Chua corsage memristor (CCM) is a voltage-controlled locally-active memristor, which has complex dynamic behaviors and potential applications in the field of neuromorphic computing. According to the DC <i>V</i>-<i>I</i> plot, the CCM can be classified as two-lobe, four-lobe, and six-lobe type. By analyzing their non-volatility and local activity, it is found that they have the same locally-active region and a common stable equilibrium. The mathematical models of the three CCMs are simplified based on the mechanism of neuromorphic behavior, namely, local activity. After the model simplification, the absolute value operation disappears, but the locally-active domain remains unchanged. For the simplified CCM, its small-signal equivalent circuit at the locally-active operating point is established, which is consistent with CCMs before being simplified. Hence, the model simplification does not change the small-signal characteristics of CCMs.</sec><sec>To further investigate the application of voltage-controlled locally-active memristor in modeling the neuromorphic behavior of neurons, the simplified CCM model is used to connect a capacitor and an inductor to construct a third-order neuron circuit. By applying theoretical analysis methods such as local activity, edge of chaos, and Lyapunov exponents, we predict the parameter domains where different neuromorphic behaviors are generated. The distribution of neuromorphic behaviors is described on a dynamic map determined by the parameters of applied voltage <i>V</i><sub>D</sub> and external inductance <i>L</i>. When the memristor is biased in the locally-active region, the system response changes among resting state, periodic spiking oscillation, and chaotic behaviors.</sec><sec>Finally, according to the simplified CCM mathematical model, the corresponding emulator circuit is designed by using three operational amplifiers, two multipliers, a current conveyor, and several resistors and capacitors. Based on the presented memristor emulator circuit, the hardware implementation of the neuron circuit is given. The experimental results verify the correctness and feasibility of the simplified CCM emulator circuit, and show that the simplified CCM-based neuron circuit can produce a variety of neuromorphic behaviors, including resting state, periodic spiking, chaotic state, bimodal response, periodic oscillation, all-or-nothing phenomenon, and spike clustering phenomenon. We expect that this work is helpful in further studying the mechanism of neuromorphic behaviors of the neuron circuit and its practical applications.</sec>

Regulation of periodic spiking by different types of calcium conductance in a mathematical model of snail RPa1 neurons

Regulation of periodic spiking by different types of calcium conductance in a mathematical model of snail RPa1 neurons

Externally-Triggered Spiking and Spiking Cluster Oscillation in Broadband Optoelectronic Oscillator

An approach to generating periodic spiking and spiking cluster in a broadband nonlinear opto-electronic oscillator (OEO) is proposed and demonstrated. Through injecting a long-duration square-wave pulse train with a repetition frequency equal to an integral multiple of the free spectral range into the OEO with a high net gain, controllable spiking and spiking cluster oscillation can be achieved when the dual-drive Mach-Zehnder modulator (DDMZM) in the OEO is biased far away from its quadrature point. The proposed scheme is verified by both numerical simulation and experiment. The results indicate that, when the DDMZM in the OEO cavity is biased near its minimum transmission point, positive spiking train with nanosecond-level pulse width is excited by the rising edge of the injected square-wave pulse train. Through further enhancing the injection strength, uniformly-spaced positive spiking fills the high-level voltage duration of each injected square-wave pulse. Hence, spiking cluster oscillation is achieved. Benefiting from its controllability, this approach can be applied in ultra-wideband (UWB) communications, electronic countermeasures, spiking coding and high-speed neuromorphic computing.

Autonomous three-dimensional oscillator with five terms: spiking oscillations generation mechanism, microcontroller implementation and controls

This paper presents a report on the microcontroller implementation of an autonomous three-dimensional oscillator with five terms (ATDOFT) and performance analysis based on partial and total amplitude controls. ATDOFT displays periodic spiking behaviors, period-tripling bifurcation to chaos, chaotic spiking attractors, coexisting attractors and bistable attractors. ATDOFT is divided into two subsystems; namely the fast and slow subsystems to investigate the mechanism of the spiking dynamics. Relying on the stability analysis based on the fast subsystem with respect to the slow variable, it is shown that the spiking oscillations present in the ATDOFT arise from the system switching between the unstable state and the stable state of the lone equilibrium point of the fast subsystem. By inserting two controller parameters into the rate equations of the ATDOFT, total and partial amplitude controls are achieved. Finally, the dynamical behaviors found in ATDOFT are validated by the microcontroller implementation.

Hopf bifurcation and phase synchronization in memristor-coupled Hindmarsh–Rose and FitzHugh–Nagumo neurons with two time delays

A memristor-coupled heterogenous neural network consisting of two-dimensional (2D) FitzHugh–Nagumo (FHN) and Hindmarsh–Rose (HR) neurons with two time delays is established. Taking the time delays as the control parameters, the existence of Hopf bifurcation near the stable equilibrium point in four cases is derived theoretically, and the validity of the Hopf bifurcation condition is verified by numerical analysis. The results show that the two time delays can make the stable equilibrium point unstable, thus leading to periodic oscillations induced by Hopf bifurcation. Furthermore, the time delays in FHN and HR neurons have different effects on the firing activity of neural network. Complex firing patterns, such as quiescent state, chaotic spiking, and periodic spiking can be induced by the time delay in FHN neuron, while the neural network only exhibits quiescent state and periodic spiking with the change of the time delay in HR neuron. Especially, phase synchronization between the heterogeneous neurons is explored, and the results show that the time delay in HR neurons has a greater effect on blocking the synchronization than the time delay in FHN neuron. Finally, the theoretical analysis is verified by circuit simulations.

Four-dimensional Hindmarsh–Rose neuron model with hidden firing multistability based on two memristors

Since memristors can be used to describe electromagnetic induction effects, this paper proposes a novel 4D HindMarsh-Rose (HR) neuron model based on two flux-controlled memristors to show complex dynamics of neuronal electrical activity. It has no equilibrium point, revealing hidden dynamical behaviors. The complex dynamics of the system are illustrated by phase portraits, the time sequences, bifurcation diagrams, and Lyapunov exponents spectra. The presented 4D HR neuron model can produce coexisting multiple hidden firing patterns, for instance, periodic spiking, chaotic spiking, transient chaotic spiking, periodic bursting, chaotic bursting, transient chaotic bursting, stochastic bursting, and transient stochastic bursting. Besides, rich nonlinear dynamics, such as anti-monotonicity and initial offset boosting, are also found. Finally, Multisim circuit simulation is performed and the results are in accordance with numerical simulation.