Three-dimensional memristive Morris–Lecar model with magnetic induction effects and its FPGA implementation

Abstract

To characterize the magnetic induction flow induced by neuron membrane potential, a three-dimensional (3D) memristive Morris–Lecar (ML) neuron model is proposed in this paper. It is achieved using a memristor induction current to replace the slow modulation current in the existing 3D ML neuron model with fast-slow structure. The magnetic induction effects on firing activities are explained by the spiking/bursting firings with period-adding bifurcation and periodic/chaotic spiking-bursting patterns, and the bifurcation mechanisms of the bursting patterns are elaborated using the fast-slow analysis method to create two bifurcation sets. In particular, the 3D memristive ML model can also exhibit the homogeneous coexisting bursting patterns when switching the memristor initial states, which are effectively illustrated by the theoretical analysis and numerical simulations. Finally, a digitally FPGA-based hardware platform is developed for the 3D memristive ML model and the experimentally measured results well verify the numerical ones.

Introduction

With the wide application of artificial neural networks, neurons and neural networks have become the research hotspot in recent years. In order to study and simulate the firing activities of neurons, researchers have proposed a large number of neuron models (Li et al. 2013). The diverse functions and topological connections of neural networks have required the design of neuron models to be higher and higher (Xu et al. 2016). Hodgkin-Huxley neuron model is one of the earlier neuron models with complex mathematical expression and too many unknown parameters, which can be used to simulate many kinds of neuron firing activities (Xu et al. 2018a, b; Hu and Liu 2019). To simplify the Hodgkin-Huxley neuron model, researchers have presented many simplified neuron models with similar features, including Morris–Lecar (ML) neuron model (Morris and Lecar 1981; Behdad et al. 2015; Lu et al. 2021; Wu et al. 2021), Hindmarsh–Rose (HR) neuron model with induced magnetic induction (Parastesh et al. 2018; Mondal et al. 2019; Bao et al. 2020a; Wang and Shi 2020; Wouapi et al. 2021), memristive Wilson neuron model (Xu et al. 2022b), Fitzhugh-Nagumo neuron model (Chen et al. 2020), Izhikevich neuron model (Izhikevich 2003), adaptive synaptic neuron model (Bao et al. 2021; Xu et al. 2022a), piece-wise linear planar neuron model (Ji et al. 2015), and bidirectional-coupled neuron model (Njitacke et al. 2022). Among them, the original two-dimensional (2D) ML neuron model is an equivalent neuromorphic circuit to reappear the voltage oscillations in calcium and potassium ion-channels of the barnacle muscle fiber, in which the time-dependent calcium and potassium conductances can be regarded as two generic memristors (Rajamani et al. 2018). Therefore, generic memristors can be used to express the time-dependent ion-channel conductances of biological neurons.

Although the generic calcium and potassium memristors are involved, the original 2D ML model cannot exhibit chaotic spiking and bursting firings that reproduce in biological neuronal system with fast and slow structures. To this end, a 3D fast-slow ML neuron model was proposed (Izhikevich 2000) and fold/Hopf and fold/homoclinic bursting firings have been disclosed thereby (Wang et al. 2008). Furthermore, with such a fast-slow neuron model, complex mode transitions from periodic/chaotic bursting to periodic/chaotic spiking or vice versa have been demonstrated (Jia et al. 2017; Zhao and Gu, 2017) and coexisting multiple-stable firing patterns have also been discovered (Bao et al. 2019). In fact, the thorough study of mode transitions and coexisting firing patterns helps to better understand the complex dynamics of the electrical activities of action potentials observed in biological experiments (Gu et al. 2014).

In addition to being used to express the ion-channel conductances, memristors can also be used to characterize the electromagnetic induction flows caused by neuron action potentials (Xu et al. 2018a, b; Usha and Subha 2019). In complex electrophysiological environments, the firing activities of biological neurons are affected by the energy consumption of electromagnetic radiation (Ma and Tang 2015). Following the theory of electromagnetic field, magnetic flux is used to decribe the effect of electromagnetic radiation (Lv et al. 2016; Wu et al. 2017), and the coupling relationship between neuron cells is euqivelently realized by memristive synapse (Xu et al. 2018a, b; Chen et al. 2019). Because of these wise considerations, plenty of memristive neuron models with magnetic induction effects have been reported in recent years (Ma et al. 2017; Ge et al. 2018; Li et al. 2022). Thus, various periodic/chaotic spiking-bursting firing behaviors were unconvered, such as hidden bursting firings, extreme multistability, spiking/bursting mode transition or pattern selection, synchronous transition, and so on. Besides, recent work showed that the memristor can also act as a synapse to reproduce the biophysical properties of the chemical synapse (Wu et al. 2022).

When the firing activities of biological neurons are related to magnetic induction flows, memristors can be introduced into the existing neuron models to simulate these magnetic induction effects. Inspired by this strategy, a novel 3D memristive synapse-based ML model was proposed by introducing a memristive synapse-induced magnetic induction current to replace the applied current in the original 2D ML model (Morris and Lecar 1981), from which periodic/chaotic spiking-bursting firings were exhibited numerically and confirmed experimentally (Bao et al. 2020c). However, when the memristive synapse possesses the tangent hyperbolic memductance, the 3D memristive synapse-based ML model cannot display homogeneous extreme multistability of coexisting bursting patterns that is encountered in the memristive HR neuron model (Bao et al. 2020b; Zhang et al. 2021). From a biological point of view, media and cells can be continuously magnetized and polarized in the process of polarization and depolarization (Ma et al. 2019). In particular, periodic firings regulate the propagation, pumping, and distribution of ions within cells, so the memristive synapse with periodic memductance can be used to estimate the polarization properties. To simulate this situation, an improved 3D memristive ML model using a memristor with sine-bounded memductance is proposed in this paper, from which homogeneous coexisting bursting patterns are thereby disclosed when switching the memristor initial states. Moreover, a digitally FPGA-based hardware platform is developed for the improved 3D memristive ML model based on a field programmable gate array (FPGA) development board (Zhang et al. 2021; Cai et al. 2022), which has strong anti-interference ability and is flexible to digital circuit design due to the advantages of FPGA technology.

Memristive ML model with magnetic induction flow

This section briefly reviews the 3D Morris–Lecar (ML) model with fast-slow structure and then proposes a 3D memristive ML model considering magnetic induction flow.

The 3D ML model with fast-slow structure

Numerous reported studies have contributed to the understanding of various properties of biological membranes. The ML model is one of those studies describing the voltage oscillations in the barnacle muscle fiber (Rajamani et al. 2018). It is a fast-spiking system described by two first-order ordinary differential equations (Morris and Lecar 1981). However, a neuron is a fast-slow burster. Using a slow linear subsystem to modulate the fast-spiking system, a 3D ML model was proposed by Izhikevich (2000). Correspondingly, the three first-order ordinary differential equations can be described by

$$\begin{array}{cc}\hfill C\frac{\text{d}V}{\text{d}t}& =g_{\text{Ca}}{M}_{\infty }(V_{\text{Ca}}-V)+g_{\text{K}}N(V_{\text{K}}-V)+g_{\text{L}}(V_{\text{L}}-V)-I\hfill \\ \hfill \frac{\text{d}N}{\text{d}t}& ={\tau }_N(N_{\infty }-N)\hfill \\ \hfill \frac{\text{d}I}{\text{d}t}& =\epsilon (V_0+V)\hfill \\ \hfill \end{array}$$ 1

Three state variables V, N, and I represent the membrane potential, opening probability of K⁺ channels, and slow modulation current, respectively. The model parameters V_Ca, V_K, and V_L denote the steady-state potentials corresponding to Ca⁺⁺, K⁺, and leak channels, respectively. The model parameters g_Ca, g_K, and g_L denote the conductances for Ca⁺⁺, K⁺, and leak pathways, respectively. The model parameter C is the membrane capacitance, V₀ is the static potential, and the model parameter ε is the very small constant to constraint the time-scale of slow modulation current.

Besides, the model parameters M_∞ and N_∞ represent the opening probabilities of Ca⁺⁺ and K⁺ channels at steady state, and the model parameter τ_N denotes the opening rate constant of K⁺ channels. These model parameters M_∞, N_∞, and τ_N are three nonlinear functions related to the membrane potential V. They can be described by

$$\begin{array}{cc}\hfill M_{\infty }& =0.5+0.5tanh[(V-V_1)/V_2]\hfill \\ \hfill N_{\infty }& =0.5+0.5tanh[(V-V_3)/V_4]\hfill \\ \hfill {\tau }_N& =(1/3)cosh[0.5(V-V_3)/V_4]\hfill \\ \hfill \end{array}$$ 2

where V₁, V₂, V₃, and V₄ are the parameters associated with M_∞ and N_∞. With such a 3D ML model given in (1), diverse parameter-dependent firing activities, such as periodic spiking/bursting and chaotic spiking/bursting, were demonstrated and their bifurcation mechanisms were elaborated in Bao et al. (2019).

The 3D memristive ML model with magnetic induction effect

It has been confirmed (Rajamani et al. 2018) that the K⁺ channel in the 3D ML model given in (1) can be imitated as a generic and volatile K⁺ channel memristor. Meanwhile, it has also been validated that memristor can be used to characterize the magnetic induction flow induced by the neuron membrane potential (Bao et al. 2020a) and the magnetic induction current generated by memristor can be severed as a slow modulation current injected into ion-channel. Thus, the 3D ML model given in (1) can be reconstructed using the magnetic induction current and a 3D memristive ML model can be thereby presented.

For the neuron membrane potential V, the magnetic induction current with an induction intensity k can be denoted as kI_M. To better characterize the magnetic induction flow, a voltage-controlled generic memristor is presented and defined by

$$\begin{array}{cc}\hfill I_M& =W(\phi )V=sin(\mathrm{\pi }\phi )V\hfill \\ \hfill \dot{\phi }& =\epsilon (V+V_0)\hfill \\ \hfill \end{array}$$ 3

where φ is a magnetic flux and W(φ) = sin(πφ) is a bounded memductance with sine nonlinearity. The parameter ε is an internal constant, indicating the time scale factor of induced electromotive force generated in finite transient period related to the characteristics of the medium (Li et al. 2022). The presented generic memristor involves a static potential V₀ that commonly exists in biological neuron (Rajamani et al. 2018). The presented generic memristor given in (3) is different from the generic memristor reported in Bao et al. (2020c). In the latter, a tangent hyperbolic memductance is employed for the generic memristor. Besides, from a biological point of view, both media and cells can be magnetized and polarized during the process of polarization and depolarization (Ma et al. 2019; Bao et al. 2020b). Therefore, the memristor with periodic memductance in (3) is used to embody the polarization properties.

In the 3D ML model given in (1), we use the magnetic induction current kI_M to replace the slow modulation current I with just opposite sign, i.e. kI_M =  − I. Indeed, this induction current flowing through memristive synapse/memristor is dependent on the intrinsic physical properties of the media/memristor. Therefore, negative or positive modulation on the excitability can be switched. Thus, a 3D memristive ML model is proposed and mathematically described by

$$\begin{array}{cc}\hfill \frac{\text{d}V}{\text{d}t}& =g_{\text{Ca}}{M}_{\infty }(V_{\text{Ca}}-V)+g_{\text{K}}N(V_{\text{K}}-V)+g_{\text{L}}(V_{\text{L}}-V)+ksin(\mathrm{\pi }\phi )V\hfill \\ \hfill \frac{\text{d}N}{\text{d}t}& ={\tau }_N(N_{\infty }-N)\hfill \\ \hfill \frac{\text{d}\phi }{\text{d}t}& =\epsilon (V_0+V)\hfill \\ \hfill \end{array}$$ 4

where the membrane capacitance C is set as 1 μF/cm² for simplicity. Referring to Bao et al. (2020c), the model parameters are determined as V_Ca = 1 mV, V_L =  − 0.5 mV, g_Ca = 1.2 mS/cm², g_K = 2 mS/cm², g_L = 0.5 mS/cm², V₀ = 0.2 mV, V₁ =  − 0.01 mV, V₂ = 0.15 mV, V₃ = 0.1 mV, and V₄ = 0.05 mV, along with the adjustable V_K (mV), k, and ε. Note that the newly added term ksin(πφ)V with positive sign is defined to ensure the induction intensity k of the magnetic induction current is positive to conform to the real physical meaning.

It should be clarified that the induction intensity k of the magnetic induction flow was not considered in Bao et al. (2020c). In this paper, however, with the consideration of the induction intensity k, the magnetic induction effect of membrane potential on the biological neuron can be studied entirely.

Equilibrium point and its stability

Let the left-hand side of the mathematical model (4) equal 0, i.e. $\text{d}V/\text{d}t=\text{d}N/\text{d}t=\text{d}\phi /\text{d}t=0$. Thus, the equilibrium point of the 3D memristive ML model can be obtained as

$$E=[-V_0,{\overline{N}}_{\infty },(1/\mathrm{\pi })\text{arcsin}(\mathrm{\Delta }/k)]$$ 5

where

$$\begin{array}{cc}\hfill {\overline{M}}_{\infty }& =0.5-0.5tanh[(V_0+V_1)/V_2]\hfill \\ \hfill {\overline{N}}_{\infty }& =0.5-0.5tanh[(V_0+V_3)/V_4]\hfill \\ \hfill {\overline{\tau }}_{\infty }& =(1/3)cosh[0.5(V_0+V_3)/V_4]\hfill \\ \hfill \end{array}$$

$$\mathrm{\Delta }=g_{\text{Ca}}{\overline{M}}_{\infty }(V_{\text{Ca}}/V_0+1)+g_{\text{K}}{\overline{N}}_{\infty }(V_{\text{K}}/V_0+1)+g_{\text{L}}(V_{\text{L}}/V_0+1)$$

Accordingly, the Jacobian matrix at the equilibrium point E is derived as

$${\mathbf{J}}_E=\left[\begin{array}{ccc}{g}_{\text{Ca}}{h}_1-g_{\text{K}}{\overline{N}}_{\infty }-g_{\text{L}}+\mathrm{\Delta }\hfill & g_{\text{K}}(V_{\text{K}}+V_0)& h_3\\ {\overline{\tau }}_{\infty }{h}_2\hfill & -{\overline{\tau }}_{\infty }& 0\\ \epsilon \hfill & 0& 0\end{array}\right]$$ 6

where

$$\begin{array}{cc}\hfill h_1& =0.5V_2^{-1}{\text{sech}}^2[(V_0+V_1)/V_2](V_{\text{Ca}}+V_0)-{\overline{M}}_{\infty }\hfill \\ \hfill h_2& =0.5V_4^{-1}{\text{sech}}^2[(V_0+V_3)/V_4]\hfill \\ \hfill h_3& =-\mathrm{\pi }k{V}_{0}cos[\text{arcsin}(\mathrm{\Delta }/k)]\hfill \\ \hfill \end{array}$$

For the determined parameters, the equilibrium point E and its eigenvalues can be numerically calculated from (5) and (6). Then, the stability for the equilibrium point can be evaluated. Denote V_K =  − 200 mV, and take the induction intensity k and inner constant ε as two adjustable parameters. When the two adjustable parameters (k, ε) are selected as different values, the equilibrium point E and its three eigenvalues along with stability are calculated from (5) and (6) and listed in Table 1. The results show that the 3D memristive ML model has two classes of equilibrium points in the parameter regions of interest, namely unstable saddle point (USP) and unstable saddle-focus (USF).

Table 1.

The equilibrium points, eigenvalues, and stabilities related to (k, ε)

(k, ε)	Equilibrium point E	Eigenvalues	Stability
(0.4, 0.02)	(− 0.2, 6.14 × 10–6, − 0.1976)	− 3.2682, 0.3889, 0.0108	USP
(0.4, 0.1)	(− 0.2, 6.14 × 10–6, − 0.1976)	− 3.2683, 0.3376, 0.0622	USP
(0.4, 0.2)	(− 0.2, 6.14 × 10–6, − 0.1976)	− 3.2684, 0.2000 ± j0.0446	USF
(1, 0.02)	(− 0.2, 6.14 × 10–6, − 0.0748)	− 3.2682, 0.3654, 0.0343	USP
(1, 0.1)	(− 0.2, 6.14 × 10–6, − 0.0748)	− 3.2686, 0.2000 ± j0.1507	USF
(1, 0.2)	(− 0.2, 6.14 × 10–6, − 0.0748)	− 3.2690, 0.2003 ± j0.2922	USF

Magnetic induction effects on firing activities

To better demonstrate the magnetic induction effects on firing activities, the initial states (V(0), N(0), φ(0)) are set as (0, 0, 0) and MATLAB fourth-order Runge–Kutta algorithm is employed for the proposed 3D memristive ML model. Note that the memristor inner constant ε is fixed as 0.02.

Magnetic induction-related bifurcation behaviors

The induction intensity k is adjusted in the region [0.3, 1.6] and the steady-state potential V_K is adjusted in the region [− 420 mV, − 180 mV]. When the parameter space constructed by k and V_K is scanned, a two-dimensional colorful diagram is measured by computing the firing spikes of membrane potential V per cycle and the result is shown in Fig. 1. The black marked by 0 represents chaotic spiking/bursting firings, the dark-red marked by 20 represents periodic bursting firings with more than 20 spikes per burst, and the other colors marked by the other numbers represent periodic spiking/bursting firings with different spikes per burst. Importantly, the chaotic spiking/bursting firings are inlayed between the periodic spiking firings and periodic bursting firings, and the periodic bursting firings are successively evolved in a period-adding bifurcation hierarchy. As can be seen, various spiking/bursting firing activities are demonstrated in Fig. 1, including periodic spiking/bursting firings, chaotic spiking/bursting firings, period-doubling bifurcation, tangent bifurcation, period-adding bifurcation, and so on. Because the reduced ML model can not exhibit chaotic dynamics (Morris and Lecar 1981; Izhikevich 2000), the chaotic spiking/bursting firing activities in the 3D memristive ML model are initiated by the magnetic induction flow of membrane potential, which just reflects the magnetic induction effect existing in biological neurons.

Fig. 1.

On the parameter space of k and V_K for fixed ε = 0.02, the two-dimensional colorful diagram for the spiking/bursting firings with period-adding bifurcation. The black represents chaotic spiking firings, the dark-red represents periodic bursting firings with more than 20 spikes per burst, and the other colors represent periodic spiking/bursting firings with different spikes per burst

The induction intensity is fixed as k = 1 and the steady-state potential is adjusted in V_K ∈ [− 420 mV, − 180 mV]. Figure 2a shows the first two Lyapunov exponents (LEs) (top) and bifurcation diagrams (bottom) of the maxima V_max of membrane potential V. When decreasing the V_K, the firing activity begins with period-1 spiking firing, and splits into period-2 spiking firing at V_K =  − 188.1 mV and period-4 spiking firing at V_K =  − 197.6 mV via twice period-doubling bifurcations. The firing activity enters into chaotic spiking firing with positive first Lyapunov exponent at V_K =  − 201.4 mV. The chaotic spiking firing ends at V_K =  − 207.4 mV via tangent bifurcation and periodic bursting firing with 3 spikes per burst then appears. Thus, a periodic window occurs in the region of V_K ∈ [− 207.4 mV, − 224.8 mV], during which a self-similar bifurcation structure can be observed. Thereafter, the firing activity goes into chaotic bursting firing with irregular spikes per burst via period-doubling bifurcations, and mutates into periodic bursting firing with 4 spikes per burst at V_K =  − 227.6 mV via tangent bifurcation. Subsequently, the firing activity maintains in periodic bursting firings in V_K ∈ [− 227.6 mV, − 420 mV] and their spikes per burst successively increase via period-adding bifurcations (Yang et al. 2018). Therefore, the presented 3D memristive ML model has dynamical transitions from periodic spiking to chaotic spiking, to chaotic bursting and to periodic bursting as decreasing the steady-state potential.

Fig. 2.

When increasing the steady-state potential V_K and induction intensity k, first two Lyapunov exponents (top) and bifurcation diagrams of the maxima V_max of membrane potential V (bottom) for fixed initial state (0, 0, 0). a Bifurcation plots for fixed k = 1 and adjusted V_K, b bifurcation plots for fixed V_K =  − 210 mV and adjusted k

Similarly, the steady-state potential is fixed as V_K =  − 210 mV and the induction intensity is adjusted in k ∈ [0.1, 1.6]. Figure 2b displays the first two LEs (top) and bifurcation diagram (bottom) of the maxima V_max of membrane potential V. When decreasing the k, the firing activity begins with period-4 spiking firing, and splits into period-8 spiking firing at k = 1.55 via period-doubling bifurcation. Thereafter, the firing activity enters into chaotic spiking firing with positive first Lyapunov exponent at k = 1.51 and mutates into maximum periodic window with period-3 at k = 1.13 via tangent bifurcation. Within the periodic window, the self-similar bifurcation structure appears. The maximum periodic window ends at k = 0.79 and the firing activity breaks into chaotic bursting firing with irregular spikes per burst therein. When further decreasing k, the periodic window with period-4 arrives at k = 0.65 and finishes at k = 0.56, and then the chaotic bursting firing arises again and quickly disappears at k = 0.49. Subsequently, the firing activity alternates between periodic and chaotic bursting firings in k ∈ [0.3, 0.49] and their spikes per burst successively increase via period-adding bifurcations. As can be observed, the 3D memristive ML model has dynamical transitions from periodic spiking to chaotic spiking, to chaotic bursting and to periodic bursting as decreasing the induction intensity.

Consequently, the results given in Fig. 2 demonstrate the rich oscillating voltage patterns in Barnacle muscle fibers. Meanwhile, they also show that the 3D memristive ML model can simulate the same firing activities as the original 3D ML model with fast-slow structure (Izhikevich 2000). From the biological point of view, this means that the 3D memristive ML model works just as well but it can better embody the magnetic induction effect in biological neurons.

Periodic/chaotic spiking-bursting patterns

According to the bifurcation plots given in Fig. 2a, for fixed k = 1, four typical values of model parameter V_K are selected as − 190 mV, − 206 mV, − 227 mV, and − 320 mV, respectively. The time-domain waveforms of periodic/chaotic spiking-bursting patterns of the membrane potential V along with the magnetic flux φ are depicted in Fig. 3.

Fig. 3.

Time-domain waveforms of periodic/chaotic spiking-bursting patterns of the membrane potential V along with the magnetic flux φ. a Periodic spiking pattern at V_K =  − 190 mV, b chaotic spiking pattern at V_K =  − 206 mV, c Chaotic bursting pattern at V_K =  − 227 mV, d periodic bursting pattern at V_K =  − 320 mV

Interestingly, the time-domain waveforms of bursting patterns in Fig. 3 show the block structured behaviors, whether they are periodic or chaotic bursting patterns. Besides, the time-domain waveforms of magnetic flux φ are concurrently given in Fig. 3, demonstrating the bounded behaviors of magnetic flux φ. In particular, when the bursting behaviors occur, the magnetic flux φ becomes a slow variable with respect to the fast variable V. Therefore, the fast-slow effect can be found in the 3D memristive ML model, similar to that existed in the original 3D ML model with fast-slow structure described in (1) (Izhikevich 2000), reflecting the availability of the 3D memristive ML model in imitating neuronal action potential. Besides, the dynamic amplitude of the bounded memductance W(φ) = sin(πφ) increases as V_K decreases, in other words, the memristor magnetic induction effect decreases as the steady-state potential V_K increases.

Correspondingly, the phase plane plots of periodic/chaotic spiking-bursting patterns in the plane of magnetic flux φ and membrane potential V are shown in Fig. 4. As can be seen, the periodic spiking pattern with two spikes per burst arrives at V_K =  − 190 mV, the chaotic spiking pattern with irregular spikes appears at V_K =  − 206 mV, the chaotic bursting pattern with irregular spikes per burst occurs at V_K =  − 227 mV, and the periodic bursting pattern with eight spikes per burst happens at V_K =  − 320 mV. Therefore, the selection of the spiking/bursting firing patterns can be effectively realized by changing the steady-state potential in the 3D memristive ML model.

Fig. 4.

Phase plane plots of periodic/chaotic spiking-bursting patterns in the plane of magnetic flux φ and membrane potential V. a Periodic spiking pattern at V_K =  − 190 mV, b chaotic spiking pattern at V_K =  − 206 mV, c chaotic bursting pattern at V_K =  − 227 mV, d periodic bursting pattern at V_K =  − 320 mV

Bifurcation mechanisms of bursting patterns

When the periodic/chaotic spiking-bursting patterns occur in the 3D memristive ML model, the magnetic flux φ becomes a slow variable with respect to the fast variable V. In this case, the 3D memristive ML model described in (4) can be divided as a fast subsystem consisting of the first two equations and a slow subsystem constructed by the last equation. Thus, the fast-slow analysis method can be used to expound the bifurcation mechanisms of bursting patterns.

The fast subsystem can be rewritten as

$$\begin{array}{cc}\hfill \frac{\text{d}V}{\text{d}t}& =g_{\text{Ca}}{M}_{\infty }(V_{\text{Ca}}-V)+g_{\text{K}}N(V_{\text{K}}-V)+g_{\text{L}}(V_{\text{L}}-V)+ksin(\mathrm{\pi }\phi )V\hfill \\ \hfill \frac{\text{d}N}{\text{d}t}& ={\tau }_N(N_{\infty }-N)\hfill \\ \hfill \end{array}$$ 7

where the term sin(πφ) in the first equation of (7) can be taken as a constant parameter with slow change.

Let the left-hand side of the mathematical model (7) equal 0, i.e.$\text{d}V/\text{d}t=\text{d}N/\text{d}t=0$, and denote

$$\begin{array}{cc}\hfill \tilde{M}& =0.5+0.5tanh[(\tilde{V}-V_1)/V_2]\hfill \\ \hfill \tilde{N}& =0.5+0.5tanh[(\tilde{V}-V_3)/V_4]\hfill \\ \hfill \tilde{\tau }& =(1/3)cosh[0.5(\tilde{V}-V_3)/V_4]\hfill \\ \hfill \end{array}$$ 8

The equilibrium point of the fast subsystem in (7) can be obtained as

$$E_{\text{FS}}=(\tilde{V},\tilde{N})$$ 9

where $\tilde{V}$ is yielded by solving the following equation

$$g_{\text{Ca}}\tilde{M}(V_{\text{Ca}}-\tilde{V})+g_{\text{K}}\tilde{N}(V_{\text{K}}-\tilde{V})+g_{\text{L}}(V_{\text{L}}-\tilde{V})+ksin(\mathrm{\pi }\phi )\tilde{V}=0$$ 10

Thus the fast-nullcline curve can be readily constructed, which is related to the slow variable φ.

Correspondingly, the Jacobian matrix at E_FS can be described as

$${\mathbf{J}}_{\text{EFS}}=\left[\begin{array}{cc}{g}_{\text{Ca}}{\tilde{h}}_1-g_{\text{K}}\tilde{N}-g_{\text{L}}+ksin(\mathrm{\pi }\phi )\hfill & g_{\text{K}}(V_{\text{K}}-\tilde{V})\\ \tilde{\tau }{\tilde{h}}_2\hfill & -\tilde{\tau }\end{array}\right]$$ 11

where

$$\begin{array}{cc}\hfill {\tilde{h}}_1& =0.5V_2^{-1}{\text{sech}}^2[(\tilde{V}-V_1)/V_2](V_{\text{Ca}}-\tilde{V})-\tilde{M}\hfill \\ \hfill {\tilde{h}}_2& =0.5V_4^{-1}{\text{sech}}^2[(\tilde{V}-V_3)/V_4]\hfill \\ \hfill \end{array}$$

And the characteristic quadratic equation is derived as

$$P_{\text{EFS}}(\lambda )={\lambda }^2+{\xi }_1\lambda +{\xi }_2$$ 12

where

$$\begin{array}{cc}\hfill {\xi }_1& =\tilde{\tau }-g_{\text{Ca}}{\tilde{h}}_1+g_{\text{K}}\tilde{N}+g_{\text{L}}-ksin(\mathrm{\pi }\phi )\hfill \\ \hfill {\xi }_2& =-\tilde{\tau }[g_{\text{Ca}}{\tilde{h}}_1-g_{\text{K}}\tilde{N}-g_{\text{L}}+g_{\text{K}}{\tilde{h}}_2(V_{\text{K}}-\tilde{V})+ksin(\mathrm{\pi }\phi )]\hfill \\ \hfill \end{array}$$

Hence, the temporary parameters ξ₁ and ξ₂ are closely related to $\tilde{V}$ and φ, indicating that the eigenvalues of (12) are dependent on the slow variable φ.

When ξ₂ = 0, a fold bifurcation (FB) occurs in the fast subsystem (7). According to the relations (10) and (12), the FB set can be obtained as

$$\text{FB:}\left\{\begin{array}{c}{g}_{\text{Ca}}\tilde{M}(V_{\text{Ca}}-\tilde{V})+g_{\text{K}}\tilde{N}(V_{\text{K}}-\tilde{V})+g_{\text{L}}(V_{\text{L}}-\tilde{V})+ksin(\mathrm{\pi }\phi )\tilde{V}=0\hfill \\ g_{\text{Ca}}{\tilde{h}}_1-g_{\text{K}}\tilde{N}-g_{\text{L}}+g_{\text{K}}{\tilde{h}}_2(V_{\text{K}}-\tilde{V})+ksin(\mathrm{\pi }\phi )=0\hfill \end{array}\right)$$ 13

And when ξ₁ = 0 and ξ₂ > 0, a Hopf bifurcation (HB) appears in the fast subsystem (7). According to the relations (10) and (12), the HB set can be solved as

$$\text{HB:}\left\{\begin{array}{c}{g}_{\text{Ca}}\tilde{M}(V_{\text{Ca}}-\tilde{V})+g_{\text{K}}\tilde{N}(V_{\text{K}}-\tilde{V})+g_{\text{L}}(V_{\text{L}}-\tilde{V})+ksin(\mathrm{\pi }\phi )\tilde{V}=0\hfill \\ \tilde{\tau }-g_{\text{Ca}}{\tilde{h}}_1+g_{\text{K}}\tilde{N}+g_{\text{L}}-ksin(\mathrm{\pi }\phi )=0\hfill \\ g_{\text{Ca}}{\tilde{h}}_1-g_{\text{K}}\tilde{N}-g_{\text{L}}+g_{\text{K}}{\tilde{h}}_2(V_{\text{K}}-\tilde{V})+ksin(\mathrm{\pi }\phi )<0\hfill \end{array}\right)$$ 14

Here the conditional equation $\tilde{\tau }=(1/3)cosh[0.5(\tilde{V}-V_3)/V_4]\ge 1/3$ should be considered. Further, the HB set in (14) can be simplified as

$$\text{HB:}\left\{\begin{array}{c}{g}_{\text{Ca}}(\tilde{M}{V}_{\text{Ca}}-\tilde{M}\tilde{V}-{\tilde{h}}_1\tilde{V})+g_{\text{K}}\tilde{N}{V}_{\text{K}}+\tilde{\tau }\tilde{V}+g_{\text{L}}{V}_{\text{L}}=0\hfill \\ \tilde{\tau }-g_{\text{Ca}}{\tilde{h}}_1+g_{\text{K}}\tilde{N}+g_{\text{L}}-ksin(\mathrm{\pi }\phi )=0\hfill \\ \tilde{\tau }+g_{\text{K}}{\tilde{h}}_2(V_{\text{K}}-\tilde{V})<0\hfill \end{array}\right)$$ 15

The model parameters are considered as V_Ca = 1 mV, V_L =  − 0.5 mV, g_Ca = 1.2 mS/cm², g_K = 2 mS/cm², g_L = 0.5 mS/cm², V₀ = 0.2 mV, V₁ =  − 0.01 mV, V₂ = 0.15 mV, V₃ = 0.1 mV, V₄ = 0.05 mV, ε = 0.02, and k = 1. When the steady-state potential is adjusted in V_K ∈ [− 420 mV, − 180 mV], the FB and HB sets described by (13) and (15) are depicted in the φ–V_K plane and their numerical curves are shown in Fig. 5. As can be revealed, Fig. 5 shows two FBs and one HB in V_K ∈ [− 420 mV, − 241 mV] but only two FBs in V_K ∈ (− 241 mV, − 180 mV]. With the bifurcation sets in Fig. 5, the bifurcation mechanisms of periodic/chaotic bursting patterns in the 3D memristive ML model are elaborated thereby.

Fig. 5.

For V_K ∈ [− 420 mV, − 180 mV], the FB and HB sets of the fast subsystem in the φ–V_K plane, where the typical model parameters are used and the two adjustable model parameters are fixed as ε = 0.02 and k = 1

When the steady-state potential is set to V_K =  − 227 mV and V_K =  − 320 mV respectively, the phase plane plots of chaotic and periodic bursting patterns in the φ− V plane and the fast-nullcline curve AB provided by the fast subsystem in the φ–$\tilde{V}$ plane are simultaneously shown in Fig. 6. Here, the fast-nullcline curve AB is depicted by the equilibrium point E_FS, the FBP and HBP represent fold and Hopf bifurcation points respectively, and blue UNF, magenta SNF, red USP, and green SNP represent unstable node-focus having two complex roots with positive parts, stable node-focus having two complex roots with negative parts, unstable saddle point with one positive root and one negative root, and stable node point with two negative roots, respectively. Thus, two types of bifurcation behaviors can be demonstrated in the fast subsystem.

Fig. 6.

Two types of bifurcation behaviors and their bifurcation mechanisms. a Chaotic fold/fold bursting pattern at V_K =  − 227 mV, b periodic Hopf/fold/fold bursting pattern at V_K =  − 320 mV

As shown in Fig. 6a, the curve AB is divided as three line segments by two FBPs (FBP1 and FBP2) with the slow evolution of variable φ, among which the FBP1 and FBP2 locate at φ =  − 0.0075 and − 0.0863 respectively, and the three line segments include blue UNF, red USP, and green SNP. Thus, a fold/fold bifurcation with chaotic bursting pattern occurs in the 3D memristive ML model. Similarly, as shown in Fig. 6b, the curve AB is divided as four line segments by one HBP and two FBPs (FBP1 and FBP2) with the slow evolution of variable φ, among which the HBP locates at φ =  − 0.0557 and the FBP1 and FBP2 locate at φ =  − 0.0398 and − 0.0867 respectively, and the four line segments contain blue UNF, magenta SNF, red USP, and green SNP. Thus, a Hopf/fold/fold bifurcation with periodic bursting pattern appears in the 3D memristive ML model.

Memristor initial-switched coexisting bursting patterns

What is significantly interesting is that the 3D memristive ML model can exhibit homogeneous coexisting bursting patterns switched by memristor initial states for a given set of model parameters. The determined model parameters can be referred to as those used in Fig. 6.

Mechanism of memristor initial-switched coexisting behaviors

Integrating the third equation of the 3D memristive ML model given in (3) from − ∞ to τ, there yields

$$\phi ={\int }_{-\infty }^{\tau }\epsilon (V_0+V)\text{d}\xi =\phi (0)+{\int }_0^{\tau }\epsilon (V_0+V)\text{d}\xi$$ 16

where $\phi (0)={\int }_{-\infty }^0\epsilon (V_0+V)\text{d}\xi$, representing the initial state of the magnetic flux φ. Thus, the reconstructed memristive ML model can be described as

$$\begin{array}{cc}\hfill \frac{\text{d}V}{\text{d}t}& =g_{\text{Ca}}{M}_{\infty }(V_{\text{Ca}}-V)+g_{\text{K}}N(V_{\text{K}}-V)+g_{\text{L}}(V_{\text{L}}-V)+ksin[\mathrm{\pi }\phi (0)+\mathrm{\pi }{\int }_0^{\tau }\epsilon (V_0+V)\text{d}\xi ]·V\hfill \\ \hfill \frac{\text{d}N}{\text{d}t}& ={\tau }_N(N_{\infty }-N)\hfill \\ \hfill \end{array}$$ 17

indicating that the initial state φ(0) can be expressed explicitly.

Suppose

$$\phi (0)={\phi }_{00}\pm 2m$$ 18

where m is a natural number and φ₀₀ is a compensation initial state that satisfies the following condition

$$-\mathrm{\pi }<{\phi }_{00}+{\int }_0^{\tau }\epsilon (V_0+V)\text{d}\xi <\mathrm{\pi }$$ 19

The sine memductance W(φ) has the property of periodic cycle, i.e.

$$\begin{array}{cc}\hfill sin[\mathrm{\pi }{\phi }_0+\mathrm{\pi }{\int }_0^{\tau }\epsilon (V_0+V)\text{d}\xi ]& =sin[{\phi }_{00}\pm 2m+\mathrm{\pi }{\int }_0^{\tau }\epsilon (V_0+V)\text{d}\xi ]\hfill \\ \hfill & =sin[{\phi }_{00}+\mathrm{\pi }{\int }_0^{\tau }\epsilon (V_0+V)\text{d}\xi ]\hfill \\ \hfill \end{array}$$

Hence, the reconstructed model described in (17) is invariant under the transformation given in (18). However, the asymptotic motions initiated from different attracting regions of memristor initial state settle on the disconnected attractors with different positions. These disconnected attractors are homogeneous and their memristor initial states corresponding to two adjacent attractors have the fixed offset 2 along the φ coordinate. Therefore, the 3D memristive ML model has the freedom for offset-switching the memristor initial state φ(0), resulting in that the infinitely many homogeneous coexisting bursting patterns appear as m tends to infinity.

Homogeneous coexisting bursting patterns

It is demonstrated from the aforementioned theoretical analysis that the memristor initial-switched coexisting behaviors emerge in the 3D memristive ML model. To display the striking memristor initial-switched coexisting phenomenon, the intuitions about two kinds of homogeneous coexisting bursting patterns are depicted by the phase plane plots in the φ–V plane, as shown in Fig. 7, where the initial states (V(0), N(0)) are fixed as (0, 0) and the memristor initial state φ(0) is set to − 2, 0, 2, and 4, respectively. As can be seen, the chaotic or periodic attractors of bursting patterns are homogeneous and their gaps between two adjacent attractors are all the fixed offset 2 along the φ coordinate, indicating the occurrence of the homogeneous extreme multi-stability.

Fig. 7.

For different memristor initial states of φ(0) with fixed (V(0), N(0)) = (0, 0), numerically simulated homogeneous coexisting bursting patterns with offset 2 along the φ coordinate. a Homogeneous coexisting chaotic bursting patterns for V_K =  − 227 mV, b homogeneous coexisting periodic bursting patterns for V_K =  − 320 mV

It should be noticed that in the homogeneous coexisting bursting patterns, each chaotic or periodic flow initiated from a certain initial state region settles on a determined attractor position. Thus, the coexisting bursting patterns are boosted to different attractor positions along the φ coordinate as the memristor initial state is changed, which is similar to those reported in Bao et al. (2020b).

The phase plane plots of homogeneous coexisting bursting patterns in Fig. 7 manifest that the coexisting behaviors are closely dependent on the memristor initial state. To measure the coexisting behaviors in the initial state planes, the attraction basins for V_K =  − 227 mV and V_K =  − 320 mV in the φ(0)–V(0) initial plane with N(0) = 0 are shown in Fig. 8, where the orange blocks represent the attracting regions of chaotic bursting patterns, the cyan blocks represent the attracting regions of periodic bursting patterns, and the dark-blue blocks represent the divergent regions of unbounded behaviors. Obviously, the two attraction basins shown in Fig. 8a, b have almost similar attracting regions with a stripe structure.

Fig. 8.

Attraction basins for V_K =  − 227 mV and V_K =  − 320 mV in the φ(0)–V(0) initial plane with N(0) = 0. a The attraction basin for V_K =  − 227 mV, b the attraction basin for V_K =  − 320 mV

Besides, the attracting regions of chaotic and periodic bursting patterns are independently disconnected and their gaps between two adjacent color blocks are all the fixed offset 2 along the φ(0) coordinate. The gap with fixed offset 2 agrees with the above theoretical analysis. Particularly, the wider the range of memristor initial state, the greater the number of the attracting regions, to infinity, indicating the coexistence of homogeneous infinitely many bursting patterns.

FPGA-based hardware experiments

In analog circuit experiments, due to model idealizations and parasitic parameters, various experimental errors common occur, leading to unbelievable experimental results. To avoid the experimental errors, many digital neuron circuits based on FPGA were developed and diverse neuron firing activities were really revealed experimentally (Cai et al. 2022; Zhang et al. 2021; Pu et al. 2021).

Discrete modeling based on fourth-order Runge–Kutta method

With the 3D memristive ML model given in (4), a discrete model can be obtained using fourth-order Runge–Kutta method. Denote V_n, N_n, and φ_n as the sampling values of V, N, and φ at the beginning of the n-th interval and V_n+1, N_n+1, and φ_n+1 as the sampling values of V, N, and φ at the beginning of the (n + 1)-th interval.

First, we define

$$\begin{array}{cc}\hfill M_1& =0.5+0.5tanh[(V_n-V_1)/V_2]\hfill \\ \hfill N_1& =0.5+0.5tanh[(V_n-V_3)/V_4]\hfill \\ \hfill {\tau }_1& =(1/3)cosh[0.5(V_n-V_3)/V_4]\hfill \\ \hfill \end{array}$$

There yields

$$\begin{array}{cc}\hfill {\delta }_{V1}& =g_{\text{Ca}}{M}_1(V_{\text{Ca}}-V_n)+g_{\text{K}}{N}_n(V_{\text{K}}-V_n)+g_{\text{L}}(V_{\text{L}}-V_n)+ksin(\mathrm{\pi }{\phi }_n)V_n\hfill \\ \hfill {\delta }_{N1}& ={\tau }_1(N_1-N_n)\hfill \\ \hfill {\delta }_{\phi 1}& =\epsilon (V_0+V_n)\hfill \\ \hfill \end{array}$$ 20

Then we define

$$\begin{array}{cc}\hfill M_2& =0.5+0.5tanh[(V_n+0.5h{\delta }_{V1}-V_1)/V_2]\hfill \\ \hfill N_2& =0.5+0.5tanh[(V_n+0.5h{\delta }_{V1}-V_3)/V_4]\hfill \\ \hfill {\tau }_2& =(1/3)cosh[0.5(V_n+0.5h{\delta }_{V1}-V_3)/V_4]\hfill \\ \hfill \end{array}$$

There gives

$$\begin{array}{cc}\hfill {\delta }_{V2}& =g_{\text{Ca}}{M}_2(V_{\text{Ca}}-V_n-0.5h{\delta }_{V1})+g_{\text{K}}(N_n+0.5h{\delta }_{N1})(V_{\text{K}}-V_n-0.5h{\delta }_{V1})\hfill \\ \hfill & +g_{\text{L}}(V_{\text{L}}-V_n-0.5h{\delta }_{V1})+ksin[\mathrm{\pi }({\phi }_n+0.5h{\delta }_{\phi 1})](V_n+0.5h{\delta }_{V1})\hfill \\ \hfill {\delta }_{N2}& ={\tau }_2(N_2-N_n-0.5h{\delta }_{N1})\hfill \\ \hfill {\delta }_{\phi 2}& =\epsilon (V_0+V_n+0.5h{\delta }_{V1})\hfill \\ \hfill \end{array}$$ 21

Further we define

$$\begin{array}{cc}\hfill M_3& =0.5+0.5tanh[(V_n+0.5h{\delta }_{V2}-V_1)/V_2]\hfill \\ \hfill N_3& =0.5+0.5tanh[(V_n+0.5h{\delta }_{V2}-V_3)/V_4]\hfill \\ \hfill {\tau }_3& =(1/3)cosh[0.5(V_n+0.5h{\delta }_{V2}-V_3)/V_4]\hfill \\ \hfill \end{array}$$

There gets

$$\begin{array}{cc}\hfill {\delta }_{V3}& =g_{\text{Ca}}{M}_3(V_{\text{Ca}}-V_n-0.5h{\delta }_{V2})+g_{\text{K}}(N_n+0.5h{\delta }_{N2})(V_{\text{K}}-V_n-0.5h{\delta }_{V2})\hfill \\ \hfill & +g_{\text{L}}(V_{\text{L}}-V_n-0.5h{\delta }_{V2})+ksin[\mathrm{\pi }({\phi }_n+0.5h{\delta }_{\phi 2})](V_n+0.5h{\delta }_{V2})\hfill \\ \hfill {\delta }_{N3}& ={\tau }_3(N_3-N_n-0.5h{\delta }_{N2})\hfill \\ \hfill {\delta }_{\phi 3}& =\epsilon (V_0+V_n+0.5h{\delta }_{V2})\hfill \\ \hfill \end{array}$$ 22

Finally, we define

$$\begin{array}{cc}\hfill M_4& =0.5+0.5tanh[(V_n+h{\delta }_{V3}-V_1)/V_2]\hfill \\ \hfill N_4& =0.5+0.5tanh[(V_n+h{\delta }_{V3}-V_3)/V_4]\hfill \\ \hfill {\tau }_4& =(1/3)cosh[0.5(V_n+h{\delta }_{V3}-V_3)/V_4]\hfill \\ \hfill \end{array}$$

There has

$$\begin{array}{cc}\hfill {\delta }_{V4}& =g_{\text{Ca}}{M}_4(V_{\text{Ca}}-V_n-h{\delta }_{V3})+g_{\text{K}}(N_n+h{\delta }_{N3})(V_{\text{K}}-V_n-h{\delta }_{V3})\hfill \\ \hfill & +g_{\text{L}}(V_{\text{L}}-V_n-h{\delta }_{V3})+ksin[\mathrm{\pi }({\phi }_n+h{\delta }_{\phi 3})](V_n+h{\delta }_{V3})\hfill \\ \hfill {\delta }_{N4}& ={\tau }_4(N_4-N_n-h{\delta }_{N3})\hfill \\ \hfill {\delta }_{\phi 4}& =\epsilon (V_0+V_n+h{\delta }_{V3})\hfill \\ \hfill \end{array}$$ 23

By combining (20), (21), (22), and (23), the discrete model of the 3D memristive ML model can be established as

$$\begin{array}{cc}\hfill V_{n+1}& =V_n+h({\delta }_{V1}+2{\delta }_{V2}+2{\delta }_{V3}+{\delta }_{V4})/6\hfill \\ \hfill N_{n+1}& =N_n+h({\delta }_{N1}+2{\delta }_{N2}+2{\delta }_{N3}+{\delta }_{N4})/6\hfill \\ \hfill {\phi }_{n+1}& ={\phi }_n+h({\delta }_{\phi 1}+2{\delta }_{\phi 2}+2{\delta }_{\phi 3}+{\delta }_{\phi 4})/6\hfill \\ \hfill \end{array}$$ 24

where h is the sampling interval, δ_Vj, δ_Nj, and δ_φj (j = 1, 2, 3, 4) are the temporary variables.

Digitally FPGA-based implementation

Xilinx XC7Z020 FPGA board is used as a digital development platform to realize the discrete model in (24). A 14-bit dual-channel AD9767 D/A converter (DAC) is installed through an expandable interface. Verilog hardware description language (HDL) is used to customize the discrete model program in Vivado compilation environment, which can provide rich intellectual property (IP) cores for floating-point operations like adder, subtractor, multiplier, exponent, and sinusoid.

The Verilog HDL program has a hierarchical structure, as shown in Fig. 9. It includes a main controller and four layers of IP instances, customized functions, algorithm iterations, and output modules. When the FPGA board is switched on, the main controller initializes the entire logic circuits and begins iterating over the initial states of V_n, N_n, and φ_n. Then, three intermediate variable units M_j, N_j, τ_j (j = 1, 2, 3, 4) are successively generated and three function units F_V, F_N, F_φ are thereby constructed using the IP instances and their nonlinear functions.

Fig. 9.

Hierarchical structure of Verilog HDL program for the FPGA-based implementation of the 3D memristive ML model. Abbreviations: Adder (ADD), Subtractor (SUB), Multiplier (MUL), Exponent (EXP), and Sinusoid (SIN)

In the program for realizing the fourth-order Runge–Kutta method, the customized function vector F = [F_V, F_N, F_φ] is loaded in the form of time multiplexing and used for deriving four vectors K₁ = [δ_V1, δ_N1, δ_φ1], K₂ = [δ_V2, δ_N2, δ_φ2], K₃ = [δ_V3, δ_N3, δ_φ3], and K₄ = [δ_V4, δ_N4, δ_φ4], respectively. The main controller manages the four-time loading sequences of F and switches the subsequences of K₁, K₂, K₃, and K₄ using the cycle counter. It takes 243 system clocks in total to complete the four-time loadings. Accordingly, Y_n+1 = [V_n+1, N_n+1, φ_n+1] can be obtained by weighing the calculation results.

To acquire the experimental results on oscilloscope, it is necessary to convert the floating-point numbers used in the calculation process into 14-bit integers, and then convert them into analog signals via the DAC module. Before being converted into integers, the experimental data streams can fully meet the input requirements of the DAC module and make full use of its output range by shifting and amplifying the original floating-point numbers. Finally, the integer data streams combined with the synchronous clock are fed into the DAC module to output the analog ones.

Hardware experimental measurements

The initial states are set as (V(0), N(0), φ(0)) = (0, 0, 0) and the sampling interval is determined as h = 0.01. When the model parameters used in the numerical results in Fig. 3 are adopted, the time-domain waveforms of the membrane potential V overlapped with the magnetic flux φ experimentally measured from the digitally FPGA-based platform is shown in Fig. 10. Accordingly, the experimentally measured phase plane plots of periodic/chaotic spiking-bursting patterns in the plane of magnetic flux φ and membrane potential V are shown in Fig. 11. The experimental results manifest that the periodic and chaotic spiking patterns occur at V_K =  − 190 mV and − 206 mV respectively, and chaotic and periodic bursting patterns appear at V_K =  − 227 mV and − 320 mV respectively. They are completely consistent with the numerical results in Figs. 3 and 4, which verify the feasibility and correctness of the FPGA-based hardware implementation.

Fig. 10.

Experimentally measured time-domain waveforms of periodic/chaotic spiking-bursting patterns of the membrane potential V along with magnetic flux φ. a periodic spiking pattern at V_K =  − 190 mV, b chaotic spiking pattern at V_K =  − 206 mV, c chaotic bursting pattern at V_K =  − 227 mV, d periodic bursting pattern at V_K =  − 320 mV

Fig. 11.

Experimentally measured phase plane plots of periodic/chaotic spiking-bursting patterns in the plane of magnetic flux φ and membrane potential V. a Periodic spiking pattern at V_K =  − 190 mV, b chaotic spiking pattern at V_K =  − 206 mV, c chaotic bursting pattern at V_K =  − 227 mV, d periodic bursting pattern at V_K =  − 320 mV

The initial states (V(0), N(0)) are fixed as (0, 0) and the memristor initial state φ(0) is set to − 2, 0, 2, and 4, respectively. When V_K =  − 227 mV and V_K =  − 320 mV, the homogeneous coexisting bursting patterns can be measured experimentally and their phase plane plots in the φ–V plane are displayed in Fig. 12. Note that the red, blue and green phase plane plots in Fig. 12a are preset with offsets of 1.9 V, − 1.9 V, − 3.8 V, respectively, along the φ coordinate; while the red, blue and green phase plane plots in Fig. 12b are preset with offsets of 1.95 V, − 1.95 V, − 3.9 V, respectively, along the φ coordinate. Therefore, the memristor initial-switched coexisting behaviors can also be observed from the digitally FPGA-based platform.

Fig. 12.

For different memristor initial states of φ(0) with fixed (V(0), N(0)) = (0, 0), experimentally measured homogeneous coexisting bursting patterns with offset 2 along the φ coordinate. a Homogeneous coexisting chaotic bursting patterns for V_K =  − 227 mV, b homogeneous coexisting periodic bursting patterns for V_K =  − 320 mV

Conclusion

This paper proposed a 3D memristive ML neuron model, which is achieved using a memristor induction current to replace the slow modulation current in the existing 3D ML neuron model with fast-slow structure. By several numerical methods, the magnetic induction effects on firing activities were explained by the spiking/bursting firings with period-adding bifurcation and periodic/chaotic spiking-bursting patterns. Using the fast-slow analysis method to create the fold/Hopf bifurcation sets, the bifurcation mechanisms of the bursting patterns were elaborated, and the chaotic fold/fold bursting pattern and periodic Hopf/fold/fold bursting pattern were discovered. In particular, through the theoretical analysis and numerical simulations, the homogeneous coexisting bursting patterns switched by the memristor initial states were revealed in the 3D memristive ML model. In addition, a digitally FPGA-based hardware platform was developed for the 3D memristive ML model, from which the experimental results were measured to verify the numerical ones. With the digital neuron circuit, various periodic/chaotic spiking-bursting patterns in neurons can be acquired physically, which will be beneficial to many applications of neuron-based artificial neural networks.

The present paper has extended the previous work on the field by considering the magnetic induction effects generated in the 3D memristive ML model. On the one hand, the current work mainly studies the signal processing of single neuron, which has potential significance to provide a reference for the design of large-scale integrated memristive neural network and an idea for realizing non-invasive control of neural circuit array using external magnetic field (Yao et al. 2020). On the other hand, the 3D memristive ML neuron model can fully express various firing patterns of biological neurons by selecting appropriate model parameters and initial states, especially the characteristics of spiking firings, which has potential significance for the design of neuromorphic spiking neural network (SNN) architectures (Rao et al. 2022). Moreover, when multiple such memristive ML models are coupled, they can express a certain size of neural network, which can certainly be used to simulate physiological and pathological processes, such as Parkinson’s diseases and epileptic seizures (Majhi et al. 2019). All these works deserve our future study.

Data availability

The data used in this research work are available from the authors by reasonably request.

Declarations

Conflict of interests

The authors declare that they have no conflict of interest.