Setting a double-capacitive neuron coupled with Josephson junction and piezoelectric source

Abstract

Perception of voice means acoustic electric conversion in the auditory system, and changes of external magnetic field can affect the neural activities by taming the channel current via some field components including memristor and Josephson junction. Combination of two capacitors via an electric component is effective to describe the physical property of artificial cell membrane, which is often used to reproduce the characteristic of electric activities in cell membrane. Involvement of two capacitive variables for two capacitors in the neural circuit can discern the effect of field diversity in the media in two sides of the cell membrane in theoretical way. A Josephson junction is used to couple a piezoelectric neural circuit composed of two capacitors, one inductor and one nonlinear resistor. Field energy is mainly kept in the capacitive and inductive components, and it is obtained and converted into dimensionless energy function. The Hamilton energy function in an equivalent auditory neuron is verified by using the Helmholtz theorem. Noisy excitation on the neural circuit can be detected via the Josephson junction channel and similar stochastic resonance is detected by regulating the noise intensity, as a result, the average energy reaches a peak value under stochastic resonance. An adaptive law controls the bifurcation parameter, which is relative to the membrane property, and energy shift controls the mode selection during continuous growth of the bifurcation parameter. That is, external energy injection derived from acoustic wave or magnetic field will control the energy level, and then suitable firing patterns are controlled effectively.

Introduction

Nervous system can perceive a variety of external signals, and these physical, chemical and even mechanic stimuli can be converted into equivalent electric currents for activating neurons in different functional regions. In a practical way, artificial neurons and neural circuits are coupled to control mechanical devices including arms and legs (Mbeunga et al. 2021; Ngongiah et al. 2023, 2024; Pearson et al. 2006; Wadden and Ekeberg 1998). The patch clamp technology provides reliable approach of experimental data for membrane potentials under different electrophysiological stimuli, and these sampled data are often used to verify the reliability of theoretical neuron models (Amiri et al. 2013; Druckmann et al. 2008; Khaliq et al. 2003; Linne and Jalonen 2014; Geit et al. 2008). Neural activities can be explored in mathematical, physical, and biological models. Some of the neuron models are helpful to clarify the dynamical mechanism of neural disease in the brain, and the relation between neural activities and neural diseases has been explored in the recent works (Liu et al. 2022; Xie et al. 2024; Yu et al. 2023a). Biological neurons show distinct adaption to external stimuli, which can be encoded and filtered within specific frequency band. For example, external stimuli beyond the threshold can wake up the quiescent neurons. Activation of autapse (Qin et al. 2014; Song et al. 2019; Wang et al. 2017; Yao et al. 2019) can excite or suppress the neural activities by shunting channel current along an auxiliary loop with adjustable time delay or feedback gain. Local distribution of autapse in a neural network can develop defects or heterogeneities, which have distinct impact on wave propagation in the neural networks (Baysal et al. 2021; Ge et al. 2019; Ma et al. 2016; Yilmaz et al. 2016). Besides the cooperation between neurons via synaptic connection (Breakspear et al. 2003; Jacquir et al. 2006; Sacu 2024) or field coupling (Lv et al. 2019; Wen et al. 2022; Zhou et al. 2022; Zhou and Wei 2021), the astrocytes (Allen 2014; Schipke and Kettenmann 2004) plays important roles in regulating neural activities and then astrocyte-coupled-neuron models (Calim et al. 2021; Petit and Magistretti 2016) are suggested to explore the mode selection of electrical activities in nervous system. From dynamical viewpoint, synaptic plasticity (Kotaleski and Blackwell 2010; O’Donnell 2023) enables neurons select time-varying synaptic intensity or coupling intensity, and the neurons are connected with adaptive growth in the coupling intensity. Two recent works (Wu et al. 2023a; Yang et al. 2023a) explained the controllability and self-adaptive property of biophysical neurons from energy aspect, they claimed that energy diversity regulates the coupling intensity between neurons (Hou et al. 2023a, 2023b; Sun et al. 2023; Wang et al. 2023; Yang and Ma 2023) and energy level in single neuron is controlled by its intrinsic energy level, which four different average energy levels account for four main firing modes (Li and Xu 2024) and energy shift is effective to control mode transition in neural activities. Different firing patterns of neurons subjecting to an electromagnetic field show that the electrochemical synapses facilitate richer variety of dynamical behavior (Zandi-Mehran et al. 2020). It is also found that the long-term dynamics is typically irregular and weakly correlated independent of the network architecture (Klinshov et al. 2024).

The work reviews the topics related to signal propagation in complex networks and describes the difference between the microscopic and the macroscopic scale (Ji et al. 2023). Analogy to the idea gas, the work hold the view that interconnected many-body systems characterized by macroscopic properties that cannot be directly deduced from those of their microscopic constituents (Artime et al. 2024). Implement of equivalent neural circuits (Wei et al. 2017; Wu et al. 2023b, 2024; Zhang et al. 2020) is effective to detect and predict possible characteristic of electrical activities as those biological neurons in the nervous system. Considering the physical property of electromagnetic field in the media beside two sides of the cell membrane, continuous pumping and stochastic diffusion of intracellular ions seem like the charge and discharge on capacitors, and the propagation of ions induces current effect for generating time-varying magnetic field. Therefore, capacitor and inductor are basic electromagnetic components for building a simple neural circuit, which can be excited to presenting similar firing patterns as biological neurons. Considering the physical effect and energy conversion ability, more functional components including nonlinear resistor (NR), memristor (Corinto et al. 2015; Jeong and Shi 2018; Mohammad et al. 2016), Josephson junction (JJ) (Finger 2000; Mishra et al. 2021), thermistor, photocell (Zhang and Ma 2021), and piezoelectric ceramic (PC) (Guo et al. 2021; Zhu et al. 2023) can be incorporated into additive branch circuits, so that physical field can be estimated in reasonable way. As a result, the memristor-coupled neural circuits (Bao et al. 2023a, 2023b; Jia et al. 2024a; Li et al. 2021; Lin et al. 2023) can be activated to obtain memristive neuron, which can estimate the electromagnetic induction and even radiation, and these memristive neurons (Xie et al. 2024; Shen et al. 2022; Wang et al. 2016; Wu et al. 2022; Yang et al. 2023b) can exchange signals under field coupling when synaptic connection are suppressed greatly.

Two sides of the cell membrane have different gradient field and energy diversity is dependent on the material property of cell membrane. In most of the previous works, these neurons prefer to use one capacitive variable for membrane potential and external stimuli are often converted into trans-membrane current with different forms even the effect of ion channels and electromagnetic induction are considered. Therefore, it is interesting to estimate the effect of material property of the cell membrane. For example, the outer membrane and inner membrane have different capacitive properties, and it can be considered as a double-layer membrane (Jia et al. 2023, 2024b), which two capacitors are connected with different components, and the difference of output voltages from the capacitors is suitable for describing the membrane potential of a neuron. In Ref (Yang et al. 2024), two linear circuits connected by a nonlinear resistor is used to describe the neural activities for a nonlinear cell membrane. In particular, Guo et al. ((2023) suggested a memristive cell membrane and the membrane parameter is regulated by the energy flow. That is, reliable neuron models should estimate all physical effects in clear way, and then energy injection can control the mode selection and collective behaviors of the neural networks (Gao et al. 2021; Hussain et al. 2022; Majhi et al. 2019; Wang et al. 2020), which developing climate states, regular patterns and effective wave propagation and energy exchange. Readers can find possible suggestions about neuro dynamics and activation of functional neural circuit from physical viewpoints in the reviews and references therein (Lin et al. 2021; Ma 2023; Ma et al. 2019; Wang and Ma 2018; Wang et al. 2019). To discover the relation between neuro dynamics and neural disease detection, intelligence of artificial neurons, please see recent review works (Liu et al. 2022; Quaranta et al. 2020; Yang et al. 2021).

In this work, a PC is used to capture external voice and the converted signals are used to excite a neural circuit coupled with a JJ, which is sensitive to changes of external magnetic field, the energy function is defined and verified from theoretical way. Similar stochastic resonance (Li et al. 2024a; Liu et al. 2024; Yu et al. 2023b) is induced, and its occurrence can be predicted by calculating the distribution of signal to noise ratio (SNR) and average energy function for this functional neuron. Finally, one membrane parameter is regulated under an adaptive growth and mode selection is controlled completely.

Model, energy and adaptive control scheme

The capacitive property accounts for the energy characteristic in a capacitor and even in cell membrane, which two sides cover the inner and external static electrical field of a biological neuron. Therefore, the circuit approach for double-layer neuron can be inspired by the Chua-memristive circuits (Chua 1971) by using two capacitors in the circuit. External stimuli just inject energy into the neuron and it is encoded via the cell membrane by generating different channel currents, which can affect the membrane potential by regulating the excitability. To simulate the effect of the magnetic field, a JJ which is formed by two overlapping superconducting films separated by a thin insulating barrier in physics (Willsch et al. 2024), is paralleled with the Chua circuit while the inductor is replaced by the PC. During energy injection or release, the energy proportion between capacitive, inductive and even memristive channels is regulated synchronously. As a result, energy balance is broken to induce suitable firing patterns and the average energy level keeps close to a constant for keeping specific firing modes with time. JJ can perceive changes of external magnetic field by inducing additive phase error, which can adjust the junction channel current effectively. Incorporating a PC into a neural circuit can capture external acoustic wave as auditory neuron. In Fig. 1, a PC neuron coupled with a JJ is suggested, and the output voltages from two capacitors are used to approach the potentials for the outer and inner cell membrane. The resistor connected to two capacitors is linear, and it indicates that the cell membrane is linear and isotropic along the membrane surface.

Fig. 1.

A PC neural circuit coupled by a JJ. PC, C₁, C₂, NR, R, R_S denotes PC source, capacitors, nonlinear resistor, linear resistors, respectively

The PC and the JJ are used to perceive external voice and changes of external magnetic field. The channel currents for the JJ and the NR are approached by

$$\left\{\begin{array}{c}i_{\mathit{JJ}}^{}=I_{C}sin\phi ;\\ i_{\mathit{NR}}^{}=-\frac{1}{\rho }\left(V-\frac{1}{2}\frac{V^2}{V_0}-\frac{1}{3}\frac{V^3}{V_0^2}\right);\\ \end{array}\right)$$ 1

where i_JJ represents the tunneling current of the JJ, I_C is the critical current and φ is the superconducting phase difference across the junction. i_NR denotes the current flowing through the NR with a smooth nonlinearity, the parameter ρ is the resistance for the I-V curve in the linear region for the device, V₀ is the reverse saturation voltage of the device, V is the voltage across the NR (Khibnik et al. 1993). When external acoustic wave is adjustable, the output voltage V_PC from the PC device in Fig. 1 can be selected as a kind of time-varying function, for simplicity, periodic PC signals can be considered as exciting voltage source. The circuit equations are obtained to explore the relation between voltages for two capacitors and the phase error across JJ in Eq. (2).

$$\left\{\begin{array}{c}{C}_1\frac{dV_{C1}^{}}{\mathit{dt}}=\frac{V_{C2}^{}-V_{C1}^{}}{R}-i_{\mathit{NR}}^{}-I_{C}sin\phi ;\\ C_2\frac{dV_{C2}^{}}{\mathit{dt}}=\frac{V_{C1}^{}-V_{C2}^{}}{R}+\frac{V_{\mathit{PC}}^{}-V_{C2}^{}}{R_S};\\ \frac{d\phi }{\mathit{dt}}=\frac{2e}{ħ}V_{C1}^{}.\\ \end{array}\right)$$ 2

where the parameters e and ħ are the electron charge and the reduced Planck constant, respectively (Josephson 1965); V_C1 is the voltage across the capacitor C₁ ( equal to the voltage of the NR); V_C2 is the voltage for the capacitor C₂. Similar scale transformation is applied to obtain equivalent and dimensionless variables and controllable parameters as follows.

$$\left\{\begin{array}{c}x=\frac{V_{C1}^{}}{V_0},y=\frac{V_{C2}^{}}{V_0},z=\phi ,\tau =\frac{t}{R{C}_2},\alpha =\frac{C_2}{C_1},\xi =\frac{R}{\rho };\\ \beta =\frac{I_{C}R}{V_0},\gamma =\frac{R}{R_S},{\mu }_S=\frac{RV_{\mathit{PC}}^{}}{R_S{V}_0},\mu =\frac{2\text{e}{V}_{0}R{C}_2}{ħ}.\\ \end{array}\right)$$ 3

The selection of membrane ratio α for capacitive parameters controls the switch between outer membrane and inner membrane because outer membrane often holds higher capacitance. Therefore, the capacitor C₁ links to outer membrane at α < 1, and C₂ is relative to inner membrane. For α > 1, it means that the outer membrane is relative to capacitor C₂ and acoustic wave is perceived by the outer membrane directly. The parameter μ_s measures the current of the PC. Inserting the dimensionless parameters and variables into Eq. (1) and Eq. (2), a PC neuron sensitive to external magnetic field is obtained by

$$\left\{\begin{array}{c}\frac{\mathit{dx}}{d\tau }=\alpha \left[\left(\xi -1\right)x+y-\frac{1}{2}\xi x^2-\frac{1}{3}\xi x^3-\beta sinz\right];\\ \frac{\mathit{dy}}{d\tau }=x-\left(1+\gamma \right)y+{\mu }_S;\\ \frac{\mathit{dz}}{d\tau }=\mu x.\\ \end{array}\right)$$ 4

Therefore, the functional neuron with double capacitive variables for cell membrane can be excited by the changes of μ_s. Any changes of magnetic field will disturb the phase error for the JJ and the variable z is modified to regulate the membrane potentials (x, y, x − y) synchronously. Indeed, the field energy is mainly kept in the capacitor, and the physical field energy and its dimensionless form are estimated by

$$\left\{\begin{array}{c}W=W_1+W_2=\frac{1}{2}{C}_1{V}_{C1}^2+\frac{1}{2}{C}_2{V}_{C2}^2;\\ H=\frac{1}{C_1{V}_0^2}\left(H_1+H_2\right)=\frac{1}{2\alpha }{x}^2+\frac{1}{2}{y}^2.\\ \end{array}\right)$$ 5

On the other hand, the Hamilton energy function for the neuron in Eq. (4) can be obtained by using the Helmholtz theorem, which the nonlinear oscillator model is rewritten in a vector form.

$$\begin{array}{cc}\hfill \left[\begin{array}{c}\frac{\mathit{dx}}{d\tau }\\ \frac{\mathit{dy}}{d\tau }\\ \frac{\mathit{dz}}{d\tau }\\ \end{array}\right]=& \left[\begin{array}{c}\alpha (\xi -1)x+\alpha y-\frac{1}{2}\alpha \xi x^2-\frac{1}{3}\alpha \xi x^3-\alpha \beta sinz\\ x-(1+\gamma )y+{\mu }_s\\ \mu x\\ \end{array}\right]\hfill \\ \hfill =& \left[\begin{array}{c}-\alpha y\\ x\\ \mu x\\ \end{array}\right]+\left[\begin{array}{c}\alpha (\xi -1)x-\frac{1}{2}\alpha \xi x^2-\frac{1}{3}\alpha \xi x^3-\alpha \beta sinz+2\alpha y\\ {\mu }_s-(1+\gamma )y\\ 0\\ \end{array}\right]\hfill \\ \hfill =& \left[\begin{array}{ccc}0& -\alpha & -\mu \alpha \\ \alpha & 0& 0\\ \mu \alpha & 0& 0\end{array}\right]\left[\begin{array}{c}\frac{x}{\alpha }\\ y\\ 0\\ \end{array}\right]+\left[\begin{array}{ccc}{\alpha }^2[\xi (-\frac{1}{2}x-\frac{1}{3}{x}^2+1)-1-\frac{\beta }{x}sinz+\frac{2y}{x}]& 0& 0\\ 0& \frac{{\mu }_s}{y}-(1+\gamma )& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}\frac{x}{\alpha }\\ y\\ 0\\ \end{array}\right].\hfill \\ \hfill \end{array}$$ 6

The energy function H for Eq. (6) meets the criterion as follows

$$\left\{\begin{array}{c}-\alpha y\frac{\partial H}{\partial x}+x\frac{\partial H}{\partial y}+\mu x\frac{\partial H}{\partial z}=0;\\ \frac{\mathit{dH}}{d\tau }=(\xi -1)x^2-\frac{1}{2}\xi x^3-\frac{1}{3}\xi x^4-\beta xsinz-(1+\gamma )y^2+{\mu }_{s}y\\ =\mathrm{\nabla }{H}^T{F}_C(x,y,z).\\ \end{array}\right)$$ 7

Indeed, the second formula in Eq. (5) satisfies the criterion in Eq. (7), and any changes in the membrane ratio α due to energy injection or mechanical stimuli will adjust the energy value and even distinct mode transition can be induced. In presence of external fluctuation of magnetic field, the phase error for the JJ will be disturbed as follows

$$\left\{\begin{array}{c}\frac{\mathit{dx}}{d\tau }=\alpha \left[\left(\xi -1\right)x+y-\frac{1}{2}\xi x^2-\frac{1}{3}\xi x^3-\beta sinz\right];\\ \frac{\mathit{dy}}{d\tau }=x-\left(1+\gamma \right)y+{\mu }_S;\\ \frac{\mathit{dz}}{d\tau }=\mu x+S_{\mathit{ext}}^{}.\\ \end{array}\right)$$ 8

In Eq. (8), the disturbance from of external magnetic field, S_ext, can be in a constant or noisy form. For noisy excitation from magnetic field, its statistical property with zero average and intensity D is given in

$$<S_{\mathit{ext}}^{}(\tau )>=0,<S_{\mathit{ext}}^{}(\tau )S_{\mathit{ext}}^{}({\tau }^{\prime })>=D\delta (\tau -{\tau }^{\prime }).$$ 9

S_ext(τ) means Gaussian white noise with zero average and intensity D. Where δ is the Dirac function. By taming the noise intensity D, SNR is changed accompanying with distinct shift in the average energy (energy levels) as follows

$$SNR=h\frac{f_p}{\mathrm{\Delta }f};<H>=\frac{1}{{\tau }_f}{\int }_0^{f}H(\tau )d\tau \approx \frac{1}{N}\sum _{i=1}^N{H}_i.$$ 10

where h, f_p and Δf measure the peak height of power spectrum, specific frequency relative to h, and peak width corresponding to half of the peak value in the power spectrum mapped from time series for membrane potential by using fast Fourier transform (FFT). The subscript i indicates ith energy value, and the running time for calculation τ_f = N*Δτ, N is iterations and Δτ means time step in numerical approach. The symbol < * > means average value for a variable within a transient period. As mentioned above, external disturbance can change the energy level and then mode transition is induced, as a result, shape deformation in the cell membrane under energy flow enables the membrane parameter growth in an adaptive way.

$$\left\{\begin{array}{c}\frac{d\alpha }{d\tau }=\sigma ·\alpha ·\vartheta (H-\lambda );\vartheta (P)=1,P\ge 0,\vartheta (P)=0,P<0;\\ H=\frac{1}{2\alpha }{x}^2+\frac{1}{2}{y}^2.\\ \end{array}\right)$$ 11

The gain σ controls the growth of intrinsic parameter for the cell membrane and ϑ(P) is a Heaviside function. When the energy level is beyond a threshold λ, the bifurcation parameter α keeps continuous growth and mode transition can be induced for keeping possible energy level. For simplicity, the PC source is controlled by a periodic acoustic wave and the converted signal is selected with μ_S = Acos(ωτ), which can inject energy and promote energy exchange in the neural circuit. Indeed, the electromagnetic field energy results from the static distribution and diffusion of intracellular ions, and any shape deformation in the cell membrane and ion channels will modify the energy level and some corresponding parameters are changed synchronously. Therefore, other controllable parameters can be applied to similar control law presented in Eq. (11), and the energy threshold or energy proportion can be used as control parameters.

Numerical results and discussion

External acoustic wave is converted into an equivalent electric stimulus on the cell membrane, and external energy is captured and encoded. The media is polarized and magnetized when electric stimuli are imposed on a neuron and cardiac tissue, and then the excitability is changed to modulate the firing patterns in neural activities. For these oscillator-like models, numerical results can be obtained by using fourth Rung-Kutta algorithm on Matlab platform, in presence of noisy disturbance, Euler forward algorithm can be used find numerical results easily. In Fig. 2, the amplitude of the PC source is adjusted for bifurcation analysis and detection of chaos by calculating the largest Lyapunov exponent (LLE).

Fig. 2.

Distribution of peak values from membrane potentials a and largest Lyapunov exponents b. Setting parameters α = 2, β = 0.8, ξ = 1.29, γ = 4.6, μ = 0.2, ω = 1.2, and initial values (0.2, 0.1, 0.1)

Mode transition is dependent on the stimulus intensity and the firing patterns can show chaos by applying suitable amplitudes in the PC forcing. When chaos appears, positive LLE is detected and the sampled time series for membrane potentials will present more dense peaks. Furthermore, the Hamilton energy for the neuron presenting different firing patterns is plotted in Fig. 3.

Fig. 3.

Evolution of membrane potential a, c, e and energy function H b, d, f in Eq. (5). For a, b periodic patterns, A = 0.01; c, d bursting patterns, A = 0.72; e, f chaotic firing, A = 1.4. Parameters are fixed at α = 2, β = 0.8, ξ = 1.29, γ = 4.6, μ = 0.2, ω = 1.2, and external forcing μ_S = Acos(ωτ)

In presence of regular firing patterns, the neuron often keeps higher average energy values by triggering periodic, spiking and even bursting in the electric activities. However, the average energy of the neuron holds lower values under chaotic discharge and the firing patterns become irregular with time. The angular frequency of the PC source is also modified to detect mode transition in the electrical activities, and the dependence of LLE on this forcing frequency is plotted in Fig. 4.

Fig. 4.

Distribution of peak values from membrane potentials a and largest Lyapunov exponents b. Setting parameters α = 2, β = 0.8, ξ = 1.29, γ = 4.6, μ = 0.2, A = 1.4

From Fig. 4, it is demonstrated that further increase of the forcing frequency will induce chaotic patterns even the LLE is beyond zero with small values. In Fig. 5, changes of membrane potential and the energy function are presented when the neuron is activated to show two kinds of firing activities.

Fig. 5.

Evolution of membrane potential a, c and energy function H b, d in Eq. (5). For a, b periodic patterns, ω = 1; c, d chaotic patterns, ω = 1.75. Setting parameters α = 2, β = 0.8, ξ = 1.29, γ = 4.6, μ = 0.2, A = 1.4

The results in Fig. 5 are similar to the results in Fig. 3, changes of membrane potentials are accompanied with energy release, and the chaotic neuron used to keep lower average energy values than a neuron presenting in periodic oscillation in the electric activities. External magnetic field can modify the neural activities by taming the phase error and its channel current along the JJ because of magnetization in the media. When the neuron is exposed to stable magnetic field, the additive phase error in the JJ is considered as a constant and the channel current shows some shifts. In Fig. 6, dependence of firing modes on constant magnetic field is calculated.

Fig. 6.

Bifurcation diagram a and LLE distribution b under constant magnetic field S_ext = G. The parameters are fixed at α = 2, ω = 1.2, β = 0.8, ξ = 1.29, γ = 4.6, μ = 0.2, A = 1.4

With increase of the intensity of external magnetic field, the neural activities show transition from periodic to chaos accompanying intermittent emergence of periodic patterns. Furthermore, two kinds of firing patterns and the evolution of corresponding energy functions with time are plotted in Fig. 7.

Fig. 7.

Evolution of membrane potential a, c and energy function H b, d in Eq. (5). For a, b periodic patterns, G = 0.72; c, d chaotic patterns, G = 2. Setting parameters α = 2, ω = 1.2, β = 0.8, ξ = 1.29, γ = 4.6, μ = 0.2, A = 1.4

During mode transition from periodic patterns to chaotic patterns, the energy in the neuron fluctuated with different amplitudes and the average energy values also show slight diversity. It is interesting to discuss similar case that external magnetic field is fluctuated with time, for example, S_ext = Bcos(fτ). As a result, the additive phase error between two terminals of JJ becomes time-varying, and the channel current sin(z) has a modulation on the excitability of the neuron. Any changes of the amplitude or frequency in S_ext = Bcos(fτ) account for the fluctuations of external magnetic field, which is converted into appropriate channel current along the JJ in the neural circuit. In Fig. 8, bifurcation analysis and changes of LLE are plotted.

Fig. 8.

Bifurcation diagram a and distribution of LLE b in presence of periodic magnetic field S_ext = Bcos(fτ). Setting parameters α = 2, ω = 1.2, β = 0.8, ξ = 1.29, γ = 4.6, μ = 0.2, A = 1.4, f = 1

Time-varying magnetic field has distinct impact on the mode transition in this PC neuron, and the JJ channel is effective to discern changes from external magnetic field by inducing more different firing patterns with time. The evolution of energy level and membrane potentials is plotted in Fig. 9.

Fig. 9.

Sampled time series for membrane potentials a, c and energy function b, d in presence of changeable magnetic field. For a, b bursting neuron, B = 1.01; c, d chaotic neuron, B = 0.07. Setting parameters α = 2, ω = 1.2, β = 0.8, ξ = 1.29, γ = 4.6, μ = 0.2, A = 1.4, f = 1

From Fig. 9, distinct mode transition predicts potential energy shift in the neuron, and the average Hamilton energy shows some diversities when time-varying magnetic field is applied to disturb the neural activities, which chaotic state is relative to lower average energy values. The frequency of external magnetic field is estimated by plotting the LLE and distribution of peak values for membrane potentials in Fig. 10.

Fig. 10.

Bifurcation diagram a and distribution of LLE b in presence of periodic magnetic field S_ext = Bcos(fτ). Setting parameters α = 2, ω = 1.2, β = 0.8, ξ = 1.29, γ = 4.6, μ = 0.2, A = 1.4, B = 0.07

When the frequency of external magnetic field is further increased, the cell membrane is excited to present chaotic characteristic and positive LLE is detected. That is, high frequency excitation from the external magnetic field can break regular firing patterns in the neural circuit and it indicates that neurons are sensitive to magnetic field stimuli with high frequencies. The changes of energy values for the neuron presenting different neural activities are presented in Fig. 11.

Fig. 11.

Sampled time series for membrane potentials a, c and energy function b, d in presence of changeable magnetic field. For a, b bursting neuron, f = 0.3; c, d chaotic neuron, f = 0.4. Setting parameters α = 2, ω = 1.2, β = 0.8, ξ = 1.29, γ = 4.6, μ = 0.2, A = 1.4, B = 0.07

The results in Fig. 11 are consistent with the showing in Fig. 10, and the selection of firing modes is relative to the energy levels. Transition from bursting states and periodic discharge to chaotic states means irregular energy release, and the chaotic neuron will present low value for average energy. It is interesting to discern the stochastic resonance by applying noisy radiation, which the external magnetic field is fluctuated in noisy term as Gaussian white noise. The distributions of SNRs and average energy < H > are estimated in Fig. 12 by selecting different values for noise intensity D.

Fig. 12.

Dependence of SNR a, c and < H > b, d on the intensity D for noisy magnetic field. For a, c noisy excitation is imposed on the inner membrane; b, d noise excites the outer membrane. Setting parameters α = 3.6, ω = 1.2, β = 0.8, ξ = 1.29, γ = 4.6, μ = 0.2, A = 1.4

From Fig. 12, a peak with maximal value can be found in the curve for SNRs and the average energy values < H > by adjusting the intensity of noisy disturbance carefully. It indicates the appearance of stochastic resonance and then the neurons show high regularity in the neural activities, and its average energy obtains a maximal value at this noise intensity. However, it requires some diversity for the noise intensity inducting SR when noise is used to excite the inner and outer membrane, respectively.

As presented in Eq. (5), the energy function H for the neuron is relative to the ratio of membrane parameters and the two capacitive variables (x, y). Any changes of α means shape deformation of the cell membrane due to energy injection or mechanical stimuli, the dependence of firing patterns and energy value on the bifurcation parameter α is plotted in Figs.13, 14 and 15 by applying different material condition with suitable gain and energy threshold in Eq. (11).

Fig. 13.

Bifurcation diagram of peak values a LLE b and average energy c < H > with changing the capacitive ratio α. Setting parameters ω = 1.2, β = 0.8, ξ = 1.29, γ = 4.6, μ = 0.2, A = 1.4, initials (0.2, 0.1, 0.1)

Fig. 14.

Evolution of membrane a, c and energy function H b, d under different capacitive ratios for α. For a, b periodic neuron, α = 1.32; c, d chaotic neuron, α = 1.46. Setting parameters ω = 1.2, β = 0.8, ξ = 1.29, γ = 4.6, μ = 0.2, A = 1.4

Fig. 15.

Evolution of membrane potential a Hamilton energy b and growth of capacitive parameter ratioα c. Setting parameters ω = 1.2, β = 0.8, ξ = 1.29, γ = 4.6, μ = 0.2, A = 1.4, σ = 0.01, λ = 0.9, and initials (0.2, 0.1, 0.1, 1.64)

It is found that continuous change of capacitive ratio relative to changes of membrane parameters will induce distinct transition in firing modes of neural activities, and the average energy also shows decrease with further increase of the ratio values (larger α). For better illumination, energy shift and changes of membrane potentials are plotted in Fig. 14 by fixing different values for capacitive ratio α.

It is confirmed that changes of the membrane parameters (C₁, C₂) will modify the capacitance ratio due to shape deformation of cell membrane, energy and its firing patterns are controlled synchronously. That is, continuous energy injection will have impact on the membrane parameters; as a result, the capacitive ratio shows possible growth in adaptive way during energy changes for selecting suitable firing modes in the neural activities. In Fig. 15, mode transition induced by the growth of membrane parameters and shift of energy level are calculated following the adaptive criterion in Eq. (11).

When energy is injected in the form of PC stimuli or noisy excitation from the external magnetic field, the firing patterns are adjusted accompanying with synchronous shift in the neuron energy. During the absorption and release of energy in the neuron, membrane parameters are adjusted to keep suitable energy level in adaptive way and then it reaches to a saturation value for keeping stable firing mode and energy level.

In a summary, functional components such as PC and JJ can be connected to capacitors for building a reliable neural circuit. From physical viewpoint, these external physical stimuli just inject energy into the neural circuit and media. Therefore, some material parameters are changed under shape deformation for keeping suitable energy level. In presence of noisy excitation, SNR can be induced to keep high regularity in neural activities, and maximal average energy value (high power). The suggestion and proposal of the adaptive law for one membrane ratio parameter accounts for the self-adaptive regulation mechanism in neural activities, that is, adaptive changes of some intrinsic parameters are effective to capture and shunt energy in the neurons for keeping suitable energy level and firing modes. In fact, other functional electric components can be connected to these neural circuits with double capacitive components for building functional neurons with double-layer membrane. For example, phototube can be used to encode external illumination by generating photocurrent (Li et al. 2024b). Applying similar scheme to other nonlinear circuits, similar functional neural circuits and functional neurons with nonlinear membranes can be created. However, it is challenge to explore similar synchronization stability and formation of spiral waves (Ding et al. 2023; Hu et al. 2024) in the network composed of these double-capacitive neurons. In practical way, synchronous incorporation of specific components including phototube, piezoelectric ceramic, JJ, thermistor and even memristor can enhance the sensing ability of these neural circuits, which can be further used to control electromechanical arms by applying different physical signals. Readers also can design similar digital circuits and approach equivalent maps for further signal processing. Energy value in the neural circuit can’t be detected in direct way and approach of energy value for the neuron model depends on synchronous confirmation of all variables. Therefore, for the mentioned adaptive control law in this work, voltage function as V² can be used to control the parameter growth because voltage and membrane potential are often detectable.

Conclusions

In this work, energy function for a PC neural circuit coupled with a JJ is defined and explained from physical viewpoint. A piezoelectric-JJ is incorporated into a neural circuit with two capacitive variables, which are relative to the potentials and electromagnetic field for two sides of the cell membrane. Stochastic resonance is induced and detected by calculating the SNR distribution and dependence of average energy on noise intensity. The average energy obtains high value accompanying with peak value for SNR at moderate noise intensity. Energy injection due to periodic or noisy disturbance can break the energy balance between outer membrane and inner membrane for the neuron, and shape deformation of cell membrane also induces shift of intrinsic parameter. Therefore, our suggested adaptive growth law for intrinsic parameters is effective to clarify the adaptive control mechanism for mode transition and energy shift under external stimuli. In this way, functional neurons have more chance and controllability to trigger most suitable neural activities and energy values in presence of multi-channel stimuli. This double-capacitive neuron is more suitable to characterize the energy and mode selection of neural activities, and self-adaption is clarified from physical aspect.

Funding

Innovative Research Group Project of the National Natural Science Foundation of China, 62361037, Chunni Wang

Data availability

The data are available upon reasonable request.

Declarations

Conflict of interest

The authors declare that they have no conflict of interest with this publication.