Energy and synchronization between two neurons with nonlinear coupling

Abstract

Consensus and synchronous firing in neural activities are relative to the physical properties of synaptic connections. For coupled neural circuits, the physical properties of coupling channels control the synchronization stability, and transient period for keeping energy diversity. Linear variable coupling results from voltage coupling via linear resistor by consuming certain Joule heat, and electric synapse coupling between neurons derives from gap junction connection under special electrophysiological condition. In this work, a voltage-controlled electric component with quadratic relation in the i–v (current–voltage) is used to connect two neural circuits composed of two variables. The energy function obtained by using Helmholtz theorem is consistent with the Hamilton energy function converted from the field energy in the neural circuit. Chaotic signals are encoded to approach a mixed signal within certain frequency band, and then its amplitude is adjusted to excite the neuron for detecting possible occurrence of nonlinear resonance. External stimuli are changed to trigger different firing modes, and nonlinear coupling activates changeable coupling intensity. It is confirmed that nonlinear coupling behaves functional regulation as hybrid synapse, and the synchronization transition between neurons can be controlled for reaching possible energy balance. The nonlinear coupling is helpful to keep energy diversity and prevent synchronous bursting because positive and negative feedback is switched with time. As a result, complete synchronization is suppressed and phase lock is controlled between neurons with energy diversity.

Introduction

For a biological neuron, inner electric field energy and magnetic field energy is exchanged during the propagation of intracellular and extracellular ions, and membrane potential is adjusted under external stimuli. For a couple or more neurons, diversity in electromagnetic field energy forces the creation and growth of synaptic connections and thus they can reach fast energy balance in the neural network (Torrealdea et al. 2006; Zhou et al. 2022a; Xie et al. 2022, 2023; Wang et al. 2022). For example, Zhou et al. (2022a) claimed that adaptive growth of electric synapse results from energy diversity between neurons. The electric synapse (Curti and O’Brien 2016; Bennett 1997; Xie et al. 2021a; Bennett and Zukin 2004; Zandi-Mehran et al. 2020) activates its coupling regulation under special condition and it becomes transient because continuous consumption of Joule heat can induce temperature effect on neural activities. Bidirectional coupling via electric synapses often provides a fast energy balance by applying variables error on the nonlinear oscillators in the form of negative feedback. From a dynamical viewpoint, the electric synapse coupling just induces linear variable coupling of membrane potentials (Bennett 2000; Zhou et al. 2021a; Velazquez and Carlen 2000; Gerasimova et al. 2015). On the other hand, the chemical synapse coupling (Balenzuela and García-Ojalvo 2005; Shafiei et al. 2020; Smith and Pereda 2003; Kundu et al. 2019; Hu and Cao 2016) keeps continuous regulation on the collective electric activities in neurons and it is approached by equivalent memristive synapse connecting two neurons (Wu et al. 2022a). Smith and Pereda (2003) confirmed that chemical synapse has impact on the activation of nearby electric synapse. From physical viewpoint, the release of neurotransmitter and activation of Calcium accounts for the functional regulation from chemical synapse, and field coupling is triggered to connect neurons for reaching energy balance between neurons. Therefore, synchronous firing patterns can be controlled under chemical synapse coupling, which is accompanied with field coupling via electromagnetic field (Yao and Wang 2022, 2021; Yao et al. 2021; Zhou et al. 2022b; Xu et al. 2019). For example, Yao and Wang (2022, 2021), Yao et al. (2021) suggested that hybrid synapse coupling can be approached by activating capacitive and inductive field coupling, which is realized by connecting capacitor, inductor and even memristor in parallel or in series. In fact, the field coupling can be considered as nonlinear coupling via hybrid synapse (Sun et al. 2013; Yu et al. 2017; Calim et al. 2020; Uzuntarla 2019; Xu et al. 2021) and it has certain advantage than the simple electric synapse coupling by consuming large Joule heat during the energy propagation along the coupling channels. Considering the physical approach and circuit implement, these hybrid synapses can be considered a kind of field coupling. In a practical way, the circuit realization and implement of hybrid synapse can be designed by using combination of capacitor, inductor, resistor and memristor and even nonlinear resistor, and nonlinear coupling is activated to connect the equivalent neural circuits. In presence of resistance of artificial synapse, the nonlinear electric component in the coupling channels can consume a little Joule heat and it also emits energy flow because it can be considered as an active component. Therefore, the channel current becomes nonlinear, and nonlinear coupling (Gieseler et al. 2014; Wang et al. 2010; Petereit and Pikovsky 2017; Wei et al. 2019; Chithra and Raja 2017) is switched to regulate the synchronous behaviors between chaotic oscillators. As reported in Wei et al. (2019), the coupling intensity is regulated in adaptive way and the stability of synchronization in the network is controllable.

Reliable neural circuits coupled by specific electric components can reproduce certain biophysical function of biological neurons in nervous system. For example, a phototube is activated to excite a simple RLC circuit (resistor–inductor–capacitor), this neural circuit is sensitive to light and can be considered as an artificial light-sensitive neuron as visual neuron (Xie et al. 2021a, b). The involvement of thermistor into nonlinear circuit can perceive external temperature because the channel current across the thermistor is dependent on the temperature and this shunted current can regulate the charge and discharge of capacitor in this circuit (Xu et al. 2020; Xu and Ma 2021). Furthermore, activation of memristive channel can enhance the biophysical function of neurons (Wu et al. 2022b, 2020) and electromagnetic field energy can be defined theoretically. In particular, electric field variable (Wu et al. 2019) is supplied into the Hindmarsh–Rose neuron model (González-Miranda 2007; Ochs and Jenderny 2021; Cai et al. 2021) and external electric field is applied to control the mode selection in electric activities. In Ref. (Cai et al. 2021), an equivalent neural circuit is proposed to mimic the dynamical property of electric activities produced in the HR neuron. A special current can well explain the enhanced firing along with seizure induced by inhibitory interneuron (Wang et al. 2023), and it is helpful to avoid seizure. The memristive neurons show distinct controllability because the external magnetic field can be captured by regulating the memristive current and the firing modes are controlled effectively (Zhang et al. 2018; Bao et al. 2021; Chen et al. 2021; Pu et al. 2021; Rajagopal et al. 2019). For example, Zhang et al. (2018) suggested a scheme to design memristive neuron with lower energy consumption. Rajagopal et al. (2019) presented a new memristive neuron with fractional order and the effect of electromagnetic induction is estimated. In addition, Josephson junction (JJ) can perceive external magnetic field, and its involvement into neural circuit can be used to control the neural behaviors and similar stochastic resonance can be induced under noisy disturbance in the magnetic field (Zhang et al. 2020a, b; Dana et al. 2006; Njitacke et al. 2022a). Considering the distinct physical properties of JJ and memristors, more additive branch circuits are connected to the neural circuits to enhance the ability for perceiving physical signals, and then these biophysical neurons become more controllable because external physical stimuli can be converted into equivalent channel currents, which regulate the membrane potential and firing modes synchronously. For more neurons, these specific components can be used as functional synapse to connect neural circuits, and the coupling channels are controllable because of nonlinear relation for the voltage and current.

Continuous energy supply and exchange are crucial for neurons in presenting kinds of firing patterns, and stable energy balance is helpful to keep synchronous electric activities (Moujahid et al. 2011; Torrealdea et al. 2009). For generic neuron model and nonlinear oscillators, Zhou et al. (2021b) explained how to approach the sole Hamilton energy function and used as appropriate Lyapunov function. For more guidance about neurodynamics from physical viewpoint, readers can refer to the recent reviews (Ma et al. 2019; Ma 2023). In this paper, a kind of nonlinear coupling is used to couple two feasible neurons, the Hamilton energy is derived and physical property of the neural circuit is explained. Two neurons are coupled via nonlinear coupling via a hybrid synapse, and the synchronization stability is discussed.

Model and scheme

For generic neuron models composed of quadratic term for membrane potentials, the combination of simple and ideal capacitor, inductor, and shunted current across nonlinear resistor with nonlinear relation between channel current and output voltage is effective to build a controllable neural circuit. To facilitate the enhancement of biophysical function of neural circuit, specific electric components are embedded into the branch circuits of RLC circuit (Resistor–inductor–capacitor) (Kyprianidis et al. 2012), see the recent review (Ma 2022, 2023). When higher order terms are included into the neuron model, it needs the involvement of similar voltage-controlled component and memristor, and energy is also shunted in these electric components. For simplicity, an improved RLC circuit is suggested in Fig. 1.

Fig. 1.

Schematic diagram for a RLC circuit. V_M denotes the voltage for the electric component M with quadratic operation on the voltage V for the capacitor, NR is a nonlinear resistor and its current is described in Eq. (3). C, L, defines capacitance for capacitor, and inductance for inductor, respectively

The external stimulus i_ext is generated from a voltage source and it is shunted into three branch circuits. The circuit equations for Fig. 1 are obtained to bridge the voltage V and channel current i_L as follows

$$\left\{\begin{array}{cc}& C\frac{\mathit{dV}}{\mathit{dt}}=i_{\mathit{ext}}-i_L-i_{\mathit{NR}};\hfill \\ \hfill & L\frac{d{i}_L}{\mathit{dt}}=V-E-R{i}_L-V_M;\hfill \\ \hfill \end{array}\right)$$ 1

The functional component M can be considered as voltage-controlled, and its voltage V_M is defined by

$$V_M=-\frac{d}{rs{V}_0}{V}^2,E=\frac{c}{\mathit{rs}}{V}_0+\lambda V_0;$$ 2

where the constant E is used to describe the reverse potential in the ion channel current. The normalized parameters (c, d, r, s, λ) are the same parameters in the Hindmarsh–Rose (HR) neuron model (Hindmarsh and Rose 1982, 1984). Inspired by the i–v relation with quadratic term for nonlinear component in Ref. (Kyprianidis et al. 2012; Rajasekar and Lakshmanan 1988), the relation of current and voltage across the NR is defined by

$$i_{\mathit{NR}}=-\frac{1}{\rho }\left(\frac{b{V}^2}{V_0}-\frac{a{V}^3}{V_0^2}\right);$$ 3

where ρ and V₀ are the resistance in the linear region and cut-off voltage in the i–v curve for NR. a and b are same as the parameters in the HR model (Hindmarsh and Rose 1982, 1984). V₀ in Eqs. (2) and (3) is the same. The electromagnetic field energy in the neural circuit, and average energy cost per time unit in the electric component M can be estimated by

$$\left\{\begin{array}{cc}& W=W_C+W_L-W_M=\frac{1}{2}C{V}^2+\frac{1}{2}L{i}_L^2+\frac{d}{rs{V}_0}{V}^2{i}_L\rho C;\hfill \\ \hfill & W_M=i_M{V}_M\rho C=-\frac{d}{rs{V}_0}{V}^2{i}_L\rho C;\hfill \\ \hfill \end{array}\right)$$ 4

That is, the field energy in the neuron is kept in capacitive and inductive forms. In this simple neuron with one capacitive variable and one inductive variable, the involvement of voltage-controlled component M into the ion channel occupies partial electric field energy in capacitive form. Capacitive energy is pumped and shunted into the voltage-controlled channel, so W_M has opposite direction of energy flow from W_C. Furthermore, these physical variables and parameters in Eqs. (1–4) are mapped into dimensionless variables by using the following scale transformation

$$\left\{\begin{array}{cc}& x=\frac{V}{V_0},w=\frac{i_L\rho }{V_0},\tau =\frac{t}{\rho C},I_{\mathit{ext}}=\frac{\rho }{V_0}{i}_{\mathit{ext}},\hfill \\ \hfill & r=\frac{\rho RC}{L},s=\frac{\rho }{R};\hfill \\ \hfill \end{array}\right)$$ 5

As a result, an equivalent neuron model is obtained by

$$\left\{\begin{array}{cc}& \dot{x}=-w-a{x}^3+b{x}^2+I_{\mathit{ext}};\hfill \\ \hfill & \dot{w}=-c+d{x}^2+rs(x-\lambda )-rw;\hfill \\ \hfill \end{array}\right)$$ 6

From dynamical viewpoint, the external current I_ext can be adjusted to trigger mode transition in the electric activities, and external field is also helpful to change the effect of reverse potential λ for regulating the firing modes. According to the definition for dimensionless variables and parameters, these normalized parameters (a, b, c, d, r, s) are associated with the properties of ion channel, and λ accounts for the resting potential of one ion channel for the neuron. In addition, the equivalent Hamilton energy H can be mapped from the field energy W by using the same scale transformation on Eq. (4), and it is defined by

$$\begin{array}{cc}\hfill W& =\frac{1}{2}C{V}_0^2{x}^2+\frac{1}{2}\frac{L}{{\rho }^{2}C}{w}^2+\frac{d}{\mathit{rs}}{x}^{2}wC{V}_0^2\hfill \\ \hfill & =\frac{C{V}_0^2}{\mathit{rs}}\left[\frac{1}{2}rs{x}^2+\frac{1}{2}{w}^2+d{x}^{2}w\right]\hfill \\ \hfill & =\frac{C{V}_0^2}{\mathit{rs}}H;\hfill \\ \hfill H& =\frac{1}{2}rs{x}^2+\frac{1}{2}{w}^2+d{x}^{2}w;\hfill \\ \hfill \end{array}$$ 7

Guided by the Helmholtz theorem (Kobe 1986), the Hamilton energy for the neuron asks for the criterion

$$\left\{\begin{array}{cc}& \mathrm{\nabla }{H}^T{F}_c=0;\mathrm{\nabla }{H}^T{F}_d=\frac{\mathit{dH}}{d\tau };\hfill \\ \hfill & \frac{\mathit{dX}}{d\tau }=F(X)=F_c(X)+F_d(X);X\subset R^N\hfill \\ \hfill \end{array}\right)$$ 8

Surely, the neuron in Eq. (6) has much similarity to the previous two-variable HR model proposed by Hindmarsh and Rose (1982). Considering the characteristic of inner field of neuron, the equivalent vector for Eq. (6) is updated by

$$\begin{array}{cc}\hfill \left(\begin{array}{c}\dot{x}\\ \dot{w}\\ \end{array}\right)& =\left(\begin{array}{cc}& -w-a{x}^3+b{x}^2+I_{\mathit{ext}}\hfill \\ \hfill & -c+d{x}^2-rw+rs(x-\lambda )\hfill \\ \hfill \end{array}\right)=[J(x,w)+R(x,w)]\mathrm{\nabla }H=F_c+F_d\hfill \\ \hfill & =\left(\begin{array}{cc}& -w-d{x}^2\hfill \\ \hfill & rsx+2dxw\hfill \\ \hfill \end{array}\right)+\left(\begin{array}{cc}& -a{x}^3+(b+d)x^2+I_{\mathit{ext}}\hfill \\ \hfill & d{x}^2-rs\lambda -c-rw-2dxw\hfill \\ \hfill \end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{cc}0\hfill & -1\\ 1\hfill & 0\end{array}\right)\left(\begin{array}{cc}& 2dxw+rsx\hfill \\ \hfill & w+d{x}^2\hfill \\ \hfill \end{array}\right)+\left(\begin{array}{cc}\frac{-a{x}^3+(b+d)x^2+I_{\mathit{ext}}}{2dxw+rsx}\hfill & 0\\ 0\hfill & \frac{d{x}^2-rs\lambda -c-rw-2dxw}{w+d{x}^2}\end{array}\right)\left(\begin{array}{cc}& 2dxw+rsx\hfill \\ \hfill & w+d{x}^2\hfill \\ \hfill \end{array}\right);\hfill \\ \hfill \end{array}$$ 9

The energy function H for Eq. (9) follows the criterion as follows

$$(-w-d{x}^2)\frac{\partial H}{\partial x}+(rsx+2dxw)\frac{\partial H}{\partial w}=0;$$ 10

As a result, an appropriate solution for the Hamilton energy is obtained by

$$H=\frac{1}{2}rs{x}^2+\frac{1}{2}{w}^2+d{x}^{2}w=H_C+H_L-H_M;$$ 11

This energy form in Eq. (11) is consistent with the energy function in Eq. (7), which is mapped from physical field energy after scale transformation. The changes of the Hamilton energy with time is confirmed by

$$\begin{array}{cc}\hfill \frac{\mathit{dH}}{d\tau }& =\mathrm{\nabla }{H}^T{F}_d=x^4(d^2-2adw-ars)+x^3[(b+d)(2dw+rs)-2d^{2}w)\hfill \\ \hfill & -d{x}^2(rs\lambda +c+rw-w)-w^2(r+2dx)-w(rs\lambda +c)+(rsx+2dxw)I_{\mathit{ext}};\hfill \\ \hfill \end{array}$$ 12

Changes in the parameters (r, s, d) have direct impact on the energy flow, which can also be controlled by external stimulus. According to Eq. (11), the Hamilton energy of the neuron is relative to the firing mode, membrane potential and the normalized parameters (r, s, d, w) directly. As defined in Eq. (11), the first term H_C and the second term H_L define electric field energy and magnetic field energy, and any changes in the excitability will modify the ratio between the two kinds of field energy.

$$H_L=\frac{1}{2}{w}^2,H_C=\frac{1}{2}rs{x}^2,P=\frac{H_C}{H_L}=\frac{rs{x}^2}{w^2};$$ 13

Periodic stimulus, chaotic series and even noise can be applied to regulate the neural activities and energy flow is shunted between magnetic field and electric field. It is interesting to discuss the mode transition when the energy proportion P is selected with different values. For two neurons, the synchronization stability is dependent on the biophysical properties of the coupling channel. Here, we consider the synchronization control for the two neurons connected by a nonlinear resistor, which the channel current has the form in Eq. (3). The coupled neural circuits are described by

$$\left\{\begin{array}{cc}& C\frac{\mathit{dV}}{\mathit{dt}}=i_{\mathit{ext}}-i_L-i_{\mathit{NR}}-i_{\mathit{coupling}};\hfill \\ \hfill & L\frac{d{i}_L}{\mathit{dt}}=V-E-R{i}_L-V_M;\hfill \\ \hfill & C\frac{d{V}^{\prime }}{\mathit{dt}}=i_{\mathit{ext}}-i_L^{\prime }-i_{\mathit{NR}}^{\prime }+i_{\mathit{coupling}};\hfill \\ \hfill & L\frac{d{i}_L^{\prime }}{\mathit{dt}}=V^{\prime }-E-R{i}_L^{\prime }-V_M^{\prime };\hfill \\ \hfill & i_{\mathit{coupling}}=-\frac{1}{\rho }\left[\frac{b{(V-V^{\prime })}^2}{V_0}-\frac{a{(V-V^{\prime })}^3}{V_0^2}\right]\hfill \\ \hfill \end{array}\right)$$ 14

By using the similar scale transformation, the dynamics for two coupled neurons can be given in the form

$$\left\{\begin{array}{cc}& \dot{x}=-w-a{x}^3+b{x}^2+I_{\mathit{ext}}+b{(x-x^{\prime })}^2-a{(x-x^{\prime })}^3;\hfill \\ \hfill & \dot{w}=-c+d{x}^2+rs(x-\lambda )-rw;\hfill \\ \hfill & {\dot{x}}^{\prime }=-w^{\prime }-a{x}^{\prime 3}+b{x}^{\prime 2}+I_{\mathit{ext}}-b{(x-x^{\prime })}^2+a{(x-x^{\prime })}^3;\hfill \\ \hfill & {\dot{w}}^{\prime }=-c+d{x}^{\prime 2}+rs(x^{\prime }-\lambda )-r{w}^{\prime };\hfill \\ \hfill \end{array}\right)$$ 15

The last two terms in the first and third formulas in Eq. (15) denote the equivalent dimensionless current across the coupling channel. The error function for states and Hamilton energy is defined respectively,

$$\left\{\begin{array}{cc}& \theta (e_x,e_w)=\sqrt{{(x-x^{\prime })}^2+{(w-w^{\prime })}^2};\hfill \\ \hfill & \mathrm{\Delta }H=\left|H_1-H_2\right|=\left|\frac{1}{2}rs{x}^2+\frac{1}{2}{w}^2+d{x}^{2}w-\frac{1}{2}rs{x}^{\prime 2}-\frac{1}{2}{w}^{\prime 2}-d{x}^{\prime 2}{w}^{\prime }\right|\hfill \\ \hfill \end{array}\right);$$ 16

In addition, phase series can be obtained by applying Hilbert transformation on the sampled time series for membrane potentials of two neurons, and then phase synchronization and phase lock between two neurons can be further verified when external stimuli are controlled to trigger different firing modes in the neurons. It is important to discuss the dynamical property of the coupling channels. In presence of linear coupling via ideal resistor with resistance R_k, the coupling intensity for two different cases (linear and nonlinear coupling) can be expressed by

$$\left\{\begin{array}{cc}{i}_{\mathit{coupling}}=k(x^{\prime }-x),k=\frac{\rho }{R_k},\hfill & \text{linear}\text{coupling}\text{via}{R}_k;\\ i_{\mathit{coupling}}^{\prime }=b{(x-x^{\prime })}^2-a{(x-x^{\prime })}^3,\hfill & \text{nonlinear}\text{coupling}\text{via}NR;\\ k^{\prime }=b(x-x^{\prime })-a{(x-x^{\prime })}^2,\hfill & \text{nonlinear}\text{coupling}\text{via}NR;\end{array}\right)$$ 17

That is, nonlinear coupling introduces time-varying coupling intensity k′ and it terminates to zero adaptively under complete synchronization or balance between membrane potentials. As described in Eq. (6), the external stimulus I_ext can be derived from periodic signal source, and the deterministic model can be excited to present different firing modes. Indeed, realistic signal source may be more complex and the neuron will be excited by mixed signals, which can be filtered from a chaotic system. For simplicity, signals from Pikovskii-Rabinovich (PR) oscillator (Pikovskii and Rabinovich 1978) are encoded to stimulate the neuron, and it is defined by

$$\left\{\begin{array}{cc}& \frac{d{x}^{\prime }}{d\tau }=y^{\prime }-\delta z^{\prime };\hfill \\ \hfill & \frac{d{y}^{\prime }}{d\tau }=-x^{\prime }+2\gamma y^{\prime }+\alpha z^{\prime }+\beta ;\hfill \\ \hfill & \frac{d{z}^{\prime }}{d\tau }=\mu (x^{\prime }+z^{\prime }-z^{\prime 3});\hfill \\ \hfill & I_{\mathit{ext}}=E_{0}sin(x^{\prime });\hfill \\ \hfill \end{array}\right)$$ 18

where x′, y′, z′ are dimensionless variables mapped from the output voltage and current in the nonlinear circuit, and it presents chaotic state at α = 0.165, β = 0, γ = 0.201, δ = 0.66, μ = 1/0.047. E₀ is a positive constant and considered as gain for control of the amplitude of the filtered chaotic signals, and similar chaotic signal from Lorenz, Rössler or other chaotic systems can be encoded for I_ext, which is effective to activate mode transition and nonlinear resonance in the electric activities. That is, the sampled time series for variable x′ are chaotic and further encoding in sine function as I_ext = E₀sin(x′) will introduce irregular stimulus on the neuron. The coefficient variability (CV) is estimated to judge the coherence degree as follows

$$CV=\frac{\sqrt{〈T^2〉-{〈T〉}^2}}{〈T〉};$$ 19

where T denotes the interspike interval in the sampled time series for membrane potential, and the symbol $〈,\ast 〉$ represents an average of the variable within certain transient period. Lower value for CV means higher coherence resonance in the neuron.

Results and discussion

The fourth order Runge–Kutta algorithm is used to approach numerical solution with time step h = 0.01. To present different firing patterns, the amplitude and angular frequency in the external stimulus u_s = Acosωτ are adjusted to control the neuron in Eq. (6) with initial value (0.02, 0.01).The parameters are selected as a = 0.52, b = 4.23, c = 2.6, d = 0.92, r = 0.119, s = 0.05,λ =  − 1.6. For coupling synchronization, the initial values for two neurons are fixed at (0.02, 0.01, 0.03, 0.02). In Fig. 2, bifurcation analysis is supplied to confirm the appearance of different firing modes.

Fig. 2.

Bifurcation of ISI for membrane potential x vs. parameters A, ω. For a ω = 0.12; b A = 8.0

By adjusting the external stimulus, this neuron is suitable to produce a variety of firing patterns including chaotic, bursting and spiking, and it means this neuron has the main dynamical characteristic as those biological neurons. Extensive numerical results confirmed that noisy disturbance accompanying periodic stimuli can generate stochastic resonance by changing the noise intensity carefully. For better showing, the electric activities are plotted and corresponding energy function is calculated to discern mode dependence on the firing modes in Fig. 2.

From Fig. 3, the energy in a single neuron is changed with the transition of firing modes, and further increasing the angular frequency of external stimulus can induce chaotic states. In presence of spiking patterns, the neuron used to keep higher energy level, while chaotic activities supports a lower average value in the Hamilton energy. Except the spiking condition, transient switch in the energy ratio is detected, and it means a fast energy release and exchange between magnetic field and electric field.

Fig. 3.

Firing patterns of membrane, Hamilton energy H in Eq. (11), energy ratio P between H_C and H_L in Eq. (13), at A = 8. For a1–a3 spiking patterns ω = 0.0001; b1–b3 bursting patterns ω = 0.02; c1–c3 periodic patterns ω = 0.05; d chaotic patterns ω = 0.08

Realistic stimuli on neurons are not distinct periodic type, it is worthy of investigating the nonlinear response when mixed signals is applied. For simplicity, chaotic signals from PR in Eq. (18) are encoded to excite the neuron by applying different amplitudes for I_ext = E₀sin(x′), the Largest Lyapunov exponent, average energy < H > , peak values from membrane potential and distribution for CV in Eq. (19) are calculated in Fig. 4.

Fig. 4.

a Bifurcation of peaks for membrane potential x vs. E₀ in I_ext = E₀sin(x′); b average energy dependence on E₀; c Largest Lyapunov exponent vs. E₀; d CV distribution versus E₀

From Fig. 4, the encoded chaotic signals E₀sin(x′) can inject stimuli as mixed signals and the excitability of the neuron can be regulated by the gain E₀ effectively, so mode transition can be controlled completely. Indeed, E₀sin(x′) can be considered as combination of periodic and stochastic disturbance, and appropriate setting for the gain E₀ can induce coherence resonance-like behavior in the neural activities. Further increasing the value for the gain E₀ prefers to impose chaotic stimulus and the neuron is excited to present chaotic firing patterns. The curve for CV distribution in Fig. 4d has no lowest value and it is in some difference from the previous curve for CV versus noise intensity, which moderate noise intensity supports coherence resonance accompanying with lowest CV value. It is interesting to clarify the energy characteristic of the neuron excited by this stimulus, and the results are illustrated in Fig. 5.

Fig. 5.

Evolution of Hamilton energy H and changes in the ratio P between H_C and H_L is plotted in presence of mixed signals I_ext = E₀sin(x′). For a1, a2 E₀ = 1.0; b1, b2 E₀ = 6.0; c1, c2 E₀ = 20.0

In fact, I_ext = E₀sin(x′) can excite the neuron as quasi-periodic signals and the neuron prefers to trigger chaotic states and it keeps lower Hamilton energy because of fast discharge. During the firing of neural activities, electric field energy will keep a lower value than the magnetic field energy except some transient period. It indicates that the channel current in the neuron is fluctuated quickly and membrane potential is regulated to release energy quickly as well.

Synaptic connection can propagate energy effectively by regulating the synaptic current, and the time-varying coupling is dependent on the energy diversity. Linear electric synapse coupling requires special electrophysiological condition and consumption of Joule heat becomes inevitable. Indeed, biological neurons prefer to trigger field coupling and nonlinear coupling by activating hybrid synapse. In Fig. 6, the synchronous response between two neurons under the same periodic stimulus is calculated.

Fig. 6.

Evolution of error function θ in Eq. (16) and phase error Δϕ for two coupled neurons in presence of periodic stimulus. For a1, a2 A = 8.0, ω = 0.0001; b1, b2 A = 8.0, ω = 0.02; c1, c2 A = 8.0, ω = 0.05; d1, d2 A = 8.0, ω = 0.08. Phase series ϕ₁ and ϕ₂ are obtained by applying Hilbert transformation on the sampled time series for the variables (x, x′), and Δϕ = ϕ₁–ϕ₂

In presence of nonlinear coupling via hybrid synapse defined in Eq. (17), two identical neurons have difficult to reach complete synchronization. However, they can reach phase lock when the external stimulus is adjusted, and it indicates that the two neurons can guided to present some suitable firing modes in the neural activities. Furthermore, the energy diversity between two coupled neurons is calculated in Fig. 7.

Fig. 7.

Evolution of energy error for two coupled neurons driven by periodic signals. For a spiking neurons, A = 8.0, ω = 0.0001; b bursting neurons, A = 8.0, ω = 0.02; c periodic neurons, A = 8.0, ω = 0.05; d chaotic neurons, A = 8.0, ω = 0.08

Two identical neurons have the same distinct period and synchronization between periodic neurons becomes easy. When neurons reach complete synchronization, energy diversity is reduced to zero and energy balance is stabilized. This nonlinear coupling just activates subthreshold coupling because the gain in the coupling term is relative to some intrinsic parameters in the neuron. Therefore, they can reach transient synchronization rather than stable complete synchronization, and energy diversity will be changeable with time. In addition, the changes of coupling term and synaptic intensity in Eq. (17) is estimated when neurons are excited to present different modes.

From Fig. 8, in presence of four different firing modes, the synaptic current becomes time-varying, and it means that two neurons keep certain diversity of membrane potential. Therefore, complete synchronization is blocked, and it is helpful to prevent bursting synchronization and the occurrence of seizure in the nervous system. Considering the difference in excitability in biological neurons, two neurons connected via hybrid synapse are excited by encoded chaotic signals with different intensities, and the results are plotted in Fig. 9.

Fig. 8.

Changes in synaptic intensity k′ along the hybrid synapse for two coupled neurons in Eq. (18). For a A = 8.0, ω = 0.0001; b A = 8.0,ω = 0.02; c A = 8.0,ω = 0.05; d A = 8.0, ω = 0.08

Fig. 9.

Evolution of error function for two coupled neurons presented in different firing modes and phase error diagram by mixed signals I_ext = E₀sin(x′). For a1, a2 E₀ = 1.0; b1, b2 E₀ = 6.0; c1, c2 E₀ = 20.0

When the mixed signals are encoded with lower gain and intensity, phase synchronization between two neurons becomes available and it means this hybrid synapse is effective to trigger synchronous firing patterns. With further increase of the gain in the mixed signals, neurons prefer to present chaotic patterns and complete synchronization becomes difficult and phase lock is also broken with time. The energy diversity between neurons driven by encoded chaotic signals is also obtained to predict whether energy balance can be realized in Fig. 10.

Fig. 10.

Evolution of energy error for two coupled neurons driven by mixed signals I_ext = E₀sin(x′). For a E₀ = 1.0; b E₀ = 6.0; c E₀ = 20.0

It is found that the two neurons show time-varying energy diversity with time, and thus energy propagation along the hybrid synapse is continued, it is helpful to find coexisting different firing patterns in the nervous systems. Furthermore, the coupling intensity for nonlinear coupling is also estimated in Fig. 11 to predict whether the two neurons keep its nonlinear coupling all the time.

Fig. 11.

Changes in synaptic intensity k′ along the hybrid synapse for two coupled neurons driven by I_ext = E₀sin(x′). For a E₀ = 1.0; b E₀ = 6.0; c E₀ = 20.0

From Fig. 11a, it means the synaptic current is terminated because two neurons have the same membrane potentials and nonlinear coupling is switched off within a transient period. Therefore, two neurons keep their own firing modes and they can reach partial synchronization. From Fig. 11b, c, the synaptic current fluctuates with time and it means the hybrid synapse keeps working for continuous energy propagation and exchange because of distinct diversity in the two neurons. In particular, appearance of negative value for the coupling intensity k′ indicates this nonlinear coupling activates positive feedback on each neuron, therefore, energy diversity and firing modes are regulated with time.

In summary, realistic biological neurons often receive mixed signals from externals stimulus and neurons in the neural networks will capture more signals from other neurons. Furthermore, these multi-channels injections are detected to compose an encoded signal within certain frequency band. When two or more neurons are excited, hybrid synapses rather than sole and ideal synapses are activated to propagate energy between neurons, transition from synchronization and desynchronization is switched when the coupling intensity along the nonlinear channel is changed between negative and positive values. As a result, two neurons can present different firing patterns and modes in the neural activities. Hybrid synapse plays an important role to keep energy diversity and these neurons are blocked to reach synchronous firing patterns, which can prevent the occurrence of seizure. In fact, hybrid synapse accounts for nonlinear coupling and it is suitable to approach close biophysical property and physical effect of realistic synapses for biological neurons. In a practical way, combination of functional electric components including memristor, nonlinear resistor, thermistor, phototube, piezoelectric component and Josephson junction can enhance the sensitivity and controllability of synaptic connection and coupling channels, the main advantage of these functional synapse is its intensity can be regulated adaptively and external stimuli can control the coupling channel directly. The energy definition within this work is defined and confirmed from physical viewpoint (Njitacke et al. 2022b), it is different from the previous energy description in Wang and Zhu (2016), Zhu et al. (2019), Wang et al. (2021) for neurons. As mentioned in our recent works, continuous energy injection and absorption will induce shape deformation, some neurons will show parameter shift to keep pace with other neurons for showing synchronous firing or desynchronization, as a result, self-adaption of biological neurons are released. Nonlinear coupling provides possible intermittent positive and negative regulation on two neurons, and this scheme can be further used to induce and control chimera in neural networks Yang et al. (2022), Kanagaraj et al. (2023), Feng et al. (2023) by developing coexistence of synchronization and non-synchronization.

Conclusions

Based on the Helmholtz theorem, an energy function for a two-variable neuron is defined from physical viewpoint and it is also confirmed by applying scale transformation on the field energy in the neural circuit composed of a voltage-controlled component. Filtered chaotic signals are used to excite the neuron for mimicking realistic stimulus. It indicates that biological neurons can be excited to present regular patterns under mixed signal matching with realistic signals within certain frequency band, and average energy and CV distributions are calculated to discern mode transition in electrical activities. Furthermore, two neural circuits are coupled by the same nonlinear resistor, and synchronization stability and phase lock are controlled by the nonlinear coupling. Under some firing modes, the coupling intensity is decreased to zero and nonlinear coupling is terminated with the same membrane potentials. In other cases, continuous nonlinear coupling contributes to phase lock or phase synchronization, and possible bursting synchronization is prevented. That is, nonlinear coupling provides effective energy exchange and supports coexistence of multiple firing modes in neurons under energy diversity. To activate the self-adaption of biological neurons, shape deformation accompanying with parameter shift becomes inevitable and then the hybrid synapse is controlled to adjust the coupling intensity for reaching next energy balance between neurons.

Data availability

Data available on appropriate request from the authors.

Declarations

Conflict of interest

The authors declare no any interest conflict with publication of this work.