Vibrational Resonance and Electrical Activity Behavior of a Fractional-Order FitzHugh–Nagumo Neuron System

Abstract

Making use of the numerical simulation method, the phenomenon of vibrational resonance and electrical activity behavior of a fractional-order FitzHugh–Nagumo neuron system excited by two-frequency periodic signals are investigated. Based on the definition and properties of the Caputo fractional derivative, the fractional L1 algorithm is applied to numerically simulate the phenomenon of vibrational resonance in the neuron system. Compared with the integer-order neuron model, the fractional-order neuron model can relax the requirement for the amplitude of the high-frequency signal and induce the phenomenon of vibrational resonance by selecting the appropriate fractional exponent. By introducing the time-delay feedback, it can be found that the vibrational resonance will occur with periods in the fractional-order neuron system, i.e., the amplitude of the low-frequency response periodically changes with the time-delay feedback. The weak low-frequency signal in the system can be significantly enhanced by selecting the appropriate time-delay parameter and the fractional exponent. In addition, the original integer-order model is extended to the fractional-order model, and the neuron system will exhibit rich dynamical behaviors, which provide a broader understanding of the neuron system.

1. Introduction

Vibrational resonance (VR) has attracted considerable attention in the field of nonlinear sciences in the last twenty years. Based on the study of stochastic resonance (SR) [1], VR is firstly proposed by Landa and McClintock [2]. When replacing the noise in SR with an appropriate high-frequency signal, the weak low-frequency signal can be greatly amplified, which is similar to the typical “Inverted Bell” resonance phenomenon in SR, and is named as VR. Compared with the noise in SR, the high-frequency signal in VR is more controllable. Biharmonic signals are common in various fields, such as acoustics [3], optics [4], engineering [5], neuroscience [6], etc. Recently, the research hotspots of VR have changed from classic bistable systems [7,8,9] to fractional-order systems [10], delay systems [11], and network dynamical systems [12].

Due to the complex definition and lack of corresponding application background, the research on fractional calculus has been limited to the field of mathematics for a long time. However, compared with integer calculus, the power-law characteristic of complex social and physical phenomena can be accurately approximated through fractional calculus. Hence, this theory is gradually used in viscoelastic materials [13], electrification process [14], control theory [15], and neuron models [16], etc. To the best of our knowledge, the research on VR for fractional-order systems is quite few, and most of which are limited to bistable and multi-stable systems [17,18,19]. However, it is found that fractional calculus has its own unique advantages in describing certain neuronal characteristics. For example, fractional-order differentiation can be used to account for the firing rate of neocortical pyramidal neurons when stimulated by sinusoidal current [20]. Anastasio et al. [21] thought that the net output of the motor neurons in the visual system is consistent with fractional-order differentiation relative to eye position. Therefore, it is of great significance to study VR in the fractional-order neuron system.

In this study, we consider the FitzHugh–Nagumo (FHN) neuron model. As one of the simplest mathematical models for disclosing the dynamical behavior of neurons, the FHN model is widely used in integer-order systems for studying VR [22,23]. In brain activities [24], the neurons may exhibit two quite different time scales, and then, it is reasonable to reveal the mechanism of weak signal detection of neuron by VR. Since time delay is inherent in the neuron system, great progress has been made in the research on the effect of time delay on VR. Adding the time-delay feedback to the recovery variable of the FHN neuron model, the phenomenon of multiple VR can be induced in the neuron [25]. It is found that the bifurcation point and equilibrium point change periodically with the increase of time delay in a fractional order quantic oscillator system, and the output can be enhanced by selecting appropriate time delay [26].

Inspired by the above-mentioned ideas, the effects of fractional order and time delay on VR and dynamical behavior are studied in the fractional-order FHN neuron model. The VR can be induced in the fractional-order neuron model without strict requirement for the amplitude of high-frequency signal. It can be found that multiple VR occurs in the neuron with the increase of delay. Compared with integer-order model, the fractional-order FHN neuron exhibits rich electrical activities. The remainder of this paper is organized as follows: In Section 2, the fractional-order FHN neuron model excited by two periodic signals is briefly introduced. In Section 3, the VR in the fractional-order FHN model without and with time delay is studied. In Section 4, the effects of fractional exponent and time delay on the dynamical behavior in the fractional-order FHN model are discussed. Several conclusions are given in Section 5.

2. Fractional-Order FitzHugh–Nagumo Neuron Model

There are three common definitions of fractional calculus, namely Riemann–Liouville (R-L) definition, Grünwald–Letnikov (G-L) definition, and the Caputo definition. With the property of supersingularity, R-L definition is mainly used for the analysis of mathematical theory and is not convenient for engineering and physical modeling. G-L definition can be regarded as the extension of the limit form of integral calculus difference definition, which is widely applied to early numerical calculation. As its initial condition is the form of integer calculus, and it has clear physical meaning, the Caputo definition is used in this paper.

The fractional derivative of univariate function $f(t)$ is defined as

$$\frac{{\mathrm{d}}^{\alpha }f\left(t\right)}{\mathrm{d}{t}^{\alpha }}={}_{t_0}D_t^{\alpha }f(t)=\frac{1}{\Gamma (n-\alpha )}{\displaystyle {\int }_{t_0}^t\frac{f^{(n)}(\tau )\mathrm{d}\tau }{{(t-\tau )}^{\alpha -n+1}}}, (n-1<\alpha <n),$$

(1)

where $\Gamma (n-\alpha )$ is the gamma function, $t_0$ and $t$ are the lower limit and upper limit of the definite integral, $n$ is the minimum positive integer greater than $\alpha$, and $f^{(n)}(\tau )$ is the $nth$ derivative of the function $f(\tau )$. Under the joint excitation of harmonic signals, the fractional-order FHN neuron model with time delay (the model comes from the model involved in Ref. [27]) is given by the following form:

$$\begin{array}{l}\epsilon \frac{{\mathrm{d}}^{\alpha }x}{\mathrm{d}{t}^{\alpha }}=x(t)-\frac{x{(t)}^3}{3}-y(t),\\ \frac{{\mathrm{d}}^{\alpha }y}{\mathrm{d}{t}^{\alpha }}=x(t)+a+f\cos (\omega t)+F\cos (\mathsf{\Omega }t)+K(y(t-\tau )-y(t)),\end{array}$$

(2)

where $x\left(t\right)$ and $y\left(t\right)$ represent the fast-varying membrane potential and the slow-varying recovery variable of neuronal cells, respectively; $\alpha$($0<\alpha \le 1$) is the fractional exponent; $K$ is the strength of time-delay feedback. $\tau \ge 0$ is the delay parameter, and Equation (2) degenerates into the fractional-order FHN neuron model without delay when $\tau =0$. $\epsilon =0.05$ is the time scale ratio, which is chosen to ensure that the membrane potential $x\left(t\right)$ evolves faster than the recovery variable $y\left(t\right)$. $f\cos (\omega t)$ and $F\cos (\mathsf{\Omega }t)$ represent the low-frequency signal and the high-frequency signal, respectively, which satisfy $f<<1$, $\omega <<\mathsf{\Omega }$. The value of $a$ determines the behavior of the system under the conditions that $f=F=\tau =0$ and $\alpha =1$. If $\left|a\right|>1.0$, the system is excitable and has only a stable fixed point; if $\left|a\right|<1.0$, a limit cycle in the system arises. Here, the parameter $a=1.05$ is chosen to make the system in the excitable state [28]. The variables in the model are dimensionless.

We consider the fractional derivative of $x\left(t\right)$ defined with the Caputo fractional derivative,

$$\frac{{\mathrm{d}}^{\alpha }x(t)}{\mathrm{d}{t}^{\alpha }}=f(x,t).$$

(3)

The discrete format of fractional-order L1 algorithm is [29]:

$$\frac{{\mathrm{d}}^{\alpha }x(t)}{\mathrm{d}{t}^{\alpha }}\approx \frac{{(\mathrm{d}t)}^{-\alpha }}{\Gamma (2-\alpha )}\left[{\displaystyle \sum _{k=0}^{N-1}\left[x(t_{k+1})-x(t_k)\right]}\left[{(N-k)}^{1-\alpha }-{(N-1-k)}^{1-\alpha }\right]\right],$$

(4)

where $t_k=k\Delta t$. Combining the right sides of Equations (3) and (4) and solving for $x$ at time $t_N$, it can be concluded that the discrete format of Equation (3) is

$$x(t_N)\approx {(\mathrm{d}t)}^{\alpha }\Gamma (2-\alpha )f(x,t)+x(t_{N-1})-\left[{\displaystyle \sum _{k=0}^{N-2}\left[x(t_{k+1})-x(t_k)\right]}\left[{(N-k)}^{1-\alpha }-{(N-1-k)}^{1-\alpha }\right]\right].$$

(5)

where the Markov term weighted by the gamma function is given by

$${(\mathrm{d}t)}^{\alpha }\Gamma (2-\alpha )f(x,t)+x(t_{N-1})$$

(6)

and the memory trace is given by

$${\displaystyle \sum _{k=0}^{N-2}\left[x(t_{k+1})-x(t_k)\right]}\left[{(N-k)}^{1-\alpha }-{(N-1-k)}^{1-\alpha }\right]$$

(7)

The memory trace integrates information of all the previous activities and has a memory effect, which is the typical property of fractional-order system. When $\alpha =1$, the memory trace has no effect, and Equation (5) degenerates into classical Euler algorithm.

According to Equation (5), the discrete format of Equation (2) can be obtained as follows:

$$\begin{array}{l}x(t_N)\approx \frac{1}{\epsilon }{(\mathrm{d}t)}^{\alpha }\Gamma (2-\alpha )(x(t_{N-1})-\frac{{(x(t_{N-1}))}^3}{3}-y(t_{N-1}))+x(t_{N-1})\\ -\left[{\displaystyle \sum _{k=0}^{N-2}\left[x(t_{k+1})-x(t_k)\right]}\left[{(N-k)}^{1-\alpha }-{(N-1-k)}^{1-\alpha }\right]\right],\\ y(t_N)\approx {(\mathrm{d}t)}^{\alpha }\Gamma (2-\alpha )(x(t_{N-1})+a+f\cos (\omega t_{N-1})+F\cos (\mathsf{\Omega }{t}_{N-1})+K(y(t_{N-m})-y(t_{N-1})))+y(t_{N-1})\\ -\left[{\displaystyle \sum _{k=0}^{N-2}\left[y(t_{k+1})-y(t_k)\right]}\left[{(N-k)}^{1-\alpha }-{(N-1-k)}^{1-\alpha }\right]\right],\end{array}$$

(8)

where $m$ is the number of discrete points caused by time delay.

3. Vibrational Resonance in the Fractional-Order FHN Neuron Model

3.1. VR in the Fractional-Order FHN Neuron Model without Time Delay

The response amplitude $Q$ at the low-frequency signal is usually used as the index to measure the VR, which is defined by:

$$Q=\sqrt{Q_s^2+Q_c^2},$$

(9)

with

$$\begin{array}{l}{Q}_s=\frac{2}{nT}{\displaystyle \underset{0}{\overset{nT}{\int }}x(t)}\sin (\omega t)\mathrm{d}t,\\ Q_c=\frac{2}{nT}{\displaystyle \underset{0}{\overset{nT}{\int }}x(t)}\cos (\omega t)\mathrm{d}t,\end{array}$$

(10)

where $T=2\pi /\omega$ and $n$ is a positive integer. For the numerical simulation, the parameters are selected as $\epsilon =0.05, K=0.2, f=0.01, n=50$.

Under fixed parameters $\omega =0.5, \mathsf{\Omega }=5$, the curves of VR in the fractional-order FHN neuron without time delay are plotted in Figure 1a. It can be seen that different fractional exponents correspond to different phenomena of the VR. In Figure 1b, with the decrease of $\alpha$, the region of the VR in the neuron changes, and the amplitude of low-frequency $Q$ corresponding to the optimal $F$ gradually decreases. Compared with the integer-order FHN neuron model, it is found that the low-frequency signal can also be amplified without strict requirement for the amplitude of high-frequency $F$ in the fractional-order neuron. For example, in Figure 1b, the integer-order FHN neuron model fails to reach the state of the VR for $F=0.15$, while for $\alpha =0.96$, the VR occurs for the same value of $F$. Hence, it is shown that the VR in the FHN neuron can be induced by adjusting appropriate fractional exponent and the high-frequency force.

In order to further discuss the effect of $\alpha$ on the VR, the response amplitude $Q_{\max }$ versus $\alpha$ for $F\in [0.1,0.25]$ is depicted in Figure 2a. For a certain range of $F$, although the response amplitude $Q_{\max }$ is not a strictly monotonic increasing function of $\alpha$, in general, with the increase of $\alpha$, the response amplitude $Q$ can be optimized under the appropriate high-frequency force. However, when the value of $F$ is fixed, the function of $Q$ versus $\alpha$ shows different monotonic characteristics, as shown in Figure 2b. For $F=0.11$, the curve of response amplitude is approximately a straight line, which indicates that VR does not occur with the change of $\alpha$, while for $F=0.15$ or $F=0.2$, the response amplitude $Q$ is a nonlinear function of $\alpha$, and the low-frequency signal can be significantly enhanced by selecting an appropriate fractional-order $\alpha$ compared with the integer-order system.

3.2. Multiple VR in Fractional-Order FHN Neuron Model with Delay

For the fractional-order FitzHugh–Nagumo neuron with time delay, the numerical result of functional curves of the response amplitude $Q$ versus $\tau$ can be obtained by combining Equation (8) with Equation (9). The response amplitude $Q$ versus time delay $\tau$ in the fractional-order FHN neuron excited by two frequency signals is given in Figure 3. Figure 3e,f are partial, enlarged views of Figure 3a,d, respectively. From Figure 3a–d, it is seen that multiple resonance occurs in the neuron with the increase of the time-delay parameter. Therefore, the response amplitude $Q$ can reach the maximum by selecting appropriate time-delay parameters, and the response amplitude $Q$ can be greatly amplified compared with the neuron without time delay. Another notable phenomenon is that the response amplitude $Q$ is periodic with the change of $\tau$. From Figure 3a–d, it is clear that that the period of response amplitude $Q$ is $2\pi /\omega$, while Figure 3e,f shows that the response amplitude $Q$ varies with another period of $2\pi /\mathsf{\Omega }$. Therefore, the response amplitude $Q$ in the fractional-order FHN neuron with time-delay feedback presents two different periods, namely fast period $2\pi /\mathsf{\Omega }$ and slow period $2\pi /\omega$, which exactly correspond to the period of high-frequency signal and low-frequency signal. Utilizing the periodicity of response amplitude $Q$ and selecting appropriate time-delay parameters are helpful to realizing the effective control of the fractional-order FHN neuron. The only regret is that the parameters involved in the system are difficult to be optimized quickly to satisfy the requirement of resonance, which is consistent with the conclusion in Ref. [30].

4. Dynamical Behavior of Fractional-Order FHN Neuron Model

4.1. Effect of the Fractional-Order

In this subsection, the dynamical behavior of the fractional-order FHN neuron without time-delay feedback is studied. Figure 4 shows the bifurcation diagram of interspike interval (ISI) of the neuron versus the fractional-order $\alpha$, where $\alpha \in \left[0.6,1\right]$. From Figure 4, when $\alpha$ is small, it can be seen that the fractional-order FHN neuron does not fire, and when the $\alpha$ is relatively large, various periodic and irregular firing patterns appear in the neuron. It can be found that the fractional-order FHN neuron displays complex dynamics by adjusting a single parameter $\alpha$, which is also observed in other neurons [31,32].

In order to intuitively depict the effect of the fractional-order $\alpha$ on the dynamical behavior of the FHN neuron, the evolution of the membrane potential $x\left(t\right)$ is given in Figure 5. From Figure 5a, when $\alpha =0.6$, the neuron is in the quiescent state. As can be seen from Figure 5b,c, when $\alpha =0.792$ and $\alpha =0.84$, the firing pattern of the neuron is period 1 bursting. Obviously, their interspike intervals are different, which are four times and two times $2\pi /\omega$, respectively. In Figure 5a, when $\alpha =0.934$, the discharge rhythm of the FHN neuron alternates between period 1 and period 2. From Figure 5a–d, it is found that the fractional-order FHN neuron is more active with the increase of $\alpha$, which is consistent with that revealed in Figure 4.

4.2. Effect of the Time Delay

In order to study the effect of time-delay feedback on the dynamical behavior of the fractional-order FHN neuron, $\alpha =0.84$ in Figure 5c is selected. Figure 6 depicts the bifurcation diagram of interspike interval (ISI) of the neuron versus the time-delay parameter $\tau$, from which it can be seen that the fractional-order FHN neuron with time-delay feedback exhibits rich dynamical behaviors. The time series of the membrane potential $x\left(t\right)$ for different values of $\tau$ are given in Figure 7. When $\tau =0$, the neuron fires with period 1, which corresponds to the case of the neuron without time delay in Figure 5c. When $\tau =5.6$, it is seen that the firing pattern of the neuron is period 2 bursting. It can be seen from Figure 7c that a new firing pattern appears in the neuron, which regularly alters between period 2 and period 1. The multiple spiking of the firing pattern also appears in the neuron, as shown in Figure 7d. Thus, bursting patterns and interspike intervals of the fractional-order FHN neuron can be controlled by adjusting $\alpha$ or $\tau$.

5. Conclusions

The vibrational resonance and electrical activity behaviors in the fractional-order FHN neuron are studied in this paper. When the original integer-order model is extended to the fractional-order model, the region of the VR will be wider, so the requirement for the amplitude of high-frequency signal can be relaxed, and then, the VR phenomenon can be induced by choosing the appropriate fractional-order $\alpha$. Introducing time-delay feedback, it is found that the phenomenon of vibrational resonance appears with two different periods in the fractional FHN neuron, which are exactly equal to the periods of two frequency signals. The fractional-order FHN neuron exhibits rich electrical activities, and its bursting patterns and interspike interval can be controlled by adjusting $\alpha$ or $\tau$.