Levy noise-induced self-induced stochastic resonance in a memristive neuron

Self-induced stochastic resonance (SISR) is a subtle resonance mechanism requiring a nontrivial scaling limit between the stochastic and the deterministic timescales of an excitable system, leading to the emergence of a limit cycle behavior which is absent without noise. All previous studies on SISR in neural systems have only considered the idealized Gaussian white noise. Moreover, these studies have ignored one electrophysiological aspect of the nerve cell: its memristive properties. In this paper, first, we show that in the excitable regime, the asymptotic matching of the Levy timescale (that follows a power law, unlike Gaussian noise that follows Kramers law) and the deterministic timescale (controlled by the singular parameter) can also induce a strong SISR. In addition, it is shown that the degree of SISR induced by Levy noise is not always higher than that of Gaussian noise. Second, we show that, for both types of noises, the two memristive properties of the neuron have opposite effects on the degree of SISR: the stronger the feedback gain parameter that controls the modulation of the membrane potential with the magnetic flux and the weaker the feedback gain parameter that controls the saturation of the magnetic flux, the higher the degree of SISR. Finally, we show that, for both types of noises, the degree of SISR in the memristive neuron is always higher than in the non-memristive neuron. Our results could find applications in designing neuromorphic circuits operating in noisy regimes.

Self-induced stochastic resonance (SISR) is a subtle resonance mechanism requiring a nontrivial scaling limit between the stochastic and the deterministic timescales of an excitable system, leading to the emergence of a limit cycle behavior which is absent without noise. All previous studies on SISR in neural systems have only considered the idealized Gaussian white noise. Moreover, these studies have ignored one electrophysiological aspect of the nerve cell: its memristive properties. In this paper, first, we show that in the excitable regime, the asymptotic matching of the Lévy timescale (that follows a power law, unlike Gaussian noise that follows Kramers' law) and the deterministic timescale (controlled by the singular parameter) can also induce a strong SISR. In addition, it is shown that the degree of SISR induced by Lévy noise is not always higher than that of Gaussian noise. Second, we show that, for both types of noises, the two memristive properties of the neuron have opposite effects on the degree of SISR: the stronger the feedback gain parameter that controls the modulation of the membrane potential with the magnetic flux and the weaker the feedback gain parameter that controls the saturation of the magnetic flux, the higher the degree of SISR. Finally, we show that, for both types of noises, the degree of SISR in the memristive neuron is always higher than in the non-memristive neuron. Our results could find applications in designing neuromorphic circuits operating in noisy regimes.

I. INTRODUCTION
Noise is ubiquitous in neural systems. Several studies have shown, both theoretically and experimentally, that noise can play a constructive role in neural information processing [1][2][3][4][5][6][7][8][9]. Noise-induced resonance mechanisms are a category of phenomena showing this constructive counter-intuitive role of noise. Several types of noiseinduced resonance mechanisms have been identified and extensively studied, particularly in neural systems. These include stochastic resonance (SR) [1, 4,10], coherence resonance (CR) [5,[11][12][13][14], inverse stochastic resonance [15][16][17][18][19], recurrence resonance [20], and self-induced stochastic resonance (SISR) [19,[21][22][23][24][25][26][27][28][29]. In this paper, we focus on the latter. SISR requires a nontrivial scaling limit between the stochastic and the deterministic timescales of an excitable system, leading to the emergence of periodic oscillations which are absent without noise. Generically, SISR occurs when a multiple-timescale excitable dynamical system is driven by a noise of weak amplitude. During SISR, the escape timescale of trajectories from one attracting region in phase space to another is distributed exponentially (with Gaussian noise) or polynomially (with Lévy noise), and the associated transition rate is governed by an activation energy (for Gaussian noise) or by a band width (for Lévy noise). Suppose the excitable system (e.g., a neuron) is placed out-of-equilibrium, and its activation energy decreases monotonically as the neuron relaxes slowly to a stable quiescent state (stable fixed * marius.yamakou@fau.de & yamakou@mis.mpg.de † tran@math.uni-leipzig.de & trandat@mis.mpg.de point); then, at a specific instant during the relaxation, the timescale of escape due to noise and the timescale of relaxation match, and the neuron fires at this point almost surely. If this activation or band width brings the neuron back out-of-equilibrium, the relaxation stage can start over again, and the scenario repeats itself indefinitely, leading to a coherent spiking activity which cannot occur without noise. SISR essentially depends on the interplay of three different timescales: the slow and fast timescales in the deterministic equation of the system, plus a third timescale characteristic to the noise.
It is important to note that the mechanism of SISR is very different from those of SR and CR. In fact, it has been shown in [22] that CR and SISR are actually two distinct mechanisms even though both lead to the emergence of weak noise-induced coherent oscillations. Moreover, in our previous work [28] (see also [30]), it has been shown that the way SISR in the first layer of a duplex neural network controls CR in the second layer, is different from the control of CR when we have CR in the first layer. This difference in the controllability of CR by SISR and CR in multiplex networks further confirms the fact that CR and SISR are actually different mechanisms. Compared to CR and SR, the conditions to be met for the mechanism of SISR are more subtle: Like CR, SISR does not require an external periodic signal as in SR. Remarkably, unlike CR, SISR does not require the system's parameters to be in the vicinity of bifurcation thresholds, making it more robust to parametric perturbations than CR. Moreover, unlike both SR and CR, SISR requires a strong timescale separation between the variables of the excitable system.
All previous investigations on SISR have treated the input noise process as solely Gaussian, with finite variance [19,[21][22][23][24][25][26][27][28][29]. In this paper, we focus on SISR in a neuron arXiv:2012.03032v1 [nlin.AO] 5 Dec 2020 with a more general input noise process: The common Gaussian fluctuation is replaced with an α-stable Lévy white noise, which for certain sets of parameters is equivalent to its standard Gaussian white noise analogue. In this respect, the study of SISR in a neuron perturbed by a Lévy white noise should reproduce the results of Gaussian white noise as a special case.
The exchange of charged ions across the membrane of the nerve cell can induce complex electromagnetic field inside and outside this membrane, and the membrane potential of neuron gets modulated by the induced electromagnetic field. Thus, by Faraday's law of electromagnetic induction, the effect of electromagnetic induction on the cell must be considered. Recently, M. Lv et al. [31] proposed a modified neural model which takes into account the effect of the magnetic field, generated by the internal bioelectricity of the nerve cell (i.e., the movement of the charged ions across the membrane on the spiking activity of the cell). In the modified (improved) neuron models, the effects of electromagnetic induction are described by using the magnetic flux. And the modulation of the membrane potential by the magnetic flux is realized by using a memristor coupling, hence the term memristive neurons [32]. The modification of the original neural models, so that they take into account these electromagnetic effects, consisted of adding a variable for the magnetic flux into the original equations.
Several studies have shown that memristive neurons can generate a rich variety of modes in electric activities by not only varying the external input current, but also by varying the magnetic flux parameters -those that control the memristive properties of the neuron [33][34][35][36][37][38][39]. It has been shown that the magnetic flux coupling between neurons can induce perfect phase synchronization of chaotic time series of membrane potentials [36]. This result basically showed that neurons exposed to their own external magnetic field can induce phase synchronization and appropriate behaviors can be selected from different magnetic flux parameter values.
It has also been shown that the magnetic field coupling can contribute to the signal exchange between neurons by triggering superposition of electric field when synapse coupling is not available [37]. Here, the contribution of field coupling from each neuron is described by introducing appropriate weight dependent on the distance between two neurons. It was found that the degree of synchronization is dependent on the intensity and weight of the field coupling and that the pattern selection of the network connected with gap junction can be modulated by this field coupling.
The memristive properties have also been shown to play a significant role in dynamics of other types of biological tissues. For example, it has been shown that target wave propagation can be blocked to stand in a local area of the cardiac tissue and the excitability of this tissue can be suppressed to approach quiescent but homogeneous state when electromagnetic flux (generated by the motion of ions across the membrane of the cardiac cell) is imposed on the cardiac tissue [38].
Moreover, it has been shown that a spiral wave can be triggered and developed by setting specific initial conditions in the cardiac tissue under the effects of magnetic flux, i.e., the tissue still support the survival of standing spiral waves under specific values of the magnetic flux parameters [39].
It is now well-accepted that the effects of the magnetic flux across the membrane of the cell should be considered when investigating the emergence of individual or collective electrical activities and wave propagation in the nerve and cardiac cells [31,38]. However, all previous studies on SISR in neural systems have been done only with non-memristive neuron models. Thus, the effects of the memristive properties of the neuron on SISR are not known. The questions we want to address in this paper are the following: (i) Can a Lévy noise with a mean exit time that follows a power law also induce SISR? (ii) Which noise induces the highest degree of SISR, Lévy or Gaussian noise? (iii) How do the memristive properties of the neuron affect the degree of SISR induced by these two types of noises?
The rest of the paper is organized as follows: In Sec. (II), we describe the mathematical equation modelling a memristive neuron driven by Lévy noise and we also determine the excitable parameter space of model in terms of the memristive parameters. Sec. (III) is devoted to the theoretical analysis of the mechanism of SISR. In Sec.
(IV), we present and discuss the numerical results. And in Sec. (V), we have summary and conclusions.

A. Model description
We consider a memristive FitzHugh-Nagumo (FHN) neuron model of type-II excitability [31,40], driven by an α-stable Lévy process, and described by the following stochastic differential equations with the deterministic velocity vector field is given by where (v, w, φ) ∈ R 3 represent the fast action potential variable v, the slow recovery current (or sodium gating) variable w that restores the resting state of the neuron, and the third variable φ is the magnetic flux across membrane which can generate additive current. The parameter 0 < ε := τ /t 1 is timescale separation ratio (also called singular parameter) between the slow timescale τ and the fast timescale t. It accounts for the slow kinetics of the sodium channel in the nerve cell and controls the main morphology of the action potential generated [41]. The constant parameter d is such d ∈ (0, 1), and c > 0 is a codimension-one Hopf bifurcation parameter.
(2) is the memory conductance of a magnetic flux-controlled memristor and it is used to describe the coupling between magnetic flux φ and membrane potential v of the neuron [42][43][44]. The memory conductance of a memristor is often described by where a and b are constant parameters. In this paper, we fix a = 0.1 and b = 0.02, to stay consistent with other works [45]. The magnetic feedback gain parameters k 1 and k 2 describe the interaction between the magnetic flux and membrane potential. More precisely, k 1 bridges the coupling and modulation on the membrane potential v from magnetic flux φ, and k 2 describes the degree of polarization and magnetization by adjusting the saturation of magnetic flux [46]. The term k 1 ρ(φ)v in Eq.
(2), therefore, describes the modulation on the membrane potential of the neuron, and it depends on the variation in the magnetic flux. Combining Faraday's law of electromagnetic induction and the basic properties of a memristor, the term k 1 ρ(φ)v is regarded as additive induction current on the membrane potential. The dependence of electric charge q on the magnetic flux φ is defined as [33,47] Moreover, because the current i is defined as the time rate of charge q, the physical significance for the term ρ(φ)v could be described as where V denotes an induced electromotive force with a feedback gain parameter k 1 . The potassium and sodium ionic currents contribute to the magnetic flux across the membrane and also to the membrane potential. This introduces a negative feedback term −k 2 φ in the third equation of Eq.

B. The excitable regime of the model
The deterministic memristive FHN neuron (i.e., the system in Eq. (1) with D = 0) with a unique and stable fixed point cannot maintain a self-sustained spiking activity. One says in this case that the neuron is in the excitable regime [53], in contrast to the oscillatory regime, where the neuron continuously spikes due to the occurrence of a bifurcation onto a limit cycle. In the excitable regime, choosing an initial condition in the basin of attraction of the unique stable fixed point will result in at most one large non-monotonic excursion into the phase space after which the trajectory returns to this fixed point and stays there until the initial conditions are changed again. The deterministic predisposition required for SISR is an excitable regime, so that during SISR, the self-sustained and coherent spike trains produced by the neuron is due only to the presence of noise and not because of the occurrence of bifurcations onto a limit cycle. This is one of the crucial differences between SISR and CR -the predisposition required for the latter mechanism is the close proximity of parameters to the bifurcation threshold, so that weak noise amplitudes can easily drive the system to this bifurcation threshold with overwhelming the dynamics stochastically [11,13,22]. In this section we determine the excitable regime of the memristive FHN neuron model. At the fixed points (v e , w e , φ e ) ∈ F ix (rest states of the neuron), the variables v(τ ), w(τ ), and φ(τ ) reach a stationary state, while the set of fixed points defined by the intersection of the nullclines as depends on the parameters c, d, k 1 , and k 2 . The sign of determines the number of fixed points. In this paper, we consider the case where we have only one stable fixed point. If ∆ > 0, we have a unique fixed point given by Moreover, in the model we arbitrarily fix d = 0.5 once and for all, and we determine the excitable regime of the model in terms of the parameter c and the two new parameters k 1 and k 2 -the magnetic gain parameters. With the fixed values of the parameters a = 0.1, b = 0.02, and d = 0.5, p and g in Eq. (12) now depend only on c, k 1 , and k 2 . We have: which are both always positive for c < 1, k 1 ≥ 0 and k 2 > 0. Hence, ∆ in Eq. (10) will always be positive for c < 1, k 1 ≥ 0 and k 2 > 0, ensuring the uniqueness of the fixed point (v e , w e , φ e ) in Eq. (11).
With initial conditions at the unique fixed point v e (c, k 1 , k 2 ), w e (c, k 1 , k 2 ), φ e (c, k 1 , k 2 ) , we numerically computed a codimension-one and codimension-two bifurcations, showing the excitable and oscillatory regimes of the memristive neuron with the respect to the parameter c in Fig. 2(a) and the magnetic gain parameters k 1 and k 2 in Fig. 2(b), respectively.
The bifurcation diagram in Fig. 2(a) shows a nonzero inter-spike interval (ISI) for 0 < c < c h , where c h = 0.875 is the super-critical Hopf bifurcation threshold. For c ≥ c h , there is no spiking, i.e., ISI = 0, indicating that the neuron is in an excitable regime at k 1 = 0.1 and k 2 = 0.1. However, it is well known that variations in these magnetic gain parameters can significantly affect the dynamical response of the neuron [46], thereby switching the neuron from an excitable to an oscillatory regime and vice versa, even when c > c h . Hence, it is important to determine the range of values of k 1 and k 2 in which the neuron will remain in the excitable regime for a particular value of c, chosen such that c h < c < 1. Fig. 2(b) shows, for c = 0.95 > c h = 0.875 (i.e., c is far enough from the bifurcation threshold and also less than one so that the stable fixed point is unique), a two-parameter space bifurcation diagram with respect to k 1 and k 2 . We also note that k 2 starts at a non-zero value, i.e., at k 2 = 0.01, to ensure that our fixed point in Eq. (11) is unique. The color-coded ISI shows the oscillatory regime in red and yellow where ISI > 0. The yellow region corresponds to few points around the origin of the (k 1 , k 2 ) plane, where ISI takes relatively large values. For example, at k 1 = 0.0361 and k 2 = 0.01 we have ISI = 10.16, and at k 1 = 0.0643 and k 2 = 0.03, ISI takes its largest value, i.e., ISI = 17.78. The black region (where ISI = 0) corresponds to the excitable regime, with the deterministic model in Eq. (1) consisting of unique and stable fixed point given by Eq. (11). Therefore, throughout this paper, we will investigate the mechanism of SISR when the neuron is in the excitable regime defined

III. THE ASYMPTOTIC MATCHING OF TIMESCALES AND SISR
Now we consider Eq.
(1) such that its deterministic version is in the excitable regime, defined by the parameters intervals and values above. To understand how noise can induced a regular escape of trajectories from the basin of attraction of the stable fixed point, leading to the emergence of a coherent spike train, we transform Eq. (1) from the slow timescale τ to the fast timescale t to obtain Eq. (13). Under this timescale transformation, the noise term is re-scaled according to the scaling law of Lévy motion. That is, if L τ is a Lévy motion, then, for every λ > 0, λ − 1 α L λτ is also a Lévy motion (i.e., they have the same distribution). We note that because of this scaling law, the term 1/ α √ ε was introduced in the noise term in Eq. (1) to guarantee that in Eq. (13), the amplitude of the noise (D) measures the relative strength of the noise term compared to the deterministic term f 1 (v, w, φ) irrespective of the value of ε.
In the adiabatic limit ε → 0, the timescale separation between v t and the two other variables w t and φ t become very large. This indicates that w t and φ t are frozen on the O(1) fast timescale, hence Eq. (13) is approximated by Eq. (14) where U k1 (v t ) is the derivative of the potential with respect v. U k1 (v) is the double-well potential with the constant solutions of the last two equations in Eq. (14) given by w and φ, respectively. This potential has, respectively, a left local minimum, a saddle, and a right local minimum at where P = 3[k 1 ρ(φ) − 1] and Q = 3w, see Fig. 3. It was shown in [54,55] that for barrier crossing phenomena driven by Lévy white noise in the double-well potential, the mean exit time from one of the wells follows a power law in the limit as D → 0 and not Kramers' law as the Gaussian white noise would do in the limit as D → 0 [27,56,57]. Following the general results in [54,55], we calculated, for the double-well potential in Eq. (15), the mean exit times of the Lévy process as: The mean exit times in Eq. (17) do depend on the location of the local minima v l and v r . Moreover, they also follows a power law with respect to the noise intensity D, which means that the jumping events of trajectories occur with a polynomially small probabilities [54,55]. On the other hand, the mean exit times of the Gaussian process follow Kramers' law [27,56], with escape events occurring with exponentially small probabilities, and are given by: where U ± are the energy barrier functions that depend, technically, on w and φ. The asymmetry of the potential in Eq. (15) is controlled only by the sign of the coefficient of the linear term, i.e., the sign of w. While the depths of the wells U ± are controlled by the value of w and more significantly, by the term k 1 ρ(φ)/2. But in the limit as ε → 0 in Eq. (13), the magnetic variable φ becomes almost constant and only the magnetic gain parameter k 1 now significantly changes the depths of the potential wells U ± . So we can drop the φ dependence in the energy barrier functions and write them as: Thus, in the Gaussian case, the trajectories surmount the potential barriers U ± , such that the mean exit times depend exponentially on the depth of the potential well. We notice in Fig. 3 that the depths of these barriers are inversely proportional to the strength of the magnetic gain parameter k 1 . Thus, a stronger magnetic flux due to a larger value of k 1 should, on average, reduce the duration of the mean exit times of the trajectory perturbed by Gaussian noise, contributing to an increase in the spiking frequency.
On the other hand, we also notice that the positions of the minima (at v l and v r , indicated by the short vertical bars in Fig. 3) from the saddle (at v m = 0.0) change with k 1 . We observe that the stronger magnetic flux k 1 , the smaller the distances of v l or v r from 0.0, which in turn shortens, on average, the duration of the mean exit times of the trajectory perturbed by Lévy noise, contributing to an increase in the spiking frequency.
From Eq. (1), the deterministic timescale at which trajectories move on the stable parts of the 2-dimensional cubic nullcline of the current model, given by w(v, φ) = − v 3 3 + (1 − k 1 ρ(φ))v (not shown), is ε −1 [27]. When there is no noise (D = 0), the neuron is in the excitable regime and as ε → 0, trajectories tend to spend a lot of time moving adiabatically along the stable parts of the 2D cubic nullcline, toward the unique stable fixed point at (v e , w e , φ e ) given by Eq. (11), where it stops and stays for ever until a new perturbation is provoked by e.g., a random process.
When noise is switched on (D = 0), it may kick a trajectory, which is moving quasi-deterministically at a timescale of ε −1 along one stable branch of the 2D cubic nullcline, to another branch and then back. This corresponds to jumps out of the left and right potential wells, thereby causing a spike -an oscillation. Depending on the type of noise perturbing the neuron, an escape from left to right (right to left) occurs at the stochastic timescale ET exit given by the first (second) equation of Eq. (17) for the Lévy process or Eq. (18) for the Gaussian process.
It has been shown that the occurrence of SISR crucially depends on the neuron's ability to asymptotically match, with probability close to unity, the deterministic timescale ε −1 (i.e., timescale at which a trajectory moves along the stable parts of the 2D cubic nullcline) and the stochastic timescale ET exit (i.e., the timescale at which this trajectory escapes from the stable parts of this nullcline) at unique exit points w − and w + located, respectively, on the left and right stable branches of the 2D cubic nullcline [19,[21][22][23][24][25][26][27][28][29].
If the deterministic timescale is shorter than the stochastic timescales (i.e., ε −1 < ET exit ), then the trajectory has no time to escape from the left and right stable branches of the cubic nullcline which respectively correspond to the left and right wells of the potential U k1 (v). Because the neuron is in an excitable regime, the trajectory gets trapped in the left well of the potential (i.e., on the left stable branch of the cubic nullcline on which the unique stable fixed point is located) for too long. In this scenario, a spike is a rare event and this could destroy the coherence of the spiking, especially for short time intervals.
On the other hand, if the deterministic timescale is longer than the stochastic timescales (i.e., ε −1 > ET exit ), then the trajectory frequently escape from the potential wells (i.e., the stable branches of the cubic nullcline). In this scenario, spiking is frequent (i.e., not rare) but incoherent because the trajectory escapes at several different points on the each of the stable branches of the cubic nullcline.
Interestingly, if at specific and unique points w − and w + on respectively the left and right stable branch of the cubic nullcline, the deterministic timescale matches the stochastic timescales (i.e., ε −1 = ET exit ), frequent and coherent spiking emerges -SISR occurs. The uniqueness of the exit points w − and w + can only be guaranteed by the monotonicity of the minima v l (w) and v r (w) in the case of Lévy noise (see Eq. (20)) and the barrier functions U − (w) and U + (w) in the case of Gaussian noise (see Eq. (21)).
In Fig. 4, we show the graphs of the functions |v l |, v r , U − , and U + with respect to w ∈ [− 2 3 , 2 3 ], where the lower and upper bounds of this interval correspond to the w-coordinate of the local minimum and maximum of the cubic nullcline, respectively. Here, we see that these functions are all monotone with respect to w ∈ [− 2 3 , 2 3 ]. Hence, frequent and coherent spiking would occur if we match the deterministic and stochastic timescales only at w − on the left stable branch and at w + on the right stable branch of the cubic nullcline, that is: for the Lévy process, and for the Gaussian process. Therefore, the occurrence of SISR (i.e., frequent and coherent spiking activity) will depend on the neurons' ability to asymptotically match the timescales by taking the following double scaling limits: for the Lévy process, and lim (ε,D)→(0,0) for the Gaussian process [21,27]. Due to the anomalous long jumps of a trajectory perturbed by a Lévy process [55,[58][59][60], this trajectory does not necessarily have to hit the saddle point at v m before escaping from the stable branches of the 2D cubic nullcline. Hence, escapes may instantaneously occur even with a very weak noise intensity. This means that the "frequent spiking" requirement of SISR can be easily achieved by a Lévy process, even with a very weak intensity. However, the "coherent spiking" requirement of SISR can only be guaranteed by the asymptotic scaling limits given in Eq. (22).
In the Gaussian case, a trajectory can only escape from a potential well after hitting the boundary at the saddle point at v m . Therefore, the "frequent spiking" condition of SISR requires that the noise intensity is not too weak (otherwise, we get a Poissonian spike train -a rare spiking event which could destroy the coherence of the spiking [27]). Moreover, we observe that the stochastic timescales of the Gaussian noise in Eq. (18) depend on the energy barrier functions U ± . If these barriers are too deep (i.e., U ± → ∞), then weak noise intensities cannot provoke escapes (at least frequently), and the trajectory will remain strapped inside a potential well. Thus, the noise has be to weak (so that the mean exit times satisfy Eq. (18)), but strong enough to able to invoke some spiking. If this Gaussian noise is strong enough to invoke spiking, then the "coherent spiking" requirement of SISR can only be guaranteed by the asymptotic scaling limits given by Eq. (23). Thus, for Lévy noise, we expect SISR to occur even at very weak noise intensities. But for Gaussian noise, we expect SISR to occur at a comparatively larger intensity.
To answer the three main questions we are interested in, we will set the memristive neuron in the excitable regime by choosing c = 0.95, a = 0.1, b = 0.02, d = 0.5, and also set the scale and the mean parameters of the Lévy process at σ = 0.5 and µ = 0.0, respectively. We also chose a sufficiently small timescale separation parameter, i.e., ε = 0.001 1, weak noise intensity, i.e., 0 < D < 1, and then numerically search for the combined values of k 1 ∈ [0.0, 2.0], k 2 ∈ [1.0, 2.0], α ∈ (0, 2], and β ∈ [−1, 1] for which the scaling limit conditions in Eq. (22) and Eq. (23) are satisfied (or at least to some degree) or not.

IV. NUMERICAL RESULTS AND DISCUSSION
To measure the degree of SISR, we use the coefficient of variation (CV ), an important statistical measure based on the time intervals between spikes [11]. From a neurobiological point of view, CV is more important than other measures (e.g., power spectral density and autocorrelation function) because it is related to the timing precision of information processing in neural systems [61]. CV uses the inter-spike intervals (ISIs) where the kth interval is the difference between two consecutive spike times t k and t k+1 of the neuron, and is defined as: where ISI and ISI 2 represent the mean and the mean squared ISIs, respectively. When CV = 1, we have Poissonian spike train (i.e., rare and incoherent spiking), and when CV > 1 we have a point process that is even more variable than a Poisson process [62]. In both these cases, the degree of SISR is quite low as the double limits in the left-hand sides of Eq. sides. In this case, we will have perfectly "deterministic" periodic spiking. In our numeical simulations, we used the fourth-order Runge-Kutta and the Janicki-Weron algorithms [63,64] to numerically integrate Eq. (13) for a very long time interval (i.e., T = 25, 000, 000 time unit which allows for the small value of ε = 0.001, the collection of sufficiently many ISIs for statistical estimate). We then average the ISIs over time and up to 20 realizations for each noise amplitude.
We note that because of the discontinuous jump property of a Lévy process (with a variance going to infinity as α → 0), trajectories can (in a short time intervalpossibly as short as the integration time step) easily escape from a basin of attraction to infinity without hitting the boundary. While with the continuous jump property (with variance finite) of a Gaussian process, trajectories must hit the boundary of the basin of attraction before escaping.
In our simulations, we imposed a simple artificial constraint on our system that controls the allowed values of the membrane potential v during the spiking process, i.e., if |v| > 3.0 (larger enough than the extreme values of v for w ∈ [− 2 3 , 2 3 ]), then v = 3.0 × sign(v). This constraint eliminates the problem of escape of trajectories to infinity when the neuron is driven by Lévy noise, see Fig. 5(c) and (d). Hence, the integration of Eq. (13) can be done with any time step when Lévy noise is used [65]. Fig. 5(a) and (c) respectively show the variation of CV with the noise intensity D for an impulsive (α = 0.1) and symmetric (β = 0.0) Lévy noise and a time series of the coherent spike trains obtained at a noise intensity which satisfies Eq. (22). The CV -curve and time series are computed in a weak magnetic flux regime (k 1 = 0.1, k 2 = 0.1) and show that as long as Eq. (22) is valid, Lévy noise can (i) induce a high degree of SISR even at very weak noise intensities (e.g., CV ≈ 0.2 at D = 1.0×10 −15 ), and (ii) induce an even higher SISR at relatively larger noise intensities (e.g., CV = 0.0014 at D ≈ 0.9).
On the other hand, Fig. 5(b) and (d) respectively show the variation of CV with the noise intensity D for Gaussian noise (α = 2.0, β = 0.0) and a time series of the coherent spike train obtained at a noise intensity which satisfies Eq. (23), in the same weak magnetic flux regime (k 1 = 0.1, k 2 = 0.1). Comparing the degree of SISR induced by a Lévy noise with parameters at α = 0.1 and β = 0.0 to that of Gaussian noise (α = 2.0, β = 0.0), we see that Lévy noise can induce a higher degree of SISR with both extremely weak and weak noise amplitudes. In Fig. 5(b) with Gaussian noise, we have a low (and almost constant) CV ≈ 0.065 only in the weak (but not too weak) noise intensities, i.e., for D ∈ (0.01, 0.1). Fig. 6(a) and (b) show minimum coefficient of variation (CV min ) against the stability index (α) and the skewness (β) parameters of the Lévy process in a weak (k 1 = 0.1 and k 2 = 0.1) and in a strong (k 1 = 2.0 and k 2 = 1.0) magnetic flux regime, respectively. In Fig. 6(a), a right-skewed, i.e., β ∈ (0.0, 1.0], Lévy process with a low stability index, In Fig. 6(b), with a strong magnetic flux regime (k 1 = 2.0, k 2 = 1.0), the variation in the degree of SISR is almost everywhere the same as in Fig. 6(a). However, in the region bounded by α ∈ [0.3, 0.8] and β ∈ [−1.0, −0.4], the data show that there is a slight difference in the degree of SISR in both panels. In these intervals, SISR is slightly more pronounced in the strong magnetic regime than in the weak one. In both panels of Fig. 6, when we have Gaussian noise (i.e., α = 2.0 and β = 0.0), we have a CV min ≈ 0.095 for weak magnetic flux and CV min ≈ 0.034 for strong magnetic flux. Later, we shall  Fig. 5(c)), hence the low value of the CV min in these regions.
In Fig. 7, we show the variation in the degree of SISR with the variations in the strengths of the magnetic gain parameters k 1 and k 2 in three specific regions of interest in Fig. 6(a): (i) when the degree of SISR is low, i.e., in the white region with α = 0.8 and β = −1.0, (ii) when the degree of SISR is high, i.e., the dark red region with α = 2.0 and β = 0.0 (i.e., Gaussian), and (iii) when the degree of SISR is very high, i.e., the black region with α = 0.1 and β = 1.0. We also note that in all the panels of Fig. 7, the magnetic gain parameter k 2 is restricted to k 2 ≥ 1.0, so that the memristive neuron always lies in the excitability region (black region) for all values of k 1 ≥ 0.0, as indicated in Fig. 2(b).
In Fig. 7(a), we can now clearly see the effects of the magnetic gain parameters on the degree of SISR when α = 0.8 and β = −1.0, corresponding, from Fig. 6(a), to a relatively large CV min ≈ 0.46. We observe that: the stronger the magnetic gain parameter k 1 -that bridges the coupling and modulation on the membrane potential v from magnetic field φ -and the weaker the parameter k 2 -that describes the degree of polarization and magnetization by adjusting the saturation of magnetic fluxthe higher the degree of SISR. In Fig. 7(a), as k 2 → 1.0 and k 2 → 2.0, the color-coded CV min goes from a white region with a relatively high value of CV min ≈ 0.23, via a yellow and a red, to a black region with the lowest CV min ≈ 0.153. Moreover, irrespective of the value of k 2 , when k 1 = 0.0, CV min takes the highest value of the panel (i.e., CV min ≈ 0.23). Further numerical simulations (not shown) indicated that this behavior is qualitatively the same for many pairs of values of α ∈ We shall see later in Fig. 7(c) that this significant improvement in the degree of SISR depends on intervals in which α and β are located.
In Fig. 7(b), we have Gaussian noise (i.e., α = 2.0 and β = 0.0) and effects of the magnetic gain parameters are qualitatively the same as in Fig. 7(a) with a Lévy noise having parameters at α = 0.8 and β = −1.0. That is, the weaker k 2 and the stronger k 1 become, the lower is CV min , on average. It is worth noting, by comparing Fig. 7(a) and (b), that even though the degree of SISR induced by Gaussian noise (α = 2.0 and β = 0.0) is higher than that induced by Lévy noise with parameters in the intervals α ∈ [0.3, 0.8], β ∈ [−1.0, −0.4], the effects of the magnetic gain parameters k 1 and k 2 on the degree of SISR is less stronger in the Gaussian case. That is, in Fig. 7(b), CV min varies in the interval [0.031, 0.093], compared to [0.153, 0.23] in Fig. 7(a)). The bigger range in the latter interval indicates the stronger effects of the magnetic gain parameters on the degree of SISR induced by Lévy noise when its parameters lie in the intervals α ∈ [0.3, 0.8] and β ∈ [−1.0, −0.4].
Moreover, it important to note that the degree of SISR in the non-memristive (k 1 = 0 = k 2 ) neuron is always lower than that in the memristive one. This result is confirmed by comparing CV min in the non-memristive FHN neuron perturbed by Gaussian noise -studied in our previous work [27] -to the memristive FHN model studied in the current paper. In the non-memristive case, the lowest CV value is always at CV ≈ 0.2, while in the memristive case, the lowest value gets even smaller, i.e., CV ≈ 0.034, especially as k 1 → 2 and k 2 → 1.
In Fig. 7(c), we have a Lévy noise with α = 0.1 and β = 1.0, which corresponds to a black region (i.e., with a high degree of SISR) in Fig. 6(a). In this case, compared to Fig. 7(a), the magnetic gain parameters do not have a significant effect on the high degree of SISR. The degree remains very high with a CV min slowing varying within an extremely thin interval of [0.001345, 0.001359], for all values of k 1 and k 2 .
In the adiabatic limit ε → 0, the fact that stronger magnetic flux k 1 can significantly improve the degree of SISR in the Gaussian and in some Lévy cases can conceptually be explained in term of the the potential landscapes in Fig. 3 and the mean exit times given by Eq. (17). In the Gaussian case, mean exit times depend exponentially on the barrier functions U ± (see Eq. (18)) which should not be too deep, so that weaker noise intensities can be sufficient to provoke jumps (spikes) from one potential well to another. So as k 1 → 2 (i.e., becomes stronger), U ± → 0 (i.e., become shallower, see Fig. 3), and the more easily weak noise intensities can provoke frequent spikes. And if this frequent spiking is combined with the scaling limits in Eq. (23), the degree of SISR gets higher (i.e., CV min → 0.0).
In the Lévy cases, mean exit times in Eq. (17) depend on the location of the minima v l and v r and hence, also on band widths of the wells (i.e., the distances from the minima v = v l and v = v r of the wells to the saddle point v = v m = 0.0; see Fig. 3 which shows a reduction in the distance between the short vertical bars all located at these minima, and the point v = 0, as k 1 increases). The shorter these band widths are (i.e., the closer v l and v r are to v m = 0.0), the shorter the mean exit times given in Eq. (17). Thus, weak noise intensities can more easily provoke frequent jumps (spikes) from one potential well to another. When this frequent spiking is combined with the scaling limits in Eq. (22), the degree of SISR gets higher.
However, when the Lévy noise becomes impulsive (i.e., as α → 0, with a variance that tends to infinity, see Fig. 1 and also [55]), the anomalous (and sometimes instantaneous) long jumps of trajectories becomes significant. In this case, the band widths which are controlled by magnetic gain parameter k 1 do not longer affect the mean exit times. Hence, as α → 0, the variation in the magnetic gain parameters should not have any significant effects on the high degree of SISR as long as Eq. (22) is satisfied. This is what we observe in Fig. 7(c) with α = 0.1 and β = 1.0.
Nevertheless, this inability to affect the degree of SISR when α → 0 depends also on the skewness of the Lévy noise. If the noise is left-skewed (as e.g., in Fig. 7(a) with β < 0), then left potential well (i.e., the left stable branch of the cubic nullcline on which the unique stable fixed point is located) is favoured compared to the right well (i.e., the right stable branch which has no fixed point). This results into trajectories staying a bit longer in this FIG. 7. Minimum coefficient of variation (CVmin) against magnetic gain parameters k1 and k2 in: (a) Lévy noise with α = 0.8 and β = −1.0 (corresponding to the white region in Fig. 6(a)); (b) Gaussian noise with α = 2.0 and β = 0.0 (corresponding to the dark red region in Fig. 6(a)); and (c) Lévy noise with α = 0.1 and β = 1.0 (corresponding to the black region in Fig. 6(a)). In (a) and (b), the stronger k1 and the weaker k2 are, the more enhanced is SISR. In(c), k1 and k2 have no significant effect on the high degree of SISR as shown by the very low and quasi-constant CVmin ≈ 0.001352. left well, provoking these intermittent intervals of subthreshold spiking which destroys the regularity of the ISIs.
In this left-skewed case, the magnetic gain parameters have significant effect on the degree of SISR as we saw in Fig. 7(a).

V. SUMMARY AND CONCLUSIONS
In this paper, we investigated and compared the mechanism of SISR induced by Lévy white noise and Gaussian white noise in a memristive FHN neuron. We showed that depending on the values of the stability index α ∈ (0, 2) and skewness parameter β ∈ [−1, 1] of a Lévy noise, the neuron could exhibit a very high degree of SISR with a minimum coefficient of variation as low as 0.0014, compared to 0.034 in the case of Gaussian noise (with α = 2, and β = 0). However, the degree of SISR induced by a Lévy noise is not always higher than that induced by the Gaussian noise. In the intervals α ∈ [0.3, 0.8] and β ∈ [−1.0, −0.4], the Lévy processes induce a lower degree of SISR (with CV min ∈ [0.2, 0.46]) than the Gaussian process which induces a higher degree of SISR with CV min = 0.034.
It is shown that, the stronger magnetic gain parameter k 1 (i.e., the parameter that bridges the coupling and modulation on membrane potential v from magnetic field φ) and the weaker k 2 (i.e., the parameter that controls the degree of polarization and magnetization by adjusting the saturation of magnetic field φ) are, the higher the degree of SISR for both Lévy and Gaussian processes. However, in the Lévy case, this effects of the magnetic gain parameters become insignificant when the process becomes impulsive (i.e., when α → 0) and right-skewed (i.e, β > 0). Moreover, it has been shown, for both types of noises, that the degree of SISR in the memristive neuron is always higher than the degree in the non-memristive neuron.
Looking forward, we must be cognizant that Lévy white noise is only one possible type of a non-Gaussian white noise which can induce SISR. The mechanism via which noise with a temporal correlation (i.e., colored noise) can induce SISR is worth investigating. The additional timescale brought about by the temporal correlation of the noise into the system, may come along with new interesting dynamics.