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## Lévy noise induced stochastic resonance in an FHN model

### Hint

### Swipe to navigate through the articles of this issue

13-01-2016

### Keywords

Mathematical model, Single cell model, Hodgkin–Huxley model, FitzHugh–Nagumo model

### Introduction

More typically, ordinary differential equations (ODEs) are used to model the behaviour of electricity in a myocardial cell. ODE models can be placed into two groups. First-generation models include behaviour described by a phenomenological model as well as some of the underlying physiology [1]. The ionic channels most responsible for generating an action potential are included in a first-generation model, but many of the finer details are simplified. Examples of first-generation models are the FitzHugh–Nagumo and Luo–Rudy Phase I models.

The FitzHugh–Nagumo model (FHN), named after Richard FitzHugh who suggested the system in 1961 and J. Nagumo who created the equivalent circuit the following year to describes a prototype of an excitable system [2].

Second-generation models include all of the detail of a firstgeneration model and as many of the finer details as possible. Examples of second-generation models are the models of Courtemanche, Winslow, and Puglisi–Bers [3].

### Materials and Methods

*Mathematical model*

In neurons, action potential transfers inside the single cell and participate in neuron-to-neuron communication. Since neural and cardiac cells have many similarities, much of the mathematics of cardiac cell modeling is drawn from the pioneering work of Hodgkin and Huxley.

*Single cell models*

The behaviour of electrical activity can be modelled with something as simple as a cubic polynomial [1,3]. Phenomenological models reproduce only the macroscopic details regarding electrical activity and do not include any of the underlying physiological details that cause the creation and behaviour of electrical activity in the heart. Polynomial equation does not model the re-polarization phase [1]. The advantage to using phenomenological models is that they can simulate an action potential with the lowest possible computational cost [4].

*Hodgkin–Huxley model*

The Hodgkin – Huxley model (HH), or conductance-based model, is a mathematical model that describes how action potentials in neurons are initiated and propagated. It is a set of nonlinear differential equations that approximates the electrical characteristics of excitable cells such as neurons and cardiac myocytes.

Alan Lloyd Hodgkin and Andrew Huxley described the model in 1952 to explain the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon [5].

This model usually represents the biophysical characteristic of the cell membrane. The lipid layer is represented by the capacitance C_{m}. Voltage-gated and leak ion channels are represented by nonlinear (gn) and linear (gL) conductance’s, respectively. The voltage source En whose voltage is determined by the ratio of the intra- and extracellular concentrations of the ionic species of interest [5].

The capacitive current Ic is defined by the rate of change of charge q at the trans membrane surface is

The charge q(t) is related to the instantaneous membrane voltage V_{m}(t) and membrane capacitance C_{m} by the relationship q = C_{m}.V_{m}. Thus the capacitive current can be rewritten as

In the Hodgkin-Huxley model of the squid axon, the ionic current I_{ion} is subdivided into three distinct components, a sodium current I_{Na}, a potassium current I_{K}, and a small leakage current IL that is primarily carried by chloride ions. So the differential equation is given by

Where I_{appl} is an externally applied current, such as might be introduced through an intracellular electrode.

The individual ionic currents I_{Na}, IK and IL represent the macroscopic currents flowing through a large population of individual ion channels. In HH-style models, the macroscopic current is assumed to be related to the membrane voltage through an Ohm’s law relationship of the form V=IR [5]. In many cases Ohm’s law described by I=GV. Where G is the conductance and given by G=1/R. So the total ionic current Iion is the algebraic sum of the individual contributions from all participating channel types found in the cell membrane is given by (46)

This expands to the following expression for the Hodgkin- Huxley model of the squid axon:

In general, the conductances are not constant values, but they can depend on other factors like membrane voltage or the intracellular calcium concentration. In order to explain their experimental data, Hodgkin and Huxley postulated that GNa and GK were voltage-dependent quantities, whereas the leakage current GL was taken to be constant. Although Hodgkin and Huxley did not describe about the properties of individual membrane channels when they developed their model, it will be convenient for us to describe the voltagedependent aspects of their model in those terms [5].

*The FitzHugh–Nagumo model*

One of the simplest single cell models is what is now called the FitzHugh–Nagumo (FHN) model. The model was originally developed as simplification of the Hodgkin–Huxley model by FitzHugh in 1961. The behaviour of a neuron after stimulation by an external input current is expressed by equation (6) and (7) [6] and expressed in the equivalent circuit by Nagumo in 1962 [7].

If the external stimulus I ext {\displaystyle I_{\text{ext}}} current Ist exceeds a certain threshold value, the system will exhibit a regular excitation. Where v is membrane voltage v {\displaystyle v}and w is a linear recovery variable and a, b, τ are model parameters. An example of values for these parameters is given in **Table 1**. These parameters may be modified to model different cell types (**Figure 1**) [6].

Parameter | Value |
---|---|

Ist | 0.32 |

a | 0.7 |

b | 0.8 |

τ | 12.5 |

**Table 1.** Parameters value of simple FitzHugh-Nagumo model [6].

**Figure 1:** Circuit diagram of the tunnel diode FitzHugh–Nagumo model.

The motivation for the FHN model was to isolate conceptually the essential mathematical properties of excitation and propagation from the electrochemical properties of sodium and potassium ion flow [6].

### Results and Discussion

Solution of equation (6) and (7) are obtained using the parameters of **Table 1**. It has been observe that every parameter has a threshold value. Below or above this threshold regular excitation is occurred. Threshold label of Ist identified in this present studied is 0.324. Threshold value of a, b, τ are 0.69, 0.79 and 14.4 respectively. The results are shown graphically on **Figures 2** to **5**.

**Figure 2:** Graph of membrane voltage for a=0.7, b=0.8, τ=12.5 and I = 0.3, 0.32 and 0.325.

**Figure 3:** Graph of membrane voltage for i=0.32, b=0.8, τ=12.5 and a = 0.72, 0.7 and 0.69.

**Figure 4:** Graph of membrane voltage for i=0.32, a=0.7, τ=12.5 and b = 0.82, 0.8 and 0.79.

**Figure 5:** Graph of membrane voltage for i=0.32, a=0.7, b=0.8 and τ = 13.2, 14.0 and 14.4.

### Conclusion

In this article, simulation of FitzHugh-Nagumo model is made. The parameters of FitzHugh-Nagumo model for regular excitation are studied. Simulation results are carried using MATLAB. The FHN model is simple to implement and computationally inexpensive but it is limited in terms of the physiological accuracy. Many attempts have been made to modify the FHN model to make it more physiologically accurate while retaining the simplicity of the original model.

### References

- Sundnes J, Lines GT, Cai X, Nielsen BF, Mardal KA, Tveito A (2006) Computing the Electrical Activity in the Heart. Springer-Verlag Berlin Heidelberg.
- William Erik Sherwood (2015) Fitzhugh–Nagumo Model. Encyclopedia of Computational Neuroscience, Springer, New York, United States.
- Dean RC. Numerical methods for simulation of electrical activity in the myocardial tissue. PhD thesis. University of Saskatchewan,Canada 2009.
- AJ Pullan, ML Buist, LK. Cheng (2005) Mathematically modelling the electrical activity of the heart: From cell to body surface and back again. World Scientific, New Jersey.
- Hodgkin AL, Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 1952; 117: 500-544.
- R. FitzHugh, Impulses and Physiological States in Theoretical Models of Nerve Membrane. Biophys J 1961; 1: 445-466.
- J Nagumo, S Arimoto, S Yoshizawa. An active pulse transmission line simulating nerve axon, Proceedings of the IRE 1962; 50 :2061–2070.

## Firing activities of a fractional-order FitzHugh-Rinzel bursting neuron model and its coupled dynamics

### Abstract

Fractional-order dynamics of excitable systems can be physically described as a memory dependent phenomenon. It can produce diverse and fascinating oscillatory patterns for certain types of neuron models. To address these characteristics, we consider a nonlinear fast-slow FitzHugh-Rinzel (FH-R) model that exhibits elliptic bursting at a fixed set of parameters with a constant input current. The generalization of this classical order model provides a wide range of neuronal responses (regular spiking, fast-spiking, bursting, mixed-mode oscillations, etc.) in understanding the single neuron dynamics. So far, it is not completely understood to what extent the fractional-order dynamics may redesign the firing properties of excitable systems. We investigate how the classical order system changes its complex dynamics and how the bursting changes to different oscillations with stability and bifurcation analysis depending on the fractional exponent (0 < *α* ≤ 1). This occurs due to the memory trace of the fractional-order dynamics. The firing frequency of the fractional-order FH-R model is less than the classical order model, although the first spike latency exists there. Further, we investigate the responses of coupled FH-R neurons with small coupling strengths that synchronize at specific fractional-orders. The interesting dynamical characteristics suggest various neurocomputational features that can be induced in this fractional-order system which enriches the functional neuronal mechanisms.

### Introduction

Collective oscillatory dynamics and synchronous activity are the fundamental phenomena in dynamical systems^{1,2}. It has both theoretical importance and biophysical significance in computational neuroscience. Mathematical biophysical models^{3,4,5,6} are the primary tools to characterize the nervous system. The foremost exciting step in neural dynamics is to understand the system architecture of individual neurons in terms of mathematical models of membrane potential. Various types of spiking and bursting are the dynamical responses of excitable cells^{7,8}. Such type of research analyzes the chaotic behavior of excitable systems. When mathematical models are described as a single neuron or network, then the nonlinear dynamical techniques are applied to study the emerging oscillatory patterns and the synchronization phenomena. The classical order dynamical models depend on the immediate previous response, however the fractional-order derivative depends on all the previous responses, so it has a memory effect^{9,10,11}. It can produce a different kinds of multiple time scale neuronal dynamics^{12}. It^{13,14,15,16} provides a wide range in understanding the rich dynamical and neuronal responses. Fractional-order calculus originated from a letter written to Leibnitz by L’Hospital^{13,14,15}. Now, it has become a promising and reliable mathematical tool that includes hereditary properties or memory dependence phenomena^{13,14,15,16,17}. The discussion of multiple time scale dynamics has been studied in some previous articles^{18,19} which have the potential importance in signal processing. It has been studied that the firing rate of neocortical pyramidal neurons with the injected applied sinusoidal current can be well approximated with fractional-order derivative^{20}.

Many researchers have worked on the fractional-order dynamical systems^{21,22}. Results show that it follows power-law dynamics^{23,24,25}. In human memory, the power-law dynamics was investigated earlier^{26,27}, the accuracy of memory dependence decays at a rate nearly equal to *t*^{−α} where \(0 < \alpha < 1\). The power-law adaptation helps in describing some dynamical behavior of biological systems^{12,28}. In recent years, fractional-order derivative has become very useful in modeling biological phenomena^{13,14,15,16,29}, viscoelastic properties of tissue^{30}, tissue electrode interface^{31}, the kinetic property of drug delivery^{32,33}, diffusion process^{34,35,36}, biophysical neuron models and neural networks^{37,38,39,40,41,42}. It has been found that cognitive behaviour can be modelled using fractional dynamics^{43}. It was observed that fractional-order dynamics is used in vestibular oculomotor system^{44} and fly motion of sensitive neurons H1^{28}. It can include the mechanism of synapses^{28} and the geometrical properties of excitable cells^{44,45}. Single neuron models are analyzed by using fractional-order dynamics such as Hodgkin-Huxley (H-H), Morris-Lecar (M-L), FitzHugh-Nagumo (FHN), Hindmarsh-Rose (H-R) models, etc^{10,39,46,47,48,49}. In this study, it has been demonstrated that the fractional-order dynamics of a solitary nerve membrane can be analyzed by using a suitable biophysical model that exhibits elliptic bursting. It reflects the rate of change of information through membrane voltage that leads to previous history-dependent activities. Different parameter regimes corresponding to qualitatively different dynamical properties are analyzed. This article investigates the FH-R neuron^{50,51,52} in a fractional-order domain to get a general idea about the spike patterns and spike frequency with stability and bifurcation scenarios. The relation between spiking and bursting is a significant question as well as a fascinating phenomenon in mathematical neuroscience, especially in neural coding. Bursting presents a recurrent transition between repetitive spiking and quiescent state. The switching phases depend on the strength of the slowly changing current stimulus to the dendrite. An exciting feature of the elliptic bursting is that the frequency of emerging spiking activity and ceasing the spiking is nonzero; at that time, the amplitude of the oscillations may be small^{53}. It was experimentally studied that this type of bursting can be found in trifacial nerves controlling the jaw movement of rodents^{54}.

It has been previously found that the slow variable in the 2D FHN model creates mathematical complexity that allows various dynamics for the membrane voltage of the neuron model including chaos. Therefore, fractional-order FH-R single neuron model has an excellent qualitative feature that exhibits many diverse oscillations of action potentials. Necessary and sufficient conditions are investigated for asymptotic stability analysis of the fractional-order commensurate FH-R model. Bifurcation shows the qualitative changes between the quiescent state and the oscillatory state^{5,6}. The fractional-order FH-R model is investigated with a certain fixed-parameter sets, and extreme numerical computations are derived for examining the dynamical characteristics with analytical analysis using the fractional exponent as a predominant parameter. As a consequence, this generalization of the classical order model can produce biophysical variability. We present the effect of the fractional-order dynamics on the synchronization criterion in different ensembles for coupled oscillators. We observed different dynamical behavior with various fractional-orders that were not present in the classical order model.

### Fractional-Order FH-R Model

FitzHugh and Rinzel introduced FH-R model (1976, in an unpublished article)^{50,52,53,55}, which is the modification of the classical FHN neuron model. The 2D FHN model^{3,4} illustrates a geometrical explanation of interesting biophysical phenomena that are relevant to neuronal excitabilities and spike generation. It exhibits continuous spiking with a specific external stimulus. However, it is not capable of generating various fascinating firing patterns produced in cortical neurons. FH-R neuron model which is the improved version of the FHN model, can produce abundant firing activities for some parameters when it is varied in a specific fixed range. The fast-slow subsystems describe the model; the fast subsystem consists of classical FHN equation^{3}. The slow subsystem is one dimensional. It is biologically plausible and computationally efficient single neuron model. The commensurate fractional-order FH-R model is described as

$$\begin{array}{rcl}\frac{{d}^{\alpha }v}{d{t}^{\alpha }} & = & v-{v}^{3}/3-w+y+I={f}_{1}(v,w,y),\\ \frac{{d}^{\alpha }w}{d{t}^{\alpha }} & = & \delta (a+v-bw)={f}_{2}(v,w,y),\\ \frac{{d}^{\alpha }y}{d{t}^{\alpha }} & = & \mu (c-v-dy)={f}_{3}(v,w,y),\end{array}$$

(1)

where *v*, *w* and *y* represent the membrane voltage, recovery variable and slow modulation of the current respectively. *I* measures the constant magnitude of external stimulus current, and *α* is the fractional exponent which ranges in the interval \((0 < \alpha \le 1)\). *a*, *b*, *c*, *d*, *δ* and *μ* are the system parameters. The system reduces to the original classical order system when \(\alpha =1\). *μ* indicates a small parameter that determines the pace of the slow system variable, *y*. The fast subsystem (*v*-*w*) presents a relaxation oscillator in the phase plane where *δ* is a small parameter. *v* is expressed in mV (millivolt) scale. Time *t* is in ms (millisecond) scale^{4,6,11}. It exhibits tonic spiking or quiescent state depending on the parameter sets for a fixed value of *I*. The parameter *a* in the 2D FHN model corresponds to the parameter *c* of the FH-R neuron model^{53,55}. If we decrease the value of *a*, it causes longer intervals between two burstings, however there exists a relatively fixed time of bursting duration. With the increasing of *a*, the interburst intervals become shorter and periodic bursting changes to tonic spiking.

The relation between injected current stimulus and membrane potential to generate an action potential, i.e., spike^{10}, was previously introduced. The ideal resistor-capacitor theory describes the passive cell membrane dynamical analysis and the non-ideal resistor-capacitor circuit diagrams can characterize the oscillatory behavior^{10,16,39,56,57}. The theory preserves the membrane voltage behavior. It plays a significant role in analyze the dielectric behavior of the cell membranes^{16,39,57}. It was observed in experimental results that fractional-order dynamics follow a general power-law relation^{10,56}. In the electrical activities of neurons, it was shown that the power-law dynamics follows \(\alpha =0.76\) and 0.86 for warm and cold frog sciatic neurons respectively^{39}. The non-ideal capacitor theory for current-voltage relation was described by a fractional-order derivative as follows, \(C\frac{{d}^{\alpha }V}{d{t}^{\alpha }}=I\) where \(0 < \alpha < 1\). It follows a power-law dynamics and preserves the memory effects in the variations of the membrane voltage^{9,10,11,16,39}. We consider this contrast in the fractional-order condition. The membrane voltages and specific membrane potential changes may instigate seizure-like activity in epilepsy^{55}. It may cause reactions in the muscles for the specific strength of the stimulus. This type of bursting phenomenon can be explored in a more general way so that it may span in different research areas^{55}. Let us study the fractional-order fast-slow system that contributes different firing activities which appears and disappears with the change of fractional-order exponents at the various set of predefined fixed parameters.

### Method

### Numerical solution scheme

To examine the fractional dynamics of FH-R model, we consider the most familiar definition of the fractional derivative in Caputo sense^{13,14,15}. Consider the fractional-order derivative of a variable x(*t*) for the fractional exponent \(\alpha \in (0,1)\) as folllows

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

(2)

using the definition, we have

$$\frac{{d}^{\alpha }{\rm{x}}}{d{t}^{\alpha }}=\frac{1}{\Gamma (1-\alpha )}\,{\int }_{0}^{t}\,{(t-\tau )}^{-\alpha }{\rm{x}}^{\prime} (\tau )d\tau ,$$

(3)

where gamma function is defined as \(\Gamma (z)={\int }_{0}^{\infty }\,{e}^{-u}{u}^{z-1}du\). An additional advantage of Caputo order derivative is that the derivative of a constant is zero. It is efficient to integrate all the previous activities of the function weighted by a function that follows power-law dynamics. Now, applying the L1 scheme^{9,41,42,58} on Eq. (3), approximating the fractional-order derivative as

$$\frac{{d}^{\alpha }{\rm{x}}}{d{t}^{\alpha }}\approx \frac{{(dt)}^{-\alpha }}{\Gamma (2-\alpha )}[\mathop{\sum }\limits_{k=0}^{N-1}\,[{\rm{x}}({t}_{k+1})-{\rm{x}}({t}_{k})]\,[{(N-k)}^{(1-\alpha )}-{(N-1-k)}^{(1-\alpha )}]],$$

(4)

and combining Eqs (2) and (4), the numerical solution of Eq. (2) can be formulated as

$$\begin{array}{rcl}{\rm{x}}({t}_{N}) & \approx & {(dt)}^{\alpha }\Gamma (2-\alpha )f({\rm{x}},t)+{\rm{x}}({t}_{N-1})\\ & & -\,[\mathop{\sum }\limits_{k=0}^{N-2}\,[{\rm{x}}({t}_{k+1})-{\rm{x}}({t}_{k})]\,[{(N-k)}^{(1-\alpha )}-{(N-1-k)}^{(1-\alpha )}]],\end{array}$$

(5)

where, *t*_{k} represents the *k*^{th} time step and \({t}_{k}=k\Delta t\). The variable x is considered as \({\rm{x}}\equiv (v,w,y)\) in our numerical results. Approximation of the fractional-order derivative for the membrane voltage (*v*(*t*)) is given by

$$\begin{array}{rcl}v({t}_{N}) & \approx & {(dt)}^{\alpha }\Gamma (2-\alpha ){f}_{1}({\rm{x}},t)+v({t}_{N-1})\\ & & -\,[\mathop{\sum }\limits_{k=0}^{N-2}\,[v({t}_{k+1})-v({t}_{k})]\,[{(N-k)}^{(1-\alpha )}-{(N-1-k)}^{(1-\alpha )}]].\end{array}$$

(6)

Similarly, we can derive numerically the expressions for other two variables (*w* and *y*) of Eq. (1). Hence the numerical solution of Eq. (2) can be summarized as the difference between the markov term weighted by the gamma function and the memory trace. Memory trace has the main functional role in the fractional-order system as it integrates all the past activities. The markov term weighted by the gamma function is given by \({(dt)}^{\alpha }\Gamma (2-\alpha )f({\rm{x}},t)+{\rm{x}}({t}_{N-1})\) and the memory trace is given by \([\mathop{\sum }\limits_{k=0}^{N-2}\,[{\rm{x}}({t}_{k+1})-{\rm{x}}({t}_{k})]\,[{(N-k)}^{(1-\alpha )}-{(N-1-k)}^{(1-\alpha )}]]\). The memory trace has no effect for \(\alpha =1\) and the fractional-order system behaves like classical order model. The nonlinearity in the memory trace increases as we decrease the fractional-order *α* from 1 and the system dynamics depends on time. The fractional-order FH-R system is numerically integrated by using this scheme. We have considered different sets of parameters as follows^{53,55}\(a=0.7,\,b=0.8,\,d=1,\,\delta =0.08\), \(c=-\,0.775\) and \(\mu =0.0001\), set I: \(I=0.3125\), set II: \(I=0.4\), set III: \(\mu =0.18\), \(I=3\), set IV: \(c=1.3\), \(\mu =0.0001\) and \(I=0.3125,\) set V: \(c=-\,0.908\), \(\mu =0.002\) and \(I=0.3125\) and remaining parameters are similar as above. We perform the analysis of FH-R model with these parameter sets. The system shows different firing patterns like elliptic bursting, tonic spiking/regular spiking, fast-spiking and high amplitude single spike with small amplitude oscillations. The different firing activities together with the mode transitions are investigated for different parameter regimes corresponding to qualitatively various dynamical behavior of a nerve cell.

### The characteristics of the fractional-order biophysical model

#### Stability analysis

The fixed points of the system (1) are derived as \({w}^{\ast }=({v}^{\ast }+a)/b\), \({y}^{\ast }=(c-{v}^{\ast })/d\) and \({v}^{\ast 3}-3{v}^{\ast }p=q\), where \(p=(1-\frac{1}{b}-\frac{1}{d})\) and \(q=(3I-\frac{3a}{b}+\frac{3c}{d})\) respectively. Depending on the nature of the discriminant of the cubic polynomial \(F({v}^{\ast })={v}^{\ast 3}-3{v}^{\ast }p=q\), the system (1) can have maximum three equilibrium states. Throughout this study, the assumption (A) \(bd < d+b\) holds (based on the numerical data).

**Proposition I**.

The cubic function *F*(*v**) is strictly increasing and there exists only one branch of equilibrium state \(E(q)=({v}^{\ast }(q),\frac{{v}^{\ast }(q)+a}{b},\frac{c-{v}^{\ast }(q)}{d})\), with \(q\in {\mathbb{R}}\), for system (1), where \({v}^{\ast }(q)={F}^{-1}(q)\).

**Proof:** We have \(F({v}^{\ast })={v}^{\ast 3}-3{v}^{\ast }p\) and \(F^{\prime} ({v}^{\ast })=3{v}^{\ast 2}-3p\). The discriminant of *F*′ is given by \(D(F^{\prime} )=\frac{36}{bd}(bd-d-b)\). Using the assumption (A), we obtain \(D(F^{\prime} ) < 0\) that implies \(F^{\prime} ({v}^{\ast }) > 0\) and the function *F* is strictly increasing (and invertible) on \({\mathbb{R}}\). Thus, it has only one real root \({v}^{\ast }(q)={F}^{-1}(q)\). The Jacobian of the system (1) at the fixed point \(E({v}^{\ast },{w}^{\ast },{y}^{\ast })\) is given by

$$J({v}^{\ast })=(\begin{array}{ccc}1-{v}^{\ast 2} & -1 & 1\\ \delta & -\delta b & 0\\ -\mu & 0 & -\mu d\end{array}).$$

The characteristic polynomial is

$$\begin{array}{rcl}Q(\lambda ) & = & {\lambda }^{3}-(1-{v}^{\ast 2}-\delta b-\mu d){\lambda }^{2}\\ & & +\,(\delta -\delta b+\mu -d\mu +bd\mu +b\delta {v}^{\ast 2}+d\mu {v}^{\ast 2})\lambda \\ & & -\,(bd\delta \mu -b\delta \mu -d\delta \mu -bd\delta \mu {v}^{\ast 2}).\end{array}$$

From assumption (A) and \({\rm{\det }}\,(J)=\mu \delta ((bd-b-d)-bd{v}^{\ast 2}) < 0\), we obtain that at least one of the roots of the characteristic polynomial *Q*(*λ*) is negative. Considering the value of the parameter \(d=1\) (which is constant and fixed for all the parameter sets), we have \(Q(-\mu )=\mu (b\delta -\mu )\). If \(\mu < b\delta \) then \(Q(-\mu ) > 0\), which implies that at least one real root of *Q*(*λ*) lies in \((-\infty ,-\,\mu )\) otherwise the root lies in \([\,-\,\mu ,0)\). We will discuss the case when \(\mu < b\delta \) for analytical treatment and proceeding in the similar way, we can also derive the analytical results for the case when \(\mu > b\delta \).

The system changes its stability through Hopf bifurcation and it occurs when the trace of the Jacobian matrix vanishes i.e., \(1-{v}_{H}^{2}-\delta b-\mu d=0\), which gives \({v}_{H}=-\,\sqrt{1-\delta b-\mu }\) (say *γ*_{1}) and \({v}_{H}=\sqrt{1-\delta b-\mu }\) (say *γ*_{2}). *v*_{H} denotes the system variable where Hopf bifurcation occurs.

**Proposition II**.

The equilibrium state *E*(*q*) of system (1) is asymptotically stable (independent of the fractional exponent, *α*) for any \(q\le F({\gamma }_{1})\) or \(q\ge F({\gamma }_{2})\).

**Proof:** Suppose if we take the situation where \(q\le F({\gamma }_{1})\), then \({v}^{\ast }={v}^{\ast }(q)={F}^{-1}(q)\le {\gamma }_{1} < 0\). Also if \(q\ge F({\gamma }_{2})\), then \({v}^{\ast }={v}^{\ast }(q)={F}^{-1}(q)\ge {\gamma }_{2}\). It can be obtained that in both the cases \(Q(1-{v}^{\ast 2}-\delta b-\mu ) < 0\). Thus, the negative real root (say *λ*_{1}) of *Q*(*λ*) lies in \((1-{v}^{\ast 2}-\delta b-\mu ,-\,\mu )\) and other two roots satisfy \({\lambda }_{2}+{\lambda }_{3}=1-{v}^{\ast 2}-\delta b-\mu -{\lambda }_{1} < 0\) and \({\lambda }_{2}{\lambda }_{3}=\frac{{\rm{\det }}(J)}{{\lambda }_{1}} > 0\) respectively. From the above discussion, we can conclude that the roots lie on the negative real axis, so the equilibrium state *E*(*q*) is asymptotically stable and independent of the fractional exponent.

**Proposition III**.

If \(q\in (F({\gamma }_{1}),F({\gamma }_{2}))\), then the equilibrium state *E*(*q*) of system (1) is asymptotically stable iff \((1-{v}^{\ast 2}-\delta b-\mu -{\lambda }_{1})\sqrt{-{\lambda }_{1}} < 2\sqrt{-det(J)}\,\cos \,\frac{\alpha \pi }{2}\), or equivalently,

$$\alpha < \frac{2}{\pi }\arccos (min(1,\,max(0,\frac{(1-{v}^{\ast 2}-\delta b-\mu -{\lambda }_{1})\sqrt{-{\lambda }_{1}}}{2\sqrt{-det(J)}}))),$$

(7)

where \({\lambda }_{1}={\lambda }_{1}(q)\in (-\infty ,-\,\mu )\) is the smallest root of the characteristic polynomial *Q*(*λ*).

**Proof:** We have already shown that the smallest root \({\lambda }_{1}={\lambda }_{1}(q)\in (-\infty ,-\,\mu )\) and the other two roots of *Q*(*λ*) satisfy \({\lambda }_{2}+{\lambda }_{3}=1-{v}^{\ast 2}-\delta b-\mu -{\lambda }_{1}\), \({\lambda }_{2}{\lambda }_{3}=\frac{{\rm{\det }}(J)}{{\lambda }_{1}} > 0\). Now, the roots *λ*_{2} and *λ*_{3} satisfy the asymptotic stability condition \(

## The FitzHugh-Nagumo (FHN) model

### The FHN system of equations for one cell

*du/dt = F(u, v) = u(1 - u)(u - a) - v,*

dv/dt = H(u, v) = ε (bu - v),

dv/dt = H(u, v) = ε (bu - v),

where

*a*is the threshold for excitation. To the right below

*u(t)*excitation (black) and

*v(t)*recovery (blue) variables are plotted. To the left the

*(u, v)*phase plane of the system is shown.

*F(u,v) = 0*null cline is the red line and

*H(u,v) = 0*null cline is the green line.

Below you can explore the model for different parameter values. The script makes 800 time steps

*dt*.

a b ε

dt u0 v0 Ymin Ymaxfield

The system makes excitation cycle and goes to the stable fixed point

*(u=0, v=0)*. Note that for small

*ε*values the excitation variable

*u*is fast with abrupt steps and the recovery

*v*is slow.

[1] FitzHugh-Nagumo model in Scholarpedia

[2] J.D. Murray *Mathematical Biology I. An Introduction*

Heart rhythms

*updated*29 Nov 2011

## Firing activities of a fractional-order FitzHugh-Rinzel bursting neuron model and its coupled dynamics

### Abstract

Fractional-order dynamics of excitable systems can be physically described as a memory dependent phenomenon. It can produce diverse and fascinating oscillatory patterns for certain types of neuron models. To address these characteristics, we consider a nonlinear fast-slow FitzHugh-Rinzel (FH-R) model that exhibits elliptic bursting at a fixed set of parameters with a constant input current. The generalization of this classical order model provides a wide range of neuronal responses (regular spiking, fast-spiking, bursting, mixed-mode oscillations, etc.) in understanding the single neuron dynamics. So far, it is not completely understood to what extent the fractional-order dynamics may redesign the firing properties of excitable systems. We investigate how the classical order system changes its complex dynamics and how the bursting changes to different oscillations with stability and bifurcation analysis depending on the fractional exponent (0 < *α* ≤ 1). This occurs due to the memory trace of the fractional-order dynamics. The firing frequency of the fractional-order FH-R model is less than the classical order model, although the first spike latency exists there. Further, we investigate the responses of coupled FH-R neurons with small coupling strengths that synchronize at specific fractional-orders. The interesting dynamical characteristics suggest various neurocomputational features that can be induced in this fractional-order system which enriches the functional neuronal mechanisms.

### Introduction

Collective oscillatory dynamics and synchronous activity are the fundamental phenomena in dynamical systems^{1,2}. It has both theoretical importance and biophysical significance in computational neuroscience. Mathematical biophysical models^{3,4,5,6} are the primary tools to characterize the nervous system. The foremost exciting step in neural dynamics is to understand the system architecture of individual neurons in terms of mathematical models of membrane potential. Various types of spiking and bursting are the dynamical responses of excitable cells^{7,8}. Such type of research analyzes the chaotic behavior of excitable systems. When mathematical models are described as a single neuron or network, then the nonlinear dynamical techniques are applied to study the emerging oscillatory patterns and the synchronization phenomena. The classical order dynamical models depend on the immediate previous response, however the fractional-order derivative depends on all the previous responses, so it has a memory effect^{9,10,11}. It can produce a different kinds of multiple time scale neuronal dynamics^{12}. It^{13,14,15,16} provides a wide range in understanding the rich dynamical and neuronal responses. Fractional-order calculus originated from a letter written to Leibnitz by L’Hospital^{13,14,15}. Now, it has become a promising and reliable mathematical tool that includes hereditary properties or memory dependence phenomena^{13,14,15,16,17}. The discussion of multiple time scale dynamics has been studied in some previous articles^{18,19} which have the potential importance in signal processing. It has been studied that the firing rate of neocortical pyramidal neurons with the injected applied sinusoidal current can be well approximated with fractional-order derivative^{20}.

Many researchers have worked on the fractional-order dynamical systems^{21,22}. Results show that it follows power-law dynamics^{23,24,25}. In human memory, the power-law dynamics was investigated earlier^{26,27}, the accuracy of memory dependence decays at a rate nearly equal to *t*^{−α} where \(0 < \alpha < 1\). The power-law adaptation helps in describing some dynamical behavior of biological systems^{12,28}. In recent years, fractional-order derivative has become very useful in modeling biological phenomena^{13,14,15,16,29}, viscoelastic properties of tissue^{30}, tissue electrode interface^{31}, the kinetic property of drug delivery^{32,33}, diffusion process^{34,35,36}, biophysical neuron models and neural networks^{37,38,39,40,41,42}. It has been found that cognitive behaviour can be modelled using fractional dynamics^{43}. It was observed that fractional-order dynamics is used in vestibular oculomotor system^{44} and fly motion of sensitive neurons H1^{28}. It can include the mechanism of synapses^{28} and the geometrical properties of excitable cells^{44,45}. Single neuron models are analyzed by using fractional-order dynamics such as Hodgkin-Huxley (H-H), Morris-Lecar (M-L), FitzHugh-Nagumo (FHN), Hindmarsh-Rose (H-R) models, etc^{10,39,46,47,48,49}. In this study, it has been demonstrated that the fractional-order dynamics of a solitary nerve membrane can be analyzed by using a suitable biophysical model that exhibits elliptic bursting. It reflects the rate of change of information through membrane voltage that leads to previous history-dependent activities. Different parameter regimes corresponding to qualitatively different dynamical properties are analyzed. This article investigates the FH-R neuron^{50,51,52} in a fractional-order domain to get a general idea about the spike patterns and spike frequency with stability and bifurcation scenarios. The relation between spiking and bursting is a significant question as well as a fascinating phenomenon in mathematical neuroscience, especially in neural coding. Bursting presents a recurrent transition between repetitive spiking and quiescent state. The switching phases depend on the strength of the slowly changing current stimulus to the dendrite. An exciting feature of the elliptic bursting is that the frequency of emerging spiking activity and ceasing the spiking is nonzero; at that time, the amplitude of the oscillations may be small^{53}. It was experimentally studied that this type of bursting can be found in capital one secured credit limit increase nerves controlling the jaw movement of rodents^{54}.

It has been previously found that the slow variable in the 2D FHN model creates mathematical complexity that allows various dynamics for the membrane voltage of the neuron model including chaos. Therefore, fractional-order FH-R single neuron model has an excellent qualitative feature that exhibits many diverse oscillations of action potentials. Necessary and sufficient conditions are investigated for asymptotic stability analysis of the fractional-order commensurate Bank of america new york ny headquarters model. Bifurcation shows the qualitative changes between the quiescent state and the oscillatory state^{5,6}. The fractional-order FH-R model is investigated with a certain fixed-parameter sets, and extreme numerical computations are derived for examining the dynamical characteristics with analytical analysis using the fractional exponent as a predominant parameter. As a consequence, this generalization of the classical order fhn model can produce biophysical variability. We present the effect of the fractional-order dynamics on the synchronization criterion in different ensembles for coupled oscillators. We observed different dynamical behavior with various fractional-orders that were not present in the classical order model.

### Fractional-Order FH-R Model

FitzHugh and Rinzel introduced FH-R model (1976, in an unpublished article)^{50,52,53,55}, which is the modification of the classical FHN neuron model. The 2D FHN model^{3,4} illustrates a geometrical explanation of interesting biophysical phenomena that are relevant to neuronal excitabilities and spike generation. It exhibits continuous spiking with a specific external stimulus. However, it is not capable of generating various fascinating firing patterns produced in cortical neurons. FH-R neuron model which is the improved version of the FHN model, can produce abundant firing activities for some parameters when it is varied in a specific fixed range. The fast-slow subsystems describe the model; the fast subsystem consists of classical FHN equation^{3}. The slow subsystem is one dimensional. It is biologically plausible and computationally efficient single neuron model. The commensurate fractional-order FH-R model is described as

$$\begin{array}{rcl}\frac{{d}^{\alpha }v}{d{t}^{\alpha }} & = & v-{v}^{3}/3-w+y+I={f}_{1}(v,w,y),\\ \frac{{d}^{\alpha }w}{d{t}^{\alpha }} & = & \delta (a+v-bw)={f}_{2}(v,w,y),\\ \frac{{d}^{\alpha }y}{d{t}^{\alpha }} & = & \mu (c-v-dy)={f}_{3}(v,w,y),\end{array}$$

(1) **fhn model**

where *v*, *w* and *y* represent the membrane voltage, recovery variable and slow modulation of the current respectively. *I* measures the constant magnitude of external stimulus current, and *α* is the fractional exponent which ranges in the interval \((0 < \alpha \le 1)\). *a*, *b*, *c*, *d*, *δ* and *μ* are the system parameters. The system reduces to the original classical order system when \(\alpha =1\). *μ* indicates a small parameter that determines the pace of the slow system variable, *y*. The fast subsystem (*v*-*w*) presents a relaxation oscillator in the phase plane where *δ* is a small parameter. *v* is expressed in mV (millivolt) scale. Time *t* is in ms (millisecond) scale^{4,6,11}. It exhibits tonic spiking or quiescent state depending on the parameter sets for a fixed value of *I*. The parameter *a* in the 2D FHN model corresponds to the parameter *c* of the FH-R neuron model^{53,55}. If we decrease the value of *a*, it causes longer intervals between two burstings, however there exists a relatively fixed time of bursting duration. With the increasing of *a*, the interburst intervals become shorter and periodic bursting changes to tonic spiking.

The relation between injected current stimulus and membrane potential to generate an action potential, i.e., spike^{10}, was previously introduced. The ideal resistor-capacitor theory describes the passive cell membrane dynamical analysis and the non-ideal resistor-capacitor circuit diagrams can characterize the oscillatory behavior^{10,16,39,56,57}. The theory preserves the membrane voltage behavior. It plays a significant role in analyze the dielectric behavior of the cell membranes^{16,39,57}. It was observed in experimental results that fractional-order dynamics follow a general power-law relation^{10,56}. In the **fhn model** activities of neurons, it was shown that the power-law dynamics follows \(\alpha =0.76\) and 0.86 for warm and cold frog sciatic neurons respectively^{39}. The non-ideal capacitor theory for current-voltage relation was described by a fractional-order derivative as follows, \(C\frac{{d}^{\alpha }V}{d{t}^{\alpha }}=I\) where \(0 < \alpha < 1\). It follows a power-law dynamics and preserves the memory effects in the variations of the membrane voltage^{9,10,11,16,39}. We consider this contrast in the fractional-order condition. The membrane voltages and specific membrane potential changes may instigate seizure-like activity in epilepsy^{55}. It may cause reactions in the muscles for the specific strength of the stimulus. This type of bursting phenomenon can be explored in a more general way so that it may span in different research areas^{55}. Let us study the fractional-order fast-slow system that contributes different firing activities which appears and disappears with the change of fractional-order exponents at the various set of predefined fixed parameters.

### Method

### Numerical solution fhn model examine the fractional dynamics of FH-R model, we consider the most familiar definition of the fractional derivative in Caputo sense^{13,14,15}. Consider the fractional-order derivative of a variable x(*t*) for the fractional exponent \(\alpha \in (0,1)\) as folllows

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

(2)

using the definition, we have

$$\frac{{d}^{\alpha }{\rm{x}}}{d{t}^{\alpha }}=\frac{1}{\Gamma (1-\alpha )}\,{\int }_{0}^{t}\,{(t-\tau )}^{-\alpha }{\rm{x}}^{\prime} (\tau )d\tau fhn model (3)

where gamma function is defined as \(\Gamma (z)={\int }_{0}^{\infty }\,{e}^{-u}{u}^{z-1}du\). An additional advantage of Caputo order derivative is that the derivative of a constant is zero. It is efficient to integrate all the previous activities of the function weighted by a function that follows power-law dynamics. Now, applying the L1 scheme^{9,41,42,58} on Eq. (3), approximating the fractional-order derivative as

$$\frac{{d}^{\alpha }{\rm{x}}}{d{t}^{\alpha }}\approx \frac{{(dt)}^{-\alpha }}{\Gamma (2-\alpha )}[\mathop{\sum }\limits_{k=0}^{N-1}\,[{\rm{x}}({t}_{k+1})-{\rm{x}}({t}_{k})]\,[{(N-k)}^{(1-\alpha )}-{(N-1-k)}^{(1-\alpha )}]],$$

(4)

and combining Eqs (2) and (4), the numerical solution of Eq. (2) can be formulated as

$$\begin{array}{rcl}{\rm{x}}({t}_{N}) & \approx & {(dt)}^{\alpha }\Gamma (2-\alpha )f({\rm{x}},t)+{\rm{x}}({t}_{N-1})\\ & & -\,[\mathop{\sum }\limits_{k=0}^{N-2}\,[{\rm{x}}({t}_{k+1})-{\rm{x}}({t}_{k})]\,[{(N-k)}^{(1-\alpha )}-{(N-1-k)}^{(1-\alpha )}]],\end{array}$$

(5)

where, *t*_{k} represents the *k*^{th} time step and \({t}_{k}=k\Delta t\). The variable x is considered as \({\rm{x}}\equiv (v,w,y)\) in our numerical results. Approximation of the fractional-order derivative for the membrane voltage (*v*(*t*)) is given by

$$\begin{array}{rcl}v({t}_{N}) & \approx & {(dt)}^{\alpha }\Gamma (2-\alpha ){f}_{1}({\rm{x}},t)+v({t}_{N-1})\\ & & -\,[\mathop{\sum }\limits_{k=0}^{N-2}\,[v({t}_{k+1})-v({t}_{k})]\,[{(N-k)}^{(1-\alpha )}-{(N-1-k)}^{(1-\alpha )}]].\end{array}$$

(6)

Similarly, we can derive numerically the expressions for other two variables (*w* and *y*) of Eq. (1). Hence the numerical solution of Eq. (2) can be summarized as the difference between the markov term weighted by the gamma function and the memory trace. Memory trace has the main functional role in the fractional-order system as it integrates all the past activities. The markov term weighted by the gamma function is given by \({(dt)}^{\alpha }\Gamma (2-\alpha )f({\rm{x}},t)+{\rm{x}}({t}_{N-1})\) and the memory trace is given by \([\mathop{\sum }\limits_{k=0}^{N-2}\,[{\rm{x}}({t}_{k+1})-{\rm{x}}({t}_{k})]\,[{(N-k)}^{(1-\alpha )}-{(N-1-k)}^{(1-\alpha )}]]\). The memory trace has no effect for \(\alpha =1\) and the fractional-order system behaves like classical order model. The nonlinearity in the memory trace increases as we decrease the fractional-order *α* from 1 and the system dynamics depends on time. The fractional-order FH-R system is numerically integrated by using this scheme. We have considered different sets of parameters as follows^{53,55}\(a=0.7,\,b=0.8,\,d=1,\,\delta =0.08\), \(c=-\,0.775\) and \(\mu =0.0001\), set I: \(I=0.3125\), set II: \(I=0.4\), set III: \(\mu =0.18\), \(I=3\), set IV: \(c=1.3\), \(\mu =0.0001\) and \(I=0.3125,\) set V: \(c=-\,0.908\), \(\mu =0.002\) and \(I=0.3125\) and remaining parameters are similar as above. We perform the analysis of FH-R model with these parameter sets. The system shows different firing patterns like elliptic bursting, tonic spiking/regular spiking, fast-spiking and high amplitude single spike with small amplitude oscillations. The different firing activities together with the mode transitions are investigated for different parameter regimes corresponding to qualitatively various dynamical behavior of a nerve cell.

### The characteristics of the fractional-order biophysical model

#### Stability analysis

The fixed points of the system (1) are derived as \({w}^{\ast }=({v}^{\ast }+a)/b\), \({y}^{\ast }=(c-{v}^{\ast })/d\) and \({v}^{\ast 3}-3{v}^{\ast }p=q\), where \(p=(1-\frac{1}{b}-\frac{1}{d})\) and \(q=(3I-\frac{3a}{b}+\frac{3c}{d})\) respectively. Depending on the nature of the discriminant of the cubic polynomial \(F({v}^{\ast })={v}^{\ast 3}-3{v}^{\ast }p=q\), the system (1) can have maximum three equilibrium states. Throughout this study, the assumption (A) \(bd < d+b\) holds (based on the numerical data).

**Proposition I**.

The cubic function *F*(*v**) is strictly increasing and there exists only one branch of equilibrium state \(E(q)=({v}^{\ast }(q),\frac{{v}^{\ast }(q)+a}{b},\frac{c-{v}^{\ast }(q)}{d})\), with \(q\in {\mathbb{R}}\), for system (1), where \({v}^{\ast }(q)={F}^{-1}(q)\).

**Proof:** We have \(F({v}^{\ast })={v}^{\ast 3}-3{v}^{\ast }p\) and \(F^{\prime} ({v}^{\ast })=3{v}^{\ast 2}-3p\). The discriminant of *F*′ is given by \(D(F^{\prime} )=\frac{36}{bd}(bd-d-b)\). Using the assumption (A), we obtain \(D(F^{\prime} ) < 0\) that implies \(F^{\prime} ({v}^{\ast }) > 0\) and the function *F* is strictly increasing (and invertible) on \({\mathbb{R}}\). Thus, it has only one one united bank houston root \({v}^{\ast }(q)={F}^{-1}(q)\). The Jacobian of the system (1) at the fixed point \(E({v}^{\ast },{w}^{\ast },{y}^{\ast })\) is given by

$$J({v}^{\ast })=(\begin{array}{ccc}1-{v}^{\ast 2} & -1 & 1\\ \delta & -\delta b & 0\\ -\mu & 0 & -\mu d\end{array}).$$

The characteristic polynomial is

$$\begin{array}{rcl}Q(\lambda ) & = & {\lambda }^{3}-(1-{v}^{\ast 2}-\delta b-\mu d){\lambda }^{2}\\ & & +\,(\delta -\delta b+\mu -d\mu +bd\mu +b\delta {v}^{\ast 2}+d\mu {v}^{\ast 2})\lambda \\ & & -\,(bd\delta \mu -b\delta \mu -d\delta \mu -bd\delta \mu {v}^{\ast 2}).\end{array}$$

From assumption (A) and \({\rm{\det }}\,(J)=\mu \delta ((bd-b-d)-bd{v}^{\ast 2}) < 0\), we obtain that at least one of the roots of the characteristic polynomial *Q*(*λ*) is negative. Considering the value of the parameter \(d=1\) (which is constant and amazon com promo code for all the parameter sets), we have \(Q(-\mu )=\mu (b\delta -\mu )\). If \(\mu < b\delta \) then \(Q(-\mu ) > 0\), which implies that at least one real root of *Q*(*λ*) lies in \((-\infty ,-\,\mu )\) otherwise the root lies in \([\,-\,\mu ,0)\). We will discuss the case when \(\mu < b\delta \) for analytical treatment and proceeding in the similar way, we can also derive the analytical results for the case when \(\mu > b\delta \).

The system changes its stability through Hopf bifurcation and it occurs when the trace of the Jacobian matrix vanishes i.e., \(1-{v}_{H}^{2}-\delta b-\mu d=0\), which gives \({v}_{H}=-\,\sqrt{1-\delta b-\mu }\) (say *γ*_{1}) and \({v}_{H}=\sqrt{1-\delta b-\mu }\) (say *γ*_{2}). *v*_{H} denotes the system variable where Hopf bifurcation occurs.

**Proposition II**.

The equilibrium state *E*(*q*) of system (1) is asymptotically stable (independent of the fractional exponent, *α*) for any \(q\le F({\gamma }_{1})\) or \(q\ge F({\gamma }_{2})\).

**Proof:** Suppose if we take the situation where \(q\le F({\gamma }_{1})\), then \({v}^{\ast }={v}^{\ast }(q)={F}^{-1}(q)\le {\gamma }_{1} < 0\). Also if \(q\ge F({\gamma }_{2})\), then \({v}^{\ast }={v}^{\ast }(q)={F}^{-1}(q)\ge {\gamma }_{2}\). It can be obtained that in both the cases \(Q(1-{v}^{\ast 2}-\delta b-\mu ) < 0\). Thus, the negative real root (say *λ*_{1}) of *Q*(*λ*) lies in \((1-{v}^{\ast 2}-\delta b-\mu ,-\,\mu )\) and other two roots satisfy \({\lambda }_{2}+{\lambda }_{3}=1-{v}^{\ast 2}-\delta b-\mu -{\lambda }_{1} < 0\) and \({\lambda }_{2}{\lambda }_{3}=\frac{{\rm{\det }}(J)}{{\lambda }_{1}} > 0\) respectively. From the above discussion, we can conclude that the roots lie on the negative real axis, so the equilibrium state *E*(*q*) is asymptotically stable and independent of the fractional exponent.

**Proposition III**.

If \(q\in (F({\gamma }_{1}),F({\gamma }_{2}))\), then the equilibrium state *E*(*q*) of system (1) is asymptotically stable iff \((1-{v}^{\ast 2}-\delta b-\mu -{\lambda }_{1})\sqrt{-{\lambda }_{1}} < 2\sqrt{-det(J)}\,\cos \,\frac{\alpha \pi }{2}\), or equivalently,

$$\alpha < \frac{2}{\pi }\arccos (min(1,\,max(0,\frac{(1-{v}^{\ast 2}-\delta b-\mu -{\lambda fhn model }_{1}}}{2\sqrt{-det(J)}}))),$$

(7)

where \({\lambda }_{1}={\lambda }_{1}(q)\in (-\infty ,-\,\mu )\) is the smallest root of the characteristic polynomial *Q*(*λ*).

**Proof:** We have already shown that the smallest root \({\lambda }_{1}={\lambda }_{1}(q)\in (-\infty ,-\,\mu )\) and the other two roots of *Q*(*λ*) satisfy \({\lambda }_{2}+{\lambda }_{3}=1-{v}^{\ast 2}-\delta b-\mu -{\lambda }_{1}\), \({\lambda }_{2}{\lambda }_{3}=\frac{{\rm{\det }}(J)}{{\lambda }_{1}} > 0\). Now, the roots *λ*_{2} and *λ*_{3} satisfy the asymptotic fhn model condition \(

## Lévy noise induced stochastic resonance in an FHN model

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13-01-2016 Issue 3/2016

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- Title
- Lévy noise induced stochastic resonance in an FHN model
- Authors:
- ZhanQing Wang

Yong Xu

Hui Yang - Publication date
- 13-01-2016
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Issue 3/2016

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## Emilie Miller Goes to L.A. for Modeling

From a young age, sophomore Emilie Miller has been a natural in front of the camera. Professional modeling, however, is something she has grown very passionate about over the past year since she signed her developmental contract with West Model Management in St. Louis.

“I went to California this summer and signed with Two Model Management, a modeling agency based in Los Angeles,” Miller said. “While I was there I took a bunch of test shots since I had just signed and the trip will hopefully open up many new opportunities and make it more likely to get paid jobs.”

This past July, Miller took the next step in her modeling career by traveling out to California with her mom, Mendi Miller, for two weeks and signing with Two Model Management. She was nervous for this specific shoot in comparison to previous shoots, because it was so different than anything she had done before. This was her first time traveling for a photo shoot instead of staying local.

“The trip gave me hope that she has potential to pursue a career in modeling,” Mendi said. “I also enjoyed being with her for so long, because we had awesome one on one time which can sometimes be difficult with three kids.”

Emilie has lots of support from her family and friends especially her mom who encouraged her to start modeling. When her close friends found out about the trip they could not have been more excited for her, but were slightly upset when they found out that she would be gone for two weeks. One of Emilie’s best friends, Sophia Gabel, who she has been playing volleyball with since fifth grade, was especially excited for her when she found out about the trip.

“I was so sad when she left, because we were together pretty much everyday,” Gabel said. “I couldn’t feel more happy for her, because she is pursuing something that she loves.”

## Understand the Dynamics of the FitzHugh-Nagumo Model with an App

In 1961, R. Fitzhugh (Ref. 1) and J. Nagumo proposed a model for emulating the current signal observed in a living organism’s excitable cells. This became known as the FitzHugh-Nagumo (FN) model of mathematical neuroscience and is a simpler version of the Hodgkin-Huxley (HH) model (Ref. 2), which demonstrates the spiking currents in neurons. In today’s blog post, we’ll examine the dynamics of the FN model by building an interactive app in the COMSOL Multiphysics® software.

### A Nerve Cell’s Action Potential

Nerve cells are separated from the extracellular region by a lipid bilayer membrane. When the cells aren’t conducting a signal, there is a potential difference of about -70 mV across the membrane. This difference is known as the cell’s resting potential. Mineral ions, such as sodium and potassium, and negatively charged protein ions, contained within the cell, maintain the resting potential. When the cell receives an external stimulus, its potential spikes toward a positive value, a process known as *depolarization*, before falling off again to the resting potential, called *repolarization*.

*Plot of a cell’s action potential.*

In one example, the concentration of the sodium ions at rest is much higher in the extracellular region than it is within the cell. The membrane contains gated channels that selectively allow the passage of ions though them. When the cell is stimulated, the sodium channels open up and there is a rush of sodium ions into the cell. This sodium “current” raises the potential of the cell, resulting in depolarization. However, since the channel gates are voltage driven, the sodium gates close after a while. The potassium channels then open up and an outbound potassium current flows, leading to the repolarization of the cell.

Hodgkin and Huxley explained this mechanism of generating action potential through mathematical equations (Ref. 2). While this was a great success in the mathematical modeling of biological phenomena, the full Hodgkin-Huxley model is quite complicated. On the other hand, the FitzHugh-Nagumo model is relatively simple, consisting of fewer parameters and only two equations. One is for the quantity *V*, which mimics the action potential, and the other is for the variable *W*, which modulates *V*.

Today, we’ll focus on the FN model, while the HH model will be a topic of discussion for a later time.

### Defining the FitzHugh-Nagumo Model

The two equations in the FN model are

\frac{dV}{dt} &=& V-V^3/3-W + I \label{FN1}

and

\frac{dW}{dt} &=& \epsilon(V+a-bW). \label{FN2}

The parameter I corresponds to an excitation while *a* and *b* are the controlling parameters of the model. The evolution of *W* is slower than that of the evolution of *V* due to the parameter ε multiplying everything on the right-hand side of the second equation. The fixed points of the FN model equations are the solutions of the following equation system,

V-V^3/3-W + I &=& 0 \label{FN3}

and

V+a-bW &=& 0. \label{FN4}

The *V*-nullcline and *W*-nullcline are the curves V-V^3/3-W + I and V+a-bW, respectively, in the VW-plane. Note that the *V*-nullcline is a cubic curve in the VW-plane and the *W*-nullcline is a straight line. The slope of fhn model line V+a-bW is controlled in such a way that the nullclines intersect at a single point, making it the system’s only fixed point.

The parameter I simply shifts *V*-nullcline up or down. Thus, changing I modulates the position of the fixed point so that different values of I cause the fixed point to be on the left, middle, or right part of the curve V-V^3/3-W + I.

### Understanding the Dynamics of the FN Model

To simulate what happens when the fixed point is in each region, we can use the *Global DAE* interface included in the base package of COMSOL Multiphysics.

The *V*-nullcline is shown in the figure below in a green color. In the region above this nullcline \frac{dV}{dt}<0, while in the region below it is positive. The *W*-nullcline is shown in red; in the region to the right of this straight line, \frac{dW}{dt}>0, and to the left, \frac{dW}{dt}<0.

Let’s first examine what happens if the fixed point is on the right side, Region 3, of the *V*-nullcline. We’ll say that when *t*, representing time, equals zero, both *V* and *W* are also at zero. In this case, both \frac{dV}{dt} and \frac{dW}{dt} are positive at and around the starting point and thus both change as time progresses. But since *V* evolves faster than *W*, *V* increases rapidly while *W* remains virtually unchanged. In the figure, we can see that this results in a near-horizontal part of the V-W curve.

As the curve approaches the *V*-nullcline, the rate of change of *V* slows down and *W* becomes more prominent. Since \frac{dW}{dt} is still positive, *W* must increase, and the curve moves upwards. The fixed point then attracts this curve and the evolution ends at the fixed point.

*Plot of the VW-plane when the fixed point is on the right side of the* V*-nullcline.*

If the fixed point is in the middle, Region 2, then what we have discussed so far holds true. The difference is that once the curve goes beyond the right knee of the *V*-nullcline, \frac{dV}{dt} becomes negative and *V* rapidly decays. While moving left, the curve crosses the red nullcline from right to left. From this point on, while both *V* and *W* diminish, the evolution of *V* dominates and the naughty america free account becomes horizontal once again.

This continues until the curve hits the left part of the *V*-nullcline. The curve begins to hug the *V*-nullcline and starts a slow downward journey. When it touches the left knee of the *V*-nullcline, it moves rapidly toward the right part of the *V*-nullcline. Note that this motion never hits the fixed point and therefore keeps repeating, which we can see in the plot below.

*Plot of the VW-plane when the fixed point is in the middle region of the* V*-nullcline.*

That leaves us with one last case to discuss — when the fixed point is on the left part, Region 1, of the *V*-nullcline. The results should look like the following plot. Note that the analyses we previously performed carry over.

*Plot of the VW-plane when the fixed point is on the left side of the* V*-nullcline. *

### Simulating the FN Model Using the Application Builder

To explore the rich dynamics of the FN model described above, we need to repeatedly change various inputs without changing the underlying model. As such, a user interface that allows us to easily change the model parameters, perform the simulation, and analyze the new results without having to navigate the Model Builder tree structure to perform these various actions is desirable.

To accomplish this, we can turn to the Application Builder. This platform allows us to create an easy-to-use simulation app that exposes all of the essential aspects of the model, while keeping the rest behind the scenes. With this app, we can rapidly change the parameters via a user-friendly interface and study the results using both static figures and animations. The app also makes it easy for students to understand the FN model’s dynamics without having to worry about creating a model.

The important parameters of the FN model, i.e. *a, b,* ε, and *I*, are displayed in the app’s *Model Parameters* section. The graphical panels display various quantities of interest such as the waveform for *V* and *W*. We display the phase plane diagram in the top-right panel along with the *V*– and *W*-nullclines. The position of the fixed point is easily identifiable from that plot. Once the simulation is fhn model, we can animate the time trajectories by choosing the animation option from the ribbon toolbar. To get a summary of the simulation parameters and results, we can select the *Simulation Report* button.

*App showing the dynamics of the FitzHugh-Nagumo model when the fixed point is in Region 2. *

We can easily reproduce the cases described in the previous section with our app. The image above, for example, shows what happens when the fixed point is in Region 2. We can easily move the fixed point to either Region 1 or 3 by making the current 0.1 or 2.5, respectively. Note that any other parameters in the app can also be changed to see if other interesting trends emerge.

The app that we’ve presented here is just one example of what you can create with the Application Builder. The design of your app, from its layout to the parameters that are included, is all up to you. The flexibility of the Application Builder enables you to add as much complexity as needed, in part thanks to the Method Editor for Java® methods. In a follow-up blog post, we’ll create an app to illustrate the dynamics of the more complicated HH model. Stay tuned!

### References

- FitzHugh R. Impulses and Physiological States in Theoretical Models of Nerve Membrane.
*Biophysical Journal*. 1961;1(6):445-466. - Hodgkin AL, Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve.
*The Journal of Physiology*. 1952;117(4):500-544.

### Further Resources on Using the Application Builder

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