# Stability Analysis for a Fractional-Order Coupled FitzHugh–Nagumo-Type Neuronal Model

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Proposition**

**1**

- All the roots of the characteristic Equation (2) are in the open left half-plane, regardless of the fractional orders ${q}_{1}$ and ${q}_{2}$, if and only if the following inequalities are satisfied:$$\gamma >0,\phantom{\rule{1.em}{0ex}}{\beta}_{1}+{\beta}_{2}>0\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}min\{1,\gamma \}+min\{{\beta}_{1},{\beta}_{2}\}>0.$$
- The characteristic Equation (2) has a root in the open right half-plane, regardless of the fractional orders ${q}_{1}$ and ${q}_{2}$, if and only if either one of the following conditions is satisfied:
- i.
- $\gamma <0$;
- ii.
- $\gamma >0$ and ${\beta}_{1}+{\beta}_{2}+\gamma +1\le 0$;
- iii.
- ${\beta}_{1}<0$, ${\beta}_{2}<0$ and ${\beta}_{1}{\beta}_{2}\ge \gamma >0$.

**Proposition**

**2**

- i.
- The characteristic Equation (2) has a pair of complex conjugated roots on the imaginary axis of the complex plane if and only if $({\beta}_{1},{\beta}_{2})\in \mathsf{\Gamma}(\gamma ;{q}_{1},{q}_{2})$.
- ii.
- All the roots of the characteristic Equation (2) are in the open left half-plane if and only if$${\beta}_{2}>{\varphi}_{\gamma ,{q}_{1},{q}_{2}}\left({\beta}_{1}\right).$$
- iii.
- If ${\beta}_{2}<{\varphi}_{\gamma ,{q}_{1},{q}_{2}}\left({\beta}_{1}\right)$, the characteristic Equation (2) has at least one root in the open right half-plane.

## 3. Fractional-Order Coupled FithHugh–Nagumo-Type Neuronal Model

#### 3.1. Existence of Equilibrium States

- Case 1:${v}_{1}^{*}={v}_{2}^{*}$

- If $4\alpha >{(1-a)}^{2}$, then system (4) has a unique equilibrium point$$({v}_{1}^{*},{v}_{2}^{*},{w}_{1}^{*},{w}_{2}^{*})=(0,0,0,0).$$
- If $4\alpha ={(1-a)}^{2}$, then system (4) has two equilibrium states$$({v}_{1}^{*},{v}_{2}^{*},{w}_{1}^{*},{w}_{2}^{*})\in \left\{(0,0,0,0),\left({\displaystyle \frac{1+a}{2}},{\displaystyle \frac{1+a}{2}},{\displaystyle \frac{\alpha (1+a)}{2}},{\displaystyle \frac{\alpha (1+a)}{2}}\right)\right\}.$$
- If $4\alpha <{(1-a)}^{2}$, then system (4) has three equilibrium states$$({v}_{1}^{*},{v}_{2}^{*},{w}_{1}^{*},{w}_{2}^{*})\in \left\{(0,0,0,0),\left(f(a,\Delta ),f(a,\Delta ),\alpha f(a,\Delta ),\alpha f(a,\Delta )\right)\right\},$$

- Case 2:${v}_{1}^{*}\ne {v}_{2}^{*}$

**Remark**

**1.**

#### 3.2. Stability of Equilibrium States

**Proposition**

**3.**

- asymptotically stable, regardless of the fractional orders ${q}_{1}$ and ${q}_{2}$, if and only if ${\mu}_{+}<{\mu}_{s}$;
- unstable, regardless of the fractional orders ${q}_{1}$ and ${q}_{2}$, if and only if ${\mu}_{+}>{\mu}_{u}$.

**Proof.**

**Remark**

**2.**

**Proposition**

**4.**

**Proof.**

**Remark**

**3.**

- if $g<\frac{{\mu}_{s}+a}{2}$, the trivial equilibrium state is asymptotically stable, regardless of the fractional orders ${q}_{1}$ and ${q}_{2}$;
- if $g>\frac{{\mu}_{u}+a}{2}$, the trivial equilibrium state is unstable, regardless of the fractional orders ${q}_{1}$ and ${q}_{2}$;
- if $g\in \left(\frac{{\mu}_{s}+a}{2},\frac{{\mu}_{u}+a}{2}\right)$, the stability of the trivial equilibrium depends on the fractional orders ${q}_{1}$ and ${q}_{2}$, as seen from Proposition 4.

## 4. Numerical Simulations

#### 4.1. Case 1: A Unique Equilibrium State

#### 4.2. Case 2: Five Coexisting Equilibrium States

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Cottone, G.; Paola, M.D.; Santoro, R. A novel exact representation of stationary colored Gaussian processes (fractional differential approach). J. Phys. A Math. Theor.
**2010**, 43, 085002. [Google Scholar] [CrossRef] - Engheia, N. On the role of fractional calculus in electromagnetic theory. IEEE Antennas Propag. Mag.
**1997**, 39, 35–46. [Google Scholar] [CrossRef] - Henry, B.; Wearne, S. Existence of Turing instabilities in a two-species fractional reaction-diffusion system. SIAM J. Appl. Math.
**2002**, 62, 870–887. [Google Scholar] [CrossRef] [Green Version] - Heymans, N.; Bauwens, J.C. Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheologica Acta
**1994**, 33, 210–219. [Google Scholar] [CrossRef] - Mainardi, F. Fractional Relaxation-Oscillation and Fractional Phenomena. Chaos Solitons Fractals
**1996**, 7, 1461–1477. [Google Scholar] [CrossRef] - Du, M.; Wang, Z.; Hu, H. Measuring memory with the order of fractional derivative. Sci. Rep.
**2013**, 3, 3431. [Google Scholar] [CrossRef] - Anastasio, T. The fractional-order dynamics of brainstem vestibulo-oculomotor neurons. Biol. Cybernet.
**1994**, 72, 69–79. [Google Scholar] [CrossRef] - Lundstrom, B.; Higgs, M.; Spain, W.; Fairhall, A. Fractional differentiation by neocortical pyramidal neurons. Nat. Neurosci.
**2008**, 11, 1335–1342. [Google Scholar] [CrossRef] - Weinberg, S.H. Membrane capacitive memory alters spiking in neurons described by the fractional-order Hodgkin-Huxley model. PLoS ONE
**2015**, 10, e0126629. [Google Scholar] [CrossRef] - Drapaca, C. Fractional calculus in neuronal electromechanics. J. Mech. Mater. Struct.
**2016**, 12, 35–55. [Google Scholar] [CrossRef] - Grevesse, T.; Dabiri, B.E.; Parker, K.K.; Gabriele, S. Opposite rheological properties of neuronal microcompartments predict axonal vulnerability in brain injury. Sci. Rep.
**2015**, 5, 9475. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Teka, W.; Marinov, T.M.; Santamaria, F. Neuronal spike timing adaptation described with a fractional leaky integrate-and-fire model. PLoS Comput. Biol.
**2014**, 10, e1003526. [Google Scholar] [CrossRef] [PubMed] - Jun, D.; Guang-Jun, Z.; Yong, X.; Hong, Y.; Jue, W. Dynamic behavior analysis of fractional-order Hindmarsh–Rose neuronal model. Cogn. Neurodyn.
**2014**, 8, 167–175. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kaslik, E. Analysis of two-and three-dimensional fractional-order Hindmarsh-Rose type neuronal models. Fract. Calc. Appl. Anal.
**2017**, 20, 623–645. [Google Scholar] [CrossRef] [Green Version] - Brandibur, O.; Kaslik, E. Stability properties of a two-dimensional system involving one Caputo derivative and applications to the investigation of a fractional-order Morris-Lecar neuronal model. Nonlinear Dyn.
**2017**, 90, 2371–2386. [Google Scholar] [CrossRef] [Green Version] - Shi, M.; Wang, Z. Abundant bursting patterns of a fractional-order Morris–Lecar neuron model. Commun. Nonlinear Sci. Numer. Simul.
**2014**, 19, 1956–1969. [Google Scholar] [CrossRef] - Upadhyay, R.K.; Mondal, A.; Teka, W.W. Fractional-order excitable neural system with bidirectional coupling. Nonlinear Dyn.
**2017**, 87, 2219–2233. [Google Scholar] [CrossRef] - Brandibur, O.; Kaslik, E. Stability of two-component incommensurate fractional-order systems and applications to the investigation of a FitzHugh-Nagumo neuronal model. Math. Methods Appl. Sci.
**2018**, 41, 7182–7194. [Google Scholar] [CrossRef] [Green Version] - Teka, W.; Stockton, D.; Santamaria, F. Power-Law Dynamics of Membrane Conductances Increase Spiking Diversity in a Hodgkin-Huxley Model. PLoS Comput. Biol.
**2016**, 12, e1004776. [Google Scholar] [CrossRef] - Majhi, S.; Bera, B.K.; Ghosh, D.; Perc, M. Chimera states in neuronal networks: A review. Phys. Life Rev.
**2019**, 28, 100–121. [Google Scholar] [CrossRef] - Guo, S.; Dai, Q.; Cheng, H.; Li, H.; Xie, F.; Yang, J. Spiral wave chimera in two-dimensional nonlocally coupled Fitzhugh–Nagumo systems. Chaos Solitons Fractals
**2018**, 114, 394–399. [Google Scholar] [CrossRef] - Schmidt, A.; Kasimatis, T.; Hizanidis, J.; Provata, A.; Hövel, P. Chimera patterns in two-dimensional networks of coupled neurons. Phys. Rev. E
**2017**, 95, 032224. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Eydam, S.; Franović, I.; Wolfrum, M. Leap-frog patterns in systems of two coupled FitzHugh-Nagumo units. Phys. Rev. E
**2019**, 99, 042207. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mao, X. Complicated dynamics of a ring of nonidentical FitzHugh–Nagumo neurons with delayed couplings. Nonlinear Dyn.
**2017**, 87, 2395–2406. [Google Scholar] [CrossRef] - Lavrova, S.; Kudryashov, N.; Sinelshchikov, D. On some properties of the coupled Fitzhugh-Nagumo equations. J. Phys. Conf. Ser.
**2019**, 1205, 012035. [Google Scholar] [CrossRef] [Green Version] - Mondal, A.; Sharma, S.K.; Upadhyay, R.K.; Mondal, A. Firing activities of a fractional-order FitzHugh-Rinzel bursting neuron model and its coupled dynamics. Sci. Rep.
**2019**, 9, 15721. [Google Scholar] [CrossRef] - Li, X.; Han, C.; Wang, Y. Novel Patterns in Fractional-in-Space Nonlinear Coupled FitzHugh–Nagumo Models with Riesz Fractional Derivative. Fractal Fract.
**2022**, 6, 136. [Google Scholar] [CrossRef] - Ramadoss, J.; Aghababaei, S.; Parastesh, F.; Rajagopal, K.; Jafari, S.; Hussain, I. Chimera state in the network of fractional-order fitzhugh–nagumo neurons. Complexity
**2021**, 2021, 2437737. [Google Scholar] [CrossRef] - Momani, S.; Freihat, A.; Al-Smadi, M. Analytical study of fractional-order multiple chaotic FitzHugh-Nagumo neurons model using multistep generalized differential transform method. Abstract Appl. Anal.
**2014**, 2014, 276279. [Google Scholar] [CrossRef] [Green Version] - Podlubny, I. Fractional Differential Equations; Academic Press: Cambridge, MA, USA, 1999. [Google Scholar]
- Diethelm, K. The Analysis of Fractional Differential Equations; Springer: Berlin/Heidelberg, Germany, 2004. [Google Scholar]
- Kilbas, A.; Srivastava, H.; Trujillo, J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives; Gordon and breach Science Publishers: Yverdon Yverdon-les-Bains, Switzerland, 1993; Volume 1. [Google Scholar]
- Li, C.; Zhang, F. A survey on the stability of fractional differential equations. Eur. Phys. J. Special Top.
**2011**, 193, 27–47. [Google Scholar] [CrossRef] - Rivero, M.; Rogosin, S.V.; Tenreiro Machado, J.A.; Trujillo, J.J. Stability of fractional order systems. Math. Prob. Eng.
**2013**, 2013, 356215. [Google Scholar] [CrossRef] - Sabatier, J.; Farges, C. On stability of commensurate fractional order systems. Int. J. Bifurc. Chaos
**2012**, 22, 1250084. [Google Scholar] [CrossRef] - Matignon, D. Stability Results For Fractional Differential Equations With Applications To Control Processing. In Proceedings of the Computational Engineering in Systems Applications, Lille, France, 9–12 July 1996; pp. 963–968. [Google Scholar]
- Cong, N.; Tuan, H.; Trinh, H. On asymptotic properties of solutions to fractional differential equations. J. Math. Anal. Appl.
**2020**, 484, 123759. [Google Scholar] [CrossRef] [Green Version] - Brandibur, O.; Kaslik, E. Exact stability and instability regions for two-dimensional linear autonomous systems of fractional-order differential equations. Fract. Calc. Appl. Anal.
**2021**, 24, 225–253. [Google Scholar] [CrossRef] - Brandibur, O.; Garrappa, R.; Kaslik, E. Stability of Systems of Fractional-Order Differential Equations with Caputo Derivatives. Mathematics
**2021**, 9, 914. [Google Scholar] [CrossRef] - Lakshmikantham, V.; Leela, S.; Devi, J.V. Theory of Fractional Dynamic Systems; Scientific Publishers: Cambridge, MA, USA, 2009. [Google Scholar]
- Brandibur, O.; Kaslik, E. Stability analysis of multi-term fractional-differential equations with three fractional derivatives. J. Math. Anal. Appl.
**2021**, 495, 124751. [Google Scholar] [CrossRef] - Doetsch, G. Introduction to the Theory and Application of the Laplace Transformation; Springer: Berlin/Heidelberg, Germany, 1974. [Google Scholar]
- Westerlund, S.; Ekstam, L. Capacitor theory. IEEE Trans. Dielectr. Electr. Insul.
**1994**, 1, 826–839. [Google Scholar] [CrossRef] - Diethelm, K.; Ford, N.; Freed, A. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn.
**2002**, 29, 3–22. [Google Scholar] [CrossRef] - Garrappa, R. Numerical solution of fractional differential equations: A survey and a software tutorial. Mathematics
**2018**, 6, 16. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Number of equilibrium states in the parametric plane $(a,\alpha )$ for different values of the parameter $g\in \{\pm 0.2,\pm 0.6,\pm 1\}$.

**Figure 2.**Evolution of the membrane potentials ${v}_{1}$ (blue) and ${v}_{2}$ (red) of the two neurons, for fixed parameter values $a=0.3$, $\u03f5=0.01$ and $\beta =0.1$, coupling coefficient $g=0.2$, fixed fractional order ${q}_{2}=1$ and ${q}_{1}\in \{0.6,0.7,0.8,0.9,1\}$ (top to bottom). Initial conditions for system (3) have been chosen in a neighborhood of the trivial equilibrium.

**Figure 3.**Evolution of the membrane potential ${v}_{1}$ for fixed parameter values $a=1.5$, $\u03f5=0.032$ and $\beta =2$, coupling coefficient $g=0.8$, fixed fractional order ${q}_{2}=1$ and ${q}_{1}\in \{0.8,0.95\}$, considering multiple initial conditions for system (3) in a neighborhood of the trivial equilibrium (corresponding solutions plotted with different colors).

**Table 1.**Equilibrium states of (3) for the parameter values: $a=1.5$, $\u03f5=0.032$ and $\beta =2$.

Equilibrium $({\mathit{v}}_{1}^{*},{\mathit{v}}_{2}^{*},{\mathit{w}}_{1}^{*},{\mathit{w}}_{2}^{*})$ | ${\mathit{\mu}}_{+}$ | Stability |
---|---|---|

$(2.01375,-0.555812,1.00687,-0.277906)$ | $-2.46674$ | asympt. stable |

$(-0.555812,2.01375,-0.277906,1.00687)$ | $-2.46674$ | asympt. stable |

$(0.38957,-0.183994,0.194785,-0.0919969)$ | $1.02553$ | unstable |

$(-0.183994,0.38957,-0.0919969,0.194785)$ | $1.02553$ | unstable |

$(0,0,0,0)$ | $0.1$ | depends on ${q}_{1},{q}_{2}$ |

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Brandibur, O.; Kaslik, E.
Stability Analysis for a Fractional-Order Coupled FitzHugh–Nagumo-Type Neuronal Model. *Fractal Fract.* **2022**, *6*, 257.
https://doi.org/10.3390/fractalfract6050257

**AMA Style**

Brandibur O, Kaslik E.
Stability Analysis for a Fractional-Order Coupled FitzHugh–Nagumo-Type Neuronal Model. *Fractal and Fractional*. 2022; 6(5):257.
https://doi.org/10.3390/fractalfract6050257

**Chicago/Turabian Style**

Brandibur, Oana, and Eva Kaslik.
2022. "Stability Analysis for a Fractional-Order Coupled FitzHugh–Nagumo-Type Neuronal Model" *Fractal and Fractional* 6, no. 5: 257.
https://doi.org/10.3390/fractalfract6050257