Abstract
A general FitzHugh–Rinzel model, able to describe several neuronal phenomena, is considered. Linear stability and Hopf bifurcations are investigated by means of the spectral equation for the ternary autonomous dynamical system and the analysis is driven by both an admissible critical point and a parameter which characterizes the system.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The physiological and chemical properties that characterize neurons make them able to receive, process and transmit electrical signals that, associated with ionic currents, cross the membrane of the neuron. These electrical signals are called nerve impulses, while the difference in electrical charge that exists between the inside and outside of the neuronal cell is called membrane potential. The variation in the membrane potential is called action potential and it travels along the axon and is transmitted unchanged to other neurons in the form of electrical impulses. In this way, information is transmitted from one neuron to another, forming what is known as synapse. This phenomenon is well known in literature and an extensive bibliography exists in regard [1,2,3]. A reference point for these studies are the works of Hodgkin and Huxley [HH], who developed the model of the propagation of an electrical signal along a squid axon (an axon so great to be called giant). Their model consists of a system of four differential equations describing the dynamics of the membrane potential and the three fundamental ionic currents: the sodium current, the potassium current and the leakage current, which is mainly due to chlorine but also considers the effect of other minor ionic currents. However, the non linearity and high dimensionality of the HH model made the analysis too complicated, so that simpler models were introduced to allow the essential aspects of the dynamics of models to be captured.
One of these models is the FitzHugh-Nagumo system (FHN) where, indicating by \( \, U(x,t)\, \) the trasmembrane potential and by \(\,W(x,t)\,\) a variable associated with the contributions to the membrane current from sodium, potassium and other ions, it is given by
Constant \( D\, > \, 0\, \) is a diffusion coefficient related to the axial current in the axon. It follows from the HH theory where, denoting by d the diameter of the axon and by \( r_i \) the resistivity, the spatial variation of the potential V gives the term \( (d/4 r_{i}) V_{xx} \) from which the term \( D\, U_{xx} \) is deduced [3]. Furthermore \( \varepsilon , \, c, \,\) and \(\, \beta \, \) are constants that characterize the model’s kinetic.
The documentation is numerous and the analysis is extensive (see, for instance, [4, 5] and references therein).
As for function \(\, f(U),\,\) it depends on the reaction kinetics of the model and can assume various expressions such as a piecewise linear form, or \( f(U) = U-U^3/3 \). Besides, more in general, function f(U) assumes the following form [1, 2]:
The cubic term is due to an instantaneous inversion of the sodium permeability and can be thought to play the same role as the m variable in the HH model, where the variable of activation of the channels of sodium is considered. Hence, a represents a threshold constant and is an excitability parameter [6]. In addition, a can take both positive and negative values (see,f.i. [7]) and cases with function a(x) are considered in [8] for inhomogeneous means.
Besides, one aspect worth noting is the existence of an equivalence between the FHN model and the third-order equation characterizing Josephson junctions in superconductivity [9,10,11]. It follows that the analysis of such models is reflected in both biological and superconducting phenomena and, in addition, in dissipative problems [12,13,14].
Similarly, in order to investigate other phenomena such as, for example, bursting oscillations, the well known system of FitzHugh–Rinzel (FHR) can be considered [15,16,17,18,19]. This model is derived from the FHN model and, unlike the latter, has an additional variable that changes periodically from a rapid spike oscillation to a silent phase during which the membrane potential changes slowly [1].
Indeed, bursting phenomena occur in various scientific fields (see, f.i. [20] and references therein), and many devices are being built to mimic the behavior of a biological synapse, suggesting that electronic synapses may be introduced in the future to directly connect neurons [21]. As a result, the FHR system is increasingly being studied to provide a mathematical description of physical phenomena occurring in organisms.
The FitzHugh–Rinzel model considered in this paper is the following one:
where the physical variables (U, W, Y) represent, respectively, the transmembrane potential, the recovery variable and the slow current in the dendrite. Moreover, the parameter \( \varepsilon \) specifies the relationship between the time constants of the activator and inhibitor [6], and c and \( \beta \) can be related to the number of cell membrane channels open to sodium and potassium ions, respectively [22]. Constant I measures the amplitude of the external stimulus current and is modulated by the variable Y on a slower time scale [1]. In addition, if \( \beta \varepsilon \) and \( \delta d \) are positive constants, they can be regarded as the coefficients of viscosity [23].
When \( k=3 \) and \( a=-1, \) (3) turns into this model:
often studied in literature (see, f.i. [15, 18, 23] and references therein).
Aim of the paper is to analyze the linear stability of the critical points of the FHR system, as well as to highlight the cases of Hopf bifurcations. Considering the spectrum equation, and its eigenvalues, stability is evaluated by the Lienard-Chipart criterion. Furthermore, for what concerns instability, showing that the problem can be expressed by way of a positive parameter R, the steady and/or oscillatory Hopf bifurcations cases are determined by means of the instability coefficient power (ICP) method introduced by Rionero (see, f.i. [23, 24] and references therein). The plan of the paper is the following one: Sect. 2 highlights some premises by which the subsequent theorems will be proved. In Sect. 3 the mathematical problem and linear operator L with its invariants is given. Finally, in Sects. 4 and 5, Hopf bifurcations driven by critical point \( \bar{U} \) and driven by coefficient \( -\eta = - \varepsilon \beta \) are evaluated.
2 Some premises
Due to the oscillatory activities of neurons, the onset of oscillatory bifurcations has gained the attention of many researchers. Regarding the study of Hopf’s bifurcations, an extensive literature exists (see, f.i. [23,24,25,26] and references therein). In order to justify the results stated here, some introductory considerations will be required.
Indeed, in relation to linear stability, according to [26] when a phenomenon is modelled by the system:
introducing a fixed solution \( \varvec{\bar{U}} \) and the perturbation \( {\textbf {u}} = {\textbf {U}}- \varvec{\bar{U}}, \) the behaviour of \( {\textbf {u}} \) is governed by:
with \({\textbf {u}}_0\) initial perturbation to \( \varvec{\bar{U}} \) and \((N{\textbf {u}})_{{\textbf {u}}_0} =0.\)
Considering the linear operator
the stability and instability of \( \varvec{\bar{U}} \) is called linear if it is evaluated via the linear system
neglecting the nonlinear contribution \(N{\textbf {u}}. \)
In this regard, some theorems can be provided.
Theorem 1
If
is the spectral equation whose eigenvalues of the \( n \, x\,n \) matrix \( ||a_{i,j} || \) are \( \lambda _i (i = 1, 2,3\ldots ,n),\) and if and only if all the eigenvalues have negative real parts, then \({\textbf {u\,=\,0}}\) is linearly globally attractive and asymptotically exponentially stable. Otherwise, if there exists at least an eigenvalue with positive real part, then \({\textbf {u\,=\,0}}\) is unstable. \(\square \)
In addition, as proved in [26], for a system formed by three equations such as the FHR model, the spectrum equation (9) of L is reduced to the following expression:
where
represent the invariants of L whose spectrum is the set \( \sigma =\{ \lambda _1, \lambda _2, \lambda _3\} \) of its eigenvalues. Moreover, connected to the invariants \( I_i (i=1,2,3), \) we can introduce the quantities:
and
and, according to [23], the following Lienard-Chipart criterion holds:
Theorem 2
If and only if
all the eigenvalues have negative real part. In particular, each of the conditions:
is necessary for all the roots to have negative real parts. Otherwise some roots will have positive real parts. \(\square \)
Moreover, taking into account that the instability can occur only via a zero eigenvalue (\( \lambda =0 \Leftrightarrow A_3=0) \) or via a pure imaginary eigenvalues, \( \lambda _{1,2} =\pm i\omega \,\,(i \,\,{imaginary unit,} \) \(\,\, \omega \in \Re ^+ )\) such that \(\mathcal{P}(i\omega ,R)=0, \) the onset of instability will be defined either as steady bifurcation or Hopf bifurcation depending on wether the instability occurs through a steady or oscillatory state [26].
When the problem at issue depends on a positive parameter R, let denote by \( R_{c_k} \) the lowest roots of value of R such that \( A_k(R)=0 \) for \( k=1,2,3. \) According to [23], it is possible to introduce the
and the following theorem holds:
Theorem 3
Let \(A_{\bar{k}}\) be the spectrum equation coefficient with the biggest ICP and let the critical point \( \bar{C} \) be linearly asymptotically stable at \(R =\bar{R} = 0.\) Then, at the growing of R from \(R = 0\), the instability occurs at \(R= R_{C_{\bar{k}}} \) and one has a steady bifurcation if \( \bar{k} =3, \) while an oscillatory bifurcation occurs at an \( R\in ]0, R_{C_{\bar{k}}}[ \) if \(k < 3\). \(\square \)
3 Mathematical model
Let consider the FHZ system (3) and assuming
it results:
If \( C=(\bar{U},\bar{W}, \bar{Y}) \) is an admissible critical point, considering:
as the perturbation vector, from (20) one obtains:
Linearizing about C, it results:
Denoting by
the linear operator, according to (11)–(12), for \( k=3, \) one has:
as the invariants of L. Besides, taking into account (13)–(15) one deduces:
and letting
one obtains
4 Hopf bifurcations driven by \( \bar{U} \)
The FHR system depends on several parameters, and according to each coefficient, various Hopf bifurcations conditions can be obtained.
In order to study Hopf bifurcations driven by critical point \( \bar{U} \), the attention is focused on
already introduced in (27), and the following theorem for linear stability can be proved:
Theorem 4
Let \( \bar{C}= (\bar{U}, \bar{W}, \bar{Y}) \) be an admissible critical point and let assume constants \( (\varepsilon ,\, \delta ,\, d,\, \beta ), \) be positive. Then, whatever the value of variable \( a \in R \) may be, if
then the critical point \( \bar{C} \) is linearly, globally attractive and asymptotically exponentially stable.
Proof
Condition (29) ensures that \( \Gamma \ge 0, \) and it is possible to prove that the positiveness of the FHR system’s constants implies that \( A_k, \, (k=0,1,2,3), \) determined in (28), are all non-negative. Moreover, they are increasing functions of \( \Gamma . \)
This ensures that conditions (16) of theorem 2 state, and hence theorem holds. \(\square \)
When conditions (29) are not satisfied, i.e the critical point \( \bar{U} \) is such that the following inequality:
holds, then it results \( \Gamma <0 \) and in this case it is possible to introduce a positive parameter R as “bifurcation parameter”. Indeed if we let:
it results:
with:
So that, denoting by \( R_{c_k} \) the lowest roots of value of R such that \( A_k=0 \) for \( k=1,2,3, \) one has:
and the following theorem holds:
Theorem 5
In the hypothesis (30), let \( R= -\Gamma = -[ \bar{U}^2 \,- 2 (a+1) \bar{U}\, + a ]>0 \) and let constants \( (\varepsilon ,\, \delta ,\, d,\,\beta ), \) be positive.
Then, at the growing of R from \( R=0, \) conditions
ensure that a simple oscillatory bifurcation occurs at a \( \bar{R} \in ]0, R_{C_1}[,\) with a frequency \(\displaystyle \frac{\varphi }{2\pi }\, \) where \( \displaystyle \varphi ^2 = \frac{A_3(\bar{R})}{ A_1(\bar{R})} = A_2 (\bar{R}). \)
If, in particular
a simple oscillatory bifurcations occurs at a \( \bar{R} \in ]0, R_{C_1}=R_{C_2} [. \)
Otherwise, if
a steady + oscillatory bifurcation appears with a frequency given by \( \varphi = (2\pi )(\sqrt{A_2})_{R_{c_1}}. \)
Moreover, if
a simple oscillatory bifurcation occurs at a \( \bar{R} \in ]0, R_{C_2}[.\)
Proof
When \( R=-\Gamma =0, \) it results:
with \( A_k >0\, (k=1,2,3) \). So, the critical point is linearly, asymptotically stable for \( R=0. \)
Besides, when inequalities (35) hold, it means that
i.e. \(A_{1}\) is the spectrum equation coefficient with the biggest instability coefficient power, so that at \( R= R_{C_1}, \) it results:
and hence, in view of the continuity of \( A_1 A_2-A_3, \) there exists a \( \bar{R} \in ]0, R_{C_1}[ \) such that
being \( \bar{R} \) the lowest root of \( A_1 A_2 = A_3\) in \( ]0, R_{C_1}[ \) and it results
and hence
with
Besides, conditions (36) imply that \( R=R_{C_1} = R_{C_2} < R_{C_3} \) that means \( A_1=A_2=0\) and
\( A_3(R_{C_1})= -\gamma \,\eta \, c_1+ c_3 \,\,\gamma \,\eta \, >0 \)
Consequently, the spectrum equation is reduced to:
and hence
This means that a simple oscillatory bifurcation occurs at a \( \bar{R} \in ]0, R_{c_1}= R_{c_2}[. \)
Instead, when (37) holds, \( R_{c_1}= R_{c_3}<R_{c_2} \); and hence one obtains \( A_1= A_3=0. \) So, from the spectrum equation it results:
and a steady (\( \lambda =0 \)) \( + \) oscillatory bifurcation of frequency \( \varphi /\pi \) with \( \varphi = (\sqrt{A_2})_{R_{c_1}} \) occurs.
Analogous results can be obtained if we suppose \( R_{c_2} \) to be the biggest ICP and hence (38) is proved, too. \(\square \)
5 Hopf bifurcations driven by \(- \eta = - \varepsilon \,\beta >0 \)
The previous bifurcation criterion required that \( \Gamma \le 0. \) In the present section, we prove that, by choosing \( \eta = \varepsilon \beta \) as bifurcating parameter and letting \( \eta \le 0, \) the Hopf bifurcation can arise with \(\Gamma \ge 0.\)
Indeed, the following theorem states:
Theorem 6
Let consider a critical point \( \bar{C} \) such that:
and let constants \( (\varepsilon ,\, \delta ,\, d ), \) be positive. Assuming \( R = - \eta = - \varepsilon \,\beta >0, \) then, at the growing of R from \( R=0, \) conditions
ensure that a simple oscillatory bifurcation occurs at a \( \bar{R} \in ]0, R_{C_1}[,\) while if
a steady+oscillatory bifurcation appears.
Moreover, if
a simple oscillatory bifurcation occurs at a \( \bar{R} \in ] 0, R_{C_2}[.\)
Proof
Condition (44) ensures that \( \Gamma , \) defined in (27), is positive. Moreover, since (28), when \( \eta =0, \) it results \( A_k>0 \,\, (k=1,2,3) \) and
So, the critical point \( \bar{C}= (\bar{U}, \bar{W}, \bar{Y}) \) is linearly, asymptotically stable for \( R = \bar{R} =0 \). In addition, denoting by
from (28) it results
and so, by retracing the analysis set forth in the previous bifurcation cases, this theorem can also be proved. \(\square \)
6 Remarks and discussion
As it is well known, the phenomenon related to Hopf bifurcations is of great importance and it is widely studied. In this paper, the FHR model (3) considered also depends on the variable a generally not present in the bifurcations studies and it generalizes the FHR system (4), which, on the contrary, is more often considered in literature.
Moreover, the results obtained [see, f.i. Theorems 4–5 and condition (30)] do not require any assumptions for the real variable a and this implies that the analysis can certainly be directed to a wider set of physical cases.
Furthermore, the equivalence that such a mathematical model creates between biological problems and superconducting processes of Josephson junctions or viscoelasticity, suggests that the analysis of such models is reflected in a large number of realistic mathematical models.
In this paper the onset of Hopf bifurcations, driven by specific parameters, is considered. In particular an analysis on the onset of steady and oscillatory bifurcations has been performed driven by both an admissible critical point \( \bar{U} \) and a coefficient characterized the FHR system.
Looking forward, in order to obtain a more comprehensive view of the stability and instability of critical points, the analysis can be extended to evaluate Hopf bifurcations driven by all other coefficients that characterize the FHR system. Moreover, it will be possible to determine explicit critical points at particular values of the FHR system variables and also evaluate the explicit value of the bifurcation parameters R.
References
Keener, J.P., Sneyd, J.: Mathematical physiology. Springer, New York (1998)
Izhikevich, E.M.: Dynamical systems in neuroscience: the geometry of excitability and bursting. The MIT Press, London (2007)
Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2070 (1962)
Kudryashov, N.A., Rybka, K.R., Sboev, A.: Analytical properties of the perturbed FitzHugh-Nagumo model. Appl. Math. Lett. 76, 142–147 (2018)
De Angelis, M.: A wave equation perturbed by viscous terms: Fast and slow times diffusion effects in a Neumann problem. Ric. Mat. 68, 237–252 (2019)
Gutman, M., Aviram, I., Rabinovitch, A.: Abnormal frequency locking and the function of the cardiac pacemaker. Phys. Rev. E 70, 037202 (2004)
Zhao, Y., Billings, S.A., Coca, D., Guo, Y., Ristic, R.I., De Matos, L.L.: Identification of a Temperature Dependent FitzHugh-Nagumo model for the Belousov-Zhabotinskii Reaction. Int. J. Bifurc. Chaos 21, 3249–3258 (2011). https://doi.org/10.1142/S0218127411030490
Dikansky, A. Fitzhugh-Nagumo equations in a nonhomogeneous medium. Discret. Contin. Dyn. Syst. 216–224 (2005)
De Angelis, M., Fiore, G.: Diffusion effects in a superconductive model. Commun. Pure Appl. Anal. 13, 217–223 (2014)
Scott, A.C.: The nonlinear universe chaos emergence. Springer, Berlin (2007)
De Angelis, M., Renno, P.: On asymptotic effects of boundary perturbations in exponentially shaped josephson junctions. Acta Appl. Math. 132, 251–259 (2014)
De Angelis, M.: On exponentially shaped Josephson junctions. Acta Appl. Math. 122, 179–189 (2012). https://doi.org/10.1007/s10440-012-9736-9
De Angelis, M., Renno, P.: Diffusion and wave behaviour in linear Voigt model. Comptes Rendus Mécanique 330(1), 21–26 (2002)
D’Anna, A., De Angelis, M., Fiore, G.: Existence and uniqueness for some 3rd order dissipative problems with various boundary conditions. Acta Appl. Math. 122, 255–267 (2012)
Kudryashov, N.A.: On integrability of the FitzHugh-Rinzel model. Russ. J. Nonlinear Dyn. 15(1), 13–19 (2019)
De Angelis, M.: A note on explicit solutions of FitzHugh-Rinzel system. Nonlinear Dyn. Syst. Theory 21(4), 360–366 (2021)
De Angelis, M.: Transport phenomena in excitable systems: existence of bounded solutions and absorbing sets. Mathematics 10(12), 2041 (2022). https://doi.org/10.3390/math10122041
Yadav, A., Swami, A.K., Srivastava, A.: Bursting and chaotic activities in the nonlinear dynamics of FitzHugh-Rinzel neuron model. IJERGS 4(3), 173–184 (2016)
De Angelis, M.: A priori estimates for solutions of FitzHugh-Rinzel system. Meccanica 57, 1035–1045 (2022). https://doi.org/10.1007/s11012-022-01489-6
De Angelis, F., De Angelis, M.: On solutions to a FitzHugh-Rinzel type model. Ric. Mat. 70, 51–65 (2021)
Corinto,F., Lanza, V., Ascoli, A., Gilli, M.:Synchronization in networks of FitzHugh-Nagumo neurons with memristor synapses. In: 20th European Conference on Circuit Theory and Design (ECCTD), pp. 608–611 (2011)
Rocsoreanu, C., Georgescu, A., Giurgiteanu, N.: vol. 10. , Germany (2012)
Rionero, S.: Longtime behaviour and bursting frequency, via a simple formula, of FitzHugh-Rinzel neurons. Rend. Lincei. Sci. Fis. e Nat. 32, 857–867 (2021). https://doi.org/10.1007/s12210-021-01023-y
Rionero, S.: Absence of subcritical instabilities and global nonlinear stability for porous ternary diffusive-convective fluid mixtures. Phys. Fluids 24, 104101 (2012)
Capone, F., Carfora, M.F., De Luca, R., Torcicollo, I.: Nonlinear stability and numerical simulations for a reaction-diffusion system modelling Allee effect on predators. Int. J. Nonlinear Sci. Numer. Simul (2021). https://doi.org/10.1515/ijnsns-2020-0015
Rionero, S.: Hopf bifurcations in dynamical systems. Ricerche Mat. 68, 811–840 (2019). https://doi.org/10.1007/s11587-019-00440-4
Acknowledgements
This paper has been performed under the auspices of the National Group of Mathematical Physics GNFM-INdAM.The author is grateful to the anonymous reviewers for their comments and suggestions.
Funding
Open access funding provided by Università degli Studi di Napoli Federico II within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declare that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
De Angelis, M. Hopf bifurcations in dynamics of excitable systems. Ricerche mat (2022). https://doi.org/10.1007/s11587-022-00742-0
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11587-022-00742-0