Hopf bifurcations in dynamics of excitable systems

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.


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/4r i )V xx from which the term D U xx is deduced [3]. Furthermore ε, c, and β 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 ε specifies the relationship between the time constants of the activator and inhibitor [6], and c and β 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 βε and δd are positive constants, they can be regarded as the coefficients of viscosity [23].
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: section 2 highlights some premises by which the subsequent theorems will be proved. In section 3 the mathematical problem and linear operator L with its invariants is given. Finally, in section 4 and section 5, Hopf bifurcations driven by critical pointŪ and driven by coefficient −η = −εβ are evaluated.

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 solutionŪ and the perturbation u = U −Ū, the behaviour of u is governed by: Considering the linear operator a i,j = const. ∈ R and independent from t, the stability and instability ofŪ is called linear if it is evaluated via the linear system In this regard, some theorems can be provided.
is the spectral equation whose eigenvalues of the n x n matrix ||a i,j || are λ i (i = 1, 2, 3..., n), and if and only if all the eigenvalues have negative real parts, then u = 0 is linearly globally attractive and asymptotically exponentially stable. Otherwise, if there exists at least an eigenvalue with positive real part, then u = 0 is unstable.
In addition, as proved in [26], for a system formed by three equations such as the FHR model, the spectrum equation (2.9) of L is reduced to the following expression: where (2.11) I 1 = a 11 + a 22 + a 33 ; represent the invariants of L whose spectrum is the set σ = {λ 1 , λ 2 , λ 3 } of its eigenvalues. Moreover, connected to the invariants I i (i = 1, 2, 3), we can introduce the quantities: (2.14) A 2 = a 11 a 12 a 21 a 22 + a 11 a 13 a 31 a 33 + a 22 a 23 a 32 a 33 = λ 1 (λ 2 + λ 3 ) + λ 2 λ 3 = I 2 and (2.15) and, according to [23], the following Lienard-Chipart criterion holds: 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.
Moreover, taking into account that the instability can occur only via a zero eigenvalue (λ = 0 ⇔ A 3 = 0) or via a pure imaginary eigenvalues, λ 1,2 = ±iω (i imaginary unit, ω ∈ ℜ + ) such that P(iω, 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 (2.18) "instability coefficient power" (ICP ) k of A k : (ICP ) k = 1 R C k and the following theorem holds: Let Ak be the spectrum equation coefficient with the biggest ICP and let the critical pointC be linearly asymptotically stable at R =R = 0. Then, at the growing of R from R = 0, the instability occurs at R = R Ck and one has a steady bifurcation ifk = 3, while an oscillatory bifurcation occurs at an

Mathematical model
is an admissible critical point, considering: as the perturbation vector, from (3.20) one obtains: Linearizing about C, it results: Denoting by the linear operator, according to (2.11)-(2.12), for k = 3, one has: as the invariants of L. Besides, taking into account (2.13)-(2.15) one deduces: and letting

Hopf bifurcations driven byŪ
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 pointŪ , the attention is focused on Γ =Ū 2 − 2(a + 1)Ū + a already introduced in (3.27), and the following theorem for linear stability can be proved:  ≤ − a 2 + a + 1 + a + 1 or U ≥ a 2 + a + 1 + a + 1, then the critical pointC is linearly, globally attractive and asymptotically exponentially stable.
Proof. Condition (4.29) ensures that Γ ≥ 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 (3.28), are all non-negative. Moreover, they are increasing functions of Γ.
This ensures that conditions (2.16) of theorem 2.2 state, and hence theorem holds.
Then, at the growing of R from R = 0, conditions ensure that a simple oscillatory bifurcation occurs at aR ∈]0, R C1 [, with a frequency ϕ 2π where If, in particular a steady+oscillatory bifurcation appears with a frequency given by ϕ = (2π)( √ A 2 ) Rc 1 .
Proof. When R = −Γ = 0, it results: 1, 2, 3). So, the critical point is linearly, asymptotically stable for R = 0. Besides, when inequalities (4.35) hold, it means that i.e. A 1 is the spectrum equation coefficient with the biggest instability coefficient power, so that at R = R C1 , it results: and hence, in view of the continuity of A 1 A 2 − A 3 , there exists aR ∈]0, R C1 [ such that Besides, conditions (4.36) imply that R = R C1 = R C2 < R C3 that means A 1 = A 2 = 0 and Consequently, the spectrum equation is reduced to: 3 ) = 0 and hence This means that a simple oscillatory bifurcation occurs at aR ∈]0, R c1 = R c2 [. Instead, when (4.37) holds, R c1 = R c3 < R c2 ; and hence one obtains A 1 = A 3 = 0. So, from the spectrum equation it results: and a steady (λ = 0) + oscillatory bifurcation of frequency ϕ/π with ϕ = ( √ A 2 ) Rc 1 occurs. Analogous results can be obtained if we suppose R c2 to be the biggest ICP and hence (4.38) is proved, too.

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 (1.3) considered also depends on the variable a generally not present in the bifurcations studies and it generalizes the FHR system (1.4), which, on the contrary, is more often considered in literature.
Moreover, the results obtained [see, f.i. Theorems 4.4-4.5 and condition (4.30)] 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 pointŪ 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.