Longtime behaviour and bursting frequency, via a simple formula, of FitzHugh–Rinzel neurons

The longtime behaviour of the FitzHugh–Rinzel (FHR) neurons and the transition to instability of the FHR steady states, are investigated. Criteria guaranteeing solutions boundedness, absorbing sets, in the energy phase space, existence and steady states instability via oscillatory bifurcations, are obtained. Denoting by λ3+∑k=13Ak(R)λ3-k=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \lambda ^{3} + \sum\nolimits_{{k = 1}}^{3} {A_{k} } (R)\lambda ^{{3 - k}} = 0 $$\end{document}, with R bifurcation parameter, the spectrum equation of a steady state m0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_0$$\end{document}, linearly asymptotically stable at certain value of R, the frequency f of an oscillatory destabilizing bifurcation (neuron bursting frequency), is shown to be f=A2(RH)2π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f=\displaystyle \frac{\sqrt{A_2(R_\mathrm{H})}}{2\pi } $$\end{document} with RH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_\mathrm{H}$$\end{document} location of R at which the bifurcation occurs. The instability coefficient power (ICP) (Rionero in Rend Fis Acc Lincei 31:985–997, 2020; Fluids 6(2):57, 2021) for the onset of oscillatory bifurcations, is introduced, proved and applied, in a new version.


Introduction
The brain contains many millions of neurons which trasmit the electrochemicals signals via the following mechanism. Any neuron has a central region (soma), the dendrites three and the axon. The dendrites are thin fibers around the soma while the axon is a long cylinder-starting from soma and ending in contact with other neurons (via the structures called synapses)-is constituted by long fibers. Each neuron performs a relative simple activity: the dendrites receive input from other neurons or external sources and elaborate an output signal which is propagated along the axon branchies, to thousands of other neurons (the branchies terminate on the dendrites or cell bodies of other neurons). Along the years, starting from about the second half of the past century Gerstener et al. (2014), various mathematical models have been introduced for modeling the bio-physical activity of neurons (Hodgin 1948;Hodgin and Huxley 1952;Izhikevich 2007;Ermentrant and Temam 2010;Gerstener et al. 2014). The FitzHugh-Rinzel model is given by Rinzel (1981Rinzel ( , 1987, Rinzel and Ermentrout (1989), Izhikevich (2004), FitzHugh (1955, 1961 with I, , a, b, , c, d real constants and In the absence of third equation and y = 0 , (1) reduces to the FitzHugh-Nagumo binary model which contains-as special case for = 1, a = b = 0-the celebrated Van der Pol oscillator (Van der Pol 1926;Hale and Kocak 1991). The variable y, introduced by Rinzel, represents the bursting (1) of potential between the dentritic spine head and the surrounding medium , y = slowly moving current in the dentrite , w = recovery variable , I = applied external current , = small parameter determining the pace of the slow .
1 3 behaviour of neurons: alternance between brief bursts of oscillator activity and quiescent period. In the present paper, we investigate the longtime behaviour of (1) solutions, the instability of steady states and the oscillatory bifurcations onset via the Instability Coefficients Power (ICP) approach (Rionero 2020(Rionero , 2021a. Because of the neurons oscillatory activity, the onset of oscillatory bifurcation has attracted the attention of many scientists {see Wojcik and Shilnikov 2011;Yadav et al. 2016;Alidousti and Khoshsiar Ghaziani 2017;Xie et al. 2018;Temam 1988 and the references therein} , but the results obtained-although of sure interest-with respect to the rich dynamics of (1) generated by the seven parameters (I, , a, b, , c, d)-appear to be longely partial. Our aim is to obtain-via the instability coefficient power (ICP) approach (Rionero 2020(Rionero , 2021a)-general criteria guaranteeing the existence of oscillatory Hopf bifurcations and an estimate of their locations (in order to obtain-via a simple closed form-the neurons activity frequency). The plane of the paper is as follows. Section 2 is devoted to boundedness and longtime behaviour of (1) solutions. In particular, the existence of absorbing sets is put in evidence. The critical points are investigated in Sect. 3 while their linear stability/ instability conditions are considered in Sect. 4. The subsequent Sect. 5 is dedicated to the bifurcation power of the spectrum equation coefficients. Successively, in Sects. 6-8 the transition to instability via Hopf bifurcations driven by the growing of the bifurcating paramters v,̄,̄ , respectively, is analyzed. In Sect. 9, the case { 2 < + 3 ,̄> 0} is investigated while the operativity of conditions (3) is checked in Sect. 10. The paper ends with: discussion, final remarks and perspectives (Sect. 11).

Boundedness and longtime dynamics
Setting in view of (1) 2 -(1) 3 one immediately realizes that > 0 and > 0 play the role of "viscosity coefficients". The following properties hold.

Property 1 Let
Then the solutions of (1) are bounded.
Proof In view of (2), (1) can be written Introducing the "energy" one has In view of one has But with positive constant, implies dE dt ≤ − ( + )v 2 + w 2 + y 2 + ( − 1)wv + (1 − )yv + ( cy + aw + Iv) + 3 4 ( + + 1) 2 . (9) and choosing the constant such that then (10) implies with In view of (12), one easily obtains and hence ◻ Remark 1 We remark that, case by case, according to the values of the parameters contained in (1), the existence of the positive constants appearing in (11), has to be verified. Only then properties 1-2 hold. We underline that (11) equivalently can be written for at least a positive , i.e. equivalently (10) In fact, (16) is implied by (17) and is not allowed by (13) 1 . (

Critical points
In view of (1), one has and the critical points are the roots of One has and hence it follows Setting one has Remark 2 We remark that 1. The DS (1)-depending on seven real parameters-has a very rich dynamics; 2. A, B-being real constants-(25) 1 admits at least one real root for any value of the parameters-given by the celebrated Cardano formula-and therefore at least ∞ 7 critical points are admissible; (21) Let be the spectrum of L, i.e. the set of the roots of (31) and recall that

The instability coefficient power method for ternary DS
We call "auxiliary coefficient" of the spectrum equation (31) the coefficient A 0 given by and remark that properties (ii) and (iii) of Sect. 4 give to the coefficients A k , k ∈ {0, 1, 2, 3} , via their becoming zero at certain values of the bifurcation parameter R, the power of driving the location and the type of the occurring bifurcation. The following property holds.
Property 3 Let a critical point m 0 be linearly, asymptotically stable at certain value R of the bifurcation parameter R and let R c k be the lowest root of A k (R) = 0 , at the increasing (decreasing) of R from R =R . Measuring the instability power of A k via the (ICP) index given by then the coefficient with the biggest (ICP) drives the occurring of a steady bifurcation for k = 3 , while k < 3 implies the occurring of an Hopf bifurcation at R 0 , lowest root of A 0 (R) = 0 and the estimates Proof Property 3 is implied by the following properties 4-5. ◻

Property 4 At a value R of the bifurcation parameter R, an Hopf bifurcation occurs if and only if
Proof (43) (47) ( + A 1 )( 2 + A 2 ) = 0 1 3 are immediately obtained. Since the general integral of the spectrum equation implies one has that, A 1 > 0 implies the existence of a simple Hopf bifurcation (SHB) of frequency coupled to a time exponentially decreasing perturbation, while A 1 = 0 implies that the Hopf bifurcation is coupled to a steady one (steady-Hopf bifurcation). ◻

Property 5 Let
lowest root of A 0 = 0 , at which the SHB occurs.

The ICP method for a bifurcation parameter R ∈] − ∞, ∞[
Since the parameters appearing in (1) have only to be real numbers, it appears necessary a formulation of the ICP method for a bifurcation parameter ∈] − ∞, ∞[. Let at a value ∈] − ∞, ∞[ of R exists a steady state m 0 . Then the following formulation of the ICP method holds.

Property 6
If, at R = , the steady state m 0 is linearly asymptotically stable, then, at the increasing (decreasing) of R from R = , the coefficient A k of the spectrum equation of m 0 , first becoming zero, drives not only the transition to instability but also the type and location of the occurring bifurcation which is a SHB if k < 3 , a steady one if k = 3 , and a Hopf bifurcation coupled to a steady one if R c k = R c 3 , k = 1 or k = 3.
Proof Introducing the parameters R,R such that it follows that R ≥ 0,R ≥ 0 respectively and the formulation of the ICP method given in property 4 can be applied with R = 0 . ◻
Proof In vire of (32)-(34), the coefficients of the spectrum equation can be written with and are-in view of (57)-increasing functions of . It follows that On the other hand and the linear stability conditions (36) are verified ∀ ≥ 0, i.e. ∀v ≥ 1 . ◻

On setting (58) becomes
and at the decreasing of v 2 from v 2 = 1 , the A k are decreasing functions of ̄.
Denoting by ̄c k the lowest root of A k (̄) and letting (57) holds, one has and it follows that

Property 8 Let (57) holds with
Then, at the growing of ̄ from ̄= 0, guarantees the existence of a ̄c ∈]0,̄c 3 [ at which an oscillatory bifurcation occurs, while guarantees that at ̄c 1 =̄c 3 an oscillatory bifurcation, coupled with a steady one, occurs.

Hopf bifurcations driven by ̄= −˛ growing
The bifurcations criteria obtained in the previous section, all require ≤ 0 . In the present section, we obtain that decreasing as bifurcating parameter and letting ≤ 0 , the Hopf bifurcation can arise with ≥ 0.

Remark 4 We remark that
implies that b is a bifurcating parameter.

Hopf bifurcations driven by ̄= −ˇ growing
A criterion analogous to the which one of property 10 can be easily obtained.

Steady bifurcations
In view of it follows that (87) A k (̄c k ) = 0. (88) implies the onset of steady bifurcations.

Applications
The boundedness and existence of absorbing sets have a basic importance for the longtime behaviour of any DS Izhikovich (2000a). Their existence for the solutions of the FHR model (1) appears to be-as far as we know-new in the existing literature, at least in the formulation given in Sect. 2. Therefore, a check on the operativity of properties 1,2 could be of relevant interest. We return on the conditions (3) and remark that are equivalent to and consider as prototypes of a general case, the following two cases In view of remark 1, we have to check on the existence of a positive value of , such that (11) holds. In both the cases (98), (99) one has

Check on the operativity of conditions (3) in the case (99)
In the case (99), one has Therefore, in the case (99), one has which implies that boundedness and existence of absorbing sets, according to properties 1-2, is guaranteed for < 0.088 .

Check on the operativity of conditions (3) in the case (100)
In the case (100), since (103) 1 is verified for r < 0.088 , the boundedness and existence of absorbing sets are guaranteed by the values of the bifurcation parameter such that i.e. and hence The case (99), with {b = 0.8, = 0.08, = 0.002} has been investigated in Yadav et al. (2016) and the case (100), with the same b and , has been investigated in Alidousti and Khoshsiar Ghaziani (2017). The procedures applied in Yadav et al. (2016), Alidousti and Khoshsiar Ghaziani (2017), are completely different of the which ones of the present paper. In particular, boundedness and existence of absorbing sets are not considered (Fig. 1).
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