Abstract
A ternary autonomous dynamical system of FitzHugh–Rinzel type is analyzed. The system, at start, is reduced to a nonlinear integro differential equation. The fundamental solution H(x, t) is explicitly determined and the initial value problem is analyzed in the whole space. The solution is expressed by means of an integral equation involving H(x, t) . Moreover, adding an extra control term, explicit solutions are achieved.
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1 Introduction
The FitzHugh–Rinzel (FHR) system [1,2,3,4] is a three-dimensional model deriving from the FitzHugh–Nagumo (FHN) model [5,6,7,8,9,10,11,12] developed to incorporate bursting phenomena of nerve cells. Indeed, a number of different cell types exhibit a behaviour characterized by brief bursts of oscillatory activity alternated by quiescient periods during which the membrane potential only changes slowly, and this behaviour is called bursting, see e.g. [13]. Accordingly, bursting oscillations are characterized by a variable of the system that changes periodically from an active phase of rapid spike oscillations to a silent phase. These phenomena are becoming increasingly important as they are being investigated in many scientific fields. Indeed, phenomena of bursting have been observed as electrical behaviours in many nerve and endocrine cells such as hippocampal and thalamic neurons, mammalian midbrain and pancreatic in \( \beta -\) cells, see e.g. [1] and references therein. Also, in the cardiovascular system, bursting oscillations are generated by the electrical activity of cardiac cells that excite the heart membrane to produce the contraction of ventricles and auricles [14]. Furthermore, bursting oscillations represent a topic of potential interest in dynamics and bifurcation mechanisms of devices and structures and in the analysis of nonlinear problems in mechanics [15,16,17,18,19]. Recent studies proved that the development of this field helps in the studying of the restoration of synaptic connections. Indeed, it seems that nanoscale memristor devices have potential to reproduce the behaviour of a biological synapse [20, 21]. This would lead in the future, also in case of traumatic lesions, to the introduction of electronic synapses to connect neurons directly.
The paper is organized as follows. In Sect. 1.1 the mathematical problem is defined and the state of the art with the aim of the paper are discussed. In Sect. 2, the explicit expression of the fundamental solution H(x, t) is achieved and some properties are proved. In Sect. 3 the integral solution for the initial value problem is given. In Sect. 4 the insertion of an extra term allows us to obtain explicit solutions for the model.
1.1 Mathematical considerations, state of the art and aim of the paper
Generally, denoting by \( D, \varepsilon , \beta , c \) constant parameters, the (FHN) model is a p.d.e. system such that
where function \(\, f(u)\,\) depends on the reaction kinetics of the model. In the literature f(u) can assume a piecewise linear form, see, e.g. [22] and reference therein, or \( f(u) = u-u^3/3\) [12]. However, in general, one has [5, 13]:
As for the FitzHugh–Rinzel model, most of the articles consider the following system characterized by three o.d.e.:
where \( I_{ext}, \varepsilon ,\beta ,c,d, h,\delta \) indicate arbitrary constants.
In this paper, in order to evaluate the contribute of a diffusion term, the following FitzHugh–Rinzel type system is considered:
Indeed, the second order term with \( D > 0 \) represents just the diffusion contribution and it can be associated to the axial current in the axon. It derives from the Hodgkin-Huxley (HH) theory for nerve membranes where, if b represents the axon diameter and \( r_i\) is the resistivity, the spatial variation in the potential V gives the term \((b/4r_i )V_{x x}\) from which term \( D\, u_{xx} \) derives [23].
Moreover it is also assumed \( \,\beta , \,\,d,\, \varepsilon , \,\,\delta \,\, \) as positive constants that together with \( c,\,\, h,\, \) characterize the model’s kinetic.
Model (4) can be considered as a two time-scale slow-fast system with two fast variables (u, w) and one slow variable (y). However, if for instance, \( \varepsilon = \delta \) the system can be considered as a two time-scale with one fast variable u and two slow variables (w, y). Otherwise, if \( \delta \) and \( \varepsilon \) have significant difference, it can also be considered as a three-time-scale system with the fast variable u, the intermediate variable and the slow variable [24].
As for function f(u) one considers the non-linear form expressed in formula (2). As a consequence, it results
Then, the system (4) becomes
Indicating by means of
the initial values, from (6)\(_{2,3} \) it follows:
Consequently, denoting the source term by
problem (6)–(7) can be modified into the following initial value problem \(\,\mathcal{P}\):
As for the state of art, mathematical considerations allow to assert that the knowledge of the fundamental solution H(x, t) related to the linear parabolic operator L :
leads to determine the solution of \(\,\mathcal{P}\). Indeed, if \(\, F(\,x,\,t,\,u\,)\,\) verifies appropriate assumptions, through the fixed point theorem, solution can be expressed by means of an integral equation, see f.i. [25, 26].
Moreover, according to [26], when operator L assumes a similar but simpler form, many properties and inequalities are achieved.
The aim of the paper is to explicitly determine the fundamental solution H(x, t) which involves naturally the diffusion constant D. Then, the initial value problem in all the space is analyzed and the solution is deduced by means of an integral equation. Moreover, using a method of travelling wave, solutions of a modified FitzHugh–Rinzel type system have been explicitly determined pointing out the influence of the diffusion parameter D.
2 Fundamental solution and its properties
Indicating by \(\,T\,\) an arbitrary positive constant, let us consider the initial- value problem (10) defined in the whole space \( \Omega _T: \)
\(\ \Omega _T =\{(x,t) : x \in \mathfrak {R}, \ \ 0 < t \le T \},\)
and let us denote by
the Laplace transform with respect to \(\,t.\,\) If \(\,\hat{H} (x,s)\,\) expresses the \( \mathcal{L}_t\) transform of the fundamental solution \(\, H ( x,t),\, \) from (10) it follows:
and formally it follows that
So that, denoting by \( J_1 (z) \,\) the Bessel function of first kind and order \(\, 1,\,\) let us consider the following functions:
Besides, by setting
and by denoting
the following theorem holds:
Theorem 1
In the half-plane \( \mathfrak {R}e \,s > \,max(\,-\,a ,\,-\beta \varepsilon ,- \delta d\,)\,\) the Laplace integral \(\,\mathcal{L }_t\, H \, \,\) converges absolutely for all \(\,x>0,\,\) and it results:
Moreover, function H(x,t) satisfies some properties typical of the fundamental solution of heat equation, such as:
-
(a)
\( H( x,t) \, \, \in C ^ {\infty }, \,\,\,\,\) \(\,\,\, t>0, \,\,\,\, x \,\,\, \in \mathfrak {R}, \)
-
(b)
for fixed \(\, t\,>\,0,\,\,\, H \,\) and its derivatives are vanishing esponentially fast as \(\, |x| \, \) tends to infinity.
-
(c)
In addition, it results \( \displaystyle \lim _{t\, \rightarrow 0}\,\,H(x,t)\,=\,0, \) for any fixed \(\, \eta \,>\, 0,\, \) uniformly for all \(\, |x| \,\ge \, \eta .\, \)
Proof
Since for all real \(\, z\,\) one has \(\, |J_1\,(z)|\, \le \, 1 , \) the Fubini–Tonelli theorem assures that
and being
it results:
where
Besides, since Fubini–Tonelli theorem and (19) one has:
from which, since (20),
is deduced.
Besides, by means of property of convolution for which \( f*g=g*f, \) since (14) and (15), property a) is evident. Moreover, properties b) and c) are proved following Theorem 3.2.1 of [25]. In particular, as for property c), for \( |x|\ge \eta \) and since \( |J_1 (z) |\le 1, \) it results:
from which property follows. \(\square \)
Now, introducing the following functions:
it results:
Moreover, by denoting
the convolution with respect to t, for \(\,t\,>\,0,\,\) as proved in [26] by means of formula (20),(21) and (24), it results:
where \(\, K_ {\varepsilon }\, \) is given by
Hence, the following theorem can be proved:
Theorem 2
For \(\,t\,>\,0,\,\) it results \( {L }\,H =0, \) i.e.
Proof
Let us consider that:
Accordingly, given relation (31), one has
Besides, considering that:
one has:
where it results:
So that, denoting by
one has:
Consequently, since for equation (31) one has \(K_ {\varepsilon } (x,t) = \, H_{1}(x,t)*e^{\,-\varepsilon \beta \,t}, \) by means of Fubini–Tonelli theorem and (29), it is proved that:
On the other hand, the convolution \( e^{-\,\delta d \,t} * H(x,t) \) is given by
with
As a consequence, it results:
Therefore, given relations (31), (40), (42), theorem holds. \(\square \)
3 Solution related to the FitzHugh–Rinzel problem
To provide the solution by means of the integral expression (13), some convolutions need to be determined.
In order to evaluate \( \int _0^t\,d\tau \, \int _\mathfrak {R}\,H(\xi ,\,\tau ) \, \,d\xi , \) let us start to observe that
with
Consequently, for (39), one has:
So that, according to (17), it results:
Now, let us evaluate
Considering (27)\(_3\), after an integration by parts, one obtains:
and, for (39), it results:
Moreover, since (17) and (47), one deduces:
Now, let us denote by
the convolution with respect to the space x, and let
Considering (42) and (48) one has:
Moreover, it results:
where
Consequently, given (9) and (13), for (45), (51), (52) and (53), it results:
and this formula, together with relations (8), allow us to determine also \(\, v(x,t) \,\) and \(\, y(x,t) \,\) in terms of the data.
4 Explicit solutions
Several methods have been developed to find exact solutions related to partial differential equations [27,28,29,30,31]. In this case, by referring to [7], an extra term is added in order to achieve some solutions. Accordingly, let us consider
where \( k\ne 0 \) and let us assume \( \varepsilon \beta = \delta \, d \) and \( f(u) = 2\,u\,(a-u)\,(u-1). \)
Under these conditions, problem (10) turns into:
where, by denoting \(\displaystyle \varphi _1 = 2\, u^2\, (\,a+1\,-u\,) + k \int _0^t\, \,e^{\,-\,\varepsilon \,\beta \,(\,t-\tau \,)}\, \, u^2(x,\tau )\,\, d\tau , \, \) it results:
and
In order to find explicit solutions, let us introduce
obtaining, from system (55), the following equation:
Now, let us consider
that is a solution of Riccati type equation:
and let us assume
Since
in order to satisfy Eq. (59), constant b needs to assume the following expression:
and, moreover, it has to be
and
So, if for instance, it is assumed \( \varepsilon \beta =1, \) \(z_0=0 \), \( \varepsilon + \delta = 0.04, k= 0.01 \) and \( C=1 \) with \( D> 0.019, \) for \( A = \sqrt{D}, \) by introducing
equation (61) gives
In Fig. 1, solutions u(z) expressed by means of formula (65) are illustrated for different values of D, by showing that the amplitude increases as \( 0<D <1 \) increases.
Remark When fast variable u simulates the membrane potential of a nerve cell, while slow variable w and superslow variable y determine the corresponding ion number densities, model (55) with its solutions can be of interest in applications to understand how impulses are propagated from one neuron to another. Moreover, as similarly underlined in [4], the knowledge of exact solutions can help in testing different applications of models in neuroscience.
References
Bertram, R., Manish, T., Butte, J., Kiemel, T., Sherman, A.: Topological and phenomenological classification of bursting oscillations. Bull. Math. Biol. 57(3), 413 (1995)
Wojcik, J., Shilnikov, A.: Voltage interval mappings for an elliptic bursting model. In: González-Aguilar, H., Ugalde, E. (eds.) Nonlinear Dynamics New Directions Theoretical Aspects, p. 219. Springer, Berlin (2015)
Zemlyanukhin, A.I., Bochkarev, A.V.: Analytical properties and solutions of the FitzHugh–Rinzel model. Rus. J. Nonlinear Dyn. 15(1), 3–12 (2019)
Kudryashov, N.A.: On integrability of the FitzHugh–Rinzel model. Rus. J. Nonlinear Dyn. 15(1), 13–19 (2019)
Izhikevich, E.M.: Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, p. 397. The MIT Press, Cambridge (2007)
De Angelis, M., Renno, P.: Asymptotic effects of boundary perturbations in excitable systems. Discrete Contin. Dyn. Syst. Ser. B 19(7), 2039–2045 (2014)
Kudryashov, N.K., Rybka, K.R., Sboev, A.: Analytical properties of the perturbed FitzHugh–Nagumo model. Appl. Math. Lett. 76, 142–147 (2018)
De Angelis, M.: On a model of superconductivity and biology. Adv. Appl. Math. Sci. 7(1), 41–50 (2010)
Rionero, S.: A rigorous reduction of the \(L^2\)-stability of the solutions to a nonlinear binary reaction–diffusion system of PDE’s to the stability of the solutions to a linear binary system of ODE’s. J. Math. Anal. Appl. 319(2), 377–397 (2006)
Rionero, S., Torcicollo, I.: On the dynamics of a nonlinear reaction–diffusion duopoly mode. Int. J. Non-Linear Mech. 99, 105–111 (2018)
De Angelis, M.: Asymptotic estimates related to an integro differential equation. Nonlinear Dyn. Syst. Theory 13(3), 217–228 (2013)
Gambino, G., Lombardo, M.C., Rubino, G., Sammartino, M.: Pattern selection in the 2D FitzHugh–Nagumo model. Ricerche di Matematica (2018). https://doi.org/10.1007/s11587
Keener, J.P., Sneyd, J.: Mathematical Physiology, p. 470. Springer, New York (1998)
Quarteroni, A., Manzoni, A., Vergara, C.: The cardiovascular system: mathematical modelling, numerical algorithms and clinical applications. Acta Numer. 26, 365–590 (2017)
De Angelis, F., Cancellara, D., Grassia, L., D’Amore, A.: The influence of loading rates on hardening effects in elasto/viscoplastic strain-hardening materials. Mech. Time-Dependent Mater. 22(4), 533–551 (2018)
Carillo, S., Chipot, M., Valente, V., Vergara Caffarelli, G.: On weak regularity requirements of the relaxation modulus in viscoelasticity. Commun. Appl. Ind. Math. 10(1), 78–87 (2019)
De Angelis, F., Cancellara, D.: Dynamic analysis and vulnerability reduction of asymmetric structures: fixed base vs base isolated system. Compos. Struct. 219, 203–220 (2019)
De Angelis, F.: Extended formulations of evolutive laws and constitutive relations in non-smooth plasticity and viscoplasticity. Compos. Struct. 193, 35–41 (2018)
Simo, H., Woafo, P.: Bursting oscillations in electromechanical systems. Mech. Res. Commun. 38, 537–541 (2011)
Juzekaeva, E., Nasretdinov, A., Battistoni, S., Berzina, T., Iannotta, S., Khazipov, R., Erokhin, V., Mukhtarov, M.: Coupling cortical neurons through electronic memristive synapse. Adv. Mater. Technol. 4(6), 1800350 (2019)
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). IEEE (2011)
De Angelis, M.: A priori estimates for excitable models. Meccanica 48(10), 2491–2496 (2013)
Murray, J.D.: Mathematical Biology I, p. 767. Springer, New York (2003)
Xie, W., Jianwen, X., Cai, L., Jin, Y.: Dynamics and geometric desingularization of the multiple time scale FitzHugh Nagumo Rinzel model with fold singularity. Commun. Nonlinear Sci. Numer. Simul. 63, 322–338 (2018)
Cannon, J.R.: The One-Dimensional Heat Equation, p. 483. Addison-Wesley Publishing Company, Boston (1984)
De Angelis, M., Renno, P.: Existence, uniqueness and a priori estimates for a non linear integro–differential equation. Ricerche di Matematica 57, 95–109 (2008)
De Angelis, M.: On the transition from parabolicity to hyperbolicity for a nonlinear equation under Neumann boundary conditions. Meccanica 53(15), 3651–3659 (2018)
Li, H., Guoa, Y.: New exact solutions to the Fitzhugh Nagumo equation. Appl. Math. Comput. 180(2), 524–528 (2006)
De Angelis, M.: A wave equation perturbed by viscous terms: fast and slow times diffusion effects in a Neumann problem. Ricerche di Matematica 68(1), 237–252 (2019)
Prinaria, B., Demontis, F., Li, S., Horikis, T.P.: Inverse scattering transform and soliton solutions for square matrix nonlinear Schrödinger equations with non-zero boundary conditions. Phys. D Nonlinear Phenom. 368, 22–49 (2018)
Kudryashov, N.K.: Asymptotic and exact solutions of the FitzHugh–Nagumo model. Regul. Chaotic Dyn. 23(2), 152–160 (2018)
Acknowledgements
The present work has been developed with the economic support of MIUR (Italian Ministry of University and Research) performing the activities of the project ARS\(01_{-} 00861\) “Integrated collaborative systems for smart factory - ICOSAF”. The paper has been performed under the auspices of G.N.F.M. of INdAM. The authors are grateful to anonymous referees for their helpful comments.
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De Angelis, F., De Angelis, M. On solutions to a FitzHugh–Rinzel type model. Ricerche mat 70, 51–65 (2021). https://doi.org/10.1007/s11587-020-00483-y
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DOI: https://doi.org/10.1007/s11587-020-00483-y