A priori estimates for solutions of FitzHugh–Rinzel system

The FitzHugh–Rinzel system is able to describe some biophysical phenomena, such as bursting oscillations, and the study of its solutions can help to better understand several behaviours of the complex dynamics of biological systems. We express the solutions by means of an integral equation involving the fundamental solution H(x, t) related to a non linear integro-differential equation. Properties of H(x, t) allow us to obtain a priori estimates for solutions determined in the whole space, showing both the influence of the initial data and the source term.

where and terms , c, d, h, , are positive constants that characterize the model's kinetics. The second order term with D > 0 can be associated to the axial current in the axon, and it derives from the Hodgkin-Huxley theory for nerve membranes. Indeed, 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 xx from which term Du xx derives (see f.i. [16]), and in [9] an analysis on contribution due to this term has been developed. Furthermore, when the fast variable u simulates the membrane potential of a nerve cell, while the slow variable w and the super-slow variable y determine the corresponding number densities of ions, the model (1) simulates the propagation of impulses from one neuron to another, and studies on solutions can help in testing the responses of the various models in neuroscience.
Several methods have been developed to find exact solutions related to partial differential equations and an extensive bibliography on the study of analytical behaviors exists (see,f.i [17][18][19][20][21][22][23]). The aim of this paper is to determine a priori estimates for the FHR solution by means of suitable properties of the fundamental solution H(x, t), showing how the effects due to the initial perturbation are vanishing when t tends to infinity, and simultaneously, as time increases, the effect of the nonlinear source remains bounded.
The paper is organized as follows: in Sect. 2 we define the mathematical problem and report some of the results already proved in [9], as well as other results well known in literature. In Sect. 3, some properties related to the fundamental solution H(x, t) are obtained and, in a subsection, some relationships on convolutions which characterize the explicit solution, are highlighted. In Sect. 4, estimates on convolution are proved and in Sect. 5, the solution is expressed by means of these particular convolution integrals. Finally, in Sect. 6, a priori estimates are showed.

Mathematical considerations
Indicating by T an arbitrary positive constant, let us consider the set: represent the initial values, then from (1) 2,3 , one deduces: Besides, letting system (1) becomes and hence, when denotes the source term, problem (6) with initial conditions (3), can be modified into the following initial value problem P: In order to determine the solution of problem (8), let us consider the following functions: where J 1 (z) is the Bessel function of first kind and order 1.
In [9] it has been verified that function H(x, t) : represents the fundamental solution of the parabolic operator and the following theorem has been proved: T h e o r e m 1 I n t h e h a l fp l a n e ℜe s > max( − a, − , − d ) the Laplace integral L t H converges absolutely for all x > 0, and it results: where Moreover, function H(x,t) satisfies some properties typical of the fundamental solution of heat equation, such as: (b) for fixed t > 0, H and its derivatives are vanishing exponentially fast as |x| tends to infinity. (c) In addition, it results lim t →0 H(x, t) = 0, for any fixed > 0, uniformly for all |x| ≥ . ◻ To obtain results of existence and uniqueness for the problem (8), the theorem of fixed point can be applied and therefore, also according to [24], for initial term and source function we shall admit: Assumption A Initial data u 0 is continuously differentiable and bounded together with its first derivative. The source term F(x, t, u) is defined and continuous on the following set: Besides, for each K > 0 and |u| < K, F( x, t, u ) is uniformly Lipschitz continuous in (x, t) for each compact set of T and it is bounded for bounded u.
Then, for all ( u 1 , u 2 ), there exists a positive constant W F such that: ◻ As a consequence, when the fundamental solution H(x, t) and source function F(x, t, u) satisfy theorem 1 and Assumption A, respectively, indicating by u(x, t) a solution of problem P, then u assumes this form: On the other hand, if u(x, t) is a continuous and bounded solution of (17), it is possible to prove that u satisfies (8).
Consequently, it is possible to conclude that

Theorem 2 Initial value problem (8) admits a unique solution only if (17) admits a unique continuous and bounded solution. ◻
Besides, by means of fixed point theorem,(and extensive proofs can be found, f.i., in [24][25][26][27][28][29]), it is possible to prove the following theorem: In this case, taking into account the source term F(x, t) defined in (7), solution (17) assumes the following form: and this formula, together with relations (4), allows us to determine also v(x, t) and y(x, t) in terms of the data.

Some properties related to H(x,t)
In order to obtain a priori estimates and asymptotic effects, some properties related to the fundamental solution H need to be evaluated.
More precisely, formula (18) shows the need to evaluate the convolution of the fundamental solution H with respect to time and space.
Consequently, this section will include a first part where two theorems involving some properties related to H(x, t) are showed, and a subsection where some premises allowing to prove properties related to convolution integrals, will be stated.
Let us start indicating by three positive functions, then the following theorem holds: (11) satisfies the following estimate:

Theorem 4 The solution function H defined in
Proof Since from (9) it results: and hence: Moreover, from (10) and by means of (22), it results: Consequently one obtains: Hence, according to (11), for (22) and (23), theorem holds. ◻ Now, let us introduce as I 0 the modified Bessel function of the first kind and order 0, and let The following theorem holds:

Theorem 5 The fundamental solution H(x, t) defined in (11) satisfies the following estimates:
Besides, indicating by one has: Proof Considering that we will firstly focus on the integral involving H 1 , and then on that involving H 2 .
Since it results: Consequently, as for it results: Now, being I 0 (|z|) < e |z| , from (36) one deduces that where l is defined in (24) 1 .

Premises on convolution integrals referring to the solution
In order to determine the estimates related to the solution, it is necessary to highlight every convolution integrals that characterize the solution itself. Therefore, in this subsection convolutions K and H will be introduced and, by means of them, solution u(x, t) will be expressed (Formula 52). Hence, let us consider and let be the convolution with respect to t.
In [9] it has been proved that: and this formula explicitly shows all the convolutions involved in the solution u(x, t).

On convolutions involving functions K and H
Formula (52) shows that an analysis of the solution directly implies estimates on both H(x, t) and on functions K , H , defined in (41) and (45). For this, let us consider A(t), B(t), C(t), (t) defined in (19) and (25), respectively. Moreover, let with q defined by (24) 2 .

Analysis of solution
In order to analyse functions u(x, t), w(x, t), and y(x, t), it appears necessary to make explicit the integrals of convolutions involving functions H and K whose estimates have been established in the previous section. Therefore, since (52), by means of convolution properties, we get: Moreover, as for functions w(x, t) and y(x, t) defined in (4), according to (43), (46) and (47), since (71), the following integrals must be considered: Remark These estimates show that the solution of the FitzHugh-Rinzel system is bounded for all t. Besides, when t tends to infinity, the effect of the non linear term (x, t) is bounded, while the effects of initial perturbances u 0 (x), w 0 (x), y 0 (x) are vanishing.