ASYMPTOTIC BEHAVIOUR OF A NEURAL FIELD LATTICE MODEL WITH DELAYS

. The asymptotic behaviour of an autonomous neural ﬁeld lattice system with delays is investigated. It is based on the Amari model, but with the Heaviside function in the interaction term replaced by a sigmoidal function. First, the lattice system is reformulated as an inﬁnite dimensional ordinary de- lay diﬀerential equation on weighted sequence state space (cid:96) 2 ρ under some appropriate assumptions. Then the global existence and uniqueness of its solution and its formulation as a semi-dynamical system on a suitable function space are established. Finally, the asymptotic behaviour of solution of the system is investigated, in particular, the existence of a global attractor is obtained.


1.
Introduction. Neural field models are often represented as evolution equations generated as continuum limits of computational models of neural fields theory. They are tissue level models that describe the spatio-temporal evolution of coarse grained variables such as synaptic or firing rate activity in populations of neurons. See Coombes et al. [4] and the literature therein. A particularly influential model is that proposed by S. Amari in [1] (see also Chapter 3 of Coombes et al. [4] by Amari): site of the neural field. Han & Kloeden [7] introduced and investigated the following lattice version of the Amari model: Delays are often included in neural field models to account for the transmission time of signals between neurons. In addition, to facilitate the analysis, the Heaviside function can be replaced by a simplifying sigmoidal function such as σ ε (x) = 1 1 + e −x/ε , x ∈ R, 0 < ε < 1. In this paper we consider the autonomous neural field lattice system with delays Throughout this paper we assume that the delays τ j > 0 are uniformly bounded, i.e., satisfy Assumption 1. There exists a constant h ∈ (0, ∞) that 0 ≤ τ i ≤ h for all i ∈ Z d . and that the interconnection matrix (k i,j ) i,j∈Z d satisfies Assumption 2. k i,j ≥ 0 for all i, j ∈ Z d and there exists a constant κ > 0 such that The main goal of this paper is to investigate asymptotic behaviour of solutions to the neural lattice system with delays (1), in particular, the attractor for the semidynamical system generated by its solutions. The initial conditions for such delay systems have the form for appropriate functions ψ i .
In particular, given a positive sequence of weights (ρ i ) i∈Z d , we consider the separable Hilbert space and norm We assume that the ρ i satisfy the following assumption.
The appropriate function space for the solutions of the lattice system with delays For a solution u(t) = (u i (t)) i∈Z d ∈ 2 ρ of (1) we denote by u t the segment of the . The corresponding initial condition (2) must then satisfy (ψ i (·)) i∈Z d ∈ C([−h, 0], 2 ρ ).
2.1. The reaction term. For any u = (u i ) i∈Z d ∈ 2 ρ , we define the operator f by To ensure that the f (u) take values in 2 ρ for every u ∈ 2 ρ and has necessary dissipative properties, we make the following standing assumptions on the f i throughout the rest of the paper.
Assumption 4. The functions f i : R → R are continuously differential with weighted equi-locally bounded derivatives, i.e., there exists a non-decreasing function L(·) ∈ C(R + , R + ) such that Assumption 6. There exist constants α > 0 and β i with β = (β i ) i∈Z d ∈ 2 ρ such that sf i (s) ≤ −α|s| 2 + β 2 i , ∀s ∈ R, ∀i ∈ Z d . It was shown in [7] that Assumption 4 implies that f i is locally Lipschitz with The following lemma from [7] states the Lipschitz and dissipative properties of the operator f . Lemma 2.1. Assume that Assumptions 4-6 hold. Then f : 2 ρ → 2 ρ is locally Lipschitz and satisfies the dissipativity condition Proof. The function σ ε takes values in the unit interval [0,1], so Then Hence it is globally Lipschitz with the Lipschitz constant L σ = 1 ε . 2.3. The forcing term. Finally, we suppose that the constant forcing term g := (g i ) i∈Z d satisfies the following assumption.
3. Existence and uniqueness of solutions. The lattice differential equation (1) can be rewritten as an infinitely dimensional ordinary differential equation where In this section we study the existence and uniqueness of solutions of the differential equation (3). To this end, we will need the following auxiliary Lemma 3.1. Let Considering only the j ∈ Z d N appearing in the sum defining Since there are a finite number of terms in the sum in the definition of K N τ,i , it follows from the elementary inequality Theorem 3.2. Suppose that Assumptions 1-7 hold. Then for each r > 0 there exists a(r) > 0 such that for every Proof.
Step 1. First, we claim that G τ (t, u t ) is well defined and bounded.
It is easy to see that G τ (t, u t ) is well defined since f (u), K τ (u t ) and g are all well defined. As for the boundedness, we denote that Since f i is locally Lipschitz and satisfies f i (0) = 0 by Assumption 4-5, we see that Then we obtain For the second term with delay, we have |K τ,i (u t )| ≤ κ by Assumption 2, which gives where we have used Assumption 3. Finally, for the last term g, Assumption 7 gives Using (5), (6) and (7) in (4) we conclude that G τ is well defined and bounded.
Step 2. Next, we claim that the maps G τ,i : By the local Lipschitz continuity of f i , which shows that this term converges to zero. Next for the second term on the right-hand side On one hand, since (10) is such an N . On the other hand, for all N . Thus K τ,i is continuous.
Using (9) and (10) in (8) Step 3. Finally, we claim the following inequality holds: where b K → 0 + as K → ∞, and C(·) > 0 is a continuous non-decreasing function. The proof is as follows.

3.2.
A prior estimate of solutions. Here we will establish some estimates of the solutions, which imply that the solutions are bounded uniformly with respect to bounded sets of initial conditions and all positive values of time.
where R j > 0, j = 1, 2, are constants depending on the parameters of the problem.
Proof. We multiply the ith component of (1) by ρ i u i (t) and sum over i to obtain 1 2 By Assumption 6 and ρ i > 0 we have Since function σ ε takes values in the unit interval, using Young's inequality we obtain The last term on the right hand of (12) satisfies In summary, collecting the inequalities above, we obtain 1 2 Integrating both sides of this differential inequality yields Let θ ∈ [−h, 0]. Replacing t by t + θ in (13) and using we obtain Finally, using that θ ∈ [−h, 0] and neglecting the negative terms yields where

Uniqueness of solutions.
Having the existence of the solution of problem (3), moreover, we now establish the uniqueness of the solution with the additional assumption that Proof. Assumption 8 implies that the operator K τ : Set w = u − v, we obtain that 1 2 Integrating from 0 to t then gives Let θ ∈ [−h, 0]. Replacing t by t+θ in the inequality above and using w(t+θ) ρ = 0 when t + θ < 0. We obtain Then take the supremum on θ, By Gronwall's inequality, we have Since w(0) = 0, we obtain that w ≡ 0.
The proof of the next corollary follows easily using (15).
Proposition 1 implies that every local solution of (1) can be extended globally, which, with the uniqueness of the solution, will allow us to define a semigroup in terms of the solution mapping and to conclude that it has a bounded absorbing set. We can thus define a semigroup of operators S : where u t is the unique solution to (3) with u 0 = ψ. The semigroup map S is continuous in its variables by Corollary 1. It also follows from inequality (11) that the semigroup has a bounded absorbing set.

Corollary 2. The bounded set defined by
is absorbing for the semigroup S.
Our aim is to study the asymptotic behaviour of solution of problem (1). In particular, we will show the existence of a global attractor. For this we will apply the following well-known results about the existence of global attractors, see [8] and [5].
Theorem 4.1. Let x → S(t, x) be continuous for any t ≥ 0. Assume that S is asymptotically compact and possesses a bounded absorbing set B 0 . Then there exists a global compact attractor A, which is the minimal closed set attracting any bounded set. If, moreover, the space X is connected and the map t → S(t, x) is continuous for any x ∈ X, then the set A is connected.

Tail estimate.
To show the asymptotic compactness of the semigroup, we need to estimate the tails of solutions of (3), i.e., their higher dimensional components, see [2].
for any initial condition ψ ∈ B and the corresponding solution u(·) of (3) with u 0 = ψ.
Proof. Define a smooth function ξ satisfying Let M be a fixed (and large) integer to be specified later, and set where | · | denotes the Euclidean norm. We multiply the ith component of (1) by ρ i v i , then summing over i ∈ Z d , and since u( First, by Assumption 6, Then, since function σ ε takes values in the unit interval, using Young's inequality, And using Young's inequality again, Inserting the estimations (17), (18) and (19) into (16), then We now estimate each term on the right hand side of the above inequality. Note that Since β = (β i ) i∈Z d ∈ 2 ρ , then for every ε > 0 there exists I 1 (ε) > 0 such that Similarly, since ρ Σ = i∈Z d ρ i < ∞, then for every ε > 0, there exists I 2 (ε) > 0 such that i∈Z d In addition, since g = (g i ) i∈Z d ∈ 2 ρ by Assumption 7, for every ε > 0 there exists Finally, for any ε > 0, choosing I(ε) := max{I 1 (ε), I 2 (ε), I 3 (ε)}, inserting the estimations (21), (22) and (23) It follows immediately from Gronwall's lemma that In a similar way as in Proposition 1 we have Thus, there exist T (ε, B) and M (ε, B) such that

4.2.
Existence of the global attractor. In order to apply Theorem 4.1, we need to prove that S generated by the delay lattice system (3) is asymptotically compact. Proof. We consider ξ n := u n tn = S(t n , ψ n ), where ψ n ∈ B, a bounded set in C([−h, 0], 2 ρ ). From (14) there is a C > 0 such that u n tn (s) ≤ C, ∀s ∈ [−h, 0], ∀n ∈ N. For fixed s ∈ [−h, 0] we can find a subsequence (which we still denote by u n ) such that u n (t n + s) ζ(s) in 2 ρ . In fact, the weak convergence here is strong, which follows from Lemma 4.2. Indeed, there exists N 1 > 0, when n ≥ N 1 , we have t n > T (where T is the constant in Lemma 4.2). Moreover, for any µ > 0 there exist K 2 (µ) and N 2 (µ) such that |i|>K2 ρ i |u n i (t n + s)| 2 < µ, Thus, {u n (t n + s)} is precompact in 2 ρ for any s ∈ [−h, 0]. Since G τ is a bounded map, Proposition 1 and the integral representation of solutions imply that Then, the Ascoli-Arzelà theorem implies that ξ n is relatively compact in C([−h, 0], 2 ρ ). Remark 2. If Assumption 8 guarranteeing uniqueness of solutions does not hold, then the lattice model (1) generates a set-valued semi-dynamical system, which can be shown to have a global attractor using essentially the same Lemmas as above.