Sigmoidal approximations of a delay neural lattice model with Heaviside functions

The approximation of Heaviside coefficient functions in delay neural lattice models with delays by sigmoidal functions is investigated. The solutions of the delay sigmoidal models are shown to converge to a solution of the delay differential inclusion as the sigmoidal parameter goes to zero. In addition, the existence of global attractors is established and compared for the various systems.


Introduction. Han & Kloeden
introduced and investigated the following infinite dimensional lattice version of the Amari neural field model [1] : where θ > 0 is a given threshold and H : R → R is the Heaviside function defined by To avoid difficulties with the Heaviside function, in [8] they replaced the Heaviside function with a sigmoidal function characterised by a small positive parameter ε such as σ ε (x) = 1 1 + e −x/ε , x ∈ R, 0 < ε < 1, and showed that the solutions of the corresponding sigmoidal lattice system converge to a solution of the lattice inclusion system with the Heaviside functions when parameter ε → 0.
Delays are often included in neural field models to account for the transmission time of signals between neurons. Systems with delay has been investigated extensively, e.g., [4,14]. Here we are interested in autonomous versions of the above neural lattice systems considered by Han & Kloeden with the inclusion of delays: and the corresponding system with a sigmoidal function instead of the Heaviside function, i.e., d dt Wang, Kloeden & Yang [14] established the existence and uniqueness of solutions of the delay sigmoidal lattice system (3) as well as the existence of a global attractor.
The main goal of this paper is to show that the solutions and the attractors of the sigmoidal lattice system (3) converge to a solution and attractor of the lattice system with Heaviside function (2) when ε → 0. This also establishes the existence of a solution of the system with Heaviside function (2) for the same initial value. The paper is organized as follows. In section 2 some necessary preliminaries and assumptions are provided. Results from [14] for the sigmoidal lattice system (3) are recalled in section 3. The Heaviside lattice system (2) is formulated as differential inclusion with delay (7) and the inflated differential inclusion (10) is established in section 4. Then in section 5, after first giving some important Lemmas, the convergence of a subsequence of solutions of the delay sigmoidal solutions (3) to a solution of the differential inclusion (6) as ε → 0 is shown. Finally, in section 6, the existence of global attractors is established for the inflated and sigmoidal systems and these attractors are compared with that of the attractor for the Heaviside system.

2.
Assumptions. We follow Han & Kloeden [7,8] and consider a weighted space of bi-infinite real valued sequences with vectorial indices i = (i 1 , · · · , i d ) ∈ Z d . In particular, given a positive sequence of weights (ρ i ) i∈Z d , we consider the separable Hilbert space 2 ρ := u = (u i ) i∈Z d : with the inner product and norm We assume that the delays τ i,j are uniformly bounded, i.e., satisfy Assumption 1. There exists a constant h ∈ (0, ∞) that 0 ≤ τ i,j ≤ h for all i, j ∈ Z d and that the weights ρ i satisfy The components of the interconnection matrix (k i,j ) i,j∈Z d are assumed to satisfy In addition, the functions f i in the interconnected terms are assumed to satisfy Assumption 4. The functions f i : R → R are continuously differentiable with 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 Finally, we suppose that the constant forcing terms g := (g i ) i∈Z d satisfy By Assumptions 4 -6 and Lemma 1 in [7] the mapping f : 2 ρ , is locally Lipschitz and satisfies the dissipativity condition f (u), u ≤ −α u 2 ρ + β 2 ρ . It also follows from Assumptions 2 and 3 that 3. The delay sigmoidal lattice system. The appropriate function space for the solutions of the lattice systems with delays (2) and (3) is the Banach space For a continuous function u : . In this section we recall the result of [14] on the existence and uniqueness of the solution of the delay sigmoidal lattice system (3). To this end, we first define the mapping Σ ε i : and then we define the mapping Σ ε : Then we can write the delay sigmoidal lattice system (3) as the following differential delay equation on 2 ρ :

XIAOLI WANG, MEIHUA YANG AND PETER E. KLOEDEN
The coefficient functions satisfy the following properties.
Proof. For each i ∈ Z d , by inequality (4), Proof. First notice that the sigmoidal function σ ε is differentiable with a uniformly bounded derivative Hence, for each i ∈ Z d , the function Σ ε i is globally Lipschitz with the Lipschitz constant √ κ ε : It thus follows that the right hand side coefficient g ε is locally Lipschitz. It will be shown in the proof of Theorem 5.5 that it also satisfies the dissipativity condition. The following results were proved in Wang, Kloeden & Yang [14].
4. Formulation of the Heaviside system as a lattice inclusion. The lattice system (2) with the Heaviside functions needs to be reformulated mathematically as an inclusion system on the state space C([−h, 0]; 2 ρ ). This requires the set-valued mapping χ on R by We define the set-valued operator Then the delay lattice differential equation system (2) can be reformulated as the lattice differential delay inclusion system: Now define the set-valued operator K on C([−h, 0], 2 ρ ) by: Similarly to the proof of Lemma 3.1 it can be shown that K takes values in 2 ρ and is uniformly bounded. The system (6) can then be reformulated as a differential delay inclusion du(t) dt ∈ G(u t ) := f (u(t)) + K(u t ) + g on the space C([−h, 0], 2 ρ ). A solution to the differential delay inclusion (7) is defined component-wise as ρ is called a solution to the differential inclusion (7) if it is an absolutely continuous function u : Let T > 0 and suppose that Assumptions 1-7 hold. Then for any initial data ψ ∈ C([−h, 0], 2 ρ ), the differential delay inclusion (7) admits a solution u t (ψ) ∈ C([−h, 0], 2 ρ ) for [0, T ] with u 0 (ψ) = ψ. We will prove this theorem by showing that a sequence of solutions to the sigmoidal system has a convergent subsequence and its limit is a solution of (7). For this we will use the inflation of the differential delay inclusion (7). 4.1. Inflated inclusion systems. Inflated systems are used to compare perturbed or approximated systems with the original system, see Kloeden & Kozyakin [11,12]. In particular, we will see that the solutions of the delay sigmoidal system (3) are solutions of an inflated system based on the differential inclusion (6). We inflate the set-valued mapping χ to give a new set-valued function χ ε defined on R by Then we inflate the lattice inclusion (6) to obtain the inflated lattice inclusion where Now define the set-valued mapping (9). Then K ε (u t ) takes values in 2 ρ and the inflated delay inclusion can be written as the following differential delay inclusion on space 2 ρ : It is clear that for Thus the solutions of the delay sigmoidal system (3) are solutions of the inflated inclusion system (10).

5.
Convergence of the delay sigmoidal solutions. Our goal in this section is to show the convergence of a subsequence of solutions of the delay sigmoidal solutions (3) to a solution of the differential inclusion (6) as ε n → 0. We first give some important lemmas. where Proof. The proof uses the inequality for any nonempty compact subsets Recall from inequality (4) that j∈Z d |k i,j | ≤ √ κρ Σ . Hence for any γ > 0 there The result then follows.
. Then for each i ∈ Z d and N ∈ N, similarly to the proof of Lemma 5.2, we obtain ) and it follows the conclusion below.
Hence, for any δ > 0, there 5.2. The convergence result. With the above preparation, we are now ready to show the convergence of a subsequence of sigmoidal solutions (5) to a solution of the differential delay inclusion system (7) as ε → 0. Consider a sequence ε m → 0 as m → ∞. For any given ψ(·) ∈ C([−h, 0], 2 ρ ). Let u εm t (ψ) be the unique solution to the ε m -sigmoidal delay lattice system, i.e., its components satisfy the delay equation: Theorem 5.5. Let T > 0, ψ ∈ C([−h, 0], 2 ρ ) and suppose that the Assumptions 1 -7 hold. Then for any sequence u εm t (ψ) of the solutions of the sigmoidal lattice systems is a solution of the delay inclusion system (7) on the interval [0, T ].
Proof. The proof is divided into four parts.
I. Componentwise convergent subsequence. First we multiply the equation (15) with u εm i (t) and use Assumption 6 to obtain 1 2 where we have used inequality (4). Hence Integrating the differential inequality (17) then gives Replacing t with t + s, s ∈ (−h, 0), we have It follows for each s ∈ [−h, 0] that This gives the uniform boundedness and equi-continuity of the sequence u εm We have shown that the derivatives sequence is uniformly bounded, which we combine with the Lipschitz continuity of f i and sigmoidal function to show that the derivatives are equicontinuous on [-h, T]. From these, it follows that there exists a subsequence of derivatives converging to a function in the space C. Actually, we can easily obtain the limit derivative is d dt u * i (·). II. Convergent subsequence in C([−h, 0], 2 ρ ). Similar to the above, multiplying the equation (15) by ρ i u εm i (t) and summing over i ∈ Z d , we obtain Replacing t with t + s as part I, then Hence for each fixed i ∈ Z d , On the other hand, since u

IV. Limit as solution of the delay lattice inclusion.
Rearranging the sigmoidal differential equaion (15) for the above convergent subsequence {u εm k (·)} k∈Z we obtain From part I, u as k → ∞ for each i ∈ Z d . We need to show Σ * i (u * t ) ∈ K i (u * t ) for each i ∈ Z d and almost every t ∈ [0, T ]. To this end, we estimate Notice that ε m k → 0. Fix an arbitrary ε ∈ [0, 1], then we can assume without loss of generality that ε m k ≤ ε for all k ∈ N, Given any T > 0, .
In addition, (iii) → 0 due to the upper semi continuity of K ε i in Lemma 5.2 and (iv) → 0 by Lemma 5.3.
Furthermore, notice that the left side of the above inequality is independent of k and m, so

Finally, equation (19) can be rewritten as
is a selector of the delay differential inclusion (6). Therefore u * t = (u * i,t ) i∈Z d is a solution of the delay differential inclusion (6) in 6. Existence and comparison of global attractors. This section consists of two parts. In the first part it is shown that the lattice inclusion system generates a set-valued semi-dynamical system with compact values on the Hilbert space C([−h, 0]; 2 ρ ), while in the second part the existence of global attractors for the various systems and the relationship between them are established.
6.1. Set-valued dynamical systems with compact values. Autonomous setvalued semi-dynamical systems, frequently called set-valued semi-groups or general dynamical systems, see e.g., Szegö & Trecanni [13], are often generated by differential inclusions or differential equations without uniqueness, cf. [2]. The extensive theory for them was mainly developed on the locally compact state space R d , but much of it holds here with modifications in the Hilbert space 2 ρ , when the system takes compact values.
Let {v m } m∈N with v m = (v m,i ) i∈Z d be an arbitrary bounded sequence with values in the attainability set Φ(τ, ψ). Then for each m ∈ N there exists a solution u m,t (ψ) of the differential delay inclusion satisfying u m,0 (ψ) = ψ and u m,τ (ψ) = v m . For each solution u m,t (ψ) = (u m,i,t (ψ)) i∈Z d given above, define A subsequence {w Nm m,τ } m∈N of {w N m,τ } N ∈N can be constructed so that Denote η m,τ = w Nm m,τ . Then by the above uniform convergence and using the same arguments as in the proof of the existence theorem in [7], there exists a subsequence {η m l ,t } l∈N of {η m,t } m∈N and a solution η * t of the differential delay inclusion with η * 0 = ψ such that It follows immediately from (22) and (23) that We conclude that every sequence {v m } m∈N in Φ(τ, ψ) has a convergent subsequence It can be assumed with loss of generality that ψ) is not upper semi-continuous. Then there exist an ε 0 > 0 and sequences t m → t in R + and ψ m → ψ in C([−h, 0]; 2 ρ ) such that Since for each m ∈ N, the set Φ(t m , ψ m ) is compact, there exists v m ∈ Φ(t m , ψ m ) such that Since v m ∈ Φ(t m , ψ m ), there exists a solution u m,t of the differential inclusion (7) such that u m,0 = ψ m and u m,tm = v m .
Closely following the proof for the compactness above and using the fact that the estimates in the proof of Theorem 1 in [7] are uniformly bounded in the initial conditions ψ m ρ ) + 1, we can construct a sequence of "truncated solutions" η m,· = w Nm m,· such that The terms η m,· of this sequence all satisfy the same estimates for the u N · as in the proof of Theorem 1 in [7]. Thus by the same arguments there, there exist a subsequence {η m l ,· } l∈N and a solution η * · of the differential inclusion (7) such that Hence where we used the equi-continuity in t of the solutions of the differential delay inclusion and t m l → t. In addition, η * t (ψ) ∈ Φ t, ψ . This means that We know by (iv) above that t → Φ(t, ψ) is upper semi-continuous for each fixed ψ ∈ C([−h, 0]; 2 ρ ), so let us suppose that t → Φ(t, ψ) is not lower semi-continuous. Then there exist an ε 0 > 0 and a sequence t n → t in R + such that Since v n ∈ Φ t, ψ , there exists a solution u n of differential delay inclusion such that u n,0 = ψ, and u n,t = v n . We denote u n,tn withv n . It is obvious that v n ∈ Φ(t n , ψ). Hence By the compactness of Φ(t, ψ), there is a convergent subsequence v n →v ∈ Φ(t, ψ). Then there exists a (possibly further) subsequence t n → t ≥ 0 with either t n ≤ t or t n ≥ t for all n .
Without loss of generality, we assume the case t n ≥ t for all n . Φ(t n , ψ) = Φ(t n −t, Φ(t, ψ)). Pickv n ∈ Φ(t n , ψ) withv n ∈ Φ(t n −t, v n ), which is compact, so there exists a convergent subsequencev n → v * . In addition, Here we have used the equi-continuity in t of the solutions of the differential delay inclusion for the same initial value (which was proved in Part II of the proof of Theorem 5.5) and t n → t. It follows thatv = v * . Hence which is impossible. Thus the set-valued mapping t → Φ(t, ψ) is continuous in t ∈ R + with respect to the Hausdorff metric.
It also follows from in the same way as in the proof of Theorem 6.2 that ε-inflated lattice differential inclusion (10) generates a set-valued dynamical systems Φ ε with values in D(C([−h, 0], 2 ρ )) for every ε ∈ [0, 1].

Existence and comparison of attractors.
To show the existence of the attractor we first need to establish the existence of the absorbing set and to show that the dynamical system is asymptotic compact in an appropriate sense. We showed in part II of the proof of Theorem 5.5 that the solutions of the various systems satisfy the inequality This holds for solutions of the sigmoidal systems and the inflated systems as well as the original delay inclusion system since the the sigmoidal functions and the set-valued mappings χ ε and χ all take values in the unit interval. Hence the closed and bounded set is a positive invariant absorbing set for the inflated systems Φ ε , for every ε ∈ [0, 1]. The following definition is taken from [7] Definition 6.3. A set-valued semi-dynamical system Φ on a Banach space X is said to be asymptotically upper-semicompact in X if every sequence y n ∈ Φ(t n , x n ) has a convergent subsequence in X whenever t n → ∞ as n → ∞ and x n ∈ D, where D is an arbitrary bounded subset of X.
The asymptotic upper-semicompactness of the inflated set-valued semi-dynamical system Φ ε on C([−h, 0]; 2 ρ ) can be shown by a very similar argument for the lattice inclusions without delay used in [7] and for the sigmoidal systems with delays (5) in [14], so the details are omitted.
Proof. The inclusion A ⊂ A ε implies that dist C([−h,0]; 2 ρ ) (A, A ε ) = 0, for all ε > 0, so it remains to show that The global attractor A ε of the set-valued dynamical system Φ ε consists of entire trajectories, i.e., continuous functions R t → u ε t ∈ C([−h, 0]; 2 ρ ) such that u ε t ∈ Φ ε (t − s, u ε s ) for all s ≤ t in R, where u ε t ∈ A ε for all t ∈ R. Suppose that the convergence (29) does not hold. Then there are η 0 and ε n ≤ ε with ε n → 0 such that Moreover, since the A εn is compact there is an a n ∈ A εn such that dist C([−h,0]; 2 ρ ) (a n , A) = dist C([−h,0]; 2 ρ ) (A εn , A) ≥ 2η 0 for all n ∈ N.
Choose an arbitrary sequence of entire solutions u εn t of Φ εn with u εn 0 = a n . These are also entire solutions of Φ ε . Applying a similar argument to that used in the proof of the corresponding theorem without delays in [8], there exists a convergent subsequence u εn t converging uniformly on any closed and bounded time interval to a continuous function u * t : R → C([−h, 0]; 2 ρ ), which is also an entire trajectory of Φ. In particular, u * 0 ∈ A. Moreover, u εn r 0 = a nr → u * 0 , so dist C([−h,0]; 2 ρ ) (a nr , A) ≤ η 0 , which is a contradiction.
In addition, the following corollary holds.