Asymptotic behaviour of a neural field lattice model with delays

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):

where $\theta > 0$ is a given threshold and $H$ $:$ $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$ is the Heaviside function.

The continuum neural models may lose their validity in capturing detailed dynamics at discrete sites when the discrete structures of neural systems become dominant. Lattice models, e.g., [2,6,9,10], can used to describe dynamics at each 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

In this paper we consider the autonomous neural field lattice system with delays

Throughout this paper we assume that the delays $\tau_\mathfrak{j}>0$ are uniformly bounded, i.e., satisfy

Assumption 1. There exists a constant $h\in(0, \infty)$ that $0 \leq\tau_\mathfrak{i}\leq h$ for all $\mathfrak{i}\in\mathbb{Z}^d.$

and that the interconnection matrix $(k_{\mathfrak{i}, \mathfrak{j}})_{\mathfrak{i}, \mathfrak{j}\in\mathbb{Z}^d}$ satisfies

Assumption 2. $k_{\mathfrak{i}, \mathfrak{j}}\geq0$ for all $\mathfrak{i}, \mathfrak{j}\in\mathbb{Z}^d$ and there exists a constant $\kappa>0$ such that $\sum\limits_{\mathfrak{j}\in\mathbb{Z}^d}k_{i, j}$ $\leq$ $\kappa$ for each $\mathfrak{i} \in\mathbb{Z}^d$.

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 $\psi_\mathfrak{i}$.

2.   Preliminaries

We follow Han & Kloeden [7] and consider a weighted space of bi-infinite real valued sequences with vectorial indices $\mathfrak{i} = (i_1, \cdots, i_d)\in\mathbb{Z}^d.$

In particular, given a positive sequence of weights $(\rho_\mathfrak{i})_{\mathfrak{i}\in\mathbb{Z}^d}$, we consider the separable Hilbert space

with the inner product

and norm

We assume that the $\rho_\mathfrak{i}$ satisfy the following assumption.

Assumption 3. $\rho_\mathfrak{i}>0$ for all $\mathfrak{i}\in\mathbb{Z}^d$ and $\rho_\Sigma: = \sum\limits_{\mathfrak{i}\in\mathbb{Z}^d}\rho_\mathfrak{i}<\infty.$

The appropriate function space for the solutions of the lattice system with delays (1) is the Banach space $\mathcal{C}([-h, 0], \ell_\rho^2)$ of continuous functions by $\mathbf{v}$ $:$ $[-h, 0]$ $\rightarrow$ $\ell_\rho^2$ with the norm

For a solution $\mathbf{u}(t) = (u_\mathfrak{i}(t))_{\mathfrak{i}\in\mathbb{Z}^d}\in\ell_\rho^2$ of (1) we denote by $\mathbf{u}_t$ the segment of the solution in $\mathcal{C}([-h, 0], \ell_\rho^2)$ defined by $\mathbf{u}_t(s) = \mathbf{u}(t+s)$ for each $s\in[-h, 0].$ The corresponding initial condition (2) must then satisfy $(\psi_\mathfrak{i}(\cdot))_{\mathfrak{i}\in \mathbb{Z}^d}$ $\in$ $\mathcal{C}([-h, 0], \ell_\rho^2).$

2.1.   The reaction term

For any $\mathbf{u} = (u_\mathfrak{i})_{\mathfrak{i}\in\mathbb{Z}^d}\in \ell_\rho^2$, we define the operator $f$ by

To ensure that the $f(\mathbf{u})$ take values in $\ell_\rho^2$ for every $\mathbf{u}\in \ell_\rho^2$ and has necessary dissipative properties, we make the following standing assumptions on the $f_\mathfrak{i}$ throughout the rest of the paper.

Assumption 4. The functions $f_\mathfrak{i}:\mathbb{R}\rightarrow \mathbb{R}$ are continuously differential with weighted equi-locally bounded derivatives, i.e., there exists a non-decreasing function $L(\cdot)\in \mathcal{C}(\mathbb{R}^+, \mathbb{R}^+)$ such that

Assumption 5. $f_\mathfrak{i}(0) = 0$ for all $\mathfrak{i}\in\mathbb{Z}^d;$

Assumption 6. There exist constants $\alpha>0$ and $\beta_\mathfrak{i}$ with $\boldsymbol\beta = (\beta_\mathfrak{i})_{\mathfrak{i}\in\mathbb{Z}^d}\in \ell_\rho^2$ such that

It was shown in [7] that Assumption 4 implies that $f_\mathfrak{i}$ is locally Lipschitz with

Since

it follows

for every $\mathbf{u} = (u_\mathfrak{i})_{\mathfrak{i}\in\mathbb{Z}^d}$ and $\mathbf{v} = (v_\mathfrak{i})_{\mathfrak{i}\in\mathbb{Z}^d}$. 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 $f:\ell_\rho^2\rightarrow \ell_\rho^2$ is locally Lipschitz and satisfies the dissipativity condition

2.2.   The interaction term

For any $\mathbf{v}$ $\in$ $\mathcal{C}([-h, 0], \ell_\rho^2)$ we define the operator $K_\tau$ by $K_\tau(\mathbf{v}) = (K_{\tau, \mathfrak{i}}(\mathbf{v}))_{\mathfrak{i}\in\mathbb{Z}^d}$ by

Lemma 2.2. The operator $K_\tau$ maps $\mathcal{C}([-h, 0], \ell_\rho^2)$ to $\ell_\rho^2$.

Proof. The function $\sigma_ \varepsilon$ takes values in the unit interval [0, 1], so

Then

Remark 1. The function $\sigma_ \varepsilon$ is differentiable with a uniformly bounded derivative

Hence it is globally Lipschitz with the Lipschitz constant $L_\sigma = \frac{1}{ \varepsilon}$.

2.3.   The forcing term

Finally, we suppose that the constant forcing term $\mathbf{g}: = ( g_\mathfrak{i})_{\mathfrak{i}\in\mathbb{Z}^d}$ satisfies the following assumption.

Assumption 7. $\mathbf{g}$ $\in$ $\ell_\rho^2$.

3.   Existence and uniqueness of solutions

The lattice differential equation (1) can be rewritten as an infinitely dimensional ordinary differential equation on $\ell_\rho^2$,

where $G_\tau(t, \mathbf{u}_t): = (G_{\tau, \mathfrak{i}}(t, \mathbf{u}_t))_{\mathfrak{i}\in\mathbb{Z}^d}.$

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 $\mathbb{Z}_N^d$ $: =$$\{\mathfrak{i} = (i_1, i_2, \cdots, i_d)\in\mathbb{Z}^d:|i_1|, |i_2|, \cdots, |i_d|\leq N\}$ and define

Lemma 3.1. The mapping $\mathbf{v} \mapsto K_{\tau, \mathfrak{i}}^N(\mathbf{v})$ is continuous from $\mathcal{C}([-h, 0], \ell_\rho^2)$ to $\mathbb{R}$ for every $\mathfrak{i}\in\mathbb{Z}^d$.

Proof. Let $\mathbf{v}^n\rightarrow\mathbf{v}^0$ in $\mathcal{C}([-h, 0], \ell_\rho^2)$. Since Assumption 1: $0\leq\tau_\mathfrak{j}\leq h$ for each $\mathfrak{j}\in\mathbb{Z}^d$, we see that $(v^n(-\tau_\mathfrak{j}))_{\mathfrak{j}\in \mathbb{Z}^d}$ $\rightarrow$ $(v^0(-\tau_\mathfrak{j}))_{\mathfrak{j}\in \mathbb{Z}^d}$ in $\ell_\rho^2$. Thus for every $\varepsilon>0$ there exist an $M(\varepsilon)>0$ such that

Considering only the $\mathfrak{j}\in\mathbb{Z}_N^d$ appearing in the sum defining $K_{\tau, \mathfrak{i}}^N$, we obtain

where $\rho_N$ $: =$ $\min_{\mathfrak{j}\in\mathbb{Z}_N^d}\rho_\mathfrak{j}.$

The mapping $x\mapsto\sigma_ \varepsilon(x-\theta)$ is continuous for all $x\in\mathbb{R}$. Since there are a finite number of terms in the sum in the definition of $K_{\tau, \mathfrak{i}}^N$, it follows from the elementary inequality

that the mapping $\mathbf{v}\rightarrow K_{\tau, \mathfrak{i}}^N(\mathbf{v})$ is continuous.

3.1.   Existence of solutions

Theorem 3.2. Suppose that Assumptions 1–7 hold. Then for each $r>0$ there exists $a(r)>0$ such that for every $\boldsymbol\psi$ $\in$ $\mathcal{C}([-h, 0], \ell_\rho^2)$ satisfying $\|\boldsymbol\psi\|_{\mathcal{C}([-h, 0], \ell_\rho^2)}$ $\leq$ $r$, the lattice delay equation (3) has at least one solution defined on $[0, a(r)]$. Moreover, the solution $u(\cdot)$ $\in$ $\mathcal{C}^1([0, a], \ell_\rho^2)$.

Proof. Step 1. First, we claim that $G_\tau(t, \mathbf{u}_t)$ is well defined and bounded.

It is easy to see that $G_\tau(t, \mathbf{u}_t)$ is well defined since $f(\mathbf{u})$, $K_\tau(\mathbf{u}_t)$ and $\mathbf{g}$ are all well defined. As for the boundedness, we denote that

Since $f_\mathfrak{i}$ is locally Lipschitz and satisfies $f_\mathfrak{i}(0)$ $=$ $0$ by Assumption 4-5, we see that

Then we obtain

For the second term with delay, we have $|K_{\tau, \mathfrak{i}}(\mathbf{u}_t)|\leq\kappa$ by Assumption 2, which gives

where we have used Assumption 3.

Finally, for the last term $\mathbf{g}$, Assumption 7 gives

Using (5), (6) and (4) in (4) we conclude that $G_\tau$ is well defined and bounded.

Step 2. Next, we claim that the maps $G_{\tau, \mathfrak{i}}:\mathcal{C}([-h, 0], \ell_\rho^2)$ $\rightarrow$ $\mathbb{R}$ are continuous for all $\mathfrak{i}\in\mathbb{Z}^d$.

We consider $\{\mathbf{u}_t^n\}_{n\in\mathbb{N}}$ $\subset$ $\mathcal{C}([-h, 0], \ell_\rho^2)$ and $\mathbf{u}_t^0$ $\in$ $\mathcal{C}([-h, 0], \ell_\rho^2)$ such that $\mathbf{u}_t^n$ $\rightarrow$$\mathbf{u}_t^0$ in $\mathcal{C}([-h, 0], \ell_\rho^2)$. Then

By the local Lipschitz continuity of $f_\mathfrak{i}$,

which shows that this term converges to zero.

Next for the second term on the right-hand side

On one hand, since $\sum\limits_{\mathfrak{j}\in\mathbb{Z}^d} k_{\mathfrak{i}, \mathfrak{j}}$ $\leq$ $\kappa$, for every $\varepsilon>0$, there exists a $N_1(\varepsilon)$ such that $\sum\limits_{\mathfrak{j}\in\mathbb{Z}^d\backslash\mathbb{Z}_N^d} k_{\mathfrak{i}, \mathfrak{j}}$ $\leq$ $4\varepsilon$ when $N\geq N_1( \varepsilon)$: we assume that the $N$ in (10) is such an $N$. On the other hand, $\sum\limits_{\mathfrak{j}\in\mathbb{Z}_N^d} k_{\mathfrak{i}, \mathfrak{j}}|u_{\mathfrak{j}}^n(t-\tau_\mathfrak{j})-u_{\mathfrak{j}}^0(t-\tau_\mathfrak{j})|$ $<$ $\varepsilon$ when $n>M(\varepsilon)$ by Lemma 3.1 for all $N$. Thus $K_{\tau, \mathfrak{i}}$ is continuous.

Using (9) and (10) in (8), we complete the proof of the claim.

Having the two steps above, by Theorem 10 in Caraballo et al. [3], for each $r>0$ there exists $a(r)>0$ such that if $\boldsymbol\psi\in\mathcal{C}([-h, 0], \ell_\rho^2)$ and $\|\boldsymbol\psi\|_{\mathcal{C}([-h, 0], \ell_\rho^2)}\leq r$, then the problem (3) has at least one solution defined on $[0, a(r)]$.

Step 3. Finally, we claim the following inequality holds:

where $b_K\rightarrow0^+$ as $K\rightarrow\infty$, and $C(\cdot)>0$ is a continuous non-decreasing function.

The proof is as follows.

By Corollary 13 in [3], we also conclude that the solution $\mathbf{u}(\cdot)$ $\in$ $\mathcal{C}^1([0, a], \ell_\rho^2)$.

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.

Proposition 1. Suppose that Assumptions 1–7 hold. Then every solution $\mathbf{u}(\cdot)$ of (3) with $\mathbf{u}_0$ $=$ $\boldsymbol\psi$ $\in$ $\mathcal{C}([-h, 0], \ell_\rho^2)$ verifies

where $R_j$ $>$ $0$, $j = 1, 2,$ are constants depending on the parameters of the problem.

Proof. We multiply the $\mathfrak{i}$th component of (1) by $\rho_\mathfrak{i} u_\mathfrak{i}(t)$ and sum over $\mathfrak{i}$ to obtain

By Assumption 6 and $\rho_\mathfrak{i}>0$ we have

so

Since function $\sigma_ \varepsilon$ takes values in the unit interval, using Young's inequality we obtain

so

The last term on the right hand of (12) satisfies

In summary, collecting the inequalities above, we obtain

Integrating both sides of this differential inequality yields

Let $\theta\in[-h, 0]$. Replacing $t$ by $t+\theta$ in (13) and using

we obtain

Finally, using that $\theta\in[-h, 0]$ and neglecting the negative terms yields

where

3.3.   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

Assumption 8. There exists a constant $\tilde{\kappa}>0$ such $\sum\limits_{\mathfrak{j}\in\mathbb{Z}^d}\frac{k_{i, j}^2}{\rho_j}\leq \tilde{\kappa}$ for each $\mathfrak{i} \in\mathbb{Z}^d$.

Lemma 3.3. Suppose that Assumptions 1–8 hold. Then the solution $\mathbf{u}$ of problem (3) is unique.

Proof. Assumption 8 implies that the operator $K_\tau:\mathcal{C}([-h, 0], \ell_\rho^2)\rightarrow \ell_\rho^2$ is Lipschitz. In fact,

Hence, suppose that we have two different solutions $\mathbf{u}$, $\mathbf{v}$ of problem (3) with the same initial condition $\mathbf{u}(s) = \mathbf{v}(s) = \boldsymbol\psi(s)$, $\forall s\in[-h, 0]$.

Set $\mathbf{w} = \mathbf{u}-\mathbf{v}$, we obtain that

where $C = \max\Big\{{{L\big(2\sqrt{\rho_\Sigma(R_1\|\boldsymbol\psi\|_{\mathcal{C}([-h, 0], \ell_\rho^2)}^2+R_2)}\big)}}, \sqrt{{{C_1}}\rho_{\Sigma}\tilde{\kappa}}L_\sigma\Big\}$.

Integrating from 0 to t then gives

Let $\theta\in[-h, 0]$. Replacing t by $t+\theta$ in the inequality above and using $\|\mathbf{w}(t+\theta)\|_\rho = 0$ when $t+\theta<0.$ We obtain

Then take the supremum on $\theta$,

By Gronwall's inequality, we have

Since $\mathbf{w}(0) = 0$, we obtain that $\mathbf{w}\equiv 0.$

The proof of the next corollary follows easily using (15).

Corollary 1. The map $(t, \boldsymbol\psi)\mapsto \mathbf{u}_t$ is continuous.

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.

4.   Asymptotic behaviour

When Assumptions 1–8 hold, Theorem 3.2 and Lemma 3.3 ensure the local existence and uniqueness of solutions of the delayed lattice system (3), while Proposition 1 shows that the solutions are, in fact, globally defined.

We can thus define a semigroup of operators $\mathcal{S}$ $:$ $\mathbb{R}^+\times\mathcal{C}([-h, 0], \ell_\rho^2)$ $\rightarrow$ $\mathcal{C}([-h, 0], \ell_\rho^2)$ by

where $\mathbf{u}_t$ is the unique solution to (3) with $\mathbf{u}_0 = \boldsymbol\psi$. The semigroup map $\mathcal{S}$ 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

with $R_0: = \sqrt{1+R_2}$, is absorbing for the semigroup $\mathcal{S}$.

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 $x\rightarrow \mathcal{S}(t, x)$ be continuous for any $t\geq0$. Assume that $\mathcal{S}$ is asymptotically compact and possesses a bounded absorbing set $B_0$. Then there exists a global compact attractor $\mathcal{A}$, which is the minimal closed set attracting any bounded set. If, moreover, the space $X$ is connected and the map $t\rightarrow \mathcal{S}(t, x)$ is continuous for any $x\in X$, then the set $\mathcal{A}$ is connected.

4.1.   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].

Lemma 4.2. Suppose that Assumptions 1–8 hold and let $B$ be a bounded set of $\mathcal{C}([-h, 0], \ell_\rho^2)$. Then, for any $\varepsilon>0$ there exist $T( \varepsilon, B)$ and $M( \varepsilon, B)$ such that

for any initial condition $\boldsymbol\psi\in B$ and the corresponding solution $\mathbf{u}(\cdot)$ of (3) with $\mathbf{u}_0$ $=$ $\boldsymbol\psi$.

Proof. Define a smooth function $\xi$ satisfying

Let $M$ be a fixed (and large) integer to be specified later, and set

where $|\cdot|$ denotes the Euclidean norm. We multiply the $\mathfrak{i}$th component of (1) by $\rho_\mathfrak{i} v_\mathfrak{i}$, then summing over $\mathfrak{i}$ $\in$ $\mathbb{Z}^d$, and since $\mathbf{u}(\cdot)$ $\in$ $\mathcal{C}^1([0, \infty], \ell_\rho^2)$, we have

First, by Assumption 6,

Then, since function $\sigma_ \varepsilon$ 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 $\boldsymbol\beta$ $=$ $(\beta_\mathfrak{i} )_{\mathfrak{i}\in\mathbb{Z}^d}$ $\in$ $\ell_\rho^2$, then for every $\varepsilon>0$ there exists $I_1( \varepsilon)>0$ such that

Similarly, since $\rho_\Sigma$ $=$ $\sum\limits_{\mathfrak{i}\in\mathbb{Z}^d}\rho_\mathfrak{i}$ $<$ $\infty$, then for every $\varepsilon>0$, there exists $I_2( \varepsilon)>0$ such that

In addition, since $\mathbf{g}$ $=$ $( g_\mathfrak{i} )_{\mathfrak{i}\in\mathbb{Z}^d}$ $\in$ $\ell_\rho^2$ by Assumption 7, for every $\varepsilon>0$ there exists $I_3( \varepsilon )>0$ such that

Finally, for any $\varepsilon>0$, choosing $I( \varepsilon ): = \max\{I_1( \varepsilon), I_2( \varepsilon), I_3( \varepsilon )\}$, inserting the estimations (21), (22) and (23) into (20) results in

It follows immediately from Gronwall's lemma that

In a similar way as in Proposition 1 we have

Thus, there exist $T(\varepsilon, B)$ and $M(\varepsilon, B)$ such that

4.2.   Existence of the global attractor

In order to apply Theorem 4.1, we need to prove that $\mathcal{S}$ generated by the delay lattice system (3) is asymptotically compact.

Lemma 4.3. Suppose that Assumptions 1–8 hold. Then the semigroup $\mathcal{S}$ is asymptotically compact.

Proof. We consider $\boldsymbol\xi^n$ $: =$ $\mathbf{u}_{t_n}^n$ $=$ $\mathcal{S}(t_n, \boldsymbol\psi^n)$, where $\boldsymbol\psi^n$ $\in$ $B$, a bounded set in $\mathcal{C}([-h, 0], \ell_\rho^2)$. From (14) there is a $C>0$ such that

For fixed $s\in[-h, 0]$ we can find a subsequence (which we still denote by $\mathbf{u}^n$) such that

In fact, the weak convergence here is strong, which follows from Lemma 4.2. Indeed, there exists $N_1>0$, when $n\geq N_1$, we have $t_n>T$ (where $T$ is the constant in Lemma 4.2). Moreover, for any $\mu>0$ there exist $K_2(\mu)$ and $N_2(\mu)$ such that

if $n\geq\max\{N_1, N_2(\mu)\}$. Hence

Thus, $\{\mathbf{u}^n(t_n+s)\}$ is precompact in $\ell_\rho^2$ for any $s\in[-h, 0].$ Since $G_\tau$ is a bounded map, Proposition 1 and the integral representation of solutions imply that

Then, the Ascoli-Arzelà theorem implies that $\boldsymbol\xi^n$ is relatively compact in $\mathcal{C}([-h, 0], \ell_\rho^2)$.

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.