Periodic measures for a neural field lattice model with state dependent superlinear noise

1.   Introduction

The objective of this paper is to investigate the existence of periodic measures for a neural field lattice model with state dependent superlinear noise on $\mathbb{Z}^{N}$ with $N\in\mathbb{N}$:

where $i = (i_{1}, i_{2}, \cdots, i_{N})\in\mathbb{Z}^{N}$, $t > 0$, and $\alpha > 0$. The variable $u_{i}$ represents the neural activity, specifically the synaptic activity of the $i$th node. The function $f_{i}:\mathbb{R}\rightarrow \mathbb{R}$ describes the continuous differentiability of neural activity attenuation for the $i$th node, and $\xi_{i, j}:\mathbb{R}\rightarrow \mathbb{R}$ is an activation function that determines a node's output based on its input. The quantity $k_{i, j}$ represents the synaptic strength from the $j$th to the $i$th node, which can have positive or negative values indicating excitation or inhibition of the $j$th neuron on the $i$th neuron, respectively. The time-dependent functions $g_{i}:\mathbb{R}\rightarrow \mathbb{R}$ and $h_{i}:\mathbb{R}\rightarrow \mathbb{R}$ describe the external forcing for drift and diffusion at the $i$th location, respectively. This is represented by a sequence of mutually independent two-sided real-valued Wiener processes $(W_{i}(t))_{i\in\mathbb{Z}^{N}}$, defining on a complete filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_{t}\}_{t\in\mathbb{R}}, P)$, and each Wiener process $W_{i}$ is associated with a superlinear state-dependent function $\lambda_{i}:\mathbb{R}\rightarrow \mathbb{R}$ within its coefficient.

Differential equations and dynamical systems play a crucial role in the mathematical modeling and analysis of aircraft design, aerospace engineering, materials science, biology, medical engineering, financial engineering, and securities markets, as well as mobile communications, aquatic communications, and other related fields. The field of dynamical systems have witnessed significant achievements by numerous esteemed scholars. Among them, the investigation of traveling wave solutions for such equations have been conducted by ^[1,2]. The examination of chaotic properties in the solutions have been carried out by ^[3,4]. The existence and uniqueness of solutions and the existence, uniqueness, and upper semi-continuity of attractors have been studied by ^[5,6,7]. Additionally, Li et al. conducted an investigation on inverse problems in predator-prey models in ^[8], while Yin et al. explored a neural network approach for the inversion of turbulence strength in ^[9]. These studies on inverse problems in mathematical physics have generated a significant impact within academic circles.

The lattice systems are commonly derived from spatial discretizations of partial differential equations. For the asymptotic behavior analysis of lattice systems, we refer readers to ^{[10,11,12,13]}. Amongst various applications, neural lattice models arising from neural networks have recently gained significant attention. These models can be broadly classified into two types: one is developed as the discretization of continuous neural field models, known as neural field lattice systems; and the other is derived as the limit of finite dimensional discrete neural networks when their sizes become increasingly large. Recent studies on neural lattice models include Faye's investigation on traveling fronts for a class of lattice neural field equations ^[14], Han and Kloeden's exploration of long-term dynamics for neural field lattice models ^[15], along with Han et al.'s examination of long-term dynamics for Hopfield-type neural lattice models ^[16], in addition to Wang et al.'s work ^[17]. Recently, Wang et al. conducted a study on the existence of weak pullback mean random attractors and invariant measures for a neural lattice model with state-dependent nonlinear noise in their work ^[18]. In addition, Caraballo et al. investigated the convergence and approximation of invariant measures for neural field lattice models under noise perturbation in their publication ^[19].

Currently, extensive research has been conducted on the dynamical behavior of differential equations driven by linear noise. To effectively handle stochastic systems with nonlinear noise, Kloeden^[20] and Wang^[21] introduced the concept of weak pullback mean random attractors. Subsequently, this concept has been widely applied in numerous studies on stochastic systems by various scholars in ^{[22,23,24,25]}. The periodic measures of stochastic differential equations have been extensively investigated by numerous experts, as documented in ^[26,27] and the references therein. Specifically, the existence and limiting behavior of periodic measures for the stochastic reaction-diffusion lattice system were examined in ^[26], considering both globally Lipschitz continuous nonlinear drift and diffusion terms. However, to the best of our knowledge, the current literature on periodic measures for the neural field lattice model with state-dependent superlinear noise is unfortunately lacking. The existence of periodic measures for the lattice systems (1.1) in $\ell^{2}$ is established through the ingenious Krylov-Bogolyubov's method, which showcases the brilliance of mathematical prowess. By employing the concept of uniform estimates on the tails of solutions, we successfully establish the tightness of a family of distribution laws of the solutions.

The paper is structured as follows: Section 2 introduces the notations and discusses the well-posedness of system (1.1). The subsequent section establishes essential uniform estimates of solutions, which play a pivotal role in demonstrating the main findings in the following section. The primary focus of Section 4 is to investigate the existence of periodic measures for system (1.1) in space $\ell^{2}$.

2.   Preliminaries

In this section, we will investigate the well-posedness of the stochastic neural field lattice system (1.1). We begin with the following Banach space $\ell^{r}$, where $\ell^{r}$ is defined by

$\ell^{r} = \Big\{u = (u_{i})_{i\in\mathbb{Z}^{N}}|u_{i}\in \mathbb{R}, \mathop{\sum}\limits_{i\in\mathbb{Z}^{N}}|u_{i}|^{r} < \infty\Big\}$ with norm $\|u\|_{r}^{r} = \mathop{\sum}\limits_{i\in\mathbb{Z}^{N}}|u_{i}|^{r}$ if $1\leq r < \infty$,

$\ell^{\infty} = \Big\{u = (u_{i})_{i\in\mathbb{Z}^{N}}|u_{i}\in \mathbb{R}, \mathop{\sup}\limits_{i\in\mathbb{Z}^{N}}|u_{i}| < \infty\Big\}$ with norm $\|u\|_{\ell^{\infty}} = \mathop{\sup}\limits_{i\in\mathbb{Z}^{N}}|u_{i}|$ if $r = \infty$.

Particularly, $\ell^{2}$ is a Hilbert space with the inner product and norm given by

For the nonlinear drift function $f_{i}\in C^{1}(\mathbb{R}, \mathbb{R})$ in system (1.1), we assume that for all $z\in\mathbb{R}$ and $i\in\mathbb{Z}^{N}$,

where $p > 2$ and $\gamma > 0$ are constants. For the sequence of continuously differentiable diffusion function $\lambda_{i}$, we assume that for every $z\in\mathbb{R}$ and $i\in\mathbb{Z}^{N}$,

where $q\in[2, p)$ is a constant. Moreover, we assume that there exists a constant $\kappa > 0$ such that

For $i, j\in\mathbb{Z}^{N}$ and $z\in\mathbb{R}$, we assume that the activation function $\xi_{i, j}$ is globally Lipschitz continuous with Lipchitz constant $L_{1}$, and there exist $a_{i, j}\in\mathbb{R}$ and $b_{i, j} > 0$ such that

In addition, we assume

Suppose $G(t), H(t):\mathbb{R}\rightarrow \ell^{2}$, $G(t) = (g_{i}(t))_{i\in\mathbb{Z}^{N}}$, $H(t) = (h_{i}(t))_{i\in\mathbb{Z}^{N}}$ are both continuous in $t\in\mathbb{R}$, which shows that for $t\in\mathbb{R}$,

In order to investigate the periodic measures of system (1.1), we assume that all given time-dependent functions are $T$-periodic in $t\in\mathbb{R}$ for some $T > 0$; that is, for all $t\in\mathbb{R}$,

If $\chi:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous $T$-periodic function, we denote

For all $u = (u_{i})_{i\in\mathbb{Z}^{N}}\in\ell^{2}$, define the operator $F$, $\Lambda$, and $\Xi$ by

By (2.3), we get that there exists $\theta_{1}\in(0, 1)$ such that for $p > 2$ and $u, v\in\ell^{2}$,

which along with $F(0)\in\ell^{2}$ and according to (2.2) implies $F(u)\in\ell^{2}$ for all $u\in\ell^{2}$. Then, $F:\ell^{2}\rightarrow \ell^{2}$ is well-defined. In addition, it follows from (2.11) that $F:\ell^{2}\rightarrow \ell^{2}$ is a locally Lipschitz continuous function; that is, for every $c\in\mathbb{N}$, there exists a constant $L_{2}(c) > 0$ such that for all $u, v \in\ell^{2}$ with $\|u\|\leq c$ and $\|v\|\leq c$,

For $u\in\ell^{2}$, by (2.6), (2.7), and (2.10), we have

In addition, for all $u, v\in\ell^{2}$, it follows from the globally Lipschitz continuity of $\xi_{i, j}$, Cauchy's inequality, and (2.6) that

which, along with (2.13), implies that $\Xi(u)$ belongs to $\ell^{2}$ and is a globally Lipschitz continuous function.

In order to rewrite the terms $\lambda_{i}(u_{i}) + h_{i}(t)$ as vectors in $\ell^{2}$, define two sequence of operators $\Lambda_{i}$ and $H_{i}$ by

where $e^{i}$ represents the infinite sequence with a value of 1 at position $i$ and a value of 0 elsewhere. Then, we can get that $\Lambda(u) = \mathop{\sum}\limits_{i\in\mathbb{Z}^{N}}\Lambda_{i}(u)$ and $H(t) = \mathop{\sum}\limits_{i\in\mathbb{Z}^{N}}H_{i}(t)$ for every $u\in \ell^{2}$ and

For $q\in[2, p)$ and $u\in\ell^{2}$, we can get from (2.4) and Young's inequality that

where $\gamma$ is the same number in (2.1). By (2.16) and $\ell^{2}\subseteq\ell^{p}$, we get that $\Lambda(u)\in\ell^{2}$ for all $u\in\ell^{2}$ and $p > 2$. Then, $\Lambda(u):\ell^{2}\rightarrow \ell^{2}$ is also well-defined. By (2.5), we get that there exists $\theta_{2}\in(0, 1)$ such that for $q\in[2, p)$ and $u, v\in \ell^{2}$,

which shows that $\Lambda:\ell^{2}\rightarrow \ell^{2}$ is a locally Lipschitz continuous function; that is, for every $c\in\mathbb{N}$, we can find a constant $L_{3}(c) > 0$ such that for all $u, v \in\ell^{2}$ with $\|u\|\leq c$ and $\|v\|\leq c$,

By the above notation, system (1.1) can be rewritten as follows: For all $t > 0$,

Let $u_{0}\in L^{2}(\Omega, \ell^{2})$ be $\mathcal{F}_{0}$-measurable. A continuous $\ell^{2}$-valued $\mathcal{F}_{t}$-adapted stochastic process $u(t)$ is called a solution of system (2.19) if $u(t) \in L^{2}(\Omega, C([0, T], \ell^{2}))\bigcap L^{p}(\Omega, L^{p}(0, T, \ell^{p}))$ for all $T > 0$, $t\geq0$ and for almost all $\omega\in\Omega$,

By (2.1)–(2.9) and the theory of functional differential equations, we can get that for any $u_{0}\in L^{2}(\Omega, \ell^{2})$, system (2.19) has local solutions $u(t) \in L^{2}(\Omega, C([0, T], \ell^{2}))\bigcap L^{p}(\Omega, L^{p}(0, T, \ell^{p}))$ for every $T > 0$. Moreover, similar to ^[24], we can get that the local solutions to system (2.19) are also global.

3.   Uniform estimates

In this section, we establish uniform estimates for the solutions to system (2.19), which play a crucial role in proving the existence of periodic measures. Specifically, we will demonstrate the compactness of a family of probability distributions related to $u(t)$ in $\ell^{2}$.

Lemma 3.1. Suppose (2.1)–(2.9) hold. Let $u_{0}\in L^{2}(\Omega, \ell^{2})$ be the initial data of system (2.19), then the solution $u(t, 0, u_{0})$ of the system (2.19) satisfies

where $M_{1}$ is a positive constant independent of $u_{0}$.

Proof. By (2.19) and Itô's formula, we get that for all $t\geq0$,

Taking the expectation, we obtain that for $t\geq0$,

By (2.1), we have

By (2.7) and Young's inequality, we get

Note that

By (2.4), we obtain

It follows from (3.3)–(3.7) and (2.8) that

which implies that for $t\geq0$,

where $C_{1} = 2\|\phi_{1}\|_{1}+\frac{p-q}{p}\Big(\frac{p\gamma}{2q}\Big)^{-\frac{q}{p-q}}4^{\frac{p}{p-q}}\|\varphi_{1}\|^{\frac{2p}{p-q}}_{\frac{2p}{p-q}} +4\|\varphi_{2}|^{2}+\frac{8\kappa\|b\|^{2}}{\alpha}+2\|\bar{H}\|^{2}+\frac{4}{\alpha}\|\bar{G}\|^{2}$. This completes the proof. □

The subsequent step involves obtaining uniform estimates on the tails of solutions to the stochastic lattice system (2.19).

Lemma 3.2. Suppose (2.1)–(2.9) hold. For compact subset $\mathcal{K}\in \ell^{2}$, there is a number $N_{0} = N_{0}(\mathcal{K})\in \mathbb{N}$ such that the solution $u(t, 0, u_{0})$ of the system (2.19) satisfies, for all $n\geq N_{0}$ and $t\geq0$,

where $u_{0}\in \mathcal{K}$ and $\|i\|: = \max_{1\leq j\leq N}|i_{j}|$.

Proof. Let $\vartheta$ be a smooth function which is defined on $\mathbb{R}$ such that $0\leq\vartheta(z)\leq1$ for all $z\in\mathbb{R}$, and

For $n\in N$, set $\vartheta_{n} = \Big(\vartheta\Big(\frac{\|i\|}{n}\Big)\Big)_{i\in\mathbb{Z}^{N}}$ and $\vartheta_{n}u = \Big(\vartheta\Big(\frac{\|i\|}{n}\Big)u_{i}\Big)_{i\in\mathbb{Z}^{N}}$. By (2.19), we have

which along with Itô's formula implies that

Then, we get that for all $t\geq0$,

By (2.1), we find

By (2.7) and Young's inequality, we have

Note that

For the last term of (3.12), by (2.4), we get

It follows from (3.12)–(3.16) and (2.8) that

which implies that

Since $\mathcal{K}$ is a compact subset of $\ell^{2}$, we get that

By $\phi_{1}\in\ell^{1}$, $\varphi_{1}\in\ell^{\frac{2p}{p-q}}$, $\varphi_{2}\in\ell^{2}$, (2.7), and (2.9), we infer that

It follows from (3.17)–(3.19) that as $n\rightarrow \infty$,

Then, for every $\varepsilon > 0$, we can find that there exists a number $N_{0} = N_{0}(\mathcal{K})\in\mathbb{N}$ such that for all $n\geq N_{0}$ and $t\geq0$,

uniformly for $u_{0}\in\mathcal{K}$ and $t\geq0$. This concludes the proof. □

Remark 1. In order to establish the existence of periodic measures for stochastic lattice system (2.19), the main challenge lies in deriving the tightness of a family of probability distributions of solutions. Our approach involves approximating solutions in $\ell^{2}$ using finite-dimensional methods. In order to achieve this, it is necessary to establish uniformly small estimates for the "tail ends" of these solutions for $t\geq0$ as stated in Lemma 3.2. For further elaboration on cutoff techniques related to estimating the "tail ends", please refer to ^[13,28].

4.   Existence of periodic measures

The primary focus of this section is to establish the existence of periodic measures for the lattice system (2.19) in $\ell^{2}$. First, we introduce the transition operators associated with the lattice system and subsequently provide evidence for the convergence and compactness properties exhibited by a family of probability distributions representing solutions to this particular lattice system.

Suppose $\psi:\ell^{2}\rightarrow \mathbb{R}$ is a bounded Borel function. For $0\leq r\leq t$, we set

In addition, for $G\in\mathcal{B}(\ell^{2})$, $0\leq r\leq t$, and $u_{0}\in \ell^{2}$, we set

where $1_{G}$ is the characteristic function of $G$. Then, we can get that $p(r, u_{0};t, \cdot)$ is the probability distribution of $u(t)$ in $\ell^{2}$. Furthermore, the transition operator $p_{0, t}$ is denoted as $p_{t}$ for the sake of convenience.

Definition 4.1. A probability measure $\mu$ on $\ell^{2}$ is called a periodic measure of lattice system (2.19) if

Now, we show the properties of transition operators $\{p_{r, t}\}_{0\leq r\leq t}$ as follows.

Lemma 4.1. Suppose (2.1)–(2.9) hold. Then, we have

(i) The family $\{p_{r, t}\}_{0\leq r\leq t}$ is Feller; that is, if $\psi:\ell^{2}\rightarrow \mathbb{R}$ is bounded and continuous, then $p_{r, t}\psi:\ell^{2}\rightarrow \mathbb{R}$ is bounded and continuous.

(ii) The family $\{p_{r, t}\}_{0\leq r\leq t}$ is T-periodic; that is,

(iii) $\{u(t, 0, u_{0})\}_{t\geq 0}$ is a $\ell^{2}$-value Markov process.

Lemma 4.2. Suppose (2.1)–(2.9) hold. Then, the family $\{\mathcal{L}(u(t, 0, u_{0})):t\geq0\}$ of the distribution laws of the solutions to system (2.19) is tight on $\ell^{2}$.

Proof. For all $t\geq0$, by Lemma 3.1 and Chebyshev's inequality, we get that there exists a constant $c_{1} > 0$ such that

Then, for each $\varepsilon > 0$, there exists a constant $R_{1} = R_{1}(\varepsilon) > 0$ such that

By Lemma 3.2, we obtain that for every $\varepsilon > 0$ and $m\in\mathbb{N}$, there exists an integer $n_{m} = n_{m}(\varepsilon, m)\geq1$ such that

Then, for all $t\geq0$ and $m\in\mathbb{N}$, we get

which shows that for all $t\geq0$,

For $\varepsilon > 0$, set $\mathcal{Z}_{\varepsilon} = \mathcal{Z}_{1, \varepsilon}\bigcap\mathcal{Z}_{2, \varepsilon}$, where

It follows from (4.1) and (4.3) that for all $t\geq0$,

Given $\epsilon > 0$, choose an interger $m_{0} = m_{0}(\epsilon)\in\mathbb{N}$ such that $2^{m_{0}} > \frac{8}{\epsilon^{2}}$. Then, by (4.5), we get that for all $v\in\mathcal{Z}_{\varepsilon}$,

The set $\{(v_{i})_{\|i\|\leq m_{0}}:v\in\mathcal{Z}_{\varepsilon}\}$ is bounded in the finite-dimensional space $R^{2m_{0}+1}$ as shown by (4.4), and therefore is pre-compact. As a result, $\{v:v\in\mathcal{Z}_{\varepsilon}\}$ has a finite open cover of balls with radius $\frac{\epsilon}{2}$, which combined with (4.7) implies that the set $\{v:v\in\mathcal{Z}_{\varepsilon}\}$ has a finite open cover of balls with radius $\epsilon$ in $\ell^{2}$. Since $\epsilon > 0$ can be chosen arbitrarily, the set $\{v:v\in\mathcal{Z}_{\varepsilon}\}$ is pre-compact in $\ell^{2}$. This completes the proof. □

Now, the main outcome of this paper has been proved by Krylov-Bogolyubov's method.

Theorem 4.1. Suppose (2.1)–(2.9) hold. Then, system (2.19) has a periodic measure on $\ell^{2}$.

Proof. For each $n\in\mathbb{N}$, the probability measure $\mu_{n}$ is defined by

It follows from Lemma 4.2 that the sequence $(\mu_{n})_{n = 1}^{\infty}$ is tight in $\ell^{2}$. Consequently, there exist a probability measure $\mu$ on $\ell^{2}$ and a subsequence (still denoted by $(\mu_{n})_{n = 1}^{\infty}$) such that

By (4.8)–(4.9) and Lemma 4.1, we can get that for every $t\geq0$ and every bounded and continuous function $\psi:\ell^{2}\rightarrow \mathbb{R}$,

which implies that $\mu$ is a periodic measure of system (2.19). This completes the proof. □

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

This work was supported by the Scientific Research and Cultivation Project of Liupanshui Normal University (No. LPSSY2023KJYBPY14).

Conflict of interest

The authors declare there is no conflict of interest.