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 delay differential equation on weighted sequence state space
Citation: Xiaoli Wang, Peter Kloeden, Meihua Yang. Asymptotic behaviour of a neural field lattice model with delays[J]. Electronic Research Archive, 2020, 28(2): 1037-1048. doi: 10.3934/era.2020056
[1] | Xiaoli Wang, Peter Kloeden, Meihua Yang . Asymptotic behaviour of a neural field lattice model with delays. Electronic Research Archive, 2020, 28(2): 1037-1048. doi: 10.3934/era.2020056 |
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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 delay differential equation on weighted sequence state space
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):
∂tu(t,x)=−u(t,x)+∫ΩK(x−y)H(u(t,y)−θ)dy,x∈Ω⊂R, |
where
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:
ddtui(t)=fi(ui(t))+∑j∈Zdki,jH(uj(t)−θ)+gi(t),i∈Zd. |
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
σε(x)=11+e−x/ε,x∈R,0<ε<1. |
In this paper we consider the autonomous neural field lattice system with delays
ddtui(t)=fi(ui(t))+∑j∈Zdki,jσε(uj(t−τj)−θ)+gi,i∈Zd. | (1) |
Throughout this paper we assume that the delays
Assumption 1. There exists a constant
and that the interconnection matrix
Assumption 2.
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
ui(s)=ψi(s),∀s∈[−h,0],i∈Zd, | (2) |
for appropriate functions
We follow Han & Kloeden [7] and consider a weighted space of bi-infinite real valued sequences with vectorial indices
In particular, given a positive sequence of weights
ℓ2ρ:={u=(ui)i∈Zd:∑i∈Zdρiu2i<∞} |
with the inner product
⟨u,v⟩:=∑i∈Zdρiuiviforu=(ui)i∈Zd,v=(vi)i∈Zd∈ℓ2ρ |
and norm
‖u‖ρ:=√∑i∈Zdρiu2i. |
We assume that the
Assumption 3.
The appropriate function space for the solutions of the lattice system with delays (1) is the Banach space
‖v‖C([−h,0],ℓ2ρ)=maxs∈[−h,0]‖v(s)‖ρ. |
For a solution
For any
f(u):=(fi(ui))i∈Zd. |
To ensure that the
Assumption 4. The functions
supi∈Zdmaxs∈[−r,r]|f′i(s)|≤L(ρir),∀r∈R+,i∈Zd; |
Assumption 5.
Assumption 6. There exist constants
sfi(s)≤−α|s|2+β2i,∀s∈R,∀i∈Zd. |
It was shown in [7] that Assumption 4 implies that
|fi(x)−fi(y)|≤L(ρi(|x|+|y|))|x−y|,∀i∈Zd,x,y∈R. |
Since
ρi|ui|≤√ρΣ(∑i∈Zdρiu2i)1/2=√ρΣ‖u‖ρ, |
it follows
|fi(ui)−fi(vi)|≤L(ρi(|ui|+|vi|))⋅|ui−vi|≤L(√ρΣ(‖u‖ρ+‖v‖ρ))⋅|ui−vi| |
for every
Lemma 2.1. Assume that Assumptions 4–6 hold. Then
⟨f(u),u⟩≤−α‖u‖2ρ+‖β‖2ρ. |
For any
Kτ,i(v)=∑j∈Zdki,jσε(vj(−τj)−θ),∀i∈Zd. |
Lemma 2.2. The operator
Proof. The function
|Kτ,i(v)|≤∑j∈Zdki,j≤κ,∀i∈Zd,v∈C([−h,0],ℓ2ρ). |
Then
‖Kτ(v)‖2ρ=∑i∈Zdρi|Kτ,i(v)|2≤κ2ρΣ<∞. |
Remark 1. The function
ddxσε(x)≤1εfor allx∈R. |
Hence it is globally Lipschitz with the Lipschitz constant
Finally, we suppose that the constant forcing term
Assumption 7.
The lattice differential equation (1) can be rewritten as an infinitely dimensional ordinary differential equation on
ddtu(t)=Gτ(t,ut):=f(u)+Kτ(ut)+g, | (3) |
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
KNτ,i(v):=∑j∈ZdNki,jσε(vj(−τj)−θ),i∈Zd. |
Lemma 3.1. The mapping
Proof. Let
∑j∈Zdρj|vnj(−τj)−v0j(−τj)|2<ε2,∀n≥M(ε). |
Considering only the
|vnj(−τj)−v0j(−τj)|<ε/√ρN,∀n≥M(ε),j∈ZdN, |
where
The mapping
|(a1+b1)−(a2+b2)|≤|a1−a2|+|b1−b2|,a1,a2,b1,b2∈R1 |
that the mapping
Theorem 3.2. Suppose that Assumptions 1–7 hold. Then for each
Proof. Step 1. First, we claim that
It is easy to see that
|Gτ,i(ut)|≤|fi(ui(t))|+|Kτ,i(ut)|+|gi|. | (4) |
Since
|fi(ui(t))|≤L(ρi|ui(t)|)⋅|ui(t)|≤L(√ρΣ‖u(t)‖ρ)⋅|ui(t)|. |
Then we obtain
(∑i∈Zdρi|fi(ui(t))|2)12≤L(√ρΣ‖u(t)‖ρ)‖u(t)‖ρ. | (5) |
For the second term with delay, we have
(∑i∈Zdρi|Kτ,i(ut)|2)12≤√ρΣκ, | (6) |
where we have used Assumption 3.
Finally, for the last term
‖g‖ρ<∞. | (7) |
Using (5), (6) and (4) in (4) we conclude that
Step 2. Next, we claim that the maps
We consider
|Gτ,i(unt)−Gτ,i(u0t)|≤|fi(uni(t))−fi(u0i(t))|+|Kτ,i(unt)−Kτ,i(u0t)|. | (8) |
By the local Lipschitz continuity of
|fi(uni(t))−fi(u0i(t))|≤L(√ρΣ(‖unt(0)‖ρ+‖u0t(0)‖ρ))⋅|uni(t)−u0i(t)|, | (9) |
which shows that this term converges to zero.
Next for the second term on the right-hand side
|Kτ,i(unt)−Kτ,i(u0t)|=|∑j∈Zdki,jσε(unj(t−τj)−θ)−∑j∈Zdki,jσε(u0j(t−τj)−θ)|=∑j∈Zdki,j|σε(unj(t−τj)−θ)−σε(u0j(t−τj)−θ)| |
≤∑j∈ZdNki,j|σε(unj(t−τj)−θ)−σε(u0j(t−τj)−θ)|+∑j∈Zd∖ZdNki,j|σε(unj(t−τj)−θ)−σε(u0j(t−τj)−θ)|≤Lσ∑j∈ZdNki,j|unj(t−τj)−u0j(t−τj)|+2∑j∈Zd∖ZdNki,j. | (10) |
On one hand, since
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
Step 3. Finally, we claim the following inequality holds:
∑|i|≥Kρi|Gτ,i(ut)|2≤C(√ρΣ‖u(t)‖ρ)(maxs∈[−h,0]∑|i|≥Kρiu2i(t+s)+bK), |
where
The proof is as follows.
∑|i|≥Kρi|Gτ,i(ut)|2≤3∑|i|≥Kρi|fi(ui(t))|2+3∑|i|≥Kρi|∑j∈Zdki,jσε(uj(t−τj)−θ)|2+3∑|i|≥Kρi|gi|2≤3∑|i|≥KρiL2(ρi|ui(t)|)|ui(t)|2+3∑|i|≥Kρi⋅κ2+3∑|i|≥Kρi|gi|2≤3L2(√ρΣ‖u(t)‖ρ)∑|i|≥Kρi|ui(t)|2+3∑|i|≥Kρi⋅κ2+3∑|i|≥Kρi|gi|2≤C(√ρΣ‖u(t)‖ρ)(maxs∈[−h,0]∑|i|≥Kρiu2i(t+s)+bK). |
By Corollary 13 in [3], we also conclude that the solution
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
‖ut‖2C([−h,0],ℓ2ρ)≤R1e−αt‖ψ‖2C([−h,0],ℓ2ρ)+R2, | (11) |
where
Proof. We multiply the
12ddt‖u‖2ρ=∑i∈Zdρiuifi(ui)+∑i∈Zd(ρiui∑j∈Zdki,jσε(uj(t−τj)−θ))+∑i∈Zdρiuigi. | (12) |
By Assumption 6 and
ρifi(ui(t))ui(t)≤−αρiu2i(t)+ρiβ2i, |
so
∑i∈Zdρifi(ui(t))ui(t)≤−α‖u(t)‖2ρ+‖β‖2ρ. |
Since function
|ρiui∑j∈Zdki,jσε(uj(t−τj)−θ)|≤|ρiui∑j∈Zdki,j|≤α4ρiu2i+1αρi(∑j∈Zdki,j)2≤α4ρiu2i+1αρiκ2, |
so
∑i∈Zd|ρiui∑j∈Zdki,jσε(uj(t−τj)−θ)|≤α4‖u(t)‖2ρ+1αρΣκ2. |
The last term on the right hand of (12) satisfies
∑i∈Zdρigiui(t)≤α4∑i∈Zdρiu2i(t)+1α∑i∈Zdρig2i≤α4‖u(t)‖2ρ+1α‖g‖2ρ. |
In summary, collecting the inequalities above, we obtain
12ddt‖u(t)‖2ρ≤−12α‖u(t)‖2ρ+‖β‖2ρ+1α(ρΣκ2+‖g‖2ρ). |
Integrating both sides of this differential inequality yields
‖u(t)‖2ρ≤‖u(0)‖2ρe−αt+2α(‖β‖2ρ+1α(ρΣκ2+‖g‖2ρ))(1−e−αt). | (13) |
Let
‖u(t+θ)‖ρ=‖ψ(t+θ)‖ρ≤‖ψ‖C([−h,0],ℓ2ρ),t+θ<0, |
we obtain
‖u(t+θ)‖2ρ≤‖ψ‖2ρe−α(t+θ)+2α(‖β‖2ρ+1α(ρΣκ2+‖g‖2ρ))(1−e−α(t+θ)). |
Finally, using that
‖ut‖2C([−h,0],ℓ2ρ)≤eαhe−αt‖ψ‖2C([−h,0],ℓ2ρ)+2α(‖β‖2ρ+1α(ρΣκ2+‖g‖2ρ))=R1e−αt‖ψ‖2C([−h,0],ℓ2ρ)+R2, | (14) |
where
R1:=eαh,R2:=2α(‖β‖2ρ+1α(ρΣκ2+‖g‖2ρ)). |
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
Lemma 3.3. Suppose that Assumptions 1–8 hold. Then the solution
Proof. Assumption 8 implies that the operator
∑i∈Zdρi|Kτ,i(ut)−Kτ,i(vt)|2≤∑i∈Zdρi|∑j∈Zdki,j(σε(uj(t−τj)−θ)−σε(vj(t−τj)−θ))|2≤∑i∈Zdρi(∑j∈Zdki,j|σε(uj(t−τj)−θ)−σε(vj(t−τj)−θ)|)2≤∑i∈Zdρi(Lσ∑j∈Zdki,j|uj(t−τj)−vj(t−τj)|)2≤∑i∈ZdρiL2σ(∑j∈Zdki,j√ρj√ρj|uj(t−τj)−vj(t−τj)|)2≤ρΣ˜κL2σ‖ut−vt‖2C([−h,0],ℓ2ρ). |
Hence, suppose that we have two different solutions
Set
12ddt‖w‖2ρ≤L(√ρΣ(‖u‖ρ+‖v‖ρ))‖w‖2ρ+‖w‖ρ√C1ρΣ˜κLσ‖wt‖C([−h,0],ℓ2ρ)≤L(2√ρΣ(R1‖ψ‖2C([−h,0],ℓ2ρ)+R2))‖w‖2ρ+‖w‖ρ√C1ρΣ˜κLσ‖wt‖C([−h,0],ℓ2ρ)≤C‖wt‖2C([−h,0],ℓ2ρ), |
where
Integrating from 0 to t then gives
‖w(t)‖2ρ≤2C∫t0‖wτ‖2C([−h,0],ℓ2ρ)dτ+‖w(0)‖2ρ. |
Let
‖w(t+θ)‖2ρ≤2C∫t+θ0‖wτ‖2C([−h,0],ℓ2ρ)dτ+‖w(0)‖2ρ. |
Then take the supremum on
‖wt‖2C([−h,0],ℓ2ρ)≤2C∫t0‖wτ‖2C([−h,0],ℓ2ρ)dτ+‖w(0)‖2ρ. |
By Gronwall's inequality, we have
‖wt‖2C([−h,0],ℓ2ρ)≤2Cte2Ct‖w(0)‖2ρ+‖w(0)‖2ρ. | (15) |
Since
The proof of the next corollary follows easily using (15).
Corollary 1. The map
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.
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
S(t,ψ)=ut, |
where
It also follows from inequality (11) that the semigroup has a bounded absorbing set.
Corollary 2. The bounded set defined by
B0:={ψ∈C([−h,0],ℓ2ρ):‖ψ‖C([−h,0],ℓ2ρ)≤R0}, |
with
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
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
maxs∈[−h,0]∑|i|>2M(ε,B)ρi|ui(t+s)|2<ε,t≥T, |
for any initial condition
Proof. Define a smooth function
ξ(s){=0, 0≤s≤1,∈[0,1], 1≤s≤2,=1, s≥2. |
Let
vi(t)=ξM(|i|)ui(t) with ξM(|i|)=ξ(|i|M), i∈Zd, |
where
12ddt∑i∈ZdρiξM(|i|)|ui(t)|2=∑i∈ZdρiξM(|i|)ui(t)dui(t)dt=∑i∈ZdρiξM(|i|)ui(t)fi(ui(t))+∑i∈ZdρiξM(|i|)ui(t)gi+∑i∈ZdρiξM(|i|)ui(t)∑j∈Zdki,jσε(uj(t−τj)−θ). | (16) |
First, by Assumption 6,
ρiξM(|i|)ui(t)fi(ui)≤−αρiξM(|i|)u2i(t)+ρiξM(|i|)β2i. | (17) |
Then, since function
|ρiξM(|i|)ui(t)∑j∈Zdki,jσε(uj(t−τj)−θ)|≤|ρiξM(|i|)ui(t)κ|≤α4ρiξM(|i|)u2i(t)+κ2αρiξM(|i|). | (18) |
And using Young's inequality again,
∑i∈ZdρiξM(|i|)giui(t)=∑|i|≥MρiξM(|i|)giui(t)≤α4∑i∈ZdρiξM(|i|)u2i(t)+1α∑|i|≥Mρig2i. | (19) |
Inserting the estimations (17), (18) and (19) into (16), then
12ddt∑i∈ZdρiξM(|i|)u2i(t)≤−12α∑i∈ZdρiξM(|i|)u2i(t)+∑i∈ZdρiξM(|i|)β2i+κ2α∑i∈ZdρiξM(|i|)+1α∑|i|≥Mρig2i. | (20) |
We now estimate each term on the right hand side of the above inequality. Note that
∑i∈ZdρiξM(|i|)β2i=∑|i|≥MρiξM(|i|)β2i≤∑|i|≥Mρiβ2i. |
Since
∑i∈ZdρiξM(|i|)β2i≤∑|i|≥Mρiβ2i<16ε whenM≥I1(ε). | (21) |
Similarly, since
∑i∈ZdρiξM(|i|)=∑|i|≥MρiξM(|i|)≤∑|i|≥Mρi<α6κ2εwhenM≥I2(ε). | (22) |
In addition, since
∑|i|≥Mρig2i≤α6ε, ∀t∈R,whenM≥I3(ε). | (23) |
Finally, for any
ddt∑i∈ZdρiξM(|i|)u2i(t)≤−α∑i∈ZdρiξM(|i|)u2i(t)+ε, ∀M≥I(ε). |
It follows immediately from Gronwall's lemma that
∑i∈ZdρiξM(|i|)u2i(t)≤e−αt∑i∈ZdρiξM(|i|)u2i(0)+εα. |
In a similar way as in Proposition 1 we have
maxs∈[−h,0]∑i∈ZdρiξM(|i|)u2i(t+s)≤eαhe−αtmaxs∈[−h,0]∑i∈ZdρiξM(|i|)ψ2i(s)+εα. |
Thus, there exist
maxs∈[−h,0]∑|i|>2Mρi|ui(t+s)|2≤εift≥T. |
In order to apply Theorem 4.1, we need to prove that
Lemma 4.3. Suppose that Assumptions 1–8 hold. Then the semigroup
Proof. We consider
‖untn(s)‖≤C,∀s∈[−h,0],∀n∈N. |
For fixed
un(tn+s)⇀ζ(s) in ℓ2ρ. |
In fact, the weak convergence here is strong, which follows from Lemma 4.2. Indeed, there exists
∑|i|>K2ρi|uni(tn+s)|2<μ,∑|i|>K2ρi|ζi(s)|2<μ,∑|i|≤K2ρi|uni(tn+s)−ζi(s)|2<μ |
if
‖un(tn+s)−ζ(s)‖2ρ≤∑|i|≤K2ρi|uni(tn+s)−ζi(s)|2+∑|i|>K2ρi|uni(tn+s)−ζi(s)|2≤∑|i|≤K2ρi|uni(tn+s)−ζi(s)|2+2∑|i|>K2ρi|uni(tn+s)|2+2∑|i|>K2ρi|ζi(s)|2<5μ. |
Thus,
‖un(tn+s)−un(tn+t)‖ρ≤∫ts‖Gτ(untn+τ)‖ρdτ≤K(t−s)if−h≤s<t≤0. |
Then, the Ascoli-Arzelà theorem implies that
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.
[1] |
Dynamics of pattern formation in lateral-inhibition type neural fields. Biol. Cybernet. (1977) 27: 77-87. ![]() |
[2] |
Attractors for lattice dynamical systems. Internat. J. Bifur. Chaos Appl. Sci. Engrg. (2001) 11: 143-153. ![]() |
[3] |
On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems. Discrete Contin. Dyn. Syst. (2014) 34: 51-77. ![]() |
[4] |
S. Coombes, P. B. Graben, R. Potthast and J. Wright, Neural Fields. Theory and Applications, Springer, Heidelberg, 2014. doi: 10.1007/978-3-642-54593-1
![]() |
[5] |
On the connectedness of attractors for dynamical systems. J. Differential Equations (1997) 133: 1-14. ![]() |
[6] |
Non-autonomous lattice systems with switching effects and delayed recovery. J. Differential Equations (2016) 261: 2986-3009. ![]() |
[7] |
Asymptotic behaviour of a neural field lattice model with a Heaviside operator. Phys. D (2019) 389: 1-12. ![]() |
[8] |
(1991) Attractors for Semigroups and Evolution Equations. Cambridge: Cambridge University Press. ![]() |
[9] |
Asymptotic behavior of non-autonomous lattice systems. J. Math. Anal. Appl. (2007) 331: 121-136. ![]() |
[10] |
S. Zhou, Attractors for first order dissipative lattice dynamical systems, Phys. D, 178 (2003) 51–61. doi: 10.1016/S0167-2789(02)00807-2
![]() |
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5. | Xiaoying Han, Peter E. Kloeden, Dynamics of a Random Hopfield Neural Lattice Model with Adaptive Synapses and Delayed Hebbian Learning, 2024, 75, 0041-5995, 1883, 10.1007/s11253-024-02298-8 | |
6. | Xiaoying Han, Peter E. Kloeden, Dynamics of a random Hopfield neural lattice model with adaptive synapses and delayed Hebbian learning, 2024, 75, 1027-3190, 1666, 10.3842/umzh.v75i12.7594 | |
7. | Tianhao Zeng, Shaoyue Mi, Dingshi Li, Dynamical behaviors of an impulsive stochastic neural field lattice model, 2024, 24, 0219-4937, 10.1142/S0219493724500126 | |
8. | Hailang Bai, Mingkai Yuan, Dexin Li, Yunshun Wu, Weak and Wasserstein convergence of periodic measures of stochastic neural field lattice models with Heaviside ’s operators and locally Lipschitz Lévy noises, 2025, 143, 10075704, 108602, 10.1016/j.cnsns.2025.108602 |