# Almost Periodic Solutions for Wilson-Cowan Type Model with Time-Varying Delays

Authors: Shasha Xie; Zhenkun Huang
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

Wilson-Cowan model of neuronal population with time-varying delays is considered in this paper. Some sufficient conditions for the existence and delay-based exponential stability of a unique almost periodic solution are established. The approaches are based on constructing Lyapunov functionals and the well-known Banach contraction mapping principle. The results are new, easily checkable, and complement existing periodic ones.


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## Body

## 1. Introduction

Consider a well-known Wilson-Cowan type model [1, 2] with time-varying delays
(1)dXP(t)dt=-XP(t)+[kP-rPXP(t)]×G[wP1XP(t-τP(t))-wN1XN(t-τN(t))+IP(t)],dXN(t)dt=-XN(t)+[kN-rNXN(t)]×G[wP2XP(t-τP(t))-wN2XN(t-τN(t))+IN(t)],
where XP(t),XN(t) represent the proportion of excitatory and inhibitory neurons firing per unit time at the instant t, respectively. rP and rN are related to the duration of the refractory period, kP and kN are constants. wP1, wN1, wP2, and wN2 are the strengths of connections between the populations. IP(t),IN(t) are the external inputs to the excitatory and the inhibitory populations. G(·) is the response function of neuronal activity and it is always assumed to be sigmoid type. τP(t), τN(t) correspond to the transmission time-varying delays.It is interesting to revisit Wilson-Cowan system on the following points.(i)
The Wilson-Cowan model has a realistic biological background which describes interactions between excitatory and inhibitory populations of neurons [1–3]. It has extensive application such as pattern analysis and image processing [4, 5].(ii)
There exists rich dynamical behavior in Wilson-Cowan model. Theoretical results about stable limit cycles, equilibria, chaos, and oscillatory activity have been reported in [2, 3, 6–9]. Recently, Decker and Noonburg [8] reported new results about the existence of three periodic solutions when each neuron was stimulated by periodical inputs. However, under time-varying (periodic or almost periodic) inputs, Wilson-Cowan model can have more complex state space and coexistence of divergent solutions and local stable solutions which could not be easily estimated by its boundary. To see this, we can refer to Figure 1 for the phase portrait of solutions of the following Wilson-Cowan type model with G(z)=tanh(z) and almost periodic inputs [10–12]:
(2)dXP(t)dt=-XP(t)+[1-XP(t)]×G[XP(t)-XN(t)+5sin2t],dXN(t)dt=-XN(t)+[1-XN(t)]×G[XP(t)-XN(t)+0.5costsint].(iii)
Few works reported almost periodicity of Wilson-Cowan type model in the literature. Under almost periodic inputs, whether there exists a unique almost periodic solution of (1) which is stable? How to estimate its located boundary? Revealing these results can give a significant insight into the complex dynamical structure of Wilson-Cowan type model.Figure 1
Coexistence of divergent and local stable solutions of (2) with almost periodic inputs.Throughout this paper, we always assume thatkP, kN, rP, rN, wP1, wP2, wN1, and wN2 are positive constants, τP(t), τN(t), IP(t), and IN(t) are almost periodic functions [12], and set
(3)τP⊤=supt∈ℝτP(t),τN⊤=supt∈ℝτN(t),IP⊤=supt∈ℝ|IP(t)|,IN⊤=supt∈ℝ|IN(t)|.Moreover, we need some basic assumptions in this paper.(H1)
G(0)=0, supv∈ℝ|G(v)|⩽Bsand there exists an L>0 such that
(4)|G(u)-G(v)|⩽L|u-v|for∀u,v∈ℝ.(H2)
The quadratic equation(K+Rx)(I+Wx)=x has a positive solution δ, where
(5)W=max{L(wP1+wN1),L(wP2+wN2)},K=max{kP,kN},R=max{rP,rN},I=max{LIP⊤,LIN⊤}.(H3)
τP(t),τN(t) are bounded and continuously differentiable with 0≤τP(t)≤τP⊤,
(6)0≤τN(t)≤τN⊤,1-τ˙P(t)>0,1-τ˙N(t)>0fort∈R.For for allu=(u1,u2)∈ℝ2, we define the norm ∥u∥=max{|u1|,|u2|}. Let B:={ψ∣ψ=(ψ1,ψ1)}, where ψ is an almost periodic function on ℝ2. For all ψ∈B, if we define induced nodule ∥ψ∥B=supt∈ℝ∥ψ(t)∥, then B is a Banach space. The initial conditions of system (1) are of the form
(7)XP(s)=ψP(s),XN(s)=ψN(s),s∈[-τ,0],
where τ=max{τP⊤,τN⊤} and ψ=(ψP,ψN)∈B.Definition 1 (see [12]).
Letu(t):ℝ→ℝn be continuous. u(t) is said to be almost periodic on ℝ if, for any ε>0, it is possible to find a real number l=l(ε)>0, and for any interval with length l(ε), there exists a number δ=δ(ε) in this interval such that |u(t+δ)-u(t)|<ε, for all t∈ℝ.
The remaining part of this paper is organized as follows. In Section2, we will derive sufficient conditions for checking the existence of almost periodic solutions. In Section 3, we present delay-based exponential stability of the unique almost periodic solution of system (1). In Section 4, we will give an example to illustrate our results obtained in the preceding sections. Concluding remarks are given in Section 5.


## 2. Existence of Almost Periodic Solutions

Theorem 2.
Suppose that(H1) and (H2) hold. If KW+R(Bs+δW)<1, then there exists a unique almost periodic solution of system (1) in the region
(8)B*={ψ∣ψ∈B,∥ψ∥B≤δ}.Proof.
For for allψ=(ψP,ψN)∈B*, we consider the almost periodic solution Xψ(t)=(XPψ(t),XNψ(t)) of the following almost periodic differential equations:
(9)dXP(t)dt=-XP(t)+[kP-rPψP(t)]×G[wP1ψP(t-τP(t))-wN1ψN(t-τN(t))+IP(t)],dXN(t)dt=-XN(t)+[kN-rNψN(t)]×G[wP2ψP(t-τP(t))-wN2ψN(t-τN(t))+IN(t)].
By almost periodicity ofτP(t), τN(t), IP(t), and IN(t) and Theorem 3.4 in [12] or [10], (9) has a unique almost periodic solution
(10)XPψ(t)=∫-∞t‍exp(-(t-s))[kP-rPψP(s)]×G[wP1ψP(s-τP(s))-wN1ψN(s-τN(s))+IP(s)]ds,XNψ(t)=∫-∞t‍exp(-(t-s))[kN-rNψN(s)]×G[wP2ψP(s-τP(s))-wN2ψN(s-τN(s))+IN(s)]ds.
Define a mappingF:B*→B by setting F(ψ)(t)=(FP(ψ)(t),FN(ψ)(t))=Xψ(t), for all ψ∈B*. Now, we prove that F is a self-mapping from B* to B*. From (10) and (H1), we obtain
(11)|FP(ψ)(t)|≤supt∈ℝ|∫-∞t‍exp(-(t-s))[kP-rPψP(s)]×G[wP1ψP(s-τP(s))-wN1ψN(s-τN(s))+IP(s)]ds{∫-∞t‍}|≤supt∈ℝ[kP+rP|ψP(t)|]×G|[wP1ψP(t-τP(t))-wN1ψN(t-τN(t))+IP(t)]|≤(kP+rP∥ψ∥B)×L(wP1∥ψ∥B+wN1∥ψ∥B+IP⊤)≤(kP+rP∥ψ∥B)×(L(wP1+wN1)∥ψ∥B+LIP⊤).
By similar estimation, we can get(12)|FN(ψ)|≤(kN+rN∥ψ∥B)×(L(wP2+wN2)∥ψ∥B+LIN⊤).
Therefore, by the above estimations and(H2), we get
(13)∥F(ψ)∥B=supt∈ℝ{|FP(ψ)|,|FN(ψ)|}≤(K+R∥ψ∥B)(W∥ψ∥B+I)≤(K+Rδ)(Wδ+I)=δ,
which implies that F(ψ)∈B*. So, the mapping F is self-mapping from B* to B*. Next, we prove that F is a contraction mapping in the region B*. For all ψ, ϕ∈B*, by (10), we have
(14)|FP(ψ)(t)-FP(ϕ)(t)|≤supt∈ℝ|∫-∞t‍exp(-(t-s))[kP-rPψP(s)]×G[wP1ψP(s-τP(s))-wN1ψN(s-τN(s))+IP(s)]ds-∫-∞t‍exp(-(t-s))[kP-rPϕP(s)]×G[wP1ϕP(s-τP(s))-wN1ϕN(s-τN(s))+IP(s)]ds{∫-∞t‍}|,
which leads to
(15)|FP(ψ)(t)-FP(ϕ)(t)|≤supt∈ℝ∫-∞t‍exp(-(t-s))≤supt∈ℝ∫-∞t‍×[kPL|wP1ψP(s-τP(s))-wN1ψN(s-τN(s))≤supt∈ℝ∫-∞t‍×kPLlL-wP1ϕP(s-τP(s))+wN1ϕN(s-τN(s))|≤supt∈ℝ∫-∞t‍+rP|ψP(s)G[wP1ψP(s-τP(s))≤supt∈ℝ∫-∞t‍+rPψP(s)G-wN1ψN(s-τN(s))≤supt∈ℝ∫-∞t‍+rPψP(s)G+IP(s)]≤supt∈ℝ∫-∞t‍+rPj-ϕP(s)G[wP1ϕP(s-τP(s))≤supt∈ℝ∫-∞t‍+rPj-ϕP(s)G-wN1ϕN(s-τN(s))≤supt∈ℝ∫-∞t‍+rPj-ϕP(s)G+IP(s){wP1ϕP(s-τP(s))}]{ψP(s)G[wP1ψP(s-τP(s))}|{kPL|wP1ψP(s-τP(s))-wN1ψN(s-τN(s))}]ds≤kPL(wP1supt∈ℝ|ψP(t)-ϕP(t)|+wN1supt∈ℝ|ψN(t)-ϕN(t)|)+supt∈ℝ∫-∞t‍exp(-(t-s))rP+supt∈ℝ∫-∞t‍×[|ψP(s)G(wP1ψP(s-τP(s))+supt∈ℝ∫-∞t‍×ψP(s)Gnnn-wN1ψN(s-τN(s))+IP(s))+supt∈ℝ∫-∞t‍×ψP(s)Gnnn-ψP(s)G(wP1ϕP(s-τP(s))+supt∈ℝ∫-∞t‍×mm-wN1ϕN(s-τN(s))+supt∈ℝ∫-∞t‍×mm+IP(s))|+supt∈ℝ∫-∞t‍×m+|ψP(s)G(wP1ϕP(s-τP(s))+supt∈ℝ∫-∞t‍×m+n-wN1ϕN(s-τN(s))+IP(s))+supt∈ℝ∫-∞t‍×m+n-ϕP(s)G(wP1ϕP(s-τP(s))+supt∈ℝ∫-∞t‍×m+n-wN1ϕN(s)+IP(s))|]|ds≤kPL(wP1+wN1)∥ψ-ϕ∥B+rP(Bs∥ψ-ϕ∥B+δL(wP1+wN1)∥ψ-ϕ∥B).
By similar argument, we can get(16)|FN(ψ)(t)-FN(ψ)(t)|≤kNL(wP2+wN2)∥ψ-ϕ∥B+rN(Bs∥ψ-ϕ∥B+δL(wP2+wN2)∥ψ-ϕ∥B).
From (15) and (16), we have
(17)∥F(ψ)-F(ψ)∥B=supt∈ℝ{|FP(ψ)-FP(ψ)|,|FN(ψ)-FN(ψ)|}≤KW∥ψ-ϕ∥+R(Bs∥ψ-ϕ∥+δW∥ψ-ϕ∥)=(KW+RBs+RδW)∥ψ-ϕ∥.
SinceKW+RBs+RδW∈(0,1), it is clear that the mapping F is a contraction. Therefore the mapping F possesses a unique fixed point X*∈B* such that FX*=X*. By (9), X* is an almost periodic solution of system (1) in B*. The proof is complete.Remark 3.
Obviously, quadratic curve𝒞(v):=RWv2+(KW+RI-1)v+KI satisfies with 𝒞(0)≥0. So, Δ:=(KW+RI-1)2-4RWKI>0 and KW+RI<1 guarantees the existence of δ in (H2) and δ lies in the following interval:
(18)[1-KW-RI-Δ2RW,1-KW-RI+Δ2RW].
By Theorem2, we know that the unique almost periodic solution depends on KW+R(Bs+δW)<1. Choosing δ=(1-(KW+RI))/2RW, we get a simple assumption as follows:(H^2)
KW+RI<1-2KWRI,RBs<(1-(KW-RI))/2,and hence it leads to a parameter-based result.Corollary 4.
Suppose that(H1) and (H^2) hold. Then there exists a unique almost periodic solution of system (1) in the region B*={ψ∣ψ∈B,∥ψ∥B≤(1-(KW+RI))/2RW}.


## 3. Delay-Based Stability of the Almost Periodic Solution

In this section, we establish locally exponential stability of the unique almost periodic solution of system (1) in the region B*, which is delay dependent.Theorem 5.
Suppose that(H1)–(H3) hold. If KW+R(Bs+δW)<1 and there exist constants ℓ1>0, ℓ2>0 such that
(19)(1-rPBs)ℓ1>supt∈ℝCP1-τ˙P(υP-1(t)),(1-rNBs)ℓ2>supt∈ℝCN1-τ˙N(υN-1(t)),
where CP:=L[(kP+rPδ)ℓ1wP1+(kN+rNδ)ℓ2wP2] and CN:=L[(kP+rPδ)ℓ1wN1+(kN+rNδ)ℓ2wN2], υP-1(t) and υN-1(t) are the inverse functions of υP(t)=t-τP(t) and υN(t)=t-τN(t), then system (1) has exactly one almost periodic solution X*(t) in the region B* which is locally exponentially stable.Proof.
From Theorem2, system (1) has a unique almost periodic solution X*(t)∈B*. Let X(t)=(XP(t),XN(t)) be an arbitrary solution of system (1) with initial value ψ=(ψP,ψN)∈B*. Set X(t)=XP(t)-XP*(t), Y(t)=XN(t)-XN*(t). By system (1), we get
(20)dX(t)dt=-X(t)+[kP-rP(X(t)+XP*(t))]×G[wP1(X(t-τP(t))+XP*(t-τP(t)))-wN1(Y(t-τN(t))+XN*(t-τN(t)))IP(t)]-[kP-rPXP*(t)]×G[wP1XP*(t-τP(t))-wN1XN*(t-τN(t))+IP(t)],dY(t)dt=-Y(t)+[kN-rN(Y(t)+XN*(t))]×G[wP2(X(t-τP(t))+XP*(t-τP(t)))-wN2(Y(t-τN(t))+XN*(t-τN(t)))+IN(t)]-[kN-rNXN*(t)]×G[wP2XP*(t-τP(t))-wN2XN*(t-τN(t))+IN(t)].
Construct the auxiliary functionsFP(u), FN(u) defined on [0,+∞) as follows:
(21)FP(u):=(u-1+rPBs)ℓ1+supt∈ℝCPeuτP⊤1-τ˙P(υP-1(t)),FN(u):=(u-1+rNBs)ℓ2+supt∈ℝCNeuτN⊤1-τ˙N(υN-1(t)).
One can easily show thatFP(u), FN(u) are well defined and continuous. Assumption (19) implies that FP(0)<0, FP(u)→+∞ as u→+∞ and FN(0)<0, FN(u)→+∞ as u→+∞. It follows that there exists a common λ>0 such that FP(λ)<0 and FN(λ)<0.
Consider the Lyapunov functional(22)V(t)=[ℓ1|X(t)|+ℓ2|Y(t)|]eλt+∫t-τP(t)t‍CP1-τ˙P(υP-1(s))|X(s)|eλ(s+τP⊤)ds+∫t-τN(t)t‍CN1-τ˙N(υN-1(s))|Y(s)|eλ(s+τN⊤)ds.
Calculating the upper right derivative ofV(t) along system (1), one has
(23)D+V(t)≤λ[ℓ1|X(t)|+ℓ2|Y(t)|]eλt+eλtℓ1[-|X(t)|+|[kP-rP(X(t)+XP*(t))]×G[wP1(X(t-τP(t))+XP*(t-τP(t)))-wN1(Y(t-τN(t))+XN*(t-τN(t)))+IP(t)]-[kP-rPXP*(t)]×G[wP1XP*(t-τP(t))-wN1XN*(t-τN(t))+IP(t)]|]+eλtℓ2[-|Y(t)|+|{-wN2XN*(t-τN(t))+IN(t)]|}[kN-rN(Y(t)+XN*(t))]×G[wP2(X(t-τP(t))+XP*(t-τP(t)))-wN2(Y(t-τN(t))+XN*(t-τN(t)))+IN(t)]-[kN-rNXN*(t)]×G[wP2XP*(t-τP(t))-wN2XN*(t-τN(t))+IN(t)]|]+CP1-τ˙P(υP-1(t))|X(t)|eλ(t+τP⊤)-CP|X(t-τP(t))|exp[λ(t-τP(t)+τP⊤)]+CN1-τ˙N(υN-1(t))|Y(t)|eλ(t+τN⊤)-CN|Y(t-τN(t))|exp[λ(t-τN(t)+τN⊤)],
which leads to
(24)D+V(t)≤[(λ-1)ℓ1eλt|X(t)|+(λ-1)ℓ2eλt|Y(t)|]+eλtℓ1[kPL(wP1|X(t-τP(t))|+eλtℓ1kkkkkjj+wN1|Y(t-τN(t))|)+rP|X(t)|Bs+rPδL(wP1|X(t-τP(t))|+wN1|Y(t-τN(t))|)]+eλtℓ2[kNL(wP2|X(t-τP(t))|eλtℓ2kkkkkkll+wN2|Y(t-τN(t))|)+rN|Y(t)|Bs+rNδL(wP2|X(t-τP(t))|+rN|Y(t)|Bs+rNδLk+wN2|Y(t-τN(t))|)]+CP1-τ˙P(υP-1(t))|X(t)|eλ(t+τP⊤)-CP|X(t-τP(t))|eλt+CN1-τ˙N(υN-1(t))|Y(t)|eλ(t+τN⊤)-CN|Y(t-τN(t))|eλt≤[(λ-1+rPBs)ℓ1+supt∈ℝCPeλτP⊤1-τ˙P(υP-1(t))]×eλt|X(t)|+[(λ-1+rNBs)ℓ2+supt∈ℝCNeλτN⊤1-τ˙N(υN-1(t))]×eλt|Y(t)|≤-c1[|X(t)|+|Y(t)|]eλt,
where c1:=-(1/2)max{FP(λ),FN(λ)}>0. We have from the above that V(t)≤V(t0) and
(25)min{ℓ1,ℓ2}eλt(|X(t)|+|Y(t)|)≤V(t)≤V(0),t≥0.
Note that(26)V(0)=[(ℓ1+ℓ2)+τP⊤sups∈[-τP⊤,0]CP1-τ˙P(υP-1(s))eλτP⊤+τN⊤sups∈[-τN⊤,0]CN1-τ˙N(υN-1(s))eλτN⊤]∥ψ-ψ*∥,
where ψ*(s)=X*(s), s∈[-τ,0]. Then there exists a positive constant M>1 such that
(27)|XP(t)-XP*(t)|≤M∥ψ-ψ*∥e-λt,|XN(t)-XN*(t)|≤M∥ψ-ψ*∥e-λt.
The proof is complete.Setδ=(1-(KW+RI))/2RW. It follows from Corollary 4 and Theorem 5 the following.Corollary 6.
Suppose that(H1), (H^2), and (H3) hold. If there exist constants ℓ1>0, ℓ2>0 such that
(28)(1-rPBs)ℓ1>supt∈ℝL(αPℓ1wP1+αNℓ2wP2)1-τ˙P(υP-1(t)),(1-rNBs)ℓ2>supt∈ℝL(αPℓ1wN1+αNℓ2wN2)1-τ˙N(υN-1(t)),
where
(29)αP:=kP+rP1-(KW+RI)2RW,αN:=kN+rN1-(KW+RI)2RW,
υ
P
-
1
(
t
) and υN-1(t) are defined as Theorem 5, then system (1) has exactly one almost periodic solution X*(t) in the region
(30)B*={ψ∣ψ∈B,∥ψ∥B≤1-(KW+RI)2RW},
which is locally exponentially stable.


## 4. An Example

In this section, we give an example to demonstrate the results obtained in previous sections. Consider a Wilson-Cowan type model with time-varying delays as follows:(31)dXP(t)dt=-XP(t)+[kP-rPXP(t)]×G[wP1XP(t-τP(t))-wN1XN(t-τN(t))+IP(t)],dXN(t)dt=-XN(t)+[kN-rNXN(t)]×G[wP2XP(t-τP(t))-wN2XN(t-τN(t))+IN(t)],
where kP=kN=1, rP=rN=0.01, wP1=wP2=wN1=wN2=0.1, τP(t)=τN(t)=0.1t+10, IP(t)=7sin7t, G(v)=tanh(v), and IN(t)=7cos2t. It is easy to calculate that
(32)K=1,W=0.2,R=0.01,I=7,υP-1(t)=υN-1(t)=(t+10)0.9,L=Bs=1,δ=182.5.It is easy to check that(H1) and (H^2) hold. By Corollary 4, (31) has a unique almost periodic solution X*(t) in region B*={ψ∣ψ∈B,∥ψ∥B≤182.5}. Setting ℓ1=ℓ2=0.5, we can check that (H3) and (28) hold and hence X*(t) is exponentially stable in B*. Figure 2 shows the transient behavior of the unique almost periodic solution (XP(t),XN(t)) in B*. Phase portrait of attractivity of XP and XN is illustrated in Figure 3.Figure 2
Transient behavior of the almost periodic solution of (31) located in B*.Figure 3
Phase portrait of attractivity of the unique almost periodic solution of (31) located in B*.


## 5. Concluding Remarks

In this paper, we investigate Wilson-Cowan type model and obtain the existence of a unique almost periodic solution and its delay-based local stability in a convex subset. Our results are new and can reduce to periodic case, hence, complement existing periodic ones [7, 8]. We point out that there will exist multiple periodic (almost periodic) solution for system (1) under suitable parameter configuration. However, it is difficult to analyze its multistability of almost periodic solution by the existing method [10, 11, 13, 14]. We leave it for interested readers.



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