Stability of the stationary solutions of neural field equations with propagation delays

3.1.1 Computation of the spectrum of A

In this section we use ${\mathbf{L}}_1$ for ${\tilde{\mathbf{L}}}_1$ for simplicity.

The spectrum $\Sigma (\mathbf{A})$ consists of those $\lambda \in \mathbf{C}$ such that the operator $\Delta (\lambda )$ of $\mathcal{L}(\mathcal{F})$ defined by $\Delta (\lambda )=\lambda \mathrm{Id}+{\mathbf{L}}_0-\mathbf{J}(\lambda )$ is non-invertible. We use the following definition:

Definition 3.2 (Characteristic values (CV))

The characteristic values of A are the λs such that$\Delta (\lambda )$has a kernel which is not reduced to 0, that is, is not injective.

It is easy to see that the CV are the eigenvalues of A.

There are various ways to compute the spectrum of an operator in infinite dimensions. They are related to how the spectrum is partitioned (for example, continuous spectrum, point spectrum…). In the case of operators which are compact perturbations of the identity such as Fredholm operators, which is the case here, there is no continuous spectrum. Hence the most convenient way for us is to compute the point spectrum and the essential spectrum (see Appendix A). This is what we achieve next.

Remark 1 In finite dimension (that is, $dim\mathcal{F}<\infty$), the spectrum of A consists only of CV. We show that this is not the case here.

Notice that most papers dealing with delayed neural field equations only compute the CV and numerically assess the linear stability (see [9, 24, 33]).

We now show that we can link the spectral properties of A to the spectral properties of ${\mathbf{L}}_{\lambda }$. This is important since the latter operator is easier to handle because it acts on a Hilbert space. We start with the following lemma (see [34] for similar results in a different setting).

Lemma 3.3$\lambda \in {\Sigma }_{\mathit{ess}}(\mathbf{A})\iff \lambda \in {\Sigma }_{\mathit{ess}}({\mathbf{L}}_{\lambda })$.

Proof Let us define the following operator. If $\lambda \in \mathbf{C}$, we define ${\mathcal{T}}_{\lambda }\in \mathcal{L}(\mathcal{C},\mathcal{F})$ by ${\mathcal{T}}_{\lambda }(\varphi )=\varphi (0)+\mathbf{L}({\int }_{\cdot }^0{e}^{\lambda (\cdot -s)}\varphi (s)ds)$$\varphi \in \mathcal{C}$. From [28], Lemma 34], ${\mathcal{T}}_{\lambda }$ is surjective and it is easy to check that $\varphi \in \mathcal{R}(\lambda \mathrm{Id}-\mathbf{A})$ iif ${\mathcal{T}}_{\lambda }(\varphi )\in \mathcal{R}(\lambda \mathrm{Id}-{\mathbf{L}}_{\lambda })$, see [28], Lemma 35]. Moreover $\mathcal{R}(\lambda \mathrm{Id}-\mathbf{A})$ is closed in $\mathcal{C}$ iff $\mathcal{R}(\lambda \mathrm{Id}-{\mathbf{L}}_{\lambda })$ is closed in $\mathcal{F}$, see [28], Lemma 36].

Let us now prove the lemma. We already know that $\mathcal{R}(\lambda \mathrm{Id}-\mathbf{A})$ is closed in $\mathcal{C}$ if $\mathcal{R}(\lambda \mathrm{Id}-{\mathbf{L}}_{\lambda })$ is closed in $\mathcal{F}$. Also, we have $\mathcal{N}(\lambda \mathrm{Id}-\mathbf{A})=\{\theta \to e^{\theta \lambda }\mathbf{U},\mathbf{U}\in \mathcal{N}(\lambda \mathrm{Id}-{\mathbf{L}}_{\lambda })\}$, hence $dim\mathcal{N}(\lambda \mathrm{Id}-\mathbf{A})<\infty$ iif $dim\mathcal{N}(\lambda \mathrm{Id}-{\mathbf{L}}_{\lambda })<\infty$. It remains to check that $codim\mathcal{R}(\lambda \mathrm{Id}-\mathbf{A})<\infty$ iif $codim\mathcal{R}(\lambda \mathrm{Id}-{\mathbf{L}}_{\lambda })<\infty$.

Suppose that $codim\mathcal{R}(\lambda \mathrm{Id}-\mathbf{A})<\infty$. There exist ${\varphi }_1,\dots ,{\varphi }_N\in \mathcal{C}$ such that $\mathcal{C}=Span({\varphi }_i)+\mathcal{R}(\lambda \mathrm{Id}-\mathbf{A})$. Consider ${\mathbf{U}}_i\equiv {\mathcal{T}}_{\lambda }({\varphi }_i)\in \mathcal{F}$. Because ${\mathcal{T}}_{\lambda }$ is surjective, for all $\mathbf{U}\in \mathcal{F}$, there exists $\psi \in \mathcal{C}$ satisfying $\mathbf{U}={\mathcal{T}}_{\lambda }(\psi )$. We write $\psi ={\sum }_{i=1}^N{x}_i{\varphi }_i+f$, $f\in \mathcal{R}(\lambda \mathrm{Id}-\mathbf{A})$. Then $\mathbf{U}={\sum }_{i=1}^N{x}_i{\mathbf{U}}_i+{\mathcal{T}}_{\lambda }(f)$ where ${\mathcal{T}}_{\lambda }(f)\in \mathcal{R}(\lambda \mathrm{Id}-{\mathbf{L}}_{\lambda })$, that is, $codim\mathcal{R}(\lambda \mathrm{Id}-{\mathbf{L}}_{\lambda })<\infty$.

Suppose that $codim\mathcal{R}(\lambda \mathrm{Id}-{\mathbf{L}}_{\lambda })<\infty$. There exist ${\mathbf{U}}_1,\dots ,{\mathbf{U}}_N\in \mathcal{F}$ such that $\mathcal{F}=Span({\mathbf{U}}_i)+\mathcal{R}(\lambda \mathrm{Id}-{\mathbf{L}}_{\lambda })$. As ${\mathcal{T}}_{\lambda }$ is surjective for all $i=1,\dots ,N$ there exists ${\varphi }_i\in \mathcal{C}$ such that ${\mathbf{U}}_i={\mathcal{T}}_{\lambda }({\varphi }_i)$. Now consider $\psi \in \mathcal{C}$. ${\mathcal{T}}_{\lambda }(\psi )$ can be written ${\mathcal{T}}_{\lambda }(\psi )={\sum }_{i=1}^N{x}_i{\mathbf{U}}_i+\tilde{\mathbf{U}}$ where $\tilde{\mathbf{U}}\in \mathcal{R}(\lambda \mathrm{Id}-{\mathbf{L}}_{\lambda })$. But $\psi -{\sum }_{i=1}^N{x}_i{\varphi }_i\in \mathcal{R}(\lambda \mathrm{Id}-\mathbf{A})$ because ${\mathcal{T}}_{\lambda }(\psi -{\sum }_{i=1}^N{x}_i{\varphi }_i)=\tilde{\mathbf{U}}\in \mathcal{R}(\lambda \mathrm{Id}-{\mathbf{L}}_{\lambda })$. It follows that $codim\mathcal{R}(\lambda \mathrm{Id}-\mathbf{A})<\infty$. □

Lemma 3.3 is the key to obtain $\Sigma (\mathbf{A})$. Note that it is true regardless of the form of L and could be applied to other types of delays in neural field equations. We now prove the important following proposition.

Let us show that ${\Sigma }_{\mathit{ess}}(-{\mathbf{L}}_0)=\Sigma (-{\mathbf{L}}_0)$. The assertion ‘⊂’ is trivial. Now if $\lambda \in \Sigma (-{\mathbf{L}}_0)$, for example, $\lambda =-l_1$, then $\lambda \mathrm{Id}+{\mathbf{L}}_0=diag(0,-l_1+l_2,\dots )$.

As an example, Figure 1 shows the first 200 eigenvalues computed for a very simple model one-dimensional model. We notice that they accumulate at $\lambda =-1$ which is the essential spectrum. These eigenvalues have been computed using TraceDDE, [36], a very efficient method for computing the CVs.

Last but not least, we can prove that the CVs are almost all, that is, except for possibly a finite number of them, located on the left part of the complex plane. This indicates that the unstable manifold is always finite dimensional for the models we are considering here.

Corollary 3.5$\mathit{Card}\Sigma (\mathbf{A})\cap \{\lambda \in \mathbb{C},\Re \lambda >-l\}<\infty$where$l={min}_i{l}_i$.

Proof If $\lambda =\rho +i\omega \in \Sigma (\mathbf{A})$ and $\rho >-l$, then λ is a CV, that is, $\mathcal{N}(\mathrm{Id}-{(\lambda \mathrm{Id}+{\mathbf{L}}_0)}^{-1}\mathbf{J}(\lambda ))\ne \varnothing$ stating that $1\in {\Sigma }_P({(\lambda \mathrm{Id}+{\mathbf{L}}_0)}^{-1}\mathbf{J}(\lambda ))$ (${\Sigma }_P$ denotes the point spectrum).

But for λ big enough since is bounded.

Hence, for λ large enough $1\notin {\Sigma }_P({(\lambda \mathrm{Id}+{\mathbf{L}}_0)}^{-1}\mathbf{J}(\lambda ))$, which holds by the spectral radius inequality. This relationship states that the CVs λ satisfying $\Re \lambda >-l$ are located in a bounded set of the right part of $\mathbb{C}$; given that the CV are isolated, there is a finite number of them. □

3.1.2 Stability results from the characteristic values

We start with a lemma stating regularity for ${(\mathbf{T}(t))}_{t\ge 0}$:

Lemma 3.6 The semigroup${(\mathbf{T}(t))}_{t\ge 0}$of (4) is norm continuous on$\mathcal{C}$for$t>{\tau }_m$.

Proof We first notice that $-{\mathbf{L}}_0$ generates a norm continuous semigroup (in fact a group) $\mathbf{S}(t)=e^{-t{\mathbf{L}}_0}$ on $\mathcal{F}$ and that ${\tilde{\mathbf{L}}}_1$ is continuous from $\mathcal{C}$ to $\mathcal{F}$. The lemma follows directly from [27], Theorem VI.6.6]. □

Using the spectrum computed in Proposition 3.4, the previous lemma and the formula (5), we can state the asymptotic stability of the linear equation (4). Notice that because of Corollary 3.5, the supremum in (5) is in fact a max.

Corollary 3.7 (Linear stability)

Zero is asymptotically stable for (4) if and only if$max\Re {\Sigma }_p(\mathbf{A})<0$.

We conclude by showing that the computation of the characteristic values of A is enough to state the stability of the stationary solution ${\mathbf{V}}^f$.

Corollary 3.8 If$max\Re {\Sigma }_p(\mathbf{A})<0$, then the persistent solution${\mathbf{V}}^f$of (3) is asymptotically stable.

Proof Using $\mathbf{U}=\mathbf{V}-{\mathbf{V}}^f$, we write (3) as $\dot{\mathbf{U}}(t)=-\mathbf{L}{\mathbf{U}}_t+G({\mathbf{U}}_t)$. The function G is $C^2$ and satisfies $G(0)=0$, $DG(0)=0$ and ${\parallel G({\mathbf{U}}_t)\parallel }_{\mathcal{C}}=O({\parallel {\mathbf{U}}_t\parallel }_{\mathcal{C}}^2)$. We next apply a variation of constant formula. In the case of delayed equations, this formula is difficult to handle because the semigroup T should act on non-continuous functions as shown by the formula ${\mathbf{U}}_t=\mathbf{T}(t)\varphi +{\int }_0^t\mathbf{T}(t-s)[X_{0}G({\mathbf{U}}_s)]ds$, where $X_0(\theta )=0$ if $\theta <0$ and $X_0(0)=1$. Note that the function $\theta \to X_0(\theta )G({\mathbf{U}}_s)$ is not continuous at $\theta =0$.

where ${\pi }_2$ is the projector on the second component.

Now we choose $\omega =-max\Re {\Sigma }_p(\mathbf{A})/2>0$ and the spectral mapping theorem implies that there exists $M>0$ such that and . It follows that ${\parallel {\mathbf{U}}_t\parallel }_{\mathcal{C}}\le M{e}^{-\omega t}({\parallel U_0\parallel }_{\mathcal{C}}+{\int }_0^t{e}^{-\omega s}{\parallel G({\mathbf{U}}_s)\parallel }_{\mathcal{F}}ds)$ and from Theorem 2.2, ${\parallel G({\mathbf{U}}_t)\parallel }_{\mathcal{C}}=O(1)$, which yields ${\parallel {\mathbf{U}}_t\parallel }_{\mathcal{C}}=O(e^{-\omega t})$ and concludes the proof. □

Finally, we can use the CVs to derive a sufficient stability result.

Proposition 3.9 If${\parallel \mathbf{J}\cdot DS({\mathbf{V}}^f)\parallel }_{{\mathbf{L}}^2({\Omega }^2,{\mathbf{R}}^{p\times p})}<{min}_i{l}_i$then${\mathbf{V}}^f$is asymptotically stable for (3).

Proof Suppose that a CV λ of positive real part exists, this gives a vector in the Kernel of $\Delta (\lambda )$. Using straightforward estimates, it implies that ${min}_i{l}_i\le {\parallel \mathbf{J}\cdot DS({\mathbf{V}}^f)\parallel }_{\mathcal{F}}$, a contradiction. □

3.1.3 Generalization of the model

In the description of our model, we have pointed out a possible generalization. It concerns the linear response of the chemical synapses, that is, the lefthand side $(\frac{d}{dt}+l_i)$ of (1). It can be replaced by a polynomial in $\frac{d}{dt}$, namely $P_i(\frac{d}{dt})$, where the zeros of the polynomials $P_i$ have negative real parts. Indeed, in this case, when J is small, the network is stable. We obtain a diagonal matrix $\mathbf{P}(\frac{d}{dt})$ such that $\mathbf{P}(0)={\mathbf{L}}_0$ and change the initial condition (as in the theory of ODEs) while the history space becomes ${\mathcal{C}}^{d_s}([-{\tau }_m,0],\mathcal{F})$ where $d_s+1={max}_{i}deg{P}_i$. Having all this in mind equation (1) writes

(6)

This indicates that the essential spectrum ${\Sigma }_{\mathit{ess}}(\mathcal{A})$ of $\mathcal{A}$ is equal to ${\bigcup }_{i}Root(P_i)$ which is located in the left side of the complex plane. Thus the point spectrum is enough to characterize the linear stability:

Proposition 3.10 If$max\Re {\Sigma }_p(\mathcal{A})<0$the persistent solution${\mathbf{V}}^f$of (6) is asymptotically stable.

Using the same proof as in [20], one can prove that $max\Re \Sigma (\mathcal{A})<0$ provided that ${\parallel \mathbf{J}\cdot DS({\mathbf{V}}^f)\parallel }_{{\mathbf{L}}^2({\Omega }^2,{\mathbf{R}}^{p\times p})}<{min}_{k\in \mathbb{N},\omega \in \mathbb{R}}|P_k(i\omega )|$.

Proposition 3.11 If${\parallel \mathbf{J}\cdot DS({\mathbf{V}}^f)\parallel }_{{\mathbf{L}}^2({\Omega }^2,{\mathbf{R}}^{p\times p})}<{min}_{k\in \mathbb{N},\omega \in \mathbb{R}}|P_k(i\omega )|$then${\mathbf{V}}^f$is asymptotically stable.