When studying a dynamical system, a good starting point is to look for invariant sets. Theorem 2.2 provides such an invariant set but it is a very large one, not sufficient to convey a good understanding of the system. Other invariant sets (included in the previous one) are stationary points. Notice that delayed and non-delayed equations share exactly the same stationary solutions, also called persistent states. We can therefore make good use of the harvest of results that are available about these persistent states which we note . Note that in most papers dealing with persistent states, the authors compute one of them and are satisfied with the study of the local dynamics around this particular stationary solution. Very few authors (we are aware only of [19, 26]) address the problem of the computation of the whole set of persistent states. Despite these efforts they have yet been unable to get a complete grasp of the global dynamics. To summarize, in order to understand the impact of the propagation delays on the solutions of the neural field equations, it is necessary to know all their stationary solutions and the dynamics in the region where these stationary solutions lie. Unfortunately such knowledge is currently not available. Hence we must be content with studying the local dynamics around each persistent state (computed, for example, with the tools of [19]) with and without propagation delays. This is already, we think, a significant step forward toward understanding delayed neural field equations.
From now on we note a persistent state of (3) and study its stability.
We can identify at least three ways to do this:
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1.
to derive a Lyapunov functional,
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2.
to use a fixed point approach,
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3.
to determine the spectrum of the infinitesimal generator associated to the linearized equation.
Previous results concerning stability bounds in delayed neural mass equations are ‘absolute’ results that do not involve the delays: they provide a sufficient condition, independent of the delays, for the stability of the fixed point (see [15, 20–22]). The bound they find is similar to our second bound in Proposition 3.13. They ‘proved’ it by showing that if the condition was satisfied, the eigenvalues of the infinitesimal generator of the semi-group of the linearized equation had negative real parts. This is not sufficient because a more complete analysis of the spectrum (for example, the essential part) is necessary as shown below in order to proof that the semi-group is exponentially bounded. In our case we prove this assertion in the case of a bounded cortex (see Section 3.1). To our knowledge it is still unknown whether this is true in the case of an infinite cortex.
These authors also provide a delay-dependent sufficient condition to guarantee that no oscillatory instabilities can appear, that is, they give a condition that forbids the existence of solutions of the form . However, this result does not give any information regarding stability of the stationary solution.
We use the second method cited above, the fixed point method, to prove a more general result which takes into account the delay terms. We also use both the second and the third method above, the spectral method, to prove the delay-independent bound from [15, 20–22]. We then evaluate the conservativeness of these two sufficient conditions. Note that the delay-independent bound has been correctly derived in [25] using the first method, the Lyapunov method. It might be of interest to explore its potential to derive a delay-dependent bound.
We write the linearized version of (3) as follows. We choose a persistent state and perform the change of variable . The linearized equation writes
(4)
where the linear operator is given by
It is also convenient to define the following operator:
3.1 Principle of linear stability analysis via characteristic values
We derive the stability of the persistent state (see [19]) for the equation (1) or equivalently (3) using the spectral properties of the infinitesimal generator. We prove that if the eigenvalues of the infinitesimal generator of the righthand side of (4) are in the left part of the complex plane, the stationary state is asymptotically stable for equation (4). This result is difficult to prove because the spectrum (the main definitions for the spectrum of a linear operator are recalled in Appendix A) of the infinitesimal generator neither reduces to the point spectrum (set of eigenvalues of finite multiplicity) nor is contained in a cone of the complex plane C (such an operator is said to be sectorial). The ‘principle of linear stability’ is the fact that the linear stability of U is inherited by the state for the nonlinear equations (1) or (3). This result is stated in the Corollaries 3.7 and 3.8.
Following [27–31], we note the strongly continuous semigroup of (4) on (see Definition A.3 in Appendix A) and A its infinitesimal generator. By definition, if U is the solution of (4) we have . In order to prove the linear stability, we need to find a condition on the spectrum of A which ensures that as .
Such a ‘principle’ of linear stability was derived in [29, 30]. Their assumptions implied that was a pure point spectrum (it contained only eigenvalues) with the effect of simplifying the study of the linear stability because, in this case, one can link estimates of the semigroup T to the spectrum of A. This is not the case here (see Proposition 3.4).
When the spectrum of the infinitesimal generator does not only contain eigenvalues, we can use the result in [27], Chapter 4, Theorem 3.10 and Corollary 3.12] for eventually norm continuous semigroups (see Definition A.4 in Appendix A) which links the growth bound of the semigroup to the spectrum of A:
Thus, U is uniformly exponentially stable for (4) if and only if
We prove in Lemma 3.6 (see below) that is eventually norm continuous. Let us start by computing the spectrum of A.
3.1.1 Computation of the spectrum of A
In this section we use for for simplicity.
Definition 3.1 We defineforby:
whereis the compact (it is a Hilbert-Schmidt operator see[32]Chapter X.2]) operator
We now apply results from the theory of delay equations in Banach spaces (see [27, 28, 31]) which give the expression of the infinitesimal generator as well as its domain of definition
The spectrum consists of those such that the operator of defined by is non-invertible. We use the following definition:
Definition 3.2 (Characteristic values (CV))
The characteristic values of A are the λs such thathas 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, ), 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 . 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.
Proof Let us define the following operator. If , we define by . From [28], Lemma 34], is surjective and it is easy to check that iif , see [28], Lemma 35]. Moreover is closed in iff is closed in , see [28], Lemma 36].
Let us now prove the lemma. We already know that is closed in if is closed in . Also, we have , hence iif . It remains to check that iif .
Suppose that . There exist such that . Consider . Because is surjective, for all , there exists satisfying . We write , . Then where , that is, .
Suppose that . There exist such that . As is surjective for all there exists such that . Now consider . can be written where . But because . It follows that . □
Lemma 3.3 is the key to obtain . 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.
Proposition 3.4 A satisfies the following properties:
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1.
.
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2.
is at most countable.
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3.
.
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4.
For , the generalized eigenspace is finite dimensional and , .
Proof
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1.
. We apply [35], Theorem IV.5.26]. It shows that the essential spectrum does not change under compact perturbation. As is compact, we find .
Let us show that . The assertion ‘⊂’ is trivial. Now if , for example, , then .
Then is closed but . Hence . Also , hence . Hence, according to Definition A.7, .
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2.
We apply [35], Theorem IV.5.33] stating (in its first part) that if is at most countable, so is .
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3.
We apply again [35], Theorem IV.5.33] stating that if is at most countable, any point in is an isolated eigenvalue with finite multiplicity.
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4.
Because , we can apply [28], Theorem 2] which precisely states this property. □
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 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 3.5where.
Proof If and , then λ is a CV, that is, stating that ( denotes the point spectrum).
But
for λ big enough since
is bounded.
Hence, for λ large enough , which holds by the spectral radius inequality. This relationship states that the CVs λ satisfying are located in a bounded set of the right part of ; 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 :
Lemma 3.6 The semigroupof (4) is norm continuous onfor.
Proof We first notice that generates a norm continuous semigroup (in fact a group) on and that is continuous from to . 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.
We conclude by showing that the computation of the characteristic values of A is enough to state the stability of the stationary solution .
Corollary 3.8 If, then the persistent solutionof (3) is asymptotically stable.
Proof Using , we write (3) as . The function G is and satisfies , and . 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 , where if and . Note that the function is not continuous at .
It is however possible (note that a regularity condition has to be verified but this is done easily in our case) to extend (see [34]) the semigroup to the space . We note this extension which has the same spectrum as . Indeed, we can consider integral solutions of (4) with initial condition in . However, as has no meaning because is not continuous in , the linear problem (4) is not well-posed in this space. This is why we have to extend the state space in order to make the linear operator in (4) continuous. Hence the correct state space is and any function is represented by the vector . The variation of constant formula becomes:
where is the projector on the second component.
Now we choose and the spectral mapping theorem implies that there exists such that
and
. It follows that and from Theorem 2.2, , which yields and concludes the proof. □
Finally, we can use the CVs to derive a sufficient stability result.
Proposition 3.9 Ifthenis asymptotically stable for (3).
Proof Suppose that a CV λ of positive real part exists, this gives a vector in the Kernel of . Using straightforward estimates, it implies that , 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 of (1). It can be replaced by a polynomial in , namely , where the zeros of the polynomials have negative real parts. Indeed, in this case, when J is small, the network is stable. We obtain a diagonal matrix such that and change the initial condition (as in the theory of ODEs) while the history space becomes where . Having all this in mind equation (1) writes
Introducing the classical variable , we rewrite (6) as
(7)
where is the Vandermonde (we put a minus sign in order to have a formulation very close to 1) matrix associated to P and , , . It appears that equation (7) has the same structure as (1): , , are bounded linear operators; we can conclude that there is a unique solution to (6). The linearized equation around a persistent states yields a strongly continuous semigroup which is eventually continuous. Hence the stability is given by the sign of where is the infinitesimal generator of . It is then routine to show that
This indicates that the essential spectrum of is equal to 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 Ifthe persistent solutionof (6) is asymptotically stable.
Using the same proof as in [20], one can prove that provided that .
Proposition 3.11 Ifthenis asymptotically stable.
3.2 Principle of linear stability analysis via fixed point theory
The idea behind this method (see [37]) is to write (4) as an integral equation. This integral equation is then interpreted as a fixed point problem. We already know that this problem has a unique solution in . However, by looking at the definition of the (Lyapunov) stability, we can express the stability as the existence of a solution of the fixed point problem in a smaller space . The existence of a solution in gives the unique solution in . Hence, the method is to provide conditions for the fixed point problem to have a solution in ; in the two cases presented below, we use the Picard fixed point theorem to obtain these conditions. Usually this method gives conditions on the averaged quantities arising in (4) whereas a Lyapunov method would give conditions on the sign of the same quantities. There is no method to be preferred, rather both of them should be applied to obtain the best bounds.
In order to be able to derive our bounds we make the further assumption that there exists a such that:
Note that the notation represents the matrix of elements .
Remark 2 For example, in the 2D one-population case for, we have.
We rewrite (4) in two different integral forms to which we apply the fixed point method. The first integral form is obtained by a straightforward use the variation-of-parameters formula. It reads
(8)
The second integral form is less obvious. Let us define
Note the slight abuse of notation, namely .
Lemma B.3 in Appendix B.2 yields the upperbound . This shows that ∀t, .
Hence we propose the second integral form:
We have the following lemma.
Lemma 3.12 The formulation (9) is equivalent to (4).
Proof The idea is to write the linearized equation as:
(10)
By the variation-of-parameters formula we have:
We then use an integration by parts:
which allows us to conclude. □
Using the two integral formulations of (4) we obtain sufficient conditions of stability, as stated in the following proposition:
Proposition 3.13 If one of the following two conditions is satisfied:
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1.
and there exist , such that
whererepresents the matrix of elements,
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2.
,
thenis asymptotically stable for (3).
Proof We start with the first condition.
The problem (4) is equivalent to solving the fixed point equation for an initial condition . Let us define with the supremum norm written , as well as
We define on .
For all we have and . We want to show that . We prove two properties.
1.
tends to zero at infinity.
Choose .
Using Corollary B.3, we have as .
Let , we also have
For the first term we write:
Similarly, for the second term we write
Now for a given we choose T large enough so that . For such a T we choose large enough so that
for . Putting all this together, for all :
From (9), it follows that when .
Since is continuous and has a limit when it is bounded and therefore .
2.
is contracting on
.
Using (9) for all we have
We conclude from Picard theorem that the operator has a unique fixed point in .
There remains to link this fixed point to the definition of stability and first show that
where is the solution of (4).
Let us choose and such that
. M exists because, by hypothesis, . We then choose δ satisfying
(11)
and such that . Next define
We already know that is a contraction on (which is a complete space). The last thing to check is , that is , . Using Lemma B.3 in Appendix B.2:
Thus has a unique fixed point in which is the solution of the linear delayed differential equation, that is,
As in implies in , we have proved the asymptotic stability for the linearized equation.
The proof of the second property is straightforward. If 0 is asymptotically stable for (4) all the CV are negative and Corollary 3.8 indicates that is asymptotically stable for (3).
The second condition says that is a contraction because
.
The asymptotic stability follows using the same arguments as in the case of . □
We next simplify the first condition of the previous proposition to make it more amenable to numerics.
Corollary 3.14 Suppose that,
.
If there exist, such that, thenis asymptotically stable.
Proof This corollary follows immediately from the following upperbound of the integral
. Then if there exists , such that , it implies that condition 1 in Proposition 3.13 is satisfied, from which the asymptotic stability of follows. □
Notice that is equivalent to . The previous corollary is useful in at least the following cases:
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If is diagonalizable, with associated eigenvalues/eigenvectors: , , then and
.
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If and the range of is finite dimensional: where is an orthonormal basis of , then and
. Let us write the matrix associated to (see above). Then is also a compact operator with finite range and
. Finally, it gives
.
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If is self-adjoint, then it is diagonalizable and we can chose , .
Remark 3 If we suppose that we have higher order time derivatives as in Section 3.1.3, we can write the linearized equation as
(12)
Suppose thatis diagonalizable then
whereand. Also notice that,
. Then using the same functionals as in the proof of Proposition 3.13, we can find two bounds for the stability of a stationary state:
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Suppose that, that is, is stable for the non-delayed equation where. If there exist, such that
.
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.
To conclude, we have found an easy-to-compute formula for the stability of the persistent state . It can indeed be cumbersome to compute the CVs of neural field equations for different parameters in order to find the region of stability whereas the evaluation of the conditions in Corollary 3.14 is very easy numerically.
The conditions in Proposition 3.13 and Corollary 3.14 define a set of parameters for which is stable. Notice that these conditions are only sufficient conditions: if they are violated, may still remain stable. In order to find out whether the persistent state is destabilized we have to look at the characteristic values. Condition 1 in Proposition 3.13 indicates that if is a stable point for the non-delayed equation (see [18]) it is also stable for the delayed-equation. Thus, according to this condition, it is not possible to destabilize a stable persistent state by the introduction of small delays, which is indeed meaningful from the biological viewpoint. Moreover this condition gives an indication of the amount of delay one can introduce without changing the stability.
Condition 2 is not very useful as it is independent of the delays: no matter what they are, the stable point will remain stable. Also, if this condition is satisfied there is a unique stationary solution (see [18]) and the dynamics is trivial, that is, converging to the unique stationary point.
3.3 Summary of the different bounds and conclusion
The next proposition summarizes the results we have obtained in Proposition 3.13 and Corollary 3.14 for the stability of a stationary solution.
Proposition 3.15 If one of the following conditions is satisfied:
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1.
There exist such that
and , such that ,
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2.
thenis asymptotically stable for (3).
The only general results known so far for the stability of the stationary solutions are those of Atay and Hutt (see, for example, [20]): they found a bound similar to condition 2 in Proposition 3.15 by using the CVs, but no proof of stability was given. Their condition involves the -norm of the connectivity function J and it was derived using the CVs in the same way as we did in the previous section. Thus our contribution with respect to condition 2 is that, once it is satisfied, the stationary solution is asymptotically stable: up until now this was numerically inferred on the basis of the CVs. We have proved it in two ways, first by using the CVs, and second by using the fixed point method which has the advantage of making the proof essentially trivial.
Condition 1 is of interest, because it allows one to find the minimal propagation delay that does not destabilize. Notice that this bound, though very easy to compute, overestimates the minimal speed. As mentioned above, the bounds in condition 1 are sufficient conditions for the stability of the stationary state . In order to evaluate the conservativeness of these bounds, we need to compare them to the stability predicted by the CVs. This is done in the next section.