Jian Wang

J Mendoza Avila

Denis Efimov

Alexander Yu Aleksandrov

Leonid Fridman

Jesus Mendoza

Conditions of self-oscillations in generalized Persidskii systems

For a class of generalized Persidskii systems, whose dynamics are described by superposition of a linear part with multiple sector nonlinearities and exogenous perturbations, the conditions of practical stability, instability and oscillatory behavior in the sense of Yakubovich are established. For this purpose the conditions of local instability at the origin and global boundedness of solutions (practical input-to-state stability) are developed in the form of linear matrix inequalities. The proposed theory is applied to investigate robustness to unmodeled dynamics of nonlinear feedback controls in linear systems, and to determine the presence of oscillations in the models of neurons.

I. INTRODUCTION

Analysis of stability in dynamical systems is a popular and complex problem, whose theory is well-developed nowadays, and it is represented by different notions and approaches [START_REF] Khalil | Nonlinear Systems[END_REF]. The principal tool to check the stability of an equilibrium or a trajectory is based on the Lyapunov function method [START_REF] Malkin | Theory of Stability of Motion[END_REF], [START_REF] Hahn | Stability of Motion[END_REF] and its extensions [START_REF] Lasalle | Stability by Liapunov's Direct Method with Aplications[END_REF]- [START_REF] Lin | A smooth converse Lyapunov theorem for robust stability[END_REF]. Despite that it has been theoretically proven that existence of an appropriate Lyapunov function is necessary and sufficient for various concepts of stability, there is a barrier for practical application of this procedure: the lack of constructive techniques of finding a needed Lyapunov function for a generic nonlinear system. As a consequence, different canonical forms of nonlinear dynamical models with suitable Lyapunov functions are proposed: Lurie systems, Lipschitz dynamics, homogeneous systems, etc.

The class of Persidskii systems can be considered in such a context. It was first studied in [START_REF] Barbashin | On construction of Lyapunov functions for nonlinear systems[END_REF] with a linear combination of the integrals of the nonlinearities as a Lyapunov function, and next in [START_REF] Persidskii | Problem of absolute stability[END_REF] the proposed kind of Lyapunov functions was complemented by a combination of the absolute values of the states. This class of models was investigated in the context of diagonal stability [START_REF] Kazkurewicz | Matrix Diagonal Stability in Systems and Computation[END_REF], neural networks [START_REF] Hopfield | Computing with neural circuits: a model[END_REF], [START_REF] Sontag | Recurrent networks[END_REF], sliding mode controls [START_REF] Aparicio | Stabilization of systems with switchings on the axis of their coordinates and its input-to-state properties[END_REF] and digital filters [START_REF] Erickson | Stability analysis of fixed-point digital filters using computer generated Lyapunov functions -Part I: Direct form and coupled form filters[END_REF]. In a recent work [START_REF] Efimov | Robust stability analysis and implementation of Persidskii systems[END_REF], for a kind of generalized Persidskii systems (the right-hand side of the dynamics includes a linear part, several sector nonlinearities and external disturbances) the input-to-state stability (ISS) conditions were proposed in the form of linear matrix inequalities (LMIs).

In many cases, the goal mode for a dynamical system is (periodic or chaotic) oscillation [START_REF] Fradkov | Introduction to oscillations and chaos[END_REF], [START_REF] Leonov | Frequency Methods in Oscillation Theory[END_REF], which can be entrained by an external force or generated by the system itself (then the latter case corresponds to self-or auto-oscillations). These oscillations can form a desired regime of operation, as in many natural systems [START_REF] Goldbeter | A model for circadian oscillations in the drosophila period protein (per)[END_REF], [START_REF] Izhikevich | Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting[END_REF], or be unwanted parasitic behavior, as chattering in sliding mode controls [START_REF] Shtessel | New approach to chattering analysis in systems with sliding modes[END_REF]- [START_REF] Levant | Chattering analysis[END_REF], but in all cases the conditions of their persistence are required for analysis and design. That is why this topic attracts many researchers, for examples see [START_REF] Fradkov | Introduction to oscillations and chaos[END_REF], [START_REF] Leonov | Frequency Methods in Oscillation Theory[END_REF], [START_REF] Goncalves | Global stability of relay feedback systems[END_REF]- [START_REF] Andrievsky | Hidden oscillations in aircraft flight control system with input saturation[END_REF] and references therein. A generic and useful theory for studying irregular oscillations was proposed by V.A. Yakubovich [START_REF] Yakubovich | Frequency oscillations conditions in nonlinear systems with stationary single nonlinearity[END_REF], [START_REF] Yakubovich | Conditions for self-induced oscillations in nonlinear systems[END_REF]. The conditions of oscillations in the sense of Yakubovich are based on existence of two Lyapunov functions [START_REF] Efimov | Oscillatority of nonlinear systems with static feedback[END_REF], [START_REF] Efimov | On conditions of oscillations and multihomogeneity[END_REF]. The first Lyapunov function ensures local instability of an equilibrium, while the second one provides attractiveness of a vicinity of that steady state, then under some mild conditions the presence of an oscillating mode can be established.

The aim of this work is to formulate the conditions of existence of oscillations in the sense of Yakubovich for the class of generalized Persidskii systems1 . To this end, the conditions of local instability at the origin and practical ISS are developed (the latter extends significantly the result obtained in [START_REF] Efimov | Robust stability analysis and implementation of Persidskii systems[END_REF]), which can be verified by solving LMIs (a constructive feature that is rarely available for strongly nonlinear systems). The analysis of oscillations is performed following [START_REF] Yakubovich | Conditions for self-induced oscillations in nonlinear systems[END_REF], [START_REF] Efimov | Oscillatority of nonlinear systems with static feedback[END_REF]. The proposed conditions are used to investigate robustness of nonlinear feedback applied to linear systems, and when a linear unmodeled dynamics appears in the control channel. Presence of such a dynamics may lead to instability of the system, then appearance of an oscillatory mode (a kind of chattering) implies that a certain closed-loop performance is preserved. Examples of application of the proposed theory to FitzHugh-Nagumo and Hindmarsh-Rose models of neurons are given.

The outline of this paper is as follows. The problem statement is given in Section II. Preliminary results are presented in Section III. Conditions of practical stability, instability and existence of oscillations are established in Section IV. Robustness against unmodeled dynamics is studied in Section V. For illustration of efficiency of the presented conditions, two neural models are considered in Section VI.

Notation • R + = {x ∈ R : x ≥ 0},
× R + → R + belongs to the class KL if β(•, s) ∈ K ∞ and β(s,
•) is a non-increasing function with lim t→+∞ β(s, t) = 0 for each fixed s ∈ R + . • The notation DV (x)v stands for the directional derivative of a continuously differentiable function V : R n → R evaluated at the point x with respect to a vector v ∈ R n . • Denote the identity matrix of dimension n × n by I n , the vector of dimension n with all elements equal 1 by 1 n . • diag{g} ∈ D n represents a diagonal matrix of dimension n × n with a vector g ∈ R n on the main diagonal, where D n ⊂ R n×n is the set of diagonal matrices. The set of diagonal matrices of dimension n × n with nonnegative elements will be denoted by

D n + = D n ∩ R n×n + . For Λ ∈ D n
+ , Λ i with i = 1, n corresponds to the i th element on the main diagonal.

• A series of integers 1, 2, ..., n is denoted by 1, n.

• In work, it is assumed that if the upper limit of a summation or a sequence is smaller than the lower one, then the corresponding terms have to be omitted and the conditions skipped.

II. PROBLEM STATEMENT

Consider the following class of extended Persidskii systems:

ẋ(t) = A 0 x(t) + M j=1 A j f j (x(t)) + d(t), t ≥ 0, (1)
where

x(t) = [x 1 (t) . . . x n (t)] ∈ R n is the state vector, x(0) ∈ R n ; and d(t) ∈ R n is the external perturbation, d ∈ L n ∞ ; f j (x) = [f j 1 (x 1 ) . . . f j n (x n )]
, f j (0) = 0, j = 1, M are continuous functions (the solutions of the system (1) exist in the forward time at least locally), the matrices A k ∈ R n×n for k = 0, M . Assumption 1. For any i = 1, n, j = 1, M :

sf j i (s) > 0 ∀s ∈ R \ {0}.
In the assumption above, it is stated that all nonlinearities belong to a sector and may take zero values at the origin only. Under this hypothesis, after a proper re-indexing and decomposition of f j in (1), there exist m ∈ {0, . . . , M } and µ ∈ {m, . . . , M } such that for all i = 1, n:

lim s→±∞ f z i (s) = ±∞, ∀z = 1, m; lim s→±∞ s 0 f z i (σ)dσ = +∞, ∀z = 1, µ.
Thus, it is supposed that some of the nonlinearities are radially unbounded, and m = 0 corresponds to the case when all nonlinearities are bounded (at least for negative or positive argument). It is also claimed that other part of these nonlinearities may have unbounded integral (clearly if m > 0, then all radially unbounded nonlinearities also have unbounded integrals and µ ≥ m due to the introduced sector condition).

If A r = 0 for all r = 0, M -1 and µ = M , then we recover the system studied by Persidskii in the conventional framework [START_REF] Persidskii | Problem of absolute stability[END_REF], where also was suggested to use a Lyapunov function

V (x) = n i=1 λ i |x i | with λ i > 0 for all i = 1, n.
Our goal is to propose conditions establishing a presence of oscillating trajectories in (1) (in the sense of Yakubovich, given below in Definition 6).

III. PRELIMINARIES

Consider a nonlinear system:

ẋ(t) = f (x(t), d(t)), t ≥ 0, (2)
where

x(t) ∈ R n is the state, d(t) ∈ R m is the external input, d ∈ L m
∞ , and f : R n+m → R n is a continuous function, f (0, 0) = 0. For initial condition x 0 ∈ R n and input d ∈ L m ∞ , define corresponding solutions by x(t, x 0 , d) for any t ≥ 0 for which the solutions exist.

A. Basic concepts

In this work we will be interested in the following stability properties [START_REF] Khalil | Nonlinear Systems[END_REF], [START_REF] Hahn | Stability of Motion[END_REF], [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF]: Definition 1. The system (2) is called input-to-state practically stable (ISpS), if there are some functions β ∈ KL, γ ∈ K and a constant c ≥ 0 such that

x(t, x 0 , d) ≤ β( x 0 , t) + γ(||d|| [0,t) ) + c ∀t ≥ 0 for any input d ∈ L m
∞ and any x 0 ∈ R n . The function γ is called the nonlinear asymptotic gain.

The system is called ISS if c = 0.

Definition 2. The system (2) is called globally practically stable if there are σ, γ ∈ K and c ≥ 0 such that

x(t, x 0 , d) ≤ σ( x 0 ) + γ(||d|| [0,t) ) + c ∀t ≥ 0 for any x 0 ∈ R n and d ∈ L m ∞ . For d = 0, (2)
is called Lagrange stable, and it is called just stable if c = 0.

It is straightforward to conclude that any ISpS system (2) is globally practically stable with σ(s) = β(s, 0). Definition 3. The system (2) is called unstable at the origin for d = 0, if there is an > 0 such that for any δ > 0, x(t , x 0 , 0) / ∈ B( ) for some x 0 ∈ B(δ) and some t ≥ 0.

These properties have the following Lyapunov/Chetaev function characterizations:

Definition 4. A smooth function V : R n → R + is called ISpS-Lyapunov function for the system (2) if there exist α 1 , α 2 , α 3 ∈ K ∞ , θ ∈ K and r ≥ 0 such that for all x ∈ R n and all d ∈ R m : α 1 ( x ) ≤ V (x) ≤ α 2 ( x ), DV (x)f (x, d) ≤ r + θ( d ) -α 3 ( x ). Such a function V is called ISS-Lyapunov function if r = 0.
For global practical stability it is enough that an ISpS-Lyapunov function exists for x > X with some X ∈ R + . Definition 5. A smooth function V : R n → R is called Chetaev function for the system (2) with d = 0 if V (0) = 0, and there exists 0 > 0 such that V + ∩ B( ) = ∅ for any ∈ (0, 0 ], where

V + = {x ∈ B( 0 ) : V (x) > 0}, and
DV (x)f (x, 0) > 0 ∀x ∈ V + .
Theorem 1.

[36] The system (2) is ISpS (ISS) if it admits an ISpS (ISS) Lyapunov function.

In the above theorem, existence of the corresponding Lyapunov function is also necessary for ISpS (ISS) property if f is assumed to be locally Lipschitz continuous. Theorem 2. [START_REF] Chetaev | The Stability of Motion[END_REF] The system (2) with d = 0 is unstable if it admits a Chetaev function.

Under additional mild restrictions, existence of a Chetaev function is also necessary for instability [START_REF] Efimov | On conditions of oscillations and multihomogeneity[END_REF].

B. Oscillations in the sense of Yakubovich

A function h : R n → R is called monotone if the condition x 1 ≤ x 1 , . . . , x n ≤ x n implies that either h(x 1 , . . . , x n ) ≤ h(x 1 , . . . , x n ) or h(x 1 , . . . , x n ) ≥ h(x 1 , . . . , x n ).
Definition 6. [START_REF] Yakubovich | Frequency oscillations conditions in nonlinear systems with stationary single nonlinearity[END_REF], [START_REF] Efimov | Oscillatority of nonlinear systems with static feedback[END_REF] For -∞ < π -< π + < +∞ a solution x(t, x 0 , 0) of the system (2) with x 0 ∈ R n and d = 0 is called [π -, π + ]-oscillation with respect to an output h(x) (where h : R n → R is a continuous monotone function) if the solution is defined for all t ≥ 0 and

lim inf t→+∞ h(x(t, x 0 , 0)) = π -, lim sup t→+∞ h(x(t, x 0 , 0)) = π + .
A solution x(t, x 0 , 0) of the system (2) is called oscillating, if there exist some output h and constants π -, π + such that the solution x(t, x 0 , 0) is [π -, π + ]-oscillation with respect to the output h. A forward complete system (2) with d = 0 is called oscillatory, if for almost all x 0 ∈ R n the solutions x(t, x 0 , 0) of the system are oscillating. The oscillatory system (2) is called uniformly oscillatory, if for almost all x 0 ∈ R n for the corresponding solutions x(t, x 0 , 0) there exist an output h and constants π -, π + independently on initial conditions.

In other words, the solution x(t, x 0 , 0) is oscillating if the output h(x(t, x 0 , 0)) is asymptotically bounded and there is no single limit value of h(x(t, x 0 , 0)) for t → +∞. Note that the term "almost all solutions" is used to emphasize that generally the system (2) has a nonempty set of equilibria. The concept of oscillations in the sense of Yakubovich is rather generic, it includes periodical oscillations (limit cycles), quasi-periodical, recurrent and chaotic trajectories.

Despite its complexity, the notion given in Definition 6 has a simple Lyapunov characterization. Theorem 3. [START_REF] Efimov | Oscillatority of nonlinear systems with static feedback[END_REF] Let the system (2) have two continuously differentiable Lyapunov functions V 1 : R n → R + and V 2 : R n → R + such that for υ 1 , υ 2 , υ 3 , υ 4 ∈ K ∞ the following inequalities are satisfied for all x ∈ R n :

υ 1 ( x ) ≤ V 1 (x) ≤ υ 2 ( x ), υ 3 ( x ) ≤ V 2 (x) ≤ υ 4 ( x ),
and for some 0

< X 1 < υ -1 1 • υ 2 • υ -1 3 • υ 4 (X 2 ) < +∞: DV 1 (x)f (x, 0) > 0 for all 0 < x < X 1 and x / ∈ Π; DV 2 (x)f (x, 0) < 0 for all x > X 2 and x /
∈ Π, where Π ⊂ R n is a set with zero Lebesgue measure containing all equilibria of the system, and

Ω ∩ Π = ∅ for Ω = {x ∈ R n : υ -1 2 • υ 1 (X 1 ) ≤ x ≤ υ -1 3 • υ 4 (X 2 )}.
Then the system (2) with d = 0 is oscillatory.

A Lyapunov function for the linearized system (2) at the origin is a candidate for V 1 (it can be replaced with Chetaev one [START_REF] Efimov | On conditions of oscillations and multihomogeneity[END_REF], as in this work). Instead of existence of the function V 2 one can require global practical stability of the system (2). In [START_REF] Efimov | Oscillatority of nonlinear systems with static feedback[END_REF], for a class of uniformly oscillatory systems (2) it is shown that these conditions are also necessary.

C. Stability of extended Persidskii systems

Stability conditions for the system (1) can be expressed in the form of LMIs: Theorem 4. [START_REF] Efimov | Robust stability analysis and implementation of Persidskii systems[END_REF] Let Assumption 1 be satisfied and there exist

P = P ∈ R n×n , Ξ k ∈ D n + for k = 0, M , Λ j ∈ D n + for j = 1, M , Υ s,j ∈ D n + for s = 0, M -1 and j = s + 1, M , Γ = Γ ∈ R n×n such that P > 0 or P ≥ 0, µ z=1 Λ z > 0; Γ > 0; Q ≤ 0; (3)
m k=0 Ξ k + 2 m s=0 m j=s+1 Υ s,j > 0,
where

Q =           Q1,1 Q1,2 Q1,3 • • • Q 1,M +1 P Q 1,2 Q2,2 Q2,3 • • • Q 2,M +1 Λ 1 Q 1,3 Q 2,3 Q3,3 • • • Q 3,M +1 Λ 2 . . . . . . . . . . . . . . . . . . Q 1,M +1 Q 2,M +1 Q 3,M +1 • • • Q M +1,M +1 Λ M P Λ 1 Λ 2 • • • Λ M -Γ           Q1,1 = A 0 P + P A0 + Ξ 0 ; Qj+1,j+1 = A j Λ j + Λ j Aj + Ξ j , j = 1, M ; Q1,j+1 = P Aj + A 0 Λ j + Υ0,j , j = 1, M ; Qs+1,j+1 = A s Λ j + Λ s Aj + Υs,j , s = 1, M -1, j = s + 1, M .
Then the system (1) is ISS.

The proof is based on the following Lyapunov function (recall that Λ j i is the i th element on the main diagonal of Λ j ):

V (x) = x P x + 2 M j=1 n i=1 Λ j i xi 0 f j i (s)ds, (4)
whose time derivative with respect to (1) admits an estimate for Q ≤ 0:

DV (x)f (x, d) ≤ -x Ξ 0 x - M j=1 f j (x) Ξ j f j (x) -2 M j=1 x Υ 0,j f j (x) -2 M -1 s=1 M j=s+1 f s (x) Υ s,j f j (x) +d Γd. (5)
The feature of the LMIs (3) consists of a fine treatment of all diagonal items in the off-diagonal blocks of Q represented by the matrices Υ s,j ≥ 0 for s = 0, M -1 and j = s + 1, M . Without introduction of Υ s,j all corresponding elements would be considered as perturbations, but actually due to the sector condition imposed on nonlinearities f j (x) in Assumption 1, these terms can be sign-definite and, hence, helpful. Remark 1. The condition Ξ 0 > 0 of Theorem 4 for the case m = 0 can be relaxed to Ξ 0 + 2 K j=1 Υ 0,j > 0 under hypothesis that the functions x i f j i (x i ) are radially unbounded for all i = 1, n and j = 1, K with some 1 ≤ K ≤ M .

If M = 1, then the system (1) approaches the Lurie form, and the related ISS analysis can be found in [START_REF] Arcak | Input-to-state stability for a class of Lurie systems[END_REF], [START_REF] Dashkovskiy | Practical examples of ISS systems[END_REF] (in the latter work a Lyapunov function similar to (4) is used).

IV. CONDITIONS OF STABILITY, INSTABILITY AND

OSCILLATIONS

In this section, first, the conditions of ISpS will be presented for (1) extending the result of Theorem 4 by using Theorem 1 and the Lyapunov function (4) (we will use these results to demonstrate global boundedness). Second, the conditions of instability of (1) with d = 0 at the origin will be given based on Theorem 2. Finally, the conditions of oscillatory behavior will be proposed combining the previously obtained results in the spirit of Theorem 3.

A. Global practical stability

To establish global practical stability property we can use the fact that it is related with ISS, whose conditions for (1) are given in Theorem 4. Note that for this property (since it is simpler than ISS) the derivative of the Lyapunov function (4) should be negative definite in x only for sufficiently big values of the state norm, and even its positive definiteness has to be ensured outside of a ball. These observations provide an insight how we are going to develop the result of Theorem 4, but before we need to introduce additional conventions on the shape and ordering of the functions f j (x), j = 1, M : The last condition implies that all f j i (x), j = 1, M have nonlinear and different asymptotic growth (this property also can be assured by putting the linear components of f j (x) in the linear part explicitly, the same for the functions of a common rate). Roughly speaking, we ask for some growth "uniqueness" of the nonlinearities f j (x), j = 1, M . Assumption 3. For any i = 1, n there exists Z i > 0 such that for all |s| ≥ Z i :

sf j i (s) > 0, ∀j = 1, M ; |f 1 i (s)| ≥ |f 2 i (s)| ≥ • • • ≥ |f Ri i (s)| ≥ |s| ≥ |f Ri+1 i (s)| ≥ • • • ≥ |f M i (s)|, where R i ∈ {0, . . . , M } is an index.
In the above inequalities, the limit terms f 0 i and f M +1 i in the cases R i = 0 and R i = M , respectively, have to be omitted. Under Assumption 2, any weighting these nonlinearities preserves the relations given in Assumption 3 with different Z i . These hypotheses are not restrictive, and in the most cases it can be respected by rearranging and reordering the nonlinearities in [START_REF] Khalil | Nonlinear Systems[END_REF]. For brevity of formulation of the result below define R = sign(min 1≤i≤n R i ) (we have the property 0 ≤ R ≤ m ≤ µ, where the indices m and µ save their meaning under Assumption 3).

Theorem 5. Let assumptions 2, 3 be satisfied and there exist indices κ ∈ {1, . . . , µ} and ι ∈ {R, . . . , m}, the matrices

P = P ∈ R n×n , Ξ k ∈ D n for k = 0, M , Λ j ∈ D n for j = 1, M ; Υ s,j ∈ D n for s = 0, M -1 and j = s + 1, M , Γ = Γ ∈ R n×n such that Λ j ≥ 0, j = 1, κ; Ξ s ≥ 0, Υ s,j ≥ 0, s = 0, ι, j = s + 1, ι; Γ > 0; P > 0 µ = 0 P ≥ 0, κ z=1 Λ z > 0 µ ≥ 1 ; ι s=0 Ξ s + 2 ι s=0 ι j=s+1 Υ s,j > 0; Q ≤ 0,
where the matrix Q is given in Theorem 4. Then the system (1) is globally practically stable.

If, additionally, Assumption 1 holds and Λ j ≥ 0, j = κ + 1, M , then the system (1) is ISpS.

The same claim can be formulated without the requirement on diagonal structure of Υ s,j for s = R, ι, j = ι + 1, M and s = ι + 1, M -1, j = s + 1, M or Ξ s for s = ι + 1, M , but at the price of a more complex writing, which is skipped.

Proof. Consider the Lyapunov function given in (4). Let us show that outside a ball around the origin, it is positive and radially unbounded, and its derivative with respect to (1) admits an estimate given in (5) that is negative and radially unbounded with respect to the state x, positive definite with respect to d and has a constant bias (as in Theorem 1).

To prove positivity of V for sufficiently big values of x , first, let us suppose that µ = 0 (hence, R = 0), then the term x P x contains the elements with the maximal amplitude and P should be positive definite (due to Assumption 2 any weighting the nonlinearities, even with negative multipliers, cannot outperform the quadratic growth in this case). Let R = 0 and µ > 0, then the matrix P can be nonnegative definite provided that the first κ biggest nonlinearities, which have unbounded integrals, form a positive definite function (that is possible for µ > 0, and if µ = 0 then the condition P > 0 becomes obligatory). In such a case there exists Z > max 1≤i≤n Z i such that V (x) takes only positive values and it is radially unbounded for all x ≥ Z (i.e., the terms with the maximal growth dominate the others for sufficiently big values of x ). For R = 1, Assumption 3 implies that in this case µ ≥ m ≥ 1. Then, either κ ≤ min 1≤i≤n R i and κ z=1 Λ z has to be positive definite (in such a case the matrix P may be arbitrary), or P has to be nonnegative definite. Again, positivity and radial unboundedness of V (x) for x ≥ Z is recovered as desired. The conditions given in the formulation of the theorem cover all these scenarios. Note that V (x) will be globally positive definite if Assumption 1 is satisfied and Λ j ≥ 0 for all j = 1, M .

To prove negative definiteness of V we will use similar arguments. The matrices Ξ s and Υ s,j for s = 0, ι and j = s + 1, ι in (5) represent the terms with biggest norm for x ≥ Z, where Z is a sufficiently big constant as before. According to conditions of the theorem they form a negative definite quadratic form with respect to f s (x) and x, and this form will dominate the rest items (the influence of them can be upper bounded by a bias q > 0 for x < Z, and for x ≥ Z they are covered by the first ι elements). For R = 0 the parameter m can take any nonnegative value, and if m = 0 (hence, ι = 0) the introduced condition leads to Ξ 0 > 0.

The obtained properties are sufficient for global practical stability (see Definition 2), and if additionally Assumption 1 is validated, Λ j ≥ 0 for j = 1, M and the function V is positive definite, then V is an ISpS-Lyapunov function, and due to Theorem 1 the system (1) is ISpS.

The imposed assumptions, 1, 2 and 3, introduce a rather complex hierarchy for the nonlinearities of the system (1). This is explained by the fact that the considered class of dynamical systems in [START_REF] Khalil | Nonlinear Systems[END_REF], and the stability problem itself, are rather generic and admit plenty of extensions or modifications. Remark 2. One such a development can be obtained by observing that the class of Persidskii systems (1) is invariant under a proper linear shift of the nonlinearities. Indeed, let for any j = 1, M there exist ϑ j ∈ R such that s f j i (s) > 0 ∀s ∈ R \ {0} for all i = 1, n, where f j (x) = f j (x)-ϑ j x, then Assumption 3 is verified for f j (x). Thus, a properly introduced auxiliary linear feedback does not change the sector property of the system nonlinearities, then the following equivalent dynamics can be studied:

ẋ(t) = Ã0 x(t) + M j=1 A j f j (x(t)) + d(t), t ≥ 0,
where Ã0 = A 0 + M j=1 ϑ j A j . Therefore, if for this new system the conditions of Theorem 5 are satisfied, then [START_REF] Khalil | Nonlinear Systems[END_REF] possesses the global practical stability property (or ISpS). The advantage of such a transformation is that the matrix Ã0 may have an improved stability characteristic (see the next section).

Remark 3. If Assumption 2 fails to satisfy, then the LMIs of Theorem 5 should be complemented by additional restrictions to guarantee that the meaningful terms with the same growth have a negative definite sum.

The results of this subsection introduce a series of tools for analysis of boundedness of trajectories in the extended Persidskii system (1), these tools can be combined and also relaxed considering particular scenarios and applications.

B. Local instability at the origin

For local instability, it is sufficient to ensure positivity of V at some vicinity of the origin and positive definiteness of the derivative V in the domain where V (x) > 0. To this end, let us introduce a reordering of the nonlinearities:

Assumption 4. Let lim sup s→0 |f j i (s)| |s| ∈ {0, +∞} and
lim sup s→0 |f j i (s)| |f k i (s)| ∈ {0, +∞} for all i = 1, n and all j = k = 1, M .
This condition reads that all f j i (x), j = 1, M have nonlinear and different local growth at the origin. Assumption 5. There exist Z > 0, η > 0 such that for all s ∈ (-Z, 0) ∪ (0, Z):

ηsf j i (s) ≥ s 0 f j i (σ)dσ > 0, ∀j = 1, M ; |f 1 i (s)| ≥ |f 2 i (s)| ≥ • • • ≥ |f Ri i (s)| ≥ |s| ≥ |f Ri+1 i (s)| ≥ • • • ≥ |f M i (s)|,
where R i ∈ {0, . . . , M } is an index for all i = 1, n.

As we can see, all restrictions are imposed for x < Z and there is no requirement on radial unboundedness of functions and integrals for a local investigation. Theorem 6. Let assumptions 4, 5 hold and there are indices κ ∈ {1, . . . , M } and ι ∈ {1, . . . , n}, the matrices

P = P ∈ R n×n ; Ξ k ∈ D n for k = 0, M ; Λ j ∈ D n for j = 1, M ; Υ s,j ∈ D n for s = 0, M -1 and j = s + 1, M such that Λ z ι ≥ 0, z = 1, κ; κ z=1 Λ z ι > 0 κ ≤ R ι P ι,ι ≥ 0, P ι,ι + κ z=1 Λ z ι > 0 κ > R ι ; Ξ 0 + ρP ≤ 0, Ξ j ≤ 0, Υ 0,j + ρηΛ j ≤ 0, j = 1, M , Υ s,j ≤ 0, s = 1, M -1, j = s + 1, M ; Q ≥ 0,
where ρ > 0 is a parameter and the matrix Q is given in Theorem 4. Then (1) with d = 0 is unstable at the origin.

Proof. Consider again the function V (x) given in (4) and its derivative with respect to (1) for the case d = 0, which admits a lower estimate for Q ≥ 0:

DV (x)f (x, 0) ≥ -x Ξ 0 x - M j=1 f j (x) Ξ j f j (x) -2 M j=1 x Υ 0,j f j (x) -2 M -1 s=1 M j=s+1 f s (x) Υ s,j f j (x).
Let us check the conditions of Theorem 2 showing that V is a Chetaev function for (1) under the hypotheses of the theorem. It is necessary to ensure that at least for one coordinate, say x ι with an index ι ∈ {1, . . . , n}, when all the coordinates x i = 0 for i = 1, n and i = ι, the function V (x) takes positive values. Indeed, if it is the case, then even for nonzero x i with i = ι it will be a small neighborhood around the origin containing the axis x ι with V (x) > 0. According to Assumption 5, the nonlinearities are sorted in the decreasing order, and if κ ≤ R ι , then the first κ nonlinearities are bigger than the linear part and other terms, Λ z ι ≥ 0 for z = 1, κ and κ z=1 Λ z ι > 0. Hence, V (x) is positive definite in x ι (we have to recall here assumptions 4 and 5). If κ > R ι , then in addition it is necessary to ensure nonnegative definiteness of P ι,ι x 2 ι , and we conclude on the same property of V . Finally, let us check that DV (x)f (x, 0) is locally positive definite in the domain where V (x) > 0. Under the conditions of theorem, DV (x)f (x, 0) ≥ ρV (x) that clearly implies the desired conclusion. The result follows by Theorem 2.

As previously for Theorem 5, the formulation of Theorem 6 admits various relaxations that are omitted for brevity (for example, the requirement on diagonal structure of Υ s,j for s = R, κ, j = κ + 1, M and s = κ + 1, M -1, j = s + 1, M , or Ξ s for s = κ + 1, M can be skipped).

C. Existence of oscillations

Following the result of Theorem 3, it is straightforward to formulate the required conditions: Theorem 7. Let the requirements of theorems 5, 6 be satisfied the system (1) with d = 0 admit the only equilibrium at the origin. Then (1) with d = 0 is oscillatory, and for any d ∈ L n ∞ the solutions stay bounded. Proof. Under these restrictions, the set Π from Theorem 3 contains just the origin. Fulfillment of the conditions of Theorem 5 ensures the global practical stability (it replaces existence of the function V 2 from Theorem 3), while Theorem 6 guarantees instability of the only steady state (it substitutes existence of the function V 1 from Theorem 3). Hence, there is a compact forward invariant set Ω ⊂ R n attracting asymptotically almost all trajectories of the system, and obviously Ω ∩ Π = ∅. Then similarly to the proof of Theorem 3, there is a coordinate x i with i ∈ {1, . . . , n} such that these trajectories are oscillating for h(x) = x i . Remark 4. It is clear that the LMIs given in theorems 5 and 6 are of the same kind, then it can be difficult to resolve them simultaneously. A way to fix this issue is to apply the linear shifts as in Remark 2 by checking, for example, the conditions of Theorem 6 for f j (x), j = 1, M and Ã0 .

Let us demonstrate the efficiency of the developed theory in applications.

V. ROBUSTNESS AGAINST UNMODELED DYNAMICS Consider a linear system:

ż(t) = Az(t) + Bu(t), (6)
where z(t) ∈ R ν and u(t) ∈ R are the state and the control vectors, ν > 0 and > 0 are some integers, A ∈ R ν×ν and B ∈ R ν× . Assume that there is a control input

u(t) = - M j=1 K j φ j (z(t)),
where M > 0 is an integer and φ j : R ν → R ν are secor nonlinear functions, φ j (z) = [φ j 1 (z 1 ) . . . φ j ν (z ν )] , φ j (0) = 0, j = 1, M ; K j ∈ R ×ν for j = 1, M ; and the closedloop system is stable and possesses a desired regulation performance (clearly it is a system in the form (1) with d = 0). Next, assume that in the control channel there is an unmodeled dynamics (actuator) represented by a linear stable filter:

u(t) = Cζ(t), υ(t) = - M j=1 K j φ j (z(t)), ζ(t) = Dζ(t) + Gυ(t), (7)
where ζ(t) ∈ R is the state of the filter for an integer > 0, υ(t) ∈ R is the control applied to the actuator but designed for the system (6), and u(t) ∈ R is the output, which is the control for the system (6);

C ∈ R × , D ∈ R × and G ∈ R × .
The closed-loop dynamics of ( 6) and ( 7) can be rewritten in the form of ( 1) with d = 0:

x = [z 1 , . . . , z ν , ζ 1 , . . . , ζ ] ∈ R n , n = ν + , f j (x) = [φ j 1 (x 1 ), . . . , φ j ν (x ν ), x ν+1 , . . . , x n ] , j = 1, M , A 0 = A BC 0 D , A j = - 0 G K j 0 , j = 1, M .
Therefore, using the result of Theorem 7 we can verify the oscillatory property of the closed-loop system under such an unmodeled actuator.

The previous problem statement is a well-studied case in the sliding mode control theory, where the observed oscillations are called "chattering" [START_REF] Shtessel | New approach to chattering analysis in systems with sliding modes[END_REF]- [START_REF] Bartolini | Chattering avoidance by second order sliding mode control[END_REF], [START_REF] Pisano | Contact force regulation in wire-actuated pantographsvia variable structure control and frequency-domain techniques[END_REF]- [START_REF] Levant | Chattering analysis[END_REF].

Example 1. Let ν = = 1, = 2 and A = 0, B = 1, C = [1 0], D = -δ δ 0 -δ , G = 0 δ , M = 1, φ 1 (z) = |z| α sign(z),
where α ∈ (0, 1) and δ > 0 are tuning parameters, K 1 > 0. Such a control is used for finite-time stabilization [START_REF] Bernuau | On homogeneity and its application in sliding mode[END_REF]. Hence,

A 0 =   0 1 0 0 -δ δ 0 0 -δ   , A 1 =   0 0 0 0 0 0 -δK 1 0 0   ,
and the matrix A 0 is not Hurwitz, it has one zero eigenvalue and two equal to -δ.

To check the conditions of Theorem 5 select

P = P > 0, Λ 1 = diag[Λ 1 1 , 0, 0] ∈ D 3 with Λ 1 1 > 0, Ξ 0 = diag[0, Ξ 0 2 , Ξ 0 3 ] ∈ D 3
+ with Ξ 0 2 > 0, Ξ 0 3 > 0, and the matrix Υ 0,1 ∈ R 3×3 in a generic form (full matrix) with requirement 6), [START_REF] Barbashin | On construction of Lyapunov functions for nonlinear systems[END_REF] for δ = 2, K 1 = 2 and α = 0.25 i.e., the first element of this matrix corresponding to the term |z| α+1 is positive. It is obvious that the Lyapunov function V (x) given in ( 4) is positive definite under introduced restrictions. It is also straightforward to verify that there exist the matrices as above such that Q ≤ 0 and in ( 5)

(Υ 0,1 ) 1,1 > 0,
DV (x)f (x, 0) ≤ -(Υ 0,1 ) 1,1 |z| α+1 -Ξ 0 2 ζ 2 1 -Ξ 0 3 ζ 2 2 + 2 (Υ 0,1 ) 2,1 |ζ 1 ||z| α + (Υ 0,1 ) 3,1 |ζ 2 ||z| α ,
then the required global boundedness of trajectories is proven. Indeed, applying Young's inequality for i = 1, 2 we obtain

2|ζ i ||z| α ≤ ε i ζ 2 i + 1 εi |z| 2α
, then ε i > 0 can be selected sufficiently small to satisfy ε i (Υ 0,1 ) i+1,1 < Ξ 0 i+1 , and |z| 2α is radially dominated by |z| α+1 for α ∈ (0, 1). Note that in the linear case, i.e., for α = 1, some additional restrictions on δ and K 1 have to be imposed to prove the boundedness.

To use Theorem 6 (see [START_REF] Fridman | Sliding Mode Control in Engineering[END_REF] for application of Chetaev function in similar context), take any ϑ 1 > 0 and shift the nonlinearity f 1 by linear feedback (as in Remark 2), then we obtain the new matrix

Ã0 = A 0 + ϑ 1 A 1 =   0 1 0 0 -δ δ -ϑ 1 δK 1 0 -δ   .
It is straightforward to verify that always there exists a value of ϑ 1 sufficiently big such that the matrix Ã0 has a positive eigenvalue in the subspace of the variable z. Since α ∈ (0, 1), then also for any value of ϑ 1 the function f j 1 (x) = f j 1 (x 1 )ϑ 1 x 1 stays in the required sector onto a vicinity of the origin. Take P

= P > 0, Λ 1 = diag[Λ 1 1 , 0, 0] ∈ D 3 + with Λ 1 1 > 0, Ξ 0 ∈ D 3 with Ξ 0
1 < 0, and Υ 0,1 ∈ D 3 with (Υ 0,1 ) 1,1 < 0, then the conditions of Theorem 6 are satisfied for ι = 1. By Theorem 7 the system is oscillatory. The results of simulation for δ = 2, K 1 = 2 and α = 0.25 are shown in Fig. 1 (in this case u = ζ 1 ).

VI. OSCILLATIONS IN NEURON MODELS

We will consider two models of neural dynamics.

A. FitzHugh-Nagumo model

This model describes a prototype of an excitable neuron, or a relaxation oscillator [START_REF] Izhikevich | Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting[END_REF]:

ẋ1 (t) = x 1 (t) - 1 3 x 3 1 (t) -x 2 (t) + I, ẋ2 (t) = 1 τ (x 1 (t) -bx 2 (t) + a),
where x 1 (t) ∈ R and x 2 (t) ∈ R are the membrane voltage and linear recovery variable, respectively; I ∈ R is a constant magnitude of external stimulus current; a ≥ 0 and b ≥ 0 are parameters (for a = b = 0 the system can be reduced to Van der Pol oscillator); t ≥ 0. If b ∈ (0, 1), then the system admits the only equilibrium with the coordinates x 10 ∈ R and x 20 ∈ R satisfying:

1 3 x 2 10 + 1 -b b x 10 = I - a b , bx 20 = x 10 + a,
and for brevity further we assume this constraint. Obviously, this system is in the form (1) with n = 2 and M = 1:

A 0 = 1 -1 1 τ -b τ , A 1 = -1 3 0 0 0 , f 1 (x) = x 3 1 x 3 2 , d = I a τ .
To check global boundedness of the solutions, Theorem 5 can be used directly with m = µ = R = 1, and in particular

V (x) = 1 2 (x 2 1 + τ x 2 2 ), V ≤ 3 2 x 2 1 - 1 3 x 4 1 - b 2 x 2 2 + 1 2 I 2 + a 2 2b
is an admissible choice. To verify local instability at the equilibrium applying Theorem 6 we will shift it to the origin via the change of coordinates e 1 (t) = x 1 (t) -x 10 , e 2 (t) = x 2 (t) -x 20 , then this error dynamics will preserve the form of the original system for d = 0 and f 1 (e) = e 1 (e 2 1 + 3x 10 (e 1 + x 10 )) e 2 (e 2 2 + 3x 20 (e 2 + x 20 ))

.

Next, for any value of x 10 and x 20 , at least in a vicinity of the origin, the sector condition is verified for this f 1 (e), then V (e) = e P e, P = P is the required Chetaev function (the Chetaev function for the linearized dynamics) that can satisfy the conditions of Theorem 6. In such a case, by Theorem 7, the system is oscillatory.

B. Hindmarsh-Rose model

This model of neuronal activity represents well the spikingbursting behavior of the membrane potential observed in vivo, and it is more complicated [START_REF] Izhikevich | Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting[END_REF]:

ẋ1 (t) = x 2 (t) -ax 3 1 (t) + bx 2 1 (t) -x 3 (t) + I, ẋ2 (t) = c -kx 2 1 (t) -x 2 (t), ẋ3 (t) = r[s(x 1 (t) -x R ) -x 3 (t)],
where again x 1 (t) ∈ R is the membrane potential, and x 2 (t), x 3 (t) ∈ R represent the transport of ions across the membrane (x 2 (t) stays for sodium and potassium ions, and it is called the spiking variable; x 3 (t) corresponds to other ions, it is the bursting variable); I ∈ R is the stimulus current, a, b, c, k, r, s, x R ∈ R + are parameters. This system can be also presented in the form [START_REF] Khalil | Nonlinear Systems[END_REF]

VII. CONCLUSIONS

The most existing works on Persidskii systems deal with their standard form (i.e., with M = 1 in (1)), and the case with M > 1 was for the first time studied recently in [START_REF] Efimov | Robust stability analysis and implementation of Persidskii systems[END_REF] (multiple nonlinearities enlarge the domain of possible applications), where a careful treatment of cross terms between f j and x in the derivative of V is developed in order to obtain non-conservative stability conditions. Following [START_REF] Efimov | Robust stability analysis and implementation of Persidskii systems[END_REF], the present work extends those results to practical ISS analysis (in Theorem 5 almost all conditions of Theorem 4 are relaxed) and to instability verification (Theorem 6) using LMIs. The obtained in Theorem 7 conditions of existence of oscillations in the sense of Yakubovich were never proposed for any kind of Persidskii systems. The results are used to investigate robustness to linear unmodeled dynamics of nonlinear controls for linear systems. The efficiency of the obtained conditions is illustrated on models of neurons.

Assumption 2 .

 2 Let lim sup s→±∞ |f j i (s)| |s| ∈ {0, +∞} and lim sup s→±∞ |f j i (s)| |f k i (s)| ∈ {0, +∞} for all i = 1, n and all j = k = 1, M .

Figure 1 .

 1 Figure 1. Simulation results of (6), (7) for δ = 2, K 1 = 2 and α = 0.25

  with n = 3 and M = 2: ) = (x 2 + x + ϑ)x and ϑ > 0 is a parameter. Let us verify only global boundedness of the solutions using Theorem 5 (the local instability can be checked by considering linearized dynamics at equilibria). Note that Assumption 2 is not satisfied in this case, then the LMIs of Theorem 5 complemented by restriction given in Remark 3 are feasible with m = µ = R = 2 and κ = ι = 1 for a = 1, b = 3, k = 0.5, s = 4, r = 10 -3 and ϑ = 0.1.

  where R is the set of real numbers. • | • | and • denote the absolute value in R and the Euclidean norm on R n , respectively; for any > 0 define an open ball around the origin by B( ) = {x ∈ R n : x < }. • For a (Lebesgue) measurable function d : R + → R m define the norm ||d|| [t0,t1) = ess sup t∈[t0,t1) d(t) , then ||d|| ∞ = ||d|| [0,+∞) and the set of d with the property ||d|| ∞ < +∞ we further denote as L m ∞ (the set of essentially bounded measurable functions). • A continuous function α : R + → R + belongs to the class K if α(0) = 0 and the function is strictly increasing. The function α : R + → R + belongs to the class K ∞ if α ∈ K and it is increasing to infinity. A continuous function β : R +

A preliminary version of this work[START_REF] Wang | On existence of oscillations in Persidskii systems[END_REF] contains sketched proofs without analysis of neural models.

This work is partially supported by 111 project No. D17019 (China), by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898, and by the Russian Foundation for Basic Research (grant no. 19-01-00146-a).


Dr Liang Chang

Nathalie Fischer-Durand

Dr Geoffrey Gontard

Prof. Dr Benoît Bertrand

Serge Thorimbert
email: serge.thorimbert@sorbonne-universite.fr

Luc Dechoux
email: luc.dechoux@sorbonne-universite.fr

A solvent-free, catalyst-free formal [3+3] cycloaddition dearomatization strategy: towards new fluorophores for biomolecules labelling

A general, sustainable dearomatization reaction for nitrogen-containing heterocycles was developed. Under solvent free conditions and without catalyst, the biorenewable methyl coumalate (MC) reacts as an efficient C3 partner to convert nine types of basic aromatic rings into their pyrido[1,2-a] fused derivatives in good to excellent yields. The fluorescence properties of some of the products were harnessed to conjugate fluorescent tags to BSA and immunoglobulin G.

 These electronic characteristics may sometimes allow, after interaction with the target, a change in the fluorescence properties. In this article we present the synthesis of such platforms with such electronics features, based on the dearomatization of pyridine or other nitrogen-based aromatics.

To access conjugated non-aromatic polycyclic scaffolds, dearomatization reactions have been continuously regarded as an attractive strategy.

challenging. Generally, the construction of such polycyclic compounds relies on the direct addition of nucleophile to preactivated pyridinium intermediates, known as the Reissert reaction. [4] Another strategy relies on the internal electrophile-triggered C-N bond forming dearomatization reactions. For instance, You and coworkers reported Ir-catalyzed intramolecular asymmetric allylic dearomatization reaction of pyridines and pyrazines, [5] quinolines and (iso)-quinolines, [6] benzoxazoles, benzothiazoles and benzimidazoles. [7] Unsworth's group reported a high-yielding silver(I)-or copper(II)-catalyzed dearomatizing intramolecular cyclization strategy, which allows the conversion of simple aromatic compounds containing ynone substituents, including indole, anisole, pyrrole and benzofuran derivatives, into functionalized quinolizinone scaffolds. [8] In these methods, allylic and propargylic substitutions are mainly achieved via N-nucleophilic aromatics with C-electrophiles. However, this method requires not only the pre-synthesis of electrophiles or their precursors, but also the use of transition metals as catalyst. In the recent years, most of the new dearomatization reactions have consisted in cycloaddition (CA) reactions or formal CA. This includes [2+2]-, [9] [3+2]-, [10] [3+3]-, [11] [4+2]-, [12] [5+1]-, [13] [4+3]-CA. [14] In most of the cases theses CA reactions concern indole derivatives as substrates.

We have shown that methyl coumalate (MC) reacted as an equivalent of enal with 1,3-dicarbonyl derivatives in formal [3+3] CA giving access to the efficient synthesis of dihydropyrans. The transformation starts by a formal Knoevenagel reaction, which implies a cascade -1,6-addition on MC -6p-electrocyclic opening -1,5-proton shift, leading to the intermediate 1-oxatriene.

The final 6p-electrocyclization gives the dihydropyrans (Scheme 1a). [15] Additional advantage of using MC as a substrate lies in its biorenewable nature, which allows the reaction to run in an environmental-friendly manner. In the context of CA, pyridines have been scarcely used, and especially for [3+3] reactions. [11] We postulated that pyridylacetates would react with MC as for the 1,3dicarbonyl derivatives to form an intermediate 1-azatriene which would undergo a 6p-electrocyclization leading to the formation of quinolizine scaffolds (Scheme 1b). Herein we describe a general, sustainable dearomative strategy using the bio-based MC for the conversion of nitrogen-based aromatic rings into their pyrido [1,2-a] fused derivatives.

Scheme 1. Formal [3+3] cycloadditions

The self-catalyzed imine-enamine tautomerism in pyridyl acetate [16] would generate the nucleophilic enamine required for the reaction to begin. Thus, the reaction could be realized in solvent-free and catalyst-free conditions. These mild reaction conditions were successfully applied to a wide range of different aza-aromatic substrates. To begin with, six-membered aza-aromatics, such as pyridine and pyrazine, afforded the corresponding quinolizine (D-2a) and pyro-pyrazine (D-2b) products in high yield (89%quant) (Scheme 2). Benzene and pyridine-fused six-membered aza-aromatics, such as quinoline, quinoxaline and naphthyridine, also underwent such reaction with the dearomative products (D-2c, D-2d, D-2e) obtained in 81-87% yield. Scheme 2. Substrate scope for six-membered aza-aromatics a) our previous work [15] O Moreover, five-membered aza-heterocycles also proved to be suitable substrates for this mild dearomatization reaction (Scheme 3). Thiazole, benzoxazole, benzothiazole and unprotected benzoimidazole were all well tolerated, providing their corresponding bi-and tri-cyclic products (D-2f-D-2i) in good to excellent yields. Crystals suitable for X-Ray diffraction were obtained for tricyclic product D-2i. Scheme 3. Substrate scope for five-membered aza-aromatics Substituted pyridines are also suitable feedstocks for this reaction, leading to corresponding quinolizine products in good to excellent yields (Scheme 4). The plausible mechanism for this reaction is illustrated in Scheme 5. The nucleophilic enol D1-a' coming from the tautomeric equilibrium of pyridyl ester D-1a undergoes a 1,6 addition on MC. [17] The resulting intermediate A could run a 6π electrocyclic ring-opening to give B, [18] followed by a 1,5-proton transfer, opening access to 1-azatriene C. We could not rule out a bimolecular elimination -protonation from A to C but the cascade via B is privileged based on the previous results we observed with Grignard reagents as nucleophiles. [17a, 18a] This transformation from starting materials D-1 to 1-azatriene C corresponds to a formal Knoevenagel reaction. A final 6-p electrocyclization furnishes the quinolizine D-2a. The presence of two EWG groups at C3 and C5 of the 1-azahexatriene accelerates and favors the 6p electrocyclic process. [19] Scheme 5: Plausible mechanism

R OEt O Formal Knoevenagel O R OEt O CO2Me HO2C 6π-electrocyclization O R OEt O CO2Me b) this work HO2C O O MeO2C N EWG Formal Knoevenagel N EWG CO2Me HO2C 6π-electrocyclization N EWG CO2Me HO2C Formal [3+3] O O MeO2C R R R Formal [3+3] Y N X CO 2 Et X = CH, N Y = CH, N O O MeO 2 C + MC neat 110 °C Y N X CO 2 Et CO 2 Me HO 2 C X = CH, N Y = CH, N N CO 2 Et CO 2 Me HO 2 C D-2a: 3 h, quant. N N CO 2 Et
In addition to be readily performed, this dearomatization reaction, also generates novel structures with potential applications in biological endeavors. The prepared molecules present a carboxylic acid function ready for the grafting onto biomolecules. Thus, the dearomatized products presenting this carboxylic acid moiety were then explored for their emission properties (table S1). It appears that while all 6-membered dearomatized products were poorly to non-emissive, 5-membered dearomatized products (D2f-i) presented interesting emission properties in the blue region (441-479 nm) with a shift of the emission wavelength depending on the second heteroatom present in the 5-membered cycle. Among the acquired products, D-2i showed the highest quantum yield (table S1), affording a potential fluorescent marker for the labelling of various biomolecules including proteins, DNA etc. [20] As a model protein, we chose bovine serum albumin (BSA) that has 35 accessible amine functions from lysine residues for potential conjugation. D-2i was first activated into its N-succinimidyl ester and added to a solution of BSA in sodium phosphate buffer pH 7.8 (Figure S1). After reaction, the conjugate was purified by extensive dialysis and ultrafiltration. The average number of fluorescent probe D-2i conjugated per BSA was evaluated by UV-Vis spectroscopy by measuring the concentrations of BSA and D-2i using calibration curves established at their maximal absorbance wavelengths (lmax) (280 and 420 nm respectively). The conjugation ratio was estimated to be 9 starting with 30 eq. of D-2i per BSA. The covalent nature of the linkage between the protein and D-2i was evidenced by performing an assay without activation of the acid with TSTU. After purification, we detected neither absorbance at 420 nm nor fluorescence emission at lex= 420 nm (figure S1, black line) showing that there are no nonspecific interactions. Conjugate BSA-(D-2i)9 showed strong fluorescence emission at 490 nm upon excitation at 420 nm in PBS pH 7.4 whereas excitation of BSA alone at 420 nm did not result in any fluorescence emission (Figure S1).

However, to ensure high selectivity and site-specificity, it has become appealing to design bio-conjugation reactions based on functional groups not naturally occurring in biomacromolecules. [21] Among these reactions, the coupling between an acylhydrazine and an aldehyde to form an acyl-hydrazone bond has been frequently used, especially for the site-selective labelling of glycoproteins. As a model of glycoprotein, we chose an immunoglobulin G (IgG). This choice was motivated by the relevance of fluorophore-antibody conjugates in the field of immunoassays. [START_REF] Christopoulos | Immunoassay[END_REF] For this purpose, the acid group of D-2i was activated into its Nsuccinimidyl ester and converted to the protected acyl-hydrazine D-2i-1 upon reaction with t-Bu-carbazate. Cleavage of the Boc group by HCl in ether/chloroform led to the acyl hydrazine hydrochloride D-2i-2 (Figure 1a). Site-selective labelling of IgG was then performed in two steps. First, the carbohydrate moieties located on the Fc region of IgG were oxidized with sodium periodate in acetate buffer (pH 4.0) in order to generate aldehyde groups. [23] Subsequent reaction with a large excess of acyl hydrazine D-2i-2 (20 eq.) in acetate buffer pH 5.5 for 18 h at 25°C afforded acyl hydrazone bonds that were reduced with NaBH3CN to form more stable acyl hydrazine bonds. The acidic conditions used for the oxidation step and the formation of the acyl hydrazone bonds ensured that aldehyde groups reacted only with acyl hydrazine groups and not with amine groups, thus avoiding protein homo-coupling side-reactions. After dialysis and ultrafiltration in order to remove unbound D-2i-2, the average number of fluorescent molecules conjugated per IgG was again evaluated by UV-Vis spectroscopy by measuring the concentrations of D-2i-2 and IgG using calibration curves established at 280 and 420 nm. This conjugation rate was estimated to be 6, which is consistent with the average number of labelling sites expected to be generated given the chosen oxidation conditions (10 mM NaIO4, pH 4,0, 1h). [21] Again, nonspecific interactions were not observed when incubating IgG and D-2i-2 without the oxidation step (i.e. no absorbance at 420 nm and no fluorescence detected after dialysis: figure 1c, black line: IgG recovered). The fluorescence emission spectrum of IgG-( D-2i-2)6 conjugate was recorded in PBS pH 7.4 upon excitation at 420 nm and showed strong fluorescence emission at 490 nm whereas excitation of IgG alone at 420 nm did not result in any fluorescence emission (Figure 1).

In summary, we outlined a novel formal [3+3] cycloaddition dearomatization reaction, that consists in a tandem formal Knoevenagel-6p-electrocyclization with an unprecedented pyrido [1,2-α] fused heterocyclic large scope. This reaction can tolerate eleven types of aza-aromatics. This protocol not only employs a biorenewable diene precursor as feedstock without the need of solvent nor catalyst, but also builds the C-C and C-N bond in one pot, meeting the requirement of green chemistry in both environmental-friendly and step-economical aspects. Furthermore, the products possess novel structures which may serve as new fluorescent dyes, providing promising probes for protein labelling and relevant biological assays.

Scheme 4 . 1 N

 41 Scheme 4. Substrate scope for pyridine derivatives.

Figure 1 .

 1 Figure 1. D-2i and derivatives in fluorescent IgG labelling.

Acknowledgements

We thank the Sorbonne Université and CNRS for funding. L.C acknowledges the China Scholarship Council (CSC) for PhD scholarship.

Keywords: dearomatization • nitrogen heterocycles • Michael addition • methyl coumalate • fluorescent probe


Tania S Vergara Gomez

Marc Dubois

Stanislav Glybovski

Benoit Larrat

Julien De Rosny

Carsten Rockstuhl

Monique Bernard

Redha Abdeddaim

Stefan Enoch

Frank Kober

Resonant and non-resonant coupled-wire coils for small-animal multinuclear imaging

Metasurface coils coupled with an unmatched loop were suggested earlier as an alternative to classical surface and volume coils, with the aim to optimize volume and sensitivity. We developed a new approach combining a commercial surface coil and a non-resonant coupled-wire structure. With this approach, the signal contribution from the driving loop can be efficiently added to that of the wire surface. The configuration was simulated, built and tested for 1 H and 19 F at 7T. The results showed that this new strategy improves the coil's sensitive volume while simultaneously maintaining high SNR.

Introduction

Commercial MR coils can be divided in two categories: volume and surface coils. Volume coils such as birdcage coils [1,2] offer large sensitive volume, but low SNR [3]. In contrast, surface coils present higher sensitivity, resulting in high SNR [4]. Tradeoff between large FOV and sensitivity has been targeted in earlier works on metamaterial RF coils coupled with a non-matched loop [5,6,7]. This configuration presented so far the best compromise between SNR and sensitive volume [7]. Nevertheless, a small surface coil [START_REF] Hoult | The NMR receiver: a description and analysis of design[END_REF] still provided the highest SNR. Therefore, the aim of this work was to investigate if advantages can be obtained by combining both strategies. We propose an elementary structure consisting of two parallel wires with adjustable length to switch between resonant and non-resonant regimes. We hypothesized that beyond their use in resonant regime, such structures can be used off-resonance and driven by commercial coils.

Methods

Three cases of coils were analyzed as single channel transmit-receive antenna for proton and fluorine imaging. Numerical simulations were carried out (Figure 1) to study the current distribution in the coils and their spatial B 1 distribution. Case (a): surface coil (loop with a matched circuit), case (b): resonant coupled-wire coil (non-matched loop with two wires of length L 0 ) and case (c): non-resonant coupled-wire coil (matched loop with two detuned wires of length L<L 0 ) were simulated.

Prototypes of the simulated combinations of coils were built and tested for proton and fluorine with a phantom and in vivo using a Bruker PharmaScan 7T MR system. A commercial 1 H birdcage coil was included in the experiments as additional reference for proton imaging.

Results

Figure 1D shows the maximum of the current amplitudes in the loop (red curve) and the coupled-wire structure (blue curve) in function of the wires length. Figure 1E shows the spatial B 1 distribution on the horizontal axis. It can be seen that the highest B 1 magnitude depends on the loop current while the enlargement of the sensitive volume depends on the wire current. Case (c) benefits from both contributions, resulting in a wide FOV and a high sensitivity.

SNR maps and profiles of the phantom are shown in Figure 2. As expected from the numerical predictions, case (c), featured a notable enlargement of the sensitive volume compared to the surface coil alone (case (a)) while keeping a high SNR.

According to Figure 3, the coils showed a similar SNR distribution for fluorine as for proton. Figures 4 and5 show the results of the in vivo experiments for 1 H and 19 F, respectively and proved that the behavior of the coils remained consistent in in vivo situations.

Discussion

The numerical studies show that when using a case (b) coil, the contribution of the loop is reduced due to the extremely low current flowing in it, resulting in a low B 1 . On the contrary, the current in the wires is almost constant, leading to a wide FOV.

Note that an attempt to benefit from both contributions by combining a matched loop with a resonant coupled-wire structure would be impeded by the strong mutual coupling [START_REF] Mispelter | NMR probeheads for biophysical and biomedical experiments: theoretical principles & practical guidelines[END_REF]. To mitigate this and still benefit from both contributions we proposed a new approach combining a detuned coupled-wire structure with a surface coil (case (c)). This strategy allowed us to increase the loop current without strongly modifying the current within the wires, resulting in a coil with higher sensitivity and large FOV.

The simulation results were validated experimentally on a phantom and on mice in vivo imaging. The obtained results perfectly corroborate the numerical predictions from Figure 1. As expected, case (b) has a homogeneous but low SNR. Case (c) preserves the high SNR of the surface coil, but a significant enlargement of the sensitive volume can be clearly seen, particularly in vivo.

Conclusion

Coupled-wire structures can be used as RF coils in both resonant and non-resonant regimes. The chosen regime will affect the current amplitude distribution within the coil and therefore the spatial distribution of the B 1 field. Experiments on phantom and in vivo confirmed the numerical predictions by coupling a carefully detuned coupled-wire structure with a commercial surface coil. It was further demonstrated that coupled-wire coils provide enough SNR to obtain 19 F images at low concentrations. We believe that such structures bring a new and flexible alternative to the design of versatile RF coils, with the aim to mitigate the conventional trade-off between FOV and SNR.

Figures

  Figures

FIGURE 1 :

 1 FIGURE 1: Numerical analysis of: (A) surface coil with a matching circuit, (B) resonant coupled-wire coil conformed by two wires with length L 0 coupled with an unmatched feed loop and (C) non-resonant coupled-wire coil that combines a matched surface coil with two wires of length L < L 0 . Each configuration of the coil was tuned and matched to 300 MHz. (D) shows the plot of the current amplitudes in the loop (red) and in the wires (blue). (E) shows the B 1 profiles obtained along a horizontal line along the wires at 1 cm above the wire plane.

FIGURE 2 :

 2 FIGURE 2: SNR maps in gray scale with coronal and sagittal orientation of the phantom. The dash lines locate the profile cuts. (A),(E) birdcage coil; (B),(F) case B; (C),(G) surface coil; (D),(H) case C. (I) Coronal-sagittal profiles obtained from the cyan line. (J) Sagittal-axial profiles obtained from the red line. (K) Axial-coronal profiles obtained from the yellow line. Noise standard deviation is given on top of each map.

FIGURE 3 :

 3 FIGURE 3: 19 F T 2 -turboRARE-3D images of the phantom in jet color map acquired with: (A) case B ; (B) surface coil and (C) case C in sagittal orientation. The images were overlaid on top of 1 H FLASH images in gray scale.

FIGURE 4 :

 4 FIGURE 4: In vivo 1 H SNR maps. Sagittal and coronal-plane images with: (A), (E) birdcage coil; (B), (F) case B; (C), (G) surface coil and (D), (H) case C.

FIGURE 5 :

 5 FIGURE 5: In vivo 19 F FLASH coronal-plane images in jet scale obtained with: (A) case B; (B) surface coil and (C) case C. The images were overlaid on top of 1 H FLASH images in gray scale.