Adrien Laurent

Hans Z Munthe-Kaas

The universal equivariance properties of exotic aromatic B-series

Keywords: Butcher series, exotic aromatic B-series, equivariance, geometric numerical integration, stochastic differential equations AMS subject classification (2020): 15A72, 37C81, 41A58, 60H35, 65C30

Exotic aromatic B-series were originally introduced for the calculation of order conditions for the high order numerical integration of ergodic stochastic differential equations in R d and on manifolds. We prove in this paper that exotic aromatic B-series satisfy a universal geometric property, namely that they are characterised by locality and orthogonal-equivariance. This characterisation confirms that exotic aromatic B-series are a fundamental geometric object that naturally generalises aromatic B-series and B-series, as they share similar equivariance properties. In addition, we classify with stronger equivariance properties the main subsets of the exotic aromatic B-series, in particular the exotic B-series. Along the analysis, we present a generalised definition of exotic aromatic trees, dual vector fields, and we explore the impact of degeneracies on the classification.

Introduction

Consider the ordinary differential equation y I ptq f pyptqq, yp0q y 0 , (1.1) where f : R d Ñ R d is a Lipschitz vector field and y 0 R d , and a one-step integrator for solving (1.1) of the form y n 1 Φpy n , hq, (1.2) where h is the timestep of the method. Following the backward error analysis idea [START_REF] Hairer | Geometric numerical integration[END_REF], in order to study the properties of the integrator (preservation of invariants or measures, order, behaviour in long-time,. . . ), it proves convenient to rewrite the scheme as the exact solution of a modified ODE ỹI ptq f pỹptqq.

For large classes of integrators, such as Runge-Kutta methods, the modified vector field f can be expressed as a formal Taylor series in f and its partial derivatives, called a B-series [START_REF] Chartier | Algebraic structures of B-series[END_REF]. The paper [START_REF] Munthe-Kaas | Aromatic Butcher series[END_REF] presents universal geometric conditions on f to show that it can be written as a B-series. More precisely, any smooth local map that is invariant under affine change of coordinates can formally be written as a B-series.

Originally introduced in [START_REF] Butcher | An algebraic theory of integration methods[END_REF][START_REF] Hairer | On the Butcher group and general multi-value methods[END_REF], Butcher series proved to be a powerful tool for the construction of numerical integrators for solving ODEs with a high order of accuracy and preserving geometric properties (see the textbooks [START_REF] Hairer | Geometric numerical integration[END_REF][START_REF] Butcher | Numerical methods for ordinary differential equations[END_REF][START_REF] Butcher | B-series: algebraic analysis of numerical methods[END_REF] and the review [START_REF] Mclachlan | Butcher series: a story of rooted trees and numerical methods for evolution equations[END_REF]). The papers [START_REF] Chartier | Preserving first integrals and volume forms of additively split systems[END_REF][START_REF] Iserles | B-series methods cannot be volumepreserving[END_REF] introduced simultaneously an extension of B-series called aromatic B-series for the study of volume preserving integrators (see also the recent works [START_REF] Bogfjellmo | Algebraic structure of aromatic B-series[END_REF][START_REF] Bogfjellmo | Using aromas to search for preserved measures and integrals in Kahan's method[END_REF][START_REF] Lejay | Constructing general rough differential equations through flow approximations[END_REF][START_REF] Laurent | The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators[END_REF][START_REF] Bronasco | The Hopf algebra structures of composition and substitution of exotic clumped S-series[END_REF]). In this context, one is interested in finding methods (1.2) satisfying divp f q 0. The aromatic B-series proves to be a crucial tool as the standard operations on vector fields, the divergence operator, and Taylor expansions rewrite conveniently in aromatic B-series. One then wonders whether Bseries and aromatic B-series are merely tools for manipulating tedious Taylor expansions or natural far-reaching algebraic objects. This question is answered in [START_REF] Mclachlan | B-series methods are exactly the affine equivariant methods[END_REF][START_REF] Munthe-Kaas | Aromatic Butcher series[END_REF] where universal geometric characterisations of B-series and aromatic B-series are given (see also [START_REF] Markl | GL n -invariant tensors and graphs[END_REF][START_REF] Mclachlan | Butcher series: a story of rooted trees and numerical methods for evolution equations[END_REF]).

In the context of stochastic differential equations (SDEs), it is known that there is no backward error analysis in the strong sense in general [START_REF] Shardlow | Modified equations for stochastic differential equations[END_REF]. However, there exists a similar idea for ergodic SDEs. Consider overdamped Langevin dynamics with Stratonovich noise of the form dY ptq Π M pY ptqqfpY ptqqdt Π M pY ptqq ¥ dW ptq, Y p0q Y 0 , (1.3) where f : R d Ñ R d is a Lipschitz vector field, Y 0 R d , Π M pxq is the orthogonal projection on the tangent bundle of M at the point x M (note that Π M pxq I d if M R d ), and W is a standard d-dimensional Brownian motion in R d on a probability space equipped with a filtration and fulfilling the usual assumptions. Under a growth assumption on f , the solution of (1.3) is ergodic, that is, it follows a deterministic distribution, called the invariant measure, in long time [START_REF] Faou | Conservative stochastic differential equations: mathematical and numerical analysis[END_REF][START_REF] Debussche | Weak backward error analysis for SDEs[END_REF][START_REF] Abdulle | High order numerical approximation of the invariant measure of ergodic SDEs[END_REF]. In [START_REF] Laurent | Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs[END_REF][START_REF] Laurent | Order conditions for sampling the invariant measure of ergodic stochastic differential equations on manifolds[END_REF][START_REF] Laurent | Algebraic Tools and Multiscale Methods for the Numerical Integration of Stochastic Evolutionary Problems[END_REF], an extension of the aromatic B-series, called exotic aromatic B-series, is introduced to write conveniently Taylor expansions (called Talay-Tubaro expansions [START_REF] Talay | Expansion of the global error for numerical schemes solving stochastic differential equations[END_REF] in this context) of the solutions of (1.3) and to build high-order approximations of the invariant measure of (1.3), with applications in molecular dynamics [START_REF] Lelièvre | Free energy computations[END_REF]. The main idea is to introduce two new types of edges to represent the Laplacian and the scalar product. Ergodic integrators for solving (1.3) have an invariant measure that can be written as the invariant measure of an exact problem of the form (1.3) with a modified vector field f that typically has the form (1.4) where f i is the vector basis of R d , the c n are real constants, and each term is summed on all involved indices. Note that the expansion (1.4) is a linear combination of monomials in the components f i and their partial derivatives, where we use pairs of indices. Note also that the power of h associated to a monomial is not given by the number of occurrences of f , in opposition to the deterministic context. For the integrators presented in [START_REF] Laurent | Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs[END_REF][START_REF] Laurent | Order conditions for sampling the invariant measure of ergodic stochastic differential equations on manifolds[END_REF] for solving (1.3), the modified vector field can be expressed as an exotic B-series [START_REF] Bronasco | The Hopf algebra structures of composition and substitution of exotic clumped S-series[END_REF] in R d (see examples in [START_REF] Laurent | Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs[END_REF]Sec. 5.1]) and as a partitioned exotic aromatic B-series on manifolds at least for the first orders [START_REF] Laurent | Order conditions for sampling the invariant measure of ergodic stochastic differential equations on manifolds[END_REF][START_REF] Bronasco | The Hopf algebra structures of composition and substitution of exotic clumped S-series[END_REF]. Moreover, the Talay-Tubaro expansions presented in [START_REF] Laurent | Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs[END_REF][START_REF] Laurent | Order conditions for sampling the invariant measure of ergodic stochastic differential equations on manifolds[END_REF] are exotic aromatic S-series [START_REF] Bronasco | Exotic B-series and S-series: algebraic structures and order conditions for invariant measure sampling[END_REF].

h f hf i f i h 2 rc 1 f i jj c 2 f i j f j c 3 f j f j f i sf i h 3 rc 4 f i jjkk c 5 f j k f i jk c 6 f k f k f j f j f i sf i . . .
For the high-order approximation of (1.3), the number of terms in the Taylor expansions explodes quickly, which makes the exotic aromatic B-series a crucial tool for the study of integrators for solving SDEs (see, for instance, the order two expansion in [24, App. D]). A natural question is the following: are exotic aromatic B-series just a technical tool used for carrying out tedious calculations? Or are they fundamental objects satisfying similar geometric properties as B-series and aromatic B-series? In this paper, we show that the exotic aromatic B-series satisfy a universal equivariance property, which justifies that the exotic aromatic B-series formalism is a natural extension of B-series and aromatic B-series. We extend this result to characterise subsets of the exotic aromatic B-series, such as the exotic B-series. This work also allows us to give a more general definition of exotic aromatic B-series, free of the degeneracies introduced in the numerical context.

The article is organized as follows. We present in Section 2 the definition of the geometric properties used in the characterisation, the general definition of the exotic aromatic B-series, and the main results of the paper. The characterisation of exotic aromatic B-series is proven in Section 3, while we derive the strong classification of exotic aromatic B-series in Section 4. We give outlooks on future works in Section 5.

Preliminaries and main results

This section is devoted to the definition of locality, equivariance and decoupling. We then give a new general definition of exotic aromatic trees and their associated elementary differential. The main results of the paper are presented in Subsection 2.3.

Locality, equivariance, and partitions

We define the geometric properties used in the characterisation of exotic aromatic B-series. A natural property of modified vector fields is locality.

Definition 2.1. Let d ¥ 0, a map φ d : XpR d q Ñ XpR d q is local if supppφ d pfqq supppf q, supppf q tx R d , f pxq $ 0u.
In [START_REF] Munthe-Kaas | Aromatic Butcher series[END_REF], the aromatic B-series are characterised by locality and a property of equivariance.

Let XpR d q be the set of smooth vector fields on R d and G be a finite dimensional Lie subgroup of the set of diffeomorphisms DiffpR d q on R d . The group G has the form H R d , where H is a subgroup of GL d pRq called the isotropy group. An element g pA, bq G acts on a vector field f XpR d q by pg ¤ f qpxq Af pA ¡1 px ¡ bqq.

The G-equivariance is the compatibility with the action of G on vector fields.

Definition 2.2. A map φ

d : XpR d q Ñ XpR d q is G-equivariant if φ d pg ¤ f q g ¤ φ d pfq, g G, f XpR d q.
In this work, we consider G H R d , where H is a matrix group called the isotropy group. If H GL d pRq, the G-equivariance is written for simplicity GL-equivariance, while we write orthogonal-equivariance if H O d pRq. The first main result of this paper is the characterization of exotic aromatic B-series with orthogonal-equivariance and locality.

The second goal of this work is to characterize the subsets of exotic aromatic B-series, in particular the exotic B-series. In this context, the dimension d ¥ 0 of the problem plays an important role, so that we rely on sequences of maps φ pφ d :

XpR d q Ñ XpR d qq d indexed by the dimension d. Such a sequence is local (respectively G-equivariant) if φ d is local (respec- tively G-equivariant) for all d.
To observe the interactions between the dimensions, the notion of equivariance is extended to affine transformations [START_REF] Mclachlan | B-series methods are exactly the affine equivariant methods[END_REF], and we refer to such a property as strong equivariance in the following.

Definition 2.3. Let the set of affine transformations

AffpR d 1 , R d 2 q ta: R d 1 Ñ R d 2 , apxq Ax b, pA, bq R d 2 ¢d 1 ¢ R d 2 u
, and HpR d 1 , R d 2 q be a subset defined for all dimensions d 1 and d 2 . A sequence of smooth maps φ pφ d :

XpR d q Ñ XpR d qq d is equivariant with respect to H if for all d 1 , d 2 , for all apxq Ax b HpR d 1 , R d 2 q and x R d 1 , φ satisfies f 2 papxqq Af 1 pxq ñ φ d 2 pf 2 qpapxqq Aφ d 1 pf 1 qpxq, f 1 XpR d 1 q, f 2 XpR d 2 q. A sequence of smooth maps φ pφ d q d is affine-equivariant if it is equivariant with respect to all affine transformations in Aff.
The different subsets of AffpR d 1 , R d 2 q that we consider are associated to the classical homogeneous spaces corresponding to the Lie-group H O d pRq:

SpR d 1 , R d 2 q tapxq Ax b AffpR d 1 , R d 2 q, A T A I d 1 u, (2.1)
GpR d 1 , R d 2 q tapxq Ax b AffpR d 1 , R d 2 q, AA T I d 2 u. (2.2)
The transformations in (2.1) are the left-orthogonal transformations and correspond to the Stiefel manifold, while the right-orthogonal transformations in (2.2) correspond to the Grassmann manifold. The associated equivariance properties are called Stiefel-equivariance and Grassmann-equivariance. A sequence φ is said to be semi-orthogonal-equivariant if it is both Stiefel-equivariant and Grassmann-equivariant. The elementary differentials associated to standard B-series keep decoupled systems decoupled. Some exotic aromatic B-series also satisfy this property, which motivates the following definition. Similarly to [START_REF] Mclachlan | B-series methods are exactly the affine equivariant methods[END_REF], for f 1 XpR d 1 q and f 2 XpR d 2 q, we use the notation

h f 1 f 2 XpR d 1 d 2 q, hpx, yq pf 1 pxq, f 2 pyqq. Definition 2.4. A sequence φ pφ d : XpR d q Ñ XpR d qq d is decoupling if for all f 1 XpR d 1 q and f 2 XpR d 2 q, φ satisfies φ d 1 d 2 pf 1 f 2 q φ d 1 pf 1 q φ d 2 pf 2 q, that is, for all x R d 1 , y R d 2 , φ d 1 d 2 ppf 1 pxq, f 2 pyqqq pφ d 1 pf 1 qpxq, φ d 2 pf 2 qpyqq. A sequence φ is trivially decoupling if for all f XpR d 1 q, φ satisfies φ d 1 d 2 pf 0q φ d 1 pfq 0.
The Stiefel-equivariance and Grassmann-equivariance properties are stronger than the orthogonal-equivariance in the following sense. The proof is omitted as it is nearly identical to [START_REF] Mclachlan | B-series methods are exactly the affine equivariant methods[END_REF]Lem. 4

Exotic aromatic trees

Introduced originally in [START_REF] Laurent | Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs[END_REF][START_REF] Laurent | Order conditions for sampling the invariant measure of ergodic stochastic differential equations on manifolds[END_REF] for the calculation of the order conditions for the approximation of ergodic stochastic differential equations, the exotic aromatic trees are an extension of aromatic trees that involve two new kind of edges: lianas and stolons. The definition we give in the present paper is a generalization of the one originally presented in [START_REF] Laurent | Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs[END_REF][START_REF] Laurent | Order conditions for sampling the invariant measure of ergodic stochastic differential equations on manifolds[END_REF] (see also [START_REF] Bronasco | Exotic B-series and S-series: algebraic structures and order conditions for invariant measure sampling[END_REF][START_REF] Bronasco | The Hopf algebra structures of composition and substitution of exotic clumped S-series[END_REF]). It reduces to the same definition under a regularity assumption discussed in Section 4.3. We choose an approach based on permutations as in [START_REF] Munthe-Kaas | Aromatic Butcher series[END_REF] (see also [START_REF] Bogfjellmo | Algebraic structure of aromatic B-series[END_REF][START_REF] Laurent | The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators[END_REF]).

Definition 2.6. We consider graphs of the form pV, A 0 , σ, τ q with V a finite set of vertices and A 0 a finite set of arrows. The vertices are indexed from 1 to |V |, and the arrows from 0 to |A|, where A A 0 zt0u. The map τ : A Ñ V is the target map. The source map is a permutation σ : V A 0 Ñ V A 0 that has no fixed points and satisfies σ ¥ σ id. Two such graphs are equivalent if there exists a bijection between their sets of nodes and arrows that are compatible with the source and target maps. An exotic aromatic tree is an equivalence class of such graphs. We denote Γ the set of exotic aromatic trees. Definition 2.6 differs from standard definitions of directed graphs as the source map usually sends arrows to nodes. The extension presented here allows arrows to be sources of arrows and vertices to be sources of vertices. If σpa 1 q a 2 , we say that the unordered tuple pa 1 , a 2 q is a liana and we represent it with a dashed edge between the two nodes τ pa 1 q and τ pa 2 q, that can be identical. If σpv 1 q v 2 , we call the unordered tuple pv 1 , v 2 q a stolon and we draw it with a double edge between v 1 and v 2 . The set of lianas is denoted L and the set of stolons is S. An exotic aromatic tree without lianas and stolons is called an aromatic tree, an extension of standard trees allowing for loops. A loop is a list of nodes pv 1 , . . . , v K q such that there is a standard edge linking v 1 to v 2 , . . . , v K to v 1 (also called K-loop in [START_REF] Iserles | B-series methods cannot be volumepreserving[END_REF]). Note that an exotic aromatic tree is an aromatic tree if and only if σpV q A 0 . In this case, Definition 2.6 reduces to an equivalent definition of the one in [START_REF] Munthe-Kaas | Aromatic Butcher series[END_REF]. We refer to Table 1 for examples.

Example. Let the exotic aromatic tree γ pV, A 0 , σ, τ q with the nodes V t1, 2, 3, 4u, the arrows A 0 t0, 1, 2, 3u, and the following source and target maps σ p0, 1qp1, 2qp2, 3qp3, 4q, τ p2, 3, 4q, where we use the notation τ pτp1q, . . . , τ p|A|qq. The tree γ has one loop p4q, one liana p1, 2q, and one stolon p2, 3q. The associated graph is the following, where we detail the vertices and arrows for clarity.

1 0 2 3
1 2 4 3
An exotic aromatic tree has a unique root r defined the following way. If σp0q V , r σp0q is the root and 0 is called a ghost arrow. If σp0q A, the liana r p0, σp0qq L is the root and is called a ghost liana. We draw the ghost arrow (respectively ghost liana) on the graphical representations of exotic aromatic trees as an edge (respectively dashed edge) with one end left unattached. In the aromatic context, the ghost arrow is usually omitted on the graphical representation, hence the name.

We say that two elements x, y V A 0 are neighbours if σpxq y or τ pxq y or τ pyq x.

This defines a notion of connectedness on exotic aromatic trees. The connected components without the root are called aromas and a finite unordered collection of aromas is a multi-aroma. The connected component with the root is a connected exotic aromatic tree. We denote Γ c the set of connected exotic aromatic trees and Γ 0 the set of multi-aromas, also represented as equivalence classes of graphs pV, A, σ, τ q without the arrow 0. The aromas are gathered in Γ 0

c . An exotic aromatic tree decomposes into a number of aromas and one connected exotic aromatic tree. This notion of connectedness is a strong motivation to understand the exotic aromatic trees as graphs and not as trees as done beforehand in the literature. If there are no lianas and no stolons, we find the standard definition of aromas and rooted trees in the context of aromatic trees.

We define an exotic tree as an exotic aromatic tree that reduces to a standard Butcher tree when removing all the lianas. Note that there is a difference between the notions of exotic trees and connected exotic aromatic trees. A connected exotic aromatic tree does not reduce to a tree in general when removing the lianas.

Example. Let the following exotic aromatic trees

γ 1 , γ 2 , γ 3 .
The exotic aromatic tree γ 1 is a disconnected aromatic tree with one aroma. The graph γ 2 is connected, but is not an exotic tree as removing the lianas of γ 2 yields γ 1 , which is not a tree. On the other hand, γ 3 is an exotic tree.

We denote the set of nodes that are the target of j arrows by V j . For a given aromatic tree γ, we define its composition κ : N Ñ N by κpjq |V j |, and its derived composition by κ I pjq jκpjq. A straightforward observation yields that the cardinals of V and A satisfy |V | |κ| and |A| |κ I |, where |κ| κp0q κp1q . . . We write Γ κ the set of exotic aromatic trees with composition κ and Γ m the set of exotic aromatic trees such that |κ| m.

Note that Γ κ is finite while Γ m is infinite for all m. Contrary to the case of Butcher trees and aromatic trees, the order of an exotic aromatic tree is not given by the number of its nodes |κ|. Then the following identity holds

|κ| |κ I | 1 2|γ|. (2.3)
We mention that in the aromatic context, the order of an aromatic tree coincides with the number of nodes and (2. (2.4) If an exotic aromatic tree satisfies (2.4), it does not imply that γ is aromatic. In fact, there exists an infinite number of exotic aromatic trees satisfying (2.4) that do not reduce to aromatic trees: the exotic aromatic trees with the same number of stolons and lianas. In the context of branched rough paths, a similar identity to (2.4) on multi-indices is used in [28, eq. ( 6.3)], and the composition κ is called the fertility in this context.

Proof of Lemma 2.7. The arrows are either part of a liana or are standard arrows whose source are nodes. We denote A 0 the latter set. Similarly, the nodes of γ Γ κ can be decomposed in two sets: the nodes that are the source of an arrow in A 0 , gathered in V , and the ones that are the source of no arrows (the stolons). We observe that

|V | |V | 2|S|, |A 0 | |A 0 | 2|L|. Each node in V is the source of a unique arrow in A 0 , so that |V | |A 0 |. Thus, we deduce |κ| |κ I | 1 |V | |A 0 | 2p|V | |L| ¡ |S|q 2|γ|,
which gives the desired identity (2.3).

In Table 1, we present the list of the exotic aromatic trees of order one and two (see also Appendix A for the order three). On the contrary of the aromatic case, there exists an infinite number of exotic aromatic trees for a given number of nodes |κ| ¡ 0. Indeed, adding any number of lianas to an exotic aromatic tree does not modify the value of |κ|, but gives a different tree.

|γ| |κ| κ κ I τ σ γ Fpγqpf q 1 1 p1q p0q p0, 1q f i f i 2 1 p0, 0, 1q p0, 0, 2q p1, 1q p0, 1qp1, 2q f i jj f i p0, 1qp2, 1q f j ij f i 2 2 p1, 1q p0, 1q p1q p0, 1qp1, 2q f i j f j f i p0, 2qp1, 1q f j j f i f i p0, 1qp1, 2q f j f j i f i 2 3 p3q p0q p0, 1qp2, 3q f i f j f j f i
Table 1: List of the exotic aromatic trees of order one and two, with their associated composition, derived composition, target map, source map, and elementary differential (see Definition 2.8). We use the notation τ pτp1q, . . . , τ p|κ 1 |qq.

Characterisation of exotic aromatic B-series

Butcher trees are used to represent elementary differentials, in order to represent conveniently Taylor expansions in numerical analysis [START_REF] Hairer | Geometric numerical integration[END_REF]. We associate an elementary differential to each exotic aromatic tree. We use the standard notation f i for the vector basis of R d .

Definition 2.8. Given a smooth vector field f XpR d q and γ pV, A 0 , σ, τ q Γ an exotic aromatic tree, the elementary differential F d pγq associated to γ is the following vector field

F d pγqpfq i1 ,...,i |κ| i 0 ,...,i |κ I | ¹ vV f iv i τ ¡1 ptvuq δ iσ f i 0 ,
where i τ ¡1 ptvuq i l 1 . . . i lm for τ ¡1 ptvuq tl 1 , . . . , l m u and δ iσ ± |γ| j1 δ ip j ,iq j for the source map σ ± |γ| j1 pp j , q j q with δ i,j 1 if i j and 0 else. The elementary differential map of an exotic aromatic tree γ is the following sequence of maps indexed by the dimension of the problem Fpγq pF d pγq:

XpR d q Ñ XpR d qq d .
The maps F d are extended by linearity to SpanpΓq.

Note that for a fixed dimension d, the elementary differential map F d is not injective in general. There can be multiple ways to write a given elementary differential with exotic aromatic trees if the dimension d is too low. For instance, in dimension d 1, all the trees with composition κ represent the same elementary differential

F 1 pγqpfq V ¹ j0 pf pjq q κpjq .
This is a strong motivation for considering sequences of maps Fpγq pF d pγqq d indexed by the dimension of the problem.

Remark 2.9. The elementary differential extends to multi-aromas γ pV, A, σ, τ q Γ 0 by

F d pγqpfq i1 ,...,i |κ| i 1 ,...,i |κ I | ¹ vV f iv i τ ¡1 ptvuq δ iσ .
Example. Consider the following exotic aromatic tree γ and its associated elementary differential

γ , Fpγq i v ,ia f i 1 f i 2 f i 3 i 1 i 2 δ i 0 ,i 1 δ i 1 ,i 2 δ i 2 ,i 3 f i 0 i ,j,k f i f k f k jj f i pf, ∆f qf.
Further examples are presented in Table 1. Note that in the elementary differentials, every index appears twice. For aromatic trees, every index appears both at the top and at the bottom, while this is not the case in general for exotic aromatic trees.

An exotic aromatic B-series is a formal series indexed over exotic aromatic trees. As we consider Taylor expansions and thus use the grading by the number of nodes, we consider series with a finite number of trees with m nodes for all m. This assumption is not required in the numerical applications [START_REF] Laurent | Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs[END_REF][START_REF] Laurent | Order conditions for sampling the invariant measure of ergodic stochastic differential equations on manifolds[END_REF] as the expansions are graded naturally by the order of the trees and not by the number of nodes. Theorem 2.11 provides a simple geometric criterion for checking whether a modified vector field corresponds to an exotic aromatic B-series. In addition, it confirms that the exotic aromatic B-series are a natural extension of the aromatic B-series as they both satisfy similar universal geometric properties [START_REF] Munthe-Kaas | Aromatic Butcher series[END_REF].

We are then interested in characterising the different subsets of exotic aromatic B-series and in particular the exotic B-series, as they play an important role in stochastic numerical analysis [START_REF] Laurent | Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs[END_REF]. We propose the following characterisation. Theorem 2.12. Let φ pφ d : XpR d q Ñ XpR d qq d be a local sequence of smooth maps. Then, the Taylor expansion of φ around the trivial vector field 0 is:

• a connected exotic aromatic B-series if and only if φ is orthogonal-equivariant and decoupling,

• a B-series with stolons if and only if φ is Stiefel-equivariant,

• an exotic B-series if and only if φ is Grassmann-equivariant,

• a B-series if and only if φ is semi-orthogonal-equivariant.

In particular, affine equivariance and semi-orthogonal-equivariance are equivalent notions as they both characterise B-series.

Remark 2.13. The decoupling property exactly corresponds to the connectedness of the graphs involved in the expansion. This link was first observed in [START_REF] Mclachlan | B-series methods are exactly the affine equivariant methods[END_REF] in the context of aromatic Bseries, where a decoupling aromatic B-series is showed to be a connected aromatic B-series, i.e., a standard B-series. Observe also that the elementary differential of an exotic aromatic tree can be factored through its connected components: let µ 1 , . . . , µ m Γ 0 c , τ Γ c and the exotic aromatic tree γ µ 1 . . . µ m τ , then

F d pγqpfqpxq F d pµ 1 qpfqpxq . . . F d pµ m qpfqpxqF d pτqpfqpxq.
(2.5)

The classification of exotic aromatic B-series is summarised in Table 2. To the best of our knowledge, the equivalence of affine-equivariance and semi-orthogonal-equivariance is a new non-trivial result. We derive in Subsection 4.3 a simplified characterisation related to the numerical analysis literature under a regularity assumption on the vector fields.

Geometric characterisation of exotic aromatic B-series

This section is devoted to the proof of Theorem 2.11. Following [START_REF] Munthe-Kaas | Aromatic Butcher series[END_REF], we first restrict our study to symmetric multilinear local equivariant maps defined on the infinite jet bundle at one point in Section 3.1. We then decompose our space into invariant tensor spaces using the invariant tensor theorem. In Section 3.2, we draw a one-to-one correspondence between tensors in the invariant spaces and exotic aromatic trees. Section

Invariant tensor spaces and transfer of geometric properties

Let φ d : XpR d q Ñ XpR d q be a smooth local G-equivariant map with G H R d . The Taylor expansion of φ d around the vector field 0 is

m¥1 1 m! D m φ d p0qpf, . . . , f q, ( 3.1)
where φ d p0q 0 by locality [START_REF] Mclachlan | B-series methods are exactly the affine equivariant methods[END_REF]Lem. 6.1]. Following the transfer argument [32, Thm. 3.9], the m-th Taylor term D m φ d p0q inherits the locality and orthogonal-equivariance properties.

Moreover, the Peetre theorem [18, § 19.9] and the equivariance [32, Thm. 5.6] allow us to assume without loss of generality that the m-th Taylor term is in the space of multilinear symmetric local H-equivariant maps:

D m φ d p0q L H pS m pM SM ¦ q, M q, where M T 0 R d R d
, the action of H on S m pM SM ¦ q is the natural action induced on tensor spaces, and for a vector space V , SV : À V j0 S j V is the symmetric algebra. This result works for any isotropy group H.

Given a composition κ : N Ñ N, we define the tensor space T κ and its symmetric counterpart S κ by

T κ M V â j0 T κpjq pM ¦ T j M q, S κ M V â j0 S κpjq pM ¦ S j M q,
and their H-invariant subspaces T H κ and S H κ . Then, [START_REF] Munthe-Kaas | Aromatic Butcher series[END_REF]Thm. 5.6] gives the isomorphism

L H pS m pM SM ¦ q, M q à |κ|m S H κ .
In the affine case H GL d pRq, it is shown in [32, Thm. 6.3] with the description of Hinvariant tensors [18, § 24.3] that L H pS m pM SM ¦ q, M q is a finite dimensional space. In the orthogonal case H O d pRq, this property does not hold in general, and we get the following instead. We write ∆ m φ d : XpR d q Ñ XpR d q the term of order m ¥ 1 in the expansion of φ d around the vector field 0, that is,

∆ m φ d pfq D m φ d p0qpf, . . . , f q.
∆ m φ d pfq f t 1 ...tm φ d ppt 1 . . . t m qfq § § § t 1 ¤¤¤tm0
. Thus, we find

∆ m φ d pf 1 f 2 q f t 1 ...tm φ d ppt 1 . . . t m qf 1 pt 1 . . . t m qf 2 q § § § t 1 ¤¤¤tm0 f t 1 ...tm φ d ppt 1 . . . t m qf 1 q φ d ppt 1 . . . t m qf 2 q % § § § t 1 ¤¤¤tm0 ∆ m φ d pf 1 q ∆ m φ d pf 2 q.
Hence the result.

Thus, we restrict our study for the rest of the paper to the ∆ m φ p∆ m φ d q d that can be expressed as finite sums of tensors in S O d pRq κ . We show in Subsection 3.2 that the expansion (3.2) corresponds to the order m term of an exotic aromatic B-series.

Correspondence between exotic aromatic trees and invariant tensors

Let us now draw a correspondence between exotic aromatic trees and tensors in S

d : SpanpΓ κ q Ñ S O d pRq κ . The map r F d is a bijection if and only if 2d ¥ |κ| |κ I | 1. Moreover, the elementary differential map F d is injective on SpanpΓ κ q if 2d ¥ |κ| |κ I | 1.
Proof. Decomposition of T κ . Following Theorem 3.1, we assume without loss of generality that |κ| |κ I | 1 2d 0 is even. We rewrite T κ as

T κ M V â j0 κpjq â i1 T j i , T j i M ¦ T j M.
We number the 2d 0 components of T κ in the following way. The copies of M ¦ in T κ are numbered in an arbitrary manner from 1 to |κ| and the copies of M from 0 to |κ I | so that

T κ M 0 V â j0 κpjq â i1 T j i .
If the numbering is given by

T j i M ¦ n M n 1 ¤ ¤ ¤ M n j ,
then we write τ pn k q n, k 1, . . . , j. This defines the target map τ : A Ñ V , the arrows A t1, . . . , |κ I |u, A 0 t0u A, and the vertices V t1, . . . , |κ|u.

Definition of ω. We denote Σ κ the set of permutations σ of the set V A 0 that have no fixed point and that satisfy σ ¥ σ id. Given σ Σ κ and the target map τ , there exists a unique exotic aromatic tree pV, A 0 , σ, τ q Γ κ according to Definition 2.6. This yields a map ω : Σ κ Ñ Γ κ . We extend this map by linearity to obtain ω : SpanpΣ κ q Ñ SpanpΓ κ q.

Definition of π. The projection map π : T κ Ñ S κ is compatible with the action of O d pRq.

Thus, it induces a surjective linear map (still denoted π for simplicity) from

T O d pRq κ to S O d pRq κ .

Definition of δ. Using the isomorphism T κ Lp Â

nV A 0 M n , Rq, we define δpσq for a permutation σ Σ κ by δpσqpvq

¹ i,jV A 0 jσpiq,i j pv i , v j q, v â nV A 0 v n â nV A 0 M n ,
where p., .q is the standard scalar product in R d and where we fixed an arbitrary total order on V A 0 . We extend δ by linearity on SpanpΣ κ q.

Intermediate diagram.

We defined the linear maps π, ω and δ. We obtain the following diagram.

T O d pRq κ S O d pRq κ SpanpΣ κ q SpanpΓ κ q π ω δ
Action of G κ . The target function τ : A Ñ V is an element of V A . We denote Σ A the set of permutations of the arrows in A, respectively Σ V the set of permutations of the nodes in V , and Σ A ¢ Σ V the permutations of V A 0 that leave 0 fixed, permute the elements in V , and the elements of A without mixing them. The action of an element of g

Σ A ¢ Σ V on ξ V A is g ¤ ξ g ¥ ξ ¥ g ¡1 .
We denote G κ the stabilizer of the target function τ , that is,

G κ tg Σ A ¢ Σ V , g ¤ τ τ u.
The permutations in G κ represent the permutations of arrows and nodes that are compatible with the target map τ .

Definition of K Σ . An element g Σ A ¢Σ V acts naturally on σ Σ κ by g ¤σ g ¥σ¥g ¡1 . We observe that ωpσ 1 q ωpσ 2 q if and only if there exists g G κ such that σ 1 g ¤ σ 2 . We define K Σ as the vector subspace of SpanpΣ κ q spanned by the g 1 ¤ σ ¡ g 2 ¤ σ for g 1 , g 2 G κ and σ Σ κ . By definition, K Σ is the kernel of ω, so that the following sequence is exact.

0 K Σ SpanpΣ κ q SpanpΓ κ q 0 ω Definition of K . Using the identification T κ Lp  nV A 0 M n , Rq, the action of an element of Σ A ¢ Σ V on T κ is pg ¤ φqpvq φp â nV A 0 v gpnq q, φ Lp â nV A 0 M n , Rq, v â nV A 0 v n .
We observe that by definition of τ , πpφ 1 q πpφ 2 q if and only if φ 1 g ¤ φ 2 with g G κ .

We define K as the vector space spanned by the

g 1 ¤ φ ¡ g 2 ¤ φ for g 1 , g 2 G κ , φ T κ .
By definition, K is the kernel of π, and is also the kernel of the restriction π :

T O d pRq κ Ñ S O d pRq κ .
We have the following exact sequence.

0 K T O d pRq κ S O d pRq κ 0 π Definition of r F d .
The action of G κ commutes with δ, that is, δpg ¤ σq g ¤ δpσq. Thus, δ induces a linear map from K Σ to K . By the fundamental theorem on homomorphisms, there exists a surjective map r δ from SpanpΣ κ q{K Σ to T O d pRq κ {K . We obtain the following diagram, where r

F d π ¥ r δ ¥ ω ¡1 is surjective. T O d pRq κ {K S O d pRq κ SpanpΣ κ q{K Σ SpanpΓ κ q π ω r δ r F d
The map r , . . . q, f XpR d q.

Thus, if r F d is bijective on SpanpΓ κ q, then F d is injective on SpanpΓ κ q.

Remark 3.4. There exists a finite number of exotic aromatic trees of composition κ, so that S

O d pRq κ is finite-dimensional. However, on the contrary of the GL-equivariance setting, there is an infinite number of exotic aromatic trees with a given number of nodes |κ|, so that L O d pRq pS m pM SM ¦ q, M q is infinite-dimensional.

Characterisation of exotic aromatic B-series

As first mentioned in [START_REF] Laurent | Algebraic Tools and Multiscale Methods for the Numerical Integration of Stochastic Evolutionary Problems[END_REF]Sec. 3.2.4], the exotic aromatic B-series satisfy geometric properties.

Proposition 3.5. The exotic aromatic B-series are local, orthogonal-equivariant, and trivially decoupling.

Proof. The locality and trivially decoupling properties are straightforward from Definition 2.8.

Let g pA, bq O d pRq R d and γ Γ, then

F d pγqpg ¤ f qpxq i v ,ia vV,aA 0 ¹ vV ķv,ka 0 vV,a 0 A a iv,kv a i τ ¡1 ptvuq ,k τ ¡1 ptvuq f kv k τ ¡1 ptvuq pA ¡1 x ¡ bqδ iσ f i 0 i0 ,kv,ka vV,aA a i 0 ,k σp0q ¹ vV f kv k τ ¡1 ptvuq pA ¡1 x ¡ bqδ kσ f i 0 pg ¤ F d pγqpfqqpxq,
where a i J ,k J ± jJ a i j ,k j and we used that A T A I d . By linearity, exotic aromatic B-series are orthogonal-equivariant.

The trivially decoupling property characterises the sequences of elementary differentials associated to exotic aromatic trees independently of the dimension d. Proposition 3.6. Let φ pF d pγ d qq d be trivially decoupling, with γ d SpanpΓ κ q, then there exists a unique γ SpanpΓ κ q such that φ Fpγq.

Proof. Let d 1 ¤ d 2 , then we have F d 2 pγ d 2 qpf 1 0q F d 1 pγ d 1 qpf 1 q 0 for f 1 XpR d 1 q
as φ is trivially decoupling. On the other hand, a close inspection of Definition 2.8 yields that F d 2 pγ d 2 qpf 1 0q F d 1 pγ d 2 qpf 1 q 0, and we deduce

F d 1 pγ d 1 q F d 1 pγ d 2 q, d 1 ¤ d 2 .
(3.3)

Let d 0 p|κ| |κ I | 1q{2. For d ¥ d 0 , equation (3.3) gives F d pγ d q F d pγ d 0 q. For d ¤ d 0 , equation (3.3) gives F d 0 pγ d 0 q F d 0 pγ d q.
As F d 0 is injective on SpanpΓ κ q, γ d γ d 0 . Thus, we obtain φ Fpγ d 0 q. The uniqueness of γ d 0 is a consequence of the injectivity of F d 0 on SpanpΓ κ q (see Theorem 3.3).

Let us now prove the characterisation of exotic aromatic B-series.

Proof of Theorem 2.11. Let φ d : XpR d q Ñ XpR d q be a local, and orthogonal-equivariant map.

Thanks to Theorem 3.1, the term of order m in the Taylor expansion (3.1) of φ d around the 0 vector field has the form (3.2). Theorem 3.3 gives the existence of γ κ,d SpanpΓ κ q such that ψ κ,d r F d pγ κ,d q. The Taylor expansion of φ d around the 0 vector field thus is the exotic aromatic B-series:

m¥1 1 m! |κ|m |κ| |κ I | 12Z F d pγ κ,d qpfqpxq.
If, in addition, φ is trivially decoupling, Proposition 3.6 gives the existence of the coefficient map b : Γ Ñ R such that φ Bpbq.

Example. Let us illustrate the tensor spaces and the different maps from the proof of Theorem 3.3 for κ p2, 0, 1q (in the spirit of the examples in [START_REF] Markl | GL n -invariant tensors and graphs[END_REF][START_REF] Munthe-Kaas | Aromatic Butcher series[END_REF]). The tensor space has the form

T κ M 0 M ¦ 1 M ¦ 2 pM ¦ 3 M 1 M 2 q.
The associated set of nodes and arrows are V t1, 2, 3u and A 0 t0, 1, 2u. The target map τ : A Ñ V is given by τ p1q τ p2q 3. The stabilizer of τ is G κ tid, p1, 2q, p1, 2q, p1, 2qp1, 2qu.

We present the output of ω, r δ and r

F for the different σ Σ κ in Table 3, where we gather together the G κ -orbits. We write r δpσq as an element of

T κ {K LpT 2 M LpT 2 M, M q, M
q{K , that is, for v, w M and a bilinear map ζ LpT 2 M, M q, we have r δpσqpv, w, ζq M . For r F, we use the identification

S κ LpS 2 M LpS 2 M, M q, M q.
Replacing v w f pxq and ζ f P pxq yields the elementary differential Fpγqpf qpxq of Definition 2.8. Note that the first two lines of Table 3 are aromatic trees, and also appear in [32, Table 2]. It can be seen directly on the associated permutations σ, as each arrow is paired with a node and vice versa.

σ Σ κ γ ωpσq r δpσqpv, w, ζq r Fpγqpv, w, ζq p0, 3qp1, 1qp2, 2q ζ i pv, wqf i p0, 3qp1, 2qp2, 1q ζ i pw, vqf i ζ i pv, wqf i p0, 1qp1, 2qp2, 3q ζ j pw, f j qv i f i p0, 1qp1, 3qp2, 2q ζ j pf j , wqv i f i p0, 2qp1, 1qp2, 3q ζ j pv, f j qw i f i p0, 2qp1, 3qp2, 1q ζ j pf j , vqw i f i 1 2 pζ j pw, f j qv i ζ j pv, f j qw i qf i p0, 3qp1, 2qp1, 2q pv, wqζ i pf j , f j qf i pv, wqζ i pf j , f j qf i p0, 1qp1, 2qp2, 3q pw, ζpf j , f j qqv i f i p0, 2qp1, 2qp1, 3q pv, ζpf j , f j qqw i f i 1 2 ppw, ζpf j , f j qqv i pv, ζpf j , f j qqw i qf i p0, 1qp2, 1qp2, 3q pw, ζpf i , vqqf i p0, 1qp2, 2qp1, 3q pv, ζpf i , wqqf i p0, 2qp1, 1qp2, 3q pw, ζpv, f i qqf i p0, 2qp1, 2qp1, 3q pv, ζpw, f i qqf i 1 2 ppw, ζpf i , vqq pv, ζpf i , wqqqf i p0, 1qp2, 3qp1, 2q pv, wqζ j pf i , f j qf i p0, 2qp1, 3qp1, 2q
pv, wqζ j pf j , f i qf i pv, wqζ j pf j , f i qf i Table 3: Outputs of the functions ω, δ and r F appearing in the proof of Theorem 3.3 for the composition κ p2, 0, 1q and target map τ p1q τ p2q 3. The bilinear map ζ is assumed symmetric in the last column. The sums on all involved indices are omitted for simplicity.

Classification of exotic aromatic B-series

This section is devoted to the proof of the stronger classification of Theorem 2.12. The proof, presented in Subsection 4.2, relies heavily on the use of new dual vector fields, that we introduce in Subsection 4.1. We present the impact of degeneracies on the classification in Subsection 4.3.

Dual vector fields

The exotic aromatic trees given in Definition 2.6 produce independent elementary differentials.

The standard method for proving this property is to consider dual vector fields, as presented in [START_REF] Hairer | Geometric numerical integration[END_REF][START_REF] Iserles | B-series methods cannot be volumepreserving[END_REF][START_REF] Mclachlan | B-series methods are exactly the affine equivariant methods[END_REF][START_REF] Laurent | Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs[END_REF]. In the proof of Theorem 2.12, we use the following new dual vector fields.

Proposition 4.1. Given an exotic aromatic tree or multi-aroma γ Γ 0 Γ, index the coordinates of R |γ| by the uplets in pV A 0 q{σ (respectively in pV Aq{σ if γ Γ 0 ), where v and σpvq are identified. This corresponds to the nodes pv, aq V that are not part of stolons, the stolons s pv 1 , v 2 q S, and the lianas l pa 1 , a 2 q L. Let θ γ be the following parameter indexed by the standard nodes, the nodes in stolons, and the arrows in lianas,

θ γ pθ V , θ S , θ L q pθ V 1 , . . . , θ V |V | , θ S 1 , . . . , θ S 2|S| , θ L 1 , . . . , θ L 2|L| q.
Define the associated vector field f pθ γ q γ XpR |γ| q by

f pθ γ q,v γ pxq θ V v ¹ τ paqv θ L a x a , f pθ γ q,s γ pxq θ S v 1 ¹ τ paqv 1 θ L a x a θ S v 2 ¹ τ paqv 2 θ L a x a , s pv 1 , v 2 q S, f pθ γ q,l γ pxq 0,
where an empty product equals 1 and θ L a 1 if a L. By convention, if γ has a root, the coordinate of the root is the first one. Let γ, γ Γ 0 Γ, then pF |γ| pγqpf pθ γ q γ qq 1

θ γ § § § θ0 p0q 0 if γ $ µγ, µ Γ 0 .
In particular, the elementary differential map F is injective on SpanpΓq. Moreover, for con-

nected graphs γ, γ Γ 0 c Γ c , we find pF |γ| pγqpf pθ γ q γ qq 1 θ γ § § § θ0 p0q σpγq $ 0 if and only if γ γ.
In the latter case, the constant σpγq is the symmetry coefficient of γ, that is, the number of bijections of the vertices and arrows of γ that preserve the graph structure.

Remark 4.2. Given an exotic aromatic tree γ µ m . . . µ 1 τ , we enforce an order on the aromas, so that µ

1 µ 2 is now different from µ 2 µ 1 if µ 1 $ µ 2 .
Consider the additional parameter θ γ pθ τ , θ µ 1 , . . . , θ µm q, where the numbering of the nodes, lianas and stolons starts with τ , and continues in order with the µ i . With this order on the aromas and the numbering of θ, the first statement of Proposition 4.1 is then replaced by

pF |γ| pγqpf pθ γ q γ qq 1 θ γ § § § θ0 p0q $ 0 if and only if γ µγ, µ Γ 0 .
Proof. Let γ pV, A, σ, τ q, γ p V , Â, σ, τ q Γ 0 Γ. Definition 2.8 rewrites as F |γ| pγqpf pθ γ q γ qpxq i:

pV Aq{σÑp V Âq{σ ¹ vV pf γ q iv i τ ¡1 ptvuq f ir i: pV Aq{σÑp V Âq{σ ¹ vV ¡ θ V iv θ L τ ¡1 pivq x τ ¡1 pivq 1 iv V0 pθ S v1 θ L τ ¡1 pv 1 q x τ ¡1 pv 1 q θ S v2 θ L τ ¡1 pv 2 q x τ ¡1 pv 2 q q1 ivpv 1 ,v 2 q Ŝ © i τ ¡1 ptvuq f ir .
where we fix f ir f 1 if γ Γ 0 . By definition of the map i, if σpxq y, σpi x q i y . Moreover, it is necessary that τ ¡1 pi v q i τ ¡1 ptvuq for v V (and analogously for v S) so that pf pθ γ q γ q iv i τ ¡1 ptvuq p0q $ 0. Thus, the map i is compatible with the source and target maps.

In particular, i sends predecessors of v to predecessors of i v .

On the other hand, the θ parameter enforces the injectivity of i and it forces i to send stolons to stolons, lianas to lianas, nodes in V to nodes in V . Thus i sends γ to a subgraph of γ. If γ $ µγ, then at least an edge is missing and F |γ| pγqpf pθ γ q γ qpxq is a non-constant polynomial in x, so that it vanishes at x 0.

If γ, γ Γ 0 c Γ c , the only maps i such that pF |γ| pγqpf pθ γ q γ qq 1 θ γ § § § θ0 p0q $ 0 are the graph isomorphisms between γ and γ. The number of such maps i is σpγq.

Example. Consider the following exotic aromatic tree with its associated vector field and elementary differential

γ , f pθ γ q γ ¤ ¥ x 1 x 2 x 3 ¤ ¥ 0 θ S 1 θ L 2 θ L 3 x 2 3 θ S 2 θ L 1 x 1 0 , pF 3 pγqpf pθ γ q γ qq 1 θ γ § § § θ0 p0q 2,
where the coordinates represent in descending order the root, the stolon, and the liana. Consider now the aroma

γ , f pθ γ q γ ¢ x 1 x 2 ¢ θ S 1 θ L 1 x 2 θ S 2 θ L 2 x 2 0 , θ γ pθ S 1 , θ S 2 , θ L 1 , θ L 2 q,
and the tree

γ , f pθ γ q γ ¢ x 1 x 2 £ θ V 1 x 2 θ V 2 , θ γ pθ V 1 , θ V 2 q.
A calculation yields

pF 2 pγqpf pθ γ q γ qq 1 pxq pθ V 1 q 2 , pF 2 pγqpf pθ γ q γ qq 1 θ γ § § § θ0 p0q 0. Note that fixing θ 1 yields pF 2 pγqpf p1q
γ qq 1 p0q 1, so that the dual vector field without the θ parameter fails to identify the difference between γ and γ. The main use of the θ parameter in the proof of Proposition 4.1 is to enforce the map i to be injective and to preserve the nature of each pair pv, σpvqq. The dual vector field without the θ parameter is not sufficient, even in the aromatic context:

γ , γ , F |γ| pγqpf p1q γ qp0q 1.
This reveals a typographical error in [START_REF] Laurent | Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs[END_REF]Rk. 4.8] where the remark only applies to exotic trees, and a minor error in [START_REF] Mclachlan | B-series methods are exactly the affine equivariant methods[END_REF]Sec. 4.2]. The further proofs of this paper can be adapted straightforwardly to fix the proofs in [START_REF] Mclachlan | B-series methods are exactly the affine equivariant methods[END_REF].

Strong characterisations

This section is devoted to the proof of Theorem 2.12. Following Subsection 3.1, as the regularity assumptions of Theorem 2.12 imply the locality, orthogonal-equivariance, and trivially decoupling properties, we work directly with φ Fpγq and γ SpanpΓq. Proof. The decoupling property is straightforward from Definition 2.8. Let γ be an exotic aromatic tree without lianas and loops, f XpR

d 1 q, f XpR d 2 q, apxq Ax b SpR d 1 , R d 2 q with d 1 ¤ d 2 .
Differentiating the identity f papxqq Af pxq gives a K,J f i K papxqq a i,k f k J pxq.

We call leaves the vertices in V that are not the target of any arrow. We say a node v has depth p if the shortest path of v to a leaf passes through p different nodes (not including the start and end points). The nodes of depth at most p are gathered in the set V ppq . As γ does not have lianas or loops, V p|γ|q V . An induction on the depth yields

F d 2 pγqp f qpapxqq i,k ¹ vV p0q a iv,kv f kv pxq ¹ vV p0q f iv i τ ¡1 ptvuq papxqqδ iσ f i 0 i,k ¹ vV p1q a p1q iv,kv a p1q k τ ¡1 ptvuq ,i τ ¡1 ptvuq f kv k τ ¡1 ptvuq pxq ¹ vV p1q f iv i τ ¡1 ptvuq papxqqδ p1q iσ δ p1q kσ f i 0 ¤ ¤ ¤ i,k ¹ vV p|γ|q a p|γ|q iv,kv a p|γ|q k τ ¡1 ptvuq ,i τ ¡1 ptvuq f kv k τ ¡1 ptvuq pxqδ p|γ|q iσ δ p|γ|q kσ f i 0 i,k ¹ vV a i 0 ,k σp0q f kv k τ ¡1 ptvuq pxqδ kσ f i 0 AF d 1 pγqpfqpxq,
where a ppq iv,kv a iv,kv if σpvq τ ¡1 pwq with w V ppq and a ppq iv,kv 1 else, δ ppq iσ , δ ppq kσ contain the indices involved in the expression at step p, and the sums are on all involved indices.

On the other hand, let the exotic tree γ, the vector fields f XpR d 1 q, f XpR d 2 q, and the affine transformation apxq Ax b GpR d 1 , R d 2 q with d 1 ¥ d 2 . We find

F d 2 pγqpgqpapxqq i,k ¹ vV f iv i τ ¡1 ptvuq papxqqδ iσ f i 0 i,k ¹ vV a i τ ¡1 ptvuqL ,k τ ¡1 ptvuqL f iv i τ ¡1 ptvuq papxqqδ iσ δ pLq kσ f i 0 ,
where we used that AA T I d 2 to add the coefficients associated to lianas and δ pLq kσ identifies the coefficients k associated to lianas. The rest of the calculation is analogous to the Stiefelequivariance case: we define the depth function on the tree without the lianas and we perform the calculation with an induction on the depth of the tree. Proposition 4.4. Assume φ Fpγq is decoupling, then γ SpanpΓ c q. Proof. Let γ SpanpΓq and γ Γ be one of its exotic aromatic trees of maximal order among the ones that have at least one aroma, so that γ cγ R, γ µ m . . . µ 1 τ, c R, R SpanpΓq.

f pθ γ q γ f pθ τ q τ f pθ µ 1 q µ 1 ¤ ¤ ¤ f pθ µm q µm XpR d q,
where the first entry of f pθ γ q γ corresponds to the root of τ . Using (2.5) and Proposition 4.1, we obtain

pF d pγqpf pθ γ q γ qq 1 θ γ § § § θ0 p0q pF d pγqpf pθ γ q γ qq 1 θ γ § § § θ0 p0q $ 0.
On the other hand, as Fpγq is decoupling, we find

pF d pγqpf pθ γ q γ qq 1 θ γ § § § θ0 p0q pF |τ| pγqpf pθ τ q τ qq 1 θ γ § § §
θ0 p0q 0, as θ γ holds more parameters than θ τ . We obtain a contradiction and γ SpanpΓ c q.

We now prove our second main result.

Proof of Theorem 2.12. Assume φ Fpγq is decoupling, then Proposition 4.4 yields the connectedness of γ. Assume in addition that φ is local and Grassmann-equivariant and that at least one of the connected exotic aromatic trees τ in γ has a stolon or a loop. Consider d 1 |τ|, index the coordinates of R d 1 by pV A 0 q{σ as in Proposition 4.1. We split the coordinates of R d 1 into x y z, where y R d 2 contains the coordinates that are not stolons or nodes in a loop and z the others. Let A R d 2 ¢d 1 be the projection matrix on R d 2 , that is, Apy zq y. Define f 1 f pθ τ q τ and f 2 pAxq Af 1 pxq. The Grassmann-equivariance property and Proposition 4.1 yield

pφ d 2 pf 2 qq 1 θ τ § § § θ0 p0q pφ |τ| pf pθ τ q τ qq 1 θ τ § § § θ0 p0q pF |τ| pτqpf pθ τ q τ qq 1 θ τ § § § θ0 p0q σpτ q $ 0. As d 1 ¡ d 2 , there is at least one θ V v or θ S v that does not appear in f 2 , but appears in θ τ .
Thus pφ d 2 pf 2 qq 1 θ τ § § § θ0 p0q 0, which brings a contradiction. Assume now that φ F pγq is Stiefel-equivariant and that at least one of the exotic aromatic trees in γ has a liana or a loop. Consider the decomposition (4.1) of γ where γ is the term of maximal order among the ones that have a liana or a loop. For d 2 |γ|, we split the coordinates of R d 2 into x y z (or x z y if the root is a ghost liana, so that the root is still in first position), where y R d 1 contains the coordinates that are not lianas or nodes in a loop and z the others. Let A R d 2 ¢d 1 be such that A T is the projection matrix on R d 1 , that is, A T py zq y. Define f 2 f pθ γ q γ and f 1 pyq A T f 2 py 0q. Proposition 4.1 and the Stiefel-equivariance give

pφ d 1 pf 1 qq 1 θ γ § § § θ0 p0q pφ d 2 pf 2 qq 1 θ γ § § § θ0 p0q $ 0.
As d 1 d 2 , there is at least a node or a liana of γ that does not appear in f 1 , but appears in θ γ . We deduce pφ d 1 pf 1 qq 1 θ γ § § § θ0 p0q 0, which brings a contradiction.

If φ is semi-orthogonal-equivariant, then γ is a linear combination of connected exotic aromatic trees without lianas, loops, and stolons, that is, a combination of standard Butcher trees.

Impact of degeneracies on the classification

In a variety of contexts, the vector field f satisfies additional regularity properties. For instance, if f is a polynomial map of order p, then all exotic aromatic trees where at least a node is the target of more than p arrows have a trivial elementary differential. We mention in particular the work [START_REF] Bogfjellmo | Using aromas to search for preserved measures and integrals in Kahan's method[END_REF] on aromatic trees for quadratic differential equations that relies on such degeneracies. In the original numerical application of the exotic aromatic B-series in molecular dynamics [START_REF] Laurent | Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs[END_REF][START_REF] Laurent | Order conditions for sampling the invariant measure of ergodic stochastic differential equations on manifolds[END_REF] (see also [START_REF] Lelièvre | Free energy computations[END_REF]), the vector fields f of interest are gradients, that is, f ∇V for a smooth function V : R d Ñ R. In [START_REF] Lelièvre | Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion[END_REF][START_REF] Duncan | Variance reduction using nonreversible Langevin samplers[END_REF][START_REF] Abdulle | Accelerated convergence to equilibrium and reduced asymptotic variance for Langevin dynamics using Stratonovich perturbations[END_REF], vector fields of the form f J∇V with J the symplectic matrix are used as perturbations to reduce the variance and accelerate the speed of convergence to equilibrium in the numerical integration of Langevin dynamics. In this section, we update the classification of Theorem 2.12 for gradient vector fields f ∇V , gathered in the set X ∇ pR d q. As discussed in [START_REF] Laurent | Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs[END_REF]Remark 4.8], the gradient property of f translates into degeneracies. Proposition 4.5 ([23, 21]). We say that two exotic aromatic trees γ 1 and γ 2 are equivalent on X ∇ pR d q, written γ 1 γ 2 , if by performing the following operations, it is possible to transform γ 1 into γ 2 :

• inversion edge-liana: A The equivalence relation preserves the composition κ of the graph. Two equivalent exotic aromatic trees represent the same elementary differential F d pγ 1 q F d pγ 2 q on X ∇ pR d q. Moreover, for any connected exotic aromatic tree, there exists a unique exotic tree in its equivalence class.

Proof. This is a direct consequence of the Schwarz theorem f i j 1 ...jq f jp j 1 ...j p¡1 ij p 1 ...jq .

Example.

The following connected exotic aromatic trees are equivalent to exotic trees: , , .

For further examples, the list of exotic aromatic trees of order 3 presented in Section A gathers the equivalent exotic aromatic trees in adjacent lines.

On X ∇ pR d q, Theorem 2.11 and Theorem 2.12 simplify into the following simpler classification, which exactly characterises the exotic trees used in numerical analysis [START_REF] Laurent | Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs[END_REF].

Theorem 4.6. Let φ pφ d : X ∇ pR d q Ñ XpR d qq d be a sequence of smooth maps. The Taylor expansion of φ around the trivial vector field 0 is an exotic B-series on X ∇ pR d q if and only if φ is local, orthogonal-equivariant, and decoupling.

Conclusion and future works

In this work, we showed that smooth local orthogonal-equivariant maps and exotic aromatic B-series represent the same object. This universal property shows that exotic aromatic Bseries are not just a tool for calculations in numerical analysis, but a natural algebraic object that is interesting in itself. The analysis relies on the invariant tensor theorem for orthogonalequivariant tensors and the Peetre theorem, but also on a new generalised construction of exotic aromatic trees. In addition, we classified the intermediate subsets of exotic aromatic B-series, and in particular the exotic B-series, with respect to strong equivariance properties. We also defined new dual vector fields and identified the effect of the degeneracies appearing in numerical analysis on the classification. A variety of theoretical and applied questions arise from the present work. There exists different extensions of B-series such as partitioned B-series or Lie-Butcher series, and a variety of equivariance properties in R d but also on manifolds. We mention in particular the equivariance with respect to symplectic tranformations. It would be interesting to link the different equivariance properties with the various B-series. This could allow us to create new extensions of B-series and to find corresponding applications in numerical analysis. For the B-series presented in this paper for instance, the B-series with stolons could be used in the study of projection methods for the approximation of ODEs on manifolds and modifications of the exotic aromatic formalism could be applied to the study of stochastic differential equations with multiplicative noise or to the creation of stochastic Lie-group methods of high weak order. This is matter for future work. Fpγqpf q 1 p0, 0, 0, 0, 1q p0, 0, 0, 0, 4q p1, 1, 1, 1q p0, 1qp1, 2qp3, 4q

f i jjkk f i p0, 1qp2, 1qp3, 4q f j ijkk f i 2 p0, 1, 1q
p0, 1, 2q p1, 1, 2q p0, 2qp1, 2qp3, 1q

f i j f j kk f i p0, 2qp1, 1qp2, 3q f i j f k jk f i p0, 3qp1, 2qp1, 2q f j i f j kk f i p0, 3qp1, 1qp2, 2q f j i f k jk f i p0, 1qp1, 2qp2, 3q f i jk f j k f i p0, 1qp3, 1qp2, 2q f k ij f j k f i p0, 1qp1, 2qp2, 3q f j ik f j k f i p0, 1qp1, 2qp3, 2q f i jj f k k f i p0, 1qp2, 1qp3, 2q f j ij f k k f i 2 p1
, 0, 0, 1q p0, 0, 0, 3q p1, 1, 1q p0, 1qp1, 2qp2, 3q 3 p1, 2q p0, 2q p1, 2q p0, 1qp1, 2qp2, 3q

f i jkk f j f i p0, 1qp2, 1qp3, 2q f j ijk f k f i p0, 1qp2, 3qp1, 2q f j ikk f j f i p0, 2qp1, 1qp2, 3q f i f j jkk f i Table 3 (
f i j f j k f k f i p0, 1qp1, 2qp2, 3q f i j f k j f k f i p0, 1qp1, 2qp2, 3q f j i f j k f k f i p0, 1qp2, 1qp2, 3q f j i f k j f k f i p0, 1qp1, 3qp2, 2q f i j f j f k k f i p0, 2qp1, 1qp2, 3q f j i f j f k k f i p0, 3qp1, 2qp2, 1q f i f j k f k j f i p0, 3qp1, 2qp1, 2q f i f j k f j k f i p0, 3qp1, 1qp2, 2q f i f j j f k k f i 3 p2
, 0, 1q p0, 0, 2q p1, 1q p0, 1qp1, 2qp2, 3q f i jj f j f j f i p0, 1qp2, 2qp1, 3q

f j ik f j f k f i p0, 1qp1, 2qp2, 3q f i jj f k f k f i p0, 1qp2, 1qp2, 3q f j ij f k f k f i p0, 2qp1, 1qp2, 3q f i f j jk f k f i p0, 3qp1, 2qp1, 2q
f i f j f j kk f i 4 p3, 1q p0, 1q p1q p0, 1qp1, 2qp3, 4q

f i j f j f k f k f i p0, 4qp1, 2qp1, 3q f i f j f j k f k f i p0, 1qp1, 2qp3, 4q f j i f j f k f k f i p0, 2qp1, 1qp3, 4q f i f j f j f k k f i 5 p5q p0q p0, 1qp2, 3qp4, 5q f i f j f j f k f k f i

Lemma 2 . 7 .

 27 Define the order 1 |γ| of an exotic aromatic tree γ Γ κ by |γ| |V | |L| ¡ |S|.

  3) becomes |γ| |κ| |κ I | 1.

Definition 2 . 10 .

 210 Given a coefficient map b : Γ Ñ R that has finite support on Γ m for m ¡ 0, the associated exotic aromatic B-series in dimension d is the following formal series B d pbq m¡0 γ Γm bpγqF d pγq. An exotic aromatic B-series is a sequence Bpbq pB d pbqq d indexed by the dimension. The first main result of this paper is the characterization of exotic aromatic B-series with orthogonal-equivariance and locality. Theorem 2.11. Let φ pφ d : XpR d q Ñ XpR d qq d be a sequence of smooth maps. Then, the Taylor expansion of φ d around the trivial vector field 0 in dimension d is an exotic aromatic B-series φ d B d pb d q if and only if φ d is local and orthogonal-equivariant. If, in addition, φ is trivially decoupling, then there exists a coefficient map b : Γ Ñ R such that φ Bpbq.

Lemma 3 . 2 .

 32 Thanks to Theorem 3.1, ∆ m φ d has the following form, ∆ m φ d pfq |κ|m |κ| |κ I | 12Z ψ κ,d pf κ q, f κ pf, . . . , f looomooon κp0q , f I , . . . , f I loooomoooon κp1q , . . . q, (3.2) where only a finite number of the ψ κ,d S O d pRq κ are non-zero. Let φ be a local orthogonal-equivariant sequence of smooth maps. The strong equivariance, decoupling, or trivially decoupling properties of φ are transferred to the Taylor terms ∆ m φ in (3.2).Proof. The proof for the transfer of equivariance properties is the same as in[START_REF] Mclachlan | B-series methods are exactly the affine equivariant methods[END_REF] Prop. 6.2]. Let us prove the transfer of the decoupling property. The transfer of the trivially decoupling property uses the same arguments. The Taylor terms of φ satisfy[20, § 5.11]

O d pRq κ .Theorem 3 . 3 .

 κ33 For a given κ, there exists a surjective linear map r

F

Proposition 4 . 3 .

 43 Connected exotic aromatic B-series are decoupling. B-series with stolons are Stiefel-equivariant and exotic B-series are Grassmann-equivariant.

(4. 1 )

 1 Without loss of generality, we assume c 1. Define f pθ γ q γ as in Proposition 4.1 and following the numbering of Remark 4.2,

  Part 1/2): List of the exotic aromatic trees of order three, with their associated composition, derived composition, target map, source map, and elementary differential (see Definition 2.8).

  .2 and 6.1]. If φ pφ d : XpR d q Ñ XpR d qq d is Grassmann-equivariant, then φ is orthogonal-equivariant, trivially decoupling, and decoupling.

	Proposition 2.5 ([30]). If φ is Stiefel-equivariant, then φ is local, orthogonal-equivariant, for clarity. semi-orthogonal-equivariance Stiefel-equivariance locality decoupling orthogonal-equivariance and trivially decoupling. We summarise the links between the different geometric properties in the following graph Grassmann-equivariance trivially decoupling

Table 2 :

 2 3.3 contains the proof of Theorem 2.11 and a clarifying example. Classification of B-series with respect to their equivariance properties (see Theorems 2.11 and 2.12).

	geometric property	associated Butcher series
	orthogonal-equivariance	exotic aromatic B-series
	GL-equivariance	aromatic B-series
	Stiefel-equivariance	B-series with stolons
	Grassmann-equivariance	exotic B-series
	affine/semi-orthogonal-equivariance	B-series

  Theorem 3.1. Let H O d pRq and m a positive integer, the following isomorphism holds L H pS m pM SM ¦ q, M q Proof. Following the description of O d pRq-invariant tensors [18, § 33.2], we deduce that T H κ is trivial when |κ| |κ I | 1 is odd. As S H κ is naturally injected into T H κ , we obtain the desired result.

	à |κ| |κ I | 12Z |κ|m	S H κ .

Table 3 (

 3 Part 2/2): List of the exotic aromatic trees of order three, with their associated composition, derived composition, target map, source map, and elementary differential (see Definition 2.8).

Department of Mathematics, University of Bergen, Bergen, Norway. Adrien.Laurent@uib.no.

Department of Mathematics and Statistics, UiT -The Arctic University of Norway, Tromsø, Norway. Hans.Munthe-Kaas@uit.no.

A similar definition of the order of an exotic aromatic tree is given in the works[23,

[START_REF] Laurent | Order conditions for sampling the invariant measure of ergodic stochastic differential equations on manifolds[END_REF]. Note that the order of an exotic aromatic tree is not the number of nodes in general.

Acknowledgements. The work of Adrien Laurent was supported by the Research Council of Norway through project 302831 "Computational Dynamics and Stochastics on Manifolds" (CODYSMA).


Arnaud Tonnelier

Exact solutions for signal propagation along an excitable transmission line

I. INTRODUCTION

Nonlinear media have been largely used for their information-processing capabilities, and, among them, excitable media have became a paradigm for performing natural or unconventional computing [START_REF] Gorecki | [END_REF]17]. A bit of information is physically represented by a pulse excitation and a group of pulses delivers a message [2]. The nonequilibrium properties of the medium can be used to generate nontrivial messages and, in spatially extended system, the question of information transmission is closely related to the existence of traveling wave solutions. The ability of discrete excitable media to support nonlinear waves is well known in biology [START_REF] Murray | Mathematical Biology[END_REF] but has also been observed in geochemical systems [START_REF] Grindrod | Patterns and waves: the theory and applications of reaction-diffusion equations[END_REF], chemical reaction kinetics [37], and in engineered systems [START_REF] Pérez-Munuzuri | [END_REF]25], to mention few of them. Traveling waves are directly involved in information processing as for instance in visual perception [38] and more broadly in the functions performed by the nervous system [1]. The temporal structure of the wave plays an important role in the neural processing of sensory stimuli. In particular, the synaptic plasticity, involved in learning and memory, crucially depends on the precise timing of pulses [22]. A fundamental issue in the study of collective dynamics is to unravel the complex relation between interactions and dynamics, specially for functionally meaningful trajectories, i.e. having non-trivial spatio-temporal profiles [20,31]. In most natural systems, the precise network structure is unknown and information about the connectivity is obtained indirectly from activity measurement [START_REF] Prasse | Proc. Natl. Acad. Sci[END_REF]. The absence or presence of a connection, or the hierarchical structure of the network, can be inferred from the dynamical response properties [3,30]. In this context, we raise the following question: given a network structure, say regular lattice, all-to-all connectivity, or random network, is it possible to derive information on connectivity strength from signal propagation? It is well documented that a weak connectivity does not allow for propagation, a phenomenon known as propagation failure. The strength of each connection shapes the dynam- * arnaud.tonnelier@inria.fr ics of propagation but finding the precise relation between traveling waves and network connectivity remains a challenging problem. Detailed models provide a comprehensive description of excitable networks but they are rarely integrable and their study reveals to be very difficult beyond numerical simulations. Valuable insights into the dynamics of nonlinear waves in excitable media have been obtained using simplified models [4,7]. Analytical results based on explicit solutions have proven useful in understanding propagation phenomena in nonlinear transmission lines [32]. In this paper we introduce a simple threshold model that captures the basic properties of excitable media with, in particular, the existence of different propagating patterns. The complete solvability of the model allows an in-depth analysis of simple waves where units along the network are sequentially activated [34] and of complex patterns having a non-trivial spatio-temporal periodicity [33]. The plan of the paper is the following. In the next section, the generic formulation of the excitable transmission line is presented. In section III preliminary results on existence and stability of traveling signals are obtained. A fully solvable model is proposed and studied in section IV where analytical expressions are derived for both the signal shape and the signal velocity.

II. THE EXCITABLE TRANSMISSION LINE

We study the signals generated and propagated by onedimensional networks where each node is described by a time-dependent state variable, noted s i (t), where i ∈ Z is the index of the node in the network and t > 0 is the time. The mechanisms of emission and transmission along the network are as follow. A node may interact with the others in the network if its activation threshold is reached. The i th node is said to be activated at time t i when s i (t i ) ≥ 1, where we set without loss of generality the threshold to unity. The threshold-crossing time t i is called indifferently the activation time, the firing time or the excitation time. The firing time is transmitted so that the dynamics of the node at location i is described by

s i (t) = j w ij e(t -t j ), (1)
where w ij is the strength of the connection from j to i and e(t) is a pulse-shape coupling function that is initiated by the arrival of a firing time (at time t j ). Its analytical expression will be specified latter. We assume that the network is homogeneous, we have w ij = w(i-j), feedforward, that is w i,j = 0 for j ≥ i, and excitatory, i.e. w ij ≥ 0. We note w j = w i,i-j and n the maximum connection length, i.e. w j = 0, for j > n. After the transmission of the firing time, the node remains quiescent, i.e. a strong refractory is assumed (the node jumps back instantly to 0). The single-activation approximation is relevant in rapid information processing tasks as, for instance, in fast image recognition where each processing unit, the neuron, can emit only one action potential [29]. This situation also occurs when the recovery operates on a time scale significantly greater than the activation as in forest fires or epidemic spread. The single-spike solution is also used to facilitate the mathematical treatment of models [35]. The excitable transmission line described by (1) shares similarities with pulse-coupled neural networks, named PCNN, that have been primarily introduced to describe the dynamics of the visual cortex [6]. It is also a particular instantiation of the spike response model [START_REF] Gerstner | Spiking neuron models[END_REF] where the emission times t i are the spike timings and the elementary signal e(t) is the excitatory postsynaptic potential. The model can be derived from an approximation of differential models of biological neural networks [33].

It allows a continuous time integration of incoming signals and therefore differs from cellular automata that are synchronously updated [39]. Determining exact solutions of the excitable system (1) relies on finding a closed-form expression for the activation times t i . We are interested in the propagation along the transmission line that is, in the trivial case, associated with the existence of an increasing sequence (t i ) where the successive activation times of the nodes along the network increase linearly, i.e. we have t i+1 -t i = 1/c where c is the transmission speed. However, more complex relation between successive firing times can exist and we are set out to construct traveling signals with a nontrivial spatiotemporal periodicity. This leads to solutions that are less understood and that correspond to an intriguing behavior closely related to the lurching waves observed in continuous excitable networks [START_REF] Golomb | Proc. Natl. Acad. Sci[END_REF][START_REF] Golomb | [END_REF][START_REF] Rinzel | [END_REF] and to multichromatic travelling waves recently found in lattice Nagumo equations [18,19]. The periodicity of these traveling patterns does not come from the periodic structure of the underlying medium, as for instance in diatomic chains [START_REF] Hoffman | [END_REF] or in synfire-chains [2], but arises as a natural property of spatially homogeneous systems. This also differs from the propagation of spatio-temporal patterns generated by the successive excitation of individual elements as in burst propagation [28]; a solution tied to intrinsic nodes properties rather than connectivity.

III. TRAVELING SIGNALS

We consider the transmission of firing sequences described by a velocity, c, and a sequence of time-intervals, noted (s k ), such that the activation times of the nodes along the lattice are given by t pi+k = (pi + k)/c + s k which characterizes the ith repetition of a sequence of length p, i.e. involving p successives nodes. We have k ∈ 0, . . . , p -1 that denotes the label of the nodes in the replaying sequence. We fix the origin of the sequence at k = 0 and we set s 0 = 0. The case s k = 0 corresponds to the propagation of simple signals where the successive activation times have a constant time shift of 1/c. The sequence (s k ) measures how the propagating signal deviates from a simple signal and we refer this solution as a composite signal to suggest that the propagation can be seen as the combination of simple signals where distinct nodes can have distinct profiles. A fundamental question is how the sequence (s k ) and the velocity c are related to the network connectivity, i.e. the weights (w j ) ? In this paper, we focus on third nearestneighbor interactions, i.e. we set n = 3, that corresponds to the simplest connectivity that allows the propagation of composite signals of length p = 2. The existence of a traveling packet of two firing times is determined by the two unknowns, c and s, such that

t i = i c , if i is odd, i c + s, if i is even, (2)
The transmission line conveys two time intervals: the natural one associated with the velocity and an additional information, coded by s. The characteristic time s is an intrinsic property of the transmission line and not a free parameter. The case s = 0 corresponds to a simple signal and the case s = 1/c is related to the propagation of a synchronous pair of firing times. Traveling signals associated with the activation times (2) have two distinct profiles, noted S 1 and S 2 , such that

s i (t) = S 1 (i -ct), if i is odd, S 2 (i -ct), if i is even, that are given by    S 1 (ξ) = w 1 e(ξ + 1 c -s) + w 2 e(ξ + 2 c ) + w 3 e(ξ + 3 c -s), S 2 (ξ) = w 1 e(ξ + 1 c + s) + w 2 e(ξ + 2 c ) + w 3 e(ξ + 3 c + s), (3)
where ξ is the traveling wave coordinate. Threshold condition for the traveling wave solutions gives

f (1/c, s) = 0, and f (1/c, -s) = 0, (4)
where f (x, y) = w 1 e(x -y) + w 2 e(2x) + w 3 e(3x -y) -1.

A solution of ( 4) is associated with a traveling signal if the admissibility conditions

S i (ξ) < 1 for ξ < 0, i = 1, 2 (5)
are fulfilled. Otherwise the solution is said to be nonadmissible. Inequalities (5) ensure that the threshold is not reached before the origin and can be interpreted as causality conditions. Stability is a key property for the robust propagation of signals along the network. We consider a small perturbation of the emitting times such that ti = t i + u i where |u i | 1. We inject this ansatz into (1) and we look for a solution of the form u 2i = λ i 1 λ i 2 and u 2i+1 = λ 1 u 2i in the linearized system. Asymptotic stability of the traveling sequence (2) holds when |λ 1 λ 2 | ≤ 1 where λ 1 and λ 2 are the roots of the multivariate polynomials The case s = 0 corresponds to simple signals that are characterized by a constant interspike interval between two consecutive nodes. Within this class, we distinguish between three different solutions reported as 3-signal, 2signal and 1-signal that are classified according to the location of the time events t i = 1/c, i = 1, 2, 3, with respect to the time-to-peak of the pulse-shape function e. If the three events fall into the increasing phase of e, the traveling signal belongs to the 3-signals class. The 1-signal is obtained if 1/c is the only time less than the timeto-peak. The intermediate solution corresponds to a 2signal. The 3-signals are fast wave where each firing time of neighboring nodes contributes to the threshold crossing so that 3-signals are strictly monotone whereas for 2-signals and 1-signals the effect of the received-first and received-second firing time, respectively, has vanished or is in a decaying phase at the threshold crossing. When s > 0 the solution corresponds to a composite signal and we distinguish between the case s < 1/c where the firing sequence is strictly increasing and s > 1/c where the sequence is non monotonic.

(a + b + c)x 2 y -axy -bx -c = 0, (6)

IV. AN EXACTLY SOLVABLE MODEL

We consider a coupling function e given by the cut-off ramp function

e(t) = t, for 0 ≤ t ≤ 1, (7)
and 0 otherwise, that can be rewritten as e(t) = tχ [0,1] (t) where χ is the indicator function. The ramp function, or rectified function, is omnipresent in digital signal processing and is frequently used to describe a progressive excitation. In our model we add a cut to describe the finite duration of excitation. The simple elementary signal (7) can be obtained as an approximation of a pulse-shape function with an instantaneous jump-like decaying phase.

Results on simple signals have been presented in [34] for a triangular coupling function with a non-instantaneous decaying phase. Here we complete and develop the analytical study with a particular attention to composite signals.

The results that we will analyze are numerically illustrated in Fig. 1 where simple signals (s = 0) and composite signals (s = 0) are investigated. Numerical simulations are made using a time stepping method where, at each time step, new emission times are found by threshold-crossing detection. The calculated emission times are propagated to neighboring elements and the network state is updated using the coupling function. In Fig. 1(a) the existence of simple signals is studied with the plot of the traveling wave speed c as a function of the total connectivity strength, |w| = i w i . When |w| is small, there is no traveling signals and propagation failure is observed. Depending on the connectivity, three branches of solutions can be found, reported as 1-signal , 2-signal and 3-signal (and noted 1S, 2S and 3S), from bottom to top, respectively. Candidate solutions for both simple and composite signals are obtained in Fig. 1(b) in the (c -1 , s) plane as the crossing points of the two level curves (4). Using (7), it is easy to see that the level curves have a graph that is composed of three distinct segments that can be determined analytically as a function of the weights, w i . Inspection of the threshold condition, shown in Fig. 1(c), reveals that a candidate solution may violate the causality criterion and, as a consequence, parts of the branches shown in Fig. 1

c p = p i=1 iw i , (8)
where p = 1, 2, 3 for p-signals. It is interesting to see that the connectivity of the transmission line can be reconstructed from the velocity of propagating signals using recursively the formula, w 1 = c 1 , w i = (c i -c i-1 )/i, i = 2, 3. Therefore the full network connectivity can be inferred from the observation of three distinct propagat- ing signals.

The threshold condition only gives candidate solutions and the causality criterion ( 5) determines their admissibility. By inspecting the maximum of the wave function S(ξ) = i w i e(ξ -i/c) in the region ξ < 0, we determine the exact conditions for the admissibility of the different simple signals. The branch associated with the fast transmission, referred to as the 3S branch, defines strictly increasing traveling wave solutions that are therefore always admissible. For the two other branches, some candidate solutions may be non-admissible as it is shown in Fig. 1 different modes of signal propagation. In region labeled A, the propagation is characterised by a non-monotonic sequence of activation times and in region B, the sequence is strictly increasing. When w 1 < w 3 , the signal speed c and the time interval s belong to region A (see Fig. 2) and can be determined analytically using the thresholdcrossing condition. We find

c = w 1 + w 3 d A , s = 3w 3 -w 1 d A , (9)
where the parameter d A depends on the connectivity and is given by

d A = 2(w 1 w 2 + w 2 w 3 + 2w 1 w 3 ).
For w 1 > w 3 , candidate solutions are located in region B and are given by

c = 3w 3 d B , s = 2w 1 + w 3 d B , (10)
where

d B = 2((w 1 + w 2 )(w 1 + w 3 ) + w 1 (w 2 + w 3 )).
Note that when w 1 = w 3 , ( 9) and ( 10) coincide. If we note c cs the velocity of a composite signal, using ( 8) and ( 9), ( 10), we have the following inequalities

c 2 ≤ c cs ≤ c 3 ,
i.e. composite signals travel at a speed intermediate between fast wave speed (velocity of 3-signals) and 2-signal speed. Using (3), we find that the activity of the nodes described by the waveform S 1 is strictly increasing whereas nodes described by S 2 has a trajectory having a jump down at ξ = 1 -3/c -s. Therefore the admissibility condition (5) for composite signals depends only on the wave function S 2 and simply reads as S 2 (1 -3/c -s) < 1 (where the left side limit of the function has to be considered). We obtain the admissibility condition

w 1 (2/c -1) + w 2 (1/c + s -1) -w 3 + 1 > 0, (11)
where (c, s) is given by ( 9) when w 1 ≤ w 3 and by [START_REF] Gerstner | Spiking neuron models[END_REF] when w 1 ≥ w 3 . The stability is determined using (6) where the coefficients of the multivariate polynomials are a = w 1 Π(1/c + s), b = w 2 Π(2/c), c = w 3 Π(3/c + s), with Π the rectangular function obtained from the derivative of the coupling function e, Π(x) = 1, for 0 < x < 1 and 0 otherwise. Investigation of the roots (see Appendix B) shows that composite signals are always stable. The boundary between the two regions, given by cs = 1 and obtained when w 1 = w 3 , is related to the propagation of a synchronous signal where a pair of identical firing times propagates along the transmission line. Synchronous solutions have frequently retained a special attention for their role in conveying and processing information. Precisely synchronized action potentials with millisecond precision can propagate in cortical neural networks [5] and are supposed to play an important role in network remodeling through spike timing-dependent plasticity [9]. Synchronous waves. Existence of a synchronous solution is only possible when w 1 = w 3 and we find a propagation velocity c = 2(w 1 +w 2 ) that defines a candidate solution when w 1 + w 2 > 1. The transmitted time-interval is given by s = 1/(2(w 1 + w 2 )) and the wave profile is described by the two following functions. The first wave function is given by

S 1 (ξ) = (w 1 + w 2 )ξ + 1,
for -2/c ≤ ξ ≤ 0 and 0 otherwise. The second wave shape is defined by

S 2 (ξ) = w 1 ξ + 2w 1 w 1 + w 2 ,
for -4/c ≤ ξ ≤ -2/c, and

S 2 (ξ) = (2w 1 + w 2 )ξ + 2w 1 w 1 + w 2 + 1,
for -2/c ≤ ξ ≤ 1 -4/c, and

S 2 (ξ) = (w 1 + w 2 )ξ + 1, for 1 -4/c ≤ ξ ≤ 0.
The waveform is strictly increasing for the nodes associated with S 1 whereas the function S 2 presents a jump of discontinuity, ∆S 2 = -w 1 , at ξ = 1 -4/c. The admissibility condition (11) reads 2w 1 + w 2 < 2. The locus of existence of synchronous waves in the (w 1 , w 2 )-parameter plane is plotted in Fig. 3 and is compared with the regions of existence of simple signals. A large part of the parameter space (w 1 , w 2 ) ∈ + corresponds to the propagation of simple signals where 3-signals are the dominant patterns. The 1-signal does not exist when w 1 = w 3 (causality criterion is violated) but 2-signals transmission is possible. Propagation failure is observed when both w 1 and w 2 are small. There exist regimes where synchronous waves can travel for parameters where simple signals are also free to travel, i.e. the transmission line presents a bistability between simple and synchronous waves, the initial stimulation determines the propagating pattern. In particular, 2-signals always coexist with synchronous signals. Interestingly there also exists a tiny region where the synchronous wave can travel in parameter regimes where simple signals fail to propagate. For instance when w 1 = w 3 = 0.3 and w 2 = 0.8, the propagation of a synchronous wave is observed whereas simple waves do not exist. As w 1 changed, different regimes can be reached. If we set w 2 = 0.8, the following sequence of bifurcations, as w 1 increased, is observed. At small w 1 values, signals do not propagate along the network. As w 1 reaches the critical value w 1 = 1 -w 2 = 0.2, synchronous waves appear. When w 1 reaches the value 3/4 -w 2 /2 = 0.35, there is a coexistence between synchronous waves and 3-signals, and subsequently when w 1 = 2 -2w 2 = 0.4 a tristability regime with the possible propagation of 2-signals.

V. DISCUSSION

Cellular automata are minimal models that describe the dynamics of excitable media [14,36] where the propagation results from a cooperative effect generated and transmitted by simple identical and connected components [39]. Despite the strong similarities with the propagating patterns generated by more realistic models, as spiking neural lattice for instance [21], cellular automata do not capture the temporally continuous properties of excitable systems and their sensitivity to the updating scheme made them inadequate for describing natural processes. In this paper, we formulated a simple phenomenological model of excitable media consisting of a continuous time-integration of incoming signals supplemented with a threshold effect. The resulting model is an informationprocessing line that allows for stable signal transmission with different velocities, as observed in one-dimensional neuronal cultures [8]. Classical solutions described by a unique waveform are obtained but more complex trajectories composed of multiple profiles also exist. Using a simple activation function the model is solvable and the propagation of composite waves is studied in detail. The stability of traveling signals with several distinct profiles is shown. The exact dependence of the traveling signals on the connectivity of the network is obtained for the propagation of a doublet including the case of synchronous propagation. We consider a network with third-nearest-neighbor coupling that allows for the temporal repetition of a doublet along the line. The doublet is not generated by the re-excitation of nodes but is an intrinsic property of the network. It would be interesting to study the propagation of larger sequences, i.e. triplet or quadruplet, that can occur in a transmission line with an extended connectivity. Simulations (not shown) reveal that long range interactions give rise to more complex signals and future works are necessary to clarify the link between connectivity footprint and signal propagation. Another further extension of this work will concern the effect of multiple-activation of nodes. We restrict our attention here to the so called one-spike framework where nodes can emit only one time and interactions between several activation-times would create a richer variety of phenomena. Fixed profiles that change cyclically are studied here but irregular patterns that evolve continually may also exist. The case of quasi periodic solutions, not studied here, could be considered to broaden possible signal transmissions in the network.

APPENDIX A

Simple waves for w 1 = w 3 . For a transmission line such that n = 3 and w 1 = w 3 , the existence of p-waves is given by the following conditions. The fast wave speed (p = 3) is given by c = 4w 1 +2w 2 and the condition c ≥ 3 reads w 2 ≥ 3/2-2w 1 . For p = 2, we have c = w 1 +w 2 and conditions 2 ≤ c ≤ 3 gives 1 -w 1 /2 ≤ w 2 ≤ 3/2 -w 1 /2. For the slow wave, i.e. p = 1, the wave speed is given by c = w 1 and we obtain the condition 1 ≤ w 1 ≤ 2. The candidate solution defines an effective solution if the admissibility condition S(ξ) < 1, for ξ < 0, is fulfilled. As previously mentioned, fast waves (3-signals) are always admissible solutions. Using max ξ<0 S(ξ) < 1, we find the following admissibility condition for 2-waves:

2w 2 2 + 2w 2 1 + 5w 1 w 2 -3w 2 -3w 1 < 0.
One can show that the condition of existence for 2-signals w 2 ≤ 3/2 -w 1 /2 is fulfilled when the admissibility condition holds. For 1-wave we find the admissibility condition w 1 < 1 that is in contradiction with the existence criterion and therefore the branch of 1-wave solutions is not admissible.

APPENDIX B

Stability analysis of composite signals. We distinguish between the two regions of existence of composite signals (see Fig. 2) Case A: w 1 ≤ w 3 . From (6), we study the roots (λ 1 , λ 2 ) of the multivariate polynomials .

It follows that |λ 1 λ 2 | < 1 and the stability of the associated traveling signal.

  (ā + b + c)xy 2 -āxy -by -c = 0, where the coefficients are given by a = w 1 e (1/c + s), b = w 2 e (2/c), c = w 3 e (3/c + s), with e the derivative of the coupling function e. Coefficients (ā, b, c) are directly obtained from (a, b, c) replacing s by -s in the corresponding expression. The trivial solution λ 1 = λ 2 = 1 is related to the translational invariance of traveling wave solutions. Traveling signals are classified into two distinct types.

  (a) do not define admissible solutions. Traveling wave solutions are shown in Fig. 1(e),(f) where depending on the initial excitation, a simple wave or a composite wave propagates along the transmission line. Simple signals. The analytical description of the speed diagram of simple waves is derived from the threshold condition i w i e(i/c) = 1 and it is easy to show that the velocity, c, as a function of the global connectivity strength, |w|, is composed of three branches. As c varied, jumps at the two critical speeds c = 3 and c = 2 indicate the transitions from 3-signal to 2-signal and from 2-signal to 1-signal, respectively. The discontinuity of the speed curve is the result of the instantaneous decrease of the coupling function. The transmission speed associated with the different branches is given by

FIG. 1 .

 1 FIG. 1. Traveling signals along the transmission line for a connectivity given by w1 = 4|w|/7, w2 = w1/2 and w3 = w2/2. (a) Speed diagram of simple signals where the velocity c of the signal is plotted as a function of the global connectivity strength |w|. The graph is made of three distinct branches, noted 3S, 2S and 1S, from top to bottom, respectively. Dotted lines are for solutions that are not admissible (see panel (c))). The vertical dotted line indicates the value |w| = 2.1 that is used in plot (b) for the graphical representation of the two level curves f (x, y) = 0 and f (x, -y) = 0 (see (4)) in the (x, y) = (1/c, s) plane. Crossing points indicate candidate solutions for traveling signals. Solutions on the x-axis, i.e. for s = 0, are associated with simple signals and the corresponding branch of the speed diagram is mentioned by the labels 3S, 2S or 1S. The label CS indicates a candidate solution for a composite signal. Some candidate solutions are non-admissible as shown in (c) for a solution on the 2S branch of the speed diagram (we use here |w| = 2.4) for which the causality criterion is violated. In (d) the signal profiles, S1 and S2, of the composite signal are shown for the solution noted CW in panel (b). Raster plots of the traveling signal for (e) a simple signal and (f) a composite signal. The two insets show the associated inter emitting-time interval, noted ∆t = ti+i -ti where i is the index of the node.

Fig. 2 .FIG. 2 .

 22 FIG.2. Locus of existence of composite signals in the (c -1 , s) plane. The grey shaded quadrilateral indicates the region where a composite signal may be found. In region A, where cs > 1, the firing sequence associated with the propagating signal is non monotonic whereas in region B the wave pattern corresponds to a strictly increasing sequence of firing times. The boundary between the two regions (cs = 1) corresponds to the propagation of a pair of synchronous firing times.

FIG. 3 .

 3 FIG.3. Regions of existence of a stable signal transmission in the (w1, w2) plane for w1 = w3. A signal can propagate stably along the transmission line for parameters in the grey regions. Different areas are depicted with different grey levels that represent simple signals, labeled 3S and 2S, and composite signals, noted sS for synchronous signal. The region 3S is a half plane delimited by the straight line 2(w1 + w2) = 3. The zone labeled 'sS' is triangular and the 2S region is triangular in shape.

R 2

 2

(w 2 + 2 -w 1 w 3 (w 1 +λ 1 λ 2 = w 2 2 -w 1 w 3 (w 1 +

 2231231 w 3 )xy 2 -w 2 y -w 3 = 0, (w 1 + w 2 )x 2 y -w 1 xy -w 2 x = 0.The trivial solutions λ 1 = λ 2 = 1 and λ 1 = 0 are handled easily and we end up withλ 1 λ 2 = w 2 w 2 )(w 2 + w 3 )and we find that |λ 1 λ 2 | < 1 is always satisfied showing the stability of composite signals.Case B: w 1 ≥ w 3 . System (6) givesxy 2 -w 1 xy -w 2 y -1w 1 + w 2 = 0, (w 1 + w 2 )x 2 y -w 1 xy -w 2 x = 0,that, after dealing with the trivial cases, leads to: w 2 )(w 1 + w 2 + w 3 )

  (a) where the 2-signal branch and the 1-signal branch corresponds to non-admissible signals when |w| is large, i.e. greater than |w| ∼ 2.2 and |w| ∼ 2.35, respectively, for parameters used in the plot. A propagating signal can be found on the p th branch of the speed curve if the following conditions hold: for p = 2, w 1 +w 2 > 2w 3 and for p = 1, w 1 > min(w 2 , w 3 ). The stability of simple waves is determined from (6) that, after factoring by λ -1, reduces to a second order polynomial and it is easy to demonstrate that simple signals are always stable. Composite signals. The transmission of a time interval different from the one trivially associated with the wave speed is less well understood and studied. Exploration of candidate solutions for system (4) using the coupling function(7) leads to a viability region, i.e. a region where admissible composite signals may be found, that can be determined explicitly in the (1/c, s) plane as depicted in