Florent Foucaud
email: florent.foucaud@gmail.fr

Shahrzad Heydarshahi
email: shahrzad.heydarshahi@univ-orleans.fr

Aline Parreau
email: aline.parreau@univ-lyon1.fr

Domination and location in twin-free digraphs

Keywords: Digraphs, Dominating sets, Locating-dominating sets, Bounds

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

A dominating set in a digraph G is a set D of vertices of G such every vertex not in D has an in-neighbour in D. The domination number γ(G) of G is the smallest size of a dominating set of G. The area of domination is one of the main topics in graph theory: see the two classic books [START_REF] Haynes | Fundamentals of domination in graphs[END_REF][START_REF]Domination in graphs: advanced topics[END_REF] on the subject. While there are hundreds of papers on domination in undirected graphs, domination in digraphs is less studied; we refer to the papers [START_REF] Fu | Dominating set and converse dominating set of a directed graph[END_REF][START_REF] Lee | Domination in digraphs[END_REF][START_REF] Meggido | On finding a minimum dominating set in a tournament[END_REF] for some examples. One particular variation of domination is the concept of locationdomination, introduced by Slater for undirected graphs in [START_REF] Slater | Dominating and reference sets in graphs[END_REF] (see also [START_REF] Rall | On location-domination numbers for certain classes of graphs[END_REF][START_REF] Slater | Domination and location in acyclic graphs[END_REF]). For a set S of vertices of a digraph G, two vertices x and y of V (G) \ S are located by S if there is a vertex of S that is an in-neighbour of exactly one vertex among x and y. The set S is a locating set of G if it locates all the pairs of V (G) \ S (but does not necessarily dominate the graph). Equivalently, every vertex not in S has a distinct set of in-neighbours in S. The location number of a graph G is the size of a smallest locating set and is denoted by loc(G). A set D of vertices of a digraph G is locating-dominating if it is both locating and dominating. The location-domination number γ L (G) of a digraph G is the smallest size of a locating-dominating set of G. Note that γ L (G) -1 ≤ loc(G) ≤ γ L (G), since at most one vertex is not dominated in a locating set.

Our goal is to investigate bounds on the location-domination number of digraphs. Such bounds are absent from the literature; in fact, the only paper on location-domination in digraphs we are aware of is [START_REF] Charon | Identifying and locating-dominating codes: NP-completeness results for directed graphs[END_REF], which deals with the computational complexity of the problem. Such bounds on digraphs have been studied for the closely related concept of identifying codes in [START_REF] Foucaud | Characterizing extremal digraphs for identifying codes and extremal cases of Bondy's theorem on induced subsets[END_REF].

We now introduce our terminology. We will assume that all the considered digraphs are loopless and have no multiple arcs. A digraph G contains a set V (G) of vertices and a set A(G) of arcs, that are ordered pairs of vertices. If there is an arc from v to w, we say that v is an in-neighbour of w, and that w is an outneighbour of v. The open in-neighbourhood and the open out-neighbourhood of a vertex v, denoted by N -(v) and N + (v), respectively, are the set of in-neighbours of v and the set of out-neighbours of v, respectively. Further, the closed in-neigbourhood of v is N -[v] = N -(v) ∪ {v} and the closed out-neigbourhood of v is N + [v] = N + (v) ∪ {v}. A source is a vertex with no in-neighbours, and a sink is a vertex with no outneighbours. Two vertices are called twins if they have the same open in-neighbourhood or the same closed in-neighbourhood. If two vertices x and y satisfy N -(x) = N -(y) ∪ {y}, they are called quasi-twins. See Figure 1 for an illustration. A directed path is a sequence of vertices where each vertex has an arc to the next vertex in the sequence. A directed cycle is a directed path where the first and the last vertices are the same. An acyclic digraph is a digraph with no directed cyle. A tournament is a digraph in which there is a unique arc between any pair of vertices. A tournament is transitive if it is acyclic. A digraph is called strongly connected if there exists a directed path in both directions between every pair of vertices. Perhaps the most classic result in domination of undirected graphs is the theorem of Ore [START_REF] Ore | Theory of graphs[END_REF] which states that any undirected graph without isolated vertices has a dominating set of size at most half the order. Such a theorem does not hold for the location-domination number of undirected graphs, for example complete graphs and stars of order n have location-domination number n -1, see [START_REF] Slater | Dominating and reference sets in graphs[END_REF]. Nevertheless, Garijo, González and Márquez have conjectured in [START_REF] Garijo | The difference between the metric dimension and the determining number of a graph[END_REF] that in the absence of twins, the upper bound of Ore's theorem also holds for the location-domination number of undirected graphs; they also proved that an upper bound of roughly two thirds the order holds in this context (see [START_REF] Foucaud | Location-domination in line graphs[END_REF][START_REF] Foucaud | Location-domination and matching in cubic graphs[END_REF][START_REF] Foucaud | Locating-dominating sets in twin-free graphs[END_REF] for further developments on this matter).

Therefore, it is natural to ask whether similar bounds exist in the case of locating-dominating sets of digraphs. However, Ore's theorem does not hold for digraphs. Indeed, not only every isolated vertex but also every source of a digraph G belongs to any dominating set of G. For example, a star of order n with n -1 sources has domination number n -1 (it is easy to see that this is the only digraph of order n with no isolated vertices and domination number n -1, see [START_REF] Heydarshahi | Location-domination in twin-free graphs and digraphs[END_REF]). Lee showed (as part of a more general result) in [START_REF] Lee | Domination in digraphs[END_REF] that every source-free digraph G of order n has domination number at most 2n 3 . Moreover, the digraph of order n = 3k consisting of k vertex-disjoint directed triangles is source-free and has domination number 2n 3 , so this bound is tight when n = 0 mod 3. Better bounds have been obtained for specific classes: every tournament of order n has domination number at most log 2 n [15] (the order is tight for random tournaments), and every strongly connected digraph of order n has domination number at most n/2 [START_REF] Lee | On the domination number of a digraphs[END_REF] (this is tight for directed cycles).

What happens for the location-domination number? A first question of interest is to determine which digraphs have largest possible location-domination number. We address this question in Section 3, where we describe the class of graphs of order n with location-domination number n -1. Of course it includes stars with n -1 sources (that have domination number n -1), bidirected complete graphs and bidirected stars (that correspond to undirected graphs with location-domination number n -1), but as we will see, there are many more examples.

In Section 4, we devise a general technique to obtain small locating-dominating sets in twin-free digraphs. This technique is a refinement of those used in [START_REF] Foucaud | Locating-dominating sets in twin-free graphs[END_REF][START_REF] Garijo | The difference between the metric dimension and the determining number of a graph[END_REF][START_REF] Heydarshahi | Location-domination in twin-free graphs and digraphs[END_REF]. We use this technique in Section 5 to show that every source-free and twin-free digraph of order n has a locating-dominating set of size at most 4n 5 . This bound is improved to 3n 4 if moreover the digraph has no quasi-twins. By adding one to these bounds, they also hold even in the presence of sources. In Section 2, we describe strongly connected twin-free and quasi-twin-free digraphs of order n with location-domination number 2(n-2)

3

. We then show in Section 6 that any tournament of order n has a locating-dominating set of size at most n 2 (this is tight for transitive tournaments and other examples).

In Section 7, we show that the same bound holds for twin-free acyclic digraphs (this is also tight, for directed paths).

We address some preliminary considerations in Section 2 and conclude the paper in Section 8.

Preliminaries

We start with some useful propositions.

The following proposition generalizes a well-known fact about locating-dominating sets of undirected graphs [START_REF] Slater | Dominating and reference sets in graphs[END_REF].

Proposition 1. Let G be a digraph with a set S of pairwise twins (open or closed) or quasi-twins. There are at least |S| -1 vertices of S in any locating-dominating set of G.

Proof. By contradiction, let D be a locating-dominating set that does not contain two mutual twins or quasitwins x and y. The vertices x and y have the same in-neighbourhood in V (G) \ {x, y}, and also in D. This is a contradiction. However, we note that, unlike twins, there cannot exist a set of three pairwise quasi-twins. Proposition 2. Let x, y, z be three vertices. If x, y are quasi-twins and y, z are quasi-twins, then x, z cannot be quasi-twins.

Proof. Suppose without loss of generality that we have the arc from x to y. There are two cases. If we have the arc from z to y, then we must have both arcs from x to z and from z to x, so x and z cannot be quasi-twins. If we have the arc from y to z, then we also have the arc from x to z. But then y is an in-neighbour of z and not of x, so x and z cannot be quasi-twins.

We now present a family of twin-free digraphs G k of order n that have location-domination number almost 2n 3 . The graph G 3 is drawn in Figure 2. Proposition 3. Let G k be the strongly connected (hence, source-free) twin-free and quasi-twin-free digraph of order n = 3k + 2 obtained from k vertex-disjoint directed triangles by adding a new vertex s that is an out-neighbour of all vertices of each triangle, a vertex t that is an in-neighbour of all vertices of each triangle, and an arc from s to t. Then, we have

γ L (G k ) = 2(n-2) 3 . Proof. To see that γ L (G k ) ≤ 2(n-2) 3
, consider the following locating-dominating set: take t, one vertex of some directed triangle, and two vertices of all the other directed triangles (see Figure 2 for an example).

To see that

γ L (G k ) ≥ 2(n-2) 3
, consider a locating-dominating set D of G k . First, each of the original directed triangle contains a vertex of D, because otherwise the three vertices in this triangle are not located. Second, there is at most one directed triangle that contains only one vertex of D, because otherwise in both triangles there is a vertex only dominated by t, and they are not located. Finally, if some triangle contains only one vertex of D, then t ∈ D because otherwise some vertex of the triangle is not dominated. So, in total there are two vertices of D for each directed triangle. Since there are n-2 3 original directed triangles, the proof is finished.

s t

Figure 2: A strongly connected twin-free and quasi-twin-free digraph of order n with location-domination number

2(n-2) 3
, with a locating-dominating set in grey.

Characterizing digraphs with location-domination number n -1

In this section, we characterize those digraphs with maximum possible location-domination and location numbers.

For every digraph G of order n, any set of size n -1 is a locating set, thus loc(G) ≤ n -1. This is not true for locating-dominating sets: consider a digraph with no arcs. However, this is the only such example: if G has some arc, then it has a locating-dominating set of size at most n -1 (consider the set obtained from V (G) by removing the endpoint of one arbitrary arc).

We say that a vertex is universal if it is an in-neighbour of all other vertices. We first characterize those digraphs of order n with loc(G) = n -1 (of course, unless it has no arc, such a digraph also satisfies γ L (G) = n -1).

Proposition 4. Let G be a digraph of order n. We have loc(G) = n -1 if and only if every vertex is either universal, or a sink.

Proof. Recall that loc(G) ≤ n -1 always holds. If some vertex x has an out-neighbour y and there exists another vertex z that is not an out-neighbour of x, then V (G) \ {y, z} is a locating set of G of size n -2. Thus, if loc(G) = n -1, then every vertex in G is either universal or a sink.

On the other hand, suppose that every vertex of G is either universal or is a sink. Let U be the set of universal vertices, and S, the set of sinks. Every vertex v is dominated by the set U ∪ {v}. Thus, if two distinct vertices are not in a locating set L, they are both dominated exactly by the vertices in L ∩ U , a contradiction. Thus we have |L| ≥ n -1.

We call a digraph G a directed star if it has a special vertex that belongs to all the arcs, and there are no isolated vertices (see Figure 3). In other words, the underlying undirected graph of G is a star.

Directed stars form another family of connected digraphs with large location-domination number.

Proposition 5. For any directed star G of order n ≥ 2, we have γ L (G) = n -1.

Proof. Let x be the centre of G (the vertex belonging to all arcs). Since n ≥ 2, there is at least one arc in G, and so γ L (G) ≤ n -1. Let D be a locating-dominating set of G. Clearly, every source of G belongs to D. If two neighbours of x do not belong to D, then they are not located. Thus, at most one neighbour of x is missing from D. In the case where exactly one such vertex is not in D, in order to have this vertex dominated, x must belong to D. This shows that γ L (G) ≥ n -1.

x Figure 3: A directed star, with a minimum locating-dominating set in grey.

The only connected undirected graphs of order n with location-domination number n -1 are stars and complete graphs [START_REF] Slater | Dominating and reference sets in graphs[END_REF] (seen as digraphs, they correspond to bidirected stars and bidirected complete graphs). As we already mentioned, there are more digraph examples. We now characterize all of them. We must now prove the converse: assume that γ L (G) = n -1 (and n ≥ 4). We must prove that (b) or (c) holds.

Case 1. Suppose first that G contains some sources, and let S be the set of these sources. If there is a vertex s in S and two vertices x and y of N + (S) (where N + (S) denotes the union of the out-neighbourhoods of all vertices in S) that are located by s, then V (G) \ {x, y} is a locating-dominating set of G of size n -2, a contradiction. Thus all the vertices of S have the same neighbourhood and G contains all the arcs between S and N + (S).

Consider now the subdigraph G of G induced by N + (S). If it has a locating set L of size |V (G )| -2 (assume that the two vertices x and y are those not in L ), then the set V (G) \ {x, y} would be a locatingdominating set of G, a contradiction. Thus, we have loc(G ) = |V (G )| -1, and by Proposition 4 the vertices of G can be partitioned into sinks of G (set S ) and universal vertices of G (set U ). We let R = V (G) \ (S ∪ S ∪ U ). If R is empty, we are done, because then G satisfies the condition (c), with S 1 = S, C = U and S 2 = S . Thus, we assume that R is nonempty. If V (G) \ S contains an arc from vertex a to vertex b ∈ R such that (S ∪ U ) \ {a} is nonempty, then we could construct a locating-dominating set of G of size n -2 from V (G) by removing b and any vertex of (U ∪ S ) \ {a}. Thus, R must be an independent set, and if there is an arc from S ∪ U to R, then |S ∪ U | = 1. But since R contains no sources, there is necessarily an arc from S ∪ U to R, and so |S ∪ U | = 1. But then, G is a directed star and (b) holds, so we are done.

Case 2. Now, we assume that G has no sources. If every vertex is either universal or a sink, G satisfies (c) (with S 1 = ∅) and we are done. Thus, we may assume that there exists a vertex x with an outneighbour y, and a third vertex z that is not an out-neighbour of x. By assumption, V (G) \ {y, z} is not a locating-dominating set; but since x locates y and z, it is a locating set, and it certainly dominates y. Thus, V (G) \ {y, z} does not dominate z, that is, y is the only in-neighbour of z (recall that z has an in-neighbour since G has no sources). Let R = V (G) \ {x, y, z}. If x had an out-neighbour t in R, then V (G) \ {t, z} would be locating-dominating, a contradiction. Similarly, if x had an in-neighbour in R, then V (G) \ {x, z} would be locating-dominating, a contradiction. If there is an arc from a vertex u to a vertex v inside R, then V (G) \ {v, z} is a locating-dominating set of G, a contradiction. Thus R is an independent set. Now, if z has no neighbour in R, G is a directed star with centre y, and we are done. Thus, z must have an out-neighbour p in R. But now, V (G) \ {p, y} is locating-dominating, a contradiction. This completes the proof.

A general method to obtain locating-dominating sets of twin-free digraphs

Note that all graphs described in Section 3 of order n ≥ 4 have twins. What happens for twin-free digraphs?

In this section, we propose a general method to obtain locating-dominating sets of twin-free digraphs, based on special dominating sets. This method was used in [START_REF] Foucaud | Locating-dominating sets in twin-free graphs[END_REF] for the case of undirected graphs (a similar argument was also used in [START_REF] Garijo | The difference between the metric dimension and the determining number of a graph[END_REF]). It was adapted to digraphs in [START_REF] Heydarshahi | Location-domination in twin-free graphs and digraphs[END_REF] for quasi-twin-free digraphs, and here we extend it to all twin-free digraphs.

First we start with some definitions.

Definition 7. Let S be a set of vertices of a digraph G. The S-partition P S of V (G) \ S is the partition of V (G) \ S where two vertices are in the same part if and only if they have the same set of in-neighbours in S. Given a vertex v ∈ S, an S-external private neighbour of v is a vertex outside S that is an out-neighbour of v but of no other vertex of S in G.

Theorem 8. Suppose that G is a twin-free digraph of order n. Let S be a dominating set of G such that the S-partition of V (G) \ S contains at least x • |S| parts (with 0 < x ≤ 1). Then, γ L (G) ≤ 2x+1 3x+1 n. Moreover, if G is also quasi-twin-free, then γ L (G) ≤ x+1 2x+1 n. Proof. Let P S = P 1 ∪ . . . ∪ P n1 ∪ Q 1 ∪ . . . ∪ Q n2 be the S-partition of V (G) \ S, where P 1 , . . . , P n1 are the parts of size 1 and Q 1 , . . . , Q n2 are the parts of size at least 2.

We assume that S is maximal with the property that P S has at least x • |S| parts (this is ensured by adding vertices to S while this property holds). Now, we let D 1 = S ∪ i P i . We have the following property of D 1 .

Claim 8.A. Two vertices in V (G) \ D 1 are located by D 1 , unless they form a pair of quasi-twins.

Proof of claim. Clearly, if two vertices are in different parts of P S , they are located by some vertices in S. Thus, by contradiction, let q 1 and q 2 be two vertices of V (G) \ D 1 belonging to some part Q i0 of P S that are not quasi-twins but are not located by D 1 . Since G is twin-free, there is a vertex q 3 in V (G) \ S that can locate q 1 and q 2 : without loss of generality q 3 is an in-neighbour of q 1 but not q 2 . By our assumption q 3 / ∈ D 1 . Now, consider S = S ∪ {q 3 } and the corresponding S -partition P S of V (G) \ S (with n 1 and n 2 defined like before). Since

q 3 ∈ i Q i , we have n 1 + n 2 ≥ n 1 + n 2 + 1 (because in P S , Q i0 has been split into two parts). So n 1 + n 2 ≥ x|S| + 1 ≥ x(|S| + 1) = x|S | (because x ≤ 1)
. This contradicts the choice of S, which we assumed to be maximal with this property. ( )

Note that |D 1 | = |S| + n 1 .
Since D 1 is a dominating set, Claim 8.A shows that in the absence of quasitwins, D 1 is locating-dominating. Next, we address the case of quasi-twins. We first prove the following fact. Proof of claim. Note that two quasi-twins x and y in V (G) \ S must belong to the same part of P S , since they have the same in-neighbours in S. Let q 1 , q 2 , q 3 , q 4 be four vertices in V (G) \ D 1 such that {q 1 , q 2 } and {q 3 , q 4 } are two distinct pairs of quasi-twins (with q 1 and q 3 being in-neighbours of q 2 and q 4 , respectively). Assume by contradiction that the two pairs are not disjoint. Then, all the vertices in the two pairs belong to the same part of P S . If q 1 = q 3 , then q 2 and q 4 must be twins (because they cannot be quasi-twins by Proposition 2), a contradiction. Similarly, if q 2 = q 4 , then, because q 1 and q 4 , and q 3 and q 4 are quasi-twins, there must be an arc from q 1 to q 3 and an arc from q 3 to q 1 . Thus, q 1 and q 3 are twins. Hence, we may assume without loss of generality that q 2 = q 3 . Then, we proceed similarly as in Claim 8.A: the set S = S ∪ {q 2 } still satisfies the property that n 1 + n 2 ≥ x|S |, contradicting the maximality of S. This proves the claim. ( ) Now, for each pair of quasi-twins in V (G) \ D 1 , we add one of them to D 1 . By Claims 8.A and 8.B, the resulting set D 1 is locating-dominating and has size at most

|S| + n 1 + (n -|S| -n 1 )/2 = (n + |S| + n 1 )/2.
Consider now the set D 2 , of size n -n 1 -n 2 , consisting of V (G) without one (arbitrary) vertex from each part of P S . It is clear that D 2 is locating-dominating: all vertices of V (G) \ D 2 are located and dominated by S.

We now assume that G has no quasi-twins: then D 1 and D 2 are two locating-dominating sets of G.

If |D 2 | ≤ x+1
2x+1 n, we are done. So, assume that

|D 2 | > x+1 2x+1 n. We have |D 2 | = n -n 1 -n 2 , so n 1 + n 2 < (1 -x+1 2x+1 )n = x 2x+1 n. Recall that |S| ≤ n1+n2 x
. Therefore,

|D 1 | = |S| + n 1 ≤ |S| + n 1 + n 2 ≤ n 1 + n 2 x + (n 1 + n 2 ) = 1 x + 1 (n 1 + n 2 ) < 1 x + 1 x 2x + 1 n = x + 1 2x + 1 n , as desired.
If G has some quasi-twins, we use the locating-dominating sets D 1 and

D 2 . Again, if |D 2 | ≤ 2x+1 3x+1 n, we are done. So, assume that |D 2 | > 2x+1 3x+1 n. Then, n 1 + n 2 < (1 -2x+1 3x+1 )n = x 3x+1 n.
We obtain:

|D 1 | ≤ |S| + n + n 1 2 ≤ |S| + n + n 1 + n 2 2 ≤ n 1 + n 2 2x + n 2 + n 1 + n 2 2 < n 6x + 2 + n 2 + x 6x + 2 n = 2x + 1 3x + 1 n ,
and the proof is finished.

Applying Theorem 8 to general twin-free digraphs

We first apply the method of Section 4 for source-free digraphs.

Proposition 9. Any source-free digraph G has a minimum-size dominating set S with at least |S|/2 distinct parts in the S-partition of V (G) \ S.

Proof. Let S be a minimum-size dominating set. We choose S with a maximum number of parts in the partition P(S) of V (G) \ S.

Let S 1 be the set of vertices of S that have at least one S-external private neighbour. Next, we choose S 2 as a minimum-size set of vertices of S \ S 1 that dominates all the vertices of (V (G) \ S) \ N + (S 1 ) (that is, S 2 dominates those vertices that are not in S and are not an S-external private neighbour of any vertex in S). Note that S 1 ∩ S 2 = ∅ and S 1 ∪ S 2 dominates V (G) \ S. Finally, we let S 3 = S \ (S 1 ∪ S 2 ).

Note that |S 2 | is at most the number of parts of P(S) containing vertices with at least two in-neighbours in S, since a vertex of S suffices to dominate the vertices of each such part. Therefore, the number of parts in P(S) is at least

|S 1 | + |S 2 |.
We will now show that we have |S 1 | + |S 2 | ≥ |S|/2, which would imply the statement. Towards a contradiction, we assume that

|S 1 | + |S 2 | < |S|/2, that is, |S 3 | > |S|/2.
Then, no vertex x of S 3 has an in-neighbour in S, for otherwise S \ {x} would be a dominating set, contradicting the minimality of S. Since G is source-free, x has an in-neighbour in V (G) \ S. Let f (x) be one arbitrary in-neighbour of x, and let S 4 = {f (x) | x ∈ S 3 }. The function f is necessarily injective: if we had f (x 1 ) = f (x 2 ) = y for two distinct vertices x 1 and x 2 of S 3 , then the set S \ {x 1 , x 2 } ∪ {y} would be a smaller dominating set than S. Thus We remark that the bound of Proposition 9 is tight by considering any digraph consisting only of vertexdisjoint directed triangles.

We obtain the immediate consequence of Proposition 9 and Theorem 8.

Corollary 10. For any source-free and twin-free digraph G of order n, we have γ L (G) ≤ 4n/5. If moreover G is quasi-twin-free, then γ L (G) ≤ 3n/4.

One can extend this result to twin-free digraphs with sources (note that there can only be one source, since multiple sources would be mutual open twins).

Corollary 11. Let G be a twin-free digraph of order n ≥ 3. Then γ L (G) ≤ 4n/5 + 1. If moreover G is quasi-twin-free, then γ L (G) ≤ 3n/4 + 1.

Proof. Let s be the unique source of G. Let I be the collection of all subsets N -(x) for x ∈ V (G). The set I has order n and thus there is a non-empty subset X of V (G) \ {s} that is not in I, and thus X is not the open in-neighbourhood of any vertex in G. Let G be the graph obtained from G in which we add all the arcs between X and s. The graph G is now source-free and twin-free. By Corollary 10, there is a locatingdominating set D of G of size at most 4n/5. Then D = D ∪ {s} is a locating-dominating set of G. Indeed, all the vertices are dominated and two vertices not in D are still located by D . Thus γ L (G) ≤ 4n/5 + 1.

For the second part, we do the same reasonning but with I containing all the subsets N -[x] \ {s} and N -(x) \ {s}, for x ∈ V (G). There are at most 2n such sets and thus again there is a nonempty set X such that adding the arcs between x and s does not create twins or quasi-twins in G. The end of the proof is the same as in the first part.

Tournaments

It is clear that there are no twins in tournaments. A weaker version of the method of Section 4 was applied in [START_REF] Heydarshahi | Location-domination in twin-free graphs and digraphs[END_REF], yielding the bound γ L (T ) ≤ 5n/6 for any tournament T . This was proved by showing that any tournament T has a dominating set S with at least |S| parts in the S-partition of V (T ) \ S.

Note that our strengthened version of this method yields the better bound γ L (T ) ≤ 4n/5 + 1, via Corollary 11.

Nevertheless, using a different technique, we next prove a much stronger bound, which, moreover, is tight. We first prove this bound in transitive tournaments (for which it is actually the exact value).

Claim 16.B.

There is no arc inside any set L i .

Claim 16.C. There is no arc from L j to L i for j > i. Now we construct a locating-dominating set D of G. For i = m, . . . , 1, we construct step by step subsets D i of D. We will also define L i = L i \ j>i D j , where at the beginning of the process (i = m) we let L m = L m . We will ensure that D i ∩ L i-1 dominates L i , and D i locates the vertices of L i . Let P 1 , . . . , P k be the L i-1 -partition of L i . In particular, vertices in this partition are twins with respect to L i-1 . For each j, let v j be any vertex of P j . By Bondy's theorem (Theorem 15), there is a set S of at most k vertices in L i-1 that locates and dominates v 1 , . . . , v k .

Again by applying Bondy's theorem, for any 1 ≤ j ≤ k, there is a set S j of |P j | -1 vertices in G that locates P j . Note that the vertices of S j belong to j<i-1 L j . Let D i = S ∪ The bound of Theorem 16 is best possible by considering directed paths (the proof of the following is easy, so we omit it). Proposition 17. For the directed path P n of order n, we have γ(P n ) = γ L (P n ) = n 2 .

Conclusion

We conclude the paper with Table 1, summarizing the known upper bounds on the domination and location-domination numbers for certain classes of digraphs. The main question arising from our results is whether every twin-free digraph of order n admits a locatingdominating set of size 2n 3 . Also, it would be interesting to determine which are the tournaments and the twin-free acyclic digraphs of order n with location-domination number exactly n 2 .

  a and b are open twins. b a (b) a and b are quasi-twins. b a (c) a and b are closed twins.

Figure 1 :

 1 Figure 1: Examples of twin vertices.

Theorem 6 .

 6 For a connected digraph G of order n ≥ 2, we have γ L (G) = n -1 if and only if at least one of the following conditions holds: (a) n = 3; (b) G is a directed star; (c) V (G) can be partitoned into three (possibly empty) sets S 1 , C and S 2 , such that S 1 and S 2 are independent sets, C is a bidirected clique, and the remaining arcs in G are all the possible arcs from S 1 to C ∪ S 2 and those from C to S 2 . Proof. Assume first that n = 2 (then G is a directed star) or n = 3. If two vertices of G are not in some locating-dominating set of G, then either these two are not located, or one of them is not dominated: a contradiction. Thus, necessarily γ L (G) = n -1 and we may assume in the remainder that n ≥ 4. By Proposition 5, (b) implies that γ L (G) = n -1. Next, assume that (c) holds. Then, G is obtained by adding the set S 1 to a digraph where all vertices are either universal (those in C) or sinks (those in S 2 ). Since all vertices in S 1 are sources, they all belong to any locating-dominating set of G. Moreover, every vertex in S 2 ∪ C is dominated by all vertices in S 1 , that is, we need a locating set of G[C ∪ S 2 ] in every locating-dominating set of G. By Proposition 4, such a set has size exactly |C| + |S 2 | -1, thus γ L (G) = n -1.

Claim 8 .

 8 B. Any two pairs of quasi-twins in V (G) \ D 1 are disjoint.

  |S 4 | = |S 3 |. Now, we let S = S 1 ∪ S 2 ∪ S 4 . Clearly, S is a dominating set of G of size |S|. Moreover, every vertex x of S 3 is an S -external private neighbour of f (x). Thus, the partition P(S ) of V (G) \ S has at least |S 3 | parts, which is more than |S|/2 = |S |/2. This contradicts the choice of S. Thus, as claimed, we have |S 1 | + |S 2 | ≥ |S|/2, which concludes the proof.

3 Figure 4 :

 34 Figure 4: The levels in the proof of Theorem 16.

2 and|D| = |D | + 1 ≤

 21 k j=1 S j . Then D i ∩ L i-1 = S dominates L i and D i locates L i .|D i | ≤ k + k j=1 (|P j | -1) = |L i |. Now we prove that D = m i=1 D i ∪{s} is a locating-dominating set. The set D is a dominating set because for any x out of D (x ∈ L i ), x is dominated by D i ∩ L i-1 .It is also locating because if there are two vertices u, v in L i , they are located by D i and if they are from different L i and L j for j < i, u ∈ L i is located by its in-neighbour in L i-1 ∩ D i .We now bound the size of D. We have |D i | ≤ |L i | at each step i = m, . . . , 1 of the construction. Thus, there are at most as many vertices in D = m i=1 D i as in V (G) \ D (D i only contains vertices that are in L j with j < i, hence V (G) \ D is exactly the set of all vertices in the sets L i ). So, if D = D (that is, D contains s), we have |D| ≤ n 2 . But if D does not contain s, we have in fact that there are at most as many vertices in D as in V (G) \ (D ∪ {s}). Thus, |D | ≤ n-1

Table 1 :

 1 Summary of known upper bounds for domination and location-domination numbers of digraphs, with references. Cells with just one number correspond to tight bounds. For every other cell, we indicate the known upper bound and the size of the largest known construction.

This author is partially supported by the ANR project HOSIGRA (ANR-17-CE40-0022) and the IFCAM project "Applications of graph homomorphisms" (MA/IFCAM/18/39).

We also ask whether Proposition 9 could be strengthened in the following sense: is it true that every source-free digraph has a minimum-size dominating set S with |S|/2 vertices in S having an S-external private neighbour? This would be best possible, and an analogue of a similar result from [1], which holds for undirected graphs (with |S|/2 replaced with |S|). Moreover, we do not know whether Proposition 9 is tight for strongly connected digraphs.

Proposition 12. For a transitive tournament T of order n, we have γ L (T ) = n 2 and loc(T ) = n 2 . Proof. Let V (T ) = {v 1 , . . . , v n } where v i has each v j with j > i as its out-neighbour. To see that γ(T ) ≤ n 2 , consider the dominating set containing all v i 's with i odd. Two vertices v i and v j (i < j) not in D are located by v j-1 . Now, let D be a locating-dominating set of T . We need v 1 in D, otherwise it is not dominated. Then, every two vertices v i , v i+1 are quasi-twins, thus by Proposition 1, one of them needs to be in D. Consider the sets {v i , v i+1 } with i even and i < n: they are disjoint, and each contains one vertex of D. There are n-1 2 such sets, so we have

For locating sets, if n is odd, consider the set L that contains all v i 's with i even. Then L is a locating set of size (n -1)/2. Since loc(T ) ≥ γ L (T ) -1, this set is optimal. If n is even, at least one vertex in any set {v i , v i+1 } with i odd must be in the locating set. This gives at least n/2 vertices and thus loc(T ) = γ L (T ).

We now extend the upper bound to any tournament.

Theorem 13. For any tournament T of order n, we have γ L (T ) ≤ n/2 and loc(T ) ≤ n/2 . Proof. We prove the result by induction. By Proposition 12, this is true for any transitive tournament, and in particular if n ≤ 2. Let n ≥ 3 and assume that this is true for any tournament of order k < n.

Let T be a tournamement of order n that is not transitive. We first find a locating set of size n/2 . Let x be any vertex. Let V 0 = N -(x) and V x = N + (x) be the {x}-partition of V (T ) \ {x}. Let n 0 and n x be the sizes of V 0 and V x , respectively. Let S 0 and S x be two optimal locating sets of the tournaments induced by V 0 and V x , respectively. By induction, S 0 and S x have size at most n 0 /2 and n x /2 . Consider the set S = S 0 ∪ S x ∪ {x}. It is a locating set of T since any pair u, v with u ∈ V 0 and v ∈ V x is located by x. Its size is at most n 0 /2 + n x /2 + 1 which is equal to n/2 if n 0 or n x is odd. In this case we are done, so we can assume that both n 0 and n x are even and thus n is odd. Since we chose an arbitrary vertex x, one can also assume that all the vertices have even out-degree and in-degree (if not, we are done).

Consider now two arbitrary vertices x and y with an arc from x to y. Let V x , V y , V xy , V 0 be the {x, y}partition of V \ {x, y} (V x contains the vertices that have x but not y in their in-neighbourhood and the other notations follow the same logic). As before, if one takes a minimum locating set in each part of the partition and add x and y, one obtains a locating set of T . If there are three odd-sized sets among the four sets, one obtains, using the induction hypothesis, a locating set of size at most n/2 . Note that if V xy has odd size, then so does V y since y has even out-degree and all its out-neighbours are in V y or V xy . Since the total number of vertices is odd, there is another odd-sized set among V 0 and V x (which must actually be V 0 ). In total, there are three odd-sized sets among the sets of the partition and thus the locating set has size at most n/2 . Thus, one can assume that for any pair of vertices of T , the number of common out-neighbours is even.

We now consider three vertices x, y and z that induce a directed triangle x → y → z → x (this exists since T is not transitive). Again, we consider the {x, y, z}-partition of V (T ) \ {x, y, z}, we consider a minimum locating set of each part and we add x and y and z to obtain a locating set. If there are four odd-sized sets in the partition, the obtained locating set has, by induction, size at most n/2 . In fact, this is always the case: if V xyz (the vertices that have x, y and z as in-neighbours) is odd-sized, then V xy is also odd-sized since V xy ∪ V xyz is exactly the set of common out-neighbours of x and y, which is even-sized. In the same way, V yz and V xz are odd-sized and we are done. If V xyz is even-sized, then using the same argument, V xy , V yz , V xz are also even-sized. But then, V x ∪ V xy ∪ V xz ∪ V xyz ∪ {y} is the out-neighbourhood of x and has even size. Thus V x is odd-sized, and similarly, V y and V z are also odd-sized. Since the order of T is odd, V 0 must also be odd-sized and we are done.

We now prove in a similar way that γ L (T ) = n/2 . If n is odd, this is clear since γ L (T ) ≤ loc(T ) + 1 ≤ n/2 by the previous result. Thus we assume that n is even. We first take an arbitrary vertex x and consider the {x}-partition V 0 ,V x . If one takes a locating-dominating set for V 0 , a locating set for V x and adds x, one obtains a locating-dominating set. Since n is even, exactly one set among V 0 and V x has odd size. If |V 0 | is even, by induction, this gives a locating-dominating set of size at most |V 0 |/2 + (|V x | -1)/2 + 1 = n/2 and we are done. Thus, one can assume that all the vertices have odd in-degree and even out-degree.

Consider now two arbitrary vertices x and y with an arc from x to y in T and the associated {x, y}partition. Taking a locating-dominating set for V 0 , locating sets for the three other parts and x and y gives a locating-dominating set. If two sets of the partition that are not V 0 are odd-sized, this gives a locatingdominating set of size n/2, and we are done. So, we can assume that among V x and V xy , exactly one set is odd-sized. Indeed, x has even out-degree and its out-neighbourhood is V x ∪ V xy ∪ {y}. Thus, if V y is oddsized, we are done. Hence, we can assume that V y is even-sized, which implies that V xy is also even-sized. In particular, we can now suppose that all pairs of vertices have a common out-neighbourhood that has even size.

Finally, consider an oriented triangle x → y → z → x and the associated {x, y, z}-partition. As before, by taking x, y, z, a locating-dominating set of V 0 and locating sets in the seven other parts, we obtain a locating-dominating set. Since n is even, there is an odd number of odd-sized sets in the partition. If there are three odd-sized sets that are not V 0 , then the total locating-dominating set has size at most n/2 (indeed, if V 0 is also odd-sized, then there will be a fourth odd-sized set that is not V 0 ). As before, if V xyz is odd-sized, then V xy , V xz , V yz are also odd-sized and we are done. If it is even-sized, since x, y and z have even out-degree, we have V x , V y and V z that are odd-sized and we are also done.

Thus, there is always a locating-dominating set of T of size n/2 .

Transitive tournaments are not the only tight example. For an integer k ≥ 1, let T k be the tournament of order 3k obtained from a collection t 1 , . . . , t k of vertex-disjoint directed triangles, with arcs going from all vertices of t i to all vertices of t j whenever i < j.

Proof. Let D be an optimal locating-dominating set of T k . There must be at least one vertex of D in any t i , otherwise the vertices of t i are not located. Furtheremore, there must be two vertices of D in t 1 to dominate the three vertices of t 1 .

Assume that there are two consecutive triangles, t i and t i+1 , each with only one vertex in D. Let d i and d i+1 be the vertices of D that are in t i and t i+1 . Then, the out-neighbour of d i in t i and the in-neighbor of d i+1 in t i+1 are not located, a contradiction.

Thus, there must be at least three vertices of D in any two consecutive triangles, and at least two vertices of D in the first triangle, which gives a total of at least k + k/2 = n/2 vertices in D.

Twin-free acyclic digraphs

Twins generalize sources (since two sources are twins) but in a twin-free digraph we may have up to one source. This allows us to consider twin-free acyclic digraphs (they have exactly one source) in this section.

We will need the following theorem of Bondy, rephrasesd for our context.

Theorem 15 (Bondy [START_REF] Bondy | Induced subsets[END_REF]). Let A, B be two disjoint sets of vertices in a digraph such that the vertices of B have distinct in-neighbourhoods in A. Then, there is a subset L ⊆ A of size at most |B| -1 such that the vertices of B have distinct in-neighbourhoods in L.

Now we can prove the following. The proof technique is similar, but more complicated, as the one used to prove the same bound for the domination number of twin-free digraphs in [START_REF] Heydarshahi | Location-domination in twin-free graphs and digraphs[END_REF].

Theorem 16. If G is a twin-free acyclic digraph of order n, then γ L (G) ≤ n 2 . Proof. Let s be the unique source. We partition the vertex set step by step. For i ≥ 0, the set L i contains all sources in G i = V (G) \ j<i L j . Since G is acyclic and has only one source all the vertices are in some L i . Let m be the last non-empty L i . See Figure 4 for a picture.

The following claims are a direct consequence of the construction.

Claim 16.A. Let v ∈ L i (i > 0). Then v has an in-neighbour in L i-1 .


published or not. The documents may come

Translation criticism in website localization: The case of corporate websites localized into Arabic

Madiha KASSAWAT, madiha.kassawat@sorbonne-nouvelle.fr Baseline criteria "fit-for-purpose" (Bowker, 2020) Readability: Ease of reading, length, word frequency (Castilho, Doherty,

-

  Lack of access to the "internal knowledge" (Pym 2004), e.g. SL -Inconsistent website updates -No source & target comparison

Gaspari, & Moorkens, 2018) Word choice, reader-orientation(Chesterman, 2016) Comprehensibility: Familiar references(Baker, 2011; Nord, 2005), Logical meaning


Loïs Naudin
email: lois.naudin@gmail.com

Juan Luis Jiménez Laredo

Nathalie Corson

A Simple Model of Non-Spiking Neurons

Keywords: Neuron model, non-spiking neurons, bifurcation, Caenorhabditis elegans, retina

Due to the ubiquity of spiking neurons in neuronal processes, various simple spiking neuron models have been proposed as an alternative to conductance-based models (a.k.a. Hodgkin-Huxley type models), known to be computationally expensive and difficult to treat mathematically. However, to the best of our knowledge, there is no equivalent in the literature of a simple and lightweight model for describing the voltage behavior of non-spiking neurons, which also are ubiquitous in a large variety of nervous tissues in both vertebrate and invertebrate species, and play a central role in information processing. This paper proposes a simple model that reproduces the experimental qualitative behavior of known types of non-spiking neurons. The proposed model, which differs fundamentally from classic simple spiking models unable to characterize non-spiking dynamics due to their intrinsic structure, is derived from the bifurcation study of conductancebased models of non-spiking neurons. Since such neurons display a high sensitivity to noise, the model aims at capturing the experimental distribution of single neuron responses rather than perfectly replicating a single given experimental voltage 1 trace. We show that such a model: (i) can be used as a building block for realistic simulations of large non-spiking neuronal networks, and (ii) is endowed with generalization capabilities, granted by design.

Introduction

Conductance-based models (CBMs), based on the Hodgkin-Huxley formalism postulated in the 1950s [START_REF] Hodgkin | A quantitative description of membrane current and its application to conduction and excitation in nerve[END_REF], are an accurate biophysical representation of the neuron in which every individual parameter and state variable have an established electrophysiological meaning. Accordingly, such models have become one of the most powerful computational approaches for understanding 'low-level' functions of neural systems [START_REF] O'leary | Computational models in the age of large datasets[END_REF].

For instance, a CBM can describe the effect of specific conductance variations on hippocampal rythms [START_REF] Aussel | A detailed anatomical and mathematical model of the hippocampal formation for the generation of sharp-wave ripples and theta-nested gamma oscillations[END_REF][START_REF] Giovannini | The can-in network: A biologically inspired model for self-sustained theta oscillations and memory maintenance in the hippocampus[END_REF] or predict important dendritic effects on neuronal processing [START_REF] Poirazi | Illuminating dendritic function with computational models[END_REF]. On the downside, however, such models are computationally expensive so that one can only simulate a handful of neurons in real time [START_REF] Izhikevich | Which model to use for cortical spiking neurons?[END_REF]. Additionally, the insights obtained from a mathematical analysis perspective are quite limited as these are high-dimensional systems, difficult to treat mathematically. Therefore, to study 'higher-level' functions of neural systems, independent of 'low-level' details-be it at the level of dendrite, neuron or neural circuit [START_REF] Marín | Optimization of efficient neuron models with realistic firing dynamics. the case of the cerebellar granule cell[END_REF][START_REF] Shiau | Interneuronal gamma in hippocampus via adaptive exponential integrate-and-fire neurons[END_REF]-various simple and computationally efficient models of spiking neurons have then been proposed. Some examples are the FitzHugh-Nagumo [START_REF] Fitzhugh | Impulses and physiological states in theoretical models of nerve membrane[END_REF], Hindmarsh-Rose [START_REF] Rose | The assembly of ionic currents in a thalamic neuron i. the three-dimensional model[END_REF], Izhikevich [START_REF] Izhikevich | Simple model of spiking neurons[END_REF] or various flavors of integrate-and-fire models [START_REF] Górski | Conductance-based adaptive exponential integrate-and-fire model[END_REF][START_REF] Izhikevich | Resonate-and-fire neurons[END_REF][START_REF] Latham | Intrinsic dynamics in neuronal networks. i. theory[END_REF][START_REF] Smith | Fourier analysis of sinusoidally driven thalamocortical relay neurons and a minimal integrate-and-fire-or-burst model[END_REF]. nervous system. Nonetheless, a large variety of nervous tissues in both vertebrate and invertebrate species have revealed that a number of sensory, inter-and motor neurons function without eliciting spikes. While spiking neurons compress continuous inputs into digital signals for transmitting information via action potentials, non-spiking neurons do not elicit action potentials and modulate analog signals through graded potential responses [START_REF] Lockery | First report of action potentials in a c. elegans neuron is premature[END_REF]. Some examples are the invertebrate and vertebrate retinal neurons [START_REF] Field | Information processing in the primate retina: Circuitry and coding[END_REF], interneurons in insects and crustaceans [START_REF] Roberts | Neurones without impulses: Their significance for vertebrate and invertebrate nervous systems[END_REF], motorneurons of the Ascaris worm [START_REF] Davis | Signaling properties of ascaris motorneurons: Graded active responses, graded synaptic transmission, and tonic transmitter release[END_REF], or most of the C. elegans neurons [START_REF] Goodman | Active currents regulate sensitivity and dynamic range in c. elegans neurons[END_REF]. Importantly, they have been found in sensorimotor and central pattern generator circuits, proven to be central in neuronal integration [START_REF] Roberts | Neurones without impulses: Their significance for vertebrate and invertebrate nervous systems[END_REF] and shown to provide a determining mechanism for the control of motor behavior [START_REF] Burrows | Proprioceptive inputs to nonspiking local interneurons contribute to local reflexes of a locust hindleg[END_REF]Laurent & Burrows, 1989a;[START_REF] Laurent | Distribution of intersegmental inputs to nonspiking local interneurons and motor neurons in the locust[END_REF].

The omnipresence and the importance of non-spiking neurons in numerous neuronal processes justify the need for a simple model capable of characterizing their behavior. To that end, [START_REF] Kuramochi | A computational model based on multi-regional calcium imaging represents the spatio-temporal dynamics in a caenorhabditis elegans sensory neuron[END_REF] proposed a simple model that describes the intracellular calcium dynamics of a non-spiking C. elegans neuron. With respect to voltage dynamics, classic simple spiking models are unfortunately unable to produce adequate non-spiking dynamics due to the topology of their phase space. In this paper, we present a simple suitable model derived from the bifurcation study of non-spiking conductance-based models, extensively used to characterize such behaviors [START_REF] Kamiyama | Simulation analysis of bandpass filtering properties of a rod photoreceptor network[END_REF][START_REF] Kourennyi | Reciprocal modulation of calcium dynamics at rod and cone photoreceptor synapses by nitric oxide[END_REF][START_REF] Naudin | On the modeling of the three types of non-spiking neurons of the caenorhabditis elegans[END_REF][START_REF] Naudin | Systematic generation of biophysically detailed models with generalization capability for non-spiking neurons[END_REF][START_REF] Nicoletti | Biophysical modeling of c. elegans neurons: Single ion currents and whole-cell dynamics of awcon and rmd[END_REF][START_REF] Publio | A realistic model of rod photoreceptor for use in a retina network model[END_REF], which is able to qualitatively reproduce the two standard voltage dynamics of non-spiking neurons: near-linear and bistable. The near-linear behavior is defined by a smooth depolarization or hyperpolarization from the resting potential, while the bistable one is characterized by nonlinear transitions between the resting potential and a depolarized potential.

In order to analyze the viability and accuracy of the approach in a realistic setting, the proposed model is used in the modeling of three non-spiking neurons of the nematode Caenorhabditis elegans (C. elegans), a well-known model organism in neuroscience [START_REF] Brenner | The genetics of caenorhabditis elegans[END_REF][START_REF] Izquierdo | Role of simulation models in understanding the generation of behavior in c. elegans[END_REF][START_REF] Sarma | Openworm: Overview and recent advances in integrative biological simulation of caenorhabditis elegans[END_REF], that predominantly transmit information through non-spiking signals [START_REF] Goodman | Active currents regulate sensitivity and dynamic range in c. elegans neurons[END_REF]. In particular, we focus on the modeling of the RIM, AIY and AFD neurons which represent, to date, the three possible forms of nonspiking neuronal responses of C. elegans (Figure 1). More specifically, RIM and AIY neurons display a near-linear behavior (Figure 1.A). While AIY is more sensitive to hyperpolarization than depolarization inputs with a transition point around -30 mV, RIM is depolarized or hyperpolarized in a smooth manner due to the lack of large sustained currents. Finally, AFD exhibits a bistable behavior with large sustained currents restricting its membrane potential to two plateaus around -80 mV and -15 mV over a range of current injections (Figure 1.B).

Finally, we compare the computational efficiency of the proposed model with respect to conductance-based models developed in [START_REF] Naudin | On the modeling of the three types of non-spiking neurons of the caenorhabditis elegans[END_REF] for the same RIM, AIY and AFD neurons, as well as many additional neuron models. In particular, we show that the computational efficiency of the proposed model is of the same order of magnitude as the integrate-and-fire and Izhikevich models that are the most computational efficient neuron models to date [START_REF] Izhikevich | Simple model of spiking neurons[END_REF].

Proposed model

The goal of a neuronal model is to be able to reproduce the electrophysiological behavior of the neuron. To that end, it is crucial to capture its right underlying bifurcation structure [START_REF] Izhikevich | Dynamical systems in neuroscience[END_REF]: the bifurcation structure describes the qualitative changes that the neuron undergoes when parameter values change. Classic simple spiking models, such as the FitzHugh-Nagumo (FitzHugh, 1961), Hindmarsh-Rose [START_REF] Rose | The assembly of ionic currents in a thalamic neuron i. the three-dimensional model[END_REF], or Izhikevich [START_REF] Izhikevich | Simple model of spiking neurons[END_REF] ones are specifically built on the basis of the bifurcation structure of spiking neurons. These display Andronov-Hopf bifurcations or saddle-node bifurcations The figure has been reproduced from [START_REF] Naudin | Systematic generation of biophysically detailed models with generalization capability for non-spiking neurons[END_REF] with the consent of the authors. responsible for spikes, and are therefore inherently unable to correctly characterize the dynamics of non-spiking neurons. Moreover, for the sake of completeness, we have tested the leaky integrate-and-fire model designed with a threshold high enough to avoid the triggering of spikes, as in [START_REF] Strohmer | Integrating nonspiking interneurons in spiking neural networks[END_REF]. This model is also unable to correctly characterize non-spiking dynamics (Figure S1). Such a result is expected because of the linear model employed, since the neuron dynamics are non-linear. Then, this section presents a simple model specifically derived from bifurcation studies of conductance-based models of non-spiking neurons [START_REF] Naudin | A generic conductance-based model of non-spiking caenorhabditis elegans neurons and its mathematical analysis[END_REF][START_REF] Naudin | Systematic generation of biophysically detailed models with generalization capability for non-spiking neurons[END_REF] that is capable of reproducing their qualitative dynamics.

Conductance-based models (CBMs). In CBMs, the dynamics of the membrane potential V is described by a general equation of the form

C dV dt = - ion I ion + I (1)
where C is the membrane capacitance, ion I ion is the total current flowing accross the cell membrane, and I is an applied current. The currents I ion take the form

I ion = g ion m a ion h b ion (V -E ion )
where m (h) denotes the probability for an activation (inactivation) gate to be in the open state; a and b are the number of activation and inactivation gates, respectively; g ion is the maximal conductance associated with ion; and E ion is the reversal potential.

Bifurcation dynamics of non-spiking CBMs. In non-spiking CBMs [START_REF] Kourennyi | Reciprocal modulation of calcium dynamics at rod and cone photoreceptor synapses by nitric oxide[END_REF][START_REF] Naudin | A generic conductance-based model of non-spiking caenorhabditis elegans neurons and its mathematical analysis[END_REF][START_REF] Naudin | On the modeling of the three types of non-spiking neurons of the caenorhabditis elegans[END_REF][START_REF] Nicoletti | Biophysical modeling of c. elegans neurons: Single ion currents and whole-cell dynamics of awcon and rmd[END_REF], the steady-state current curve I ∞ determines the number of equilibra of the system and their values, as well as the bifurcations of the resting state along with the values to which they occur. It takes the general form

I ∞ (V ) = ion I ion∞ (V ) where I ion∞ (V ) = g ion m a ion∞ (V )h b ion∞ (V )(V -E ion ) (2) with x ∞ (V ) = 1 1 + exp V x 1/2 -V k x , x ∈ {m, h} .
where V x 1/2 and k x are constant parameters. Any stationary point of gating variables x ∈ {m, h} must satisfy x * = x ∞ (V * ). Replacing this into the first equation on V , fixed points V * of such models satisfy the equation

I ∞ (V * ) = I.
(3)

In other words, equilibria V * correspond to the intersection between the steady-state curve I ∞ and a horizontal line I = c where c is a constant. There are two standard steady-state curves I ∞ , monotonic and cubic (Figure 2), each involving fundamentally different neurocomputational properties for non-spiking neurons:

• As shown in Figure 2.A, CBMs with a monotonic steady-state current only have one equilibrium for any value of I. This unique equilibrium is stable for non-spiking CBMs (see [START_REF] Naudin | A generic conductance-based model of non-spiking caenorhabditis elegans neurons and its mathematical analysis[END_REF] for rigorous mathematical proofs). Thus, non-spiking neurons with such a steady-state current display a near-linear behavior characterized by smooth depolarizations or hyperpolarizations from the resting potential, such as the RIM and AIY neurons (Figure 1.A).

• As shown in Figure 2.B, a N-shaped curve leads to a saddle-node bifurcation. When I = c 2 , there are 3 equilibria, noted V c 2 1 * , V c 2 2 * and V c 2 3 * . Increasing I results in coalescence of two equilibria (the stable V c 2 1 * with the unstable V c 2 2 * ). The value I = c 3 , at which the equilibria coalesce, is called the bifurcation value. For this value of I, there exist 2 equilibria. For any value of I > c 3 , the system has only one equilibrium (e.g. I = c 4 or I = c 5 ), that is stable [START_REF] Naudin | A generic conductance-based model of non-spiking caenorhabditis elegans neurons and its mathematical analysis[END_REF]. In summary, when the parameter I increases, a stable and an unstable equilibrium approach, coalesce, and then annihilate each other. Non-spiking neurons with an N-shaped steady-state current display a bistable behavior characterized by a voltage jump between the resting potential and a depolarized potential of higher voltage, such as the AFD neuron (Figure 1.B).

Based on previous properties, it can be stated that the steady-state current determines the bifurcation structure of non-spiking neurons and the equilibrium values of their graded responses to particular stimuli.

The proposed model. Let V represent the membrane potential of a neuron. We propose the following 1-D model of the general form

τ dV dt = -f (V ) + I (4)
with f a cubic function which reads as

f (V ) = aV 3 + bV 2 + cV + d. (5)
The function f plays the same role in the dynamics of the model ( 4) as the steady-state current I ∞ in conductance-based models (1). Indeed, fixed points V * of model (4) satisfy

f (V * ) = I
so that the shape of f determines the neuro-computational features of the non-spiking model: a monotonic shape involves a near-linear behavior of the model, while an N-shape implies a bistable one with the occurrence of two saddle-node bifurcations. Therefore, the model proposes a simple cubic expression (5) thats plays the same role as the complex steady-state Parameter estimation. Parameters a, b, c and d have been estimated from the 16 points of the steady-state current using differential evolution algorithm (DE) [START_REF] Storn | Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces[END_REF].

DE is an effective method in the case of parameter estimation of neuronal models [START_REF] Buhry | Automated parameter estimation of the hodgkin-huxley model using the differential evolution algorithm: Application to neuromimetic analog integrated circuits[END_REF][START_REF] Buhry | Global parameter estimation of an hodgkin-huxley formalism using membrane voltage recordings: Application to neuro-mimetic analog integrated circuits[END_REF][START_REF] Naudin | On the modeling of the three types of non-spiking neurons of the caenorhabditis elegans[END_REF], and superior to other optimization methods such as genetic algorithms, simulated annealing and particle swarm optimization in terms of convergence speed, simulation time, and minimization of the cost function [START_REF] Buhry | Automated parameter estimation of the hodgkin-huxley model using the differential evolution algorithm: Application to neuromimetic analog integrated circuits[END_REF][START_REF] Buhry | Parameter estimation of the hodgkin-huxley model using metaheuristics: Application to neuromimetic analog integrated circuits[END_REF]. Moreover, the root-mean-square error normalized to standard deviation (STD) is used as estimator of the model quality [START_REF] Naudin | Systematic generation of biophysically detailed models with generalization capability for non-spiking neurons[END_REF]. Finally, parameter τ is set to the value determined in [START_REF] Naudin | On the modeling of the three types of non-spiking neurons of the caenorhabditis elegans[END_REF] and [START_REF] Naudin | Systematic generation of biophysically detailed models with generalization capability for non-spiking neurons[END_REF], estimated to fit experimental voltage traces; however, it can be directly hand-tuned.

Experimental variability. For a single target neuron, repetitions of the same external stimulation result in variable electrical responses [START_REF] Marder | Multiple models to capture the variability in biological neurons and networks[END_REF] due to multiple sources of intrinsic and extrinsic noise [START_REF] Destexhe | Neuronal noise[END_REF][START_REF] Faisal | Noise in the nervous system[END_REF]. This is why numerous works [START_REF] Druckmann | A novel multiple objective optimization framework for constraining conductance-based neuron models by experimental data[END_REF][START_REF] Gouwens | Systematic generation of biophysically detailed models for diverse cortical neuron types[END_REF][START_REF] Markram | Reconstruction and simulation of neocortical microcircuitry[END_REF] capture the experimental distribution of spiking neuron responses as a more accurate means to describe the neuronal behavior than perfectly replicate a single given experimental voltage trace. This approach seems to be even more suited to non-spiking neurons, which are known to be more sensitive to noise than spiking neurons [START_REF] Sarpeshkar | Analog versus digital: Extrapolating from electronics to neurobiology[END_REF]. As the steady-state current determines the bifurcation structure of non-spiking neurons and the equilibrium values of their graded responses to particular stimuli, parameters a, b, c and d of the function f defined in equation ( 5) are estimated to fit the mean of the experimental steady-state current (Figure 1). In this way, the proposed model captures the trial-to-trial variability of a single non-spiking neuron.

Fundamental differences from simple spiking neuron models and its non-spiking variants. The steady-state current forms the backbone of the dynamics of non-spiking neurons by determining: (i) all bifurcations of the resting state and the values at which they occur, and (ii) the equilibrium values of their graded responses to particular stimuli. This is not the case for the spiking neurons for which the steady-state current provides only information about possible bifurcations of the resting state. Indeed, when the equilibrium loses stability, characteristic of Andronov-Hopf bifurcations responsible for spikes, cannot be inferred from the steady-state current [START_REF] Izhikevich | Dynamical systems in neuroscience[END_REF]. This is why the existing classic simple spiking models were not derived directly from the steady-state current. This explains why these models are inadequate to produce acceptable non-spiking dynamics. Thus, the building of the proposed non-spiking model differs fundamentally from that of existing simple spiking models.

Results

This section presents the results of the optimization procedure for the three neurons under study and shows how, despite the simplicity of the model, it is able to reproduce bistable and near-linear behaviors with high accuracy. Then, we compare the computational efficiency of the proposed model with respect to conductance-based models developed in [START_REF] Naudin | On the modeling of the three types of non-spiking neurons of the caenorhabditis elegans[END_REF] for the same RIM, AIY and AFD neurons, and many additional types of neuron models. Finally, we implement a canonical example of a non-spiking network of two coupled neurons to demonstrate the ability of the proposed model to explain 'high-level' phenomena occurring in non-spiking nervous tissues.

Results of the optimization procedure

In order to fit the mean of the experimental steady-state current of RIM, AIY and AFD (Figure 1), parameters a, b, c and d of equation ( 5) are estimated using DE. The parameter values τ are obtained from [START_REF] Naudin | On the modeling of the three types of non-spiking neurons of the caenorhabditis elegans[END_REF] and [START_REF] Naudin | Systematic generation of biophysically detailed models with generalization capability for non-spiking neurons[END_REF] Bistable behavior -AFD. Figure 3 shows the dynamics of the proposed model for the AFD neuron. A high fitting quality of the N-shaped experimental steady-state current mean is obtained (Figure 3.A). Moreover, Figure 3.B shows that the proposed model displays an underlying bifurcation structure specific to non-spiking bistable neurons [START_REF] Izhikevich | Dynamical systems in neuroscience[END_REF][START_REF] Naudin | A generic conductance-based model of non-spiking caenorhabditis elegans neurons and its mathematical analysis[END_REF][START_REF] Naudin | Systematic generation of biophysically detailed models with generalization capability for non-spiking neurons[END_REF], with the occurence of two saddle-node bifurcations.

In particular, the voltage jump occurs at I ≈ 3.1 pA, exactly as in the CBM relative to AFD neuron published in [START_REF] Naudin | Systematic generation of biophysically detailed models with generalization capability for non-spiking neurons[END_REF]. Therefore, the model reproduces the essential features of the AFD neuron, making it adequate for the description of its bistable behavior. Finally, Figure 3

Computational efficiency.

Model simulation. To evaluate the computational efficiency of the model simulation, we compare the number of floating point operations (FLOPs), such as addition, multiplication or division, required to simulate 1ms (FLOPMS) of different neuron models. Note that the values of FLOPMS are related to the simulation time and not to the running time, which may vary widely across programming languages and CPU architectures. The simulation of the proposed model only takes 13 floating point operations to simulate the span of 1 ms in the neuron. These results are of the same order of magnitude as the integrate-and-fire and Izhikevich models that are the most computational efficient neuron models to date [START_REF] Izhikevich | Which model to use for cortical spiking neurons?[END_REF]. In order to establish comparisons, the CBMs relative to RIM, AIY and AFD neurons (see Appendix A) published in [START_REF] Naudin | On the modeling of the three types of non-spiking neurons of the caenorhabditis elegans[END_REF] have a computational efficiency similar to the Hodgkin-Huxley model which requires about 1200 FLOPMS [START_REF] Izhikevich | Which model to use for cortical spiking neurons?[END_REF].

That shows the large superiority of the proposed model when compared to the CBMs proposed in [START_REF] Naudin | On the modeling of the three types of non-spiking neurons of the caenorhabditis elegans[END_REF] in terms of computational efficiency. Model optimization. As discussed in Section 2, the steady-state current determines the bifurcation structure of non-spiking neurons as well as the equilibrium values of their graded responses. In this way, all parameters of the proposed model are directly estimated from steady-state current data. On the contrary, the same procedure cannot be applied to estimate the parameters of a CBM as the parametrization space is larger and requires voltage trace data (see, e.g., Appendix A). The consequence is that the optimization process of the proposed model is computationally much more efficient than that of a CBM: while a costfunction call in the proposed model requires evaluation of around 16 points (number of points of the steady-state current, see Figure 1), a CBM requires to evaluate, as in Naudin et al.

(2020), around 137500 points per function call (11 voltage traces × 12500 points per trace).

In practical terms, this means that estimating all the parameters of a CBM using DE is in the order of some days of computing [START_REF] Naudin | On the modeling of the three types of non-spiking neurons of the caenorhabditis elegans[END_REF], while estimating all the parameters of the proposed model in this paper took only a few seconds. It is outside the scope of this paper to establish a quantitative computational comparison between the optimization times in both models, as that would require to replicate the experiments in a controlled computational environment. However, we can safely conclude that the optimization process of the proposed model is computationally more efficient than that of a CBM in several orders of magnitude.

Model building. Another advantage of the proposed model is that it requires less biophysical information for its conception than a CBM. In particular, a CBM imposes a detailed modeling of the different ion channels present in the cell membrane, for which it is necessary to have contrasted biological data obtained in the laboratory. Such data are often very time-consuming and hard to obtain experimentally, as reported for the C. elegans neurons [START_REF] Goodman | Electrophysiological methods for caenorhabditis elegans neurobiology[END_REF] due to their small size and the difficulty of dissecting a one-millimeter-long worm. That is not the case of the proposed model as it relies only on the steady-state current data and not on the biophysical information of the ion channels.

Generalization capability of the proposed model

A good neuronal model should be able to predict the neuronal activity based on data that were not used for parameter tuning [START_REF] Gerstner | How good are neuron models[END_REF]. Indeed, capturing the neuron behavior for some inputs may not be sufficient to predict acceptable responses to different novel stimuli not used during the models' building [START_REF] Druckmann | Automated parameter constraining of single-neuron models[END_REF][START_REF] Druckmann | Effective stimuli for constructing reliable neuron models[END_REF][START_REF] Naudin | Systematic generation of biophysically detailed models with generalization capability for non-spiking neurons[END_REF]. A way to endow a neuron model with generalization capabilities is to reproduce the bifurcation structure of the neuron [START_REF] Izhikevich | Dynamical systems in neuroscience[END_REF][START_REF] Naudin | Systematic generation of biophysically detailed models with generalization capability for non-spiking neurons[END_REF].

We stated in Section 2 that the steady-state current determines the bifurcation structure of non-spiking neurons and the equilibrium values of their graded responses to particular stimuli.

Since the parameters of the proposed model are determined from the fitting of the steady-state current, the model will acquire the generalization capability directly from the optimization process. While this is granted in the proposed model by design, it is usually not the case in CBM where capturing the bifurcation structure is not the primary objective. Figure S3 shows a representative example of the deterioration of the steady-state current of a non-spiking CBM, as well as its dramatic implication on the neuron dynamics and its generalization capability. [START_REF] Naudin | Systematic generation of biophysically detailed models with generalization capability for non-spiking neurons[END_REF] propose in the case of non-spiking neurons to capture it by considering the steady-state current as an additional objective to be reproduced, in addition to fitting the membrane potential evolution. Such a multi-objective approach is successful but takes a very long time (about ten days). Therefore, the optimization and building process of the proposed model are efficient in the sense that it endows the models with generalization capabilities without requiring additional procedures with a high computational cost.

Implementation of a canonical non-spiking neuronal network

This section aims to demonstrate the capability of the proposed model to explain 'high-level' experimental phenomena occurring in non-spiking nervous tissues. To that end, we implement a canonical non-spiking neural network of two coupled neurons through an excitatory synapse.

This network represents a bistable sensory neuron (such as AFD) that propagates electrical signals to a postsynaptic near-linear neuron (such as RIM). The synaptic transmission is considered graded as it is the case in non-spiking nervous tissues, such as in the vertebrate retina [START_REF] Tsukamoto | Microcircuits for night vision in mouse retina[END_REF], the Ascaris [START_REF] Davis | Signaling properties of ascaris motorneurons: Graded active responses, graded synaptic transmission, and tonic transmitter release[END_REF] and C. elegans [START_REF] Lindsay | Optogenetic analysis of synaptic transmission in the central nervous system of the nematode caenorhabditis elegans[END_REF]) worms. Following Wicks et al. (1996), the mathematical expression of the network is given by

         τ 1 dV 1 dt = -a 1 V 3 1 -b 1 V 2 1 -c 1 V 1 -d 1 + I τ 2 dV 2 dt = -a 2 V 3 2 -b 2 V 2 2 -c 2 V 2 -d 2 -g syn g ∞ (V 1 )(V 2 -0)
where V 1 represents the membrane potential of the presynaptic bistable sensory neuron,

V 2 the membrane potential of the postsynaptic near-linear neuron, I the external sensory input current, g syn the maximal postsynaptic membrane conductance for the synapse, and V → g ∞ (V ) a sigmoid function relating presynaptic and postsynaptic membrane potential.

Its expression reads as

g ∞ (V ) = 1 1 + exp V -V rest V slope
with V rest the presynaptic resting potential, and V slope is the slope factor of the sigmoid function g ∞ . In the following, we set V slope = 15 mV, g syn = 0.6 nS and V rest = -76 mV (Wicks et al., 1996).

The purpose is to monitor the dynamics of the RIM neuron when it receives inputs from a bistable neuron. To that end, the AFD sensory neuron is stimulated with a series of current injections starting from -15 pA and increasing to 35 pA by 5 pA increments. As can be seen in Figure 5, the near-linear RIM neuron (Figure 5.A) becomes bistable under the effect of the coupling with the bistable AFD neuron (Figure 5.B,left). Transitions between down-states and up-states are apparent in the bimodal distribution of the membrane potential (Figure 5.B, right). Thus, it can be concluded that the connectivity of a neuron affects strongly the nature of its behavior: for instance, a near-linear neuron can become bistable when coupled with another bistable neuron. Among others, this can explain the discrepancy between the near-linear electrophysiological behavior of the RIM neuron (Figure 1.A) and its bistable calcium dynamics when exposed to odor stimuli [START_REF] Gordus | Feedback from network states generates variability in a probabilistic olfactory circuit[END_REF]. Furthermore, these results illustrate the importance of the synaptic dynamics in conferring distinct up-and down-states to non-spiking neurons. Whole-brain calcium imaging of neuronal activity in C.

elegans reveals that several dozens of non-spiking neurons have collective activity with such correlated up-and down-states [START_REF] Prevedel | Simultaneous whole-animal 3d imaging of neuronal activity using light-field microscopy[END_REF][START_REF] Schrödel | Brain-wide 3d imaging of neuronal activity in caenorhabditis elegans with sculpted light[END_REF]. The results presented in this paper suggest that the proposed model may be used to gain insights into synaptic dynamics in these network states.

Discussion

Various computationally efficient and simple spiking models have been proposed in the literature as an alternative to Hodgkin-Huxley type models. Such models have been valuable to gain insight into 'higher-level' functions of neural systems. Unfortunately, these simple models have a bifurcation structure that differs fundamentally from that of non-spiking neurons. This makes them inadequate for producing acceptable non-spiking dynamics, which are also ubiquitous in neuronal processes. To the best of our knowledge, this paper is a first attempt to develop a simple model specifically built on the basis of the bifurcation structure of non-spiking neurons. This model was then shown to have a high computational efficiency, to be able to accurately capture the individual experimental qualitative behavior of known types of non-spikings neurons, to generalize neuronal responses, and to explain and reproduce 'high-level' experimental phenomena in non-spiking networks. This section aims at discussing both the biological and the modeling implications of such a model from two perspectives: the modeling of the C. elegans neuronal network and retinal networks.

Implications to the modeling of the C. elegans neuronal network. The C. elegans nervous system is only composed of 302 neurons and about 7000 synaptic connections, and its connectome has been fully mapped [START_REF] White | The structure of the nervous system of the nematode caenorhabditis elegans[END_REF]. Despite this relative simplicity, the nematode shares many features with the complex human nervous system using similar neurotransmitters, ion channels and developmental genes, as well as similar neuronal principles to produce behavior [START_REF] Chalasani | Dissecting a circuit for olfactory behaviour in caenorhabditis elegans[END_REF][START_REF] Liu | Gabaergic motor neurons bias locomotor decisionmaking in c. elegans[END_REF]. For these reasons, it has become the ideal candidate to be modeled to investigate how behavior emerges from underlying physiological processes [START_REF] Costalago-Meruelo | Emulation of chemical stimulus triggered head movement in the c. elegans nematode[END_REF][START_REF] Izquierdo | Role of simulation models in understanding the generation of behavior in c. elegans[END_REF][START_REF] Lanza | A recurrent neural network model of c. elegans responses to aversive stimuli[END_REF][START_REF] Olivares | A neuromechanical model of multiple network oscillators for forward locomotion in c. elegans[END_REF][START_REF] Szigeti | Openworm: An open-science approach to modeling caenorhabditis elegans[END_REF]. However, as pointed out by [START_REF] Sarma | Openworm: Overview and recent advances in integrative biological simulation of caenorhabditis elegans[END_REF], such modeling studies do not take into account the specificity of C. elegans neuronal dynamics. Building models adapted to the nematode neuron dynamics is a key remaining component to make C. elegans nervous system modeling adequate for biological research [START_REF] Sarma | Openworm: Overview and recent advances in integrative biological simulation of caenorhabditis elegans[END_REF]. The simple and computationally efficient model proposed in this paper, capable of reproducing the known neuronal diversity of C. elegans neurons, could serve as a building block to study 'higher-level' functions of the C. elegans neuronal activity, such as determining the nature (excitatory or inhibitory) and strength of its synaptic connections. Indeed, such information is not revealed by the connectome [START_REF] Kopell | Beyond the connectome: The dynome[END_REF] and is of crucial importance to understand the flow of information within the nematode's nervous system [START_REF] Bargmann | From the connectome to brain function[END_REF]. Some computational works [START_REF] Costalago-Meruelo | Emulation of chemical stimulus triggered head movement in the c. elegans nematode[END_REF][START_REF] Lanza | A recurrent neural network model of c. elegans responses to aversive stimuli[END_REF][START_REF] Olivares | A neuromechanical model of multiple network oscillators for forward locomotion in c. elegans[END_REF][START_REF] Portegys | Training sensory-motor behavior in the connectome of an artificial c. elegans[END_REF]Wicks et al., 1996) estimate the synaptic polarity using an evolutionary approach in which the algorithm determines both the nature and the strength of connections to obtain observable, realistic worm behavior.

However, the networks studied are composed of neuron models that are not representative of the C. elegans neuron dynamics. For instance, [START_REF] Portegys | Training sensory-motor behavior in the connectome of an artificial c. elegans[END_REF], [START_REF] Costalago-Meruelo | Emulation of chemical stimulus triggered head movement in the c. elegans nematode[END_REF] and [START_REF] Lanza | A recurrent neural network model of c. elegans responses to aversive stimuli[END_REF] consider a homogeneous spiking model for the entire network although the C. elegans neurons are highly heterogeneous and contain non-spiking neurons [START_REF] Goodman | Active currents regulate sensitivity and dynamic range in c. elegans neurons[END_REF]. Therefore, even if the macroscopic behavior of C. elegans is successfully reproduced, the results on the synaptic polarity and their strength may not be biologically adequate. We suggest that the proposed model could provide a simple and efficient way to build models adapted to the C. elegans neuronal dynamics in order to make C. elegans modeling studies adequate for biological research.

Implications to the modeling of the retinal network. Due to the common origins of the retina and the brain, it is suggested that the neuronal activity in the retina and the brain correlates, so that changes in the retina may indicate structural and functional changes in the brain. Many authors then consider "the retina as a window to the brain" [START_REF] London | The retina as a window to the brain-from eye research to cns disorders[END_REF]. This is why the modeling of the retinal network could be a valuable tool to investigate various neuronal phenomena. Interestingly, the neuronal characteristics of the retina are similar to those of C. elegans in several aspects. First, the worm olfactory system and the retinal visual system of vertebrates follow the same general principles to process sensory information [START_REF] Chalasani | Dissecting a circuit for olfactory behaviour in caenorhabditis elegans[END_REF]. Second, the retina neurons release neurotransmitter in a graded and tonic manner [START_REF] Tsukamoto | Microcircuits for night vision in mouse retina[END_REF][START_REF] Werblin | Organization of the retina of the mudpuppy, necturus maculosus. ii. intracellular recording[END_REF] as the C. elegans [START_REF] Lindsay | Optogenetic analysis of synaptic transmission in the central nervous system of the nematode caenorhabditis elegans[END_REF]. Third, non-spiking neurons are also ubiquitous in the retina.

The retinal circuit is composed of several major classes of neurons: rod and cone photoreceptors, and horizontal, bipolar, amacrine and retinal ganglion cells (RCGs). Only RCGs and some amacrine cells trigger action potentials, while the other cells transmit visual information using graded potentials. More specifically, the cone [START_REF] Kourennyi | Reciprocal modulation of calcium dynamics at rod and cone photoreceptor synapses by nitric oxide[END_REF], horizontal [START_REF] Aoyama | Ionic current model of rabbit retinal horizontal cell[END_REF] and bipolar cells [START_REF] Schilardi | Two functional classes of rod bipolar cells in the healthy and degenerated optogenetically treated murine retina[END_REF][START_REF] Usui | Ionic current model of bipolar cells in the lower vertebrate retina[END_REF] exhibit a bistable behavior (such as the C. elegans AFD neuron considered in this paper), while the rod [START_REF] Kourennyi | Reciprocal modulation of calcium dynamics at rod and cone photoreceptor synapses by nitric oxide[END_REF] and amacrine cells display a near-linear one (such as RIM and AIY). These neurons number in the tens of millions [START_REF] Field | Information processing in the primate retina: Circuitry and coding[END_REF]. Moreover, each major retinal cell class consists of multiple cell types distinguished by variable response properties [START_REF] Field | Information processing in the primate retina: Circuitry and coding[END_REF]. From a general point of view, this neuron-to-neuron variability has been shown to play a paramount role in network performance [START_REF] Berry Ii | Functional diversity in the retina improves the population code[END_REF][START_REF] Lengler | Reliable neuronal systems: The importance of heterogeneity[END_REF][START_REF] Padmanabhan | Intrinsic biophysical diversity decorrelates neuronal firing while increasing information content[END_REF][START_REF] Shamir | Implications of neuronal diversity on population coding[END_REF]. This demonstrates the importance of generating many models to capture every individual neuron behavior within the biological population, rather than using a unique model for the entire population [START_REF] Marder | Multiple models to capture the variability in biological neurons and networks[END_REF]. Based on the fitting of the steady-state current that takes only a few seconds, the proposed model makes possible the generation of multiple models to capture the substantial neuron-to-neuron variability of retinal cells.

A Appendix: Conductance-based models of the RIM,

AIY and AFD neurons

In previous works [START_REF] Naudin | On the modeling of the three types of non-spiking neurons of the caenorhabditis elegans[END_REF][START_REF] Naudin | Systematic generation of biophysically detailed models with generalization capability for non-spiking neurons[END_REF], we determined the most suitable conductance-based models to the electrophysiology of the RIM, AIY and AFD neurons based on current biological research and a series of in-silico experiments.

Neuron models. For the RIM and AFD neurons, the I Ca,p + I Kir + I K,t + I L -model was obtained which reads as

                                     C V = -g Ca m Ca (V -E Ca ) -g Kir h Kir∞ (V )(V -E K ) -g K m K h K (V -E K ) -g L (V -E L ) + I ṁ Ca = m Ca∞ (V ) -m Ca τ m Ca , m Ca∞ (V ) = 1 + exp V m Ca 1/2 -V k m Ca -1 ṁK = m K∞ (V ) -m K τ m K , m K∞ (V ) = 1 + exp V m K 1/2 -V k m K -1 ḣ K = h K∞ (V ) -h K τ h K , h K∞ (V ) = 1 + exp V h K 1/2 -V k h K -1 (6)
whereas for the AIY neuron, the I Ca,t + I Kir + I K,p + I L -model was selected taking the form

                                     C V = -g Ca m Ca h Ca (V -E Ca ) -g Kir h Kir∞ (V )(V -E K ) -g K m K (V -E K ) -g L (V -E L ) + I ṁ Ca = m Ca∞ (V ) -m Ca τ m Ca , m Ca∞ (V ) = 1 + exp V m Ca 1/2 -V k m Ca -1 ḣ Ca = h Ca∞ (V ) -h Ca τ h Ca , h Ca∞ (V ) = 1 + exp V h Ca 1/2 -V k h Ca -1 ṁK = m K∞ (V ) -m K τ m K , m K∞ (V ) = 1 + exp V m K 1/2 -V k m K -1 (7) I ∞ (V H ) = g Ca m Ca∞ (V H )(V H -E Ca ) + g Kir h Kir∞ (V H )(V H -E K ) + g K m K∞ (V H )h K∞ (V H )(V H -E K ) + g L (V H -E L )
whereas for the system of equations ( 7), it is defined as

I ∞ (V H ) = g Ca m Ca∞ (V H )h Ca∞ (V H )(V H -E Ca ) + g Kir h Kir∞ (V H )(V H -E K ) + g K m K∞ (V H )(V H -E K ) + g L (V H -E L )
B Appendix: Supplementary figures

Figure 1 :

 1 Figure 1: In-vivo recordings of three different non-spiking neurons of C. elegans which represent, to date, the three forms of possible non-spiking neuronal responses of the nematode. (Top) Example of the evolution of membrane potential for a series of current injections, in the space of 5 seconds, starting from -15 pA and increasing to 35 pA by 5 pA increments. (Middle) Example of the evolution of the total ion currents of the different neurons when their membrane potentials are clamped at a fixed value, in spans of 0.5 seconds, from -100 mV and increasing to 50 mV by 10 mV increments. (Bottom) I-V relationships obtained from averaged voltage-clamp recordings (RIM: n = 3; AIY: n = 7; AFD: n = 3). Peak currents are measured by the absolute maximum amplitude of currents within the first 100 ms of each voltage step onset, while steady-state currents are measured by the averaged currents of the last 50 ms of each voltage step. (A) Near-linear neuron behavior with monotonic steady-state current. (B) Bistable neuron behavior with a N-shaped steady-state current.

Figure 2 :

 2 Figure 2: Two typical shapes of the steady-state current V → I ∞ (V ), in red. Intersections of I ∞ and horizontal line I = c (with c constant) correspond to equilibria of the system. We denote stable equilibria as filled circles , unstable equilibria as open circles and saddle-node equilibria as . (A) (Top) Monotonic steady-state current. V c i * , i = 1, . . . , 4, correspond to equilibria for a current injection I = c i , respectively. (Bottom) The resulting near-linear voltage dynamics with the four equilibria V c i * showed in (Top). (B) (Top) Nshaped steady-state current. The number of equilibria of the system depends on the value of I. (Bottom) The resulting bistable voltage dynamics with four equilibria V c i * showed in (Top).

Figure 3 :

 3 Figure 3: Optimization results for the AFD bistable neuron. (A) Mean of the experimental steady-state current (represented by green circles) with respective error bars, and the estimated steady-state current (represented by blue crosses) given by the equation (5) with parameter values displayed in Table 1. (B) Bifurcation diagram. The bifurcation structure is typical of bistable non-spiking neurons (Naudin et al., 2021) with the occurence of two saddle-node bifurcations (at I ≈ 2.625 pA and I ≈ 3.124 pA). (C) Example of experimental responses in AFD (represented in green), and the respective dynamics of the proposed model (represented in blue) representing the variability of the AFD neuron responses. The series of current injections start at -15 pA and increase to 35 pA by increments of 5 pA.

Figure 4 :

 4 Figure 4: Optimization results for the RIM and AIY near-linear neurons. (A) (Left) Mean of the experimental steady-state current (represented by green circles) with respective error bars, and the estimated steady-state current (represented by blue crosses) given by the equation (5) with parameter values displayed in Table 1. (Right) Function I → V * (I) for all I ∈ [-15 pA; 35 pA]. In both cases, the function is monotonic increasing, meaning that V * increases as I does which is consistent with the experimental behavior displayed in Figure 1. (B) (Left) Example of experimental responses in RIM and AIY, represented in green. (Right) Respective dynamics of the proposed model, represented in blue, representing the mean of their neuronal responses. The series of current injections start from -15 pA and increase to 35 pA by 5 pA increments.

Figure 5 :

 5 Figure 5: Voltage dynamics of the RIM neuron. (A) Evolution of the near-linear membrane potential of the RIM neuron for a series of injection currents I, from -15 pA to 35 pA by 5 pA increments. (B) (Left) Evolution of voltages of the RIM neuron when its presynaptic AFD bistable neuron is stimulated by current steps, from -15 pA to 35 pA by 5 pA increments. AFD and RIM are coupled through an excitatory synapse. Synaptic parameters: g syn = 0.6 nS (Wicks et al., 1996) and V slope = 15 mV. (Right) Corresponding histogram of the membrane potential calculated from 50 ds (5000 ms) of simulation.

  Figure S1: (Left) Examples of experimental membrane potentials of RIM, AIY and AFD neurons induced by a series of current steps starting from -15 pA and increasing to 35 pA in 5 pA increments (Q. Liu et al., 2018). (Right) Membrane potentials of the best solution of the leaky integrate-and-fire model designed with a threshold high enough to avoid the triggering of spikes, as in Strohmer et al. (2021). The solutions are obtained from the fitting of experimental voltage traces in (Left) using differential evolution algorithm. The series of current injections is the same than in (Left).

Figure S3 :

 S3 Figure S3: Example of dramatic implications of the deterioration of the steady-state current on the neuron dynamics and its generalization ability. (A) Experimental data (represented in green) and I Ca,p + I Kir + I K,t + I L -model (represented in blue) overlap for a series of current injections starting from -15 pA and increasing to 25 pA by 5 pA increments. (B) Experimental steady-state currents (represented by green circles) and estimated steady-state currents (represented by blue crosses) resulting from the fitting of membrane potential evolution in (A). Red lines delineate the interval [-15 pA, 25 pA]. In this interval, the experimental steady-state current is accurately reproduced, but the model completely deteriorates for values higher than 25 pA. (C) Dark blue curves represent the evolution of membrane potential for the same values of current injection than in (A) (i.e. stimuli starting from -15 pA and increasing to 25 pA by 5 pA increments), whereas light blue ones represent the drastic nonphysiological change of voltage traces for novel stimuli (30 pA and 35 pA). Note the difference of scale regarding y-axis between (A) and (C). (D) Bifurcation diagram. Four saddle-node bifurcations occur at I ≈ -0.66 pA, I ≈ 1.36 pA, I ≈ 3.19 pA and I ≈ 28.4 pA. This figure has been extracted from Naudin et al. (2022) with the consent of the authors.

Table 1 :

 1 in which they are determined to fit experimental voltage traces. The parameter values are shown in Table1. Parameter values of the model (4) for the different neurons. Results for the parameters a, b, c and d are obtained using differential evolution. The parameter τ is expressed in ds (10 -1 s) and is obtained from[START_REF] Naudin | On the modeling of the three types of non-spiking neurons of the caenorhabditis elegans[END_REF],[START_REF] Naudin | Systematic generation of biophysically detailed models with generalization capability for non-spiking neurons[END_REF].

	Neuron	Parameters a	b	c	d	τ (10 -1 s)
	RIM	0.000024 0.0036 0.31	7.22	0.042
	AIY	0.000044 0.0093 0.773 20.38 0.04
	AFD	0.00033	0.048	2.31	38.99 0.06

Acknowledgements

This work was supported by the ERDF (XTerm Project), Normandie Region (France) and by The LMAH, FR-CNRS-3335.

Code Accessibility

The code of differential evolution algorithm for parameter estimation is available at https: //github.com/lois76/codeDEnewGPmodel. Raw electrophysiological recording traces are available into Mendeley at https://doi.org/10.17632/tngf9w3pgd.1

Declaration of Competing Interests

The authors declare no competing interests.