A Barbier-Chebbah

O Bénichou

R Voituriez

Self-interacting random walks : aging, exploration and first-passage times

Self-interacting random walks : aging, exploration and first-passage times

I. INTRODUCTION

Random walk theory provides a natural framework to model transport processes at all scales. Beyond the historical examples provided by particle transport in simple fluids at the molecular and supramolecular scales [START_REF] Van Kampen | Stochastic Processes in Physics and Chemistry[END_REF][START_REF] Hughes | Random Walks and Random Environments[END_REF][START_REF] Klafter | First steps in random walks: From Tools to Applications[END_REF], it has also proved more recently to powerfully describe the dynamics of more complex, passive or active, larger scale systems -ranging from polymers, molecular motors or self-propelled colloids to cells or animals, whose dynamics take place in potentially complex environments [START_REF] Bouchaud | Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications[END_REF][START_REF] Ben-Avraham | Diffusion and reactions in fractals and disordered systems[END_REF][START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractionnal dynamics approach[END_REF][START_REF] Burioni | Random walks on graphs: ideas, techniques and results[END_REF][START_REF] Romanczuk | Active brownian particles from individual to collective stochastic dynamics[END_REF][START_REF] Metzler | Anomalous diffusion models and their properties: nonstationarity, non-ergodicity, and ageing at the centenary of single particle tracking[END_REF][START_REF] Bechinger | Active particles in complex and crowded environments[END_REF]. In the latter case, the coupling of the internal degrees of freedom of the random walker to those of the environment generically leads to complex correlations and require a non Markovian description of the evolution over time of the position X(t) of the random walker. Taking into account such memory effects remains a theoretical challenge even if several examples of model systems have been analyzed [START_REF] Masoliver | Firstpassage times for non-markovian processes: Correlated impacts on bound processes[END_REF][START_REF] Bicout | Absorption of a randomly accelerated particle: gambler's ruin in a different game[END_REF][START_REF] Schütz | Elephants can always remember: Exact long-range memory effects in a nonmarkovian random walk[END_REF][START_REF] Guérin | Nonmarkovian reaction kinetics[END_REF][START_REF] Boyer | Solvable random-walk model with memory and its relations with markovian models of anomalous diffusion[END_REF][START_REF] Falcón-Cortés | Localization transition induced by learning in random searches[END_REF][START_REF] Guérin | Mean first-passage times of non-markovian random walkers in confinement[END_REF][START_REF] Meyer | Optimal non-markovian search strategies[END_REF].

In this paper we focus on a broad class of such non Markovian random walkers, where memory effects emerge from the interaction of the random walker at time t with the territory that it has visited at earlier times t < t [START_REF] Amit | Asymptotic behavior of the "true" self-avoiding walk[END_REF][START_REF] Peliti | Random walks with memory[END_REF][START_REF] Ottinger | The generalised true self-avoiding walk-a model with continuously variable exponent ν[END_REF][START_REF] Tóth | Self-Interacting Random Motions[END_REF][START_REF] Sapozhnikov | Self-attracting walk with ν < 1/2[END_REF][START_REF] Pemantle | A survey of random processes with reinforcement[END_REF][START_REF] Foster | Reinforced walks in two and three dimensions[END_REF][START_REF] Ordemann | Structural properties of self-attracting walks[END_REF][START_REF] Davis | Reinforced random walk[END_REF]. This class of self-interacting random walks has clear applications in a broad range of examples where a random walker modifies locally its environment -leaving behind footprints along its path, and in turn responds to its own footprints [START_REF] Kranz | Effective dynamics of microorganisms that interact with their own trail[END_REF][START_REF] Kranz | Trail-mediated selfinteraction[END_REF]. Such behaviours have been reported for ants depositing pheromones along their path [START_REF] Dussutour | Optimal traffic organization in ants under crowded conditions[END_REF], larger territorial animals [START_REF] Giuggioli | Animal interactions and the emergence of territoriality[END_REF], and have been identified quantitatively in the case of living cells that chemically modify and remodel the extra-cellular matrix [START_REF] Alessandro | Cell migration guided by long-lived spatial memory[END_REF][START_REF] Flyvbjerg | Past attractions set future course[END_REF].

More precisely, self-interacting random walks can be defined as nearest neighbor random walks on a ddimensional lattice, for which the probability to jump to a neighboring site i at time t is proportional to a weight function w(n i ) that depends on the number of previous visits n i of the random walker to site i up to time t (see Fig. 1). Writing w(n) = e -V (n) , the process has the following clear interpretation : upon visiting site i, the random walker deposits a signal that in turn modifies the local energy landscape V experienced by the walker. Of note, in contrast to autochemotactic or autophoretic systems [START_REF] Pohl | Dynamic clustering and chemotactic collapse of self-phoretic active particles[END_REF][START_REF] Theurkauff | Dynamic clustering in active colloidal suspensions with chemical signaling[END_REF], the deposited signal is assumed to be static and permanent, but not diffusive, which leads as we argue below to long lived memory effects. To cover a broad spectrum of possible behaviours, we will consider both attractive (V decreasing) and repulsive (V increasing) self-interacting random walks, with effective potentials ranging from linear (V (n) = βn) to bounded (V (n) = βH(n)), where H(n) denotes the Heaviside function [START_REF] Tóth | Self-Interacting Random Motions[END_REF][START_REF] Pemantle | A survey of random processes with reinforcement[END_REF][START_REF] Foster | Reinforced walks in two and three dimensions[END_REF][START_REF] Ordemann | Structural properties of self-attracting walks[END_REF][START_REF] Davis | Reinforced random walk[END_REF][START_REF] Stevens | Aggregation, Blowup, and Collapse: The ABC's of Taxis in Reinforced Random Walks[END_REF].

Despite their relevance in various contexts, the properties of self-interacting random walks remain poorly understood, even if significant results have been obtained in the mathematical [START_REF] Tóth | Self-Interacting Random Motions[END_REF][START_REF] Pemantle | A survey of random processes with reinforcement[END_REF][START_REF] Davis | Reinforced random walk[END_REF][START_REF] Horváth | Diffusive limits for the true (or myopic) self-avoiding random walks and selfrepellent brownian polymers in d ≥ 3[END_REF] and physical communities [19-21, 23, 25, 26, 28, 29, 38-41]. This stems from the strongly non Markovian nature of self-interacting random walks, whose dynamics depends on the set of number of visits (or local times) {n i } i∈Z d at all sites i of the lattice at time t, and therefore on the full trajectory {X(t )} t ≤t of the random walker up to time t. This dependence leads Figure 1. Aging and first-passage times for self-interacting random walks in infinite space and in confined domains. Examples of 2d self-interacting trajectories (TSAW model, see text) a. The random path from t = 0 to T generates the local energy landscape V ((x, y), t) (proportional to the total number of visits to (x, y) until t), plotted along the vertical axis z. This dynamics can lead to long range memory effects and aging at all time scales : the trajectory after T plotted in c. explicitly depends on the full territory visited by the walker until T . In confined domains b., the statistics of visits to a given site is radically modified by confinement : the dynamics of self-interacting random walks is thus geometry dependent. d. In this article, we aim at quantifying space exploration and search kinetics of confined self-interacting random walks.

to memory effects at all time scales, which can have important consequences depending on the potential V (n) and space dimension d, such as anomalous diffusiondefined as the anomalous scaling of the mean squared displacement (MSD) :

X 2 (t) ∝ t→∞ t 2/dw (1)
with the walk dimension d w = 2, or aging -that can be defined as the dependence of increments

∆ 2 (T, t) ≡ (X(T + t) -X(T )) 2 ≡ 2D(T, t)t 2/dw (2)
on the observation time T , where D(T, t) is the effective time dependent diffusion coefficient. As we recapitulate below, the analytical determination of d w remains a theoretical challenge ; so far it has been obtained analytically or numerically for different examples of V (n) and d, but even its numerical determination remains debated for attractive linear V (n) for d = 2. In turn, the aging properties of ∆ 2 (T, t) have not been studied until recently [START_REF] Barbier-Chebbah | Anomalous persistence exponents for normal yet aging diffusion[END_REF] and will be analyzed in this paper.

A central question that arises in random walk theory is the quantification of space exploration by a random walker [START_REF] Redner | A Guide to First-Passage Processes[END_REF][START_REF] Metzler | First passage problems: recent advances[END_REF][START_REF] Bénichou | Intermittent search strategies[END_REF][START_REF] Viswanathan | Levy flights and superdiffusion in the context of biological encounters and random searches[END_REF]. Beyond the MSD and increments of the position, which provide a first quantification of the dynamics of spreading in space, several observables have been proposed to quantify space exploration. Among those, the first-passage time (FPT) and its distribution have proved to play a key role [START_REF] Guérin | Mean first-passage times of non-markovian random walkers in confinement[END_REF][START_REF] Redner | A Guide to First-Passage Processes[END_REF][START_REF] Metzler | First passage problems: recent advances[END_REF][START_REF] Condamin | First-passage times in complex scale-invariant media[END_REF][START_REF] Bénichou | Geometry-controlled kinetics[END_REF][START_REF] Mattos | First passages in bounded domains: When is the mean first passage time meaningful?[END_REF][START_REF] Bénichou | From first-passage times of random walks in confinement to geometry-controlled kinetics, From first-passage times of random walks in confinement to geometry-controlled kinetics[END_REF][START_REF] Levernier | Universal first-passage statistics in aging media[END_REF][START_REF] Giuggioli | Exact spatiotemporal dynamics of confined lattice random walks in arbitrary dimensions: A century after smoluchowski and polya[END_REF]. Indeed, beyond being a prominent technical tool of random walk theory that gives access to various observables, it quantifies the kinetics of general target search problems at all time scales, and as such has a broad range of applications from diffusion limited reactions to animal foraging behaviour.

In infinite space, the first-passage statistics to a target follow two very distinct behaviours depending on the so-called type of the random walk [START_REF] Hughes | Random Walks and Random Environments[END_REF]. In the compact or recurrent case, the survival probability S(t), i.e. the probability that the target has not been found until time t typically vanishes at long time scales as

S(t) ∝ t→∞ t -θ , ( 3
)
where θ is the persistence exponent, which has been the focus of numerous studies [START_REF] Bray | Persistence and first-passage properties in nonequilibrium systems[END_REF]. In the non compact or transient case, the survival probability admits a non zero large time limit, which defines the hitting probability Π [START_REF] Hughes | Random Walks and Random Environments[END_REF][START_REF] Redner | A Guide to First-Passage Processes[END_REF] according to S → t→∞ 1 -Π. In turn, the hitting probability is expected to decrease with the distance r from the starting position of the random walk and the target radius a according to

Π ∝ r→∞ (a/r) ψ . (4)
The corresponding transience exponent ψ was recently introduced in [START_REF] Levernier | Universal first-passage statistics in aging media[END_REF][START_REF] Levernier | Universality classes of hitting probabilities of jump processes[END_REF] and parallels the persistence exponent of recurrent processes. In spite of their pivotal role in quantifying first-passage properties of random walks, determining analytically the exponents θ, ψ for general non Markovian, aging processes remains a theoretical challenge [START_REF] Bray | Persistence and first-passage properties in nonequilibrium systems[END_REF][START_REF] Levernier | Survival probability of stochastic processes beyond persistence exponents[END_REF]. In particular, they remain unknown analytically for most examples of self-interacting processes, with the exception of [START_REF] Barbier-Chebbah | Anomalous persistence exponents for normal yet aging diffusion[END_REF] ; they will be analyzed numerically and analytically in this paper.

In the case of geometrically confined spaces, which is relevant to most of practical situations where the space accessible to the random walker is ultimately bounded, space exploration is known to be radically different. In particular, a target is eventually found with probability one for both compact and non compact processes, and the broad tails of the FPT distribution are generally suppressed [START_REF] Bénichou | Geometry-controlled kinetics[END_REF][START_REF] Mattos | First passages in bounded domains: When is the mean first passage time meaningful?[END_REF][START_REF] Levernier | Universal first-passage statistics in aging media[END_REF][START_REF] Meyer | Universality classes of first-passage-time distribution in confined media[END_REF]. FPT statistics in confinement has been the subject of intense activity over the last decade, and general results have been obtained for general scale-invariant Markovian processes [START_REF] Bénichou | Geometry-controlled kinetics[END_REF][START_REF] Meyer | Universality classes of first-passage-time distribution in confined media[END_REF] or Gaussian non Markovian processes [START_REF] Guérin | Mean first-passage times of non-markovian random walkers in confinement[END_REF]. Notably, more recently a universal scaling form of the FPT distribution was derived in the limit of large confining volume for a class of scale-invariant non Markovian processes that display power law aging [START_REF] Levernier | Universal first-passage statistics in aging media[END_REF] ; importantly, the scaling of the FPT distribution in confinement was found in this case to be fully determined asymptotically by the exponents d w , θ, ψ (in addition to the space dimension d), which are all defined in infinite space and independent of the geometric confinement.

Quantifying space exploration of self-interacting random walks in confined geometry brings in this context a new conceptual challenge. Indeed, qualitatively it is expected that geometric confinement will modify the statistics of the numbers of visits {n i } at the sites i of the confined domain over time and thus the effective potential V (n i ) experienced by the walker, thereby impacting the very dynamics of the process, as compared to that in infinite space. This can be illustrated by the simple example of the normal random walk for d = 3 : in infinite space, the mean number of visits n i to any site i converges to a finite value for t → ∞, whereas it diverges as n i ∝ t in a confined domain. In the case of self-interacting random walks, the kinetics of space exploration thus directly feeds-back to the dynamics of the process in a geometry dependent manner, which suggests that key intrinsic features of the dynamics, such as increments (quantified by D(T, t) and d w ), persistence and transience exponents d w , d f could in fact be different in confined and infinite geometries. This in particular makes earlier approaches to determine FPT statistics inapplicable, and calls for new theoretical tools to quantify the space exploration of confined self-interacting random walks ; this is at the core of this paper.

Our findings can be summarized as follows. We show that universal scaling forms of the FPT distributions of general self-interacting random walks in confinement can be derived in the large volume limit, by generalizing the approach introduced in [START_REF] Levernier | Universal first-passage statistics in aging media[END_REF]. Because of the intrinsic aging properties of self-interacting random walks, different cases emerge depending on the preparation protocol. For "fresh" initial conditions, for which the random walker starts the search for a target in a domain that has never been explored, we find that the exponents d w , θ, ψ that determine the FPT distribution are generally identical to those defined in infinite space : in other words, the FPT distribution in confinement can be asymptotically predicted from the knowledge of he process in infinite space only. This is quite remarkable because, as we show, geometric confinement ultimately deeply modifies the dynamics of the process and can even change the corresponding exponents d w,c , θ c , ψ c defined in confinement. In contrast, for aged initial conditions, for which the random walker has been extensively wandering in the domain before the search starts, the exponents that determine the FPT distribution are those defined in confinement d w,c , θ c , ψ c , and can thus be different from the classical infinite space exponents d w , θ, ψ. In that case, the process in confinement must therefore be characterized to determine the FPT distribution. In all cases, scaling functions are not universal and are process dependent. This analysis is made possible by a systematic quantitative characterization of the aging properties (quantified by D(T, t) and d w ) and exponents θ or ψ of self-interacting random walks in both confined and infinite geometries, which highlights the impact of geometric confinement on their dynamics. Finally, this paper thus proposes a unified, quantitative analysis of aging, exploration and FPT statistics of self-interacting random walks in confined and infinite geometries.

The paper is organized as follows. First, we briefly define the main classes of attractive and repelling selfinteracting walks and recall their walk dimension d w when it is known. In particular we provide a general criterion for attractive walks that leads to bounded exploration (d w = ∞), or to the regime of strong attraction (θ = ∞, to be defined below) ; in this case the FPT problem in confinement is trivially equivalent in the large volume limit to the problem in infinite space. Second, we characterize quantitatively the increments and their aging behavior, as well as the persistence and transience exponents θ, ψ in both confined and infinite geometries. Third, based on this analysis, we derive the asymptotic FPT distribution in confinement for both compact and non compact self-interacting random walks for fresh initial conditions. Last, we discuss the impact of aging on FPT statistics by analyzing the case of aged initial conditions, which allows us to assess the impact of memory effects on target search kinetics of self-interacting random walks.

II. DEFINITIONS AND MAIN CLASSES OF SELF-INTERACTING RANDOM WALKS

As stated above, self-interacting random walks can be defined as nearest neighbor random walks on a ddimensional lattice, for which the probability to jump to a neighboring site i at time t is proportional to a weight function w(n i ) that depends on the number of previous visits n i of the random walker to site i up to time t. Denoting w(n) = e -V (n) , the process has the following clear interpretation : upon visiting site i, the random walker deposits a signal that in turn modifies the local energy landscape V experienced by the walker. Different classes of random walks are obtained depending on the choice of weight function w(n) ; we remind below the main known results concerning the MSD of these processes.

A. The True Self Avoiding Walk (TSAW) :

w(n) ∝ e -βn
In this model [START_REF] Amit | Asymptotic behavior of the "true" self-avoiding walk[END_REF][START_REF] Tóth | Self-Interacting Random Motions[END_REF][START_REF] Pietronero | Critical dimensionality and exponent of the "true" self-avoiding walk[END_REF][START_REF] Obukhov | Renormalisation of the true self-avoiding walk[END_REF] the effective potential V (n) depends linearly on the local time n. For β < 0 the interaction is attractive, and leads (almost surely), as we show below, to the complete trapping of the random walker on a finite set of sites for all d, and thus formally to d w = ∞:

X(t) 2 → t→∞ C, (5)
where C is a d dependent constant. For β > 0 the interaction is repulsive and the random walker qualitatively avoids its own path. It has been shown that this leads to the following scaling of the MSD for t → ∞ [START_REF] Amit | Asymptotic behavior of the "true" self-avoiding walk[END_REF][START_REF] Grassberger | Self-Trapping Self-Repelling Random Walks[END_REF][START_REF] Pietronero | Critical dimensionality and exponent of the "true" self-avoiding walk[END_REF][START_REF] Obukhov | Renormalisation of the true self-avoiding walk[END_REF]:

• d = 1 : X(t) 2 ∝ t 4 3 , d w = 3 2 • d = 2 : X(t) 2 ∝ t ln(t) α , d w = 2, α ≈ 0.5 • d = 3 : X(t) 2 ∝ t, d w = 2
Of note, the scaling of the MSD is thus anomalous (superdiffusive) for d ≤ 2 because of self-repulsion, while it is diffusive for d > 2.

B. The Sub-Exponential Self Repelling Walk (SESRW) :

w(n) ∝ e -βn k
This model [START_REF] Ottinger | The generalised true self-avoiding walk-a model with continuously variable exponent ν[END_REF][START_REF] Tóth | Self-Interacting Random Motions[END_REF][START_REF] Toth | The "True" Self-Avoiding Walk with Bond Repulsion on Z: Limit Theorems[END_REF] extends the TSAW to effective potentials V (n) that depend sublinearly on the local time n: V (n) = βn k with 0 < k < 1. Similarly to the TSAW, in the attractive case (β < 0), the random walker is (almost surely) trapped for all d, and thus d w = ∞. For β > 0, the effect of self avoidance is clearly weaker than for the the TSAW ; it has however been shown to still lead to superdiffusion for d = 1 [START_REF] Ottinger | The generalised true self-avoiding walk-a model with continuously variable exponent ν[END_REF][START_REF] Toth | The "True" Self-Avoiding Walk with Bond Repulsion on Z: Limit Theorems[END_REF]. This can be summarized as follows :

• d = 1 : X(t) 2 ∝ t 2(1+k) (2+k) , d w = 2+k 1+k • d = 2 : X(t) 2 ∝ t ln(t) α k , d w = 2, α k ≥ 0 • d = 3 : X(t) 2 ∝ t, d w = 2 C. The self attractive random walk (SATW) : w(n) ∝ e -βH(n)
In this model [23-27, 59, 60], the effect of self interaction is assumed to saturate with the number of visits, so that the effective potential V (n) is bounded for n → ∞. For the sake of simplicity, it is assumed in the SATW model that V (n) = βH(n), with H(0) = 0 and H(n ≥ 1) = 1. Note that the SATW can thus be seen as the k → 0 limit of the SESRW defined above. For β > 0, self-avoidance is insufficient to modify the scaling of the MSD, which remain diffusive for all d :

X(t) 2 ∝ t, d w = 2. ( 6
)
In the attractive case β < 0, the random walker is never trapped. For d = 1 the MSD satisfies [START_REF] Sapozhnikov | Self-attracting walk with ν < 1/2[END_REF][START_REF] Prasad | Diffusive behaviour of self-attractive walks[END_REF] :

X(t) 2 ∝ t, d w = 2, (7)
while for d = 3 different behaviors emerge depending on the value of the parameter β

• |β| < |β c | : X(t) 2 ∝ t, d w = 2 • |β| > |β c | : X(t) 2 ∝ t 1/2 , d w = 4.
For d = 2, the scaling of the MSD is still debated [START_REF] Foster | Reinforced walks in two and three dimensions[END_REF][START_REF] Ordemann | Structural properties of self-attracting walks[END_REF]; while the existence of a subdiffusive regime with d w = 3 is consistently observed numerically, the existence of a transition for a critical value β c = 0 to a diffusive regime with d w = 2 for |β| < |β c | has been proposed, but was later questioned in [START_REF] Foster | Reinforced walks in two and three dimensions[END_REF]. Aging of the increments for the subdiffusive SATW (dw = 3 for d = 2) normalized by the expected subdiffusive scaling at long times. Each curve corresponds to a fixed value of T . Note that the increments are diffusive for t T . d. In the subdiffusive regime, the SATW performs an extremely compact exploration of space : the survival probability S(t) decays faster than any powerlaw (θ = ∞).

III. ATTRACTIVE SELF-INTERACTING RANDOM WALKS: TRAPPING AND SUBDIFFUSION

Qualitatively, attractive self-interacting random walks are attracted by their own path. Strikingly, this can lead to the full trapping of the walker within a finite set of sites in the t → ∞ limit, and therefore to a bounded MSD. This effect was demonstrated mathematically [START_REF] Davis | Reinforced random walk[END_REF][START_REF] Volkov | Phase Transition in Vertex-Reinforced Random Walks on Z with Non-linear Reinforcement[END_REF] for 1-dimensional attractive self-interacting random walks and later generalized to arbitrary d [START_REF] Stevens | Aggregation, Blowup, and Collapse: The ABC's of Taxis in Reinforced Random Walks[END_REF][START_REF] Cotar | Edge-and vertex-reinforced random walks with super-linear reinforcement on infinite graphs[END_REF][START_REF] Sellke | Reinforced random walk on the d-dimensional integer lattice[END_REF][START_REF] Basdevant | Localization of a vertex reinforced random walk on z with sub-linear weight[END_REF] : more precisely, these results state that if ∞ n=1 w(n) -1 = ∞ the random walker is free and will visit infinitely many sites of the lattice (note that the limit case w(n) ∝ 1/n must be discussed independently). Conversely, for ∞ n=1 w(n) -1 < ∞, the random walker visits only a finite set of sites and the MSD is bounded (see Supplementary Information SI). This yields immediately that attractive TSAW and SESRW lead to the full trapping of the ran-dom walker and to a bounded MSD for all d (see Fig. 2 ). Among the classes of attractive self-interacting random walks introduced above, the only case that leads to a non trivial exploration of space is thus the SATW, for which the MSD diverges for t → ∞ (see Fig. 2). Despite the diverging MSD, the effect of attractive self interactions can still have important consequences on the dynamics of space exploration; in particular, for d = 2 and d = 3 (for |β| > |β c |) the process is subdiffusive [START_REF] Foster | Reinforced walks in two and three dimensions[END_REF] and we find that the survival probability S(t) in infinite space decays faster than any power-law, so that θ = ∞ (see Fig. 2 and SI). In this case, determining the FPT distribution starting at a distance r from the target F (t, r, R) in confined domains of volume V ∝ R d is straightforward in the large volume limit because all moments of F (t, r, R) have a finite limit, so that:

F (t, r, R) ∼ t→∞ - dS dt . (8)
F (t, r, R) is thus asymptotically independent of R. Defining the rescaled variable η = t/r dw , a scaling argument finally indicates that its asymptotic distribution can be simply written:

F (η, r, R) = h(η) (9)
where h is an undetermined scaling function. In the rest of this paper, we thus focus on diffusive attractive and all repulsive self-interacting random walks, for which determining the FPT distribution F (t, r, R) in confined domains is non trivial.

IV. IMPACT OF CONFINEMENT ON INCREMENTS, θ, ψ

In this section, we characterize quantitatively the exploration properties of diffusive attractive self-interacting random walks and repulsive self-interacting random walks. We focus on the following observables : increments, and survival probability characterized by θ (for compact processes) and ψ (for non compact processes) in both infinite and confined geometries. We show numerically and provide heuristic arguments to justify that geometric confinement can deeply and non locally modify the dynamics of the process, beyond imposing locally reflecting boundary conditions. As can be expected, it is useful to analyse separately compact and non compact processes. While this property is known to impact many properties of random walks, it is expected to play a prominent role in the case of self-interacting random walks, whose dynamics is controlled by the number of visits n at each site.

A. Compact (recurrent) processes

The compact case is exemplified by the 1d (repulsive) TSAW, the 1d (repulsive) SERW and the 1d (attractive or repulsive) SATW. In the compact case, the mean number of visits n i to a given site diverges with time T by definition even in infinite space. The local energy landscape V (n i ) experienced by the random walker therefore depends on the observation time T at all time scales. We argue below that this leads to aging of the increments in infinite space at all time scales, ie a dependence on T of the effective diffusion coefficient D(T, t) defined in (2) for all T . In a confined domain, the dynamics of the random walk starting typically from the bulk is not modified by confinement up to an observation time T ∼ R dw , where R is the typical linear size of the domain; in this regime we therefore expect the increments to be identical in both confined and infinite geometries (note that for the same reason, in confined domains the analysis of increments is restricted to t R dw ). For T R dw , confinement does modify the statistics of visits to a given site; however the number of visits to a given site still diverges with time T , even if the explicit dependence on T is different in confinement and in infinite space. Aging of the increments is thus expected in confinement as well.

To make this analysis quantitative, it is useful to write V (n i ) as a Taylor series:

V (n i ) = V (n) + p≥1 V (p) (n) (δn i ) p p! , (10)
where n denotes the number of visites to a given site averaged over a spatial scale l T 1/dw and n i = n + δn i . The very definition of the dynamics of self-interacting random walks shows that it depends only on the spatial fluctuations of V ; increments are thus independent of the site independent contribution V (n) for t l dw T . In the case of the SATW and the SESRW, one has V (p) (n) ∝ nk-p by definition (we remind that for the SESRW V ∝ n k , where k → 0 yields the SATW) ; in addition, a mean field argument (see SI) yields the scaling δn i ∝ n(1-k)/2 . This shows that in the regime 1 t T all site dependent terms V (p) (n)(δn i ) p /p! for p ≥ 1 appearing in (10) vanish in the limit T → ∞ in both confined and infinite geometries because n → ∞. Selfinteractions are thus eventually negligible in this limit: the SATW and the SESRW are equivalent to a simple random walk and one has identically ∆ 2 (T, t) ∼ t in both confined and infinite geometries. In the case of the TSAW, one has V (n) = β and V (p) (n) = 0 for p > 1, independently of n. Using in addition the fact that the spatial fluctuations δn i reach a steady state in the limit n → ∞ (see [START_REF] Freund | How uniformly a random walker covers a finite lattice[END_REF] and SI), this shows that in the regime 1 t T the dynamics of increments is identical in both confined and infinite cases ; it can be shown to satisfy ∆ 2 (T, t) ∝ t 2/dw . Last, in the regime 1 T t, one recovers the scaling of the MSD in all cases : ∆ 2 (T, t) ∝ t 2/dw . These results can be recapitulated for all examples by the following scaling forms, which are identical in con-fined and infinite geometries :

1 t T : ∆ 2 (T, t) ∼ 2D < (t)t 2/dw 1 T t : ∆ 2 (T, t) ∼ 2D > t 2/dw , (11)
where the constant D > and function D < (t) are process dependent. Numerical simulations confirm this analysis in all examples of compact self-interacting random walks (see Fig. 3) : increments display aging (as seen by a dependence of ∆ 2 on the observation time T ), and their dynamics is found to be the same in infinite space and in confined domains in both regimes t, T R dw and T R dw , t R dw . In contrast to the dynamics of increments, we now argue that the persistence exponent θ can be modified by confinement. Following [START_REF] Krug | Persistence exponents for fluctuating interfaces[END_REF], we introduce here the persistence exponent θ c in confinement that can be defined by

S(t|T ) ∝ t -θc (12)
for T R dw and 1 t R dw , where S(t|T ) denotes the (survival) probability that the random walker has not reached the target between T and T + t. It is known that θ depends on the dynamics of increments at all time scales [START_REF] Bray | Persistence and first-passage properties in nonequilibrium systems[END_REF], and not only on their long time asymptotics. The exponents θ and θ c can thus be different, as was earlier found in [START_REF] Krug | Persistence exponents for fluctuating interfaces[END_REF] for models of fluctuating interfaces, because θ involves the dynamics of increments at all time scales T, t, while the definition of θ c only involves the time scales t R dw and T R dw . This is straightforwardly confirmed in the case of the 1d SATW. It is clear that for T R dw , the confined SATW is equivalent to a simple random walk (in this regime all sites have been visited and V (n i ) = β for all sites), so that θ c = 1/2; in contrast, it was shown recently that in infinite space one has θ = e -β /2 [START_REF] Barbier-Chebbah | Anomalous persistence exponents for normal yet aging diffusion[END_REF]. In the case of the SERW, the above analysis shows that in the regime T 1, the process is also equivalent to a simple random walk, so that θ c = 1/2 ; in contrast, we find numerically θ = θ c . Note however that it is found numerically that θ c ≈ θ ≈ 1 -1/d w for the TSAW for all β > 0.

B. Non compact (transient) processes

The non compact case is exemplified by the 3d (repulsive) TSAW, the 3d (repulsive) SERW and the 3d (diffusive attractive or repulsive) SATW. In the non compact case, in infinite space, a random walker visits only a fraction of sites, and ultimately only makes on average a finite number of visits to a given site i. The local energy landscape V (n i ) therefore reaches a stationary state at large observation time T , so that aging, if any, is expected to be transient : D(T, t) is asymptotically independent of T for T 1. This is indeed observed numerically in all examples of non compact self-interacting random walks: increments display weak aging at short time scales t, and cross-over to diffusive increments with numerically close diffusion coefficients at larger t for all observation times T (see Fig. 4). The effect of sel-interaction is thus moderate for non compact self-interacting random walks, which are all eventually diffusive. This can be heuristically justified as follows: at time t, the typical volume covered scales as t d/dw , while the number of visited sites scales as t, so that the local fraction of sites where the local energy landscape is non zero eventually vanishes for t → ∞ as t 1-d/dw . Self interactions are thus negligible in the large time limit for non compact processes, which are diffusive in this limit (note however that the diffusion coefficient is non trivial and depends on the small t dynamics).

The case of confined geometries is radically different for non compact processes, because confinement leads to a divergence of the number of visits to a given site, and has thus important consequences at time scales T R dw . In this regime, the above reasoning developed after [START_REF] Bechinger | Active particles in complex and crowded environments[END_REF] for compact processes in fact applies also to confined non compact processes, because the locally averaged number of visits n diverges in both cases. In particular, this yields similarly that in the regime 1 t T both the non compact SATW and the non compact SESRW are equivalent to a simple random walk, so that ∆ 2 (T, t) ∼ t. In the case of the confined non compact TSAW, one finds numerically (see also [START_REF] Amit | Asymptotic behavior of the "true" self-avoiding walk[END_REF][START_REF] Horváth | Diffusive limits for the true (or myopic) self-avoiding random walks and selfrepellent brownian polymers in d ≥ 3[END_REF] and SI for an heuristic argument) that the spatial fluctuations δn i reach a steady state in the limit T → ∞. This, together with [START_REF] Bechinger | Active particles in complex and crowded environments[END_REF], allows us to conclude that in this limit increments are similar (scaling wise) to the infinite space case and thus diffusive and independent of T (see SI).

Finally, for all confined non compact self-interacting random walks, these results can be recapitulated as follows for R 1:

T R dw : ∆ 2 (T, t) ∼ T 1 2D < (t)t, D < (t) → t 1 D < T R dw : ∆ 2 (T, t) ∼ t 1 2D > (T )t, D > (T ) → T 1 D > (13)
where D < , D > are constants. The first regime T R dw is the same in confined and infinite geometries, while the second regime T R dw is controlled by geometric confinement. Numerical simulations confirm this analysis in all examples of non compact self-interacting random walks (see Fig. 4) : increments, even if always asymptotically diffusive, are found numerically in all examples to be quantitatively different for confined and non confined non compact self-interacting random walks.

Last, for the sake of completeness, we note that similarly to the persistence exponent in the compact case (see [START_REF] Bicout | Absorption of a randomly accelerated particle: gambler's ruin in a different game[END_REF]), the transience exponent ψ c can be defined in confinement according to :

S(t|T ) ∝ a→0 a r ψc (14)
for T R dw and a dw t R dw . While, in principle ψ c can be different from its infinite space counterpart ψ, our above analysis showed that all examples of non compact self-interacting processes that we analysed are diffusive Figure 3. aging and first-passage properties for compact self-interacting random walks in infinite space and in confined geometries. aging of the increments for the 1d TSAW (a.), the 1d SATW (b.) and the 1d SESRW (c.) in both infinite space and confined domains (increments are normalized by the expected scaling for T t). The dynamic is found to be identical in infinite space (triangle) and in confined domains (stars) in both regimes t, T R dw and T R dw , t R dw . The TSAW is superdiffusive at all times scales and displays aging ; the SATW is diffusive at all times scales and displays aging. In contrast, the SESRW is diffusive for T t, but superdiffusive for t T with dw = (2 + k)/(k + 1). Persistence exponents in infinite space (θ) and in confined domains (θc) for the 1d TSAW (d.), the 1d SESRW (e.) and for the 1d SATW (f.). The persistence exponent is modified by confinement for the SATW and the SESRW models, but is found numerically to be unchanged for the TSAW.

and independent of T for t 1 in the limit T → ∞ in both confined and non confined cases; this suggests that ψ = ψ c = 1, which is consistent with our numerical simulations (see Fig. 4).

To summarize this section, we have showed quantitatively that geometric confinement can deeply and non locally modify the dynamics of self-interacting random walks, beyond imposing locally reflecting boundary conditions. In the compact case, increments remain unchanged (in the regime t R dw ) in confined and unconfined geometries, but the persistence exponent can be modified. In the non compact case, increments remain asymptotically diffusive in both cases, but their dynamis quantitatively modified by geometric confinement ; in turn, it is found that the transience exponent is unchanged.

V. FPT DISTRIBUTION IN CONFINED DOMAINS

The above analysis of increments and exponents θ and ψ shows that these observables can be impacted by confinement. Turning to the analysis of FPT properties of self-interacting random walks in confinement, one therefore needs to develop a new methodology. Indeed, so far, available methods to determine FPT statistics in confinement [START_REF] Levernier | Universal first-passage statistics in aging media[END_REF] rely implicitly on the hypothesis that increments and exponents θ and ψ, which are the key quantities defining the universality classes of FPT statistics in confinement, are not modified by confinement. Below, we extend the method developed originally in [START_REF] Levernier | Universal first-passage statistics in aging media[END_REF] to the case of self-interacting random walks by taking explicitly into account the impact of confinement on the dynamics. In this section, we consider the case of "fresh" initial conditions : at t = 0 the random walker, confined in a domain of volume V = R d with reflecting walls, starts at a distance r from the target of radius a, and the number of visits to all sites i of the domain is set to n i = 0. As stated in introduction, we focus on diffusive attractive and repulsive self-interacting random-walks, and consider separately the cases of compact processes (for which the survival probability S(t) has a power-law decay in infinite space) and non compact processes ; the case of marginal exploration (2d processes with d w = 2) is discussed in SI.

A. Compact (recurrent) case

We sketch in this section the derivation of the asymptotic FPT distribution F (t, r, R) for compact selfinteracting random walks in the large volume limit R → Figure 4. Aging and first-passage properties for non-compact self-interacting random walks in infinite space and in confined geometries. Aging of the increments for the 3d TSAW (a.), the 3d SESRW (b.) and the 3d SATW (c.) in infinite space (increments are normalized by the expected diffusive scaling at long times). Of note, the increments are stationary at timescales T 1. In contrast, in confined geometries, aging occurs at longer time scales R d , for the TSAW (d.), the SESRW (e.) and the SATW (f.) (increments are normalized by the expected diffusive scaling at long times). g. For non-compact random walks (here the 3d diffusive SATW), the survival probability tends for → ∞ to a non-zero value 1 -Π, which defines the hitting probability that depends on the initial distance to the target r and the target radius a. Hitting probability and transience exponent in infinite space (h.) and in confined domains (i.). Numerical simulations (symbols) and power law fits (plain lines). Our numerical results indicate ψ = ψc = 1 for the TSAW, the SESRW and the SATW, in agreement with the asymptotic diffusive behaviour of non compact self-interacting random walks.

∞. For compact processes the FPT distribution is independent of the target linear size a for r a ; we focus on this regime below. Following [START_REF] Levernier | Universal first-passage statistics in aging media[END_REF], F (t, r, R) can be written as a partition over trajectories that either hit the reflecting boundary before the target (with probability π(r, R) and conditional FPT distribution to the target F b (t, r, R)) or hit the target before the boundary (with probability π(r, R) and conditional FPT distribution to the target F t (t, r, R)):

F (t, r, R) = πF b (t, r, R) + (1 -π)F t (t, r, R). ( 15
)
Importantly, the weight 1 -π of trajectories that hit the target first can be expressed in the limit R → ∞ (with r fixed) in terms of the FPT distribution in infinite space

F ∞ (t, r) : π(r, R) ∝ R r ∞ R dw F ∞ (t, r)dt, (16)
which expresses the fact that most trajectories that hit the target before the boundary yield a FPT smaller than the timescale R dw . Making use of the definition of θ for processes in infinite space, we then obtain from dimensional analysis

F ∞ (t, r) ∝ r dwθ t θ+1 (17)
in the regime 1 t R dw , which yields from ( 16):

π(r, R) ∝ R r r R dwθ . ( 18
)
We stress that here the persistence exponent θ is defined in infinite and not in confined geometry. Next, the above argument leading to ( 16) also implies that

F t (t, r, R) ∝ Θ(t/R dw )F ∞ (t, r) ∝ Θ(t/R dw ) r dwθ t θ+1 , (19)
where Θ denotes a step function with Θ(x 1) = 1 and Θ(x 1) = 0. At this stage, the conditional FPT distribution F b (t, r, R) remains to be determined. By definition, this quantity involves trajectories that interact with the domain boundary. However, our analysis above shows that the increments of compact processes are identical in confined and infinite geometries. In the limit R → ∞ with r fixed, F b (t, r, R) can thus depend only on the time scales t and R dw ; dimensional analysis then yields the following scaling form:

F b (t, r, R) ∼ g(t/R dw )/t, (20)
where g is an undetermined function that depends on the process. Finally, it is convenient to introduce the rescaled variable η = t/R dw , and write, from ( 15),( 18),( 19) ,( 20) its asymptotic distribution for R → ∞ for η > 0 with r fixed :

F (η, r, R) = r R dwθ h(η) (21)
where h is an undetermined function that depends on the process. Finally, this explicitly captures the dependence of the FPT distribution on the geometrical parameters r, R, and therefore of all its moments (when they exist).

In particular, the mean FPT can be readily derived and satisfies:

T ∝ R dw(1-θ) r dwθ . (22)
The mean FPT thus scales non linearly with the confining volume V ∼ R d (because one has θ = 1 -d/d w ) for SESRW and SATW, as was found for other examples of aging processes ; notably, this scaling is linear for the TSAW. Strikingly, the asymptotic form of the FPT distribution ( 21) is comparable to that obtained in [START_REF] Levernier | Universal first-passage statistics in aging media[END_REF], and can be determined solely from the knowledge of d w , θ, which are defined in infinite space. This holds even if the dynamics of the process is ultimately impacted by the geometric confinement, as we have shown above -this result is in particular independent of the persistence exponent in confinement θ c . Fig. 5 shows an excellent quantitative agreement between numerical simulations and this analytical result. The data collapse of the properly rescaled FPT distribution shows that our approach fully captures its dependence on both r and R for all examples of compact self-interacting random walks that we have studied.

B. Non compact (transient) case

We now turn to the non-compact case. As opposed to the compact case, in the regime r a that we consider below, the FPT distribution depends on a. Following [START_REF] Levernier | Universal first-passage statistics in aging media[END_REF], we call excursion a fraction of trajectory that starts from the sphere S of radius R/2 centered on the target, next hits the boundary and eventually returns to S. The FPT distribution can then be written as a partition over the number n of excursions before the first-passage to the target, where we introduce Φ n (t) as the corresponding conditional FPT distribution :

F (t, a, r, R) = p 0 Φ 0 (t) + (1 -p 0 ) ∞ n=1 Φ n (t)P (n). (23)
Here p 0 ∼ (a/r) ψ is the probability to hit the target before the boundary starting from r, and P (n) the probability that the target is reached for the first time during the n th excursion. This can be written

P (n) = p n n-1 k=1 (1 -p k ) (24)
where p k is the probability that the target is found during the k th excursion, knowing that is has not been found before. Our analysis of increments and transience exponents ψ, ψ c above (see ( 13)) shows that, in confinement, non compact self-interacting random walks are diffusive for t 1 in both regimes T R dw (with diffusion coefficient D < ) and T R dw (with diffusion coefficient D > ). We thus denote by D n ≡ D > + δD n the effective diffusion coefficient during the n th excursion, which verifies |δD n | ≤ |D < -D > | and δD n → 0 for n 1. In addition, one has ψ c = ψ. We can thus write p k ∼ (C > + δC k )(a/R) ψ , where δC k → 0 for k 1. Note that here we have implicitly assumed (and checked numerically, see SI) that the conditional probability p k behaves as the unconditional probability that the target is found during the k th excursion. Last, a scaling argument (see SI) shows that

Φ n (t) = 1 t φ(t/t n ), (25)
where t n is the typical time elapsed before the n th excursion, which verifies

t n = R dw n-1 k=1 1 D k . ( 26
)
Finally, taking the R → ∞ limit in [START_REF] Sapozhnikov | Self-attracting walk with ν < 1/2[END_REF] with r, a fixed, one finds that the rescaled variable η = t/R d admits asymptotically the following distribution for η = 0 (see SI):

F (η, r, R) = 1 -C a R ψ h(η) ( 27
)
where h is an undetermined process dependent scaling function -not necessarily exponential -and C a process dependent constant. Similarly to the compact case, this explicitly captures the dependence of the FPT distribution on the geometrical parameters r, R, and therefore of all its moments (when they exist). In particular, the mean FPT is given by :

T ∼ R d a ψ 1 -C a R ψ . (28)
In contrast to the compact case, the mean FPT thus scales linearly with the confining volume V ∼ R d . Remarkably, the asymptotic form of the FPT distribution ( 21) is comparable to that obtained in [START_REF] Levernier | Universal first-passage statistics in aging media[END_REF] in absence of power-law aging, and can be determined solely from the knowledge of d w , ψ, which are defined in infinite space. This holds even if the dynamics of the process is impacted by the geometric confinement, as we have shown above. However, geometric confinement does not change the diffusive scaling of non compact self-interacting walks ; [START_REF] Davis | Reinforced random walk[END_REF] shows that this is sufficient to preserve the dependence on r, R of the FPT distribution in confinement. Fig. 6 shows an excellent quantitative agreement between numerical simulations and this analytical result. The data collapse of the properly rescaled FPT distribution shows that our approach fully captures its dependence on both r and R for all examples of non compact self-interacting random walks that we have studied.

VI. AGED INITIAL CONDITIONS

In this section, we analyse the impact of initial conditions on the FPT statistics of confined self-interacting random walks. As we have shown above, the dynamics of self-interacting random walks display aging properties, which can depend on geometric confinement. In other words, the dynamics is different if the random walk starts at T = 0 (fresh initial conditions studied above, for which the number of visits to any site i of the domain is set to n i = 0) or at T R dw (aged initial conditions, for which n i 1). We show that the FPT distribution can be readily obtained for aged initial conditions by adapting the approach developed above for fresh initial conditions, and highlight the impact of initial conditions.

A. Compact (recurrent) case

For aged initial conditions, because T R dw , the only relevant regime is t T . In this regime we have found the following behaviour of the increments:

∆ 2 (T, t) ∼ 2D < (t)t 2/dw ∝ t 2/dw,c (29)
where the effective walk dimension d w,c can be different from d w (see SESRW in Fig. 3). The relevant persistent exponent is clearly θ c in this regime. All steps leading to the derivation of the FPT distribution (see previous section) can then be reproduced. It is found that the rescaled variable η = t/R dw,c is asymptotically distributed according to :

Fc (η, r, R) = r R dw,cθc h c (η) (30)
where h c is an undetermined function that depends on the process. Initial conditions can thus deeply impact the FPT distribution, and even its scaling form : they can modify the walk dimension d w,c , the persistence exponent θ c , and the scaling function h c . This result is confirmed by numerical simulations (see Fig. 7). For the SATW, one has θ c = θ and d w,c = d w = 2 while for the SESRW one has θ c = θ and d w,c = d w ; the scaling of the FPT distribution is thus modified by initial conditions for these processes. In contrast, for the TSAW one has θ c = θ and d w,c = d w and the scaling of the FPT distribution is not modified by initial conditions.

B. Non compact (transient) case

In the regime T R dw and 1 t T , we have found the following diffusive scaling of increments for non compact self-interacting random walks:

∆ 2 (T, t) ∼ t 1 2D > t. (31)
In addition, we have shown that ψ c = ψ. All steps leading to the derivation of the FPT distribution (see previous section) can then be straightforwardly reproduced.

It is found that

Fc (η, r, R) = 1 -C a R ψ h c (η) (32)
where η = t/R d and h c is an undetermined function that depends on the process. In the case of non compact selfinteracting random walks, initial conditions thus do not modify the scaling of the FPT distribution ; they however can change the scaling function h c . This result is confirmed by numerical simulations (see Fig. 7).

VII. DISCUSSION AND CONCLUSION

A. Summary of the results

Our joint analytical and numerical analysis shows finally that long range memory effects can have deep consequences on the dynamics of generic self-interacting random walks ; they can induce aging (quantified by D(T, t) and d w ) and non trivial persistence and transience exponents θ and ψ, which we characterized quantitatively. In striking contrast with other non Markovian processes, we have shown that geometric confinement can strongly modify the dynamic properties of self-interacting random walks, beyond imposing locally reflecting boundary conditions : the dynamics of increments can be modified (in the non compact case), as well as persistent exponents (in the compact case).

Based on this systematic quantitative analysis, we have shown that universal scaling forms of the FPT distributions of general self-interacting random walks in confinement can be derived in the large volume limit, by generalizing the approach introduced in [START_REF] Levernier | Universal first-passage statistics in aging media[END_REF]. For "fresh" initial conditions, we find that the FPT distribution in confinement can be asymptotically predicted from the knowledge of the process in infinite space only (via the infinite space exponents d w , θ, ψ) : geometric confinement ultimately does modify the dynamics of the process and even changes the corresponding exponents d w,c , θ c , ψ c defined in confinement, but this occurs only at timescales larger than the typical FPT, and thus only mildly impacts the FPT statistics. In contrast, for aged initial conditions the exponents that determine the FPT distribution are those defined in confinement d w,c , θ c , ψ c , and can thus be different from the classical infinite space exponents d w , θ, ψ. In that case, the process in confinement must therefore be characterized to determine the FPT distribution.

B. Search efficiency of self-interacting random walkers

These results allow us to assess the efficiency of space exploration of self-interacting random walks, and in particular to discuss the impact of memory effects on target search kinetics.

In infinite space, attractive self-interactions (β < 0) can have drastic consequences on space exploration : for bounded effective interaction potentials V (n) (SATW), the random walk is subdiffusive for d = 2, 3 and |β| > |β| c and characterised by θ = ∞, so that all moments of the FPT to a target are finite. In this case memory effects thus give a decisive advantage to attractive SATW (|β| > |β| c ) as compared to normal random walks (β = 0) or repulsive self-interacting walks (β > 0).

In confined domains, the discussion is very different. If no prior information on the target position is available, the relevant observable to quantify the search kinetics is the position averaged mean FPT ¯ T . For compact processes, our results yield ¯ T ∝ R dw . Search kinetics is thus enhanced by lowering d w , which amounts to maximising the scaling of the MSD with time. In that case, memory effects give a decisive advantage to repulsive 1d TSAW and 1d SESRW, which both show a superdiffusive exponent d w < 2 for all values of β > 0. For non compact processes, we obtained ¯ T ∝ R d , which is consistent with the large time diffusive limit of non compact (repulsive or attractive) self-interacting random walks. The scaling of ¯ T with R is thus independent of memory effects, which however modify the effective diffusion coefficient and are thus favorable in the repulsive case.

If the starting distance r from the target is known, the full FPT distribution is needed to analyse the search kinetics. For compact processes, our results [START_REF] Ottinger | The generalised true self-avoiding walk-a model with continuously variable exponent ν[END_REF] show that the set of trajectories that hit the target can be decomposed into a set of fast trajectories, with timescale ∝ r dw and weight 1 -α(r/R) dwθ (where α is a constant), and a set of slow trajectories that typically hit the domain boundaries before the target, with timescale ∝ R dw and weight ∝ (r/R) dwθ . The exponents d w , θ thus appear as key parameters that control the respective weight of fast . Asymptotic FPT distribution of self-interacting random walks in confined domains for aged initial conditions. The search process starts at t = 0, but the random walker is assumed to have explored the domain from t = -T to t = 0. S(t) is the survival probability of the random walker at time t. The scaling of S(t) with geometrical parameters is deduced from [START_REF] Dussutour | Optimal traffic organization in ants under crowded conditions[END_REF] and [START_REF] Alessandro | Cell migration guided by long-lived spatial memory[END_REF], for compact and non-compact processes respectively. The collapse of numerical simulations after rescaling captures the dependence of the FPT distribution on geometrical parameters. Simulations are performed in 1d and 3d boxes of size R with reflecting boundary conditions for fixed r and a. Compact cases : a. 1d TSAW with β 1.0; b. 1d SESRW with = 1.0 and k = 0.5. The FPT distribution of the simple random walk is added for comparison (dashed curve); c. 1d SATW repulsive with β = 3.0 Non-compact cases : d. 3d TSAW with β = 1.0; e. 3d SESRW with β = 1.0 and k = 0.5; f. 3d SATW with β = 1.0. and slow trajectories, as well as the typical timescale of slow trajectories. For random processes with stationary increments, it has been proposed that both exponents are not independent and satisfy θ = 1 -d/d w [START_REF] Levernier | Universal first-passage statistics in aging media[END_REF][START_REF] Bray | Persistence and first-passage properties in nonequilibrium systems[END_REF][START_REF] Meroz | Distribution of first-passage times to specific targets on compactly explored fractal structures[END_REF] ; in that case, increasing the weight of direct, fast trajectories by increasing d w comes at the cost of increasing the timescale of indirect trajectories. This is also the case of the 1d repulsive TSAW, for which we found numerically θ = 1 -d/d w . In the case of the 1d SESRW and the 1d SATW however, we found that d w and θ are independent, with a dependence of θ only on the coupling parameter β. This shows that repulsive self-interactions can be favorable for large starting distances because they diminish the timescale of indirect trajectories by lowering d w (1d TSAW and 1d SESRW) ; they however in all cases reduce the weight of direct trajectories and are thus detrimental at short distances. In turn, attractive interactions (SATW) are detrimental for d = 2, 3 because they increase the timescale of indirect trajectories by increasing d w (subdiffusive SATW), while they preserve the diffusive scaling for d = 1 ; they can however significantly increase the weight of direct trajectories by increasing θ (d = 1, 2, 3), and are thus favorable at short distances. Finally, in the non compact case, our results [START_REF] Davis | Reinforced random walk[END_REF] show that the FPT statistics is characterized by the single time scale R d as long as r a. As in the case of the position averaged mean FPT, memory effects modify only the effective diffusion coefficient ; they are thus favorable in the repulsive case, but do not impact scaling properties of the FPT distribution.

Finally, this analysis shows that memory effects induced self-interactions can have a deep impact on space exploration, as quantified by various observables. Qualitatively, attractive self-interactions have dramatic effects and can lead to subdiffusion with compact exploration, which is favorable for local exploration, and even to selftrapping. Repulsive self-interactions have important effects for compact random walks, for which they modify the walk dimension d w and thus the scaling of the position averaged mean FPT with the size of the confining domain ; this is thus favorable for global exploration of confined domains with no prior informations on the target position.

Figure 2 .

 2 Figure 2. General properties of attractive self-interacting random walks. a. Example of trapped trajectory performed by a 1d attractive TSAW random walker (blue), compared to a diffusive trajectory of a 1d attractive SATW random walker (red). b. For d > 1, the SATW is subdiffusive for |β| > |βc| (blue sample trajectory), and diffusive for |β| < |βc| (green sample trajectory). Here d = 3 and thus |βc| = 0 [25]. c. Aging of the increments for the subdiffusive SATW (dw = 3 for d = 2) normalized by the expected subdiffusive scaling at long times. Each curve corresponds to a fixed value of T . Note that the increments are diffusive for t T . d. In the subdiffusive regime, the SATW performs an extremely compact exploration of space : the survival probability S(t) decays faster than any powerlaw (θ = ∞).

Figure 5 .

 5 Figure 5. Asymptotic FPT distribution of compact selfinteracting random walks in confined domains for fresh initial conditions. Here S(t) is the survival probability of the random walker, whose scaling with geometrical parameters is deduced from[START_REF] Ottinger | The generalised true self-avoiding walk-a model with continuously variable exponent ν[END_REF]. The collapse of numerical simulations after rescaling for different values of geometric parameters captures the dependence of the FPT distribution on the geometric parameters r, R. Simulations are performed in 1D boxes of size R with reflecting boundary conditions. a. 1D TSAW with dw = 3/2 (independent of β). b. 1d SESRW with β = 0.5 and k = 0.5. 1d repulsive SATW with (1 + e -β ) -1 = 0.7 ( c.) and (1 + e -β ) -1 = 0.4 (d.).

Figure 6 .

 6 Figure 6. Asymptotic FPT distribution of non compact selfinteracting random walks in confined domains for fresh initial conditions. Here S(t) is the survival probability of the random walker, whose scaling with geometrical parameters is deduced from[START_REF] Davis | Reinforced random walk[END_REF]. The collapse of numerical simulations after rescaling for different values of the geometrical parameters captures the dependence of the FPT distribution on r, R. Simulations are performed in 3d boxes of size R with reflecting boundary conditions and the constant C is measured numerically. a. 3d TSAW with β = 1.0. b. 3d SESRW with β = 1.0 and k = 0.5. 3d SATW repulsive with β = 3.0 (c) and with β = -1.0 (d).

Figure 7

 7 Figure 7. Asymptotic FPT distribution of self-interacting random walks in confined domains for aged initial conditions. The search process starts at t = 0, but the random walker is assumed to have explored the domain from t = -T to t = 0. S(t) is the survival probability of the random walker at time t. The scaling of S(t) with geometrical parameters is deduced from[START_REF] Dussutour | Optimal traffic organization in ants under crowded conditions[END_REF] and[START_REF] Alessandro | Cell migration guided by long-lived spatial memory[END_REF], for compact and non-compact processes respectively. The collapse of numerical simulations after rescaling captures the dependence of the FPT distribution on geometrical parameters. Simulations are performed in 1d and 3d boxes of size R with reflecting boundary conditions for fixed r and a. Compact cases : a. 1d TSAW with β 1.0; b. 1d SESRW with = 1.0 and k = 0.5. The FPT distribution of the simple random walk is added for comparison (dashed curve); c. 1d SATW repulsive with β = 3.0 Non-compact cases : d. 3d TSAW with β = 1.0; e. 3d SESRW with β = 1.0 and k = 0.5; f. 3d SATW with β = 1.0.


PharmD Aurélie Hanin

PharmD Jérôme Alexandre Denis

MD, PhD Valerio Frazzini

MD Louis Cousyn

PharmD Françoise Imbert-Bismut

PharmD Benoit Rucheton

PharmD Dominique Bonnefont-Rousselot

Clemence Marois

MD, PhD Virginie Lambrecq

MD Sophie Demeret

MD Clémence Marois

MD, PhD Vincent Navarro
email: vincent.navarro@aphp.fr

Vincent Pr

Navarro

Clémence Marois

Neuron Specific Enolase, S100-beta protein and progranulin as diagnostic biomarkers of status epilepticus

Keywords: Status epilepticus -Diagnosis -Etiology -Neuron Specific Enolase -S100beta protein -Progranulin Status epilepticus -Diagnosis -Etiology -Neuron Specific Enolase -S100-beta protein -Progranulin CONT = Control patients, CSF = Cerebrospinal Fluid, EEG = electroencephalogram, EPI = pharmacoresistant epileptic patients, ICU = Intensive Care Unit

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Status epilepticus (SE) is a life-threatening prolonged epileptic seizure, related to various etiologies and with heterogeneous clinical presentations [START_REF] Trinka | 25 years of advances in the definition, classification and treatment of status epilepticus[END_REF]. Sustained SE may lead to brain injury and is associated with worse clinical outcomes and higher mortality [START_REF] Alkhachroum | Markers in Status Epilepticus Prognosis[END_REF]. The rapid diagnosis of SE is fundamental to urgently initiate antiepileptic treatments [START_REF] Claassen | Emergency Neurological Life Support: Status Epilepticus[END_REF]. Diagnosis of convulsive SE is mostly based on clinical examination. In contrast, the diagnosis of nonconvulsive SE requires an electroencephalogram (EEG) recording [START_REF] Leitinger | Salzburg Consensus Criteria for Non-Convulsive Status Epilepticus--approach to clinical application[END_REF]. Here, we looked for the impact of SE on three biochemical markers levels and we wondered if these biochemical markers could be used to identify difficult-to-diagnose form of SE.

We considered Neuron Specific Enolase (NSE), S100-beta protein (S100B) and progranulin, which have previously been proposed as SE biomarkers [START_REF] Hanin | Cerebrospinal fluid and blood biomarkers of status epilepticus[END_REF]. They reflected respectively the neuronal injury (NSE), the astroglial injury (S100B) and the activation of microglia (progranulin) that occur in SE. Increased cerebrospinal fluid (CSF) NSE levels were reported in patients with cryptogenic or remote SE [START_REF] Correale | Status epilepticus increases CSF levels of neuron-specific enolase and alters the blood-brain barrier[END_REF]. Increased serum NSE levels were also reported in patients with sustained SE with a peak between 24 and 48 hours after the SE onset [START_REF] Degiorgio | Serum neuron-specific enolase in human status epilepticus[END_REF][START_REF] Degiorgio | Serum neuron-specific enolase in the major subtypes of status epilepticus[END_REF].

Despite good characteristics (good neuronal specificity, stability in the body fluids, a rapid and reliable biochemical assessment), NSE is not currently used as a diagnosis SE biomarker. This can be explained as most previous studies were conducted on too small patient cohorts (between 2 to 31 patients) precluding the utilization of NSE as a validated diagnosis biomarker for SE [START_REF] Gnanapavan | Guidelines for uniform reporting of body fluid biomarker studies in neurologic disorders[END_REF]. S100B was mostly studied in isolated seizures but increased CSF S100B levels were reported in a rat model of SE [START_REF] Vizuete | Phase-Dependent Astroglial Alterations in Li-Pilocarpine-Induced Status Epilepticus in Young Rats[END_REF][START_REF] Freund | Predictive value of S100-B and copeptin for outcomes following seizure: the BISTRO International Cohort Study[END_REF][START_REF] Asadollahi | The diagnostic value of serum UCHL-1 and S100-B levels in differentiate epileptic seizures from psychogenic attacks[END_REF]. S100B was not previously studied in human SE.

Nevertheless, its rapid assessment and its ability to distinguish isolated epileptic seizures from psychogenic non-epileptic seizures make S100B a promising biomarker for SE diagnosis [START_REF] Asadollahi | The diagnostic value of serum UCHL-1 and S100-B levels in differentiate epileptic seizures from psychogenic attacks[END_REF].

A higher expression of progranulin was observed in activated microglia and macrophages following pilocarpine-induced SE in rats [START_REF] Zhu | Progranulin promotes activation of microglia/macrophage after pilocarpine-induced status epilepticus[END_REF]. However, progranulin levels in CSF were not significantly increased compared to patients who underwent a lumbar puncture after a single tonic-clonic seizure [START_REF] Huchtemann | Progranulin levels in status epilepticus as a marker of neuronal recovery and neuroprotection[END_REF].

Here, we evaluated the diagnosis value of these three biomarkers in a large cohort of patients. They were measured in blood samples from SE patients, control patients and patients with pharmacoresistant epilepsy as well as in CSF samples from some SE patients and control patients.

Firstly, we assessed the ability of NSE, S100B and progranulin to identify SE patients. Secondly, we investigated the impact of SE etiology, SE semiology and the delay of blood or CSF collection after the ongoing SE onset on the biomarker's levels

Methods

Study design, setting and participants

We prospectively enrolled adult patients admitted in intensive care unit (ICU) and neurology units of the Pitié-Salpêtrière Hospital, from February 2013 to March 2021. The protocol was approved by our local ethic committee (2012, CPP Paris-VI) and sponsored by INSERM (C16- [START_REF] Chou | Unruptured Intracranial Aneurysms and SAH CDE Project Investigators (2019) Biospecimens and Molecular and Cellular Biomarkers in Aneurysmal Subarachnoid Hemorrhage Studies: Common Data Elements and Standard Reporting Recommendations[END_REF]20152482). Patients or relatives were informed and gave their consent. The study design and report are in accordance with the STROBE statement [START_REF] Cuschieri | The STROBE guidelines[END_REF].

Eligible SE patients were patients aged at least 18 years, with an ongoing SE, either a generalized convulsive SE, either a focal convulsive or focal non-convulsive SE diagnosed according to the International League Against Epilepsy criteria [START_REF] Trinka | 25 years of advances in the definition, classification and treatment of status epilepticus[END_REF]. The SE end was defined as the absence of seizures after the anesthetic's withdrawal. Physicians from ICU and neurology units excluded: patients with a subtle SE defined by minor and erratic myoclonic movements in patients with severely impaired consciousness; patients with post-anoxic SE; patients whose SE was linked to a pathological condition, such as trauma or subarachnoid aneurysmal hemorrhage, who needed immediate surgery and for whom the biomarkers could be elevated due to the underlying disease [START_REF] Chou | Unruptured Intracranial Aneurysms and SAH CDE Project Investigators (2019) Biospecimens and Molecular and Cellular Biomarkers in Aneurysmal Subarachnoid Hemorrhage Studies: Common Data Elements and Standard Reporting Recommendations[END_REF][START_REF] Helbok | Cerebrospinal fluid and brain extracellular fluid in severe brain trauma[END_REF]. Patients for whose parent, guardian or other reliable person refused permission or patients who refused themselves permission were also excluded.

Enrolled patients were divided into four groups according to their SE etiology: acute, progressive, remote and cryptogenic [18]. The type of SE was further categorized into three groups: non-refractory SE (NRSE), refractory SE (RSE) and prolonged super-refractory SE (PSRSE) [START_REF] Hirsch | Proposed consensus definitions for new-onset refractory status epilepticus (NORSE), febrile infection-related epilepsy syndrome (FIRES), and related conditions[END_REF]. The RSE was defined as a failure of at least two appropriately selected and dosed parenteral medications including a benzodiazepine, without any specific SE duration [START_REF] Hirsch | Proposed consensus definitions for new-onset refractory status epilepticus (NORSE), febrile infection-related epilepsy syndrome (FIRES), and related conditions[END_REF]. The PSRSE was defined as a RSE that persists for more than 7 days, with ongoing need for anesthetics [START_REF] Hirsch | Proposed consensus definitions for new-onset refractory status epilepticus (NORSE), febrile infection-related epilepsy syndrome (FIRES), and related conditions[END_REF]. The type of SE was further categorized into three groups according to the SE semiology: generalized convulsive SE, focal convulsive SE and focal non-convulsive SE. None of the patients had a generalized non-convulsive SE.

Eligible patients with pharmacoresistant epilepsy (EPI) were aged at least 18 years and hospitalized for a video-EEG monitoring pre-surgical evaluation (n=17) or for a resection of the epileptic focus (n=39). Patients who refused permission and patients for whose the date of the last seizure cannot be reported were excluded.

Eligible control patients (CONT) were aged at least 18 years and hospitalized in neurology units for non-acute disorders and for whom a clinical neurological evaluation and a lumbar puncture were performed. Patients with previous history of epilepsy or patients whose symptoms were linked to an acute pathological condition (trauma, hemorrhage) were excluded as well as patients with an abnormal lumbar puncture (i.e. pleocytosis, increased protein levels, intrathecal synthesis). The diagnosis for these patients were psychogenic disorders or noninflammatory peripheral neuropathies.

Biochemical assays

Blood and CSF samples were centrifuged at 3500 rpm for 10 min. Hemolytic samples (hemoglobin concentration above 47 mg/dL) were excluded.

The NSE and S100B assays were performed using respectively immunofluorimetric assays and electrochemiluminometric sandwich immunoassays (Kryptor®, Brahms and Modular®E170, Roche Diagnostics). The lowest detections were 0.8 ng/mL for NSE and 0.005 ng/mL for S100B. The coefficients of variation (CV) were found to be lower than 5% for all controls used. Progranulin measurements were performed using the progranulin-human-ELISA kit (Adipogen, Coger SAS, France). All the samples were analyzed in duplicated and progranulin measurements was repeated when CV between duplicated measures were higher than 15%.

Statistical analysis

Analyses were performed with R Software (R.3.5.0).

We first performed Kruskal-Wallis test to evaluate subgroups impact. Then, pairwise comparisons were performed using Student or Mann-Whitney when appropriate. The Benjamini-Hochberg test procedure was used to correct for multiple comparisons. Correlations between blood levels and the time between SE onset and blood sampling were evaluated by calculating respective Spearman's rho values and their level of significance. To assess the ability of biomarkers to diagnose SE, we computed the area under the receiver operating characteristics (ROC) curve and reported sensitivity and specificity for the best cut-off which was defined accordingly to the Youden's index [START_REF] Fluss | Estimation of the Youden Index and its associated cutoff point[END_REF]. Quantitative variables were expressed as mean with standard deviation. All statistical were two-sided with a type I error rate of 5%.

Results

The study cohort is represented in Fig. 1A. Main demographic data are shown in Table .1. Blood samples were collected for 36 CONT patients, 56 EPI patients and 82 SE patients. CSF samples were collected for 38 CONT patients and 34 SE patients.

Blood samples and CSF samples were collected respectively on average 7 (±12) and 11 (±17) days after the SE onset. All the blood and CSF samples were collected in patients with an ongoing SE. Twenty-nine EPI patients (52%) showed a seizure in the last forty-eight hours before the blood collection.

NSE, S100B and progranulin as diagnostic biomarkers of SE

Blood NSE, S100B and progranulin levels

Serum NSE levels were significantly higher in SE patients when compared with EPI patients and showed a non-significant increase as compared with CONT patients (Fig. 1B, Table .2).

Both serum S100B and plasma progranulin levels were higher in SE patients when compared with EPI patients and CONT patients (Fig. 1B, Table .2). No significant difference of blood levels was found between CONT patients and EPI patients for any of the three biomarkers (Table .2).

CSF NSE, S100B and progranulin levels and CSF/blood ratios

Both CSF NSE and progranulin levels were significantly higher in SE patients when compared with CONT patients (Fig. 1C, Table .2). No significant difference was found for CSF S100B levels (Table .2). Similarly, we observed in SE patients an increase in the CSF/blood ratios for progranulin, while the CSF/blood ratios tended to increase for NSE and no difference was found for CSF/blood S100B ratios (Table .2).

SE diagnosis value

The ROC curves revealed good accuracy at detecting SE for: (i) serum S100B levels (AUC 0.748 [CI 95% 0.669-0.827]), with a sensitivity of 59.7%, specificity of 90.2%, positive predictive value (PPV) of 83.6% and negative predictive value (NPV) of 72.8% when using the cut-off of 0.09 ng/mL; and (ii) plasma progranulin levels (AUC 0.756 [CI 95% 0.681-0.831]), with a sensitivity of 57.3%, specificity of 85.4%, PPV of 79.7% and NPV of 66.7% when using the cut-off of 110 ng/mL (Fig. 1D). Serum NSE levels can detect SE with a lower discrimination (AUC 0.624 [CI 95% 0.536-0.713]).

Biomarkers levels according to SE refractoriness, SE semiology and SE etiology

SE refractoriness

We next wondered if blood biomarkers levels differed according to SE refractoriness and to the delay between the ongoing SE onset and the blood collection. We found that only plasma progranulin levels differed according to the SE refractoriness (p<0.001). Plasma progranulin levels were significantly higher for PSRSE patients when compared with RSE patients (177.1±95.0 versus 107.7±43.0, p<0.001) and NRSE patients (177.1±95.0 versus 137.5±63.2, p=0.030). Similarly, plasma progranulin levels were found correlated with the ongoing SE duration before the blood collection (rho=0.360, p=0.0032) while no correlation was found for serum NSE and S100B levels (Fig. 2A). We did not look for the impact of SE refractoriness on CSF biomarkers levels as lumbar puncture were mostly done for refractory SE.

SE semiology

We then researched if these three biomarkers behave differentially according to SE semiology.

Forty-two (51%) patients had a generalized convulsive SE, 33 (40%) had a focal convulsive SE and 7 (9%) had a focal non-convulsive SE. The blood or the CSF levels were not significantly different according to SE semiology for all the three markers (data not shown).

SE etiology

We then researched if these three biomarkers behave differentially according to SE etiology.

The SE etiologies were classified as acute for 28 patients (34%), progressive for 20 patients (24%), remote for 24 patients (29%) and cryptogenic for 10 patients (12%). The CSF levels were not significantly different according to SE etiology for all the three markers (data not shown). Serum NSE levels were not significantly different according to SE etiology whereas serum S100B and plasma progranulin levels tended to increase in acute SE (Fig. 2B). We observed a high heterogeneity of serum S100B levels (0.03-0.70 ng/mL) and plasma progranulin levels (77-490 ng/mL) for the patients with acute etiology (Fig. 2B). It might be explained by the heterogeneity of the underlying etiologies (encephalitis n=19, posterior reversible encephalopathy syndrome n=3, metabolic disturbances n=2, alcohol disorders n=2, subdural hematoma n=2).

Discussion

The identification of suitable biomarkers could help physicians to establish a more accurate diagnosis quickly, especially for difficult-to-diagnose form of SE.

The SE patients had higher CSF NSE levels when compared with control patients and higher serum levels when compared with pharmacoresistant epileptic patients. These results are in accordance with previous studies [START_REF] Degiorgio | Serum neuron-specific enolase in human status epilepticus[END_REF][START_REF] Degiorgio | Serum neuron-specific enolase in the major subtypes of status epilepticus[END_REF] but were obtained in a larger cohort which support their accuracy. We showed, for the first time, that SE patients had higher S100B levels when compared with control (in serum and CSF samples) and pharmacoresistant epilepsy patients (in serum samples). Similarly, SE patients had higher progranulin levels when compared with control patients (in plasma and CSF samples) and pharmacoresistant epilepsy patients (in plasma samples).

We may wonder if the increase of the biomarkers levels is related to SE onset, and therefore biomarkers could be used for SE diagnosis; and/or if it reflects brain damage associated to SE.

A mixed scenario can be considered with an unequal distribution according to the biomarkers.

The NSE is a world-renowned biomarker to assess brain damage in acute neurological disorders or after cardiac arrest [START_REF] Chou | Unruptured Intracranial Aneurysms and SAH CDE Project Investigators (2019) Biospecimens and Molecular and Cellular Biomarkers in Aneurysmal Subarachnoid Hemorrhage Studies: Common Data Elements and Standard Reporting Recommendations[END_REF][START_REF] Helbok | Cerebrospinal fluid and brain extracellular fluid in severe brain trauma[END_REF][START_REF] Chou | Participants in the International Multi-disciplinary Consensus Conference on the Multimodality Monitoring (2014) Monitoring biomarkers of cellular injury and death in acute brain injury[END_REF][START_REF] Tawk | The Relationship Between Serum Neuron-Specific Enolase Levels and Severity of Bleeding and Functional Outcomes in Patients With Nontraumatic Subarachnoid Hemorrhage[END_REF]. Increased NSE levels following SE may be related to neuronal death and NSE could be preferentially used as a SE prognosis biomarker.

Nevertheless, several arguments suggest that NSE could also be a SE diagnosis biomarker. Firstly, while brain damage increased with the SE duration, we did not observe a correlation between the serum NSE levels and the ongoing SE duration before the blood collection.

Secondly, serum NSE levels were not significantly different according to SE semiology while generalized convulsive SE are known to be more severe than focal convulsive SE. Thirdly, we previously reported that patients with seizures had higher blood NSE levels when compared with patients with periodic discharges at EEG despite periodic discharges were associated with higher brain damage [START_REF] Hanin | Serum neuron-specific enolase: a new tool for seizure risk monitoring after status epilepticus[END_REF][START_REF] Jaitly | Prognostic value of EEG monitoring after status epilepticus: a prospective adult study[END_REF]. It suggested that increased NSE levels may be related to seizures onset and NSE could be used as a SE diagnosis biomarker. The S100B is a marker of astroglial injury and found increased in several acute neurological disorders [START_REF] Helbok | Cerebrospinal fluid and brain extracellular fluid in severe brain trauma[END_REF][START_REF] Chou | Participants in the International Multi-disciplinary Consensus Conference on the Multimodality Monitoring (2014) Monitoring biomarkers of cellular injury and death in acute brain injury[END_REF][START_REF] Sanchez-Peña | S100B as an additional prognostic marker in subarachnoid aneurysmal hemorrhage[END_REF]. While it was reported a quickly and progressive gliosis after SE, the absence of correlation between serum S100B levels and the ongoing SE duration before the blood collection suggests that the increase of S100B levels could be merely related to SE onset and therefore this biomarker could be used for SE diagnosis.

In contrast, plasma progranulin levels were higher in patients for whom the blood was collected several days after SE onset and therefore for PSRSE patients. The increase of the progranulin levels occurs primarily in activated microglia [START_REF] Zhu | Progranulin promotes activation of microglia/macrophage after pilocarpine-induced status epilepticus[END_REF]. However, the microglial activation only occurs few days after the SE onset which could explain the increased delay. The sustained SE may enhance the microglial activation and inflammation processes and therefore, the increase of plasma progranulin levels could reflect the SE duration and subsequent consequences.

Plasma progranulin levels might further testify the brain damage than the occurrence of SE.

Nonetheless, as a microglial activation frequently occurs in SE of autoimmune origin [START_REF] Tan | Inflammation, ictogenesis, and epileptogenesis: An exploration through human disease[END_REF], we may wonder if this biomarker might be also used to identify patients with SE of autoimmune origin and therefore guide the early treatment strategy. Despite these three biomarkers were all increased in the blood of SE patients, and NSE and S100B increased could be related to SE onset, we believe that S100B is the most promising marker for SE diagnosis. Indeed, S100B had a higher AUC value than NSE, and it could be quickly assessed contrary to progranulin. We identified a cut-off at 0.09 ng/mL, which was accurately associated with a SE diagnosis for 84% of the patients. As control and pharmacoresistant epilepsy patients rarely had serum S100B levels above 0.09 ng/mL (less than 10% in our cohort), it might be interesting to use serum S100B levels as an emergency triage marker. If a patient without any acute brain injury (trauma, hemorrhage) is admitted to the emergency department with convulsive movements, confusion or decreased alertness and has serum S100B levels above 0.09 ng/mL, clinicians may be prompted to keep the patient in the hospital for medical and neurological observation and to request an emergency EEG.

Nonetheless, we believe that clinicians will not rely solely on this biomarker to decide to decide on a return home. Our study highlights the potential of these biomarkers for SE diagnosis and allows to propose a cut-off for the clinical use of S100B.

We found no association between the SE etiology and the levels of serum NSE, as previously reported [START_REF] Degiorgio | Serum neuron-specific enolase in the major subtypes of status epilepticus[END_REF]. However, we observed a high heterogeneity in S100B and progranulin levels for patients with an acute SE. It could be explained by a heterogeneity of SE etiologies classified as an acute SE accordingly to the International League Against Epilepsy Classification [18]. If this first classification is useful to categorize SE, it could be relevant to add biochemical biomarkers to this classification in order to distinguish acute SE with or without inflammatory underlying causes.

We investigated the potential of NSE, S100B and progranulin as SE biomarkers in a large cohort of patients for the first time. Although our population was heterogenous for SE etiology and for the delay of enrollment after the SE onset, we found promising results for the three biomarkers. We confirmed the increase of NSE levels in SE patients whatever the SE etiology.

Serum S100B appears to be the most promising biomarker for SE diagnosis by using the cutoff at 0.09 ng/mL. Plasma progranulin could be an interesting biomarker to reflect inflammatory processes.

There are some limitations for this study: it was conducted in a single center, most SE patients were enrolled in the Neuro-Intensive care unit specialized for the management of prolonged super-refractory SE and therefore, the collection of blood and CSF samples was delayed for patients who were previously managed in another hospital. Due to the small number of patients, we were not able to specifically analyze the interest of blood and CSF biomarkers for nonconvulsive SE patients. Nonetheless, the strengths of our study were to collect samples from a large cohort of patients and to provide cut-off for clinical use. Further studies are needed to investigate the diagnosis potential of the three biomarkers for non-convulsive SE and to investigate the ability of progranulin to specifically distinguish inflammation processes related to a sustained SE (i.e. consequences of SE) from inflammation processes involved in SE development (i.e. causes of SE) [START_REF] Abboud | Autoimmune encephalitis: proposed best practice recommendations for diagnosis and acute management[END_REF]. The Fig. 1B and Fig. 1C represent respectively the distribution of NSE levels (up), S100B levels (middle) and progranulin levels (down) in blood (Fig. 1B) and in CSF (Fig. 1C). The violin plots show the density of the data at different levels. The white point represents the median of the data, the dark box indicates the interquartile range and the two end points represent respectively the maximum and the minimum values (expect for outliers).

Statements and Declarations

The Fig. 1D shows the ROC curves for serum NSE (red), serum S100B (blue) and plasma progranulin (black) to identify patients with SE. The table below represents the values of sensitivity (Se), specificity (Sp), positive predictive value (PPV) and negative predictive value (NPV) for the best cut-off of each biological marker, defined according to the Youden's index.

Fig.2 Progranulin as a biomarker of inflammatory status epilepticus

The Fig. 2A shows the relations between serum NSE levels (up), serum S100B levels (middle) and plasma progranulin levels (down) and the time between the SE onset and the blood sampling. The correlations were assessed with the Spearman test.

The Fig. 2B represents the distribution of serum NSE levels (up), serum S100B levels (middle) and plasma progranulin levels (down) according to the SE etiologies. The white points represent the median of the data, the dark boxes indicate the interquartile ranges and the two end points represent respectively the maximum and the minimum values (expect for outliers). Progranulin (ng/mL)

Table.1 Demographic characteristics of the study cohort.

Fig. 1

 1 Fig.1 Neuron Specific Enolase, S100-beta protein and progranulin as diagnosis biomarkers of status epilepticusThe Fig.1Arepresents the number of patients for each group (CONT, EPI, SE) and each biological compartment (blood, CSF).

  Abbreviations: CONT = Control patients; EPI = pharmacoresistant Epileptic patients; F = Female; M = Male; SE = Status Epilepticus.

Table 2 :

 2 Blood, CSF, CSF/blood ratio of NSE, S100B and progranulin in SE patients, control patients and EPI patients.Data are represented as mean ± standard deviation (sd).

Fig. 1

 1 Fig.1

Fig. 1 Serum

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Fig. 2

 2 Fig.2

Funding

This work received support from the "Investissements d'avenir" program ANR-10-IAIHU-06, from the "Fondation pour la Recherche Médicale" (FDM20170839111) and from the Fondation Assitance Publique-Hôpitaux de Paris (EPIRES-Marie Laure PLV Merchandising).

Data availability

All anonymized data are available on request.

Availability of data and material

Hanin et al. 12

All data are available on request upon the corresponding author.

Code availability

Not applicable.

Summary

Status epilepticus (SE) is a life-threatening prolonged epileptic seizure. A rapid diagnosis is fundamental to initiate antiepileptic treatment and to prevent the development of neurological sequels. Several serum and cerebrospinal fluid biomarkers have been proposed to help in the diagnosis of SE. Nevertheless, previous studies were conducted on too small patient cohorts, precluding the utilization of interesting biomarkers for the SE diagnosis.

Here, we aimed to assess the ability of Neuron Specific Enolase (NSE), S100-beta protein (S100B) and progranulin to help in the diagnosis of SE in a large cohort of patients (36 control patients, 56 patients with pharmacoresistant epilepsy and 82 SE patients).

Blood NSE, S100B and progranulin levels were higher in SE patients when compared with control patients or patients with pharmacoresistant epilepsy. Both NSE and progranulin levels were higher in cerebrospinal fluid from SE patients when compared with control patients. The receiver operating characteristics curves revealed good accuracy at detecting SE for serum S100B (AUC 0.748) and plasma progranulin (AUC 0.756). The performances were lower for serum NSE (AUC 0.624). Eighty-four percent of patients with serum S100B levels above 0.09 ng/mL presented with a SE whereas nighty percent of patients without SE had serum S100B levels lower than 0.09 ng/mL. Serum S100B levels were not significantly different according

Conflicts of interest

Vincent Navarro reports personal fees from UCB Pharma, EISAI, GW Pharma and LivaNova,

Ethics approval

The protocol was approved by our local (2012, CPP Paris-VI) and by the INSERM ethic committees (C16-16, 20152482). The study was performed in accordance with the ethical standards as laid down in the 1964 Declaration of Helsinki and its later amendments or comparable ethical standards.

Consent to participate

Patients or relatives were informed and give their consent.

Consent for publication

Patients or relatives were informed and give their consent.

Abbreviations: AU = Arbitrary Units, CONT = Control patients, EPI = pharmacoresistant Epileptic patients, SE = Status Epilepticus.

Specificity