M Boutahar

M Royer-Carenzi

Identifying trend nature in time series using autocorrelation functions and R-routines based on stationarity tests

Keywords: time series, stationarity, unit root test, Dickey-Fuller, quadratic trend

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Introduction

Time series non-stationarity can originate form various sources: either from a trend component or from a seasonal or even a cyclical component. In this paper, we will be interested in the non-stationarity caused by a trend. There are two kinds of trends: either a deterministic trend which can be modeled by some function of time (polynomial trend is generally considered), or a stochastic trend which presents unit roots. Deterministic and stochastic trends are two specific models suggested by Nelson and Plosser in [START_REF] Nelson | Trends and random walks in macroeconomics time series : some evidences and applications[END_REF]. ,d)

Deterministic trend (DetS

Z t = a 0 + a 1 t + • • • + a d t d + B t (1) Stochastic trend (StoS,d) ∆ d (Z t ) = B t , (2)
where we take a d = 0, ∆ is the differencing operator and (B t ) t is a moving average process

B t = j∈Z b j E t-j , (H1)
where (E t ) t is a sequence of identically distributed and independent centered variables, such that

E(E 2k t ) < ∞, for some k ≥ 2, ( H2
)
CONTACT M. Boutahar. Email: mohamed.boutahar@univ-amu.fr We denote (B t ) t as (SN), for Stationary Noise. It's well known that causal and invertible ARMA pocesses satisfy Hypotheses (H1) and (H3). When B t = E t , it is called (WN) for white noise. In this case, the associated models defined in Equations 1 and 2, are referred as (DetW,d) and (StoW,d).

When modeling time series, specially for macroeconomic and financial data, it is very important to identify the nature of the trend: deterministic or stochastic. Indeed, every type of trend induces specific behaviors, that we can illustrate with moments properties. Let us consider a (DetS,1) model associated to a white noise, then IE(Z t ) = a 0 +a 1 t, and var(Z t ) = σ 2 B , providing a stationary variance but a non-stationary mean. On the contrary, under a (StoS,1) model with Z 0 = 0, we have IE(Z t ) = 0, and var(Z t ) = tσ 2 B , providing the opposite feature. Thus the source of non-stationarity differs with trend nature. Consequently identifying the correct trend is fundamental. Globally, time series with a deterministic trend always revert to the trend in the long run (the effects of shocks are eventually eliminated); and the forecast confident intervals have constant width. On the contrary, time series with a stochastic trend never recover from shocks to the mean (the effects of shocks are permanent); and the forecast confident intervals grow with the horizon. Several authors (see for instance [START_REF] Chan | A Note on Trend Removal Methods : The Case of Polynomial Regression versus Variate Differencing[END_REF] and [START_REF] Nelson | Spurious Periodicity in Inappropriately Detrended Time Series[END_REF]) studied the consequences of an inappropriate modeling choice, underlying the importance of developing procedures able to produce a reliable classification.

Autocorrelation function properties have been widely studied, and are helpful to specify accurate models for stationary time series. The theoretical autocorrelation function at lag h (|h| < n) is estimated from data (Z 1 , • • • , Z n ) with the random variable

Ξ(h) = n-h t=1 (Z t+h -Z)(Z t -Z) n t=1 (Z t -Z) 2 , (3)
where

Z = n t=1 Z t n ,
is the random mean. We show that autocorrelation functions also have interesting properties in our framework, since they permit to identify the presence of a trend in a time series. Nevertheless it can not be used to distinguish between either a deterministic or a stochastic trend. Dickey and Fuller developed a unit root test that is an essential tool in time series modeling [START_REF] Dickey | Distribution of the Estimators for Autoregressive Time Series with a Unit Root[END_REF]. The authors considered the following models

M 1 : Z t = ρ Z t-1 + E t M 2 : Z t = a 0 + ρ Z t-1 + E t M 3 : Z t = a 0 + a 1 t + ρ Z t-1 + E t ,
where (E t ) t is a white noise. Dickey-Fuller procedure permits to test the null hypothesis of a unit root (ρ = 1) against the alternative hypothesis of a stationary AR(1) model without drift (resp. with drift, resp. with trend), accordingly to model M 1 (resp. M 2 , resp. M 3 ). Note that a time series with a (linear) deterministic trend, as defined in (DetW,1), is included in M 3 model, by taking ρ = 0. Consequently, under a (linear) deterministic trend, Dickey-Fuller unit-root test, led in M 3 framework, usually rejects the null hypothesis. On the other hand, (StoW,1)-time series are included in M 1 model, with ρ = 1, so that (H 0 ), tested in M 1 framework, is usually not rejected, accordingly with the rejection rate.

Several strategies based on Dickey-Fuller unit root-test have been developed ([8, 20, 10]), and they mostly appear to be appropriate to choose between a linear deterministic trend or a stochastic trend. But all the observed deterministic trends are not linear, trends with higher degrees can be involved. In this case, Ertur noticed in [START_REF] Ertur | Méthodologies de test de la racine unitaire[END_REF] that the usual strategies fail in detecting a quadratic deterministic trend. Indeed, under a quadratic trend for instance, Dickey-Fuller test generally concludes to the presence of a (spurious) unit root, even using model M 3 that allows a linear trend.

In this paper, we aim to include polynomial trends with a degree higher than one. Let us introduce the general model :

M 3,d : Z t = a 0 + a 1 t + • • • + a d t d + ρ Z t-1 + B t (a d = 0) .
In [START_REF] Ouliaris | Testing for a Unit Root in the Presence of a Maintained Trend[END_REF], Ouliaris, Park and Phillips developed a test that corrects the bias caused by high degree trend when testing for a unit root. We included this test in a strategy, that correctly identifies either a deterministic trend or a stochastic one.

In Section 2, we detail autocorrelation functions convergence when time series are driven by a trend, either deterministic or stochastic. This result permits to detect the presence of a trend, but without precise identification. In Section 3, we study existing strategies based on Dickey-Fuller tests, and analyze their performance in classifying models (DetW,1), (DetW,2), (StoW,1) and (StoW,2). Note that we also consider the simplest model (WN) as a null-model. Next, we provide a new strategy, based on Trend Diagnosis Tests (TDT), able to identify trends type, even when (WN) is replaced by a (SN) underlying process, and when higher-degrees d are considered. Finally, we apply our strategy on real data such as abortion rate in Montreal, Québec, and also on the CO2 atmospheric concentration. All the functions are implemented in R language, and they are available at the website: www.i2m.univ-amu.fr/perso/manuela.royer-carenzi/AnnexesR.TrendTS/TrendTS.html

Trend detection

It is essential to start by plotting the graph of the series in order to visualize its evolution, but the presence or absence of a trend is sometimes difficult to detect from the plot. Autocorrelation appears to be a powerful tool in the detection of a trend insofar as its behavior is specific in the presence of a trend. We recall that theoretical autocorrelations ρ(h) = cor(Z t , Z t+h ) are well-defined only if process (Z t ) t is square-integrable and stationary. However the associated random variables Ξ(h), as defined in Equation 3, can be computed for any observed time series

(Z 1 , • • • , Z n ).

Autocorrelation functions behavior for (WN)-time series

Theorem 2.1.

Let (Z t ) t be a white noise. Then

√ n t (Ξ(1), Ξ(2), • • • , Ξ(r)) L ------→ n→+∞ N r ( t (0, • • • , 0), Id r ) ,
where t v denotes the transpose of vector v, and Id r is the identity r × r-matrix.

Theorem 2.1 is a particular case of Theorems 7.2.1. or 7.2.2. in [START_REF] Pj Brockwell | Time series : Theory and methods[END_REF], that require Hypotheses (H1), (H3-a) and either Hypothesis (H2) or (H3-c). Thus, we get that the random variables Ξ(1), Ξ(2), • • • , • • • , Ξ(r) are asymptotically independent and identically distributed as Gaussian random variables with zero mean and variance 1/n. Consequently, for any fixed lag h = 1, • • • , r, for large n, the sample autocorrelation function √ n ρ(h) is expected to be a realization of a standard Gaussian, that is to be valued in the interval [-1.96, 1.96], with 95% coverage. Thus, sample autocorrelation functions are used to assess for white noise. But even when the underlying process is a white noise, several autocorrelations among ρ(1), • • • , ρ(r) may lie out of the interval [-1.96/ √ n, 1.96/ √ n]. The asymptotic independence property for variables Ξ(h) implies that, when sample size n is large, the number of observed autocorrelation functions out of this interval behaves as a Binomial B(r, 0.05) distribution. We developed an R function, based on the classical acf() function, that takes into account the multiple testing paradigm by incorporating the binomial exact test and Sidak correction for white noise diagnosis. Our function is called acfG() and is available at our website.

As an example, we simulate a white noise, and compute sample autocorrelation functions with function acfG(). Figure 1 shows that 3 values lie out of the interval [-1.96/ √ n, 1.96/ √ n], plotted with blue dashed lines. But binomial exact test (p-value = 0.1159) confirms that such a number remains consistent with white noise hypothesis. Moreover, a second set of interval, computed with Sidak correction, is provided, and plotted with red dotted lines. If at least one sample autocorrelation function lies out this global interval, then white noise hypothesis is rejected. In our simulation, white noise diagnosis is confirmed, both by graphics and by Binomial's test. Let (Z t ) t be a stochastic process such that Z t = d j=0 a j t j + B t , where a d = 0 and (B t ) t is (SN) satisfying Hypotheses (H1) to (H3). Then

Ξ(h) IP ------→ n→+∞ 1 , ∀ h = 0 .
Proof is given in Appendix A. Figure 2 (Left) illustrates the slow decreasing behavior of sample autocorrelation functions when time series are driven by a deterministic (DetS,2) trend. Let (Z t ) t be a stochastic process such that ∆ d (Z t ) = E t , with Z t = 0, for any t ≤ 0 and and (B t ) t is (SN) satisfying Hypotheses (H1) to (H3). Then

Ξ(h) IP ------→ n→+∞ 1 , ∀ h = 0 . Proof is given in Appendix B.

Trend-nature identification

In the previous section, we showed that the autocorrelation functions, computed from variables Z 1 , • • • , Z n , have a particular asymptotic behavior in presence of a trend. But Figure 2 illustrates that the behavior is similar either for a deterministic or a stochastic trend. Consequently, a deeper study has to be led in order to specify the type of trend. We explore Dickey-Fuller unit-root testing diagnosis.

Dickey-Fuller tests failure

In [START_REF] Dickey | Distribution of the Estimators for Autoregressive Time Series with a Unit Root[END_REF], Dickey and Fuller introduced unit root tests adapted to models M 1 , M 2 and M 3 . Every time the null and the alternative hypothesis are mathematically expressed in the same way:

(H 0 ) : ρ = 1 against (H 1 ) : |ρ| < 1.
But the alternative hypothesis interpretation depends on the considered model.

(M 1 ) (H 1 ) : (Z t ) t is a stationary and centered AR(1) process; (M 2 ) (H 1 ) : (Z t ) t is a stationary not-centered AR(1) process; (M 3 ) (H 1 ) : (Z t ) t is (linear-)trend stationary (TS).

And test statistics do not have the same expression, possibly leading to opposing conclusions, even on the same data. To distinguish between the null or the alternative hypothesis, one has to use the suitable test statistics, adapted to every model.

In [START_REF] Dickey | Distribution of the Estimators for Autoregressive Time Series with a Unit Root[END_REF], Dickey and Fuller also developed joined tests :

Test Φ 1 under (M 2 ) (H 0 ) : (a 0 , ρ) = (0, 1) ; Test Φ 2 under (M 3 ) (H 0 ) : (a 0 , a 1 , ρ) = (0, 0, 1) ; Test Φ 3 under (M 3 ) (H 0 ) : (a 0 , a 1 , ρ) = (a 0 , 0, 1) .

Simulation study

We perform some simulations to explore the behavior of Dickey-Fuller based-tests, according to data generating process. We set (E t ) t as a white noise with variance σ 2 E . We consider processes (Z t ) t successively driven by one of the following models :

(WN) Z t = a 0 + E t called "white noise with constant", (DetW,1)

Z t = a 0 + a 1 t + E t called "linear-trend stationary", (DetW,2) Z t = a 0 + a 1 t + a 2 t 2 + E t called "quadratic-trend stationary", (StoW,1) ∆(Z t ) = E t called "difference stationary", (StoW,2) ∆ 2 (Z t ) = E t called "second-order difference stationary".
Models (StoW,1) and (StoW,2) do contain a unit root and hence are not stationary, whereas models (DetW,1) and (DetW,2) are not stationary although they do not contain any unit root. Logically, Dickey-Fuller test should not reject the null hypothesis for almost realizations driven from models (StoW,1) and (StoW,2), precisely with a rate (1-α)%, where α stands for the significance level. Reciprocally, under (DetW,1) and (DetW,2) models, the null hypothesis should be rejected. For simulations, we set n = 300, a 0 = 5, a 1 = 1, a 2 = 1. Random generations of E t were taken from Gaussian centered variables with standard deviation σ E , a fixed value among {0.5, 1, 3, 5, 10, 20, 30, 50, 100, 200, 300, 500}. We ran 5 000 simulations from every model, and successively applied Dickey-Fuller tests, dedicated to models M 1 , M 2 and M 3 . Table 1 presents the results when σ E = 10, and contains the rate of null-hypothesis rejecting, crossing every data generating process (DGP) with every stationarity test, under a standard level α = 5%.

It appears that Dickey-Fuller diagnosis is accurate for (WN) and (StoW,1) models. As expected, only the convenient model (M 3 ), provides a correct answer for (DetW,1). And in Figure C1 from Appendix C, we illustrate that diagnosis is unclear for (DetW,1) realizations when using any test under models (M 1 ) or (M 2 ). Indeed, results vary greatly according to parameter σ E , taking value in the set {0.5, 1, 3, 5, 10, 20, 30, 50, 100, 200, 300, 500}. Surprisingly, unit root is far to be correctly detected in (StoW,2) realizations. Finally, diagnosis is mainly incorrect for model (DetW,2), since unit roots are systematically detected, whereas we have ρ = 0. Thus, Dickey-Fuller-based tests fail in diagnosing unit root for quadratic trends models.

Dickey-Fuller-based strategies

Several strategies based on Dickey-Fuller tests have been developed ([8, 20, 10]). From Table 1, we can deduce that all the strategies permit to discriminate between first-order deterministic or stochastic trend, but they do not plan to integrate second-order trends. Thus, the most simple strategy proposed in [START_REF] Dickey | Unit Roots in Time Series Models: Tests and Implications[END_REF] falsely classifies (DetW,2) model as a random walk, whereas Perron predicts a linear-trend stationary process [START_REF] Perron | Trends and Random Walk in Macroeconomics Time Series : Further Evidence from a New Approach[END_REF]. The most advanced strategy given in [START_REF] Dolado | Cointegration and unit roots[END_REF] nearly identifies (DetW,2) processes, by describing them as

∆(Z t ) = β 0 + β 1 t + E t , instead of ∆(Z t ) = β 0 + β 1 t + ∆(E t ), where ∆(Z t ) = Z t -Z t-1 is the differentiated series.
But diagnosis is mainly incorrect for (StoW,2) processes. Surprisingly, only the most simple strategy suggested in [START_REF] Dickey | Unit Roots in Time Series Models: Tests and Implications[END_REF] detects unit root in half realizations, and predicts either a linear-trend stationary process or a stationary AR(1) process otherwise. And other strategies predict lineartrend stationary process. Consequently, it appears useful to elaborate a new strategy to identify not only one-degree trends, but also higher-degree ones. (stationarity around a linear deterministic trend). Here, we refer to the level-stationarity test.

Simulation study for OPP and KPSS tests

Models (StoS,1) and (StoS,2) do contain a unit root and hence are not stationary, whereas models (DetS,1) and (DetS,2) are not stationary although they do not contain any unit root. Logically, KPSS's test should reject the null hypothesis for most realizations driven from all these models. Hence KPSS's test does not appear to be a good candidate for trend-nature identification. Nonetheless, KPSS's test reveals heterogeneous behaviors when applied to the differentiated series ∆(Z t ). Indeed, processes (Z t ) driven from models (DetS,1) and (StoS,1) do become stationary as soon as they are differentiated.

(SN)

Z t = B t =⇒ ∆(Z t ) = B t -B t-1 , (4)
(DetS,1)

Z t = a 0 + a 1 t + B t =⇒ ∆(Z t ) = a 1 + B t -B t-1 , (5)
(DetS,2)

Z t = a 0 + a 1 t + a 2 t 2 + B t =⇒ ∆(Z t ) = a 1 + 2 a 2 t + B t -B t-1 , (6)
(StoS,1)

Z t = Z t-1 + B t =⇒ ∆(Z t ) = B t , (7)
(StoS,2)

Z t = 2 Z t-1 -Z t-2 + B t =⇒ ∆(Z t ) = ∆(Z t-1 ) + B t . (8)
Moreover OPP's test should reject the null hypothesis only for realizations driven from models (DetS,1) and (DetS,2), vice versa for (StoS,1) and (StoS,2). Applied to the differentiated series, OPP's test should not reject the null hypothesis only for realizations initially driven from models (StoS,2). We ran the suggested testing procedure on 5 000 simulations using every data generating process among (SN), (DetS,1), (DetS,2), (StoS,1) and (StoS,2), when the associated stationary noise is either a (WN), or (SN) such as a MA(2) or an ARMA(1,1) centered, causal and invertible process. We took again parameters values as a 0 = 5, a 1 = 1, a 2 = 1, n = 300. Table 2 presents the null-hypothesis rejection rate when B t = E t where σ E varies in {0.5, 1, 3, 5, 10, 20, 30, 50, 100, 200, 300, 500}.

Table 2 shows that KPSS and OPP tests perform as expected, not only for simulations driven from one-order trends, but also for quadratic trends. Moreover, results remain identical for any value of parameter σ E , varying from 0.5 to 500, such as shown in Figure D1, Appendix D. We also present results for simulations with underlying stationary noises (B t ) t , such as a MA(2) and an ARMA [START_REF] Akaike | Information theory as an extension of the maximum likelihood principle[END_REF][START_REF] Akaike | Information theory as an extension of the maximum likelihood principle[END_REF]. In this case, both Supplementary Figures S1 andS2 show the same behavior for KPSS and OPP tests, whatever σ E value. We ran again 5 000 simulations under every data generating process among (WN), (DetW,1), (DetW,2), (StoW,1) and (StoW,2), with a 0 = 5, a 1 = 1, a 2 = 1, n = 300 and σ E , taking value in the set {0.5, 1, 3, 5, 10, 20, 30, 50, 100, 200, 300, 500}. Then we applied successively tests i) to iv) to any simulation, using a risk α = 5%. In Table 3, every column sums the diagnosis percentage associated to the corresponding DGP. The percentage in bold refers to the expected diagnosis. For example, if the DGP is (WN), only two sets of responses were obtained to TDT : Alt/Alt/Null/Null for 95.247% of (WN) simulations (percentage written in bold), and Alt/Alt/Alt/Null for the other 4.753% simulations. Note that the two TDT diagnosis associated to (WN) only differ due to KPSS response on the initial series. The two obtained percentages are consistent with the Type I error for this test, when α is fixed to 5%. In the same way, if the DGP is (StoW,1), several responses are possible. Note that among tests i) to iv), only the first and the last one are led under the Null hypothesis. Consequently, when α = 5%, each of tests i) to iv) may approximately produce the Null response in 95% simulations, and the Alt response in the other 5%. This theoretically yields to percentages 95% × 95% = 90.25% (StoW,1)-simulations with response Null/ -/ -/Null 95% × 5% = 4.75% (StoW,1)-simulations with response Null/ -/ -/Alt 95% × 5% = 4.75% (StoW,1)-simulations with response Alt/ -/ -/Null 5% × 5% = 0.25% (StoW,1)-simulations with response Alt/ -/ -/Alt

We do observe almost these expected percentages by summing the corresponding TDT percentages in Table 3, column (StoW,1). For instance, in relation to the expected diagnosis Null/ -/ -/Null, we observed either Null/Alt/Alt/Null or Null/Alt/Null/Null diagnosis for (StoW,1)-simulations, with respective percentages 86.162% and 3.216%, summing to 89.378%, that is very close to the expected 90% percentage. But the interest of Table 3 lies is in its reverse reading. Let us associate a DGP to a TDT diagnosis. As an example, since none simulation under (WN), (DetW,1), (DetW,2) or (StoW,1) DGP led to responses Null/Null/Alt/Alt, then if one obtains such a response on its time series, this means that a (StoW,2) model is suitable. Alternately Alt/Alt/Alt/Null diagnosis could lead either to a (WN), a (DetW,1) or a (StoW,1) model. But referring to occurrence percentages, 99.993/(4.753 + 99.993 + 5.822) = 90.436% of simulations with TDT Alt/Alt/Alt/Null are produced by a (DetW,1) DGP, only 4.299% by a (WN) and 5.266% by a (StoW,1). Then a (DetW,1) model appears as the best candidate, but (WN) and (StoW,1) models can not be totally excluded. As shown in Figure E1, Appendix E, the relative occurrence of these three models for diagnosis Alt/Alt/Alt/Null remains the same, whatever σ E value in the set {0.5, 1, 3, 5, 10, 20, 30, 50, 100, 200, 300, 500}. In this precise case, even if a linear trend is present, the relevance of either a (WN) or a (DetW,1) model depends on the intensity of the trend in relation to the variance of the associated noise. The convenient choice can be ruled out by previously computing autocorrelation functions, with acfG() function. If several models remain acceptable, then we suggest to construct and compare them.

In order to evaluate the effect of autocorrelation on TDT strategy, we also run simulations with autocorrelated noises. In other words, we replaced E t either by a MA(2) process or by an ARMA(1,1) process, denoted as B t , see Supplementary. Percentage of TDT diagnosis associated to every Data Generating Process, driven by (SN), are given in Supplementary Tables S1 andS2. Percentage diagnosis remains similar regardless of the underlying model for B t . From Table 3 and Supplementary Tables S1 andS2, we suggest to associate each model with some sets of responses to TDT. More specifically, with diagnoses ranked by risk of occurrence : Figure E1 and Supplementary Figures S3 andS4 illustrate that this classification remains stable as σ E varies, whenever the underlying noise is (WN) or (SN). We implemented this diagnosis strategy as the R function trend.diag.tests(), with an argument nb.mod taking values among c("single", "multiple"), specifying either if only the main model is returned or if all possibilities are returned, even the most occasional one. In this case, all the suggested models can be constructed and compared.

• Alt/Alt/Null

Higher-degree trends

Actually, it is possible to detect higher-degree trends, either deterministic (DetS,d) or stochastic (StoS,d), with d = 3, 4, 5, by iterating OPP and KPPS tests on the successive differentiated series. More precisely,

Step 0 :

Compute sample autocorrelation functions with acfG() in order to distinguish between a white noise and a time series with a trend. If the series is driven by a trend, then run the following steps.

Step 1 :

Run OPP test on the given time series. We implemented this diagnosis strategy as the R function trend.diag.high(). When applying this strategy, a model is suggested, leading either to Equation 1 or 2, with parameters to be determined. In particular, process (B t ) t is rarely a white noise, and should rather be modeled by an ARMA(p,q) process. The validity of the global model has to be confirmed with residuals diagnosis. Table 4 shows that our strategy TDT applied to abortion data suggests a (StoS,2) model. Both functions trend.diag.tests() and trend.diag.high() confirm this diagnosis. Dickey-Fuller tests rather suggest a (StoS,1) model. Indeed, most Dickey-Fuller tests do not reject the null for the initial series, but they are unable to detect a unit root in the differentiated series. We construct both a (StoS,1) and a (StoS,2) model, as in Equation 2, by modeling the stationary process (B t ) t by the first clearly valid model among all ARMA(p,q) with p, q ≤ 2, sorted by minimizing Schwarz's Bayesian Criterion [START_REF] Schwartz | Null hypothesis rejection rate for Dickey-Fuller tests, with respect to the underlying generating process used for simulations[END_REF]:

• (B t ) t is AR(1) for (StoS,2) model, suggested by our strategy TDT.

• (B t ) t is AR(2) for (StoS,1) model, suggested by Dickey-Fuller tests.

In Table 5, we compare both models relevance in terms of information criteria such as AIC, BIC, AICc [START_REF] Akaike | Information theory as an extension of the maximum likelihood principle[END_REF][START_REF] Schwartz | Null hypothesis rejection rate for Dickey-Fuller tests, with respect to the underlying generating process used for simulations[END_REF][START_REF] Hurvich | Bias of the corrected AIC criterion for underfitted regression and time series models[END_REF] and prediction criterion computed between the observed series and predictions for the last 4 values(almost 10%), such as Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE). Table 5 shows that the (StoS,2) model provided by our strategy is the most suitable.

Figure 4 plots forecasts for abortion data. Residuals being normally distributed, we plot both forecasts and 80% and 95% prediction intervals, computed from gaussian quantiles.

Evolution of atmospheric CO2 concentration

Since 1959, atmospheric CO2 concentrations (ppm) has been measured monthly in situ air measurements, at Mauna Loa, Observatory, Hawaï see [START_REF] Keeling | Exchanges of atmospheric co2 and 13 co2 with the terrestrial biosphere and oceans from 1978 to 2000. i. global aspects[END_REF]. Figure 5 reveals that the global average concentration of atmospheric carbon dioxide has a clear increasing trend, and also a seasonal monthly component.

Our strategy does not take into account seasonal component, then we deseasonalize, by regressing the CO2 series on the seasonal dummy variables and by retaining the residuals from this regression. Autocorrelation functions plotted in Figure 6 (Right) show that the remaining series is driven by a trend. From Table 6, we see that Dickey-Fuller tests applied to the deseasonalized series clearly suggests a (StoS,1) model, whereas our TDT strategy produces responses Alt/Alt/Alt/Alt to tests i) to iv), suggesting rather a (DetS,2) model. We construct both a (StoS,1) and a (DetS,2) model, as in Equations 1 and 2, by modeling the stationary process (B t ) t by the first valid model among all SARMA(p,q)(P,Q) [START_REF] Hardy | An introduction to the theory of numbers, 4 th with corrections[END_REF] with p, q, P, Q ≤ 2, sorted by minimizing Schwarz's Bayesian Criterion [START_REF] Schwartz | Null hypothesis rejection rate for Dickey-Fuller tests, with respect to the underlying generating process used for simulations[END_REF]: [START_REF] Hardy | An introduction to the theory of numbers, 4 th with corrections[END_REF] for (DetS,2) model, suggested by our strategy TDT.

• (B t ) t is SARMA(1,2)(1,1)
• (B t ) t is SARMA(1,0)(1,1) [START_REF] Hardy | An introduction to the theory of numbers, 4 th with corrections[END_REF] for (StoS,1) model, suggested by Dickey-Fuller tests.

In Table 7, we compare both models relevance in terms of information criteria such as AIC, BIC, AICc [START_REF] Akaike | Information theory as an extension of the maximum likelihood principle[END_REF][START_REF] Schwartz | Null hypothesis rejection rate for Dickey-Fuller tests, with respect to the underlying generating process used for simulations[END_REF][START_REF] Hurvich | Bias of the corrected AIC criterion for underfitted regression and time series models[END_REF] and prediction criterion computed between the observed series and predictions for the last six years (almost 10%), such as RMSE and MAPE. Table 7 shows that the (DetS,2) model provided by our strategy is the most suitable.

Figure 7 shows that forecasts for CO2 atmospheric concentration maintain the same trajectory, with great accuracy. Indeed, prediction intervals are so thin that they are hardly visible.

Conclusion

We present a strategy to detect and identify trend component in time series. We recall that, as a first analysis, visualizing time series plot is indispensable, since it may already suggest the presence of a trend. Then trend can be confirmed by analyzing autocorrelation functions with acfG(). Next, whence a trend is detected, it remains to identify its nature. Indeed deterministic or stochastic trends do not produce the same forecasts.

From Dickey-Fuller tests behavior, it appears useful to elaborate a new strategy to identify not only one-degree trends, but also higher-degree ones. For instance diagnosis for data generated from a (StoS,2) model results misleading. Indeed, the tests being ran on a single realization, the conclusion provided by each test will not necessary correspond to the majority response, that was obtained over a large number of simulations. Thus realizations of a (StoS,2) process can be falsely inferred as generated from either a (DetS,1) or a (DetS,2) or a (StoS,1) model. Then we propose a new strategy, involving other stationarity tests.

In this paper, we suggest a strategy based on OPP and KPSS tests, called TDT, and implemented in diag.trend.tests() and diag.trend.high() functions, in order to select between either a (DetS,d) or a (StoS,d) generating process. Our TDT strategy was assessed on simulations, and compared with Dickey-Fuller tests diagnosis. Of course, both procedures, TDT and Dickey-Fuller tests, may result in the same model suggestion. But diagnosis can be different, especially for time series with a quadratic trend, that reduces Dickey-Fuller tests reliability. Hence when both procedures suggest different models, both model candidates should be computed, validated and compared. Applied on two real data sets, our automatic TDT strategy provides a better model than the one provided using the classical Dickey-Fuller tests. Thus we recommend TDT strategy to improve and facilitate modeling procedure.

In order to study the asymptotic behavior of variables Ξ(h), estimators of the theoretical autocorrelation function for stationary square-integrable processes, we first compute variables Γ(h), estimators of the theoretical autocovariance function cov(Z t , Z t+h ). Definition A.2. From random variables (Z 1 , • • • , Z n ), we define the autocovariance function as

Γ(h) = n-h k=1 (Z k+h -Z)(Z k -Z) n , |h| ≤ n.
Using Equation A3, we can express Γ(h), by summing from k = 1 to k = n -h the products of

Z k -Z = d j=0 a j k j - a d d + 1 n d + o(n d ) + B k -B with Z k+h -Z = d j=0 a j (k + h) j - a d d + 1 n d + o(n d ) + B k+h -B .
We have to study every term, and clarify its asymptotic behavior.

a) First, we consider all product terms involving either B or n-h k=1 Bk n , or n-h k=1 Bk+h n . Let us denote by T a the sum of all these terms. Since (B t ) t is (SN) satisfying Hypotheses (H1) to (H3), then we can apply the weak law of large numbers for moving averages (see [START_REF] Pj Brockwell | Time series : Theory and methods[END_REF] do multiply either themselves or polynomials with degree ≤ d, then these product terms converge IP to 0, as soon as they are divided by n d .

T a = o IP (n d ) (A4)
b) Next, we study the behavior of the following terms :

T b,1 = n-h k=1 B k B k+h n , T b,2 = n-h k=1 d j=0 a j k j B k+h n , T b,3 = n-h k=1 d j=0 a j (k + h) j B k n .
Applying again the weak law of large numbers for moving averages (see [START_REF] Pj Brockwell | Time series : Theory and methods[END_REF], Prop 7.3.5), we obtain the IP-convergence of term T b,1 to γ B (h), and then

T b,1 = o IP (n d )
On the other hand, we need Cauchy-Schwarz's inequality to study terms T b,2 and T b,3 in the same way. We get

T b,2 = d j=0 a j n-h k=1 k j √ n × B k+h √ n ≤ d j=0 a j   n-h k=1 k 2j n 1/2 × n-h k=1 B 2 k+h n 1/2   ≤ n -h n n-h k=1 B 2 k+h n -h 1/2 d j=0 a j n 2j 2j + 1 + o(n 2j ) 1/2
.

The weak law of large numbers for moving averages and Prop 7. We get the same formula for T d3 . In addition, Equation A2 also provides i) to iv) to any simulation, using a risk α = 5%. Given a data generating process, several diagnosis were observed, but not all of them. In the paper, Table 3 shows TDT diagnosis, when gathering simulations associated to a fixed DGP, whatever σ E value in {0.5, 1, 3, 5, 10, 20, 30, 50, 100, 200, 300, 500}. Here, Figure E1 illustrates the stability of the classification associated to every model as σ E varies.

Note that the classification remains stable when σ E keeps growing. But when noise intensity is too high in relation to the linear coefficient a 1 , the trend becomes imperceptible, and KPSS test sometimes fails to reject the null for several (detT,1) simulations with σ E > 300. Whereas Alt/Alt/Alt/Null diagnosis is accurately associated to almost 99.9% of (detT,1) simulations while σ E ≤ 300, 83.6% of (detT,1) simulations with σ E = 500 have the convenient diagnosis Alt/Alt/Alt/Null, but the 16.4% other simulations are associated to diagnosis Alt/Alt/Null/Null, that is accurate for (WN). And the confusion between (detT,1) and (WN) naturally increases with σ E . In this case, the true model (detT,1) might no longer be the most suitable for the series.

  and where parameters b j (j ∈ Z) satisfyj∈Z |b j | < ∞ j∈Z b j = 0

Figure 1 .

 1 Figure 1.: Sample autocorrelation functions for a white noise simulation.

2. 2 .

 2 Autocorrelation functions for (DetS,d)-time series Theorem 2.2.

2. 3 .

 3 Autocorrelation functions for (StoW,d)-time series Theorem 2.3.

Figure 2 (

 2 Right) illustrates the slow decreasing behavior of sample autocorrelation functions when time series are driven by a stochastic (StoW,2) trend.

Figure 2 .

 2 Figure 2.: Sample autocorrelation functions for simulations with either a deterministic (Left) or a stochastic trend (Right).

4. 2 . 1 .

 21 Terminated pregnancies in Québec, Canada Since 1969, abortion is legal in Canada at all stages of pregnancy. Since 1971, la Régie de l'assurance maladie du Québec provides the number of voluntarily terminated pregnancies in Québec, Canada, plotted in Figure 3 (Left). Autocorrelation functions, given in Figure 3 (Right), suggest that data are driven by a trend.

Figure 3 .

 3 Figure 3.: Abortion data evolution (Left) and autocorrelation functions (Right).

Figure 4 .

 4 Figure 4.: Predictions for abortion series with TDT model, where prediction intervals colored in steel blue (resp. light grey) represent 80% (resp. 95%) confidence level.

Figure 5 .

 5 Figure 5.: CO2 atmospheric concentration evolution.

Figure 6 .

 6 Figure 6.: CO2 deseasonalized series evolution (Left) and associated autocorrelation functions (Right).

Figure 7 .

 7 Figure 7.: Predictions for CO2 atmospheric concentration with TDT model, where prediction intervals colored in steel blue (resp. light grey) represent 80% (resp. 95%) confidence level.

2 d 1 )

 21 3.5 in[START_REF] Pj Brockwell | Time series : Theory and methods[END_REF] imply the IPconvergence of the left hand term to γ B (0) 1/2 = σ B . In addition, the right hand term is o(n d+1 ). Consequently,T b = T b,1 + T b,2 + T b,3 = o IP (n d+1 ) (A5)c) Let us denote by T c all the product terms involving o(n d ), not studied yet. From Equation A1, we get that o(n d ) multiplies polynomials with degree ≤ d. Then all product terms converge to 0, as soon as they are divided by n d . Consequently, we haveT c = o(n 2d ) . (A6) d) It remains to specify terms with polynomial products, in order to explicit the leading coefficient. Let us consider T d,1 = a All the terms T d,1 to T d,4 contain a leading term, associated to degree 2d. There is nothing to do for T d,1 . Equation A2 provides T d,2 = -2 n 2d + o(n 2d ) .

  n 2d + o(n 2d ) .

Finally,T

  d = T d,1 + T d,2 + T d,3 + T d,4

Figure

  Figure E1.: (First part) Percentage of simulations with the associated diagnosis for TDT tests, versus every data generating process. All the simulations, driven with a (WN), where σ E takes successive values in {0.5, 1, 3, 5, 10, 20, 30, 50, 100, 200, 300}, are gathered.

Figure

  Figure E1.: (Last part) Percentage of simulations with the associated diagnosis for TDT tests, versus every data generating process. All the simulations, driven with a (WN), where σ E takes successive values in {0.5, 1, 3, 5, 10, 20, 30, 50, 100, 200, 300}, are gathered.

Table 1 .

 1 : Null-hypothesis rejection rate (%) in Dickey-Fuller tests, when σ E = 10.

		DGP a
	Test	(WN) (DetW,1) (DetW,2) (StoW,1) (StoW,2)
	Test ρ under (M 1 )	100	0	0	5.06	0
	Test ρ under (M 2 )	100	0	0	4.96	12.88
	Test ρ under (M 3 )	100	100	0	5.04	32.4
	Test Φ 1 under (M 2 )	100	0	100	4.86	95.46
	Test Φ 2 under (M 3 )	100	100	100	4.92	99.08
	Test Φ 3 under (M 3 )	100	100	100	5.1	90.4

a Data Generating Process.

Table 2 .

 2 : Null-hypothesis rejection rate (%) for KPSS and OPP stationarity tests, for DGP simulations when the underlying process is a white noise (E t ) t . We vary σ E on the set {0.5, 1, 3, 5, 10, 20, 30, 50, 100, 200, 300, 500}. The final rejection rate is computed as the average of the rejection rates obtained for each σ E .

	) stationary

3.2. Accurate behavior for OPP and KPSS tests 3.2.1. Additional stationarity tests 3.2.1.1. OPP stationarity test.

In

[START_REF] Ouliaris | Testing for a Unit Root in the Presence of a Maintained Trend[END_REF]

, Ouliaris, Park, and Phillips generalized Dickey-Fuller unit-root tests ρ to models M 3,d with polynomial trends, where d = 2, 3, 4 or 5. We denote this general test by OPP. Note that the invariance principle for partial sums, required in OPP test, applies to (SNassociated to an underlying (SN). Contrary to Dickey-Fuller tests, KPSS test takes the presence of unit root as the alternative hypothesis, and the stationarity as the null hypothesis. Actually, KPSS test can consider as null-hypothesis either level-stationarity or trend-stationarity a Data Generating Process.

  OPP test to series Z t ; ii) OPP test to series ∆(Z t ) ; iii) KPSS test to series Z t ; iv) KPSS test to serie ∆(Z t ). We call Trend Diagnosis Tests (TDT) the set of responses to tests i) to iv) computed on a time series. Let us denote by Null, the case where the null hypothesis can not be rejected, and by Alt otherwise. From Equations 4 to 8, we provide the expected diagnosis led by TDT strategy for every model. Hence, (SN) expected diagnosis : Alt/Alt/Null/Null ,

	4. Applications
	4.1. A new strategy for trend identification
	4.1.1. TDT strategy
	We suggest to apply the following tests successively :
	i) (DetS,1) expected diagnosis : Alt/Alt/Alt/Null ,
	(DetS,2) expected diagnosis : Alt/Alt/Alt/Alt ,

(StoS,1) expected diagnosis : Null/Alt/Alt/Null , (StoS,2) expected diagnosis : Null/Null/Alt/Alt .

Table 3 .

 3 : Percentage of Trend Diagnosis Tests (TDT) associated to every Data Generating Process (DGP)when σ E takes values in {0.5, 1, 3, 5, 10, 20, 30, 50, 100, 200, 300}.

		DGP a
	TDT b	(WN)	(DetW,1) (DetW,2) (StoW,1) (StoW,2)
	Alt/Alt/Alt/Alt	0	0	100 c	0.438	0
	Alt/Alt/Null/Alt	0	0	0	0	0
	Alt/Null/Alt/Alt	0	0	0	0	0
	Alt/Null/Null/Alt	0	0	0	0	0
	Null/Alt/Alt/Alt	0	0	0	4.273	6.213
	Null/Alt/Null/Alt	0	0	0	0.005	0
	Null/Null/Alt/Alt	0	0	0	0	90.427
	Null/Null/Null/Alt	0	0	0	0	0.007
	Alt/Alt/Alt/Null	4.753	99.993	0	5.822	0
	Alt/Alt/Null/Null	95.247	0.007	0	0.084	0
	Alt/Null/Alt/Null	0	0	0	0	0
	Alt/Null/Null/Null	0	0	0	0	0
	Null/Alt/Alt/Null	0	0	0	86.162	0.089
	Null/Alt/Null/Null	0	0	0	3.216	0
	Null/Null/Alt/Null	0	0	0	0	3.220
	Null/Null/Null/Null	0	0	0	0	0.015
	Total percentage	100	100	100	100	100

a Data Generating Process b Trend Diagnosis Tests c Bold font highlights the expected TDT diagnosis associated to every DGP.

  If the null is rejected, we identify a (DetS,d) model, otherwise (StoS,d), with d ≥ 1. It remains to precise d. If a (StoS,d) model is detected in Step 1, differentiate the current time series, and apply OPP test. Iterate this step until the null is rejected. Then d corresponds to the number of necessary differentiations. Case 2b : If a (DetS,d) model is detected in Step 1, differentiate the current time series, and apply KPSS test. Iterate this step until the null is rejected. Then d corresponds to the number of necessary differentiations.

	Step 2 :
	Case 2a :

Table 4 .

 4 : p-values provided by several tests on the initial and the differentiated abortion series.

		Test
	Series	OPP	KPSS	ρ under (M 1 ) ρ under (M 2 ) ρ under (M 3 )
	Zt	0.2	0.01	0.9784	0.011	0.99
	∆(Zt) 0.0852	0.02	0.01	0.01	0.01
	4.2. Application on data

Table 5 .

 5 : Models comparison for abortion series.

		Criterion
	Model	AIC	AICc	BIC	RMSE MAPE
	TDT a : (StoS,2) with AR(1) 695.4	695.7	698.9	206.2	0.731
	DF b : (StoS,1) with AR(2)	710.1	710.8	715.4	382	1.469

a Trend Diagnosis Tests. b Dickey-Fuller tests.

Table 6 .

 6 : p-values provided by several tests on CO2 deseasonalized series.

		Test
	Series	OPP KPSS	ρ under (M 1 ) ρ under (M 2 ) ρ under (M 3 )
	Zt	0.01	0.01	0.9791	0.99	0.5465
	∆(Zt)	0.01	0.01	0.01	0.01	0.01

Table 7 .

 7 : Models comparison for CO2 deseasonalized series.

	Criterion

a Trend Diagnosis Tests. b Dickey-Fuller tests.

Appendix A. Sample autocorrelation functions behavior for (DetS,d) models -Proof of Theorem 2.2

We consider the polynomial case with degree d ≥ 1. Let us define as S j (n) the sum of the j-th power of the first n integers:

From Faulhaber's formula, detailed in [START_REF] Conway | The book of numbers[END_REF], we know that

where B p are rational numbers called Bernoulli numbers.

Lemma A.1. For any j ∈ IN * , the sum of the j-th power of the first n integers, S j (n), is a (j + 1)-degree polynomial with leading coefficient 1 j+1 . More precisely,

where o(f ) is one of the Landau symbols, as defined in [START_REF] Hardy | An introduction to the theory of numbers, 4 th with corrections[END_REF], p. 7-8. o(f ) is called "little-O of f ", and expresses the convergence to 0 of a given function, when it is divided by f .

Then, We just give the proof for d = 1, since the general case can be deduced using the decomposition technique suggested in [START_REF] Chan | Limiting distributions of least squares estimates of unstable autoregressive processes[END_REF].

We first differentiate the initial series at a given lag h :

Let L denotes de lag operator i.e. LX t = X t-1 , hence V k,h can be written as

where A is the following polynomial

where

Consequently the process (V t,h ) is also a moving average, by straightforward calculations we can show that (V t,h ) satisfies Hypotheses (H1) to (H3).

Let us set

where [x] is the integer part of x. Then using [START_REF] Boutahar | Comparison of non-parametric and semi-parametric tests in detecting long memory[END_REF] and Theorem 2 in [START_REF] Davydov | The invariance principle of stationary processes, Theory of probability and its application XV[END_REF], we get the weak convergence :

V n,h (.)

where D[0, 1] is the set of càdlàg functions with Skorokhod topology, and where (W t ) t is a standard Brownian motion. Moreover f B and f V are the spectral densities associated to processes (B t ) and (V t,h ) :

We also define

We recall that

Then by weak convergence continuity, we obtain that

Autocorrelation function definition was given in Equation 3. We deduce that

Then,

Prokhorov theorem ([21]

) permits to deduce that n ( Ξ(h) -1) = O IP (1).

Appendix C. Boxplot of null-hypothesis rejection rate when σ E varies -Complement to Table 1 In the main paper, Table 1 shows results for σ E = 10. Here, Figure C1 displays results when all the simulations with σ E taking successive values in Appendix D. Boxplot of null-hypothesis rejection rate when σ E varies, and when noise is (WN) -Complement to Table 2 In the main paper, Table 2 shows that KPSS and OPP tests perform accurately on (WN), (DetW,1), (DetW,2), (StoW,1) and (StoW,2) simulations. In Table, 2, σ E successively takes values in the set {0.5, 1, 3, 5, 10, 20, 30, 50, 100, 200, 300, 500} and the final rejection rate is computed by gathering all the simulations obtained for each σ E . Here, Figure D1 illustrates the stability of testing procedure for every data generating process as σ E varies. Note that an outlier is observed when applying KPSS test to (DetS,1) simulations. This means that KPSS test generally rejects the null hypothesis, as expected, in almost all cases. Actually, KPSS test sometimes fails to reject the null for several (DetS,1) simulations with σ E = 500, that is to say when noise intensity is too high in relation to the linear coefficient a 1 , so that the trend becomes imperceptible. Thus σ E = 500 is above the high-limit for noise intensity. 3 We ran 5 000 simulations under every data generating process among (WN), (DetW,1), (DetW,2), (StoW,1) and (StoW,2), with a 0 = 5, a 1 = 1, a 2 = 1, n = 300 and σ E , taking value in the set {0.5, 1, 3, 5, 10, 20, 30, 50, 100, 200, 300}. Then we applied successively tests


Ameĺie Beduer

†, ‡,∥ Florent Seichepine, †,⊥,@,# Emmanuel Flahaut

†,⊥,@,# Isabelle Loubinoux

∇,○ Laurence Vaysse

Christophe Vieu

Elucidation of the Role of Carbon Nanotube Patterns on the Development of Cultured Neuronal Cells

Carbon nanotubes (CNTs) promise various novel neural biomedical applications for interfacing neurons with electronic devices or to design appropriate biomaterials for tissue regeneration. In this study, we use a new methodology to pattern SiO 2 cell culture surfaces with double-walled carbon nanotubes (DWNTs). In contrast to homogeneous surfaces, patterned surfaces allow us to investigate new phenomena about the interactions between neural cells and CNTs. Our results demonstrate that thin layers of DWNTs can serve as effective substrates for neural cell culture. Growing neurons sense the physical and chemical properties of the local substrate in a contact-dependent manner and retrieve essential guidance cues. Cells exhibit comparable adhesion and differentiation scores on homogeneous CNT layers and on a homogeneous control SiO 2 surface. Conversely, on patterned surfaces, it is found that cells preferentially grow on CNT patterns and that neurites are guided by micrometric CNT patterns. To further elucidate this observation, we investigate the interactions between CNTs and proteins that are contained in the cell culture medium by using quartz crystal microbalance measurements. Finally, we show that protein adsorption is enhanced on CNT features and that this effect is thickness dependent. CNTs seem to act as a sponge for culture medium elements, possibly explaining the selectivity in cell growth localization and differentiation.

INTRODUCTION

Carbon Nanotubes (CNTs) have different properties, such as extraordinary strength, toughness, electrical conductivity, and specific surface area, which make them excellent candidates for interfacing with neural systems for the development of biocompatible, durable, and robust neuroprosthetic devices. CNTs have been studied as possible substrates for neuronal cell adhesion. [START_REF] Hu | Polyethyleneimine functionalized single-walled carbon nanotubes as a substrate for neuronal growth[END_REF][START_REF] Hu | Chemically functionalized carbon nanotubes as substrates for neuronal growth[END_REF] Studies have demonstrated that CNT surfaces act as an extremely efficient biocompatible substrate on which neurons adhere and proliferate. CNT homogeneous films have also been shown to influence neurite branching, length, and density. [START_REF] Webster | Nano-biotechnology: Carbon nanofibres as improved neural and orthopaedic implants[END_REF][START_REF] Keefer | Carbon nanotube coating improves neuronal recordings[END_REF] In vivo, CNTs have been very recently investigated as scaffold for stem cell therapy because of their favorable electrical properties, and it has been shown that CNTs could reduce the size of a damaged brain area of a rat. [START_REF] Lee | Amine-modified single-walled carbon nanotubes protect neurons from injury in a rat stroke model[END_REF] In addition to dx.doi.org/10.1021/la304278n | being biocompatible, [START_REF] Hu | Chemically functionalized carbon nanotubes as substrates for neuronal growth[END_REF][START_REF] Webster | Nano-biotechnology: Carbon nanofibres as improved neural and orthopaedic implants[END_REF][START_REF] Smart | The biocompatibility of carbon nanotubes[END_REF][START_REF] Shein | Engineered neuronal circuits shaped and interfaced with carbon nanotube microelectrode arrays[END_REF] CNTs are electrically conducting (they can behave like a metal or like a semiconductor, depending on their structure) and can be integrated into microfabricated devices. This panel of properties opens up new and exciting prospects. A recent development in CNT-based devices is represented by the design of multi-electrode arrays (MEA) to both electrically stimulate and record signals from neurons. Such MEAs were made by synthesizing islands of CNTs on lithographically defined, conductive substrates. Reports indicate the possibility of directly stimulating isolated neurons via CNTs in culture. [START_REF] Keefer | Carbon nanotube coating improves neuronal recordings[END_REF][START_REF] Gheith | Stimulation of neural cells by lateral currents in conductive layer-by-layer films of single-walled carbon nanotubes[END_REF] It was also demonstrated that coating of metal electrodes with CNTs led to low-impedance electrodes, [START_REF] Keefer | Carbon nanotube coating improves neuronal recordings[END_REF] and that CNT layers could be used as thin films to enhance or support neuronal excitation. [START_REF] Webster | Nano-biotechnology: Carbon nanofibres as improved neural and orthopaedic implants[END_REF][START_REF] Keefer | Carbon nanotube coating improves neuronal recordings[END_REF][START_REF] Lee | Amine-modified single-walled carbon nanotubes protect neurons from injury in a rat stroke model[END_REF][START_REF] Fabro | Spinal cord explants use carbon nanotube interfaces to enhance neurite outgrowth and to fortify synaptic inputs[END_REF] Chemically modified CNTs are also being used as microelectrode neural interfaces. CNTs can drastically increase the charge injection capacity which can permit the reduction of the electrode size, while providing biocompatible surface and reducing inflammation reaction. [START_REF] Gabay | Electro-chemical and biological properties of carbon nanotube based multi-electrode arrays[END_REF][START_REF] Jakobs | Engineered selforganization of neural networks using carbon nanotube clusters[END_REF] Lovat et al., by culturing neurons on multiwalled carbon nanotubes substrates, [START_REF] Lovat | Carbon nanotube substrates boost neuronal electrical signaling[END_REF] demonstrated that CNTs improved neural signal transfer while supporting dendritic elongation and cell adhesion. These authors observed an increase in the frequency of spontaneous post-synaptic currents and action potential firings, but details of the electrical features of such coupling are still lacking. Indeed, at the cell/CNTs interface, various effects are combined because of the complex and numerous properties of CNTs (conductivity, topography, chemical functionalization, hydrophobicity, and young modulus), but all these phenomena are still not completely understood.

From several studies involving cell culture on CNT layers, it is becoming clear that nanotopography exhibited by CNT layers stimulates behavioral changes in cells and plays a critical role in modifying cell development and proliferation, as well as the strength of adhesion to substrates. [START_REF] Hu | Polyethyleneimine functionalized single-walled carbon nanotubes as a substrate for neuronal growth[END_REF][START_REF] Hu | Chemically functionalized carbon nanotubes as substrates for neuronal growth[END_REF][START_REF] Mattson | Molecular functionalization of carbon nanotubes and use as substrates for neuronal growth[END_REF] Nano-and microtopography have been recognized as fundamental parameters in the design of bioinspired materials with controlled adhesion. [START_REF] Brunetti | Neurons sense nanoscale roughness with nanometer sensitivity[END_REF][START_REF] Beduer | Engineering of adult human neural stem cells differentiation through surface micropatterning[END_REF] However, the results presented in the literature for cell adhesion on nonstructured randomly rough surfaces, which represents the majority of natural surfaces, remain controversial. Some studies have documented a decrease in proliferation and adhesion with the increase in surface roughness, [START_REF] Kunzler | Systematic study of osteoblast response to nanotopography by means of nanoparticle-density gradients[END_REF] whereas others have shown the opposite by using others cell types. [START_REF] Gentile | Cells preferentially grow on rough substrates[END_REF] Most of the studies related to the culture of neural cells on CNT substrates used homogeneous surfaces and compared several surface types. [START_REF] Agarwal | Interfacing live cells with nanocarbon substrates[END_REF] However, patterned surfaces are a key design to study and interpret cell surface interactions. Only a few studies have reported experiments using patterned surfaces, for example, to study the cellular anchorage mechanism and the role of the nanoscale neurite-CNT interaction [START_REF] Sorkin | Process entanglement as a neuronal anchorage mechanism to rough surfaces[END_REF][START_REF] Abdullah | Aligned, isotropic and patterned carbon nanotube substrates that control the growth and alignment of chinese hamster ovary cells[END_REF][START_REF] Zhang | Guided neurite growth on patterned carbon nanotubes[END_REF][START_REF] Malarkey | Carbon nanotubes in neuroscience[END_REF] or to form engineered, electrically viable networks of neurons. [START_REF] Zhang | Guided neurite growth on patterned carbon nanotubes[END_REF] Micropatterning of CNTs using lithography to create self-organizing networks has been demonstrated. [START_REF] Jakobs | Engineered selforganization of neural networks using carbon nanotube clusters[END_REF] Gabay et al. used a polydimethylsiloxane stamp to deposit nanoparticules onto a quartz surface to generate catalytic islands upon which CNTs were grown via chemical vapor deposition. [START_REF] Fabro | Spinal cord explants use carbon nanotube interfaces to enhance neurite outgrowth and to fortify synaptic inputs[END_REF] When neurons were cultured onto these patterns, neurons adhered essentially on CNT regions. Sorkin et al [START_REF] Sorkin | Compact self-wiring in cultured neural networks[END_REF] used a similar approach and created patterned networks to study functional neuron networking with CNT discs serving as electrical connections for sensing or stimulation. The patterning approach is thus very useful for the generation and investigation of neural networks in vitro. However, most of these studies involving CNT patterning used substrates that were initially nonpermissive for cell attachment and growth. These particular conditions could explain the entire selectivity of cells for the CNT regions. In other words, it is the difference in cell attachment on the different surfaces that caused the preferential localization of the cells on CNT patterns. In the present work, we used an original patterning method that combined spray-coating of CNTs and microcontact printing to create a new configuration for cells, in which they could choose between two permissive surface types: SiO 2 and CNTs. We think that by using this kind of patterned substrate to culture neural cells, we can learn more about many phenomena involved in the interactions between CNTs and neural cells. To explain the observed highest affinity of cells for CNTs, we decided to investigate the adsorption of proteins contained in the cell culture medium to CNT surface. Indeed, several studies have demonstrated the role of nanotopography for protein adsorption. [START_REF] Kotov | Nanomaterials for neural interfaces[END_REF] To explore in details the role of CNTs on protein adsorption, we used a quartz-crystal microbalance (QCM). This instrument determines adsorption events on a piezoelectric quartz sensor through the measurement of the mechanical resonance frequency shift due to mass changes. [START_REF] Kanazawa | Frequency of a quartz microbalance in contact with liquid[END_REF][START_REF] Cooper | A survey of the 2001 to 2005 quartz crystal microbalance biosensor literature: Applications of acoustic physics to the analysis of biomolecular interactions[END_REF] By coating the sensing electrode with a thin CNT layer, we could adapt a QCM technique to quantify protein adsorption on this kind of surface and demonstrate a "porous reservoir effect" of CNT surfaces.

MATERIALS AND METHODS

Fabrication of Engineered Substrates for 2D Cell

Culture on CNT Patterns. Preparation of CNTs Suspensions. CNTs used in this work were prepared in-house by catalytic chemical vapor deposition (CCVD) of CH 4 at 1000°C (H 2 :CH 4 atmosphere), as reported earlier. [START_REF] Flahaut | Gram-scale CCVD synthesis of double-walled carbon nanotubes[END_REF] After catalyst removal by dissolution with a concentrated aqueous HCl solution, the CNTs were filtered and washed with deionised water (0.45 μm polypropylene filtration membrane) until neutral. The sample contained approximately 80% of the double-walled nanotubes (DWNTs), the rest being mainly single-walled nanotubes (∼15%) and triple-walled nanotubes. In order to obtain a stable CNT suspension and to avoid nanotubes agglomeration, we added a noncytotoxic dispersing agent, the caraboxymethylcellulose (CMC). DWNTs and CMC were mixed with ultrapure water with a mass ratio of 1:10 (CNTs: 0.1% and CMC: 1%). The CNT concentration of this suspension corresponds to 1 μg/ mL. The suspension was sonicated for 30 min (Sonics Vibra Cell) at a power of 150 W, while cooling in an ice bath. The mixture, appearing like a stable black suspension, however, was centrifuged (16 000 rpm for 30 minutes) in order to eliminate the largest agglomerates.

SiO 2 Substrate Patterning with CNTs using Soft Lithography. SiO 2 surfaces were patterned with CNTs using an adapted soft lithography method. The first step of this method consisted in inking a microstructured polydimethylsiloxane (PDMS) stamp with CNTs. Details of this process can be found in the Supporting Information. In order to generate dense and homogeneous layers of CNTs on the PDMS stamp surface, we used a spray-coating technique. This technique is cost-effective and scalable to a large area. [START_REF] Majumder | Insights into the physics of spray coating of SWNT films[END_REF][START_REF] Beduer | Multi-scale engineering for neuronal cell growth and differentiation[END_REF] Spraycoating was carried out for 10 min to cover a 2 × 2 cm microstructured PDMS surface and obtain a dense CNT layer, 100 nm in thickness. [START_REF] Beduer | A simple and versatile micor contact printing method for generating carbon nanotubes patterns on various substrates[END_REF] By controlling the spray-coating duration, it was possible to control the CNT layer thickness. Once the stamp has been covered with a thin CNT layer, it was brought into contact with an ethanol wet SiO 2 surface. The stamp was then peeled away, and we obtained reproducibly largescale patterned SiO 2 surfaces as shown in Figure 1. After fabrication, all surfaces were dipped in a water bath and heated at 60°C for 2 h to remove residual surfactant. We finally obtained micrometric patterns made of CNTs, having a controlled thickness on an SiO 2 surface, and suitable for cell culture experiments. The thin layers of CNTs patterned following this μCP process exhibited a metallic behavior. The resistivity is on the order of 0.1 Ω cm.

This process was applied to create all surfaces used in this work. For surfaces covered with a continuous CNT layer, we used a flat PDMS stamp, while a microstructured PDMS stamp was used to produce patterned surfaces. Two types of patterns were used in this work: the first one consisted of arrays of micrometric lines measuring 20 μm in width and 8 mm in length, and the second one consisted in micrometric lines measuring between 7 μm and 20 μm in width and turning at an angle of 90°. Various DWNT layer thicknesses (between 20 and 100 nm) were used, as mentioned for each experiment. Details of the process can be found in the Supporting Information.

2.2. Cell Culture. Neuroblastoma mouse cells (neuro2a) were grown in DMEM medium supplemented with 10% fetal bovine serum (PAA Laboratories) and 1% penicillin streptomycin (GIBCO) in petri dishes. Cells were subcultured twice a week and maintained at 37°C and 5% CO 2 . All reported experiments were performed using cells with less than twenty passages. Neuro2a cells were seeded on substrates at a density of 1.2 × 10 4 cells/cm 2 . Cells were incubated for 24 h and then differentiated by switching the DMEM with 10% serum medium to a medium of DMEM with 1% BSA (bovine serum albumin, Euromedex). Cells were then maintained at 37°C for 48 h before being fixed with 3.5% paraformaldehyde (Sigma) for characterization.

2.3. Sample Characterization. AFM Measurements. Atomic force microscopy (NanoWizard II, JPK instruments) was used for imaging CNTs and for measuring the height of CNT layers and the roughness of the patterned surface. All images and measurements were performed in a dry environment at room temperature in contact mode over a sampling of 50 × 50 μm 2 for roughness measurements. Multiple measurements were made in different scan directions. At least three images were recorded per sample. The images had a resolution of 256 × 256 pixels and were acquired at a scanning rate of about 1 Hz.

SEM Observations. A scanning electron microscope (SEM) (Hitachi S-4800) was used to characterize both patterned surfaces alone and patterned surfaces with cultured cells to investigate cell and neurite localization. Substrates were rinsed with a phosphate-buffered saline solution, dried, and directly used for SEM observation. For patterned surfaces with cultured cells, quantifications were realized by counting at least 300 cells on four randomly chosen fields (200× magnification).

Fluorescence Microscopy and Image Analysis. To observe cells and quantify their behavior, the actin cytoskeleton and cell nucleus were stained. Cells were fixed for 1 hour in a 3.7% (wt/vol) solution with formaldehyde, containing 15 mM sucrose at room temperature. After permeabilization with Triton X100, F-actin was stained with tetramethylrhodamine (TRITC)-conjugated phalloidin (molecular probes), at a dilution of 1:200. The cell nucleus was stained with 4′-6-diamidine-2-phenyl indole (DAPI) used in a 1:100 ratio. Fluorescence images were acquired using a Leica fluorescent microscope with a 20× objective. Images of at least 300 cells on four randomly chosen observation fields were captured and analyzed for each experimental condition. Experiments have been repeated at least three times. Neuro2a cells were considered as differentiated when they developed at least one neurite measuring at least 20 μm in length.

Statistical Analysis. Statistical analysis was based on at least three independent series of experiments. A t test or a one-way ANOVA analysis was performed using GraphPad Prism. Statistical significance is indicated as follows: ***p < 0.001, **p < 0.01, and *p < 0.05.

2.4. Quartz Crystal Microbalance Measurements. For quartz crystal microbalance measurements, we used an E4 quartz crystal microbalance with dissipation monitoring (QCM-D) from Q-Sense. Two separated identical flow chambers were used. One chamber contained virgin SiO 2 quartz, and the second chamber contained SiO 2 quartz covered with a continuous CNT layer. CNT layers of various thicknesses were used. The resonance frequency baseline signal was stabilized before the injection of culture medium elements into the chambers. The 7th resonance frequency was chosen for data analysis. The experiments were conducted at the temperature used to culture neural cells (i.e., 37°C).

A simplified version of the Sauerbrey equation can be used with respect to the type of crystal used in the study. The additional mass per unit area (in ng/cm 2 ) is Δm = -(17.7/n) × Δf, where n = 7 (7th order) and Δf is the absolute value of the negative resonance frequency shift measured in real time, which gives a variation of 2.53 ng/cm 2 for a Δf of 1 Hz.

2.5. Effects of CNTs on Neural Cells Proliferation. Neuro2a cells were cultured in parallel on a plane SiO 2 surface and on a plane SiO 2 surface coated with a homogeneous 60 ± 10 nm thick layer of CNTs, hereafter called a CNT surface. In this case, a pure solution of fetal bovine serum (FBS) was incubated prior to the cell culture on both surfaces for 3 h. After removal of the FBS solution, surfaces were air dried. Cells were seeded on surfaces at an initial density of 1.2 × 10 4 cells/cm 2 and cultured as usual. For a quantitative analysis of proliferation, cell densities were counted each day for 4 days by optical microscopy (5× objective) on four randomly chosen fields. Experiments were independently repeated three times.

RESULTS

The primary aim of the present study was to investigate the effects of CNTs on neural cell development and to elucidate their role in the cell culture process. For this, we first observed cell behavior of cells cultured in parallel on nonpatterned surfaces made of virgin silicon dioxide surfaces (SiO 2 ) or consisting of a dense and thin layer of CNTs printed over a SiO 2 surface (SiO 2 /CNTs). CNT layers were made by using a spray-coating technique. In comparison to other coating techniques such as solution casting [START_REF] Meitl | Solution casting and transfer printing single-walled carbon nanotube films[END_REF] or dipping, [START_REF] Rahy | Coating of carbon nanotubes on flexible substrate and its adhesion study[END_REF] the spraycoating process developed here allowed for denser CNT layers to be obtained in a single step, with a substrate surface totally covered with CNTs (see the Supporting Information). We thus cultured in parallel and in the same conditions neuro2a cells on a SiO 2 surface, a permissive substrate for neuro2a cells (control), or a thin and continuous CNT layer. No supplementary coating of the surfaces was realized. The affinity between cells and surfaces was evaluated by quantifying adherent cell density and cell differentiation after 3 days of culture. As can be seen in Figure 2, on homogeneous surfaces, these two parameters are comparable regardless of the surface composition. The proportion of differentiated cells raised comparable scores, 92 ± 4% and 95 ± 3%, respectively, for the SiO 2 and CNT surfaces.

We then investigated the influence of CNTs on neural cell morphology, in particular by considering their neurite number. We thus distinguished neural cell populations developing 1 neurite, 2 or 3 neurites, 4 or 5 neurites, and finally 5 or more neurites. Figure 3 shows that when cultured on a SiO 2 surface, a large majority of cells exhibited only 2 or 3 neurites (69 ± 9%) and that only 14 ± 8% of the neural cells developed 4 or 5 neurites. On the other hand, when cultured on a dense and continuous CNT layer, neural cells globally exhibited more neurites. Namely, the mean neurites number per neuron raised 3.3 ± 0.3 for CNT layer against only 2.2 ± 0.3 neurites per neuron for the SiO 2 surfaces. The proportion of neural cells exhibiting 4 or 5 neurites raised 40 ± 7%, whereas the proportion of neural cells with 2 or 3 neurites was only 43 ± 9% (against 69 ± 9% on virgin SiO 2, p = 0.005). No significant effect was observed concerning the neurites length (data not shown).

Both surface types exhibited equivalent affinity for neuro2a neural cells in term of cell adhesion and differentiation. However, major differences were observed in the developed neurites number. To further investigate the interactions between neural cells and CNTs, we decided to culture those cells in the presence of both surface types by means of patterned surfaces, closely juxtaposing the SiO 2 areas and CNT areas. To create a competition situation between CNTs and SiO 2 , we cultured neural cells on SiO 2 surfaces, patterned through a simple and rapid printing process previously developed. The developed method allowed us to very rapidly process large scale surfaces, thus giving the opportunity to study cell behavior over a large population. Moreover, in contrast with some already described methods, [START_REF] Hu | Chemically functionalized carbon nanotubes as substrates for neuronal growth[END_REF][START_REF] Jakobs | Engineered selforganization of neural networks using carbon nanotube clusters[END_REF][START_REF] Jang | Directional neurite growth using carbon nanotube patterned substrates as a biomimetic cue[END_REF][START_REF] Park | Carbon nanotube monolayer patterns for directed growth of mesenchymal stem cells[END_REF] this simple soft lithography process is particularly compatible with the cell culture, as no toxic solvents were used at any step, surfaces could easily be sterilized before starting the cell culture, and CNT patterns were not removed during the whole culture process without any previous chemical functionalization needed.

The patterned surfaces thus obtained were in the plane configuration of CNTs. Directly after cell seeding, cells were randomly distributed and covered the entire surface homogeneously, independent of the presence of the CNT patterns (see the Supporting information). After three days of culturing, the overall cell density and differentiation rate were comparable to those obtained on nonpatterned substrates, as is shown in Figure 2 (panles a and b). We obtained, namely, a cell density of 80 cells/mm 2 and a global differentiation score of 82 ± 7% for SiO 2 /CNT patterned surfaces. But surprisingly, by looking further in depth, we observed that a huge majority of cells developed precisely on the CNT patterns, as illustrated in Figure 4. Moreover, we observed that the neurites were guided in the direction of the printed CNT patterns. Neurites developed and extended in the direction of the CNT lines and were even able to follow lines exhibiting a corner of 90°, as shown in Figure 4.

SEM observations have been complemented by optical fluorescent imagery, and quantitative analyses are presented in Figure 5. In fact, 84 ± 5% of cells were found on CNT patterns in a competition situation (Figure 5a) and cells located on CNT patterns exhibited a higher differentiation rate than cells located on the neighboring SiO 2 areas (Figure 5b). A proportion of the differentiated cells raised 86 ± 4% for cells located on CNT patterns. For cells grown on SiO 2 nearby zones, this score is only of 55 ± 4%, whereas 92 ± 4% of cells were found to be differentiated on a homogeneous SiO 2 surface (see Figure 2b). This observation underlined that cells behaved differently with the proximity of CNT features.

In a second step, we investigated the influence of cell body localization on neurite behavior. For this, we distinguished the cell population with the cell body located on the CNT features (denoted "A" in Figure 4a) from cells with bodies located on the SiO 2 features (denoted "B" in Figure 4a). For cells having their body located on the CNT features (population A), 85 ± 5% of their neurites were located on the CNT features as well. Only 15 ± 5% of neurites escaped CNT features to grow on neighboring SiO 2 areas. On the other hand, for cells having their body on SiO 2 features, 54 ± 3% of neurites developed on SiO 2 and other neurites reached a neighboring CNT feature and grew further along it (see Figure 5, panels c and d).

All these observations lead to the conclusion that in the presence of these two kinds of surfaces (SiO 2 and CNTs) close together, the neural cells, neuro2a, preferentially developed on the CNT lines. One possible hypothesis for the selective localization of cells on the CNT patterns is the higher affinity of CNTs for the proteins of the culture medium compared to SiO 2 , thus creating a more favorable environment for cell growth and neurite development. To investigate the possible interaction between proteins of the culture medium and CNTs, we performed QCM-D analysis of adsorption of various elements such as BSA and serum (FBS).

To investigate the precise role of the CNT layer on this phenomenon, QCM-D experiments were carried out using virgin SiO 2 quartz crystals and various SiO 2 quartz coated with continuous CNT layers of different thicknesses. All CNT layers were first characterized by AFM. They had a comparable surface roughness (Ra = 8.5 ± 2 nm) and measured between 20 and 100 nm in height. Bovine serum albumin (BSA), at a concentration of 10 μg/mL, was then incubated on both quartz crystal types. In our cell culture experiments, BSA was used at this concentration to induce neural cell differentiation. Figure 6a shows typical QCM traces obtained when incubating substrates with BSA. As soon as BSA entered the chambers, a frequency shift was detected, denoting adsorption events on both surfaces. This frequency shift was always significantly greater for the quartz covered with CNTs. Thus, we observed a clear and reproducible difference between the two surface types (SiO 2 and CNTs). Namely, CNT surfaces exhibiting higher frequency shifts adsorbed more BSA than virgin SiO 2 quartz.

Frequency shift (and thus mass adsorption) was found to be directly linked with the CNT layer thickness (Figure 6, panels a and b). For a layer thickness between 20 and 70 nm, we noted a proportional relation between the adsorbed mass of proteins and the layer thickness (correlation factor is 0.98) (Figure 6b). However, for layers thicker than 70 nm, the adsorbed mass did not increase any more, as if the layers were saturated and could not adsorb additional proteins. After incubation of BSA on both surface types, we washed the quartz with PBS in order to know if BSA proteins were strongly attached to the surface. We observed a low desorption of BSA for both surface types (see the Supporting Information). However, the washing process only removed loosely attached proteins, and a significant amount of BSA proteins remained adsorbed on the surfaces. Once again, the amount of BSA proteins found on the surface after incubation and rinsing was significantly higher on the CNT surfaces compared to the SiO 2 surfaces (see Figure S1 of the Supporting Information).

In order to confirm the increasing adsorption on the CNTs, we carried out the same experiments with other molecules, such as poly-L-lysine, which is very often used to enhance cell adhesion and fetal bovine serum, which is commonly added to the cell culture medium or an antibody (donkey-anti-goat IgG monoclonal antibody). Table 1 summarizes the obtained results. We always observed that the amount of molecules adsorbed on the CNT surfaces was significantly higher than on the SiO 2 surfaces, whatever the incubated solution. The frequency shift difference between SiO 2 and CNT surfaces was particularly high for antibodies and poly-L-lysine (around 4 times higher).

In order to illustrate the enhanced adsorption of CNTs and investigate its effect on the cell culture, we compared cell proliferation on both CNT and SiO 2 surfaces. For this specific investigation, both surfaces were used either uncoated or coated with fetal bovine serum. Figure 7 represents proliferation curves obtained for those four surfaces after 4 day long cultures. We clearly observed that proliferation was accelerated on the coated CNT surface in comparison with the uncoated CNT surface. After 4 days of culture, the cell number on the coated CNT surface was more than twice higher (2.4) than on all other surfaces. The same coating realized on SiO 2 did not lead to comparable increasing proliferation, underlying that CNTs adsorbed a higher quantity of FBS are responsible for this high proliferation rate.

DISCUSSION

In the first step, we noted that CNTs drastically increased the average number of neurites per cell. The number of cells equipped with 4 or 5 neurites was three times higher on CNTs than on a SiO 2 surface. Studies performed using as-produced CNTs did not always show similar results. For example, Mattson et al. [START_REF] Mattson | Molecular functionalization of carbon nanotubes and use as substrates for neuronal growth[END_REF] found that surfaces of unmodified nanotubes were favorable for neurites outgrowth but did not promote neurite branching, suggesting that adhesion of growth cones to the carbon surface was relatively weak. In contrast, neurons grown on CNTs coated with a bioactive molecule, such as 4hydroxynonenal [START_REF] Mattson | Molecular functionalization of carbon nanotubes and use as substrates for neuronal growth[END_REF] or nerve growth factors, [START_REF] Matsumoto | Neurite outgrowths of neurons with neurotrophincoated carbon nanotubes[END_REF] elaborate multiple neurites, which exhibit extensive branching. Matsumoto et al. [START_REF] Matsumoto | Neurite outgrowths of neurons with neurotrophincoated carbon nanotubes[END_REF] showed that CNTs coated with NGF promoted neurite outgrowths in the same manner as soluble NGF. Hu et al. [START_REF] Hu | Chemically functionalized carbon nanotubes as substrates for neuronal growth[END_REF] showed that multiwalled carbon nanotubes (MWNTs) were also a permissive substrate for rat hippocampal neurons. In this study, the authors showed that both polyethyleneimine and asprepared MWNTs supported neuronal viability and permitted neurite outgrowth. However, as-produced MWNTs were found to reduce the initiation of neurite outgrowth, characterized by the reduction of the growth cone number and the neurite number per neuron. The studies discussed above focused on investigating CNT-neuron interactions by varying the surface chemistry of CNT substrates. Mechanical and electrical properties of CNT substrates such as surface roughness and conductivity, respectively, could also modulate and increase neuronal growth and neurite outgrowth. [START_REF] Sorkin | Process entanglement as a neuronal anchorage mechanism to rough surfaces[END_REF][START_REF] Zhang | Guided neurite growth on patterned carbon nanotubes[END_REF][START_REF] Malarkey | Conductive single-walled carbon nanotube substrates modulate neuronal growth[END_REF] In our case, CNTs were used as-produced, and no supplementary coating was performed previous to the cell culture. CNT layers exhibited a higher roughness than the SiO 2 surface, which could partially explain the increased neurites number we obtained, compared to our control surface. By patterning, we created a situation in which cells have the choice to develop either on SiO 2 features or on CNT features. This original experimental design turned out to reveal new effects that could not be anticipated from cell culture experiments achieved on nonpatterned surfaces. Cell culture results on those patterned surfaces showed, firstly, that the global adherent cell number was comparable to the scores obtained for nonpatterned surfaces. However, some selectivity of the CNT features was observed, and cells were found to develop preferentially on the CNT features. Directly after cell seeding, cells were randomly distributed (see the Supporting Information). However, during the first day of the culture, cells had migrated, and cell spreading was observed preferentially on CNT features. This means that the cells were able to explore their neighboring environment. Most of the cells were located on CNT features and exhibited a higher differentiation rate compared to residual cells growing on SiO 2 neighboring areas, which were not able to spread and migrate enough for joining adjacent CNT features. In conclusion, our investigation of patterned CNT/SiO 2 surfaces revealed that CNTs appeared to be more attractive for cell adhesion, growth, and differentiation.

In the same way, we have observed that neurites were guided by the CNT patterns. Neurites elongated along patterned CNT lines. Even for the CNT patterns lines turning at an angle of 90°, neurites followed pattern geometry. This indicated that the attractive effect of CNTs happened at two different length scales: firstly, at the cell body scale, explaining cell localization selectivity and secondly, at the neurite scale, during growth. Those guiding effects of CNT features suggest that neurites were confronted with the same choice of cell body. They explored their local environment and finally grew on the most favorable surface. This explanation is further supported by the fact that even cells which were not able to develop on CNTs very often exhibited neurites on the CNT patterns. One origin of this selective effect could be that the CNT features exhibited a higher surface roughness than the neighboring SiO 2 features, as was often mentioned in the literature. [START_REF] Biggs | Nanotopographical modification: A regulator of cellular function through focal adhesions[END_REF][START_REF] Voge | Carbon nanotubes in neural interfacing applications[END_REF][START_REF] Gentile | Cells preferentially grow on rough substrates[END_REF] This effect of CNTs on neuronal adhesion, for example, was explored in a recent study by Sorkin et al., [START_REF] Sorkin | Process entanglement as a neuronal anchorage mechanism to rough surfaces[END_REF] who cultured neurons from embryonic rat cortices on CNT features that consisted of micrometric discs. Cells displayed a preference to grow on CNT discs, and a close examination revealed extensive curling of the neurites around CNTs. The authors explained that CNT discs allowed entanglement of thin neurites with relatively similar diameters to CNTs, and that such an entanglement may represent an anchoring mechanism allowing neurons to attach to rough surfaces. Gentile et al. [START_REF] Gentile | Cells preferentially grow on rough substrates[END_REF] recently reported that cells preferentially grow on etched silicon rough surfaces. An increased proliferation rate was obtained for moderately rough surfaces (Ra: ∼10-45 nm). According to them, this could be attributed to several possible mechanisms, among which is the increased effective surface energy over nonplanar substrates. Cell stable adhesion and consequent proliferation would be energetically favorable on rough surfaces. Therefore, the theory of adhesion of elastic solids (here the cells) on randomly rough surfaces could explain the observed preferential stable adhesion and cell localization. [START_REF] Gentile | Cells preferentially grow on rough substrates[END_REF][START_REF] Persson | Adhesion between an elastic body and a randomly rough hard surface[END_REF] Furthermore, a rough surface allowed a higher number of possibilities for cells to construct adhesion complexes with the surface. [START_REF] Brunetti | Neurons sense nanoscale roughness with nanometer sensitivity[END_REF][START_REF] Gentile | Cells preferentially grow on rough substrates[END_REF][START_REF] Gittens | The effects of combined micron-/submicron-scale surface roughness and nanoscale features on cell proliferation and differentiation[END_REF] The processes that mediate the cellular adhesion to nanoscale surface structures are, however, not well understood and may be a direct result of the influence of the surface topography/roughness, as discussed above, or an indirect one. Our results showed that CNT features could act as porous reservoirs for various elements (proteins, growth factors, etc.) contained in the culture medium. Once again, this latter effect may originate from the enhanced roughness of the CNT surfaces. These results are in accordance with Chung et al., [START_REF] Chung | An electrode of quartz crystal microbalance decorated with CNT/ chitosan/fibronectin for investigating early adhesion and deforming morphology of rat mesenchymal stem cells[END_REF] who found that quartz electrodes equipped with CNTs had an increased specific surface area, leading to enhanced adsorption of chitosan. However, we showed that protein adsorption was affected by the CNT layer thickness (see Figure 6b). By increasing the CNT layer thickness exhibiting the same surface roughness (see Figure 6c), we increased protein adsorption. We were even able to evidence a linear relation between CNT layer thickness and adsorbed protein mass. However, this phenomenon seemed to be limited by protein diffusion inside the CNT network. Indeed, for CNT layer thickness above 70 nm, proteins could not diffuse deeper in the CNT network, leading to a kind of saturation. The same observations were made for various types of proteins. This result clearly shows that the enhancement of protein adsorption on the CNT layers is dominated by a bulk effect rather than a pure surface effect. Indeed, different layers of the same surface roughness exhibited increased adsorption with increasing thickness. Thus, we think that the dominant effect responsible for this enhanced protein adsorption is the generation of a porous network of nanoscale elements acting as a reservoir of proteins rather than a pure roughness induced effect. Finally, we have shown that the CNT layers could be charged with proteins and that cells, in the presence of CNT patterns, could be able to sense a surface concentration gradient. This surface gradient on patterned substrates could explain why cells chose to grow preferentially on CNT features, taking advantage of the numerous proteins adsorbed on the CNT patterns. This interpretation is further supported by our observation that cells duplicated more rapidly when CNT layers were charged with serum previously to the cell culture. Previous reports on how surface topography or roughness affect protein adsorption are numerous, see ref 42, for example. Dynamic interactions between nanotopography and proteins are complex, due to the combination of attractive and repulsive forces which are governed by local changes in surface properties. [START_REF] Roach | Modern biomaterials: A reviewbulk properties and implications of surface modifications[END_REF][START_REF] Wilson | Mediation of biomaterial-cell interactions by adsorbed proteins: A review[END_REF][START_REF] Woo | Nano-fibrous scaffolding architecture selectively enhances protein adsorption contributing to cell attachment[END_REF] The overall effect of surface nanoscale features is not fully understood, according to reports available in the literature. [START_REF] Lord | Influence of nanoscale surface topography on protein adsorption and cellular response[END_REF][START_REF] Shen | Induced stepwise conformational change of human serum albumin on carbon nanotube surfaces[END_REF][START_REF] Rechendorff | Enhancement of protein adsorption induced by surface roughness[END_REF] In our case, CNT surfaces always lead to an increased mass adsorption with various ranges of proteins, as antibodies or bovine serum albumin, suggesting that the CNT patterns could act as local porous reservoirs for active proteins of the culture medium. This could explain all of our results obtained on the cell culture.

CONCLUSION

Patterning SiO 2 surfaces with CNTs created a new original situation for cell culture experiments and revealed new phenomena on cell interactions with CNTs. We have evidenced that CNTs enhanced protein adsorption, leading to protein gradients on the patterned surface. Patterned substrates thus exhibited two types of surfaces that are different in terms of composition, roughness, and protein amount. Our results show that neural cells preferentially adhere to the CNT patterns.

CNTs thus appear to be the most convenient surface for cell growth and differentiation, in comparison with the SiO 2 surface. This conclusion could not be obtained by culturing cells on homogeneous surface types consisting either of CNTs or of SiO 2 . Namely, cells were able to adapt to SiO 2 and exhibited comparable adhesion and development on SiO 2 and on CNTs. Finally, this shows that patterning can be useful to determine the most appropriate growth surface for cells. It would be interesting to investigate the distance over which cells are able to migrate in order to grow on a neighbouring appropriate CNT surface. Thus, we could obtain information on the balance between cell mobility and surface attraction. Among the possible mechanisms capable to explain the preferential adhesion, differentiation, and growth of neural cells on CNT patterns, this paper highlights the existence of an indirect cell/ CNT interaction mediated by the preferential adsorption of culture medium proteins on CNT networks.

■ ASSOCIATED CONTENT

* S Supporting Information

The fabrication process of CNT micropatterned surfaces and typical QCM-D responses to BSA solution incubation are detailed in a supporting pdf file. This material is available free of charge via the Internet at http://pubs.acs.org.

Figure 1 .

 1 Figure 1. Typical surfaces prepared for a 2D cell culture. (a and b) SEM images of a patterned SiO 2 substrate after printing CNTs. The printed features are 7 to 20 μm wide lines. (a) Periodic arrays of line and space CNT arrays. (b) Specific arrays of lines with 90°corners. CNTs appear bright under SEM inspection. (c and d) AFM characterization of typical CNT patterns created by microcontact printing. (c) AFM image of CNT lines on a SiO 2 surface (height signal). (d) AFM image showing details inside a CNT pattern at the nanoscale (vertical deflection signal). There is no CNT ordering inside a micropattern.

Figure 2 .

 2 Figure 2. Influence of CNTs on neuronal cell (neuro2a) density and differentiation. Neuro2a cells were cultured for 3 days on a SiO 2 nonpatterned surface, on thin nonpatterned layers of CNTs, or on SiO 2 /CNTs line and space patterns. (a) Cell density (DAPI-stained nuclei per mm 2 ) on the different surfaces. (b) Percentage of differentiated neural cells (percentage of neuro2a cells exhibiting at least one neurite measuring at least 20 μm) on the different surfaces. Data represent the mean values (± SE) coming from observations of at least 200 cells on four randomly chosen imaging fields obtained for three independent experiments.

Figure 3 .

 3 Figure 3. Histogram representing the proportion of differentiated neuronal cells equipped with only one neurite, two or three neurites, four or five neurites, or more than five neurites for the two types of unpatterned surfaces: SiO 2 and CNT thin layer. For each condition, around 400 cells were observed.

Figure 4 .

 4 Figure 4. Neuro2a cells culture on a SiO 2 /CNTs micropatterned surface. (a) SEM images of neuro2a cells cultured on patterned surfaces after 2 days of differentiation. Arrows point to neurites developed on CNT patterns. The letter A indicates a specific cell body on a CNT feature, and the letter B points out a specific cell body outside of a CNT feature (on a SiO 2 feature). (b) Optical fluorescence image of neural cells grown on CNT patterns (similar to Figure 1, right) after phalloidin staining. Note that neurites follow the CNT lines turning at an angle of 90°.

Figure 5 .

 5 Figure 5. Influence of SiO 2 /CNTs micropatterns on neuro2a cell behavior. (a) Detailed distribution of cells inside a SiO 2 /CNT pattern (18 μm line and 22 μm space). (b) Detailed differentiation scores of cells inside a SiO 2 /CNT pattern. (c and d) Detailed neurite outgrowth on patterned surfaces. (c) Quantification of neurite localization inside SiO 2 /CNT patterns, as a function of the localization of the cell body. Values represent mean (± SE) of three independent experiments. At least 300 neurites were observed for each experiment. (d) Schematic representation of various situations observed for neural cell culture on the SiO 2 /CNT patterned surfaces (CNTs are shown in black). Cell bodies are mainly located on the CNT patterns. Neurites turned out to develop preferentially on CNT patterns and followed the direction of the CNT lines. Cells located on SiO 2 developed less neurites, which were either localized on the SiO 2 regions or on the CNT patterns and aligned by the CNT lines.

Figure 6 .

 6 Figure 6. QCM-D investigation of BSA adsorption on unpatterned CNT thin layers of increasing thickness, printed on a SiO 2 surface. (a) Typical QCM-D frequency responses to the incubation of a BSA solution (10 μg/mL diluted in PBS) for virgin SiO 2 quartz (light blue curve) and for SiO 2 quartz covered with continuous CNT layers of various heights. BSA was introduced at t = 9 min following a baseline preincubation with PBS/BSA-free solution. The curves display the 7th resonance frequency shift, and the corresponding surface density of adsorbed mass is displayed on the right vertical axis. (b) Evolution of the final frequency shift (after 200 minutes of incubation) measured by QCM-D. (c) Evolution of the CNT layer roughness against the thickness of the CNT layer measured by AFM.

Figure 7 .

 7 Figure 7. Cell density curves for neuro2a cells on SiO 2 , SiO 2 coated with fetal bovine serum, CNT layers (layer thickness is 80 ± 5 nm), and CNT layers coated with fetal bovine serum. Each data point is the average cell density from three experiments. Error bars indicate the standard error (SE).

Table 1 .

 1 Relative Frequency Shifts Measured After the Incubation (200 minutes) of Various Solutions a

	entity	ΔF CNTs/ΔF SiO 2
	BSA (10 μg/mL)	1.3
	poly-L-lysine (40 μg/mL)	4
	foetal bovine serum	1.6
	antibody (40 μg/mL)	4.3
	a The ratio represents the final frequency shift measured by QCM-D
	on a CNT thin layer (layer thickness is 80 ± 5 nm) printed on a SiO2
	quartz, divided by the same quantity measured on a virgin SiO2 quartz.

■ AUTHOR INFORMATION

Corresponding Author *E-mail: abeduer@laas.fr. Tel: +33 5 61 33 63 75. Fax: +33 5 61 33 62 08. Address: LAAS-CNRS Ameĺie Beduer, Groupe NBS, 7 Avenue du Colonel Roche, 31077 Toulouse Cedex 4, France.

Notes

The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

The authors acknowledge the technical staff of the clean room facility of LAAS-CNRS for their assistance in silicon master fabrication for the micro-contact printing process.