Carlos Granero-Belinchon

Stéphane G Roux

Nicolas B Garnier

Information theory for non-stationary processes with stationary increments

published or not. The documents may come

Information theory for non-stationary processes with stationary increments

to analyse a non-stationary signal with stationary centered increments. We describe a time-averaged framework that gathers all available data points in a time window representing a single realization, whether it is the full experimental time duration, or just a fraction of it [START_REF] Vu | Information In The Non-Stationary Case[END_REF].

The present paper is organized as follows. In section 2, we present the general framework of information theory for a non-stationary signal, and our new framework that exploit time averages. We then give a particular emphasis on self-similar processes. In section 3, we report a benchmarking of our framework in the special case of Gaussian self-similar signals, a model situation where it is possible to obtain analytical developments. In section 4, we explore the case of non-Gaussian self-similar processes. Finally, in section 5, we drop the hypothesis of self-similarity and we apply our framework to a multifractal process.

Information theory for non-stationary processes

Non-stationary processes with stationary increments

In this article, we consider non-stationary processes with stationary increments. Such a process can be written as a motion M = {m t } t∈R obtained by integrating a stationary noise W = {w t } t∈R :

m t = m 0 + t 0 w t dt . ( 1
)
where m 0 and w 0 are the values at time t = 0, both of which can be set to 0 without loss of generality. Nowadays signals are recorded and stored on digital media, which amounts to consider in practice a set of data sampled at discrete times t k where k ∈ N + . We further assume that the signals are equi-sampled, i.e., dt is constant and we choose dt = 1. So we consider in this article discrete time processes and we express them as motions M = {m t } t∈N + obtained by integrating a stationary noise W = {w t } t∈N + according to:

m t = m 0 + t ∑ k=1 w k , t > 0 , (2)
where again m 0 = w 0 = 0. Eq.( 2) can also be replaced by

m t = m t-1 + w t , t > 0 . (3)
If the noise W is not centered, i.e., has a statistical mean E(W) = β = 0, we introduce the centered noise w t = w tβ. The equations for the motion M read:

m t = m 0 + βt + t ∑ k=1 w k , t > 0 , (4)
= m t-1 + β + w t , t > 0 .

(

) 5
The process M can be interpreted as a motion built on the stationary centered noise W together with an additive deterministic drift, which is the linear trend βt.

In this article, we study motions without trend, so we impose that the noise W is centered, i.e., that its statistical mean E(W) = β = 0. Besides the simple centering of the increments W → W -E(W), any detrending method can be applied to M, e.g., using moving averages. As a consequence, the motion M is centered: its statistical mean is E{m t } = m 0 = 0 at all times t > 0. Nevertheless, its variance, and all its higher order moments, may depend on time: the motion M is a non-stationary process with stationary increments. Typical examples of such processes are Brownian motion and fractional Brownian motion [START_REF] Mandelbrot | Fractional brownian motions fractional noises and applications[END_REF], both of which have a variance that evolves with time.

General framework

For a generic non-stationary process X t = {x t } t∈R , the probability density function (PDF) p x t (x t ) changes with time. The information theory framework can be applied to each random variable x t , i.e., at each time t. To do so, the PDF of x t needs to be estimated at each time t, which in practice requires to have many realizations available [START_REF] Gómez Herrero | Assessing coupling dynamics from an ensemble of time series[END_REF].

To analyze the temporal dynamics of a random processes at a given time t, we consider the m-dimensional vector obtained with the Takens time-embedding procedure [START_REF] Takens | Detecting strange attractors in turbulence[END_REF]:

x (m,τ) t = x t , x t-τ , • • • , x t-(m-1)τ . ( 6
)
The embedding dimension m controls the order of the statistics that are considered, and the delay τ defines a time scale. We define below some information theory quantities that are functionals of the m-point joint-distributions p

x (m,τ) t (x (m,τ) t
), in order to characterize linear and non-linear temporal dynamics.

Shannon entropy

The entropy of x (m,τ) t is:

H(x (m,τ) t ) = - R m p x (m,τ) t (x) log(p x (m,τ) t (x))dx . (7)
This quantity depends on time t, as well as on embedding parameter m and delay τ. We further

note it H (m,τ) t (X) = H(x (m,τ) t
), where the index t indicates the time and the parameters (m, τ) are indicated as upper indices. It measures the amount of information characterizing the m-dimensional PDF of the process X at time t sampled at scale τ. When m = 1, the entropy does not depend on τ and does not probe the dynamics of the process; we then note it H t (X), dropping the (m = 1, τ) upper indices. However, for embedding dimension m > 1 the entropy depends on the linear and non-linear dynamics of the process. Indeed, the entropy involves arbitrarily high order moments of the joint PDF

p x (m,τ) t (x (m,τ) t
). As usual, the entropy does not depend on the first moment of the distribution.

Using the time-increments of size τ, δ τ x t ≡ x tx t-τ , it can be shown (see appendix A) that the amount of information measured by H (m,τ) t (X) is the same as the amount of information in the vector

x(m,τ) t ≡ (x t , δ τ x t , δ τ x t-τ , ..., δ τ x t-(m-2)τ ), i.e., H (m,τ) t (X) = H(x (m,τ) t ) (8) = H x t , δ τ x t , δ τ x t-τ , ..., δ τ x t-(m-2)τ .
For processes with stationary increments, the marginal distribution of x t may be strongly time-dependent, but the marginal distributions of any increment is time-independent. Eq.( 8) thus suggests that the time-dependence of H (m,τ) t (X) originates mainly from x t , the first component of the rewritten embedded vector xm,τ

t . Nevertheless, it should be observed that although the m -1 increments, considered by themselves, have a stationary dependence structure, the covariance of x t with any of the increments is a priori non-stationary.

Mutual information and auto-mutual information

The mutual information MI measures the amount of information shared by two processes. For two non-stationary time-embedded vectors x (m,τ) t 1 and y

(n,τ) t 2
, it is defined as :

MI(x (m,τ) t 1 , y (n,τ) t 2 ) = H t 1 (x (m,τ) t 1 ) + H t 2 (y (n,τ) t 2 ) -H(x (m,τ) t 1 , y (n,τ) t 2 ) . (9)
In the following, we use auto-mutual information I (m,n,τ) t (X) to measure, for a single process X, the shared information between two successive time-embedded vectors of dimension m and n [START_REF] Granero-Belinchon | Information Theory to Probe Intrapartum Fetal Heart Rate Dynamics[END_REF]:

I (m,n,τ) t (X) = MI(x (n,τ) t , x (m,τ) t-nτ ) . ( 10
)
Auto-mutual information defined in [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF] probes the dynamics of the process X t at time t by measuring the dependencies between two consecutive chunks of m and n points sampled every τ.

Entropy rate

The entropy rate, or entropy gain [START_REF] Crutchfield | Regularities unseen, randomness observed: The entropy convergence hierarchy[END_REF], of order m at time t measures the increase of Shannon entropy when the embedding dimension is increased from m to m + 1. It is defined as the variation of Shannon entropy between x (m,τ) t-τ and x (m+1,τ) t , two successive time-embedded versions of the process X:

h (m,τ) t (X) = H (m+1,τ) t (X) -H (m,τ) t-τ (X) (11)
= H t (X) -I (m,1,τ) t (X) . (12)
Within the general framework, the entropy, mutual information and entropy rate are well defined at any time t for a non-stationary process. Although this framework can formally be used to analyze non-stationary processes at any time t, in practice it is often impossible to assess statistics at a fixed time t, as the number of available realizations from real world datasets may be very small. To overcome this issue, we propose in the next section another framework that considers averages over a finite and possibly large time window, which represents for example the duration of an experimental measurement.

Practical time-averaged framework

We now focus on non-stationary processes with stationary increments. We develop in this section a pragmatic approach which can be applied when a single time trace of a non-stationary signal is available.

We first present a very formal perspective that defines a time-averaged PDF of a non-stationary process. We then propose a practical approach which uses a very simple estimation of such a time-average PDF. We finally use this practical approach to define all the information quantities that we are interested in.

Time-averaged framework

Using a formal perspective, we consider the global statistics of the dataset, when forgetting its time dynamics, and we formally consider the time-averaged probability density function in the time window [t 0 , t 0 + T] :

pT,t 0 ,x (m,τ) (x) = 1 T t 0 +T t 0 p x (m,τ) t (x)dt . ( 13
)
Because of the time-average, this probability density function doesn't depend on a single time t but on the starting time t 0 and the duration T of the time window.

In the case of a stationary process, the PDF p x (m,τ) t (x) is independent of t, so the PDF pT,t 0 ,x (m,τ) (x) is independent of t 0 and T.

In the case of a non-stationary process with stationary centered increments, the dependence on t 0 only appears on the mean of the time-averaged PDF pT,t 0 ,x (m,τ) (x). As a consequence, since the Shannon entropy does not depend on the mean, none of the information theoretic quantities depends on t 0 .

In the case of a non-stationary process with stationary but non-centered increments, there is a drift: the first moment of p x (m,τ) t (x) evolves linearly with time. When integrated in time in eq.( 13), this induces a deformation of the time-averaged PDF pT,t 0 ,x (m,τ) (x), which affects a priori moments of any order. As a consequence, the Shannon entropy is then expected to depend on t 0 .

In the following, we focus on non-stationary processes with stationary centered increments, described in section 2.1.

Practical framework

In practice, given a time series of length T, we propose to very roughly approximate the PDF pT,t 0 ,x (m,τ) defined in [START_REF] Vu | Information In The Non-Stationary Case[END_REF] with the normalized histogram pT,t 0 of all data points x (m,τ) t , t ∈ [t 0 , t 0 + T] available in the time window. This is a very strong assumption, as pT,t 0 ,x (m,τ) is a priori very different from any p x (m,τ) t , and a priori very different from the histogram pT,t 0 constructed after cumulating all the available data in the interval. This pragmatic approach comes down to treat the set of available data points in the time interval exactly in the same way as if it was a set of data points originating from a stationary, albeit unknown, process and then estimate its PDF.

In the following, we drop the hat in the notations, and consider only the ersatz probabilities p in place of the time-averaged probabilities p. As we discuss later in section 6, if several experimental realizations are available, it is of course possible to use them to enhance the estimation of the time-averaged PDF.

Information theory quantities in the practical framework

Given a time series of length T, and considering the ersatz PDFs pT,t 0 ,x (m,τ) , we define H(m,τ)

T (X)
the time-averaged Shannon entropy, Ī(m,n,τ) T (X) the time-averaged auto-mutual information and h(m,τ)(X) T the time-averaged entropy rate, as described below.

Ersatz Shannon entropy

We define the ersatz entropy of the time-embedded signal as the entropy of the time-averaged PDF pT,t 0 ,x (m,τ) :

H(m,τ) T (X) = - R m pT,t 0 ,x (m,τ) (x) log( pT,t 0 ,x (m,τ) (x))dx . (14)
H(m,τ) T (X) gives the amount of information of the set of values of the signal x (m,τ) t in the time interval [t 0 , t 0 + T] and hence it can be interpreted as the total information characterizing the temporal trajectory {x (m,τ) t , t ∈ [t 0 , t 0 + T]} of the process. If the process has stationary centered increments, the total amount of information in the trajectory depends only on its length T, and not on its starting time t 0 . In that sense, the ersatz entropy H(m,τ)

T is not stationary. Using the rewriting (8), we argue that this dependence in T originates from x t -the first component of the vector xm,τ t -which has a time-dependent marginal distribution. Because the m -1 other components of xm,τ t are increments, they have by hypothesis a stationary dependence structure. So increasing the embedding dimension does not impact the dependence of the ersatz entropy on the window size T, but only its dependence on the increments size τ.

Auto-mutual information

We define the ersatz auto-mutual information as:

Ī(m,n,τ) T (X) = H(m,τ) T (X) + H(n,τ) T (X) - H(m+n,τ) T (X) . ( 15
)
Entropy rate

We define the ersatz entropy rate over a time interval of size T as:

h(m,τ) T (X) = H(m+1,τ) T (X) - H(m,τ) T (X) (16) = HT (X) - Ī(m,1,τ) T (X) . ( 17
)
From ( 16), we may expect a cancelation of the main dependence in T which is the same for H(m+1,τ) T should be stationary, in the sense that it should not depend on the length T of the time interval that is considered.

If the available samples span a very large time window, one may consider using multiple non-overlapping time windows of size T starting at various times. Because of the stationarity and zero-mean of the increments and hence the independence of the ersatz quantities on t 0 , it is possible to average the different estimations of the ersatz quantities obtained in each window. It is also possible to use all the non-overlapping windows to populate the histogram and thus enhance the estimation of the time-averaged PDF. Each of these two operations will increase the statistics and hence improve the estimation.

Self-similar processes

In this section, we focus on the special case of self similar processes, i.e., signals which exhibit monofractal scale invariance [START_REF] Mandelbrot | The fractal geometry of nature[END_REF]. Such processes have been used as a satisfying first approximation to model or describe very various phenomena, such as ionic transport [START_REF] Mauritz | Dielectric relaxation studies of ion motions in electrolyte-containing perfluorosulfonate ionomers: 4. long-range ion transport[END_REF], fluid turbulence [START_REF] Chevillard | A phenomenological theory of Eulerian and Lagrangian velocity fluctuations in turbulent flows[END_REF], climate [START_REF] Kavvas | Introduction to the focus issue: physics of scaling and self-similarity in hydrologic dynamics, hydrodynamics and climate[END_REF], river flows [START_REF] Rigon | On Hack's law[END_REF], cloud structure [START_REF] Gotoh | A fractal dimensional analysis on the cloud shape parameters of cumulus over land[END_REF] or earthquakes [START_REF] Console | Bath's law and the self-similarity of earthquakes[END_REF], as well as neural signals [START_REF] Ivanov | Levels of complexity in scale-invariant neural signals[END_REF], stock markets [START_REF] Drozdz | Imprints of log-periodic self-similarity in the stock market[END_REF][START_REF] Cont | Scaling in stock market data: stable laws and beyond[END_REF], texture patterns [START_REF] Uhl | A systematic evaluation of the scale invariance of texture recognition methods[END_REF] or internet traffic [START_REF] Chakraborty | Self-similar and fractal nature of internet traffic[END_REF]. A process X t is monofractal scale-invariant if there exists a real number H such that for all a ∈ R + * , the probability density functions of x at and a H x t are equivalent. H is called the Hurst exponent. If H < 0, the process is stationary and called a fractional noise. If 0 ≤ H < 1, the process is non-stationary with stationary increments. The case H = 1/2 corresponds to the traditional Brownian motion.

Assuming x t=0 = 0, the scale invariance property can be expressed as [START_REF] Flandrin | Wavelet analysis and synthesis of fractional Brownian motion[END_REF]:

p x at (x) = 1 a H p x t x a H . ( 18
)
The scale invariance property of a process X t transfers to its increments, as well as any of its time-embedded version:

p x (m,aτ) at (x) = 1 a mH p x (m,τ) t x a H . ( 19
)
This relation allows to express the non-stationary PDF of x (m,τ) t at any time t as a function of the PDF at unit-time (t=1). This is done by using the factor a = 1/t in eq.( 19), i.e., by rescaling each coordinate of the embedded vector by the factor t H .

Using eq.( 13), it is straightforward to see that the scale invariant property of the form ( 19) is also valid for the time-averaged PDF pT,t 0 ,x (m,τ) (x).

Because of its definition (2) as a cumulative sum of a noise, a motion can be seen as accumulating the correlations between successive points of the noise. When performing a time-embedding, the particular case τ = 1 is interesting: considering the relation [START_REF] Yaglom | Correlation theory of processes with random stationary nth increments[END_REF], we may expect that the information contained in the time-embedded motion m (m,τ=1) t is closely related to the information contained in the time-embedded noise w (m,τ=1) t . This is not the case anymore when τ ≥ 2.

Fractional Brownian motion

The Fractional Brownian motion (fBm) was proposed by Mandelbrot and Van Ness [START_REF] Mandelbrot | Fractional brownian motions fractional noises and applications[END_REF] and quickly became a benchmark for self-similarity and long-range dependence. The fBm is the only Gaussian self-similar process with stationary increments. It is characterized by its Hurst exponent, H.

The fBm is a motion, obtained by integrating according to (2) a fractional Gaussian noise (fGn), defined as a centered Gaussian process with the correlation structure

c fGn (τ) = σ 2 1 2 (τ -1) 2H -2τ 2H + (τ + 1) 2H . ( 20
)
The fGn is a stationary noise with the standard deviation σ 1 . It is scale-invariant with a Hurst exponent H -1.

The non-stationary covariance structure of the fBm B reads

E{B t B t-τ } = σ 2 1 2 t 2H + (t -τ) 2H -τ 2H , (21)
where τ < t.

General framework

We show below how the theoretical information quantities depend on time t and delay τ. We start from the relation ( 8) between the entropy of the time-embedded vector and the entropy of the increments and we normalize each component of the vector

x(m,τ) t by its standard deviation. The standard deviation σ t of the motion x t evolves with time as σ t = σ 1 t H , while the standard deviation σ τ of the increments x tx t-τ is independent of t, thanks to the stationarity of the increments, and evolves with the size of the increment as σ τ = σ 1 τ H . So we have:

H (m,τ) t (X) = H x t /σ t , δ τ x t /σ τ , ..., δ τ x t-(m-2)τ /σ τ + ln σ t + (m -1) ln σ τ , (22)
= H x t /t H , δ τ x t /τ H , ..., δ τ x t-(m-2)τ /τ H + H ln t + (m -1)H ln τ + m ln σ 1 . ( 23
)
We then use the scaling law [START_REF] Granero-Belinchon | Information Theory to Probe Intrapartum Fetal Heart Rate Dynamics[END_REF] for a = 1/t to relate the joint probability at a given time t to the joint probability at unit-time t = 1, which leads to:

H x t /t H , δ τ x t /τ H , ..., δ τ x t-(m-2)τ /τ H = H x 1 , δ τ/t x 1 , ..., δ τ/t x 1-(m-2)τ/t -m ln σ 1 (24) = H x(m,τ/t) t=1 -m ln σ 1 . (25)
Using (8) again at time t = 1, we have H

x(m,τ/t) t=1 = H (m,τ/t) t=1
(X), so we can express the time-dependent Shannon entropy (7) for self-similar processes as:

H (m,τ) t (X) = H (m,τ/t) t=1 (X) + H ln t + (m -1)H ln τ . ( 26
)
The entropy rate can be rewritten with ( 11) and ( 26) as:

h (m,τ) t (X) = h (m,τ/t) t=1 (X) + H ln τ , (27)
where h

(m,τ/t) 1 (X) is the entropy rate at time t = 1, using the rescaled time delay τ/t.

Although the two quantities H

(m,τ/t) t=1

(X) and h (m,τ/t) t=1

(X) are considered at a fixed time t = 1, they still depend on t via the delay τ/t. Because τ/t is small as soon as t τ, we expect that the dependence of the entropy H (m,τ) t (X) on time t is mainly in H ln t, and that the entropy rate is almost time-independent.

Fractional Brownian motion

The PDF p B t of the fBm is Gaussian at any time t, so we can express its Shannon entropy and entropy rate at time t by using eq.( 26) and the expression of the Shannon entropy of a Gaussian multivariate process [START_REF] Zografos | Expressions for Rényi and Shannon entropies for multivariate distributions[END_REF]. We obtain the following approximated expressions:

H (m,τ) t (B) mH FBM 1 + H ln t + (m -1)H ln τ (28) h (m,τ) t (B) H FBM 1 + H ln τ , ( 29
)
where

H FBM 1 ≡ 1 2 ln 2πeσ 2
1 is the entropy of the fBm at unit-time. These formulae are exact for m = 1, 2, but for m ≥ 3, constant terms as well as corrections in τ/t have been omitted for clarity.

Practical time-averaged framework

For a generic self-similar process, we are not able to derive any analytical results in the practical time-averaged framework. Nevertheless, the behaviors expected for a generic non-stationary process with stationary increments are holding: i) the ersatz entropy H(m,τ)

T is not stationary, in the sense that it depends on the length T of the time-interval, ii) the ersatz entropy rate h(m,τ)

T is stationary.

Fractional Brownian motion

The ersatz entropy of the fBm over a time window of size T τ can be expressed by averaging its covariance structure on a time window of size T. We obtain [START_REF] Granero-Belinchon | Scaling of information in turbulence[END_REF]:

H(1,τ) T (B) = H FBM 1 + H ln T . ( 30
)
The entropy of the fBm thus increases linearly with the logarithm of the window size T. The larger the time window, the more there is information in the trajectory. The auto-mutual information of the fBm can be derived in the same way using (15) for T τ:

Ī(1,1,τ) T (B) = -H ln τ T + C τ T , (31)
where C τ T is a correction in τ/T that reads

C τ T = 1 2 ln τ 2T + 1 2H+1 ( τ T + 1 2H+1 ) (32) = - 2H + 1 4 τ T + O τ T 2 . ( 33
)
The ersatz mutual information depends logarithmically on the scale τ and the window size T. The larger the window-size T or the smaller the scale τ, the stronger the dependencies.

The ersatz entropy rate of order m = 1 is obtained by combining [START_REF] Cont | Scaling in stock market data: stable laws and beyond[END_REF] and [START_REF] Uhl | A systematic evaluation of the scale invariance of texture recognition methods[END_REF] according to [START_REF] Mandelbrot | Fractional brownian motions fractional noises and applications[END_REF]:

h(1,τ) T (B) = H FBM 1 + H ln τ -C τ T (34)
which is independent of T up to corrections in τ/T, while being linear in ln(τ) with a constant slope H. The correction -C(τ/T) in eq.( 34) is positive, see eq. [START_REF] Flandrin | Wavelet analysis and synthesis of fractional Brownian motion[END_REF]. Comparing ( 28) with [START_REF] Cont | Scaling in stock market data: stable laws and beyond[END_REF] shows that for the fBm, the ersatz entropy dependence on T is exactly the same as the entropy dependence on t. Comparing ( 29) with [START_REF] Zografos | Expressions for Rényi and Shannon entropies for multivariate distributions[END_REF] shows that the entropy rate and the ersatz entropy rate do not depend on t or T up to corrective terms that are negligible if the scale τ is not too large. We also see explicitly that both quantities evolves with the scale τ in H ln τ, again up to corrections of order τ/t and τ/T.

The example of the fBm suggests that for a scale-invariant process the evolution of any information theory quantity with the scale τ is the same within the practical time-averaged framework or the general framework. We push this analysis further in the next sections, by exploring if this property holds when the process is non-Gaussian.

Benchmarking the practical framework with the fBm

We focus in this section on the fractional Brownian motion, for which analytical expressions were derived in the previous sections. We use the fBm not only to benchmark our estimators of information theory quantities, but also to illustrate the use of the practical framework and the expected behavior of the ersatz quantities when used on a self-similar process of Hurst exponent H.

Characterization of the estimates

Data

To obtain a fBm, we integrate a fractional Gaussian noise (fGn). We use circulant matrix method [START_REF] Helgason | Synthesis of multivariate stationary series with prescribed marginal distributions and covariance using circulant matrix embedding[END_REF] to impose the correlation structure of the fGn [START_REF] Crutchfield | Regularities unseen, randomness observed: The entropy convergence hierarchy[END_REF] . Then, we center and normalize the noise such that the standard deviation, σ fGn , is equal to one. We then take the cumulative sum to obtain the fBm. Through all this article, H = 0.7 for all the processes used to illustrate our results, but we have checked that they hold for any other value 0 < H < 1.

Procedure

We estimate the Shannon entropy H(m,τ)

T with our own implementation of the k-nearest neighbors estimate from Kozachenko and Leonenko [START_REF] Kozachenko | Sample estimate of entropy of a random vector[END_REF]. We estimate the auto-mutual information Ī(m,p,τ)

T with the algorithm provided by Kraskov, Stogbauer and Grassberger [START_REF] Kraskov | Estimating mutual information[END_REF]. This estimator is also based on a nearest neighbors search and it provides -amongst several good properties -a build-in cancellation of the bias difference originating from each of the two arguments. In the following, we note k the number of neighbors, which is the only parameter of the estimators. The entropy rate h(m,τ)

T is then computed using eq.( 17).

We generate for each motion a set of 100 independent realizations of fixed size T with a Hurst exponent H = 0.7. We compute averages of the estimates on the realizations and use the standard deviation as error bars in the different graphs.

In subsections 3.1.3 and 3.1.4, we characterize respectively the bias and standard deviation (std) over realizations of our estimators of entropy, auto-mutual information and entropy rate.

Convergence / bias

We detail here how the ersatz entropy rate evolves with T and k. We report in Fig. 1a our results for all possible values of the couples (log 2 (T), k) ∈ [9, ..., 17] × [4, ..., 18], while τ is set to 1 here. According to eq.( 34), the ersatz entropy rate of the fBm converges for large T to the value H fBm 1 (horizontal black line in Fig. 1a) thanks to the vanishing of the correction term C(τ/T), according to [START_REF] Flandrin | Wavelet analysis and synthesis of fractional Brownian motion[END_REF].

Fig. 1a can be interpreted as describing the behavior of the bias of the estimator. This bias vanishes non-monotonically as k/T 1 m+1 . When k/T 1 m+1 is reduced, first the bias is positive and diminishes toward negative values and then converges to zero. This behavior was previously reported for the k-nn mutual information estimator applied for stationary processes [START_REF] Granero-Belinchón | Probing high-order dependencies with information theory[END_REF][START_REF] Kraskov | Estimating mutual information[END_REF][START_REF] Gao | Demystifying Fixed k-Nearest Neighbor Information Estimators[END_REF], and we confirm it is valid for the fBm. for the fBm (a, in black) and for the Hermitian (b, in blue) and even-Hermitian (c, in red) log-normal processes.

k/(T 1/2 ) h(1,1) T k/(T 1/2 ) k/(T 1/2 ) (a) (b) (c) 0
We observed the same convergence for a large range of scales τ > 1: the ersatz entropy rate then converges to H 1 fBm + H ln τ for large T with the same behavior of the bias.

Standard deviation of the estimates

We present in Fig. 2a the evolution of the standard deviation of the ersatz entropy, mutual information and entropy rate with T for τ = 1. The standard deviation of both the entropy and mutual information is large, and does not decrease when T -and hence the number of samples -increases. On the contrary, the standard deviation of the entropy rate is much smaller and decreases when T increases. We attribute this feature to the dependence of the quantities on the observation time T, see eqs. [START_REF] Cont | Scaling in stock market data: stable laws and beyond[END_REF] and [START_REF] Uhl | A systematic evaluation of the scale invariance of texture recognition methods[END_REF] for the fBm. While HT and ĪT increase as ln T, this is not the case for hT which is independent on T (up to small corrections, negligible for smallish τ). Although it is difficult to explain why the standard deviation of the entropy and mutual information remain constant when T increases, it seems that this results from a balance between the non-stationarity (in ln T) and the increased statistics. On the contrary, for the entropy rate which is stationary, the decrease of the std is as expected.

As a conclusion, both the bias and the standard deviation of the ersatz entropy rate increase when k increases or T decreases and can be made arbitrarily small by increasing the window size T. In the remainder of this article, we choose k = 5 and when studying the behavior of information theoretic quantities on the scale τ, we set T = 2 16 .

Dependence on times T and τ

In this section, we present a detailed numerical study of the ersatz entropy, auto-mutual information and entropy rate of the fBm with H = 0.7. In particular, we present a quantitative comparison with the analytical expressions [START_REF] Ivanov | Levels of complexity in scale-invariant neural signals[END_REF][START_REF] Drozdz | Imprints of log-periodic self-similarity in the stock market[END_REF] in the general framework, as well as with analytical expressions [START_REF] Cont | Scaling in stock market data: stable laws and beyond[END_REF][START_REF] Uhl | A systematic evaluation of the scale invariance of texture recognition methods[END_REF][START_REF] Zografos | Expressions for Rényi and Shannon entropies for multivariate distributions[END_REF] in the practical framework for the fBm. These comparisons allow: first, to validate the analytical expressions obtained for fBm in the practical framework, and second to show that the information theoretic quantities in the practical framework evolve in T and τ exactly as their counterparts evolve in the general framework in t and τ. To compare analytical and numerical results, we vary the window size T, the scale τ and the embedding dimension m. T (stars), for τ = 1, as functions of T, for the fBm (a, black) the Hermitian (b, in blue) and even-Hermitian (c, in red) log-normal processes.

Entropy and auto-mutual information

Dependence on T

The left column of Fig. 3 shows the ersatz Shannon entropy H(m,τ) T (Fig. 3a) and auto-mutual information Ī(m,1,τ) T (Fig. 3c) at a given scale τ = 1, as a function of ln T. The evolution of these two quantities for m = 1 is very close to H ln T, which is represented by a continuous black line. This is in agreement with eq.( 30) and eq. [START_REF] Uhl | A systematic evaluation of the scale invariance of texture recognition methods[END_REF]. For m > 1, we obtain in the practical framework the behaviors predicted in the general framework, replacing t by T in the equations. We observe that the auto-mutual information does not depend on the embedding dimension m, while the entropy does, with an offset that seems to depend linearly on m. The dependence of the entropy and the auto-mutual information on the time window T is the signature of the non-stationarity of the signal.

Dependence on τ

The right column of Fig. 3 shows the ersatz Shannon entropy and auto-mutual information for a fixed window size T = 2 16 when varying the scale parameter τ. The ersatz Shannon entropy behaves as (m -1)H ln(τ), see Fig. 3b, in agreement with eq.( 26) or eq.( 28). The ersatz auto-mutual information behaves as -H ln(τ) for any embedding m, see Fig. 3d, in agreement with eq.( 31), thus suggesting this formula is valid for any embedding dimension.

Stationarity of the entropy rate

Fig. 4a shows that the ersatz entropy rate h(m=1,τ)

T with embedding dimension m = 1 is almost constant when T is varied. For embedding dimensions m > 1, there is a small variation, of about 15%, much smaller than the 200% variation observed for either the entropy or the auto-mutual information (Fig. 3a,c) on the same range of T. This small dependence on T can be due to the correction in eq.( 34), which may depend on m. We argue that it is mostly due to bias, which increases with the embedding dimension. Indeed, we observe that the entropy rate seems to converge for larger T to the same value close to H fBm 1 for all m. As a larger T corresponds to a larger sampling of the statistics, the bias is reduced, as reported in Fig. 1. Moreover, for m = 1, eq.( 34) predicts a positive correction that vanishes when T is large: on the contrary we observe a convergence to a value lower than H fBm 1 which hints H(m,1) that the bias is negative and larger than the theoretical correction. This suggests that the form of eq.( 34) is still valid for embedding dimensions m > 1.

T (a) Ī(m,1,1) T (c) ln(T ) (b) m = 1 m = 2 m = 3 m = 4 (d) ln(τ ) 0 1 2 3 4 5

Entropy rate dependence on scale τ

Fig. 4b shows that for a fixed window size T = 2 16 the ersatz entropy rate is proportional to H ln(τ). We have added a black line defined by the linear function H FBM 1 + H ln τ, as suggested by eq.( 34) without the corrective term. This black line perfectly describes the evolution of the entropy rate with the scale τ, which is independent on the embedding dimension m.

To observe the finer evolution of the entropy rate on the scale τ, we subtract the main contribution H ln τ to the entropy rate and we plot h(m,τ)

T -H ln τ for different embedding dimensions in Fig. 5. We observe a slight increase, which is larger for larger embedding dimensions. For m = 1, the correction term can be evaluated from eq. [START_REF] Chakraborty | Self-similar and fractal nature of internet traffic[END_REF], and is at most 2.10 -3 , and does not account for the evolution reported here, which is probably due to the bias which increases when the number of points -which is proportional to T/τ -decreases and when the embedding m increases.

For a scale invariant self similar process, the standard deviation σ τ of the increments of size τ behaves as σ τ = σ 1 τ H . Subtracting H ln τ amounts to subtracting ln σ τ : for each scale τ, this corresponds to normalizing the down-sampled data (taking one point every τ points) by the standard deviation σ τ of the increments of size τ. When the Hurst exponent is a priori unknown, σ τ can be computed, and used to compute the main contributionln(σ τ ); thus the fine evolution of the entropy rate with τ can be used as a tool to probe the deviation from the self similarity assumption, which is interesting for multifractal signals. T for a motion built from a Hermitian (blue) or even-Hermitian (red) log-normal noise, as a function of a) the time window size T or b) the time scale τ. Results for the fBm (from Fig. 4 with m = 1) are reported in black for comparison. T = 2 16 and k = 5. The horizontal lines in (a) indicates the entropy H 1 of the noise (in black for the fBm and in red and blue for a log-normal process).

The evolution of h(1,τ)

T with the time scale τ is presented in Fig. 6b). In the same way as for the fBm, we again observe a large increase, almost proportional to ln τ. Because this strong tendency originates from the increase of the standard deviation σ τ of the increments of size τ when τ increases, we again normalize the entropy rate by subtracting ln σ τ = H ln τ. Results are presented in Fig. 7, together with results for the fBm with m = 1 for comparison.

The normalized ersatz entropy rate of the motion built from the even-Hermitian log-normal noise appears as almost independent of τ. This behavior is identical to the one observed for the fBm, but the remaining constant value is different (H fBm 1 or H ln 1 ). The ersatz entropy rate of the fBm (in black) and the even-Hermitian motion (in red) both behaves exactly as H ln τ, which is the expected behavior for a self-similar process, see eq.( 27). On the contrary, the motion built with the Hermitian transformation of rank 1 exhibits an additional variation in τ: the normalized entropy rate h(m,τ)

T ln(σ τ ) evolves from the value H ln 1 at τ = 1 -expected for the motion built with a log-normal noise and obtained for the even-Hermitian process at any τ -up to the value H fBm 1 -expected for a Gaussian process, and obtained for the fBm at any τ.

As a conclusion, one can estimate the Hurst exponent of a perfectly self-similar process as the slope of the linear fit in ln τ of the ersatz entropy rate. This is a valid approach for the fBm and the motion built from the noise constructed with the even-Hermitian transformation, because the ersatz entropy rate then behaves linearly in ln τ. On the contrary, the motion built using an hermitian transformation of rank 1 does not appear as perfectly self-similar. This can be indeed verified by plotting the normalized PDFs (setting the standard deviation to unity) of the increments m tm t-τ of the motions for various values of τ. As can be seen in Fig. 8 the PDFs of the increments of the "standard log-normal process" varies with the scale τ, while these of the "even-Hermitian motion" remain identical. For τ = 1, the increments are nothing but the log-normal noises, which are log-normal, as prescribed. For large τ, the increments of the "even-Hermitian motion" remain log-normal, while the increments of the standard log-normal motion" deforms and seems to become more Gaussian. The ersatz entropy rate catches this fine evolution perfectly.

Application of the practical framework to a multifractal process

We now explore the proposed time-averaged framework on the Multifractal Random Walk, to illustrate how it performs on a multifractal process. The multifractal random walk (MRW) [START_REF] Bacry | Multifractal random walk[END_REF][START_REF] Bacry | Multifractal stationary random measures and multifractal random walk with log-infinitely divisible scaling laws[END_REF] h(1,τ) is a popular multiplicative cascade process widely used to model systems that exhibit multifractal properties [START_REF] Delour | Intermittency of 1D velocity spatial profiles in turbulence: a magnitude cumulant analysis[END_REF]. Like the fBm, the MRW is a motion obtained by integrating -again with eq. ( 2) -a stationary noise W MRW = {w MRW t } t∈R such that

w MRW t = w fGn t e ω t (35)
where W fGn = {w fGn t } t∈R is a fGn with parameter H fGn and Ω = {ω t } t∈R is a Gaussian random process, independent of X fGn with a correlation function

c ω (τ) = -c 2 log L |τ| + 1 if |τ| < L (36)
= 0 otherwise where L is the integral scale, set here to L = T.

The MRW is a scale invariant process: the power spectrum of its time-derivative W MRW behaves as a power law with an exponent 2(H fGnc 2 ) + 1, which would be the Hurst exponent obtained for a fGn with parameter H fGnc 2 . Any moment of order q of the increments of size τ behaves as a power law of τ with the exponent ζ(q). Contrary to the fBm, the MRW is not exactly self similar and exhibits intermittency: ζ(q) = H fGn q -c 2 2 q 2 is not a linear function of q, as expected for a self-similar process. As a consequence, the shape of the PDF of the increments depends on the scale.

We choose the parameter H fGn such that the power spectrum of the noise W MRW is identical to the one of the fBm used in the former sections, i.e., H fGn = 0.7 + c 2 . We set the parameter c 2 = 0.025, a value widely used to model the intermittency of Eulerian turbulent velocity field [START_REF] Chevillard | A phenomenological theory of Eulerian and Lagrangian velocity fluctuations in turbulent flows[END_REF].

Figure 9 compares the evolution of the PDF of the increments of the fBm and the MRW. As expected, no change is observed for the fBm, while the PDF of the MRW has wider tails for smaller τ. The fBm is perfectly self-similar, while the MRW exhibits intermittency [START_REF] Granero-Belinchón | Kullback-Leibler divergence measure of intermittency: Application to turbulence[END_REF]: the PDF of its increments T and the embedding time scale τ are varied: we obtained analytical expressions for embedding dimension m = 1, and confirmed them numerically for m ≥ 1. Besides the fBm, we reported numerical observations for various self similar or multifractal processes. The ersatz entropy H(m,τ)

T always diverges logarithmically in T while the ersatz entropy rate h(m,τ)

T always behaves as almost independent of T. The examination of how the ersatz entropy rate h(m,τ)

T depends on the scale τ provides a fine exploration of either the self-similarity or the multifractality of the process.

This exploration of the multifractality of a non-stationary process with stationary increments using the ersatz entropy rate h(m,τ) T (M) gives a viewpoint very similar to the one reported when analyzing the increments of the process with the regular Shanon entropy, as reported in [START_REF] Granero-Belinchón | Kullback-Leibler divergence measure of intermittency: Application to turbulence[END_REF]. We are currently investigating how to relate quantitatively the two approaches.

In the same vein, the ersatz entropy rate allowed us to discriminate two different non-stationary processes, and obtain fine differences in their self similarity properties (figure 7), in close relation to a method using the entropy rate of the increments of the signal, as exposed in [START_REF] Granero-Belinchón | Probing high-order dependencies with information theory[END_REF]. A possible connection is also under investigation.

Through all this article, we have estimated the ersatz quantities of a process on a single trajectory [t 0 ; t 0 + T] of this process; this situation corresponds to the worst case scenario where only a single realization of the process is know. If enough experimental data are available, one can improve the estimation of the ersatz quantities in two ways. First, if the same experiment has been conducted multiple times, and thus multiple realizations are available over the time interval [t 0 ; t 0 + T], one can use all these independent realizations to enhance the estimation of the time-averaged PDF. Second, if a single but long enough realization of size T is available, one can split it into multiple time intervals [kT; (k + 1)T], k ∈ [0.. T /T ] and the use these intervals as independent realizations as in the first case. This later situation is made possible by the assumption that the increments of the signal are not only stationary, but also centered.

.

  As a consequence, the ersatz entropy rate h(m,τ)

Figure 1 .

 1 Figure 1. Dependence of h(1,τ) T (for τ = 1)) on k/T 1 2for the fBm (a, in black) and for the Hermitian (b, in blue) and even-Hermitian (c, in red) log-normal processes.

Figure 2 .

 2 Figure 2. Standard deviations of H(1,τ) T (triangles), Ī(1,1,τ) T

Figure 3 .Figure 4 . 1 +Figure 5 .

 3415 Figure 3. (a) Entropy H(m,τ) T

  Figure 6.

Figure 7 .

 7 Figure 7. Ersatz entropy rate h(m=1,τ) T ln(σ τ ) for motions built on Hermitian (blue) or even-Hermitian (red) log-normal noise, together with results for the fBm (black) as a function of ln τ. T = 2 16 and k = 5. The straight lines in (b) indicate the theoretical values of the entropy of the processes.

Figure 8 .

 8 Figure 8. PDF of the increments of the (a) Hermitian and (b) even-Hermitian log-normal motions of size τ = 2 j , from j = 0 (bottom) up to j = 6 (up). Curves have been arbitrarily shifted on the Y-axis for clarity.

Figure 10 .Figure 11 .

 1011 Figure 10. Ersatz entropy rate h(m=1,τ) T of a MRW with H = 0.7. (a): as a function of the window size T for fixed τ = 1. (b): as a function of the scale τ for T = 2 16 . The horizontal line in (a) indicates the numerical value H MRW 1

Acknowledgments:

The authors wish to thank L. Chevillard for stimulating discussions. This work was supported by the LABEX iMUST (ANR-10-LABX-0064) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

Application of the practical framework to non-Gaussian self-similar processes

In this section, we turn to non-Gaussian processes and describe the results obtained in the time-average framework generalized in this larger class of processes.

Procedure

We construct two different motions, in the very same way as we did for the fBm. We integrated two log-normal noises synthesized with the same log-normal marginal distribution and with the same correlation function [START_REF] Crutchfield | Regularities unseen, randomness observed: The entropy convergence hierarchy[END_REF] as the fGn, but different dependance structure. To generate these noises, we use the methodology proposed in [START_REF] Helgason | Synthesis of multivariate stationary series with prescribed marginal distributions and covariance using circulant matrix embedding[END_REF] to obtain the log-normal marginal by applying two different transformations to the cumulative distribution function F Z of a Gaussian white noise Z: the Hermitian transformation of rank 1 ( f 1 (z) = F -1 (F Z (z))) and the even-Hermitian transformation of rank 2 :

), where F is the cumulative distribution function of the targeted log-normal distribution. This synthesis is performed with the toolbox provided at www.hermir.org. Once the two log-normal noises have been generated, they are integrated using eq.( 2) to obtain two non-stationary scale invariant processes with non-Gaussian statistics.

The dependence structures of the two log-normal noises were previously studied in detail [START_REF] Granero-Belinchón | Probing high-order dependencies with information theory[END_REF]: while the correlation function is the same for the two noises -and identical to the targeted one of the fBm given by (20) -the complete dependence structure was shown to be different.

To study these two non-stationary and non-Gaussian motions, we use again realizations of T = 2 16 points, k = 5 neighbors and we focus on the case where embedding dimension m = 1 and Hurst exponent H = 0.7.

Bias and standard deviation

We report in Fig. 1b and 1c the evolution of the ersatz entropy rate of the Hermitian and the even-Hermitian log-normal processes in function of k T 1/2 . We observe exactly the same behavior as for the fBm: the entropy rate converges to H ln 1 , the entropy of the log-normal process at unit-time 1 (horizontal blue/red line in Fig. 1b,c), which then gives an estimation of the bias of our estimator, which appears to be the same as for the fBm.

We report in Fig. 2b and Fig. 2c the behavior of the standard deviation of the estimators. Again, exactly as for the fBm, the standard deviation is large for the ersatz entropy and the ersatz auto-mutual information, while it is much smaller for the ersatz entropy rate.

Again, both the bias and the standard deviation of the entropy rate increase when k increases or T decreases and can be made arbitrarily small by increasing T. These results do not depend on the marginal distribution: they have been obtained not only for the fBm with Gaussian statistics, but also for two motions built on log-normal noises.

Dependence on times T and τ

The evolution of h(1,τ)

T on the time window size T for the two motions is presented in Fig. 6a). As it was the case for the fBm, h

T depends only weakly on T, and seems to converge for larger T to the value H ln 1 , up to a small corrective term.

1 if X is a log-normal process of mean µ and standard deviation σ, then the process log X is Gaussian with the mean

and the standard deviation σ = 2 log 1 + σ 2 µ 2 and the entropy of X can be expressed as [START_REF] Granero-Belinchón | Probing high-order dependencies with information theory[END_REF]: is deformed when the scale τ of the increments is varied, although no analytical expression of the PDF is available.

We apply our practical framework and plot in Fig. 10a) the evolution of the ersatz entropy rate of the MRW with T. Again, the entropy rate seems to be independent of T. We nevertheless observe a small tendency to increase towards the value H MRW 1 the entropy of the MRW at unit-time. Here, because there is no analytical expression of the PDF, H MRW 1 cannot be derived analytically and we numericaly estimate its value.

The dependence in τ is plotted in Fig. 10b). We again observe a strong linear evolution of the ersatz entropy rate in H ln τ. After subtracting this strong tendency (Fig. 11), we still observe an evolution with τ, but this evolution appears much weaker than for the Hermitian log-normal (blue curve in Fig. 7). Indeed, the deformation of the PDFs of the increments when varying τ is much slower for the MRW (Fig. 9b) than for the Hermitian log-normal (Fig. 8a).

Discussion and Conclusions

We proposed a new framework in information theory to analyze a non-stationary process by considering it as resulting from a gedanken stationary process and estimating the PDF by cumulating all available samples in a time interval of size T. This framework hence considers a PDF obtained by time-averaging over a time window [t 0 : t 0 + T], and then proceeds to compute the associated information theory quantities. In particular, the ersatz entropy HT (X) that is then defined can be interpreted as the amount of information characterizing the complete trajectory {X t , t ∈ [t 0 , t 0 + T]} of the process X. If we assume that the increments of X are stationary and centered, then HT (X) and all other information ersatz theoretical quantities depend only on the duration T and not of the first time t 0 .

We illustrated our approach by focusing first on a model system: the fractional Brownian motion. We derived in this context the analytical expressions of the ersatz entropy, ersatz auto-mutual information and ersatz entropy rate, which allowed a pedagogical description of our new information theory quantities. We also reported how the ersatz quantities behave when the time-interval size

Appendix Entropy of a time-embedded signal

The time-embedded vector x (m,τ) t (eq. 6) can be mapped into the vector x(m,τ) t ≡ (x t , δ τ x t , δ τ x t-τ , ..., δ τ x t-(m-2)τ ) by the linear transformation Q m :

where Q m is the band matrix defined as: ), which proves [START_REF] Yaglom | Correlation theory of processes with random stationary nth increments[END_REF].


Matteo Conforti
email: matteo.conforti@ing.unibs.it

F Baronio

Stefano Trillo

Dispersive shock waves in phase

come

Parametric nonlinearities characteristic of quadratic media can exactly balance dispersion (and or diffraction) to form solitons [1,2]. Conversely, in such media, the phenomena occurring when nonlinearities overwhelm dispersion, are left mostly unexplored. Studies performed in this regime for cubic Kerr media show that dispersive (or collisionless) shock waves (also known as undular bore) develop in settings ruled by the defocusing nonlinear Schrödinger equation (NLSE). The distinctive feature of this phenomenon is the onset of oscillations determined by the regularizing action of dispersion over infinite gradients produced by the nonlinear response [3,4]. This process has been observed in fibers [5] and more recently in the spatial domain [6][7][8] where suitable generalizations of the NLSE account for the specific nature (saturable, nonlocal, etc.) of cubic nonlinearities. On the other hand, a properly generalized NLSE can also describe the propagation of light pulses in second-harmonic generation (SHG) in the so-called cascading limit [1]. In this framework, formation of steep gradients are attributed to self-steepening terms that stem from corrections to the main cascading term (i.e., the effective nonlinear index due to repeated up-and down-conversions), whose effects have been investigated theoretically and experimentally [1,[9][10][11][12].

In this letter we show, however, that dispersive shock formation can occur regardless of such corrections, driven by the leading-order index change due to mismatched SHG, when the group-velocity dispersion (GVD) is sufficiently weak. To this end we introduce a hydrodynamic reduction of the standard SHG model that we perform by starting from a suitable renormalization of the latter. We validate the outcome of such an approach by means of numerical analysis of the original model, predicting that this regime is accessible in practice. In this context, the self-steepening term that arises from group-velocity mismatch (GVM) [9][10][11] appears to play only a perturbative role that modifies the symmetry of the wave-breaking process.

We consider type I mixing of pulse envelopes w(Z, T ) and v(Z, T ) (electric fields in V/m) at fundamental frequency (FF) and its second harmonic (SH), respectively, which travel along Z in a normally dispersive quadratic medium. In the retarded time frame T moving with the FF group-velocity, the model that governs SHG reads as [1,2,11] iw Z -

β ′′ 1 2 w T T + χvw * e -i∆kZ = 0, iv Z + iδv T - β ′′ 2 2 v T T + χw 2 e i∆kZ = 0, (1)
where

β ′′ 1,2 stands for GVDs, δ = β ′ 2 -β ′ 1 is the GVM (β ′ 1,2
are the inverse group velocities), ∆k is the wave-number mismatch, and χ is the effective nonlinearity. We introduce scaled variables suited to investigate the weakly dispersing regime: z = Z/L, t = T /T 0 , a 1 = w/ √ P , a 2 = v/ √ P exp(-i∆kZ), with L = √ L D L N L , L N L = |∆k|/(χ 2 P ) and L D = T 2 0 /β ′′ 1 being the nonlinear and dispersive lengths associated with input pulse duration T 0 and peak FF field √ P . In terms of such variables, introducing also the small parameter ε = L N L /L D as well as δk = ∆kL, γ = β ′′ 2 /β ′′ 1 , and η = L/L GV M , where

L GV M = T 0 /δ, we cast Eqs. (1) in dimensionless form iεa 1z -ε 2 2 a 1tt + |δk|ε a 2 a * 1 = 0, iεa 2z + iεηa 2t -γ ε 2 2 a 2tt -εδka 2 + |δk|ε a 2 1 = 0.(2)
We consider the standard case such that only a FF pulse is launched. By using the SH asymptotic expansion

a 2 = ∞ n=0 a (n)
2 /|δk| n , and the method of repeated substitution [1], retaining leading-order terms, we arrive at a single evolution equation (obtained at n = 1 order):

iεu z - ε 2 2 u tt + s|u| 2 u + 2iεσ|u| 2 u t = 0, (3)
where, for sake of clarity we defined u = a 1 , s = sign(δk) and σ = η/(ε|δk|). In the following we consider effective self-focusing nonlinearities (s = 1), which yields the integrable defocusing NLSE in the limit of null GVM (σ = 0). When σ = 0, Eq. ( 3) is also integrable, being equivalent to Chen-Lee-Liu equation (CLLE) [12][13][14]. We em-phasize that the validity of Eq. ( 3) entails weak overall FF to SH conversion, which, assuming small GVM and GVD, requires |δk| ≫ 4/ε. If we take |δk| = M 4/ε, M ≫ 1, then σ = η/(4M ). For ε ≪ 1, the leading order system (geometric optics or hydrodynamic limit), that stems from the WKB transformation u(z, t) = ρ(z, t) exp[iS(z, t)/ε] applied to Eq. ( 3), takes the form of the following first-order quasi-linear system of equations:

ρ z + (vρ + σρ 2 ) t = 0 v z + vv t + ρ t + 2σ(ρv) t = 0 ( 4
)
where v = -S t is the equivalent fluid velocity. Equations (4) entail that a smooth input pulse breaks at finite distance z s [4, [14][15][16]. At variance with the case of a single hyperbolic equation where such distance can be evaluated a priori, Eqs. ( 4) allow us to give only a lower and upper bound (i.e. a capture window) for z s . This can be done by applying a well-known criterion by Lax [14,16], which requires Eqs. ( 4) to be cast in diagonal form r ± z + λ ± r ± t = 0 in terms of Riemann invariants r ± = ±(v + 2σρ) + ρ(1 + 2σv), with corresponding eigenvelocities λ ± = v + 2σρ ± 2 ρ(1 + 2σv) (also expressed through Riemann variables using identical formulas of the NLSE case: λ -= (-3r -+ r + )/4, λ + = (-r -+ 3r + )/4). We focus henceforth on the initial condition a 1 (0, t) = ρ(0, t) = ν + (1 -ν) exp(-t 2 ), v(0, t) = 0 representing an unchirped Gaussian input pulse on a weak pedestal ν ≪ 1, for which the results are summarized in Fig. 1. For null background (ν = 0) and σ = 0 we find z s ∈ [ √ e/3, √ 2e/3], while the shock distance turns out to decrease with σ [see Fig. 1(a)]. A similar trend is obtained in the presence of pedestal, as displayed in Fig. 1(b) for ν = 0.1, which shows also how the distance z s computed via direct numerical integration of Eqs. ( 4) is correctly captured by the bounds from the Lax criterion. We then proceeded to verify the shock formation process via numerical integration of the original model (2). The whole physical parameter space can be explored by varying the GVD parameter ε and the GVM parameter η. We report examples for a fixed ε = 0.05 and δk = 200/ε (i.e. M = 50 ≫ 1) which ensures to be 3). The solid blue curve is the input. Here η = σ = 0, ε = 0.05. in the cascading limit. Without loss of generality we present results for ν = 0.1 (pedestal power equal to 1 % of the peak power) which, though not essential for the physics, enhances the visibility of post-breaking oscillations. First, we consider the case of zero GVM (η = 0), setting γ = 2 to account for the (usually) higher GVD at SH. The evolution of the FF field displayed in Fig. 2(a) shows symmetric steepening on both the leading and the trailing edges, until beyond the breaking distance z s ∼ 1 (in good agreement with z s predicted from the hydrodynamic limit), undular bores appear, filling extending portions of time with fast oscillations. The SH component shown in Fig. 2(b) remains always small, though it shows a complicated behavior dominated by periodic back-conversions and a sort of temporal focusing at short distance, which is found to be ruled by the GVD at SH. For comparison Fig. 2(c) reports the corresponding evolution according to Eq. ( 3), which, as shown, turns out to be in satisfactory agreement with that of Fig. 2(a), as also witnessed by the snapshots displayed in Fig. 2(d).

Next, we consider the case of non-vanishing GVM, taking η = 30, γ = 2, for which the evolution of the FF and SH fields are shown in Fig. 3(a,b). In this case, the GVM leads to break the symmetry in time, enhancing the steepening effect on the trailing edge of the pulse which turns out to break at a shorter distance z ≃ 0.8 compared with the GVM-free case. This is in agreement with the estimates from the hydrodynamic limit (4) [see Fig. 1(b)], and is satisfactorily reproduced by the reduced CLLE [Eq. 3], whose dynamics is reported in Fig. 3(c) (in this case σ = 0.15). Furthermore, the leading edge also undergoes a secondary breaking at a longer distance (z ≃ 1.1), where the trailing edge already exhibits a welldeveloped dispersive shock wave. This strong asymmetry is also clear from the snapshots shown in Fig. 3(d). Note also that the SH evolution exhibits weak initial emission of non-phase-matched component traveling with natural SH group velocity, while the rest of the SH field follows almost adiabatically the FF. An accurate comparison between the output fields obtained from Eqs. (2) and Eq.

(3) shows slight discrepancies for higher values of |σ|. In fact, our numerics suggest that the reduction of Eqs.

(2) to Eq. (3) following from truncation at order n = 1, holds for small |σ| (roughly |σ| < 0.15 for ε = 0.05), i.e. when GVM is small enough with respect to mismatch. A better accuracy up to the shock point can be obtained even for larger σ by including other terms in Eq. (3) [we find the most significant to be -2sεσ 2 (uu t ) t u * arising at order n = 2, details will be given elsewhere]. However, beyond z s , even improved reductions break down for high values of |σ|. This can be explained as follows: on one hand, shock entails pulse spectral broadening; on the other hand, GVM introduces a frequency dependence of the effective phase-matching (ω P M ≈ δk/η). When the spectrum broadens enough to reach the effective phase-matching frequency ω P M , portions of the FF spectrum are phase-matched and give strong conversion which make the single-field reduction to break down.

Finally, one can address the observability of the process by returning to dimensional quantities (peak power and duration, characteristic length L), as follows:

P = ∆k 2 χ 2 |δk|ε , T 0 = 1 χε |∆k|β ′′ 1 P , L = |δk| |∆k| . (5)
In order to observe the dynamical interplay between GVM and GVD illustrated in Fig. 3, we can consider for instance a periodically poled stoichiometric lithium tantalate sample [17], which has been successfully exploited to study femtosecond pulse propagation with strong phase-and group-velocity mismatch [11]. A poling period of Λ = 12.66µm at FF wavelength of 1400 nm gives a residual mismatch ∆k = 1250 cm -1 , and dis-persion parameters β ′′ 1 = 0.1 ps 2 /m, δ = 324 ps/m [17]. Using χ = 1.43•10 -5 m/V (plane-wave limit), the values ε = 0.05 and δk = 4000 yield T 0 ≈ 260 fs, L ≈ 3.2 cm, and an input peak intensity I ≈ 100 GW/cm 2 . The latter compares favorably with the values reported in the literature [10], and will be further reduced with nonnegligible contribution from cubic nonlinearities which is of the self-focusing type, or in waveguide geometries. Additionally we get η = 40 and correspondingly σ = 0.2.

In conclusion, we have reported a hydrodynamic reduction describing shock formation in highly mismatched SHG. Oscillatory wave-breaking driven by the interplay of cascading and GVD has been demonstrated numerically and predicted to be observable. The perturbative role of GVM-induced self-steepening is expected to cause an asymmetry in the shock process. Future work will address how this scenario changes in the (possibly phase-matched) strong conversion regime.

Funding from PRIN 2009 project (No. 2009P3K72Z) is acknowledged.

Fig. 1 .

 1 Fig. 1. Shock capture window (shaded area) vs. GVM parameter σ: (a) Gaussian input; (b) Gaussian on pedestal, ν = 0.1. The bullets give the shock distance obtained via direct numerical integration of Eqs. (4).

Fig. 2 .

 2 Fig. 2. (a-b) Color level plot of FF field |a 1 | (a) and SH field (b), ruled by Eqs. (2); (c) same for FF field |u| from Eq. (3); (d) Snapshots of output field moduli |a 1 | (FF, solid black) and |a 2 | (SH, solid red) from Eqs. (2), compared with |u| (FF, dashed red) from Eq. (3). The solid blue curve is the input. Here η = σ = 0, ε = 0.05.

Fig. 3 .

 3 Fig. 3. As in Fig. 2, with non vanishing GVM such that η = 30 [Eqs. (2)] and σ = 0.15 [Eq. (3)].

Informational Fourth Page

In this section, please provide full versions of citations to assist reviewers and editors (OL publishes a short form of citations) or any other information that would aid the peer-review process.


I Mazurenko

V P Hitaishi

E Lojou

Recent advances in surface chemistry of electrodes to promote direct enzymatic bioelectrocatalysis

Keywords: Enzyme, catalysis, direct electron transfer, electrode chemical modification, enzyme engineering

Redox enzymes catalyze major reactions in microorganisms to supply energy for life. Their use in electrochemical biodevices requires their integration on electrodes, while maintaining their activity and optimizing their stability. In return, such applicative development puts forward the knowledge on involved catalytic mechanisms, providing a direct electrode connection of the enzyme is fulfilled. Enzymes being large molecules with active site embedded in an insulating moiety, direct bioelectrocatalysis supposes strategies for specific orientation of the enzyme to be developed. In this review, we summarize recent advances during the past three years in the chemical modification of electrodes favoring direct electrocatalysis. We present the different methodologies used according to the electrode materials, including metals, carbon-based electrodes or porous structures, and discuss the gained insights into bioelectrocatalysis. We especially focus on enzyme engineering which appears as an emerging strategy for enzyme anchoring. Remaining challenges will be discussed with regards to these last findings.

Introduction

Redox enzymes are key elements in microorganisms where they catalyze crucial reactions at active sites composed of metals (Cu, Ni, Fe, Mo, W, etc.) or organic compounds (flavin adenine dinucleotide as an example). In addition, cofactors are maturated in the way to act as electronic relays enabling fast electron transfer between the active site, which can be buried in the insulating protein moiety, and the physiological partner or the substrate. Use of these enzymes as bioelectrocatalysts in biodevices (biosensors, bioreactors or enzymatic fuel cells), imposes their immobilization on electrodes, while maintaining their activity. When the enzyme is electrically connected to the electrode without the help of redox mediators acting as shuttles, the direct electron transfer (DET) process will allow not only the fundamental study of enzymatic catalysis, but also will decrease the overvoltages, increase the interfacial electron transfer (ET) rate, and simplify the future device [START_REF] Milton | Direct enzymatic bioelectrocatalysis: differentiating between myth and reality[END_REF]. Because enzymes are usually in the diameter range 5-10 nm, direct wiring supposes the orientation of the protein that places the active center or an electronic relay at a tunneling distance of the electrode (typically less than 2 nm). Within this objective, electrode chemical functionalization aims: i) to adsorb or graft the enzyme specifically or preferentially via the surface anchors thanks to suitable interactions, hence to succeed in DET, ii) to enhance the loading of enzymes, hence to enhance the catalytic current and eventually get the knowledge of the electroactive enzyme amount, iii) to narrow the orientation, hence to enhance the ET rate, iv) to maintain enzyme conformation, hence to enhance the electrocatalytic activity and stability. Generally speaking, three main ways are followed for DET-favored enzyme attachment: physical adsorption (electrostatic, hydrophobic, - interactions), chemical linkage (amide, imine, maleimide, click chemistry), and host-guest interactions [START_REF] Nm | Direct electron transfer between a site-specific pyrene-modified laccase and carbon nanotube/gold nanoparticle supramolecular assemblies for bioelectrocatalytic dioxygen reduction[END_REF]. Many works are still devoted to the effect of electrostatic interactions on DET, which is one of the main factor controlling protein-protein recognition, hence protein-electrode interactions. Although the existence of a strong dipole moment in the protein may narrow down the distribution of orientations to a set of prevailed ones [START_REF] Mazurenko | How the intricate interactions between carbon nanotubes and two bilirubin oxidases control direct and mediated O2 reduction[END_REF], it is unlikely to achieve a 100% ordering of adsorbed proteins in a desired orientation. Furthermore, immobilization through electrostatic forces should not be sufficient for long term stability of enzyme based bioelectrodes, and hence, more specific attachment should be searched.

Previous reviews on enzyme/electrode interactions provided some key elements [START_REF] Yates | Methodologies for "Wiring" Redox Proteins/Enzymes to Electrode Surfaces[END_REF][START_REF] Jenner | Electrochemistry of surface-confined enzymes: Inspiration, insight and opportunity for sustainable biotechnology[END_REF]. The current review focuses on the development and emerging concepts reported during the last 3 years. We will divide our discussion depending on the type of electrode which is used and the corresponding available modifications. We will extend our discussion on engineering of enzymes allowing specific attachment on electrodes, an emerging domain in bioelectrocatalysis. We intend to provide the reader with the most relevant elements required to succeed in DET with all the enzymes newly identified.

Modification of metallic surfaces

Metallic electrodes used for bioelectrocatalysis are mainly made of Au, although Ag, Cu, Ni, Fe and metal oxides or alloys are also reported. Metal modification is expected to prevent change in enzyme conformation induced by certain amino acids interaction with the bare metal [START_REF] Pankratov | Interfacial behavior and activity of laccase and bilirubin oxidase on bare gold surfaces[END_REF]. Self-assembled-monolayers (SAMs), most often based on Au-S bonds, on metal surfaces are ideal tools to understand the role of surface hydrophobicity or charges on the efficiency of the electrocatalysis as highlighted by studies on copper efflux oxidase [START_REF] Chumillas | Comprehensive Study of the Enzymatic Catalysis of the Electrochemical Oxygen Reduction Reaction (ORR) by Immobilized Copper Efflux Oxidase (CueO) From Escherichia coli[END_REF],

cellobiose dehydrogenase [START_REF] Lamberg | Electrical activity of cellobiose dehydrogenase adsorbed on thiols: Influence of charge and hydrophobicity[END_REF], or human sulfite oxidase [START_REF] Kalimuthu | Chitosan-Promoted Direct Electrochemistry of Human Sulfite Oxidase[END_REF]. The additional advantage is the possibility to couple electrochemistry with analytical surface methods, such as quartz crystal microbalance (QCM), surface plasmon resonance (SPR), ellipsometry, surface enhanced infrared absorption (SEIRA), polarization modulation-infrared reflection-adsorption spectroscopy (PMIRRAS), surface-enhanced Raman spectroscopy (SERS) etc. Hitaishi et al.

provided an in-depth analysis of bilirubin oxidase (BOD) immobilization on various SAMs by such coupling [START_REF] Hitaishi | Electrostatic-Driven Activity, Loading, Dynamics, and Stability of a Redox Enzyme on Functionalized-Gold Electrodes for Bioelectrocatalysis[END_REF]. Varying the pH of M. verrucaria BOD adsorption or the pH of electroactivity, this study allowed a full correlation between enzyme loading, conformation and electroactivity. New fundamental concepts driving enzyme immobilization were demonstrated: i) the global charge of the enzyme drives its loading on the interface; ii) the dipole moment of the enzyme induced by anisotropy of charges, and the environment charge around the first electron acceptor (Cu T1), drive the enzyme orientation on the surface; iii) higher enzyme coverage does not necessarily translate into higher specific activity; iv) change in local pH induces dynamics of reorientation of enzymes on electrodes. All-atom molecular dynamic simulation confirms the electrostatic model for BOD, and highlights preserved conformation of the enzyme in the immobilized state [START_REF] Yang | Bilirubin Oxidase Adsorption onto Charged Self-Assembled Monolayers: Insights from Multiscale Simulations[END_REF]. Mixed SAMs can be used to tune the number of surface functionalities and their distribution [START_REF] Lacour | Formation Kinetics of Mixed Self-Assembled Monolayers of Alkanethiols on GaAs(100)[END_REF]. Covalent immobilization can be realized by amide coupling between corresponding carboxylic and amine functions on the enzyme and the SAM. Discrepancy exists concerning induced stabilization of the catalytic signal, mainly depending on the way the grafting is carried out [START_REF] Gutierrez-Sanchez | Efficiency of Enzymatic O2 Reduction by Myrothecium verrucaria Bilirubin Oxidase Probed by Surface Plasmon Resonance, PMIRRAS, and Electrochemistry[END_REF].

Diazonium salt reduction is another strategy to functionalize gold electrode, leading to even more stable modification [START_REF] Guo | Spectroscopic Identification of the Au-C Bond Formation upon Electroreduction of an Aryl Diazonium Salt on Gold[END_REF]. Enzymes can then be either physically adsorbed or covalently attached by methods described for SAMs. The main challenge is to achieve a strict monolayer formation enabling fast DET between enzymes and the electrode [START_REF] Lopez | Use of Selective Redox Cross-Inhibitors for the Control of Organic Layer Formation Obtained via Diazonium Salt Reduction[END_REF]. Using a radical scavenger to decrease multilayer formation, and coupling SEIRA to electrochemistry, the O 2tolerant hydrogenase from Ralstonia eutropha was shown to adsorb on the surface while directly oxidizing H 2 [START_REF] Harris | In Situ Spectroelectrochemical Studies into the Formation and Stability of Robust Diazonium-Derived Interfaces on Gold Electrodes for the Immobilization of an Oxygen-Tolerant Hydrogenase[END_REF].

Chemical modification of metal nanoparticles (NPs)

NPs are ideal matrixes for enzyme immobilization as their size and shape may tune enzyme adsorption, loading and wiring [START_REF] Bollella | The influence of the shape of Au nanoparticles on the catalytic current of fructose dehydrogenase[END_REF]. Functionalization of NPs in the way similar to chemical modification carried out on flat metal electrodes serves to specifically tune the adsorption of enzymes for DET. Electrostatic, hydrophobic, - interactions, and eventually combination of them help in narrowing the distribution of orientations [START_REF] Takahashi | Improved direct electron transfer-type bioelectrocatalysis of bilirubin oxidase using thiol-modified gold nanoparticles on mesoporous carbon electrode[END_REF]. As in the case of flat metal, AuNP functionalization allows covalent binding of the enzymes [START_REF] Ratautas | Bioanode with alcohol dehydrogenase undergoing a direct electron transfer on functionalized gold nanoparticles for an application in biofuel cells for glycerol conversion[END_REF]. Interestingly, by coupling electrochemistry to SERS it was demonstrated that different ET pathway induced by a different orientation of Didymocrea sp. J6 Lac could be obtained depending on whether AuNPs are modified by SAMs or not [START_REF] Dagys | Oxygen electroreduction catalysed by laccase wired to gold nanoparticles via the trinuclear copper cluster[END_REF].

NPs other than AuNPs shed new lights on bioelectrocatalysis. TiO 2 NPs coated with carboxylic-functionalized Ag nanoclusters wired carbon monoxide dehydrogenase via surface Cys residues for efficient photoelectrochemical conversion of CO 2 to CO [START_REF] Zhang | Fast and Selective Photoreduction of CO2 to CO Catalyzed by a Complex of Carbon Monoxide Dehydrogenase, TiO2, and Ag Nanoclusters[END_REF]. Magnetic NPs (MNP), such as those based on iron oxides, served as a platform for direct bioelectrocatalysis.

In addition MNPs enabled disassembling of enzymes through external magnetic stimulus, allowing the refreshment of the bioelectrodes [START_REF] Trifonov | Power Generation by Selective Self-Assembly of Biocatalysts[END_REF]. Multidisciplinary analysis of the conformation of the Aspergillus sp. Lac [START_REF] Rodriguez-Padron | Highly efficient direct oxygen electro-reduction by partially unfolded laccases immobilized on waste-derived magnetically separable nanoparticles[END_REF] on MNPs revealed structural changes in the environment of Cu T1. Electrocatalysis was not hampered, but spatial hindrance between Cu sites and MNPs were reduced instead, suggesting importance of structure flexibility contrary to the admitted rule that change in conformation induces lost of activity.

Chemical modification of carbon nanostructures

Carbon nanostructures, such as carbon particles, carbon nanofibers (CNF), carbon nanotubes (CNTs), eventually arranged as buckypaper films, or graphene have been the subject of intensive research toward efficient redox enzyme immobilization. Many recent reviews are available that the reader can look at [START_REF] Mazurenko | Recent developments in high surface area bioelectrodes for enzymatic fuel cells[END_REF][START_REF] Gross | Buckypaper bioelectrodes: emerging materials for implantable and wearable biofuel cells[END_REF][START_REF] Reeve | Enzymes as modular catalysts for redox half-reactions in H-2-powered chemical synthesis: from biology to technology[END_REF]. The interest in carbon structures is the wide range of available chemical surface functionalities, from aromatic to carboxylic ones, and their easy to tune chemistry thanks to wall reactivity, allowing physical or covalent attachment of enzymes [START_REF] Gupta | Recent advances in carbon nanotube based electrochemical biosensors[END_REF]. Atomic oxygen/carbon ratio of CNF [START_REF] Komori | Controlled direct electron transfer kinetics of fructose dehydrogenase at cup-stacked carbon nanofibers[END_REF], or length of CNTs [START_REF] Xia | Factors affecting the interaction between carbon nanotubes and redox enzymes in direct electron transfer-type bioelectrocatalysis[END_REF] can be varied to modify the surface charge, hence the orientation of the enzyme through electrostatic interactions. Integral membrane proteins can directly exchange electrons with the electrode when incorporated in hydrophobic CNF networks [START_REF] Wang | Bacterial Respiratory Chain Diversity Reveals a Cytochrome c Oxidase Reducing O-2 at Low Overpotentials[END_REF]. Carbon surfaces can also be modified through deposition of polyelectrolyte as thin films [START_REF] Tang | Three-Dimensional Sulfite Oxidase Bioanodes Based on Graphene Functionalized Carbon Paper for Sulfite/O-2 Biofuel Cells[END_REF]. Further, diazonium salt reduction is the method of choice to graft different functions on the carbon surface to drive specific interactions with regions of enzymes involved in DET [START_REF] Bollella | Enhanced Direct Electron Transfer of Fructose Dehydrogenase Rationally Immobilized on a 2-Aminoanthracene Diazonium Cation Grafted Single-Walled Carbon Nanotube Based Electrode[END_REF][START_REF] Gentil | Oriented Immobilization of NiFeSe Hydrogenases on Covalently and Noncovalently Functionalized Carbon Nanotubes for H-2/Air Enzymatic Fuel Cells[END_REF][START_REF] Gentil | Direct Electrochemistry of Bilirubin Oxidase from Magnaporthe orizae on Covalently-Functionalized MWCNT for the Design of High-Performance Oxygen-Reducing Biocathodes[END_REF]. Different ET pathways respectively through Cu T1 or TNC of M. verrucaria BOD were suggested according to the charge of the function generated by diazonium salt reduction [START_REF] Lalaoui | Diazonium Functionalisation of Carbon Nanotubes for Specific Orientation of Multicopper Oxidases: Controlling Electron Entry Points and Oxygen Diffusion to the Enzyme[END_REF].

Functionalities on the carbon surface can alternatively be induced by amine electrooxydation generating a nitrogen-carbon bond [START_REF] Xia | Enhanced direct electron transfer-type bioelectrocatalysis of bilirubin oxidase on negatively charged aromatic compound-modified carbon electrode[END_REF]. Finally, modified-CNT and NP-based electrodes can be combined, as demonstrated by Minteer and coworkers for laccase and BOD-based O 2 reduction [START_REF] Neto | Hybrid Bioelectrocatalytic Reduction of Oxygen at Anthracene-modified Multi-walled Carbon Nanotubes Decorated with Ni90Pd10 Nanoparticles[END_REF], and by Leopold et al. for galactose oxidase [START_REF] Wayu | Functionalized carbon nanotube adsorption interfaces for electron transfer studies of galactose oxidase[END_REF].

Chemical modification of porous structures

There is a tremendous interest in the design of hierarchical porous materials, including carbon-and Au-based ones, due to their use in a wide range of applications [START_REF] Yang | Hierarchically porous materials: synthesis strategies and structure design[END_REF]. Porous structures enhance enzyme loading and may alleviate the orientation requirement by providing multipoint attachment. The pore size is one of the main factor in that case, and it was formalized that pores with size close to enzyme size were the most suitable for high ET rate [START_REF] Sakai | Assembly of direct-electron-transfer-type bioelectrodes with high performance[END_REF][START_REF] Mazurenko | Pore size effect of MgO-templated carbon on enzymatic H2 oxidation by the hyperthermophilic hydrogenase from Aquifex aeolicus[END_REF]. However, in most reported cases, pore size is significantly larger that enzyme dimension, and suitable chemistry of the pore walls is required for desired orientation [START_REF] Xiao | Nanoporous Gold-Based Biofuel Cells on Contact Lenses[END_REF]. A full analysis has been provided by Kano' group underlining that, in addition to pore curvature and edge effects, electrostatic modulations induced by chemical modification of pores of Ketjen Black affect the efficiency of the bioelectrocatalysis [START_REF] Sakai | Assembly of direct-electron-transfer-type bioelectrodes with high performance[END_REF]. Further, potential below and above the pzc of the electrode can be applied not only to force penetration of the enzyme in the pores, but also to tune the charge of the pore walls for favored enzyme orientation [START_REF] Lopez | Potential pulseassisted immobilization of Myrothecium verrucaria bilirubin oxidase at planar and nanoporous gold electrodes[END_REF].

Complex architectures are nowadays designed to provide new insight in direct bioelectrocatalysis. Functionalized AuNPs embedded in mesoporous carbon matrixes were shown to induce productive interactions between pyridine moiety on AuNPs and FeS site on formate dehydrogenase favoring direct formate/CO 2 conversion [START_REF] Sakai | Direct electron transfer-type four-way bioelectrocatalysis of CO2/formate and NAD(+)/NADH redox couples by tungstencontaining formate dehydrogenase adsorbed on gold nanoparticle-embedded mesoporous carbon electrodes modified with 4-mercaptopyridine[END_REF]. Porous graphite electrode was used as a platform for amorphous carbon nitride deposition (a-CNx) [START_REF] Blout | Orientation of a Trametes versicolor laccase on amorphous carbon nitride coated graphite electrodes for improved electroreduction of dioxygen to water[END_REF] presenting a versatile surface chemistry. Trametes versicolor Lac was bound to such electrode surface either through amide or imine coupling, and the two binding methods were combined.

4-fold enhancement of the catalytic current was observed. Despite a full coverage of the surface by Lac, electrochemistry coupled to AFM and XPS demonstrated that less than 10% of immobilized enzymes exchanges electrons with the electrode. Such a low percentage of enzymes participating to the catalysis was also calculated in the case of both hydrogenase and BOD on functionalized CNTs [START_REF] Mazurenko | Impact of substrate diffusion and enzyme distribution in 3D-porous electrodes: a combined electrochemical and modelling study of a thermostable H-2/O-2 enzymatic fuel cell[END_REF]. These data demonstrate that enzyme immobilization is far from being optimum, and novel immobilization strategies are required.

As an example of one promising direction, an original construction was proposed to get effective direct bioelectrocatalytic oxidation of glucose. It is based on the enzymatic reduction of platinum nanoclusters by glucose oxidase inside carbon mesopores, allowing inside-out electrical wiring of FAD active site for direct glucose oxidation at a potential of -80 mV vs Ag/AgCl [START_REF] Trifonov | Enzymatic self-wiring in nanopores and its application in direct electron transfer biofuel cells[END_REF]. A biofuel cell was constructed based on this new concept at the anode, displaying a power density around 50 µW.cm -2 and retaining 80% original power after several days. However, considering enzymes are used as alternatives to platinum catalysts, this concept, although an elegant idea, has to shift toward the use of non noble metals.

Protein engineering for specific recognition of functionalized electrodes

Protein engineering is a growing research direction toward efficient and controlled DET for bioelectrocatalysis. Such procedure takes benefit of surface modification strategies, in such a way the engineered protein must specifically recognize chemical functions on the electrode.

His-Tags are classical tools initially developed to facilitate protein purification that can be also used to anchor enzymes on electrodes [START_REF] Amano | Construction of a biointerface on a carbon nanotube surface for efficient electron transfer[END_REF]. Such modification may however decrease DET because of the length of the tag and its almost exclusive positioning on C-ter or N-ter sequences, thus restricting the modulation of orientation [START_REF] Takamura | Improvement in Electron Transfer Efficiency Between Multicopper Oxidase and Electrode by Immobilization of Directly Oriented Enzyme Molecules[END_REF]. Alternatively, coupling between single amino acid residues present or introduced on the surface of proteins and corresponding chemical functions on electrodes can be realized [START_REF] Nm | Direct electron transfer between a site-specific pyrene-modified laccase and carbon nanotube/gold nanoparticle supramolecular assemblies for bioelectrocatalytic dioxygen reduction[END_REF]. In most cases, cysteine (Cys) is the targeted residue. Al-Lolage et al. introduced a single Cys residue by site-directed mutagenesis at different locations in the flavodehydrogenase domain of cellobiose dehydrogenase [START_REF] Meneghello | Studying Direct Electron Transfer by Site-Directed Immobilization of Cellobiose Dehydrogenase[END_REF][START_REF] Ma | Direct Electron-Transfer Anisotropy of a Site-Specifically Immobilized Cellobiose Dehydrogenase[END_REF][START_REF] Meneghello | A Flexible Method for the Stable, Covalent Immobilization of Enzymes at Electrode Surfaces[END_REF].

After covalent attachment of the various mutants through maleimide groups on the electrode, it was demonstrated that DET occurs through the cytochrome domain, whose mobility is utmost crucial for electron transfer. The coverage of maleimide functions was restricted to 10% of the surface to favor the access and the coupling of enzymes. A ratio of 4-5 was however still obtained between the deposited enzymes and the electroactive ones. A similar protocol was carried out in Magnaporthe oryzae BOD, highlighting the key role of enzyme orientation and distance between the Cu T1 and the electrode on DET [START_REF] Al-Lolage | Site-Directed Immobilization of Bilirubin Oxidase for Electrocatalytic Oxygen Reduction[END_REF]. In a similar work, engineered M. oryzae BOD was immobilized by Au-S bond formation in a macroporous gold electrode [START_REF] Zhang | Rational Design of Enzyme-Modified Electrodes for Optimized Bioelectrocatalytic Activity[END_REF]. The stability of the bioelectrode was enhanced compared to the nonengineered enzyme. Interestingly, the occupation of the whole internal pore volume by the enzyme was demonstrated using confocal fluorescent microcopy. A single Cys residue was also introduced in E. coli [NiFe] hydrogenase close to the surface FeS cluster, after removal of the naturally existing Cys. Specific attachment to Ag nanoclusters allowed enhanced photoproduction of H 2 [START_REF] Zhang | Direct visible light activation of a surface cysteineengineered NiFe -hydrogenase by silver nanoclusters[END_REF].

Non canonical amino acids (NAA) can be incorporated in any place of a redox enzyme, contrary to His-tag [START_REF] Yanez-Sedeno | Copper(I)-Catalyzed Click Chemistry as a Tool for the Functionalization of Nanomaterials and the Preparation of Electrochemical (Bio)Sensors[END_REF]. NAA library offers a great versatility already largely exploited in protein-protein interaction biological studies. However, although prone to provide new insight on DET pathways, the use of NAA for enzyme anchoring is currently scarce in bioelectrocatalysis. Schlesinger et al. wired copper efflux oxidase through propargyl-L-lysine NAA incorporated at different distances from the Cu T1 [START_REF] Schlesinger | Electron transfer rate analysis of a sitespecifically wired copper oxidase[END_REF]. The carbon electrode was chemically modified by pyrene-azide derivative for - stacking on the electrode on one hand and for Huisgen copper catalyzed azide-alkyne click reaction on the other hand. Effect of the length of the linker and positions of the NAA on the ET rate was discussed. A similar strategy was employed to wire tryptophan oxidase [START_REF] Xia | S-Click Reaction for Isotropic Orientation of Oxidases on Electrodes to Promote Electron Transfer at Low Potentials[END_REF]. An alternative close strategy is to introduce peptides in enzymes for specific binding to electrodes [START_REF] Zernia | Surface-Binding Peptide Facilitates Electricity-Driven NADPH-Free Cytochrome P450 Catalysis[END_REF]. The influence on DET of distances between actives sites and electrodes, but also of enzyme-enzyme neighboring that could induce steric hindrance or conformation change were reported. Lee et al. [START_REF] Lee | Significant enhancement of direct electric communication across enzyme-electrode interface via nano-patterning of synthetic glucose dehydrogenase on spatially tunable gold nanoparticle (AuNP)-modified electrode[END_REF] designed NP nanopatterns with NP sizes between 10 and 80 nm, enabling to tune their special organization on the surface. An engineered glucose dehydrogenase (GDH) was bound to NPs through a fused specific peptide. Although no clear catalytic waves of glucose oxidation were visible on the voltammograms,, this strategy should be relevant in the future to investigate the effect of coverage on other enzymatic systems bioelectrocatalysis.

Conclusion and Future directions

Attractive features exhibited by redox enzymes in solution, such as selectivity and high activity of catalysis, have prompted numerous attempts to convert enzymes into heterogeneous catalysts. Gold and carbon seem to be the most suitable electrode materials and even simple adsorption on them have often allowed to address electrochemically many redox enzymes. However, if one wants to strictly control enzyme immobilization, to perform more precise enzymatic studies or to unveil the applied potential of enzymatic catalysts, an additional functionalization is required.

In the quest for the optimum bioelectrode in terms of catalytic efficiency and stability, chemical modification of the electrochemical interface has been the mostly investigated.

Although, the general rules have been now identified to succeed in promoting DET, many enzymes are still not addressable by direct electrochemistry or display a very low interfacial ET rate. New approaches are further required for such enzymes, e.g. for some carbon monoxide dehydrogenases or [MoFe] nitrogenase [START_REF] Milton | Nitrogenase bioelectrocatalysis: heterogeneous ammonia and hydrogen production by MoFe protein[END_REF] which would be very attractive enzymes to convert CO 2 or N 2 . Extended interdisciplinary collaboration between chemist and biochemist communities should allow in the future to take advantage of the tremendous possibility offered by genetic enzyme engineering, as illustrated in this review by the recent results obtained with non natural amino acids inserted in specific location of an enzyme. As one of the examples, specific enzyme recognition by surface groups with controlled density might be used for enzyme patterning or cascades on the surface and to study the effect of molecular crowdedness. Furthermore, careful control of the immobilization chemistry will allow the development of the renewable bioelectrodes that, while still in its infancy, may help to overcome the inherent instability of redox enzymes.

Advanced methods to get new insights into enzyme amount, conformation and activities distribution are required especially coupling electrochemistry to spectroscopies or microscopies [START_REF] Kornienko | Advancing Techniques for Investigating the Enzyme-Electrode Interface[END_REF][START_REF] Zigah | Micro-and Nanoscopic Imaging of Enzymatic Electrodes: A Review[END_REF]. Ultimately, control of enzyme binding on electrodes will help the development of single-enzyme electrochemistry [START_REF] Baker | Perspective and Prospectus on Single-Entity Electrochemistry[END_REF].

Figure 1 .

 1 Figure 1. Examples of redox enzymes and their active centers addressable by DET.

Figure 2 .

 2 Figure 2. Most used electrode modification methods to promote direct electron transfer. (A) Reactions used to introduce the required functionalities to the electrode surface; (B) Reactions used to couple enzymes to the electrode surface

Figure 3 .

 3 Figure 3. Examples of surface modification to promote direct electron transfer of multicopper oxidases: (A) Magnaporthe oryzae BOD attached to the electrode surface through maleimide coupling at different positions and corresponding (green) direct and (red) mediated catalytic currents of oxygen reduction. Adapted from [53]; (B) Catalytic oxygen reduction by Magnaporthe oryzae BOD adsorbed on non-modified and modified by diazonium salt grafting MWCNTs. Adapted from [33]; (C) Catalytic oxygen reduction by Myrothecium verrucaria BOD adsorbed on GCE modified by different functions using amine electrooxidation. Dashed lines correspond to non-modified GCE. Adapted from [36]; (D) (black lines) Direct and (red lines) mediated catalytic oxygen reduction by Myrothecium verrucaria BOD adsorbed on negatively (6-MHA), neutral (BT) or positively (4-ATP) charged SAMs. Adapted from [10].

Acknowledgement

This work was financially supported by ANR (ENZYMOR-ANR-16-CE05-0024), and by Aix Marseille University for funding V.P. Hitaishi.