Tobias Kühn
email: tobias.kuhn@inserm.fr

Rémi Monasson

Information content in continuous attractor neural networks is preserved in the presence of moderate disordered background connectivity

Continuous attractor neural networks (CANN) form an appealing conceptual model for the storage of information in the brain. However a drawback of CANN is that they require finely tuned interactions. We here study the effect of quenched noise in the interactions on the coding of positional information within CANN. Using the replica method we compute the Fisher information for a network with position-dependent input and recurrent connections composed of a short-range (in space) and a disordered component. We find that the loss in positional information is small for not too large disorder strength, indicating that CANN have a regime in which the advantageous effects of local connectivity on information storage outweigh the detrimental ones. Furthermore, a substantial part of this information can be extracted with a simple linear readout.

I. INTRODUCTION

The ring attractor neural network was proposed by Amari in the 70's as a practical way to memorize a collective variable within a noisy neural population [START_REF] Amari | Dynamics of pattern formation in lateral-inhibition type neural fields[END_REF]. This work opened the way to various theoretical applications of the concept of continuous attractor neural networks (CANN), e.g. in the contexts of the orientational tuning [START_REF] Ben-Yishai | Theory of orientation tuning in visual cortex[END_REF] or hippocampal place cells [START_REF] Tsodyks | Associative memory and hippocampal place cells[END_REF], as well as to extensions, in particular to the case of multiple attractor embeddings [START_REF] Battaglia | Attractor neural networks storing multiple space representations: a model for hippocampal place fields[END_REF][START_REF] Romani | Continuous attractors with morphed/correlated maps[END_REF][START_REF] Battista | Capacity-resolution trade-off in the optimal learning of multiple low-dimensional manifolds by attractor neural networks[END_REF]. While indirect evidence for the existence of CANN could be found in recordings of activity in the hippocampus [START_REF] Posani | Integration and multiplexing of positional and contextual information by the hippocampal network[END_REF], in the enthorinal [START_REF] Yoon | Specific evidence of low-dimensional continuous attractor dynamics in grid cells[END_REF] and the prefrontal cortex [START_REF] Wimmer | Bump attractor dynamics in prefrontal cortex explains behavioral precision in spatial working memory[END_REF], a direct and beautiful observation of ring attractor coding for head direction was obtained only recently in the ellipsoid body of the fly [START_REF] Kim | Ring attractor dynamics in the drosophila central brain[END_REF].

From a theoretical point of view, CANN models rely on recurrent excitatory interactions between neurons active for similar values of the encoded variable, e.g. the position of the animal in physical space, together with a longrange inhibition preventing all cells to be active together. This combination of local positive interactions and global inhibition creates a localized bump of activity, whose center of mass reliably represents the collective variable. In this regard, a crucial condition is that the bump can be easily moved (under weak external, "sensory", inputs) to span the continuous set of values of the variable. This condition imposes that the short-range connections are finely tuned, so that the model be effectively translation invariant.

When the finite-tuning condition breaks down, e.g. due to random modulations of the interactions, the bump can get stuck in the absence of neural noise [START_REF] Tsodyks | Associative memory and hippocampal place cells[END_REF]. In practice, quenched noise in the interactions can come from imperfect learning of one environment, or from interferences resulting from other information encoded (maps, objects distorting the map locally, ...). Quantifying the loss in the accuracy of information storage resulting from heterogeneities in the interactions is an important issue.

We address this question here in the framework of decoding of information, based on analytical and numerical calculations. We propose an analytically tractable model of binary (active/silent) neurons receiving position-dependent inputs, and connected to each other through spatially coherent and short-range interactions, on top of a disordered and incoherent background. Using the replica method we compute the Fisher information in the high-dimensional neural activity about the encoded position as a function of the intensity of disordered interactions. This quantity was identified as a measure that is both relatively easy to compute for many systems and objectively quantifies the information contained in the neural activity about the stimulus (an orientation or a point in space) [START_REF] Seung | Simple models for reading neuronal population codes[END_REF]. It is more appropriate for this quantification than, for example, the readout of the center of mass of a bump of activity [START_REF] Pouget | Narrow Versus Wide Tuning Curves: What's Best for a Population Code?[END_REF]. Yet, the Fisher information is not an information measure in the sense of Shannon. From this point of view, the mutual information between the stimulus and the neural activity is the quantity we are eventually interested in [START_REF] Cover | Elements of Information Theory[END_REF]. However, the latter is a global quantity, integrated over all possible stimuli and its computation is generally more difficult than that of the Fisher information, which puts restrictions on the system it can be calculated for [START_REF] Sompolinsky | Population coding in neuronal systems with correlated noise[END_REF]. In the thermodynamic limit, the mutual information can be obtained from the Fisher information under the condition that FIG. 1: Scheme of the network model employed in this work. The bell-shaped curves represent the space sensitive single-neuron (feedforward) input and the red circle a symbolic inhibitory neuron that ensures that the summed activity of all neurons is constant.

the correlations are not too strong [START_REF] Brunel | Mutual information, fisher information, and population coding[END_REF]. We have explicitly checked that this prerequisite is fulfilled by our model, so that the computation of the Fisher information is actually sufficient. For a current review discussing both measures of information and their use in neuroscience, see e.g. [START_REF] Kriegeskorte | Neural tuning and representational geometry[END_REF].

The paper is organized as follows. In sec. II, our model is introduced. We establish the Fisher information as a means to quantify the information contained in the neural activity about the stimulus, together with its relation to other information-theoretic measures, compute it in the thermodynamic (mean-field) limit and derive its analytical properties in the limiting case of weak connection strengths. In sec. III, we validate our mean-field results by means of Monte-Carlo calulations, study the dependence of the Fisher information on changes in the recurrent and feed-forward connectivity and how precise a linear readout can be compared to the bound predicted by the Fisher information. In sec. IV, we put our results into context and give an outlook to possible future directions.

II. MODEL AND METHODS

A. Distribution of neural activities

We model neurons as binary units, taking the value 1 when active and 0 when inactive. Each neuron is receiving a 'sensory' input, whose value depends on the mismatch between the position r i in physical space it is maximally responding to and the position of the 'animal', see fig.

(1) for a scheme of our setup. The probability distribution of activities is governed by a Boltzmann law P (n|ξ) ∼ e -E [n] and the energy

E [n] = - 1 2 i =j (J ij + K ij ) n i n j - i n i U (ξ -r i ) , (1)
where

K ij = K (r i -r j )
is the local part of the interaction that we assume to be decaying in space with a typical length scale w rec and strength K rec . J is the disordered part of the connectivity with the statistics

J ij = 0, J ij J i j = g 2 2N δ ii δ jj (2)
and U mimics the space-dependent input, which we model by

U (∆x) = U inp exp - ∆x 2 w 2 inp . (3)
We are assuming periodic boundary conditions. As a simple way to model global inhibition, we impose the constraint that the summed activity is fixed to

i n i = f • N = M , where f ∈ [0, 1].
Our model is closely related to that of [START_REF] Monasson | Crosstalk and transitions between multiple spatial maps in an attractor neural network model of the hippocampus: Phase diagram[END_REF], where the case of a proper continuous attractor neural network (CANN) is studied, so the connectivity matrix is composed of a sum of local connectivities in different environments. However, in this study we are interested in the effect of the disorder on the information content in the neural activity for a single environment. Also, the presence of other maps is not the only source of disorder as there is always some variability in the connectivity. Therefore, to simplify the setup, we content ourselves with approximating the disordered contribution to the connections as Gaussian, which also corresponds to the high-temperature behaviour of the disorder in [START_REF] Monasson | Crosstalk and transitions between multiple spatial maps in an attractor neural network model of the hippocampus: Phase diagram[END_REF].

B. Fisher information and mutual information

We now want to quantify the amount of information contained in the neural activity. One possibility to do so is to compute the Fisher information for a given stimulus ξ,

I n (ξ) := - ∂ 2 ∂ξ 2 ln P (n|ξ) n , (4)
a standard measure for the quantification of information in neural populations [START_REF] Pouget | Narrow Versus Wide Tuning Curves: What's Best for a Population Code?[END_REF][START_REF] Kriegeskorte | Neural tuning and representational geometry[END_REF]. According to the Cramér-Rao bound, its inverse gives a lower bound on the variance of any unbiased estimator of ξ [START_REF] Cover | Elements of Information Theory[END_REF]. We will discuss this relation in greater depth in section III C. Furthermore, in the thermodynamic limit, that we are interested in, it also determines the mutual information, a connection first established in [START_REF] Brunel | Mutual information, fisher information, and population coding[END_REF] and later refined in [START_REF] Wei | Mutual Information, Fisher Information, and Efficient Coding[END_REF]. The mutual information is given by the decrease of the entropy of the neural activity due to the knowledge of the stimulus, concretely

I MI := - n P (n) ln P (n) + dξ p (ξ) n P (n|ξ) ln P (n|ξ)
where P (n) := dξ p (ξ) P (n|ξ) .

In [START_REF] Brunel | Mutual information, fisher information, and population coding[END_REF] (their eq. ( 13)), the relation

I MI = -dξ p (ξ) ln p (ξ) + dξ p (ξ) ln I n (ξ) 2πe + O 1 N (5)
was derived for an ensemble of neurons with fixed covariances, without disorder. However, in this study we are limiting ourselves to the saddle-point approximation of the Fisher information, which is valid up to corrections of order 1/N as well. So the presence of disorder does not change much and we obtain that

I MI J = -dξ p (ξ) ln p (ξ) + dξ p (ξ) ln I n (ξ) J 2πe + O 1 N , (6)
where . . . J is indicating the average over the disordered connectivity J. In appendix D, we rederive this relation, for unconnected neurons, but more directly than in [START_REF] Brunel | Mutual information, fisher information, and population coding[END_REF].

Hereafter, we will focus on the Fisher information, which is easier to obtain than the mutual information that we get for free due to eq. ( 6). Determining the Fisher information for our model, we obtain from eq. ( 4) after some lines of computation, detailed in appendix B,

I n (ξ) = i,j U (ξ -r i ) n i n j n -n i n n j n J U (ξ -r j ) (7) = U (ξ -r) T CU (ξ -r) , (8)
where C denotes the disorder-averaged covariance matrix of n. Conditioned on one realization of the disorder furthermore, we have introduced the thermal average

f (n) n := 1 Z J (ξ) n f (n) e -E[n] , (9)
for some function f , together with the partition function

Z J (ξ) := n e -E[n] . (10)
Eq. ( 7) can be brought into a more familiar form by noting that

∂ ∂ξ T (ξ -r i ) := ∂ ∂ξ n i n = n i   j U (ξ -r j ) (n j -n j )   n = CU i (11) ⇔ U (ξ -r i ) = C -1 T i , (12)
where we have introduced the tuning curve T of neuron i indicating its average activity given the input ξ. With this, the Fisher information can be written as [START_REF] Sompolinsky | Population coding in neuronal systems with correlated noise[END_REF] I

n (ξ) = T (ξ) T C -1 T (ξ) . ( 13
)
This form is more handy when dealing with experimental data because the tuning curve is (in principle) a directly measurable quantity, whereas U is not. For our purposes, however, the form of eq. ( 7) is more practical because there, the only quantity depending on the disorder is the covariance matrix. We therefore only have to compute the disorder average of the covariance matrix, which we will tackle in the following.

C. Disorder-averaged statistics

As usual for disordered systems [START_REF] Mézard | Spin Glass Theory and Beyond[END_REF], we determine the statistics from the logarithm of the partition function, the cumulant-generating functional (or Gibbs free energy)

W (h) J = dJ P (J) ln   n, i ni=M e 1 2 i =j (Jij +Kij )ninj + i ni[U(ξ-ri)+hi]   . (14)
The computation of W proceeds along the classical lines [START_REF] Sherrington | Solvable model of a spin-glass[END_REF], with the difference that we have a local connectivity. We therefore do not only introduce the Gaussian helping field q to decouple the four-point terms emerging from the disorder average, but also a space-dependent (also Gaussian) order parameter φ x to decouple the local term n T Kn.

As apparent from the saddle-point equations ( 18) and ( 19), q quantifies the population-averaged variance of the activity and φ x the population-averaged input to the neuron with place field at position x. Furthermore, due to the restriction on the summed activity, we introduce the Lagrange multiplier λ. As derived in appendix B, we obtain the disorder-averaged cumulant-generating function in the thermodynamic limit N → ∞ W (h) J = extr q,q,ψ,φ,λ

1 2 N g 2 q 2 - 1 2 N g 2 q2 - 1 2 φ T K -1 φ -N λ -g 2 (q -q) f. (15)
+

y 1 √ 2π dt y e - t 2 y 2 x ln 1 + e φx+txg √ 2q+U(ξ-rx)+λ+hx (16)
=:G (h, φ, q) , (17)
where the "extr." implies a supremum over q, q, ψ and φ and an infimum over λ. We comment on the latter point in appendix A 1. As detailed in appendix A 2, we obtain the saddle-point equations

q = dx Dt x 1 1 + e -(φx+txg √ 2q+U(ξ-x)+λ) 2 , (18)
φ x = dy K (x -y) Dt y 1 1 + e -(φy+tyg √ 2q+U(ξ-y)+λ) . ( 19
)
and

f = dx Dt x 1 1 + e -(φx+txg √ 2q+U(ξ-x)+λ) , (20)
with the Gaussian measure

Dt = 1 √ 2π dte -t 2 2 . ( 21
)
The entire statistics of our system can now be determined by taking derivatives of G with respect to h, which is set to 0 afterwards. We have to calculate the total derivative, also taking into account the h-dependence of q, φ and λ, which in turn, by the implicit-function theorem, we obtain by taking the total derivative with respect to h of the their saddle-point-equations. This yields

1 N d 2 dh 2 W f (h) J = ∂ 2 G ∂h 2 -    ∂ 2 G ∂h∂φ ∂ 2 G ∂h∂q ∂ 2 G ∂h∂λ    T    ∂ 2 G ∂φ 2 ∂ 2 G ∂φ∂q ∂ 2 G ∂φ∂λ ∂ 2 G ∂q∂φ ∂ 2 G ∂q 2 ∂ 2 G ∂q∂λ ∂ 2 G ∂λ∂φ ∂ 2 G ∂λ∂q ∂ 2 G ∂λ 2    -1    ∂ 2 G ∂φ∂h ∂ 2 G ∂q∂h ∂ 2 G ∂λ∂h    . ( 22
)
Evaluating this expression, we obtain

C = V + V K eff V + C indirect (23) = V (1 -KV ) -1 + C indirect , ( 24
)
where V is the diagonal matrix with the disorder-averaged single-neuron variances

V xy = δ xy v x , (25)
where

v x := Dt ∂m x ∂φ x = Dt m x (1 -m x ) , (26)
m x is the magnetization conditioned on the Gaussian helping variable t,

m x := 1 1 + e -[φx+t g √ 2q+U(ξ-x)+λ] , (27)
the effective local connectivity K eff is given by K eff -1

xy := -∂ 2 G
∂φx∂φy and fulfills the Dyson equation

K eff xy = K xy + dz K xz v z K eff zy , (28)
and C indirect emerges from the remaining part of the Hessian in eq. [START_REF] Moser | Grid cells and cortical representation[END_REF]. It results in an subleading contribution to the Fisher information (see below), so we give its precise form only in appendix B.

D. Disorder-averaged Fisher information

The Fisher information per neuron averaged over the disorder now reads

I n (ξ) = x y U (ξ -x) v x δ x,y + v x K eff xy v y + C indirect xy U (ξ -y) (29)
In fig. [START_REF] Battista | Capacity-resolution trade-off in the optimal learning of multiple low-dimensional manifolds by attractor neural networks[END_REF] in the appendix, we show these three contributions separately for the parameters used for fig.

(3). The first term stems from the single-neuron variances and is therefore also present without network (if present the variances are effected by the network, though). The third is always negligible, which intuitively makes sense because it emerges from the indirect h-dependence of the free energy via g and λ, which are both spatially unstructured. The second term emerges from the (positive) local interactions and also contributes positively. In order to gain a better intuition for where this term comes from, it is useful to re-derive the expression for the Fisher information using eq. ( 13), limiting ourselves to the case without disorder. By some lines of rearrangements, shown in appendix B 2, we derive that the vector of the derivatives of the tuning curves can be expressed as

T = V (1 + K eff V ) U . ( 30
)
0.0 0.2 0.4 0.6 0.8 1.0 place-field center x 0.14 0.16 0.18 Combining this expression with eq. ( 13) and C = V + V K eff V , we are getting back eq. ( 29) (without the contribution of the disorder) after canceling a factor (V + V K eff V ), as expected. However, it is also insightful to write down the expression before this cancellation,

Activity ρ(x) a K rec =0.0 K rec =1.2 K rec =1.4 K rec =1.6 K rec =1
I n = =(T ) T U T (V + V K eff V ) =C -1 [V + V K eff V ] -1 =T (V + V K eff V ) U , (31)
because it gives an intuition about how the local connectivity shapes the Fisher information: first, it modifies the tuning curves, which is captured by the term V K eff V U , second it introduces cross-covariances, which is captured by the term V K eff V contributing to the covariance. As apparent from fig.

(2), panel a, the tuning curves are sharpened with increasing K rec , which is reflected by the fact that the direct contribution of the cross-covariances to the Fisher information is positive, see fig. [START_REF] Battista | Capacity-resolution trade-off in the optimal learning of multiple low-dimensional manifolds by attractor neural networks[END_REF]. The cross cavariances, in turn, are detrimental in our case: they reduce the Fisher information, as apparent from panel b in fig. [START_REF] Ben-Yishai | Theory of orientation tuning in visual cortex[END_REF]. We can study further eq. ( 29) analytically, which amounts to examining the saddle-point equations ( 18) to [START_REF] Sherrington | Solvable model of a spin-glass[END_REF]. In particular, we can do this in the limiting case g → 0. In this limit, the Gaussian integrals get trivial. As detailed in appendix A 3, we can use for the study of the derivatives of q, φ and λ that, in eqs. [START_REF] Wei | Mutual Information, Fisher Information, and Efficient Coding[END_REF] to [START_REF] Sherrington | Solvable model of a spin-glass[END_REF], these quantities are (implicitly) given by

0 = ∂ ∂ [{φ x } x , λ, q] G g (g, q, φ, λ) , (32)
with G as given in eq. ( 17) and therefore, by the implicit-function theorem,

∂ ∂g   {φ x } x λ q   = - ∂ 2 ∂ [{φ x } x , λ, q] 2 G g [q, φ, λ] -1 ∂ 2 ∂g∂ [{φ x } x , λ, q] G g [q, φ, λ] . (33)
For a reasonable choice of the parameters, also guaranteeing the stability of the saddle-point solution,

∂ 2 ∂[{φx} x ,λ,q]
2 G g is invertible (which we as well check numerically by computing it explicitly in appendix B). The partial derivative of ∂ {φx} x ,λ,q G with respect to g vanishes in the limit g → 0, as we show in appendix A 3. Therefore, the derivatives of q, φ and λ with respect to g go to 0 for vanishing g. This results carries over to higher-order cumulants and to the cumulant-generating functional itself. Because the Fisher information depends on g only via these quantities, its derivative vanishes in the limit of g → 0:

lim g→0 ∂ ∂g I n (ξ) = 0. ( 34
)
The derivatives with respect to the strength of the local interaction, K rec , however, in general do not vanish for vanishing connectivity. Therefore even small connection strengths will have a (beneficial) effect, as seen before.

III. NUMERICAL VALIDATION OF MEAN-FIELD RESULTS AND APPLICATIONS A. Monte Carlo simulation

To validate our computation derived for the thermodynamic limit, we perform Monte-Carlo simulations for multiple sets of parameters, see fig. [START_REF] Tsodyks | Associative memory and hippocampal place cells[END_REF].We use a standard Metropolis algorithm, taking into account the condition of a fixed total activity by flipping always two spins, in opposite direction, as suggested in [START_REF] Monasson | Crosstalk and transitions between multiple spatial maps in an attractor neural network model of the hippocampus: Phase diagram[END_REF]. The code for these simulations, alongside with the implementation of the mean-field results is provided in [21]. The results confirm our mean-field result that the disorder diminishes the Fisher information, but quite slowly if the disorder is on a moderate level, as predicted by eq. (34) and visible in fig. [START_REF] Tsodyks | Associative memory and hippocampal place cells[END_REF].

B. Influence of the network on the Fisher information

In an attractor network, disorder may result from the presence of other maps stored in the same network. Therefore it scales in the same way as the spatially dependent part of the connectivity. It is thus interesting to examine the behavior of the Fisher information when both parts of the connectivity are scaled by the same factor r:

K → r • K, J → r • J. ( 35
)
Because the derivative of the expression for the Fisher information with respect to the disorder strength g vanishes for g = 0, the effect of the local part of the connectivity dominates for small synaptic strength: the Fisher information initially increases. This can be understood from what we have derived before: increasing the local connectivity sharpens the tuning curves and therefore increases the signal. This effect is diminished (but not cancelled) by the introduction of covariances between the neurons (see also [START_REF] Pouget | Narrow Versus Wide Tuning Curves: What's Best for a Population Code?[END_REF]). For larger scaling factors, however, this overall beneficial effect is wiped out by the disorder, whose detrimental effect eventually dominates, fig. ( 4), panel a. Finally, we ask if we can keep the Fisher information constant by increasing the recurrent weights when the input gets weaker. We have plotted lines of constant Fisher information for varying strength of the input and the recurrent connections in fig. [START_REF] Battaglia | Attractor neural networks storing multiple space representations: a model for hippocampal place fields[END_REF], panel b. We can indeed make up to a decrease in the input by strengthening local connections, even though of course only in a limited range.

C. Linear readout

To put our results into context and convey a more intuitive understanding, we briefly discuss what one can learn from the Fisher information about the accuracy of a linear readout. g (approximately the first two data points at 0.1 and 0.5), this estimate is approximately unbiased and therefore only there, the Cramér-Rao bound guaranties that the Fisher information is an upper bound. Note that in this plot, different to all the others, we are plotting the total Fisher information for all 100 neurons, not the averaged, single-neuron analog. Parameters as in fig. 3, panel a.

We have fitted a readout vector w to the activity as measured after the thermalization process (so the random initial conditions should not play a role) and computed the squared residual error as res (ξ) ξ := min

w w • n -ξ 2 2 n ξ = ξ 2 ξ -2w T train C test ξ,n + w T train C test n,n w train ( 36
)
where

w train = C train n,n + λ • 1 -1 C train ξ,n , (37)
where . . . n and . . . ξ denote the thermal average over the configurations n and the average over the distribution of the stimulus ξ, respectively and λ denotes the strength of the L 2 regularization that we impose. We might expect to obtain an upper bound for the accuracy of the linear estimator by the Cramér-Rao bound. However, due to the periodic boundary conditions, the estimate from the linear readout ξ est. = w • n will be biased. This is particularly apparent at the borders 0 and 1, where the estimate will always be 1 2 , corresponding to random guessing. The farther away from the them the stimulus is situated, the less pronounced the effect becomes. We have therefore limited the fitted stimuli to ξ ∈ [0.4, 0.6]. However, even in this regime, the linear readout gets biased in the highly disordered regime, so that there the Cramér-Rao bound only applies in its generalized form [START_REF] Cover | Elements of Information Theory[END_REF], their eq. ( 12.333):

(w • n -ξ) 2 n ≥ (1 + b (ξ)) 2 I n (ξ) + b (ξ) 2 , (38)
where b (ξ) := w • n ξ -ξ is the bias of the linear estimator. In case of random guessing, in particular, b (ξ) = -1, so that the error is solely determined by the square of the bias. Therefore, the bound given by the Fisher information in fig.

(5) is only meaningful for small disorder (g ∼ 0.1, 0.5), whereas for greater disorder, it is invalidated by the bias, up to the point where the linear readout basically generates a random guess (for g ∼ 2). In the low-disorder regime, however, we observe that the error of the linear readout is not far from the optimal case given by the Cramér-Rao bound.

IV. DISCUSSION AND OUTLOOK

In this work, we have studied an attractor-inspired neural network with a connectivity consisting of two parts: (1) excitatory couplings between neurons that are similarly tuned to stimuli, and (2) a quenched random background without local tuning. We have studied the influence of both of these contributions on the information about the stimulus contained in the neural activity through the analytical computation of the Fisher information. As expected the local part of the connectivity enhances the information content, whereas the disordered part degrades it. However, the latter effect is mild. By fitting a linear readout to estimate the driving stimulus we show that the Fisher information is not only a formal estimate of the information contained in the neural activity, but also gives a useful bound on how much of it can be extracted with simple decoders.

It has been recognized for a long time that the presence of disorder in the interactions could impact the information stored in attractor networks in the form of patterns. In the case of CANNs this translates into a breaking of the translational symmetry of bump-like solutions, see for instance [START_REF] Tsodyks | Associative memory and hippocampal place cells[END_REF] and box 3 in [START_REF] Moser | Grid cells and cortical representation[END_REF]. However, despite the loss of translation invariance, noise in the neural activity (controlled here by the inverse of the coupling strength, playing the role of temperature in statistical mechanics) may be sufficient to move the bump [START_REF] Monasson | Crosstalk and transitions between multiple spatial maps in an attractor neural network model of the hippocampus: Collective motion of the activity[END_REF]. We here show that the disorder does not wipe out all information in the attractor network. In particular, the Fisher information is robust to the introduction of disorder, staying constant to first order in g. Consequently, globally enhancing the connectivity strength (the local and the disordered part by the same factor, as in eq. ( 35)) initially always has a beneficial effect, which is overtaken by the effect of the disorder only for larger connectivity. In real biological networks, the connectivity is not fixed, but builds up during development, partially through learning [START_REF] Richter | Understanding neural circuit development through theory and models[END_REF]. One might therefore speculate if this process is optimized for the synaptic strengths to eventually match this sweet spot.

Similarly, we can ask for the information-theoretical implication of the development of tuning curves as observed in the visual cortex. In young animals, recurrent connections between similarly tuned neurons get enhanced, other weakened [START_REF] Ko | The emergence of functional microcircuits in visual cortex[END_REF], and the orientation selectivity is sharpened [START_REF] Sadeh | Emergence of functional specificity in balanced networks with synaptic plasticity[END_REF]. According to our analysis, these changes in recurrent connectivity could compensate a decrease in strength of sensory inputs [START_REF] Lim | Mechanisms underlying sharpening of visual response dynamics with familiarity[END_REF], see fig.

(4)(b).

While our model allows for this kind of qualitative considerations, it is minimal in the sense that it contains the ingredients needed to study the effects we are interested in in their simplest form. This of course limits biological plausibility and calls for enhancements. From a technical point of view, we did of course not consider all possible scenarios. We give below an outlook on possible further directions, starting with the technical aspects.

The results we have obtained here hold for binary-valued neurons. From a neuroscience perspective, this assumption can be interpreted as follows. Consider the neural population activity in a time-bin of duration ∆t. If ∆t is small enough (in practice, not larger than the inverse of the typical firing rate), it is likely that each neuron i has fired at most once in a time bin. We can therefore represent its activity through a binary variable n i equal to 0 in the absence of spike, or to 1 if a spike has been emitted. This point of view was adopted in data-driven models of the hippocampal activity [START_REF] Posani | Integration and multiplexing of positional and contextual information by the hippocampal network[END_REF]. If the duration ∆t is large, the binary hypothesis breaks down and continuous models, taking into account the real-value nature of firing rates, should be considered. Continuous-attractor models with real-valued neurons show similar -albeit for some parameters a bit richer [START_REF] Treves | Threshold-linear formal neurons in auto-associative nets[END_REF] -behavior as models with binary neurons [START_REF] Tsodyks | Associative memory and hippocampal place cells[END_REF], so we do not expect our conclusions to be qualitatively modified in that setting. In addition, binary patterns represent an important limiting case: they may maximize the retrievable information, in the case of random memories, as compared to more complex ones [START_REF] Treves | Graded-response neurons and information encodings in autoassociative memories[END_REF].

Then, we have limited ourselves to a parameter regime far away from phase transitions. In particular we have not considered the low-temperature/high-connectivity regime. The analysis of this case (without the computation of the Fisher information) was carried out by one of us and collaborators in a series of papers [START_REF] Monasson | Crosstalk and transitions between multiple spatial maps in an attractor neural network model of the hippocampus: Phase diagram[END_REF][START_REF] Monasson | Crosstalk and transitions between multiple spatial maps in an attractor neural network model of the hippocampus: Collective motion of the activity[END_REF][START_REF] Monasson | Transitions between spatial attractors in place-cell models[END_REF], in the specific case of background noise due to a extensive number αN of alternative attractors embedded in the network. Although the quenched noise distribution in this case was not Gaussian we do not expect the phase diagram to significantly change. Based on these previous works we can thus build educated guesses on what to expect in terms of information theory. Note that, in terms of scaling, the square of the disorder strength used here, g 2 , roughly corresponds to the load α in [START_REF] Monasson | Crosstalk and transitions between multiple spatial maps in an attractor neural network model of the hippocampus: Phase diagram[END_REF] (compare, e.g. their saddle-point equations, eq. ( 28) to ours, eqs. ( 18) to [START_REF] Sherrington | Solvable model of a spin-glass[END_REF]).

Analog to [START_REF] Monasson | Crosstalk and transitions between multiple spatial maps in an attractor neural network model of the hippocampus: Phase diagram[END_REF], we expect a glassy phase for large disorder and weak local connectivity, a ferromagnetic ("bump") phase for weak disorder and strong local connectivity and a paramagnetic phase in case both contributions are small (large-temperature regime). In the present study, we have basically stayed in the latter regime (however, the activity was still bump-like due to the feed-forward input). This limitation has also allowed us to stick to the replica-symmetric solution of the saddle-point equations -an assumption that might not be satisfied at very low temperature and strong enough disorder; however, the comparison with the results of Monte-Carlo simulations in fig.

(3) led to reasonable results. This is expected; for spin-glass models, the effect of replica-symmetry breaking is typically rather limited, in particular close to the Almeida-Thouless line. In order to study the glassy regime, corresponding to a network load α beyond the critical value, we expect that replica symmetry breaking has to be considered. However, as for the Fisher information, we expect it to be approximately zero in the glassy state because the activity would not show any dependence on space anymore (besides a residual one due to the input), and the disorder-averaged singleneuron variances vanish for large disorder (and all other cumulants as well). As for the paramagnetic to ferromagnetic transition, we expect a qualitative change in the shape of the neural activities, with considerably sharper bumps on the ferromagnetic side of the transition. Due to the mechanisms discussed around eq. ( 13), we expect a corresponding steep increase in the Fisher information.

On a more biological side, we have made, for convenience, several unrealistic assumptions that could be waived. The receptive fields in our setup, for example, have all the same shape and size and are evenly spaced -in reality there is of course variability in shape and they are scattered in the environment. In particular, place fields may cluster near new objects [START_REF] Bourboulou | Dynamic control of hippocampal spatial coding resolution by local visual cues[END_REF], suggesting the importance of taking into account inhomogeneous densities. These features could rather easily be included in our framework, at least if the probability distributions for the single-neuron properties are independent. The additions suggested above only require another average over them (see also appendix D). Then, though we have here studied a one-dimensional stimulus, the features determining the Fisher information, such as the sharpness of the tuning curve, can also be defined in higher dimensions, and we expect qualitatively the same results in that case. Last of all, it would be interesting to better understand how much of the information we have estimated can be extracted from the neural population in practice, beyond the linear readout mechanism considered here.

As mentioned in the main text, we fix the summed activity of all neurons to a certain number M , mimicking the effect of a global inhibition. This determines the partition function, which reads

Z M [h] = n, i ni=M exp   i<j, K ij n i n j + N i=1 h i n i   .
Explicitly performing the spin sums under this limitation is difficult, so we introduce the Fourier series with Z M [j] as coefficients:

U k [h] := N M =0 e i2πkM Z M [h] = N M =0 e i2πkM n δ M, i ni exp   i<j, K ij n i n j + N i=1 h i n i   = n exp   i<j, K ij n i n j + N i=1 h i n i   N M =0 e i2πkM δ M, i ni = n exp   i<j, K ij n i n j + N i=1 (h i + i2πk) n i   .
Applying the transform to obtain the Fourier coefficients from a periodic function, we get

Z M [h] = 1 0 dk e -i2πkM U k [j] = 1 0 dk s exp   i<j, K ij n i n j -i2πk M - N i=1 n i + N i=1 h i n i   .
For N 1, we can evaluate the k-integral in saddle-point approximation, so that in this limit, we replace the partition function by its "grandcanonical" counterpart

Z f,gc [h] := inf λ   n exp   i<j, K ij n i n j + N i=1 h i n i + λ N i=1 n i -N f     ,
where we have introduced f := M N and λ = i2πk. Note that even though the stationary value of λ is real, we are integrating it along the imaginary axis (using what is known as Bromwich contour, see e.g. [START_REF] Touchette | The large deviation approach to statistical mechanics[END_REF], appendix C), as always when the integration variable has been introduced as Lagrance multiplyer. Varying λ, we are therefore looking for the infimum, not the supremum. We obtain the "grand-canonical" cumlant-generating function that we will work with:

W λ [h] := ln   s exp   i<j, K ij n i n j + N i=1 [h i n i + λ (n i -f )]     λ h,j,K (f ) such that ∂W λ [h] ∂λ = N f.

Now we decouple the interacting term by means of a Gaussian helping field exp

  1 2 i =j, K ij n i n j   = 1 (2π) N 2 det (K) dφ e -1 2 φ T K -1 φ+ i φini
and replace the φ-integral by another saddle-point approximation:

W [h] = sup φ inf λ - 1 2 φ T K -1 φ + i ln 1 + e hi+λ+φi -λf
For the examples shown in the figures, we are assuming a rectangular shape for K,

K (r i -r j ) = K rec , for |r i -r j | ≤ w rec 0, else, (A1)
but this choice is only made for convenience, the theoretical results extend to general shapes.

Incorporating disorder

Drawing random connections in addition to the spatially ordered ones additionally modifies the extremizing probability distribution and introduces more contributions to the pairwise covariances. We would like to compute the quenched average of the cumulant-generating functional, so

W (h) J = dJ P (J) ln n e 1 2 i =j (Jij +Kij )ninj + i (niU(ξ-ri)+hini) (A2) = lim n→0 dJ P (J) -1 + n 1 ,...,n n e i =j (Jij +Kij ) n α=1 n α i n α j + i n α i (U(ξ-ri)+hi) n , (A3)
where we have used the replica trick to represent the logarithm [START_REF] Mézard | Spin Glass Theory and Beyond[END_REF]. As indicated in the main text, eq. ( 2), we assume that the couplings are uncorrelated and Gaussian, so that, after the standard procedure of introducing appropriate helping fields, we obtain

dJ P (J) e i =j Jij n α=1 n α i n α j (A4) = exp   - 1 2 g 2 N i α,β n α i n β i   α √ N g √ 2π dq α exp - 1 2 N g 2 q2 α + qα i n α i (A5) × α =β √ N g √ 2π dq αβ exp - 1 2 N g 2 q 2 αβ + q αβ i n α i n β i . ( A6
)
We combine this result with the contribution from the network without disorder, but local connectivity and solve the resulting integral, assuming replica-symmetry in q and φ. The validity of the assumption of replica symmetry is validated numerically by comparing our theoretical results with the outcomes of Monte-Carlo computations and discussed in sec. IV. Dropping subleading terms in N , we solve the integrals in saddle-point approximation and obtain for W :

W (h) J = lim n→0 extr q,q,φ - 1 n + e -1 2 N g 2 n(n-1)q 2 -1 2 N g 2 nq 2 e -1 2 nφ T K -1 φ n (A7) ×   l,γ 1
n γ l =0   k e g 2 q α =β n α k n β k e i g 2 q i α n α i + n α=1 n α i φ i + i n α=1 n α i (U(ξ-ri)+hi)   (A8) = 1 2 N g 2 q 2 - 1 2 N g 2 q2 - 1 2 φ T K -1 φ (A9) + k 1 √ 2π dt k e -t 2 k 2 i ln 1 + e φ i +tig √ 2q+U(ξ-ri)+g 2 (q-q)+hi . (A10)
Taking now into account the restriction on the total activity in addition, the mean-field cumulant-generating functional reads

W (h) J = extr q,q,φ,λ 1 2 N g 2 q 2 - 1 2 N g 2 q2 - 1 2 φ T K -1 φ -N λf (A11) + k 1 √ 2π dt k e -t 2 k 2 i ln 1 + e φ i +tig √ 2q+(U(ξ-ri))+g 2 (q-q)+λ+hi . ( A12
)
Because we are evaluating this quantity only at its extremal values, we are free to express it in shifted coordinates, λ + g 2 (q -q) → λ, in order to simplify our expressions and to get rid of q, so that we obtain

W I (h) J = extr q,q,ψ,φ,λ 1 2 N g 2 q 2 - 1 2 N g 2 q2 - 1 2 φ T K -1 φ -N λ -g 2 (q -q) f. (A13) + k 1 √ 2π dt k e -t 2 k 2 i ln 1 + e φ i +tig √ 2q+U(ξ-ri)+λ+hi (A14) =:G g (h, φ, q, λ) , (A15)
the latter two corresponding to eqs. ( 12) -( 13) in [START_REF] Monasson | Crosstalk and transitions between multiple spatial maps in an attractor neural network model of the hippocampus: Phase diagram[END_REF]. In particular, all auxiliary fields have a well-behaved limit for g → 0. Furthermore, from eq. (A20), we obtain the derivatives of the auxiliary variables with repect to g to be given by

∂ ∂g   {φ x } x λ q   = ∂ 2 ∂ [{φ x } x , λ, q] 2 G [q, φ, λ] -1 ∂ 2 ∂g∂ [{φ x } x , λ, q] G g [q, φ, λ] . (A24) Further differentiating ∂ ∂[{φx} x ,λ,q] G g [q, φ, λ] with respect to g yields ∂ 2 ∂g∂ [{φ x } x , λ, q] G g [q, φ, λ] =   dy K xy Dt y t y √ 2q • m y (1 -2m y ) x dy Dt y t y √ 2q • m y (1 -2m y ) dx Dt y t y √ 2q • m y (1 -2m y ) (1 -m y )   g=0 = 0, (A25)
with m x as introduced in eq. ( 27). The last equality in eq. (A25) holds because for g = 0, m x is independent of t x and the remaining t x -integrand is antisymmetric. To obtain the derivatives of the order parameters at g = 0, we therefore only have to check that differentiating ∂ {φx} x ,λ,q G with respect to q, φ x and λ once more yields a regular Hessian. We obtain

∂ 2 ∂ [{φ x } x , λ, q] 2 G g [q, φ, λ] =   -K -1 xy + δ xy v y v x g 2 κ 3 x v y dz v z g 2 dz κ 3 z g 2 κ 3 y g 2 dz κ 3 z g 2 + g 4 dz κ 4 z   , (A26)
where we have omitted the t x -dependence of m x for brevity and have introduced the higher-order cumulants

κ 3 x := ∂ 2 m x ∂φ 2 x = Dt m x (1 -2m x ) (1 -m x ) (A27) κ 4 x := ∂ 3 m x ∂φ 3 i = Dt m x (1 -m x ) 1 -6m x + 6m 2 x , (A28)
(the fourth-order one for later use). It is not apparent why the Hessian should have a zero mode -indeed, this would mean in particular that the saddle-point approximation is not well-defined. So as long as we trust the saddle-point approximation, we also know that the derivatives of the order-parameters with respect to g vanish for g = 0. Also, in appendix B, we numerically confirm that the Hessian does not have zero modes for g → 0.

and

Z J (ξ) = n e i niU(ξ-ri)+ i<j (Jij +Kij )ninj .
For the Fisher information, we need the second derivative of the logarithm of P with respect to ξ:

- ∂ 2 ∂ξ 2 ln (P (n|ξ)) = - ∂ 2 ∂ξ 2 i n i U (ξ -r i ) + ∂ 2 ∂ξ 2 ln Z J (ξ) = - ∂ ∂ξ i n i U (ξ -r i ) + ∂ ∂ξ ∂ ∂ξ Z J (ξ) Z J (ξ) = - i n i U (ξ -r i ) + ∂ 2 ∂ξ 2 Z J (ξ) Z J (ξ) - ∂ ∂ξ Z J (ξ) Z J (ξ) 2 .
Upon averaging over the neurons states of the neuron n i , we obtain

I n (ξ) = - 1 Z J (ξ) n e i niU(ξ-ri)+ i<j Jij ninj i n i U (ξ -r i ) + 1 Z J (ξ) n e i niU(ξ-ri)+ i<j Jij ninj   i n i U (ξ -r i ) + i n i U (ξ -r i ) 2   - 1 Z J (ξ) n e i niU(ξ-ri)+ i<j Jij ninj i n i U (ξ -r i ) 2 J (B1) = i,j U (ξ -r i ) n i n j n -n i n n j n J U (ξ -r j ) , (B2)
where we have used the usual thermal average

f (n) n := 1 Z J (ξ) n f (n) e i niU(ξ-ri)+ i<j Jij ninj . (B3)
As indicated before, to determine the Fisher information, we therefore just have to compute the covariance matrix, which we achieve by differentiating the cumulant-generating functional twice with respect to h, considering all indirect dependencies via the auxiliary fields (evaluated at their respective saddle-point values). Taking into account both the fixed total activity and the disorder, the cumulant-generating functional is given by eq. (A15). Formally differentiating this expression yields We obtain the derivatives of q and φ by taking the total derivatives of their defining equations, i.e. (A16) and (A19), which yields

d 2 dh 2 W f (h) J = ∂ 2 G ∂h 2 + 2 ∂ 2 G
0 = d dh ∂ ∂q G (h, φ, q, λ) = ∂ 2 G ∂q 2 ∂q ∂h + ∂ 2 G ∂λ∂q ∂λ ∂h + ∂ 2 G ∂φ∂q ∂φ ∂h + ∂ 2 G ∂q∂h (B7) 0 = d dh ∂ ∂λ G (h, φ, q, λ) = ∂ 2 G ∂q∂λ ∂q ∂h + ∂ 2 G ∂λ 2 ∂λ ∂h + ∂ 2 G ∂φ∂λ ∂φ ∂h + ∂ 2 G ∂λ∂h (B8) 0 = d dh ∂ ∂φ G (h, φ, q, λ) = ∂ 2 G ∂q∂φ ∂q ∂h + ∂ 2 G ∂λ∂φ ∂λ ∂h + ∂ 2 G ∂φ 2 ∂φ ∂h + ∂ 2 G ∂φ∂h , (B9)
so that we obtain after inserting into (B6)

d 2 dh 2 W f (h) J = ∂ 2 G ∂h 2 -    ∂ 2 G ∂h∂φ ∂ 2 G ∂h∂q ∂ 2 G ∂h∂λ    T    ∂ 2 G ∂φ 2 ∂ 2 G ∂φ∂q ∂ 2 G ∂φ∂λ ∂ 2 G ∂q∂φ ∂ 2 G ∂q 2 ∂ 2 G ∂q∂λ ∂ 2 G ∂λ∂φ ∂ 2 G ∂λ∂q ∂ 2 G ∂λ 2    -1    ∂ 2 G ∂φ∂h ∂ 2 G ∂q∂h ∂ 2 G ∂λ∂h    . (B10)
In order to compactly write down the entries of the matrix and the vectors above, we introduce the effective local connectivity

K -1 eff xy := - ∂ 2 G ∂φ x ∂φ y ⇔ ∂ 2 G ∂φ∂φ -1 xy = -(K eff ) xy , (B11)
which fulfills the Dyson equation whose concrete form we obtain by performing the derivatives of G explicitly:

K -1 eff xy = K -1 xy -δ xy v x (B12) ⇔ K eff xy = K xy + K xz v z K eff zy . (B13)
Using the identities derived in appendix C, in particular eq. (C9), we can note the final form of the covariance matrix:

C = V + V K eff V -(1 N + V K eff ) gκ 3 , v S -1 gκ 3 v (1 N + K eff V ) , ( B14
)
where V is the diagonal matrix with the disorder-averaged variances v i and

S = N + g 2 i κ 4 i g i κ 3 i g i κ 3 i i v i + gκ 3 v K eff gκ 3 , v (B15)
The Fisher information, finally is then given by

I n (ξ) = x,y U (ξ -x) C xy U (ξ -y) . (B16)
The last term from B14 contributes to the Fisher information with a term containing (twice) the expression The space-dependence of the contribution from the covariance is mostly determined by the shape of U (multiplications by K or K eff merely smear it out), so that multiplication with its derivative with U is well approximated by a spatial derivative and summation over space and therefore yields a contribution close to 0. This part of the covariance therefore only yields subleading contributions to the Fisher information (see fig. ( 6)) and we can neglect it in the analysis. This makes sense because it emerges from the source-dependence of q and λ, the auxiliary variables representing the disorder and the global inhibition, which are global quantities. It is therefore expected that their contribution to the spatial information is negligible.

gκ 3 v (1 N + K eff V ) U (

Analysis of the covariance matrix

Having an analytical expression for the covariance matrix at hand, we can investigate its behavior for special cases, in particular around g = 0. Because it depends on g only via the cumulants v, κ 3 and κ 4 , we primarily have to examine their behavior near g = 0. We observe that

d dg v x = ∂ ∂g v x + ∂v x ∂q ∂q ∂g + dy ∂v x ∂φ y ∂φ y ∂g + ∂v x ∂λ ∂λ ∂g (B18) = ∂ ∂g Dt m x (1 -m x ) = Dt t 2q m x 1 -3m x + 2m 2 x g=0 = 0, (B19)
where we have used the result from appendix (A 3) that the derivatives of the auxiliary variables with respect to g vanish as g goes to 0. Again because m x does not depend on t x for g = 0 and the remaining integral over t x is antisymmetric, it also yields 0. With the same argument, the derivatives of the other cumulants vanish as well. Therefore, the linear orders of all g-dependent quantities, that the covariance C depends on, vanish. Thus, the derivatives of the covariances and of the Fisher information equal 0 for g = 0 as well, as apparent from the plots in figure (3).

Relating inputs and tuning curves by means of K eff

Without disorder, the tuning curve T in the thermodynamic limit is given by

T x = dy K (x -y) 1 1 + e -(φy+U(ξ-y)+λ) (B20) φ y = dx K (y -z) T z (B21)
and therefore, we can write for its derivative

T = T (1 -T) U + KT (B22) ⇔ [1 -T (1 -T) K] T = T (1 -T) U (B23) T = (1 -V K) -1 V U = V -1 -K -1 U (B24) ⇔ T = V 1 + K V -1 -K -1 U , (B25)
where we have abbreviated

V ij = δ ij f i (1 -f i ).
We furthermore have

K eff = K + KV K eff (B26) ⇔ (1 -KV ) K eff = K (B27) ⇔ K eff = (1 -KV ) -1 K = V -1 V -1 -K -1 K (B28) ⇔ K eff = K V -1 -K -1 V -1 (B29) ⇔ K eff V = K V -1 -K -1 , (B30)
where we obtained the second-to-last equivalence by transposing. Inserting this expression into eq. (B25), we arrive at eq. ( 30).

Appendix C: Matrix-vector calculus 1. Inversion of a matrix with blocks on the diagonal of the sizes N and M

Assume we have a matrix of the form

U := A b b T a , (C1)
where

A ∈ R N ×N , b ∈ R N ×M , a ∈ R M ×M , (C2)
a and A are symmetric and A is invertible. To invert it, we make the ansatz

V := C d d T c . (C3)
Multiplying U and V , we obtain the conditions

AC + bd T = 1 N (C4) Ad + bc = 0 (C5) b T C + ad T = 0 (C6) b T d + ac = 1 M (C7)
Solving (C5) for d and inserting into (C7), we obtain

c = a -b T A -1 b -1 and d = -A -1 b a -b T A -1 b -1 .
Solving (C4) for C and inserting the results gained until here, we obtain

C = A -1 + A -1 b a -b T A -1 b -1 A -1 b T .
Plugging these results into the left-hand side of (C6), which we did not use so far, we obtain

b T A -1 + A -1 b a -b T A -1 b -1 A -1 b T -a A -1 b a -b T A -1 b -1 T = A -1 b T + b T A -1 b -a + a a -b T A -1 b -1 A -1 b T -a a -b T A -1 b -1 A -1 b T =a a -b T A -1 b -1 A -1 b T -a a -b T A -1 b -1 A -1 b T = 0,
therefore our ansatz is consistent. Summarizing, we can write the inverse of U as

U -1 = A -1 0 0 0 + -A -1 b 1 M a -b T A -1 b -1 -A -1 b T , 1 M . (C8)

Vector-matrix-vectors multiplication

Calculating cross-covariances, we are interested in calculating objects of the type

B b A b b T a -1 B b T .
Making use of (C8), we then obtain

B b A b b T a -1 B b T =BA -1 B + 1 N -BA -1 b a -b T A -1 b -1 b T 1 N -A -1 B (C9)
Appendix D: Relating Fisher and mutual information for uncoupled neurons with inhomogeneously distributed place fields

Here, we consider independent neurons, but allow variability in the tuning curves T. The probability distribution of the neural population is then given by

P (n) = N i=1
[n i T i (ξ) + (1 -n i ) (1 -T i (ξ))] P ξ (ξ) .

(D1)

To compute the mutual information, we first need to compute the entropy of this distribution, which is given by

h uncond = -lim N →∞ 1 N n P (n) ln (P (n)) T ,
where we denote by . . . T the average over the variability of the tuning curves. The tricky part here is that we average over a logarithm, a complication that we deal with by introducing replicas (n + 1 in this case because of the prefactor P (n), compare [START_REF] Nadal | Information processing by a perceptron in an unsupervised learning task[END_REF]), which leads to (1 -T (ξ α ))

h uncond = -lim N →∞ 1 N lim
T .

An obvious idea is now to evaluate eq. (D3) in saddle-point approximation, as also shown in [START_REF] Sompolinsky | Population coding in neuronal systems with correlated noise[END_REF][START_REF] Brunel | Mutual information, fisher information, and population coding[END_REF]. This is indeed what we will do, but with a small twist because one of the eigenvalues of the Hessian of G T vanishes for n → 0. However, this replicon mode can be identified to be the one corresponding to the replica-symmetric direction. This allows us to transform the n + 1-dimensional integral over the ξ α such that the first coordinate corresponds to the replica-symmetric direction (1, . . . , 1) and the other n are orthogonal to it. Like this, we can perform the integral over the first coordinate exactly and only the orthogonal directions are evaluated in saddle-point approximation.

Having determined the unconditioned entropy in this way, we obtain the mutual information MI = h uncond -h cond by subtracting

h cond = -lim N →∞
1 N n dξ P (n|ξ) ln (P (n|ξ)) T (D4) from the unconditioned entropy. Performing the limit of k → 0 in eq. (D3), we see that, to zeroth order, h uncond equals h cond , so that the mutual information is, to first order, given by the one-loop correction

I MI = dξP ξ (ξ) 1 2 ln - N λ 1,k=0 T (ξ) 2π - 1 2 -ln (P ξ (ξ)) + O 1 N , (D5)
where λ 1,k=0 T is the n-fold degenerate eigenvalue of G T at k = 0 (see below). Note that our computation neither requires the introduction of helping fields to perform the average over the state space of the neural population, as [START_REF] Brunel | Mutual information, fisher information, and population coding[END_REF], nor do we assume it to be normally distributed, as in [START_REF] Sompolinsky | Population coding in neuronal systems with correlated noise[END_REF]. However, in return, we are assuming the neurons to be independent, which limits the applicability of our approach. What is left to do is the computation of the Hessian of G T . On the replica-symmetric line, we only have two values for its entries, the diagonal and the off-diagonal. We calculate The eigenvalues of the Hessian of G T are given by

∂ 2 G T ∂ξ 2 α ξ0=•••=ξn=ξ = n=0,1 k γ=0,γ =α [nT (ξ γ ) (1 -n) (1 -T (ξ γ ))] (2n -1) T (ξ α )
λ 0 (ξ) = ∂ 2 G ∂ξ 2 α ξ0=•••=ξn=ξ + n ∂ 2 G ∂ξ α ∂ξ β ξ0=•••=ξn=ξ λ 1 (ξ) = ∂ 2 G ∂ξ 2 α ξ0=•••=ξn=ξ - ∂ 2 G ∂ξ α ∂ξ β ξ0=•••=ξn=ξ ,

FIG. 2 :

 2 FIG. 2: Panel a: tuning curves for a network without disorder in the connectivity for different strengths of local interactions. Panel b: corresponding change in the Fisher information; the dotted lines show eq. (31) with the term V K eff V in the middle part, constituting C -1 , removed. Parameters: g = 0, w two point = 0.1, U inp = 0.2, w one point = 0.07, f = 0.15.

FIG. 3 :

 3 FIG. 3: The Fisher information per neuron in dependence of the disorder, comparison with Monte-Carlo simulations. In panel a, the network is strongly input driven, and in panel b only weakly. Parameters, panel a: K rec = 5, w rec = 0.1, U inp = 2.25, w inp = 0.07, f = 0.15; panel b: K rec = 20, U inp = 0.2, other parameters as in panel a.

FIG. 4 :FIG. 5 :

 45 FIG.4: Interplay of feedforward, local and disordered recurrent input shaping the Fisher information. For panel a, we scale the synapses according to eq. (35), keeping the other parameters fixed, for panel b, we keep the the Fisher information constant, varying U inp and K rec concertedly. Parameters, panel a: K max rec = 8, w rec = 0.1, U inp = 0.2, w inp = 0.07, f = 0.15, g max = 0.16 panel b: g = 5, other parameters as in panel a.

FIG. 6 :

 6 FIG.6: The Fisher information per neuron in dependence of the disorder, contributions from different parts of covariance as defined in eq. (23). Parameters as in fig.(3).

(

  dξ α P ξ (ξ α )) (G T (ξ)) N -1 ,(D3)where we have introducedG T (ξ) := n α=0 T (ξ α ) + n α=0

1 ( 1 [

 11 2n -1) [nT (ξ) + (1 -n) (1 -T (ξ))] k T (ξ) =α,β [nT (ξ γ ) + (1 -n) (1 -T (ξ γ ))] (2n -1) 2 T (ξ α ) T (ξ β ) nT (ξ) + (1 -n) (1 -T (ξ))] k-1 [T (ξ)]

ACKNOWLEDGMENTS

TK thanks Ulisse Ferrari and Gabriel Mahuas for many very insightful discussions. This work was partly funded by the Human Frontier Science Program RGP0057/2016 grant and TK by a short-term postdoc fellowship of the German Academic Exchange Service (DAAD).

Appendix A: Computing the cumulant-generating function

As indicated in the main text, computing the Fisher information basically amounts to calculating the covariance matrix of the activity, which we obtain from the disorder-average of the cumulant-generating functional defined in eq. ( 14), or in other words, of the logarithm of the partition function. In the following, we will explain step by step how to take into account its features, starting with the fixed total activity, then including the disordered part of the connectivity and finally deriving and analysing the saddle-point equations for this case in order to compute the cumulant-generating functional in the thermodynamic limit.

which leads to the saddle-point equations

dy. (A19)

Drawing the limit of N → ∞, we have turned the sums over neuron sites into integrals over space, which we indicate by renaming the indices to x and y instead of i and j. Finally, setting h = 0 and using eq. (A18) to simplify eq. (A16), we obtain the final saddle-point equations as given in the main text, eqs. ( 18) to [START_REF] Sherrington | Solvable model of a spin-glass[END_REF]. Note that this simplification is valid for the saddle-point values q, {φ x } x and λ -however, when taking further derivatives of G with respect to h (as necessary to determine covariances), we have to assume general q, {φ x } x and λ (not as given in the saddle point) and therefore have to use the right-hand side of eq. (A16), and not of eq. ( 18).

3. Analysis of the saddle-point equations in the limit g → 0

The quantities q, φ x and λ are implicitly given by

For g = 0, the integrands in the saddle-point equations ( 18) to ( 20) get independent of t and we can perform the Gaussian integrals, so that we obtain q = dx 1

1 + e -(φx+U(ξ-x)t)+λ) 2 , (A21)

(A22)

Appendix B: Computing the Fisher information First, we convince ourselves that computing the Fisher information simply amounts to computing the covariance matrix. Consider the probability distribution for the neural network state n, conditioned on the stimulus ξ, P (n|ξ), given by

where the first one is non-degenerate, whereas the second one is n-fold degenerate. Inserting eqs. (D8) and (D12), we obtain that, for k = 0,

where the latter expression equals minus the Fisher information I n (ξ) for the stimulus ξ. Therefore, inserting this result into eq. (D5), we finally obtain

as expected according to [START_REF] Brunel | Mutual information, fisher information, and population coding[END_REF].


Cyrille Chazallon
email: cyrille.chazallon@insa-strasbourg.fr

Cédric Barazzutti

Hervé Pelletier

Mai-Lan Nguyen

Pierre Hornych

Saida Mouhoubi

Daniel Doligez

Journal of Testing and Evaluation

Keywords: glass fiber grids, indentation test, direct tension test

The sentence beginning "The tests consisted in building test sections…" has been edited for clarity.

Introduction

Geogrids are generally used for the rehabilitation of cracked pavements (semirigid pavements, asphalt pavements, and flexible pavements), but nowadays, they are also included in new pavements. To ensure an efficient reinforcement and a reliable design of the pavement incorporating a grid, it is required to know the residual mechanical properties of the geogrid after its implementation and compaction of the above asphalt concrete layer and the level of damage of the grid. When

C1

w

C2

e perform a bibliographic study on asphalt concrete reinforced by grids, one can observe that the damage of the grid is not taken into account in the modellings that are performed either analytically or with finite element methods, as shown in Graziani et al. [START_REF] Graziani | Structural Response of Grid-Reinforced Bituminous Pavements[END_REF] and Khodaii and Fallah. [START_REF] Fallah | Developing a Fatigue Fracture Model for Asphalt Overlay Reinforced with Geogrid[END_REF] Recently some studies from a field test or laboratory compaction test underline that multiple damage exist, which are revealed in the analysis of the tensile strength and of the stiffness of the recovered grids, as shown in Gonzalez-Torre et al., [START_REF] Gonzalez-Torre | Damage Evaluation during Installation of Geosynthetics Used in Asphalt Pavements[END_REF] van Rompu et al., [START_REF] Van Rompu | Procédé de renforcement d'enrobés Colgrill R : qualification des grilles en laboratoire et exemple de réalisation[END_REF] Chazallon et al., [START_REF] Chazallon | In Situ Damage Evaluation of Geogrid Used in Asphalt Concrete Pavement[END_REF] and Chazallon et al. [START_REF] Chazallon | Laboratory and In Situ Damage Evaluation of Geogrid Used in Asphalt Concrete Pavement[END_REF] . This damage depends on the hardness of the coating and the shape of the yarns. Nowadays, the main grids that are used for pavement reinforcement are cost-effective grids made of glass fibers. If coated glass fiber grids are subjected to less damage than the uncoated ones, it is crucial to be able first to quantify their damage level in strength and stiffness and then to know how to reproduce it with laboratory conditions in order to improve the grids design: coating and shape of the yarns.

This article presents the results of the national French project SolDuGri dealing with pavement reinforcement by geogrids and more precisely the reproduction in laboratory of the in situ damage concerning the mechanical strength and the stiffness modulus. In this project, full-scale tests have been performed to evaluate the in situ damage of geogrids in asphalt concrete pavements. The

C3

tests consisted in building test sections with four different types of geogrids and two different compacted asphalt concrete layers located above the geogrids. After construction, the geogrids have been recovered from different locations on the field sections, where they had been subjected to compaction only, or also to construction traffic, and their residual mechanical properties have been evaluated in the laboratory. We will present the materials used, the tested sections, and then the tensile tests on recovered grids. In parallel, we try to reproduce in the laboratory the damage observed on in situ recovered grid using a modified experimental indentation setup. For this, we present the first results of laboratory indentation tests using specific indenters whose geometries have been determined from asphalt concrete surface shapes in contact with the grids. To estimate the loss after damage by indentation, the experimental data deduced from ultimate tensile tests using ASTM D 6637/D 6637M, Standard Test Method for Determining Tensile Properties of Geogrids by the Single or Multi-Rib Tensile Method, [START_REF]Standard Test Method for Determining Tensile Properties of Geogrids by the Single or Multi-Rib Tensile Method[END_REF] were compared to results obtained on new grids without any damage. [START_REF] Chazallon | Laboratory and In Situ Damage Evaluation of Geogrid Used in Asphalt Concrete Pavement[END_REF] We can notice that the mechanical properties are affected by the indentation effect, especially the resistance strength that is slightly reduced.

Materials Used

SEMICOARSE ASPHALT CONCRETE MATERIAL For

C4

this test, the asphalt concrete material chosen is a standard semicoarse asphalt concrete (self-compacting asphalt concrete [SCAC] "AC AQ4 10 surf 35/50" according to NF EN 13108-1, Mélanges bitumineux -Spécifications des matériaux -Partie 1 : enrobés bitumineux 8 ) designed with the crushed aggregates in the area where the experimentation was to be carried out. The binder is a 35/50 penetration grade bitumen according to NF EN 12591, Bitumes et liants bitumineux -Spécifications des bitumes routiers. [START_REF]Bitumes et liants bitumineux -Spécifications des bitumes routiers[END_REF] The stiffness modulus, measured by two-points bending tests, is 12,945 MPa at 15°C and 10 Hz for the hot mix (160°C).

These asphalt concretes have been used in their usual range of thickness: 5 to 7 cm. The tack coat between the asphalt layers, including the grid impregnation, was a classical cationic rapid setting bitumen emulsion, classified as C69B3 according to NF EN 13808, Bitumes et liants bitumineux -Cadre de spécifications pour les émulsions cationiques de liants bitumineux. [START_REF]Bitumes et liants bitumineux -Cadre de spécifications pour les émulsions cationiques de liants bitumineux[END_REF] GLASS FIBER GRIDS Four different grids, made of glass fiber coated with two different resins, of different hardness, were chosen for the study. The first two grids (G1 and G2) had a strength of 20 kN/m for a strain of 1 % and were coated with a stiff resin and a soft resin, respectively. The two other grids (G3 and G4) had a 40 kN/m strength at 1 % strain, stiff and soft resins, respectively.

Tested Sections In

C5

total, five test sections were built for the SolDuGri project on an existing subgrade with a load-bearing capacity higher than 100 MPa (falling weight deflectometer tests). One of these sections, 15 m long, was devoted to the study of grid damage after construction. This section included the four different types of geogrids (G1, G3, G2, and G4).

A leveled asphalt concrete layer was firstly realized on the existing subgrade. Four SCAC layers, each 5 cm thick, were then laid one after the other to constitute the pavement test section (fig. 1).

The 20-cm-thick pavement structure was chosen because a thick structure was required for other laboratory studies planned in the SolDuGri project. [START_REF] Godard | SolDuGri Project : For Sustainable Reinforcements of Infrastructures with Glass Fiber Grids[END_REF] Void content measurements by a Troxler device on the surface of the second and fourth layers indicated mean void contents of 5.7 % and 5.8 %, respectively, on sections made of hot mix.

The different geogrids were placed on the surface of the second SCAC layer. To study the effect of different surface conditions on the damage of the geogrids, on one half of the section, the surface of the second SCAC layer was milled before placing the grids (to simulate the reinforcement of an old pavement-surface B), and the second part was compacted normally to simulate a new construction (surface A). On both parts, a tack coat with 500 g/m 2 of residual bitumen was applied to ensure impregnation of the geogrids. A tack coat with 300 g/m 2 of residual bitumen was applied on the other interfaces. The full-scale test sections were built using standard roadwork equipment (fig. 2) as used on real construction sites. The C6 grids could be so affected by three types of impact during the construction: movements of pneumatic tires of delivery trucks and movements of metallic caterpillars of paving machines directly on some grids, as well as the indentation of aggregates because of the compaction on the whole surface of grids. The positions of traffic, i.e., truck tires and metallic caterpillars of paving machines, on the pavement during each phase of construction were precisely measured and reported for the investigation.

For the study of the geogrid damage after construction, a specific procedure of placement of the geogrids was adopted. The four different types of grids were cut in small sections, 72 cm by 72 cm large. After construction, when the pavement was completely cooled, the geogrids were recovered from different locations on the field sections, where they had been subjected to compaction only, or also to construction traffic, by the trucks and by the finisher. Care was taken to minimize the damage of the grids during recovery. Each grid section FIG. 1 Transverse profile of test sections.

Journal of Testing and Evaluation

CHAZALLON ET AL. ON REPRODUCTION OF GEOGRID IN SITU DAMAGE was located and sawed in a block of about 64 by 64 cm 2 . Each block was then placed inside a large oven and heated to 40°C for about 5 to 7 h. The recovered geogrid sections were finally marked to identify their type and their location on the field section, and they were sent to the laboratory for evaluation of their residual mechanical properties.

Although great care was taken for the recovering of the geogrid sections, 100 % of the yarns could not be recovered properly without any damage because of the strong bond between the geogrids and the asphalt concrete layers. After a visual examination, all the grids that have yarns that were cut or seemed to have been torn or bent during the extraction were eliminated. Only grids that had been damaged directly by the pavement construction process were considered for laboratory evaluation.

Tension Tests on Recovered Grids

In total, 32 grid sections were placed in the test section and have been recovered. A total of 27 have been tested until now. Each section was trafficked on more than 50 % of its surface by the wheels of the delivery trucks or the metallic caterpillars of the finisher while paving the test section. Concerning the tension tests, slightly less than 50 % of the yarns had been damaged only by the compaction, although a little more than 50 % had been damaged by both compaction and trafficking by the caterpillars or the wheels.

During recovery, the loading received by each grid was carefully identified (see Table 1), and three different loading processes were distinguished:

-Grids submitted only to compaction of the upper layers without any other loading (no loading).

-Grids submitted to compaction and to loading by the wheels of the asphalt truck (tire).

-Grids submitted to compaction and loading by the finisher's caterpillars. Because of the narrow asphalt layers, a small finisher with an overall width of the caterpillars of 1.5 m was used. This width was smaller than the spacing of the truck wheels. Therefore, the zones loaded by the caterpillars were different from the zones loaded by the tires. 7 (0/7) Each recovered grid was cut into individual warp or filling yarns, which were submitted to tension tests. The yarn samples were 29 cm long and were tested at 20°C. For each yarn, the secant modulus was determined between the maximum force and 100 N (small strain tests were also performed). The strains were measured with the press strain measurement device and a local extensometer for the small strain tests. The same tests have also been performed on new grids coming from the manufacturer for comparison. [START_REF] Chazallon | In Situ Damage Evaluation of Geogrid Used in Asphalt Concrete Pavement[END_REF][START_REF] Gharbi | Wedge Splitting Characterisation of the Bond between Asphalt Layers Reinforced with Glass Fibre Grid[END_REF] Globally, after 361 traction tests, these tests give the following results: one observes residual strengths varying between 20 and 55 % of initial tensile strengths and residual secant moduli at the maximum strength between 68 and 95 % of initial secant moduli at the maximum strength. [START_REF] Chazallon | Laboratory and In Situ Damage Evaluation of Geogrid Used in Asphalt Concrete Pavement[END_REF]

Reproduction of In Situ Damage of Geogrids

In order to reproduce in the laboratory the damage observed on recovered geogrid by compaction and trafficking, the first step has consisted in the accurate analysis of the topography of real asphalt concrete blocks coming from in situ test sections using numerical microscopy (HIROX RH 2000), as described in figure 3A.

The surface (15 by 15 cm 2 ) of each asphalt concrete block that has been in contact with the geogrid has been observed. These asphalt concrete blocks have been extracted from hot mixed asphalt concrete slabs compacted in the laboratory with the same AC 10 surf 35/50 (same density, binder, aggregates), which were used for the in situ test section. By analyzing several cross profiles in two in-plane directions (see in fig. 3A bottom), specific peaks associated with surface asperities have been identified and then geometrically characterized (shape, characteristic lengths such as height, edge length, edge radius, opening angle). Two specific peak geometries are involved for each block surface: conical and V shaped.

Figure 3B shows the evolution of the measured open angle as a function of the measured curvature radius for the block whose surface topography is depicted in figure 3A. From these experimental data, a statistical analysis has been performed for eight real concrete blocks, as depicted in figure 3C for the measured open angle. Two average values can be extracted from statistical analysis: average values of open angle close to 110°and of radius curvature of 0.7 mm. From these two average values, a wide range of conical and V-shaped indenters have then been designed. About 45 indenter geometries have been manufactured with open angles varying from 90°to 130°, two values of tip radius less than 0.1 and 0.7 mm, and for V-shaped indenters, with edge lengths varying between 1 and 14 mm, as depicted in figure 4. After machining, each indenter tip has been controlled using numerical microscope observations (fig. 4B and4C). The material used for indenters is a hardened nickel-chromium steel to avoid or limit their plastic deformation during indentation tests.

To realize indentation tests, an universal Bruker's UMT Tribolab experimental setup has been equipped with built-in sample holders to fix the tested geogrid yarns (fig. 5). Two options are integrated to this sample holder: (i) a heating system to perform indentation tests both at room temperature and also at 100°C (±5°C), and (ii) a specific plate in order to control the orientation of the geogrid during indentation tests, especially when using V-shaped indenters.

For all indentation tests, the indenter tip is pushed in contact to the surface of the geogrid yarns. An initial preload of 2 N is applied, and then a linear increasing load is imposed up to a maximal value of 900 N at a loading rate of 20 N • s -1 . After a holding time of 5 s, the load is linearly decreased from 900 to 0 N at an unloading rate of 20 N • s -1 . For each of the testing conditions and for each sample, indentation experiments have been replicated three or five times as a function of the observed experimental scattering. Because of the stiffness of the experimental setup, the measured penetration depth has to be corrected as a function of the normal applied load (F z ). The determination of the stiffness is a standard procedure for load and displacement-sensing indentation experiments. Indentation tests are performed by increasing progressively the normal load applied in a hard sample, whose mechanical properties are well known. [START_REF] Oliver | An Improved Technique for Determining Hardness and Elastic Modulus Using Load and Displacement Sensing Indentation Experiments[END_REF] Figure 6 exhibits experimental load-displacement curves obtained as a function of indenter geometries and the tested temperature for a given grid (the penetration depth ratio [pdr] corresponds to the ratio between the effective penetration depth and the real thickness of the tested yarn). It is interesting to note the characteristic curve describing the applied load as a function of the penetration depth that does not follow a classical kick's law (F z = C • h 2 , with F z , the normal applied load, h, the effective penetration depth, and C, a constant function of the indenter tip geometry and the hardness of the tested material), especially at room temperature. For shallow indentation depth, the tip penetrates rapidly into the grid (with a horizontal slope), and at a specific penetration For a given indenter geometry, figure 7 shows the mechanical response of the different warp yarns related to the characteristic microstructure of the grid. The C7 resistance to penetration measured by indentation experiments is quite different and directly related to the resin used or the density of glass fibers embedded in resin, or both. Moreover, it is not obvious to observe that the mechanical response to the indentation experiment of warp yarns depends on the geometry of the indenter (in fig. 7, the edge length of V-shaped indenter; in fig. 8, FIG. 4 Examples of V-shaped indenters with an edge length of 10 mm with different open angles manufactured in hardened nickel-chromium steel (A). Projected profiles of conical (B) and V-shaped (C) indenters measured using a numerical microscope in order to control their geometrical properties after machining.

FIG. 5 Experimental indentation setup (UMT Tribolab, Bruker) with a focus on the indenter and sample holders.

Journal of Testing and Evaluation the conical-shaped indenter) but also on the testing temperature. ESEM AQ5 observations have been performed on the residual imprint left on the surface using a V-shaped indenter (fig. 9). It is interesting to note that the glass fibers are cut in a brittle way by the indenter edge. The damage generated by the indentation process propagates deeper than the near surface region. It is interesting to note that any pop-ins cannot be observed on the recorded loading curve. Theses pop-ins are in general related to crack formation under the tip during penetration. This phenomenon can be explained by the fact that the cutting process of individual glass fiber does not liberate much fracture energy and cannot be detected on the macroscopic load-displacement curve.

In order to reproduce effects of compaction and trafficking on the mechanical properties of geogrid damage, we have performed, using two V-shaped indenters with an aperture of 90°and with an edge length of 6 mm and 1 mm, respectively, one indentation in the middle of 5 new warp yarns, G1, at an elevated temperature (105°C) and Journal of Testing and Evaluation The mechanical properties are affected by the indentation effect, especially the resistance strength that is slightly reduced, as described in Table 2, divided by 2 for indentation with a V-shaped indenter 3D-6-90 at 60 % of the thickness of a geogrid. With 3D-6-90 indenters and 105°C indentations, it was possible to partly reproduce the in situ residual strength, as previously mentioned, where residual strengths are varying between 20 and 55 % of initial tensile strengths, and residual secant moduli (moduli at the maximum strength) between 68 and 95 % of initial secant moduli at the maximum strength. [START_REF] Chazallon | Laboratory and In Situ Damage Evaluation of Geogrid Used in Asphalt Concrete Pavement[END_REF] The edge length seems to be an important parameter. Figure 10 shows residual tensile specimens after loading to rupture. Delaminations of the glass fibers (fig. 10A) with 7D-1-90 and a pdr of 40 % and brittle fracture with either a longitudinal propagation of the main crack (fig. 10B) with 7D-1-90 and a pdr of 60 % or a local rupture (fig. 10C) with 3D-6-90 and a pdr of 60 % are the three schemes of fracture under tensile loading, as observed also on the in situ recovered geogrid (fig. 11). FIG. 9 ESEM observations of residual imprint after indentation test using a V-shaped indenter geometry.

TABLE 2

Tensile strength and secant modulus after indentation tests on G1 warp yarns, as a function of the tip geometry and the imposed penetration depth (3D-6-90: V-shaped indenter with open angle of 90°and edge length of 6 mm; 7D-1-90: V-shaped indenter with open angle of 90°and edge length of 1 mm)

Conclusion

The indentation test results are encouraging, and the range of the residual strengths and secant moduli are in good agreement with what has been observed from 361 laboratory direct tensile tests performed on recovered in situ grids. These results prove that laboratory indentation tests whose indenter characteristics are determined from field measurement can be used to estimate the loss of mechanical performances because of in situ trafficking and the compaction process. This study shows that indentation tests can be a useful technique to characterize and to Journal of Testing and Evaluation evaluate the grid quality and that this characterization has a direct correlation with the in situ good performance of the glass fiber grids, which is the consequence.

The analysis of the influence of the V-shaped indenters' aperture and the influence of the conical shape for all warp and filling yarns of the grids will be the next step. A better understanding of the influence of the coating and multiple indentations on the residual strength or the stiffness modulus, or both, of the yarns is required. For that, we think that the plastic work of each indentation could be calculated. It would be helpful to determine the influence of the hardness of the coating and the indentation direction (parallel, 45°or orthogonal to the yarns), which generates more damage when the yarns are subjected to multiple indentations.

FIG. 2

 2 FIG.2Construction of the full-scale test sections and trafficking of construction equipment on the grids.

  FIG. 3 Three-dimensional surface topography obtained by numeric microscope HIROX of a real concrete asphalt block (A) and corresponding statistical analysis showing the evolution of the open angle as a function of the curvature radius (B) and the distribution of measured open angle of asperities (C).

FIG. 6

 6 FIG. 6 Examples of loaddisplacement curves obtained at room temperature (A) and at T = 100°C (B) using V-shaped indenters with different edge lengths and open tip angles.

  FIG. 7 Examples of loaddisplacement curves obtained on different tested grids (warp yarns) using a V-shaped indenter with a 14-mm edge length and 130°o pen angle at room temperature (A) and at T = 100°(B).

FIG. 8

 8 FIG. 8 Examples of loaddisplacement curves obtained on different tested grids (warp yarns) using a conical indenter with 130°open angle at room temperature (A) and at T = 100°(B).

FIG. 10

 10 FIG. 10 Optical observations of the residual tensile specimen after indentation and then after tension test to rupture (A)with 7D-1-90 and a pdr of 40 % (B) with 7D-1-90 and a pdr of 60 % (C) with 3D-6-90 and a pdr of 60 %.

  FIG. 11 Optical observations of the residual tensile recovered specimen and then after tension test to rupture (A), delamination (B), local rupture (C), and (D) longitudinal propagation of the main crack.

TABLE 1

 1 Number of warp and filling yarns tested per loading type and per support type (A/B)

	Grid	G1 (A/B)	G2 (A/B)	G3 (A/B)	G4 (A/B)
	No loading (NL)	51 (32/19)	29 (16/13)	31 (4/27)	25 (4/21)
	Tires (T)	51 (27/24)	35 (17/18)	45 (14/31)	35 (12/23)
	Caterpillars (C)	21 (7/14)	10 (5/5)	21 (4/17)

Table 2

 2 for the 20 indented new warp yarns.

IFSTTAR, Centre de Nantes, Route de Bouaye, CS4,

Bouguenais CEDEX, France 4 ICube UMR7357, CNRS, INSA de Strasbourg, Université de Strasbourg, de 24 Blvd. de la Victoire, 67084 Strasbourg CEDEX, France 5 6D Solutions, 17 Place Xavier Ricard, 69110 Sainte Foy lès Lyon, France

ACKNOWLEDGMENTS

The work presented in this article was sponsored by the French National Research Agency (ANR-SolDuGri project ANR-14-CE22-0019).

We note that Ref. 11 could not be verified. Please check and confirm that the information con-