Hussam Al Daas

Grey Ballard
email: ballard@wfu.edu

Laura Grigori
email: laura.grigori@inria.fr

Kathryn Rouse
email: kathryn.rouse@inmar.com.

Hussam Al Daas
email: hussam.al-daas@stfc.ac.uk

Suraj Kumar
email: suraj.kumar@inria.fr

Tight Memory-Independent Parallel Matrix Multiplication Communication Lower Bounds

Communication lower bounds have long been established for matrix multiplication algorithms. However, most methods of asymptotic analysis have either ignored the constant factors or not obtained the tightest possible values. Recent work has demonstrated that more careful analysis improves the best known constants for some classical matrix multiplication lower bounds and helps to identify more efficient algorithms that match the leading-order terms in the lower bounds exactly and improve practical performance. The main result of this work is the establishment of memory-independent communication lower bounds with tight constants for parallel matrix multiplication. Our constants improve on previous work in each of three cases that depend on the relative sizes of the aspect ratios of the matrices.

INTRODUCTION

The cost of communication relative to computation continues to grow, so the time complexity of an algorithm must account for both the computation it performs and the data that it communicates. Communication lower bounds for computations set targets for efficient algorithms and spur algorithmic development. Matrix multiplication is one of the most fundamental computations, and its I/O complexity on sequential machines and parallel communication costs have been well studied over decades [START_REF] Aggarwal | Communication complexity of PRAMs[END_REF][START_REF] Demmel | Communication-Optimal Parallel Recursive Rectangular Matrix Multiplication[END_REF][START_REF] Hong | I/O complexity: The red-blue pebble game[END_REF][START_REF] Irony | Communication lower bounds for distributed-memory matrix multiplication[END_REF][START_REF] Scquizzato | Communication Lower Bounds for Distributed-Memory Computations[END_REF].

The earliest results established asymptotic lower bounds, ignoring constant factors and lower order terms. Because of the ubiquity of matrix multiplication in high performance computations, more recent attempts have tightened the analysis to obtain the best constant factors [17,[START_REF] Olivry | Automated Derivation of Parametric Data Movement Lower Bounds for Affine Programs[END_REF][START_REF] Smith | A Tight I/O Lower Bound for Matrix Multiplication[END_REF]. These improvements in the lower bound also helped identify the best performing sequential and parallel algorithms that can be further tuned for high performance in settings where even small constant factors make significant differences. We review these advances and other related work in § 2.

The main result of this paper is the establishment of tight constants for memory-independent communication lower bounds for parallel classical matrix multiplication. In the context of a distributed-memory parallel machine model (see § 3.1), these bounds apply even when the local memory is infinite, and they are the tightest bounds in many cases when the memory is limited. Demmel et al. [START_REF] Demmel | Communication-Optimal Parallel Recursive Rectangular Matrix Multiplication[END_REF] prove asymptotic bounds for general rectangular matrix multiplication and show that three different bounds are asymptotically tight in separate cases that depend on the relative sizes of matrix dimensions and the number of processors. Our main result, Theorem 1 in § 4, reproduces those asymptotic bounds and improves the constants in all three cases. Further, in § 5, we analyze a known algorithm to show that it attains the lower bounds exactly, proving that the constants we obtain are tight. We present a comparison to previous work in Tab. 1 and discuss it in detail in § 6.

We believe one of the main features of our lower bound result is the simplicity of the proof technique, which makes a unified argument that applies to all three cases. The key idea is to cast the lower bound as the solution to a constrained optimization problem (see Lemma 5) whose objective function is the sum of variables that correspond to the amount of data of each matrix required by a single processor's computation. The constraints include the well-known Loomis-Whitney inequality [START_REF] Loomis | An inequality related to the isoperimetric inequality[END_REF] as well as new lower bounds on individual array access (see Lemma 4) that are necessary to establish separate results for the three cases. All of the complexity of the three cases, including establishing the thresholds between cases and the leading terms in each case, are confined to a single optimization problem. We use fundamental results from convex optimization (reviewed in § 3.2) to solve the problem analytically. This unified argument is elegant, it improves previous results to obtain tight constants, and it can be applied more generally to other computations that have iteration spaces with uneven dimensions.

RELATED WORK

Memory-Dependent Bounds for Matrix Multiplication

The first communication lower bound for matrix multiplication was established by Hong and Kung [START_REF] Hong | I/O complexity: The red-blue pebble game[END_REF], who obtain the result using computation directed acyclic graph (CDAG) analysis that multiplication of square n × n matrices on a machine with cache size M requires Ω(n 3 / √ M) words of communication. Irony, Toledo, and Tiskin [START_REF] Irony | Communication lower bounds for distributed-memory matrix multiplication[END_REF] reproduce the result using a geometric proof based on the Loomis-Whitney inequality (Lemma 1), show it applies to general rectangular matrices (replacing n 3 with n 1 n 2 n 3 ), and obtain an explicit constant of (1/2) 3/2 ≈ .35. They also observe that the result is easily extended to the distributed memory parallel computing model (as described in § 3.1) under mild assumptions by dividing the bound by the number of processors P. We refer to such bounds as "memory-dependent, "

following [START_REF] Ballard | Brief announcement: strong scaling of matrix multiplication algorithms and memory-independent communication lower bounds[END_REF], where the cache size M is interpreted as the size of each processor's local memory. Later, Dongarra et al. [START_REF] Dongarra | Revisiting Matrix Product on Master-Worker Platforms[END_REF] tightened the constant for sequential and parallel memory-dependent bounds to (3/2) 3/2 ≈ 1.84. More recently, Smith et al. [START_REF] Smith | A Tight I/O Lower Bound for Matrix Multiplication[END_REF] prove the constant of 2 and show that it is obtained by an optimal sequential algorithm and is therefore tight. Both of these results are proved using the Loomis-Whitney inequality. Kwasniewski et al. [17] use CDAG analysis to obtain the same constant of 2 and show that it is tight in the parallel case (when memory is limited) by providing an optimal algorithm.

Bounds for Other Computations

Hong and Kung's proof technique can be applied to a more general set of computations, including the FFT [START_REF] Hong | I/O complexity: The red-blue pebble game[END_REF]. Ballard et al. [START_REF] Ballard | Graph expansion and communication costs of fast matrix multiplication[END_REF] use the proof technique of [START_REF] Irony | Communication lower bounds for distributed-memory matrix multiplication[END_REF] to generalize the lower bound (with the same explicit constant) to other linear algebra computations such as LU, Cholesky, and QR factorizations. The constants of the lower bounds for these and other computations are tightened by Olivry et al. [START_REF] Olivry | Automated Derivation of Parametric Data Movement Lower Bounds for Affine Programs[END_REF], including reproducing the constant of 2 for matrix multiplication.

Kwasniewski et al. [START_REF] Kwasniewski | On the Parallel I/O Optimality of Linear Algebra Kernels: Near-Optimal Matrix Factorizations[END_REF] also obtain tighter constants for LU and Cholesky factorizations using CDAG analysis. Christ et al. [START_REF] Christ | Communication Lower Bounds and Optimal Algorithms for Programs That Reference Arrays -Part 1[END_REF] show that a generalization of the Loomis-Whitney inequality can be used to prove communication lower bounds for a much wider set of computations, but the asymptotic bounds do not include explicit constants. This approach is applied to a tensor computation by Ballard and Rouse [START_REF] Ballard | Communication Lower Bounds for Matricized Tensor Times Khatri-Rao Product[END_REF][START_REF] Ballard | General Memory-Independent Lower Bound for MTTKRP[END_REF].

Memory-Independent Bounds for Parallel Matrix Multiplication

This section describes the related work that focuses on the topic of this paper. Aggarwal, Chandra, and Snir [START_REF] Aggarwal | Communication complexity of PRAMs[END_REF] extend Hong and Kung's result for matrix multiplication to the LPRAM parallel model, which closely resembles the model we consider with the exception that there exists a global shared memory where the inputs initially reside and where the output must be stored at the end of the computation. Communication bounds in the related BSP parallel model are also memory independent, and Scquizzato and Silvestri [START_REF] Scquizzato | Communication Lower Bounds for Distributed-Memory Computations[END_REF] establish the same asymptotic lower bounds for matrix multiplication in that model. In addition to proving bounds for sequential matrix multiplication and the associated memory-dependent bound for parallel matrix multiplication, Irony, Toledo, and Tiskin [START_REF] Irony | Communication lower bounds for distributed-memory matrix multiplication[END_REF] prove also that a parallel algorithm must communicate Ω(n 2 /P 2/3 ) words, and they provide explicit constants in their analysis.

Note that the size of the local memory M does not appear in this bound. Ballard et al. [START_REF] Ballard | Brief announcement: strong scaling of matrix multiplication algorithms and memory-independent communication lower bounds[END_REF] reproduce this result for classical matrix multiplication as well as prove similar results for fast matrix multiplication. They distinguish between memory-dependent bounds (results described in § 2.1) and memory-independent bounds for parallel algorithms, and they show the two bounds relate and affect strong scaling behavior. In particular, when M ≫ n 2 /P 2/3 (or equivalently P ≫ n 3 /M 3/2 ), the memory-dependent bound is unattainable because the memory-independent bound is larger.

Demmel et al. [START_REF] Demmel | Communication-Optimal Parallel Recursive Rectangular Matrix Multiplication[END_REF] extend the memory-independent results to the rectangular case (multiplying matrices of dimensions n 1 × n 2 and n 2 × n 3 ), showing that three different bounds apply that depend on the relative sizes of the three dimensions and the number of processors, and their proof provides explicit constants. For one of the cases, and for a restricted class of parallelizations, Kwasniewski et al. [17] prove a tighter constant and show that it is attainable by an optimal algorithm.

We summarize the constants obtained by these previous works and compare them to our results in Tab. 1. Further details of the comparison are given in § 6.1. Following Theorem 1, the table assumes m = max{n 1 , n 2 , n 3 }, n = median{n 1 , n 2 , n 3 }, and k = min{n 1 , n 2 , n 3 }.

Communication-Optimal Parallel Matrix Multiplication Algorithms

Both theoretical and practical algorithms that attain the communication lower bounds have been proposed for various computation models and implemented on many different types of parallel systems. The idea of "3D algorithms" for matrix multiplication was developed soon after communication lower bounds were established; see [START_REF] Agarwal | A three-dimensional approach to parallel matrix multiplication[END_REF][START_REF] Aggarwal | Communication complexity of PRAMs[END_REF]7,[START_REF] Johnsson | Minimizing the communication time for matrix multiplication on multiprocessors[END_REF] for a few examples. These algorithms partition the 3D iteration space of matrix multiplication in each of the three dimensions and assign subblocks across a 3D logical grid of processors. McColl and Tiskin [START_REF] Mccoll | Memory-Efficient Matrix Multiplication in the BSP Model[END_REF] and Demmel et al. [START_REF] Demmel | Communication-Optimal Parallel Recursive Rectangular Matrix Multiplication[END_REF] present recursive algorithms that effectively achieve similar 3D logical processor grid for square and general rectangular problems, respectively. High-performance implementations of these algorithms on today's supercomputers demonstrate that these algorithms are indeed practical and outperform standard library implementations [17,[START_REF] Solomonik | Communication-Optimal Parallel 2.5D Matrix Multiplication and LU Factorization Algorithms[END_REF].

PRELIMINARIES

Parallel Computation Model

We consider the α-β-γ parallel machine model [START_REF] Chan | Collective communication: theory, practice, and experience[END_REF][START_REF] Thakur | Optimization of Collective Communication Operations in MPICH[END_REF]. In this model, each of P processors has its own local memory of size M and can compute only with data in its local memory. The processors can communicate data to and from other processors via messages that are sent over a fully connected network (i.e., each pair of processors has a dedicated link so that there is no contention on the network). Further, we assume the links are bidirectional so that a pair of processors can exchange data with no contention. Each processor can send and receive at most one message at the same time. The cost of communication is a function of two parameters α and β, where α is the per-message latency cost and β is the per-word bandwidth cost. A message of w words sent from one processor to another costs α + βw. The parameter γ is the cost to perform a single arithmetic operation. For dense matrix multiplication when sufficiently large local memory is available, bandwidth cost nearly always dominates latency cost. Hence, we focus on the bandwidth cost in this work.

In our model, the communication cost of an algorithm is counted along the critical path of the algorithm so that if two pairs of processors are communicating messages simultaneously, the communication cost is that of the largest message.

In this work, we focus on memory-independent analysis, so the local memory size M can be assumed to be infinite. We consider limited-memory scenarios in § 6.2.

Fundamental Results

In this section we collect the fundamental existing results we use to prove our main result, Theorem 1. The first lemma is a geometric inequality that has been used before in establishing communication lower bounds for matrix multiplication [START_REF] Ballard | Graph expansion and communication costs of fast matrix multiplication[END_REF][START_REF] Demmel | Communication-Optimal Parallel Recursive Rectangular Matrix Multiplication[END_REF][START_REF] Irony | Communication lower bounds for distributed-memory matrix multiplication[END_REF]. We use it to relate the computation performed by a processor to the data it must access.

Lemma 1 (Loomis-Whitney [START_REF] Loomis | An inequality related to the isoperimetric inequality[END_REF]). Let V be a finite set of lattice points in R 3 , i.e., points (i, j, k) with integer coordinates. Let ϕ i (V ) be the projection of V in the i-direction, i.e., all points (j, k) such that there exists an i so that (i, j, k) ∈ V . Define ϕ j (V ) and ϕ k (V ) similarly. Then

|V | ≤ |ϕ i (V )| • |ϕ j (V )| • |ϕ k (V )|,
where | • | denotes the cardinality of a set.

The next set of definitions and lemmas allow us to solve the key constrained optimization problem (Lemma 5) analytically. We first remind the reader of the definitions of differentiable convex and quasiconvex functions and of the Karush-Kuhn-Tucker (KKT) conditions. Here and throughout, we use boldface to indicate vectors and matrices and subscripts to index them, so that x i is the ith element of x, for example.

Definition 1 ([8, eq. (3.2)]). A differentiable function f : R d → R is convex if its domain is a convex set and for all x, y ∈ dom f , f (y) ≥ f (x) + ⟨∇f (x), y -x⟩. Definition 2 ([8, eq. (3.20)]). A differentiable function д : R d → R is quasiconvex if its domain is a convex set and for all x, y ∈ dom д, д(y) ≤ д(x) implies that ⟨∇д(x), y -x⟩ ≤ 0.
Definition 3 ([8, eq. (5.49)]). Consider an optimization problem of the form

min x f (x) subject to д(x) ≤ 0 (1)
where f : R d → R and д : R d → R c are both differentiable. Define the dual variables µ ∈ R c , and let J д be the Jacobian of д. The Karush-Kuhn-Tucker (KKT) conditions of (x, µ) are as follows:

• Primal feasibility: д(x) ≤ 0;

• Dual feasibility: µ ≥ 0;

• Stationarity: ∇f (x) + µ • J д (x) = 0; • Complementary slackness: µ i д i (x) = 0 for all i ∈ {1, . . . , c}.
The next two results establish that our key optimization problem in Lemma 5 can be solved analytically using the KKT conditions. While the results are not novel, we provide proofs for completeness.

Lemma 2 ([6, Lemma 2.2]). The function д 0 (x) = Lx 1 x 2 x 3 , for some constant L, is quasiconvex in the positive octant.

Proof. Let x, y be points in the positive octant with д 0 (y) ≤ д 0 (x). Then y 1 y 2 y 3 ≥ x 1 x 2 x 3 . Applying the inequality of arithmetic and geometric means (AM-GM inequality) to the values y 1 /x 1 , y 2 /x 2 , y 3 /x 3 (which are all positive), we have 1 3

y 1 x 1 + y 2 x 2 + y 3 x 3 ≥ y 1 y 2 y 3 x 1 x 2 x 3 1/3 ≥ 1. ( 2
)
Then ∇д 0 (x) = -x 2 x 3 -x 1 x 3 -x 1 x 2 , and

⟨∇д 0 (x), y -x⟩ = 3x 1 x 2 x 3 -y 1 x 2 x 3 -x 1 y 2 x 3 -x 1 x 2 y 3 = 3x 1 x 2 x 3 1 - 1 3 y 1 x 1 + y 2 x 2 + y 3 x 3 ≤ 0,
where the last inequality follows from eq. ( 2). Then by Def. 2, д 0 is quasiconvex on the positive octant. □ Lemma 3. Consider an optimization problem of the form given in eq. (1). If f is a convex function and each д i is a quasiconvex function, then the KKT conditions are sufficient for optimality.

Proof. Suppose x * and µ * satisfy the KKT conditions given in Def. 3. If µ * = 0, then by stationarity, ∇f (x * ) = 0.

Then the convexity of f (Def. 1) implies

f (x) ≥ f (x * ) + ⟨∇f (x * ), x -x * ⟩ = f (x * )
for all x ∈ dom f , which implies that x * is a global optimum. Now suppose µ * 0, then without loss of generality (and by dual feasibility) there exists m ≤ c such that µ * i > 0 for 1 ≤ i ≤ m and µ * i = 0 for m < i ≤ c. Complementary slackness implies that д i (x * ) = 0 for 1 ≤ i ≤ m. Consider any (primal) feasible x ∈ dom f . Then д i (x) ≤ 0 for all i, and thus

д i (x) ≤ д i (x * ) for 1 ≤ i ≤ m. By quasiconvexity of д i (Def. 2), this implies ⟨∇д i (x * ), x -x * ⟩ ≤ 0.
Stationarity implies that ∇f (x * ) = -m i=1 µ * i ∇д i (x * ), and thus

⟨∇f (x * ), x -x * ⟩ = - m i=1 µ * i ⟨∇д i (x * ), x -x * ⟩ ≥ 0.
By convexity of f (Def. 1), we therefore have

f (x) ≥ f (x * ) + ⟨∇f (x * ), x -x * ⟩ ≥ f (x * ),
and thus x * is a global optimum. □

MAIN LOWER BOUND RESULT

Lower Bounds on Individual Array Access

The following lemma establishes lower bounds on the number of elements of each individual matrix a processor must access based on the number of computations a given element is involved with. This result is used to establish a set of constraints in the key optimization problem used in the proof of Theorem 1.

Lemma 4. Given a parallel matrix multiplication algorithm that multiplies an n 1 × n 2 matrix A by an n 2 × n 3 matrix B using P processors, any processor that performs at least 1/Pth of the scalar multiplications must access at least n 1 n 2 /P elements of A and at least n 2 n 3 /P elements of B and also compute contributions to at least

n 1 n 3 /P elements of C = A • B.
Proof. The total number of scalar multiplications that must be computed is n 1 n 2 n 3 . Consider a processor that computes at least 1/Pth of these computations. Each element of A is involved in n 3 multiplications. If the processor accesses fewer than n 1 n 2 /P elements of A, then it would perform fewer than n 3 • n 1 n 2 /P scalar multiplications, which is a contradiction. Likewise, each element of B is involved in n 1 multiplications. If the processor accesses fewer than n 2 n 3 /P elements of B, then it would perform fewer than n 1 • n 2 n 3 /P scalar multiplications, which is a contradiction.

Finally, each element of C is the sum of n 2 scalar multiplications. If the processor computes contributions to fewer than n 1 n 3 /P elements of C, then it would perform fewer than n 2 • n 1 n 3 /P scalar multiplications, which is again a contradiction. □

Key Optimization Problem

The following lemma is the crux of the proof of our main result (Theorem 1). We state the optimization problem abstractly here, but it may be useful to have the following intuition: the variable vector x represents the sizes of the projections of the computation assigned to a single processor onto the three matrices, where x 1 corresponds to the smallest matrix and x 3 corresponds to the largest matrix. In order to design a communication-efficient algorithm, we wish to minimize the sum of the sizes of these projections subject to the constraints of matrix multiplication (and the processor performing 1/Pth of the computation), as specified by the Loomis-Whitney inequality (Lemma 1) and Lemma 4. A more rigorous argument that any parallel matrix multiplication algorithm is subject to these constraints is given in Theorem 1.

We are able to solve this optimization problem analytically using properties of convex optimization (Lemma 3). The three cases of the solution correspond to how many of the individual variable constraints are tight. When none of them is tight, we can minimize the sum of variables subject to the bound on their product by setting them all equal to each other (Case 3). However, when the individual variable constraints make this solution infeasible, those become active and the free variables must be adjusted (Cases 1 and 2).

Lemma 5. Consider the following optimization problem:

min x ∈R 3 x 1 + x 2 + x 3 such that mnk P 2 ≤ x 1 x 2 x 3 nk P ≤ x 1 mk P ≤ x 2 mn P ≤ x 3 ,
where m ≥ n ≥ k ≥ 1 and P ≥ 1. The optimal solution x * depends on the relative values of the constraints, yielding three cases:

(

) if P ≤ m n , then x * 1 = nk, x * 2 = mk P , x * 3 = mn P ; (2) if m n ≤ P ≤ mn k 2 , then x * 1 = x * 2 = mnk 2 P 1/2 , x * 3 = mn P ; (3) if mn k 2 ≤ P, then x * 1 = x * 2 = x * 3 = mnk P 2 3 . 1
This can be visualized as follows:

P 1 m n mn k 2 x * 1 = nk x * 2 = mk P x * 3 = mn P x * 1 = x * 2 = mnk 2 P 1/2 x * 3 = mn P x * 1 = x * 2 = x * 3 = mnk P 2/3
Proof. By Lemma 3, we can establish the optimality of the solution for each case by verifying that there exist dual variables such that the KKT conditions specified in Def. 3 are satisfied. This optimization problem fits the assumptions of Lemma 3 because the objective function and all but the first constraint are affine functions, which are convex and quasiconvex, and the first constraint is quasiconvex on the positive octant (which contains the intersection of the affine constraints) by Lemma 2.

To match standard notation (and that of Lemma 3), we let

f (x) = x 1 + x 2 + x 3 and д(x) =           (mnk/P) 2 -x 1 x 2 x 3 nk/P -x 1 mk/P -x 2 mn/P -x 3          
.

Thus the gradient of the objective function is ∇f (x) = 1 1 1 and the Jacobian of the constraint function is

J д (x) =           -x 2 x 3 -x 1 x 3 -x 1 x 2 -1 0 0 0 -1 0 0 0 -1          
.

Case 1 (P ≤ n m ). We let x * = nk mk and verify the KKT conditions. Primal feasibility is immediate, and dual feasibility follows from P ≤ m n ≤ m k , the condition of this case and by the assumption n ≥ k. Stationarity follows from direct verification that , where the first two inequalities are implied by the assumption m ≥ n ≥ k and the last follows from mn k 2 ≤ P, the condition of this case. Dual feasibility is immediate, and stationarity is directly verified. Complementary slackness is satisfied because the 1st constraint is tight for x * and µ * 1 is the only nonzero. Note that the optimal solutions coincide at boundary points between cases so that the values are continuous as P varies. □

µ * • J д (x * ) = -1 -1 -1 .

Complementary

Communication Lower Bound

We now state our main result, memory-independent communication lower bounds for general matrix multiplication with tight constants. After the general result, we also present a corollary for square matrix multiplication. The tightness of the constants in the lower bound is proved in § 5.

Theorem 1. Consider a classical matrix multiplication computation involving matrices of size n 1 × n 2 and n 2 × n 3 . Let m = max{n 1 , n 2 , n 3 }, n = median{n 1 , n 2 , n 3 }, and k = min{n 1 , n 2 , n 3 }, so that m ≥ n ≥ k. Any parallel algorithm using P processors that starts with one copy of the two input matrices and ends with one copy of the output matrix and load balances either the computation or the data must communicate at least Dmn + mk + nk P words of data, where

D =              mn+mk P + nk if 1 ≤ P ≤ m n 2 mnk 2 P 1/2 + mn P if m n ≤ P ≤ mn k 2 3 mnk P 2/3 if mn k 2 ≤ P .
Proof. To establish the lower bound, we focus on a single processor. If the algorithm load balances the computation, then every processor performs mnk/P scalar multiplications, and there exists some processor whose input data at the start of the algorithm plus output data at the end of the algorithm must be at most (mn + mk + nk)/P words of data (otherwise the algorithm would either start with more than one copy of the input matrices or end with more than one copy of the output matrix). If the algorithm load balances the data, then every processor starts and end with a total of (mn + mk + nk)/P words, and some processor must perform at least mnk/P scalar multiplications (otherwise fewer than mnk multiplications are performed). In either case, there exists a processor that performs at least mnk/P multiplications and has access to at most (mn + mk + nk)/P data.

Let F be the set of multiplications assigned to this processor, so that |F | ≥ mnk/P. Each element of F can be indexed by three indices (i 1 , i 2 , i 3 ) and corresponds to the multiplication of A(i 1 , i 2 ) with B(i 2 , i 3 ), which contributes to the result C(i 1 , i 3 ). Let ϕ A (F ) be the projection of the set F onto the matrix A, so that ϕ A (F ) are the entries of A required for the processor to perform the scalar multiplications in F . Here, elements of ϕ A (F ) can be indexed by two indices:

ϕ A (F ) = {(i 1 , i 2 ) : ∃ i 3 s.t. (i 1 , i 2 , i 3 ) ∈ F }. Define ϕ B (F )
implies that |ϕ A (F )| • |ϕ B (F )| • |ϕ C (F )| ≥ |F | ≥ n 1 n 2 n 3 P = mnk P ,
and the lower bound on the projections from Lemma 4 means

|ϕ A (F )| ≥ n 1 n 2 P , |ϕ B (F )| ≥ n 2 n 3 P , |ϕ C (F )| ≥ n 1 n 3 P .
For any algorithm, the processor's projections must satisfy these constraints, so the sum of their sizes must be at least the minimum value of optimization problem. Then by Lemma 5 (and assigning the projections to x 1 , x 2 , x 3 appropriately based on the relative sizes of n 1 , n 2 , n 3 ), the result follows. □

We also state the result for square matrix multiplication, which is a direct application of Theorem 1 with n 1 = n 2 = n 3 .

Corollary 2. Consider a classical matrix multiplication computation involving two matrices of size n × n. Any parallel algorithm using P processors that starts with one copy of the input data and ends with one copy of the output data and load

C C1 1 C1 2 C1 3 C2 1 C2 3 C3 1 C3 2 C3 3 A A 11 A 12 A 13 A 21 A 23 A 31 A 32 A 33 B B 1 1 B 1 2 B 1 3 B 2 1 B 2 3 B 3 1 B 3 2 B 3 3
Fig. 1. Visualization of Alg. 1 with a 3 × 3 × 3 processor grid. The 3D iteration space is mapped onto the processor grid, and the matrices are mapped to the faces of the grid. The dark highlighting corresponds to the input data initially owned and the output data finally owned by processor (1, 3, 1), and the light highlighting signifies the data of other processors it uses to perform the local computation. The arrows show the sets of processors involved in the three collective operations involving processor (

the output matrix. Lines 3, 4 specify simultaneous All-Gathers across sets of p 3 , p 1 processors, respectively, and Line 8 specifies simultaneous Reduce-Scatters across sets of p 2 processors. Note that if p k = 1 for any k = 1, 2, 3, then the corresponding collective can be ignored as no communication occurs. The difference between Alg. 1 and [1, Algorithm 1] is the Reduce-Scatter collective, which replaces the All-to-All collective and has smaller latency cost.

We assume that bandwidth-optimal algorithms, such as bidirectional exchange or recursive doubling/halving, are used for the All-Gather and Reduce-Scatter collectives. The optimal communication cost of these collectives on p processors is (1 -1 p )w, where w is the words of data in each processor after All-Gather or before Reduce-Scatter collective [START_REF] Chan | Collective communication: theory, practice, and experience[END_REF][START_REF] Thakur | Optimization of Collective Communication Operations in MPICH[END_REF]. Each processor also performs (1 -1 p )w computations for the Reduce-Scatter collective. p 3 ) n 1 n 2 p 1 p 2 and (1 -1 p 1 ) n 2 n 3 p 2 p 3 , respectively, to accomplish All-Gather operations, and the communication cost of performing the Reduce-Scatter operation in Line 8 is

(1 -1 p 2 ) n 1 n 3 p 1 p 3 . Note that if p k = 1 for any k = 1, 2, 3
, then the cost of the corresponding collective reduces to 0. Thus the overall cost of Algorithm 1 is

n 1 n 2 p 1 p 2 + n 2 n 3 p 2 p 3 + n 1 n 3 p 1 p 3 - n 1 n 2 + n 2 n 3 + n 1 n 3 P . (3)
Due to Reduce-Scatter operation, each processor also performs (1 -1 p 2 ) n 1 n 3 p 1 p 3 computations, which is dominated by the n 1 n 2 n 3 P computations of Line 6.

Optimal Processor Grid Selection

The communication cost of Algorithm 1, given by eq. ( 3), depends on the processor grid dimensions. Here we discuss how to select the processor grid dimensions such that the lower bound on communication given in Theorem 1 is attained by Alg. 1 given the matrices dimensions n 1 , n 2 and n 3 and the number of processors P. As before, we let m, n, k represent the maximum, median, and minimum values of the three dimensions. Letting p 1 , p 2 , p 3 be the grid dimensions, we similarly define p, q, r to be the processor grid dimensions corresponding to matrix dimensions m, n, k, respectively.

Because the order of processor grid dimensions will be chosen to be consistent with the matrix dimensions, we will have p ≥ q ≥ r . To demonstrate the tightness of the lower bound, the analysis below assumes that the processor grid dimensions divide the matrices dimensions.

Following Theorem 1, depending on the relative sizes of the aspect ratios among matrix dimensions and the number of processors, we encounter three cases that correspond to 3D, 2D, and 1D processor grids. That is, when p i = 1 for one value of i, then the processor grid is effectively 2D, and when p i = 1 for two values of i, the processor grid is effectively 1D. In the following we show how to obtain the grid dimensions and show that Algorithm 1 attains the communication lower bound given in Theorem 1 in each case.

First, suppose P ≤ m n . In this case, we set r = q = 1, and set p = P to obtain a 1D grid. From eq. ( 3), Algorithm 1 has communication cost mn + mk P + nkmn + mk

+ nk P = 1 - 1 P nk,
which matches the 1st case of Theorem 1. Now suppose m n < P ≤ mn k 2 . We set r = 1, and set p and q such that m p = n q , yielding p = P mn 1/2 m and

q = P mn 1/2
n. Note that the assumption m n < P is required so that q > 1, and p > 1 also follows. Our analysis also assumes that p and q are integers, which is sufficient to show that the lower bound is tight in general as there are an infinite number of dimensions for which the assumption holds. In this case, we have a 2D processor grid, and Finally, suppose mn k 2 < P. As suggested in [START_REF] Agarwal | A three-dimensional approach to parallel matrix multiplication[END_REF], we set the grid dimensions such that m p = n q = k r . That is, r = P mnk 1/3 k, q = P mnk 1/3 n, and p = P mnk 1/3 m. Note that the assumption mn k 2 < P is required so that r > 1 (which also implies q > 1 and p > 1). This assumption was implicit in the analysis of [START_REF] Agarwal | A three-dimensional approach to parallel matrix multiplication[END_REF]. Again, we assume that p, q, r are integers. Thus, we have a 3D processor grid and a communication cost of Comparing the obtained communication cost in each case with the lower bound obtained in Theorem 1 we conclude that Algorithm 1 is optimal given the grid dimensions are selected as above.

Optimal Processor Grid Examples

Figure 2 illustrates each of the three cases for a fixed set of matrix dimensions. Here we consider multiplying a 9600×2400 matrix A with a 2400 × 600 B so that C is 9600 × 600, so in our notation with m ≥ n ≥ k, A is m × n, B is n × k, and C is m × k. The 3D m × n × k iteration space is visualized with faces corresponding to correctly oriented matrices. In this example, we consider P ∈ {3, 36, 512}.

With 3 processors, we fall into the 1st case, as P ≤ m n = 4, and the optimal processor grid is 3 × 1 × 1, which is 1D as shown in Fig. 2a. Note that the computation assigned to each processor is not a cube in this case, that is, m p n q k r . The only data that must be communicated are entries of B, though all processors need all of B.

When P = 36, we fall into the 2nd case, and the optimal processor grid is 2D: 12 × 3 × 1, as shown in Fig. 2b. Here we see that the iteration space assigned to each processor is 800 × 800 × 600, so m p = n q k r . In this case, entries of B and C must be communicated, but each entry of A is required by only one processor.

Finally, for P = 512, we satisfy P > mn k 2 = 64 and fall into the 3rd case. The optimal processor grid is shown in Fig. 2c to be 32 × 8 × 2, and we see that the local computation of each processor is a cube: m p = n q = k r . For a 3D grid, entries of all 3 matrices are communicated.

CONCLUSION

Theorem 1 establishes memory-independent communication lower bounds for parallel matrix multiplication. By casting the lower bound on accessed data as the solution to a constrained optimization problem, we are able to obtain a result with explicit constants spanning over three scenarios that depend on the relative sizes of the matrix aspect ratios and the number of processors. Algorithm 1 demonstrates that the constants established in Theorem 1 are tight, as the algorithm is general enough to be applied in each of the three scenarios by tuning the processor grid. As we discuss below, our lower bound proof technique tightens the constants proved in earlier work, and we believe it can be generalized to improve known communication lower bounds for other computations.

Comparison to Existing Results

We now provide full details of the constants presented in Tab. 1, and compare the previous results with the constants of Theorem 1. The first row of the table gives the constant from the proof by Aggarwal, Chandra, and Snir [2, Theorem 2.3]. While the result is stated asymptotically, an explicit constant is given in a key lemma ([2, Lemma 2.2]) used in the proof, from which we can derive the constant for the main result.

The second row of the table corresponds to the work of Irony, Toledo, and Tiskin [START_REF] Irony | Communication lower bounds for distributed-memory matrix multiplication[END_REF], who establish the first parallel bounds for matrix multiplication. Their memory-independent bound is stated for square matrices with a parametrized prefactor corresponding to the amount of local memory available [START_REF] Irony | Communication lower bounds for distributed-memory matrix multiplication[END_REF]Theorem 5.1]. If we generalize it straightforwardly to rectangular dimensions and minimize the prefactor over any amount of local memory, then we obtain a bound of at least 1/2 • (mnk/P) 2/3 , which is asymptotically tight for mn/k 2 ≤ P. They do not provide any tighter results for P < mn/k 2 .

The third row of the table corresponds to the results of Demmel et al. [START_REF] Demmel | Communication-Optimal Parallel Recursive Rectangular Matrix Multiplication[END_REF]. This work was the first to establish bounds for small values of P and identify the three cases of asymptotic expressions. Theorem 1 obtains the same cases and leading order terms (up to constant factors) [11, Table I], and we present the explicit constant factors for leading order terms derived in [11, Section II.B]. We note that the boundaries between cases differ by a constant in that paper, which we do not reflect in Tab. 1. Compared to these results, Theorem 1 establishes a tighter constant in all three cases. We note that Kwasniewski et al. claim a combined result of memory-dependent and memory-independent parallel bounds [17,Theorem 2]. The memory-independent term has a constant that matches the 3rd case of Theorem 1.

However, the proof includes a restrictive assumption on parallelization strategies, requiring that each processor is assigned a set of domains that are subblocks of the iteration space with dimensions a × a × b for some a, b, and therefore does not apply to all parallel algorithms.

Limited-Memory Scenarios

The local memory required by Alg. 1 matches the amount of communication performed plus the data already owned by the processor, which is given by the positive terms in eq. ( 3) and matches the value of D in Theorem 1 with the optimal processor grid. Note that the local memory M must be large enough to store the inputs and output matrices, so M ≥ (mn + mk + nk)/P. When 1D or 2D processor grids are used, the local memory required is no more than a constant more than the minimum required to store the problem. Further, Alg. 1 can be adapted to reduce the temporary memory required to a negligible amount at the expense of higher latency cost but without affecting the bandwidth cost, and thus the algorithmic approach can be used even in extremely limited memory scenarios. In the case of 3D processor grids, however, the temporary memory used by Alg. 1 asymptotically dominates the minimum required, and thus the algorithm cannot be applied in limited-memory scenarios. Reducing the memory footprint in this case necessarily increases the bandwidth cost. Algorithms that smoothly trade off memory for communication savings in these limited memory scenarios are well studied [START_REF] Ballard | Brief announcement: strong scaling of matrix multiplication algorithms and memory-independent communication lower bounds[END_REF]17,[START_REF] Mccoll | Memory-Efficient Matrix Multiplication in the BSP Model[END_REF][START_REF] Solomonik | Communication-Optimal Parallel 2.5D Matrix Multiplication and LU Factorization Algorithms[END_REF].

From the lower bound point of view, while Theorem 1 is always valid, it may not be the tightest bound in limitedmemory scenarios. The memory-dependent bound with leading term 2mnk/(P √ M) (see [17,[START_REF] Olivry | Automated Derivation of Parametric Data Movement Lower Bounds for Affine Programs[END_REF][START_REF] Smith | A Tight I/O Lower Bound for Matrix Multiplication[END_REF] and discussion in § 2.1) can be larger. In particular, this occurs when mn/k 2 < P ≤ 8/27 • mnk/M 3/2 , and the memory-dependent bound dominates the memory-independent bound of 3(mnk/P) 2/3 in that case. This scenario implies that M < 4/9•(mnk/P) 2/3 , in which case the temporary space required by Alg. 1 exceeds the available memory. Thus, the tightness of Theorem 1 for mn/k 2 < P requires an assumption of sufficient memory.

When P ≤ mn/k 2 , the memory-independent bounds in the first two cases of Theorem 1 are always tight, with no assumption on local memory size. That is, the memory-dependent bound never dominates the memory-independent bound. Consider the 2nd case, so that m/n ≤ P ≤ mn/k 2 and the memory-independent bound is 2(mnk 2 /P) 1/2 . Because the local memory must be large enough to store the largest matrix as well as the other two matrices, we have M > mn/P. This implies 2mnk/(P √ M) < 2(mnk 2 /P) 1/2 , so the memory-independent bound dominates.

Suppose further that P ≤ m n . In this case, the leading-order term of the memory-independent bound is nk. This 1st-case bound dominates the 2nd-case bound, which dominates the memory-dependent bound from the argument above. Comparison of the full bounds of the 1st and 2nd cases simplifies to 2(mnk 2 /P) 1/2 ≤ mk/P + nk, which holds by the arithmetic-geometric mean inequality.

Extensions

The proof technique we use to obtain Theorem 1 is more generally applicable. The basic approach of defining a constrained optimization problem to minimize the sum amount of data accessed subject to constraints on that data that depend on the nature of the computation has been used before for matrix multiplication [START_REF] Smith | A Tight I/O Lower Bound for Matrix Multiplication[END_REF] and for tensor computations [START_REF] Ballard | Communication Lower Bounds for Matricized Tensor Times Khatri-Rao Product[END_REF][START_REF] Ballard | General Memory-Independent Lower Bound for MTTKRP[END_REF]. The key to the results presented in this work is the imposition of lower bound constraints on the data accessed in each individual array given by Lemma 4. These lower bounds become active when the aspect ratios of the matrices are large relative to the number of processors and allow for tighter lower bounds in those cases. The argument given in Lemma 4 is not specific to matrix multiplication, it depends only on the number of operations a given word of data is involved in, so it can be applied to many other computations that have iteration spaces with uneven dimensions. We believe this will yield new or tighter parallel communication bounds in several cases.

  and ϕ C (F ) similarly. The processor must access all of the elements in ϕ A (F ), ϕ B (F ), and ϕ C (F ) in order to perform all the scalar multiplications in F . Because the processor starts and ends with at most (mn + mk + nk)/P data, the communication performed by the processor is at least|ϕ A (F )| + |ϕ B (F )| + |ϕ C (F )| -mn + mk + nk P ,which is a lower bound on the communication along the critical path of the algorithm.In order to lower bound |ϕA (F )| + |ϕ B (F )| + |ϕ C (F )|,we form a constrained minimization problem with this expression as the objective function and constraints derived from Lemmas 1 and 4. The Loomis-Whitney inequality (Lemma 1)

2 Fig. 2 .

 22 Fig. 2. Example parallelizations of iteration space of multiplication of a 9600 × 2400 matrix A with a 2400 × 600 matrix B Hence the communication costs of Lines 3, 4 in Algorithm 1 are (1 -1p 3 ) n 1 n 2 p 1 p 2 and (1 -1 p 1 ) n 2 n 3 p 2 p 3 , respectively, to accomplish All-Gather operations, and the communication cost of performing the Reduce-Scatter operation in Line 8 is(1 -1 p 2 ) n 1 n 3 p 1 p 3 . Note that if p k = 1 for any k = 1, 2, 3, then the cost of the corresponding collective reduces to 0. Thus the overall cost of Algorithm 1 isn 1 n 2 p 1 p 2 + n 2 n 3 p 2 p 3 + n 1 n 3 p 1 p 3 -n 1 n 2 + n 2 n 3 + n 1 n 3 P .(3)

3 -

 3 mn + mk + nk P , which matches the 3rd case of Theorem 1.

Table 1 .

 1 Summary of explicit constants of leading term of parallel memory-independent rectangular matrix multiplication communication lower bounds for multiplication dimensions m ≥ n ≥ k and P processors

		1 ≤ P ≤ m n	m n ≤ P ≤ mn k 2	mn k 2 ≤ P
	Leading term	nk	P mnk 2	1/2	P mnk	2/3
					2/3
	[2]	-	-	1 2		≈ .63
	[14]	-	-		1 2 = .5
			1/2
	[11]	16 25 = .64	2 3	≈ .82	1
	Theorem 1	1	2		3

  slackness is satisfied because the only nonzero dual variables are µ *

									1 , µ * 3 , and µ * 4 , and the 1st, 3rd, and
	4th constraints are tight.
	Case 2 ( m n ≤ P ≤ mn k 2 ). We let
	x * = mnk 2 P	1/2	mnk 2 P	1/2	mn P
	and
	µ * =	P mnk 2/3	3/2	mn 0 0 1 -P k 2	1/2
	and verify the KKT conditions. The primal feasibility of x 1 = x 2 is satisfied because
			nk P	≤	mk P	≤	P mnk 2	1/2
	Case 3 ( mn k 2 ≤ P). We let
	P x * = mnk	2/3	P mnk	2/3	P mnk	2/3
	and
		µ * =	P mnk	4/3		0 0 0
	and verify the KKT conditions. We first consider the primal feasibility conditions. We have
		nk P	≤	mk P	≤	mn P	≤	P mnk	2/3

where the first inequality follows from the assumption m ≥ n and the second inequality follows from m/n ≤ P (one condition of this case). The other constraints are clearly satisfied. Dual feasibility requires that 1-(Pk 2 /mn) 1/2 ≥ 0, which is satisfied because P ≤ mn/k 2 (the other condition of this case). Stationarity can be directly verified. Complementary slackness is satisfied because the 1st and 4th constraints are both tight for x * , corresponding to the only nonzeros in µ * .

ACKNOWLEDGMENTS

This work is supported by the National Science Foundation under Grant No. CCF-1942892 and OAC-2106920. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant agreement No. 810367).

balances either the computation or the data must communicate at least 3 n 2 P 2/3 -3 n 2 P words of data.

OPTIMAL PARALLEL ALGORITHM

In this section we present an optimal parallel algorithm (Alg. 1) to show that the lower bound (including the constants) is tight. The idea is to organize the processors into a 3D processor grid and assign the computation of the matrix multiplication (a 3D iteration space) to processors according to their location in the grid. The algorithm is not new, but we present it here in full detail for completeness and to provide the complete analysis, which has not appeared before. In particular, Alg. 1 is nearly identical to the one proposed by Aggarwal et al. [START_REF] Agarwal | A three-dimensional approach to parallel matrix multiplication[END_REF], though they use the LPRAM model and analyze only the case where P is large. In the LPRAM model, for example, processors can read concurrently from a global shared memory, while in the α-β-γ model, the data is distributed across local memories and is shared via collectives like All-Gathers. Demmel et al. [START_REF] Demmel | Communication-Optimal Parallel Recursive Rectangular Matrix Multiplication[END_REF] present and analyze their recursive algorithm to show its asymptotic optimality in all three cases, but they do not track constants. See § 2.4 for more discussion of previous work on optimal parallel algorithms.

Consider the multiplication of an n 1 × n 2 matrix A with an n 2 × n 3 matrix B, and let C = A • B. Algorithm 1 organizes the P processors into a 3-dimensional p 1 × p 2 × p 3 logical processor grid with p 1 p 2 p 3 = P. Note that one or two of the processor grid dimensions may be equal to 1, which simplifies to a 2-or 1-dimensional grid. A processor coordinate is

The basic idea of the algorithm is to perform two collective operations, All-Gathers, so that each processor receives the input data it needs to perform all of its computation (in an All-Gather, all the processors involved end up with the union of the input data that starts on each processor). The result of each local computation must be summed with all other contributions to the same output matrix entries from other processors, and the summations are performed via a Reduce-Scatter collective operation (in a Reduce-Scatter, the sum of the input data from all processors is computed so that it ends up evenly distributed across processors).

Algorithm 1 Comm-Optimal Parallel Matrix Multiplication

Algorithm 1 imposes requirements on the initial distribution of the input matrices and the final distribution of the output. These conditions do not always hold in practice, but to show that the lower bound (which makes no assumption on data distribution except that only 1 copy of the input exists at the start of the computation) is tight, we allow the algorithm to specify its distributions. For simplicity of explanation, we assume that p 1 , p 2 , p 3 evenly divide n 1 , n 2 , n 3 , respectively. We use the notation


Amaia Nuñez-Del Moral

Paula C Bianchi

Iria Brocos-Mosquera

Augusto Anesio

Paola Palombo

Rosana Camarini

Fabio C Cruz

Luis F Callado

Vincent Vialou
email: cent.vialou@inserm.fr

Amaia M Erdozain
email: amaiaerdozain@ehu.eus

The matricellular protein hevin is involved in alcohol use disorder

Keywords: hevin, astrocytes, alcohol use disorder, human post-mortem brain, Sparcl1 34 35 38

Astrocytic-secreted matricellular proteins have been shown to influence various aspects of synaptic function. More recently, they have been found altered in animal models of psychiatric disorders such as drug addiction. Hevin (also known as Sparc-like 1) is a matricellular protein highly expressed in the adult brain that has been implicated in resilience to stress, suggesting a role in motivated behaviors. To address the possible role of hevin in drug addiction, we quantified its expression in human postmortem brains and in animal models of alcohol abuse. Hevin mRNA and protein ex-21 pression were analyzed in postmortem human brain of subjects with an antemortem 22 diagnosis of alcohol use disorder (AUD, n = 25) and controls (n = 25). All the studied 23 brain regions (prefrontal cortex, hippocampus, caudate nucleus and cerebellum) in AUD 24 subjects showed an increase in hevin levels either at mRNA or/and protein levels. To test 25 if this alteration was the result of alcohol exposure or indicative of a susceptibility factor 26 to alcohol consumption, mice were exposed to different regimens of intraperitoneal al-27 cohol administration. Hevin protein expression was increased in the nucleus accumbens 28 after alcohol withdrawal and reinstatement. The role of hevin in AUD was determined 29 using an RNA interference strategy to downregulate hevin expression in nucleus 30 accumbens astrocytes, which led to increased ethanol consumption. Additionally, etha-31 nol reinstatement after withdrawal increased hevin levels in blood plasma. Altogether, 32 these results support a novel role for hevin in the neurobiology of AUD.

33

INTRODUCTION

Alcohol use disorder (AUD) is one of the psychiatric diseases that most negatively 37 affect global health, society, and the economy [START_REF] Heilig | Addiction as a brain disease revised: why it still matters, and the need for consilience[END_REF][START_REF] Rehm | Global burden of disease and injury and economic cost attributable to alcohol use and alcohol-use disorders[END_REF]. Ethanol is considered the most dan-impairments, learning and memory deficits, and a shift from positive to negative reinforcement [START_REF] Abrahao | Alcohol and the brain: neuronal molecular targets, synapses, and circuits[END_REF][START_REF] Koob | Neurobiology of addiction: a neurocircuitry analysis[END_REF]. An essential aspect of AUD is the high rate of relapse, depicted as recurrent abstinence periods followed by reinstatement of consumption. This pathological pattern of alcohol consumption might take part in the progression and deepening of the disorder [START_REF] Becker | Alcohol dependence, withdrawal, and relapse[END_REF][START_REF] Koob | Plasticity of reward neurocircuitry and the 'dark side' of drug addiction[END_REF][START_REF] Sliedrecht | Alcohol use disorder relapse factors: A systematic review[END_REF]. Ethanol's effects on the brain involve alterations in synaptic plasticity and morphological reorganization of circuits implicated in reward, stress and executive functions. Although the contribution of various molecular mediators, including monoamines, neuropeptides, and stress hormones, has been recognized [START_REF] Koob | Neurobiology of addiction: a neurocircuitry analysis[END_REF], no key genetic switch mediating significant traits of the AUD phenotype has been characterized.

Astrocytes play critical roles in central nervous system homeostasis and are implicated in the pathogenesis of neurological and psychiatric conditions, including drug dependence [START_REF] Nagai | Hyperactivity with disrupted attention by activation of an astrocyte synaptogenic cue[END_REF][START_REF] Wang | Astrocytes in cocaine addiction and beyond[END_REF]. In regard to AUD, activation of mouse cortical astrocytes by chemogenetic approaches increases alcohol consumption and alters its locomotor and sedative effects [START_REF] Erickson | Cortical astrocytes regulate ethanol consumption and intoxication in mice[END_REF]. Likewise, activation of astrocytes in the nucleus accumbens (NAc) potentiates the rewarding properties of alcohol and decreases alcohol self-administration after prolonged abstinence [START_REF] Bull | Rat nucleus accumbens core astrocytes modulate reward and the motivation to self-administer ethanol after abstinence[END_REF]. Chronic alcohol exposure profoundly modifies the transcriptomic profile of astrocytes in vitro, in utero and in vivo, with changes in the expression of genes involved in cellular differentiation, metabolism, gene transcription, extracellular matrix composition and synaptogenesis [START_REF] Erickson | Astrocyte-specific transcriptome responses to chronic ethanol consumption[END_REF][START_REF] Erickson | Glial gene networks associated with alcohol dependence[END_REF][START_REF] Trindade | Developmental alcohol exposure leads to a persistent change on astrocyte secretome[END_REF][START_REF] Pignataro | Brief alcohol exposure alters transcription in astrocytes via the heat shock pathway[END_REF]. Astrocytic synaptogenic factors promote excitatory synapse formation, maturation, and refinement during development [START_REF] Nagai | Hyperactivity with disrupted attention by activation of an astrocyte synaptogenic cue[END_REF][START_REF] Christopherson | Thrombospondins are astrocyte-secreted proteins that promote CNS synaptogenesis[END_REF][START_REF] Gan | SPARCL1 promotes excitatory but not inhibitory synapse formation and function independent of neurexins and neuroligins[END_REF][START_REF] Kucukdereli | Control of excitatory CNS synaptogenesis by astrocyte-secreted proteins Hevin and SPARC[END_REF][START_REF] Murphy-Ullrich | Revisiting the matricellular concept[END_REF][START_REF] Singh | Astrocytes assemble thalamocortical synapses by bridging Nrx1α and NL1 via hevin[END_REF]. In particular, hevin, a matricellular protein with synaptogenic properties secreted by astrocytes, is induced in prenatal and postnatal animal models of AUD [START_REF] Trindade | Developmental alcohol exposure leads to a persistent change on astrocyte secretome[END_REF][START_REF]Molecular neuropathology of astrocytes and oligodendrocytes in alcohol use disorders[END_REF].

Hevin (also known as SPARC-like 1 [SPARCL1], SC1, ECM2 or Mast9) specifically promotes the formation and maintenance of thalamocortical excitatory synapses, by linking presynaptic neurexin-1α and postsynaptic neuroligin-1B [START_REF] Gan | SPARCL1 promotes excitatory but not inhibitory synapse formation and function independent of neurexins and neuroligins[END_REF][START_REF] Kucukdereli | Control of excitatory CNS synaptogenesis by astrocyte-secreted proteins Hevin and SPARC[END_REF][START_REF] Singh | Astrocytes assemble thalamocortical synapses by bridging Nrx1α and NL1 via hevin[END_REF][START_REF] Risher | Astrocytes refine cortical connectivity at dendritic spines[END_REF]. It is highly expressed in human and mouse adult brain [START_REF] Girard | Modulation of endothelial cell adhesion by hevin, an acidic protein associated with high endothelial venules[END_REF][START_REF] Johnston | Molecular cloning of SC1: a putative brain extracellular matrix glycoprotein showing partial similarity to osteonectin/BM40/SPARC[END_REF][START_REF] Lively | The extracellular matrix protein SC1/hevin localizes to multivesicular bodies in bergmann glial fibers in the adult rat cerebellum[END_REF][START_REF] Mongrédien | Cartography of hevin-expressing cells in the adult brain reveals prominent expression in astrocytes and parvalbumin neurons[END_REF][START_REF] Nuñez-Delmoral | Characterization of hevin (SPARCL1) immunoreactivity in postmortem human brain homogenates[END_REF][START_REF] Soderling | Cloning and expression of murine SC1, a gene product homologous to SPARC[END_REF], in astrocytes and parvalbumin interneurons, and in a restricted number of other neuronal subpopulations, including glutamatergic neurons in cortical and subcortical regions [START_REF] Mongrédien | Cartography of hevin-expressing cells in the adult brain reveals prominent expression in astrocytes and parvalbumin neurons[END_REF]. Importantly, hevin has been implicated in resilience to stress, showing an antidepressant-like effect in a model of social defeat [START_REF] Vialou | ΔFosB in brain reward circuits mediates resilience to stress and antidepressant responses[END_REF].

Given the effects of alcohol on the limbic system, and the role of hevin in structural and synaptic plasticity, we explored the role of hevin in AUD. The pattern of hevin expression was assessed in human cortico-limbic structures associated with AUD in postmortem brains from AUD subjects. We also assessed hevin levels in the brain and blood of mice exposed to different regimens of intraperitoneal alcohol administration. In addition, we evaluated the contribution of astrocytic hevin in the nucleus accumbens to ethanol preference and consumption by downregulating its expression using RNA interference.

MATERIALS AND METHODS

Human samples

Human samples were obtained at autopsy in the Basque Institute of Legal Medicine (Bilbao, Spain). Brain dissections were performed at the moment of the autopsy for the collection of PFC, HIP, caudate nucleus (CAU) and CB samples, which were immediately stored at -70°C until assay. A blood toxicological study by the National Institute of Toxicology (Madrid), using different standard methods (including radioimmunoassay, enzymatic immunoassay, high-performance liquid chromatography and gas chromatography-mass spectrometry) determined the presence or not of ethanol, antidepressants, antipsychotics and psychotropic drugs at the time of death. The study included 75 subjects divided into three experimental groups: 25 subjects with an antemortem diagnosis of AUD, 25 subjects with an antemortem diagnosis of major depression and 25 control subjects, with no antemortem history of any neuropsychiatric disease. The antemortem diagnoses were performed by psychiatrists based on DMS-IV and ICD-10 criteria. The depression group was included as a control of disease-specific changes. Each subject with AUD was matched by age, sex and postmortem delay with a control and depression subject; none of these parameters were statistically different between groups (p>0.05, one-way ANOVA). A full description of the demographic characteristics of all the subjects is shown in Supplementary Table 1. Samples from matched AUD, depression and control subjects were processed and tested in parallel. The study was performed in compliance with legal policy and ethical review boards for postmortem brain studies.

Animals and treatments

All experiments were performed in strict conformity with the European Union laws and policies for use of animals in neuroscience research (European Committee Council Directive 2010/63/EU). C57BL/6J mice were housed in groups of five animals per cage under standard conditions 23°C and a 12 h light/dark cycle, with food and water provided ad libitum. Experiments were performed on 2 to 4 month old mice.

Mice were exposed to four different ethanol regimens (Fig. 3A) [START_REF] Bahi | Involvement of nucleus accumbens dopamine D1 receptors in ethanol drinking, ethanol-induced conditioned place preference, and ethanol-induced psychomotor sensitization in mice[END_REF]. They were injected intraperitoneally (i.p.) either with saline or ethanol for 13 days, not injected for the following 3 days (withdrawal), and then given a challenge dose of saline or ethanol (reinstatement). The four resulting experimental groups were as follows: 1) saline -saline (S-S), 2) saline -ethanol (S-E), 3) ethanol-saline (E-S) and 4) ethanol-ethanol (E-E). A rewarding dose of ethanol, 1.75 g ethanol/kg, or 0.9% NaCl solution were used [START_REF] Nocjar | Ethanol consumption and place-preference conditioning in the alcohol-preferring C57BL/6 mouse: relationship with motor activity patterns[END_REF].

Mice were killed 24 h after the last injection, and brain dissections were performed immediately after. Five brain areas were collected with bilateral punches: frontal cortex (FC), amygdala (AMY), hippocampus (HIP), dorsal striatum (CPu) and nucleus accumbens (NAc). Blood was collected in EDTA-coated tubes and centrifuged to obtain plasma. All the samples were stored at -70°C until their preparation for Western blot assays.

For hevin downregulation in astrocytes of the NAc, mice were anesthetized with an i.p. mixture containing 100 mg/kg of ketamine and 10 mg/kg of xylazine, placed in a stereotaxic frame, and injected with the adenovirus AAV2.5-GFAP-EmGFP-miR-hevin (9.4x10 12 vg/ml) or the control viral vector AAV5.GFAP.eGFP.WPRE.hGH (9.1x10 12 vg/ml, University of North Carolina, USA). Coordinates (with respect to bregma) used to target the NAc were: anteroposterior +1.6, mediolateral +1.45, and dorsoventral -4.3, with an angle of 10° from the midline. The virus was injected at a volume of 0.5 µl, at a rate of 0.1 µl/min. Behavioral tests were performed 3 weeks after surgery. The correct injection site and the expression of the transgene were confirmed by immunohistochemistry.

Sample preparation

For Western blot experiments both human and animal brain samples were processed as previously described [START_REF] Nuñez-Delmoral | Characterization of hevin (SPARCL1) immunoreactivity in postmortem human brain homogenates[END_REF], with the following modifications: 1) in human brain samples the nuclear fraction was removed for all of the total homogenates, and 2) in mice brain punches the homogenization was performed by sonication (QSONICA Q55), due to the small sample volume. Protein content was measured by the Bradford method [START_REF] Bradford | A rapid and sensitive method for the quantitation of microgram quantities of protein utilizing the principle of protein-dye binding[END_REF] and the samples were stored at -70°C until analyzed by Western blot. Mouse plasma samples were homogenized in buffer supplemented with protease and phosphatase inhibitors (5 mM Tris-HCl pH 7.4, 50 µL/g of Sigma protease inhibitor cocktail, 5 mM Na3VO4 and 10 mM NaF) before sonication (QSONICA Q55) and stored at -70°C until analyzed by Western blot.

Real-time reverse transcription polymerase chain reaction (RT-qPCR)

mRNA extraction, cDNA synthesis and qPCR of the four human brain regions were performed as previously described [START_REF] Muguruza | Evaluation of 5-HT2A and mGlu2/3 receptors in postmortem prefrontal cortex of subjects with major depressive disorder: Effect of antidepressant treatment[END_REF] with minor modifications. The qPCR reactions were performed on a StepOne TM system using TaqMan TM Fast Universal Master Mix (Thermo Fisher) and the specific gene probes for hevin and the housekeeping genes glyceraldehyde 3-phosphate dehydrogenase (GAPDH) and ribosomal protein S13 (RPS13) (TaqMan ® gene expression assay Hs00949886_m1, Hs99999905_m1 and Hs01945436_u1, respectively). Hevin mRNA expression was normalized to the expression of the GAPDH and RPS13 housekeeping genes and with a reference sample (pool of all samples for each brain region); and determined by the comparative Ct method (ΔΔCt), where ΔΔCt = (Ct (hevin)sample -Ct (housekeeping)sample)-(Ct (hevin)reference sample -Ct (housekeeping) reference sample) using StepOne TM Software v2.1. A negative internal control (mQH2O) was also included. The qPCRs were run in triplicates for each cDNA.

Western blot

Western blot experiments were performed as previously described [START_REF] Nuñez-Delmoral | Characterization of hevin (SPARCL1) immunoreactivity in postmortem human brain homogenates[END_REF]. In human and mouse brain samples hevin immunoreactivity was normalized against the β-actin signal. For mouse plasma samples, the Ponceau S staining confirmed the equal amount of protein load. In all cases, a standard pool of total homogenate (from human or mouse origin, respectively) was used as an external reference sample. The primary and secondary antibodies used for the detection of hevin and β-actin proteins are shown in Supplementary Table 2.

Intermittent ethanol access schedule (IEA) in hevin knockdown mice

In order to measure the alcohol intake in the active period of the mice, one week before starting the experiment, the light cycle was changed, with lights off at 9 a.m. and on at 9 p.m., and both hevin knockdown (KD, n = 7) and control (n = 7) mice were individually housed, with water and food ad libitum. Mice were first habituated to two bottles of water for 3 days. IEA was performed for a total of 18 days [START_REF] Portero-Tresserra | Maternal separation increases alcohol-drinking behaviour and reduces endocannabinoid levels in the mouse striatum and prefrontal cortex[END_REF]; water and ethanol bottles were placed in each cage every day at 9 a.m. and removed at 5 p.m.. The consumed volume was measured after 4 h and 8 h of the beginning of the experiment, and the bottles' position was switched every day to avoid possible bottle side preference bias. Ethanol concentration (w/v) was increased every 3 days in order to induce an alcohol intake behavior. The initial ethanol concentration was 5%, followed by 10%, 15%, 20%, 3 days of ethanol withdrawal, and 10% of ethanol as the reinstatement concentration. Ethanol solutions were prepared by mixing 96% (v/v) ethanol with tap water. Mice were weighted every 3 days. Ethanol preference (percentage of ethanol consumed from the total volume for each measurement; > 50% was considered as a preference for ethanol) and consumption (g ethanol/kg mouse ingested for each measurement) were calculated for each mouse.

Immunohistochemistry

Mice were anesthetized with a lethal dose of pentobarbital at 150 mg/kg i.p. and perfused intracardially with 4% paraformaldehyde in a phosphate-buffered solution.

Brains were removed and stored at 4°C in 4% paraformaldehyde until use. Brains were cut on a vibratome in coronal sections of 40 µm. Slices were incubated in a blocking buffer containing 0.2% gelatin and 0.25% triton in PBS for 1 h at room temperature, then treated with antibodies against hevin (mSPARCL1 AF2836, 1:2000, R&D) or anti-GFP (A010-pGFP-5, 1:2000, Badrilla) overnight at 4°C. After washing, slices were incubated with a Cyanine Cy™3 conjugated secondary antibody for 2 h at room temperature (Jackson Immunoresearch, USA). Slices were incubated with DAPI (4',6-diamidino-2-fenilindol, D9542, 1:15000, Sigma-Aldrich) 20 min at room temperature before mounting with Fluoromount-G. All the slices were scanned on a NanoZoomer 2.0-HT (Hamamatsu Photonics, Hamamatsu City, Japan) for quantification, as previously described [START_REF] Mongrédien | Cartography of hevin-expressing cells in the adult brain reveals prominent expression in astrocytes and parvalbumin neurons[END_REF].

Statistical analysis

Statistical analysis was performed using GraphPad Prism 9®. Comparison of hevin mRNA and protein levels in the human brain between the AUD, depression and control groups was performed using one-way ANOVA followed by Dunnett's multiple comparisons test. Taking into account the large sample size (n = 25) and the high inter-subject variability in human brain morphology and configuration, data was considered normally distributed [START_REF] Dean | Using CNS autopsy tissue in psychiatric research: a practical guide[END_REF][START_REF] Seghier | Interpreting and utilising intersubject variability in brain function[END_REF]. Data obtained in animal studies did not pass the D'Agostino and

Pearson normality test and F-test for the equality of variances, so non-parametric statistical tests were used. Specifically, biochemical analyses in mouse brains and plasma were performed using the Kruskal-Wallis test followed by Dunn's multiple comparisons test.

Differences between hevin KD mice and controls in ethanol preference and consumption

were assessed with the Mann-Whitney test and the Friedman test was used for ethanol concentration data. Results were considered statistically significant when the p-value was <0.05.

RESULTS

Hevin expression in postmortem human brains of subjects with AUD

Hevin mRNA and protein expression levels were measured in four human brain areas (PFC, HIP, CAU and CB) from 75 subjects. Individuals were divided according to their antemortem psychiatric history: AUD (n = 25), depression (n = 25) and control (n = 25) groups. Interestingly, in all the studied brain areas an increase in hevin mRNA was detected in the AUD group compared to the controls (PFC: 59% increase, p = 0.0028; HIP: 72% increase, p = 0.0058; CAU: 92% increase, p = 0.0028 and CB: 70% increase, p = 0.0344). No significant differences were detected between the depression and control groups (Fig. 1A-D). To further elucidate whether hevin mRNA increase paralleled an enhancement in the translation process, the protein expression levels were also determined. Both previously described ~130 kDa and ~100 kDa hevin bands [START_REF] Kucukdereli | Control of excitatory CNS synaptogenesis by astrocyte-secreted proteins Hevin and SPARC[END_REF][START_REF] Nuñez-Delmoral | Characterization of hevin (SPARCL1) immunoreactivity in postmortem human brain homogenates[END_REF][START_REF] Brekken | expression and characterization of murine Hevin (SC1), a member of the SPARC family of matricellular proteins[END_REF][START_REF] Weaver | Processing of the matricellular protein hevin in mouse brain is dependent on ADAMTS4[END_REF] were measured in total homogenates of PFC, HIP, CAU and CB. In accordance to the alterations observed in hevin mRNA, an increase in the ~130 kDa and ~100 kDa forms was detected in PFC and CB of AUD subjects in comparison to controls (PFC: 25% increase in ~130 kDa band, p = 0.0015 and 46% increase in ~100 kDa band, p < 0.0001; CB: 32% increase in ~130 kDa band, p = 0.0344 and 38% increase in ~100 kDa band, p = 0,013) (Fig. 2A,D). In the HIP, only the ~100 kDa band was significantly increased in AUD (56% increase, p = 0.00384), although the ~130 kDa band showed a similar trend (Fig. 2B). In contrast, in the CAU, no significant differences were observed between both groups in the expression of any of the bands (Fig. 2C). No correlation was found between ethanol levels detected in the blood at time of death and levels of brain hevin mRNA or protein expression. Finally, the depression group did not show significant differences in the levels of protein expression in comparison to the control group in any brain area, as observed for the mRNA levels (Fig. 2A-D).

Hevin protein expression in mouse brain after ethanol administration

Modeling the complex facets of AUD in animals is challenging. Repeated i.p. administration of ethanol in mice induces behavioral sensitization, a model associated with relapse to drug-seeking behavior [START_REF] Fish | Repeated Alcohol: Behavioral Sensitization and Alcohol-Heightened Aggression in Mice[END_REF]. Here, hevin expression was evaluated in several mice brain regions (FC, AMY, HIP, CPu and NAc) after different ethanol regimens. To control ethanol plasma levels, mice were injected i.p. with ethanol (13 days) followed by a withdrawal period (3 days) and a challenge dose of ethanol or saline. In contrast to the human brain, mice did not express the ~100 kDa band, or its immunoreactive signal was too low to be analyzed, as described before [START_REF] Weaver | Processing of the matricellular protein hevin in mouse brain is dependent on ADAMTS4[END_REF][START_REF] Weaver | Proteolysis of the matricellular protein hevin by matrix metalloproteinase-3 produces a SPARC-like fragment (SLF) associated with neovasculature in a murine glioma model[END_REF]. Therefore, the present data corre-spond only to the ~130 kDa band. Twenty-four h after an ethanol challenge in animals injected repeatedly with ethanol, an increase in hevin protein expression was observed in the NAc in comparison to the other 3 treatments (saline, one single dose of ethanol and withdrawal period after repeated injections) (74% increase vs S-S, p = 0.0041; 68% increase vs S-E, p = 0.0232, and 98% increase vs E-S, p = 0.0007) (Fig. 3A). Acute ethanol injection reduced hevin protein levels in the AMY (54% decrease vs S-S, p = 0.0471 (Fig. 3B). No significant changes were observed in the other brain regions.

Hevin protein expression in mouse plasma after ethanol treatment

Hevin expression levels were also analyzed in mouse plasma after chronic ethanol treatment. The ~100 kDa band was not identifiable in these samples. Ethanol challenge after withdrawal produced higher levels of the hevin ~130 kDa band in comparison to the saline challenge after withdrawal (65% increase vs E-S, p = 0.0396) (Fig. 4.). An additional band with a molecular weight around 47 kDa was also analyzed due to the high immunoreactivity signal of this band in most of the samples. Plasma showed high variability in the expression levels of the 47 kDa band among mice, with some of them expressing high levels of this form (i.e. two mice from the E-S group), and almost undetectable levels for others. As a result, no differences were detected in hevin expression levels between the ethanol treatments for the 47 kDa band (Fig. 4.). In order to assess if the expression of the lower band was the result of the cleavage of the 130 kDa band, the ratio of the immunoreactivity of both bands was also calculated (130 kDa/47 kDa). No significant differences were detected between treatment groups in the 130 kDa/47 kDa ratio (Fig. 4.). To test the role of hevin in the long-term adaptation to ethanol, we downregulated hevin expression in mouse NAc astrocytes using an RNA interference strategy, and tested the consequences of this cell-specific and regional downregulation of hevin on the consumption of ethanol in an intermittent ethanol access schedule (IEA). The downregulation of hevin in the astrocytes of the NAc was verified by immunohistochemistry of hevin protein and GFP in brain slices after the IEA schedule was completed.

Immunohistochemistry assays confirmed a complete deletion of hevin expression in NAc astrocytes from mice infected with the miRNA-containing viruses (Fig. 5A).

Mice showed a high preference for the initial low dose of the ethanol solution (Fig. 5B). Preference for the ethanol bottle (calculated as the percentage of ethanol consumed from the total volume) was similar between hevin KD mice and control mice for every dose during the entire procedure (Fig. 5B). When measuring total alcohol consumption (ingested ethanol expressed as g ethanol/kg), hevin downregulation in NAc astrocytes did not affect consumption of ethanol at low and medium concentrations (5%, 10% and 15%, w/v) (Fig. 5C). However, when exposed to a higher concentration of ethanol (20%), hevin KD mice showed an increased consumption compared to the lowest dose (5%), while control animals maintained a similar intake over time (Fig. 5C). After three days of withdrawal, hevin KD mice showed a higher relapse to consuming 10% ethanol than controls (Fig. 5C). Interestingly, these effects were only observed in the first 4 h of ethanol access. The last 4 h of the IEA experiment (from 4 h to 8 h) revealed that both groups, hevin KD and control mice, drank as much water as ethanol. Altogether, these data demonstrate that hevin KD animals consume more ethanol at the highest concentration and after ethanol withdrawal, while having a similar preference for ethanol compared to control mice.

DISCUSSION

AUD is characterized by two main features, the loss of control over the consumption of alcoholic beverages, and the relentless necessity to consume alcohol during abstinence, known as craving [START_REF] Spanagel | Alcohol: neurobiology of alcohol addiction[END_REF][START_REF] Weiss | Behavioral neurobiology of alcohol addiction: recent advances and challenges[END_REF]. However, their neurobiological bases remain mainly unknown.

Postmortem brain samples from subjects with a reliable antemortem diagnosis of AUD, and appropriately matched control subjects, provide a unique opportunity to search for any clinical relevance of altered molecular expression of different proteins in patients with AUD [START_REF] Mccullumsmith | Postmortem brain: an underutilized substrate for studying severe mental illness[END_REF]. In this line, the principal finding of the present study is that the matricellular protein hevin is overexpressed in the brain of subjects with AUD.

Human brain regions analyzed in this study were selected based on their implication in different aspects of AUD and on tissue availability. The PFC was chosen for its role in executive functions, such as control, planning and inhibition of behaviors; the HIP for its important role in the association of particular environments and emotional states with the effect of the drug; the CAU, for its involvement in cognition, learning, memory, and feedback processing; and the CB, for its role in movement and balance incoordination after alcohol consumption, in addition to its role in cognition, emotion and drug addiction [START_REF] Koob | Neurobiology of addiction: a neurocircuitry analysis[END_REF]. All the studied regions showed an increase in hevin levels either at the mRNA and/or protein levels in AUD subjects. This is the first evidence in humans for the role of hevin in the pathophysiology of AUD. Even if the mechanisms underlying this observation or its consequences are difficult to determine in humans, several hypotheses can be postulated. Hevin promotes the formation and maintenance of excitatory synapses specifically and modulates the refinement and stabilization of dendritic spines [START_REF] Kucukdereli | Control of excitatory CNS synaptogenesis by astrocyte-secreted proteins Hevin and SPARC[END_REF][START_REF] Singh | Astrocytes assemble thalamocortical synapses by bridging Nrx1α and NL1 via hevin[END_REF][START_REF] Risher | Astrocytes refine cortical connectivity at dendritic spines[END_REF][START_REF] Gan | Specific factors in blood from young but not old mice directly promote synapse formation and NMDA-receptor recruitment[END_REF]; therefore, the upregulation of hevin expression could be implicated in the profound structural synaptic rearrangement/adaptation that occurs in AUD [START_REF] Avchalumov | Synaptic plasticity and its modulation by alcohol[END_REF][START_REF] Gass | Neurochemical and neurostructural plasticity in alcoholism[END_REF]. Indeed, we previously found an important loss in neuronal structural proteins in AUD subjects, and other authors have reported that AUD subjects also suffer neuronal and dendritic spine loss [START_REF] Gass | Neurochemical and neurostructural plasticity in alcoholism[END_REF][START_REF] Ferrer | Decreased numbers of dendritic spines on cortical pyramidal neurons in human chronic alcoholism[END_REF][START_REF] Erdozain | Alcohol-related brain damage in humans[END_REF][START_REF] Labisso | The Loss of α-and β-tubulin proteins are a pathological hallmark of chronic alcohol consumption and natural brain ageing[END_REF][START_REF] Bengochea | Effect of chronic alcoholism on the human hippocampus[END_REF]. The increase in hevin could thus be an adaptative mechanism triggered to counteract the effects of ethanol on neuronal structure. Furthermore, at a molecular level, hevin increases NMDA receptor expression and function [START_REF] Gan | SPARCL1 promotes excitatory but not inhibitory synapse formation and function independent of neurexins and neuroligins[END_REF][START_REF] Singh | Astrocytes assemble thalamocortical synapses by bridging Nrx1α and NL1 via hevin[END_REF][START_REF] Gan | Specific factors in blood from young but not old mice directly promote synapse formation and NMDA-receptor recruitment[END_REF].

Hence, hevin could also mediate part of the synaptic plasticity occurring after chronic ethanol, such as long-term potentiation and NMDA receptor dependent plasticity, described in preclinical studies as well as in postmortem human brains from subjects with AUD [START_REF] Avchalumov | Synaptic plasticity and its modulation by alcohol[END_REF][START_REF] Freund | Glutamate receptors in the frontal cortex of alcoholics[END_REF][START_REF] Michaelis | Glutamate receptor changes in brain synaptic membranes during chronic alcohol intake[END_REF][START_REF] Loheswaran | Impairment of Neuroplasticity in the Dorsolateral Prefrontal Cortex by Alcohol[END_REF]. In our study we did not find any difference in hevin expression levels between sexes. As a control for disease specificity, a group of subjects with antemortem diagnoses of depression was added. Hevin expression was unaltered in the depression group in our study, consistent with a previous study [START_REF] Teyssier | Activation of a ΔFOSB dependent gene expression pattern in the dorsolateral prefrontal cortex of patients with major depressive disorder[END_REF]. These results in the human brain raise the question of whether hevin overexpression is induced by ethanol or if it is a pre-AUD state, suggesting vulnerability.

We used preclinical models of AUD and compared the effects of chronic alcohol withdrawal, chronic alcohol relapse and acute alcohol administration on hevin protein levels in mouse brain. We selected cortico-limbic brain regions involved in the neurocircuitry of drug addiction, in particular ventral and dorsal striatum, PFC, HIP and AMY [START_REF] Koob | Neurobiology of addiction: a neurocircuitry analysis[END_REF]. Because cerebellum has been classically associated to movement and balance incoordination after long-term alcohol consumption, we did not include this region in our mouse model. In contrast to human data, we observed hevin protein expression changes only in the AMY and NAc. In the AMY, a decrease in hevin levels was observed after a single dose of ethanol, but not after chronic ethanol administration. In this context, and according to the possible role of hevin in regulating glutamatergic signaling and neuroplasticity in response to ethanol, acute and low doses of ethanol decrease excitability and increase dendritic spines in the rodent AMY, coinciding with anxiolytic effects, while repeated withdrawal normalizes dendritic arborization, causing tolerance to ethanol anxiolysis [START_REF] Harrison | Effects of acute alcohol on excitability in the CNS[END_REF][START_REF] Pandey | Effector immediate-early gene arc in the amygdala plays a critical role in alcoholism[END_REF][START_REF] You | Reversal of deficits in dendritic spines, BDNF and Arc expression in the amygdala during alcohol dependence by HDAC inhibitor treatment[END_REF]. In contrast, in the NAc, hevin was overexpressed only with a challenging dose after withdrawal, while no change was observed after acute ethanol injection. Chronic ethanol intake and withdrawal produce alterations in the rodent NAc in dendritic spine number, structure and organization, and increase NMDA and AMPA receptor expression and function, which suggests that the increase in hevin observed in the NAc could be a compensatory response to the ethanol-related glutamatergic inhibition [START_REF] Zhou | Chronic alcohol drinking alters neuronal dendritic spines in the brain reward center nucleus accumbens[END_REF][START_REF] Peterson | Effects of ethanol exposure and withdrawal on dendritic morphology and spine density in the nucleus accumbens core and shell[END_REF][START_REF] Uys | Chronic intermittent ethanol exposure and withdrawal leads to adaptations in nucleus accumbens core postsynaptic density proteome and dendritic spines[END_REF][START_REF] Ji | Dopamine receptors differentially control binge alcohol drinking-mediated synaptic plasticity of the core nucleus accumbens direct and indirect pathways[END_REF][START_REF] Laguesse | Prosapip1-dependent synaptic adaptations in the nucleus accumbens drive alcohol intake, seeking and reward[END_REF][START_REF] Kircher | Ethanol experience enhances glutamatergic ventral hippocampal inputs to d1 receptor-expressing medium spiny neurons in the nucleus accumbens shell[END_REF]. Overall, while changes in hevin levels in the AMY could be implicated in the acute alcohol experience, the increased hevin expression in the NAc after chronic ethanol exposure suggests that hevin might also participate in the long-term adaptation to ethanol. As noted, mouse and human brains showed a different pattern of hevin expression in several regions (i.e. PFC, HIP) after alcohol exposure. This difference could be explained by two essential factors largely discussed in the literature: 1) the different regimens of alcohol exposure (time course, total drug intake and probable withdrawal period) which raises the question of what is the necessary cumulative effect of ethanol to cause neuroplasticity, and 2) different psychosocial and environmental stressors [START_REF] Koob | Neurocircuitry of addiction[END_REF].

To test the potential role of hevin on alcohol consumption, we used an RNA interference strategy to downregulate hevin expression in NAc astrocytes, the main cell type expressing hevin [START_REF] Mongrédien | Cartography of hevin-expressing cells in the adult brain reveals prominent expression in astrocytes and parvalbumin neurons[END_REF], and evaluated the consequences of hevin manipulation in the IEA model of voluntary ethanol intake. Our results indicate a disrupted reward system when hevin is downregulated. Specifically, we detected a higher ethanol consumption in hevin KD mice in two cases: i) at high ethanol concentrations (20%), which were administered after 15 days of intake, and ii) after three days of withdrawal with an intermediate ethanol concentration (10%). These findings suggest that ethanol addiction and withdrawal are associated with dysregulated hevin signaling. Since hevin increases excitatory synaptic plasticity, knocking-down hevin might disrupt the glutamatergic plasticity involved in the rewarding effects of ethanol [START_REF] Griffin | Increased extracellular glutamate in the nucleus accumbens promotes excessive ethanol drinking in ethanol dependent mice[END_REF], making it necessary to increase alcohol consumption in order to reach the same hedonic response [START_REF] Cofresí | Evidence for incentive salience sensitization as a pathway to alcohol use disorder[END_REF]. Nevertheless, the ethanol preference analysis revealed that both hevin KD and control mice drank as much water as ethanol, showing no differences in the preference at any dose or time. We suggest that mice undergo autoregulation in the water intake probably to avoid the dehydration produced by the higher ethanol intake. Despite the inherent limitations of experimental models of psychiatric disorders, the present model contributes to a better understanding of the underlying plasticity of alcohol relapse.

Finally, we tested whether hevin levels were altered in the plasma of mice after chronic alcohol withdrawal, chronic alcohol relapse or acute alcohol administration. Interestingly, a very similar expression profile was obtained after the different ethanol treatments in plasma samples compared to the NAc, with increased hevin expression only with a challenge dose after withdrawal. The origin of hevin protein in plasma is still uncertain. Hevin is not expressed by any blood cell type [START_REF] Nuñez-Delmoral | Characterization of hevin (SPARCL1) immunoreactivity in postmortem human brain homogenates[END_REF], but is expressed in other tissues than the brain, such as lung [START_REF] Bendik | Characterization of MAST9/Hevin, a SPARC-like protein, that is down-regulated in non-small cell lung cancer[END_REF], endothelial cells of high endothelial venules [START_REF] Girard | Modulation of endothelial cell adhesion by hevin, an acidic protein associated with high endothelial venules[END_REF][START_REF] Girard | Cloning from purified high endothelial venule cells of hevin, a close relative of the antiadhesive extracellular matrix protein SPARC[END_REF] and stomach [START_REF] Klingler | Species-, organ-and cell-type-dependent expression of SPARCL1 in human and mouse tissues[END_REF]. The increase in hevin in the plasma of ethanol-treated mice suggests that hevin may be a component of the response involved in remodeling events associated with neuronal degeneration following neural injury, as it has been proposed in neurodegenerative diseases and several cancers [START_REF] Lau | SPARC and hevin expression correlate with tumour angiogenesis in hepatocellular carcinoma[END_REF][START_REF] Yin | Neuronal pentraxin receptor in cerebrospinal fluid as a potential biomarker for neurodegenerative diseases[END_REF]. In animal models of epilepsy, ischemia and injury, hevin is induced in reactive astrocytes, suggesting that it might participate in neuronal remodeling in pathological conditions in the adult stage [START_REF] Lively | SC1/Hevin and Reactive Gliosis after Transient Ischemic Stroke in Young and Aged Rats[END_REF][START_REF] Lively | The Extracellular Matrix Protein SC1/Hevin Localizes to Excitatory Synapses Following Status Epilepticus in the Rat Lithium-Pilocarpine Seizure Model[END_REF].

As a whole, this study sheds light on the role of hevin in the pathophysiology of AUD. In humans, we previously found that hevin is expressed in astrocytes and some glutamatergic and GABAergic neurons [START_REF] Mongrédien | Cartography of hevin-expressing cells in the adult brain reveals prominent expression in astrocytes and parvalbumin neurons[END_REF]. The source of hevin overexpression in AUD subjects is unknown. Our cell-specific RNA interference strategy in mice shows that hevin expression in NAc astrocytes is crucial for increased voluntary ethanol consumption. As described above, we hypothesize that hevin could take part in the acquisition of the addiction behavior by strengthening and promoting specific synaptic signaling and/or modulating neuronal plasticity/adaptation in brain areas involved in AUD.

In conclusion, our postmortem brain analysis revealed an increase in hevin levels in various brain regions in subjects with AUD. In order to explore whether this increase was a consequence of alcohol intake, we used a mouse model of chronic ethanol administration and found that only chronic ethanol relapse increases hevin in the brain reward center, the NAc. In addition, we evaluated whether alterations in hevin expression affected alcohol-related behavior in mice, and revealed that hevin downregulation in NAc astrocytes, where the vast majority of hevin is expressed in this region, modulates alcohol consumption. Overall, this study provides the first evidence for the role of hevin in alcohol consumption and addiction.

Figure 1 .

 1 Figure 1. Hevin mRNA expression in postmortem human brains of alcohol use disorder (AUD) subjects. Relative hevin mRNA levels were quantified by RT-qPCR in postmortem brain samples from subjects with AUD, depression and controls in the prefrontal cortex (PFC) (A), hippocampus (HIP) (B), caudate nucleus (CAU) (C) and cerebellum (CB) (D). n = 25 per group. One-way ANOVA followed by Dunnett's post-hoc test, *p< 0.05, **p< 0.01. Samples were run in triplicate. Bars show mean ± SEM, dots represent the average value of each individual.

Figure 2 .

 2 Figure 2. Protein expression of hevin in postmortem human brains of alcohol use disorder (AUD) subjects. Hevin protein expression in postmortem brain samples from subjects with AUD, depression (DEP) and controls (CONT) in the prefrontal cortex (PFC) (A), hippocampus (HIP) (B), caudate nucleus (CAU) (C) and cerebellum (CB) (D). Representative Western blot images for each brain region are shown, along with the graphics representing the actin-normalized immunoreactivity (IR) of both the ~130 kDa and ~100 kDa hevin bands. n = 25 per group. One-way ANOVA followed by Dunnett's post-hoc test, *p< 0.05, **p< 0.01, ****p< 0.0001. Experiments were performed in triplicate. Bars show mean ± SEM, dots represent the average value of each individual.

Figure 3 .

 3 Figure 3. Hevin protein expression in the brain of mice exposed to different regimens of ethanol. A) Schematic figure showing the four ethanol treatments performed. (B-F) Representative Western blot images and the graphics representing the actin-normalized immunoreactivity (IR) of the ~130 kDa band in the four treatment groups (S-S: saline -saline, S-E: saline -ethanol, E-S: ethanol -saline and E-E: ethanol -ethanol) in the following brain regions: B) nucleus accumbens (NAc), C) amygdala (AMY), D) frontal cortex (FC), E) hippocampus (HIP), and F) dorsal striatum (CPu). n = 5-15 per group. Kruskal-Wallis followed by Dunn's post-hoc test, *p< 0.05, **p< 0.01, ***p< 0.001. Experiments were performed in triplicate. Bars show mean ± SEM, dots represent the average value of each individual.

Figure 4 .

 4 Figure 4. Hevin protein expression in mouse plasma after different regimens of ethanol. A representative Western blot of hevin in mouse plasma and its Ponceau S stain images, along with the graphics representing the actin-normalized immunoreactivity of the ~130 kDa and ~47 kDa hevin bands and the 130 kDa/47 kDa band ratio in the four treatment groups (S-S: saline -saline, S-E: saline -ethanol, E-S: ethanol -saline and E-E: ethanol -ethanol). n = 7-8 per group. Kruskal-Wallis followed by Dunn's post-hoc test, *p< 0.05. Experiments were performed in triplicate. Bars show mean ± SEM, dots represent the average value of each individual.

Figure 5 .

 5 Figure 5. Effect of hevin protein downregulation in nucleus accumbens (NAc) astrocytes on ethanol consumption in mice. A) Immunohistochemical validation of hevin downregulation in NAc astrocytes after injection of miRNA-containing viruses. Hevin immunofluorescence, in red, is

FUNDING:

  AUTHOR CONTRIBUTION: AME: VV, LFC and FCC made substantial contributions to the conception of the work and to the search for funding. A.N.-dM, PCB, IBM, AA, PP and RC participated on the acquisition of the data. A.N.-dM and PCB performed the data analysis. A.N.-dM, PCB AME, VV, LFC and FCC contributed to the interpretation of data. AME, VV, A.N.-dM and LFC drafted the paper, and all the authors gave their final approval of the version to be published. This work was supported by ERAB (The European Foundation for Alcohol Research, EA 18 19); Centro de Investigación Biomédica en Red de Salud Mental (CIBERSAM), Spain; the Basque Government (IT1512-22); Fundación Vital; and FP7 Marie Curie Actions Career Integration Grant (FP7-PEOPLE-2013-CIG #618807). A.N.-dM. is a recipient of a predoctoral fellowship from the Basque Government and received an EMBO short-term fellowship.

COMPETING INTERESTS:

The authors have nothing to disclose.