Vikas Vikram Singh

Abdel Lisser

Monika Arora

Zero-sum games with distributionally robust chance constraints

Keywords: Distributionally robust chance constraints, Zero-sum game, Saddle point equilibrium, Second-order cone program

We consider a two-player zero-sum game with random linear constraints whose distributions are known to belong to moments based uncertainty sets. We show that a saddle point equilibrium problem is equivalent to a primal-dual pair of second-order cone programs.

Introduction

A two-player zero-sum game is defined using a single payoff function where one player plays the role of maximizer and another player plays the role of minimizer. More commonly, a zero-sum game is introduced with a payoff matrix where the rows and the columns are the actions of player 1 and player 2, respectively. A saddle point equilibrium (SPE) is the solution concept to study the zero-sum games and it exists in the mixed strategies [START_REF] Neumann | On the theory of games[END_REF]. Dantzig and later Adler showed the equivalence between linear programming problems and two-player zero-sum games [START_REF] Adler | The equivalence of linear programs and zero-sum games[END_REF][START_REF] Dantzig | A proof of the equivalence of the programming problem and the game problem[END_REF]. Charnes [START_REF] Charnes | Constrained games and linear programming[END_REF] generalized the zero-sum game considered in [START_REF] Neumann | On the theory of games[END_REF] by introducing linear inequality constraints on the mixed strategies of both Email addresses: vikassingh@iitd.ac.in (Vikas Vikram Singh), abdel.lisser@l2s.centralesupelec.fr (Abdel Lisser), monika@iiitd.ac.in (Monika Arora)

the players and called it a constrained zero-sum game. An SPE of a constrained zero-sum game can be obtained from the optimal solutions of a primal-dual pair of linear programs [START_REF] Charnes | Constrained games and linear programming[END_REF]. Singh and Lisser [START_REF] Singh | A second-order cone programming formulation for zero sum game with chance constraints[END_REF] considered a stochastic version of constrained zero-sum game considered by Charnes [START_REF] Charnes | Constrained games and linear programming[END_REF] where the mixed strategies of each player are restricted by random linear inequality constraints which are modelled using chance constraints. When the random constraint vectors follow a multivariate elliptically symmetric distribution, the zero-sum game problem is equivalent to a primal-dual pair of second-order cone programs(SOCPs) [START_REF] Singh | A second-order cone programming formulation for zero sum game with chance constraints[END_REF].

Nash equilibrium is the generalization of SPE and it is used as a solution concept for the general-sum games. Under certain conditions on payoff functions and strategy sets there always exists a Nash equilibrium [START_REF] Debreu | A social equilibrium existence theorem[END_REF]. The general-sum games under uncertainties are considered in the literature [START_REF] Ravat | On the characterization of solution sets of smooth and nonsmooth convex stochastic Nash games[END_REF][START_REF] Singh | Existence of Nash equilibrium for chanceconstrained games[END_REF][START_REF] Singh | Distributionally robust chanceconstrained games: Existence and characterization of Nash equilibrium[END_REF][START_REF] Shen | Games with distributionally robust joint chance constraints[END_REF][START_REF] Singh | An equivalent mathematical program for games with random constraints[END_REF] which capture both risk neutral and risk averse situations.

In this paper, we consider a more general two player zero-sum game as compared to [START_REF] Singh | A second-order cone programming formulation for zero sum game with chance constraints[END_REF]. Unlike in [START_REF] Singh | A second-order cone programming formulation for zero sum game with chance constraints[END_REF], the strategy set of each player is defined by a compact polyhedral set which is further restricted by some random linear inequalities and the information on the distribution of the random constraint vectors is not exactly known. We consider two different uncertainty sets based on the partial information on the mean vectors and covariance matrices of the random constraint vectors. We show that there exists an SPE of the game Z α and an SPE problem is equivalent to a primal-dual pair of SOCPs.

The rest of the paper is organized as follows. The definition of a distributionally robust zero-sum game is given in Section 2. Section 3 presents the reformulation of distributionally robust chance constraints as second order cone constraints under two different uncertainty sets. Section 4 outlines a primal-dual pair of SOCPs whose optimal solutions constitute an SPE of the game.

The model

We consider a two player zero-sum game where each player has continuous

strategy set. Let C 1 ∈ R K1×m , C 2 ∈ R K2×n , d 1 ∈ R K1 and d 2 ∈ R K2 . We consider X = {x ∈ R m | C 1 x = d 1 , x ≥ 0} and Y = {y ∈ R n | C 2 y = d 2 , y ≥ 0}
as the strategy sets of player 1 and player 2, respectively. We assume that X and Y are compact sets. Let u : X × Y → R be a payoff function associated to the zero-sum game and we assume that player 1 (resp.player 2) is interested in maximizing (resp. minimizing) u(x, y) for a fixed strategy y (resp. x) of player 2 (resp. player 1). For a given strategy pair (x, y) ∈ X × Y , the payoff function u(x, y) is given by

u(x, y) = x T Gy + g T x + h T y, (2.1)
where G ∈ R m×n , g ∈ R m and h ∈ R n . The first term of (2.1) results from the interaction between both the players whereas the second and third term represents the individual impact of player 1 and player 2 on the game, respectively.

The strategy sets are often restricted by random linear constraints which are modelled using chance constraints. The chance constraint based strategy sets appear in many practical problems, e.g., risk constraints in portfolio optimization [START_REF] Ji | Risk-budgeting multi-portfolio optimization with portfolio and marginal risk constraints[END_REF]. In this paper, we consider the case where the strategies of player 1 satisfy the following random linear constraints,

(a 1 k ) T x ≤ b 1 k , k = 1, 2, . . . , p, (2.2)
whilst the strategies of player 2 satisfy the following random linear constraints (a 2 l ) T y ≥ b 2 l , l = 1, 2, . . . , q.

(2.3)

Let I 1 = {1, 2, . . . , p} and I 2 = {1, 2, . . . , q} be the index sets for the constraints of player 1 and player 2, respectively. For each k ∈ I 1 and l ∈ I 2 , the vectors a 1 k and a 2 l are random vectors defined on a probability space (Ω, F, P). We consider the case where the only information we have about the distributions of a 1 k and a 2 l is that they belong to some uncertainty sets D 1 k and D 2 l , respectively. The uncertainty sets D 1 k and D 2 l , are constructed based on the partially available information on the distributions of a 1 k and a 2 l , respectively. Using the worst case approach, the random linear constraints (2.2) and (2.3) can be formulated as distributionally robust chance constraints given by inf

F 1 k ∈D 1 k P (a 1 k ) T x ≤ b 1 k ≥ α 1 k , ∀ k ∈ I 1 , (2.4)
and inf

F 2 l ∈D 2 l P (-a 2 l ) T y ≤ -b 2 l ≥ α 2 l , ∀ l ∈ I 2 , (2.5)
where α 1 k and α 2 l are the confidence levels of player 1 and player 2 for kth and lth constraints, respectively. Therefore, for a given α 1 = (α 1 k ) k∈I1 and α 2 = (α 2 l ) l∈I2 , the feasible strategy sets of player 1 and player 2 are given by

S 1 α 1 = x ∈ X| inf F 1 k ∈D 1 k P{(a 1 k ) T x ≤ b 1 k } ≥ α 1 k , ∀ k ∈ I 1 , (2.6)
and

S 2 α 2 = y ∈ Y | inf F 2 l ∈D 2 l P{(-a 2 l ) T y ≤ -b 2 l } ≥ α 2 l , ∀ l ∈ I 2 . (2.7)
We call the zero-sum game with the strategy set S 1 α 1 for player 1 and the strategy set S 2 α 2 for player 2 as a distributionally robust zero-sum game. We denote this game by Z α . A strategy pair (x * , y

* ) ∈ S 1 α 1 × S 2 α 2 is called an SPE of the game Z α at α = (α 1 , α 2 ) ∈ [0, 1] p × [0, 1] q , if u(x, y * ) ≤ u(x * , y * ) ≤ u(x * , y), ∀ x ∈ S 1 α 1 , y ∈ S 2 α 2 .

Reformulation of distributionally robust chance constraints

We consider two different uncertainty sets based on the partial information about the mean vectors and covariance matrices of the random constraint vec-

tors a i k , i = 1, 2, k ∈ I i .
For each uncertainty set, distributionally robust chance constraints (2.4) and (2.5) are reformulated as second-order cone constraints.

Moments based uncertainty sets

For each player i, i = 1, 2, we consider the case where the mean vector and covariance matrix of the random vector a i k for all k ∈ I i are known to belong to polytopes U µ i k and U Σ i k , respectively. We consider polytopes

U µ i k = Conv(µ i k1 , µ i k2 , . . . , µ i kM ) and U Σ i k = Conv(Σ i k1 , Σ i k2 , . . . , Σ i kM )
, where Σ i kj ≻ 0, j = 1, 2, . . . M ; Conv denotes the convex hull and Σ i kj ≻ 0 implies that Σ i kj is a positive definite matrix. For each i = 1, 2, and k ∈ I i , the uncertainty set for the distribution of a i k is defined by

D i k µ i k , Σ i k =    F i k E F i k a i k ∈ U µ i k COV F i k [a i k ] ∈ U Σ i k    , (3.1)
where E F i k and COV F i k are expectation and covariance operator under probability distribution F i k , respectively. The uncertainty set (3.1) is considered in [START_REF] El-Ghaoui | Worst-case value-at-risk and robust portfolio optimization: A conic programming approach[END_REF]. As for the second uncertainty set, we consider the case where the mean vector of a i k lies in an ellipsoid of size γ i k1 ≥ 0 centered at µ i k and the covariance matrix of a i k lies in a positive semidefinite cone defined with a linear matrix inequality. It is defined by

D i k (µ i k , Σ i k ) =    F i k E F i k [a i k ] -µ i k ⊤ Σ i k -1 E F i k [a i k ] -µ i k ≤ γ i k1 , COV F i k [a i k ] ⪯ γ i k2 Σ i k    , (3.2)
where Σ i k ≻ 0 and γ i k2 > 0; for the given matrices B 1 and

B 2 , B 1 ⪯ B 2 implies that B 2 -B 1 is a positive semidefinite matrix. The uncertainty set (3.2) is considered in [14].

Second-order cone constraint reformulation

We show that the distributionally robust chance constraints (2.4) and (2.

(µ 1 kj ) T x + α 1 k 1 -α 1 k ||(Σ 1 kj ) 1 2 x|| ≤ b 1 k , ∀ j = 1, 2, . . . , M, k ∈ I 1 , (3.3)
-(µ 2 kj ) T y + α 2 k 1 -α 2 k ||(Σ 2 kj ) 1 2 y|| ≤ -b 2 k , ∀ j = 1, 2, . . . , M, k ∈ I 2 . (3.4)
Proof. Based on the structure of uncertainty set (3.1), (2.4) can be written as inf

(µ,Σ)∈U 1 k inf F 1 k ∈D(µ,Σ) P (a 1 k ) T x ≤ b 1 k ≥ α 1 k ,
where

D(µ, Σ) = F 1 k E F 1 k [a 1 k ] = µ, COV F 1 k [a 1 k ] = Σ , and
U 1 k = (µ, Σ) µ ∈ U µ 1 k , Σ ∈ U Σ 1 k .
The bound of one-sided Chebyshev inequality can be achieved by a two-point distribution given by equation ( 2) of [START_REF] Rujeerapaiboon | Chebyshev inequalities for products of random variables[END_REF]. Therefore, we have inf

F 1 k ∈D(µ,Σ) P (a 1 k ) T x ≤ b 1 k =        1 - 1 1+ (µ T x-b 1 k ) 2 (x T Σx) , if µ T x ≤ b 1 k , 0, otherwise.
For the case µ

T x > b 1 k , inf F 1 k ∈D(µ,Σ) P a 1 k x ≤ b 1 k = 0,
which makes constraint (2.4) infeasible for any α 1 > 0. Therefore, for x ∈ S 1 α1 the condition µ T x ≤ b 1 k always holds and the constraint (2.4) is equivalent to inf

(µ,Σ)∈U 1 k 1 - 1 1 + (µ T x -b 1 k ) 2 /(x T Σx) ≥ α 1 k ,
which can be reformulated as min

µ∈U µ 1 k b 1 k -µ T x max Σ∈U Σ 1 k √ x T Σx ≥ α 1 k 1 -α 1 k . ( 3.5)
The above inequality (3.5) can be reformulated as (3.3). Similarly, we can show that (2.5) is equivalent to (3.4). □ Lemma 3.2. For each i = 1, 2, and k ∈ I i , let the distribution F i k of random vector a i k , lies in the uncertainty set D i k µ i k , Σ i k defined by (3.2). Then, the constraints (2.4) and (2.5) are equivalent to (3.6) and (3.7), respectively, given by

(µ 1 k ) T x + α 1 k 1 -α 1 k γ 1 k2 + γ 1 k1 Σ 1 k 1 2 x ≤ b 1 k , ∀ k ∈ I 1 , (3.6)
-(µ 2 k ) T y + α 2 k 1 -α 2 k γ 2 k2 + γ 2 k1 Σ 2 k 1 2 y ≤ -b 2 k ∀ k ∈ I 2 . (3.7)
Proof. Based on the structure of the uncertainty set (3.2), the constraint (2.4) can be written as inf

(µ,Σ)∈ Ũ1 k inf F 1 k ∈D(µ,Σ) P a 1 k x ≤ b 1 k ≥ α 1 k ,
where

D(µ, Σ) = F 1 k E F 1 k [a 1 k ] = µ, COV F 1 k [a 1 k ] = Σ and Ũ1 k = (µ, Σ) µ -µ 1 k ⊤ Σ 1 k -1 µ -µ 1 k ≤ γ 1 k1 , Σ ⪯ γ 1 k2 Σ 1 k .
Using the similar arguments as in the Lemma 3.1, the constraint (2.4) is equiv-

alent to b 1 k + v 1 (x) v 2 (x) ≥ α 1 k 1 -α 1 k , (3.8)
where

v 1 (x) =      min µ -µ T x s.t. µ -µ 1 k ⊤ Σ 1 k -1 µ -µ 1 k ≤ γ 1 k1 , (3.9)
v 2 (x) =      max Σ x T Σx s.t. Σ ⪯ γ 1 k2 Σ 1 k .
Let β ≥ 0 be a Lagrange multiplier associated with the constraint of optimization problem (3.9). By applying the KKT conditions, the optimal solution of (3.9) is given by µ

= µ 1 k + √ γ 1 k1 Σ 1 k x √ x T Σ 1 k x
and the associated Lagrange multiplier is given by β =

x T Σ 1 k x 4γ 1 k1
. Therefore, the corresponding optimal value

v 1 (x) = -(µ 1 k ) T x -γ 1 k1 x T Σ 1 k x. Since, u T Σu ≤ u T γ 1 k2 Σ 1 k u for any u ∈ R n , then, v 2 (x) = γ 1 k2 x T Σ 1 k x.
Therefore, using (3.8), (2.4) is equivalent to (3.6). Similarly, we can show that (2.5) is equivalent to (3.7). □

The reformulation of feasible strategy sets (2.6) and (2.7) for uncertainty sets (3.1) and (3.2) can be written as

S 1 α 1 = x ∈ X | (µ 1 kj ) T x + κ α 1 k ||(Σ 1 kj ) 1 2 x|| ≤ b 1 k , ∀ j = 1, 2, . . . , M, k ∈ I 1 , (3.10)
and For We assume that the strategy sets (3.10) and (3.11) satisfy the strict feasibility condition given by Assumption 3.3.

S 2 α 2 = y ∈ Y | -(µ 2 lj ) T y + κ α 2 l ||(Σ 2 lj ) 1 2 y|| ≤ -b 2 l , ∀ j = 1, 2, . . . , M, l ∈ I 2 . . (3.11) For each i = 1, 2, k ∈ I i , if κ α i k = α i k 1-α i k , ( 3
each i = 1, 2, k ∈ I i , if κ α i k = α i k 1-α i k γ i k2 + γ i k1 ,

Assumption 3.3.

1. There exists an x ∈ S 1 α 1 such that the inequality constraints of S 1 α 1 defined by (3.10) are strictly satisfied.

2. There exists an y ∈ S 2 α 2 such that the inequality constraints of S 2 α 2 defined by (3.11) are strictly satisfied.

The conditions given in Assumption 3.3 are Slater's condition which are sufficient for strong duality in a convex optimization problem. We use these conditions in order to derive equivalent SOCPs for the zero-sum game Z α .

Existence and characterization of saddle point equilibrium

In this section, we show that there exists an SPE of the game Z α if the distributions of the random constraint vectors of both the players belong to the uncertainty sets defined in Section 3.1. We further propose a primal-dual pair of SOCPs whose optimal solutions constitute an SPE of the game Z α . Theorem 4.1. Consider the game Z α where the distributions of the random constraint vectors a i k , k ∈ I i , i = 1, 2, belong to the uncertainty sets described in Section 3.1. Then, there exists an SPE of the game for all α ∈ (0, 1) p × (0, 1) q .

Proof. Let α ∈ (0, 1) p × (0, 1) q . For uncertainty sets described in Section 3.1, it follows from Lemma 3.1 and Lemma 3.2 that the strategy sets S 1 α 1 and S 2

Equivalent primal-dual pair of second-order cone programs

From the minimax theorem [START_REF] Neumann | On the theory of games[END_REF], (x * , y * ) is an SPE for the game Z α if and only if

x * ∈ arg max

x∈S 1 α 1 min y∈S 2 α 2 u(x, y), (4.1)
y * ∈ arg min y∈S 2 α 2 max x∈S 1 α 1 u(x, y). (4.2)
We start with problem min y∈S 2 α 2 max x∈S 1 α 1 u(x, y). The inner optimization problem max x∈S 1 α 1 u(x, y) can be equivalently written as max

x,t 1 k,j x T Gy + g T x + h T y s.t. (i) -x T µ 1 k,j -κ α 1 k t 1 k,j + b 1 k ≥ 0, ∀ j = 1, 2 . . . , M, k ∈ I 1 , (ii) t 1 k,j -Σ 1 k,j 1 2 x = 0, ∀ j = 1, 2 . . . , M, k ∈ I 1 , (iii) C 1 x = d 1 , x r ≥ 0, ∀ r = 1, 2, . . . , m. (4.3)
Let

λ 1 = λ 1 k,j M j=1,k∈I1 ∈ R M p , δ 1 k,j ∈ R m for all k ∈ I 1 , j = 1, 2 .
. . M , and ν 1 ∈ R K1 be the Lagrange multipliers of constraints (i), (ii), and equality constraints given in (iii) of (4.3), respectively. Then, the Lagrangian dual problem of the SOCP (4.3) can be written as

min λ1≥0, δ 1 k,j , ν 1 max x≥0, t 1 k,j x T Gy + g T x + h T y + k∈I1 M j=1 λ 1 k,j -x T µ 1 k,j -κ α 1 k t 1 k,j + b 1 k + (δ 1 k,j ) T t 1 k,j -Σ 1 k,j 1 2 x + (ν 1 ) T (d 1 -C 1 x) = min λ1≥0,δ 1 k,j ,ν 1 max x≥0 x T Gy - k∈I1 M j=1 λ 1 k,j µ 1 k,j + Σ 1 k,j 1 2 δ 1 k,j -(C 1 ) T ν 1 + g + k∈I1 M j=1 max t 1 k,j (δ 1 k,j ) T t 1 k,j -κ α 1 k λ 1 k,j t 1 k,j + h T y + (ν 1 ) T d 1 + k∈I1 M j=1 λ 1 k,j b 1 k .
The inner maximization problems in the above Lagrangian dual problem will be unbounded unless we have the following dual constraints

Gy - k∈I1 M j=1 λ 1 k,j µ 1 k,j + Σ 1 k,j 1 2 δ 1 k,j -(C 1 ) T ν 1 + g ≤ 0, ||δ 1 k,j || ≤ κ α 1 k λ 1 k,j , ∀ k ∈ I 1 , j = 1, 2 . . . , M.
Under Assumption 3.3, the Lagrangian dual problem of (4.3) has zero duality gap [START_REF] Boyd | Convex Optimization[END_REF]. Therefore, the problem min y∈S 2 α 2 max x∈S 1 α 1 u(x, y) is equivalent to the following SOCP min Similarly, problem max x∈S 1 α 1 min y∈S 2 α 2 u(x, y) is equivalent to the following SOCP max

y, ν 1 , δ 1 k,j , λ 1 k,j h T y + (ν 1 ) T d 1 + k∈I1 M j=1 λ 1 k,j b 1 k s.t. (i) Gy - k∈I1 M j=1 λ 1 k,j µ 1 k,j + Σ 1 k,j 1 2 δ 1 k,j -(C 1 ) T ν 1 + g ≤ 0, (ii) -(µ 2 lj ) T y + κ α 2 l ||(Σ 2 lj ) 1 2 y|| ≤ -b 2 l , ∀ j = 1, 2, . . . , M, l ∈ I 2 , (iii) ||δ 1 k,j || ≤ κ α 1 k λ 1 k,j , λ 1 k,j ≥ 0, ∀ k ∈ I 1 , j = 1, 2 . . . , M, (iv)
x, ν 2 , δ 2 l,j , λ 2 l,j g T x + (ν 2 ) T d 2 + l∈I2 M j=1 λ 2 l,j b 2 l s.t. (i) G T x - l∈I2 M j=1 λ 2 l,j µ 2 l,j + Σ 2 l,j 1 2 δ 2 l,j -(C 2 ) T ν 2 + h ≥ 0, (ii) (µ 1 kj ) T x + κ α 1 k ||(Σ 1 kj ) 1 2 x|| ≤ b 1 k , ∀ j = 1, 2, . . . , M, k ∈ I 1 , (iii) ||δ 2 l,j || ≤ κ α 2 l λ 2 l,j , λ 2 l,j ≥ 0, ∀ l ∈ I 2 , j = 1, 2, . . . M (iv) C 1 x = d 1 , x r ≥ 0, ∀ r = 1, 2, . . . , m. (4.5)
It follows from the duality theory of SOCPs that (4.4) and (4.5) form a primaldual pair [START_REF] Boyd | Convex Optimization[END_REF].

Remark 4.2. For each i = 1, 2, and k For each i = 1, 2, and k Next, we show that the equivalence between the optimal solutions of (4.4)-(4.5)

∈ I i , if κ α i k = α i k 1-α i k , (4
∈ I i , if κ α i k = α i k 1-α i k γ i k2 + γ i
and an SPE of the game Z α .

Theorem 4.3. Consider the zero-sum game Z α where the feasible strategy sets of player 1 and player 2 are given by (3.10) and (3.11), respectively. Let Assumption 3.3 holds. Then, for a given α ∈ (0, 1) p × (0, 1) q , (x * , y * ) is an SPE of the game Z α if and only if there exists (ν 1 * , (δ 1 * k,j ) k,j , λ 1 * ) and (ν 2 * , (δ 2 * l,j ) l,j , λ 2 * ) such that (y * , ν 1 * , (δ 1 * k,j ) k,j , λ 1 * ) and (x * , ν 2 * , (δ 2 * l,j ) l,j , λ 2 * ) are optimal solutions of (4.4) and (4.5), respectively.

Proof. Let (x * , y * ) be an SPE of the game Z α . Then, x * and y * are the solutions of (4.1) and (4.2), respectively. Therefore, there exists (ν 1 * , (δ 1 * k,j ) k,j , λ 1 * ) and (ν 2 * , (δ 2 * l,j ) l,j , λ 2 * ) such that (y * , ν 1 * , (δ 1 * k,j ) k,j , λ 1 * ) and (x * , ν 2 * , (δ 2 * l,j ) l,j , λ 2 * ) are optimal solutions of (4.4) and (4.5) respectively. Let (y * , ν 1 * , (δ 1 * k,j ) k,j , λ 1 * ) and (x * , ν 2 * , (δ 2 * l,j ) l,j , λ 2 * ) be optimal solutions of (4.4) and (4.5), respectively. Under Assumption 3. Consider the constraint (i) of (4.4) at optimal solution (y * , ν 1 * , (δ 1 * k,j ) k,j , λ 1 * ) and multiply it by x T , where x ∈ S 1 α1 . Then, by using Cauchy-Schwartz inequality, we have (4.9)

It follows from (4.7), (4.8), and (4.9) that (x * , y * ) is an SPE of the game Z α . □

Lemma 3 . 1 .

 31 [START_REF] Singh | A second-order cone programming formulation for zero sum game with chance constraints[END_REF] are equivalent to second-order cone constraints for the uncertainty sets defined by (3.1) and (3.2). For each i = 1, 2, and k ∈ I i , let the distribution F i k of random vector a i k , lies in uncertainty set D i k µ i k , Σ i k defined by (3.1). Then, the constraints (2.4) and (2.5) are equivalent to (3.3) and (3.4), respectively, given by

  .10) and(3.11) represent the reformulations of (2.6) and (2.7) under uncertainty set defined by (3.1), respectively.

  and M = 1, (3.10) and (3.11) represent the reformulations of (2.6) and (2.7) under uncertainty set defined by (3.2), respectively.

α 2 areα 1 and S 2 α 2 are

 22 given by (3.10) and(3.11), respectively. It is easy to see that S 1 convex and compact sets. The function u(x, y) is a bilinear and continuous function. Hence, there exists an SPE from the minimax theorem[START_REF] Neumann | On the theory of games[END_REF].

C 2 y

 2 = d 2 , y s ≥ 0, ∀ s = 1, 2, . . . , n. (4.4)

. 4 )

 4 and (4.5) represent the primal-dual pair of SOCPs for the uncertainty sets defined by (3.1).

k1 and M = 1 ,( 4 . 4 )

 144 and (4.5) represent the primal-dual pair of SOCPs for the uncertainty set defined by (3.2).

  3, (4.4) and (4.5) are strictly feasible. Therefore, strong duality holds for primal-dual pair (4.4)-(4.5). Then, we haveg T x * + (ν 2 * ) T d 2 + l∈I2 M j=1 λ 2 * l,j b 2 l = h T y * + (ν 1 * ) T d 1 +

xb 2 lb 1 k

 21 T Gy * + g T x + h T y * ≤ h T y * + (ν 1 * ) T d 1 + x * T Gy + g T x * + h T y ≥ g T x * + (ν 2 * ) T d 2 + , ∀ y ∈ S 2 α2 . (4.8)Take x = x * and y = y * in (4.7) and (4.8), then from (4.6), we getu(x * , y * ) = h T y * + (ν 1 * ) T d 1 + = g T x * + (ν 2 * ) T d 2 +

Acknowledgement This research was supported by DST/CEFIPRA Project No. IFC/4117/DST-CNRS-5th call/2017-18/2 and CNRS Project No. AR/SB:2018-07-440.


Fragueiro¹ ; Claire

Cury¹ ; Federica

Santacroce² ; Ford

Burles³ ; Giuseppe

Iaria³ ; Giorgia

Committeri²

Agustina Fragueiro
email: agustina.fragueiro@inria.fr

Medial positioning of the hippocampus and hippocampal fissure volume in Developmental Topographical Disorientation

Keywords: Developmental Topographical Disorientation and Incomplete Hippocampal Inversion hippocampus, incomplete hippocampal inversion, developmental topographical disorientation, spatial navigation, hippocampal fissure, subiculum, subfields automatic segmentation

Developmental Topographical Disorientation (DTD) refers to the lifelong inability to orient by means of cognitive maps in familiar surroundings despite otherwise well-preserved general cognitive functions, and the absence of any acquired brain injury or neurological condition. While reduced functional connectivity between the hippocampus and other brain regions has been reported in DTD individuals, no structural differences in grey matter tissue for the whole brain neither for the hippocampus were detected. Considering that the human hippocampus is the main structure associated with cognitive map-based navigation, here, we investigated differences in morphological and morphometric hippocampal features between individuals affected by DTD (N=20) and healthy controls (N=238). Specifically, we focused on a developmental anomaly of the hippocampus that is characterized by the incomplete infolding of hippocampal subfields during foetal development, giving the hippocampus a more round or pyramidal shape, called Incomplete Hippocampal Inversion (IHI). We rated IHI according to standard criteria and extracted hippocampal subfield volumes after FreeSurfer's automatic segmentation.

We observed similar IHI prevalence in the group of individuals with DTD with respect to the control population. Neither differences in whole hippocampal nor major hippocampal subfield volumes have been observed between groups. However, when assessing the IHI independent criteria, we observed that the hippocampus in the DTD group is more medially positioned comparing to the control group. In addition, we observed bigger hippocampal fissure volume for the DTD comparing to the control group. Both of these findings were stronger for the right hippocampus comparing to the left.

Our results provide new insights regarding the hippocampal morphology of individuals affected by DTD, highlighting the role of structural anomalies during early prenatal development in line with the developmental nature of the spatial disorientation deficit.

Introduction

Developmental Topographical Disorientation (DTD) refers to the lifelong inability to orient in familiar surroundings despite otherwise well-preserved general cognitive functions, and the absence of any acquired brain injury or neurological condition [START_REF] Iaria | Developmental Topographical Disorientation[END_REF]. Individuals with DTD are unable to orient by means of cognitive maps in both ecological [START_REF] Bianchini | Developmental topographical disorientation in a healthy subject[END_REF][START_REF] Iaria | Developmental topographical disorientation: Case one[END_REF] and virtual [START_REF] Bianchini | Where Am I? A new case of developmental topographical disorientation[END_REF][START_REF] Iaria | Developmental topographical disorientation: A newly discovered cognitive disorder[END_REF] environments. This inability to orient by means of cognitive maps suggests that the neural mechanisms related to the presence of DTD may involve the hippocampus, a region in the medial temporal lobe that is critical for the formation and use of cognitive maps in both human and non-human animals [START_REF] Bottini | Knowledge Across Reference Frames: Cognitive Maps and Image Spaces[END_REF][START_REF] Hartley | The well-worn route and the path less traveled: Distinct neural bases of route following and wayfinding in humans[END_REF][START_REF] Iaria | Retrosplenial and hippocampal brain regions in human navigation: Complementary functional contributions to the formation and use of cognitive maps: Retrosplenial cortex and hippocampus in navigation[END_REF][START_REF] O'keefe | The Hippocampus as a Cognitive Map[END_REF].

Few studies have investigated the functional and structural properties of the hippocampus in individuals affected by DTD. Iaria and colleagues (2014) reported reduced functional connectivity between the right hippocampus and the prefrontal cortex in this population. In addition, a DTD case study using functional and diffusion tensor imaging [START_REF] Kim | A Neural Basis for Developmental Topographic Disorientation[END_REF] has reported an atypical response profile in the retrosplenial cortex that presented with a decreased functional coupling with the parahippocampal place area, which was, however, not related to compromised structural connectivity. Furthermore, no structural differences in grey matter tissue for the whole brain neither for the hippocampus were detected when using voxel-based morphometry [START_REF] Iaria | Developmental topographical disorientation and decreased hippocampal functional connectivity: Dtd and Hippocampal Functional Connectivity[END_REF]. Thus, while reduced functional connectivity has been observed in DTD, no structural differences have been reported till now.

On the other hand, there is evidence of a developmental anomaly of the hippocampus that is characterized by the incomplete infolding of hippocampal subfields before gestational week 21 [START_REF] Bajic | Incomplete inversion of the hippocampus-A common developmental anomaly[END_REF][START_REF] Fu | Hippocampal Malrotation: A Genetic Developmental Anomaly Related to Epilepsy?[END_REF]. This structural anomaly during development gives the hippocampus a more round or pyramidal shape, and is referred to as hippocampal malrotation or incomplete hippocampal inversion (IHI). As the development stopped, IHI hippocampi resemble the foetal shape and remain more medially located with a deeper collateral sulcus. Along normal development, before week 16, the subiculum region is larger and more medially positioned, and subfields CA1, CA2 and CA3 are arranged linearly before adopting the characteristic arc shape. Only by weeks 18-21, the dentate gyrus and CA embed into the temporal lobe, with CA1-3 subfields forming an arc [START_REF] Kier | Embryology of the Human Fetal Hippocampus: MR Imaging, Anatomy, and Histology[END_REF]. The incompletion of this process results in the IHI characteristic shape, which occurs predominantly in the left hippocampus [START_REF] Cury | Incomplete Hippocampal Inversion: A Comprehensive MRI Study of Over 2000 Subjects[END_REF]. Although it is considered one extreme of the normal phenotypic spectrum, it is more prevalent in epilepsy and it is thought to be a factor of susceptibility for hippocampal sclerosis [START_REF] Bastos-Leite | Analysis of shape and positioning of the hippocampal formation: An MRI study in patients with partial epilepsy and healthy controls[END_REF]. It has been associated with larger subiculum volumes and smaller CA1 volumes in healthy young adults [START_REF] Fragueiro | Incomplete Hippocampal Inversion and Hippocampal Subfield Volumes: Implementation and Inter-Reliability of Automatic Segmentation[END_REF], to smaller CA1 and right CA3 volumes in healthy aging [START_REF] Colenutt | Incomplete Hippocampal Inversion and Its Relationship to Hippocampal Subfield Volumes and Aging: Incomplete Hippocampal Inversion and Aging[END_REF], and to extra-hippocampal morphological changes in sulci mainly located in occipito-temporal cortices [START_REF] Cury | Incomplete Hippocampal Inversion: A Comprehensive MRI Study of Over 2000 Subjects[END_REF]. A genomewide association study on adolescence evidenced correlations between IHI and genetic traits associated either with intelligence or education attainment [START_REF] Cury | Genome wide association study of incomplete hippocampal inversion in adolescents[END_REF], however, it remains unclear whether IHI is related to cognitive impairments, in particular to those related to hippocampal function [START_REF] Stiers | Hippocampal malrotation in pediatric patients with epilepsy associated with complex prefrontal dysfunction[END_REF]. Interestingly, it has been proposed that IHI could be a genetic developmental imaging feature that may represent a more diffuse but subtle structural atypicality during brain development [START_REF] Fu | Hippocampal Malrotation: A Genetic Developmental Anomaly Related to Epilepsy?[END_REF].

Here, we investigated differences in morphological and morphometric hippocampal features between individuals affected by DTD and healthy controls. Considering that the hippocampus is the main structure involved in the generation and use of cognitive maps, as well as, that DTD and IHI are both developmental features, we proposed to explore if there may be an association among their presence. To this aim, we rated IHI according to [START_REF] Cury | Incomplete Hippocampal Inversion: A Comprehensive MRI Study of Over 2000 Subjects[END_REF] criteria and we extracted hippocampal subfield volumes after their automatic segmentation to compare DTD to control individuals.

Materials and methods

Participants

2.1.1 DTD group. The DTD group included 20 females right-handed individuals (age range 25 to 64 years) recruited locally through a public advertising campaign. These individuals reported no psychiatric or neurological disorders, they all had normal or corrected-to-normal vision, and were all diagnosed with DTD following the criteria described in previous studies [START_REF] Burles | Behavioural and cognitive mechanisms of Developmental Topographical Disorientation[END_REF]. These criteria are: (1) having experienced the inability of orient, and therefore getting lost, from childhood, (2) experiencing getting lost in extremely familiar surroundings, (3) despite having no other cognitive complaints, and (4) in the absence of any other cognitive or neurological condition. All participants provided written informed consent as approved by the local research ethics board of the University of Calgary (REB14-2176). These individuals were included in the study following a visual examination for quality issues. 2.1.2 Healthy Control group 1 (HC1). The first HC group consisted of 220 healthy females, 213 of them with an age range = 26 to 30 years and 7 of them with an age over 36. All of them are part of the Human Connectome Dataset WU-Minn [START_REF] Van Essen | The WU-Minn Human Connectome Project: An overview[END_REF]. These individuals were included in the study following exclusion of participants with quality control issues (as reported on the database), major neurological diseases, and psychiatric or medical disorders.

Healthy Control group 2 (HC2).

To exclude scanner-related differences between groups, we included 18 females (age range 25 to 64) whose images were acquired with the same protocol that the DTD group. These individuals reported no psychiatric or neurological disorders, they all had normal or correctedto-normal vision, and were assessed as no-DTD according to the criteria described in previous studies [START_REF] Burles | Behavioural and cognitive mechanisms of Developmental Topographical Disorientation[END_REF]. All participants provided written informed consent as approved by the local research ethics board of the University of Calgary (REB14-2176). These individuals were included in the study following a visual examination for quality issues. A summary with the descriptive information of all groups can be found in Table 1.

MRI data

2.2.1 DTD and HC2 databases. Participants were scanned using a 3T GE Discovery 750 MRI with an eight-channel head coil. Twenty-two individuals (DTD=9, HC2=13) were scanned in 2011 and 2013, and anatomical images were acquired using a T1-weighted SPGR sequence (Minimum Full TE, 11° flip angle, 1 mm isotropic voxels, 25.6 cm FOV). Sixteen additional individuals (DTD=11, HC2=5) were scanned in the same scanner in 2017, with anatomical images acquired using a T1-weighted fast SPGR sequence (Minimum Full TE, 10° flip angle, 1 mm isotropic voxels, 25.6 cm FOV, ARC factor 2). 2.2.2 HC1 database. All HC participants were scanned on a customized Siemens 3T "Connectome Skyra" housed at Washington University in St. Louis, using a standard 32-channel Siemens head coil and a body transmission coil designed by Siemens using the gradients of the WU-Minn and MGH-UCLA Connectome scanners. T1-weighted 3D MPRAGE were acquired with TR=2400 ms, TE=2.14 ms, TI=1000 ms, 8° flip angle, FOV=224x224, 0.7 mm isotropic voxel, bandwidth=210 Hz/px, iPAT=2, and acquisition time=7:40 (min:sec).

Incomplete Hippocampal Inversion rating

T1-weighted images were linearly transformed to MNI using flirt in FSL 6.0.5.1 in order to perform the assessment with a standardized orientation between subjects. The software medInria (https://med.inria.fr/) [START_REF] Vichot | Cardiac Interventional Guidance using Multimodal Data Processing and Visualisation: medInria as an Interoperability Platform[END_REF] was used to rate IHI, according to the criteria reported in [START_REF] Cury | Incomplete Hippocampal Inversion: A Comprehensive MRI Study of Over 2000 Subjects[END_REF], in 220 brains from the HC1, 18 brains from the HC2 and 20 from the DTD group. Note that all individual criteria C1, C2, C3, and C5 were rated independently. C1 assesses the roundness and verticality of the hippocampal body, with a score of 0 for flat and horizontal hippocampi and, on the other extreme, a maximum score of 2 for an oval and vertical hippocampus. C2 assesses the verticality and depth of the collateral sulcus relatively to the size of the hippocampus, with a score of 0 for a horizontal collateral sulcus not crossing the lateral limit of the hippocampus, and a maximum score of 2 if the sulcus surpasses the limit of the hippocampus and has a vertical orientation. C3 judges the medial positioning of the hippocampus, assessed by the length of the part of the subiculum that is not covered by the dentate gyrus relatively to the lateral part of the CA/subiculum that is covered by the dentate gyrus. Even in particular cases of variability in subfields location or orientation, the two segments were always defined orthogonally to the brain midline at the level of the superior bend of the subiculum. In addition, C3 considers whether there is visible presence of cerebrospinal fluid in the temporal horn of the lateral ventricle laterally to the hippocampus, or not, with higher scores for the first case (i.e. visible cerebrospinal fluid). C5 evaluates the depth of both the collateral and the occipitotemporal sulcus with respect of the subiculum, with a score of 0 if none of the sulci exceeds the level of the subiculum and a maximum score of 2 if at least one of the sulci exceeds vertically the superior limit of the subiculum. A total IHI score (going from 0 to 8) was calculated for each hemisphere by summing all criteria. If the total score exceeded 3.75, the hemisphere was classified as IHI. Criteria 4 was not rated as recommended in Cury et al 2015, since this criterion is not well defined in the literature and therefore not comparable across studies.

Before starting the rating, we evaluated intra-and inter-rater agreement. For intra-rater coherence, AF rated the same database of 30 subjects randomly extracted from the Human Connectome Project database (age 26-30) with a distance of 15 days. For inter-rater agreement, CC rated the same 30 subjects. AF and FS blindly rated 40 participants (20 from the DTD database and 20 randomly selected from the HC1 database). Finally, AF rated the 18 participants from the HC2. Kappa scores were calculated for C5, and kappa weight scores were calculated for criteria C1, C2, and C3. All intra-and inter-rater scores were assessed independently for each hemisphere and all of them were above 0.65 (Table 2).

Hippocampal subfields segmentation

The tool for segmentation of hippocampal subfields offered by FreeSurfer 7 was run using the T1-weighted scans after applying recon-all method. All process steps were applied, including motion correction, intensity normalization, Talairach transformation, among others (for the complete details see https://surfer.nmr.mgh.harvard.edu/fswiki/recon-all). The tool (segmentHA_T1) segments the different subfields by using a Bayesian inference approach based on image intensities and prior knowledge of a probabilistic atlas which was generated from in vivo manual segmentations and ultra-high-resolution ex vivo MRI data of the hippocampal substructures [START_REF] Iglesias | Bayesian longitudinal segmentation of hippocampal substructures in brain MRI using subject-specific atlases[END_REF].

The segmented subfields with this method followed the head-body-tail subdivision [START_REF] Iglesias | A computational atlas of the hippocampal formation using ex vivo , ultra-high resolution MRI: Application to adaptive segmentation of in vivo MRI[END_REF] and are: CA1 body, CA1 head, CA2/3 head, CA2/3 body, CA4 head, CA4 body, fimbria, granule cell layer of dentate gyrus (GC-ML-DG) head, GC-ML-DG body, hippocampus-amygdala-transition-area (HATA), tail, hippocampal fissure, molecular layer head, molecular layer body, parasubiculum, presubiculum head, presubiculum body, subiculum head, subiculum body (Figure 1). Volumes in mm 3 for each subfield were obtained through FreeSurfer (quantifyHAsubregions) and corrected by whole brain volume calculated on FSL (fslstats).

2.5 Pre-analyses for sanity check 2.5.1 Age-effect control: Data were analysed with the Kolmogorov-Smirnov Test. As not all variables were normally distributed, we conducted non-parametric statistics. As the HC1 database provided age-range but not each individual subject age, the following analyses on age were conducted on participants from the HC2 (N=18) and the DTD (N=20) groups. To test for an age effect on IHI, we conducted Spearman's correlations between age and IHI global scores for each hemisphere independently. To test for an age effect on volumes, we conducted Spearman's correlations between age and volumes for the following subfields of interest: CA1, CA2+3+4+DG, complete subiculum (including presubiculum, parasubiculum and subiculum), hippocampal fissure, and tail. Head and body sections were added together for each subfield of interest. Correlations between age and whole hippocampal head, whole hippocampal body and whole hippocampal volume were also tested. 2.5.2 Scanner-effect on IHI: To test for scanner related differences that may affect IHI rating, we first compared HC1 (N=220) and HC2 (N=18) groups. Mann-Whitney tests for independent samples between groups were conducted for all criteria independently and for the global IHI score. Each hemisphere was tested independently. P-values were corrected for multiple comparisons using Bonferroni correction (x10 tests).

Scanner-effect on volumes:

To test for scanner-related differences that may impact the automatic segmentation of hippocampal subfields (for examples of segmentations for all three groups see Supplementary Figure 1), we conducted Mann-Whitney tests between HC1 (N=219, as segmentation failed for 1 subject) and HC2 (N=18) for the subfields of interest (CA1, CA2+3+4+DG, complete subiculum, hippocampal fissure, and tail), for whole hippocampal head, whole hippocampal body and whole hippocampus volume. Each hemisphere was tested independently. P-values were corrected for multiple comparisons using Bonferroni correction (x16 tests).

Considering that IHI may impact hippocampal automatic segmentation [START_REF] Kim | Automatic hippocampal segmentation in temporal lobe epilepsy: Impact of developmental abnormalities[END_REF], we finally repeated the same analyses but excluding IHI hippocampi for each hemisphere independently. We conducted Mann-Whitney tests between HC1 (N=177) and HC2 (N=14) for the left hemisphere after excluding left IHI participants. The same analyses were conducted for the right hemisphere after excluding right IHI hippocampi (HC1=205, HC2=16). P-values were corrected for multiple comparisons using Bonferroni correction (x16 tests).

Pre-analyses results

2.6.1 Age-effect. No significant correlation was observed between age and IHI scores for any hemisphere (Figure 2A), neither between age and any subfield or whole hippocampal volumes (Figure 2B). The complete list of correlation's coefficients and uncorrected p-values is reported in Table 3. 2.6.2 Scanner-effect on IHI. In the HC1 database (N=213, age range = 26 to 30 years; N=7, age>36), we rated as IHI 19% (N=42) of hippocampi in the left hemisphere and 6% (N=14) of hippocampi in the right hemisphere; a 4% of brains were identified as IHI bilaterally. In the HC2 group (N=18, age 25 to 64), we rated as IHI 22% (N=4) of hippocampi in the left hemisphere and 11% (N=2) of hippocampi in the right hemisphere; a 5% of brains were identified as IHI bilaterally. The rates of IHI prevalence in both HC1 and HC2 groups are similar to those previously reported in the healthy population in other studies [START_REF] Cury | Incomplete Hippocampal Inversion: A Comprehensive MRI Study of Over 2000 Subjects[END_REF][START_REF] Colenutt | Incomplete Hippocampal Inversion and Its Relationship to Hippocampal Subfield Volumes and Aging: Incomplete Hippocampal Inversion and Aging[END_REF]. No significant difference was evidenced between HC1 and HC2 for any independent criterion (C1, C2, C3, C5) neither for the global IHI score (Left IHI score: U=1830, corrected p>1; Right IHI score: U=1784.5, corrected p>1) (Figure 3A). For the subsequent analyses on IHI, we merged HC1 and HC2 in one healthy control group (HC, N=238). 2.6.3 Scanner-effect on volumes. We observed a difference between HC1 and HC2 groups for the right subiculum (U=1108, corrected p=0.032) and a strong tendency for the left fissure (U=1145, corrected p=0.0501) (Figure 5A). When repeating the same analyses but excluding IHI hippocampi for each hemisphere independently, no significant difference was observed (Right subiculum: U=1014, corrected p=0.353; Left fissure: U=706, corrected p=0.237). For the subsequent analyses on volumes, we merged HC1 and HC2 in one healthy control group, first without excluding IHI (HC, N=237), but after we repeated the analysis excluding IHI for each hemisphere independently (left: DTD=15, HC=191; right: DTD=18, HC=221).

Conclusion on the pre-analyses for sanity check.

After conducting these pre-analyses for a sanity check of our data, we safely merged HC1 and HC2 in one HC group including a broader age-range and images acquired with the two different scanners. Furthermore, due to the tendency for a scanner-effect observed for the subiculum and fissure, if any difference between HC and DTD groups would be found for these subfields, analyses would be replicated including only HC2 as control group, as it shares the same scanner than the DTD group.

Statistical Analyses

2.7.1 IHI scores. We conducted Mann-Whitney tests for independent samples between HC (N=238) and DTD (N=20) groups for all criteria independently and for the global IHI score. Hemispheres were tested independently. P-values were corrected for multiple comparisons using Bonferroni correction (x10 tests). 2.7.2 Volumes. We conducted Mann-Whitney tests comparing the whole HC (N=237) group with the DTD (N=20) group for the subfields of interest (CA1, CA2+3+4+DG, subiculum, hippocampal fissure, and tail), whole hippocampal head, whole hippocampal body and whole hippocampal volume. Each hemisphere was tested independently. P-values were corrected for multiple comparisons using Bonferroni correction (x16 tests).

In addition, we repeated the same analyses but excluding IHI hippocampi for each hemisphere independently. We conducted Mann-Whitney tests between HC (N=191) and DTD (N=15) for the left hemisphere after excluding left IHI participants. The same analyses were conducted for the right hemisphere after excluding right IHI hippocampi (HC=221, DTD=18). P-values were corrected for multiple comparisons using Bonferroni correction (x16 tests).

Finally, considering the tendency observed for a scanner effect in the left hippocampal fissure segmentation, we conducted Mann-Whitney tests to compare fissure volumes between both groups acquired with the same scanner, thus, between DTD (N=20) and HC2 (N=18). Each hemisphere was tested independently. P-values were Bonferroni corrected (x2 tests).

All statistical analyses on this study, for both IHI and volumes, were conducted on IBM SPSS Statistics 25. Violin plots (figures 3, 5 and 6), boxplots (figures 5 and 6) and dot plots (figure 6) were performed using ggplot2 [START_REF] Wickham | ggplot2: Elegant Graphics for Data Analysis[END_REF] in RStudio running on R version 4.2.1 (2022-06-23 ucrt).

Results

IHI occurrence in HC and DTD group

For the DTD database (N=20), we rated as IHI 25% (N=5) of hippocampi in the left hemisphere and 10% (N=2) of hippocampi in the right hemisphere; a 10% of brains were identified as bilateral IHI. The rates of IHI prevalence in the DTD group are similar to both control groups (HC1 and HC2) and to those previously reported in the healthy population in other studies [START_REF] Cury | Incomplete Hippocampal Inversion: A Comprehensive MRI Study of Over 2000 Subjects[END_REF][START_REF] Colenutt | Incomplete Hippocampal Inversion and Its Relationship to Hippocampal Subfield Volumes and Aging: Incomplete Hippocampal Inversion and Aging[END_REF].

When observing each criteria independently, a significant difference was evidenced between HC and DTD groups for C3 in both the right (U=1297.5, corrected p=0.003) and left hippocampus (U=1472, corrected p=0.027) (Figure 3B), meaning that the hippocampus is more medially positioned for DTD participants as compared to HC (Figure 4), with a more significant effect for the right hemisphere comparing to the left. No significant differences were evidenced for criteria 1, 2 or 5, neither for the total IHI score.

Volumetric measures in HC and DTD groups

When comparing volumes between the HC and the DTD groups, significant differences were observed only for the hippocampal fissure bilaterally (Left: U=771, corrected p=0.000009; Right: U=793, corrected p=0.00001), with higher fissure volumes for the DTD group (Figure 5B). No significant differences were evidenced for the other subfields of interest, neither for whole head, body or whole hippocampal volumes.

Due to the scanner effect previously observed for some specific subfields (right subiculum and left fissure), and taking into consideration that IHI may affect hippocampal automatic segmentation [START_REF] Kim | Automatic hippocampal segmentation in temporal lobe epilepsy: Impact of developmental abnormalities[END_REF], we repeated the analyses comparing HC and DTD groups but now excluding IHI hippocampi independently for each hemisphere. The difference for hippocampal fissure volume remained significant for both hemispheres (Left: U=467, corrected p=0.0002; Right: U=699, corrected p=0.00008).

Finally, when comparing fissure volumes between the two groups acquired with the same scanners (DTD vs HC2), we observed that the difference remained significant for the right hippocampal fissure (U=86, corrected p=0.01) but not for the left (U=125, corrected p=0.224) (Figure 6).

Discussion

We observed similar IHI prevalence in a group of individuals with DTD with respect to a control population. Neither differences in whole hippocampal or major hippocampal subfield (head, body, CA, subiculum, tail) volumes have been evidenced between groups. Thus, in agreement with previous studies, no major morphological nor volumetric differences are reported for DTD subjects. However, when assessing IHI criteria independently, we observed that the hippocampus in the DTD group is more medially positioned comparing to the control group, meaning a shorter distance of the medial part of the subiculum not covered by the dentate gyrus relatively to the more lateral part of CA/subiculum that is covered by the dentate gyrus [START_REF] Cury | Incomplete Hippocampal Inversion: A Comprehensive MRI Study of Over 2000 Subjects[END_REF]. Volumetric analyses allowed us to determine that this medial positioning is not associated to a smaller subiculum structure, which has the same volume between groups. Interestingly, this specific medial portion of the hippocampal formation (i.e. the subiculum), located between the hippocampus proper and the parahippocampal region, has been defined as the "heart of the extended hippocampal system" [START_REF] Aggleton | The subiculum: The heart of the extended hippocampal system[END_REF], as it is the main origin for the largest array of extrinsic hippocampal efferents, be they cortical or subcortical, in rodents [START_REF] Witter | Cortico-hippocampal communication by way of parallel parahippocampal-subicular pathways[END_REF] and primates [START_REF] Aggleton | Multiple anatomical systems embedded within the primate medial temporal lobe: Implications for hippocampal function[END_REF]. Thus, the subiculum is the principal source of many of the hippocampal projections to cortical targets that are critically implicated in multiple forms of spatial cognition and memory [START_REF] Morris | Ibotenate Lesions of Hippocampus and/or Subiculum: Dissociating Components of Allocentric Spatial Learning[END_REF]Naber et al., 2000;[START_REF] Sun | CA1-projecting subiculum neurons facilitate object-place learning[END_REF]. As well, it influences the hippocampus directly via inputs to the CA1, and indirectly via projections to the entorhinal cortex [START_REF] Commins | Physiological evidence for a possible projection from dorsal subiculum to hippocampal area CA1[END_REF][START_REF] Swanson | An autoradiographic study of the organization of the efferet connections of the hippocampal formation in the rat[END_REF].

The subiculum connections display a complex topography that has be divided in "proximal" and "distal" subiculum, distinguishing locations within the transverse plane of the subiculum with different functions [START_REF] Aggleton | The subiculum: The heart of the extended hippocampal system[END_REF][START_REF] Witter | Cortico-hippocampal communication by way of parallel parahippocampal-subicular pathways[END_REF]. This subdivision of the subiculum, assumes that the dentate gyrus lies at the core of the structure, and if the structure is unrolled, "proximal" refers to tissue closer to the centre of the hippocampus reaching the border of CA1, while "distal" refers to the further tissue that limits with the presubiculum. The proximal subiculum is preferentially linked with the perirhinal cortex, lateral entorhinal cortex in rats or rostral and medial entorhinal cortex in primates, prefrontal cortex, amygdala, and nucleus accumbens, and it is more involved in processing object-based information. Meanwhile, the distal subiculum (located more medially, right before the presubiculum) is reciprocally interconnected with the medial entorhinal cortex in rats, the caudal and lateral entorhinal cortex in primates, and parahippocampal cortices. This distal portion of the subiculum has been involved in processing spatial information, reflecting their parahippocampal cortex connections [START_REF] Aggleton | Multiple anatomical systems embedded within the primate medial temporal lobe: Implications for hippocampal function[END_REF][START_REF] Witter | Cortico-hippocampal communication by way of parallel parahippocampal-subicular pathways[END_REF].

In the same line, the CA1/proximal subiculum and the distal subiculum/presubiculum areas have been proposed to be 'hotspots' of anatomical connectivity for multiple cortical areas in both non-human [START_REF] Kravitz | A new neural framework for visuospatial processing[END_REF] and human [START_REF] Dalton | New insights into anatomical connectivity along the anterior-posterior axis of the human hippocampus using in vivo quantitative fibre tracking[END_REF] primates. A parieto-medial temporal pathway supporting spatial navigation has been indeed described as emerging from the caudal portion of the inferior parietal lobule and terminating in the parahippocampal cortex and hippocampus, through both direct and indirect projections [START_REF] Kravitz | A new neural framework for visuospatial processing[END_REF]. The direct projections arrive first to the CA1/prosubiculum, second to the pre/parasubiculum, and third, to the posterior parahippocampal areas. The indirect projections, instead, connect the same source to the same targets, relaying through the posterior cingulate and retrosplenial cortices [START_REF] Kravitz | A new neural framework for visuospatial processing[END_REF]. A recent study on structural connectivity in humans provided further evidence of the existence of discrete hubs of dense anatomical connectivity along the anteriorposterior axis of the hippocampus, one corresponding to the CA1/proximal subiculum, and another more medial corresponding to the distal subiculum/presubiculum [START_REF] Dalton | New insights into anatomical connectivity along the anterior-posterior axis of the human hippocampus using in vivo quantitative fibre tracking[END_REF].

Accumulating evidence from neuroimaging studies has specifically involved the medial hippocampal hub (i.e. distal subiculum/pre/parasubiculum), which is targeted by visuospatial processing areas from the medial parietal and occipital cortices, in scene-based cognitive processing [START_REF] Dalton | The pre/parasubiculum: A hippocampal hub for scene-based cognition[END_REF]. Functional MRI studies have evidenced that the subiculum, plays a unique role in visuospatial cognition, with the anteromedial subiculum involved in the discrimination of complex scenes from different viewpoints [START_REF] Hodgetts | Ultra-High-Field fMRI Reveals a Role for the Subiculum in Scene Perceptual Discrimination[END_REF], and the distal portion of the subiculum implicated in scene processing and scene construction [START_REF] Dalton | Differentiable Processing of Objects, Associations, and Scenes within the Hippocampus[END_REF], mainly due to its functional connections with the parahippocampal cortex [START_REF] Grande | Transversal functional connectivity and scene-specific processing in the human entorhinal-hippocampal circuitry[END_REF]. Interestingly, individuals with DTD suffer only from spatial orientation and spatial memory problems, but they do not suffer from general memory problems [START_REF] Iaria | Developmental topographical disorientation: A newly discovered cognitive disorder[END_REF], and the more medial part of the subiculum (or the distal subiculum/presubiculum), which has been rated as shorter in length (or more compress) in the DTD group comparing to the control group, has been specifically linked to visuospatial information processing. Therefore, our finding aligns with the description of different hubs of anatomical connectivity along the hippocampus associated with different cortical inputs/outputs, in agreement with studies that have emphasized the importance of considering the hippocampus as a heterogeneous structure [START_REF] Aggleton | The subiculum: The heart of the extended hippocampal system[END_REF][START_REF] Dalton | New insights into anatomical connectivity along the anterior-posterior axis of the human hippocampus using in vivo quantitative fibre tracking[END_REF][START_REF] Kravitz | A new neural framework for visuospatial processing[END_REF].

The cortico-hippocampal-parahippocampal communication has been proposed to be crucial for the construction and use of cognitive maps [START_REF] Epstein | The cognitive map in humans: Spatial navigation and beyond[END_REF]. By accessing to allocentric and egocentric streams of spatial information, the parahippocampal cortex enables the construction of global representations of spatial environments [START_REF] Weniger | The human parahippocampal cortex subserves egocentric spatial learning during navigation in a virtual maze[END_REF]. Increased activity exclusively within the parahippocampal cortex, and more pronounced in the right parahippocampus, was reported during learning in a virtual maze [START_REF] Weniger | The human parahippocampal cortex subserves egocentric spatial learning during navigation in a virtual maze[END_REF]. Furthermore, a lesion study found deficits in virtual maze learning only after right parahippocampal gyrus removal covering the perirhinal, entorhinal and posterior parahippocampal cortices [START_REF] Weniger | Posterior parahippocampal gyrus lesions in the human impair egocentric learning in a virtual environment[END_REF]. These studies stand a main role for the right parahippocampal region in spatial navigation and spatial memory, comparing to the left. Our result of a more medial positioning for the hippocampus in the DTD group, with stronger results for the right hippocampus comparing to the left, encourages volumetric and shape analyses on related medial temporal lobe structures, specifically on the subiculum and parahippocampal regions, in individuals reporting DTD. Thus, further research may investigate if the medial position of the hippocampus may be linked to atypical subiculum shape or atypical shape/volumes of parahippocampal structures. In addition, it would be of interest for further research to study if this specific morphological feature may be associated with the reduced functional connectivity between the right hippocampus and the prefrontal cortex already reported in DTD individuals [START_REF] Iaria | Developmental topographical disorientation and decreased hippocampal functional connectivity: Dtd and Hippocampal Functional Connectivity[END_REF].

In a second group of analyses, we observed higher hippocampal fissure volumes for the DTD comparing to the control group, mainly for the right hemisphere. The hippocampal fissure is a sulcus around which the hippocampal infolding occurs. It begins as a small indentation at the 10th week of foetal development, but between the 18th and the 21st week, it is obliterated almost completely by the fusion between CA and DG [START_REF] Humphrey | The development of the human hippocampal fissure[END_REF]. The FreeSurfer method that we used to segment hippocampal subfields allowed us to obtain hippocampal fissure volumes, including the residual hippocampal cavities, which are cysts resulting from lack of hippocampal fissure obliteration that are most commonly localized laterally, at the apex of the hippocampal fold, between the CA and the DG. Hippocampal cavities mainly reflect cerebrospinal fluid collection [START_REF] Yoneoka | Criteria for normalcy of cavities observed within the adult hippocampus: High-resolution magnetic resonance imaging study on a 3.0-T system[END_REF] and, although their presence is common on MRI scans, their clinical significance is unclear. An increased prevalence of hippocampal cavities has been reported in patients with memory disorders, and their prevalence was reported to increase with age in healthy subjects [START_REF] Maller | Hippocampal sulcal cavities: Prevalence, risk factors and relationship to memory impairment[END_REF]. On the contrary, other studies have reported a lack of association between the presence of hippocampal cavities and age, sex, or general cognitive functioning (i.e. declarative memory, executive functioning, working memory, verbal fluency, attention) [START_REF] Blom | Hippocampal sulcal cavities: Prevalence, risk factors and association with cognitive performance. The SMART-Medea study and PREDICT-MR study[END_REF][START_REF] Uttner | Hippocampal cavities are not associated with cognitive impairment in transient global amnesia[END_REF]. One study has reported an enlargement of the hippocampal sulcus associated with medial temporal lobe atrophy in Alzheimer disease, but not enlargement of the hippocampal cavities [START_REF] Bastos-Leite | Analysis of shape and positioning of the hippocampal formation: An MRI study in patients with partial epilepsy and healthy controls[END_REF].

Interestingly, the criteria 3 used in our study for IHI rating, considers not only the medial positioning of the hippocampus, but also if there is presence of cerebrospinal fluid in the temporal horn of the lateral ventricle or not, with higher scores for the first case [START_REF] Cury | Incomplete Hippocampal Inversion: A Comprehensive MRI Study of Over 2000 Subjects[END_REF]. Thus, the DTD group presented a more medial positioning of the hippocampus but also the presence of cerebrospinal fluid in the temporal horn laterally to the hippocampus. Therefore, increased cerebrospinal fluid was observed in the DTD group on larger hippocampal sulcus and cavities corresponding to the hippocampal fissure, but also inside the temporal horn laterally to the hippocampus.

A limitation of our study is the use of different scanners in which the images for the DTD and the big majority of the control group were acquired, which may have an impact on the automatic segmentation output. To address this issue and reduce the possibility of observing a confounding effect, we first applied the same recon-all processing to all images, and secondly, we repeated our analyses reducing factors that may introduce a segmentation bias by excluding IHI hippocampi. Finally, we included in our study a control group in which the images were acquired with the same scanner used in the DTD group. Although we have applied all these measures to control image processing quality, we acknowledge that the automatic segmentation of the hippocampal fissure remains particularly vulnerable to signal loss and distortion due to its small size, especially in lower contrast images, leading to lower reliability [START_REF] Brown | Test-retest reliability of FreeSurfer automated hippocampal subfield segmentation within and across scanners[END_REF]. As such, the findings reported in our study regarding the difference in hippocampal fissure volume between the DTD and the control group should be taken cautiously and encourage further research on this specific subfield through a manual segmentation approach and possibly with ultra-high definition images.

A second aspect that we would like to discuss, regards the definition of the most medial edge of the subiculum for the assessment of criteria 3. In the rating used in our study, we sat the medial edge of the medial segment at the level of the superior bend of the subiculum, and not necessarily where the cortical tissue of the temporal lobe reaches the absolute most medial extreme, which in some specific morphological variations may correspond to a lower portion of the presubiculum. Given the degree of individual variability in morphology of the medial edge of the hippocampus, we considered important to specify the criteria applied in our rating, thus, setting the two segments always orthogonally to the brain midline at the level of the superior bend of the subiculum, even in particular cases of individual variability in subfield location/orientation. Finally, we understand that the small number of participants in the DTD group may be considered as a limitation. To address the small size of both the DTD and the HC2 groups, we opted to increase the power in our Mann Whitney analyses by including a bigger control group (merging HC1 and HC2) as representative of the general population. The broad age-range of the DTD group was taken into consideration by including also a broad age-range in the control group, after having excluded the possibility of an age-effect with the pre-analysis specifically conducted on our data.

Overall, the findings described in this study provide with the very first evidence of morphological markers in the brain of individuals affected by DTD. This is of significant impact given that, to date, there is no evidence in the literature of any neurological structural property of the brain that may confirm the presence of DTD as described by individuals experiencing this lifelong condition. A neurological marker that would confirm the presence of DTD not only helps individuals in the identification of the condition but will improve significantly the enrolment of DTDs across studies investing any aspect of this condition. As such, we are expecting that our findings will serve as stepping stones for future studies investigating the mechanisms of DTDs and their potential treatments.

In conclusion, our results provide new insights regarding the hippocampal morphology of individuals affected by DTD, whose hippocampus appear more medially positioned and present enlarged hippocampal fissure volume, thus increased intra-hippocampal cerebrospinal fluid, comparing to a control population. In line with the developmental (i.e. not acquired) nature of the spatial disorientation deficit, the present data underly the role of hippocampal structural anomalies during early prenatal development.

Figure 1 .

 1 Figure 1. Example of subfields segmentation at the level of the hippocampal body. In this brain, the left hippocampus was rated as IHI.

Figure 2 .

 2 Figure 2. Correlation plots between age and: A) left and right global IHI scores (Left: rho=-13.4, p=0.423; Right: rho=0.059, p=0.725); B) left and right whole hippocampal volumes (Left: rho=0.095, p=0.569; Right: rho=0.055, p=0.742), including participants from the HC2 and DTD groups.

Figure 3 .

 3 Figure 3. Violin plots depicting the differences in score for each IHI independent criterion between: A) HC1 and HC2, and B) HC and DTD groups. The black line represents the quantile 0.5, while the black point corresponds to the mean. *p<0.05, **p<0.01, Bonferroni corrected.

Figure 4 .

 4 Figure 4. Example of three right hippocampi for criteria 3, all of them after linear MNI transformation. A) Image from the HC1 database with a score of 0 for criteria 3 on the right hemisphere; B) image from the HC2 database with a score of 0 for criteria 3; C) image from the DTD database with a score of 1.5 for criteria 3, as the hippocampus is more medially positioned and the temporal horn of the lateral ventricle is filled with cerebrospinal fluid. Orange arrows indicate the part of the subiculum that is not covered by the dentate gyrus. Blue arrows indicate the part of CA/subiculum that is covered by the dentate gyrus. The red arrow indicates presence of cerebrospinal fluid in the temporal horn.

Figure 5 .

 5 Figure 5. Violin plots overlapped with the corresponding boxplots comparing hippocampal subfield volumes between: A) HC1 and HC2, and B) HC and DTD groups. BV=brain volume. * p≤0.05, *** p<0.001, Bonferroni corrected.

Table 1 .

 1 Descriptive information of all the groups.

	Group N	Age (years)	Behavioural assessment	MRI scanner
	DTD	20 females	(range=25 to 64)	No psychiatric or neurological	3T GE Discovery 750
			mean=47.15±11.86	disorders, assessed as DTD.
	HC1	220 females N=213 (range=26-30)	Human Connectome Dataset WU-	Siemens 3T
			N=7 (range > 36)	Minn. No major neurological	"Connectome Skyra"
				diseases, psychiatric or medical
				disorders.
	HC2	18 females	(range=25 to 64)	No psychiatric or neurological	3T GE Discovery 750
			mean=39.61±13.74	disorders, assessed as no-DTD.

Table 2 .

 2 Mean kappa scores and confidence intervals across all criteria for intra-and inter-rater agreement evaluation.

	AF -AF	AF -CC	AF -FS

Table 3 .

 3 Correlation scores between age and volumes, and between age and IHI global scores.

	Age	Volumes	Spearman's rho	Uncorrected p-value
	Left	CA1	0.028	0.870
		CA2/3/4/DG	0.121	0.468
		Subiculum	0.066	0.695
		Fissure	0.060	0.721
		Tail	0.178	0.286
		Hippocampal body	0.284	0.084
		Hippocampal head	-0.152	0.361
		Whole hippocampus	0.095	0.569
	Right	CA1	-0.073	0.664
		CA2/3/4/DG	-0.136	0.417
		Subiculum	0.185	0.265
		Fissure	0.203	0.221
		Tail	0.156	0.350
		Hippocampal body	0.223	0.179
		Hippocampal head	-0.165	0.323
		Whole hippocampus	0.055	0.742
		Incomplete Hippocampal Inversion (IHI)	Spearman's rho	Uncorrected p-value
	Left	Global IHI score	-0.134	0.423
	Right	Global IHI score	0.059	0.725

Data corresponding to the first control group are available in open access by the Human Connectome Dataset WU


Maryse Charpentier-Noyer

Pierre Nicolle

Olivier Payrastre

Hugo Marchal

François Bouttier

Eric Gaume

Relevance of using ensemble forecasts of flash-flood impacts for an emergency service: an evaluation for the October 2018 flood event in the Aude river basin, France

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

Relevance of using ensemble forecasts of flash-flood impacts for an emergency service: an evaluation for the October 2018 flood event in the Aude river basin, France

Maryse Charpentier-Noyer 1,2 , Pierre Nicolle 1 , Olivier Payrastre 1 , Hugo Marchal 3 , François Bouttier 3 , Eric Gaume 1 (maryse.charpentier@developpement-durable.fr)

1 Univ. Gustave Eiffel, GERS-LEE , 2 DREAL Centre , 3 CNRM, Météo-France

Results

Context

  Comparative evaluation of 3 rainfall ensemble forecasting products: Arome-PE, pepi & pertDpepi (experimental products) Evaluation based on effectiveness of SDIS emergency service during the october 2018 Aude flood (Agent-based model) Hydrometeorological flash flood impact chain to forecast number of inundated important proportion of flood damages and fatalities Limited predictability, fast dynamic high specific discharge, high spatio-temporal variability  low anticipation capacity for flood emergency management Methodology Number of inundated and rescued buildings Number of mobilized rescue teams Cost: nb of buildings non rescued in time Number of inundated and rescued buildings Number of mobilized rescue teams Cost: nb of buildings non rescued in time Interest of ABM to evaluate ensemble forecasts for decision making and crisis management Capacity to represent complex dynamics of rescue operations in fast evolving and uncertain situation Better representation compared to conventional economic value approaches Anticipation using rainfall forecasts increases rescue operation efficiency but limited by intervention capacities, loss of time due to forecasts uncertainties (false alarms), and location of main impacts Agents: firemen rescue teams: available, travelling; arrived at site, in operation Decision criteria: based on expected level of flood impacts -intervention threshold: from 25 forecasted inundated buildings -intervention capacity: 100 buildings/team -reallocation of rescue teams if rescue operations achieved Cost function: damage level based on the delay between onset flooded building and rescue team arrival 14th to 16th October 2018 flood of the Aude river Heavy precipitation event: 300 mm locally (return period > 100 years), high spatio-temporal variability Large flood: 14 deaths, around 100 injuries and 130-180 million euros damages Charpentier-Noyer, M., Peredo, D., Fleury, A., Marchal, H., Bouttier, F., Gaume, E., Nicolle, P., Payrastre, O., and Ramos, M.-H.: A methodological framework for the evaluation of short-range flash-flood hydrometeorological forecasts at the event scale, Natural Hazards and Earth System Sciences, 23, 2001-2029, https://doi.org/10.5194/nhess-23-2001-2023, 2023. Charpentier-Noyer, M., Payrastre, O., Nicolle, P., Marchal, H., Bouttier, F., Gaume, E.: An evaluation of short-range flash flood ensemble forecasts based on the agent-based modeling Precipitation Obs (QPE): high-resolution radar Antilope Forecasts (QPF): short-range (0-6h) ensemble forecasts (AROME-EPS, pepi, pertDpepi) Fire brigade location, rescue team number, location of rescue operations Key take away Potential value of hydrometeorological forecasts : timely assistance and cost reducing Optimistic case Constraint on the number of assisted flooded buildings  Higher costs