domain: info math

text:
Geovani Rizk
email: geovani.rizk@dauphine.psl.eu

Igor Colin
email: igor.colin@huawei.com

Albert Thomas
email: albert.thomas@huawei.com

Rida Laraki
email: rida.laraki@lamsade.dauphine.fr

Yann Chevaleyre
email: yann.chevaleyre@lamsade.dauphine.fr

An ↵-No-Regret Algorithm For Graphical Bilinear Bandits

We propose the first regret-based approach to the Graphical Bilinear Bandits problem, where n agents in a graph play a stochastic bilinear bandit game with each of their neighbors. This setting reveals a combinatorial NP-hard problem that prevents the use of any existing regret-based algorithm in the (bi-)linear bandit literature. In this paper, we fill this gap and present the first regret-based algorithm for graphical bilinear bandits using the principle of optimism in the face of uncertainty. Theoretical analysis of this new method yields an upper bound of Õ( p T ) on the ↵-regret and evidences the impact of the graph structure on the rate of convergence. Finally, we show through various experiments the validity of our approach.

Introduction

In this paper, we are interested in solving centralized multi-agent problems that involve interactions between agents. For instance, consider the problem of the configuration of antennas in a wireless cellular network [START_REF] Siomina | Automated optimization of service coverage and base station antenna configuration in UMTS networks[END_REF] or the adjustment of turbine blades in a wind farm [START_REF] Van Dijk | Yaw-misalignment and its impact on wind turbine loads and wind farm power output[END_REF][START_REF] Bargiacchi | Learning to coordinate with coordination graphs in repeated single-stage multi-agent decision problems[END_REF]. Choosing a parameter for an antenna (respectively adjusting a turbine blade) has an impact on its signal quality (respectively its energy collection efficiency) but also on the one of its neighboring antennas due to signal interference (respectively its neighboring turbine blade due to wind turbulence). By considering each antenna or turbine blade as an agent, these problems can be modeled as a multi-agent multi-armed bandit problem [START_REF] Bargiacchi | Learning to coordinate with coordination graphs in repeated single-stage multi-agent decision problems[END_REF] with the knowledge of a coordination graph [START_REF] Guestrin | Coordinated reinforcement learning[END_REF] where each node represents an agent and each edge represents an interaction between two agents. As a natural extension of the well-studied linear bandit to this multi-agent setting we focus on the graphical bilinear bandit setting introduced in [START_REF] Rizk | Best arm identification in graphical bilinear bandits[END_REF]: at each round, a central entity chooses an action for each agent and observes a bilinear reward for each couple of neighbors in the graph that is function of the actions chosen by the two neighboring agents.

The objective of the learner can either be to maximize the cumulative reward, which requires a trade-off between exploration and exploitation, or to identify the best action without being concerned about suboptimal rewards obtained during the learning process, known as the pure exploration setting [Bubeck et al., 2009, Audibert and[START_REF] Audibert | Best arm identification in multi-armed bandits[END_REF]. For graphical bilinear bandits, while the latter has already been addressed in [START_REF] Rizk | Best arm identification in graphical bilinear bandits[END_REF], maximizing the cumulative reward (and thus minimizing the associated regret) remains an open question that we address in this paper.

Contribution. Given the NP-Hardness property of the problem and the impossibility of designing an algorithm that has sublinear regret with respect to the horizon T , we propose the first regret-based algorithm for the graphical bilinear bandit problem that gives a guarantee on the ↵-regret of Õ( p T ) where ↵ (greater than 1/2) is a problem-dependent parameter and Õ hides the logarithmic factors. The problem as well as the notion of ↵-regret are introduced in Section 2. We provide a first algorithm and establish theoretical guarantees in Section 3. In Section 4, we identify the problem-dependent parameters that one could use to improve both theoretical guarantees and empirical performances of the algorithm. Accordingly, we state a refined bound that significantly improves ↵. Finally, Section 5 gives an experimental overview of the various theoretical bounds established for the regret.

Related work. Centralized multi-agent bandit problems where the learner has to choose the actions of all the agents at each round implies to deal with parallelizing the learning process on the agents. In the context of linear rewards where all the agents share the same reward function, [START_REF] Chan | Parallelizing contextual linear bandits[END_REF] give a detailed analysis of the problem and show that a sublinear regret in T can be reached but with an additional cost specific to the parallelization. The main difference with their work is that they consider independent agents, whereas we assume interactions between the agents.

Choosing an action for each node of a graph to then observe a global reward over the graph is a combinatorial bandit problem [START_REF] Cesa-Bianchi | Combinatorial bandits[END_REF], the number of joint arms scaling exponentially with the number of nodes. This has been studied in the regret-based context [START_REF] Chen | Combinatorial multi-armed bandit: General framework and applications[END_REF][START_REF] Perrault | Statistical efficiency of thompson sampling for combinatorial semi-bandits[END_REF] where the notion of ↵-regret is introduced to cope with the NP-Hardness of the problem. Most papers assume the availability of an oracle to solve the inner combinatorial optimization problem at each round. In this paper, we do not make this assumption and we present an algorithm that takes into account the graph structure, which has not been considered in the combinatorial framework.

Bandit problems in graphs have been studied from several angles. For instance, on the one hand, in [START_REF] Valko | Spectral bandits for smooth graph functions[END_REF], [START_REF] Mannor | From bandits to experts: On the value of side-observations[END_REF], each node of the graph is an arm of the bandit problem, which, if pulled, gives information on its neighboring arms. On the other hand, in [START_REF] Cesa-Bianchi | A gang of bandits[END_REF] each node is an instance of a linear bandit and the neighboring nodes are assumed to have similar unknown regression coefficients. In this setting, the reward observed for each agent only depends on its chosen action which does not take into consideration agent interactions.

The graphical bilinear bandit setting has been first established in [START_REF] Rizk | Best arm identification in graphical bilinear bandits[END_REF] where they tackle the problem of identifying the 1/2-best joint arm (which has an associated global reward equal to at least 1/2 of the optimal one). They study the pure exploration setting while we tackle the regret-based one. To the best of our knowledge, our paper is the first one to provide regret-based algorithms for the graphical bilinear bandits setting.

Problem setting

Let G = (V, E) be the directed graph defined by V the finite set of nodes representing the agents and E the set of edges representing the agent interactions. We assume that if (i, j) 2 E then (j, i) 2 E.1 The Graphical Bilinear Bandit with a graph G consists in the following sequential problem [START_REF] Rizk | Best arm identification in graphical bilinear bandits[END_REF]: at each round t, each agent i 2 V chooses an arm x (i) t in a finite arm set X and receives for each of its neighbors j a bilinear reward

y (i,j) t = x (i)> t M ? x (j) t + ⌘ (i,j) t , (1)
where M ? 2 R d⇥d is an unknown matrix, and ⌘ (i,j) t a zero-mean -sub-Gaussian random variable. 2

For all agent i 2 V , we denote N i the set of its neighboring agents. Let n = Card (V ) denote the number of nodes, m = Card (E) the number of edges and K = Card (X ) the number of arms. We assume that for all x 2 X , kxk 2  L. Let kM ? k F be the Frobenius norm of M ? with kM ? k F  S. We assume that the expected reward of each edge is positive which gives for all (x, x 0 ) 2 X 2 , 0  x > M ? x 0  LS.

For each round t, we define the global reward y t = P (i,j)2E y (i,j) t as the sum of the local rewards obtained over the graph at round t, and ⇣ x

(1) ? , . . . , x

(n) ?

⌘ = arg max (x (1) ,...,x (n) ) P (i,j)2E x (i)> M ? x (j) as the optimal joint arm that maximizes the expected global reward. The corresponding global pseudo-regret over T rounds is then defined as follows:

R(T ) = T X t=1 2 4 X (i,j)2E x (i)> ? M ? x (j) ? X (i,j)2E x (i)> t M ? x (j) t 3 5 .
We know from [START_REF] Rizk | Best arm identification in graphical bilinear bandits[END_REF] that finding ⇣ x

(1) ? , . . . , x

(n) ?

⌘ is NP-Hard with respect to the number of agents n.

Proposition 2.1 [START_REF] Rizk | Best arm identification in graphical bilinear bandits[END_REF] Theorem 4.1). Consider a given matrix M ? 2 R d⇥d and a finite arm set X ⇢ R d . Unless P=NP, there is no polynomial time algorithm guaranteed to find the optimal solution of max (x (1) ,...,x (n) )2X n X (i,j)2E

x (i)> M ? x (j) .

Hence, the objective of designing an algorithm with a sublinear regret in T is not feasible in polynomial time with respect to n. However, some NP-hard problems are ↵-approximable (for some ↵ 2 (0, 1]), which means that there exists a polynomial-time algorithm guaranteed to produce solutions whose values are at least ↵ times the value of the optimal solution. For these problems, it makes sense to consider the ↵-pseudo-regret as in [START_REF] Chen | Combinatorial multi-armed bandit: General framework and applications[END_REF] which is defined for all ↵ 2 (0, 1] as follows

R ↵ (T ) , T X t=1 2 4 X (i,j)2E ↵x (i)> ? M ? x (j) ? X (i,j)2E x (i)> t M ? x (j) t 3 5 ,
and set the objective of designing an algorithm with a sublinear ↵-regret.

The linear form. The reward can either be viewed as a bilinear reward as shown in (1) or as a linear reward in higher dimension [START_REF] Jun | Bilinear bandits with low-rank structure[END_REF] with y

(i,j) t = D vec ⇣ x (i) t x (j)> t ⌘ , vec (M ? ) E + ⌘ (i,j) t
where for any matrix A 2 R d⇥d , vec (A) denotes the vector in R d 2 which is the concatenation of all the columns of A, and h•, •i refers to the dot product between two vectors.

To simplify the notation, let us refer to any x 2 X as a node-arm and define the arm set Z = {vec (xx 0> )|(x, x 0 ) 2 X } where any z 2 Z will be referred as an edge-arm. If the arm x (i) t 2 X represents the node-arm allocated to the node i 2 V at time t, for each edge (i, j) 2 E we will denote the associated edge-arm by z

(i,j) t := vec ⇣ x (i) t x (j)> t ⌘
2 Z and define ✓ ? = vec (M ? ) the vectorized 2 One may wish that the agent's reward i also takes into account the quality of its chosen arm alone (i.e., not compared to the one chosen by its neighbors). We discuss this case in the appendix and show that if we add a linear term as a function of x (i) t to the bilinear reward, the new reward can still be reduced to a simple bilinear reward with an unknown matrix M? 2 R (d+1)⇥(d+1) . version of the unknown matrix M ? with k✓ ? k 2  S. With those notations, the (now) linear reward can be rewritten as follows:

y (i,j) t = D z (i,j) t , ✓ ? E + ⌘ (i,j) t . (2)
When needed, for any node-arms (x, x 0 ) 2 X 2 we use the abbreviation z xx 0 , vec(xx 0> ). Finally for any vector x 2 R d and a symmetric positive-definite matrix A 2 R d⇥d , we define kxk A , p x > Ax.

3 Optimism in the face of uncertainty for Graphical Bilinear Bandit

Let us first notice that given the linear reward (2) that can be observed for each of the m edges (i, j) 2 E, the central entity is trying to solve m parallel and dependent linear bandit problems. The parallel aspect comes from the fact that at each round t the learner has to allocate all the m edge-arms of the graph (one for each edge (i, j) in E) before receiving the corresponding m linear rewards. The dependent aspect comes from the fact that the choice of most of the edge-arms are depending on other edge-arms. Indeed, two adjacency edges (i, j) and (j, k) share the node j, hence their allocated edge-arms must share the node-arm x (j) t . In this paper, we choose to design an algorithm based on the principle of optimism in the face of uncertainty [START_REF] Auer | Finite-time analysis of the multiarmed bandit problem[END_REF], and in the case of a linear reward [START_REF] Li | A contextual-bandit approach to personalized news article recommendation[END_REF], [START_REF] Abbasi-Yadkori | Improved algorithms for linear stochastic bandits[END_REF], we need to maintain an estimator of the true parameter ✓ ? . To do so, let us define for all rounds t 2 {1, . . . , T } the OLS estimator of ✓ ? as follows:

✓t = A 1 t b t , (3)
where,

A t = I d 2 + t X s=1 X (i,j)2E z (i,j) s z (i,j)> s ,
with > 0 a regularization parameter and

b t = t X s=1 X (i,j)2E z (i,j) s y (i,j) s .
We define also the confidence set

C t ( ) = ( ✓ : k✓ ✓t k A 1 t  s d 2 log ✓ 1 + tmL 2 / ◆ + p S ) ,
where with probability 1 , we have that ✓ ? 2 C t ( ) for all t 2 {1, . . . , T }, and 2 (0, 1].

Since the graphical bilinear bandit can be seen as a linear bandit in dimension d 2 and with K n possible arms, 3 the idea of our method is to overcome the NP-Hard optimization problem that one would have to solve at each round when directly applying the OFUL algorithm as defined in Abbasi-Yadkori et al. [2011]. Indeed, at each round, instead of looking for the best optimistic joint arm

(x (1) t , . . . , x (n) t ) = arg max (x (1) ,...,x (n) )2X n max ✓2Ct 1 ( ) * X (i,j)2E z x (i) x (j) , ✓ + , 3
It can be seen as a linear bandit problem because the goal of the central entity is to maximize at each round t the expected global reward

P (i,j)2E x (i)> t M?x (j) t = P (i,j)2E z (i,j)> t ✓? = D P (i,j)2E z (i,j) t , ✓? E Algorithm 1: Adaptation of OFUL algorithm Graphical Bilinear Bandit Input :graph G = (V, E), node-arm set X (V 1 , V 2 ) = Approx-MAX-CUT(G) for t = 1 to T do // Find the optimistic best couple of node-arms ⇣ x t , x 0 t , ✓t 1 ⌘ = arg max (x,x 0 ,✓)2X 2 ⇥Ct 1 hz xx 0 + z x 0 x , ✓i; // Allocate x t and x 0 t in the graph x (i) t = x t for all i in V 1 ; x (i) t = x 0
t for all i in V 2 ; Obtain for all (i, j) in E the rewards y

(i,j) t ; Compute ✓t as in (3) end return ✓t Algorithm 2: Approx-MAX-CUT Input :G = (V, E) Set V 1 = ;, V 2 = ; for i in V do n 1 = Card ({(i, j) 2 E | j 2 V 1 }); n 2 = Card ({(i, j) 2 E | j 2 V 2 }); if n 1 > n 2 then V 2 V 2 [ {i} else V 1 V 1 [ {i}; end return (V 1 , V 2 )
we consider the couple of node-arms (x ? , x 0 ? ) = arg max (x,x 0 )2X 2 hz xx 0 + z x 0 x , ✓ ? i maximizing the reward along one edge. 4 Given the optimal joint arm (x , ✓ ? i  hz x?x 0 ? + z x 0 ? x? , ✓ ? i .

Hence, an alternative objective is to construct as many edge-arms z x?x 0 ? and z x 0 ? x? as possible in the graph. Naturally, we do not have access to ✓ ? , so we use the principle of optimism in the face of uncertainty, which is to find the couple (x t , x 0 t ) such that

(x t , x 0 t ) = arg max (x,x 0 )2X 2 max ✓2Ct 1 ( ) hz xx 0 + z x 0 x , ✓i ,
and then allocate the node-arms to maximize the number of locally-optimal edge-arms z xtx 0 t and z x 0 t xt . The motivation of this alternative goal can be understood thanks to (4) where we know that the associated expected reward is better than the one obtained with the edge-arms created with the optimal joint arms. Of course, one can also easily understand that for a general type of graph it is not possible to allocate those edge-arms everywhere (i.e., to all the edges). For example take a complete graph of three nodes, allocating the edge-arms z xtx 0 t and z x 0 t xt is equivalent to allocate x t and x 0 t to the nodes. By doing so, two nodes will pull x t and the third one will pull x 0 t , which makes it inevitable to allocate sub-optimal and unwanted edge-arms of the form z xtxt .

The main concern with this method is to control how many unwanted edge-arms are drawn in each round (relative to the total number m of edges) in order to minimize their impact on the resulting regret. Assigning x t to a subset of nodes and x 0 t to the complementary is equivalent to cutting the graph into two pieces and creating two distinct sets of nodes

V 1 and V 2 such that V = V 1 [ V 2 and V 1 \ V 2 = ;
. Pulling the maximum amount of optimal edge-arms thus boils down to finding a cut passing through the maximum number of edges.

problem is known as the Max-Cut problem, which is also NP-Hard. However, the extensive attention this problem has received allows us to use one of the many approximation algorithms (see, e.g., Algorithm 2) which are guaranteed to yield a cut passing through at least a given fraction of the edges in the graph. 5Proposition 3.1. Given a graph G = (V, E), Algorithm 2 returns a couple

(V 1 , V 2 ) such that Card ({(i, j) 2 E | (i 2 V 1 ^j 2 V 2 ) _ (i 2 V 2 ^j 2 V 1 )}) m 2 .
Using Algorithm 2 we can thus divide the set of nodes V into two subsets V 1 and V 2 with the guarantee that at least m/2 edges connect the two subsets. Combining this algorithm with the principle of optimism in the face of uncertainty for linear bandits Abbasi-Yadkori et al.

[2011] we obtain Algorithm 1 for which we can state the following guarantees on the ↵-regret.

Theorem 3.2. Given the Graphical Bilinear Bandit problem defined in Section 2, let 0   1 be a problem-dependent parameter defined by

= min x2X hz xx , ✓ ? i 1 m P (i,j)2E D z (i,j) ?
, ✓ ? E 0 , and set ↵ = 1+ 2 , then the ↵-regret of Algorithm 1 satisfies

R ↵ (T )  Õ ⇣⇣ d 2 + S p ⌘ p T m max (2, (LS) 2 ) ⌘ + LSm ⇠ d 2 log 2 ✓ T mL 2 / ◆⇡ ,
where Õ hides the logarithmic factors.

One can notice that the first term of the regret-bound matches the one of a standard linear bandit that pulls sequentially T m edge-arms. The second term captures the cost of parallelizing m draws of edge-arms per round. Indeed, the intuition behind this term is that the couple (x t , x 0 t ) chosen at round t (and thus after having already pulled tm edge-arms and received tm rewards) is relevant to pull the (tm + 1)-th edge-arm but not necessarily the (m 1) edge-arms that follows (from the (tm + 2)-th to the (tm + m)-th one) since the reward associated to the (tm + 1)-th edge-arms could have led to change the central entity choice if it had been done sequentially. In [START_REF] Chan | Parallelizing contextual linear bandits[END_REF], they characterize this phenomenon and show that this potential change in choice occurs less and less often as we pull arms and get rewards, hence the dependence in O(log(T m))).

What is and what value can we expect? This parameter measures what minimum gain with respect to the optimal reward one could get by constructing the edge-arms of the form z xx . For example, if there exists x 0 2 X such that hz x0x0 , ✓ ? i = 0, then = 0 as well and we are in the worst case scenario where we can only have a guarantee on an ↵-regret with ↵ = 1/2. In practice, having = 0 is reached when given the couple (x ? , x 0 ? ) = arg max (x,x 0 )2X 2 hz xx 0 + z x 0 x , ✓ ? i, this x 0 is either x ? or x 0 ? . In other words, if the unwanted edge-arms associated to the couple (x ? , x 0 ? ) gives low rewards, then the guarantee on the regret will be badly impacted. Hence we can wonder how we can prevent this phenomenon. We answer this question in the next section by taking into account both the proportion of undesirable edge arms allocated and their potential rewards at the selection of the pair (x t , x 0 t ) in order to improve in practice but also theoretically the performance of the algorithm.

Improved Algorithm for Graphical Bilinear Bandits

In this section, we want to capitalize on Algorithm 1 and its 1+ 2 -no-regret property to optimize the allocation of the arms x t and x 0 t such that the unwanted and suboptimal arms z xtxt and z x 0 t x 0 t penalize as less as possible the reward. Indeed, in Algorithm 1, the of the couple (x t , x 0 t ) is driven by the optimistic reward of the edge-arms z xtx 0 t and z xtx 0 t but not on the one of the unwanted arms z xtxt and z x 0 t x 0 t . Here, the first improvement that we can provide is to include them in the selection of the couple (x t , x 0 t ) as follows

(x t , x 0 t ) = arg max (x,x 0 )2X 2 max ✓2Ct 1 ( ) hz xx 0 + z x 0 x + z xx + z x 0 x 0 , ✓i .
However, this selection suggests that the number of unwanted edge-arms z xtxt and z x 0 t x 0 t will be equal to the number of optimal edge-arm z xtx 0 t and z x 0 t xt (in the sense of optimism), which is in general not true. 6 Hence, one has to take into consideration the proportion of edges that will be allocated by suboptimal and unwanted edge-arms or by optimistically optimal edge-arms.

To do so, let us define m 1 (respectively m 2 ) the number of edges that goes from nodes in V 1 (respectively V 2 ) to nodes in V 1 as well (respectively V 2 ) and m 1!2 (respectively m 2!1 ) the number of edges that goes from nodes in V 1 (respectively V 2 ) to nodes in V 2 (respectively V 1 ). One can notice that by definition of the edge set E, we have m 1!2 = m 2!1 and that the total number of edges m = m 1!2 + m 2!1 + m 1 + m 2 . By choosing the couple (x t , x 0 t ) as follows

(x t , x 0 t ) = arg max (x,x 0 )2X 2 max ✓2Ct 1 ( ) hm 1!2 • z xx 0 + m 2!1 • z x 0 x + m 1 • z xx + m 2 • z x 0 x 0 , ✓i ,
we are optimizing the total optimistic reward that one would obtain when allocating only two arms (x, x 0 ) 2 X 2 in the graph. This strategy is described in Algorithm 3.

Algorithm 3: Improved OFUL for Graphical Bilinear Bandits

Input :graph G = (V, E), node-arm set X (V 1 , V 2 ) = Approx-MAX-CUT(G); m 1!2 = m 2!1 = Card ({(i, j) 2 E|i 2 V 1 ^j 2 V 2 }); m 1 = Card ({(i, j) 2 E|i 2 V 1 ^j 2 V 1 }); m 2 = Card ({(i, j) 2 E|i 2 V 2 ^j 2 V 2 }); for t = 1 to T do ⇣ x t , x 0 t , ✓t 1 ⌘ = arg max (x,x 0 ,✓)2X 2 ⇥Ct 1 hm 1!2 • z xx 0 + m 2!1 • z x 0 x + m 1 • z xx + m 2 • z x 0 x 0 , ✓i; x (i) t = x t for all i in V 1 ; x (i) t = x 0
t for all i in V 2 ; Obtain for all (i, j) in E the rewards y (i,j) t ; Compute ✓t as in (3) end return ✓t Before stating the guarantees on the ↵-regret of this improved algorithm, let us recall that we defined (x ? , x 0 ? ) = arg max (x,x 0 )2X hz xx 0 + z x 0 x , ✓ ? i and let us define the couple (x ? , x0 ? ) such that

(x ? , x0 ? ) = arg max (x,x 0 )2X hm 1!2 • z xx 0 + m 2!1 • z x 0 x + m 1 • z xx + m 2 • z x 0 x 0 , ✓ ? i .
We define 0 to be the expected reward difference of allocating (x ? , x0 ? ) instead of (x ? , x 0 ? ),

= hm 1!2 z x? x0 ? z x?x 0 ? + m 2!1 z x0 ? x? z x 0 ? x? + m 1 (z x? x? z x?x? ) + m 2 z x0 ? x0 ? z x 0 ? x 0 ? , ✓ ? i .
The new guarantees that we get on the ↵-regret of Algorithm 3 are stated in the following theorem. Theorem 4.1. Given the Graphical Bilinear Bandit problem defined as in Section 2, let be defined as in Theorem 3.2, let 0  ✏  1 2 be a problem dependent parameter that measures the gain of optimizing on the suboptimal and unwanted arms defined as:

✏ = P (i,j)2E hz (i,j) ?
, ✓ ? i

,

and set ↵ = 1 ⇥ m1+m2 m (1 ) ✏ ⇤ where ↵ 1/2 by construction, then the ↵-regret of Algorithm 3 satisfies R ↵ (T )  Õ ⇣⇣ d 2 + S p ⌘ p T m max (2, (LS) 2 ) ⌘ + LSm ⇠ d 2 log 2 ✓ T mL 2 / ◆⇡ ,
where Õ hides the logarithmic factors.

Here one can see that the improvement happens in the ↵ of the ↵-regret. Let us analyze more precisely this term. First of all, notice that in a complete graph (which is the worst case scenario in terms of graph type and guarantees of the cut), we get ↵ 1+ 2 + ✏ which shows that optimizing over the suboptimal arms already improves our bound by ✏. On the contrary, in the most favorable graphs, which are bipartite graphs (i.e., graphs where all the m edges goes from nodes in V 1 to nodes in V 2 or vice versa), we have m 2 + m 1 = 0 and ✏ = 0 which gives ↵ = 1 and makes the Algorithm 3 a no-regret algorithm. What may also be of interest is to understand how ↵ varies with respect to , ✏ and the quantity m 1 and m 2 for graphs that are between a complete graph and a bipartite graph. We investigate experimentally this dependency in Section 5.

Experiments -Influence of the problem parameters on the regret

In this section, we give some insights on the problem-dependent parameters and ✏ and the corresponding ↵. Let ↵ 1 and ↵ 2 be the ↵ stated respectively in Theorem 3.2 and Theorem 4.1. In the first experiment, we show the dependence of ↵ 1 and ↵ 2 on the graph type and the chosen approximation algorithm for the max-cut problem with respect to and ✏. We also highlight the differences between the two parameters ↵ 1 and ↵ 2 and the significant improvement in guarantees that one can obtain using Algorithm 3 depending on the type of the graph. The results are presented in Table 1.

One can notice that the complete graph seems to be the one that gives the worst guarantee on the ↵-regret with respect to ✏ and . Thus, we conducted a second experiment where we consider the worst case scenario in terms of the graph type -e.g., the complete graph-and where there is n = 10 agents. The second experiment studies the variation of ✏ and with respect to the unknown parameter matrix M ? . To design such experiments, we consider the arm-set X as the vectors (e 1 , . . . , e d ) of canonical base in R d , which implies by construction that the arm-set Z contains the vectors (e 1 , . . . , e d 2 ) of the canonical base in R d 2 . We generate the matrix M ? randomly in the following way: first, all elements of the matrix are drawn i.i.d. from a standard normal distribution, and then we take the absolute value of each of these elements to ensure that the matrix only contains positive numbers. The choice of the vectors of the canonical base as the arm-set allows us to modify the matrix M ? and to illustrate in a simple way the dependence on and ✏. Consider the best couple (i ?

, j ? ) = arg max (i,j)2{1,...,d} 2 hz eiej + z ej ei , ✓ ? i, we want to see how the reward of the suboptimal and z e i ? e i ? , the closer ⇣ is to 1 the higher the reward of the unwanted arms. The dimension d of the arm-set is 10 (which gives linear reward with unknown parameter ✓ ? of dimension 100).

The plotted curve represents the average value of the parameters over 100 different matrices M ? initiated randomly with positive values.

Figure 2: Fraction of the global reward obtained at each round by applying the Algorithm 1, Algorithm 3 and the Explore-Then-commit algorithm (here named GBB-BAI) using the exploration strategy in [START_REF] Rizk | Best arm identification in graphical bilinear bandits[END_REF]. We use a complete graph of 5 nodes, we run the experiment on 5 different matrices as in Figure 1 with ⇣ = 0 and run it 10 different times to plot the average fraction of the global reward edge-arms z e i ? e i ? and z e j ? e j ? impact the value of , ✏ and thus ↵. Notice that the reward associated to the edge-arm z e i ? e i ? (respectively z e j ? e j ? ) is M ?i ? i ? (respectively M ?j ? j ? ). Hence we define 0  ⇣ < 1 and set M ?i ? i ? = M ?j ? j ? = ⇣ ⇥ 1 2 (M ?i ? j ? + M ?j ? i ? ). We study the variation of , ✏, ↵ 1 and ↵ 2 with respect to ⇣. The results are presented in Figure 1. One can see that when the associated rewards of z e i ? e i ? and z e j ? e j ? are low (thus is low and ✏ high), Algorithm 3 gives a much better guarantees than Algorithm 1 since it will focus on other node-arms than e i ? and e j ? that will give a higher global reward. Moreover, even when the unwanted edge-arm gives a high reward, the guarantees on the regret of Algorithm 3 are still stronger because it takes into consideration the quantities m 1 and m 2 of the constructed suboptimal edge-arms.

Finally, we design a last experiment that compares in practice the performance of Algorithm 1 and Algorithm 3 with the Explore-Then-Commit algorithm by using the exploration strategy designed in [START_REF] Rizk | Best arm identification in graphical bilinear bandits[END_REF] during the exploration phase, and by allocating the nodes in V 1 and V 2 with the best estimated couple (x, x 0 ) = arg max (x,x 0 ) hz xx 0 + z x 0 x , ✓t i during the commit phase.

Conclusion & discussions

In this paper, we presented the first regret-based algorithm for the stochastic graphical bilinear bandit problem with guarantees on the ↵-regret. We showed that by exploiting the graph structure and the typology of the problem, one can both improve the performance in practice and have a better theoretical guarantee on its ↵-regret. Also, we showed experimentally that our algorithm achieves a better performance than Explore-Then-Exploit on our synthetic datasets. The method presented in this article can be extended in many ways. First, one can consider cutting the graph into 3 or more pieces, which is equivalent to approximating the problem of a Max-k-Cut [START_REF] Frieze | Improved approximation algorithms for max k-cut and max bisection[END_REF] with k 3. With the knowledge of such a partition of nodes V 1 , . . . , V k , one may want to look for a k-tuple of node-arms maximizing the optimistic allocated reward rather than a pair, therefore introducing an elegant tradeoff between the optimality of the solution and the computational complexity of the arms allocation. One could also study this problem in the adversarial setting, in particular adapting adversarial linear bandit algorithms to our case. Finally, our setting could be extended to the case where each agent has its own reward matrix.

Figure 1 :

 1 Figure1: Variation of ✏, , ↵ 1 and ↵ 2 with respect to the parameter ⇣. The closer ⇣ is to 0 the lower the reward of the unwanted arms z e i ? e i ? and z e i ? e i ? , the closer ⇣ is to 1 the higher the reward of the unwanted arms. The dimension d of the arm-set is 10 (which gives linear reward with unknown parameter ✓ ? of dimension 100). The plotted curve represents the average value of the parameters over 100 different matrices M ? initiated randomly with positive values.

1.

  For all authors... (a) the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes] (b) Did you describe the limitations of your work? [Yes] (c) Did you discuss any potential negative societal impacts of your work? [No] (d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] 2. If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] In the appendix section 3. If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No] The code will be released if the paper is accepted to the conference (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] All the details are explained again in the supplementary material (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] Also in the appendix 4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators? [No] (b) Did you mention the license of the assets? [No] (c) Did you include any new assets either in the supplemental material or as a URL? [No] (d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [No] (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] 5. If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable? [No] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [No] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [No]

Table

  Value of several parameters with respect to the type of graph. Experiments were performed on graphs of n = 100 nodes, and results for the random graph are averaged over 100 draws.

			Graph types
		Complete	Random	Circle	Star	Matching
	m1+m2 m	0.495	0.453	0.01	0	0
	↵1		0.5 + 0.5
	↵2	0.505 + 0.495 + ✏	0.547 + 0.453 + ✏	0.99 + 0.01 + ✏	1	1

36th Conference on Neural Information Processing Systems (NeurIPS 2022).

One could define an undirected graph instead. However, we consider that interactions between two neighbors are not necessarily symmetrical, with respect to the obtained rewards, so we choose to keep the directed graph to emphasize this asymmetry.

The reward of an edge here denotes the sum of the rewards of the two corresponding directed edges in E.

Most of the guarantees for the approximation of the Max-Cut problem are stated with respect to the optimal Max-Cut solution, which is not exactly the guarantee we are looking for: we need a guarantee as a proportion of the total number of edges. We thus have to be careful on the algorithm we choose.

It is for instance true in complete graphs with an even number of nodes n but it is already false when the number of nodes become odd.


Jonathan Curot
email: jonathan.curot@cnrs.fr

Luc Valton

Emmanuel J Barbeau

Is iEEG-based cognitive neuroscience research clinically relevant? Examination of three "neuromemes"

Much progress has been made in the field of cognitive neuroscience thanks to intracerebral EEG (iEEG) research, largely due to the possibility of directly recording brain activity with unsurpassed spatial and temporal precision while patients perform cognitive tasks. However, do these patients gain anything from the time and effort they devote to this endeavour? In this chapter, we focus on three neuromemes, the "eloquent cortex", "localisationism" and the "nociferous cortex" to provide possible answers to this question. We discuss the value of these neuromemes and show that clinical care of epilepsy and iEEG-based cognitive neuroscience are consubstantial in the sense that iEEG during epilepsy assessment provides an understanding of physiological processes of the healthy brain; but also, that cognitive iEEG research in epileptic patients has a direct impact on semiology and curative neurosurgery. Last, we highlight how recent cognitive iEEG research provides insights into interictal complaints and could improve identification of the epileptogenic zone.

Introduction

Concept neurons and the famous Jennifer Aniston cell [START_REF] Quiroga | Invariant visual representation by single neurons in the human brain[END_REF][START_REF] Bausch | Concept neurons in the human medial temporal lobe flexibly represent abstract relations between concepts[END_REF][START_REF] Reddy | Concept Cells through Associative Learning of High-Level Representations[END_REF], memories induced by direct electrical brain stimulation (EBS) as if a neurostimulator could replace a madeleine [START_REF] Bartolomei | The "Proust phenomenon": Odor-evoked autobiographical memories triggered by direct amygdala stimulation in human[END_REF]… Neuroscience based on intracerebral EEG (iEEG) led to fundamental scientific discoveries that are widely popularised. Some of them became mainstream and culturally shared. Compared to scalp EEG or functional MRI, iEEG is recorded with unsurpassed spatial and temporal precision. Such an approach has been used in a large number of cognitive studies that take advantage of iEEG to investigate the electrophysiological correlates of cognition in humans (for recent reviews, see [START_REF] Jacobs | Direct brain recordings fuel advances in cognitive electrophysiology[END_REF][START_REF] Mukamel | Human intracranial recordings and cognitive neuroscience[END_REF][START_REF] Parvizi | Promises and limitations of human intracranial electroencephalography[END_REF][START_REF] Jerbi | Decoding cognitive states and motor intentions from intracranial EEG: How promising is high-frequency brain activity for brain-machine interfaces?[END_REF]). Although there is no doubt that iEEG research has significantly contributed to cognitive neuroscience, it would only be fair to ask the opposite. Has iEEG-based cognitive neuroscience contributed to the clinical workup of epileptic patients and their welfare? After all, many patients have contributed time to cognitive iEEG studies with the sincere hope it would help other patients like them in the future.

In this chapter, we will focus on three aspects that are commonplace concepts used every day by clinicians, electrophysiologists and neurosurgeons with epileptic patients in their care: the "eloquent cortex" [START_REF] Fried | The Myth of Eloquent Cortex, or What is Non-Eloquent Cortex?[END_REF], "localisationism" [START_REF] Duffau | The death of localizationism: The concepts of functional connectome and neuroplasticity deciphered by awake mapping, and their implications for best care of brain-damaged patients[END_REF] and the "nociferous cortex" [START_REF] Kleen | The nociferous influence of interictal discharges on memory[END_REF]. We suggest that these concepts are in fact "neuromemes", exchangeable cognitive units that spread easily from one individual to another and which can be maintained throughout time [START_REF] Blackmore | The meme machine[END_REF][START_REF] Dawkins | Memes: the new replicators[END_REF]. Memes gather ideas, behaviours or styles, with a common goal to easily replicate within a group. Some of these memes directly concern the epilepsy pathology as analysed recently by Baxendale [START_REF] Baxendale | Epilepsy: Lessons for clinicians from popular memes on social media[END_REF]. Neuromyths have been the focus of other authors such as Devinsky [START_REF] Devinsky | The myth of silent cortex and the morbidity of epileptogenic tissue: Implications for temporal lobectomy[END_REF] who commented on three: "epilepsy is a static disorder with minimal morbidity and mortality; epileptogenic tissue impairs only the functions of the seizure focus; and the anterior temporal lobes contain areas of nonfunctional, ''silent'' cortex." Hence, the medical culture, like any other culture, is rich with easily replicated and sustainable concepts and theories, i.e., "neuromemes". Although the term "neuromyths" has infiltrated the neuroscience community [START_REF] Dekker | Neuromyths in education: Prevalence and predictors of misconceptions among teachers[END_REF][START_REF] Lilienfeld | Memory scrutinized through electrical brain stimulation: A review of 80 years of experiential phenomena[END_REF][18], "neuromemes" seems to be a more appropriate term in the context of this chapter because neuromemes are not systematically wrong or lack rationality.

However, one characteristic of a neuromeme is that it is loosely defined. It is this malleability that makes it so useful. It can be used in many situations and can be adapted to the needs, circumstances or even the epoch. However, this represents a paradox since medicine usually relies on the efficiency of precisely defined concepts. This suggests that there is a high risk of such neuromemes causing a bias or polluting clinical practice.

In this chapter, we will show that intracranial explorations have played an important role in the genesis of neuromemes. However, we will also show that as a result of the progress made in the field, neuromemes can now be examined and refined for the benefit of patients.

iEEG research: a cocoon for neuromemes

Paradoxically, while iEEG has been the historical nest for the creation or enrichment of some neuromemes, and neuromyths for that matter, it can also simultaneously play a significant role in confirming or invalidating them. First, we will review each of our selected neuromemes, which are closely related to Penfield's work either in terms of their origin or significant promotion.

The eloquent cortex

The concept of the "eloquent cortex" did not originate with Penfield, but is derived from his work. In fact, Drake was very likely the first to use the term "eloquent" after a series of lectures by Penfield in the sixties in a paper on the surgical treatment of arteriovenous malformation [START_REF] Fried | The Myth of Eloquent Cortex, or What is Non-Eloquent Cortex?[END_REF][START_REF] Kahn | Eloquent: History of a word's adoption into the neurosurgical lexicon[END_REF][START_REF] Drake | Cerebral arteriovenous malformations: considerations for and experience with surgical treatment in 166 cases[END_REF]. One literal definition of "eloquent" that can be found in the dictionary is "clearly expressing or indicating something". In the epilepsy literature, it is defined as the "cortex related reproducibly to a given function." [START_REF] Rosenow | Presurgical evaluation of epilepsy[END_REF].

"Cortical stimulation allows the most precise localization of eloquent cortex" [START_REF] Schevon | Pediatric language mapping: Sensitivity of neurostimulation and Wada testing in epilepsy surgery[END_REF] or "Fastripples near the resection and in distant pathologic areas could have changed the resection in 8 patients without harming functionally eloquent areas" [START_REF] Van't Klooster | Residual fast ripples in the intraoperative corticogram predict epilepsy surgery outcome[END_REF] are just two of the countless occurrences of the term "eloquent" (see [START_REF] Kahn | Eloquent: History of a word's adoption into the neurosurgical lexicon[END_REF]). In the current scientific literature, particularly in the context of preoperative functional mapping, the term "silent cortex" appeared in opposition to the parts of the cortex that were not "eloquent", which is consistent with previous ideas concerning brain equipotentiality [START_REF] Devinsky | The myth of silent cortex and the morbidity of epileptogenic tissue: Implications for temporal lobectomy[END_REF]. However, in 1993, Fried questioned the myth of the eloquent cortex and highlighted the danger of such terminology "which replaces the neurological reality it sets out to simplify" [START_REF] Fried | The Myth of Eloquent Cortex, or What is Non-Eloquent Cortex?[END_REF]. In 2005, Devinsky tried to dismiss the myth of the silent cortex based on the example of temporal lobe surgery: "Because the normal brain does not contain functionless, "silent" areas, the procedure can have negative as well as positive cognitive or behavioural consequences" [START_REF] Devinsky | The myth of silent cortex and the morbidity of epileptogenic tissue: Implications for temporal lobectomy[END_REF]. This is obviously an important issue for patients undergoing neurosurgery.

Localisationism

According to the localisationist theory, the brain is conceived as a mosaic of relatively independent functional regions, i.e., each part of the brain is associated with a particular function, in keeping with the seminal suggestions by Franz Gall in the early 19th century. This view appeared to be supported by later discoveries, for example, of Broca's area and the motor and sensitive homunculus proposed by Penfield. Let us take the example of Penfield's interpretative cortex [START_REF] Penfield | The interpretive cortex: The stream of consciousness in the human brain can be electrically reactivated[END_REF]. The experiential phenomena that Penfield reported were always produced by electrical stimulations applied to a large area of the temporal neocortex, which included the superior, lateral and inferomedial temporal lobes [START_REF] Penfield | The brain's record of auditory and visual experience: a final summary and discussion[END_REF]. He named this area the interpretative cortex since, contrary to the pericentral and visual regions that he also studied, only stimulation here led to flashbacks of past experiences or to interpretative illusions of the present situation [START_REF] Penfield | The brain's record of auditory and visual experience: a final summary and discussion[END_REF][START_REF] Penfield | Some Mechanisms of Consciousness Discovered During Electrical Stimulation of the Brain[END_REF]. His surgical exploration technique restricted his application of electrical stimulation to only the surface of the temporal cortex. Despite a few attempts at deep stimulation, he only exceptionally observed such responses by stimulations of the temporal uncus, and never by stimulations of the hippocampus. However, the subsequent history of intracranial EEG research clearly demonstrated that the interpretative cortex was not by any means the only region that induces experiential memory phenomena when stimulated and that the memory network extends far beyond [18,[START_REF] Bancaud | Anatomical origin of deja vu and vivid "memories" in human temporal lobe epilepsy[END_REF]. This implies that cognitive functions depend on networks of brain areas [START_REF] Barbeau | Recollection of vivid memories after perirhinal region stimulations: synchronization in the theta range of spatially distributed brain areas[END_REF], the complexity of which is probably only barely understood at present [START_REF] Sporns | Contributions and challenges for network models in cognitive neuroscience[END_REF] (see also Chapters 37 and 39).

While no clinician will deny that complex cognitive functions are subtended by distributed networks [START_REF] Varela | The brainweb: phase synchronization and large-scale integration[END_REF][START_REF] Bressler | Large-scale brain networks in cognition: emerging methods and principles[END_REF], daily clinical thinking remains contaminated by the localizationist approach, from which the idea of the "eloquent cortex" is derived [START_REF] Devinsky | The myth of silent cortex and the morbidity of epileptogenic tissue: Implications for temporal lobectomy[END_REF]. This impacts the quality of patient care because it oversimplifies neurosurgery planning and in particular its consequences.

The nociferous cortex

This is perhaps the least identifiable of the three neuromemes we selected. In 1954, Penfield and Jasper suggested the concept of a "nociferous cortex" to describe the notion that the epileptic focus, now more appropriately defined as the epileptogenic zone, may be deleterious for widespread areas beyond the epileptogenic zone and may disrupt normal processes [START_REF] Penfield | Epilepsy and the functional anatomy of the human brain[END_REF] (see also Chapter 5). The "nociferous cortex" in epilepsy refers to the fact that local brain damage can have remote consequences, a secular concept, which could date back to the time of Brown Sequard in the middle of the 19th century [START_REF] Fornito | The connectomics of brain disorders[END_REF][START_REF] Koehler | Brown-Séquard and cerebral localization as illustrated by his ideas on aphasia[END_REF]. It is, for example, closely related to the concept of diaschisis described by Von Monakow [START_REF] Luauté | Von Monakow's diaschisis. History and future of a discovery[END_REF][START_REF] Pearce | Von Monakow and diaschisis[END_REF]. The distributed effects of focal lesions on brain dynamics, such as diaschisis, compensation or transneuronal degeneration, are nowadays widely supported by the study of brain connectomics and are applicable to a wide range of diseases [START_REF] Sporns | Contributions and challenges for network models in cognitive neuroscience[END_REF][START_REF] Fornito | The connectomics of brain disorders[END_REF][START_REF] Sporns | The human connectome: A structural description of the human brain[END_REF].

In epilepsy, the "nociferous concept" was updated about 25 years ago thanks to cognitive psychology. Hermann and Seidenberg demonstrated executive disorders in temporal lobe epilepsy, whereas executive functions are thought to depend primarily on the prefrontal lobes [START_REF] Hermann | Executive System Dysfunction in Temporal Lobe Epilepsy: Effects of Nociferous Cortex versus Hippocampal Pathology[END_REF]. In 2014, Coan et al. revealed decreased grey matter in extratemporal regions in patients suffering from temporal lobe epilepsy [START_REF] Coan | Frequent seizures are associated with a network of gray matter atrophy in temporal lobe epilepsy with or without hippocampal sclerosis[END_REF]. The influence of such ideas is now growing in the field of epilepsy as a result of a series of recent intracranial EEG-based neuroscience studies that demonstrate the remote influence of epileptic activities on brain functions [START_REF] Ung | Interictal epileptiform activity outside the seizure onset zone impacts cognition[END_REF][START_REF] Henin | Spatiotemporal dynamics between interictal epileptiform discharges and ripples during associative memory processing[END_REF][START_REF] Gelinas | Interictal epileptiform discharges induce hippocampal-cortical coupling in temporal lobe epilepsy[END_REF][START_REF] Dahal | Interictal epileptiform discharges shape large-scale intercortical communication[END_REF]. Unlike the "localisationism" and "eloquent cortex" neuromemes, the meme of the nociferous cortex holds up only if networks are considered [START_REF] Gilliam | Connectionology" provides Further evidence for nociferous epileptic cortex[END_REF]. This may be the reason why it is the only one of the three selected neuromemes that appears to have some validity. It is likely that in the future, the strong interest in network neuroscience will contribute to improving the cognitive status of epileptic patients, particularly following neurosurgery.

Dispelling the myth of the eloquent vs the silent cortex

Many years after Drake (1963), Spetzler and Martin (1986) defined the eloquent cortex as "areas that speak to readily identifiable neurological function" and, "if injured, result in a disabling neurologic deficit" [START_REF] Kahn | Eloquent: History of a word's adoption into the neurosurgical lexicon[END_REF][START_REF] Spetzler | A proposed grading system for arteriovenous malformations[END_REF].

At first glance, the etymological choice of "eloquent" to name the primary cortices that induce obvious and easily reproducible symptoms after EBS appears sound. EBS can trigger oral and written language symptoms [START_REF] Bédos Ulvin | Intracerebral stimulation of left and right ventral temporal cortex during object naming[END_REF][START_REF] Ojemann | The neurobiology of language and verbal memory: Observations from awake neurosurgery[END_REF][START_REF] Salanova | Long-term efficacy and safety of thalamic stimulation for drug-resistant partial epilepsy[END_REF][START_REF] Young | A Review of Cortical and Subcortical Stimulation Mapping for Language[END_REF], motor [START_REF] Signorelli | Intraoperative electrical stimulation mapping as an aid for surgery of intracranial lesions involving motor areas in children[END_REF][START_REF] Fried | Functional Organization of Human Supplementary Studied by Electrical Stimulation Motor Cortex[END_REF][START_REF] Usui | Extraoperative functional mapping of motor areas in epileptic patients by high-frequency cortical stimulation[END_REF] or sensory manifestations [START_REF] Jonas | Right hemispheric dominance of visual phenomena evoked by intracerebral stimulation of the human visual cortex[END_REF], during either awake surgery [START_REF] Duffau | Awake surgery for nonlanguage mapping[END_REF][START_REF] Mandonnet | Direct electrical stimulation as an input gate into brain functional networks: principles, advantages and limitations[END_REF] or intracerebral explorations [START_REF] Selimbeyoglu | Electrical stimulation of the human brain: perceptual and behavioral phenomena reported in the old and new literature[END_REF][START_REF] Trébuchon | Electrical Stimulation for Seizure Induction and Functional Mapping in Stereoelectroencephalography[END_REF][START_REF] Jonas | Prédiction du devenir fonctionnel postopératoire en chirurgie de l'épilepsie grâce aux stimulations électriques corticales[END_REF].

Moreover, there seems to be "silent" areas that do not respond to electrical brain stimulation: for instance, the posterior cingulate cortex where EBS never induces memory or other symptoms, whereas stimulation of surrounding areas can easily trigger motor, vestibular and visual symptoms [18,[START_REF] Foster | Direct cortical stimulation of human posteromedial cortex[END_REF]. A similar phenomenon can be observed in the orbitofrontal cortex, a ghost area, almost absent from the EBS literature [START_REF] Selimbeyoglu | Electrical stimulation of the human brain: perceptual and behavioral phenomena reported in the old and new literature[END_REF].

Therefore, "silent" and "eloquent" do not appear to be inappropriate terminology.

Unfortunately, the term "eloquent" has since been extensively used and deviated from the original (literal) definition [START_REF] Kahn | Eloquent: History of a word's adoption into the neurosurgical lexicon[END_REF]. In neurosurgery, it is now a synonym of "which should be preserved" and the silent cortex is that which can be easily resected without visible or disabling deficit. In that respect, numerous articles and reviews have suggested that EBS is a good functional mapping method to determine the "eloquent cortex" before epilepsy surgery (e.g., [START_REF] Jonas | Language Mapping Using Stereo Electroencephalography: A Review and Expert Opinion[END_REF][START_REF] Nandakumar | Automated eloquent cortex localization in brain tumor patients using multi-task graph neural networks[END_REF]). The premise is that if resecting the eloquent cortex is avoided there is a reasonable likelihood that the patient will not suffer from cognitive deficits following neurosurgery. Despite the rise of functional imaging, EBS is still considered the gold standard for mapping "functional areas" and predicting functional deficits. However, there are several conceptual difficulties with this approach.

(1) How can the limits of eloquent and silent brain areas be defined? This question is particularly relevant since a series of studies demonstrated that EBS effects are dependent on the parameters [START_REF] Kahane | Manifestations électrocliniques induites par la stimulation électrique intracérébrale par "chocs" dans les épilepsies temporales[END_REF][START_REF] Roux | Variability of intraoperative electrostimulation parameters in conscious individuals: language cortex[END_REF][START_REF] Mohan | The effects of direct brain stimulation in humans depend on frequency, amplitude, and white-matter proximity[END_REF][START_REF] Zangaladze | The effectiveness of low-frequency stimulation for mapping cortical function[END_REF][START_REF] Paulk | Local and distant cortical responses to single pulse intracranial stimulation in the human brain are differentially modulated by specific stimulation parameters[END_REF], that EBS effects may not be reproducible despite using the same parameters [18], that optimal parameters are not known [START_REF] Marti | Extraoperative electrical stimulation mapping in epilepsy presurgical evaluation: a proposal and review of the literature[END_REF] and that the understanding of the physiological mechanisms underlying EBS is limited [START_REF] Borchers | Direct electrical stimulation of human cortex -the gold standard for mapping brain functions?[END_REF].

Should EBS be accompanied by afterdischarges [START_REF] Halgren | Recall deficits produced by afterdischarges in the human hippocampal formation and amygdala[END_REF], or should they remain under the threshold for induction [START_REF] Suthana | Memory Enhancement and Deep-Brain Stimulation of the Entorhinal Area[END_REF][START_REF] Coleshill | Material-Specific Recognition Memory Deficits Elicited by Unilateral Hippocampal Electrical Stimulation[END_REF]?

(2) The term "silent cortex" is still inappropriately used for areas where no obvious symptoms were elicited during functional mapping, while their resection may lead to deficits after surgery. For example, who is able to predict spatial and navigation deficits [START_REF] Kolarik | Close but no cigar: Spatial precision deficits following medial temporal lobe lesions provide novel insight into theoretical models of navigation and memory[END_REF] or de novo psychiatric symptoms [START_REF] Yrondi | Post-traumatic Stress Disorder With Flashbacks of an Old Childhood Memory Triggered by Right Temporal Lobe Epilepsy Surgery in Adulthood[END_REF][START_REF] Cleary | Predicting and preventing psychopathology following temporal lobe epilepsy surgery[END_REF] that occur after right anterior and medial temporal lobectomy guided by EBS? Do we know what type of postoperative cognitive decline to expect and how this can be adequately assessed? At present there are too few studies regarding memory, social cognition and psychiatric consequences of right anterior temporal lobe surgery to conclude that this is a silent cortical area, if such a notion is even possible. The situation is just as unclear for other regions such as the cingulate or the prefrontal cortex.

(3) What is the value of the "eloquent cortex" when the patient is no longer eloquent? Not all patients can report their symptoms, due to either ictal amnesia or aphasia, especially in temporal lobe epilepsy [START_REF] Unterberger | Epileptic aphasia -A critical appraisal[END_REF].

(4) Are "non-eloquent" and "silent" synonymous? Does "not responding to EBS" really mean constantly and reliably silent areas? Aren't "silent" areas just "shy" areas?

Patients are not always assisted to verbalise subjective experiences through specific appropriate questions, especially when the temporal lobes are explored. In case of neuropsychological disturbances, such experiences are only revealed if an appropriate cognitive task is performed at the time of stimulation [START_REF] Halgren | Recall deficits produced by afterdischarges in the human hippocampal formation and amygdala[END_REF][START_REF] Morris | Hippocampal electrical stimulation and localisation of long-term episodic memory[END_REF]. Coleshill et al. [START_REF] Coleshill | Material-Specific Recognition Memory Deficits Elicited by Unilateral Hippocampal Electrical Stimulation[END_REF] for example, were able to demonstrate material-specific recognition memory deficits in some patients by combining unilateral electrical stimulations in the hippocampus synchronised with the presentation of verbal or visual stimuli. To date, no such routine is standardised and EBS still cannot replace the Wada test. The aim of the Wada test is to assess the hemispheric lateralization of speech or memory during preoperative evaluation of refractory epilepsy in order to avoid a risk of sequela. It consists of an intracarotid injection of an anaesthetic drug (barbiturate or propofol), which transiently inhibits the ipsilateral cerebral hemisphere in order to assess the activity of the contralateral one [START_REF] Curot | Bilateral Wada test: Amobarbital or propofol?[END_REF].

This also implies that theoretically, different tasks must be carried out for a single stimulation of a given region in order to test the role of this region in different functions and to determine with certainty whether the errors observed in the tests are indeed related to the EBS (and not to the preoperative cognitive state of the patient). For instance, the procedures to evaluate language are not standardised and each centre uses its own battery of tests (image naming, reading, spontaneous speech, repetition, auditory comprehension, etc.). Usually, only one of these tasks is performed for the sake of time, which strongly limits the understanding of the links between the stimulated area and its precise function in language.

(5) Cognitive functions are supported by large-scale networks. All cognitive processes are due to the emergence of a distributed, specific, transient, and synchronous neural assembly characterised by the level of synchrony of its components [START_REF] Varela | The brainweb: phase synchronization and large-scale integration[END_REF]. There is dense interregional connectivity, which makes each brain region part of an extensive network [START_REF] Sporns | Contributions and challenges for network models in cognitive neuroscience[END_REF][START_REF] Sporns | The human connectome: A structural description of the human brain[END_REF]. These large-scale brain networks consist of nodes that share many of the same connections, sometimes reciprocally [START_REF] Bressler | Large-scale brain networks in cognition: emerging methods and principles[END_REF][START_REF] Mesulam | Defining Neurocognitive Networks in the BOLD New World of Computed Connectivity[END_REF]. There is now evidence that EBS does not only have a local effect at the EBS site but also long-range effects. Intracranial implantations are governed by hypotheses about the epileptogenic network [START_REF] Isnard | French guidelines on stereoelectroencephalography (SEEG)[END_REF][START_REF] Bartolomei | Defining epileptogenic networks : Contribution of SEEG and signal analysis[END_REF] and sample only a fraction of the brain. Functional and physiological processes are not considered in the surgical plan. For instance, language investigation would theoretically require bilateral implantation and should cover a large fronto-temporo-parietal surface [START_REF] Friederici | The language network[END_REF], which is generally not the case, for an accurate mapping of the patient's specific language network. Indeed, about 40% of patients show a decline in image naming after left temporal lobe neurosurgery [START_REF] Busch | Effect of invasive EEG monitoring on cognitive outcome after left temporal lobe epilepsy surgery[END_REF], whereas the preservation of regions essential for naming, for example by limiting the posterior ventral extent of the resection, could help preventing such outcome. EBS effects propagate far away from the electrode [START_REF] Histed | Direct Activation of Sparse, Distributed Populations of Cortical Neurons by Electrical Microstimulation[END_REF], with the capacity to inhibit distant cortical regions through afferents connections [START_REF] Logothetis | The effects of electrical microstimulation on cortical signal propagation[END_REF]. In addition, networks can only be activated by some of their hubs and not by all regions involved in the network [START_REF] Jacobs | Explaining How Brain Stimulation Can Evoke Memories[END_REF][START_REF] Kim | Network-based brain stimulation selectively impairs spatial retrieval[END_REF][START_REF] Kim | A network approach for modulating memory processes via direct and indirect brain stimulation: Toward a causal approach for the neural basis of memory[END_REF]. This is the case for memory. The rhinal cortex seems to be the "gatekeeper" of memory networks [START_REF] Fernandez | The rhinal cortex: "gatekeeper" of the declarative memory system[END_REF], the site where EBS most frequently induces memories. There are also "closed doors" to neuromodulation, such as the posterior cingulate region, where EBS never induces memory phenomenon [18], even though it is a region that is known to play a role in memory. Hubs are defined as "nodes occupying a central position in the overall organization of a network", having a key role in information integration, making them "points of vulnerability that are susceptible to disconnection and dysfunction in brain disorders" [START_REF] Van Den Heuvel | Network hubs in the human brain[END_REF]. However, hubs should not be conceived of as an updated and more timely formulation of the concept of "eloquent cortex". The properties of a network cannot be limited to its hubs and the use of this term should be restricted to the context of the definition above to avoid to go back to poorly defined terminology.

In conclusion, there are alternatives to EBS such as using ERPs [START_REF] Grunwald | Prediction of postoperative seizure control by hippocampal event-related potentials[END_REF][START_REF] Elger | Human temporal lobe potentials in verbal learning and memory processes[END_REF] or highfrequency activity induced by specific cognitive tasks [START_REF] Kambara | Presurgical language mapping using event-related high-gamma activity: The Detroit procedure[END_REF][START_REF] Jerbi | Task-related gamma-band dynamics from an intracerebral perspective: Review and implications for surface EEG and MEG[END_REF]. Some authors have suggested that these approaches could be a means of reducing the duration of the EBS procedure, which would be more efficient and at the same time more comfortable for the patient and save time for the clinician. They could also be a complement to EBS to preselect the sites to stimulate [START_REF] Cuisenier | Relationship between direct cortical stimulation and induced high-frequency activity for language mapping during SEEG recording[END_REF].

While EBS is probably one of the major sources of the misleading "eloquent" vs "silent" cortices concepts, an analysis of the limits of EBS demonstrates that the term "eloquent" should be avoided in clinical practice. New ways of thinking about the possible postoperative deficits can be imagined. iEEG-based cognitive neuroscience suggests that standard, multimodal, cognitive procedures should be performed that are tailored to each location and if possible standardised among centres, to better assess the predictive value of EBS in postoperative deficits. A collective effort to this end, guided by recent progress in the methods and concepts developed recently in the framework of iEEG-based cognitive neuroscience, would clearly benefit epileptic patients.

Eliminating the implicit dogma of "localisationism"

Although clinicians may still, often implicitly or unconsciously, use localisationism to interpret brain-behaviour relationships, it is undeniable that modern neuroscience has considerably modified clinical thinking by introducing the idea that it is probable that networks are the underpinning of the cognitive or behavioural symptoms that occur during seizures.

EBS in the areas of epileptogenic networks

One major contribution of iEEG-based cognitive science is the improvement in knowledge about seizure semiology, initiated by scalp EEG [START_REF] Gastaut | Epileptic seizures; clinical and electrographic features, diagnosis and treatment[END_REF]. This was achieved through analysis of the sequence of symptoms that occur at the onset of seizures, that coincided temporally with the iEEG recorded in the different brain areas that were sampled [START_REF] Bancaud | La stéréo-électroencéphalographie dans l'épilepsie : informations neurophysiopathologiques apportées par l'investigation fonctionnelle stéreotaxique[END_REF]. Semiology is one of the keys to clinical reasoning in epilepsy and to the nosology and syndromic classification of epilepsy [97]. Semiology is the basis for building hypotheses on the epileptogenic network. It significantly influences the strategy of electrode implantation in patients who will benefit from an invasive pre-surgical assessment. It may also drive tailored neuromodulation treatments in the future as has recently been suggested [START_REF] Ren | Subthalamic Nucleus Stimulation Modulates Motor Epileptic Activity in Humans[END_REF][START_REF] Filipescu | The effect of medial pulvinar stimulation on temporal lobe seizures[END_REF].

A part of the diagnostic arsenal in epileptology is video-EEG, especially during phase 2 (intracranial EEG recording), and it has simply revolutionised the understanding of seizure semiology. The primary contribution of intracranial EEG to the understanding of semiology is related to the possibility of simultaneously recording iEEG and analysing symptomatology on video during seizures and then analysing the networks underlying each symptom through the correlation of the symptoms to the distribution of the concomitant ictal discharge on EEG. The symptomatogenic zone ("the area of cortex that, when activated by an epileptiform discharge, produces the ictal symptom") should be dissociated from the epileptogenic zone [START_REF] Rosenow | Presurgical evaluation of epilepsy[END_REF] (see also Chapter 2). Clinical symptoms have also been studied beyond a simple visual analysis of the regions concerned by the propagation of the epileptic discharge. This is particularly true of cognitive symptoms, for which a panel of spectral or network analyses have been

EBS in "healthy areas"

The contribution of intracranial EEG goes beyond simply understanding the sequence of symptoms that occurs during seizures. In reality, the direct benefit for a patient who agrees to participate in a cognitive protocol during his or her own intracranial exploration is not obvious and most of the time is not measurable.

Nevertheless, the data acquired during each of these procedures improve clinicians' knowledge of the brain networks that underly normal cognition (see also Chapter 6), which incidentally influences their ability to draw anatomical-functional conclusions from the seizure semiology. The possibility to identify the "normal" networks that underly specific cognitive functions through the study of functional or effective coupling (i.e., using coherence or other measures) or cortico-cortical evoked potential (CCEP; [104,105]; see also Chapter 39) improves the overall knowledge of clinicians. Long-term collaboration between clinicians and cognitive neuroscience researchers, and other neuroscience methods such as functional MRI constantly, if only gradually, increase the quality of interpretation of the symptoms reported by patients or observed during seizures [106].

In sum, the age-old tension between segregation and integration in the field of neuroscience also pervades the field of epilepsy. However, results clearly show that the network paradigm is a more accurate description of how the brain works and is also more nuanced (see also Chapters 37 and 39). It opens the way to other critical questions for patients, such as how these networks reorganise in the context of the pathology or how brain plasticity can be preserved or promoted after neurosurgery. The network approach is fully embedded in the notion of the "nociferous cortex" as will be seen in the next section.

Updating and validating the "nociferous cortex" concept

The concept that a pathological area can impact remote healthy areas is not necessarily intuitive. However, it fits perfectly with the view that the brain relies on dynamic network activity. A series of recent studies contributed to renewed interest in this concept. For example, it was shown that interictal epileptic discharges can alter largescale networks beyond the epileptogenic zone [START_REF] Ung | Interictal epileptiform activity outside the seizure onset zone impacts cognition[END_REF]107], either by pathological coupling (e.g. spikes pathologically coupled to the spindles involved in memory consolidation [START_REF] Gelinas | Interictal epileptiform discharges induce hippocampal-cortical coupling in temporal lobe epilepsy[END_REF][START_REF] Dahal | Interictal epileptiform discharges shape large-scale intercortical communication[END_REF] or by remote inhibition of healthy tissue (e.g. fast-ripples, a local phenomenon, that inhibits distant neurons outside the epileptogenic zone [108]. With regards to the nociferous cortex, combining the study of interictal epileptic activities and cognitive tasks is clinically relevant in different ways.

An impact on antiseizure medicine

The nociferous concept can help clinicians to determine how to adjust an antiseizure medicine, not only in terms of seizure persistence but also in relation to residual interictal cognitive complaints. More than 50% of patients with epilepsy complain that their memory moderately or severely limits their daily life [109]. In a recent study, we showed that approximately 75% of a large population of patients with temporal lobe epilepsy with initial subjective memory complaints had objective memory impairment after 3 weeks, regardless of whether they had a brain lesion, the type of lesion or how they responded to antiseizure medicine [110]. Such results suggest that there are other factors that explain cognitive alterations in epileptic patients, the best candidate being interictal epileptic discharges (IED). Some iEEG studies have provided convincing evidence that IED can disrupt cognitive processes, particularly those involved in memory [START_REF] Ung | Interictal epileptiform activity outside the seizure onset zone impacts cognition[END_REF]111] (see Chapter 5). Beyond correlational results, it is suggested that IED disrupt memory processes because they occur at key moments (encoding and recall, [112], retrieval and maintenance, [107]), most notably because they directly disrupt the physiological interactions between neocortical (spindles) and hippocampal oscillations (ripples) necessary for memory consolidation. IEDs lead to pathological coupling with spindles [START_REF] Gelinas | Interictal epileptiform discharges induce hippocampal-cortical coupling in temporal lobe epilepsy[END_REF] or they interfere with hippocampal physiological ripples during encoding and recall [START_REF] Henin | Spatiotemporal dynamics between interictal epileptiform discharges and ripples during associative memory processing[END_REF]. Such studies are almost impossible to conduct with scalp EEG or functional MRI. This clearly reinforces the importance of treating certain epileptic patients who are apparently free of seizures but who have a persistent interictal cognitive complaint probably due to IED [113].

A fingerprint of the EZ/non-EZ?

Recent studies have led to a renewed interest in cognitive tasks as markers of the EZ. To be succinct, the question is whether examination of the iEEG recorded during cognitive tasks can inform clinicians about the EZ or the epileptogenic network. A few studies suggest that seizure onset regions are dysfunctional at baseline and are functionally disconnected from the other functional network hubs [114,115]. IED (interictal epileptic spikes, interictal high frequency oscillations) during cognitive processes [107,[116][117][118] or coherence analyses during cognitive paradigms [119], or even resting state analyses may reveal these fingerprints. IEDs that occur in the areas where a seizure starts do not impact cognitive scores, whereas IEDs recorded contralateral to the SOZ or bilaterally doubled the risk of errors on a short-term memory task where letters within a sequence had to be recognized a few seconds after encoding [107]. A network synchronized in the theta-band activated during episodic memory was not found in the electrodes that belong to the seizure-onset zone [119]. The seizure onset and non-seizure onset zones may also show different electrophysical patterns during interictal cognitive tasks [118]. Increased relative entropic differences between the SOZ and non-SOZ were observed during a verbal memory task while the number of IEDs decreased during the task only in the non-SOZ, suggesting that iEEG recordings coupled with cognitive tasks might be able to unmask interictal fingerprint patterns specific to the EZ [118]. Similarly, cognitive tasks may induce a differential "reactivity" of interictal epileptic activities within or outside the EZ: for instance, an oddball task induced a significant decrease relative to high frequency oscillation rates, in particular ripples (see also Chapter 33), within the epileptic, but not in the non-epileptic hippocampus [116,117]. Transient suppression of ripples in the seconds following the stimuli presentation was also only observed in the non-epileptic hippocampus. This approach, although promising, is still in its infancy and is dominated by verbal [START_REF] Ung | Interictal epileptiform activity outside the seizure onset zone impacts cognition[END_REF]118] or visual [120] memory tasks, attentional tasks [116, 117] or short-term memory tasks [107]. Long-term memory tasks, especially episodic, are rare [119]). It is currently mostly relevant for EZ located in the left, dominant hemisphere, whereas only a few results are available for the right hemisphere. What is most notable is that there are still discrepancies and incongruences concerning the influence of cognitive tasks on epileptic activities according to the brain area. Results even appear to differ significantly within the same study [116,117]. However, the study of IED during cognitive tasks could be of value to optimise presurgical assessment by distinguishing EZ and non-EZ structures, completing the information derived from the examination of spontaneous seizures.

Although the concept of the nociferous cortex is somewhat outdated and rarely used, it conveys the important, yet often overlooked idea that epileptic networks may impact healthy networks and brain areas. It may help to explain some of the cognitive deficits exhibited by patients beyond simply relating them to the EZ or lesions. It might also help to provide a more nuanced and complete pattern of the electrophysiological alterations in a given patient.

Conclusion

Epilepsy and cognition are intricately intertwined in several ways. While epilepsy is related to basic dysfunctions of neural assemblies, its expression is often in the form of cognitive, affective or behavioural symptoms. In addition, epileptic patients often suffer from cognitive deficits. Last but not least, an important goal in patient management is preventing an impact on cognition by antiseizure medicine or neurosurgery. The variety of types of epilepsy also makes this specific brain disease particularly abundant in terms of cognitive symptoms and profiles (see also Chapter 3), unlike other brain diseases which usually have more stereotyped cognitive profiles.

To return to the question raised in this chapter: has iEEG-based cognitive neuroscience contributed to the clinical workup of epileptic patients and their welfare?

We think this is undoubtedly so. It guides the localisation of the epileptogenic network through a thorough analysis of the symptoms and subsequently often helps to clarify the type of epilepsy. It obviously guides intracerebral electrode planning and neurosurgery.

It is also an important factor in improving the understanding of the cognitive deficits that patients experience. Directly and indirectly, iEEG studies have significantly contributed to improvements in these areas. Table 1 provides an overview of the contribution of iEEG cognitive research to epilepsy management.

Cognitive iEEG research helps to question some neuromemes that, although widely disseminated in the field of epilepsy, appear inadequate or limiting, for example, the "eloquent cortex" and "localisationism", especially since clinicians may be unaware of their influence. Given the highly detailed cognitive paradigms currently available, these loosely defined concepts should be refined. However, this can only be done through a collective effort of clinicians and researchers across many epilepsy centres, because the task is tremendous, requires agreement and guidelines, and probably the inclusion or large groups of patients to be able to update these concepts. In contrast, the "nociferous" neuromeme appears to fit more naturally into clinical practice as we have seen, with current trends of trying to assess the impact of the epilepsy network on the global functioning of the brain. Conversely, iEEG can reveal that what appears at first sight to be nociferous can also turn out to be functional and integrated with cognitive networks, as recently described for heterotopia with similar electrophysiological responses to the healthy cortex during an attentional task [121]. In any event, identifying neuromemes in epilepsy could be highly useful as many people in the field are unaware of them and therefore do not question the validity of some of the concepts they use every day.

In general, cognitive iEEG has helped to decrease the stigmatisation of patients.

For example, an often-overlooked importance of iEEG is that it is continuously recorded for several days or even weeks. Therefore, it can be correlated with the fleeting, highly

  proposed. For instance, coherence analyses (seeChapter 36) during ictal humming demonstrated the activation of a network involving the superior temporal and inferior frontal gyrus[100]. Faced with such results, any clinician who observes early ictal humming in the seizure semiology will pay specific attention to these brain areas when analysing an MRI in order to identify any structural abnormality or will want to target these two areas during the presurgical workup. Fear is another example. Fear networks identified by coherence analyses during seizures [101, 102] revealed the synchronisation of the amygdala, ventral medial PFC, cingulate and dorsomedial prefrontal cortex during fear [103]. Electrode implantations, as well as structural brain MRI to identify possible lesions, could therefore be driven by such results and target these regions in patients who are frightened while having a seizure. Such a paradigm becomes particularly important to understand the semiology of brain areas for which EBS does not or rarely induces symptoms. By improving knowledge of the semiology of seizures, iEEG-based cognitive neuroscience can certainly play a role in the electrode implantation strategy.

Table : Aspects of patients' epilepsy impacted by iEEG cognitive neuroscience. CCEP: cortico-cortical evoked potential; ERP: evoked related potential; EZ: epileptogenic zone; IED: interictal epileptic discharge.

 :

			• More relevant EZ	• Interictal epileptic activity
			identification, theoretically	reactivity (e.g., ripple, fast-
		Epileptogenic network mapping	better post-surgical outcome	ripple, IEDs) during cognitive tasks • Fingerprints during resting
	Aspect	Clinical benefits	Consequence for patients	Example of IEEG methods state or cognitive tasks
			• More accurate	• Verbatim analyses (e.g.,
			classification of syndromes • Complementary tool for	diversity of experiential • Correlation between IED
	Interictal cognitive complaint	More detailed understanding of symptoms, refinement of symptoms Impact of IED on cognition	• Improved epilepsy indication of antiseizure diagnostic medicine • Better knowledge of daily consequences of seizures • Less stigmatisation • Curing cognitive deficits (interictal cognitive complaints) by adjusting treatment, when patients	phenomena) rates and cognitive processes • Video analyses or performance • Behavioural scales (e.g., graduation of consciousness) • Anatomical and functional correlations (concomitant • Correlation between IED rate during sleep and performance • Interferences of IED with
			seem seizure free	video & iEEG) cognitive processes and
				physiological oscillations
			• Help to focus the MRI	• iEEG visual analyses
	Semiology		analysis on specific areas, • Better identification and help to identify lesions with information of the possible	• Network analyses during • Cognitive task proposed to symptoms patients during
			low visibility postoperative deficits	ECoG/SEEG/awake
			• Identification of targets for • Better identification and	craniotomy
	General knowledge about the brain	Identification of the networks that underly specific symptoms Brain activity in healthy areas during cognitive tasks	intracranial implantation, information of the more relevant sample of consequences of epileptic intracranial recordings activities on cognition • Distinction between • Questioning neuromemes symptomatogenic vs epileptogenic zone • Choice of tailored neuromodulation treatments (in the future) • Extrapolation to other brain diseases	• Therapeutic neuromodulation (e.g. memory modulation in Alzheimer's disease, etc.)
			• Valuation of the patient
			• Identification of functional / dysfunctional brain areas, not necessarily mapping the EZ • Information about the who becomes a participant in understanding his own pathology and who contributes to neuroscience research	• EBS that induces symptom • EBS during cognitive tasks, causal stimulation • High gamma activity activated by cognitive task
	Presurgical assessment	Functional mapping	consequences of the neurosurgery, • Prediction of deficit(s)	• ERP to specific cognitive tasks • CCEPs (functional
			• Clearer information to	tractography)
			patients
			• Guiding the boundaries of
			the surgical resection

subjective phenomena that sometimes occur during seizures but which are impossible to replicate using experiments in the lab. As such, iEEG is unsurpassed. For example, converging iEEG data demonstrate a pivotal involvement of the hippocampus and rhinal cortex in experiential memory phenomena such as "déjà-vu", which has helped to discard the interpretation of déjà-vu as an unconscious fantasy or being related to a past-life experience [122,123]. Only intracranial EEG can capture the semiological diversity of such subjective phenomena such as déjà-rêvé [124], or the degree of consciousness alterations [125], while improving our understanding of their neural correlates.

6.