Miklós Molnár
email: molnar@lirmm.fr

On the Protection of Multicast Trees in All Optical Networks Using the NEPC Strategy

Keywords: all optical network, multicasting, fault-tolerant routing, pcycle, optical constraint

Due to the huge volume of the carried data in all optical networks, the ability of these networks to cope with failures becomes crucial. One of the proposed mechanism is the protection of the routes (lightpaths and light trees) with the help of precomputed ans reserved backup routes permitting to rapidly change the routes in the case of failures. The protection with p-cycles is an efficient method, which was originally proposed to protect on-cycle and straddling links. Recently the cyclic protection was extended to protect nodes in lightpaths and in light trees. As light trees can contain some branching nodes where the incoming light is spitted to several outgoing sub trees, the protection of these kind of trees is more delicate. In opaque networks where the switches may be virtual sources, efficient cycle-based protection schemes can be computed since each node can be the source of newly inserted light branches on an arbitrary precomputed cycle. Contrarily, with the apparition of optical switches and all optical solutions in core networks, the protection scheme should correspond to stringent optical constraints. Namely, the branching nodes of a protection scheme must correspond to available splitters. That is, the protection of multicast trees is limited in all optical domains. Our paper aims with the formulation of the potential protection possibilities and give the conditions to apply NEPC based protection.

Introduction

On the one hand, due to they huge capacity and robustness, all optical networks are becoming increasingly important in core networks. On the other hand, in recent multimedia applications, the communications should not be interrupted for a long time by a failure of a link or a switch. A failure can implicate important packet losses. Since the duplication of communications using independent routes is expensive, shared cold protection mechanism is often proposed as solution. With route restoration, when a failure is detected, a new route to the destination is requested, computed and configured in a reactive manner [START_REF] Ramamurthy | Survivable WDM Mesh Networks, Part II-Restoration[END_REF]. This solution may correspond to a high recovery delay. In the case of precomputed protections, the routers and switches reroute the communications to the preplanned backup routes in case of a failure [START_REF] Ramamurthy | Survivable WDM Mesh Networks, Part I-Protection[END_REF]. So, this solution is faster. To successfully redirect the affected traffic, sufficient capacity should be allocated to precomputed backup routes. Protection can be dedicated to protect some privileged targets (links and/or nodes) and can correspond to a shared protection scheme, where a same backup possibility may protect against different failures. Often, a given backup resource may protect several primary target but only one target at a time. In [START_REF] Ramamurthy | Survivable WDM Mesh Networks[END_REF]the authors state that shared protection provides significant savings over dedicated protection but shared protection is more susceptible to multiple link failures than dedicated protection.

Often, to characterize the quality of a protection, two measures of a solution are considered: the recovery delay and the number of failures managed [START_REF] Duato | A Theory of Fault-Tolerant Routing in Wormhole Networks[END_REF]. Moreover, the efficiency of a protection scheme can be given by the ratio of the protected primary communication capacity and the needed backup capacity.

It is therefore critical to reduce the recovery delay as far as possible. A good candidate for this reduction is a protection scheme based on small cycles permitting fast local recoveries in the case of failures. Trivially, a single cycle can not protect against multiple failures but a set of cycles can.

Grover and Stamatelakis proposed in [START_REF] Grover | Cycle-Oriented Distributed Preconfiguration: Ring-like Speed with Mesh-like Capacity for Self-planning Network Restoration[END_REF] a protection methods based on pre-configured protection cycles (or p-cycles) for WDM networks. The p-cycles offer the advantages of both ring and mesh protection schemes. Namely, restoration time can be fast as in ring protection, and protection efficiency can be high as in mesh-protection schemes. p-cycles may protect not only on-cycle links but also straddling links of the cycle. Moreover, cycles can be applied to perform an efficient shared protection. To protect a traffic pattern, a set of p-cycles should be computed. Since the optimal p-cycle set design is NP-hard [START_REF] Grover | Cycle-Oriented Distributed Preconfiguration: Ring-like Speed with Mesh-like Capacity for Self-planning Network Restoration[END_REF], several heuristic design algorithms were proposed [START_REF] Grover | Advances in optical network design with p-cycles: Joint optimization and pre-selection of candidate p-cycles[END_REF] [START_REF] Doucette | Algorithmic Approaches for Efficient Enumeration of Candidate p-Cycles and Capacitated p-Cycle Network Design[END_REF].

Often the design of pre-configured protection cycle can be made at the same time that the primary route design and the optimization can concern both the primary and the backup route design. In these cases the optimizations are inseparable and we talk about joint optimization of the (protected) routing scheme. Consequently if the optimization of the protection scheme follows the primary route computation, the optimization is a non-joint optimization.

Link failures are more frequent, but generally node failures impact several communications traversing the failed node. A node failure can be equivalent to several link failures. Dedicated protection schemes which offer separated solutions for link and node failure recovery are expensive. Accordingly, solutions proposing a combined link and node failure recovery are more interesting. A good protection scheme offers solutions against both link and node failures. The extension of p-cycle based schemes for node failure protection is one of the ideas providing a combined protection. If an on-cycle node is failed, then the cycle may trivially be used to protect the communications traversing it. The p-cycle concept was extended to path segment protection against possible node failures in [START_REF] Shen | Extending the p-cycle concept to path segment protection for span and node failure recovery[END_REF] where a capacity optimization model was also developed. The authors found that using p-cycles only a small additional spare capacity is needed to achieve a good node-failure protection. Node-encircling p-cycles (NEPCs) may protect node failures as it is explained in a first model of the paper [START_REF] Doucette | Combined Node and Span Protection Strategies with Node-Encircling p-Cycles[END_REF]. A single set of NEPCs was proposed for combined node and span protection. NEPCs can be shared by several nodes, and each node may use as many different NEPCs as needed for a capacity-efficient usage of the backup capacities. A different, failure-independent path-protecting p-cycle scheme (FIPP) was proposed in [START_REF] Kodian | Failure-Independent Path-Protecting p-Cycles: Efficient and Simple Fully Preconnected Optical-Path Protection[END_REF], that extends the p-cycle concept into a path-oriented protection. In this manner, FIPP p-cycles become similar in capacity efficiency to shared-backup path protection (SBPP, cf. [START_REF] Kini | Shared backup label switched path restoration[END_REF]), supporting failure-independent end-node activation and control against either span or node failure with fully pre-connected protection paths.

Our study focuses on the applicability of the NEPC strategy to protect multicast light trees.

Protection of Multicast Trees

Multicast protection schemes are classified in [START_REF] Zhong | Optimized Designs of p-Cycles for Survivable Multicast Sessions in Optical WDM Networks[END_REF] into five major schemes: tree-based, ring-based, path-based and segment-based protections.

A trivial solution for the protection of a multicast tree (originally in ATM networks) has been proposed in [START_REF] Wu | Backup VP planning for multicast connections in ATM networks[END_REF] based on the extraction of all the paths from the root to each destination in the multicast tree and then protect each paths with a unicast protection scheme. This solution is easy to realize in optical networks but is very expensive since the shared protection of the common part of the paths is not resolved.

A protection scheme of a multicast session against any single link failure in an optical network has been proposed in [START_REF] Singhal | Provisioning of Survivable Multicast Sessions Against Single Link Failures in Optical WDM Mesh Networks[END_REF]. A path-pair (disjoint primary and backup paths) is computed to each destination by the optimal path-pair-based shared disjoint paths (OPP-SDP) algorithm. The paths can share edges with already-existing path pairs and also with other edges on the primary tree. An improved version called "cross-sharing" has been proposed in [START_REF] Singhal | Cross-sharing vs. self-sharing trees for protecting multicast sessions in mesh networks[END_REF] which improves the backup sharing. It allows to share available backup capacities of multiple multicast sessions. Portions of backup paths of a given multicast session not only self-shares with its primary tree but also cross-shares with the portions of backup tree of other sessions.

A different solution for tree protection is to complete the primary tree by links producing redundancies and cycles and protecting all links and nodes in the primary tree. In [START_REF] Fei | A "Dual-Tree" Scheme for Fault-Tolerant Multicast[END_REF], a dual-tree scheme for multicast fault-tolerant is proposed. In this protection technique, the primary tree leafs are interconnected without the use of any link or inner node of the primary tree. The obtained redundant scheme contains a spanning tree which can be used even if there are failures.

The computation of edge-disjoint spanning trees for primary and backup routes is also a natural idea [START_REF] Yener | Design and performance of convergence routing on multiple spanning trees[END_REF]. In this case, the backup spanning tree remains unaffected even if the primary tree is failed. The topological constraint for this protection scheme is very hard and necessitates the four-connectedness of the topological graph. Moreover, the edge-disjoint approach does not guarantee vertex-disjoint trees.

ITo facilitate the redundant multicast routing the authors in [START_REF] Mdard | Redundant trees for preplanned recovery in arbitrary vertex-redundant or edge-redundant graphs[END_REF] propose to design two directed arc-disjoint trees (a "multitree"). In this scheme, if any vertex or edge in the graph leaves, each destination vertex remains to be connected to the source by at least one of the directed trees. Despite the advantages and the simplicity of this proposition, there are some disadvantages: the backup capacity is quasi-equivalent with the primary capacity and the redirection procedure may be complicated and may go back to the source.

Often, to formulate protection propositions for optical multicasting, it is supposed that every switch is of full wavelength conversion capability and optical multicasting capability (cf. the condition in [START_REF] Zhang | p-cycle based tree protection of optical multicast traffic for combined link and node failure recovery in WDM mesh networks[END_REF]). These conditions are always true in opaque optical networks where every node may be a virtual source. It can choose the wavelength arbitrary to continue the multicast forwarding in every outgoing fiber. These conditions are not true in all optical networks. We discuss the specificities of all optical networks in Section 4.

As it is detailed in the next section, p-cycles can also be used with success for multicast tree protection.

Cycle Based Link/Node Protection of Optical Multicast Trees

The multicast tree protection by cycles is based on the fact that if a link or a node in the tree fails, the neighbor nodes of the failed element may inform the network control plan and this latter can initiate to switch to backup pre-configured cycles. The procedure must reconnect the disconnected parts of the failed trees by forming a new multicast trees (or an equivalent multicast route, cf. later). The most significant references on cycle based solutions are presented in the following. F. Zhang and W. D. Zhong proposed the application of p-cycles to dynamic provisioning for multicast sessions in WDM networks in [START_REF] Zhang | Applying p-Cycles in Dynamic Provisioning of Survivable Multicast Sessions in Optical WDM Networks[END_REF]. The design of p-cycle based protection for static survivable multicast sessions was discussed in [START_REF] Zhong | Optimized Designs of p-Cycles for Survivable Multicast Sessions in Optical WDM Networks[END_REF] where several joint and non-joint optimizations and heuristic algorithms are compared. In these works, only link failures were considered. In [START_REF] Zhong | Optimized Designs of p-Cycles for Survivable Multicast Sessions in Optical WDM Networks[END_REF] and [START_REF] Zhang | Optimizations of $p$-Cycle-Based Protection of Optical Multicast Sessions[END_REF] Integer Linear Programming (ILP) based methods and ILP based heuristics were formulated for optimal and quasi-optimal p-cycle protection schemes respectively and both in joint optimization and non-joint optimization cases.

The node encircling p-cycle technique may be applied with success even if the node is a branching node of a multicast light tree. The paper [START_REF] Zhang | Optimizations of $p$-Cycle-Based Protection of Optical Multicast Sessions[END_REF] proposed a p-cycle based solution for combined node and link failure recovery which can be applied to protect light trees. Further node and link failure protection schemes for multicast traffic protection using the p-cycle concept were proposed and analyzed in [START_REF] Zhang | Performance Evaluation of Optical Multicast Protection Approaches for Combined Node and Link Failure Recovery[END_REF].

An efficient heuristic algorithm for p-cycle based multicast tree protection has been proposed in [START_REF] Zhang | p-cycle based tree protection of optical multicast traffic for combined link and node failure recovery in WDM mesh networks[END_REF]. In this proposed version, p-cycles protect light trees on an end-to-end basis, instead of protecting each link and node. In the case of a link failure, trees which are arc-disjoint with the cycle, having the source and the destination nodes on the cycle can be protected. The solution permits the recovery against intermediate node failures. The authors also discuss the conditions how the affected trees can share the protection cycles. Since the end-to-end protection of large trees needs large p-cycles, and it may be difficult to find this kind of p-cycles, the paper propose a tree partition algorithm to solve the protection of sub trees instead of the whole tree.

In the presented p-cycle based protection schemes it is always assumed that every node is of full wavelength conversion capability and optical multicasting capability. Moreover the light trees are directed and the cycles should complete the trees in the case of a failure by directed paths. The most sensible and complicated case corresponds to the failure of a branching node of a light tree. The basic idea of the propositions based on node encircling can be resumed as follows (such a situation is shown in Figure 1). The cycle P encircles the node b which is a branching node in the directed tree T : there is an ancestor node of b in the tree which is also on the cycle (it is the node a) and all of the sub trees of b are rooted by an on-cycle node (the nodes c, d and e in our example). When a failure of b is occurred, a new connected light tree can be obtained using the recovery capacity of the cycle P as it is indicated in the figure (an arbitrary directed path on the cycle spanning the successor nodes of b is presented in the example). Notice that the network is an opaque network and the nodes c, d and e have multicasting capability and can be virtual sources of the corresponding sub trees. Originally, the node encircling p-cycle (NEPC) concepts was proposed in [START_REF] Doucette | Combined Node and Span Protection Strategies with Node-Encircling p-Cycles[END_REF]. The cycle presented in Figure 1 is considered as a simple NEPC. The authors state if the network graph is at least two-connected, it is always possible to draw at least one logically encircling non-simple cycle which can protect a node failure. Such a non-simple cycle P = (a, c, d, e, d, f, a) is presented in Figure 2. This kind of non-simple NEPC was proposed for node protection in light-paths, but the schema can be adapted for light trees as it is illustrated by the second part of the figure. This figure shows the use of a directed non-simple walk (a, f, d, e, d) belonging to the non-simple cycle to establish the protection against the failure of node b. Even if the same wavelength is used in the multicast tree and in the backup multicast route, this wavelength can cross a switch from different incoming ports to different outgoing ports, and the illustrated non-trivial multicast route can be established after the recovery. The finally obtained backup structure is not a light tree but a light hierarchy which has been proposed in [START_REF] Zhou | Light-Hierarchy: The Optimal Structure for Multicast Routing in WDM Mesh Networks[END_REF]. Trivially, in an opaque network a p-cycle may recover the failure of a branching node, even if the failed node is on the cycle. The condition of the recovery is that all sub trees of the failed node must contain a common node (successor of the protected branching node) on the cycle. Such a situation is illustrated by Figure 3 .. can contain internal nodes which are not destinations nor branching nodes. To simplify, we suppose that they are simple links. To obtain more large protected parts, a p-cycle can offer the protection against the failure of nodes and links of an encircled sub tree as it is indicated in Figure 4 (in this case, the cycle is not an NEPC encircling only one node). Notice that the encircled sub tree must be free of destinations. If there is a destination in the sub tree, the recovery based on the cycle and regarding the direction of the links in the sub tree can not assume the connection to this destination node. If the widest, the cycle based protection can be applied from the source node and can be concerned the destinations on the cycle as it is the case of the failure independent path-protecting (FIPP) p-cycle approaches applied for multicast trees [START_REF] Zhang | Performance Evaluation of Optical Multicast Protection Approaches for Combined Node and Link Failure Recovery[END_REF]. In all cases, the encircled sub tree can not contain any destination, only the on-cycle destinations and the destinations that are outside the cycle can simply be protected.

Obviously, more the protected part and the cycle are wide, more the reconfiguration is complicated and slow. Moreover, only the intermediate node failures can be recovered by the cycle and source and destination node failures cannot be restored by any protection algorithm. The source node protection necessitates the duplication of the source as it is explained in [START_REF] Zhang | Extending p-cycles to Source Failure Recovery for Optical Multicast Media Traffic[END_REF].

For the previously presented cycle-based protections, opaque networks with multicast capable nodes were supposed. In the following, we analyze

Protection of Branching Nodes of Light Trees under Optical Constraints

In transparent all optical networks, where the nodes are optical switches and the O/E/O conversion in these switches is not desirable, the multicast routing must satisfy additional optical constraints.

-All of the optical switches can not split the incoming light. To perform the multicast routing with a light tree, the branching nodes of the tree must coincide with switches having special light splitters. -In addition to the uniqueness of the wavelength in the fibers, the same wavelength should be used along the light-paths and light trees. The wavelength can only be changed if there is a wavelength converter in the traversed optical switch. Since our analyzed problem focuses on the sparse splitting constraints for the multicast routing, without loss of generality, we suppose that there are no converters in the network (in the recent study, the conflicts on the wavelength assignment are not analyzed).

Here, we focus on the first, strong constraint corresponding to the splitting capacity of the optical switches. In all optical networks, a node capable to split the incoming light is called as a Multicast Capable (MC) node, otherwise it is a Multicast Incapable (MI) node [START_REF] Zhang | Constrained Multicast Routing in WDM Networks with Sparse Light Splitting[END_REF]. One can suppose that any node (also an MI node) at least has the Tap and Continue ability to tap into the incoming signal for local usage and forward it to only one output [START_REF] Ali | Cost-effective implementation of multicasting in wavelengthrouted networks[END_REF]. So, any node can be an intermediate destination on a light tree but only the MC nodes can be branching nodes.

Several propositions have been formulated to compute light trees corresponding to these constraints (cf. Reroute-to-Source, Reroute-to-Any, Member-First and Member-Only algorithms in [START_REF] Zhang | Constrained Multicast Routing in WDM Networks with Sparse Light Splitting[END_REF]). Typically, if there is not a single light tree satisfying the constraints, a set of trees (a "lightforest") is computed corresponding to them.

As a result of the constraints, not only the primary light trees but also the backup routes obtained after the recovery must satisfy the mentioned conditions. That is, after the recovery, the eventual branching nodes of the new light tree must correspond to splitters (to simplify, at a first time, we suppose that the new optical route after the recovery also corresponds to a light tree).

Unfortunately, the splitters may be sparse in an all optical network. In some cases, depending on the network topology, on the availability of the splitters and on the location of the primary light trees, some of the latter can not be protected by the NEPC technique as it is illustrated by Figure 5. In the figures, the MI nodes are represented by squares and the MC nodes are circles. Since the nodes a, c and e are MI nodes, a new optical multicast route covering all of the sub trees of the node b and respecting the constraints on the splitters can not be built when the node b is failed.

Figure 6 presents two different repartitions of the multicast capable nodes in the same network and for the same light tree. In these new situations, the protection with the help of the cycle P against the failure of node b is possible due to the connected MC nodes. Our principal question is the following: When is it possible the protection of a branching node of a light tree using the capacity of an node encircling p-cycle? The following proposition gives a necessary and sufficient condition for this protection. Remember, there is no wavelength conversion in the network and the same wavelength should be used after the recovery.

Let G = (V, E) an undirected graph representing the all optical network topology with two type of nodes. Let T = (W, F ) be a light tree and b a branching node in the tree. Let P an NEPC encircling b in the topology graph. To simplify we note also by P the set of nodes on the cycle P . Let S = W ∪P \{b} the set of nodes in the intersection of the tree Notice that the p-cycle may contain several nodes which are not in S. These nodes are only relay nodes in the case of a failure recovery and so they are not relevant for us. Our analysis focuses on the nodes in S.

At first, we give the condition to protect a branching node with a simple NEPC against a failure. Proposition 1. A simple NEPC P can protect against the failure of the node b, iff it contains a path P P covering the node set S such that P P does not contains any internal node of type M I belonging to S.

Proof. Let us suppose that to protect the tree T against the failure of b, a sub-path P P of the cycle P is used after the recovery. In the new optical multicast route (in the new light tree), trivially, the extremities of the path P P have a degree 2 (and so these nodes can be MI or MC nodes) but the internal nodes belonging to S should be branching nodes having one or more sub trees which should be connected to the source).

The condition is necessary. Let us suppose that all paths in P and covering the node set S contain an MI type internal node. Without loss of generality, let P ′ be such a path and let v ∈ S the MI type internal node in this path. v separates the nodes of S into two non empty node sets and the path P ′ into two sub paths. To obtain the connectivity of the two sub paths and also the sub trees rooted in v, v should have a degree 3 or more after the recovery, but v is of MI.

The condition is sufficient. If there is a path P P such that each internal node of P P belonging to S is an MC node, then these nodes can be the new branching nodes of the backup multicast structure.

A non-simple NEPC can extend the protection against a branching node failure of a multicast tree. The difference between simple and nonsimple cycles is that non-simple cycles MAY contain several times a same node (a same node can have several occurrences in the cycle). So, the connected subsets of the non-simple cycle are not obligatory paths but may be walks returning several times at a same node. To satisfy the constraints on the availability of splitters, all the node occurrences in the walks should be considered. Figure 7 illustrates two cases, where nonsimple NEPC can be used for the protection. In the first example, the non-simple NEPC P = (a, c, d, e, d, f, a) can protect the branching node b (all of the neighbor nodes of b are on this non-simple cycle). In this cycle, two fibers between the nodes d and e are reserved. The nodes a and d are MI nodes. There is no simple path in the cycle satisfying the previous condition (cf. Proposition 1). The figure indicate how the walk (a,c,d,e,d) in this cycle can be used to create a backup light hierarchy to recover the failure. As it was shown in the simple case, the extremities of the walk (the nodes a and d) have a degree 2. The particularity of this protection is that the node d is present twice in the walk. Beyond the occurrence corresponding to an extremity of the applied walk, the other occurrence of this node is a simple relay node in the obtained light hierarchy. The degree 2 of each occurrence corresponds to the fact that it is an MI node. After the failure recovery, the optical switch d is traversed twice: at first from c to e and secondly from e to its single existing sub tree. Of course, as d is an MI node, it can have only one sub tree in the primary tree. Finally, the degree of each node occurrence corresponds to the type of the given node (MC or MI).

The second example in the figure proposes an other non-simple NEPC P = (a, c, d, f, d, e, d, a) using two fibers between the nodes d and e and also between d and f . To recover the failure of the node b in the indicated tree (b has four successors in the tree: c, d, e and f ), for instance, the walk (a,c,d,f,d,e,d) can be proposed. The walk returns three times to the node d and each occurrence of d in the resulted light hierarchy has a degree 2. These returns using the walk P P are also illustrated in Figure 8. The generalization of the results is summarized by the following. In other world: an MI type node can be repeated in the walk P P several times, if it also has an extremity occurrence in the walk. Proof. Let us suppose that to protect the tree T against the failure of b, a walk P P in the non-simple cycle P is applied. Remember that in a walk, a node may appear several times (several occurrences of the node may belong to the walk).

In the new optical multicast route (in the new obtained light hierarchy), trivially, the extremities of the walk P P have a degree 2. The internal nodes of P P belonging to S are either branching MC nodes having one or more sub trees or they are simple relay nodes.

Let s i nS be an internal relay node, and let T s its sub tree in the original multicast tree. Let us suppose that s has a relay node occurrence in a path P P . There is two cases:

1. This node s has an occurrence in the walk located at the extremity of the walk. This extremity node occurrence can connect the sub tree T s to the new multicast route. 2. There is no extremity node occurrence corresponding to s. In this case, the sub tree T s can not be connected to the new multicast route.

The condition is necessary (cf. 2). The condition is sufficient. If it is true, all of the sub trees belonging to the nodes in S can be connected to the new multicast route.

Trivially, as a walk has two extremities, only occurrences of these two nodes (if they are MI nodes) can be repeated in the walk used to recover the node failure.

A Particular Protection Scheme Using Non-Simple Cycles

Non-simple cycles extend the protection possibilities of branching nodes in all optical light trees. In the previously presented cycle-based protection schemes, the cycle (simple or not) is used to the failure recovery in the following manner. If a node failure occurs, then a sequence of some consecutive nodes in the cycle (i.e. a walk) is used to re-connect the sub trees of the failed optical multicast tree. Let P = (x 1 , x 2 , x 3 , . . . , x n , x 1 ) a cycle (represented by a circular list of nodes) with n node occurrences (if the cycle is a non-simple cycle, then several occurrences correspond to a same node of the topology graph). The walk used for the protection can be an arbitrary sequence of these node occurrences of the circular list satisfying the previously mentioned conditions. Eventual returns to some nodes in the cycle permit a different and more flexible (but more complicated) protection of the encircled branching node. In the following, we present a special case based on a particular non-simple cycle. This example also illustrates the protection capabilities of the non-simple cycles.

A particular protection scheme can be proposed when a non-simple NEPC is composed using two fibers (one in both directions) between the connected node pairs in the cycle. Figure 9 shows such an NEPC and its proposed utilization for the failure recovery. There are multiple MI nodes in the cycle, but MC and MI type nodes alternate which permit the protection using the presented NEPC in a particular manner.

In the following proposition, we suppose that the links belonging to a p-cycle can be used arbitrarily for the protection. That is, an arbitrary connected subset of the links (and not only a path or a walk) can serve the protection. Proposition 3. Let a non-simple NEPC P be systematically composed from two fibers between the concerned (neighbor) nodes. It can protect against the failure of the encircled node b if it contains a path P P covering the node set S of neighbors such that the path P P does not contain a sequence of three successive internal MI nodes belonging to S.

Proof. Each node s in S has at least one sub tree T s rooted in s which must be re-connected to the source after the failure recovery. Trivially, if s is an MI node, it has only one sub tree in the original multicast tree. Moreover, each node occurrence of an MI node can have a degree 2 in the recovery structure. Contrarily, the MC nodes can be branching nodes after the recovery. The capacities in the mentioned NEPC can be allocated to the failure recovery as follows. From the source, the visited MC nodes belonging to S are branching nodes. Let us suppose that an internal MI type node s is a relay node in the NEPC. Remember that the unique sub tree T s can not be connected to the backup structure by this node occurrence of s. To this, the backup structure must return to s from an other visited node. If P P contains three successive internal MI nodes belonging to S, and there is no MC node between them, the return to the middle MI node in S is not possible using the cycle. Let s 1 , s 2 and s 3 the three successive MI nodes in the cycle (cf. Figure 10). In the best case, the sub trees of s 1 and s 3 are connected to the recovery structure using incoming links (l 01 and l 43 ) as it is illustrated. Moreover, a path from s 1 to s 3 must be used to connect the two parts of the multicast routes. Since s 1 , s 2 and s 3 are MI nodes, the middle node s 2 can be a relay node in this path but can not be branching node. Its sub tree can not be connected to the multicast route using the in-cycle links.

If there is at least one MC node (named t) either from s i to the next MI node in S or before s i , then this node t becomes a branching node and the not yet used path from t to s i can connect the sub tree of s i to the backup structure. The particularity of this protection is that it uses not only a path or a walk in the cycle but also other related links which are connected to the basic path. So, the configuration of the protection concernes not only the configuration of the nodes following the walk or the path scheme but it is completed with some additional configurations. The obtained backup structure is a light hierarchy using the capacities of different in-cycle links. In this study, we presented the conditions how NEPC can be used to protect light trees against branching node failures in all optical WDM networks. The most important contribution of this work is simple: the backup multicast route must correspond to the well known optical constraints. We analyzed the constraint imposed by spare splitting capacities in the network. The branching nodes of the backup route must be MC nodes.

We also presented how non simple NEPCs can be applied to the failure recovery. Often, this application leaves to the formulation of backup light hierarchies. They extend the protection possibilities but the configuration of these hierarchies is more complicated.

The analysis of the protection of multicast routes in all optical networks needs intensive future works. In real all optical networks, all of the feasible light trees can not be protected against branching node failures. A deep analysis of network topologies from the point of view of the node protection is needed. The protection strategy can be different from the NEPC based protection. Different strategies should be analyzed.

Fig. 1 .

 1 Fig. 1. A cycle P may protect against the failure of a branching node b in a multicast tree in an opaque optical network

Fig. 2 .

 2 Fig. 2. A non-simple NEPC which may protect against the failure of a branching node b

  (the cycle P in dotted line passes through the node b). Property 1. A cycle is an NEPC (simple or not) and can be used to recover the failure of the encircled node b iff all the neighbor nodes of b are in the cycle.

Fig. 3 .

 3 Fig.3. In an opaque optical network, cycle P may protect against the failure of an on-cycle branching node

Fig. 4 .

 4 Fig. 4. A cycle P may protect against the failure of any link and node of a sub tree limited by it

Fig. 5 .Fig. 6 .

 56 Fig. 5. A configuration of MC and MI type nodes on the cycle P which does not permit the protection against the failure of the branching node b of a multicast tree

Fig. 7 .

 7 Fig. 7. Non-simple NEPCs permit the protection against the failure of the branching node b using walks on the NEPC

Fig. 8 .

 8 Fig. 8. The detailed walk of the second example

Fig. 9 .

 9 Fig.9. A particular non-simple NEPC and its usage for the protection against the failure of the branching node b using a particular walk on the NEPC

Figure 11 Fig. 10 .

 1110 Figure 11 illustrates two cycles: one with a feasible and another with a not feasible sequence of nodes.

Fig. 11 .

 11 Fig. 11. A feasible and a not feasible sequence of nodes in an NEPC using a backup hierarchy on the cycle


Frederic Abergel
email: frederic.abergel@ecp.fr

A NONLINEAR PARTIAL INTEGRO-DIFFERENTIAL EQUATION FROM MATHEMATICAL FINANCE

Keywords: 2000 Mathematics Subject Classification. Primary: 35A01, 35K55, 35R09, 47G20; Secondary: 60H15 Analysis of pde's, nonlinear parabolic partial integro-differential equation, short-time existence, fixed point methods, local and stochastic volatility models, calibration of vanillas

Consistently fitting vanilla option surfaces is an important issue when it comes to modeling in finance. As far as local and stochastic volatility models are concerned, this problem boils down to the resolution of a nonlinear integro-differential pde. The non-locality of this equation stems from the quotient of two integral terms and is not defined for all bounded continuous functions. In this paper, we use a fixed point argument and suitable a priori estimates to prove short-time existence of solutions for this equation.

1. Introduction. Financial modelling has been an area of extremely rapid growth in the past 30 years, and some extremely interesting mathematical challenges have emerged. One of the utmost importance for real-life applications to derivatives trading is that of calibration. Similar to common situations in many areas of physics and engineering, once a model has been suggested, its parameters have to be estimated using external data. In the case of derivative modelling, those data are the liquid (tradable) options, generally known as the vanilla products. It is well known since the pioneering work of Litzenberger and Breeden [START_REF] Breeden | State Contingent Prices Implicit in Option Prices[END_REF] and its celebrated extension by Bruno Dupire [START_REF] Dupire | Pricing and Hedging with Smiles[END_REF] that the knowledge of market data such as the prices of vanilla options across all strikes and maturities is equivalent to the knowledge of the riskneutral marginals of the underlying stock distribution, and moreover, that there is a unique one-dimensional driftless diffusion which recovers exactly such marginals. However, it has also been well-known for almost as many years that the evolution in time of the so-called local volatility is not stable, thereby leading researchers and financial engineers to look for a more robust, stochastic volatility type of modelling. In this paper, we consider the calibration problem for a generic stochastic volatility model: more precisely, we address the issue of calibrating to market data a generic model with a stochastic component and a local component for the volatility process. Such models are very useful in practice, since they offer both the flexibility and realistic dynamics of stochastic volatility models, and the exact calibration properties of local volatility models. In mathematical terms, the problem we consider is a non linear partial integro-differential equation for which we are able to prove short-time existence of classical solutions under suitable assumptions. The paper is organized as follows: Section 2 is devoted to the mathematical formulation of the problem. Section 3, to notations and statement of the main result. In Section 4, we recall some important technical results stemming from the general theory of parabolic pdes. Section 5 contains the proof of the main result. Finally, Section 6 is a short conclusion.

2.

Local and Stochastic Volatility model.

Diffusion equations.

The LSV model is an extension of Dupire's local volatility model. In the simplest situation -the two-dimensional case -the dynamics of the model are given by the following system of SDEs

dS t S t = a(t, S t )b(Y t )dB 1 t + rdt dY t = α(t, Y t )dB 2 t + ξ(t)dt
Here, (S t , t ≥ 0) is the stock price process and (Y t , t ≥ 0) the stochastic component of the volatility. The function b simply transforms that factor into a proper volatility. a is the local volatility part of the model, choosing its value properly will enable us to calibrate the vanillas of the model. α is the volatility of the volatility factor and µ and ξ are drift terms that may depend on the state variables and on time. B 1 and B 2 are standard brownian motions with correlation ρ.

2.2.

The LSV model calibration. In order to fit the vanillas of this model, we write the Kolmogorov forward equation on the joint density p(t, S, y) of the couple

(S t , Y t ) ∂p ∂t - ∂ 2 ∂S 2 ( 1 2 a 2 b 2 S 2 p) - ∂ 2 ∂S∂y (ρabαSp) - ∂ 2 ∂y 2 ( 1 2 α 2 p)) + ∂ ∂S (rSp) + ∂ ∂y (ξp) + rp = 0 p(S, y, 0) = δ(S = S 0 , y = y 0 )
with (S 0 , y 0 ) the initial conditions. Taking q = pdy the marginal of S, we get the equation

∂q ∂t - ∂ 2 ∂S 2 ( 1 2 a 2 S 2 ( b 2 pdy)) + ∂ ∂S (rSq) + rq = 0
Using Dupire's results from [START_REF] Dupire | Pricing and Hedging with Smiles[END_REF], we know that q has to solve the following equation in order to fit perfectly the vanillas of the market

∂q ∂t - ∂ 2 ∂S 2 ( 1 2 σ 2 D S 2 q) + ∂ ∂S (rSq) + rq = 0
where σ D is Dupire's local volatility and contains the information about the vanillas we want to reproduce. We identify the terms in this last formula. This gives us the value of a 2 (t, S) = σ 2 D (t, S) q b 2 pdy = σ 2 D (t, S) pdy b 2 pdy . Eventually, the joint density that calibrates the smile of our model is solution of the nonlinear partial integro-differential equation

∂p ∂t - ∂ 2 ∂S 2 ( 1 2 σ 2 D b 2 S 2 pdy b 2 pdy p) - ∂ 2 ∂S∂y (ρσ D bαS( pdy b 2 pdy ) 1 2 p) - ∂ 2 ∂y 2 ( 1 2 α 2 p)) + ∂ ∂S (rSp) + ∂ ∂y (ξp) + rp = 0
The rest of this paper is devoted to the study of a more general n-dimensional version of this equation.

3. Generalized equation and notations. Throughout this article, we denote by 0 < t ≤ T the time-variable and by x = (x 1 , x 2 , ..., x n ) ∈ Ω ⊂ R n the ndimensional space variable where Ω is an open subset with a sufficiently smooth boundary (we will precise this notion later). When we consider the equation from a financial point of view, the first-variable x 1 stands for the spot and the last n -1 for the volatility. Hence, we write S = x 1 , y = (x 2 , ..., x n ). We also denote by

D T =]0, T [ Ö Ω
O(p) := ∂p ∂t - ∂ 2 ∂S 2 (ρ 11 α 2 1 I(p)p) - n i=2 ∂ 2 ∂S∂x i (ρ 1i α 1 α i I(p)p) - n i,j=2 ∂ 2 ∂x i ∂x j (ρ ij α i α j p) + n i=1 ∂ ∂x i (β i p) + γp = 0 on D T ∪ B T (1)
where (ρ ij ) 1≤i,j≤n is a correlation matrix ie is symetric positive definite and verifies: ρ ii = 1 2 for all i and -1 2 < ρ ij < 1 2 for i = j. We add the boundary condition p = Ψ on B ∪ C T with Ψ constant on C T . We also let p 0 denote the function p 0 (t, S, y) = Ψ(S, y). The complexity of this equation stems from the following integral term:

I(p)(t, S) = Ω S y p(t, S, x 2 , ..., x n )dx 2 ...dx n Ω S y b 2 (x 2 , ..., x n )p(t, S, x 2 , ..., x n )dx 2 ...dx n = Ω S y p(t, S, y)dy Ω S y b 2 (y)p(t, S, y)dy
(1) belongs to the class of nonlinear, parabolic and nonlocal equations. An interesting reference concerning that kind of equations is [START_REF] Alibaud | Existence, uniqueness and regularity for nonlinear parabolic equations with nonlocal terms equations with nonlocal terms[END_REF]. However, our case doesn't fall under the scope of that paper: the operator I(.) is not defined on C b (D T ). Let us now make a few remarks about our particular equation.

Remark 1. In the case of an equation with no I(p) term, it becomes a classic linear equation of parabolic type. That kind of equation has been properly solved for quite some time now, see [START_REF] Friedman | Partial differential equations of parabolic type[END_REF] or [START_REF] Ladyzhenskaya | Linear and quasilinear equations of parabolic type[END_REF]. When b is constant, the problem is reduced to the previous remark. This observation is the key to our resolution method. First, we suppose that b does not vary too much and approximate the nonlocal term I(p) with a suitable constant. We then isolate the error made during this process in the second term and use a fixed point method to solve the new equation.

In order to use this method and the results from [START_REF] Friedman | Partial differential equations of parabolic type[END_REF], one has to assume that the coefficients of the equation belong to Holder spaces H k,h,h/2 (we shall define them in the preliminaries).

Remark 2. The question whether I is properly defined is natural. To answer it, we have to prove that Ω S y b 2 (y)p(t, S, y)dy is bounded away from 0. We assume that b is non-negative and that Ψ the initial condition is strictly positive. By restricting ourselves to short times, we are sure that p is not too far from its initial condition and thus is strictly positive.

Remark 3. From a financial viewpoint, it is natural to consider a domain Ω cylindrical with respect to the spot. However, since it is quite challenging to study a PDE on a domain with corners, we reduce our study to domains with an S-section depending on S.

Assumptions. The theorem we shall prove requires the following assumptions on b and Ψ.

(H1) b ∈ C 1 (R n-1 ), ∃(δ 1 , δ 2 ) ∈ R 2 , 0 < δ 1 ≤ b ≤ δ 2 on R n-1 (H2) ∀2 ≤ i ≤ n, | ∂b 2
∂xi | ≤ b * on Ω y where b * is a constant we will choose later (H3) Ψ is strictly positive and in H 2,h,h/2 . This gives us two results on p 0 . First, p 0 belongs to H 2,h,h/2 (D T ) and second

0 < p 0 = inf p 0 ≤ sup p 0 = p 0 (H4) O(Ψ) = 0 on ∂B in a sense described

in the preliminaries

Under the previous assumptions, we have the following result: Theorem 3.1. If the α i belong to H 2,h,h/2 (D T ), are positive and bounded away from 0 by a stricly positive constant e, if the β i are in H 1,h,h/2 (D T ) and if γ belongs to H 0,h,h/2 (D T ), then, for b * small enough, there exists 0 < T * ≤ T and a solution of the equation (1) on D T * ∪ B T * .

The rest of the paper will be devoted to the proof of Theorem 3.1 above. We start with the basic mathematical notions it requires.

Preliminaries.

4.1. Holder Spaces. In this section, we write ∂ xi and ∂ ∂xi without distinction. Let us start as in [START_REF] Friedman | Partial differential equations of parabolic type[END_REF] and [START_REF] Ladyzhenskaya | Linear and quasilinear equations of parabolic type[END_REF] with the following notion of distance

d(P, Q) = [|x - x | 2 + |t -t |] 1/2
where P = (t, x) and Q = (t , x ) belong to D T and |x| is the norm of the n-dimensionnal vector x. Given such a metric d, we can define the concept of h-Holder continuity (with h a number between 0 and 1 strictly). For a function u, we write:

|u| D T 0 = sup D T |u| H D T h (u) = sup P,Q∈D T |u(P ) -u(Q)| d(P, Q) h |u| D T h = |u| D T 0 + H D T h (u) |u| D T h < ∞ if and only if u is uniformly holder (exponent h) in D T .
We denote by H 0,h,h/2 (D T ) the set of all functions u for which |u| D T h < ∞. Now, if all the derivatives used in the equation exist, we write for k ≤ 2:

|u| D T k+h = |u| D T h + Σ|∂ x u| D T h + ... + Σ|∂ k x u| D T h + |∂ t u| D T h ( 2
)
where the sums are taken over all the partial derivatives of the indicated order. We denote by H k,h,h/2 (D T ) the set of all functions u for which |u| D T k+h < ∞. It is a Banach space and an algebra with the norm given by definition 2. Indeed, for all u,v in H k,h,h/2 (D T ), we have:

|uv| D T k+h ≤ |u| D T k+h |v| D T k+h (3)
We can now make the assumptions about D T more precise: for every point Q of C T , there exists an (n+1)-dimensional neighborhood V such that V ∩ C T can be represented, for some i (1 ≤ i ≤ n), in the form x i = r(t, x 1 , ..., x i-1 , x i+1 , ..., x n ) with r, ∂ x r, ∂ 2 xx r, ∂ t r Holder continuous (exponent h) and ∂ 2 xt r, ∂ 2 tt r simply continous. We also have to consider functions ψ defined on B ∪ C T . Such a function ψ is said to belong to H k,h,h/2 if there exists a Ψ in H k,h,h/2 (D T ) such that Ψ = ψ on B ∪ C T . We then define |ψ| k+h = inf |Ψ| D T k+h where the inf is taken with respect to all the Ψ's in H k,h,h/2 (D T ) which coincide with ψ on B ∪ C T . This process defines a norm on H k,h,h/2 . 4.2. Linear equations of parabolic type. The following results will be useful in the proof of our result. They concern the PDE:

Lu := ∂u ∂t - n i,j=1 a ij (x, t) ∂ 2 u ∂x i ∂x j + n i=1 b i (x, t) ∂u ∂x i +c(x, t)u =f (x, t) on D T ∪ B T u = ψ on B ∪ C T (4)
We shall need the assumptions:

• the coefficients of the operator L belong to H 0,h,h/2 (D T ), let K 1 be a bound on their norm

• for all (x, t) in D T and for all ξ ∈ R n , n i,j=1

a ij (x, t)ξ i ξ j ≥ K 2 | ξ | 2 (K 2 > 0) • ψ ∈ H 2,h,h/2 and |f | D T h < ∞
In addition, given the assumption about D T , if we consider a function ψ ∈ H 2,h,h/2 , for any extension Ψ of ψ, ∂ t Ψ is uniquely defined (by continuity) on the boundary ∂B of B, and the definition is independent of Ψ. We denote this function (on ∂B) by ∂ t ψ. The other terms of Lψ are also uniquely defined (by continuity) on ∂B. Thus, the quantity Lψ is well-defined on ∂B.

Theorem 4.1. Under the previous assumptions and if Lψ = f on ∂B, there exists a unique solution of the equation 4, this solution belongs to H 2,h,h/2 (D T ) and we have the Schauder inequality (with K H 2 depending only on K 1 , on K 2 , on h and on D T ) Proof. The first part of the result is classic, its proof can be found in [START_REF] Friedman | Partial differential equations of parabolic type[END_REF]. As to the result with ψ = 0, which is more original, one needs a result from [START_REF] Ladyzhenskaya | Linear and quasilinear equations of parabolic type[END_REF] about volume potentials and representation of solutions of parabolic equations. It is the theorem (16.2) of section IV.16 we shall use. One reads that the solution of the equation 4 with ψ = 0 can be written as

|u| D t 2+h ≤ K H 2 (|ψ| 2+h + |f | D t h ) (5) Furthermore, if ψ = 0,
u(x, t) = t 0 dτ Ω G(x, z, t, τ )f (z, τ )dz
where G is the Green's function for the operator L and verifies

|G(x, y, t, τ )| ≤ K(t -τ ) -n 2 exp(-K |x -y| 2 t -τ )
with K and K two constants depending on the data of the problem. Using both these results, we get, for all t ≤ t and x ∈ Ω

|u(x, t )| ≤ t|f | D t 0 t 0 dτ Ω K(t -τ ) -n 2 -1 exp(-K |x -y| 2 t -τ )dz ≤ t|f | D t 0 K 0
where K 0 depends on K 1 , on K 2 , on h and on D T .

5. Proof of Theorem 3.1. We are interested in the equation

∂p ∂t - ∂ 2 ∂S 2 (ρ 11 α 2 1 I(p)p) - n i=2 ∂ 2 ∂S∂x i (ρ 1i α 1 α i I(p)p) - n i,j=2 ∂ 2 ∂x i ∂x j (ρ ij α i α j p) + n i=1 ∂ ∂x i (β i p) + γp = 0 on D T ∪ B T
and want to prove the theorem: If the α i belong to H 2,h,h/2 (D T ), are positive and bounded away from 0 by a stricly positive constant e, if the β i are in H 1,h,h/2 (D T ) and if γ belongs to H 0,h,h/2 (D T ), then, for b * small enough, there exists 0 < T * ≤ T and a solution of the equation ( 1) on D T * ∪ B T * .

Proof.

The assumption (H2) gives us some control over the variations of b. Let us denote by b = b(y 0 ) a strictly positive value taken by b (with y 0 ∈ Ω S y for some arbitrary S ∈ Ω S ). We use the assumption on b to approximate the integral term I(p) with 1/b 2 , the gap between those two quantities is quantified with the Lemma 5.1. There exists a constant K b (depending only on h, n, δ 1 , δ 2 , p 0 and Ω) and a polynomial function P strictly positive and increasing on R * + such that ∀p ∈ H 2,h,h/2 (D T ) verifiying p 0 ≤ p, we have

|I(p) - 1 b 2 | D t 2+h + | I(p) - 1 b | D t 2+h ≤ b * K b P (|p| D t 2+h ).
Remark 4. As a consequence of this lemma, we see that ∀p ∈ H 2,h,h/2 (D T ) verifiying p 0 ≤ p, I(p) belongs to H 2,h,h/2 (D T ).

We then write the equation as

∂p ∂t - ∂ 2 ∂S 2 (ρ 11 α 2 1 1 b 2 p) - n i=2 ∂ 2 ∂S∂x i (ρ 1i α 1 α i 1 b p) - n i,j=2 ∂ 2 ∂x i ∂x j (ρ ij α i α j p) + n i=1 ∂ ∂x i (β i p) + γp = ∂ 2 ∂S 2 (ρ 11 α 2 1 (I(p) - 1 b 2 )p) + n i=2 ∂ 2 ∂S∂x i (ρ 1i α 1 α i ( I(p) - 1 b )p)
To solve this equation, we apply a fixed point method and use the lemma 5.1 to get an upper bound on the second term. We take a real number x ≥ |p 0 | D T 2+h and t ∈ R * + and let X t x denote the set

X t x = {p ∈ H 2,h,h/2 (D t ), |p| D t 2+h ≤ x, p 0 2 ≤ p ≤ p 0 + p 0 2 , p = Ψ on B ∪ C T }
The set X t x clearly contains the function p 0 . We then consider the application M which takes a function u ∈ X and sends it on v ∈ H 2,h,h/2 (D T ) solution of the equation

O v := ∂v ∂t - ∂ 2 ∂S 2 (ρ 11 α 2 1 1 b 2 v) - n i=2 ∂ 2 ∂S∂x i (ρ 1i α 1 α i 1 b v) - n i,j=2 ∂ 2 ∂x i ∂x j (ρ ij α i α j v) + n i=1 ∂ ∂x i (β i v) + γv = ∂ 2 ∂S 2 (ρ 11 α 2 1 (I(u) - 1 b 2 )u) + n i=2 ∂ 2 ∂S∂x i (ρ 1i α 1 α i ( I(u) - 1 b )u)
with the boundary condition v = Ψ on B ∪ C T . The existence of v is given by Theorem 4.1. Indeed, the coefficients of this equation belong to the appropriate spaces and because of (H4) the necessary condition

O ψ = ∂ 2 ∂S 2 (ρ 11 α 2 1 (I(Ψ) - 1 b 2 )Ψ) + n i=2 ∂ 2 ∂S∂x i (ρ 1i α 1 α i ( I(Ψ) - 1 b )Ψ)
on ∂B is verified. It remains to prove that this operator is elliptic: let (ξ i ) 1≤i≤n be n real numbers and (t, S, y) ∈ D T . We write

f 1 = α1 b and f i = α i for i ≥ 2, we have n i,j=1 ρ ij f i (t, S, y)f j (t, S, y)ξ i ξ j ≥ K ρ n i=1 f 2 i (t, S, y)ξ 2 i ≥ K ρ e 2 n i=1 ξ 2 i
where the existence of K ρ is a consequence of ρ being a positive definite matrix. This proves the ellipticity of the operator, v exists and belongs to H 2,h,h/2 (D T ). We now want to show that for suitable x and t, v belongs to X t x ie that

|v| D t 2+h ≤ x p 0 2 ≤ v ≤ p 0 + p 0 2
For the first inequality, we apply 5

|v| D t 2+h ≤ K H 2 (|ψ| D t 2+h + | ∂ 2 ∂S 2 (ρ 11 α 2 1 (I(u) - 1 b 2 )u) + n i=2 ∂ 2 ∂S∂x i (ρ 1i α 1 α i ( I(u) - 1 b )u)| D t h ) ≤ K H 2 (|ψ| D T 2+h + |ρ 11 α 2 1 (I(u) - 1 b 2 )u| D t 2+h + n i=2 |ρ 1i α 1 α i ( I(u) - 1 b )u| D t 2+h ) ≤ K H 2 (|ψ| D T 2+h + (|ρ 11 α 2 1 | D T 2+h + n i=2 |ρ 1i α 1 α i | D T 2+h ) (|I(u) - 1 b 2 | D t 2+h + | I(u) - 1 b | D t 2+h )|u| D t 2+h ) ≤ K H 2 (|ψ| D T 2+h + b * K P (|u| D t 2+h )|u| D t 2+h )
where

K = (|ρ 11 α 2 1 | D T 2+h + n i=2 |ρ 1i α 1 α i | D T 2+h
)K b (we apply lemma 5.1 for the last line). We remember that u belongs to X t x , thus |u| D t 2+h ≤ x and then

|v| D t 2+h ≤ K H 2 (|ψ| D T 2+h + b * K P (x)x) Taking x * = max(K H 2 (|ψ| D T 2+h + 1), |p 0 | D T 2+h ) and b * ≤ 1 K P (x * )x * , 7 gives us |v| D T 2+h ≤ x * . It remains to prove that p0 2 ≤ v ≤ p 0 + p0 2 . Let us write ṽ = v -p 0 . It is clear that ṽ verifies O ṽ = O p 0 + ∂ 2 ∂S 2 (ρ 11 α 2 1 (I(u) - 1 b 2 )u) + n i=2 ∂ 2 ∂S∂x i (ρ 1i α 1 α i ( I(u) - 1 b )u)
on D T ∪ B T with ṽ = 0 on B ∪ C T (here we use the fact that Ψ is constant on C T ).

We now apply the second part of Theorem 4.1, the inequality 6, to this function ṽ

|ṽ| D t 0 ≤ tK 0 |O p 0 + ∂ 2 ∂S 2 (ρ 11 α 2 1 (I(u) - 1 b 2 )u) + n i=2 ∂ 2 ∂S∂x i (ρ 1i α 1 α i ( I(u) - 1 b )u)| D t 0 ≤ tK 0 (|O p 0 | D T 0 + | ∂ 2 ∂S 2 (ρ 11 α 2 1 (I(u) - 1 b 2 )u) + n i=2 ∂ 2 ∂S∂x i (ρ 1i α 1 α i ( I(u) - 1 b )u)| D t h ) ≤ tK 0 (|O p 0 | D T 0 + 1) Taking T * K 0 (|O p 0 | D T 0 + 1) = p0 2 , we get |ṽ| D t 0 ≤ p0 2 .
Eventually, since v = p 0 + ṽ, the last inequality is proved and v belongs to X T *

x * . The application M maps X T *

x * into itself.

Using this statement, we construct a bounded sequence (p n ) n∈N of functions belonging to X T *

x *

• p 0 has been previously defined

•
| ∂ 2 pn ∂xi∂xj (x, t) -∂ 2 pn ∂xi∂xj (x , t ) | (| x -x | 2 + | t -t |) α/2 ; (x, t), (x , t ) ∈ D T * } ≤ x We have sup{ | ∂ 2 p ∂xi∂xj (x, t) -∂ 2 p ∂xi∂xj (x , t ) | (| x -x | 2 + | t -t |) α/2 , (x, t); (x , t ) ∈ D T * } ≤ x
And this computation being true for all the derivatives appearing in the norm H 2,h,h/2 , we find that p ∈ H 2,h,h/2 . The last step of the proof is to take the limit in 7. The only result needed is I(p n ) → I(p). Since p n ∈ X T * x * , the denominator is bounded away from 0. Two applications of the dominated convergence theorem give us the convergence we need. Thus, it is possible to take the limit in 7 which gives us

∂p ∂t - ∂ 2 ∂S 2 (ρ 11 α 2 1 1 b 2 p) - n i=2 ∂ 2 ∂S∂x i (ρ 1i α 1 α i 1 b p) - n i,j=2 ∂ 2 ∂x i ∂x j (ρ ij α i α j p) + n i=1 ∂ ∂x i (β i p) + γp = ∂ 2 ∂S 2 (ρ 11 α 2 1 (I(p) - 1 b 2 )p) + n i=2 ∂ 2 ∂S∂x i (ρ 1i α 1 α i ( I(p) - 1 b )p) gives us | p(b 2 -b 2 )dy| D T h ≤ b * K|p| D t h And since | p(b 2 -b 2 )dy| D T 2+h =| p(b 2 -b 2 )dy| D T h + | ∂p ∂t (b 2 -b 2 )dy| D T h + | ∂p ∂S (b 2 -b 2 )dy| D T h + | ∂ 2 p ∂S 2 (b 2 -b 2 )dy| D T h
We get from the previous computation

| p(b 2 -b 2 )dy| D T 2+h ≤ b * K|p| D t 2+h
We now have to find a bound on

| 1 b 2 pdy | D T 2+h . Since p belongs to X T * x * , we have | 1 b 2 pdy | D T 0 ≤ 2 δ 2 1 p0V (Ω)
. Now, let (t, S) and (t , S ) belong to ]0, T [ Ö Ω S . We write , we find the same kind of results for the second term involved in the lemma. This concludes the proof.

Conclusion.

In this paper, we have shown that the equation driving the calibration problem for local and stochastic volatility models is well-posed in the case of suitably regularized initial conditions. It is however clear that the solution of the full Kolmogorov equation with Dirac initial condition is not obtained as a consequence of Theorem 1 : possible extensions of our results towards this direction are currently being explored. Let us also mention that a generalization of Theorem 1 to multidimensional correlation calibration has already been investigated and will be presented in [1]

  the domain of definition and by B = {0} Ö Ω, B T = {T } Ö Ω and C T =]0, T [ Ö ∂Ω the different parts of the boundary. Given the particular part played by the spot, we consider Ω S = {S ∈ R/∃y ∈ R n-1 , (S, y) ∈ Ω} and ∀S ∈ Ω S , Ω S y = {y ∈ R n-1 /(S, y) ∈ Ω}. We are interested in the following equation:

0 ( 6 )

 06 we can write a bound containing the time on the supremum of the solution |u| D t 0 ≤ tK 0 |f | D t and K 0 only depends on K, on K and on Ω.

  by induction, we write p n+1 = M (p n ) By construction, we have ∀n ∈ N, |p n | D T 2+h ≤ x * . Repeated applications of the Ascoli-Arzel theorem give us a function p ∈ C 2 (D T * ) limit in C 2 (D T * ) of a subsequence of p n . Since sup{

b 2 1 b 2 b 2 5 ) 2 √

 212252 (y)p(S , t , y)dy -Ω S y b 2 (y)p(S, t, y)dy| Ω S y b 2 (y)p(S, t, y)dy Ω S y b 2 (y)p(S , t , y)dy≤ KH D th (p)D((t, S), (t , S )) We used the same kind of arguments than earlier, K denotes here another constant depending on δ 1 , δ 2 , p 0 and Ω. This gives us| 1 b 2 pdy | D T h ≤ K(1 + |p| D T h ).As for derivatives of pdy , for instance with respect to S, we have ∂p ∂S dy( b 2 pdy) 2 | D T h ≤ K(1 + |p| D T h ) 2 |p| D T 1+hThe same kind of computation is true for the derivative of second order| ∂ ∂S (-b 2 ∂p ∂S dy ( b 2 pdy) 2 )| D T h = | 2( b 2 ∂p ∂S dy) 2 ( b 2 pdy) 3 -b 2 ∂ 2 p ∂S 2 dy ( b 2 pdy) 2 | D T h ≤ K[(1 + |p| D T h ) 3 (|p| D T 1+h ) 2 + (1 + |p| D T h ) 2 |p| D T 2+h ] Eventually, we get | 1 b 2 pdy | D T 2+h ≤ K(1 + |p| D T 2+h + (|p| D T 2+h ) 2 + (|p| D T 2+h ) 3 + (|p| D T 2+h ) 4 + (|p| D T 2+h )Combining this result with the previous computations, we find|I(p) -1 b 2 | D T 2+h ≤ b * KP (|p| D t 2+h )with P a polynomial function of degree 6, strictly positive on R * + . Now, writing I(p) -1 b = I(p)-1 b I(p)+ 1 b

That concludes the proof of the theorem, p is solution of our equation.

Let us now prove lemma 5.1.

Proof. By definition, we have:

Let us compute one after another the terms appearing in this norm (we remember that those functions only depend on t and S). Let (t, S) belong to ]0, T [ Ö Ω S . We have

here and in the rest of the proof, K stands for some constant depending only on the data of the problem (Ω, n, δ 1 ...) but not on p nor on b * . We get the last line from the following computation where y = (x 2 , ., x n ) and y 0 = (x 2 , ., x n )


Sybil Rose Stacpoole

Bilada Bilican

Daniel Webber

Aryna Luzhynskaya

Xiaoling He

Alastair Compston

Ragnhildur Karadottir

Robin A Franklin

Siddharthan Chandran

Robin Jm Franklin

Derivation of neural

Keywords: Physiological normoxia, hES-NPCs Hypoxia, Neural Stem Cells, Neural Conversion, Neural Differentiation, Embryonic stem cells

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

Introduction

The capacity of human embryonic stem cells (hESC) to generate defined neuronal and glial lineages offers a major opportunity to study neurodevelopment and model neurological disease in vitro, as well as having potential direct therapeutic applications in the field of regenerative neurology. Notwithstanding major advances in hES neural specification and differentiation over the last decade, there remain significant challenges to overcome before the promise of hESCs for neurological diseases can be fully realised [START_REF] Zhao | Hypoxia-driven proliferation of embryonic neural stem/progenitor cells--role of hypoxia-inducible transcription factor-1alpha[END_REF][START_REF] Erceg | Human embryonic stem cell differentiation toward regional specific neural precursors[END_REF][START_REF] Hedlund | Neuronal cell replacement in Parkinson's disease[END_REF][START_REF] Conti | Neural stem cell systems: physiological players or in vitro entities?[END_REF]. These include the need to optimise survival, fate and function of neural derivatives upon both neural conversion and long-term differentiation in vitro and in vivo.

Neural stem cells (NSCs) can be readily generated from ESCs by culture in defined conditions in the absence of extrinsic signals; the so-called default model of neuralisation [START_REF] Munoz-Sanjuan | Neural induction, the default model and embryonic stem cells[END_REF]. However, significant cell death is observed under such serum-free, defined conditions [START_REF] Smukler | Embryonic stem cells assume a primitive neural stem cell fate in the absence of extrinsic influences[END_REF]. The mechanism through which the cells die involves both apoptotic and parthanatic pathways [START_REF] Smukler | Embryonic stem cells assume a primitive neural stem cell fate in the absence of extrinsic influences[END_REF][START_REF] Clarke | Low oxygen enhances primitive and definitive neural stem cell colony formation by inhibiting distinct cell death pathways[END_REF][START_REF] Chen | Oxygen tension regulates survival and fate of mouse central nervous system precursors at multiple levels[END_REF][START_REF] Pistollato | Oxygen tension controls the expansion of human CNS precursors and the generation of astrocytes and oligodendrocytes[END_REF][START_REF] Cimadamore | Nicotinamide rescues human embryonic stem cell-derived neuroectoderm from parthanatic cell death[END_REF], accompanied by the generation of reactive oxygen species (ROS) [START_REF] Cimadamore | Nicotinamide rescues human embryonic stem cell-derived neuroectoderm from parthanatic cell death[END_REF]. Consequently, neuralisation protocols often contain antioxidants, which may increase the propensity to accumulate genetic mutations, or involve co-culture with stromal feeder layers [START_REF] Erceg | Human embryonic stem cell differentiation toward regional specific neural precursors[END_REF][START_REF] Smukler | Embryonic stem cells assume a primitive neural stem cell fate in the absence of extrinsic influences[END_REF][START_REF] Clarke | Low oxygen enhances primitive and definitive neural stem cell colony formation by inhibiting distinct cell death pathways[END_REF][START_REF] Cimadamore | Nicotinamide rescues human embryonic stem cell-derived neuroectoderm from parthanatic cell death[END_REF][START_REF] Perrier | Derivation of midbrain dopamine neurons from human embryonic stem cells[END_REF][START_REF] Li | Physiological levels of reactive oxygen species are required to maintain genomic stability in stem cells[END_REF]. In addition, antagonism of the TGFβ signalling pathway has been shown to augment the efficiency of neural conversion and thereby increase survival; however, this can also influence the default identity of neural progenitors and potentially limit their ability to be directed towards defined cell types [START_REF] Chambers | Highly efficient neural conversion of human ES and iPS cells by dual inhibition of SMAD signaling[END_REF][START_REF] Patani | Activin/Nodal inhibition alone accelerates highly efficient neural conversion from human embryonic stem cells and imposes a caudal positional identity[END_REF].

The importance of ROS in mediating cell death during neural conversion under routine culture at oxygen (O 2 ) levels of 20%, which is far removed from than that found under physiological conditions in the central nervous system (CNS), suggests higher oxygen tension may be deleterious to neural specification and differentiation [START_REF] Clarke | Low oxygen enhances primitive and definitive neural stem cell colony formation by inhibiting distinct cell death pathways[END_REF][START_REF] Cimadamore | Nicotinamide rescues human embryonic stem cell-derived neuroectoderm from parthanatic cell death[END_REF]. In the CNS, oxygen levels vary from 8% at the pia to 0.55% in the midbrain, with measurements from the human brain recording a mean level of 3.2% at 22-27mm below the dura and 4.4% at 7-12 mm [START_REF] Erecinska | Tissue oxygen tension and brain sensitivity to hypoxia[END_REF][START_REF] Dings | Clinical experience with 118 brain tissue oxygen partial pressure catheter probes[END_REF]. Overall, the mean tissue level of oxygen in adult organs is about 3%, although it is considerably less in the developing embryo where stem cells abound [START_REF] Csete | Oxygen in the cultivation of stem cells[END_REF]. There is a growing literature around the critical influence of oxygen levels on stem cell fate, proliferation and survival (7-10, 12, 17-27).

Furthermore, oxygen has been proposed to act as a developmental morphogen [START_REF] Simon | The role of oxygen availability in embryonic development and stem cell function[END_REF]; low oxygen promotes tyrosine hydroxylase positive dopaminergic neurons from midbrain NSCs and oligodendrocyte differentiation from human fetal neural precursors (NPCs) [START_REF] Pistollato | Oxygen tension controls the expansion of human CNS precursors and the generation of astrocytes and oligodendrocytes[END_REF][START_REF] Studer | Enhanced proliferation, survival, and dopaminergic differentiation of CNS precursors in lowered oxygen[END_REF][START_REF] Akundi | Hypoxia alters cell cycle regulatory protein expression and induces premature maturation of oligodendrocyte precursor cells[END_REF]. In addition, oxygen tension is thought to be tightly regulated in the stem cell niche and it is thought that changes in the partial pressure of oxygen (pO 2 ) contribute to the mobilisation of stem cells in an injury response [START_REF] Parmar | Distribution of hematopoietic stem cells in the bone marrow according to regional hypoxia[END_REF][START_REF] Simsek | The distinct metabolic profile of hematopoietic stem cells reflects their location in a hypoxic niche[END_REF][START_REF] Mohyeldin | Oxygen in stem cell biology: a critical component of the stem cell niche[END_REF].

In view of the importance of low pO 2 in maintenance of pluripotency, mediated in part through Notch signalling and upregulation of Oct-4, it remains unclear as to whether low O 2 interferes with both neural conversion of human ESCs and subsequent neuronal differentiation of hESC-derived NPCs [START_REF] Gustafsson | Hypoxia requires notch signaling to maintain the undifferentiated cell state[END_REF][START_REF] Covello | HIF-2alpha regulates Oct-4: effects of hypoxia on stem cell function, embryonic development, and tumor growth[END_REF]. Mouse ES studies suggest that culture at 4% O 2 does not limit neural conversion or terminal differentiation [START_REF] Clarke | Low oxygen enhances primitive and definitive neural stem cell colony formation by inhibiting distinct cell death pathways[END_REF]. Furthermore, our understanding of the effect of low, physiological levels of oxygen on human ESCderived neuronal subtype specification, as well as long-term differentiation and function, is incomplete. One prediction from rodent and human fetal literature is that low oxygen could enable longer-term culture of differentiated progeny [START_REF] Li | Physiological hypoxia promotes survival of cultured cortical neurons[END_REF]. A benefit of longerterm culture under physiological oxygen levels is that this would allow more accurate disease modelling paradigms, particularly for neurodegenerative diseases in which reactive oxygen species and oxidative stress have been widely postulated to play a role in cell death [START_REF] Trotti | Glutamate transporters are oxidant-vulnerable: a molecular link between oxidative and excitotoxic neurodegeneration?[END_REF][START_REF] Behl | Oxidative nerve cell death in Alzheimer's disease and stroke: antioxidants as neuroprotective compounds[END_REF]. Moreover, for both disease modelling and pre-clinical assessments, a key functional assay of neuronal derivatives requires transplantation.

Given that routine transplantation studies cause, in effect, a stress challenge consequent on an oxygen switch from 20% to approximately 3-4% upon transplantation, it would be of considerable value to model the effect of such a "switch" in vitro.

Against this background, we sought to investigate whether 3% O 2 can allow the generation of NPCs from hESCs, the timescale involved and whether the resultant NPCs can undergo regional specification and functional maturation. Finally, we examine the survival of these NPCs following transfer from a 20% to 3% O 2 environment, providing an in vitro model of the oxygen challenge that occurs during transplantation.

Results

NPCs can be reliably and efficiently derived from hESCs in a 3% O 2 environment.

In order to address whether hESC-NPCs could be efficiently derived in low oxygen conditions, feeder-free hESCs, grown in a chemically defined medium (CDM) [START_REF] Johansson | Evidence for involvement of activin A and bone morphogenetic protein 4 in mammalian mesoderm and hematopoietic development[END_REF][START_REF] Vallier | Activin/Nodal and FGF pathways cooperate to maintain pluripotency of human embryonic stem cells[END_REF][START_REF] Joannides | A scaleable and defined system for generating neural stem cells from human embryonic stem cells[END_REF] at 20% O 2 , were enzymatically detached and transferred to suspension culture at 3% O 2 , along with removal of activin and FGF-2. A pimonidazole binding assay was used to biochemically confirm growth of cells at low oxygen; pimonidazole adducts on the surface of hypoxic cells, binding most efficiently at a pO 2 < 10 mmHg (Fig. 1D) [START_REF] Arteel | Evidence that hypoxia markers detect oxygen gradients in liver: pimonidazole and retrograde perfusion of rat liver[END_REF]. Over 14 days, efficient neural conversion was confirmed by quantitative immunolabelling that revealed loss of expression of the pluripotent marker OCT4 (1.1 +/-0.7% positive) with concomitant upregulation of the neuroepithelial markers SOX1 (98.7 +/-0.5%) and NESTIN (97.4 +/-0.3%), and maintenance of the stem cell marker SOX2 (Fig. 1A-C). There was no significant difference between the efficiency of neural conversion at 3% and 20% oxygen, with a neural identity acquired by D14 in both instances (Fig. 1C), consistent with previous reports of a two week timescale for neural conversion of hESCs at 20% O 2 [START_REF] Patani | Activin/Nodal inhibition alone accelerates highly efficient neural conversion from human embryonic stem cells and imposes a caudal positional identity[END_REF][START_REF] Joannides | A scaleable and defined system for generating neural stem cells from human embryonic stem cells[END_REF][START_REF] Yan | Directed differentiation of dopaminergic neuronal subtypes from human embryonic stem cells[END_REF][START_REF] Li | Directed differentiation of ventral spinal progenitors and motor neurons from human embryonic stem cells by small molecules[END_REF]. Neural conversion at 3% O 2 was robust, highly reliable and reproducible across two independent hESC lines, irrespective of whether feeder-dependent (H9 n=5; HUES9 n=10) or feeder-free (H9 n=10).

Improved survival with neuralisation of hESCs at 3% O 2 compared to 20% O 2 .

Having established that hES-NPCs can be reliably generated at 3% O 2 , this system was compared to the standard of 20% O 2 . Cultures appeared healthier at 3% O 2 , with more rounded, brighter spheres and fewer dead cells (Supp. Fig. 1A). Growth curves confirmed a significant increase in numbers of NPCs generated at 3% compared to 20% O 2 , under basal culture conditions (Fig. 2A; day 21, p = 0.00115). Furthermore, at D14, a greater proportion of cells grown at 3% O 2 (82.9%) were viable compared to 20% O 2 (66.7%), demonstrated though propidium iodide (PI) and annexin V exclusion (Fig. 2B).

The protective effect of 3% O 2 could be partly reproduced by the addition of the antioxidant N-acetyl-l-cysteine (1μM) to 20% O 2 cultures, resulting in an increase of viable cells to 75.5% at D14 (supplementary Fig. 1B).

The intracellular response to low oxygen is coordinated by the hypoxia inducible factors (HIFs), with low oxygen stabilising the alpha subunit, allowing for a rapid response to changes in oxygen that does not rely upon alterations in mRNA expression [START_REF] Semenza | Life with oxygen[END_REF]. A time-course analysis revealed stabilisation of HIF-1α protein was transient: it appeared within 6hrs, was maximal at 24hrs and became undetectable by 6 days (Fig. 2C). By contrast, HIF-2α stabilisation demonstrated a delayed kinetic compared to HIF-1α, protein levels being upregulated as HIF-1α is downregulated at day 3, with HIF-2α expression persistent at D14 (Fig. 2C).

hESC-NPCs derived at 3% O 2 display tri-lineage phenotypic potential and such cultures can be maintained for up to 3 months in the absence of exogenous growth factors.

In order to address whether low oxygen levels would prevent differentiation of hESCderived NPCs [START_REF] Gustafsson | Hypoxia requires notch signaling to maintain the undifferentiated cell state[END_REF], cells were plated on poly-d-lysine -laminin coated coverslips in the absence of FGF-2 at day 30 and maintained long-term at 3% oxygen. The neuronal marker β-III tubulin was observed by 24hrs, synapsin at 48hrs and the more mature neuronal marker Map2a+b was expressed by 5 days post plating (Fig. 3A). To confirm functional maturation, electrophysiological studies were undertaken that revealed Map2+ βIII tubulin+ neurons generated action potentials, as early as 10 days post plating (n=6/6, Fig. 3D). Recordings performed 30 days after plating demonstrated more mature-looking action potentials (n=25/27), which were inhibited by the addition of the voltage-gated sodium channel blocker tetrodotoxin (TTX) (n=16, Fig. 3B). The frequency of these action potentials depended on the amount of current injected (Fig. 3C), as expected for functional neurons [START_REF] Moe | Multipotent progenitor cells from the adult human brain: neurophysiological differentiation to mature neurons[END_REF]. There was also evidence of spontaneous activity in neurons after 30 days of differentiation at 3% O 2 (Fig. 3F). These findings were remarkably similar to the electrophysiological maturation of neurons observed at 20% O 2 (supplementary Fig 2 ), clearly demonstrating that there was no maturation block at low oxygen.

In addition to neurons, long-term differentiated cultures at 3% O 2 consistently contained GFAP positive astrocytes and O4/MBP expressing oligodendrocytes (Fig. 3F).

Furthermore, and in contrast to this and previous studies at 20% oxygen, we found that healthy differentiated cultures could be maintained for at least 3 months in basal B27 supplemented medium in the absence of exogenous neurotrophins [START_REF] Erceg | Human embryonic stem cell differentiation toward regional specific neural precursors[END_REF][START_REF] Hanson | Cyclic AMP elevation is sufficient to promote the survival of spinal motor neurons in vitro[END_REF].

Directed differentiation of 3% O 2 -derived NPCs into midbrain dopaminergic and spinal motor neurons.

A key aspect of hES-NPCs is their competence to respond to developmental cues that direct differentiation into defined cell types. In order to determine whether 3% O 2 derived NPCs retain this competence, we next applied existing protocols for directed differentiation of hES-NPCs into spinal motor neurons and midbrain dopaminergic neurons [START_REF] Perrier | Derivation of midbrain dopamine neurons from human embryonic stem cells[END_REF][START_REF] Yan | Directed differentiation of dopaminergic neuronal subtypes from human embryonic stem cells[END_REF][START_REF] Li | Directed differentiation of ventral spinal progenitors and motor neurons from human embryonic stem cells by small molecules[END_REF]. Motor neuron specification was achieved through sequential application of 0.1μM retinoic acid (RA) for 1 week to specify a caudal, neuronal identity, followed by a further week of RA with 1μM purmorphamine to ventralise the cells. RT-PCR confirmed induction of HOXB4 and MASH1 with upregulation of PAX6, followed by induction of OLIG2 and NKX6.1 (Fig. 4A). Immuno-labelling demonstrated expression of HB9, a transcription factor specifically expressed by postmitotic motor neurons (Fig. 4B,C) and, importantly, ChAT expression was also observed by 10 days (Fig. 4D).

Similarly, midbrain dopaminergic specification was achieved following sequential application of 100ng/ml FGF-8 for 1 week and FGF-8 with 1μM purmorphamine for 1-2 weeks. PCR characterisation showed expression of the midbrain marker EN1, along with PITX3 and NURR1, which are required for the development of substantia nigra dopaminergic neurons (Fig. 4E). Immuno-labelling also revealed EN1 and tyrosinehydroxylase (TH) positive neurons (Fig. 4F). A proportion of TH+ neurons also stained for MAP2a+b, and coexpressed vesicular monoamine transporter (VMAT2), which is required for packaging dopamine into subcellular compartments in monoamine neurons (Fig. 4G,H). As with non-patterned neurons, specified dopaminergic and motor neurons were readily cultured for at least 30 days without the requirement for exogenous BDNF, GDNF, IGF, ascorbic acid or cAMP at 3% oxygen.

The relative efficiency of induction of midbrain dopaminergic and spinal motor neurons at 3% versus 20% oxygen was addressed by quantitative RT-PCR analysis of expression of OLIG2 and EN-1, which are specific progenitor markers of motor neurons and midbrain dopaminergic neurons respectively. This revealed a 2-fold increase in the expression of OLIG2 at 3% oxygen, and furthermore, a 5-fold increase in EN-1 induction at day 28 (Fig 4A andE).

In vitro switch simulation of transplantation from 20% to 3% O 2 leads to NPC death, whilst maintenance at 3% O 2 is protective.

To model the oxygen challenge that occurs with transplantation, in vitro 'switch' experiments were performed. Single cell dissociates from day 30 NPCs derived at both 3% and 20% O 2 were shown to be comparable by FACS analysis, that demonstrated uniform co-expression of MUSASHI and SOX2: 99.6% (3%) and 99.8% (20% O 2 ) (Fig. 5A) with absence of pluripotency and other germ layer markers (supp. Fig 3). PI exclusion without annexin V binding also demonstrated comparable levels of viability at day 30 of 3% and 20% O 2 -derived NPCs (Fig 5B). This finding is consistent with the view that it is the neuralisation process itself that imposes maximal stress [START_REF] Cimadamore | Nicotinamide rescues human embryonic stem cell-derived neuroectoderm from parthanatic cell death[END_REF], with subsequent longer-term maintenance of NPCs being more readily achieved under either 3% or 20% O 2 .

Having confirmed equivalence in terms of neural identity and viability, NPCs were then plated as dissociated single cells in the absence of growth factors and maintained at either 20% or 3% O 2 , or switched from 20% to 3% O 2 . Ethidium bromide (dead cell) and calcein (live cell) staining at 48hrs showed that the survival of differentiating cells swapped from 20% to 3% O 2 was much worse than cells maintained at 3% O 2 (57 +/-2.4 % vs 79.9 +/-2.4% p = 0.0001), with cells remaining at 20% O 2 intermediate between the two groups (67+/-0.7% vs 79.9 +/-2.4%, p = 0.0033) (Fig. 5C). Together these findings strongly suggest that switching cells from high to low oxygen levels results in significant cell death that can be prevented by maintaining cells throughout at physiological levels of oxygen.

Discussion

We report that a physiological, 3% O 2 environment does not present a barrier to the generation of tripotential NPCs from human ESCs, nor to their specification into midbrain dopaminergic and spinal motor neurons, the efficiency of which is markedly enhanced at low oxygen. Furthermore, compared to basal conditions at 20% oxygen, the application of a defined, feeder-free neuralising system to this low oxygen environment results in the generation of greater numbers of NPCs, and, upon differentiation, allows the establishment of mixed cultures of neurons and glia that can be maintained for at least three months, without the requirement for exogenous growth factors. Significant cell death was observed on switching differentiating NPCs from a high to low oxygen environment, modelling the oxygen challenge presented by transplantation.

The cellular response to low oxygen is co-ordinated by the three HIF isoforms, with HIF1α believed to be the global regulator of the hypoxic response . HIFs are heterodimeric DNA binding proteins with α and β subunits. At 20% O 2 , the α subunits are degraded, whereas at low O 2 , HIFα is stabilised, allowing binding to the HIFβ/ARNT subunit and activation of target genes, which are involved in a myriad of diverse functions, including metabolism, angiogenesis, survival and migration [START_REF] Semenza | Life with oxygen[END_REF].

Previous reports that HIF-1α interacts with Notch [START_REF] Gustafsson | Hypoxia requires notch signaling to maintain the undifferentiated cell state[END_REF], and that HIF-2α regulates Oct4 [START_REF] Covello | HIF-2alpha regulates Oct-4: effects of hypoxia on stem cell function, embryonic development, and tumor growth[END_REF], might suggest that low O 2 would maintain pluripotency in ESCs and present a barrier to efficient neural induction and terminal differentiation. However, we observed that at 3% O 2 , efficient neuralisation was complete by 14 days, a timescale comparable to that at 20% O 2 [START_REF] Patani | Activin/Nodal inhibition alone accelerates highly efficient neural conversion from human embryonic stem cells and imposes a caudal positional identity[END_REF][START_REF] Joannides | A scaleable and defined system for generating neural stem cells from human embryonic stem cells[END_REF], indicating that a low O 2 environment does not adversely affect acquisition of a neural fate. Additionally, under basal conditions, significantly more NPCs were generated at 3% than 20% O 2 , with a greater proportion of viable cells; a finding consistent with a previous study based on feeder and matrigel-maintained human ES cells, reporting a decrease in parthanatic cell death in neurectoderm derived at 3% O 2 [START_REF] Cimadamore | Nicotinamide rescues human embryonic stem cell-derived neuroectoderm from parthanatic cell death[END_REF]. In agreement with studies on mouse ES cells [START_REF] Clarke | Low oxygen enhances primitive and definitive neural stem cell colony formation by inhibiting distinct cell death pathways[END_REF] and cortical NSCs [START_REF] Chen | Oxygen tension regulates survival and fate of mouse central nervous system precursors at multiple levels[END_REF][START_REF] Pistollato | Oxygen tension controls the expansion of human CNS precursors and the generation of astrocytes and oligodendrocytes[END_REF], the addition of N-acetyl-l-cysteine (NAC) to neuralising conditions at 20% O 2 could partly reproduce the beneficial effect of low O 2 , suggesting that reactive oxygen species contribute to cell death during neural conversion. Furthermore 3% O 2 did not prevent or delay neuronal or glial differentiation of hESC-NPCs, and in particular, the speed of electrophysiological maturation of neurons was remarkably similar in both the low and high oxygen environments.

The finding of an interaction between HIF-1α and Notch, promoting the stem cell state and inhibiting differentiation (including into β-III tubulin positive neurons), was based on observations on the embryonic carcinoma line P19, myogenic C2C12 cells and embryonic rat NSCs cultured at 1% oxygen, for a matter of hours [START_REF] Gustafsson | Hypoxia requires notch signaling to maintain the undifferentiated cell state[END_REF]. This is a rather more extreme pO 2 than that found physiologically, and the time course examined does not exclude a later down regulation of HIF-1α. Indeed, we found that HIF-1α was only transiently stabilised, appearing within 6 hours and disappearing completely by 6 days, correlating to the lag phase in the growth curve at 3% O 2 . In the majority of previous reports, the time course of the HIF response has not been fully examined, simply demonstrating the stabilisation of HIF-1α upon transfer to low O 2 conditions at up to 3 days [START_REF] Zhao | Hypoxia-driven proliferation of embryonic neural stem/progenitor cells--role of hypoxia-inducible transcription factor-1alpha[END_REF][START_REF] Gustafsson | Hypoxia requires notch signaling to maintain the undifferentiated cell state[END_REF][START_REF] Akundi | Hypoxia alters cell cycle regulatory protein expression and induces premature maturation of oligodendrocyte precursor cells[END_REF], but the later disappearance of HIF-1α has been commented upon twice before, in studies based on hESCs [START_REF] Forristal | Hypoxia inducible factors regulate pluripotency and proliferation in human embryonic stem cells cultured at reduced oxygen tensions[END_REF][START_REF] Cameron | Activation of hypoxic response in human embryonic stem cell-derived embryoid bodies[END_REF]. We propose that this transient HIF-1α stabilisation represents an adaptive response to the low O 2 , whereas HIF-2α maintains a physiological response [START_REF] Forristal | Hypoxia inducible factors regulate pluripotency and proliferation in human embryonic stem cells cultured at reduced oxygen tensions[END_REF]. This later appearance and persistence of HIF-2α may also contribute to the increased numbers of NPCs generated at 3% O 2 : HIF-2α inhibits the p53 pathway and also regulates SOD1, SOD2, GPX1 and CATALASE expression [START_REF] Bertout | HIF2alpha inhibition promotes p53 pathway activity, tumor cell death, and radiation responses[END_REF], so is well placed to modulate the survival of NPCs. Moreover, HIF-2α appears to have a critical role in the proliferation of neuroblastoma cells [START_REF] Bertout | HIF2alpha inhibition promotes p53 pathway activity, tumor cell death, and radiation responses[END_REF], and the ability of our system to isolate the downstream effects of HIF-2α from HIF-1α could provide further insights into this observation as well as into the mechanisms of maintenance of endogenous NSCs.

In addition to our finding of efficient neural conversion and tri-lineage differentiation we also observed that 3% O 2 allowed long-term maintenance of healthy, mixed differentiated cultures for over 3 months, in the absence of any exogenous neurotrophins that are typically required in cultures differentiated at 20% O 2 for considerably shorter periods such as to 28 days [START_REF] Erceg | Human embryonic stem cell differentiation toward regional specific neural precursors[END_REF][START_REF] Hanson | Cyclic AMP elevation is sufficient to promote the survival of spinal motor neurons in vitro[END_REF]. This observation is supported by a report that mouse cortical neurons thrive at 1% O 2 , with enhanced survival at 7-14 days in comparison to those at 20% O 2 [START_REF] Li | Physiological hypoxia promotes survival of cultured cortical neurons[END_REF]. The establishment of viable long-term cultures provides a unique opportunity to study the development of human neurons and glia over a much greater time course than previously possible. Taken alongside our report of successful neuronal sub-type specification at 3% O 2 to both midbrain dopaminergic and spinal motor neurons, this system will provide a more physiologically relevant model to investigate disease processes in vitro, a major avenue of research for both ES and induced pluripotent stem cell (iPS) neural cell derivatives. This is of particular importance to both Motor Neuron and Parkinson's disease, where oxidative stress has been implicated in neuronal injury [START_REF] Trotti | Glutamate transporters are oxidant-vulnerable: a molecular link between oxidative and excitotoxic neurodegeneration?[END_REF][START_REF] Behl | Oxidative nerve cell death in Alzheimer's disease and stroke: antioxidants as neuroprotective compounds[END_REF].

The finding that an in vitro simulation of the oxygen challenge that occurs in cell transplantation studies resulted in significant cell death, in contrast to maintenance at 3% oxygen, is of considerable interest, and is reflected in the emerging concept of 'hypoxic' preconditioning prior to transplanting stem cells or their derivatives [START_REF] Rosova | Hypoxic preconditioning results in increased motility and improved therapeutic potential of human mesenchymal stem cells[END_REF]. Whilst neuronal and glial populations cultured at 20% O 2 do survive and achieve functional improvements after transplantation (3), our findings warrant comparison of survival and phenotypic potential of transplanted NPCs derived from high and low oxygen environments.

Conclusion:

A gradual shift in the long held view that low oxygen equates to hypoxia has led to the realisation that in fact it often represents in situ normoxia [START_REF] Ivanovic | Hypoxia or in situ normoxia: The stem cell paradigm[END_REF]. The 3% O 2 system described in this study affords a novel approach for the generation of functional, defined cell types for in vitro and in vivo disease modeling and provides a platform for future studies exploring the therapeutic effects of cell based therapies for neurological disorders.

Materials and methods

Cell culture

H9 hESCs (Madison, WI) were maintained in feeder-free conditions in CDM (50% IMDM (Invitrogen), 50% F12 (Invitrogen), insulin 7µg/ml (Roche), transferrin 30µg/ml (Roche), bovine serum albumin 5mg/ml (Sigma), 1% lipid 100x (Invitrogen) and monthioglycerol 450µM (Sigma)), supplemented with 12ng/ml FGF-2 (R&D Systems) and 10ng/ml Activin, between passages 82-96. All cultures were supplemented with penicillin and streptomycin (Invitrogen). 6 well plates (Nunc) were coated overnight with MEF-medium and colonies were passaged with collagenase 1mg/ml (Invitrogen) every 3-5 days. HUES-9 cells (hES facility, Harvard University, Cambridge, MA), between passages 30-40, were grown on irradiated mouse embryonic fibroblast feeders, supplemented with 10ng/ml FGF-2, 10ng/ml activin and 10ng/ml insulin.

For neural conversion, colonies were lifted off with liberase 125µg/ml (Roche), incubation time 15-20mins, allowed to settle in 15ml tubes and rinsed with CDM before chopping with a McIlwain Tissue Chopper (Mickle Engineering, Gomshall, UK) at 120µm distances in two directions perpendicular to each other . Resulting cellular aggregates were then grown in repellent tissue culture flasks (Nunc) at a density of approximately 200,000 cells/ml of CDM (+/-1µM N-acetyl cysteine (Sigma) at 20% O 2 ), in the absence of growth factors. At this stage, cells were cultured either in a standard 20% O 2 and 5% CO 2 incubator or in a 3% O 2 and 5% CO 2 incubator, with oxygen displaced by nitrogen. The resultant spheres were fed every other day (50% media change) and chopped again at day 10 prior to transfer to an orbital shaker, to prevent aggregation. From D12 onward 20ng/ml FGF-2 plus heparin 5mg/ml (Sigma) was added to the basal CDM, with media changes every 2-3 days and mechanical passaging approximately every 10 -14 days. For regional specification, FGF-2 was added between days 12-15 and morphogens applied as described in the text. For terminal differentiation, NPCs were plated onto poly-D-lysine / laminin 10µg/ml (Sigma) coated coverslips and cultured in DMEM / 2% B27 (Invitrogen)/ 1% penicillinstreptomycin for up to 3 months, with a 50% medium change every 2-3 days. Tetrodotoxin (Tocris) was applied as indicated in the text; all other reagents come from Sigma.

Quantification and Statistical Analysis

All experiments were performed at least three times, unless otherwise stated. A student's unpaired t-test was used for statistical analysis. P values of <0.05 were considered significant; data are presented as mean ± standard error of the mean (SEM).

Titles and Legends to Figures:

Figure 1 :Figure 2 :

 12 Figure 1: hESCs can generate NPCs in 3% O 2 .

Figure 3 :

 3 Figure 3: NPCs generated at 3% O 2 differentiate and mature into glia and electrically

Figure 4 .

 4 Figure 4. 3% NPCs can be differentiated into midbrain dopaminergic and spinal motor

Acknowledgements:

We would like to thank James Raleigh for his generous gift of pimonidazole (Hypoxyprobe) and antibody, Ludovic Vallier for his kind provision of feeder-free H9 ES cultures and Rickie Patani, Kristine Westmore and David Story for valuable technical assistance. This work was supported by the MS Society UK, Evelyn Trust, MRC (UK), Wellcome Trust (AL) and Royal Society (RK). SS is supported by a Sir David Walker Fellowship, a joint Medical Research Council and Multiple Sclerosis Society Clinical Research Training Fellowship (no. G0800487) and a Raymond and Beverly Sackler Studentship.

Oxygen switch experiments

NPCs were dissociated with Accutase (Sigma) and plated at 40,000 cells per coverslip in 30µl plating medium, to allow adherence. 500µl plating medium was added after 30mins. For live-dead staining, cells plated for 48hrs were incubated for 10mins on ice with 4µM of calcein and ethidium bromide (Invitrogen) in dPBS (Invitrogen). 4 random fields from each of 3 coverslips in each group were counted (on an inverted microscope), on 3 occasions.

RNA isolation, RT-PCR and Immunoblotting

These were carried out according to standard procedures (detailed in 'Supplementary Information'). Primer sequences are contained in supplementary table 1.

Immunocytochemistry and Flow Cytometry:

Immunocytochemistry and flow cytometry were performed using standard protocols (see 'Supplementary Information'). Details of primary antibodies used are contained in Supplementary Table 2.

Electrophysiology

Whole-cell current-clamping of neurons was performed at room temperature, using glass micro electrodes of 3-6 MΩ resistance containing an internal solution consisting of 130 mM potassium gluconate, 4 mM NaCl, 10 mM HEPES, 10 mM BAPTA, 4 mM MgATP, 0.5 mM Na 2 GTP, 0.5 mM CaCl 2 , 2 mM K-Lucifer yellow (pH adjusted to 7.3 with KOH). Series resistance was 6-14 MΩ. Cultures were superfused with HEPESbuffered external solution containing 144 mM NaCl, 2.5 mM KCl, 10 mM HEPES, 1 mM NaH 2 PO4, 2.5 mM CaCl 2 , 10 mM glucose, 2mMgCl 2 , pH set to 7.35 with NaOH, bubbled with medical oxygen. Traces were corrected for -14mV junction potential.

Conflict of Interest:

The authors declare no conflict of interest.

Supplementary material:

Supplementary information is available at the website of Cell Death and Differentiation. Representative ethidium bromide and calcein staining of NPCs plated for differentiation for 48hrs at 3% O 2 in the absence of growth factors; cells from the same field of view are shown. Cell counts showed that these differentiating cultures survived better at 3% than at 20% oxygen, with those that were swapped from 20% to 3% faring worst of all. P-values * = 0.0033, ** = 0.0024, *** = 0.0001. Supplementary Figure 1: (A): The difference in viability between the NPCs derived at 3% and 20% oxygen was apparent through direct observation, with cells grown at 3% appearing more rounded and phase bright, with fewer dead cells in the background. (B): Addition of 1μM NAC to the 20% cells was protective, with 75.5% viable cells at D14, falling in between the 20% and 3% population. The cell only sample (in the absence of annexin V and PI) is included as a negative control. Supplementary Figure 2: Electrophysiological maturation at 20% oxygen.