Benoit Combemale
email: benoit.combemale@irisa.fr

Olivier Barais
email: barais@irisa.fr

Omar Alam
email: omar.alam@mail.mcgill.cajoerg.kienzle@mcgill.ca

Jörg Kienzle

Using CVL to Operationalize Product Line Development with Reusable Aspect Models

This paper proposes a software design modelling approach that uses the Common Variability Language (CVL) to specify and resolve the variability of a software design, and the aspect-oriented modelling technique Reusable Aspect Models (RAM) to specify and then compose the detailed structural and behavioural design models corresponding to the chosen variants. This makes it possible to 1) exploit the advanced modularization capabilities of RAM to specify a complex, detailed design concern and its variants by means of a set of interdependent aspect models; 2) use CVL to provide an easy-to-use product-line interface for the design concern; 3) automatically generate a detailed design model for a chosen variant using a custom generic CVL derivation operator and the RAM weaver.

INTRODUCTION

A well-established and convenient practice in variability management is to provide a specification of the variability in terms of features separately from the associated artifacts that provide an implementations of the actual reusable assets. While some de-facto standards such as feature diagrams [START_REF] Kang | Feature-oriented domain analysis (foda)[END_REF] are widely used to represent commonality (i.e., common properties) and variability (i.e., differences) of a system in terms of features, they still have to rely on specific operators to modularly implement and then compose the reusable aspects.

In this paper, we propose an approach combining the Common Variability Language (CVL) [START_REF] Fleurey | Standardizing variability -challenges and solutions[END_REF] to specify and resolve the variability of a software design, and the aspectoriented modelling technique Reusable Aspect Models (RAM) [START_REF] Kienzle | Aspect-Oriented Multi-View Modeling[END_REF] to implement and compose reusable object-oriented software design aspects. We use RAM to describe and compose the assets, while the feature model and its resolution (which are currently not explicit in RAM) are made explicit using CVL.

The contribution of this paper is therefore twofold: On the one hand we show how CVL can be used to extend an existing approach for AOM with well-established practices coming from the variability management community. On the other hand we illustrate the use of CVL with specific modularity and composition operators tailored to work with an aspect-oriented modelling technique. The derivation operator of CVL is specialized to work with RAM, resulting in an implementation of a generic opaque variation point that can produce the composition directives allowing the RAM weaver to produce a woven model corresponding to the chosen configuration.

The remainder of the paper is structured as follows: section 2 presents an overview of the proposed approach; section 3 illustrates the details of the approach by means of a software design concern product line for workflow executions; section 4 presents related work and the last section draws some conclusions.

APPROACH OVERVIEW: COMBINING CVL AND RAM

The approach we propose is based on RAM and CVL. Reusable Aspect Models (RAM) [START_REF] Kienzle | Aspect-Oriented Multi-View Modeling[END_REF] is a modelling approach that allows a designer to specify models that define the structure and behaviour of recurring design solutions. RAM models are inherently reusable, which means that it is possible to customize the generic design solution models to application-specific needs when applying them in a software design of a specific application. Currently, the RAM tool comes with a growing library of reusable design models, including models for low-level utility concerns, design patterns, network communication, workflow definition and execution, and transactions.

In general, there is no one single good way to solve a specific design problem. This is why RAM supports the definition of families of interrelated design models -in RAM terminology called concerns -that describe different variations of how to address a design problem. Typically, at the core of such a design concern is at least one aspect model that encapsulates the structure and behaviour common to all variations. Additional structure and behavioural properties covering variations of the design are modelled within extensions to that core model.

The Common Variability Language (CVL) 1 [4] is a domain-independent language for specifying and resolving variability over any instance of any MOF-compliant metamodel. Inspired by feature model, CVL contains several layer. The Variability Abstraction Model (VAM ) is in charge of expressing the variability in terms of a tree-based structure. The core concepts of the VAM are the variability specifications (VSpecs). The VSpecs are nodes of the VAM and can be divided into three kinds: Choices, Variables and Classifiers. The Choices are VSpecs that can be resolved to yes or no (through ChoiceResolution), Variables are VSpecs that requires a value for being resolved(VariableValue) and Classifiers are VSpecs that imply the creation of instances and then providing per-instance resolutions (VInstances). In this paper, we mainly use the Choices VSpecs, which can be intuitively compared to features, which can or cannot be selected during the product derivation (yes/no decision). Besides the VAM, CVL also contains a Variability Realization Model (VRM). This model provide a binding between the base model and the VAM. It makes possible to specify the changes in the base model implied by the VSpec resolutions. These changes are expressed as Variation Points in the VRM. The Variation Points capture the derivation semantics, i.e. the actions to perform during the Derivation. Finally, CVL model contains resolution models to fix the variability capture in the VAM.

In this paper, we propose an approach combining CVL to specify and resolve the variability, and RAM to implement and compose reusable object-oriented software aspects. We use RAM to describe and compose the assets while the feature model and its resolution (which are currently not explicit in RAM) are made explicit using CVL.

The global approach is a two-level process: first the reusable aspects are capitalized and their possible combinations are captured in a variability model. Second, the variability model is used to select an expected set of features (aka configuration) from which a woven model is produced by composition of the suitable reusable aspects.

In practice, as illustrated in Figure 1, the approach is divided into the following five steps: implementation of the reusable aspects using RAM; specification of the variability (called variability abstract model ) using the choice diagram proposed by CVL; resolution of the variability by selecting a set of features (called resolution model ) using CVL; derivation of the composition directives using the generic CVL derivation operator, with a dedicated opaque variation point that we propose (and include in the variability realization model ); composition of the corresponding reusable aspects as describe in the composition directives with the RAM weaver.

From a methodological perspective, we also distinguish two roles for users of our approach:

• Design Concern Expert. The design concern expert knows the domain captured in the reusable aspects and knows their possible combination. This person is thus in charge of leveraging this domain to model one or several reusable aspects with RAM (step in Figure 1), and the respective variability with CVL (step in Figure 1).

• Application Engineer. The application engineer create models in the application domain. These users, through their modelling activities, can select the expected features with CVL (step in Figure 1), and then automatically derive the composition directives (step in Figure 1) used by RAM to automatically generate the corresponding woven model (step in Figure 1).

APPROACH DETAILS: THE WORKFLOW CASE STUDY

A workflow is a set of operations that need to be completed in a certain order to fulfill a goal or task. For example, workflows have been used in software engineering to describe how a system under development is to interact with its environment.

At run-time, a system that is supposed to behave according to a workflow specification needs to incorporate a workflow execution engine in its design. To facilitate this, we have designed a reusable workflow design concern in RAM that provides such a functionality. Since workflows can be of varying sophistication, we designed a product line of workflow execution engines, which allows the designer to choose the most appropriate configuration for his specific application.

In this section of the paper we illustrate our process in detail by means of the workflow design concern case study. The following subsections correspond to the steps outlined in section 2.

Designing Reusable Assets using RAM

This subsection outlines the detailed design of the aspect models that are part of a RAM design concern. For space reasons, all presented models are simplified versions of the real RAM workflow design concern models: only the classes relevant to the definition of the different kind of nodes found in a workflow are shown. Structure related to the execution of the workflow, as well as all the sequence diagrams describing the behaviour of the design have been omitted. The in-

The Core Workflow Aspect Model

Figure 2 shows the WorkFlow aspect which defines the minimal model elements found in every workflow. It states that a basic workflow is composed of nodes, which can be sequence or control flow nodes. A special sequence node is the StartNode, a special control flow node is the EndNode. The WorkFlowNode has two abstract methods, depositToken() and addNextNode(WorkFlowNode n), which are implemented differently by the two subclasses. This allows other parts of the system to treat workflow nodes in a uniform way. For example, the work flow execution engine (not shown for space reasons) can deposit a token into any kind of node in order to execute it, whether it is a SequenceNode or a Con-trolFlowNode. The design that encodes that a Workflow is composed of zero or more WorkflowNodes is implemented by another reusable aspect model, ZeroToMany. The instantiation directive on the bottom of Figure 2 instantiations that tell the RAM weaver how to compose ZeroToMany with Workflow.

With this base workflow aspect, a user can build very simple, sequential workflows. Application-specific actions are to be designed by extending |CustomizableNode, a class that executes the execute method when a token is deposited before scheduling the next node.

Workflow Extensions

The RAM workflow concern provides additional aspect models that define more elaborate control flows. For instance, there are control flow nodes that have multiple successor nodes, where each outgoing path is named using a string. Figure 3 shows an aspect called OutPath that extends the WorkFlow aspect and defines the structure needed for control flow nodes with more than one named outgoing path in the class |CFNWithOutPath. A new kind of named sequence node is introduced, OutpathNode, and a new kind of control flow node, |CFNWithOutpath. Internally, the Map aspect is reused to define a hash table that maps strings to OutpathNode as shown by the instantiation directives at the The RAM workflow design concern defines many other workflow extensions, which can unfortunately not be shown here for space reasons. They are:

• ConditionalExecution, which allows for selective execution of workflows;

• Synchronization, which allows concurrent workflows to wait for each other;

• Conditional Synchronization, which allows concurrent workflows to wait for each other conditionally;

• Timed Synchronizaton which allows concurrent workflows to wait for each other until a timer expires;

• Input, which allows workflows to wait for input messages coming from the network;

• Output, which allows workflows to send output messages to the network;

• Nesting, which makes hierarchies of workflows possible.

Each extension is designed in one aspect model that extends the base workflow model, and optionally depends on other models to provide lower-level functionality. The left hand side of Figure 5 shows an overview of all the aspect models of the workflow design concern and their dependencies (shown using black straight arrows). The aforementioned extensions are above the Workflow aspect, since they add structure and behaviour to the latter. At the bottom of the figure, below the Workflow aspect, are all the low-level design models implementing design patterns (e.g. Singleton), recurring data structures (e.g. ZeroToMany, Stack or Map), or utility aspects (e.g. NetworkCommand ).

Specifying the Variability using CVL

To use a RAM design concern within an application model, the designer first needs to perform careful tradeoff analysis that takes into account the functional and non-functional requirements of the application under development to determine the desired concern variation. Then the designer needs to add instantiation directives into the application model that instantiate the RAM aspect models that correspond to the desired design concern variation. As a result, the RAM weaver can then compose the design concern models with the application model to yield the complete design model.

Unfortunately, this reuse process is quite cumbersome for the designer. In the current version of RAM, a design concern family does not have a well-defined user interface. The designer is confronted with a collection of interdependent aspect models as shown on the left hand side of Figure 5. Once the designer has determined which features of the design concern are relevant to her, she must determine which RAM models contain the design of these features, and then she must manually instantiate them.

To ease the task of the application developer, we propose to use the CVL Variation Model (VAM) to present a simpleto-use, feature-oriented view of a RAM design concern to the user. It encodes the set of choices and the constraints between choices. For the workflow example, we obtain the choice model as depicted on the right hand side of Figure 5. This choice model is quite simple, it contains a root choice W orkf low with eight optional sub-choices and one mandatory choice saveM odel.

The mapping between the choice model and the RAM aspects, illustrated using red arrows in Figure 5, is designed in the CVL Variability Realization Model (VRM). The VRM contains ten Opaque Variation Points (OVP). An OVP is a black box variation point whose behaviour is defined with an action language expression specified in the CVL model. In our CVL implementation, we currently support OVPs defined in Groovy 3 , in Javascript or in Kermeta [START_REF] Jézéquel | Model Driven Language Engineering with Kermeta[END_REF]. With these action languages, the designer can modify the base model directly. Each variation point has access to a context that contains the list of objects to remove (toRemove), the list of objectHandles associated with this variation point (ctx), the list of variables and their associated value defined in the the resolution model (args), and a map of key/value pairs that variation points can use to pass data to subsequent variation points (map).

3 http://groovy.codehaus.org/ Among the ten OVPs, there are only three different types. The first one is bound to the root choice and contains the action language expression to create a RAM aspect that defines the composition directives. The second one is bound to the SaveM odel choice and contains the action language expression to save the final model. The eight remaining OVPs are bound to RAM aspects and contains the code to create the composition directive in the root aspect previously created. The action language expressions for these three types of OVPs are shown in Listing 1. a s p e c t c r e a t e . setName ( a r g s . g e t ( "name " ) ) ; 4 maps . put ( " NewAspect " , a s p e c t c r e a t e ) ; 56// E x p r e s s i o n f o r t h e e i g h t OVP a s s o c i a t e d t o RAM a p s e c t s 7

Listing 1: Worflow OVPs in Groovy

A s p e c t n e w a s p e c t = maps . g e t ( " NewAspect " ) ; 8

I n s t a n t i a t i o n i n s t = ca . m c g i l l . c s . s e l . ram . RamFactory . eINSTANCE . c r e a t e I n s t a n t i a t i o n ( ) ; 9 i n s t . s e t T y p e ( ca . m c g i l l . c s . s e l . ram . I n s t a n t i a t i o n T y p e .EXTENDS) ; 10 i n s t . s e t E x t e r n a l A s p e c t ( c t x . g e t ( 0 ) ) ; 11 n e w a s p e c t . g e t I n s t a n t i a t i o n s ( ) . add ( i n s t ) ; 12 ------------------------------------------

Specifying the Resolution using CVL

With the CVL VAM model, the application engineer can do the selection according to the variability captured in the VAM model. For this point, we automatically generate an initial solution for the resolution model according to the choice model constraints (cardinalities, isImpliedByP arent, Def aultResolution, . . . ). The application engineer can change this choice resolution decision using a graphical tool as shown in Figure 6. In this example, we take the decision to select

Derivation of a Software Design using CVL

The tool screenshot shown in Figure 7 illustrates the entire derivation process. Arrows a) and b) depict the VAM model. Arrow c) shows a choice resolution. Arrow d) shows an OVP which specifies the mapping to a specific RAM model (object handle shown in arrow e)). Based on this model, the derivation engine executes the code of the OVPs selected by the choice resolution. The expression for the OVP bound to the RAM aspect is generic and can be used as such for new aspects than can be composed. Only the object handle that maps to the concrete aspect must be respecified. The CVL derivation engine and the workflow models can be downloaded on our website5 .

Composing the Reusable Assets using RAM

Using the generated aspect model as input, the RAM weaver recursively composes all dependent aspect models to generate a complete design concern model that corresponds to the selected configuration. In our case we selected the ParallelExecution, ConditionalExecution and Synchronization variants. As a result, the RAM weaver will compose ParallelExecution, ConditionalExecution, Synchronization, as well as the dependent aspects Outpath, Inpath, Key-Counter, Named, ZeroToMany, and Map.

The resulting structural model is shown in Figure 8

Discussion on the Use of OVPs

We could probably design the VRM to use other kinds of variation points such as ObjectSubstitution or FragmentSubstitution. Nevertheless, in combining AOM and CVL, the composition complexity is primarily in the aspect weaver. Consequently, the derivation engine remains simple. Consistency checking would, for instance, be part of the aspect weaver, undisturbed by the presence of the OVP. Additionally, we were able to build a generic OVP to create the aspect composition directives. Checking this OVP only needs to be done once. The CVL model could be further improved by using ConfigurableUnit and ConfigurableUnitUsage to avoid the eight repetitions of the same OVP. This will be investigated in future work.

RELATED WORK

This section discusses related work on aspect-oriented modelling and variability modelling approaches.

Because composing models by hand is a cumbersome task, tools and approaches have been proposed that automate significant parts of model composition [START_REF] Jeanneret | An analysis of model composition approaches[END_REF]. The early aspect [START_REF] Pinto | Early aspects at icse 2008: workshop on aspect-oriented requirements engineering and architecture design[END_REF] and aspect-oriented modelling [START_REF]Aspect-Oriented Modeling Workshop Series[END_REF] workshop series introduced many aspect-oriented modelling (AOM) approaches that differ on (i) model composition activities, (ii) the types of inputs required other than the 2 models to be composed and (iii) the types of models that can be com-posed. We chose RAM in this work because of its ability to compose structural and behavioural models and because the existence of large aspect-oriented models that capture variabilities. Indeed, RAM has been applied to model many software designs concerns. The biggest design concern is the AspectOptima case study, a transaction support middleware product line [START_REF] Kienzle | Aspect-Oriented Multi-View Modeling[END_REF][START_REF] Kienzle | AspectOPTIMA: A Case Study on Aspect Dependencies and Interactions[END_REF]. AspectOptima offers support for multiple transaction models (flat, nested, multithreaded and open multithreaded transactions), different concurrency control strategies (pessimistic lock-based and optimistic time stamp-based), and different update strategies (inlace and deferred update). However, even if we used RAM in this paper, the proposed approach should also work for other AOM approaches.

Many formalisms were proposed in the past decade for variability modeling. For an exhaustive overview, we refer the readers to the literature reviews that gathered variability modeling approaches [START_REF] Pohl | Variability management in software product line engineering[END_REF][START_REF] Hubaux | A preliminary review on the application of feature diagrams in practice[END_REF][START_REF] Benavides | Automated Analysis of Feature Models 20 Years Later : A Literature Review 6[END_REF][START_REF] Rabiser | Requirements for product derivation support: Results from a systematic literature review and an expert survey[END_REF][START_REF] Chen | A systematic review of evaluation of variability management approaches in software product lines[END_REF]. All formalisms for variability modeling could be used following the approach we introduce in this paper. In our case, we use the choice diagram proposed by CVL, very similar to an attributed feature diagram with cardinalities.

More recently, few works deal with the binding between the feature in the variability modeling and the actual assets. Let us cite for example FeatureMapper [START_REF] Heidenreich | FeatureMapper: Mapping Features to Models[END_REF], which is a tool for combining SPL and MDE that makes it possible to bind a feature to a design model. Relying on CVL, we use in our approach the provided action language to describe in the realization model the binding between the features in the choice model and the actual RAM assets. A dedicated derivation operator is automatically obtained by implementing a dedicated opaque variation point in the CVL generic derivation operator.

Recently, several works have shown the benefits of coupling aspect-oriented modelling approaches and variability approaches. Voelter et al. [START_REF] Voelter | Product line implementation using aspect-oriented and model-driven software development[END_REF] was the first to combine AOM and MDE techniques to achieve an explicit separation of concerns in software product lines. In the domain of software architecture, we cite Morin et al. [START_REF] Morin | Taming dynamically adaptive systems using models and aspects[END_REF] and Parra et al. [START_REF] Parra | Unifying design and runtime software adaptation using aspect models[END_REF] that combine architecture aspect models and feature models to ease the design of adaptive systems. In [START_REF] Perrouin | Reconciling Automation and Flexibility in Product Derivation[END_REF], Perrouin et al. proposes to specify variants by means of model fragments and the product derivation process consists in merging those fragments together.

CONCLUSION

In the Reusable Aspect Models approach, a design concern is a collection of interrelated aspect models describing a family of design solutions for a specific design problem. This paper showed how we used CVL to specify an easy-touse product line interface for RAM design concerns. When faced with a specific design problem for which a RAM design concern exists, an application developer can consult the variation model provided by CVL to get an overview of all possible design choices. After making her choice, she passes the resulting resolution model to the CVL derivation engine, which knows about how to map features to RAM aspect models. Based on this knowledge, the derivation engine automatically creates an aspect that instantiates all the aspect models that are needed for the designer. Using this aspect, the RAM weaver generates a complete model of the chosen design concern configuration.

Figure 1 :

 1 Figure 1: Approach Overview Combining CVL and RAM

Figure 2 :

 2 Figure 2: The Base Workflow Aspect Model

2Figure 4 :

 4 Figure 3: The Outpath Aspect Model

Figure 5 :

 5 Figure 5: The Workflow Design Concern (RAM Models and Dependencies) and the CVL Variation Model

1

  // E x p r e s s i o n f o r CreateAspectWorkflow OVP 2 A s p e c t a s p e c t c r e a t e=ca . m c g i l l . c s . s e l . ram . RamFactory . eINSTANCE . c r e a t e A s p e c t ( ) ; 3

  --13 // E x p r e s s i o n f o r SaveModel OVP 14 A s p e c t a s p e c t t o S a v e = maps . g e t ( " NewAspect " ) ; 15 R e s o u r c e S e t r e s o u r c e S e t = new R e s o u r c e S e t I m p l ( ) ; 16 r e s o u r c e S e t . g e t R e s o u r c e F a c t o r y R e g i s t r y ( ) . getExtensionToFactoryMap ( ) . put ( 17 "ram " , new XMIResourceFactoryImpl ( ) ) ; 18 URI f i l e U R I = URI . c r e a t e F i l e U R I (new j a v a . i o . F i l e ( a r g s . g e t ( "modelname " ) . g e t A b s o l u t e P a t h ( ) ) ; 19 R e s o u r c e r e s o u r c e = r e s o u r c e S e t . c r e a t e R e s o u r c e ( f i l e U R I ) ; 20 r e s o u r c e . g e t C o n t e n t s ( ) . add ( a s p e c t t o S a v e ) ; 21 r e s o u r c e . s a v e ( j a v a . u t i l . C o l l e c t i o n s .EMPTY MAP) ;

Figure 6 :

 6 Figure 6: The Workflow Resolution model

6 .

 6 Since Outpath is instantiated both by ParallelExecution and ConditionalExecution, there are also two instances of Map that map Strings to WorkflowNodes. Map is used a third time in Synchronization to map InpathNodes to Integers.

Figure 7 : 1 Figure 8 :

 718 Figure 7: The Workflow CVL model

CVL is currently a proposal submitted to OMG. Cf. http: //variabilitymodeling.org.

An EMF resources is a container of persisted model elements

[START_REF] Heidenreich | FeatureMapper: Mapping Features to Models[END_REF] http://www.irisa.fr/triskell/perso_pro/obarais/ pmwiki.php?n=App.

VARY2012[START_REF] Hubaux | A preliminary review on the application of feature diagrams in practice[END_REF] Again, only the classes directly related to the different kind of workflow nodes are shown. The structure pertaining to workflow execution (executors, execution contexts and parameters) have been omitted for space reasons.

Acknowledgements

This work has been partially supported by VaryMDE, a collaboration between Inria and Thales Research & Technology.


Simona Dobre

Thierry Bastogne

Christophe Profeta

Muriel Barberi-Heyob

Alain Richard

Limits of variance-based sensitivity analysis for nonidentifiability testing in high dimensional dynamic models

Keywords: identifiability, sensitivity analysis, nonlinear systems, dynamical systems, systems biology

In systems biology, a common approach to model biological processes is to use large systems of nonlinear differential equations. The associated parameter estimation problem then requires a prior handling of the global identifiability question in a realistic experimental framework. The lack of a method able to solve this issue has indirectly encouraged the use of global sensitivity analysis to select the subset of parameters to estimate. Nevertheless, the links between these two global analyses are not yet fully explored.

The present work reveals new bridges between sensitivity analyses and global non-identifiability, through the use of functions derived from the Sobol' high dimensional representation of the model output. We particularly specify limits of variance-based sensitivity tools to completely conclude on global non-identifiability of parameters in a given experimental context.

Introduction

In systems biology, the inference of biological networks from quantitative properties of their elementary constituents is a major area of research [START_REF] Cassman | Systems Biology: International Research and Development[END_REF][START_REF] Kitano | Foundations of systems biology[END_REF]. This raises particular challenges such as the identification of highdimensional nonlinear dynamical systems and more precisely the analysis of their parameter identifiability [START_REF] Mirsky | Automatic Control in Systems Biology[END_REF][START_REF] Raue | Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood[END_REF][START_REF] Stelling | Robustness vs. identifiability of regulatory modules?[END_REF].

Different classifications of parameter identifiability definitions exist. We refer herein to three classes (see Fig. 1): a priori identifiability, a posteriori identifiability and practical identifiability. The first class, known also as the theoretical or structural identifiability of model parameters, examines the question of existence and uniqueness of a solution to the parameter estimation problem [START_REF] Walter | Identification of Parametric Models from experimental data[END_REF], in an idealized framework where (1) the system and model have identical structure (no characterization error); [START_REF] Brun | Practical identifiability of ASM2d parameters -systematic selection and tuning of parameter subsets[END_REF] the data are noise-free, and (3) the input signals and measurement times can be chosen at will. However, this is only a necessary condition which can not guarantee successfull parameter estimation from real data. The second class, namely the a posteriori identifiability only considers the first two working assumptions and is a particular case of the output distinguishability [START_REF] Grewal | Identifiability of linear and nonlinear dynamical systems[END_REF] for a finite collection of noise-free observations and a given input signal. The last class, practical identifiability, only relies on the first hypothesis and accounts for the noise factor but is generally established for a given estimation criterion [START_REF] Dochain | Structural identifiability of biokinetic models of activated sludge respiration[END_REF][START_REF] Vanrolleghem | Practical identifiability of a biokinetic model of activated sludge respiration[END_REF]. For that reason, this class of identifiability is often linked to the theory of optimization in mathematics.

Sensitivity analysis of the model output with respect to changes in model parameters is another technique widely used in system modeling to discriminate influential and non influential parameters [START_REF] Saltelli | Global sensitivity analysis : The primer[END_REF][START_REF] Streif | Systems Analysis of Biological Networks, chapter Steady state sensitivity analysis of biochemical reaction networks. A brief review and new methods[END_REF]. Dynamic sensitivity analysis has already been applied to biological networks for various purposes such as experimental design [START_REF] Schlosser | Experimental design for parameter estimation through sensitivity analysis[END_REF], parameter estimation [START_REF] Miller | Sensitivity analysis and parameter estimation in dynamic modeling of chemical kinetics[END_REF] or the analysis of oscillatory systems [START_REF] Rand | Mapping global sensitivity of cellular network dynamics: sensitivity heat maps and a global summation law[END_REF][START_REF] Zak | Sensitivity analysis of oscillatory (bio) chemical systems[END_REF].

Several investigations on the connections between dynamic sensitivity and parameter identifiability analyses have been carried out [START_REF] Brun | Practical identifiability of ASM2d parameters -systematic selection and tuning of parameter subsets[END_REF][START_REF] Stelling | Robustness vs. identifiability of regulatory modules?[END_REF][START_REF] Yue | Insights into the behaviour of systems biology models from dynamic sensitivity and identifiability analysis: a case study of a nf-κb signalling pathway[END_REF], but the latter were only focused on local analysis.

In genomics, proteomics or metabolomics, biological parameters may vary widely within different ranges. As a consequence, global a posteriori identifiability needs to be addressed. Unfortunately, there is no technique able to assess the global identifiability condition in a given experimental context, i.e. when the input signal and the sampling conditions are imposed by the experimental context. Consequently, authors generally prefer to apply global sensitivity analysis techniques without solid justifications related to identifiability. Indeed, while the relationship between local sensitivity and identifiability analysis, through the Fisher information matrix, is clearly established [START_REF] Brun | Practical identifiability analysys of large environmental simulation models[END_REF][START_REF] Yao | Modeling ethylene/butene copolymerization with multi-site catalysts: Parameter estimability and experimental design[END_REF][START_REF] Zak | Importance of input perturbations and stochastic gene expression in the reverse engineering of genetic regulatory networks: insights from an identifiability analysis of an in silico network[END_REF], the link between global studies is less obvious. As a matter of fact, only insensitive parameters are generally considered as being nonidentifiable. This is not surprising since global sensitivity measures usually serve as model reduction principles (before parameter estimation) or in tandem with uncertainty analysis for model robustness analysis [START_REF] Saltelli | Global sensitivity analysis : The primer[END_REF]. However, sensitive parameters could also be non-identifiable.

Identifiability Local Global a priori identifiability a posteriori identifiability practical identifiability

Choice of the model structure The objective of this paper is to present new results on the connections between global a posteriori identifiability and global dynamic sensitivity analysis. This study is structured around the Sobol' decomposition1 method [START_REF] Sobol | Global sensitivity indices for nonlinear mathematical models and their Monte-Carlo estimates[END_REF]. Specific functions, entitled Ψ and Ω-functions, derived from the Sobol' high dimensional model respresentation, are introduced. Their linear time-dependence and injectivity are examined, and their consequences on the non-identifiability of parameters are discussed. We show that variance-based sensitivity analysis can be used to test only one out of three causes of nonidentifiability in a given experimental context, where the input signal and the mesurement times are imposed. We also point out that the conclusions on parameter nonidentifiability, in the case of colinear sensitivity measures, must be treated with caution.

This paper is structured as follows: a priori and a posteriori identifiabilities are firstly defined. The global sensitivity analysis based on the Sobol' high dimensional model representation is then briefly introduced in Section 3. Finally, the main contributions of this study are presented in Section 4 in which the links between sensitivity and global identifiability analyses are decomposed, in a theoretical framework.

Identifiability analysis

Let us consider a dynamic system described by a nonlinear state-space model defined as follows:

d dt x (t) = f (x (t) , u(t), t, p); x (0) = x 0 (p) y (t, p) = h (x(t), p) (1)
where x ∈ R nx , u ∈ R nu and y ∈ R ny denote the state, input and output vectors respectively. The variable x 0 is the initial value of the state vector, p ∈ R n is the vector of model parameters and t is the time variable. f (•) and h (•) contain the state and output equations respectively.

In a priori identifiability, the solution uniqueness of the parameter estimation problem is assessed in an idealized framework [START_REF] Walter | Identification of Parametric Models from experimental data[END_REF]. However, in experimental biology, the input design is often subject to economical and/or technical constraints and the number of observations is frequently limited to a few data points collected at time instants {t k } ∈ T, k = 0, . . . , N -1. In such restrictive experimental frameworks, even if a parameter is a priori identifiable, it may not be so in practice, due to a lack of information in the available observations. The a posteriori identifiability condition can be stated as follows:

given a parametric model structure with given input signals u and initial conditions x 0 , a parameter p i , with i ∈ 1, . . . , n is a posteriori identifiable, if for almost all p * ∈ P ⊂ R n with n the number of model parameters, the following condition is satisfied

y (t k , p) = y (t k , p * ) ∀t k ∈ T =⇒ p i = p * i . (2)
The local a posteriori identifiability, corresponding to p * ∈ V (p) where V (p) denotes the neighborhood of p, is not considered here since it is not relevant to biological models, for which parameters may vary over wide ranges. Several methods exist to analyze global a priori identifiability, based on state isomorphisms [START_REF] Peeters | Identifiability of homogeneous systems using the state isomorphism approach[END_REF], differential algebra [START_REF] Audoly | Global identifiability of nonlinear models of biological systems[END_REF][START_REF] Ljung | On global identifiability for arbitrary model parametrizations[END_REF][START_REF] Saccomani | Some results on parameter identification of nonlinear systems[END_REF][START_REF] Saccomani | Parameter identifiability of nonlinear systems: the role of initial conditions[END_REF] or power series expansions [START_REF] Pohjanpalo | System identifiability based on power series expansion of the solution[END_REF][START_REF] Walter | Identification of Parametric Models from experimental data[END_REF].

Unfortunately, there is no available approach to assess a posteriori global identifiability. This lack of a practical solution has encouraged researchers in systems biology to use techniques of global sensitivity analysis [START_REF] Kontoravdi | Application of global sensitivity analysis to determine goals for design of experiments: An example study on antibody-producing cell cultures[END_REF][START_REF] Peters | Selection of morris trajectories for initial sensitivity analysis[END_REF].

Global sensitivity analysis

Several categories of sensitivity analysis methods already exist in previously published studies [START_REF] Saltelli | Global sensitivity analysis : The primer[END_REF]. Herein, we only focus on variance-based global methods and more precisely on the Sobol' sensitivity method [START_REF] Sobol | Global sensitivity indices for nonlinear mathematical models and their Monte-Carlo estimates[END_REF]. This method allows the computation of the output's sensitivity with respect to the variation of model parameters over the entire parametric domain. The sensitivity measure related to a certain parameter is evaluated while varying all other parameters as well, revealing thus any existing interaction. Hereafter, we apply this method to dynamic systems, adjusting the terminology from sensitivity indices for static systems to sensitivity functions for dynamic ones.

Hypotheses of the method:

(H 1 ) the n parameters are considered as i.i.d. random variables uniformly distributed over the ndimensional unit cube I n = [0, 1] n , i.e., p i ∼ U [0, 1], ∀i ∈ {1, . . . , n}; (H 2 ) y(t, p) is continuously differentiable and square integrable;

(H 3 ) when analyzing the global sensitivity analysis w.r.t. parameters, all other computational factors which could affect the model output, such as the simulation method, the sampling time, the input signal, etc., are not modified during the analysis.

As global sensitivity analysis with respect to parameters is generally computed independently for each model output, for the sake of readability we will consider hereafter that we are dealing with a single output model. Furthermore, knowing that we are working with dynamical systems and that we are interested in expressing the variance and sensitivity functions for specific time measurements, all the equations will be written for a specific time instant t k ∈ T (but are, of course, valid for every

t k ∈ T).
The main idea behind Sobol' decomposition is the expansion of the model output variable y(t k , p) into summands of increasing dimensionality, namely

y (t k , p) = y 0 (t k ) + n i=1 y i (t k , p i ) + n-1 i=1 n j>i y i,j (t k , p i , p j ) + . . . + y 1,...,n (t k , p 1 , . . . , p n ) (3)
∀t k ∈ T. For (3) to hold, this decomposition has the following properties:

(P 1 ) the term y 0 (t k ) must be constant w.r.t. model parameters p, i.e.

y 0 (t k ) = I n y(t k , p)dp (H1) = E [y]
with E [•] being the expectation operator. (P 2 ) the integrals of every summand over any of its dependent parameters must be zero, i.e.

V (t k ) = I n y 2 (t k , p) -y 2 0 (t k ) dp = I n n i=1 y 2 i (t k , p i ) + n-1 i=1 n j>i y 2 i,j (t k , p i , p j ) + + . . . + y 2 1,...,n (t k , p 1 , . . . , p n ) dp
Based on the hypothesis (H 1 ) and the property (P 1 ), we can further write:

V (t k ) = n i=1 V i (t k ) + n-1 i=1 n j>i V i,j (t k ) + . . . + V 1,...,n (t k ).
This equation is also known as the ANOVA-HDMR decomposition, where

V i represents the variance of the output due to the direct effect of the parameter p i , defined as

V i (t k ) = V [y i (t k , p i )] = V pi E [y |p i = p * i ] (4) with E [y |p i = p * i ]
the conditional expectation of y over all the parameters but p i which is fixed to a particular value p * i and V pi [•] the variance over the distribution of p i . V i,j represents the joint effect of the parameters p i and p j on the output y and is defined as

V i,j (t k ) = V [y i,j (t k , p i , p j )]. (5)
Analogous equations can be written for the other high-order terms in (4).

Dividing both sides of the ANOVA-HDMR decomposition in (4) by

V (t k ) gives 1 = n i=1 S i (t k ) + n-1 i=1 n j>i S i,j (t k ) + . . . + S 1,...,n (t k ) (6)
where

• S i (t k )
is the i th first-order sensitivity function (known also as the marginal sensitivity function), defined by

S i (t k ) = V i (t k )/V (t k ),
and representing the single effect of the parameter p i on the variance of the output; • and S i1,...,ir (t k ) is the r th order sensitivity function,

S i1,...,ir (t k ) = V i1,...,ir (t k )/V (t k )
representing the sensibility w.r.t. a group of parameters in interaction (also called the collective sensitivity function).

Homma and Saltelli have introduced in [START_REF] Homma | Importance measures in global sensitivity analysis of non linear models[END_REF] the definition of total sensitivity functions, regrouping the sensitivity of the model output with respect to the influence of a parameter in the different forms (direct and interactions with other parameters). These functions are defined as

S Ti (t k ) = V Ti (t k )/V (t k ), (7)
where V Ti (t k ) is the total variance in relation to the influence of parameter p i , i.e

V Ti (t k ) = V i (t k ) + n j=1 V i,j (t k ) + . . . + V 1,...,n (t k ). ( 8
)

Implications of global sensitivity and identifiability analyses

There is little reference in literature for the connections between global sensitivity analysis results (more precisely, variance-based analysis results) and parameter (non-) identifiability. Mainly, they refer to a general acknowledged link : null total sensitivity also implies the non-identifiability of the parameter in question. Nevertheless, a 'sensitive' parameter could also be nonidentifiable in practice.

This section gathers the principal theoretical results. First of all, we will introduce new functions denoted by Ψ, derived from the Sobol' decomposition and employed to characterize the complete action on the model output of a parameter p i . Then, studying the properties of the Ψ functions, we will illustrate three causes that lead to the lack of identifiability:

(a) null total sensitivity function, S Ti ; (b) non-injectivity of the Ψ i functions w.r.t. parameter p i ; (c) colinear sensitivity functions.

In order to illustrate this point, several academic examples will be introduced.

Definition of Ψ functions

Let us consider the Ψ i function representing the total effect on the model output of the parameter p i . In other terms, this function gathers all the Sobol' decomposition terms in eq. ( 3) involving the index i:

Ψ i (t k , p) = y i (t k , p i ) + n j =i y i,j (t k , p i , p j ) + . . . + +y 1,...,i,...,n (t k , p 1 , . . . , p i , . . . , p n ). ( 9
)
These functions provide the means of analyzing properties like insensitivity, non-injectivity of the model output w.r.t. the parameter p i , and the compensation of the actions of two parameters p i and p j .

Taking into account the properties (P 1 ) and (P 2 ) of the Sobol' expansion (3), we have the following property, for all t k ∈ T,

1 0 Ψ i (t k , p) dp i = 0.
Furthermore, the total variance function w.r.t. the parameter p i , can be also expressed as

V Ti (t k ) = I n Ψ 2 i (t k , p) dp. ( 10
)

Case (a): null total sensitivity functions

Even if the link between null total sensitivity functions and parameter non-identifiability is generally acknowledged, we formally address it herein with two propositions.

Proposition 4.1 A total sensitivity function, S Ti (t k ), is null iff the function Ψ i (t k , p) is also null, and that for all time measurements t k ∈ T.

Proof Variance and sensitivity functions are all positive definite. Therefore, a null total variance V Ti (t k ), or total sensitivity function S Ti (t k ), for all time measurements, will also imply the nullity of all the terms involving the index i in the Sobol' decomposition (in eq. ( 3)) [START_REF] Sobol | Global sensitivity indices for nonlinear mathematical models[END_REF], i.e.

y i (t k , p i ) = y i,j (t k , p i , p j ) = . . . = y 1,...,n (t k , p) = 0.
Considering the special construction of the Ψ i function, defined in eq. ( 9), we can conclude to the nullity of this function for all the time measurements. In practice, the parameters with a total sensitivity function inferior (for all time measurements) to an empirical threshold, are generally considered as globally nonidentifiable [START_REF] Kontoravdi | Application of global sensitivity analysis to determine goals for design of experiments: An example study on antibody-producing cell cultures[END_REF][START_REF] Saltelli | Global sensitivity analysis : The primer[END_REF][START_REF] Sobol | Global sensitivity indices for nonlinear mathematical models and their Monte-Carlo estimates[END_REF].

Example 1

Let us consider a three-parameter model defined as

y (t k , p) = (2p 1 + 4p 2 sin (2πt k )) • e -0.1t k + e -p3t 2 k (11)
with p ∈ I 3 , to illustrate the link between a null (or inferior to an empirical threshold) total sensitivity function and parameter non-identifiability. Hence, the Sobol' decomposition terms are

y 1 (t k , p 1 ) = (2p 1 -1) • e -0.1t k y 2 (t k , p 2 ) = 2 • (2p 2 -1) • sin (2πt k ) • e -0.1t k y 3 (t k , p 3 ) = e -p3t 2 k + 1 t 2 k • e -t 2 k -1
whereas all the high order summands are, in this case, null.

An a priori identifiability analysis 2 (through a Taylor series approach) concludes to the global identifiability of the three parameters in a theoretical context. The total variance functions, obtained by computer algebra, are

V T1 (t k ) = 0.33 • e -0.2t k V T2 (t k ) = 1.33 • sin (2πt k ) 2 • e -0.2t k V T3 (t k ) = 0.5 • 1 t 2 • 1 -e -2t 2 k -1 t 4 • 1 -e -t 2 k 2 .
These functions are plotted in Fig. 2, page 5, with three different sampling strategies for t k ∈ [0, 10]. This example illustrates that experimental factors as measurement sampling rates may cause a lack of parameter identifiability in practice. This latter was detected by null (or inferior to an empirical threshold) total variance functions.

Remark The converse implication of the Prop. 4.2 is false: a non-identifiable parameter can have a non null total sensitivity function. This is described in Fig. 3 by the dotted arrow 1.

Case (b): non-injectivity of the Ψ-functions

Let us consider a not null Ψ i function, non-injective w.r.t. parameter p i . In this context, the following proposition can be formulated :

Proposition 4.
Ψ i (t k , p ∼i , p i ) = Ψ i (t k , p ∼i , p * i ) , ∀t k ∈ T. (12)
Therefore, setting p = [p ∼i , p i ] ∈ P and p * = [p ∼i , p * i ] ∈ P, it follows that

y (t k , p) = y 0 (t k ) + Ψ i (t k , p) + Ψ ∼i (t k , p ∼i ) = y 0 (t k ) + Ψ i (t k , p * ) + Ψ ∼i (t k , p * ∼i ) = y (t k , p * ) ,
hence the non-identifiability of the parameter p i .

Example 2

Let us consider the following model,

y (t k , p) = (p 2 3 -2p 2 p 3 + 1 4 ) • (1 -e -t k ) + p 1 1 -e -t k p2
whereas the Ψ 3 function can be written as

Ψ 3 (t k , p) = p 2 3 -2p 2 p 3 + p 2 - 1 3 • 1 -e -t k .
Thus, y (t k , p), respectively Ψ 3 (t k , p), are both noninjective w.r.t. parameter p 3 since, for all t k ∈ T we can write

y (t k , p 1 , p 2 , p 3 ) = y (t k , p 1 , p 2 , 2p 2 -p 3 ) Ψ 3 (t k , p 1 , p 2 , p 3 ) = Ψ 3 (t k , p 1 , p 2 , 2p 2 -p 3 ) .
It is worth pointing out that the parameters p 1 and p 2 are globally identifiable.

Remark The converse implication of the Prop. 4.3, represented in Fig. 3 by the dotted arrow 2, is false: a nonidentifiable parameter can have an injective Ψ i function.

Indeed the lack of identifiability can be due to different causes and, among others, to the insensitivity of the model output with respect to parameters as emphasized in section 4.2.

Limit 1: This remark raises another question about the link of a non-injective output with respect to a parameter and the (total) sensitivity functions relative to the parameter in question. The total variance w.r.t. the parameter

p i is V Ti (t k ) = I n Ψ 2 i (t k , p) dp.
The integral operator prevents to detect the non-injectivity of Ψ-functions from the analysis of the associated sensitivity functions. This first limit about the use of the variance-based global sensitivity approaches to address non-identifiability questions is described in Fig. 3 by the dotted arrow L 1 .

Case (c): colinear sensitivity functions

If the connection between a null sensitivity function S Ti and the non-identifiability of the parameter p i seems quite straightforward, the link between colinear sensitivity functions and parameter non-identifiability is less obvious. We illustrate herein a class of colinear sensitivity functions related to non-identifiable parameters, more precisely when two Ψ i and Ψ j functions are colinear w.r.t. time.

Throughout this subsection, we will explain the different facets of the presented problem through the use of a simple example introduced below.

Example 3

Consider the following academic example

y (t k , p) = p 2 • p 3 • 1 -e -t k + p 1 • p 4 • 1-e -t k •p2 (13) with p ∈ [0, 1]
4 . In this case, the parameters p 1 and p 4 can not be mutually (a priori globally) identifiable (see section 4.4.3) and the sensitivity study reveals two colinear sensitivity functions, S T1 and S T4 (see section 4.4.4).

Definition of Ω-functions

The compensation effects study of two parameters, p i and p j , involves the use of the Ψ i and Ψ j functions, which are further decomposed as

Ψ i (t k , p) = Ω i (t k , p ∼j ) + Ω i,j (t k , p) Ψ j (t k , p) = Ω j (t k , p ∼i ) + Ω i,j (t k , p) (14)
where i = j, and Ω i represents the influence on the output y of the parameter p i independently of p j ; taking into account the property (P 2 ) of the decomposition (3), Ω i (t, p ∼j ) can be also defined as

Ω i (t k , p ∼j ) = 1 0 Ψ i (t k , p) dp j ; (15)
Ω i,j is its complementary effect on y (corresponding to the combined action on y of p i and p j ), i.e.

3 Ω i,j (t k , p) = Ψ i (t k , p) -Ω i (t k , p ∼j ) . (16)
We address here only the time dependence between only two Ψ functions, but it could be further developed in 3 Our main aim is to express the compensation on the model output y of the effects of parameters pi and pj. As the two Ψ-functions have a common part, which depends mutually on the two parameters in question, we have decomposed Ψ as the sum of two complementary Ω-functions.

order to consider the compensation on the model output of more than two parameters.

In the case of Ω-functions, we have the following properties, for l ∈ {i, j},

1 0 Ω i (t k , p ∼j ) dp i = 0; 1 0 Ω j (t k , p ∼i ) dp j = 0 1 0 Ω i,j (t k , p) dp l = 0
i.e. the three Ω-functions are orthogonal with respect to the parameters p i and p j (due to the orthogonality of the Sobol' decomposition terms in eq. ( 3)).

On the linear dependence of Ω-functions

We are interested in the description of the consequences of the time dependency of two distinct Ψ functions, on parameter non-identifiability and sensitivity colinearity.

Remark Considering three Ω-functions: Ω i , Ω j and Ω i,j , for reasons of simplicity we shall say "dependence", but in fact we mean the pairwise linear dependence of these functions.

Proposition 4.4

The time dependence of the functions Ω i (t k , p ∼j ), Ω i,j (t k , p) and Ω j (t k , p ∼i ), implies that they can be factorized as

Ω i (t k , p ∼j ) = h i (p ∼j ) • g (t k , p ∼i,j ) Ω j (t k , p ∼i ) = h j (p ∼i ) • g (t k , p ∼i,j ) Ω i,j (t k , p) = h i,j (p) • g (t k , p ∼i,j ) .
Proof Let us consider the linear dependence, with respect to time, of the functions Ω i (t k , p ∼j ) and Ω j (t k , p ∼i )

c i (p) • Ω i (t k , p ∼j ) + c j (p) • Ω j (t k , p ∼i ) = 0, ∀t k ∈ T
where c i (p) and c j (p) are non-null coefficients (parameter functions). Based on this equation, we can write the relationship between the Ω i (t k , p ∼j ) and Ω j (t k , p ∼i ) functions :

Ω i (t k , p ∼j ) = - c j (p) c i (p) • Ω j (t k , p ∼i ) (17) Ω j (t k , p ∼i ) = - c i (p) c j (p) • Ω i (t k , p ∼j ) ( 18
)
By fixing p j = 0 in (17), we can express Ω i as

Ω i (t k , p ∼j ) = - c j (p ∼j , p j = 0) c i (p ∼j , p j = 0) • Ω j (t k , p ∼i,j , p j = 0)
and furthermore as

Ω i (t k , p ∼j ) = h i (p ∼j ) • g (t k , p ∼i,j ) (19)
with g (t k , p ∼i,j ) = Ω j (t k , p ∼i,j , p j = 0) and h i (p ∼j ) = -c j (p ∼j , p j = 0)/c i (p ∼j , p j = 0).

Based on ( 18) and ( 19), and by fixing p i = 0, Ω j can be written as

Ω j (t k , p ∼i ) = h j (p ∼i ) • g (t k , p ∼i,j ) (20)
with h j (p ∼i ) defined as

h j (p ∼i ) = - c i (p ∼i , p i = 0) • h i (p ∼i,j , p i = 0) c j (p ∼i , p i = 0) .
Let us now consider the linear dependence, w.r.t. time, of the functions Ω i (t k , p ∼j ) (or equivalently Ω j (t k , p ∼i ))

and Ω i,j (t k , p)

d i (p) • Ω i (t k , p ∼j ) + d i,j (p) • Ω i,j (t k , p) = 0, ∀t k ∈ T
where d i (p) and d i,j (p) are two non-null coefficients (parameter functions). Taking into account the factorization of the Ω i (t k , p ∼j ) function in [START_REF] Raue | Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood[END_REF], we can write

Ω i,j (t k , p) = - d i (p) • h i (p ∼j ) d i,j (p) • g (t k , p ∼i,j ) ,
that is to say, the factorization of the Ω i,j function proposed in the Prop. 4.4, with h i,j (p) defined by

h i,j (p) = - d i (p) • h i (p ∼j ) d i,j (p) .
Remark If we further assume that the functions Ω i (t k , p ∼j ), Ω j (t k , p ∼i ) and Ω i,j (t k , p) are also p ∼i,j dependent, then, the same pattern of proof implies that:

Ω i (t k , p ∼j ) = h i (p i ) • g (t k , p ∼i,j ) Ω j (t k , p ∼i ) = h j (p j ) • g (t k , p ∼i,j ) Ω i,j (t k , p) = h i,j (p i , p j ) • g (t k , p ∼i,j ) . ( 21
)
This is a stronger condition than the time dependence of Ω-functions, and it will be employed to show the colinearity of sensitivity measures.

Example 3 (continued)

In order to analyze the compensation effects on the model output in (13) of the parameters p 1 and p 4 , we must firstly construct the Ψ functions

Ψ 1 (t k , p) = p 4 (2p 1 -1) (1 -e -t k •p2 ) 2 (22) Ψ 4 (t k , p) = p 1 (2p 4 -1) (1 -e -t k p2 ) 2 (23)
which can be further developed respectively as the sum of two complementary Ω-functions 25)

Ω 1 (t k , p ∼4 ) = (2p 1 -1) (1 -e -t k p2 ) 4 (24) Ω 4 (t k , p ∼1 ) = (2p 4 -1) (1 -e -t k p2 ) 4 (
Ω 1,4 (t k , p) = (2p 1 -1) (2p 4 -1) (1 -e -t k p2 ) 4 . ( 26
)
As it can be observed, the functions Ω 1 , Ω 4 and Ω 1,4 , are time and p ∼1,4 dependent, and furthermore they can be factorized as seen in eq. ( 21).

On the time linear dependence of Ω-functions and non-identifiability of p i or p j

The consequence of the time linear dependence of Ωfunctions on the parameter non-identifiability is stated below.

Proposition 4.5 The linear dependence, w.r.t. time, of the functions Ω i (t k , p ∼j ), Ω i,j (t k , p) and Ω j (t k , p ∼i ) implies the non-identifiability of p i or p j .

Proof Let us write y (t k , p) as

y(t k , p) = y 0 (t k ) + Ω i (t k , p ∼j ) + Ω i,j (t k , p) + +Ω j (t k , p ∼i ) + Ψ ∼i,j (t k , p ∼i,j )
where the function Ψ ∼i,j regroups all the terms in Sobol' expansion involving the parameters p ∼i,j . The time dependence of the functions Ω i (t k , p ∼j ), Ω i,j (t k , p) and Ω j (t k , p ∼i ) implies the factorisation of these functions, as seen in Proposition 4.4. Then, we can write y (t k , p) as

y(t k , p) = y 0 (t k ) + h (p) • g (t, p ∼i,j ) + Ψ ∼i,j (t k , p ∼i,j ) with h (p) = h i (p ∼j ) + h j (p ∼i ) + h i,j (p). Hence, for almost all p * = [p ∼i,j , p * i , p * j ] ∈ P, y (t k , p) = y (t k , p * ) ∀t k ∈ T,
implies that h(p) = h(p * ), i.e., one equation with two unknown parameters, and an infinity of possible (p i , p j ) solutions. As a consequence, it is impossible to estimate both p i and p j .

Example 3 (continued)

Since Ω 1 , Ω 4 and Ω 1,4 , defined in ( 24)-( 26), are linearly dependent w.r.t. time, from Prop. 4.5, p 1 and p 4 are mutually non-identifiable. This fact could also be seen through a Taylor series approach as stated below.

Considering the model expressed in [START_REF] Mirsky | Automatic Control in Systems Biology[END_REF], the successive derivatives of y at t 0 = 0 + satisfy a 0 (p) = 0, a 1 (p) = The second and the third parameter, which take the same values in the two solutions, are globally identifiable.

p 2 p 3 + p 1 p 4 p 2 , a 2 (p) = -p 2 p 3 -p 1 p 4 p 2 2 etc. It is easy to show that a k (p) = a k (p * ) , k = 1, 2, . . . ,
The other two, which can each take two values, are only locally identifiable.

4.4.4

On the time linear dependence of Ω-functions and total sensitivity functions Proposition 4.6 Assume that for all t k ∈ T, the functions Ω i (t k , p ∼j ), Ω i,j (t k , p) and Ω j (t k , p ∼i ) are time and p ∼i,j dependent. Then, the total sensitivity functions V Ti (t k ) and V Tj (t k ) (defined in Section 4.1) are colinear.

Proof Given Proposition 4.4 and the definition of the functions Ψ i and Ψ j , we can write:

Ψ i (t k , p) = (h i (p i ) + h i,j (p i , p j )) • g (t k , p ∼i,j ) Ψ j (t k , p) = (h j (p j ) + h i,j (p i , p j )) • g (t k , p ∼i,j ) .
Total variance functions can thus be written as

V Ti (t k ) = 1 0 h 2 i (p i ) dp i + I 2 h 2 i,j (p i , p j ) dp i,j • • I n-2 g 2 (t k , p ∼i,j ) dp ∼i,j V Tj (t k ) = 1 0 h 2 j (p j ) dp j + I 2 h 2 i,j (p i , p j ) dp i,j • • I n-2 g 2 (t k , p ∼i,j ) dp ∼i,j .
It is clear that V Ti (t k ) and V Tj (t k ) are time-colinear. Furthermore, as V (t k ) is a non-null function, this directly implies the colinearity of the total sensitivity functions S T1 (t k ) and S T2 (t k ) as well.

Remark By integrating over the parameters p ∼i,j the equation of (time and p ∼i,j ) dependence between Ω i (t k , p ∼j ) and Ω j (t k , p ∼i )

α i (p i , p j ) • Ω i (t k , p ∼j ) + α j (p i , p j ) • Ω j (t k , p ∼i ) = 0
we obtain the linear dependence w.r.t. time of y i and y j functions

α i (p i , p j ) • y i (t k , p i ) + α j (p i , p j ) • y j (t k , p j ) = 0 as I n-2 Ω i (t k , p ∼j ) dp ∼i,j = y i (t k , p i )
I n-2 Ω j (t k , p ∼j ) dp ∼i,j = y j (t k , p j ) due to the property (P 2 ) of the Sobol' decomposition (3). We can then conclude (by the same pattern of proof as in Prop. 4.4 and 4.6) to the colinearity of the first-order variance functions, V i and V j (as defined in (4)).

Example 3 (continued)

Let us continue the analysis of the model defined in [START_REF] Mirsky | Automatic Control in Systems Biology[END_REF]. Firstly, recall that the functions Ω 1 , Ω 1,4 and Ω 4 are linear dependent w.r.t. time as emphasized in ( 24)- [START_REF] Im Sobol | Sensitivity analysis for non-linear mathematical models[END_REF]. The first-order and total variances of the model output w.r.t. the parameters p 1 and p 4 are defined as

V 1 (t k ) = V 4 (t k ) = (1-tk-e -t k ) 2 48•t 2 k V T1 (t k ) = V T4 (t k ) = 4•e -t k -e -2t k +2•t k -3 72•t k whereas the second-order variance function V 1,4 (t k ) is V 1,4 (t k ) = (1 -t k -e -t k ) 2 48 • t 2 k .
This example shows that time and p ∼i,j dependent Ωfunctions lead to colinear (first-order and total) sensitivity functions.

Limit 2: Conversely to the general idea in local identifiability analysis -where the non-identifiable parameters correspond to null or colinear local sensitivity functions -in global sensitivity analysis colinear sensitivity functions will not lead necessarily to parameter nonidentifiability. Indeed, the converse of the Proposition 4.6 is not always true. This second limit, noted L 2 in Fig. 3, prevents to conclude surely on non-identifiability from the colinear analysis of sensitivity functions. Hereafter, we give two counterexamples in order to illustrate these cases.

Counterexample 1

Let us consider the two parameter model represented by Furthermore, the Ω-functions, as defined in section 4.4.1, can be expressed as

y(t k , p) = e -(p1+p2
Ω 1 (t k , p ∼2 ) = y 1 (t k , p 1 ) = (tke -p 1 t k -1+e -t k )(1-e -t k ) t 2 k Ω 2 (t k , p ∼1 ) = y 2 (t k , p 2 ) = (tke -p 2 t k -1+e -t k )(1-e -t k ) t 2 k Ω 1,2 (t k , p) = y 1,2 (t k , p 1 , p 2 ) = e -(p1+p2)•t k - (tke -p 1 t k +t k e -p 2 t k -1+e -t k )(1-e -t k ) t 2 k
whereas the variance functions becomes

V 1 (t k ) = (1-e -t k ) 3 (-2+2e -t k +t k +t k e -t k ) 2t 4 k V T1 (t k ) = (1-e -t k ) 2 (1+e -t k )(tk+tke -t k +2e -t k -2) 4t 3 k . with V 2 (t k ) = V 1 (t k ) and V T2 (t k ) = V T1 (t k ).
As shown by previous equations, the functions Ω 1 and Ω 2 are not colinear despite the equality of variance functions. This counterexample shows that the colinearity of sensitivity functions has other causes than the time dependence of Ω-functions.

Counterexample 2

Let us consider a two-parameter generic model decomposed as

y (t k , p) = y 0 (t k ) + y 1 (t k , p 1 ) + y 2 (t k , p 2 ) + y 1,2 (t k , p) . We denote by V 1 (t k ) = 1 0 y 2 1 (t k , p 1 ) dp 1 , V 2 (t k ) = 1 0 y 2 2 (t k , p 2 ) dp 2 and V 1,2 (t k ) = I 2 y 2 1,2 (t k , p 1 , p 2 )
dp, the first-order, respectively second-order variance functions. Consider the following modified model

ỹ (t k , p) = y 1 (t k , p 1 ) V 1 (t k ) + y 2 (t k , p 2 ) V 2 (t k ) + y 1,2 (t k , p) V 1,2 (t k ) . (28)
In this case, for non-null functions y 1 , y 2 and y 1,2 , we obtain the following variances:

Ṽ1 (t k ) = 1 = Ṽ2 (t k ) = Ṽ1,2 (t k ) (29)
and

ṼT1 (t k ) = 2 = ṼT2 (t k ) . ( 30
)
whereas the Ω-functions, defined in an equivalent manner as previously, are not necessarily linear dependent w.r.t. time.

For example, let us consider the following model with two globally identifiable parameters

y (t k , p) = 2p 1 + e p2t k - e t k -1 t k -1
having the following decomposition terms

y 1 (t k , p 1 ) = 2p 1 -1; y 2 (t k , p 2 ) = e p2t k - e t k -1 t k
with null y 0 and y 1,2 functions. The modified model, constructed as in [START_REF] Streif | Systems Analysis of Biological Networks, chapter Steady state sensitivity analysis of biochemical reaction networks. A brief review and new methods[END_REF], will be composed into

ỹ1 (t k , p 1 ) = √ 3 (2p 1 -1) ; ỹ2 (t k , p 2 ) = e p2t k -e t k -1 t k V 2 (t k ) with V 2 (t k ) = e 2t k -1 2t k - (e t k -1) 2 t 2 k
. As the ỹ1,2 function is null for all t k ∈ T, the Ω-functions are expressed as: Ω1 (t, p ∼2 ) = ỹ1 (t, p 1 ), Ω2 (t, p ∼1 ) = ỹ2 (t, p 2 ) and Ω1,2 (t, p) = 0. In this case, the three Ω-functions are not time linearly dependent, despite the colinearity of variance functions as stressed in ( 29) and ( 30). This illustrates the fact that the converse of Prop. 4.6 is not always true and it proves that colinear (first-order and total) sensitivity functions could be associated also to identifiable parameters.

Discussions

Summary of results

Let us give a brief summary of the results concerning the connections between parameter non-identifiability and output sensitivity (illustrated in Fig. 3):

• the first relationship between the two notions corresponds to the generally acknowledged association between insensitive parameters, i.e., null total sensitivity functions, and non-identifiable parameters (expressed through propositions 4.1 and 4.2); • secondly, it was shown that the non-injectivity of Ψfunctions also leads to the non-identifiability conclusion (Prop. 4.3); • as shown in Propositions 4.4 to 4.6, the linear dependence w.r.t. time and p ∼i,j of three Ω-functions (Ω i , Ω j and Ω i,j ) implies both the non-identifiability of the parameters in question (Prop. 4.5) and the colinearity of their total sensitivity functions (Prop. 4.6).

Fig. 3 emphasizes the central role played by the Ω and Ψ functions in the relationships between identifiability and sensitivity.

Limits of sensitivity analysis for inferring nonidentifiability

The converse of the proposition 4.1, represented by the arrow 4 in Fig. 3, is true, i.e. a null Ψ i function will also imply a null total sensitivity function, S Ti (t k ), ∀t k ∈ T.

Properties of and functions

Null functions However, the converse of Propositions 4.2, 4.3 and 4.5 are false (represented by the dashed arrows 1, 2 and 3). They only represent necessary identifiability conditions. Similarly, the converse of the Proposition 4.6 is not valid, since the colinearity of the (first-order and total) sensitivity functions can be caused by other sources than the linear dependence w.r.t. time of the Ω-functions (as seen in the Counterexamples 1 and 2).

As summarized in Fig. 3, two limits, L 1 and L 2 , prevent to conclude on global non-identifiability from the analysis of global sensitivity functions. The first one is due to the impossibility to test the non-injectivity from the variance functions. The second one shows that the colinearity analysis of sensitivity functions, usually employed in local identifiability analysis, is not always valid. Finally, testing low-sensitizing parameters associated with sensitivity functions lower to an empirical threshold is the only property that leads in practice to conclude surely on global a posteriori non-identifiability.

Conclusion

The lack of available methods to test the global a posteriori identifiability of parameters in high dimensional dynamic models probably explains the success of global sensitivity techniques, particulary in systems biology and more precisely in the identification of metabolic pathways. Nevertheless, the links between these two analyses are not yet fully explored. This present work provides new insights into the relationships between them. We show that the lack of identifiability may be due to three causes: insensitivity, colinearity, or the non-injectivity of the functions involved in the global sensitivity analysis. While the first is in truth a general acknowledged association, between insensitive parameters and non-identifiable parameters, the other two are less straightforward. Indeed, these two points correspond to two limits which prevent a sure conclusion on global non-identifiability. The first limit is due to the impossibility of testing the non-injectivity from the variance functions. The second shows that, conversely to what is admitted in local identifiability, colinear analysis of sensitivity functions is no longer valid. Thus, the conclusions about parameter non-identifiability drawn from the analysis of global sensitivity functions must be carefully analyzed. output. They could be regarded as a promising perspective to solve the global identifiability issue in practice. If the explicit expression of the output variable is known, those functions can be determined by computer algebra. Otherwise, their determination in a given experimental framework is another challenge in perspective.

  k th time instant, with k = 0, . . . , N -1 p ∈ P ⊂ R n the parameters of the model p∼i the vector composed by all the parameters except pi u ∈ R nu input signals x ∈ R nx state variables y ∈ R ny output variables E [•] the expectation operator V [•] the variance operator Vi 1 ,...,ir (t k ) the r th -order variance of the output y with respect to collective effect of the parameters pi 1 , . . ., pi r Si 1 ,...,ir (t k ) the r th -order sensitivity of the output y with respect to collective effect of the parameters pi 1 , . . ., pi r I n = [0, 1] n n-dimensional unit hypercube Ψi (t k , p) the total effect on the model output y of the parameter pi Ωi (t k , p∼j) represents the influence on the output y of the parameter pi independently of pj

Fig. 1 .

 1 Fig. 1. Classification of identifiability definitions, where M denotes the model structure, u the input signals, t k the time measurements and v k the output noise

Fig. 2 .

 2 Fig. 2. Total variance functions of the model expressed in (11), for three different time-sampling strategies: (a) t k = 0.01 • k; (b) t k = (1 + 6k)/12 and (c) t k = 0.5 • k.

Fig. 3 .

 3 Fig. 3. Schematic representation of relationships between non-identifiability and sensitivity. The link between Ψ or Ω-functions and parameter non-identifiability is represented through propositions 4.2, 4.3 and 4.5, whereas their converses are false (represented by the dashed arrows 1, 2, 3). The link between Ψ or Ω-functions and variance or sensitivity functions is described through propositions 4.1 and 4.6 and the arrow 5 (this arrow represents the fact that for a non-injective function Ψ, we can compute its total variance with respect to the parameter in question). The false implication L1, represents the fact that the non-injectivity of the model output w.r.t. a parameter can not be tested from global sensitivity measures. The converse of the proposition 4.1 (arrow 4) is true, while that of the proposition 4.6 is not always true (arrow L2). In an experimental framework, a total sensitivity function inferior to an empirical threshold implies the non-identifiability of the parameter in question, whereas the colinearity of total (or first order) sensitivity function must be treated with caution, since it does not allow to conclude surely on the non-identifiability, conversely to what is admitted in local identifiability. The available tools for the computation of variance/sensitivity functions are represented on the right side of the figure.

  Proposition 4.2 The nullity of the total sensitivity function S Ti , for all time measurements, implies also the non-identifiability of parameter p i . Proof As seen above, a null total sensitivity function, S Ti (t k ), ∀t k ∈ T, implies the nullity of the Ψ i function. This means that we can express the model output variable as a function of n -1 parameters, p ∼i . In other words, p i has no influence on the output and there is at least two distinct parameter values, p, p

* ∈ P, such as

y (t k , p) = y (t k , p * ) ∀t k ∈ T, with p * = [p 1 , . . . , p * i , . . . , p n ].

The parameter p i is thus non-identifiable.

•

  In the upper figure (2.a), we propose a suited sampling rate, t k = 0.01•k. A first conclusion that can be drawn from this figure is that even in a best case scenario p 3 is a poor-sensitive parameter as V T3 (t k ) < 0.1 and S T3 (t k ) < 0.2. • In the middle figure (2.b), we choose measurement time instants so as to get colinear total variance functions, V T1 (t k ) and V T2 (t k ). For such sampling instants sin (2πt k )

2

= 1/4, and y becomes

y (t k , p) = (2p 1 ± 2p 2 ) • e -0.1t k + e -p3t

2 k indicating the non-identifiablity of both p 1 and p 2 . • The bottom figure (2.c) exploits the total variance function for t k = 0.5 • k. In this case, for all t k , the total variance V T2 (t k ) is null, leaving only one sensitive parameter: p 1 .

3

  The non-injectivity w.r.t. p i of the function Ψ i (t k , p) (independently of t k ), implies the non-identifiability of the parameter p i .Proof Let p ∼i be fixed, with p ∼i the vector composed by all the parameters except p i . As Ψ i (t k , p) is a noninjective function w.r.t. parameter p i , there exists p i = p * i such that,

  , p) = y 0 (t k )+y 1 (t k , p 1 )+y 2 (t k , p 2 )+y 1,2 (t k , p 1 , p 2 ) .

		)•t k	(27)
	with p ∈ [0, 1]	2 and the following Sobol' decomposition
	y(t k

Decomposition known as the Hoeffding decomposition[START_REF] Hoeffding | A class of statistics with asymptotically normal distribution[END_REF], HDMR (high dimensional model representation) expansion[START_REF] Rabitz | General foundations of highdimensional model representations[END_REF] or more recently, as the Sobol' decomposition[START_REF] Im Sobol | Sensitivity analysis for non-linear mathematical models[END_REF].

For space reasons, we did not include herein the details about this identifiability study.

This study also brings out the central role of some functions, entitled Ω and Ψ functions, in both sensitivity and identifiability analyses. These functions are derived from Sobol's high dimensional representation of the model