Alexandre Genadot
email: algenadot@gmail.com.

Èle Mich

Thieullen

AVERAGING FOR A FULLY-COUPLED PIECEWISE DETER-MINISTIC MARKOV PROCESS IN INFINITE DIMENSIONS

Keywords: piecewise deterministic Markov processes, fully-coupled systems, averaging principle, reaction diffusion equations, slow-fast systems, Markov chains, neuron models, Hodgkin-Huxley model. 2010 Mathematics Subject Classification: Primary 60B12, 60J75, 35K57 Secondary 92C20, 92C45

. This model describes the propagation of an action potential along the axon of a neuron at the scale of ion channels. Mathematically, this model is a fully-coupled Piecewise Deterministic Markov Process (PDMP) in infinite dimensions. We introduce two time scales in this model in considering that some ion channels open and close at faster jump rates than others. We perform a slow-fast analysis of this model and prove that asymptotically this 'two-time-scales' model reduces to the so called averaged model which is still a PDMP in infinite dimensions for which we provide effective evolution equations and jump rates.

Introduction

The Hodgkin-Huxley model is one of the most studied models in neuroscience since its creation in the early 50's by Hodgkin and Huxley [20]. This deterministic model was first created in order to describe the propagation of an action potential or nerve impulse along the axon of a neuron of the giant squid. In our paper, we will consider a mathematical generalization of the Hodgkin-Huxley model investigated numerically in [START_REF] Faisal | Ion-Channel Noise Places Limits on the Miniaturization of the Brain's Wiring[END_REF] and analytically in [START_REF] Austin | The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism[END_REF]. This model, called the spatial stochastic Hodgkin-Huxley model in the sequel, is made of a partial differential equation (PDE) describing the evolution of the variation of the potential across the membrane, coupled with a continuous time Markov chain which describes the dynamics of ion channels which are present all along the axon. It is a Piecewise Deterministic Markov Process (PDMP) in infinite dimensions (see [START_REF] Buckwar | An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution[END_REF][START_REF] Pakdaman | Asymptotic expansion and central limit theorem for multiscale piecewise deterministic Markov processes[END_REF] for PDMP in infinite dimensions and also [START_REF] Costa | Stability and Ergodicity of Piecewise Deterministic Markov Processes[END_REF][START_REF] Costa | Singular Perturbation for the discounted continuous control of piecewise deterministic Markov processes[END_REF][START_REF] Crudu | Convergence of Stochastic Gene Networks to Hybrid Piecewise Deterministic Processes[END_REF][START_REF] Fonbona | Quantitative Estimates for the Long Time Behavior of a PDMP Describing the Movement of Bacteria[END_REF] and references therein for PDMP in finite dimension). Moreover it is fully-coupled in the sense that the evolution of the potential depends on the kinetics of ion channels and vice-versa. The role of ion channels is fundamental because they allow and amplify the propagation of an action potential. We introduce different time scales in this model considering that some ion channels open and close at faster rates than others. We perform a slow-fast analysis of this model and prove that asymptotically it reduces to the so called averaged model which is still a PDMP in infinite dimensions for which we provide effective evolution equations and jump rates. We thus reduce the complexity of the original model by simplifying the kinetics of ion channels. To the best of our knowledge no averaging results are available for PDMP in infinite dimensions.

Even if the Hodgkin-Huxley model was first introduced in neurophysiology, its paradigm is commonly used to describe the behavior of cardiac cells, see for instance [START_REF] Flaim | Contributions of sustained I Na and I Kv43 to transmural heterogeneity of early repolarization and arrhythmogenesis in canine left ventricular myocytes[END_REF][START_REF] Greenstein | Mechanisms of excitation-contraction coupling in an integrative model of the cardiac ventricular myocyte[END_REF]. It was at first deterministic. In order to include more variability, better describe the axon itself and a certain class of phenomena, stochastic versions of this model have been proposed since the 90's, see for instance [START_REF] Chow | Spontaneous action potentials due to channel fluctuations[END_REF][START_REF] Defelice | Chaotic states in a random world: relationship between the nonlinear differential equations of excitability and the stochastic properties of ion channels[END_REF][START_REF] Faisal | Ion-Channel Noise Places Limits on the Miniaturization of the Brain's Wiring[END_REF][START_REF] Fox | Stochastic Versions of the Hodgkin-Huxley Equations[END_REF], introducing in the model two kinds of randomness. One is to include in the Hodgkin-Huxley equations an external noise modeling in this way the multitude of impulses received by one single neuron through its connections to the extraordinary complex system which is the brain. The other is to introduce in the Hodgkin-Huxley equations the intrinsic noise of the neuron due to the presence of ion channels which are intrinsically stochastic entities. The spatial stochastic Hodgkin-Huxley model which deals with the intrinsic noise of neurons pertains to the latter class.

The generalized Hodgkin-Huxley equations of [START_REF] Austin | The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism[END_REF] belong also to a larger class of stochastic systems describing the evolution of a macroscopic variable, here the membrane potential, coupled with a Markovian dynamic modeling the kinetics of smaller entities, here ion channels. Between two successive jumps of the Markovian kinetics, the macroscopic variable follows a deterministic evolution. These models are known mathematically under the name of Piecewise Deterministic Markov Processes (PDMP) and are often called stochastic hybrid systems in the applications. The theory of PDMP in finite dimension (dealing with ordinary differential equations coupled with Markovian kinetics) has been developed in [START_REF] Davis | Piecewise-deterministic Markov processes: a general class of non diffusion stochastic models[END_REF][START_REF] Davis | Markov models and optimization[END_REF]. The extension to infinite dimensions (dealing with PDEs coupled with Markovian kinetics) has been performed in [START_REF] Buckwar | An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution[END_REF]. The spatial stochastic Hodgkin-Huxley model belongs to the class of PDMP in infinite dimensions since the equation describing the evolution of the membrane potential is a PDE.

Reducing the complexity of a model is of first importance since it allows to perform more tractable mathematics and numerical simulations and in this way, to go further in the understanding of the model. The reduction can be achieved by two different means. The first is to consider that the number or the population of ion channels is large enough to apply a law of large numbers argument so that we can replace the empirical proportion of ion channels in a given state by the averaged theoretical proportion of ion channels in this state. This is achieved by letting the number of ion channels go to infinity, the corresponding law of large numbers is proved in [START_REF] Austin | The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism[END_REF] and in the context of PDMP in [START_REF] Riedler | Limit Theorems for Infinite Dimensional Piecewise Deterministic Processes and Applications to Stochastic Neuron Models[END_REF]. The second way to reduce the complexity of the model, the one we will be interested in in this paper, is to consider that the model has intrinsically two time scales. It is a quite natural approach in this context, see for instance [START_REF] Fitzhugh | Thresholds and plateaus in the Hodgkin-Huxley nerve equations[END_REF][START_REF] Hille | Ionic Channels of Excitable Membranes[END_REF].

In the present paper the number of ion channels is fixed and we consider that some ion channels have a faster dynamic than others. We perform then a slow-fast mathematical analysis of this 'two-time-scales' system. We obtain a reduced system in which the intrinsic variability of ion channels is still taken into account but with a simplified kinetic. Unlike the main result of [START_REF] Austin | The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism[END_REF], after reduction, we remain here at a stochastic level. The methods used in this paper are known mathematically under the name of stochastic averaging. As far as we know, these methods have never been applied to a PDMP in infinite dimensions which we are dealing with in the present paper. For PDMPs in finite dimension, a theory of averaging has been developed in [START_REF] Faggionato | Averaging and large deviation principles for fully-coupled piecewise deterministic Markov processes[END_REF] and central limit theorem has been proved in [START_REF] Pakdaman | Asymptotic expansion and central limit theorem for multiscale piecewise deterministic Markov processes[END_REF].

In the infinite dimensional framework which we consider in this paper we have been able to reduce the complexity of a fully-coupled PDMP in infinite dimensions by using averaging methods. The natural step after this averaging result is to prove the central limit theorem associated to it. We wish to investigate this in a future work.

We conclude this introduction with the plan of the paper. In Section 2 we introduce the spatial stochastic Hodgkin-Huxley model and the formalism of PDMP in infinite dimensions. Then we introduce the two time scales in the model and prove a crucial result for the sequel. We finish this section by presenting the main assumptions on the model and our main result. Section 3 is devoted to the proof of our main result. In Section 4 we apply our result to a Hodgkin-Huxley type model as an example. At the end of the paper, an appendix provides an important lemma proved in [START_REF] Austin | The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism[END_REF], two results about tightness and the parameter values used in the simulation presented in Section 4.

Basic notations

We set I = [0, 1]. Let us define:

f H = I (f (x)) 2 + (f (x)) 2 dx f L 2 (I) = I (f (x)) 2 dx
In this paper we will work with the following triplet of Banach spaces H 1 0 (I) ⊂ L 2 (I) ⊂ H -1 (I). H 1 0 (I) will be denoted by H, it denotes the completion of the set of C ∞ functions with compact support on I with respect to the norm • H . L 2 (I) endowed with the norm • L 2 (I) is the usual space of measurable and square integrable functions with respect to the Lebesgue measure on I. H -1 (I) is the dual space of H and will be denoted by H * .

We recall here a few basic results about these three spaces. L 2 (I) and H are two separable Hilbert spaces endowed with their usual scalar products denoted respectively by (•, •) L 2 (I) and (•, •). We denote by < •, • > the duality pairing between H and H * .

For an integer k ≥ 1 we define the following functions on I:

e k (•) = √ 2 1 + (kπ) 2 sin(kπ•), f k (•) = √ 2 sin(kπ•) The family {f k , k ≥ 1} (resp. {e k , k ≥ 1}) is a Hilbert basis of L 2 (I) (resp. H).
In L 2 (I) (resp. H), the Laplacian with zero Dirichlet boundary conditions has the following spectral decomposition:

∆u = - k≥1 (kπ) 2 (u, f k ) L 2 (I) f k   resp. ∆u = - k≥1 (kπ) 2 (u, e k )e k   for u in the domain D L 2 (I) (∆) = {u ∈ L 2 (I); k≥1 k 4 (u, f k ) 2 L 2 (I) < ∞} (resp. D(∆) = {u ∈ H; k≥1 k 4 (u, e k ) 2 < ∞}).
We refer the reader to chapter 1, section 1.3 of [START_REF] Henry | Geometric Theory of Semilinear Parabolic Equations[END_REF] for more details.

We denote by C P the (Poincaré) constant such that, for all u ∈ H we have:

sup I |u| ≤ C P u H
The embeddings H ⊂ L 2 (I) ⊂ H * are continuous and dense. Moreover, for any h ∈ L 2 (I) and any u ∈ H: < h, u >≡ (h, u) L 2 (I) .

We say that a function f : H → R has a Fréchet derivative in u ∈ H if there exists a bounded linear operator T u : H → R such that:

lim h→0 f (u + h) -f (u) -T u (h) h H = 0
We then write df du [u] for the operator T u . For example, the square of the • H norm and the Dirac distribution in x ∈ I are Fréchet differentiable on H. For all u ∈ H:

d • 2 H du [u](h) = 2(u, h), dδ x du [u](h) = h(x)
for all h ∈ H. Fréchet differentiation is stable by sum and product.

Statement of the model and results

The spatial stochastic Hodgkin-Huxley model

We introduce here the stochastic Hodgkin-Huxley equations considered in [START_REF] Austin | The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism[END_REF]. Basically this spatial stochastic Hodgkin-Huxley model describes the propagation of an action potential along an axon. The axon is the part of a neuron which transmits the information received from the soma to another neuron on long distances: the length of the axon is big relative to the size of the soma. Along the axon are the ion channels which amplify and allow the propagation of the received impulse. For mathematical convenience, we assume that the axon is a segment of length one and we denote it by I = [0, 1]. The ion channels are assumed to be in number N ≥ 1 and regularly placed along the axon at the place i N for i ∈ N ∩ N I. This distribution of the channels is certainly unrealistic but we assume it to fix the ideas and in accordance with [START_REF] Austin | The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism[END_REF]. However the mathematics are the same if we consider any finite subset of I instead of 1 N (N ∩ N I). Each ion channel can be in a state ξ ∈ E where E is a finite states space. For example, for the Hodgkin-Huxley model, a state can be : "receptive to sodium ions and open" (see [START_REF] Hille | Ionic Channels of Excitable Membranes[END_REF]).

The ion channels switch between states according to a continuous time Markov chain whose jump intensities depend on the local potential of the membrane, that is why the model is said to be fully-coupled. For any states ξ, ζ ∈ E we define by α ξ,ζ the jump intensity (or rate function) from the state ξ to the state ζ. It is a real valued function of a real variable and supposed to be, as is its derivative, Lipschitz-continuous. We assume moreover that: 0 ≤ α ξ,ζ ≤ α + for any ξ, ζ ∈ E and either α ξ,ζ is constant equal to zero or is strictly positive bounded below by a strictly positive constant α -. That is, the non-zero rate functions are bounded below and above by strictly positive constants. For a given channel, the rate function describes the rate at which it switches from one state to another.

A possible configuration of all the N ion channels is denoted by an element r = (r(i), i ∈ N ∩ N I) of R = E N∩N I : r(i) is the state of the channel which is at the position i N for i ∈ N ∩ N I. The channels, or stochastic processes r(i), are supposed to evolve independently over infinitesimal time-scales. Denoting by u t ( i N ) the local potential at point i N at time t, we have:

P(r t+h (i) = ζ|r t (i) = ξ) = α ξ,ζ u t i N h + o(h) (1)
For any ξ ∈ E we also define the maximal conductance c ξ and the steady state potentials v ξ of a channel in state ξ which are both constants, the first being positive.

The membrane potential u t (x) along the axon evolves according to the following PDE:

∂ t u t = ∆u t + 1 N ξ∈E i∈N∩N I c ξ 1 ξ (r t (i))(v ξ -u t ( i N ))δ i N (2)
We will assume zero Dirichlet boundary conditions for this PDE (clamped axon). We are interested in the process (u t , r t ) t∈[0,T ] .

We recall here the result of [START_REF] Austin | The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism[END_REF] which states that there exists a stochastic process satisfying equations ( 2) and [START_REF] Austin | The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism[END_REF]. Let u 0 be in D(∆) such that min ξ∈E v ξ ≤ u 0 ≤ max ξ∈E v ξ , the initial potential of the axon. Let q 0 ∈ R be the initial configuration of ion channels. Proposition 1. ( [START_REF] Austin | The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism[END_REF].) Fix N ≥ 1 and let (Ω, F, (F t ) 0≤t≤T , P) be a filtered probability space satisfying the usual conditions. There exist a pair (u t , r t ) 0≤t≤T of càdlàg adapted stochastic processes such that each sample path of u is a continuous map from [0, T ] to H and r t is in R for all t ∈ [0, T ] and satisfying:

• (Regularity) The map t → ∂ t u t lies in L 2 ([0, T ], H * ) almost surely.
• (Dynamic: PDE)

∂ t u t = ∆u t + 1 N ξ∈E i∈N∩N I c ξ 1 ξ (r t (i))(v ξ -u t ( i N ))δ i N ∀t ∈ [0, T ], P-almost surely.
• (Dynamic: jump)

P(r t+h (i) = ζ|r t (i) = ξ) = α ξ,ζ u t i N h + o(h)
• (Initial condition: PDE) u 0 given.

• (Initial condition: jump) q 0 given.

• (Boundary conditions: PDE only)

u t (0) = u t (1) = 0, ∀t ∈ [0, T ].
Moreover there exists a constant C such that u t ∞ ≤ C for all t ∈ [0, T ] and ω.

In [START_REF] Austin | The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism[END_REF] the author proves that when the number of ion channels increases to infinity, the above model converges, in a sense, toward a deterministic model. In our paper, unlike in [START_REF] Austin | The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism[END_REF] we will work with a fixed number of ion channels but introduce two time scales in the model in order to perform a slow-fast analysis.

The spatial stochastic Hodgkin-Huxley model as a Piecewise Deterministic Markov Process in infinite dimensions

The paper [START_REF] Buckwar | An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution[END_REF] extends the theory of finite dimensional Piecewise Deterministic Markov Processes (PDMP) introduced in [START_REF] Davis | Piecewise-deterministic Markov processes: a general class of non diffusion stochastic models[END_REF] and [START_REF] Davis | Markov models and optimization[END_REF] to PDMP in infinite dimensions. The results stated here come from section 2.3 and 3.1 of [START_REF] Buckwar | An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution[END_REF] adapted to our particular notations.

For all r ∈ R we define the following function on H:

G r (u) = 1 N ξ∈E i∈N∩N I c ξ 1 ξ (r(i))(v ξ -u( i N ))δ i N (3)
which is H * valued. We do not write the dependence in N in the notation of G r (u) since, unlike in [START_REF] Austin | The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism[END_REF], N is a fixed parameter here. The stochastic process (u t , r t ) t∈[0,T ] can be described in the following way. We start with a given initial potential v 0 and ion channels configuration q 0 . Then the u component evolves according to the evolution equation:

∂ t u t = ∆u t + G q0 (u t )
until the first jump of the r component at time τ 1 which occurs according to the transition rate function Λ :

H × R → R + given for (u, r) ∈ H × R by: Λ(u, r) = i∈N∩N I ξ =r(i) α r(i),ξ (u( i N ))
according to [START_REF] Austin | The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism[END_REF]. A new value q 1 for r is then chosen according to a transition measure Q from H × R to P(R) (the set of all probability measures on R). This transition measure is also given by the jump distribution of the ion channels. For (u, r) ∈ H × R and r ∈ R which differs from r only by the component r(i 0 ):

Q(u, r)({r }) = α r(i0),r (i0) (u( i0 N )) Λ(u, r)
If r differs from r by two or more components then Q(u, r)({r }) = 0.

Then u evolves according to the "updated" evolution equation:

∂ t u t = ∆u t + G q1 (u t )
with initial condition u τ1 and Dirichlet boundary conditions, until the second jump of the r component. And so on. This description justifies the term "Piecewise Deterministic Process": the dynamic of the process is indeed purely deterministic between the jumps. As shown in [START_REF] Buckwar | An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution[END_REF] in Theorem 6, the equations of Proposition 1 define a standard Piecewise Deterministic Process (PDP) in the sense of Definition 3 of [START_REF] Buckwar | An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution[END_REF]. We recall briefly here what this means technically:

• for a given ion channel configuration r ∈ R, the PDE:

∂ t u t = ∆u t + G r (u t )
with zero Dirichlet boundary conditions admits a unique global (weak) solution continuous on H for every initial value u 0 ∈ H denoted by ψ r (t, x).

• The number of state switches in the (r t ) t∈[0,T ] component during any finite time interval is finite almost surely.

• The transition rate Λ from one state of R to another is a measurable function from H × R to R + and for all (u, r) ∈ H × R, the transition rate as a function of time is integrable over every finite time interval but diverges to ∞ when it is integrated over R + (the expected number of jumps tends to ∞ when the time horizon increases).

• The transition measure Q from one state of R to another is such that the mapping (u, r) → Q(u, r)(R) is measurable for every R ⊂ R and Q(u, r)({r}) = 0 for all (u, r) ∈ H × R.

In fact the stochastic process (u t , r t ) t∈[0,T ] is more than a piecewise deterministic process, it is a piecewise deterministic Markov process.

Theorem 1.

1. There exists a filtered probability space satisfying the usual conditions such that our infinite-dimensional standard PDP is a homogeneous Markov process on H × R.

Locally bounded measurable functions on H × R which are absolutely continuous

as maps t → f (ψ r (t, x), r) for all (x,r) are in the domain D(A) of the extended generator. The extended generator is given for almost all t by:

Af (u t , r t ) = d dt f (u t , r t ) + Bf (u t , r t ) (4)
where

Bf (u t , r t ) = i∈N∩N I ζ∈E [f (u t , r t (r t (i) → ζ)) -f (u t , r t )]α rt(i),ζ (u t ( i N )) with r t (r t (i) → ζ) is the component of R with r t (r t (i) → ζ)(j) equals to r t (j) if j = i and to ζ if j = i. The notation d dt f (u t , r t ) means that the function s → f (u s , r) is differentiated at s = t,
where u is the solution of the PDE of Proposition 1 such that u t = u t and with the channel state r held fixed equal to r t .

Remark 1. It is not usual to write a generator only along paths (u t , r t ) t≥0 as we do. We would expect for a generator an analytical expression of the form Af (u, r) for any (u, r) ∈ H × R. It is in fact possible to obtain such an expression for more regular function f . The part of the generator related to the continuous Markov chain r takes the form Bf (u, r) with no changes but the derivative d dt f (u, r) can then be expressed thanks to the Fréchet derivative of f , see [START_REF] Buckwar | An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution[END_REF] for more details. We will not use this refinement in the sequel.

Knowing that a stochastic process is Markovian is crucial for its study since a lot of mathematical tools have been developed for Markovian processes, see for example [START_REF] Ethier | Markov Processes: Characterization and Convergence[END_REF].

Singularly perturbed model and main results

2.3.1.

Introduction of two time scales in the model. We introduce now two time scales in the model. Indeed, we consider that the Hodgkin-Huxley model is a slow-fast system: some states of the ion channels communicate faster between them than others, see for example [START_REF] Hille | Ionic Channels of Excitable Membranes[END_REF]. Mathematically, this leads to introduce a small parameter ε > 0 in our equation. For the states which communicate at a faster rate, we say that they communicate at the usual rate divided by ε.

We can then consider different classes of states or partition the state space E in states which communicate at a high rate. This kind of description is very classical, see for example [START_REF] Faggionato | Averaging and large deviation principles for fully-coupled piecewise deterministic Markov processes[END_REF]. We regroup our states in classes making a partition of the state space E into:

E = E 1 • • • E l
where l ∈ {1, 2, • • • } is the number of classes. Inside a class E j , the states communicate faster at jump rates of order 1 ε . States of different classes communicate at the usual rate of order 1. For ε > 0 fixed, we denote by (u ε , r ε ) the modification of the PDMP introduced in the previous section with now two time scales. Its generator is, for f ∈ D(A):

Af (u ε t , r ε t ) = d dt f (u ε t , r ε t ) + B ε f (u ε t , r ε t ) (5)
B ε is the component of the generator related to the continuous time Markov chain r ε . According to our slow-fast description we have the two time scales decomposition of this generator:

B ε = 1 ε B + B
where the "fast" generator B is given by:

Bf (u ε t , r ε t ) = i∈N∩N I l j=1 1 Ej (r ε t (i)) ζ∈Ej [f (u ε t , r ε t (r ε t (i) → ζ) -f (u ε t , r ε t )]α r ε t (i),ζ (u ε t ( i N ))
and the "slow" generator is given by:

Bf (u ε t , r ε t ) = i∈N∩N I l j=1 1 Ej (r ε t (i)) ζ / ∈Ej [f (u ε t , r ε t (r ε t (i) → ζ) -f (u ε t , r ε t )]α r ε t (i),ζ (u ε t ( i N ))
For y ∈ R fixed and g : R × E → R, we denote by B j (y), j ∈ {1, • • • , l} the following generator:

B j (y)g(ξ) = 1 Ej (ξ) ζ∈Ej [g(y, ζ) -g(y, ξ)]α ξ,ζ (y)
which will be of interest in the sequel.

Uniform boundedness.

Here is a crucial result for the proof of our main result. Its proof takes back the argument developed in [START_REF] Austin | The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism[END_REF] but for the sake of completeness and given the intensive use of this proposition in the sequel, we provide a short proof.

Proposition 2. For any T > 0, there is a deterministic constant C > 0 independent of ε ∈]0, 1] such that:

sup t∈[0,T ] u ε t H ≤ C almost surely.
In the proof of this proposition, we will use the following representation of a solution of the PDE part of our PDMP. We say that u ε is a mild solution to (2) if:

u ε t = e ∆t u 0 + t 0 e ∆(t-s) G r ε s (u ε s )ds, t ≥ 0 (6)
almost surely where (e ∆t , t ≥ 0) is the semi-group associated to ∆ in H. For any u ∈ H: e ∆t u = k≥1 e -(kπ) 2 t (u, e k )e k , t > 0 Let us mention that the mild formulation (6) holds also up to a bounded stopping time τ , this will be useful later in the paper.

Proof. We work ω by ω. Using the mild formulation (6), we see that:

u ε t H ≤ e ∆t u 0 H + 1 N ξ∈E i∈N∩N I t 0 c ξ 1 ξ (r ε s (i))v ξ e ∆(t-s) δ i N ds H + 1 N ξ∈E i∈N∩N I t 0 c ξ u ε s ( i N )e ∆(t-s) δ i N ds H
By Lemma 2 of Appendix A the first of the above sums is bounded by a fixed constant which can be chosen independent of t ∈ [0, T ]. For the second sum, we break the integral in two pieces. For all γ > 0 by Lemma 2, we can choose η > 0 so small that:

t t-η c ξ u ε s ( i N )e ∆(t-s) δ i N ds H ≤ 1 2 γ max s∈[0,t] |c ξ u ε s ( i N )| Since |c ξ u ε s ( i N )| ≤ max c ξ u ε s ∞ ≤ max c ξ C P u ε s H , choosing γ such that γ max c ξ C P < 1 we have: t t-η c ξ u ε s ( i N )e ∆(t-s) δ i N ds H ≤ 1 2 max s∈[0,t] u ε s H
Now the second estimates of Lemma 2 gives:

t-η 0 c ξ u ε s ( i N )e ∆(t-s) δ i N ds H ≤ C 2 (η) max ξ∈E |c ξ | t-η 0 u ε s ∞ ds ≤ C P C 2 (η) max ξ∈E |c ξ | t-η 0 max σ∈[0,s] u ε σ H ds
Reassembling the above inequalities, the end of the proof is a classical application of the Gronwall's lemma. We find finally that max s∈[0,t] u ε t H is bounded by a fixed constant independent of t ∈ [0, T ], ω and ε ∈]0, 1]. 2.3.3. Main assumptions and averaged equation. We make the following assumptions. For any y ∈ R fixed, and any j ∈ {1, • • • , l}, the fast generator B j (y) is weakly irreducible on E j , i.e has a unique quasi-stationary distribution denoted by µ j (y). This quasi-stationary distribution is supposed to be, as is its derivative, Lipschitzcontinuous in its argument y (see section 2.3.4 for more details).

Following [START_REF] Yin | Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach[END_REF], the states in E j can be considered as equivalent. For any i ∈ N ∩ N I we define the new stochastic process (r t ) t≥0 by rε t (i) = j when r ε t (i) ∈ E j and abbreviate E j by j . We then have an aggregate process rε (i) with values in {1, • • • , l}. It is not a Markov process but we have the following proposition:

Proposition 3. ([28].) For any y ∈ R, i ∈ N ∩ N I and g : {1, • • • , l} → R, rε (i)
converges weakly when ε goes to 0 to a Markov process r(i) generated by:

B(y)g(r(i)) = l j=1 1 j (r(i)) l k=1,k =j (g(k) -g(j)) ξ∈E k ζ∈Ej α ζ,ξ (y)µ j (ζ)
We then average the function G r (u) against the quasi-invariant distributions. That is we consider that each pack of states E j has reached its stationary behavior. For any r ∈ R = {1, • • • , l} N∩N I we define the averaged function by:

F r (u) = 1 N i∈N∩N I l j=1 1 j (r(i)) ζ∈Ej c ζ µ j (u( i N ))(ζ)(v ζ -u( i N ))δ i N (7)
Therefore, we call averaged equation of (2) the following PDE:

∂ t u t = ∆u t + F rt (u t ) (8)
with zero Dirichlet boundary condition and initial condition u 0 and q0 (where q0 is the aggregation of the initial channel configuration q 0 ). In equation ( 8), the process (r t ) t∈[0,T ] evolves, each coordinate independently over infinitesimal time-scales, according to the averaged jump-rates between the subsets E j of E.

Proposition 4. For any T > 0 there exists a probability space satisfying the usual conditions such that equation (8) defines a PDMP (u t , rt ) t∈[0,T ] in infinite dimensions in the sense of [START_REF] Buckwar | An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution[END_REF]. Moreover, there is a constant C such that:

sup t∈[0,T ] u t H ≤ C and u ∈ C([0, T ], H) almost-surely.
Proof. The equation ( 8) is of the form (2) except that the indicator function is replaced by the probabilities µ j . This changes nothing to the mathematics and the arguments developed in [START_REF] Austin | The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism[END_REF] or [START_REF] Buckwar | An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution[END_REF] still apply. We refer the reader to [START_REF] Austin | The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism[END_REF] and [START_REF] Buckwar | An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution[END_REF] for more details.

We can now state our main result which states that our approximation (8) is valid.

Theorem 2. The stochastic process u ε solution of (2) converges in law to a solution of (8) when ε goes to 0.

More precisely, we will prove the following proposition: H) and any accumulation point u verifies:

Proposition 5. (u ε , ε ∈]0, 1]) is tight in C([0, T ],
(u t , φ) L 2 (I) = (u 0 , φ) L 2 (I) + t 0 (u s , φ ) L 2 (I) ds + t 0 < F rs (u s ), φ > ds (9)
for all φ in C 2 0 (I). Moreover the accumulation point is unique up to indistinguishability. 2.3.4. Plan of the campaign. To make the proof of our main result easier to read, we will consider in a first step that all the states communicate at a fast rate. This is called the all-fast case. We will prove our main result in detail in this case and then give the key points for the validity of our proof in the general case. In the all-fast case there is a unique class E and the generator has the simplest form:

Bf (u, r) = i∈N∩N I ζ∈E [f (u, r(r(i) → ζ)) -f (u, r)]α r(i),ζ (u( i N )) (10)
We make the following assumptions: for any i ∈ N ∩ N I with u ∈ H fixed, the Markov process r(i) has a unique stationary distribution µ(u( i N )). Then the process (r(i), i ∈ N ∩ N I) has the following stationary distribution:

µ(u) = i∈N∩N I µ(u( i N ))
We assume that the rate functions are Lipschitz continuous. The average (generalized) function is then:

F (u) = R G r (u)µ(u)(dr) (11) = 1 N ξ∈E i∈N∩N I c ξ µ(u( i N ))(ξ)(v ξ -u( i N ))δ i N if u is held fixed.
Is this hypothesis reasonable? For the most classical model in neurosciences, the rate functions are indeed, as well as their derivatives, Lipschitz continuous. See for example the rate functions in classical Hodgkin-Huxley model [START_REF] Hodgkin | A quantitative description of membrane current and its application to conductation and excitation in nerve[END_REF] (see also the Appendix C).

Let us show briefly that in this case the ergodic measure µ is Lipschitz continuous in its argument. Let i be in N ∩ N I and the membrane potential at i N be fixed equal to x ∈ R, the state space of the continuous time Markov chain r(i) is E which is a finite set. We set |E| = m. We can then enumerate the elements of E :

E = {ξ 1 , • • • , ξ m }.
The generator of r(i) is then the matrix :

J(x) = (α ξi,ξj (x)) 1≤i,j≤m
with α ξi,ξi (x) =j =i α ξi,ξj (x). Therefore, if we assumed that our continuous time Markov chain is ergodic, it has an invariant measure which is of the form:

µ(x)(ξ) = α ξ,ξ (x) α ζ,ζ (x)
The sums and products involved here are calculated over subsets of E which are given by resolving the equation µ(x) T J = 0. µ is then Lipschitz continuous in x since each α ξ,ζ is Lipschitz continuous in x and bounded.

The assumption of independence of each coordinate of the process r over infinitesimal time-scales is important to have a simple expression of the invariant measure of the process r and simplify the mathematics. It is possibly unrealistic from the biological point of view but is often assumed in mathematical neurosciences.

The plan of the campaign to prove our main result (2) in the all-fast case is the following. We want to prove the convergence in law of a family of càdlàg stochastic processes with values in a Hilbert space towards another, here deterministic, process. There is a well known strategy to do that:

• prove the tightness of the family

• identify the limit

In our case, tightness (see subsection 3.1) will follow roughly from the uniform boundedness of the family (u ε , ε ∈]0, 1]) obtained in Proposition 2. The identification of the limit will follow on one side from tightness and on the other side from some classical argument in the theory of averaging for stochastic processes, see for example [START_REF] Pavliotis | Multiscale Methods, Averaging and Homogenization[END_REF]. We will introduce a Poisson equation which will enable us to control some problematic terms (see Proposition 6) and identify the limit (subsection 3.1). We will then move to the general case with multiple classes (in subsection 3.2) with an emphasis on each point which could give an issue and show how things work in this case.

Proof of the main result

The all-fast case

We want to prove Proposition 5. Let us recall that in the all-fast case F r in (9) reduces to F in [START_REF] Faggionato | Averaging and large deviation principles for fully-coupled piecewise deterministic Markov processes[END_REF].

Let us outline the strategy of the proof following the plan of the campaign announced in Section 2.3.4.

• Tightness.

1. Thanks to Proposition 2, use the Markov inequality to prove that Aldous's condition and property [START_REF] Métivier | Convergence faible et principe d'invariance pour des martingales à valeurs dans des espaces de Sobolev[END_REF] of Theorems 3 and 4 in Appendix B hold. 2. For t ∈ [0, T ] fixed, find the finite dimensional space L δ,η of ( 23), Theorem 4 Appendix B using a truncation of the solution u ε t in the mild formulation. Bound E(d(u ε t , L δ,η ) > η) and use the Markov inequality.

3. Obtain tightness in D([0, T ], H) by Theorems 3 and 4 Appendix B and conclude that tightness in C([0, T ], H) holds as well thanks to the regularity of the solution u ε for any ε ∈]0, 1].

• Identification of the limit.

1. Begin by showing that the following property is sufficient:

lim ε→0 E t 0 < G r ε s (u ε s ) -F (u ε s ), φ > ds = 0 ∀φ ∈ C 2 0 (I)
To prove this property, use the tightness of the family (u ε , ε ∈]0, 1]).

Replace the term

< G r ε s (u ε s ) -F (u ε s ), φ > by Bf (u ε s , r ε s )
where f is the solution of a Poisson equation. Show that this solution exists and has some nice properties.

Write the semi-martingale expansion of

t 0 < G r ε s (u ε s ) -F (u ε s ), φ > ds.
Bound its martingale part thanks to the bound for the bracket of the martingale.

4. Bound its finite variation part thanks to the nice properties of f and the uniform bound from Proposition 2.

5. Aggregate all these bounds together to obtain that

E t 0 < G r ε s (u ε s ) -F (u ε s ), φ > ds 2
is bounded by a constant times ε.

6. Conclude that (9) holds and implies uniqueness up to indistinguishability.

Tightness. STEP 1 (Property [START_REF] Métivier | Convergence faible et principe d'invariance pour des martingales à valeurs dans des espaces de Sobolev[END_REF] and Aldous condition from Appendix B). We begin with the Aldous condition. Let τ be a stopping time and θ such that τ + θ < T . We need to control the size of u ε τ +θ -u ε τ H . At first we work deterministically. We break the difference u ε τ +θ -u ε τ 2

H into three pieces using the mild formulation of u ε in (6):

u ε τ +θ -u ε τ 2 H ≤ 4 e ∆(τ +θ) u 0 -e ∆τ u 0 2 H + 4 τ +θ τ e ∆(τ +θ-s) G r ε s (u ε s )ds 2 H + 4 τ 0 e ∆(τ +θ-s) -e ∆(τ -s) G r ε s (u ε s )ds 2 H
Let us show now that the supremum over θ ∈]0, η[ of each term on the right hand side of the above inequality goes to 0 with η (we take η > 0). We start with the third term:

τ 0 e ∆(τ +θ-s) -e ∆(τ -s) G r ε s (u ε s )ds 2 H Notice that: e ∆(τ +θ-s) -e ∆(τ -s) G r ε s (u ε s ) = k≥1 e -(kπ) 2 (τ -s) e -(kπ) 2 θ -1 1 N ξ∈E i∈N∩N I c ξ 1 ξ (r ε s (i))(v ξ -u ε s ( i N ))(1 + (kπ) 2 )e k i N e k
where we have used the fact that < e k , δ i N >= (1 + (kπ) 2 )e k ( i N ) ( recall that e k = √ 2 √ 1+(kπ) 2 sin(kπ•)). Thus we obtain:

τ 0 e ∆(τ +θ-s) -e ∆(τ -s) G r ε s (u ε s )ds 2 H ≤ 4|E| 2 max ξ∈E c 2 ξ (max ξ∈E v 2 ξ + C 2 ) k≥1 τ 0 e -(kπ) 2 (τ -s) e -(kπ) 2 θ -1 1 + (kπ) 2 ds 2 = 4|E| 2 max ξ∈E c 2 ξ (max ξ∈E v 2 ξ + C 2 ) k≥1 e -(kπ) 2 θ -1 2 (1 + (kπ) 2 ) 1 -e -(kπ) 2 τ 2 (kπ) 4 ≤ 4|E| 2 max ξ∈E c 2 ξ (max ξ∈E v 2 ξ + C 2 ) k≥1 e -(kπ) 2 θ -1 2 1 + (kπ) 2 (kπ) 4
where C is the constant of Proposition 2. By dominated convergence, we see that:

lim η→0 sup θ∈]0,η[ k≥1 e -(kπ) 2 θ -1 2 1 + (kπ) 2 (kπ) 4 ≤ lim η→0 k≥1 e -(kπ) 2 η -1 2 1 + (kπ) 2 (kπ) 4 = 0
For the second term as before, we can show that:

τ +θ τ e ∆(τ +θ-s) G r ε s (u ε s )ds 2 H ≤ 4|E| 2 max ξ∈E c 2 ξ (max ξ∈E v 2 ξ + C 2 ) k≥1 τ +θ τ e -(kπ) 2 (τ +θ-s) 1 + (kπ) 2 ds 2 ≤ 4|E| 2 max ξ∈E c 2 ξ (max ξ∈E v 2 ξ + C 2 ) k≥1 (1 + (kπ) 2 ) 1 -e -(kπ) 2 θ 2 (kπ) 4
By dominated convergence, we see that:

lim η→0 sup θ∈]0,η[ k≥1 (1 + (kπ) 2 ) 1 -e -(kπ) 2 θ 2 (kπ) 4 ≤ lim η→0 k≥1 (1 + (kπ) 2 ) 1 -e -(kπ) 2 η 2 (kπ) 4 = 0
For the first term we have, by the Bessel-Parseval equality:

e ∆(τ +θ) u 0 -e ∆τ u 0 2 H = k≥1 e -(kπ) 2 τ (e -(kπ) 2 θ -1)(u 0 , e k )e k 2 H = k≥1 e -2(kπ) 2 τ (e -(kπ) 2 θ -1) 2 (u 0 , e k ) 2 ≤ k≥1 (e -(kπ) 2 θ -1) 2 (u 0 , e k ) 2
By dominated convergence, we see that:

lim η→0 sup θ∈]0,η[ k≥1 (e -(kπ) 2 θ -1) 2 (u 0 , e k ) 2 ≤ lim η→0 k≥1 (e -(kπ) 2 η -1) 2 (u 0 , e k ) 2 = 0
Combining the results obtained for the three terms we have:

lim η→0 sup θ∈]0,η[ u ε τ +θ -u ε τ 2 H = 0 uniformly in ε ∈]0, 1].
Therefore, for all M, δ > 0, using the Chebyshev inequality, we can choose η so small that:

sup ε∈]0,1] sup θ∈]0,η[ P( u ε τ +θ -u ε τ H ≥ M ) ≤ sup ε∈]0,1] sup θ∈]0,η[ E( u ε τ +θ -u ε τ 2 H ) M 2 < δ
The first condition [START_REF] Métivier | Convergence faible et principe d'invariance pour des martingales à valeurs dans des espaces de Sobolev[END_REF], Theorem 4 Appendix B is verified by the application of the Markov inequality. Indeed, for any t ∈ [0, T ] and δ > 0 by Proposition 2 there exists a constant C > 0 independent of ε ∈]0, 1] and t ∈ [0, T ] such that:

sup ε∈]0,1] P( u ε t H > ρ) ≤ 1 ρ sup ε∈]0,1] E u ε t H ≤ C ρ < δ
for any ρ > 0 large enough. STEP 2 (Truncation). We have to show that for any δ, η > 0 we can find ε 0 > 0 and a space L δ,η such that:

sup ε∈]0,ε0] P(d(u ε t , L δ,η ) > η) ≤ δ where d(u ε t , L δ,η ) = inf v∈L δ,η u ε t -v H . We fix t ∈ [0, T ].
Recalling the mild representation (6), we have:

u ε t = e ∆t u 0 + t 0 1 N ξ∈E i∈N∩N I c ξ 1 ξ (r ε s (i))(v ξ -u ε s ( i N ))e ∆(t-s) δ i N ds
where, using the explicit expression of the semi-group e ∆t , e ∆(t-s)

δ i N is equal to k≥1 e -(kπ) 2 (t-s) (1 + (kπ) 2 )e k ( i N )e k .
I c ξ 1 ξ (r ε s (i))(v ξ -u ε s ( i N ))e ∆(t-s) p δ i N ds
For any δ, η > 0 we can find p independent of ε such that u ε t -u ε,p t H ≤ C ηδ with the constant C deterministic and independent of δ, η and ε ∈]0, 1]. Indeed we can easily show, as in STEP 1, that:

u ε t -u ε,p t 2 H ≤ 2 ∞ k=p+1 e -(kπ) 2 t (u 0 , e k )e k 2 H + 2C ∞ k=p+1 (1 + (kπ) 2 ) (1 -e -(kπ) 2 t ) 2 (kπ) 4
which is independent of ε ∈]0, 1]. The convergence of each series, uniformly in ε, enables us to choose a suitable p independent of ε. Let us denote

L p = span{e i , 1 ≤ i ≤ p}. We choose L δ,η = L p . Since u ε,p t ∈ L δ,η = L p E(d(u ε t , L δ,η )) ≤ E( u ε t -u ε,p t H ) ≤ C ηδ
Markov's inequality gives us :

P(d(u ε t , L δ,η ) > η) ≤ C δ with δ independent of ε ∈]0, 1].
STEP 3 (Tightness). Therefore using Theorems 3 and 4 of Appendix B, (

u ε , ε ∈ ]0, 1]) is tight in D([0, T ], H). Since we know that for each ε ∈]0, 1], u ε is in C([0, T ], H) we have in fact that (u ε , ε ∈]0, 1]) is tight in C([0, T ], H).
Identification of the limit. STEP 1 (Reduction of the problem). We know that the family (u ε , ε ∈]0, 1]) is tight in C([0, T ], H). We still denote by u ε a converging subsequence of u ε and we denote by u the corresponding accumulation point. We want to show that for all φ in C 2 0 (I):

(u t , φ) L 2 (I) = (u 0 , φ) L 2 (I) + t 0 (u s , φ ) L 2 (I) ds + t 0 < F (u s ), φ > ds
almost surely. We will show that:

E (u t , φ) L 2 (I) -(u 0 , φ) L 2 (I) - t 0 (u s , φ ) L 2 (I) ds - t 0 < F (u s ), φ > ds = 0 (12)
Notice that:

E (u ε t , φ) L 2 (I) -(u 0 , φ) L 2 (I) - t 0 (u ε s , φ ) L 2 (I) ds - t 0 < F (u ε s ), φ > ds = E t 0 < G r ε s (u ε s ) -F (u ε s ), φ > ds
Since the function: H) and the arguments (u ε , ε ∈]0, 1]) and u are uniformly bounded in ε in C([0, T ], H), by convergence in law we have that:

h(v) = (v t , φ) L 2 (I) -(v 0 , φ) L 2 (I) - t 0 (v s , φ ) L 2 (I) ds - t 0 < F (v s ), φ > ds is continuous on C([0, T ],
lim ε→0 E (u ε t , φ) L 2 (I) -(u 0 , φ) L 2 (I) - t 0 (u ε s , φ ) L 2 (I) ds - t 0 < F (u ε s ), φ > ds = E (u t , φ) L 2 (I) -(u 0 , φ) L 2 (I) - t 0 (u s , φ ) L 2 (I) ds - t 0 < F (u s ), φ > ds Therefore (12) will follow if lim ε→0 E t 0 < G r ε s (u ε s ) -F (u ε s )
, φ > ds = 0. We will show more precisely that:

lim ε→0 E t 0 < G r ε s (u ε s ) -F (u ε s ), φ > ds 2 = 0
STEP 2 (Poisson equation). We introduce the following Poisson equation on f :

H × R → R: Bf (u, r) = < G r (u) -F (u), φ > (13) R f (u, r)µ(u)(dr) = 0
with B the generator given by equation [START_REF] Ethier | Markov Processes: Characterization and Convergence[END_REF]:

Bf (u, r) = i∈N∩N I ζ∈E [f (u, r(r(i) → ζ)) -f (u, r)]α r(i),ζ (u( i N ))
Proposition 6. The Poisson equation ( 13) has a unique solution f which is measurable and locally Lipschitz continuous in its first variable with respect to the • H norm. For all fixed r ∈ R, the map u ∈ H → f (u, r) has a Fréchet derivative denoted by

df du [u, r]. Moreover sup s∈[0,T ] |f (u ε s , r ε s )| and sup s∈[0,T ] df du [u ε s , r ε s ] H *
are both bounded almost surely by constants independent of ε.

We begin the proof of the above Proposition by the following Lemma.

Lemma 1. Let B ⊂ H be a bounded domain and r be in R. The map u ∈ H →< G r (u) -F (u), φ > is:

1. bounded on B by a constant independent of r.

2. Lipschitz continuous on B with Lipschitz constant independent of r.

Fréchet differentiable on B with Fréchet derivative in

u ∈ H denoted by df du [u, r]. Moreover the Fréchet derivative is bounded in • H * norm uniformly in u ∈ B and r ∈ R. Proof. Recall that: < G r (u) -F (u), φ >= 1 N ξ∈E i∈N∩N I c ξ (1 ξ (r(i)) -µ(u( i N ))(ξ))(v ξ -u( i N ))φ( i N )
Since for ξ ∈ E, µ(•)(ξ) is bounded and Lipschitz continuous on R, the two first points follow. For the third point we note that the map

u ∈ H →< G r (u) -F (u), φ > is a linear combination of the Fréchet differentiable functions u ∈ H → µ u i N (ξ), u → u i
N and of the product of these two functions. The Fréchet derivative in u ∈ H against h ∈ H is given by:

- 1 N ξ∈E i∈N∩N I c ξ [µ (u( i N ))(ξ)(v ξ -u( i N )) + (1 ξ (r(i)) -µ(u( i N ))(ξ))]φ( i N )h( i N )
which gives us the boundedness property uniformly in r ∈ R and u ∈ B.

Proof of Proposition 6. For u ∈ H fixed, B(u, •) is an operator on R R which is a space of finite dimension. The Fredholm alternative in such a space is Im(B) = (ker(B * )) ⊥ . Moreover, by the ergodicity assumption, for any fixed u ∈ H, ker(B * (u, •)) = span(µ(u)). Therefore equation [START_REF] Fitzhugh | Thresholds and plateaus in the Hodgkin-Huxley nerve equations[END_REF] with u ∈ H fixed has a solution if and only if: R µ(u)(dr) < G r (u) -F (u), φ >= 0 This latter equality holds by the definition of F and the bi-linearity of the duality product < •, • >. Moreover, we can then always choose f satisfying R f (u, r)µ(u)(dr) = 0 by taking its projection on (ker(B * )) ⊥ . Thus we have a solution f (u, •) for any u ∈ H fixed. Uniqueness of f follows easily from the condition R f (u, r)µ(u)(dr) = 0. Recall that for all ε ∈]0, 1], u ε ∈ B where B = {u ∈ H; u ≤ C} where C is the deterministic constant given by Proposition 2. Then the desired properties of f follow from Lemma 1 and the fact that for a bounded domain

B ∈ H, each function u ∈ B → α r(i),ζ (u( i N )
) is bounded below and above by strictly positive constants, Lipschitz continuous and Fréchet differentiable with Fréchet derivative uniformly bounded in u, ζ and r. STEP 3 (Bound the martingale part). Our previous result implies that f ∈ D(A) and:

M ε t = f (u ε t , r ε t ) -f (u 0 , r 0 ) - t 0 Af (u ε s , r ε s )ds (14)
defines a square-integrable martingale, see for example Ethier and Kurtz [START_REF] Ethier | Markov Processes: Characterization and Convergence[END_REF], chapter 4 Proposition 1.7.

Proposition 7. We have :

< M ε > t = 1 ε t 0 i∈N∩N I ζ =rs(i) [f (u ε s , r s (r s (i) → ζ)) -f (u ε s , r ε s )] 2 α rs(i),ζ (u ε s ( i N ))ds (15)
Proof. It is a classical proof in stochastic calculus (see for example [START_REF] Ethier | Markov Processes: Characterization and Convergence[END_REF], Chapter 1, Problem 29). The Itô and the Dynkin formulas give two distinct decompositions of the squared process f 2 (u ε t , r ε t ) t≥0 in a semi-martingale. The uniqueness of the Doob-Meyer decomposition of a semi-martingale enables us to identify the bracket of our martingale.

We have the following semi-martingale decomposition:

t 0 < G r ε s (u ε s ) -F (u ε s ), φ > ds = εf (u ε t , r ε t ) -εf (u 0 , r 0 ) -ε t 0 d ds f (u ε s , r ε s )ds -εM ε t (16)
Recall that we are interested in bounding

E t 0 < G r ε s (u ε s ) -F (u ε s ), φ > ds 2 . Since sup s∈[0,T ] |f (u ε s , r ε s )| is bounded independently of ε ∈]0, 1]
, denoting by εg one of the two first terms of ( 16) we have E|εg| 2 ≤ C ε 2 where C is a constant independent of ε ∈]0, 1]. For the martingale term, by the Itô isometry:

E|M ε t | 2 = E < M ε > t ≤ 1 ε C since ε < M ε > t is bounded uniformly in t ∈ [0,
T ] and ε ∈]0, 1] thanks to the bounds on f and each α ξ,ζ . More precisely we have:

< M ε > t = 1 ε t 0 i∈N∩N I ζ =rs(i) [f (u ε s , r s (r s (i) → ζ)) -f (u ε s , r ε s )] 2 α rs(i),ζ (u ε s ( i N ))ds ≤ 4T ε α + N |E| sup s∈[0,T ] |f (u ε s , r ε s )| 2
STEP 4 (Bound the finite variation part). It remains to bound the third term of our semi-martingale decomposition. Proposition 8. There exists a constant C independent of ε ∈]0, 1] such that:

T 0 | d dt f (u ε t , r ε t )|dt ≤ C almost surely.
Proof. Recall first in Theorem 1 the meaning of the notation d dt f (u ε t , r ε t ). Recall that the map u → f (u, r) for r ∈ R fixed, is Fréchet differentiable with Fréchet derivative in u denoted by df du [u, r] which is a bounded linear form on H. By the Riesz representation theorem there exists f u (u, r) ∈ H such that:

df du [u, r](h) = (f u (u, r), h) H ∀h ∈ H Moreover the correspondence is isometric: df du [u, r] H * = f u (u, r) H .
We then have (see Theorem 4 of [START_REF] Buckwar | An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution[END_REF]):

d dt f (u ε t , r ε t ) =< ∆u ε t + G r ε t (u ε t ), f u (u ε t , r ε t ) >
By Proposition 6, there exists a constant C 1 independent of ε ∈]0, 1] and t ∈ [0, T ] such that:

f u (u ε t , r ε t ) H = df du [u ε t , r ε t ] H * ≤ C 1 Therefore: | d dt f (u ε t , r ε t )| ≤ C 1 ∆u ε t + G r ε t (u ε t ) H * It remains to show that ∆u ε t + G r ε t (u ε t ) H * is bounded uniformly in ε and t ∈ [0, T ].
For φ ∈ H we have, denoting by D the derivative with respect to x:

| < ∆u ε t + G r ε t (u ε t ), φ > | = -< Du ε t , Dφ > + 1 N ξ∈E i∈N∩N I c ξ 1 ξ (r(i))(v ξ -u ε t ( i N ))φ i N ≤ |(Du ε t , Dφ) L 2 (I) | + |E| max ξ∈E c ξ (max ξ∈E |v ξ | + C)C P φ H ≤ Du ε t L 2 (I) Dφ L 2 (I) + |E| max ξ∈E c ξ (max ξ∈E |v ξ | + C)C P φ H ≤ u ε t H φ H + |E| max ξ∈E c ξ (max ξ∈E |v ξ | + C)C P φ H ≤ (C + |E| max ξ∈E c ξ (max ξ∈E |v ξ | + C)C P ) φ H
where C is the deterministic constant given by Proposition 2. Thus:

∆u ε t + G r ε t (u ε t ) H * ≤ C + |E| max ξ∈E c ξ (max ξ∈E |v ξ | + C)C P
This ends the proof.

STEP 5 (All bounds together). Assembling all the bounds of the different terms we see that:

E t 0 < G r ε s (u ε s ) -F (u ε s ), φ > ds 2 ≤ Cε
with the constant C independent of ε ∈]0, 1]. It remains to let ε go to 0 to conclude the proof of the identification of the limit. STEP 6 (Uniqueness). We consider u 1 and u 2 two possible accumulation points verifying, for i = 1, 2:

u i t = e ∆t u 0 + t 0 e ∆(t-s) F (u i s )ds
for all t ∈ [0, T ] almost surely. We can show easily that for any t ∈ [0, T ]:

E( sup s∈[0,t] u 1 s -u 2 s H ) ≤ C t 0 E( sup l∈[0,s] u 1 l -u 2 l H )dl
with the constant C independent of t ∈ [0, T ]. Therefore, by application of the Gronwall's lemma we obtain that E(sup t∈[0,T ] u 1 t -u 2 t H ) = 0.

The general case

We now consider the original case with different state classes E 1 , • • • , E l . The first thing to do is to prove the tightness of the family ((u ε , rε ), ε ∈]0, 1]) in D([0, T ], H × {1, • • • , l} |N∩N I| ) (see section 2.3.3 for the definition of rε ). We apply a comparison argument. We notice that:

P(r ε t+h (i) = ζ|r ε t (i) = ξ) = 1 ε α ξ,ζ u t i N h + o(h) if ξ, ζ are in the same class E k α ξ,ζ u t i N h + o(h) otherwise
For ξ, ζ ∈ E we set:

λ ξ,ζ = 1 ε α + if ξ, ζ are in the same class E k α + otherwise
We denote by r ε,Max the associated jump process with constant rates λ ξ,ζ . We have the following stochastic domination:

P(r ε t+h (i) = ζ|r ε t (i) = ξ) ≤ P(r ε,Max t+h (i) = ζ|r ε,Max t (i) = ξ)
We construct the process rε,Max as rε by aggregation. The sequence (r ε,Max , ε ∈]0, 1]) is tight in D([0, T ], {1, • • • , l} |N∩N I| ), see for instance [START_REF] Yin | Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach[END_REF]. Therefore, by comparison, the sequence (r ε , ε ∈]0, 1]) is also tight in D([0, T ], {1, • • • , l} |N∩N I| ), see for instance [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF]. Moreover, the sequence (u ε , ε ∈]0, 1]) is tight in D([0, T ], H) by applying the arguments developed for the "all-fast" case. Endowing H × {1, • • • , l} |N∩N I| with the product topology we see that the sequence ((u ε , rε ),

ε ∈]0, 1]) is tight in D([0, T ], H × {1, • • • , l} |N∩N I| ).
We must now deal with the identification of the limit. There are six steps in the proof of the identification of the limit, we have to check that these six steps generalize to the general case (jumping slow-fast case).

In STEP 1, by the same arguments, we obtain that it is sufficient to show that:

lim ε→0 E t 0 < G r ε s (u ε s ) -F rε s (u ε s ), φ > ds 2 = 0
In STEP 2, the Poisson equation becomes:

Bf (u, r) = < G r (u) -F r (u), φ > (17)
where B is now the "fast" part of the generator (f does not depend on r explicitly because r is constructed from r). A configuration r for the ion channels is now:

r ∈ E j1 × • • • × E j N -1 where (j 1 , • • • , j N -1 ) ∈ {1, • • • , l} N -1 (noting that |N ∩ N I| = N -1).
Thats is each channel is in one of the class E j for j ∈ {1, • • • , l}. Then, since for fixed u the quasistationary measure associated to the class

E j k for k ∈ {1, • • • , N -1} is µ j k (u), we
have that the kernel of B * is spanned by:

{µ j1 (u) × • • • × µ j N -1 (u), (j 1 , • • • , j N -1 ) ∈ {1, • • • , l} N -1 }
when u held fixed. Then, by the Fredholm alternative and the definition of the averaged function F r (u), the Poisson equation ( 17) has a solution. Uniqueness follows by the projection condition:

R f (u, r)µ j1 (u)(dr)ו • •×µ j N -1 (u)(dr) = 0, ∀(j 1 , • • • , j N -1 ) ∈ {1, • • • , l} N -1 (18)
The proof of the last three steps is then exactly the same as in the "all fast" case.

m 0 h 0 3ε -1 am ε -1 bm m 1 h 0 2ε -1 am 2ε -1 bm m 2 h 0 ε -1 am 3ε -1 bm m 3 h 0 b h a h b h a h b h a h b h a h m 0 h 1 3ε -1 am ε -1 bm m 1 h 1 2ε -1 am 2ε -1 bm m 2 h 1 ε -1 am 3ε -1 bm m 3 h 1

Example

In this section we give a concrete example where our result allows to reduce the complexity of a neuronal model staying nevertheless at a stochastic level.

We consider a usual Hodgkin-Huxley model but, to be very straightforward in the application of our result, we only consider the sodium current i.e. all the ion channels are sodium channels. It is still a case of interest since sodium channels are involved in the increasing phase of an action potential. At a fixed potential, the kinetic of a sodium channel is described by Figure 1 which represents the states and the jump rates of a continuous time Markov chain denoted by r ε (i) for the channel at position i N for i ∈ N ∩ N I. A sodium channel can be in 8 different states denoted by m i h j for i ∈ {0, 1, 2, 3} and j ∈ {0, 1}, the state m 3 h 1 is the only "open" state for the sodium channel (see [START_REF] Hille | Ionic Channels of Excitable Membranes[END_REF] for more details). We simply write a m , b m and a h , b h for a m (u), b m (u) and a h (u), b h (u) (see Appendix C for more details on the rate functions). m is the fast variable and h the slow variable for the kinetic of a sodium channel.

Each channel can therefore be in one of these eight states divided in two classes :

E = E 0 E 1 where E 0 = {m 0 h 0 , m 1 h 0 , m 2 h 0 , m 3 h 0 } and E 1 = {m 0 h 1 , m 1 h 1 , m 2 h 1 , m 3 h 1 }.
Inside these two classes the states communicate at fast rates and transition between these two classes occurs at slow rates. A sodium channel is open if and only if it is in the state m 3 h 1 . We then define c m3h1 = c Na , v m3h1 = v Na and c ξ = 0, v ξ = 0 when ξ ∈ E \ {m 3 h 1 }.

The equation [START_REF] Buckwar | An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution[END_REF] describing the evolution of the potential, for a number N of ion channels, is here given by :

∂ t u ε = K∆u ε + 1 N i∈N∩N I c Na 1 m3h1 (r ε (i))(v Na -u( i N ))δ i N (19)
The global variable u represents the difference of potential between the inside and the outside of the axon membrane. We see that Equation ( 19) is indeed of the form that we have studied in this paper. K is a constant related to the radius and the internal conductivity of the axon (see Appendix C). Let us compute the different generators and quasi-invariant measures associated to this model according to our description. The two "fast" generators are the same and are given for u( i N ) fixed (again, we do not make the dependence appear), by :

B j =     -3a m 3a m 0 0 b m -b m -2a m 2a m 0 0 2b m -2b m -a m a m 0 0 3b m -3b m    
with j = 0 or 1. We can compute the associated quasi-invariant measure µ j (u( i N ))

(j = 0 or 1), the only term of interest for us is :

µ 1 (u( i N ))(m 3 h 1 ) = 1 1 + bm(u( i N )) am(u( i N )) 3
The reader familiar with the classical Hodgkin-Huxley model will notice that µ 1 (u( i N ))(m 3 h 1 ) is in fact equal to the steady-state function associated to the ODE describing the motion of the m-gates in the classical deterministic Hodgkin-Huxley description, see for example [START_REF] Hodgkin | A quantitative description of membrane current and its application to conductation and excitation in nerve[END_REF].

The asymptotic aggregated Markov chain r is valued in the two-state space {0, 1}. According to Proposition 3 its generator is:

-a h (u( i N )) a h (u( i N )) b h (u( i N )) -b h (u( i N ))
Therefore, according to Theorem 2, the reduced model is described by the following PDE coupled with the continuous-time Markov chain r :

∂ t u = K∆u + 1 N i∈N∩N I 1 1 (r(i))c Na µ 1 (u( i N ))(m 3 h 1 )(v Na -u( i N ))δ i N (20)
We display a realization of the averaged piecewise deterministic Markov process in Figure 2. To perform the numerical simulation (in C) we extended Riedler's work (cf. [START_REF] Riedler | Almost Sure Convergence of Numerical Approximations for Piecewise Deterministic Markov Processes[END_REF]) to our particular framework. In [START_REF] Riedler | Almost Sure Convergence of Numerical Approximations for Piecewise Deterministic Markov Processes[END_REF], an algorithm (Algorithm A2) is proposed to simulate the trajectory of a PDMP in finite dimension. The idea is to simulate the jumping part of the PDMP and to use an accurate method to simulate the ODE constituting the deterministic part of the PDMP. The kinetic of the jumping component for PDMP in finite and infinite dimensions have the same form. Here we simulate the jumping part of our infinite dimensional PDMP following [START_REF] Riedler | Almost Sure Convergence of Numerical Approximations for Piecewise Deterministic Markov Processes[END_REF] and simulate the PDE between successive jumps by a deterministic scheme. For the PDE, we used an explicit finite difference Euler scheme in space and time. Simulating the jumps of an inhomogeneous time Markov chain is classical and there exist a lot of efficient numerical schemes for PDE. Therefore the described method is natural. This argument is of course not a proof but gives an heuristic interpretation of our approach.

Concluding remarks

In this paper, we apply the methods of stochastic averaging to a fully coupled PDMP in infinite dimensions. To the best of our knowledge it is the first time that such a problem is considered. The two main difficulties are due, on one hand to the infinite dimensional setting and on the other hand to the fact that the system is fully-coupled. Regarding the fully-coupled property, our task was made easier by the uniform bound on the macroscopic component of the PDMP and on the jump rates of the continuous time Markov chain. As an application, we have shown how to reduce the complexity of a generalized Hodgkin-Huxley model which can be used as a paradigm in the modelling of the propagation of an action potential for neurons but also for cardiac cells.

Let us make some comments about a possible extension of our work to other forms of generalized Hodgkin-Huxley models. Firstly, to choose that the ion channels are uniformly distributed along the axon ( i N for i ∈ N ∩ N I) is just a simple way to fix ideas. We could also choose any distribution in a finite subset N of I without changing anything to our work except the notations. Secondly, in [START_REF] Buckwar | An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution[END_REF], the authors noticed that it can be of interest to replace the Dirac point masses δ i N of the spatial stochastic Hodgkin-Huxley model by functions of the form |U ( i N )| -1 1 U ( i N ) (•) where U is a neighborhood of the point i N . Indeed, the concentration of ions is essentially homogeneous in a spatially extended domain around an open channel. Along the same line, one can argue that we could replace the Dirac point masses δ i N by any smoother approximation φ i N . The mathematics become then a little easier. The methods presented in this paper apply to both cases. The next step after the present averaging result is to prove the associated central limit theorem. That is to say give more precision on how the original model converges in law to the averaged model.

Another line of work can be to consider that not only some ion channels but also the membrane potential have a faster kinetic. It seems that this situation is closer to the real behavior of the nerve impulse [START_REF] Hille | Ionic Channels of Excitable Membranes[END_REF]. This last question is related to the study of invariant measures for PDMP and the necessity to find explicit form or bounds for these invariant measures. These will be the object of a future work. The maximal conductance associated to sodium ions is c Na = 120 mS.cm -2 and the potential at rest is v Na = 115 mV. The constant K is given by K = a 2R where a is the radius of the axon, a = 0.0238 cm and R is the internal resistance of the axon, R = 34.5 Ω.cm, these data are classical, see for example [START_REF] Hodgkin | A quantitative description of membrane current and its application to conductation and excitation in nerve[END_REF]. We have used an input on the potential equal to µ = 6.7 all along the time on the segment [0, 0.1] of the axon.

  We define, for f ∈ H and x ∈ I: e ∆t p f := p k=1 e -(kπ) 2 t (f, e k )e k and e ∆t p δ x := p k=1 e -(kπ) 2 t (1 + (kπ) 2 )e k (x)e k ,

Figure 1 :

 1 Figure 1: Kinetic of sodium channels.

Figure 2 :

 2 Figure2: Simulation of the action potential against space (vertical axis) and time (horizontal axis). The averaged model (20) is displayed for a number of ion channels fixed to 250. We see what we expect from the partial differential equation: a traveling wave connecting the initial state 0 to the stable state vNa.

  u ε t , L δ,η ) > η) ≤ δ (23)whered(u ε t , L δ,η ) = inf v∈,L δ,η u ε t -v H .Appendix C. Functions and data for the simulation For the simulation, the jump rate functions are given, for real u, by :a m (u) = 0.1(25 -u) e 2.5-0.1u -1 b m (u) = 4e

Acknowledgments : the authors warmly thank the anonymous referee for his/her helpful comments, suggestions and corrections which led to a significant improvement of the previous versions of the manuscript.

This work has been supported by the Dynamics, ANR-09-BLAN-0008-01.

Appendix A. Austin's lemma about the semi-group e ∆t

We recall here a lemma in [START_REF] Austin | The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism[END_REF]. Note that the last part of conclusion 1. of the following lemma is contained in the proof of the corresponding lemma in [START_REF] Austin | The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism[END_REF].

Lemma 2. Let y be in the interior of I. Then e ∆t δ y is a smooth function on I vanishing at the end-points for any t > 0. Furthermore:

1. there is some constant C 1 > 0, depending on T but otherwise not on t ∈ [0, T ],

such that for any continuous function u : [0, t] → R we have that the function

is in H and satisfies the estimate

Moreover for any η > 0 we can choose ε > 0 so small that:

for any fixed ε > 0 there is some constant C 2 (ε), depending on ε and T but otherwise not on t ∈ [0, T ], such that for any continuous function u : [0, t] → R we have

Appendix B. Two theorems about tightness in Hilbert spaces

We recall here two important results of [START_REF] Métivier | Convergence faible et principe d'invariance pour des martingales à valeurs dans des espaces de Sobolev[END_REF] about tightness in Hilbert spaces with notations and choice of spaces adapted to our need. Theorem 3. (General criterion for tightness.) Let us assume that (u ε , ε ∈]0, 1]) satisfies Aldous's condition which means that for any δ, M > 0, there exist η, ε 0 > 0 such that for all stopping times τ such that τ + η < T :

If moreover, for each t ∈ [0, T ] fixed the family

Theorem 4. (Tightness in a Hilbert space.) H is a separable Hilbert space endowed with a basis {e k , k ≥ 1}. We denote by L k for k ≥ 1:

Then (u ε t , ε ∈]0, 1]) is tight in H if, and only if, for any δ, η > 0 there is ρ, ε 0 > 0 and L δ,η ⊂ {L k , k ≥ 1} such that :


Jean-Charles Naud
email: jean-charles.naud@irisa.fr

Daniel Ménard
email: menard@irisa.fr

Numerical Accuracy Evaluation for Polynomial Computation

Keywords: fixed-point, quantization, analytic approach, noise, correlation virgule fixe, quantification, approche analytique, correlation

Fixed-point conversion requires fast analytical methods to evaluate the accuracy degradation due to quantization noises. Usually, analytical methods do not consider the correlation between quantization noises. Correlation between quantization noises occurs when a data is quantized several times. This report explained, through an example, the methodology used in the ID.Fix tool to support correlation. To decrease the complexity, a method to group together several quantization noises inside a same noise source is described. The maximal relative estimation error obtained with the proposed approach is less than 2%.

Introduction

Fixed-point arithmetic is widely used in embedded systems to reduce implementation costs like execution time, area and power consumption. Fixed-point conversion is composed of two main steps corresponding to dynamic range evaluation and word-length (WL) optimization. The aim of WL optimization is to minimize the implementation cost as long as the effects of finite precision are acceptable. This optimization process is based on an iterative procedure where the numerical accuracy is evaluated a great number of times. Thus, efficient methods are required to evaluate this numerical accuracy to limit the optimization time.

To evaluate numerical accuracy, approaches based on fixed-point simulations are generic, but they also lead to long execution times. Thus, the search space is drastically limited and suboptimal solutions are obtained. Analytic methods reduce significantly the evaluation time by providing the mathematical expression of a metric equivalent to the numerical accuracy. The output quantization noise power is widely used as a relevant metric for evaluating the numerical accuracy.

Usually, analytical methods do not consider the correlation between quantization noises. Correlation between quantization noises occurs when a data is quantized several times. In this report, the expressions of the correlation and the covariance, considering the number of eliminated bits, are presented for truncation and rounding. Then, this report explained, through an example, the methodology used in the ID.Fix tool to support correlation.

Methodology description

Problem description

Correlation between QNSs occurs when a data x 0 is quantized several times. Figure 1 shows such an example where x i is the data after each quantization Q i with i equal to 1 or 2. Q i leads to an unavoidable quantization error e i between the values of the data x i and x 0 . e i can be assimilated to a noise source and we denote E i as the random variable corresponding to this error. Let w i denote the fractional part word-length of the data x i and q i the quantization step associated to x i . q i = 2 -wi with w i the weight of the least significant bit. The number of bits eliminated during the quantization process Q i is defined as k i . The relation between the quantization step q i and q 0 is q i = 2 ki q 0 , with i equal to 1 or 2. If x 0 is in infinite precision q 0 is equal to zero and k 1 and k 2 tend to infinity. Let X i denote the set containing all the values that can be represented in the fixed-point format after the quantization Q i .

Given that k 2 bits are common between the QNSs e 1 and e 2 , these QNSs are correlated. Let y denote the output of the global targeted system. Let H i denote the system having e i as input and y as output. As the QNS e i propagates through the system H i , correlation between different QNSs e i obviously influences the output noise power and has therefore to be considered for a precise numerical accuracy evaluation.

Quantization Mode Description

In this section, the probability density function and the statistical moments of the QNSs generated during a quantization process are presented for the truncation and rounding quantization modes in the case of two's complement coding. The quantization process Q 1 presented in Figure 1 is under consideration. The quantization error e 1 resulting from the quantization process Q 1 is defined as

e 1 = x 0 -x 1 . (1)
By using Widrow's model [START_REF] Widrow | Quantization Noise: Roundoff Error in Digital Computation, Signal Processing, Control, and Communications[END_REF]2], e 1 can be assimilated to an additive white noise, uniformly distributed, which is uncorrelated to the signal.

Truncation

In the case of truncation, the data x 0 is always rounded towards the lower value available in the set X 1 and becomes

x 1 = ⌊x 0 • q -1 1 ⌋ • q 1 = t • q 1 ∀x 0 ∈ [t • q 1 , (t + 1) • q 1 [ (2)
with ⌊•⌋ the floor function defined as ⌊x 0 ⌋ = max (n ∈ Z| n ≤ x 0 ) and with q 1 the quantization step. The probability density function (PDF) of the QNS p E1 (e 1 ) is given by (3) with δ the Kronecker delta.

p E1 (e 1 ) = 1 2 k1 2 k 1 -1 j=0 δ(e 1 -j • q 0 ) (3)

Rounding

Rounding quantization mode rounds the value x 0 to the nearest value available in the set X 1 as

x 1 = x 0 + 1 2 q 1 • q -1 1 • q 1 . (4)
The midpoint q m = (t + 1 2 ) • q 1 between t • q 1 and (t + 1) • q 1 is always rounded up to the higher value (t + 1) • q 1 . For this quantization mode, the PDF p E1 (e 1 ) is given by

p E1 (e 1 ) = 1 2 k1 2 k 1 -1 -1 j=-2 k 1 -1
δ(e 1j • q 0 .).

(5)

From [3] and [START_REF] Menard | Quantization Mode Opportunities in Fixed-Point System Design[END_REF], mean and variance expressions are given in Table 1 for each quantization mode Q 1 . If x 0 has a continuous amplitude, as in analog-to-digital conversion, k 1 is considered as +∞.

Correlation and covariance expressions

In this section, the expressions of the correlation and the covariance between two QNSs e 1 and e 2 , resulting from the quantization of one unique data x 0 as presented in Figure 1, are determined. The covariance is used in the expression of the global output quantization noise power to improve the quality of the noise power estimation. The reasoning to determine the correlation and covariance expressions is detailed in the following with k 1 ≥ k 2 and for the case Table 1: Mean and variance for the two quantization modes

Quantization mode Q 1 Mean Variance Truncation q1 2 • (1 -2 -k1 ) q 2 1 12 • (1 -2 -2k1 ) Rounding -q1 2 • (2 -k1 ) q 2 1 12 • (1 -2 -2k1 )
Figure 2: PDF of the QNSs e 1 and e 2 for

Q 1 = T , Q 2 = R, k 1 = 3 and k 2 = 2. q 0 is the quantization step.
where Q 1 is a truncation (T) and Q 2 a rounding (R). The two discrete PDFs p E1 and p E2 of respectively the QNSs e 1 and e 2 are presented in Figure 2 for the case of k 1 = 3 and k 2 = 2.

Let E 1 and E 2 denote the discrete random variables corresponding to the QNSs. The correlation

E [E 1 • E 2 ] is determined from p E1,E2 the joint probability density function between E 1 and E 2 as E [E 1 • E 2 ] = i j
i.q 0 .j.q 0 .p E1,E2 (e 1 = i.q 0 , e 2 = j.q 0 ) (

where i and j enumerate the possible events of e 1 and e 2 . The joint probability density function p E1,E2 is obtained from p E1|E2 the conditional probability of

E 1 given E 2 as p E1,E2 (e 1 , e 2 ) = p E1|E2 (e 1 |e 2 ) • p E2 (e 2 ), (7)
For each value of e 2 , 2 k1-k2 values are obtained for e 1 , thus

p E1|E2 = 1 2 k1-k2 .   2 k 2 -1 -1 j=0 δ(e 2 -jq 0 ) 2 k 1 -k 2 -1 t=0 δ (e 1 -e 2 -t • q 2 )) (8) + -1 j=-2 k 2 -1
δ(e 2jq 0 )

2 k 1 -k 2 -1 t=0 δ (e 1 -e 2 -(t + 1) • q 2 )  
To illustrate equation 8, the example with k 1 = 3 and k 2 = 2 is considered. Thus, the different cases for the quantization error e 1 and e 2 are provided in Table 2.

The columns bit 2, bit 1 and bit 0 specify the three LSB of the quantized data (x 0 ). The weight of the bit 0 is q 0 . In our case, e 2 is due to the elimination of 2 bits with rounding quantization mode. The figure 3 presents the different values of the error e 2 . The result is shown in Figure 4. Red Dirac and green Dirac correspond respectively to the first part (two first summations) and the second part (two last summations) of the previous expression. The amplitude of the different Dirac are the same and equal to From eq. 7 and 6, the correlation between E 1 and E 2 becomes

Inria bit 2 bit 1 bit 0 e 1 e 2 0 0 0 0 0 0 0 1 q 0 q 0 0 1 0 2q 0 -2q 0 0 1 1 3q 0 -q 0 1 0 0 4q 0 0 1 0 1 5q 0 q 0 1 1 0 6q 0 -2q 0 1 1 1 7q 0 -q 0
1 2 k 1 -k 2 .
E [E 1 .E 2 ] = q 2 0 2 k1 2 k 1 -k 2 -1 t=0
2 k 2 -1 -1 j=0 j(j + t.2 k2 ) + (j -2 k2-1 )(j + 2 k2-1 + t.2 k2 ) = - q 2 2 24 + q 2 0 6 - q 0 .q 1 4 . (9)
The covariance cov(

E 1 , E 2 ) is determined from the correlation term E [E 1 .E 2 ] as cov(E 1 , E 2 ) = E [E 1 • E 2 ] -E [E 1 ] • E [E 2 ] . (10)
Q 1 Q 2 Condition E[E 1 .E 2 ] cov(E 1 , E 2 ) T T k 1 ≥ k 2 q 2 2 12 + q 2 0 6 -q0.q2 4 -q0.q1 4 + q1.q2 4 q 2 2 12 - q 2 0 12 T R k 1 ≥ k 2 - q 2 2 24 + q 2 0 6 -q0.q1 4 - q 2 2 24 - q 2 0 12 R T k 1 > k 2 q 2 2 12 + q 2 0 6 -q0.q2 4
q 2 2 12 - q 2 0 12 R R k 1 = k 2 q 2 2 12 + q 2 0 6 q 2 2 12 - q 2 0 12 R R k 1 > k 2 - q 2 2 24 + q 2 0 6 - q 2 2 24 - q 2 0 12
The term

E [E 1 ] .E [E 2
] can be computed from the equations given in Table 1 and is equal to

E [E 1 ] .E [E 2 ] = q 1 2 1 -2 -k1 q 2 2 (-2 -k2 ) = q 2 0 -q 0 .q 1 4 (11)
Thus, from eq. 9 and 11, the expression of the covariance becomes

cov(E 1 , E 2 ) = - q 2 2 24 - q 2 0 12 (12)
Given that k 1 ≥ k 2 , the term q 1 is eliminated because just the k 2 least significant bits are common between e 1 and e 2 . The correlation and covariance expressions are given in Table 4 for the different quantization modes Q i corresponding to truncation (T) or rounding (R) and for different conditions on k 1 and k 2 . If x 0 is in infinite precision, q 0 is equal to 0.

Output Quantization Noise Power

Different models have been proposed to estimate the power of the quantization noise at the output of a system [5, 6, 7, 8]. These approaches do not consider the correlation between quantization noises, but they can be easily extended to integrate this correlation to improve the estimation quality.

From [6], the output quantization noise e y is the sum of the contributions of the N QNSs e i

e y (n) = N i=1 ∞ t=0 h i (t, n) • e i (n -t) (13)
where h i corresponds to the time-varying impulse response of the system H i between e i and the output y. The term t represents the delay and n the time.

The power P ey of the output quantization noise is obtained by determining the second order moment of e y with a similar derivation as in [6]. From (14), the expression of P ey is

P ey = E E 2 y = E   N i=1 ∞ t=0 h i (t, n) • e i (n -t) 2   Inria E E 2 y = E   N i=1 N j=1 ∞ t=0 ∞ v=0 h i (t, n) • h j (v, n) • e i (n -t) • e j (n -v)   = E N i=1 ∞ t=0 h 2 i (t, n) • e 2 i (n -t) + N i=1 ∞ t=0 ∞ v=0 v =t h i (t, n) • h i (v, n) • e i (n -t) • e i (n -v) + N i=1 N j=1 j =i ∞ t=0 h i (t, n) • h j (t, n) • e i (n -t) • e j (n -t) + N i=1 N j=1 j =i ∞ t=0 ∞ v=0 v =t h i (t, n) • h j (v, n) • e i (n -t) • e j (n -v)     (14)
The different quantization noises e i are assumed to be white so the autocorrelation is equal to

E[e i (n -t)e i (n -v)] = σ 2 i δ(t -v) + µ 2 i ( 15
)
and the intercorrelation

E[e i (n -t)e j (n -v)] = cov(E i , E j )δ(t -v) + µ i µ j (16) E E 2 y = N i=1 ∞ t=0 E h 2 i (t, n) • (σ 2 i + µ 2 i ) + N i=1 ∞ t=0 ∞ v=0 v =t E [h i (t, n) • h i (v, n)] • µ 2 i + N i=1 N j=1 j =i ∞ t=0 E [h i (t, n) • h j (t, n)] • (µ i • µ j + cov(E i , E j )) + N i=1 N j=1 j =i ∞ t=0 ∞ v=0 v =t E [h i (t, n) • h j (v, n)] • µ i • µ j ( 17
)
P ey = E E 2 y = N i=1 K i • σ 2 i + N i=1 N j=1 L ij • µ i • µ j + N i=1 N j=1 j =i M ij • cov(E i , E j ) RR n°7878
with

K i = ∞ t=0 E h 2 i (t, n) , L ij = ∞ t=0 ∞ v=0 E [h i (t, n)h j (v, n)] , M ij = ∞ t=0 E [h i (t, n)h j (t, n)] , (18)
and where K i , L ij and M ij are constant terms depending only of the system in infinite precision, which can thus be determined only once. The variance σ 2 i , the mean µ i and the covariance cov(E i , E j ) between QNSs depend on the quantization modes, the number of bit w i for the fractional part and the number of bits eliminated k i for each data x i .

Application description

To illustrate the proposed approach, the computation of a 3 rd order polynomial using the Horner scheme is presented. The Signal Flow Graph is presented in figure 5. The expression of the output y is

y(n) = a 0 + x(n).(a 1 + x(n).(a 2 + x(n).(a 3 + x(n)))) (19)

Quantizations noise modelling

A quantization noise is generated when the fractional word length (w F P ) of a data is reduced. Each quantization error is modelled as a noise with the mean µ and the variance σ. A quantization noise can be modelled with the quantization step q, the quantization mode (t Q ) (truncation (T) or rounding (R)) and the number of eliminated bits k between the old and the new format.

Noise source

The noise sources b allow grouping together different quantization noises e in the signal flow graph. Given that the complexity of the method depends on the number of quantization noise, the quantization noises are grouped together as much as possible to decrease the number of noise sources proceeded by the method. The figure 6 shows, in the general case, the potential quantization noises (I, II, III, IV) associated with the noise source b. The w F P of inputs and output of operation P 1 are specified with w F P P 1,0 , w F P P 1,1 and w F P P 1,2 . P i and D i represent respectively the number associated to the operations and the data. Two cases are identified. In the first case, there is a word-length constraint w F P D1 applied on data D 1 . The quantization noises are I,II and III. In the second case, there is no restriction on w F P D1 . In this case, only quantization noise I and IV are considered.

For the example of polynomial computation, the noise sources are inserted as show in figure 7.

Noise model propagation

The noise model propagation of each operator is inserted. An operation with two inputs x and y and the output z is considered. Let e x , e y and e z denotes the quantization noises associated with the input and the output. For the addition/subtraction, the output noise expression is :

e z = e x ± e y
For the multiplication, the output noise expression is :

e z = e x y ± e y x
Constant value are not considered to generate a quantization noise. By using previous expression, the noise data flow graph is generated as show in figure 8.

Determination of the impulse response of the system

Let H i denote the system having the noise source b i as input and y as output. Let h i (t, n) denote the impulse response of H i . The term t represents the delay and n the time. In our case, there is no delay, so h i = 0, ∀t = 0. The expressions of the different impulse response h i are :

h 1 (0, n) = a 3 * x 2 (n) h 2 (0, n) = (a 3 * x(n) + a 2 ) • x(n) h 3 (0, n) = (a 3 * x(n) + a 2 ) • x(n) + a 1 h 4 (0, n) = x 2 (n) h 5 (0, n) = x 2 (n) h 6 (0, n) = x(n) h 7 (0, n) = x(n) h 8 (0, n) = 1 h 9 (0, n) = 1 (20)
The matrices K, L et M are computed only one time with the followers equations.

K i = ∞ t=0 E h 2 i (t, n) (21)
L ij = ∞ t=0 ∞ v=0 E [h i (t, n)h j (v, n)] (22)
M ij = ∞ t=0 E [h i (t, n)h j (t, n)] (23)
The N -length vector K is equal to

Inria               E (a 3 * x 2 (n)) 2 E ((a 3 * x(n) + a 2 ) • x(n)) 2 E ((a 3 * x(n) + a 2 ) • x(n) + a 1 ) 2 E x 4 (n) E x 4 (n) E x 2 (n) E x 2 (n) 1 1              
The different element of the N × N matrix L are To decrease the computation time, the symmetry of L and M is used. Moreover, the M ij terms are not computed if there is no correlation between b i and b j . When there is a correlation between b i and b j , each noise source b i or b j is a single quantization noise. In our case, for the matrix M , only the terms M 12 , M 13 and M 23 have to be computed (M 21 , M 31 and M 32 are deduced by symmetry).

L = E (a 3 * x 2 (n)) 2 L 12 = L = E (a 3 * x 2 (n)) • ((a 3 * x(n) + a 2 ) • x(n)) L 13 = L = E (a 3 * x 2 (n)) • ((a 3 * x(n) + a 2 ) • x(n) + a 1 ) L 14 = L = E (a 3 * x 2 (n)) • x 2 (n)) L 15 = L = E (a 3 * x 2 (n)) • x 2 (n)) L 16 = L = E (a 3 * x 2 (n)) • x(n)) L 17 = L = E (a 3 * x 2 (n)) • x(n)) L 18 = L = E a 3 * x 2 (n) L 19 = L = E a 3 * x 2 (n) L = E ((a 3 * x(n) + a 2 ) • x(n)) 2 L 23 = L = E [((a 3 * x(n) + a 2 ) • x(n)) • ((a 3 * x + a 2 ) • x(n) + a 1 )]] L 24 = L = E ((a 3 * x(n) + a 2 ) • x(n)) • x 2 (n) L 25 = L = E ((a 3 * x(n) + a 2 ) • x(n)) • x 2 (n) L 26 = L = E [((a 3 * x(n) + a 2 ) • x(n)) • x(n)] L 27 = L = E [((a 3 * x(n) + a 2 ) • x(n)) • x(n)] L 28 = L = E [(a 3 * x(n) + a 2 ) • x(n)] L 29 = L = E [(a 3 * x(n) + a 2 ) • x(n)] L = E ((a 3 * x(n) + a 2 ) • x(n) + a 1 ) 2 L 34 = L = E ((a 3 * x(n) + a 2 ) • x(n) + a 1 ) • x 2 (n) L 35 = L = E ((a 3 * x(n) + a 2 ) • x(n) + a 1 ) • x 2 (n) L 36 = L = E [((a 3 * x(n) + a 2 ) • x(n) + a 1 ) • x(n)] L 37 = L = E [((a 3 * x(n) + a 2 ) • x(n) + a 1 ) • x(n)] L 38 = L = E [(a 3 * x(n) + a 2 ) • x(n) + a 1 ] L 39 = L = E [(a 3 * x(n) + a 2 ) • x(n) + a 1 ] L = E (x 2 (n)) 2 L 45 = L = E (x 2 (n)) 2 L 46 = L = E (x 2 (n)) • x(n) L 47 = L = E (x 2 (n)) • x(n) L 48 = L = E x 2 (n) L 49 = L = E x 2 (n) L = E (x 2 (n)) 2 L 56 = L = E (x 2 (n)) • x(n) L 57 = L = E (x 2 (n)) • x(n) L 58 = L = E x 2 (n) L 59 = L = E x 2 (n) Inria

Noise power expression

The global noise power expression is

P by = N i=1 K i .σ 2 i + N i=1 N j=1 L ij • µ i .µ j + N i=1 N j=1,j =i M ij • cov(B i , B j ) (25)
The variable of this expression are σ 2 i , µ i and cov(B i , B j ) which depends on the data wordlength and the quantization modes. The cov(B i , B j ) term are compute with the table 4.

Let q 0 is the quantization step of the initial data (x in this example). Let q 1 and q 2 are the quantization step of data quantized and k 1 and k 2 the number of eliminated bits. If the initial data x is in infinite precision q 0 is equal to zero and k 1 and k 2 tend to infinity.

Figure 1 :

 1 Figure 1: Quantized data representation

Figure 3 :

 3 Figure 3: Different values of the error e 2 for the rounding quantization with k 2 = 2

Figure 4 :

 4 Figure 4: Joint probability p E1|E2(e1,e2)

Figure 5 :

 5 Figure 5: 3 rd order polynomial using the Horner scheme

Figure 6 :

 6 Figure 6: Quantization noise in an noise source

Figure 7 :

 7 Figure 7: Noise sources insertion

Figure 8 :

 8 Figure 8: Noise models insertion

LL 88 = 1 L 1 L 99 = 1

 88111 66 = E x 2 (n)L 67 = L 76 = E x 2 (n) L 68 = L 86 = E [x(n)] L 69 = L 96 = E [x(n)] L 77 = E x 2 (n) L 78 = L 87 = E [x(n)] L 79 = L 97 = E [x(n)] 89 = L 98 = The different element of the N × N matrix M are M 12 = M 21 = E (a 3 * x 2 (n)) • ((a 3 * x(n) + a 2 ) • x(n)) M 13 = M 31 = E (a 3 * x 2 (n)) • ((a 3 * x(n) + a 2 ) • x(n) + a 1 ) M 23 = M 32 = E [((a 3 * x(n) + a 2 ) • x(n)) • ((a 3 * x + a 2 ) • x(n) + a 1 )] others = 0(24)

Table 2 :

 2 Values of the eliminated bits during the quantization and associated error e 1 and e 2 .

Table 3 :

 3 Correlation and covariance expressions for the different quantization modes of truncation (T) or rounding (R), and for different conditions on k 1 and k 2

RR n°7878

Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr ISSN 0249-6399

Table 4: Covariance expressions for the different quantization modes of truncation (T) or rounding (R), and for different conditions on k 1 and k 2

Conclusion

In the context of numerical accuracy evaluation of fixed-point systems, the expressions of the correlation and the covariance between QNSs resulting from the quantization of one unique data has been proposed in this report. The expression of the global output quantization noise integrates correlation between QNSs, which improves the quality of the estimation of the output quantization noise compared to existing approaches. The noise power in the case of a third-order polynomial computation has been described in this report.