Constance Hammond
email: hammond@inmed.univ-mrs.fr

Rachida Ammari

Bernard Bioulac

Liliana Garcia

Latest view on the mechanism of action of deep brain stimulation

Keywords: DBS, electrophysiology, Parkinson, dystonia, OCD DBS, electrophysiology, antidromic spikes, Parkinson, Dystonia, OCD

Introduction

Deep brain stimulation (DBS) has the potential to provide substantial benefit for various neurologic and neuropsychiatric diseases. DBS is an intracerebral, extracellular stimulation consisting of short pulses (in the order of 100 µs) regularly applied at a frequency of at least 100 Hz over a period of several years. First tested in ventral thalamic nuclei to alleviate essential tremor [START_REF] Benabid | Long-term suppression of tremor by chronic stimulation of the ventral intermediate thalamic nucleus[END_REF] , it is now widely used in the internal pallidal segment (GPi) or subthalamic nucleus (STN) for Parkinson's disease [START_REF] Limousin | Effect of parkinsonian signs and symptoms of bilateral subthalamic nucleus stimulation[END_REF][START_REF] Benabid | Deep brain stimulation for Parkinson's disease[END_REF][START_REF] Rezai | Deep brain stimulation for Parkinson's disease: surgical issues[END_REF] , in the GPi for generalized dystonia [START_REF] Kumar | Globus pallidus deep brain stimulation for generalized dystonia: clinical and PET investigation[END_REF][START_REF] Coubes | Electrical stimulation of the globus pallidus internus in patients with primary generalized dystonia: long-term results[END_REF] , and more recently for other diseases such as treatment-resistant obsessive compulsive disorder (OCD) [START_REF] Aouizerate | Deep brain stimulation of the ventral caudate nucleus in the treatment of obsessive-compulsive disorder and major depression. Case report[END_REF][START_REF] Lipsman | Deep brain stimulation for treatment-refractory obsessivecompulsive disorder: the search for a valid target[END_REF] , Tourette syndrome [START_REF] Houeto | Tourette's syndrome and deep brain stimulation[END_REF][START_REF] Visser-Vandewalle | DBS in tourette syndrome: rationale, current status and future prospects[END_REF] and depression [START_REF] Ressler | Targeting abnormal neural circuits in mood and anxiety disorders: from the laboratory to the clinic[END_REF] . Sites of stimulation are located inside the cortico-basal ganglia-thalamo-cortical loops, in motor or limbic regions, depending on the clinical signs.

Two major explanations have been proposed for the mechanism of action of DBS: (1) it silences stimulated neurons (i.e., it is equivalent to a local lesion) or (2) It introduces a new activity in the network. The first theory is based on the assumption that a group of neurons is responsible for the pathological activity of the whole network. Therefore silencing these neurons will immediately suppress the pathological activity. The second explanation leads to a different hypothesis: The whole network malfunctions for complex reasons but the injection in one point of a DBS-driven activity that propagates and consequently attenuates the pathological activity in many nuclei, may be beneficial as long as it does not activate too many undesirable regions. The clarification of the mechanisms of action of DBS is imperative to avoid implanting electrodes in regions having a low impact on clinical signs and/or leading to incapacitating side effects. In this review, we focus on results obtained within the last four years from multiunit and single cell electrophysiological recordings.

MECHANISMS OF STN-DBS IN THE CASE OF PARKINSON'S DISEASE

Increased synchronization and the appearance of oscillations in the activity patterns of populations of STN and GP neurons but also in motor cortical networks are salient aspects of Parkinsonism (PD). Pathological synchronization has been observed in human PD patients [START_REF] Priori | Rhythm-specific pharmacological modulation of subthalamic activity in Parkinson's disease[END_REF] , 1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine (MPTP)-treated monkeys [START_REF] Raz | Firing patterns and correlations of spontaneous discharge of pallidal neurons in the normal and the tremulous 1-methyl-4-phenyl-1, 2,3,6-tetrahydropyridine vervet model of parkinsonism [In Process Citation[END_REF] and rodents with 6hydroxydopamine (6-OHDA) lesions [START_REF] Sharott | Dopamine depletion increases the power and coherence of beta-oscillations in the cerebral cortex and subthalamic nucleus of the awake rat[END_REF] , suggesting functional alterations in the basal ganglia network. In particular increased coherence in the beta-band (13-30 Hz) is correlated with severity of symptoms in humans [START_REF] Gatev | Oscillations in the basal ganglia under normal conditions and in movement disorders[END_REF][START_REF] Uhlhaas | Neural synchrony in brain disorders: relevance for cognitive dysfunctions and pathophysiology[END_REF][START_REF] Hammond | Pathological synchronization in Parkinson's disease: networks, models and treatments[END_REF] .

Mechanisms of STN-DBS have long been reduced to a lesion-like or inhibition hypothesis until in 2003, Hashimoto et al [START_REF] Hashimoto | Stimulation of the subthalamic nucleus changes the firing pattern of pallidal neurons[END_REF] and ourselves [START_REF] Garcia | Dual effect of high-frequency stimulation on subthalamic neuron activity[END_REF] from data obtained in two totally different preparations (MPTP-treated monkeys and slices from reserpine-treated rats, respectively) introduced the concept that high frequency stimulation drives neuronal activity in a pattern locked to the stimulation frequency or its sub harmonics and consequently erases pathological activity. This DBS-driven pattern could show alternated periods of activity and pauses and was observed only when stimulation parameters were clinically efficient [START_REF] Hashimoto | Stimulation of the subthalamic nucleus changes the firing pattern of pallidal neurons[END_REF] or close to the clinical ones [START_REF] Garcia | The impact of high frequency stimulation parameters on the pattern of discharge of subthalamic neurons[END_REF] . On the light of studies performed during the late four years we will see whether this hypothesis is still valid and how it can be generalized.

The striatal network, the extrastriatal network and the place of the STN

Two networks can be distinguished within the basal ganglia, the striatal and the extrastriatal networks (Fig. 1). The striatal network lies in a single nucleus in rodents (the striatum), or in two nuclei in primates (caudate and putamen). It consists of GABAergic and cholinergic interneurons and GABAergic projection neurons. In contrast, the extrastriatal network is made up by five different neuronal populations, scattered in five different nuclei, the globus pallidus (external and internal), the subthalamic nucleus (STN) and the substantia nigra pars reticulata (SNr) and compacta (SNc). The link between the striatal and extrastriatal Electrophysiologists are used to study synaptic potentials or currents in response to single stimulations but here the question is far more complex mainly for technical reasons as recordings have to be maintained for minutes or hours and spikes identified among artifacts. If all studies on DBS mechanisms test high frequency (100-180 Hz) stimuli, they rarely apply them for long durations. Considering synaptic plasticity (potentiation or depression) that usually occurs in synaptic transmission after tetanic stimulation [START_REF] Luscher | Synaptic plasticity and dynamic modulation of the postsynaptic membrane[END_REF] , ultrashort (ms, s) and long (days, years) duration stimulations should evoke very different responses. Therefore, to evaluate the electrophysiological effects of STN-DBS, a compromise would be to stimulate and record for at least 30-60 minutes. To compare results from different studies, intensity of stimulation is not as informative as current density (mA/cm 2 ). This parameter depends on the diameter of the stimulation electrode or contact and is not always available from papers. For this reason we never mention here the value of current intensity or density.

Is STN-DBS noxious to STN neurons?

Before analyzing the electrophysiological effect of STN-DBS, the first point to verify is the extent of STN lesion due to the chronic presence of the DBS electrode and stimulation.

The classical Medtronic electrode was implanted in one STN in control or MPTP-treated monkeys, and the stimulation continuously applied for 7 months (pulses at 130 Hz, 60 µs duration). Cell counts performed in Nissl-stained coronal sections of the STN showed that the STN having the implant had only 5% difference in total cell number compared to the side that did not have the implant [START_REF] Wallace | Survival of midbrain dopaminergic cells after lesion or deep brain stimulation of the subthalamic nucleus in MPTP-treated monkeys[END_REF] . Therefore, the chronically-implanted DBS electrode does not induce local degenerescence and the beneficial effects of STN-DBS are not mediated by a lesion of the STN.

Does STN-DBS lock the electrophysiological activity of STN neurons to harmonics of the stimulation frequency?

Studies of the effect of STN-DBS locally on STN neurons gave the most controversial data. This probably results from the difficulty to separate the very short latency evoked spikes from the stimulation artifacts or as we will see later on the difference between cell body and axonal activity in some conditions. Benazzouz group recorded inhibition of STN neurons during STN-DBS in 6-OH DA rats [START_REF] Tai | Electrophysiological and metabolic evidence that high-frequency stimulation of the subthalamic nucleus bridles neuronal activity in the subthalamic nucleus and the substantia nigra reticulata[END_REF] but recently described that this inhibition only lasted 4 ms after each stimulus when the interval between two stimuli was fixed at 7.7 ms (130 Hz) [START_REF] Meissner | Subthalamic high frequency stimulation resets subthalamic firing and reduces abnormal oscillations[END_REF] .

In patients, STN-DBS of short durations decreased STN neurons activity [START_REF] Filali | Stimulation-induced inhibition of neuronal firing in human subthalamic nucleus[END_REF] and changed the firing pattern of some STN neurons via a non-identified mechanism [START_REF] Welter | Effects of high-frequency stimulation on subthalamic neuronal activity in parkinsonian patients[END_REF] . In slices STN activity is very low compared to that in vivo and does not show the pathological alterations resulting from dopamine depletion probably because the basal ganglia network is absent in coronal slices. Slices have however the advantage of allowing intracellular recordings and precise analysis of the correspondence between spikes and stimuli. In our experiments, all STN spontaneous spikes disappeared during STN-HFS and were replaced by spikes evoked by and locked to stimuli (Fig. 2A). The STN firing pattern under HFS consisted of trains of evoked spikes in the gamma range frequency that totally erased spontaneous STN activity [START_REF] Garcia | Dual effect of high-frequency stimulation on subthalamic neuron activity[END_REF][START_REF] Garcia | The impact of high frequency stimulation parameters on the pattern of discharge of subthalamic neurons[END_REF] . Interestingly, the fixed latency of DBS-driven spikes (close to 0 ms), the presence of the initial segment-somatodendritic (IS-SD) break in the recordings (Fig. 2A right) and the lack of effect of blockers of synaptic transmission, all strongly suggested that HFS-evoked spikes were antidromic (antidromic spikes propagate toward cell bodies i.e. in the direction opposite to physiological spikes that propagate in the orthodromic direction toward axon terminals). STN-DBS thus preferentially activated STN efferent axons probably at the level of their initial segment or first Ranvier node. Evoked antidromically propagated to STN somas where they were recorded (Fig. 1A, inset STN).

Antidromic spikes by colliding with orthodromic, spontaneous ones erased STN spontaneous activity.

One possible explanation for the controversial results obtained in vivo and in vitro is that action potentials evoked in axons by local DBS, inefficiently invade cell bodies in the antidromic direction due to geometric ratio. DBS would therefore lead to active axons and silent somas [START_REF] Mcintyre | Cellular effects of deep brain stimulation: modelbased analysis of activation and inhibition[END_REF][START_REF] Chomiak | Axonal and somatic filtering of antidromically evoked cortical excitation by simulated deep brain stimulation in rat brain[END_REF] . Since extracellular microelectrode recordings are biased toward recording action potentials from cell bodies rather than axons, this would result in the appearance of decreased activity within the stimulated structure though efferent axons are excited (see discussion in [START_REF] Montgomery | Effects of GPi stimulation on human thalamic neuronal activity[END_REF] ). In contrast intracellular or juxtacellular recordings record axonal spikes

Does STN-DBS antidromically activate afferent neurons to the STN?

A DBS electrode in a nucleus not only activates the efferent axons of local neurons but also the afferent axons. STN-DBS thus evokes antidromic spikes in neurons of the motor cortex and GP (Fig. 1). Axons passing through or near the STN can also be activated. Due to the low probability of antidromic invasion of somas, antidromic spikes are usually evoked in a subset of afferent neurons only.

So, a subset of layer V/VI neurons of motor cortex, that project to STN 31 displayed antidromic spikes (latency around 2 ms) whose frequency decreased with time to a steady state at around 40 Hz in response to 120 Hz STN stimulation [START_REF] Li | Resonant antidromic cortical circuit activation as a consequence of high-frequency subthalamic deep-brain stimulation[END_REF] (Fig. 2B). Recordings of short latency evoked potentials over the motor cortex during STN-DBS also indicated that axons were most likely activated [START_REF] Baker | Subthalamic nucleus deep brain stimulus evoked potentials: physiological and therapeutic implications[END_REF] . As glutamatergic cortico-subthalamic neurons give off many axon collaterals in deep and superficial layers [START_REF] Cowan | Spontaneous firing patterns and axonal projections of single corticostriatal neurons in the rat medial agranular cortex[END_REF] that contact other projection neurons and local GABAergic interneurons, antidromic invasion of a subset of neurons may retrogradely affect cortical circuits in complex ways (Fig. 1 inset cortex). But antidromic excitation of cortical networks may not be very efficient. For example, whereas the fast conducting branches in the highly myelinated brainstem region follow high frequency stimulation, the slower conducting fibres in the poorly myelinated thalamic region fail to transmit consecutive antidromic spikes and maintain a steady low-frequency (6-12 Hz) spike output during the stimulation [START_REF] Chomiak | Axonal and somatic filtering of antidromically evoked cortical excitation by simulated deep brain stimulation in rat brain[END_REF] . STN-HFS in slices (130 Hz, 90 µs) evoked antidromic spikes in GPe neurons with a mean latency of around 2 ms (Fig. 2C) that erased the ongoing activity in the recorded neurons. This may have consequences on neighbouring GPe neurons if the antidromic spikes propagate in the complex network of local GABAergic collaterals that synapse onto other GP neurons [START_REF] Sadek | A single-cell analysis of intrinsic connectivity in the rat globus pallidus[END_REF] (Fig. 1 inset GP). [START_REF] Maurice | Spontaneous and evoked activity of substantia nigra pars reticulata neurons during high-frequency stimulation of the subthalamic nucleus[END_REF] , probably as a result of the activation of ascending SNr axons. This decreased the spontaneous activity of the antidromically activated SNr neurons and inhibited other SNr neurons as shown in recordings [START_REF] Maurice | Spontaneous and evoked activity of substantia nigra pars reticulata neurons during high-frequency stimulation of the subthalamic nucleus[END_REF] probably via the activation of the complex network of intranigral collaterals between GABAergic SNr cells [START_REF] Mailly | Three-dimensional organization of the recurrent axon collateral network of the substantia nigra pars reticulata neurons in the rat[END_REF] . The same experiment has not been performed for SNc neurons.

In conclusion, STN-DBS antidromically activates subsets of neurons in the different nuclei that send axons to or close to the stimulated site. In that context, STN-DBS probably also antidromically activates PPN neurons that project to STN, though short trains of stimulation have only been tested at present [START_REF] Hammond | Anatomical and electrophysiological studies on the reciprocal projections between the subthalamic nucleus and nucleus tegmenti pedunculopontinus in the rat[END_REF][START_REF] Florio | High-frequency stimulation of the subthalamic nucleus modulates the activity of pedunculopontine neurons through direct activation of excitatory fibres as well as through indirect activation of inhibitory pallidal fibres in the rat[END_REF] . Once evoked, antidromic axonal spikes propagate to their corresponding cell bodies at the stimulation frequency or its sub harmonics, may also propagate in recurrent axonal collaterals and activate synaptic transmission that impinges onto other projection neurons (GP, SNr) or local interneurons (see Fig. 1, insets cortex and GPe). The overall result on a network is difficult to guess as it needs to know the probability of propagation of antidromic spikes in axonal branches and how these spikes evoke synaptic responses at a long term.

Does STN-DBS orthodromically induce a complex locking of GP and SNr neuronal activity?

From their point of initiation, axonal spikes also propagate in the orthodromic direction toward axon terminals, i.e. along STN efferent axons (Fig. 1). Do these spikes evoke postsynaptic responses? STN-DBS parameters that improved spontaneous movements and muscle tone (130, 210 µs, 25 s-5 min) in MPTP-treated monkeys, evoked stimulus-locked double excitatory responses at latencies around 4 and 6 ms in the globus pallidus (GPe and GPi) [START_REF] Hashimoto | Stimulation of the subthalamic nucleus changes the firing pattern of pallidal neurons[END_REF] (Fig. 3A). It thus shifted the firing pattern of GPe and GPi neurons, from irregular to In PD patients under surgical procedure, at parameters (130 Hz, 60 µs) that induce clinical improvement of rigidity and finger tapping, STN-DBS increased mean spike frequency of SNr neurons and evoked short latency (4 ms) excitatory responses.

Autocorrelograms demonstrated the presence of a periodic spiking at 130 Hz. In parallel the firing pattern changed from irregular to a 'grouped' pattern consisting of groups of spikes separated by longer periods of pauses [START_REF] Galati | Biochemical and electrophysiological changes of substantia nigra pars reticulata driven by subthalamic stimulation in patients with Parkinson's disease[END_REF] . In another study STN-DBS (140 Hz, 60 µs) evoked in SNr neurons a three phase sequence, inhibition (0-2 ms) -excitation (2-4 ms) -inhibition (4-7 ms) after the stimulation pulse. There was a 51% decrease in the percentage of the spikes contributing to bursts and a 70% decrease in the mean duration of bursting mode activity [START_REF] Maltete | Subthalamic stimulation and neuronal activity in the substantia nigra in Parkinson's disease[END_REF] .

Therefore, STN-DBS eliminates in some GP and SNr neurons the 'pathological' information encoded by the basal ganglia during PD and replaces it by a stimulus-driven firing pattern that is likely to contribute to the clinical motor improvements. This new activity would result, at least in part, from the activation of STN efferent fibres. The multiphasic pattern of the responses (alternated periods of excitation and inhibition) led some authors to suggest a participation of polysynaptic activity. This has to be confirmed since high frequency stimulations usually suppress polysynaptic responses. The exact mechanism of the complex responses recorded during short term STN-DBS or GP-DBS is not yet elucidated.

Does STN-DBS act on the survival and dopamine metabolism of remaining SNc neurons?

The rationale for testing the effect of STN-DBS on SNc neurons survival is the non tested hypothesis that glutamatergic inputs to SNc neurons, namely those from the STN [START_REF] Hammond | Electrophysiological demonstration of an excitatory subthalamonigral pathway in the rat[END_REF][START_REF] Parent | Functional anatomy of the basal ganglia. II. The place of subthalamic nucleus and external pallidum in basal ganglia circuitry[END_REF] and PPN [START_REF] Hammond | Anatomical and electrophysiological studies on the reciprocal projections between the subthalamic nucleus and nucleus tegmenti pedunculopontinus in the rat[END_REF][START_REF] Charara | Glutamatergic inputs from the pedunculopontine nucleus to midbrain dopaminergic neurons in primates: Phaseolus vulgaris-leucoagglutinin anterograde labeling combined with postembedding glutamate and GABA immunohistochemistry[END_REF] would be excitotoxic to SNc dopaminergic neurons that have not yet degenerated, since both inputs are overactive in PD. The question then arised on whether STN-DBS protects remaining dopaminergic neurons from degeneration. This has been Hz, 1 h per day over a period of three months). This amount of STN modulation was sufficient to obtain a significant rescue of SNc dopaminergic neurons from cell death.

Bilateral STN-DBS not only had a protective effect on the number of TH positive neurons but also on the total number of neurons in the SNc. It could be argued that this effect resulted from a non identical retrograde degeneration of SNc dopaminergic neurons in the different lesioned rats and thus did not result from STN-DBS. For this reason, to mimic the clinical situation and to be able to observe neuroprotection, Benabid's group performed a subacute model of MPTP treatment in primates and induced a symmetrical 50% reduction of Nisslstained and TH positive cells in the two SNc. They applied a unilateral STN-DBS after MPTP treatment and compared the number of Nissl-stained and TH-positive SNc cells between each side of the brain in two animals, the non DBS side serving as a control. They found around 20% more dopaminergic neurons in the SNc of the side that underwent DBS, compared to the non-DBS side. When the DBS electrode was located outside the STN, the difference between both sides was not significant [START_REF] Wallace | Survival of midbrain dopaminergic cells after lesion or deep brain stimulation of the subthalamic nucleus in MPTP-treated monkeys[END_REF] . Therefore STN-DBS may have offered neuroprotection to nigral dopaminergic neurons that would have degenerated as part of the disease process.

Several studies reported an excellent clinical outcome of STN-DBS in L-dopa responsive forms of Parkinson's disease and STN-DBS allows the discontinuation of L-Dopa

or equivalent treatment or large reductions in daily dose [START_REF] Moro | Chronic subthalamic nucleus stimulation reduces medication requirements in Parkinson's disease[END_REF] in contrast to GPi-DBS. The question therefore aroused whether STN-DBS favours dopamine release in the striatum. Savasta's group tested this hypothesis by measuring the extracellular content of dopamine and its metabolites in the striatum of rats that underwent a partial 6-OHDA lesion of one SNc [START_REF] Lacombe | High-frequency stimulation of the subthalamic nucleus prolongs the increase in striatal dopamine induced by acute l-3,4dihydroxyphenylalanine in dopaminergic denervated rats[END_REF] .

After a delay of three weeks to allow the degeneration of 70 % of dopaminergic nigrostriatal L-dopa-induced increase of dopamine during the stimulation period and for the following 2.5 h after the end of stimulation. In contrast, in intact animals, L-dopa failed to enhance the extracellular dopamine levels during the stimulation period. This suggests that STN-DBS interacts in a synergistic manner with L-dopa. The underlying mechanisms have not been elucidated. The functional tone of the nigrostriatal DA system can be regulated at two sites, via inhibitory autoreceptors located on presynaptic dopaminergic terminals and controlling the synthesis or release of DA, and via receptors on the soma or dendrites of these neurones, involved in regulating impulse flow [START_REF] Bunney | Comparison of effects of L-dopa, amphetamine and apomorphine on firing rate of rat dopaminergic neurones[END_REF] . It seems plausible that STN-DBS has effects on the inputs coming from the STN to the somatodendritic receptors of SNc neurons. A simple explanation would be that STN-DBS acts by directly modulating the firing rate of dopaminergic neurones [START_REF] Benazzouz | High frequency stimulation of the STN influences the activity of dopamine neurons in the rat[END_REF] . This, together with a decrease of inhibitory feedback regulation with the progression of nigrostriatal degeneration could stabilize the DA levels in the striatum in synergy with L-dopa.

STN-DBS regularizes neuronal activity in the extrastriatal network as well as in the motor cortex and amplifies L-dopa treatment

To conclude, STN-DBS due to the central position of the STN in the basal ganglia has multiple effects. It introduces a stimulation-locked, intermittent activity in many sites of the basal ganglia network (STN, motor cortex, GPe, GPi, SNr), mainly via antidromically propagated DBS-driven spikes and thus decreases ongoing pathological activity at these sites. This may also be valid for PPN and the good results obtained on axial motor signs with STN- [START_REF] Stefani | Bilateral deep brain stimulation of the pedunculopontine and subthalamic nuclei in severe Parkinson's disease[END_REF] , may result from a better regularization of PPN discharge pattern. Direct activation of passing fibres dorsal to the STN and in particular nigrostriatal and pallido-thalamic axons may also participate to the beneficial effect as the best position of the DBS electrode active contact is in the dorsal part of the STN. Via a still unknown mechanism, STN-DBS protects dopaminergic neurons that have not yet degenerated and ameliorates the endogenous release of dopamine in the striatum when applied concomitantly with L-dopa treatment. This last hypothesis on dopamine release is strengthened by the fact that only patients responding to L-dopa are good candidates for STN-DBS. Conversely, this may imply that STN is not a good target for DBS in the case of dopamine-independent disorders.

DBS for other neurological disorders DBS in motor cortico-basal ganglia-thalamocortical loop for essential tremor

DBS of ventral nuclei of the thalamus can dramatically relieve essential tremor in the majority of patients [START_REF] Benabid | Long-term suppression of tremor by chronic stimulation of the ventral intermediate thalamic nucleus[END_REF][START_REF] Hariz | Multicentre European study of thalamic stimulation for Parkinsonian tremor, a 6-year follow-up[END_REF] . Essential tremor is thought to arise from dysfunction of the glutamatergic olivocerebellar pathway which projects to ventral thalamic (VL) nuclei [START_REF] Deuschl | Pathophysiology of nonparkinsonian tremors[END_REF] . VL-HFS in rat brain slices silenced thalamic relay neurons after a transient period of intense depolarization [START_REF] Anderson | Mechanisms of deep brain stimulation: an intracellular study in rat thalamus[END_REF] . The authors hypothesized that VL-HFS introduced a functional deafferentation of stimulated neurons, thereby stopping tremor from propagating to thalamo-cortical loops. To test whether this depression of afferent synaptic transmission is selective, they stimulated at 5 Hz in two different loci within the VL to mimic afferent stimuli at tremor frequency. Both stimulations evoked excitatory postsynaptic potentials (EPSPs) at 5 Hz in the recorded VL neuron. A concomitant short duration HFS (125 Hz, for 10 s) in one locus only totally suppressed the 5 Hz EPSPs in the HFS-stimulated pathway but not in the non-stimulated one, suggesting that HFS selectively disrupts afferent synaptic transmission [START_REF] Anderson | Selective attenuation of afferent synaptic transmission as a mechanism of thalamic deep brain stimulation-induced tremor arrest[END_REF] . Neuronal activity is altered in basal ganglia and ventral thalamic nuclei in dystonia [START_REF] Lozano | Globus pallidus internus pallidotomy for generalized dystonia[END_REF] .

The firing pattern of GPi neurons known to be regular in monkeys [START_REF] Wichmann | Comparison of MPTPinduced changes in spontaneous neuronal discharge in the internal pallidal segment and in the substantia nigra pars reticulata in primates[END_REF] consists in patients of irregular grouped discharges with intermittent pauses and a third of the neurons discharge at the frequency of the electromyogram [START_REF] Zhuang | Neuronal activity in the basal ganglia and thalamus in patients with dystonia[END_REF][START_REF] Tang | Neuronal firing rates and patterns in the globus pallidus internus of patients with cervical dystonia differ from those with Parkinson's disease[END_REF] . Neurons in ventral oralis posterior / intermediate nuclei of the thalamus (Vop/Vim) have a sustained activity at 130-150 Hz, organized in bursts lasting from 500 ms to 5 s and recurring at a frequency similar to that of dystonia frequency [START_REF] Zhuang | Neuronal activity in the basal ganglia and thalamus in patients with dystonia[END_REF] .

DBS for medically intractable forms of dystonia has been tested in GPi [START_REF] Coubes | Treatment of DYT1-generalised dystonia by stimulation of the internal globus pallidus[END_REF] , STN [START_REF] Benabid | Deep brain stimulation of the corpus luysi (subthalamic nucleus) and other targets in Parkinson's disease. Extension to new indications such as dystonia and epilepsy[END_REF] and VL nuclei of the thalamus [START_REF] Vercueil | Deep brain stimulation in the treatment of severe dystonia[END_REF] . GPi-DBS is now currently used for primary generalized DYT-1 positive dystonia and idiopathic cervical dystonia [START_REF] Coubes | Electrical stimulation of the globus pallidus internus in patients with primary generalized dystonia: long-term results[END_REF][START_REF] Kupsch | Pallidal deep-brain stimulation in primary generalized or segmental dystonia[END_REF][START_REF] Vidailhet | Bilateral, pallidal, deepbrain stimulation in primary generalised dystonia: a prospective 3 year follow-up study[END_REF] . In contrast to Parkinson's disease, the beneficial effects of DBS in dystonia are not immediate but progressive over weeks to months. However, recordings in patients can only be performed during the surgical procedure, i.e. at t0, or in control animals, owing to the lack of reliable animal models of dystonia.

During short duration GPi-DBS, 50 to 70% Vop neurons of the thalamus reduced their average discharge frequency with a delay of a few ms in control monkeys [START_REF] Anderson | Effects of high-frequency stimulation in the internal globus pallidus on the activity of thalamic neurons in the awake monkey[END_REF] or dystonic patients [START_REF] Montgomery | Effects of GPi stimulation on human thalamic neuronal activity[END_REF] , suggesting that DBS activates GPi efferent axons that are GABAergic and inhibitory onto thalamic neurons (Fig. 3B). Moreover, 88% of Vop neurons were antidromically activated with a 1 ms latency probably as a result of the activation of axons originating in Vop and passing in the vicinity of the GPi-DBS electrode (Fig. 2E).

DBS in limbic cortico-basal ganglia-thalamocortical loop for obsessive compulsive disorder

Obsessive compulsive disorder has been consistently associated with metabolic hyperactivity in the caudate nucleus, medial thalamus, and orbitofrontal cortex in patients at rest [START_REF] Baxter | Local cerebral glucose metabolic rates in obsessive-compulsive disorder. A comparison with rates in unipolar depression and in normal controls[END_REF][START_REF] Swedo | Cerebral glucose metabolism in childhood-onset obsessive-compulsive disorder[END_REF][START_REF] Saxena | Neuroimaging and frontal-subcortical circuitry in obsessive-compulsive disorder[END_REF] .

Recently, a dramatic increase in neuronal activity of the ventral caudate nucleus was identified and correlated to the patients' self-evaluated obsessions [START_REF] Guehl | Neuronal Correlates of Obsessions in the Caudate Nucleus[END_REF] . DBS of the ventral anterior internal capsule [START_REF] Nuttin | Electrical stimulation in anterior limbs of internal capsules in patients with obsessive-compulsive disorder[END_REF][START_REF] Van Kuyck | Behavioural and physiological effects of electrical stimulation in the nucleus accumbens: a review[END_REF] or accumbens [START_REF] Aouizerate | Deep brain stimulation of the ventral caudate nucleus in the treatment of obsessive-compulsive disorder and major depression. Case report[END_REF] are therapeutic approaches for treatment-resistant DBS of the accumbens (130 Hz, 200 µs, during 30 min) in control rats induced the inhibition of nearly all the recorded orbitofrontal neurons probably as a result of the antidromic activation of cortico-accumbens axons and other corticofugal axons [START_REF] Mccracken | High-frequency deep brain stimulation of the nucleus accumbens region suppresses neuronal activity and selectively modulates afferent drive in rat orbitofrontal cortex in vivo[END_REF] . The authors suggest that antidromic spikes propagate in axonal collaterals of cortical neurons and thus evoke inhibitory responses in neighbouring neurons via GABAergic interneurons (Fig. 1, inset cortex). But this has still to be demonstrated as antidromic axonal spikes often inefficiently invade axon collaterals and somas [START_REF] Chomiak | Axonal and somatic filtering of antidromically evoked cortical excitation by simulated deep brain stimulation in rat brain[END_REF] .

CONCLUSION

Electrical stimulation of a nucleus with short duration pulses (less than 1 ms) preferentially activates axons rather than somas [START_REF] Ranck | Which elements are excited in electrical stimulation of mammalian central nervous system: a review[END_REF][START_REF] Nowak | Axons, but not cell bodies, are activated by electrical stimulation in cortical gray matter. I. Evidence from chronaxie measurements[END_REF] . This results in the generation of axonal spikes and antidromic activation of local cell bodies via their efferent axons as well as distant cell bodies that send axons to the stimulated structure (Fig. 1). DBS-driven antidromic spikes collide with spontaneous orthodromic ones leading to the blockade of ongoing (pathological) activity. This dual effect has been clearly shown in the STN [START_REF] Garcia | Dual effect of high-frequency stimulation on subthalamic neuron activity[END_REF] , motor cortex [START_REF] Li | Resonant antidromic cortical circuit activation as a consequence of high-frequency subthalamic deep-brain stimulation[END_REF] , GPe-GPi (Ammari et al personal communication) and SNr 36 during STN-DBS, in ventral neurons of the thalamus during GPi-DBS [START_REF] Montgomery | Effects of GPi stimulation on human thalamic neuronal activity[END_REF] and suggested in the orbitofrontal cortex during accumbens-DBS [START_REF] Mccracken | High-frequency deep brain stimulation of the nucleus accumbens region suppresses neuronal activity and selectively modulates afferent drive in rat orbitofrontal cortex in vivo[END_REF] . An additional complication stems from the fact that activated axons also propagate spikes in the orthodromic direction and give rise to sustained neurotransmitter release [START_REF] Windels | Influence of the frequency parameter on extracellular glutamate and gamma-aminobutyric acid in substantia nigra and globus pallidus during electrical stimulation of subthalamic nucleus in rats[END_REF][START_REF] Windels | Pallidal origin of GABA release within the substantia nigra pars reticulata during high-frequency stimulation of the subthalamic nucleus[END_REF][START_REF] Lee | High-frequency stimulation of the subthalamic nucleus increases glutamate in the subthalamic nucleus of rats as demonstrated by in vivo enzyme-linked glutamate sensor[END_REF] .

How postsynaptic responses (glutamatergic or GABAergic) follow a high frequency and long duration stimulation such as DBS is a question that still remains open, as the electrophysiological studies performed so far have only focused on relatively short term stimulations [START_REF] Hashimoto | Stimulation of the subthalamic nucleus changes the firing pattern of pallidal neurons[END_REF][START_REF] Montgomery | Effects of GPi stimulation on human thalamic neuronal activity[END_REF][START_REF] Bar-Gad | Complex locking rather than complete cessation of neuronal activity in the globus pallidus of a 1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine-treated primate in response to pallidal microstimulation[END_REF][START_REF] Anderson | Effects of high-frequency stimulation in the internal globus pallidus on the activity of thalamic neurons in the awake monkey[END_REF] . The overall consequence of DBS on stimulated networks appears to be the generation of a new activity, locked to the stimulation but with a lower frequency and interspersed pauses. This DBS-driven activity significantly attenuates the spontaneous pathological one, exacerbates the responsiveness to L-dopa and reverses several markers to control [START_REF] Lacombe | High-frequency stimulation of the subthalamic nucleus prolongs the increase in striatal dopamine induced by acute l-3,4dihydroxyphenylalanine in dopaminergic denervated rats[END_REF]82,83 , yet preserves the transmission of cortical information [START_REF] Degos | Neuroleptic-induced catalepsy: electrophysiological mechanisms of functional recovery induced by high-frequency stimulation of the subthalamic nucleus[END_REF][START_REF] Mccracken | High-frequency deep brain stimulation of the nucleus accumbens region suppresses neuronal activity and selectively modulates afferent drive in rat orbitofrontal cortex in vivo[END_REF] .

  by the projection neurons of the striatum, the GABAergic medium spiny neurons and in the reverse direction by the dopaminergic neurons of the SNc. The large reduction of the latter, due to the degeneration of nigro-striatal neurons, and the consequent loss of dopamine in the striatum, leads to 'typical' PD.STN occupies a strategic position inside the extrastriatal network as STN neurons are the only glutamatergic neurons, they receive afferents from motor-related cortical areas, they project to all nuclei of the extrastriatal network including SNc dopaminergic neurons and are reciprocally connected with brainstem neurons of the pedunculopontine nucleus (PPN)[START_REF] Bolam | Synaptic organisation of the basal ganglia[END_REF] .Preparations and parameters of stimulation used to study STN-DBS are diverseDifferent types of preparations have been used to study DBS mechanisms, from anesthetized in vivo models of PD to in vitro slices. Each preparation has its own advantages and pitfalls but their combination should allow understanding DBS mechanisms as long as we keep aware of the limitations of the technique used. In the present review DBS refers to high frequency stimulations in vivo and HFS to that in vitro.The question of the mechanisms of action of DBS relies on the analysis of what does ahigh frequency and long duration stimulation of neuronal elements (DBS is applied for years).

  level of the initial segment (IS spikes, Fig.2A middle) and higher intensities of stimulation would compensate for the geometric ratio.

  also activates axons passing through and near the STN. Thus short STN-DBS (130 Hz, 60 µs, 30 s trains) in control rats in vivo antidromically activated a subpopulation of SNr neurons with a latency of around 1 ms

  and regular. The most probable mechanism is the orthodromic activation of STN efferent axons projecting to GP, the release of glutamate and the monosynaptic excitation of postsynaptic GPe or GPi neurons. In contrast, in control monkeys, bursts of 100 Hz stimuli (10 pulses) induced powerful excitatory responses in the GPe but inhibition in the GPi attributed to the activation of the disynaptic STN-GPe-GPi pathway[START_REF] Kita | Balance of monosynaptic excitatory and disynaptic inhibitory responses of the globus pallidus induced after stimulation of the subthalamic nucleus in the monkey[END_REF] . Do GPi-DBS and STN-DBS have similar effects on GPi neurons since both ameliorate clinical signs of PD 41 ? Bar-Gad et al[START_REF] Bar-Gad | Complex locking rather than complete cessation of neuronal activity in the globus pallidus of a 1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine-treated primate in response to pallidal microstimulation[END_REF] recorded the activity of GPe or GPi neurons in MPTPtreated monkeys in response to microstimulations applied in GPe or GPi (135 Hz, 200 µs, 600-3000 trains of 10 stimuli separated by 500 ms). They report in both GP a double excitation with latencies of 3 and 6 ms, separated by a short period of inhibition. Overall 70% of the neurons displayed locked activity i.e. they lost their basic firing pattern and switched to a predicted, orderly discharge that was locked to the stimulus. Again, a complex locking of target neurons activity is recorded. Does STN-DBS similarly affect GP and SNr neurons? Deniau's group studied the spontaneous and evoked SNr activity before and during STN-DBS in control rats or rats treated with neuroleptics to block dopaminergic transmission. In control rats, STN-DBS (130 Hz, 60 µs, 30 s) evoked antidromic spikes and inhibition as previously seen but also orthodromic spikes (excitation) in SNr[START_REF] Maurice | Spontaneous and evoked activity of substantia nigra pars reticulata neurons during high-frequency stimulation of the subthalamic nucleus[END_REF] . The short latency excitation is likely to result from the orthodromic activation of the excitatory STN-SNr glutamatergic axon terminals. In cataleptic rats, at parameters that reversed the catalepsy (130 Hz, 60-80 µs), STN-DBS regularized the pattern of discharge of STN neurons as it significantly decreased the number of neurons exhibiting burst discharges and reduced the number of bursts emitted by bursting neurons. It also reversed to control the classical triphasic response to motor cortex stimulation[START_REF] Degos | Neuroleptic-induced catalepsy: electrophysiological mechanisms of functional recovery induced by high-frequency stimulation of the subthalamic nucleus[END_REF] . Intracellular recordings from neurons in the pars compacta of the SN (SNc) STN-DBS in slices revealed increased generation of EPSPs and increased frequency of action potentials[START_REF] Lee | Neurotransmitter release from high-frequency stimulation of the subthalamic nucleus[END_REF] .

  and primate models of PD. Temel et al 50 injected 6-OHDA at four sites in both striatum of rats. During the phase of ongoing neurodegeneration in the SNc, half of the lesioned rats were treated with bilateral STN-DBS (pulse width at 60 µs, frequency at 130

  dorsolateral part of the striatum, they implanted the stimulation electrode in the STN and the microdialysis probe in the striatum, both ipsilateral to the lesion. The i.p. injection of L-dopa (50 mg/kg) increased by around 3 times the content of extracellular dopamine in the lesioned striatum measured with HPLC one hour after the injection. STN-DBS at clinical parameters (130 Hz, 60 µs, during one hour) amplified by around 100% this

  with PPN-DBS

15 DBS

 15 in motor cortico-basal ganglia-thalamocortical loop for dystonia

  mechanisms were studied with imaging techniques in patients and electrophysiological techniques in control rats as robust animal models of OCD are lacking.

LegendsFigure 1 :

 1 Figure 1: STN and the striatal and extrastriatal networks

Figure 2 DBS 25 Figure 3 :Figure 1 :Figure 2 Figure 3 :Figure 1 :Figure 2 Figure 3 :

 2253123123 Figure 2 DBS-driven antidromic spikes

Acknowledgements

This work was supported by Institut National de la Recherche Médicale (Inserm), Fondation de France and Association France Parkinson.


Z Ammari
email: zied.ammari@u-cergy.fr

F Nier

Mean field limit for bosons and propagation of Wigner measures

Keywords: 2000 Mathematics subject classification: 81S30, 81S05, 81T10, 35Q55

We consider the N-body Schrödinger dynamics of bosons in the mean field limit with a bounded pair-interaction potential. According to the previous work [AmNi], the mean field limit is translated into a semiclassical problem with a small parameter ε → 0, after introducing an εdependent bosonic quantization. The limit is expressed as a push-forward by a nonlinear flow (e.g. Hartree) of the associated Wigner measures. These object and their basic properties were introduced in [AmNi] in the infinite dimensional setting. The additional result presented here states that the transport by the nonlinear flow holds for rather general class of quantum states in their mean field limit.

Introduction

The mathematical analysis of the mean field limit of the N-body quantum dynamics of bosons started with the work of [Hep] and [GiVe]. Since, the problem has experienced intensive investigations using mainly the so-called BBGKY hierarchy method explained in [Spo]. Interest was focused on studying the cases of singular interaction potential (see for example [BGM], [EY], [BEGMY], [ESY]).

Recently, a new method was given in [FGS] (see also [FKP]) for a scalar bounded potential which inspires this work. The convergence of the quantum dynamics are typically tested on the above quoted articles, either on coherent states or on Hermite states. Even when such specific choices are avoided, the convergence on arbitrary states still has to be studied.

In the work [AmNi], Wigner measures were extended to the infinite dimensional setting, as Borel probability measures under general assumptions. It was also explained how previous weak formulations of the mean field limit are contained in the definition of these asymptotic Wigner measures, after a reformulation of the N-body problem as a semiclassical problem with the small parameter ε = 1 N → 0. The basic properties of these Wigner measures were considered and they were used to check that the mean field dynamics for the coherent states and Hermite states are essentially equivalent.

In this paper, the problem of the mean field dynamics is considered under some restrictive assumptions on the initial data. The convergence of N-body Schrödinger dynamics of bosons in the mean field limit will be proved for a class of density operator sequences, which contains all the common examples. Remember that contrary to the finite dimensional case no natural pseudodifferential calculus can be deformed by arbitrary nonlinear flows, and the propagation of Wigner measures as dual objects cannot be straightforward in the infinite dimensional case. The limit is expressed as push-forward by a nonlinear flow (e.g. Hartree) of Wigner measures associated with the sequence of density operators. The result holds here when the pair interaction potential is bounded on L 2 (R 2d

x,y ). This can be considered as a regular case and subsequent work will be devoted to more singular cases like in [FKS] with a Coulombic interaction V (xy) = 1 |x-y| or in the derivation of cubic nonlinear Schrödinger equations with V (xy) = δ (xy) like in the [ESY].

Since in the literature the non relativistic and the semi-relativistic dynamics of bosons were both studied (see [ElSc]), an abstract setting for the linear part of the flow seems relevant. Examples are reviewed in the last section.

We keep the same notations as in [AmNi]. The phase-space, a complex separable Hilbert space, is denoted by Z with the scalar product ., . . The symmetric Fock space on Z is denoted by H and n Z is the n-fold symmetric (Hilbert) tensor product, so that H = ⊕ n∈N n Z as a Hilbert direct sum. Algebraic direct sums or tensor products are denoted with a alg superscript. Hence H 0 = ⊕ alg n∈N n Z denotes the subspace of vectors with a finite number of particles. For any p, q ∈ N, the space P p,q (Z ) of complex-valued polynomials on Z is defined with the following continuity condition: b ∈ P p,q (Z ) iff there exists a unique b ∈ L ( p Z , q Z ) such that:

b(z) = z ⊗q , bz ⊗p .
The subspace of P p,q (Z ) made of polynomials b such that b is a compact operator is denoted by P ∞ p,q (Z ). The Wick monomial of symbol b ∈ P p,q (Z ) is the linear operator b Wick : H 0 → H 0 defined as follows:

b Wick | n Z = 1 [p,+∞) (n) n!(n + q -p)! (n -p)! ε p+q 2 S n-p+q b ⊗ I n-p Z ,
where S n is the symmetrization orthogonal projection from ⊗ n Z onto n Z . Remark that b Wick depends on the scaling parameter ε.

Consider a polynomial Q ∈ P 2,2 (Z ) such that Q ∈ L ( 2 Z
) is bounded symmetric. The many-body quantum Hamiltonian of bosons is a self-adjoint operator on H having the general shape:

H ε = dΓ(A) + Q Wick , (1)
where A is a given self-adjoint operator on Z . The time evolution of the quantum system is given by U ε (t) = e -i t ε H ε and U 0 ε (t) = e -i t ε dΓ(A) for the free motion. The commutation [Q Wick , N] = 0 with the number operator N = dΓ(1) = |z| 2 Wick , ensures the essential self-adjointness of H ε on D(dΓ(A)) ∩ H 0 and the fact that both dynamics preserve the number. Now we turn to the description of the nonlinear classical dynamics analogues of (1). Let us first recall some notations from [AmNi]. Polynomials in P p,q (Z ) admit Fréchet differentials. For b ∈ P p,q (Z ), set

∂ z b(z)[u] = ∂r b(z + ru) |r=0 , ∂ z b(z)[u] = ∂ r b(z + ru) |r=0 ,
where ∂r , ∂ r are the usual derivatives over C. Moreover, ∂ k z b(z) naturally belongs to ( k Z ) * (i.e.: k-linear symmetric functionals) while ∂ j z b(z) is identified via the scalar product with an element of j Z , for any fixed z ∈ Z . For b i ∈ P p i ,q i (Z ), i = 1, 2 and k ∈ N, set

∂ k z b 1 . ∂ k z b 2 (z) = ∂ k z b 1 (z), ∂ k z b 2 (z) ( k Z ) * , k Z ∈ P p 1 +p 2 -k,q 1 +q 2 -k (Z ) .
The multiple Poisson brackets are defined by

{b 1 , b 2 } (k) = ∂ k z b 1 .∂ k z b 2 -∂ k z b 2 .∂ k z b 1 , {b 1 , b 2 } = {b 1 , b 2 } (1) .
The energy functional

h(z) = z, Az + Q(z) , z ∈ D(A),
has the associated vector field X : D(A) → Z , X(z) = Az + ∂ zQ(z) and the nonlinear field equation

i∂ t z t = X(z t )
with initial condition z 0 = z ∈ D(A). For our purpose, we only need the integral form of the later equation

z t = e -itA z -i t 0 e -i(t-s)A ∂ zQ(z s ) ds, for z ∈ Z . (2)
The standard fixed point argument implies that (2) admits a unique global C 0 -flow on Z which is denoted by

F : R × Z → Z (i.e.: F is a C 0 -map satisfying F t+s (z) = F t • F s (z) and F t (z) solves (2)
for any z ∈ Z ). While considering the evolution of the Wick symbols, the action of the free flow e -itA will be summarized by the next notation :

b t = b • e -itA : Z ∋ z → b t (z) = b(e -itA z) , b t ∈ ⊕ alg p,q∈N P p,q (Z ) , (3)
for any b ∈ ⊕ alg p,q∈N P p,q (Z ) and any t ∈ R. Moreover, if z t solves (2), and Q t is defined according to (3), then w t = e itA z t solves the differential equation

d dt w t = -i∂ zQ t (w t ) .
Therefore for any b ∈ P p,q (Z ), the following identity holds

d dt b(w t ) = ∂ zb(w t )[-i∂ zQ t (w t )] + ∂ z b(w t )[-i∂ zQ t (w t )] = i{Q t , b}(w t ).
This yields for any z ∈ Z and b ∈ ⊕ alg p,q∈N P p,q (Z ), the Duhamel formula

b • F t (z) = b t (z) + i t 0 {Q, b t-t 1 } • F t 1 (z) dt 1 , (4)
by observing that

{Q t 1 , b}(w t 1 ) = {Q, b -t 1 }(z t 1 ).

Results

While introducing or using Wigner measures, all the arguments are carried out with extracted sequences (or subsequences) (ε n ) n∈N such that lim n→∞ ε n = 0, instead of considering a non countable range (0, ε), ε > 0, of values for the small parameter ε. Without loss of generality (see [AmNi]) one can consider a countable family (ρ ε n ) n∈N of density matrices, ρ ε n ≥ 0, Tr [ρ ε n ] = 1, and test them with ε n -quantized (Wick, Weyl or anti-Wick) observables before taking the limit ε n → 0. For the sake of conciseness, the ε or ε n parameter does not appear in the notations of quantized observables.

The first condition which characterizes our class of ε n -dependent density matrices reads:

∃λ > 0 : ∀k ∈ N, Tr[N k ρ ε n ] ≤ λ k uniformly in n ∈ N , (N = N ε n ) . ( H0
)
Wigner measures were constructed in [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF]Corollary 6.14] for the sequence (ρ ε n ) n∈N . Possibly extracting a subsequence still denoted (ε n ) n∈N , there exists a Borel probability measure µ called Wigner measure such that:

lim ε n →0 Tr[ρ ε n b Wick ] = Z b(z) dµ(z) , for any b ∈ ⊕ alg α,β ∈N P ∞ α,β (Z ) , (5)
with again b Wick = b Wick ε n . The statement (5) does not hold in general for all b ∈ ⊕ alg α,β ∈N P α,β (Z ) and counterexamples exhibiting the phenomenon of dimensional defect of compactness were given in [AmNi]. The extension of (5) to the larger class of symbols ⊕ alg α,β ∈N P α,β (Z ) depends on the sequence (ρ ε n ) n∈N and it turns out to be an important fact when studying the mean field limit. In the following, a sequence (ρ ε n ) n∈N with a single Wigner measure µ will have the property (P) when:

lim ε n →0 Tr[ρ ε n b Wick ] = Z b(z) dµ(z) , for any b ∈ ⊕ alg α,β ∈N P α,β (Z ) . (P)
Here is the main theorem.

Theorem 2.1 Let the sequence (ρ ε n ) n∈N of density matrices, ρ ε n ≥ 0, Tr [ρ ε n ] = 1, lim n→∞ ε n = 0,
satisfy (H0) and (P). Then the limit Remark 2.2 Since F is a C 0 -map the r.h.s. of ( 6) can be written as

lim n→∞ Tr[ρ ε n e i t εn H εn b Wick e -i t εn H εn ] = Z (b • F t )(z) dµ , (6
Z (b • F t )(z) dµ = Z b(z) dµ t ,
where µ t is a push-forward measure defined by µ t (B) = µ(F -t (B)), for any Borel set B.

We refer the reader to [AmNi] for the definition of Weyl observables and the Schwartz class of cylindrical functions S cyl (Z ).

Corollary 2.3 Let the sequence (ρ ε n ) n∈N of density matrices, ρ ε n ≥ 0, Tr [ρ ε n ] = 1, lim n→∞ ε n = 0,
satisfy (H0) and (P). Then the limit

lim ε n →0 Tr[ρ ε n e i t εn H εn b Weyl e -i t εn H εn ] = Z b • F t (z) dµ , (7)
holds for any b ∈ S cyl (Z ) and any t ∈ R.

Proof. A consequence of Thm. 2.1 and [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF]Prop. 6.15] is that the sequence

ρ ε n (t) = U ε n (t)ρ ε n U ε n (t) *
admits a single Wigner measure given by µ t . Hence, by definition lim

ε n →0 Tr[ρ ε n (t) b Weyl ] = lim ε n →0 pZ F [b](ξ ) Tr[ρ ε n (t) W ( √ 2πξ )] L p (dξ ) = pZ F [b](ξ ) Z e 2πiRe(z,ξ ) dµ t (z) L p (dξ ) .
Another formulation states that the Wigner measure µ t satisfies a transport equation in an integral form.

Corollary 2.4 Let (ρ ε n (t)) n∈N be as above and let µ t denote its Wigner measure. Then t ∈ R → µ t is a solution to the transport equation:

µ t (b) = µ 0 t (b) + i t 0 µ s ({Q, b t-s }) ds , (8)
for any b ∈ ⊕ alg p,q∈N P(Z ) and where µ 0 t (B) = µ(e -itA B) for any borel set B . Proof. The relation ( 8) is given by testing (4) on µ = µ 0 .

3 Criteria for the property (P)

In the following, two conditions which ensure the property (P) are formulated. Recall that for any P ∈ L (Z ) the operator Γ(P) acting on H is defined by

Γ(P) | n Z = P ⊗ P• • • ⊗ P
and Γ(P) is an orthogonal projector if P is too. The first criterion is a 'tightness' assumption with respect to the trace norm of the state

∀η > 0, ∃P ∈ L (Z ) finite rank orthogonal projector , ∀n ∈ N : Tr[(1 -Γ(P))ρ ε n ] < η (T ) .
The dual version is formulated as an equicontinuity assumption with respect to the Wick symbols:

∀p, q ∈ N, ∀η > 0, ∃W 0 ⊂ L ( p Z , q Z ) ∀ b ∈ W 0 , ∀n ∈ N : Tr[ρ ε n b Wick ] < η , (E)
where W 0 is a neighborhood of zero in L ( p Z , q Z ) endowed with the σ -weak topology.

Lemma 3.1 Assume that (ρ ε n ) n∈N satisfies (H0). Then (i) (T ) ⇒ (P), (ii) (E) ⇒ (P).

Proof. We aim to prove (P) for b ∈ P p,q (Z ).

(i) Start with Tr[ρ ε n b Wick ] = Tr[ρ ε n Γ(P)b Wick Γ(P)] + Tr[ρ ε n (1 -Γ(P))b Wick Γ(P)] + Tr[ρ ε n Γ(P)b Wick (1 -Γ(P))] + Tr[ρ ε n (1 -Γ(P))b Wick (1 -Γ(P))]
Estimate all the terms containing (1 -Γ(P)) in a similar way. For example, we have

Tr[ρ ε n (1 -Γ(P))b Wick Γ(P)] = Tr[ N p+q 2 ρ ε n (1 -Γ(P))b Wick N -p+q 2 Γ(P)] (9) ≤ C p,q (b) N p+q 2 ρ 1/2 ε n ρ 1/2 ε n (1 -Γ(P)) 1 (10) ≤ C p,q (b) N p+q 2 ρ ε n N p+q 2 1/2 1 (1 -Γ(P))ρ ε n (1 -Γ(P)) 1/2 1 (11) ≤ Cp,q (b)Tr[ρ ε n (1 -Γ(P))] 1/2 . ( 12
)
First ( 10) comes from the number estimate b Wick N -p+q 2 ≤ C p,q (b) then Cauchy-Schwarz inequality yield (11). The last estimate ( 12) is possible with (H0). Remark that Γ(P)b Wick Γ(P) = Γ(P)b(Pz) Wick Γ(P) and that the polynomial b(Pz) ∈ P ∞ p,q (Z ) when P is finite rank orthogonal projector. The hypothesis (T ) and the above argument allow to approximate Tr

[ρ ε n b Wick ] by the quantity Tr[ρ ε n b(Pz) Wick ] using η/3 argument. Now, write Tr[ρ ε n b Wick ] - Z b(z)dµ ≤ Tr[ρ ε n b Wick -b(Pz) Wick ] + Tr[ρ ε n b(Pz) Wick ] - Z b(Pz)dµ + Z [b(Pz) -b(z)]dµ .

So, the property (T ) and (H0) implies (P).

(ii) There exists a sequence b κ ∈ P ∞ p,q (Z ) such that bκ converges in the σ -weak topology to b. We have

Tr[ρ ε n b Wick ] - Z b(z)dµ ≤ Tr[ρ ε n b Wick -b Wick κ ] + Tr[ρ ε n b κ (z) Wick ] - Z b κ (z)dµ + Z [b κ (z) -b(z)]dµ . (13)
So, (P) holds by an η/3 argument and using respectively (E), (5) and dominated convergence for each term in the (r.h.s.) of ( 13).

Remark 3.2

1) The space of bounded operators L ( p Z , q Z ) endowed with the σ -weak topology is not a Baire space when Z is infinite dimensional. Otherwise, (E) and hence (P) would be fulfilled by any sequence (ρ ε n ) n∈N satisfying (H0), according to Banach-Steinhaus Theorem (Uniform Boundedness Principle).

2) The hypothesis (H0) in the above lemma, can be replaced by the weaker statement (see [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF]Prop.6.15])

∃C > 0 : ∀k ∈ N, Tr[N k ρ ε n N k ] ≤ C(Ck) k
uniformly in ε n . This can be interpreted as an analyticity property of t

C (m) r (t m , • • • ,t 1 ,t) = 1 2 r ∑ ♯{i: γ i =2}=r {Q t m , • • • , {Q t 1 , b t } (γ 1 ) • • • } (γ n ) γ i ∈{1,2} ∈ P p-r+m,q-r+m (Z ) . ( 14
)
Note that for shortness the dependence of C We collect some statements from [AmNi]. Remember that b denotes the operator b =

∂ q z ∂ p z q!p! b(z) ∈ L ( p Z , p Z ) associated with b ∈ P p,q (Z ). Lemma 4.2 Let b ∈ P p,q (Z ). (i) The following inequality holds true {Q s , b t } (2) L ( p Z , q Z ) ≤ 2[p(p -1) + q(q -1)] | Q| | b| L ( p Z , q Z ) .
(ii) For any m ∈ N and r ∈ {0, 1, . . . , m}, we have

C (m) r L ( p+m-r Z , q+m-r Z ) ≤ 2 2m-r ( m r ) (p + m -r) 2r (p + m -r -1)! (p -1)! | Q| m | b| L ( p Z , q Z ) ,
when p ≥ q with a similar expression when q ≥ p (replace (p + mr, p -1) with (q + mr, q -1)) .

Proof. See [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF]Lemma 5.8,5.9].

Lemma 4.3 For any δ > 0 there exists T > 0 such that for all 0 < t < T :

∞ ∑ m=0 δ m t 0 dt 1 • • • t m-1 0 dt m C (m) 0 (t m , . . . ,t 1 ,t) L ( p+m Z , q+m Z ) < ∞ (15)
Proof. It is enough to bound (15) in the case p ≥ q. Using Lemma 4.2 (iii) with r = 0, we obtain

∞ ∑ m=0 δ m t 0 dt 1 • • • t m-1 0 dt m C (m) 0 (t m , . . . ,t 1 ,t) ≤ 2 p-1 | b| ∞ ∑ m=0 2 3 δ t | Q| m .
The r.h.s. is finite whenever 0

< t < T = (2 3 δ | Q|) -1 .

Proof of Thm. 2.1

First consider the following expansion proved in [AmNi, (50)-( 52)] for any positive integer M:

U ε (t) * b Wick U ε (t) = M-1 ∑ m=0 i m t 0 dt 1 • • • t m-1 0 dt m C (m) 0 (t m , • • • ,t 1 ,t) Wick + ε 2 M ∑ m=1 i m t 0 dt 1 • • • t m-1 0 dt m U ε (t m ) * U 0 ε (t m ) {Q t m ,C (m-1) 0 (t m-1 , • • • ,t 1 ,t)} (2) Wick U 0 ε (t m ) * U ε (t m ) +i M t 0 dt 1 • • • t M-1 0 dt M U ε (t M ) * U 0 ε (t M ) C (M) 0 (t M , • • • ,t 1 ,t) Wick U 0 ε (t M ) * U ε (t M ) ,
where the equality holds in L ( s Z , s+q-p Z ) for any s ∈ N, s ≥ qp. Multiplying on the left the above identity by ρ ε n and then using number estimates with the help of (H0), yields an identity on L 1 (H ) on which we take the trace. This leads to

Tr[ρ ε n U ε n (t) * b Wick U ε n (t)] = M-1 ∑ m=0 i m t 0 dt 1 • • • t m-1 0 dt m Tr ρ ε n C (m) 0 (t m , • • • ,t 1 ,t) Wick (16) + ε n 2 M ∑ m=1 i m t 0 dt 1 • • • t m-1 0 dt m Tr ρ ε n U ε n (t m ) * U 0 ε n (t m ) {Q t m ,C (m-1) 0 (t m-1 , • • • ,t 1 ,t)} (2) Wick U 0 ε n (t m ) * U ε n (t m ) (17) +i M t 0 dt 1 • • • t M-1 0 dt M Tr ρ ε n U ε n (t M ) * U 0 ε n (t M ) C (M) 0 (t M , • • • ,t 1 ,t) Wick U 0 ε n (t M ) * U ε n (t M ) . ( 18
)
The interchange of trace and integrals on the r.h.s. is justified by the bounds on Lemma 4.2. Lemma 4.3 implies that the term of ( 16) and ( 17) are bounded by

A m = λ m+ p+q 2 sign(t) m t 0 dt 1 • • • t m-1 0 dt m C (m) 0 B m = ε n Q (p + q + m -1) 2 λ m-1+ p+q 2 sign(t) m t 0 dt 1 • • • t m-1 0 dt m C (m-1) 0
while the remainder ( 18) is estimated by

|(18)| ≤ sign(t) M t 0 dt 1 • • • t M-1 0 dt M C (M) 0 = C M .
By Lemma 4.2, the series ∑ ∞ m=0 A m and ∑ ∞ m=0 B m converge as soon as |t| < T 0 = (2 3 λ | Q|) -1 while lim M→∞ C M = 0. Hence the relation ( 16)(17)(18) holds with M = ∞ with a vanishing third term and a second term bounded by ∑ ∞ m=0 B m = O(ε n ). Therefore, we obtain lim

ε n →0 Tr[ρ ε n U ε n (t) * b Wick U ε n (t)] - ∞ ∑ m=0 i m t 0 dt 1 • • • t m-1 0 dt m Tr ρ ε n C (m) 0 (t m , • • • ,t 1 ,t) Wick = 0.
Owing to the condition (P) which provides the pointwise convergence and the uniform bound of ∑ ∞ m=0 A m , the Lebesgue's convergence theorem implies lim

ε n →0 ∞ ∑ m=0 i m t 0 dt 1 • • • t m-1 0 dt m Tr ρ ε n C (m) 0 (t m , • • • ,t 1 ,t) Wick = ∞ ∑ m=0 i m t 0 dt 1 • • • t m-1 0 dt m Z C (m) 0 (t m , • • • ,t 1 ,t; z) dµ . (19)
Now, we interchange the sum over m and the integrals on (t 1 , • • • ,t m ,t) with the integral over Z on (19) simply with a Fubini argument based on the absolute convergence (written here for t > 0):

∞ ∑ m=0 t 0 dt 1 • • • t m-1 0 dt m Z C (m) 0 (t m , • • • ,t 1 ,t; z) dµ ≤ ∞ ∑ m=0 Z |z| p+q+2m dµ t 0 dt 1 • • • t m-1 0 dt m C (m) 0 (t m , • • • ,t 1 ,t) .
Again (H0) and (P) imply that for all k ∈ N there exists λ > 0 such that

Z |z| 2k dµ = lim ε n →0 Tr[ρ ε n (|z| 2k ) Wick ] = lim ε n →0 Tr[ρ ε n N k ] ≤ λ k .
Hence, Lemma 4.3 yields for |t| < T 0 : lim

ε n →0 Tr[ρ ε n U ε n (t) * b Wick U ε n (t)] = ∞ ∑ m=0 i m t 0 dt 1 • • • t m-1 0 dt m Z C (m) 0 (t m , • • • ,t 1 ,t; z)dµ = Z ∞ ∑ m=0 i m t 0 dt 1 • • • t m-1 0 dt m C (m) 0 (t m , • • • ,t 1 ,t; z) dµ , 7
where the integrand

∞ ∑ m=0 i m t 0 dt 1 • • • t m-1 0 dt m C (m) 0 (z) is a convergent series in L 1 (µ)
. The last step is the identification of the limit with the r.h.s. of (6). Indeed, an iteration of (4) reads

b(z t ) = b t (z) + i t 0 {Q t 1 , b t }(z) dt 1 + i 2 t 0 dt 1 t 1 0 dt 2 {Q t 2 , {Q t 1 , b t }}(e it 2 A z t 2 ) ,
after setting z t = F t (z) and defining the Wick symbols b t and Q t according to (3). By induction we obtain for any M > 1:

b • F t (z) = b t (z) + M-1 ∑ m=1 i m t 0 dt 1 • • • t m-1 0 dt m C (m) 0 (t m , • • • ,t 1 ,t; z) + i M t 0 dt 1 • • • t M-1 0 dt M C (M) 0 (t M , • • • ,t 1 ,t; e it M A z t M ) .
An integration with respect to the measure µ leads to

Z b • F t (z) dµ = M-1 ∑ n=0 i n t 0 dt 1 • • • t n-1 0 dt n Z C (n) 0 (t n , • • • ,t 1 ,t; z)dµ + i M t 0 dt 1 • • • t M-1 0 dt M Z C (M) 0 (t M , • • • ,t 1 ,t; e it M A z t M ) dµ .
Again the uniform estimate ∑ ∞ m=0 A m when |t| < T 0 and lim M→∞ C M = 0, allow to take the limit as

M → ∞. This implies for |t| < T 0 Z b • F t (z) dµ = ∞ ∑ m=0 i m t 0 dt 1 • • • t m-1 0 dt m Z C (m) 0 (t m , • • • ,t 1 ,t; z) dµ.
This proves the result for |t| < T 0 and it is extended to any time by the next iteration argument. Indeed, it is clear that ρ

ε n (t) = U ε n (t)ρ ε n U ε n (t) * satisfies (H0) since U ε n (t) commute with N.
The property (P) holds for ρ ε n (t) when |t| < T 0 by Remark 2.2 and Corollary 2.3. For t, s such that |t|, |s| < T 0 , the sequence (ρ ε n (t)) n∈N satisfies (H0) and (P). Therefore, the result for short times yields lim (20)

ε n →0 Tr[ρ ε n (t)U ε n (s) * b Wick U ε n (s)] = Z b • F s (z) dµ t = Z b • F t+s (z) dµ.
Moreover, the arguments used in the proof of Thm. 2.1 can not ensure the pointwise absolute convergence of the r.h.s. ( 20) for all z ∈ Z .

Examples

Models:

M1) Let Z = L 2 (R d , dx), A = D 2
x +U(x) self-adjoint and Q is a multiplication operator by 1 2 V (xy)

with V ∈ L ∞ (R d ). M2) Let Z = L 2 (R d , dx), A = D 2
x + m 2 + U(x) self-adjoint and Q as above. M3) When Z = C d ∼ R 2d

x,ξ , one recovers the standard semiclassical limit problem and the condition (P) is always satisfied if (H0) is satisfied. We refer for example the reader to [CRR] [Ger] [GMMP] [HMR] [LiPa] [Mar] [Rob] for various results about this topic.

Density operator Sequences:

1) Every sequence (ρ ε n ) n∈N valued in a compact set of the Banach space of trace class operators has the Wigner measure δ 0 . If in addition (ρ ε n ) n∈N satisfies (H0) then (P) holds true.

2) Let (ρ ε n ) n∈N as in 1) and satisfying (H0) and let (z n ) n∈N be a sequence of Z such that lim n→∞ |z n -z| = 0. Then ρε n = W (

√ 2 iε z n )ρ ε n W (- √ 2
iε z n ) admits the unique Wigner measure µ = δ z where z and (P) holds true. The push-forward measure is µ t = δ z t .

3) Let (z n ) n∈N be a sequence valued in a compact set of Z . So ρ ε

n = |z ⊗[ε -1 n ] n z ⊗[ε -1 n ] n
| satisfies (H0) and the property (P) and admits the Wigner measures 1 2π π 0 δ e iθ z dθ where z is any cluster point of (z n ) n∈N . Several other examples can be obtained by superposition, see [AmNi]. 4) Let (z n ) n∈N be a sequence such that |z n | = 1 in Z converging weakly to 0. Then (P) fails for ρ ε n = |E(z n ) E(z n )| with E(z n ) = W ( √ 2 iε z n )|Ω , although (H0) holds.

  ) holds for any t ∈ R and any b ∈ ⊕ alg α,β ∈N P α,β (Z ) with b Wick = b Wick ε n .

  m , • • • ,t 1 ,t) on b is not made explicit on the notation and even sometimes we will write C

Remark 4 . 4

 44 As by product we have for any b∈ ⊕ alg α,β ∈N P α,β (Z ) b • F t (z) = L 1 (µ)m , • • • ,t 1 ,t; z) .