Philippe Couratier
email: philippe.couratier@unilim.fr

Géraldine Lautrette

Jaime Andres Luna

Philippe Corcia

Phenotypic variability in Amyotrophic Lateral Sclerosis

Keywords: Amyotrophic Lateral Sclerosis, phenotype, upper motor neuron, lower motor neuron, Transactivation Response DNA Binding Protein

Clinically, ALS phenotypes depend on the areas of the body that are affected, the different degrees of involvement of upper and lower motor neurons, the degrees of involvement of other systems, particularly cognition and behavior, and rates of progression.

Phenotypic variability of ALS is characteristic and can be declined on the distribution of motor manifestations but also on the presence of extra-motor signs present in a variable manner in ALS patients. Neuropathologically, ALS is defined by the loss of UMN and LMN and the presence of two representative motor neuronal cytoplasmic inclusions, Bunina bodies and 43 kDa Transactivation Response DNA Binding Protein (TDP-43) -positive cytoplasmic inclusions. The distribution of cytopathology and neuronal loss in patients is variable and this variability is directly related to phenotypic variability. Key regulators of phenotypic variability in ALS have not been determined. The functional decrement of TDP-43, and region-specific neuronal susceptibility to ALS, may be involved. Due to the selective vulnerability among different neuronal systems, lesions are multicentric, region-oriented, and progress at different rates. They may vary from patient to patient, which may be linked to the clinicopathological variability across patients.

Introduction

The concept of amyotrophic lateral sclerosis (ALS) was first established by Jean-Martin Charcot of the Salpêtrière Hospital. The name ALS was speculatively derived by Charcot who assumed that spinal cord degeneration would be a primary lesion and that the amyotrophy associated with anterior horn motor neuron degeneration could be secondary to spinal cord degeneration [START_REF] Charcot | Amyotrophies spinales deuteropathiques sclérose latérale amyotrophique[END_REF]. However, regardless of this hypothesis, the name "ALS" with only three words with the implication that amyotrophy and spinal cord degeneration coexisted was accepted and persists to this day [START_REF] Goetz | Amyotrophic lateral sclerosis: Early contributions of Jean-Martin Charcot[END_REF]. The main defining characteristic of ALS is progressive weakness due to neurodegeneration of the upper motor neuron (UMN) and lower motor neuron (LMN). Clinically, ALS is defined by a history of weakness over time and space, and by an examination showing evidence of upper and lower motor neuron dysfunction in one or more areas of the body. Neuropathologically, ALS is defined by the loss of UMN and LMN and the presence of two representative motor neuronal cytoplasmic inclusions, Bunina bodies and 43 kDa Transactivation Response DNA Binding Protein (TDP-43)positive cytoplasmic inclusions [START_REF] Okamoto | Intracytoplasmic inclusions (Bunina bodies) in amyotrophic lateral sclerosis[END_REF][START_REF] Neumann | Ubiquitinated TDP-43 in frontotemporal lobar degeneration and amyotrophic lateral sclerosis[END_REF]. The distribution of cytopathology and neuronal loss in patients is variable and this variability is directly related to phenotypic variability. Clinically, ALS phenotypes depend on the areas of the body that are affected, the different degrees of involvement of UMN and LMN, the degrees of involvement of other systems, particularly cognition and behavior, and rates of progression. Phenotypic variability of ALS is characteristic and can be declined on the distribution of motor manifestations but also on the presence of extra-motor signs present in a variable manner in ALS patients (Table 1). In this article, the clinical and neuropathological archetypes of ALS will be reviewed.

Clinical phenotypes of ALS

Based on body region of involvement

The archetypic form of ALS: spinal form

The major type of ALS has a spinal onset, known as Charcot's type. This type begins with asymmetric weakness in a limb around the age of 60 years. Most patients demonstrate a gradual worsening of weakness within a year, which progresses to a contralateral limb or to other spinal and/or bulbar areas. On neurological examination, muscle atrophy with fasciculations and hyperreflexia of limbs are detected. Babinski's sign is occasionally positive.

Typically, hyperreflexia is more evident in the lower limbs than in the upper limbs. As muscle atrophy progresses, the pyramidal signs are less evident on clinical examination. Some young patients may develop a predominant upper motor neuron form with spasticity. Patients may face a life-threatening condition as respiratory muscle weakness progresses and mean survival time has been reported to be approximately 3 years [START_REF] Oskarsson | Amyotrophic lateral sclerosis: An update for 2018[END_REF].

Flail arm phenotype

The term flail arm syndrome (FAS) is most common, but it is also referred to as Vulpian-Bernhardt's type, neurogenic man-in-a-barrel syndrome, or scapulohumeral form of ALS. Patients with FAS present with predominantly proximal, progressive and symmetric wasting and paresis of the upper limb muscles, while lower limbs and bulbar muscles are spared [START_REF] Couratier | Clinical features of flail arm syndrome[END_REF]. Upper motor neuron signs in the legs are occasionally present [START_REF] Hu | Flail arm syndrome: A distinctive variant of amyotrophic lateral sclerosis[END_REF]. Wijesekera et al. defined FAS as weakness of both upper limbs without bulbar and lower-limb symptoms for a period of at least 12 months from disease onset [START_REF] Wijesekera | Natural history and clinical features of the flail arm and flail leg ALS variants[END_REF]. Chio et al. demonstrated on a population-based study that FAS was relative rare and more common in men (incidence rates, 0.28 in men and 0.07 in women), with a men to women rate ratio of 4:1 and mean age at onset was 62.6 years (SD 11.8). In this phenotype, Frontotemporal-Dementia (FTD) is very rare (1.4%). Flail arm phenotype is relatively benign, with a median survival time of 4.0 years and a 10-year survival rate of 17% [START_REF] Chio | Phenotypic heterogeneity of amyotrophic lateral sclerosis: a population based study[END_REF].

Flail leg phenotype

Some ALS patients have weakness confined to the lumbosacral spinal cord region. This subtype is known as flail leg syndrome, Marie-Patrikios' type, or a peroneal form of ALS [START_REF] Wijesekera | Natural history and clinical features of the flail arm and flail leg ALS variants[END_REF]. Flail leg accounts for between 2.5-6.3% of motor neuron disease, and has a similar mean age of symptom onset to classic ALS. This phenotype has a similar incidence in the two genders. Onset is asymmetric in about half of patients, but typically progresses to include both lower extremities, and muscle stretch reflexes are absent or diminished. The definitions for Flail leg differ by case series but common features include insidious onset of weakness isolated to the legs, decreased or absent reflexes at presentation and symptoms confined to one spinal region for 12-24 months [START_REF] Dimachkie | Leg amyotrophic diplegia: prevalence and pattern of weakness at US neuromuscular centers[END_REF]. About half of patients show an initial pattern of weakness described as pelviperoneal, with sparing of the quadriceps and ankle plantarflexors.

The remaining patients show either diffuse weakness, or a distal pattern of weakness. The median survival time (3.0 years) and the 10-year survival rate (13%) of this phenotype are similar to those of classic ALS. FTD is present in 4% of patients with this phenotype.

Bulbar form : progressive bulbar palsy (PBP)

This phenotype is characterized by the onset of dysarthria or dysphagia with bulbar muscle atrophy. The first report of the bulbar type of ALS was likely carried out by Gombault, who was a student of Charcot [START_REF] Gombault | Sclérose symétrique des cordons latéraux de la moelle et des pyramides antérieurses dans le bulbe[END_REF]. Bulbar phenotype accounts for 20% of patients with ALS [START_REF] Winnen | The phenotypic variability of amyotrophic lateral sclerosis[END_REF]. Some patients have isolated bulbar palsy (IBP) marked by speech abnormalities or dysphagia but lack bulbar muscle atrophy or fasciculation. This condition should not be diagnosed as PBP; instead, it should be considered pseudobulbar palsy in the strictest sense. IBP patients are predominantly female and are characterized by the presence of UMN bulbar symptoms such as spastic dysarthria and emotional lability. In contrast, more typical bulbar ALS patients exhibit mixed or flaccid dysarthria, more prominent tongue fasciculations and more prominent limb involvement. Although progressive tongue atrophy and fasciculations are relatively specific to ALS, they must be distinguished from Kennedy's disease [START_REF] Garg | Differentiating lower motor neuron syndromes[END_REF]. There is an increase in frequency of bulbar phenotype with increasing age [START_REF] Chio | Phenotypic heterogeneity of amyotrophic lateral sclerosis: a population based study[END_REF]. The bulbar type of ALS with a mean survival of 2 years has poorer prognosis compared to that of the limb onset. This is because the patients are more likely to develop aspiration pneumonia and malnutrition due to dysphagia [START_REF] Chio | Phenotypic heterogeneity of amyotrophic lateral sclerosis: a population based study[END_REF].

Respiratory phenotype.

This is the rarest phenotype (annual incidence rate: men, 0.06/100,000; women, 0,01/100,000). Its median survival time is 1.4 years and no patient with this phenotype survive up to 10 years [START_REF] Chio | Phenotypic heterogeneity of amyotrophic lateral sclerosis: a population based study[END_REF].

Based on Level of involvement of LMN and UMN

Progressive muscular atrophy (PMA)

PMA is a rare, sporadic, adult-onset, clinically isolated LMN syndrome due to the degeneration of LMN, including anterior horn cells and brainstem motor nuclei. It is clinically characterized by progressive flaccid weakness, muscle atrophy, fasciculations, and reduced or absent tendon reflexes. The term PMA was first coined by the French neurologists Aran and Duchenne in 1850 to describe patients with progressive muscle atrophy. In 1853, Cruveilhier provided the first evidence of PMA being a neurogenic disorder based on the atrophy of the ventral spinal roots and the motor nerves found on autopsy of Aran's patients [START_REF] Aran | Recherches sur une maladie non encore décrite du système musculaire: Atrophie musculaire progressive[END_REF][START_REF] Cruveilhier | Sur la paralysie musculaire progressive amyotrophique[END_REF]. It has been recognized that a substantial number of patients with the initial diagnosis of PMA progress to a diagnosis of ALS through the development of UMN signs or may have UMN pathology at autopsy despite the absence of clinical UMN features during their lifetime [START_REF] Ince | Corticospinal tract degeneration in the progressive muscular atrophy variant of ALS[END_REF].

These observations support the notion of PMA belonging to an ALS spectrum rather than being a unique variant of MNDs. However, there remains a significant proportion of patients with PMA who have no clinical or subclinical evidence of UMN dysfunction, supporting the existence of PMA as a separate entity. At present, the term PMA is reserved for sporadic patients with MND with pure LMN findings on examination, who may or may not later develop clinically defined UMN features. Patients who subsequently develop UMN signs are reclassified as having ALS. Patients with PMA have LMN features, namely, progressive flaccid weakness, muscle atrophy, fasciculations, and hyporeflexia or areflexia. Weakness and atrophy typically starts in distal limb muscles in an asymmetric manner following neuropathy pattern and then spreads over months and years. There is a mean delay of approximately 23 months between the onset and the diagnosis. Symmetric proximal limb weakness (myopathy pattern) occurs in only 20% of patients. Bulbar muscles are generally spared at onset but may be involved in up to 40% of patients within a median of 19 months from onset of limb weakness. Patients with bulbar involvement are more likely to progress to ALS or run a relentless course, as seen in ALS. It is uncommon for respiratory muscles to be involved at the onset of PMA [START_REF] Kim | Study of 962 patients indicates progressive muscular atrophy is a form of ALS[END_REF]. The rate of progression in patients with PMA varies from slow (over years and decades) to very rapid (months to a year). The median survival duration after onset in patients with PMA is about 12 months longer than in patients with ALS (48.3 vs 36 months) [START_REF] Visser | Disease course and prognostic factors of progressive muscular atrophy[END_REF].

Primary lateral sclerosis

In 1865, Charcot reported on a patient with isolated degeneration of the spinal lateral cord who clinically manifested severe spastic tetraparesis [START_REF] Charcot | Sclérose des cordons latéraux de la moelle épiniére chez une femme hystérique atteinte de contracture permanente des quatre membres[END_REF]. Erb described the clinical and pathological features, naming this condition PLS [START_REF] Erb | Uber einen wenig bekannten spinalen Symptomencomplex[END_REF]. Patients with PLS present with upper motor neuron signs only. Therefore, they do not meet the clinical criteria of Awaji [START_REF] Boekestein | Sensitivity and specificity of the 'Awaji' electrodiagnostic criteria for amyotrophic lateral sclerosis: Retrospective comparison of the Awaji and revised El Escorial criteria for ALS[END_REF]. Many patients with PLS develop lower motor neuron signs within 4 years of onset [START_REF] Finegan | Primary lateral sclerosis: A distinct entity or part of the ALS spectrum? Amyotroph Lateral[END_REF][START_REF] Tartaglia | Differentiation between primary lateral sclerosis and amyotrophic lateral sclerosis: Examination of symptoms and signs at disease onset and during follow-up[END_REF][START_REF] Gordon | The natural history of primary lateral sclerosis[END_REF] and some patients manifest with frontotemporal dementia [START_REF] Kosaka | Primary lateral sclerosis: Upper-motor-predominant amyotrophic lateral sclerosis with frontotemporal lobar degeneration-immunohistochemical and biochemical analyses of TDP-43[END_REF]. PLS constitutes approximately 1-5% of all cases of ALS. Mean survival time is reported to be approximately 8 years [START_REF] Finegan | Primary lateral sclerosis: A distinct entity or part of the ALS spectrum? Amyotroph Lateral[END_REF]. The differences and similarities between PLS and ALS are still not well understood, but recent reports suggest that the majority of clinical features of PLS resemble those of ALS [START_REF] Wais | The concept and diagnostic criteria of primary lateral sclerosis[END_REF]. Thus, if PLS is seemingly regarded as a subtype of ALS, at least for the first four years of followup, examination by needle electromyography is required to assess LMN involvement. New criteria have been established in 2020 [START_REF] Turner | Primary lateral sclerosis: consensus diagnostic criteria[END_REF].

Hemiplegic form

Some patients manifest with unilateral upper motor neuron involvement. This extremely rare clinical syndrome of UMN-predominant (1% of all patients with ALS), progressive hemiparesis was first described by American neurologist Charles Karsner Mills [START_REF] Mills | A case of unilateral progressive ascending paral-ysis, probably representing a new form of degenerative dis-ease[END_REF]. This form slowly develops ipsilateral involvement and spreads to eventually involve the contralateral side. After a variable period, lower motor neuron signs will gradually be evident, suggestive of ALS [START_REF] Jaiser | Mills' syndrome revisited[END_REF]. Some patients with this variant could develop frontotemporal degeneration [START_REF] Doran | Mills syndrome with dementia: Broadening the phenotype of FTD/MND[END_REF]. Estimated survival periods are variable, but are usually longer than 10 years [START_REF] Jaiser | Mills' syndrome revisited[END_REF].

Based on involvement of non-motor regions

ALS and frontotemporal dementia (FTD) spectrum

The overlap of FTD and ALS has been well documented in FTD patients with comorbid motor neuron degeneration and in ALS patients with frontotemporal dysfunction [START_REF] Kiernan | Frontal lobe atrophy in motor neuron diseases[END_REF][START_REF] Abe | Cognitive function in amyotrophic lateral sclerosis[END_REF][START_REF] Lomen-Hoerth | The overlap of amyotrophic lateral sclerosis and frontotemporal dementia[END_REF][START_REF] Lomen-Hoerth | Are amyotrophic lateral sclerosis patients cognitively normal?[END_REF]. Cognitive impairment has been reported in approximately 50% of patients with ALS [START_REF] Ringholz | Prevalence and patterns of cognitive impairment in sporadic ALS[END_REF]. Of these patients, 10-20% manifest with FTD [START_REF] Rippon | An observational study of cognitive impairment in amyotrophic lateral sclerosis[END_REF][START_REF] Phukan | Cognitive impairment in amyotrophic lateral sclerosis[END_REF]. Cognitive decline in ALS is variable and is characterized by personality changes, irritability, social misconduct, executive dysfunction, language problems, and memory impairments [START_REF] Beeldman | The cognitive profile of ALS: A systematic review and meta-analysis update[END_REF]. Behavioral or cognitive abnormalities may be subtle leading to identify behaviorally impaired (ALS-bvi) and cognitively impaired (ALS-ci) ALS patients as compared to patients without deficit, according to the revised criteria for the diagnosis of frontotemporal cognitive and behavioural syndromes in ALS [START_REF] Strong | Amyotrophic lateral sclerosisfrontotemporal spectrum disorder (ALS-FTSD): Revised diagnostic criteria[END_REF]. The Edinburgh team proposed a new scale, ECAS avoiding the potential interaction of motor deficit with cognitive-behavioural assessment [START_REF] Abrahams | Screening for cognition and behaviour changes in ALS Amyotrophic Lateral Sclerosis and Frontotemporal[END_REF]. In ALS-FTLD patients, frontotemporal atrophy may be detected by magnetic resonance imaging of the brain with decreased cerebral blood flow in single photon emission computed tomography [START_REF] Waldemar | Focal reductions of cerebral blood flow in amyotrophic lateral sclerosis: A [99mTc]-d,l-HMPAO SPECT study[END_REF]. Neuropathologically, spreading of TDP-43 pathology and neuronal loss are involved in the frontotemporal cortices and subcortical regions, known as FTLD-TDP.

ALS and parkinsonism

In the Western Pacific area (West New Guinea, the Mariana Islands and the Kii peninsula), several endemic foci with a high incidence of ALS and PDC have been reported. Epidemiological investigations begun in 1960 have revealed the incidence of ALS in West New Guinea to be surprisingly high, at approximately 1.4%. This incidence was ten times higher than that in Guam or Kii and was estimated to be a 100 times higher than that in the continental United States [START_REF] Garruto | Amyotrophic lateral sclerosis in the Mariana Islands[END_REF]. The clinical phenotype of ALS in ALS/PDC resembles that of sporadic ALS. Phenotypic variations exist, such as PBP or PMA types. ALS/PDC has a similar neuropathology to that of sporadic ALS, characterized by the presence of Bunina bodies and TDP-43-positive cytoplasmic inclusions. The key clinical features of PDC are a lack of spontaneity and decrease in verbal frequency. Akinesia and rigidity are typically observed as parkinsonian signs in patients with PDC. The neuropathology of PDC is characterized by frontotemporal atrophy and the presence of neurofibrillary tangles (NFTs) positive for both 3-and 4-repeat tau and the absence of senile plaques [START_REF] Itoh | Biochemical and ultrastructural study of neurofibrillary tangles in amyotrophic lateral sclerosis/parkinsonism-dementia complex in the Kii peninsula of Japan[END_REF]. Interestingly, the abundance of NFTs was well observed not only in Chamorro patients with PDC but also in healthy Chamorro. In contrast, the number of NFTs in the brains of Kii ALS/PDC varied, whereas, in the brains of non-ALS/PDC patients from the village, the number of NFTs was similar to that in Japanese controls from other areas. A previous study revealed that the coexistence of TDP-43, tau, and α-synuclein is a key feature of Western Pacific ALS/PDC [START_REF] Mimuro | Amyotrophic lateral sclerosis and parkinsonism-dementia complex of the Hohara focus of the Kii Peninsula: A multiple proteinopathy?[END_REF].

Besides these endemic focal cases, an association between sporadic and familial Parkinson's disease (PD) and ALS, i.e., Brait-Fahn-Schwartz disease, has been proposed as a syndrome characterized by the co-presence of these two disorders without dementia or dysautonomia.

Parkinsonian symptoms and signs have been described in cross-sectional studies in ALS patients, with frequency from 5 to 17% [START_REF] Pradat | Extrapyramidal stifness in patients with amyotrophic lateral sclerosis[END_REF][START_REF] Pupillo | Extrapyramidal and cognitive signs in amyotrophic lateral sclerosis: a population based cross-sectional study[END_REF] Backward falls, impaired postural reflexes, retropulsion, bradykinesia, and decreased arm swing in ALS have been reported in early-stage ALS and are often linked to basal ganglia alterations [START_REF] Desai | Extrapyramidal involvement in amyotrophic lateral sclerosis: backward falls and retropulsion[END_REF]. A recent paper confirmed gait initiation in ALS patients with postural instability [START_REF] Feron | Extrapyramidal deficits in ALS: a combined biomechanical and neuroimaging study[END_REF]. On the neuropathologic level, the demonstration of a pallido-nigro-luysian (PNL) degeneration with TDP-43 pathology remains rare. Some patients manifest with parkinsonism [START_REF] Gray | Luyso-pallido-nigral atrophy and amyotrophic lateral sclerosis[END_REF][START_REF] Bergmann | Motor neuron disease with pallido-luysio-nigral atrophy[END_REF] whereas some do not manifest these symptoms [START_REF] Miki | Sporadic amyotrophic lateral sclerosis with pallidonigro-luysian degeneration: A TDP-43 immunohistochemical study[END_REF] and undetectable parkinsonism in patients with PNL involvement may be masked by motor paralysis. A recent case report described increased glial TDP-43 pathology in the PNL system and brainstem motor neurons in facial-onset sensory motor neuropathy, suggesting that the molecular pathogenesis in PNL-involved type and sporadic ALS/FTLD spectrum differs [START_REF] Rossor | TDP43 pathology in the brain, spinal cord, and dorsal root ganglia of a patient with FOSMN[END_REF]. Nigral degeneration in patients with ALS had been recognized as a rare condition [START_REF] Bonduelle | Amyotrophic lateral sclerosis[END_REF]. In 1993, Kato et al. performed a clinicopathological retrospective analysis of 15 patients with sporadic ALS, finding a decreased number of nigral melanin-containing neurons in seven patients (46.7%) and they described supranuclear ophthalmoparesis in four patients [START_REF] Kato | Diminution of dopaminergic neurons in the substantia nigra of sporadic amyotrophic lateral sclerosis[END_REF]. Nishihira et al. reported TDP-43 pathology in the substantia nigra of 16 out of 35 patients (45.7%) with ALS, in addition to nigral neuronal loss in 12 out of 35 patients (34.3%) [START_REF] Nishihira | Sporadic amyotrophic lateral sclerosis: Two pathological patterns shown by analysis of distribution of TDP-43-immunoreactive neuronal and glial cytolasmic inclusions[END_REF]. Nigral degeneration may be related to cognitive dysfunction with concomitant degeneration of frontotemporal cortices, hippocampus, and striatum. Further investigation should be performed to clarify the clinicopathological relationship of nigral degeneration in patients with ALS.

ALS with cerebellar degeneration

Cerebellar ataxia is a very rare symptom in patients with ALS. The ataxia could occur in some patients with C9orf72 mutation. Polyglutamine (polyQ) expansion in ataxin-2 (ATXN2) is known to cause the onset of spinocerebellar ataxia 2 (SCA2). In 2010, intermediate-length expansions (27-33 Qs) of polyQ was identified as a potent risk factor for ALS [START_REF] Elden | Ataxin-2 intermediate-length polyglutamine expansions are associated with increased risk for ALS[END_REF]. Both phenotypes of SCA and ALS may coexist in the same family [START_REF] Bäumer | FTLD-ALS of TDP-43 type and SCA2 in a family with a full ataxin-2 polyglutamine expansion[END_REF]. A case report demonstrated that the initial manifestation was SCA followed by motor neuron signs 16 years later [START_REF] Nanetti | Rare association of motor neuron disease and spinocerebellar ataxia type 2 (SCA2): A new case and review of the literature[END_REF]. Motor neurons in patients with ALS that have ATXN2 expansions [START_REF] Wais | The concept and diagnostic criteria of primary lateral sclerosis[END_REF][START_REF] Turner | Primary lateral sclerosis: consensus diagnostic criteria[END_REF][START_REF] Mills | A case of unilateral progressive ascending paral-ysis, probably representing a new form of degenerative dis-ease[END_REF][START_REF] Jaiser | Mills' syndrome revisited[END_REF][START_REF] Doran | Mills syndrome with dementia: Broadening the phenotype of FTD/MND[END_REF][START_REF] Kiernan | Frontal lobe atrophy in motor neuron diseases[END_REF][START_REF] Abe | Cognitive function in amyotrophic lateral sclerosis[END_REF] Qs) may contain filamentous inclusions, such as skein-like inclusions, while those in ALS without ATXN2 expansion (22-24 Qs) may contain large rounded inclusions [START_REF] Hart | Distinct TDP-43 pathology in ALS patients with ataxin 2 intermediate-length polyQ expansions[END_REF].

ALS with vacuolar degeneration of cerebral white matter

In the majority of patients with ALS, ocular movement is preserved [START_REF] Leveille | Eye movements in amyotrophic lateral sclerosis[END_REF]. However, Takeda et al. previously reported that approximately 10% of all ALS patients with subcortical white matter vacuolation presented with vertical ophthalmoparesis and suggested that the symptoms could be related to impairments in the upper motor neuron system innervating nuclei of external ocular muscles [START_REF] Takeda | Supranuclear ophthalmoparesis and vacuolar degeneration of the cerebral white matter in amyotrophic lateral sclerosis: A clinicopathological study[END_REF].

ALS with autonomic dysfunction

Autonomic dysfunction in ALS is a rare symptom, at least in the early stage of illness [START_REF] Mannen | Preservation of a certain motoneurone group of the sacral cord in amyotrophic lateral sclerosis: Its clinical significance[END_REF]. However, literature increasingly revealed the manifestation of urinary impairments in patients with ALS. Arlandis et al. reported that approximately 26% of patients with ALS demonstrated symptomatic urinary urgency [START_REF] Arlandis | Urodynamic findings in amyotrophic lateral sclerosis patients with lower urinary tract symptoms: Results from a pilot study[END_REF]. With disease progression, Onuf's nucleus may be involved and the number of neurons decreases by almost half in the advanced phase of illness [START_REF] Takeda | Dendritic retraction, but not atrophy, is consistent in amyotrophic lateral sclerosis-comparison between Onuf's neurons and other sacral motor neurons[END_REF]. Thus, patients with ALS may manifest with urinary impairments more frequently than previously thought. A subset of patients treated with tracheostomy-positive pressure ventilation (TPPV) may develop abnormal ocular movements and autonomic impairments [START_REF] Atsuta | Age at onset influences on wide-ranged clinical features of sporadic amyotrophic lateral sclerosis[END_REF].

ALS with sensory symptoms

In conjunction with the progressive damage of the corticospinal tract (CST), autopsy cases and animal studies showed involvement of sensory pathways further confirmed by morphometric measures in the somatosensory cortex as well as functional imaging and structural imaging in the white matter. Moreover, electrophysiological measurements in ALS patients showed sensory symptoms and lower amplitude of compound action potential amplitudes of the sural nerve [START_REF] Hammad | Clinical electrophysiologic and pathologic evidence for sensory abnormalities in ALS[END_REF].

Factors that influence phenotypic variability

Lessons from neuropathology: the role of TDP-43 inclusions

Upper motor neuron degeneration is based on histological demonstration of neuronophagia of Betz cells and myelin pallor of the pyramidal tract. Historically it was proposed that pyramidal degeneration progresses from its distal site, associated with the dying back phenomenon [START_REF] Ince | Motor neuron disorders[END_REF]. In fact, the site of origin is increasingly recognized as being corticofugal, which is a dying-forward process primarily starting in the corticomotoneuronal system.

The dyingforward hypothesis proposes that glutamate excitotoxicity at the level of the cortical motorneuron ultimately results in anterior horn cell metabolic deficit.

The involvement of lower motor neurons is characterized by neuronal loss and dendritic retraction of spinal and brainstem motor neurons. The cytopathology common to patients with ALS consists of the formation of inclusions in motor neurons, including Bunina bodies and TDP-43-positive cytoplasmic inclusions. Bunina bodies, first reported in 1962 in patients with familial ALS are immunoreactive to anti-cystatin C [START_REF] Okamoto | Bunina bodies in amyotrophic lateral sclerosis immunostained with rabbit anti-cystatin C serum[END_REF] and anti-transferrin antibodies [START_REF] Mizuno | Transferrin localizes in Bunina bodies in amyotrophic lateral sclerosis[END_REF].

The origin of Bunina bodies is not fully known, but it is suggested that they derived from lysosomes or the endoplasmic reticulum [START_REF] Sasaki | Ultrastructural study of Bunina bodies in the anterior horn neurons of patients with amyotrophic lateral sclerosis[END_REF]. TDP-43-positive intracytoplasmic inclusions in spinal and brainstem motor neurons appear in various forms. They can be classified into three categories: dot-like inclusions, skein-like inclusions, and round inclusions. These inclusions were identified in 1988 as ubiquitin-positive cytoplasmic inclusions [START_REF] Chou | Motoneuron inclusions in ALS are heavily ubiquitinated[END_REF]. The first identification of TDP-43 as a major component of ubiquitin-positive inclusions was reported in 2006 by Neumann et al. and Arai et al. [START_REF] Arai | TDP-43 is a component of ubiquitin-positive tau-negative inclusions in frontotemporal lobar degeneration and amyotrophic lateral sclerosis[END_REF]. The most prominent feature of TDP-43 cytopathology is the mislocalization of TDP-43 characterized by loss of native TDP-43 in the nucleus and abnormal aggregation of TDP-43 in the cytoplasm. TDP-43 in cerebral neurons is deposited in a morphologically different manner from that in spinal motor neurons. In the cerebrum, many cytoplasmic inclusions positive for TDP-43 are present in small neurons.

They are deposited in rounded, perinuclear, or curved shapes. The cerebral pathology of ALS resembles that of FTLD with motor neuron disease (FTLD-MND), with the majority of abnormal TDP-43 aggregation present in the neuronal cytoplasm and minimal aggregation in neurites and nuclei [START_REF] Mackenzie | A harmonized classification system for FTLD-TDP pathology[END_REF]. FTLD with mutations in progranulin (FTLD-GRN) is characterized by short neuritic deposition of TDP-43 and cytoplasmic aggregation of TDP-43. In C9orf72mutated FTLD-MND patients, a type B TDP-43 pathology is mostly found but some reported type A [START_REF] Mackenzie | Reappraisal of TDP-43 pathology in FTLD-U subtypes[END_REF]. Although a genetic mutation is not the only factor determining the topography of TDP-43 deposition, knowing the type is sometimes useful for understanding the relationship between genetic factors and clinical conditions in patients with ALS and FTLD. However, some patients cannot be classified into any subtype. It remains unknown how subcortical TDP-43 pathologies differ from one another [START_REF] Nishihira | Revisiting the utility of TDP-43 immunoreactive (TDP-43-ir) pathology to classify FTLD-TDP subtypes[END_REF]. Further analyses will be needed to identify cases determining the morphology of TDP-43 aggregation.

The regional discrepancy in relationship of TDP-43 accumulation and neuronal loss

The relationship between inclusion cytopathology and neuronal loss is inconsistent based on the involved area. In the spinal anterior horn or hypoglossal nucleus, the prevalence of TDP-43 pathology and neuronal loss may almost be equivalent. The second most representative regions in which the prevalence of TDP-43 aggregation and neuronal loss are equivalent are the amygdala, orbitofrontal cortex, entorhinal cortex, and frontotemporal cortices. In contrast, there are areas in which the degree of neuronal loss is very mild despite the frequent detection of TDP-43 aggregation. These include granular cells of the dentate gyrus, inferior olivary nucleus, red nucleus, and brainstem reticular formation. The neurons least sensitive for TDP-43 aggregation relative to neuronal loss are pyramidal neurons (Betz cells). It has been reported that despite the evident loss of native TDP-43 from the nucleus, abnormal aggregation in the cytoplasm is relatively rare in these neurons [START_REF] Braak | Pathological TDP-43 changes in Betz cells differ from those in bulbar and spinal α-motoneurons in sporadic amyotrophic lateral sclerosis[END_REF]. The inconsistent relationship between TDP-43 pathology and neuronal loss obscures how TDP-43 contributes to neuronal degeneration linked to neuronal loss. These discrepancies suggest that neuronal loss is plausible to be linked to decrement of the normal function of TDP-43 rather than gain of toxic function of TDP-43 and there are differences in susceptibility to TDP-43 dysfunction among the neuronal system [START_REF] Nana | Neurons selectively targeted in frontotemporal dementia reveal early stage TDP-43 pathobiology[END_REF].

Variability of disease progression in ALS

The selective vulnerability of distinct regions of the brain and spinal cord is a critical factor. The sites where ALS lesions begin and how they spread into the central nervous system have not yet been fully elucidated. Lesions may occur in one or more of susceptible areas and progress locally at various rates in each region [START_REF] Sekiguchi | Spreading of amyotrophic lateral sclerosis lesions-multifocal hits and local propagation?[END_REF]. So, a different histological susceptibility, resulting in abnormal TDP-43 function among neuronal groups, or a different cellular susceptibility associated with cellular characteristics, may at least partly explain patterns of onset and progression. A systemic TDP-43 dysfunction but also region-specific susceptibility of the motor neurons and frontotemporal cortices could contribute to lesion formation and its progression [START_REF] Ragagnin | Motor neuron susceptibility in ALS/FTD[END_REF]. The focal susceptibility related to onset may not be random, as it is determined by the age at onset and gender: bulbar onset ALS has been shown to increase with advancing years and flail arm syndrome has been shown to affect mostly men [START_REF] Chio | Phenotypic heterogeneity of amyotrophic lateral sclerosis: a population based study[END_REF]. Thus, ALS lesions seem to be multicentric, region-oriented, and have different rates of progression and phenotypic variability is based on these underlying pathological features.

Role of genetic factors

Sporadic ALS (SALS) occurs in 90% of cases, and familial ALS (FALS) in 10%, are essentially indistinguishable from each other by phenotype. In populations of European origin, the 4 major ALS causing genes are C9orf72, SOD1, TARDBP and FUS and concern two third of FALS [START_REF] Millecamps | TARDBP, and FUS mutations in familial amyotrophic lateral sclerosis: genotype-phenotype correlations[END_REF][START_REF] Millecamps | Phenotype difference between ALS patients with expanded repeats in C9ORF72 and patients with mutations in other ALS-related genes[END_REF]. For these genes it is possible to establish genotype-phenotype correlations. C9orf72 is the major genetic cause of disease over 40 years of age, whereas FUS is the the major cause for the juvenile ALS and for disease occurring in younger adults.

Trio analyses allows to confirm that de novo occurrence was preponderant in the FUS-linked SALS. For all major ALS causing genes, penetrance is incomplete. Clinically, ALS patients with SOD1 mutations frequently present with lower limbs onset, predominant lower motor neuron signs and very rarely a frontal cognitive dysfunction is associated. Disease progression appears to be bimodal in SOD1-ALS patients, with some patients showing either rather rapid (<3 years, fast progressors) or rather slow (>7 years, slow progressors) disease course. Some SOD1 mutations, such as the G41S mutation, are associated with a particularly short survival, of around one year following the diagnosis; at the opposite, the N139D mutation leads to a long disease duration of more than 10 years. Moreover, the disease course can also be different among members of the same family carrying the same mutation, as D83G with a disease course of several months to more than 10 years. In TARDBP-mutated patients, disease onset appears to be predominantly in the upper limbs, frequently associated with marked bulbar dysfunction and with FTD. FUS-related ALS phenotype is severe, with early onset and fast progression of disease leading to shorter survival. C9orf72-linked patients have more frequent bulbar onset, disease starts later in life, compared to FUS or SOD1, and is more frequently associated with FTD. Prognosis in terms of survival appears to be pejorative, compared to SOD1 or TARDBP mutation carriers.

Conclusion

The highly distinctive molecular neuropathological subtypes of ALS do not completely correlate with the various clinical phenotypes. This point is crucial as it raises the following question: is ALS a single disease with common biological mechanisms or is it several diseases with different mechanisms? Advances in genetics have shown that different genetic mutations can induce common phenotypes. This suggests that ALS is more like a syndrome. Key regulators of phenotypic variability in ALS have not been determined. The functional decrement of TDP-43 and region-specific neuronal susceptibility to ALS may be involved. Due to the selective vulnerability among different neuronal systems, lesions are multicentric, region-oriented, and progress at different rates. They may vary from patient to patient, which may be linked to the clinicopathological variability across patients.

Table 1 :

 1 Motor and extra motor variability in ALS

	Phenotype	Prevalence
	Motor variability
	Bulbar onset	Bulbar form	20%
	Spinal onset	Spinal form	60%
	Lower motor neuron dominant	Progressive muscular atrophy Flail arm syndrome Flail leg syndrome	≈5-10% ≈10% ≈5%
	Upper motor neuron dominant	Hemiplegic form	1%
		Upper-motorneuron dominant ALS
		Primary Lateral sclerosis	1-5% of ALS
	Extra
	motorvariability
	Cognition and	ALS/FTLD	50% of ALS
	Behavior		patients
	Extrapyramidal signs	ALS-Parkinsonism	≈5-10%
		ALS with pallido-nigro-luysian	rare
		degeneration
	Cerebellar	ALS-spino cerebellar ataxia	rare
	degeneration	ALS patients with C9ORF72 mutation
	Neurogenic bladder	Primary Lateral Sclerosis	50% of PLS
			patients
		ALS	rare
	Sensory signs		Variable
	Abnormal ocular	Advanced phase of ALS patients
	movement	ALS with vacuolar degeneration of the	rare
		cerebral white matter
		ALS with multisystem degeneration


Emil Garnell

Bekir Aksoy

Corinne Rouby

Herbert Shea

Olivier Doaré

Geometric optimization of dielectric elastomer electrodes for dynamic applications

Keywords: Dielectric elastomer, modal analysis, parametric optimization

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

Introduction

Dielectric elastomers (DEs) are active materials capable of large electrically-triggered deformations (up to 500 % area strain [START_REF] Pelrine | High-Speed Electrically Actuated Elastomers with Strain Greater Than 100%[END_REF][START_REF] Goh | Electrically-Induced Actuation of Acrylic-Based Dielectric Elastomers in Excess of 500% Strain[END_REF]). They consist of a soft elastomer membrane sandwiched between compliant and stretchable 5 electrodes, forming a deformable capacitor. When a high voltage is applied between the electrodes, an electrostatic pressure squeezes the membrane whose area therefore increases [START_REF] Suo | Theory of dielectric elastomers[END_REF], see fig. 1. This mechanism is very fast, and can be effective for frequencies up to 16 kHz [START_REF] Hosoya | Hemispherical breathing mode speaker using a dielectric elastomer actuator[END_REF] especially 10 if silicone is used as the membrane material [START_REF] Rosset | The need for speed[END_REF]. DEs have been investigated for a wide range of applications including soft robotics [START_REF] Gu | Soft wall-climbing robots[END_REF][START_REF] Chen | Controlled flight of a 450 microrobot powered by soft artificial muscles[END_REF], artificial muscles [START_REF] Duduta | Realizing the potential of dielectric elastomer artificial mus-455 cles[END_REF], micro-pumps [START_REF] Loverich | Concepts for a new class of 460 all-polymer micropumps[END_REF][START_REF] Cao | A magnetically coupled dielectric elastomer pump for soft robotics[END_REF], vibration absorbers [START_REF] Lu | Electrically tunable and 470 broader-band sound absorption by using micro-perforated dielectric elastomer actuator[END_REF], and loudspeakers [START_REF] Keplinger | Stretchable, Transparent, Ionic Conduc-475 tors[END_REF].

Electrodes

Dielectric elastomer membrane (a) (b) As explained in fig. 1 the membrane is compressed by 15 the electrostatic pressure only between the electrodes, and the rest of the membrane is passive. It is therefore possible to tune the actuation by patterning the electrode, and choosing cleverly where electrodes should be deposited.

URL: olivier.doare@ensta-paris.fr (Olivier Doaré)

This idea has been exploited by several research groups,

20
for very diverse objectives. Conn et al. [START_REF] Conn | Towards holonomic electro-elastomer actuators with six degrees of freedom[END_REF] patterned the electrodes on a cone DE actuator to generate a rotating motion during actuation. Zou et al. [START_REF] Zou | Active Shape Control and Phase Coexistence of Dielectric Elastomer Membrane With Patterned Electrodes[END_REF] studied the influence of the electrode shape on the static deformation of an inflated DE membrane. Langham et al. [START_REF] Langham | Modeling shape selection of buckled dielectric elastomers[END_REF] analysed

25
the buckling of DE structures, and in particular the influence of the electrode shape on the type of buckling. They showed that the buckling wavelength can be controlled by the aspect ratio of an annular electrode on a circular membrane.

30

By patterning multiple independently addressable electrodes on both sides of the elastomer membrane, different actuation shapes can be obtained. This principle has been used for instance to manufacture rotating motors [START_REF] Anderson | A thin membrane ar-490 tificial muscle rotary motor[END_REF][START_REF] Rosset | Towards fast, reliable, and manufacturable DEAs: Miniaturized motor and Rupert the rolling robot[END_REF] by using three or four independent electrodes, and to make 35 reconfigurable surfaces [START_REF] Hajiesmaili | Reconfigurable shape-morphing dielectric elastomers using spatially varying electric fields[END_REF].

Patterned electrodes have also been widely investigated for soft grippers using the electro-adhesion principle [START_REF] Shintake | Soft Robotic Grippers[END_REF]. Strong fringing fields are obtained by using several electrodes polarized with opposite voltages. Shintake et al. 40 [START_REF] Shintake | Versatile Soft Grippers with Intrinsic Electroadhesion Based on Multifunctional Polymer Actuators[END_REF] used electrode patterning techniques for DE actuators to combine electro-adhesion and actuator capabilities. Similarly, Gao et al. [START_REF] Gao | Elastic Electroadhesion with Rapid Release by Integrated Resonant Vibration[END_REF] used patterned electrodes to actuate a soft gripper at resonance to improve the release. A prototype of a wall-climbing robot has been developed 45 by Gu et al., using a DE actuator and electro-adhesion obtained with patterned electrodes [START_REF] Gu | Soft wall-climbing robots[END_REF].

The above-mentioned applications of electrode patterning for DE actuators are for static [START_REF] Shintake | Versatile Soft Grippers with Intrinsic Electroadhesion Based on Multifunctional Polymer Actuators[END_REF][START_REF] Shintake | Soft Robotic Grippers[END_REF], or dynamic cases [START_REF] Gu | Soft wall-climbing robots[END_REF], in which the kinetic energy of the DEA can be neglected compared either to the kinetic energy of a heavier biasing load, or to the potential elastic energy of biasing elements. However, because of their quick response, DE actuators are also used in high frequency applications where the DE membrane itself exhibits a modal behavior [START_REF] Hosoya | Hemispherical breathing mode speaker using a dielectric elastomer actuator[END_REF][START_REF] Zhu | Resonant behavior of a membrane of a dielectric elastomer[END_REF][START_REF] Fox | Electric field-induced surface transformations and experimental dynamic characteristics of dielectric elastomer membranes[END_REF]. Few studies focus on the relation between the electrode shape and the dynamic behavior of DE structures.

For other electro-active materials, such as piezo-electric materials for example, the optimal placement of the electrodes for vibration control has been investigated [START_REF] Clark | Optimal placement of piezoelectric actuators and polyvinylidene fluoride error sensors in active structural acoustic control approaches[END_REF]. Such studies are closely related to the modeshapes analysis of the structure, as playing on the location of the excitation will increase or decrease the modal forces on the different eigenmodes. This idea has been exploited to perform modal control on flat loudspeakers: Doaré et al. [START_REF] Doaré | Design of a Circular Clamped Plate Excited by a Voice Coil and Piezoelectric Patches Used as a Loudspeaker[END_REF] used piezoelectric patches to cancel the modal forces of undesirables modes, and Jiang et al. [START_REF] Jiang | Sound radiation of panelform loudspeaker using flat voice coil for excitation[END_REF] optimized the location of several voice-coils to minimize the contribution of unwanted modes. In the present study, we focus on a given DE actuator geometry, namely an inflated DE loudspeaker (see fig. 2). When the DE membrane is inflated, the increase in area when the high voltage is applied is converted to an increase of volume of the inflated balloon [START_REF] Keplinger | Harnessing snap-through instability in soft dielectrics to achieve giant voltage-triggered deformation[END_REF][START_REF] Zhu | Resonant behavior of a membrane of a dielectric elastomer[END_REF]. If an oscillating voltage is applied, an acoustic volume source is obtained. This configuration has been widely studied for acoustic applications [START_REF] Heydt | Acoustical performance of an electrostrictive polymer film loudspeaker[END_REF][START_REF] Hosoya | Hemispherical breathing mode speaker using a dielectric elastomer actuator[END_REF], and we have previously developed a numerical model to compute its sound radiation [START_REF] Garnell | Dynamics and sound radiation of a dielectric elastomer membrane[END_REF][START_REF] Garnell | Coupled vibro-acoustic modeling of a dielectric elastomer loudspeaker[END_REF].

cm

Here, we show that the shape of the electrodes can be tuned to adjust the acoustic frequency response of the structure, by playing on the modal forces. A finite element model is used to obtain a modal description of the structure, which is then fed into a global optimization algorithm, to improve a simple optimization objective: maximizing or minimizing the contribution of a given eigenmode to the acoustic radiation of the loudspeaker. The article is organized as follows: first a quick overview of the finite element model of the DE loudspeaker is presented, and the optimization algorithm is introduced. A prototype with the optimal electrodes is then manufac-tured, and the process to deposit the patterned electrodes is presented in section 3. The effect of the optimization on the radiated acoustic pressure is finally analysed in the results section 4, and perspectives are discussed.

Optimization procedure

The optimization procedure presented in this article is based on a finite element model of the inflated DE loudspeaker shown in fig. 2, which is described in details in [START_REF] Garnell | Dynamics and sound radiation of a dielectric elastomer membrane[END_REF][START_REF] Garnell | Coupled vibro-acoustic modeling of a dielectric elastomer loudspeaker[END_REF]. In the following only a short overview of this 100 model is given, to provide the key elements which are needed for the optimization.

Model of the inflated DE loudspeaker

To simplify the notations in the whole article, all variables are non-dimensional.

105

We consider an elastomer membrane coated by electrodes on a portion of its surface described by the electrode indicative function Γ , which equals unity when an electrode is present, and zero otherwise. The membrane is then placed over a closed cavity, pre-stretched, and in-110 flated with a static pressure (see figs. 2 and 3). A high voltage u is applied between the electrodes to vibrate the membrane and radiate sound.

To study this setup, a reference configuration is defined, where the membrane is flat and at rest, and The membrane behavior is modeled by a hyper-elastic Gent law [START_REF] Gent | A New Constitutive Relation for Rubber[END_REF], and the electromechanical coupling appears as an added stress (σ Max = u 2 /h 2 where is the elastomer dielectric permittivity) in the constitutive relations relating the stress to the deformation of the membrane [START_REF] Suo | Theory of dielectric elastomers[END_REF]. Note that the electrostatic stress is proportional to the voltage squared, which creates a large non-linearity. For loudspeaker applications, a linear relation between the input signal and the radiated pressure is desired, to avoid harmonic distortion. In order to obtain a linear excitation, the loudspeaker is driven with the following voltage [START_REF] Heydt | Acoustical performance of an electrostrictive polymer film loudspeaker[END_REF]:

u(t) = U 1 + w(t) , with |w(t)| < 1 , (1)
where U is a static bias voltage, and w the audio signal. This implies that electrostatic stress is proportional to w.

The coupling between the membrane vibrations and acoustics needs to be taken into account because the membrane is very thin, and the added mass effect which arises when the membrane vibrates the surrounding air is strong [START_REF] Garnell | Coupled vibro-acoustic modeling of a dielectric elastomer loudspeaker[END_REF]. To this aim, a fully coupled model of the membrane dynamics and acoustics inside and outside of the cavity is setup in the open source finite element software FreeFEM [START_REF] Hecht | New development in freefem++[END_REF]. The mesh density is 12 elements per flexural or acoustic wavelength (respectively for the membrane and for acoustics) at the highest studied frequency, with a conforming mesh at the interface. The free-field radiation boundary condition is implemented by using perfectly matched layers (PMLs) [START_REF] Berenger | A perfectly matched layer for the absorption of electromagnetic waves[END_REF]. The system of coupled governing equations can finally be written in the following form after discretization by finite elements:

(-ω 2 M + K)X = F , (2)
where ω is the frequency of the electrical excitation, X = [x, q i , q e ] is the vector gathering all unknowns, where q i = p i /ω 2 and q e = p e /ω 2 are the fluid displacement poten-125 tials. M and K are the total mass and stiffness matrices, and F the force vector, generated by the electrostatic stress. Two modelling choices (frequency-independent PMLs and hysteretic damping) imply that the obtained mass and stiffness matrices M and K are frequency-130 independent. Without the forcing term F , eq. ( 2) is therefore a linear eigenvalue problem, which can be solved by standard eigenvalue solvers. Modal expansions of the displacement of the membrane x, and of the radiated pressure p e are obtained:

x = n F n ω 2 n -ω 2 Ψ x n , p e = n ω 2 F n ω 2 n -ω 2 Ψ e n , (3)
where F n is the modal force on mode n, ω n the eigenfrequency, Ψ x n and Ψ e n the parts of the coupled modeshape 135 n containing the structural and exterior acoustics degrees of freedom respectively. We are interested in the pressure at a given receiver position x r , so by defining the modal participation factor A n = F n Ψ e n (x r ), the pressure radiated at the receiver location reads:

p e (x r ) = n ω 2 A n ω 2 n -ω 2 . ( 4
)
The contribution of mode n to the radiated pressure at x r is thus proportional to A n .

Definition of the optimization problem 140

In the present study, we aim at optimizing the shape of the electrode to improve objective functions which will be defined in section 2.3. The prototype is axisymmetric, and only axisymmetric electrodes will be considered in the following. The electrode shape is thus parametrized by a 145 set of radii, defined in fig. 4.

Membrane Electrode

(a) (b) For the electrode shown in fig. 4(b), the electrode indicative function Γ is defined as follows:

Γ (R) = 1 for R ≤ R 1 or R 2 < R ≤ R 3 . 0 otherwise. (5)
and the electrode radii can vary in the range:

R i-1 < R i ≤ R i+1 . (6)
In the following, the radii of the electrodes will be optimized to minimize the cost functions introduced in the following section 2.3. The definition of the optimization parameters given in fig. 4 is unpractical, as it defines a bounded optimization problem, where the bounds for one parameter depends on the values of the other parameters. In order to overcome this difficulty, relative radii are defined as:

Ři = R i R i+1 for i < N R , Ři = R i for i = N R , (7)
where N R is the number of optimized radii. All Ři vary between 0 and 1. We define for the following a vector Ř containing all the optimized radii Ři : Ř = [ Ř1 , Ř2 , Ř3 , ...].

Objective functions 150

In order to validate the optimization method, simple objective functions will be used. The contribution of the second membrane mode to the radiated pressure will be either maximized or minimized. These goals have been chosen because they yield clear results for relatively simple 155 electrode shapes, as it will be shown in section 4.

To maximize the contribution of the second mode to the total sound radiation compared to the other modes, we define from eq. ( 4) the following objective function which should be minimized [START_REF] Doaré | Radiation optimization of piezoelectric plates[END_REF]:

I = n =2 |A n | 2 |A 2 | 2 . ( 8
)
If on the other hand the radiation of the second mode should be minimized, the following objective function should be used:

J = |A 2 | 2 n =2 |A n | 2 . (9)

Optimization algorithm

The general principle of the optimization algorithm is that the time-consuming operations, namely computing the modal parameters (eigenfrequencies, eigenmodes and modal loss factors) should be performed only once. To this end, we make the following assumption:

Assumption. The electrodes have a negligible influence on the eigenfrequencies, modal damping and modeshapes of the inflated DE membrane.

This presumes that the mass and stiffness of the electrodes are negligible compared to those of the membrane. As a result, the modal parameters do not change when the electrode shape is varied during the optimization process. The electrode shape only changes the modal forces, which can be computed quickly using the following formula [START_REF] Garnell | Dynamics and sound radiation of a dielectric elastomer membrane[END_REF]:

F n ( Ř) = 1 0 Γ Ř h rλ 2 ∂Ψ x n ∂R • t + λ 1 Ψ x n • e r λ 1 λ 2 dR , (10)
Inserting this expression in the definition of A n yields:

A n ( Ř) = 1 0 g n (R)Γ Ř(R)dR . ( 11
)
where we have defined g n as:

g n (R) = Ψ e n (x r ) 1 h rλ 2 ∂Ψ x n ∂R • t + λ 1 Ψ x n • e r λ 1 λ 2 . ( 12
)
All the terms appearing in eq. ( 12) are independent of the electrode shape, so g n only needs to be computed once. The cost functions I and J defined by eqs. ( 8) and ( 9) depend only on the modal participation factors A n ( Ř), which can be computed quickly as eq. ( 11) is only a summation.

The optimization is performed as follows, and is described in fig. 5: 1. Compute the modal parameters using the model. In the present study we used the MultiStart algorithm from the Matlab Global Optimization Toolbox. This algorithm 180 runs a local minimization on a randomly chosen set of starting points, and the smallest minimum retained at the end. There is no guaranty that the obtained minimum is the global minimum, but the more starting points are used, the higher is the chance to find the global minimum. The 185 optimization has been run with increasing number of starting points, and when the obtained minimum was found to be independent of the number of starting points it is considered to be a good estimation of the global minimum. Finally 10 3 starting points are used.

190

The optimization process described above yields as a result the optimal radii of the electrodes. The results obtained with the two cost functions I and J are given in table 1.

Table 1: Optimal electrode radii, obtained for the two cost functions. In the initial design the electrode covers the whole membrane.

N • Goal

Cost function

Optimal radii Ropt

Electrode shape The finite element model can then be used to compute 195 the frequency response of the loudspeaker with these optimal electrodes, which provides a first validation of the optimization procedure. To validate further the proposed algorithm, prototypes with the optimal electrodes are manufactured, and their frequency response is measured.

1 Initial design - [1, 1, 1]

Manufacturing process

In this section, the manufacturing process to build DE loudspeaker prototypes with optimal electrodes is described. First, general design considerations on the electrodes connections to the electrical supply are discussed, and the process to deposit the electrodes is then presented.

Electrode design

As explained in section 2.2, the electrode consists in several concentric rings. However, these different rings must be connected to the electrical supply, so connections need to be designed and added.

The electrostatic pressure which deforms the membrane applies only on areas of the membrane which are covered on both sides by electrodes. If an electrode is present only on one side, or if there is no electrode on neither sides, no electrostatic pressure applies. This can be exploited to design the connections, by extending the electrodes while making sure that they are overlapping only in the desired areas.

Silicone membrane

Electrodes

Top electrode Bottom electrode Active area

Holding frames (PMMA) Aluminium tape The chosen electrode design is shown in fig. 6, where the desired active area is a disk. To connect the disk to the power supply, six radial electrodes are added between the disk and the frame, where aluminum tape is placed in contact with the electrode. The connections of the top and bottom electrodes are rotated by 30°, so that they are not superimposed. This way, electrodes are present on both sides of the membrane only in the central disk, which is the desired active area. The electrodes are not perfectly conductive [START_REF] Mccoul | Recent Advances in Stretchable and Transparent Electronic Materials[END_REF], so the connections add electrical resistance to the electrical circuit, which reduces the bandwidth in which the actuator can operate. The resistance created by the electrode connections is minimized in the designed proposed in fig. 6 by using all the available membrane area as connections: by increasing the number and the width of the connections, the connection resistance decreases. With the chosen electrode design and a surface resistivity of 10 kΩ/sq, the electrical cutting frequency lies around 1/2πR e C ≈ 10 kHz, where R e and C are the lumped resistance and capacitance of the membrane.

Electrical connections

The system should remain as axisymmetric as possible, which is why we decided to use many small connections instead of a large one.

Pad-printing the electrodes

To deposit with good accuracy the electrodes on the di-245 electric membrane, the pad-printing method proposed by Rosset et al. [START_REF] Rosset | Fabrication Process of Silicone-based Dielectric Elastomer Actuators[END_REF] has been used. The dielectric membrane is a 50 µm thick silicone film (Silpuran Film 2030 from Wacker Chemie AG). A small prestretch of 1.1 is applied to keep the membrane flat during the fabrication. The elec-250 trodes are made of carbon-black (Ketjenblack EC-600JD from Akzo Nobel N.V) loaded silicone (Silbione LSR 4305 from Elkem Silicones), which yields compliant and durable electrodes [START_REF] Rosset | Fabrication Process of Silicone-based Dielectric Elastomer Actuators[END_REF].

Electrode ink Mylar mask The pad-printing process is described by the schemat-255 ics in fig. 7. First, a cliché is manufactured. It consists in a metal plate where a disk is etched. The disk diameter corresponds to the largest electrode diameter that can be deposited (40 mm in our case). When the ink reservoir of the pad-printer is slid over the cliché, ink stays in 260 the etched area, and will be picked up by the soft silicone stamp.

To pattern the electrode, a thin Mylar film is laser-cut to form a mask corresponding to the shape of the electrode. It is placed on the membrane before the pad applies the 265 ink. The electrostatic adhesion between the mylar and the silicone membrane prevents the ink from penetrating under the mask. Pictures of the membranes with the three different types of electrodes defined in table 1 which have been manufac-270 tured at the LMTS by the method described above are shown in fig. 8. The pad printing process yields precise electrodes, with clean edges.

The thickness of the electrodes (and consequently their surface resistivity) can be controlled by repeating the padprinting (stages 1 to 7 in fig. 7). With a single printing, the obtained electrodes are approximately 3 µm-thick, which is small compared to the thickness of the membrane (this estimation is obtained from the mass added during the pad-printing process).

Results and discussion

Experimental validation of the optimal designs

Given the chosen cost functions I and J, the electrode shape has been optimized and resulted in the electrodes defined in table 1. The optimal electrodes have been manufactured following the process described in the previous section 2. In the present section, the sound radiation obtained with the three electrode designs is analysed to assess the efficiency of the optimization process.

All acoustical measurements have been performed in 290 the anechoic chamber of ENSTA, which is 3×3×3 m 3 large and specified down to 120 Hz. The pressure is measured on axis at 1 m, and the transfer function between the electrical excitation w and the radiated pressure is obtained by the exponential swept sine method [START_REF] Farina | Simultaneous Measurement of Impulse Response and Distortion with a Swept-Sine Technique[END_REF].

The measured transfer functions are plotted in fig. 9 together with the results of the model, for the three electrode designs.

The global shape of the radiated pressure is the same for the three electrode shapes: at low frequencies (500 Hz to 3000 Hz), the response is dominated by the membrane modes. At high frequencies (above 3 kHz), large peaks can be observed. They are caused by the acoustic resonances inside the cavity on which the membrane is inflated [START_REF] Garnell | Coupled vibro-acoustic modeling of a dielectric elastomer loudspeaker[END_REF] (see figs. 2 and3).

When the electrode occupies the whole membrane (electrode 1), the acoustic radiation is dominated by the second membrane mode, at 750 Hz. When the radiation of this second mode is maximized (electrode 2), the difference between the peak level of mode 2 and of the neighboring 310 modes increases further. In particular, the radiation of the third mode around 1000 Hz has decreased. Moreover, the overall radiated level decreased, as a consequence of a smaller electrode area (36 % of the initial electrode area) which reduces the electrostatic excitation.

315

The third electrode design also succeeds is satisfying its goal: minimizing the radiation of the second mode. The amplitude of the second peak in fig. 9(c) is similar to the surrounding modes 1, 3, 4 and 5. Compared to the initial design fig. 9(a), the difference in amplitude between 320 the second peak and the neighboring peaks has decreased by 15 dB. Of course, the electrode area is even smaller than electrode 2 (10 % of the initial area, see fig. 8), so the This can be understood by looking at fig. 10, where the functions g n defined by eq. ( 12) are plotted for the first 4 modes. We recall that the contribution of a given mode n to the radiated pressure is proportional to A n , which is the integral of g n on portions of the membrane where an electrode is present on both sides (see eq. ( 11)).

Figure 10 shows that g 2 is always positive, so the integral of g 2 times the indicative function of the electrode Γ cannot be cancelled, except if there is no electrode. As a consequence, whatever the electrode shape, mode 2 will contribute to the sound radiation. Figure 10 also helps to interpret the optimal electrode shapes: for example when the radiation of the second mode is maximized, the optimal electrode is a ring around radius R ≈ 0.6. This corresponds to a radius where g 3 and g 4 change signs, so the integral of these functions on this ring will be very small. At the same time, R ≈ 0.6 corresponds to electrode locations where g 2 is very large.

Discussion

In the previous section, we showed that the optimization is limited by the shape of the modal participation functions g n (see fig. 10). The contribution of the second mode cannot be reduced to zero since the function g 2 is positive for all radii.

One option to further decrease the contribution of mode 2 to the sound radiation could be to excite one of the electrode rings out of phase. Indeed, this is equivalent to setting the electrode indicative function to 1 on the electrodes actuated in phase, and to -1 on the electrodes actuated out of phase.

To test this idea, we perform an optimization of the cost function J, for an electrode with 5 radii (three rings), and set the second ring out of phase (see fig. 11(b)). This electrode is denoted electrode 4.

The numerical results presented in fig. 11 show that the optimization with the second electrode out of phase succeeds in completely canceling the contribution of mode 2 to sound radiation, which is now dominated by the contributions of modes 3 and 4, around 1 kHz.

Possible applications of this kind of optimizations may be to control the directivity of DE loudspeakers. Indeed, each membrane mode has a specific directivity pattern, 370 so by controlling which mode dominates the frequency response the directivity can be controlled. For example, for the optimization presented in fig. 11, the mode 2 has a more omnidirectional radiation pattern than higher order modes. As a consequence, decreasing the 375 contribution of mode 2 to the radiation increases the nearfield directionality of the loudspeaker, as shown in fig. 12. This is particularly visible at high frequencies, because the loudspeaker is acoustically compact at low frequencies below approximately 3 kHz (its size is small compared to the acoustical wavelength) and radiates as a monopole. Near field directivity control may be useful for headphones applications for example.

This simple example highlights the wide range of tuning possibilities which is opened by electrode shape optimization.

Conclusion

In this article, we investigated the influence of the electrode shape on the dynamics of dielectric elastomer actuators. We focused on an inflated DE loudspeaker to demonstrate that the electrode can be optimized to tune the dynamic behavior.

An optimization method based on a finite element model of the loudspeaker has been presented, and two objective functions have been considered, aiming at either increasing or decreasing the contribution of the second membrane mode to the sound radiation. It has been shown experimentally that the optimization succeeds in improving the considered objectives.

In the present article we focused on simple cost functions which yield clear and easily interpreted results, and demonstrated that the sound radiation can be tuned by adjusting the electrode shape. More acoustically relevant cost functions could then be considered, such as minimizing the deviation of the sound pressure level from its mean value [START_REF] Lu | Optimization of orthotropic distributed-mode loudspeaker using attached masses and multi-exciters[END_REF], which could help flattening the acoustic frequency response.

If several electrodes are used, numerical simulations suggest that the optimization results could be improved by playing on the phase at which the different electrodes are actuated. One may imagine that the different electrodes could be excited with different amplitudes, and with arbitrary phase between each of them. The excitation of the different electrodes could also be frequency-dependent, and the optimization then becomes a combined filter design and modal control problem.

The large design freedom provided by the patterning of the electrodes of DE actuators still remains to be explored, and we believe accurate models of the dynamics of DE actuators may help investigating this path.

Figure 1 :

 1 Figure 1: Principle of dielectric elastomer actuators. (a) Initial state. (b) Deformed state, when a high voltage u is applied.

Figure 2 :

 2 Figure 2: Picture of an inflated DE loudspeaker prototype during acoustical measurements in the anechoic chamber of ENSTA.

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115Figure 3 :

 3 Figure 3: Model of an inflated DE membrane, and definition of the needed variables. (a) Reference configuration: the membrane is flat and at rest. Variables in the reference configuration are written with capital letters. (b) Deformed configuration: the membrane is inflated with the pressure p i and a high voltage u is applied.

Figure 4 :

 4 Figure 4: Definition of the optimized parameters: the nondimensional radii R i of the different electrode rings. Here, an electrode with 3 radii is shown. (a) Initial electrode (where the second electrode between R 2 and R 3 is infinitely narrow). (b) Optimized electrode.

2 .Figure 5 :

 25 Figure 5: Flowchart of the optimization procedure.

Figure 6 :

 6 Figure 6: Design of the electrodes and of the electrical connections. The active area is enclosed by the red dashed circle.

Figure 7 :

 7 Figure 7: Pad-printing procedure used to apply the soft conductive electrodes. The steps 1-8 are then repeated on the other side of the membrane to apply the second electrode.

Figure 8 :

 8 Figure 8: Manufactured membranes with the three electrode shapes defined in table 1. The active areas are enclosed by the red dashed circles. (a) Electrode 1: initial design. (b) Electrode 2: maximize radiation of mode 2. (c) Electrode 3: minimize radiation of mode 2.

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2 Figure 9 :

 29 Figure 9: Transfer function between the radiated acoustic pressure at 1 m on axis and the excitation signal. The electrode shape is shown in the bottom right corner. (a) Electrode 1. (b) Electrode 2. (c) Electrode 3.

Figure 10 :

 10 Figure 10: Modal participation factor as a function of the membrane radius for the first four modes.

2 Figure 11 :

 211 Figure 11: (a) Computed radiation of the DE loudspeaker with electrode 4, for cost function J. (b) Optimal electrode shape. The second electrode ring (hatched) is driven out of phase of the two other rings.

Figure 12 :

 12 Figure 12: Near field directivity of the inflated DE loudspeaker, computed by the finite element model on a circle of 5 cm diameter centered on the membrane, and normalized by the level on axis. (a) Initial design, with electrode 1 (see table 1). (b) Optimal electrode obtained when the second electrode ring is actuated out of phase, shown in fig. 11(b).

Acknowledgements

The authors acknowledge the support of the French National Research Agency within the project SMArT (ANR-15-CE08-0007-02) and of the Swiss National Science Foundation, grant 200020 184661.