Dr Dominic H Ffytche
email: d.ffytche@iop.kcl.ac.uk

Dr M Catani

Disorders of visual perception

Keywords:

Visual perceptual disorders are often presented as a disparate group of neurological deficits with little consideration given to the wide range of visual symptoms found in psychiatric and neurodevelopmental disease. Here we attempt a functional anatomical classification of all disorders linked to visual perception, whatever the clinical context in which they arise, including those disorders that bridge vision, emotion, memory, language and action. Guided by clinical and neuroimaging evidence, visual perceptual disorders are classified by the functional anatomical networks likely to be involved and the class of underlying dysfunction, whether topological (a localised deficit or region of hyperfunction) or hodological (a disconnection or hyperconnection). The wider perspective forces us to consider what visual functions underlie a range of symptoms sidelined by previous classificatory schemes and helps generate novel hypotheses for further research in the area.

Introduction

If visual perception is the day-to-day experience of 'seeing' -the conscious, visual experience of the world we have when we open our eyes -a visual perceptual disorder might be considered a deviation from this experience. Current classificatory schemes focus on disorders traditionally considered neurological (i.e. based on static visual deficits typically related to brain lesions), relating each disorder to a specific brain region or its connections (for examples see 1 and 2 ). Visual symptoms such as visual hallucinations, illusions and visual perceptual distortions or symptoms more likely to be found in psychiatric or neurodevelopmental disorders are typically only mentioned in passing or not at all in such schemes. This omission is understandable given that these types of symptom are often transient, and thus difficult to study directly, and may occur without an obvious associated brain lesion. Here we attempt to incorporate both 'neurological' and 'psychiatric' disorders into a single classificatory scheme using an anatomical network approach. Our aim is to provide an anatomically-structured overview of visual perceptual dysfunction across all clinical contexts that we hope will provoke further research.

A note on terminology

The scope of our classification is set by the way we have interpreted the three constituent terms of 'visual perceptual disorder'. i) Visual The disorders classified here are visual in the sense that they are selective for vision without equivalent changes in other sensory modalities. For example, visual hypo-emotionality (a loss of emotional tone for visual stimuli) implies that, in some cases at least, emotions may be preserved for auditory and other sensory stimuli. ii) Perceptual If visual perception is the day-to-day experience of 'seeing', by this definition, its disorders might include, for example, a blurring of vision through loss of acuity. To limit the scope of the review we have focussed on symptoms attributable to dysfunction within cortico-cortical networks including those that bridge visual perception with memory, attention, emotion and action. We make a distinction between visual perception and visual imagery as the two differ in terms of their underlying neurobiology (see [START_REF]Visual hallucinatory syndromes: past, present, and future[END_REF] ). Visual percepts are seen in the external world and are outside volitional control (one cannot change the percept by force of will) whereas visual imagery is seen in the mind's eye and is under volitional control. We consider visual hallucinations and illusions as varieties of visual perceptual experience as they are closely related to visual perception in terms of underlying brain activity [START_REF]Visual hallucinatory syndromes: past, present, and future[END_REF] . Phenomena with elements of both perception and imagery are more difficult to classify. For example, experiences referred to as pseudohallucinations or incomplete hallucinations (see [START_REF] Blom | A dictionary of hallucinations[END_REF] for other uses of these terms) are seen in the mind's eye but are outside volitional control. We have highlighted such ambiguous experiences in the classification and, where appropriate, presented both perceptual and imagery viewpoints. iii) Disorder We use the term disorder for continuity with previous clinical literature, although much of the classification relates to perceptual symptoms that do not imply a specific disease or pathology.

The hodotopic framework

We have classified visual perceptual disorders using the general anatomical framework we set out in 2005 as an update of classical associationist and neo-associationist accounts of higher cognitive function and dysfunction [START_REF] Catani | The rises and falls of disconnection syndromes[END_REF] . Recent advances in functional neuroanatomy, using techniques such as functional magnetic resonance imaging (fMRI) and diffusion tensor tractography (DT-tractography), provide complementary views of higher function and the pathological mechanisms that generate cognitive and behavioural symptoms. One view, termed topological (from topos = place), relates to discrete cortical locations. The other, termed hodological (from hodos = path), relates to connections between locations. Both views are combined in a generic neuroanatomical model applicable to all higher functions and dysfunctions -the hodotopic framework [START_REF] Catani | The rises and falls of disconnection syndromes[END_REF] . From the perspective of the hodotopic framework, each higher function relates to a specific network of cortical locations and white-matter connections. Topological disorders are localised within a cortical area and result in a loss (deficit) or an increase in function (hyperfunction). Hodological disorders relate to connections between areas, including those in which function in one brain region is altered by changes in another, spatially remote, region. We use von Monakow's term diaschisis to refer to such remote effects [START_REF] Von Monakow | Diaschisis (translated from 'Diaschisis' Die Lokalisation im Grosshirn und der abbau der Funktion durch korticale Herede[END_REF] ; however, the term is not intended here to carry its original implication of

Classification method

We have allocated each visual perceptual symptom to a visual network and one of four classes of dysfunction (topological versus hodological; hyper-versus hypofunction/connection). For some symptoms, functional imaging, DTI-tractography or lesion evidence is available to help identify the appropriate location in the table (e.g. a localised increase in activity within a specific brain region). However, for many we have been forced to make an educated guess through: i) analogies to related symptoms for which evidence is available and ii) a neurophenomenological approach (see [START_REF]Visual hallucinatory syndromes: past, present, and future[END_REF][START_REF] Pribram | Visual hallucinations and the Charles Bonnet Syndrome[END_REF] ). This approach is based on the body of evidence suggesting that some, as yet undefined, aspect of activity within a specialised cortical region (resulting in an increase in imaging signal [START_REF]Neural codes for conscious vision[END_REF] ) contributes directly to visual perceptual experience (e.g. the percept of a face is associated with increased activity within face-specialised cortex [START_REF] Moutoussis | The relationship between cortical activation and perception investigated with invisible stimuli[END_REF] ). Where evidence is available, this principle seems also to apply to hallucinatory, illusory and synaesthetic percepts (see [START_REF]Visual hallucinatory syndromes: past, present, and future[END_REF] ). Thus the content of visual perceptual experience (whether an object, a face, motion etc.), localises activity to a particular brain network, irrespective of whether the experience is a normal percept, illusion, hallucination etc.

A taxonomy of disordered visual perception

Table 1 and Figure 1 outline a tentative classification of visual perceptual disorders.

Topological disorders of visual perception

Deficits -Prototypical disorders of this nature are characterised by the selective loss of a specific visual perceptual function e.g. achromatopsia [START_REF] Zeki | A century of cerebral achromatopsia[END_REF] (selective loss of colour vision following lesions of colour-specialised cortex) and akinetopsia [START_REF] Zeki | Cerebral akinetopsia (visual motion blindness) -A review[END_REF] (selective loss of motion vision following lesions of lateral occipital motion-specialised cortex; the motion deficit is not absolute with a relative sparing of the perception of slow movement [START_REF] Zihl | Selective disturbance of movement vision after bilateral brain damage[END_REF] ).

Lesions of V1 result in a scotoma restricted to a single hemifield (a hemianopia if the lesion involves the entire extent of V1 in one hemisphere). Lesions of ventral V2 / V3 are associated with upper quadrantic field defects while those of dorsal V2 / V3 are associated with inferior quadrantic field detects [START_REF] Horton | Quadrantic visual field defects: a hallmark of lesions in extrastriate (V2/V3) cortex[END_REF] . Scotomata in the context of migraine aura relate to regions of cortical spreading depression-like activity in early visual areas (e.g. V1, V2 and V3 [START_REF] Hadjikhani | Mechanisms of migraine aura revealed by functional MRI in human visual cortex[END_REF] ).

Face perception deficits (prosopagnosia) have a variety of subtypes likely to reflect the involvement of different functional modules within the face processing system, although disconnection mechanisms may also be involved [START_REF] Fox | Disconnection in prosopagnosia and face processing[END_REF] .

An inability to recognise familiar places (environmental agnosia) is associated with medial temporal lobe lesions [START_REF] Landis | Loss of topographic familiarity. An environmental agnosia[END_REF] and can be considered a topological deficit of cortex specialised for the visual environment (parahippocampal place area [START_REF] Epstein | A cortical representation of the local visual environment[END_REF] ) or a hodological disconnection of visual and memory representations of places [START_REF] Landis | Loss of topographic familiarity. An environmental agnosia[END_REF] .

Size and object distortions are referred to collectively as metamorphopsias. Objects appearing larger or smaller (macropsia and micropsia respectively) have been attributed to a dysfunction of object constancy within the temporal lobe (the integration of object distance with retinal visual angle to estimate size), although it is unclear whether this is best conceived as a topological dysfunction (i.e. a deficit or hyper-function of a cortical region specialised for constancy) or a hodological dysfunction (i.e. a disconnection or hyper-connection of relevant specialised modules). The same mechanism may be responsible for objects appearing nearer or further away (pelopsia and teleopsia), although such symptoms can also be interpreted as disorders of depth perception (and are also referred to as enhanced stereopsis [START_REF] Blom | A dictionary of hallucinations[END_REF] ). Lesions of lateral occipito-temporal cortex may cause micropsia in one hemifield [START_REF] Cohen | Selective deficit of visual size perception: two cases of hemimicropsia[END_REF] or bilaterally [START_REF] Ceriani | Seeing objects smaller than they are: micropsia following right temporo-parietal infarction[END_REF] , although as object constancy may be preserved [START_REF] Ceriani | Seeing objects smaller than they are: micropsia following right temporo-parietal infarction[END_REF] , such cases may relate to a deficit of object processing rather than constancy.

Apperceptive agnosia and cortical blindness describe diffuse visual loss affecting many visual domains. They occur in the context of extensive occipital lesions and are likely to involve both topological and hodological dysfunction. Several subtypes of disordered form perception have been described (see [START_REF] Riddoch | Visual agnosia[END_REF] for review), each likely to depend on the form perception sub-module(s) involved in the lesion. Scieropia refers to darkening of vision and follows diffuse occipital damage (see [START_REF] Wapner | Visual agnosia in an artist[END_REF] for an example: a patient whose vision was dimmed as if seeing everything at twilight). The same symptom occurring without other visual impairments (scierneuropsia) was described in the psychoanalytic literature [START_REF] Martin | On scierneuropsia, a previously unnamed psychogenic visual disturbance[END_REF] and is likely to be an example of de-realisation (see below).

The Riddoch syndrome is a variant of cortical blindness in which motion-specialised functional networks remain intact. Subjects are able to consciously perceive and discriminate visual motion in their otherwise blind visual field when stimulated with fast motion [START_REF] Zeki | The Riddoch syndrome: insights into the neurobiology of conscious vision[END_REF] , while under other stimulus conditions report visual percepts without the ability to discriminate (gnosanopsia) or have intact discrimination ability without visual percepts (blindsight or agnosopsia) [START_REF] Zeki | The Riddoch syndrome: insights into the neurobiology of conscious vision[END_REF] .

In visuo-spatial neglect, patients have a deficiency of orienting attention to one hemifield (usually left) and (may) fail to perceive stimuli in this region when rival stimuli are present in the non-neglected hemifield (extinction). The disorder has been interpreted as topological (related to the cortical modules involved in visuospatial processing and attentionsee [START_REF] Mort | The anatomy of visual neglect[END_REF] for review of regions implicated) but hodological mechanisms have also been suggested [START_REF] Thiebaut De Schotten | Direct evidence for a parietal-frontal pathway subserving spatial awareness in humans[END_REF] . Simultanagnosia (an inability to sustain attention across different locations in the visual field) leads to objects appearing fragmented or disappearing from view [START_REF] Rizzo | Simultanagnosia: a defect of sustained attention yields insights on visual information processing[END_REF] with shrinkage of the visual attentional field [START_REF] Michel | Seeing without the occipito-parietal cortex: Simultagnosia as a shrinkage of the attentional visual field[END_REF] . The disorder is likely to relate to parietal lobe specialisations for spatial visual attention, although it is unclear whether the disorder is best conceived as topological or hodological. Rapid fading of vision found with occipito-parietal lesions [START_REF] Holliday | Rapid fading of visual sensations in a subject with a parietal-occipital tumour[END_REF] is likely to be caused by the same mechanism as simultanagnosia.

Selective lesions of the amygdala are associated with perceptual deficits of emotional facial expression [START_REF] Adolphs | Impaired recognition of emotion in facial expressions following bilateral damage to the human amygdala[END_REF] with similar deficits described in autism spectrum disorder and psychopathy.

Visual imagery may be selectively impaired (i.e. with intact auditory, olfactory, gustatory etc. imagery -see [START_REF] Basso | Loss of mental imagery: a case study[END_REF] for review of case reports) with a relative preservation of visual perception. Loss of visual imagery may also be accompanied by a specific loss of visual dreaming (visual anoneria [START_REF] Hobson | Dreaming and the brain: toward a cognitive neuroscience of conscious states[END_REF] ) or a more general loss of dreams in all sensory modalities (Charcot-Wilbrand syndrome [START_REF] Bischof | Total dream loss: a distinct neuropsychological dysfunction after bilateral PCA stroke[END_REF] ). Such disorders can be interpreted as topological deficits of visual imagery generation or hodological disconnections within visual memory networks.

Hyperfunction -Topological disorders of hyperfunction are related to pathophysiological mechanisms such as local irritation (e.g. in epilepsy or at the margins of a cortical lesion) and de-afferentation (e.g. caused by eye disease [Charles Bonnet Syndrome] or a visual pathway lesion; the mechanism by which loss of visual input results in increased cortical activity is unclear but may relate to changes in thalamocortical circuitry [START_REF]The hodology of hallucinations[END_REF] ).

Hallucinations linked to V1 and surrounding areas (V2 / V3) are of simple featureless forms, colours and television-like static (visual snow). These phenomena are termed phosphenes or photopsias by some authors although they are perhaps better referred to as simple hallucinations given that they differ from other types of visual hallucination only in the location of cortex involved [START_REF]The hodology of hallucinations[END_REF] . In migraine aura, zigzag lines and patterns (teichopsia [START_REF] Plant | The fortification spectra of migraine[END_REF] ) relate to the margins of a region of cortical spreading depression-like activity in early visual cortex [START_REF] Hadjikhani | Mechanisms of migraine aura revealed by functional MRI in human visual cortex[END_REF] .

Colour, object and landscape hallucinations are each linked to their respective region of cortical specialisation [START_REF]The hodology of hallucinations[END_REF][START_REF]The anatomy of conscious vision: an fMRI study of visual hallucinations[END_REF][START_REF] Oertel | Visual hallucinations in schizophrenia investigated with functional magnetic resonance imaging[END_REF] and the same is likely to be the case for motion hallucinations and passage hallucinations (visual hallucinations of persons or animals that pass sideways out of the visual field). Increased vividness of an attribute (e.g. increased vividness for colours: hyperchromatopsia [START_REF] Howard | The perceptual consequences of visual loss: positive pathologies of vision[END_REF] ) is thought to relate to increased activity within the relevant specialised region (e.g. colour-specialised cortex for hyperchromatopsia). The origin of lilliputian hallucinations (small figures, often in costume) is unclear but has been classified here as related to object specialisations. The reduced size of such hallucinations may relate to visual constancy mechanisms [START_REF] Howard | The perceptual consequences of visual loss: positive pathologies of vision[END_REF] .

Text or letter-string hallucinations lacking semantic content are linked to the cortical area specialised for word forms, while those with semantic content (visual command hallucinations and visual verbal hallucinations 4 ) are linked to higher-level representations of visual language [START_REF] Lappin | Visual command hallucinations in a patient with pure alexia[END_REF] . Illusions of distorted faces (prosopometamorphopsia) relate to activity on the lateral convexity of the occipital lobe [START_REF] Heo | Single-photon emission computed tomography in a patient with ictal metamorphopsia[END_REF] , a region specialised for face features (the occipital face area) and the same region is likely to account for hallucinations of distorted faces.

Hallucinations of normal (undistorted) faces relate to activity within ventral occipito-temporal cortex (fusiform face area) [START_REF]The anatomy of conscious vision: an fMRI study of visual hallucinations[END_REF] and the same region is likely to account for facial intermetamorphosis (a change in the visually perceived identity of a face).

A range of symptoms collectively termed palinopsia [START_REF] Howard | The perceptual consequences of visual loss: positive pathologies of vision[END_REF][START_REF] Critchley | Types of visual perseveration: "paliopsia" and "illusory visual spread[END_REF] have been linked to dysfunction within parietal lobe co-ordinate systems [START_REF] Santhouse | Visual hallucinatory syndromes and the anatomy of the visual brain[END_REF] . They include: i) polyopia (perceiving multiple copies of the same object often arranged in rows and columns; if the number of copies is large [e.g. > 100] it is referred to as entomopia 42 ); ii) visual perseveration (an object remaining fixed in retinal co-ordinates with eye movements); iii) delayed palinopsia (an object returning to field of view); iv) illusory visual spread (the spread of a pattern from an object to its surroundings). Parietal lobe regions are also likely to underlie trailing phenomenon (a variety of visual perseveration in which a series of discontinuous stationary images trail behind a moving object) and positive afterimages (afterimages in which objects are seen in their true as opposed to complementary colours).

Patients with Anton's syndrome (visual anosognosia [denial of blindness] in the context of cortical blindness) report visual experiences that are traditionally interpreted as confabulations (false memories or false reports of visual perceptual experience) or visual imagery experiences [START_REF] Goldenberg | Imagery without perception--a case study of anosognosia for cortical blindness[END_REF] . These symptoms are likely to be the result of topological hyperfunction within visual memory systems. Flashbulb memories (detail-perfect memories of events that at the time of witnessing evoked a high level of surprise, consequentiality and/or emotional arousal [START_REF] Brown | Flashbulb memories[END_REF] ) and memory hallucinations (fantastic false memories conjured up retrospectively [START_REF] Blom | A dictionary of hallucinations[END_REF] ) may relate to a similar hyperfunction of visual memory systems (and/or visual emotion systems in the case of flashbulb memories). Re-perceptive hallucinations involve the replay of memory fragments (also termed experiential hallucinations [START_REF] Blom | A dictionary of hallucinations[END_REF] ); however, it is unclear whether such experiences are visual imagery or visual percepts [START_REF] Gloor | The role of the limbic system in experiential phenomena of temporal lobe epilepsy[END_REF] . If visual imagery they are likely to involve hyperfunction of memory systems, if visual percepts they are likely to involve hyperconnection between visual memory networks and occipital cortex.

Autoscopic phenomena describe a range of experiences in which the self is duplicated in external space. In autoscopy, the duplicate self is seen in the external world. In out-of-body experience (OBE) the physical body is seen from the perspective of the duplicated external self. In heautoscopy the hallucinated 'double' is identified as oneself, despite the lack of an exact physical resemblance, and subjective viewpoint may change between the double and physical self in rapid alternation. These phenomena are thought to relate to the disintegration of visual, proprioceptive, tactile and vestibular modalities and have been linked to transient dysfunction in the region of the temporoparietal junction (TPJ) [START_REF] Blanke | Out-of-body experience and autoscopy of neurological origin[END_REF] . Whether this dysfunction is better conceived as topological hyperfunction of one or more of the sensory modalities involved or a hodological connectivity increase between specialised regions is unclear. Visual extracampine hallucinations (hallucinations perceived in impossible locations outside the field of view) could be considered related phenomena in that they involve an alteration in the reconstruction of visual space and body schema. Other disorders likely to be related to hyperfunction or hyperconnection within the TPJ and related regions include: i) the oculogravic illusion (an illusory displacement of objects co-occurring with a change in gravity as in a diving airplane 47 ); ii) the oculogyral illusion (the apparent movement of a stationary light in a dark room during fixation following a period of rotation 47 ); iii) visual hallucinations occurring in association with caloric stimulation [START_REF] Kolev | Visual hallucinations evoked by caloric vestibular stimulation in normal humans[END_REF] ; iv) Zingerle's automatosis in which visual hallucinations occur during transient motor and vestibular dysfunction 3 (e.g. compulsive posturing, torticollis, choreo-athetosis); v) the transient intensification of visual patterns and text, often in association with neuroleptic induced oculogyric crises (paroxysmal perceptual alteration [START_REF] Uchida | Recurrent episodes of perceptual alteration in patients treated with antipsychotic agents[END_REF] ).

Hodological disorders of visual perception

Disconnection -Disconnection of cortical visual networks from sub-cortical inputs as a result of visual pathway lesions or eye disease is associated with simple and complex visual hallucinations and visual illusions [START_REF]Visual hallucinatory syndromes: past, present, and future[END_REF][START_REF] Pribram | Visual hallucinations and the Charles Bonnet Syndrome[END_REF] (the Charles Bonnet Syndrome -the change in cortical function secondary to eye disease might be considered a form of diaschisis). Brainstem lesions are associated with peduncular hallucinations (complex hallucinations associated with eye movement disorders and changes in arousal or sleep [START_REF] Benke | Peduncular hallucinosis: a syndrome of impaired reality monitoring[END_REF] ).

Depth vision (visual space) involves a number of dissociable (see [START_REF] Danta | Hemisphere function and binocular depth perception[END_REF] ) functions, some related to perceptual aspects of depth (e.g. stereopsis [binocular vision] and object constancy), others related to action aspects of depth (e.g. reaching behaviour). One aspect of depth vision (stereopsis) is lost in the central visual field (but not the visual periphery) following splenial section [START_REF] Mitchell | Binocular depth perception and the corpus callosum[END_REF] .

In cinematographic vision, moving objects appear as a series of static freeze frames at different spatial locations [START_REF] Blom | A dictionary of hallucinations[END_REF] . The disorder may relate to the disconnection of motion specialised cortex and spatial co-ordinate frames in the parietal lobe.

Associative object agnosia (loss of object meaning with intact object perception) is thought to relate to the disconnection of visual and semantic regions (see [START_REF] Riddoch | Visual agnosia[END_REF] for review). Specific deficits of colour or object naming (colour anomia, object anomia) are thought to relate to the disconnection of visual and language areas while optic aphasia (an inability to name objects by sight with preserved ability to name by touch or from description) is thought to follow disconnection of left-hemisphere verbal semantic and right-hemisphere visual semantic representations [START_REF] Luzzatti | A functional model of visuo-verbal disconnection and the neuroanatomical constraints of optic aphasia[END_REF] .

Pure alexia (a deficit of reading words with preserved writing and letter recognition) and left hemialexia (a reading deficit restricted to the left hemifield) have been interpreted as hodological dysfunctions (a disconnection of a left-hemisphere region specialised for word forms from visual inputs) and topological dysfunctions (a deficit of the word form area) (see [START_REF] Epelbaum | Pure alexia as a disconnection syndrome: new diffusion imaging evidence for an old concept[END_REF] for review). In surface dyslexia, the link between word forms and semantics is impaired so that reading occurs through phonology with characteristic errors for irregular words [START_REF] Coltheart | Surface dyslexia[END_REF] . The disorder might be considered a disconnection of regions specialised for word forms from regions specialised for visual semantics or a topological deficit of visual semantic specialisations.

The action aspect of depth vision (reaching behaviour) is dysfunctional in optic ataxia (impaired reaching with intact motor and visual function) and is thought to reflect an inability to integrate different co-ordinate frames within the parietal lobe [START_REF] Jackson | There may be more to reaching than meets the eye: re-thinking optic ataxia[END_REF] . The failure of integration might reflect a hodological disconnection of the relevant reference frames or a topological deficit of an area specialised for their integration. A variant (crossed optic ataxia) in which reaching into contra-lateral hemispace is impaired (e.g. reaching with the right hand for an object located on the left side) is thought to relate to the disconnection of the two hemispheres with a combination of splenial and parietal lesions [START_REF] Ferro | Crossed optic ataxia: possible role of the dorsal splenium[END_REF] .

Oculomotor apraxia (a disorder of gaze in which subjects are unable to voluntarily move gaze from one object to another with preserved reflex gaze shifts) can be interpreted as a disconnection of voluntary saccadic systems in the frontal lobe from reflex saccadic systems in the parietal lobe (a network classified here as related to visual attention).

Denial of cortical blindness (Anton's syndrome) might be considered a disconnection of visual cortex from body schema representations in the parietal lobe (classified here as related to visual attention). An inverse Anton's syndrome (denial of seeing) has been interpreted as a disconnection of parietal attentional mechanisms and visual cortex [START_REF] Hartmann | Denial of visual perception[END_REF] . Denial of seeing specific objects reported in hypnosis (termed negative hallucination or scotomization [START_REF] Blom | A dictionary of hallucinations[END_REF] ) is likely to relate to the same mechanism.

In reduplicative disorders, a familiar person, place or object is perceived as replaced by a duplicate person, place or object [START_REF] Young | Reduplication of visual stimuli[END_REF] (termed Capgras syndrome when involving a person).

Reduplicative phenomena are thought to relate to the disconnection of visual and affective or memory regions [START_REF] Ellis | Accounting for delusional misidentifications[END_REF] . Similar disconnection accounts are given for de-realisation (the feeling that experiences [typically visual] seem strange or unreal), visual hypoemotionality (where visual experience lacks emotional tone) [START_REF] Sierra | Separating depersonalisation and derealisation: the relevance of the "lesion method[END_REF] and visual amnesia (a deficit of registering visual experiences in short-term memory) [START_REF] Ross | Sensory-specific and fractional disorders of recent memory in man. I. Isolated loss of visual recent memory[END_REF] .

Visual alloaesthesia involves the transposition, rotation, tilting or inversion of the visual field (also referred to as reversal of vision metamorphopsia [START_REF] River | Reversal of vision metamorphopsia: clinical and anatomical characteristics[END_REF] ) and has variants where the visual environment (but not the visual field) is rotated [START_REF] Girkin | Central disorders of vision in humans[END_REF] . The mechanism behind these disturbances is unclear but may relate to a disconnection of vestibular and visual inputs, transcallosal disconnection of visual and parietal inputs or topological lesions affecting regions specialised for the integration of these modalities 2 . The disorders are associated with occipito-parietal lesions that spare the optic radiations and with brainstem pathology [START_REF] River | Reversal of vision metamorphopsia: clinical and anatomical characteristics[END_REF] . Deficits of visual axis perception (e.g. judgment of true vertical or horizontal) are associated with lesions of the anterior parietal lobe and have been attributed to a disconnection of vestibular inputs and their cortical representation or topological deficits of vestibular representations in the parietal lobe [START_REF] Von Cramon | On the cerebral organization of elementary visuospatial perception[END_REF] .

The inability to mimic meaningless or meaningful actions or postures (visual imitative apraxia) is likely to reflect disconnection of visual and motor representations [START_REF] Heilman | The disconnection apraxias[END_REF] . Utilisation behaviour (automatic manual exploration and use of objects placed within the field of view and reach) has been interpreted as a deficit of frontal inhibitory function on parietal-led exploratory behaviour [START_REF] Lhermitte | Utilization behaviour' and its relation to lesions of the frontal lobes[END_REF] and might be considered a form of diaschisis.

Hyperconnection -Of the four classes of dysfunction, hodological hyperconnection is the most speculative; however, there is indirect evidence that connectivity increases are linked to perceptual symptoms in the auditory domain [START_REF] Hubl | Pathways that make voices: white matter changes in auditory hallucinations[END_REF] . Increased connectivity might lead to perceptual symptoms through, for example, enhanced feed-forward or feed-back processing or resonant activation through ephaptic (non-synaptic) transmission. The prototypical disorder in this class is synaesthesia. In the clinical domain, synaesthesia has traditionally referred to a percept in one sense modality evoking a parallel hallucinatory percept in a different sense modality, e.g. percepts of taste evoking visual percepts of colour (coloured taste synaesthesia). More recently, work in the cognitive sciences has lifted the requirement of synaesthetic experience to cross modalities through the study of, for example, coloured grapheme synaesthesia (visual percepts of text evoking visual percepts of colours). The induced synaesthetic experience may be perceived in the external world (projector synaesthesia) or inside the mind's eye (associator synaesthesia) [START_REF] Dixon | Not all synaesthetes are created equal: projector versus associator synaesthetes[END_REF] . Other parallel sensory experiences recognised clinically include reflex hallucinations (a sensory stimulus in one modality evokes a hallucination in another e.g. seeing a butterfly whenever the thumb is touched) and functional hallucinations (hallucinations are triggered in the same modality as the inducing stimulus e.g. hearing voices in parallel with the sound of running water). Both clinical symptoms would be classed as synaesthetic by contemporary cognitive science. We would argue the content of the parallel experience in such disorders is defined by the cortical specialisation of the area with increased connectivity. Thus, for example, synaesthesia, reflex or functional hallucinations that involve colour relate to the connectivity of colour-specialised cortex, while experiences involving spatial location (i.e. number forms -a variety of synaesthesia in which numbers and other ordinal lists [e.g. days of the week] are located at specific spatial locations [START_REF] Sagiv | What is the relationship between synaesthesia and visuo-spatial number forms?[END_REF] ) relate to the connectivity of spatial representations in the parietal lobe.

Pareidolia refers to images of objects or faces seen in random visual patterns (e.g. cloud formations or tree bark). The experience is an illusion in which the inducing visual stimulus is typically of simpler form than the illusory percept and can be considered the result of hyperconnection between visual areas. The Frégoli syndrome, in which unfamiliar people are perceived as familiar (typically as a person in disguise with malevolent intent), can be conceived as hyperconnection within visual emotional or memory networks. In PTSD flashbacks, patients relive stressful events, typically with multiple sensory modalities involved and a strong emotional component. The flashbacks might be considered the result of hodological hyper-connection between visual and emotional systems or topological hyper-function of visual memory systems.

The Perky effect originally referred to an ambiguity between visual imagery and visual perception such that normal subjects were unable to detect the presence of a real visual stimulus when asked to form a visual image of it. More recently the term has been used to describe the reduction of visual perceptual sensitivity during visual imagery [START_REF] Segal | The perky effect revisited: blocking of visual signals by imagery[END_REF] . The effect is likely to relate to connectivity between visual memory networks and occipital cortex. Eidetic imagery (sometimes referred to as photographic memory) [START_REF] Gray | The enigmatic eidetic image: a critical examination of methods, data, and theories[END_REF] is likely to relate to increased connectivity between these regions.

Visual phobias and visual objects appearing beautiful and comforting (kalopsia) or ugly and menacing (kakopsia) [START_REF] Critchley | The parietal lobes[END_REF] might be considered the result of hodological hyperconnection between visual emotional networks and occipital cortex or topological hyperfunction of visual specialisations within the amygdala.

Dazzle (painful brightness of visual stimuli) is thought to be a visual variant of a thalamic pain syndrome [START_REF] Cummings | A thalamic syndrome?[END_REF] and can be interpreted as hodological hyperconnection of the thalamus and visual cortex. The same mechanism may account for the intense photophobia of children with cortical visual impairment and the paradoxical worsening of their visual acuity (as measured by the evoked response) for lighter (higher luminance) compared to darker (lower luminance) visual stimuli [START_REF] Good | Sweep visual evoked potential grating acuity thresholds paradoxically improve in low-luminance conditions in children with cortical visual impairment[END_REF] .

Limitations

The classificatory scheme has a number of limitations and is perhaps better viewed as a structured review of symptoms from which to generate and test further experimental hypotheses than a definitive pathophysiological taxonomy or clinical aide memoire. (i) It does not easily accommodate disorders involving combinations of networks or combinations of topological and hodological dysfunction (the latter marked with an asterisk in the Table ).

Typical lesions have both hodological and topological effects so that pure disorders of one or other class may be largely theoretical. (ii) The classification deconstructs traditional clinical syndromes by placing their component parts in different locations, losing associations between symptoms related to the anatomical proximity of the cortical regions underlying them. For example, in migraine aura, the same region of cortical spreading depression may result in a hyper-functional symptom at its margins (teichopsia) and a deficit in its centre (a scotoma) but each is placed in a different location in the table. Similarly the Balint syndrome (simultanagnosia, optic ataxia and oculomotor apraxia in association with bi-parietal lesions [START_REF] Rizzo | Psychoanatomical substrates of Balint's syndrome[END_REF] ) is spread across the table. (iii) Hyper-and hypo-function are convenient concepts to map on the statistical signal increases and decreases of imaging and tractography data. However, the neurophysiological meaning of such changes is unspecified. An increase in activity as measured by fMRI may have a number of neurophysiological causes (e.g. a change in excitatory input or inhibitory processing, a shift from burst to tonic mode firing [see [START_REF]The hodology of hallucinations[END_REF] ], neuromodulatory inputs etc.). Similarly DT-tractography dissects virtual connections between brain regions and different causes of micro-structural white-matter change may lead to the same change in connectivity. By focussing on increases or decreases in imaging signal, the classification is likely to miss key pathophysiological distinctions. (iv) The classification does not easily accommodate interactions between pre-disposing factors and changes occurring at the time of a symptom. The Charles Bonnet Syndrome, for example, appears both as a hodological disconnection (because of the predisposing factor of loss of visual inputs from the eye) and as topological hyperfunction (because of the localised increases in fMRI signal during hallucinations).

Clinical implications

Despite its limitations, the classificatory scheme and its underlying assumptions are of relevance to current clinical practice in a number of areas. First it suggests that, in the same way that seizure semiology is used in the context of epilepsy, the contents of hallucinations and illusions can be used as localising signs. By indicating increased activity in a specific functionally specialised cortical area, different types of visual hallucinations (e.g. a face, colour or geometrical pattern) point to specific cortical locations, although not the location of the pathology causing the activity increase. Second, the classification highlights the fact that many symptoms held traditionally as distinct, appear closely related, if not indistinguishable, when considered from the perspective of their underlying mechanism. For example, synaesthesia, functional hallucinations and reflex hallucinations are all likely to relate to the same underlying pathophysiology. Third it predicts disorders which are, as yet, undescribed, or at least not recognised as distinct clinical entities. For example, disorders classified here as hyperfunction of spatial co-ordinate frames within the parietal lobe (polyopia, visual perseveration, palinopsia etc.) do not have obvious deficit or disconnection counterparts. It is conceivable that their natural counterparts are, in fact, spatial neglect, simultanagnosia, oculomotor apraxia, classified here are as related to visual attention, as all are linked to the parietal lobe and dorsal stream. Finally, the classification highlights a number of areas which pose a challenge to visual neuroscience. For example, lilliputian hallucinations are not easily accommodated within current models of object perception, while polyopia and extracampine hallucinations are not easily accommodated within current models of spatial vision. In pointing to deficiencies in such models, visual perceptual disorders advance our understanding of normal visual function.

Conclusions

The last decade has seen advances in white-matter and functional imaging (including EEG imaging) that allow new classes of clinically relevant questions to be answered (see [START_REF] Catani | On 'the study of the nervous system and behaviour[END_REF] ). We envisage that such imaging techniques, applied to single case and group studies, will be used to test and expand the classification presented here in the future (see [START_REF] Catani | The rises and falls of disconnection syndromes[END_REF][START_REF] Catani | On 'the study of the nervous system and behaviour[END_REF] ). For now, we hope that our broad overview of visual perceptual disorders will provide a fresh perspective from which to challenge and refine the neuroscience of visual perception and visual perceptual dysfunction. (see [START_REF] Pribram | Visual hallucinations and the Charles Bonnet Syndrome[END_REF] ). The predominant pathway from the eye to V1 relays through the lateral geniculate nucleus (LGN), although a clinically important pathway passes directly from the LGN or pulvinar to motion specialised cortex on the lateral occipital surface [START_REF] Zeki | The Riddoch syndrome: insights into the neurobiology of conscious vision[END_REF] . The detailed anatomy of short association fibres within and between occipital regions and from the occipital lobe to the pulvinar has yet to be characterised in man. Chains of U-shaped fibres and parallel long association pathways passing to the parietal lobe form a cortical network (dorsal stream) [START_REF] Catani | Beyond localization: from hodology to function[END_REF] thought to underlie spatial aspects of vision while equivalent pathways to the temporal lobe form a cortical network (ventral stream) thought to underlie object vision, colours, faces and visual language. The inferior longitudinal fasciculus (ILF) carries reciprocal connections between the occipital and medial / lateral anterior temporal lobes related to visual emotion and visual memory [START_REF] Catani | Occipito-temporal connections in the human brain[END_REF] . The inferior fronto-occipital fasciculus (IFOF) connects territories in the occipital lobe and ventral frontal lobe [START_REF] Catani | Beyond localization: from hodology to function[END_REF] . Connections between the visual parietal lobe and frontal cortex are carried in the superior longitudinal fasciculus (SLF) while those between occipital and limbic cortex are carried in the cingulum and ILF. The SLF and IFOF are likely to relate to functions such as visual attention, visual working memory, visual praxis, visualvestibular function and eye movement control. Cortico-cortical visual connections between hemispheres pass predominantly through the splenium of the corpus callosum.

Figure legend

  transient neurological shock. Connections between areas can be decreased (disconnection) or increased (hyperconnection) in hodological disorders. Deficit and disconnection disorders of visual perception are found in stroke, intracranial compressing masses, neurosurgical procedures, demyelinating disorders, neurodevelopmental conditions (e.g. autism), neurodegenerative disease, schizophrenia and depression. Hyperfunction and hyperconnection disorders of visual perception are found in ophthalmic disease, neurodegenerative disease, Parkinson's disease, psychosis, epilepsy and as normal phenomena in relation to sleep or in the context of psychedelic drugs, alcohol or medication side-effects.

Figure 1

 1 Figure 1 The anatomy of visual networks. The upper left panel shows the major associative tracts with connection to the occipital lobe in the human brain. SLF = Superior longitudinal fasciculus; ILF = Interior longitudinal fasciculus; IFOF = Inferior longitudinal fasciculus. The Brodmann areas corresponding to the cortical projections of each fibre tract are given in the lower left panel. The right panels show Brodmann areas colour coded on lateral (upper) and medial (lower) views of the brain. Images modified from 77 . The primary visual cortex (V1, area 17) is located on the polar and medial surface of the occipital lobe. V2 is adjacent to V1 and is divided on the medial occipital surface into a dorsal portion and a ventral portion. V3 is adjacent to V2 and is similarly divided. The occipital lobe, lateral and ventral occipitotemporal cortex, limbic cortex and the parietal lobe contain regions specialised for different visual perceptual attributes including: familiar faces, colours, extended landscapes, objects, text and visual textures (ventral occipito-temporal cortex) and visual motion, body parts, face features, intermediate-level object processing and eye and mouth movements (posterior-lateral cortex)


Silvia Guerrera

MD Maria Laura Stromillo

MD Andrea Mignarri

MD Marco Battaglini

Silvia Marino

MD Carol Di Perri

MD Antonio Federico

MD Maria Teresa Dotti

MD Nicola De Stefano

PhD S Marino#

Clinical Relevance of Brain Volume Changes in Patients with Cerebrotendinous Xanthomatosis

Keywords: cerebrotendinous xanthomatosis, brain, magnetic resonance, therapy, atrophy

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

Introduction

Cerebrotendinous xanthomatosis (CTX) is a rare autosomal recessive disease characterised by the abnormal deposition of cholestanol and cholesterol in multiple tissues, including the central nervous system (CNS) 1;2 . In CTX patients, the systemic defective activity of the mitochondrial enzyme sterol 27-hydroxylase results in a defective bile acid synthesis (with increases in cholic acid and decreases in chenodeoxycholic acid) and abnormally high levels of cholestanol in plasma, urine, feces and other tissues [START_REF] Dotti | Mitochondrial enzyme deficiency in cerebrotendinous xanthomatosis[END_REF][START_REF] Salen | Biochemical Abnormalities in Cerebrotendinous Xanthomatosis[END_REF][START_REF] Stefano | Magnetic resonance imaging and spectroscopic changes in brains of patients with cerebrotendinous xanthomatosis[END_REF] . This abnormal metabolism is considered responsible for the clinical manifestations, which may involve different organ systems but is usually dominated by the classical triad of the disease: juvenile cataracts, tendon xanthomas and progressive neurological impairment 6;7 . These clinical manifestations are crucial to address the diagnosis, which needs to be confirmed biochemically (by the increased serum level of cholestanol) [START_REF] Wolthers | Diagnosis of cerebrotendinous xanthomatosis (CTX) and effect of chenodeoxycholic acid therapy by analisis of urine using capillary gas chromatography[END_REF] and genetically (by molecular genetic defects in the sterol 27-hydroxylase gene) 6;7 . An early diagnosis is imperative in these patients, as the pharmacological treatment with chenodeoxycholic acid (or with a combination of chenodeoxycholic acid and HMG-CoA reductase inhibitors) has been shown to slow or even reverse the progression of the disease [START_REF] Berginer | Long-term treatment of cerebrotendinous xanthomatosis with chenodeoxycholic acid[END_REF] .

Conventional magnetic resonance imaging (MRI) can be helpful in CTX patients with incomplete or peculiar clinical manifestations as the presence of the bilateral abnormal signal of the dentate nuclei and their surrounding white matter is considered a characteristic feature of the disease 5;10;11 . Additional MRI findings are white matter abnormalities as well as cerebral and cerebellar atrophy [START_REF] Berginer | Magnetic resonance imaging in cerebrotendinous xanthomatosis: a prospective clinical and neuroradiological study[END_REF] . Each of these MR features could be accurately assessed and quantified by using modern neuro-imaging quantitative tools [START_REF] Smith | Advances in functional and structural MR image analysis and implementation as FSL[END_REF] , which allow computed volume estimation of the entire brain as well as specific brain tissues and regions [START_REF] Smith | Accurate, robust and automated longitudinal and crosssectional brain change analysis[END_REF] . These measures have been used recently to demonstrate that focal and diffuse abnormalities can be found in the brain of patients with several neurological disorders since the very early clinical stages have clinical relevance and can be useful for monitoring patients' clinical evolution [START_REF] Fox | Serial magnetic resonance imaging of cerebral atrophy in preclinical Alzheimer's disease [letter][END_REF][START_REF] Dotti | Cortical damage in brains of patients with adult-form of myotonic dystrophy type1 and no or minimal MRI abnormalities[END_REF][START_REF] Stromillo | Structural and metabolic brain abnormalities in preclinical cerebral autosomal dominant arteriopathy with subcortical infarcts and leucoencephalopathy[END_REF][START_REF] Mitsumoto | Quantitative objective markers for upper and lower motor neuron dysfunction in ALS[END_REF][START_REF] Filippi | In vivo tissue characterization of multiple sclerosis and other white matter diseases using magnetic resonance based techniques[END_REF] .

All the points mentioned above are potentially important in a chronic and severe, albeit treatable, disease such as CTX. Thus, we performed a volumetric MRI analysis in a relatively large CTX population with the aim to quantify the extent of brain damage in different brain regions and tissues, to estimate, in a small subgroup of these patients, their longitudinal changes and to assess whether these quantitative MR measures might be related to patients' clinical status.

Methods

Study Population

Twenty-four CTX patients (13 women and 11 men, mean age 39.7 years, range 15-62 years) from eighteen Italian families and 22 demographically matched healthy controls (HC) (13 women and 9 men, mean age 39.2 years, range 22-56 years) were studied. All patients met the clinical criteria for CTX, had a blood level of cholestanol greater than 1 mg/dL and showed DNA point mutations or deletions in the sterol 27-hydroxylase gene. At study entry, 18 CTX patients had been treated with chenodeoxycholic acid (750 mg/day) for several years, while 5 CTX patients were not taking any medication at the time of the MRI examination. These 5 CTX patients started therapy with chenodeoxycholic acid (750 mg/day) at the time of the MRI exam and underwent a second MRI exam after a median follow-up time of 2 years (range 1.9-3.2 years), For each patient, neurological evaluation was performed within 24 hours of the MR examination by an experienced observer (MTD) who was kept blinded to the MRI results. This clinical evaluation included the full neurological examination, the assessment of clinical disability by using the modified Rankin Scale (RS) [START_REF] Bonita | Modification of Rankin Scale: Recovery of motor function after stroke[END_REF] , the assessment of the cerebellar function by using the functional system score (CB-FSS) [START_REF] Kurtzke | Rating neurologic impairment in multiple sclerosis: an expanded disability status scale (EDSS)[END_REF] , and the evaluation of the overall cognitive performance by the Mini-Mental-Status Examination (MMSE) [START_REF] Cockrell | Mini-Mental State Examination (MMSE)[END_REF] . The latter could not be assessed in 1 patient. The study was approved by the Ethics Committee of the Faculty of Medicine of the University of Siena and informed written consent was obtained from all participating subjects.

MRI examination

All subjects were examined using an identical MR protocol. Acquisitions of brain were obtained in a single session using a Philips Gyroscan operating at 1.5 T (Philips Medical Systems, Best, The Netherlands). A sagittal survey image was used to identify the anterior commissure (AC) and posterior commissure (PC). A dual-echo, turbo spin-echo sequence (TR/TE1/TE2= 2075/30/90 ms, 256x256 matrix, 1 signal average, 250mm field of view, 50 contiguous 3mm slices) yielding proton density (PD) weighted and T 2 -weighted (T 2 W) images was acquired in the transverse plane parallel to the line connecting the AC and PC. Fluid-attenuated inversion recovery (FLAIR) images (TR= 9000 ms; TE= 150 ms; IT= 2725, 50 contiguous 3mm slices) were acquired in the same direction.

Subsequently, T 1 -weighted (T 1 W) 3D gradient echo images (TR/TE=35 ms/10, 256x256 matrix, 1 signal average, 250mm field of view, 50 contiguous 3mm slices) were acquired. These were oriented to exactly match the PD/T 2 and FLAIR acquisitions. No major hardware upgrades were carried out on the scanner during the time of the study, and monthly quality assurance sessions confirmed the stability of measurements throughout the duration of the study.

MRI data analysis

Global and Regional Brain Volume measurements. Brain parenchyma volumes were measured on T 1 -W images of CTX patients and HC by using the cross-sectional version of the SIENA software, SIENAX (part of FSL 4.0 -http://www.fmrib.ox.ac.uk/fsl/). SIENAX allows global measures of normalized brain volume (NBV) as well as selective measurement of normalized cortical volumes (NCV) [START_REF] Smith | Accurate, robust and automated longitudinal and crosssectional brain change analysis[END_REF] . In the five CTX patients who underwent the second MR examination, longitudinal (two time-points) percentage brain volume change (PBVC) was estimated on T1-weighted images by using SIENA [START_REF] Smith | Advances in functional and structural MR image analysis and implementation as FSL[END_REF] . Changes in PBVC were annualised (PBVC/y) to account for differences in the length of follow-up between the two scans across the 5 patients.

For volume measurement of specific cortical regions, predefined standard space masks can be used to select the cortical gray matter of these regions (see structural atlas in http://www.fmrib.ox.ac.uk/fsl/) and estimate volumes normalized for subject's head size. We considered here normalized volumes of frontal (NFV), temporal (NTV), parietal (NPV, which included the insula) and occipital (NOV) lobes of both hemispheres. Furthermore, with an identical approach, we estimated the normalized volume of the cerebellum (NCbV). In this case, when necessary, the cerebellum mask of the subject was manually edited by an expert user (SG).

Dentate nuclei and surrounding white matter measurement. Volume quantification of the abnormal signal detected in the dentate nuclei and surrounding white matter (DV) was performed in each subject by a single observer (SG) employing a segmentation technique based on user-supervised local thresholding. For the DLV classification, lesion borders were determined primarily on PD images, but information from T1-W, T2-W and FLAIR images was also considered as the software used (Jim 4.0, Xinapse System, Leicester, UK) offered the ability to toggle between the PD, T2-W, FLAIR and T1-W images, providing the operator with convenient access to the information in all data sets while defining lesions. The value of total DV was calculated by multiplying lesion area by slice thickness.

Statistical analysis

The MR measures of the CTX population were compared to those of the HC group by using the non-parametric Mann-Whitney rank sum test. Although there were no differences in age between the groups, age was included as covariate in the measurement of group differences.

Relationships of MR metrics with clinical and demographic indices were assessed using the nonparametric Spearman rank order correlation. Values were considered significant at the 0.05 level.

The SYSTAT software version 9 running on Windows (copyright SPSS Inc.1988) was used to performed statistical calculations.

Results

Clinical features

The group of CTX patients had wide ranges of overall disability (RS: median 2, range 0-5), cerebellar functional status (CB-FSS: median 1.0, range 0-3) and overall cognitive performance (MMSE: median 22, range 15-28). As expected in a slowly progressive, chronic disorder such as CTX, measures of RS and MMSE were closely correlated with age (r= 0.73 for RS and -0.68 for MMSE, p<0.01 for both).

Global and Regional Brain Volume measurements

As expected, all CTX patients showed bilateral abnormalities in the dentate nuclei and surrounding white matter. The DV values ranged 0.16-9.85 cm [START_REF] Dotti | Mitochondrial enzyme deficiency in cerebrotendinous xanthomatosis[END_REF] .

Measurements of NBV were significantly lower in CTX patients than HC (1510±137 cm 3 in CTX, 1626±56 cm 3 in HC, p<0.001, see Figure 1). Also, NCbV values were significantly lower in CTX patients than in HC (142±25 cm 3 in CTX, 161±15 cm 3 in HC, p <0.005, see Figure 1). Brain tissue specific measures showed that NCV were significantly lower in CTX patients than HC (577±101 cm 3 in CTX, 668±33 cm 3 in HC, p<0.0001, see Figure 2). In contrast, differences in NWMV between the two groups were not significant (820±68 cm 3 in CTX, 850±44 cm 3 in HC, p=0.1).

Finally, the regional analysis revealed that the decrease in neocortical volume was widespread in CTX patients, being significantly lower in the CTX group than in the HC group in each of the measured cortical regions (NFV: 203±38 cm 3 in CTX, 235±13 cm 3 in HC; NPV: 155±29 cm 3 in CTX, 183±12 cm 3 in HC; NTV: 142±25 cm 3 in CTX, 159±7 cm 3 in HC; NOV: 78±13 cm 3 in CTX, 89±6 cm 3 in HC; <0.01 for all regions, see Figure 2). Interestingly, a single-subject analysis of the different brain regions showed that, in CTX patients, values were at least two standard deviations below the mean of the HC in 20/24 of the NPV, 15/24 of the NFV and 14/24 of both NTV and NOV. In contrast, values were at least two standard deviations below the HC mean only in 8/24 of the NCbV.

Correlations between age, clinical scores and MR measures.

In CTX patients, increasing age was closely correlated with decreases in NBV (r= -0.69, p <0.0001) and even more closely with decrease in NCV (r= -0.90, p <0.0001 (Figure 3). A similarly close correlation with age was found in each cortical region measurement. Age was also correlated, but to a lesser extent, with measures of NCbV (r= -0.52, p <0.01, see Figure 3).

A close negative correlation was also found between RS scores and NBV (r= -0.62, p<0.01), NCV (r= -0.65, p<0.001) and between CB-FSS scores and NCbV values (r= -0.58, p <0.01).There was also a close relationship between decreasing values of MMSE and decreases in both NBV (r= 0.50, p<0.01), NCV (r= 0.60, p<0.01). As expected, there was no correlation between NCbV and MMSE (r= 0.25, p> 0.1).

Finally, no significant correlations were found between measures of DV and age, clinical scores and MRI measures, with the exception of a moderate negative correlation with NCbV values, (r= -0.44, p<0.03).

Longitudinal analysis

As mentioned before, in the 5 CTX patients who began treatment with chenodeoxycholic acid at the time of the first MRI scan, conventional MRI and clinical evaluation were performed after about 2 years. All 5 patients were clinically stable during the follow-up period, maintaining the same baseline RS values. In contrast, there was an overall decrease of brain volume during the follow-up (mean±sd PVBC/y = -1.1±0.2%). This value was almost twice lower in the youngest patient than in the oldest patient (-0.68% versus -1.25%).

Discussion

This is the first study attempting a quantification of global and regional brain tissues in CTX patients by means of a computed quantitative analysis of MR images. In this relatively large patient population, we found a diffuse decrease in total brain volumes and, at a tissue type analysis, this decrease was very pronounced in the cortical grey matter and not significant in the white matter.

These results add further evidence to the hypothesis that CTX is a primary neuronal disorder where the enzymatic defect causes an abnormal accumulation of metabolites in the brain leading to a progressive impairment of the neuron metabolic apparatus and the consequent irreversible neuroaxonopathy. 23;24 In most of the CTX patients studied here, the grey matter atrophy could be documented in all neocortical regions, with particular preference for the fronto-parietal cortex. In addition, these cortical volume decreases correlated very closely with both increasing age and measures of overall disability and cognition. This is in line with the nearly constant presence of the slowly progressive sensory-motor impairment and behavioural disturbances in the clinical feature of the CTX patients and provides clear evidence for the clinical relevance of the cortical pathology as detected by quantitative MR metrics. Also, these results add to previous findings similarly reporting close correlations between clinical disability and MR measures of brain tissue damage in smaller populations of CTX patients 5;25 , stressing the importance of tissue specific quantitative MR metrics as measures of disease outcome.

In the population studied here, the quantitative analysis of cerebellar volumes also did correlate closely with specific measures of cerebellar impairment. However, it is interesting to note that, at a single-subject evaluation, a significant decrease in cerebellar volume was found only in 8/24 CTX patients and correlated only moderately with the patients' clinical status. This is particularly intriguing as common neurological symptoms and signs do include cerebellar dysfunction and cerebellar atrophy are generally considered as one of the most common findings in previous qualitative MR studies of patients with CTX. 5;11;12;26 It seems, instead, that in CTX patients significant cerebellar pathology occurs at later disease stages than generally thought and, most likely, later than in fronto-parietal cortical regions. Further studies should be carried out to confirm this important result. These might take advantage of high resolution, isotropic MR images, which were not used in the present study.

In a small subgroup of 5 CTX patients who started the classical CTX treatment at baseline, conventional MRI and clinical evaluation were repeated after about 2 years. These patients were all clinically stable during the follow-up period, but showed a sustained decrease in the overall brain volume (PVBC/y of about -1%). When compared to recent literature data using the same method for the longitudinal assessment of brain volume changes, this value is about 4 times higher than the PBVC/y of normal controls [START_REF] Giorgio | Brain atrophy assessment in multiple sclerosis: importance and limitations[END_REF] and about twice that of patients with CADASIL or multiple sclerosis 27;28 . The small number of patients studied and the lack of data on an untreated patient group do not allow for general conclusions on this important issue. However, data reported here suggest that despite therapy initiation the complex pathological process leading to atrophy seems to continue. This is in agreement with the notion that the pathological process underlying this rare neurological disease is severe and its reversibility is questionable [START_REF] De La Fuente | Cerebrotendinous xanthomatosis: neuropathological findings[END_REF] . This reinforces the idea that the classical therapy of this disorder is effective only if it is started very early, before the abnormal accumulation of cholesterol and cholestanol leads to irreversible tissue damage. The presence, in our small patient group, of very different PBVC/y values in the youngest (patient 9: -0.68%) and in the oldest (Patient 11: -1.25%) seems to be in line with this.

To conclude, MR-based measurements of brain volume changes provided in vivo estimates of CTX brain damage that closely correlated with the patients' clinical status. It also provided further in vivo evidence that the involvement of the cortical grey matter, rather than that of the white matter, is a clinical relevant aspect of the pathology of this rare metabolic disorder. MR-based, quantitative metrics allow accurate and objective estimation in lesioned and normal appearing brain. They should be used together, in multiparametric, MR-based long-term studies to monitor disease evolution and improve our understanding of the pathophysiology of rare and complex neurological disorders such as CTX.

Figure 1 :

 1 Figure 1: Box plots comparing MR measurements of normalized brain volume (NBV), normalized cortical volume (NCV) and normalized cerebellar volume (NCbV) in 22 healthy controls (HC) and 24 patients with cerebrotendinous xanthomatosis (CTX). All measurements of CTX patients are significantly (NBV: p<0.001, NCV: p<0.0001 and NCbV: p<0.005) lower than those of HC.

Figure 2 :

 2 Figure 2: Box plots comparing MR measurements of normalized volumes of the frontal (NFV) parietal (NPV) temporal (NTV) and occipital (NOV) lobes in 22 healthy controls (HC) and 24 patients with cerebrotendinous xanthomatosis (CTX). All measurements of CTX patients are significantly lower than those of HC (p<0.01 for all regions).

Figure 3 .

 3 Figure 3. Scatterplots of the correlation of normalized cortical volume (NCV) with age (r= -0.90, p<0.0001), Mini-Mental-Status Examination (MMSE: r= -0.60, p<0 0.1) and modified Rankin Scale (RS: r= -0.65, p <0.001). Values are relative to the whole group of CTX patients (MMSE value was not available for 1 patient).


Alfredo Eisinberg

Giuseppe Fedele

Domenico Frascino

Kepler's equation and limit cycles in a class of PWM feedback control systems

Keywords: Pulse width modulation, describing function, Kapteyn series, Bessel functions

come

Introduction

Pulse width modulated (PWM) switching converters are widely used in industry and in consumer products and an extensive literature exists on this topic (see [START_REF] Gelig | Stability and Oscillations of Nonlinear Pulse-Modulated Systems[END_REF] and the references therein). PWM systems constitute a class of systems with discontinuous controls in which the control actions are determined on the basis of the error signal: in each sampling period, characteristic of the modulator, the control signal is enabled only for a fraction of time proportional to the error magnitude. The main advantages of this particular control mode are related to the power amplifier efficiency, since it essentially operates as an on-off device, and to the low cost for the engineering realization. Its relay-type electronic switching leads to amplifiers with reduced size, weight and power dissipation [START_REF] Tsypkin | Relay Control Systems[END_REF]. However, the analysis of the PWM feedback control system can be extremely difficult, since the modulating device creates an effect equivalent to a non-linear non-instantaneous block [START_REF] Cava | Stability analysis of PWM feedback control system with PID regulators[END_REF]. The discontinuity control analysis can be approached by working with an approximate model obtained by the state-space averaging method which leads to a greater understanding of the dynamic performance of PWM converters although it is valid only for high frequency switching [START_REF] Middlebrook | A general unified approach to modeling switching-converter power stages[END_REF]5]. In the relevant literature, local and global stability problems have been considered for PWM feedback control systems with different regulators [START_REF] Hou | Stability analysis of pulse-width-modulated feedback systems[END_REF]. On the other hand, the describing function method has widely been used to determine the limit cycle and the dynamical behavior for the nonlinear systems [START_REF] Gelb | Multiple-input describing functions and nonlinear system design[END_REF][START_REF] Taylor | Describing Functions[END_REF][START_REF] Kienitz | On the Implementation of the Eigenvalue Method for Limit Cycle Determination in Nonlinear Systems[END_REF][START_REF] Somieski | An Eigenvalue Method for Calculation of Stability and Limit Cycles in Nonlinear Systems[END_REF][START_REF] Sun | Existence and uniqueness of limit cycle for a class of nonlinear discrete-time systems[END_REF][START_REF] Zhang | Linear estimate of the number of limit cycles for a class of non-linear systems[END_REF]. As it is well known, finding conditions to predict and to eliminate limit cycles, the so-called ripple instability, for these control systems, is an important task. Many authors have detailed various approaches for the exact determination of limit cycles in PWM systems [START_REF] Delfeld | Analysis of pulse-width-modulated control systems[END_REF][START_REF] Peterchev | Quantization resolution and limit cycling in digitally controlled PWM converters[END_REF][START_REF] Abdelnour | Application of describing function in the transient response analysis of a three term fuzzy controller[END_REF][START_REF] Heyns | Describing function based analysis of a nonlinear hydraulic transmission line[END_REF]. The use of describing function theory analysis of PWM systems without a reference input is well documented in [START_REF] Gelb | Multiple-input describing functions and nonlinear system design[END_REF]. The existing approaches substantially use the same logic: the shape of limit cycle mode, which defines the form of the nonlinearity output, is postulated; this nonlinearity output signal is processed through the linear part of the system to the nonlinearity input, finally, conditions which will produce the postulated output of the nonlinearity are sought. The system is then analyzed as a linear block interconnected with a nonlinear block described by its amplitude dependent transfer function. The resulting equations, for the amplitude and frequency, of presumed limit cycles are solved directly by a graphical procedure through the intersections of the frequency-response of the linear device and the critical locus (i.e, the inverse of the describing function) of the nonlinear ones in the Nyquist plane, otherwise by solving the nonlinear equations or a parameter optimization problem [START_REF] Gelb | Multiple-input describing functions and nonlinear system design[END_REF][START_REF] Soto | Non-linear oscillations in nonperiodic sampling systems[END_REF][START_REF] Williamson | Describing function analysis and oscillations in non-linear networks[END_REF]. The advantage of the describing function method is that it can be viewed as a kind of harmonic balance method [5]. On the other hand, the fundamental limitation is that the form of the signal in input to the nonlinearity must be guessed in advance. Another difficulty is related to the possibility to have multiple solutions since the formulation of the problem, using the describing function, leads to a simultaneous set of nonlinear algebraic relations to be solved. Moreover, since the method is approximate, another limitation is that it is only able to indicate where probably a limit cycle can occur. Finally, the plant should be low-pass to attenuate higher harmonics that is to say the analysis with this method has to be cautious. Several types of describing functions have been derived. For instance, the case of multiple input nonlinearities was discussed in [START_REF] Gelb | Multiple-input describing functions and nonlinear system design[END_REF], while the case of random inputs describing functions can be found in [START_REF] Taylor | Describing Functions[END_REF][START_REF] Lim | Quasi-linear analysis of oscillating servo-systems with random inputs[END_REF].

The aim and the organization of the paper

The aim of this paper is twofold. First, an interesting relation between the switching instants in the PWM modulator and the eccentric anomaly of the planet orbit around the sun is pointed out. Second, a sufficient condition to avoid limit cycles predicted by the dual-input describing function in a class of PWM feedback control systems is given. The method used herein reconsiders the dual-input describing function approach on the basis of an exact spectral analysis of the PWM output, and extends the results in [START_REF] Gelb | Multiple-input describing functions and nonlinear system design[END_REF] introducing a constant reference input. Moreover, in order to avoid probable limit cycles of period multiple of that characteristic of the modulator, a simple stability criterion is proposed. Finally, some simulation results, which confirm the validity of the proposed criterion, are shown. In particular, for a first order plant it will be formally proved that the describing function stability criterion ensures the local stability of an equilibrium point, while for a second order plant a Monte-Carlo simulation has confirmed that the selection of the modulator parameters, according to the proposed criterion, gives an effective method to avoid limit cycles and to ensure the local stability. The idea behind the proposed approach is that the implicit equation of the unknown PWM switching instants, when the input to the modulator is a biased sinusoidal signal due to the presence of a limit cycle, can be related to the Kepler's equation governing the trajectory of a planet around the sun [START_REF] Colwell | Solving Kepler's equation[END_REF]. The solution of the Kepler's problem allows to obtain the switching instants of the PWM output in terms of Kapteyn series as well as its Fourier coefficients and then to find an explicit expression of the describing function. The paper is organized as follows. Section 2 describes the main aspects of the Kepler's problem with a solution in terms of Kapteyn series; Sections 3 and 4 introduce the model of the PWM closed-loop system and an analysis for the local stability, respectively. In Section 5 an overview of dual-input describing function and main results are given. In order to corroborate the theoretical analysis, simulation results are reported in Section 6. Finally Section 7 is devoted to conclusions. Proofs of some results are reported in the Appendix.

Kepler's equation

Many problems in Celestial Mechanics require a solution to Kepler's equation. The Kepler transcendental equation

E -ǫ sin(E) = M, (1)
which links the eccentric anomaly of elliptic motion E, mean anomaly M and eccentricity ǫ, is of critical importance in gravitational mechanics. The basic physical meaning of this equation is better explained by Fig. 1 in which is depicted an ellipse, with eccentricity ǫ, that is the orbit of a body moving about the stationary gravitating center placed in focus of the ellipse S. Denote by C and A the center and pericenter of the ellipse, respectively. Construct also a circle with its center at point C and with a radius equal to the major semiaxis of the ellipse. At some time let the position of the rotating body be determined by point P . From P we drop a perpendicular to the major axis of the ellipse and denote the foot of the perpendicular by letter R. Extend this perpendicular to intersect the circle at point Q. Then the angle ∠ACQ is just the eccentric anomaly E. Suppose that the planet P , having passed through perihelion A, is at position P after elapsed time t, it is possible to express the polar coordinates of P , (r, v), relative to the sun in terms of t. Kepler's equation relates E to time t by means of a quantity

M = 2πt Ξ , ( 2
)
where Ξ is the time required for the planet to complete one trip in its orbit around the sun. The quantity M represents the average angular speed of the radius vector SP . Classical formulas to find r and v are

r = a (1 -ǫ cos(E)) , (3)
and

tan v 2 = 1 + ǫ 1 -ǫ tan E 2 . (4)
Therefore, if Ξ and M are known, it is possible to solve the Kepler's equation for E and to determine position (r, v) at time t by using eqs. ( 3) and (4). Although there exists a unique solution of the Kepler's problem, there are many methods to describe or approximate it [START_REF] Boyd | Rootfinding for a transcendental equation without a first guess: Polynomialization of Kepler's equation through Chebyshev polynomial expansion of the sine[END_REF][START_REF] Dubinov | Explicit Solution of the Kepler Equation[END_REF][START_REF] Feinstein | Dynamic discretization method for solving Kepler's equation[END_REF][START_REF] Mortari | Solving Kepler's Equation using Bézier curves[END_REF]. The history of the search for exact and approximate solutions of the Kepler's equation, along with the analytical and numerical methods used for this purpose, is well covered in [START_REF] Colwell | Solving Kepler's equation[END_REF]. A solution in terms of Bessel functions of the first kind:

Jn(x) = 1 π π 0 cos(x sin(θ) -nθ)dθ, (5)
was investigated by Kapteyn in his memoir in 1893 [START_REF] Kapteyn | Recherches sur les fonctions de Fourier-Bessel[END_REF], where he studied the possibility of representing functions by series of the form

f (x) = ∞ n=0 hn(f )Jn(nx), (6)
where hn(f ) are coefficients depending on the function f (x). Series in Eq. ( 6) are now called Kapteyn series.

Let E = g(M ) be the solution of the Kepler's equation and express the quantity g(M ) -M in the Fourier sine-series

g(M ) -M = ǫ sin(E) = ∞ n=1 An sin(nM ), (7)
where

An = 2 π π 0 [g(M ) -M ] sin(nM )dM = - 2 nπ π 0 [g(M ) -M ] d (cos(nM )) = - 2 nπ {[g(M ) -M ] cos(nM )} π 0 + 2 nπ π 0 cos(nM ) g ′ (M ) -1 dM (8) = 2 nπ π 0 cos(nM )g ′ (M )dM = 2 nπ π 0 cos(nM )d(g(M )). Since M = E -ǫ sin(E) = g(M ) -ǫ sin(g(M )), then An = 2 nπ 2π 0 cos [ng(M ) -nǫ sin(g(M ))] d(g(M )) = 2 n 1 π π 0 cos (nE -nǫ sin(E)) dE (9) = 2 n Jn(nǫ).
Hence the solution of (1) in terms of Kapteyn series gives

E = M + ∞ n=1 2 n Jn(nǫ) sin(nM ). ( 10
)
The series on the right side rapidly converges when ǫ < 1, and it is still convergent when ǫ = 1 [START_REF] Watson | A treatise on the theory of the Bessel function[END_REF].

System modeling

The system which will be analyzed is depicted in Fig. 2. The plant is assumed to be linear, time-invariant with low-pass characteristic. The modulator output consists of a piecewise constant function in two subintervals, [kT, kT + τ k ) and (kT + τ k , kT + T ), where τ k ∈ [0, T ] represents the switching instant in the interval [kT, kT + T ) and T is the constant sampling period. Fig. 3 shows the unipolar lead-type modulator output which is characterized by values equal to +M or -M in the first subinterval (according to the sign of the modulator input σ(t) valued for t = kT ) and equal to zero in the second subinterval:

u(t) = M sgn[σ(kT )], kT ≤ t ≤ kT + τ k , 0, kT + τ k < t < kT + T. (11) 6
- Ep kT kT + T kT + τ k σ(kT ) - 6 M u(t)
Fig. 3 Lead-type PWM modulator.

More general modulation strategies are described in [START_REF] Skoog | Generalized pulse-modulated feedback systems: Norms, gains, lipschitz constants, and stability[END_REF]. As far as the evaluation of τ k is concerned, its value is uniquely determined by a logic comparison between the input σ(t) and a sawtooth carrier c(t), of period T , which has the following expression:

c(t) = sgn[σ(kT )] Ep T (t -kT ), kT ≤ t ≤ kT + T, (12)
where Ep > 0 is the sawtooth amplitude. For the natural sampling modulator, the relationship enabling τ k to be determined is:

σ(kT + τ k ) = sgn[σ(kT )]Ep τ k T ( 13
)
where it is evident that τ k is implicitly determined by the input behavior in the interval [kT, kT + τ k ), and it is assumed that there is a unique solution τ k (see Remark 1 for details about this assumption).

Local stability and assumptions

The equations describing the dynamical behavior of the control system are:

   ẋ(t) = Ax(t) + bu(t), y(t) = c T x(t), σ(t) = r -y(t) (14)
where x ∈ R n is the state, y ∈ R and u ∈ R (described by Eq. ( 11)) are respectively the output and the input of the plant, r ∈ R is the constant reference input, A ∈ R n×n , b, c ∈ R n×1 . Moreover it will be assumed that A is a full-rank matrix.

Remark 1 It has been assumed that Eq. ( 13) has a unique solution τ k belonging to [0, T ]. Obviously, this hypothesis is not sustainable from a mathematical point of view, unless a set of input functions, which guarantees this hypothesis, is considered. However, such assumption does not represent a limitation if the sampling period is suitably chosen taking into account the time constants of the controlled process, in such a way to guarantee the uniqueness of the solution. In addition we can consider a set of input functions such that the unique solution of Eq. ( 13) belongs to [0, T ] (no modulation saturation), with the only implication that a small signal analysis of the control loop needs to be performed. In the implementation of the PWM modulator logic, the value of τ k is assumed equal to T if no solution exists in the interval [kT, kT + T ].

For t ∈ [kT, kT + τ k ] one has:

x(t) = e A(t-kT ) x(kT ) + t kT e A(t-ξ) bu(ξ)dξ. (15)
By substituting the expression of u(t) given in Eq. ( 11) and by using standard algebraic manipulation, it follows:

x(t) = e A(t-kT ) x(kT ) + M sgn [σ(kT )] A -1 e A(t-kT ) -I b, ( 16
)
where I is the identity matrix of dimension n.

In the second subinterval t ∈ [kT + τ k , kT + T ), the state system is described as:

x(t) = e A(t-kT -τ k ) x(kT + τ k ). ( 17
)
Therefore from Eq. ( 16) it is possible to get the state value in the switching instant kT + τ k :

x(kT + τ k ) = e Aτ k x(kT ) + M sgn [σ(kT )] A -1 e Aτ k -I b, (18)
which can be substituted in Eq. ( 17):

x(kT + T ) = e AT x(kT ) + M sgn [σ(kT )] A -1 e AT -e A(T -τ k ) b. (19)
Since

σ(t) = r -c T x(t), (20)
then Eq. ( 13) can be rewritten as:

c T x(kT + τ k ) + Ep T τ k sgn [σ(kT + τ k )] -r = 0. ( 21
)
Note that, by the hypothesis of a unique switching instant in the interval [kT, kT + T ), it must be that sgn [σ(kT + τ k )] = sgn [σ(kT )], therefore Eq. ( 21) becomes:

c T e Aτ k x(kT ) + M sgn[σ(kT )]c T A -1 e Aτ k -I b + Ep T τ k sgn[σ(kT )] -r = 0. ( 22
)
Eqs. ( 19) and ( 22) describe the dynamic of the PWM feedback control system. Let us suppose that an equilibrium point exists. This means that, after a transient, the state at the beginning of the period is equal to the state at the end of the period. Moreover, if r is a constant reference input, then the value of τ k , the value of the error function at the beginning of the period, i.e. σ(kT ), and consequently sgn[σ(kT )], must be constant for each period k. Indicating with (x∞, τ∞) the equilibrium point, it will be

x(kT + T ) = x(kT ) = x∞, τ k = τ∞, ∀k sgn[σ(kT )] = sgn ∞ . (23)
Eqs. ( 19) and ( 22) with ( 23) become respectively

x∞ = M sgn ∞ A -1 I -e AT -1 (e AT -e A(T -τ∞) )b, (24)
and

M c T A -1 I -e AT -1 (e Aτ∞ -I)b + Ep T τ∞ -r sgn ∞ = 0. ( 25
)
Remark 2 Note that solutions of Eqs. ( 24) and [START_REF] Kapteyn | Recherches sur les fonctions de Fourier-Bessel[END_REF] give formal conditions to the existence of equilibrium points. In [START_REF] Hou | Stability analysis of pulse-width-modulated feedback systems[END_REF] sufficient conditions for the uniform asymptotic stability in the large of the trivial solution x∞ = 0 (r = 0) are investigated.

In order to obtain the necessary and sufficient condition for local stability, we consider the linearized discrete model in the neighborhood of the equilibrium point (x∞, τ∞) [START_REF] Eisinberg | Local stability of a class of PWM feedback control systems[END_REF]:

dx(kT + T ) = F dx(kT ), F = e AT -1 L e AT bc T , (26)
where

L = c T I -e AT -1 (e Aτ∞ -e AT )b + Ep T M . ( 27
)
For the sake of the reader, Eqs. ( 26) and ( 27) are derived in the Appendix A. Hence, the equilibrium point is locally asymptotically stable if and only if the eigenvalues of the matrix F belong to the unit circle. Therefore, by Eqs. ( 26)- [START_REF] Skoog | Generalized pulse-modulated feedback systems: Norms, gains, lipschitz constants, and stability[END_REF], it is possible to obtain the critical value of Ep as a function of T and τ∞ ∈ [0, T ]. Note however that our interest is in a value of Ep which ensures the local stability, for all values of τ k in [0, T ]. Such value will be compared with that one given by the dual-input describing function method, which is valid for all values of τ k in [0, T ] and for every constant reference input r.

Overview of dual-input describing function and main results

Suppose that a limit cycle takes place in the PWM closed-loop depicted in Fig. 2. On the basis of the well known peculiarities of the subharmonic modulation technique applied to the inverters, the PWM input σ(t) and output u(t) are periodic of period N T with N positive integer [START_REF] Gelb | Multiple-input describing functions and nonlinear system design[END_REF][START_REF] Balestrino | On the ordinary and modified subharmonic control[END_REF]. Therefore the switching process generates a sinusoidal input to the PWM modulator of amplitude µEp and frequency ωc = 2π N T with a DC offset µ 0 Ep: σ(t) = µ 0 Ep + µEp sin (ωct + φ) .

In Eq. ( 28) we consider µ < µ 0 < 1 -µ so that the modulation saturation is avoided.

Remark 3 We assume that the system is operating in a condition of limit cycle in the neighborhood of an equilibrium point and that such oscillation does not involve a change of sign in the logic modulator. Therefore sgn[σ(kT )] is assumed equal to 1 in the sequel. Although such assumption should seem restrictive, however, as it will be evident in the simulation results, it does not affect the prediction capability of the method. Moreover it allows to simplify the expression of the describing function of the PWM modulator, giving the possibility of further theoretical investigations. By using Fourier series, this periodic function u(t) can be expanded as

u(t) = b 0 + ∞ n=1 [bn cos (nωct) + an sin (nωct)] ( 29
)
where the Fourier coefficients are determined by

b 0 = 1 N T N T 0 u(t)dt, bn = 2 N T N T 0 u(t) cos (nωct) dt, an = 2 N T N T 0 u(t) sin (nωct) dt. (30)
The fundamental component of u(t) is described by

u(t) ≈ b 0 + b 1 cos(ωct) + a 1 sin(ωct) = b 0 + U 1 sin(ωct + φ 1 ) (31)
where U 1 = a 2 1 + b 2 1 and φ 1 = tan -1 (b 1 /a 1 ). The dual-input describing function of the nonlinear element is the complex ratio between the fundamental component of the non linear elements and the input [START_REF] Mahalanabis | On the dual-input describing function of a nonlinear element[END_REF], i.e.

D 0 = b0 µ0Ep , D 1 = U1e jφ 1 µEpe jφ = a1+jb1 µEpe jφ . (32)
Eq. ( 13), particularized to the input σ(t) of Eq. ( 28), gives the following expression of the switching instant τ k in the generic interval [kT, kT + T ):

τ k T = µ 0 + µ sin 2π N k + 2π N τ k T + φ . ( 33
)
Eq. ( 33) can be rewritten as: where

E k -ǫ sin(E k ) = M k (
E k = 2π N k + 2π N τ k T + φ, (35)
ǫ = 2π N µ (36)
and

M k = 2π N µ 0 + 2π N k + φ. ( 37
)
Remark 4 From Eq. ( 36) it is necessary that µ ≤ min 1, N 2π to guarantee ǫ ≤ 1. Such an assumption does not affect the validity of the proposed approach because, as it will be clear afterwards, the sufficient conditions, provided by the method, refer to the onset of the limit cycle, i.e. µ ≈ 0.

Remark 5 It is important to observe that Eq. ( 34) is related to the problem of finding the abscissa corresponding to the intersection of the function sin(E k ) with the line E k -M k ǫ k (see Fig. 4). Therefore, any given problem conforming this geometrical structure, can be expressed through an expansion in terms of Kapteyn series. Eq. ( 34) is formally equivalent to the Kepler's equation [START_REF] Gelig | Stability and Oscillations of Nonlinear Pulse-Modulated Systems[END_REF]. Hence the solution of (34) in terms of Kapteyn series gives:

E k = M k + 2 ∞ n=1 1 n Jn(nǫ) sin(nM k ). (38)
From Eq. ( 38), an explicit relation of τ k /T in terms of Kapteyn series can be obtained:

τ k T = µ 0 + N π ∞ n=1 1 n Jn(nǫ) sin(nM k ). (39)
Taking into account the expression of u(t) which can be compactly rewritten as

u(t) = M N -1 k=0 (δ -1 (t -kT ) -δ -1 (t -kT -τ k )) ,
with δ -1 (t) = 1, t ≥ 0, 0, otherwise , the mean value b 0 and the first coefficients b 1 and a 1 of u(t) become:

b 0 = M N N -1 k=0 τ k T = M µ 0 + M π ∞ p=1 1 p J pN (pN ǫ) sin [p(N φ + 2µ 0 π)] , (40)
b 1 = 2M N T N -1 k=0 kT +τ k kT cos 2π N T t dt = 2M π N -1 k=0 sin πτ k N T cos π N T (2kT + τ k ) , (41)
a 1 = 2M N T N -1 k=0 kT +τ k kT sin 2π N T t dt = 2M π N -1 k=0 sin πτ k N T sin π N T (2kT + τ k ) .
(42) Let us consider the complex variable c 1 = a 1 + jb 1 which can be rewritten as:

c 1 = - M π e jφ N -1 k=0 [cos(E k ) -j sin(E k )] . (43)
Considering the following expressions associated with the Kepler-Bessel expansions [START_REF] Watson | A treatise on the theory of the Bessel function[END_REF]:

cos(E k ) = - ǫ 2 + ∞ n=1 1 n [J n-1 (nǫ) -J n+1 (nǫ)] cos(nM k ), (44)
sin

(E k ) = ∞ n=1 1 n [J n-1 (nǫ) + J n+1 (nǫ)] sin(nM k ), (45)
the Eq. (43) becomes:

c 1 = - N ǫ 2 + N -1 k=0 ∞ n=1 1 n [J n-1 (nǫ) -J n+1 (nǫ)] cos(nM k ) -j [J n-1 (nǫ) + J n+1 (nǫ)] sin(nM k ) .
(46) By using the following relations, (see Lemma 1 in Appendix 7 for a sketch of proof):

N -1 k=0 cos(nM k ) = N cos(p(N φ + 2µ 0 π)), n = pN, p ∈ N, N -1 k=0 sin(nM k ) = N sin(p(N φ + 2µ 0 π)), n = pN, p ∈ N, (47)
NODY9712_source.tex; 8/04/2010; 8:26 p. 12 the Eq. ( 46) is simplified as:

c 1 = M π e jφ N 2 ǫ + ∞ p=1 1 p J pN +1 (pN ǫ)e jp(2µ0π+N φ) - ∞ p=1 1 p J pN -1 (pN ǫ)e -jp(2µ0π+N φ) .
(48) Finally, by using Eq. ( 48) and the property of Bessel functions [START_REF] Watson | A treatise on the theory of the Bessel function[END_REF]

Jn(z) z = 1 2n (J n-1 (z) + J n+1 (z)) , (49)
an expression of D 1 is obtained:

D 1 = 2M Ep 1 2 + ∞ p=1 1 2pN +2 J pN (2πµp) + J pN +2 (2πµp) e jp(2µ0π+N φ) + - ∞ p=1 1 2pN -2 J pN -2 (2πµp) + J pN (2πµp) e -jp(2µ0π+N φ) . (50)
The next theorem gives a sufficient condition for the ripple stability of a PWM closed-loop control system. Theorem 1 A sufficient condition which guarantees the absence of solution of the harmonic balance equation and, therefore, the absence of oscillations, predicted by the dual-input describing function, of period N T with N ≥ 2, for the PWM feedback control system, is

G j π T < Ep 2M . (51)
Proof Let us consider the auxiliary variables α = 2πµ and β = 2µ 0 π + N φ. The function D 1 can be rewritten as

D 1 = u(α, β) + jv(α, β), (52)
where

u(α, β) = 2M Ep 1 2 + ∞ p=1 cos(pβ) J pN (pα) + J pN +2 (pα) 2pN + 2 - J pN -2 (pα) + J pN (pα) 2pN -2 , (53) and v(α, β) = 2M Ep ∞ p=1 sin(pβ) J pN (pα) + J pN +2 (pα) 2pN + 2 + J pN -2 (pα) + J pN (pα) 2pN -2 .
(54) If |D 1 |max denotes the maximum absolute value of D 1 , then a sufficient condition for the PWM feedback control system to be free of oscillations with period N T is

G j 2π N T < 1 |D 1 |max . ( 55
)
Since The describing function is tested for a first order controlled plant

|D 1 | = u(α, β) 2 + v(α, β) 2 ,
G(s) = 1 sγ + 1 .
The inequality in Eq. ( 51), solved for Ep, gives the critical value of Ep, namely Ep df :

Ep df = 2 π 2 γ 2 T 2 + 1 . ( 59
)
By Eqs. ( 26) and ( 27) the expressions of N and F are respectively

L = 1 -e T -τ∞ γ γ 1 -e T γ + Ep T , (60)
F = e -T γ     1 - 1 1-e T -τ∞ γ 1-e T γ + Epγ T     . (61)
Note that the derivative of F respect to τ∞:

∂F ∂τ∞ = - T 2 γ e τ∞ γ e T γ -1 Epγe T +τ∞ γ + T e T γ -(Epγ + T )e τ∞ γ 2 (62)
is less than zero for each τ∞ ∈ [0, T ]. Therefore the equilibrium point is locally stable if and only if the two following inequality are satisfied:

lim τ∞→0 F = Epγ Epγ + T e -T γ < 1, ( 63
) lim τ∞→T F = Epγ -T Epγ e -T γ > -1. (64)
Since Eq. ( 63) is satisfied ∀γ > 0, then from Eq. (64) the critical value of Ep, namely Ep ls which assures the local stability of the equilibrium point is:

Ep ls = T γ 1 + e T γ . (65)
The ratio between Ep ls and Ep df can be rewritten as:

Ep ls Ep df = x 1 + π 2 x 2 2 (1 + e x ) (66)
with x = T γ , and it is easy to show that Ep ls < Ep df for all x ≥ 0. Therefore the proposed criterion ensures the local stability of the equilibrium point for each ratio A second order plant characterized by the following transfer function

G(s) = sξ 3 + 1 (sξ 1 + 1)(sξ 2 + 1)
with ξ 1 , ξ 2 , ξ 3 ∈ [0, 1] and ξ 1 > ξ 2 , is considered. The value of the modulator sampling period T is randomly chosen in [0, min(ξ 2 , ξ 3 )/2] in view of the fact that PWM systems are usually designed with a sampling period short compared with the response time of the linear part of the system, so the ripple at sampling frequency is well attenuated at the output of the system [START_REF] Gelb | Multiple-input describing functions and nonlinear system design[END_REF]. 10 Monte-Carlo simulations with 10 3 plants have been conducted. In each simulation a value of Ep = ρEp df with ρ ∈ [0.1, 1] was chosen and the number of trials in which the proposed criterion ensures the local stability is stored. Table 1 shows, for each simulation, the probability to have a local stable PWM closed-loop process, for each ratio τ∞ T , when Ep = ρEp df . The obtained results show that the proposed criterion, even if approximate in nature due to the describing function approach, gives an effective method to avoid limit cycles and to ensure the local stability.

Concluding remarks

The first contribution of this paper is represented by an expression of the dual-input describing function, in terms of Bessel functions, for a class of PWM modulator. The second important result is a sufficient condition to ensure the absence of limit cycles predicted by the dual-input describing function of period multiple of that characteristic of the modulator. All the results proposed are related to an equivalence between the switching instants of the modulator and the eccentricity of a planet orbit around the sun, described by Kepler's equation. Numerical simulations have shown the effectiveness of the condition in the case of a first and second order plant. With similar technicalities, the proof of the second summation in Eq. ( 47) can be approached.

Fig. 1

 1 Fig. 1 Graphical representation of orbital elements.

Fig. 4

 4 Fig. 4 Graphic solution of the Kepler's equation.

  then the values of α and β which maximize |D 1 | are obtained by seeking the values that satisfy the following two equations:

  Therefore, Eq. (26) becomes dx(kT + T ) = e AT dx(kT ) + M sgn ∞ e A(T -τ∞) b dτ k .(68)Moreover, applying the implicit functions theorem to Eq. (27), one hasc T e Aτ∞ dx(kT ) + c T Ae Aτ∞ x∞ + M sgn ∞ c T e Aτ∞ b + Ep T sgn ∞ dτ k = 0. (69)From Eq. (69), it is possible to find the value of dτ k as function of dx(kT ):dτ k = -c T e Aτ∞ c T Ae Aτ∞ x∞ + M sgn ∞ c T e Aτ∞ b + Ep T sgn ∞ dx(kT ). (70)By substituting (70) in (68)dx(kT + T ) = e AT dx(kT ) -e AT bc T c T Ae Aτ∞ x∞ M sgn ∞ + c T e Aτ∞ b + account the expression of x∞, it follows that dx(kT + T ) = e AT dx(kT ) + e AT bc T L dx(kT ),(72)whereL = c T Ie AT -1 e AT e Aτ∞ -I b + c T e Aτ∞ b + Ep T M . (73)Multiplying the second term of the previous expression byIe AT -1 Ie AT L = c T Ie AT -1 e AT e Aτ∞ -I b + c T Ie AT -1 Ie AT e Aτ∞ b + Ep T M ,(74)it follows thatL = c T Ie AT -1 e Aτ∞ -e AT b + k ) = N sin [p(N φ + 2µ 0 π)] , n = pN, p ∈ N. (76)Proof Taking into account the expression of M k , χ can be rewritten as: previous equations, it follows that n must be equal to pN with p ∈ N, -1) p(N-1) N cos(pN π) cos(pπ) = N, (82) from which the proof follows.

  34)

	NODY9712_source.tex; 8/04/2010; 8:26	p. 10

Table 1

 1 Probability to have a local stable process for different values of ρ.

	ρ	0.1	0.2	0.3	0.4	0.5
	Prob.	0.5806	0.6325	0.7119	0.7764	0.8467
	ρ	0.6	0.7	0.8	0.9	1.0
	Prob.	0.9064	0.9652	0.9936	0.9997	1.0000

NODY9712_source.tex; 8/04/2010; 8:26 p. 9

By simple computations, the following non trivial solutions are obtained:

In Eq. ( 56), H is the Hessian matrix obtained for each stationary point after standard but tedious manipulations by using the property of the Bessel functions [START_REF] Watson | A treatise on the theory of the Bessel function[END_REF]

and by evaluating the second-order partial derivatives of |D 1 | respect to α and β in the stationary point.

On the basis of the above considerations, since the unique maximum of |D 1 | is for N = 2, then one can state that the following condition

is sufficient to ensure the absence of limit cycles of period N T , N ≥ 2 for the considered control system.

Remark 6 Note that the expression of D 0 in Eq. (32) for N = 2, α = 0 and β = π gives the value of µ 0 :

where G(0) = -c T A -1 b is the static gain of the process.

Simulation results

In this section the effectiveness of the proposed stability criterion is investigated through comparisons with local stability conditions. In the following, without loss of generality, we consider M = 1; moreover the process time constants will be normalized in the interval [0, 1].

Appendix A

To obtain the linearized model on the equilibrium point (x∞, τ∞) (see Eqs. ( 26) and ( 27)) let us consider the first-term truncated