Jiazi Yi

Eddy Cizeron

Salima Hamma

Benoît Parrein

Pascal Lesage

Implementation of Multipath and Multiple Description Coding in OLSR

Keywords: hoc networks, link state protocol, multipath routing, multiple description coding

In this paper we discussed the application and the implementation of multipath routing and multiple description coding (MDC) extension of OLSR, called MP-OLSR. It is based on the link state algorithm and employs periodic exchange of messages to maintain topology information of the networks. In the mean time, it updates the routing table in an on-demand scheme and forwards the packets in multiple paths which have been determined at the source. If a link failure is detected, the algorithm recovers the route automatically. Concerning the instability of the wireless networks, the multiple description coding is used to improve reliability of the network transmission, and several methods are proposed to allocate the redundancy in different paths. The simulation in NS2 shows that the new protocol can effectively improve the performance of the networks. The implementation of MP-OLSR is also proposed in the end.

I. INTRODUCTION

In recent years, more and more multipath routing protocols are proposed. These protocols consist of finding multiple routes between a source and destination node. The multipath routing could offer several benefits: load balancing, fault-tolerance, higher aggregate bandwidth, lower end-to-end delay, effectively alleviate congestion and bottlenecks [START_REF] Mueller | Multipath Routing in Mobile Ad Hoc Networks: Issues and Challenges[END_REF] and security.

In the literature, multipath routing protocols are often used for backup routes. Otherwise, if the goal is the repartition of information, the implementation is generally based on pure source routing. Several multipath protocols have been introduced in the work of [START_REF] Lee | AODV-BR: Backup Routing in Ad hoc Network[END_REF] and [START_REF] Lee | Split Multi-Path Routing with Maximally Disjoint Paths in Ad Hoc Networks[END_REF]. And a source routing multipath OLSR is presented in [START_REF] Zhou | A novel routing protocol for ad hoc sensor networks using multiple disjoint paths[END_REF] by using the shortest path algorithm. However, the suppression of nodes in multiple calls of Dijkstra algorithm This work is supported by the French program RNRT (Réseau National de Recherche en Télécommunications) under the project SEREADMO (http://www.sereadmo. org). This is a proceeding of 4 th OLSR Introp/Workshop, Ottawa, Canada.

could not work for sparse networks. Furthermore, strict node-disjoint multiple paths are not suitable for partition or fusion of group of nodes that can imply temporary a single link for connection.

In this study, we discuses a new multipath routing protocol called MP-OLSR based on OLSR [START_REF] Jacquet | Optimized Link State Routing Protocol for Ad Hoc Networks[END_REF] [START_REF] Clausen | Optimized Link State Routing Protocol[END_REF] to provide fault-tolerance, higher aggregate bandwidth and load balancing. Furthermore, the multiple description coding (MDC) provides an additional benefit to the multipath routing. By adding some overheads to each packet, which is calculated as a linear function of the original packets, the resulting packets are geometrically projected into smaller blocks and distributed into available paths. With MDC, given the failure probabilities of the paths, it is possible to find the optimal way to fragment and then distribute the blocks to the paths so that the probability of reconstructing the original information at the destination is maximized.

The remainder of the paper is organized as follows. The specification of the protocol is presented in Section II in detail. In Section III we introduce the application of Multiple Description Coding in the MP-OLSR. The simulation model and performance results are demonstrated in Section IV, and a real implementation is studied in Section V. At last, we conclude the paper in Section VI.

II. MP-OLSR SPECIFICATION

The MP-OLSR can be regarded as a hybrid multipath routing protocol. It sends out HELLO messages and TC messages periodically to be aware of the network topology, just like OLSR. However, MP-OLSR does not always keep a routing table. It only computes the routes when there are data packets need to be sent out.

The core functioning of MP-OLSR has two main parts: topology sensing and routes computation. The topology sensing is to make the nodes get to the topology information of the network, which includes link sensing, neighbor detection and topology discovery. This part gets benefit from MPRs as well as OLSR. The routes computation uses the Multipath Dijkstra Algorithm to populate the multipath based on the information get from the 44306 Nantes cedex 3 -France {jiazi.yi, eddy.cizeron, salima.hamma, benoit.parrein}@univ-nantes.fr * Société Keosys, 1, impasse Augustin Fresnel, 44815 Saint Herblain, France {ple@keosys.com} topology sensing. The source route (the hops from the source to the destination) will be saved in the header of the data packets. The intermediate nodes just read the packet header and forward the packet to the next hop. Furthermore, to overcome some drawbacks of the source routing, the route recovery is introduced.

A. Topology Sensing

The topology sensing is to make the nodes get to know the topology information of the network, which includes link sensing, neighbor detection and topology discovery. This part gets benefit from MPRs as well as OLSR to minimize the flooding of broadcast packets in the network by reducing duplicate retransmissions in the same region.

An adaptation of OLSR for the multipath routing is that in the TC message, the protocol not only include the links between local node and MPR, but the links to all the neighbors so that each node can have better information about the networks topology to construct disjoint multipath. In the simulation, it works well at low data rate. But in the scenario of high data rate, it will cause more congestion because this method will increase the size of TC message. However, with the message combining mechanism in OLSRv2, this problem can be eased.

B. Routes Computation

Contrary to classical OLSR, routes are not renewed each time a node receives a new routing message, but in an on-demand scheme, in order to avoid the loud computation of several routes for every possible destination. When a given source must send packets, the route computation procedure uses the algorithm shown in Figure 1.

The general principle of this algorithm is at step i to look for the shortest path P i to the destination d. Then the edges in P i or pointing to P i have their cost increased in order to prevent the next steps to use similar path. f p is used to increase costs of arcs belonging to the previously path P i (or which opposite arcs belong to it). This encourages future paths to use different arcs but not different vertices. f e is used to increase costs of the arcs who lead to vertices of the previous path P i . We can choose different f p and f e to get link-disjoint path or node-disjoint routes as necessary.

In Figure 1, Dijkstra(G,n) is the standard Dijkstra's algorithm which provides the source tree of shortest paths from vertex n in graph G; GetPath(ST,d) is the function that extracts the shortest-path to n from the source tree ST; Reverse(e) gives the opposite edge of e ; Head(e) provides the vertex edge e points to.

Another possible solution is instead of increasing the cost of the arc, we delete the related node from the node set so that we can get totally node-disjoint routes, which is the ideal case [START_REF] Zhou | A novel routing protocol for ad hoc sensor networks using multiple disjoint paths[END_REF]. However, in the real scenario, especially the cases that the nodes are sparse, the delete of some "key" nodes might prevent from finding new route.

Figure 2 gives the simulation results of the algorithm in the scenario of 300 nodes. As shown in the figure, by using the Multipath Dijkstra Algorithm, we can get three node-disjoint routes. When the number of routes required comes to ten, we get ten optimal routes (some nodes might be shared when node-disjoint routes are unavailable) according to the cost functions. In this article, only three or four routes are used.

C. Route Recovery

In the classical OLSR, the hop-by-hop routing is used, which means when a packet reaches an intermediate node, the protocol will check the routing table of the local node and then forward the packet to the next hop.

In contrast, in MP-OLSR, we use the semi-source routing approach. It will help the source node keep good control of the packets which will be forwarded in the multipath. However, in the mean time, the pure source routing might cause two problems: Firstly, the information in the source node might be not new enough because it needs time to flood the topology control messages to the whole network. It means when computing the routes, the source node might use the links that does not exist anymore. Secondly, even when the information in the source node is updated, the topology might change during the forwarding of the packet. Both of them will cause the failure of the packets forwarding.

To solve these problems, the route recovery is used: before a medium node trying to forward the packet, the node first check if the next hop in the source route of the packet is one of its neighbors. If yes, the packet is forwarded as it should be. If no, then it's possible that the "next hop" has moved out of the transmission range of the node. Then it is necessary to recompute the route and forward the packet through the new route.

III. THE MULTIPLE DESCRIPTION CODING

By adding redundancy to information streams and splitting them up into several sub-streams, we can improve the integrity of data, especially by sending these sub-streams along different paths from the source to the destination. This kind of transformation is called Multiple Description Coding (see [9] for a detailed review).

Given a piece of information I, a multiple description coding method generates N independently communicable packets (P 1 , P 2 ,…, P N ). Each description P i is generally much smaller than the original information. In the MDC scheme, the more descriptions are received, the closer to I is the reconstructed information Î. In our case, we just assume that it exists an integer M (0<M≤N) such that every subset of descriptions containing at least M different descriptions is sufficient to rebuild entirely I. The scheme can be easily extended to gracefully reconstruction suitable for image and video services [START_REF] Benoît Parrein | Priority image and video encoding transmission based on a discrete radon transform[END_REF]. Thus, the higher is M, the lower is the redundancy. In particular, M = 1 (respectively M =N) corresponds to the case where

A. Mojette Transform

In MP-OSLR, the Mojette transform [START_REF] Guédon | The Mojette Transform: The First Ten Years[END_REF] is used to produce different projections of the original information, each one being sent along a specific path. This discrete form of the Radon transform only requires the addition operation and is exactly invertible. The linear integration of the discrete 2D function f(k,l) is obtained via the Mojette transform over a set of I pre-defined angles, 1 tan ( )

i i q I p - = .
The pairs of integers defining the angles, (p i ,q i ) = q, and since linear integration is directionally independent, q i is restricted to to ensure [0, ] i q p Î . Assuring a Dirac pixel model the linear integrations become sums over the pixels centered on the lines b = q i kp i l. The Mojette projection operator is defined as

1 1 0 0 { ( , )} ( , , ) ( , ) ( )
Q P i i i i k l M f k l Proj p q b f k l b p l q k d - - = = = = + -

邋

where ( ) d h is the Krönecker function, i.e., ( ) B. Geometrical Buffer However, applying MDC coding to original packets may significantly increase the total number of packets that are transmitted in the network, which may results in new congestion. A possible solution consists in setting up a sending buffer and then performing the MDC on its content (that corresponds to a group of original packets). Furthermore, it could be suitable that MDC is in s systematic construction, which means that first M packets are exact copy of input packets. Incoming packets are thus geometrically buffered and redundant projections are computed. Systematic construction means that when no packet loss occurs, the complexity is null at the decoding side.

d h =1 if 0 h = , otherwise ( ) d h =0.

C. Redundancy Allocation

The redundancy of MDC must be adapted to the transmission environment. It is preferred to transmit packets with more redundancy along the more stable paths to guaranty the final decoding. The metrics can be measured in different layers of the OSI model.

In the physical layer, the link quality based on Bit Error Rate (BER) can be measured and propagated in the network to help decide the cost of the links for the multipath Dijkstra Algorithm. For simulation purposes, it assumes a precise model of propagation that can be given by ray tracing model [START_REF] Stepanov | On the impact of a more realistic physical layer on MANET simulations results[END_REF].

In the networks layer, we can monitor the length of the FIFO buffer B at each node and the redundancy is related to possible congestion status. In [START_REF] Moid | Heuristics for Jointly Optimizing FEC and ARQ for Video Streaming over IEEE 802.11 WLAN[END_REF], a simple scheme can improve efficiently the packet delivery ratio but for one path and one hop i.e. the access point. Confident by this result, we propose a heuristic, which is very closed but based on the maximum length of the buffers monitored for each path. Given the max redundancy K max , the redundancy to be applied to M different paths is max max

1 max max( ) 1 max( ) (1 ) i i m i i B B K K B B = - = - å
where max(B i ) is the longest buffer length in path i and B max is the buffer size.

Another redundancy allocation scheme is introduced in [START_REF] Benoît Parrein | Priority image and video encoding transmission based on a discrete radon transform[END_REF] at the application layer dedicated to image and video transmission, which is to maximize the expected quality E[Q] at the decoding stage:

0 [ ] [ ] L j j j E Q Q Prob X r = = D å
where Q D is the quality increment, X is the random variable representing the number of received projections and r is the function that maps the sub-streams to the corresponding number of projections needed for reconstruction. The allocation redundancy results from the maximization of the expected quality. Metric closed to the human perception are used in [START_REF] Benoît Parrein | Priority image and video encoding transmission based on a discrete radon transform[END_REF].

IV. SIMULATION AND PERFORMANCE ANALYSIS

A. Environment and Assumption

The proposed algorithm is simulated on NS2. The channel capacity of mobile hosts was set to 11Mbps. A two-ray ground reflection model, which considers both the direct and a ground reflection path, was used as radio propagation model. We use the DCF (distributed coordination function) of IEEE 802.11 for wireless LAN as the MAC layer protocol. It has functionality to notify the network layer about link breakage. In the simulations, there are 50 nodes move in a 1000m×1000m square region and 30 CBR sources are included in the transmission.

B. Simulation results

We compared the performance of OLSR and MP-OLSR in the simulations. Except for the difference of the protocols, two different kinds of strategy to discover the link failures are considered. The first one is the proactive way: when a node does not receive HELLO messages continuously from its former neighbors, it considers that the wireless link between them is broken and will delete the corresponding node from the neighbor set. The second strategy is to use link-layer feedback. When a node is trying to forward a packet to the next hop that has been out of its transmission range, its link layer will drop the packet because of MAC_RET (REtry Timeout). In the mean time, the link layer will give a feedback to the routing layer and inform the lost of the link. We also compared both of the strategies in the simulation.

The simulations are taken in Figure 4 shows the delivery ratio of the simulations. As we can see from the figure, the OLSR with feedback has better delivery ratio than the original OLSR. This is because the protocol with feedback could detect a link failure as soon as the first packet is dropped and recomputed the routing table. In the contrast, the original OLSR tends to continue to send the packets through a failed link until it finds out that it is not able to receive a HELLO message from the original neighbor. Both with the link layer feedback, the SR-MPOLSR's delivery ratio is about ten percent lower than the OLSR. This is because of the drawbacks of source routing that have been mentioned in the previous section. However, the RE-MPOLSR gives better delivery ratio thanks to the route recovery.

Figure 5 End to end delay

The simulation of MP-OLSR with MDC is also taken. The gain of delivery ratio is very little (one or two percent) compared with the RE-MPOLSR in the scenario of 50 nodes. This is because given the low density of the nodes, sometimes the multiple paths will share a lot of links or even there is just no route at all. Then it is hard to make benefits from the MDC. In fact, the MDC tends to be more suitable for large and dense networks.

Figure 6 gives the delivery ratio in the scenario of 100 nodes. This figure compares 3 protocols: RE-MPOLSR, OLSR Feedback and MDC-OLSR in which the number of descriptions is 4 (i.e. number of routes is 4) and the number of useful descriptions in the reconstruction of initial information is 2. So with higher density of the nodes, MP-OLSR with MDC will have more reliable data transmission by offering better delivery ratio. It is about 10% for speed range between 6m/s and 10m/s.

Figure 6 Delivery ratio in the scenario of 100 nodes

V. IMPLEMENTATION OF MP-OLSR

The real implementation is also taken into consideration on the Linux platform based on olsrd [START_REF] Olsrd | an adhoc wireless mesh routing daemon[END_REF]. The OLSR protocol acts as an application upon the Linux kernel and always provides the next hop information to the destination for the IP routing protocol. However, in MP-OLSR, it is possible to have different next hop for the same destination. So we modified the IP routing: whenever there is a route request, it will trigger the MP-OLSR to provide the information of the next hop, and this information will be injected to the IP routing table. Furthermore, the Mojette transform and the associated security protocol SERANO [START_REF] Grall | Sépcification du protocole de sécurité SEREADMO[END_REF] is also located in the protocol stack, as shown in Figure 7. The anonymous forwarding for multiple hops (particularly the source and the destination), the integrity control and the cypherment procedures can be provided by SERANO stack. The choice of entire security solution needs to switch off route recovery.

Figure 7 OLSR and MP-OLSR in Linux

VI. CONCLUSIONS AND FUTURE WORKS

In this paper, we discussed a multipath extension to OLSR. Our Multipath Dijkstra Algorithm is used to discover the disjoint routes and the source routing with route recovery is performed to forward the packets. In addition, the MDC is applied to enhance the reliability of the transmission. Security functions can be implemented via dedicated SERANO stack.

The future research includes refining the incremental functions of f p and f e to make them adaptive to the specific network and optimize the redundancy allocation for MDC in the data transmission. More largely, more precise simulations of physical layer within NS2 are engaged.

Figure 1 Figure 2

 12 Figure 1 The Multipath Dijkstra Algorithm

An example of projections is given in Figure 3 .

 3 The inverse transform can use Classical Mojette Inversion (CMI) method. For a detailed review, please refer to[START_REF] Pierre Guédon | Internet distributed image information system[END_REF]. Both direct and inverse Mojette operators have linear complexity in number of projections N and number of information elements I, i.e. in O(IN).

Figure 3 Four projections of a 4

 34 Figure 3 Four projections of a 4*4 image f(k,l), Proj(-1,1,b), Proj(0,1,b), Proj(1,1,b) and Proj(2,1,b)

  4 protocols: The original OLSR: It is the original single path routing protocol. It discovers the link failure in a proactive way. The implementation of UM-OLSR [12] is used here; OLSR with link layer feedback [12]; SR-MPOLSR: MP-OLSR with link layer feedback, use source routing only. The incremental functions are: f p (c) = f e (c) = 2c ; RE-MPOLSR: MP-OLSR with route recovery and link layer feedback. The incremental functions are: f p (c) = f e (c) = 2c.

Figure 4

 4 Figure 4 Delivery Ratio

Figure 5

 5 Figure5shows the average end-to-end delay. It includes the queue delay in every node and the propagation delay from the source to the destination. The multipath routing could effectively reduce the queue delay because the traffic is distributed in different routes and gives the minimum delay.


Ana Luísa Marques

Natasha L Taylor

Daniel Roquet

Steven Beze

Carine Chassain

Bruno Pereira

Claire O'callaghan

Simon J G Lewis

Franck Durif

MD Ana Marques
email: armarques@chu-clermontferrand.fr.

Structural and Functional Correlates of Hallucinations and Illusions in Parkinson's Disease

Keywords: Parkinson's disease, hallucinations, illusions, neuroimaging, visuo-perception

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

INTRODUCTION 33

Visual hallucinations (VH), defined as a percep-34 tion without existing stimuli, are one of the most 35 frequent non-motor symptoms of Parkinson's disease middle and superior frontal gyri [START_REF] Ramírez-Ruiz | Brain response to complex visual stimuli in Parkinson's patients with hallucinations: a functional magnetic resonance imaging study[END_REF]. More recently, 81 the role of abnormal connectivity of the default mode 82 network (namely in regions such as middle frontal 83 gyrus, posterior cingulate and precuneus) and dorsal attention network in the pathophysiology of VH in PD has been suggested [START_REF] Shine | Abnormal connectivity between the default mode and the visual system underlies the manifestation of visual hallucinations in Parkinson's disease: a task-based fMRI study[END_REF][START_REF] Yao | Resting activity in visual and corticostriatal pathways in Parkinson's disease with hallucinations[END_REF]. However, those studies did not discriminate between VH and VI, which are not differentiated in the majority of studies, which could explain some of the reported discrepancies. Indeed, less work has been focused on VI in PD and very few studies so far have addressed the question of brain changes that could be specifically related to either VI or VH [4, [START_REF] Ishioka | Illusory misidentifications and cortical hypometabolism in Parkinson's disease[END_REF][START_REF] Marques | Visual hallucinations and illusions in Parkinson's disease: the role of ocular pathology[END_REF][START_REF] Nishio | Defining visual illusions in Parkinson's disease: Kinetopsia and object misidentification illusions[END_REF].

One study has shown that the metabolic pattern observed in PD patients with kinetopsia (illusion of movement) was similar to the one observed in PD patients with VH, whereas the misidentification of objects had a different pattern that only partially overlapped with VH [4]. Recently, our own team demonstrated that PD patients with VH had generalized retinal and brain atrophy compared to those with VI, regardless of the disease duration [START_REF] Marques | Visual hallucinations and illusions in Parkinson's disease: the role of ocular pathology[END_REF]. This finding is in line with another study reporting decreased age-adjusted global grey matter volume in Parkinsonian patients with VH compared to those with VI [START_REF] Barrell | Valida-779 tion of a Psycho-Sensory hAllucinations Scale (PSAS) in 780 schizophrenia and Parkinson's disease[END_REF]. Yet, these observations failed to resolve whether the brain changes underpinning PD-H simply represent progression from more subtle modifications in PD-I or if these phenomena arise from discrete mechanisms.

We hypothesized that the existence of overlapping but graded patterns of structural and functional abnormalities in PD with VI and VH would argue for a continuum between these entities, whereas distinct functional connectivity features would suggest different neural underpinnings for the phenomena. To test this hypothesis, we assessed group-wise local changes of voxel values throughout the entire brain, and we analyzed the resting state functional connectivity associated with VH and VI in PD, focusing on the following regions: lateral occipital cortex (midlevel visual region involved in shape and object identification) [25], lingual gyrus (supplementary visual cortex) [26], and occipital fusiform gyrus (responsible for high-level visual processing, and for memory multisensory integration and perception)

[27], amygdala and insula (involved in emotional processing and emotional modulation of perception) [28], hippocampus (involved in encoding and retrieval of memories, conscious observation and anticipation, and mental imagery) [29], precuneus (involved visuo-spatial imagery, self-processing and consciousness) [30], and inferior/middle/superior frontal gyrus (IFG, MFG, SFG) (involved in retrieval of autobiographical memories and reality defined as a visual perception in the absence of an 171 external stimulus [START_REF] Barrell | Valida-779 tion of a Psycho-Sensory hAllucinations Scale (PSAS) in 780 schizophrenia and Parkinson's disease[END_REF]. We only considered visual misperception as VI and we distinguished them from 173 sense of presence and passage hallucinations. Among

Clinical and neuropsychological assessments

The characteristics and severity of visual hallucinations or illusions were assessed according to the Psycho-Sensory hAllucinations Scale (PSAS) [39]. The presence of other modalities such as sense of presence, passage hallucinations, auditory, olfactory and cenesthesis hallucinations was also noted. We assessed the duration and severity of PD (according to Movement Disorder Society -Unified Parkinson's Disease Rating Scale (MDS-UPDRS) [START_REF]Movement Disorder Society Task Force on Rating Scales for 783 Parkinson's Disease (2003) The Unified Parkinson's Dis-784 ease Rating Scale (UPDRS): status and recommendations[END_REF] and Hoehn & Yahr stage [START_REF] Hoehn | Parkinsonism: onset, progres-787 sion and mortality[END_REF], and noted the dopaminergic treatment doses (expressed as total Levodopa equivalent dose (LED)) [START_REF] Tomlinson | Systematic review of levodopa dose 790 equivalency reporting in Parkinson's disease[END_REF], as well as other treatments users (antidepressants, benzodiazepine, cholinesterase inhibitors, antiepileptics).

Cognitive profile (including Mattis dementia rating scale [START_REF] Mattis | Dementia Rating Scale Professional Manual[END_REF], and the copy of the complex figure of Rey-Osterrieth [START_REF] Shin | Clinical and empirical applications of the Rey-Osterrieth Complex Figure Test[END_REF]), sleep quality (Parkinson's disease sleep scale (PDSS-2) [START_REF] Chaudhuri | The Parkinson's disease sleep scale: a new instrument for assessing sleep and nocturnal disability in Parkinson's disease[END_REF], probable REM Sleep Behavior Disorder (screened with REM Sleep Behavior Disorder Single-Question (RBD-1Q) [START_REF] Postuma | A single-question screen for rapid eye movement sleep behavior disorder: a multicenter validation study[END_REF], and excessive daytime sleepiness (Epworth scale) [START_REF] Johns | A new method for measuring daytime sleepiness: the Epworth sleepiness scale[END_REF] were assessed. Subjective ophthalmological complaints (vision loss, visual field impairment, metamorphopsia, halos, diplopia, color vision complaint, epiphora, eye pains) were noted during a semi structured interview with an ophthalmologist. All clinical and neuropsychological assessments, as well as neuroimaging, were performed with participants on their regular antiparkinsonian treatment.

Imaging acquisition and analysis

Volumetric brain analysis

Pre-processing (Supplementary Material) and analysis were performed using Statistical Parametric Fisher-transformed correlation coefficient values: for the PD-I > PD-H contrast, positive beta values suggest a stronger connectivity between the seed and these regions in PD-I compared to PD-H, while negative beta values suggest a lower connectivity between the seed and these regions in PD-I compared to PD-H; for the PD-I > PD-C contrast, positive beta values suggest a stronger connectivity between the seed and these regions in PD-I compared to PD-C, while negative beta values suggest a lower connectivity between the seed and these regions in PD-I compared to PD-C; for the PD-H > PD-C contrast, positive beta values suggest a stronger connectivity between the seed and these regions in PD-H compared to PD-C, while negative beta values suggest a lower connectivity between the seed and these regions in PD-H compared to PD-C; T values, represent the size of the difference relative to the variation in the sample data; x, y, z (mm), coordinates in MNI space. Age, disease duration, MRI manufacturer, and the presence of benzodiazepine and antidepressant treatment were included as covariates.

parahippocampal region compared to PD-C, with no 346 difference between PD-I and PD-H (Table 2, Fig. 1). functional hyper-connectivity between ILO and ante-362 rior cingulate and paracingulate.

347

Distinct features between PD-I and PD-H. PD-I

363

No other between group differences were reported 364 for other seeds.

365

Average effect size for each group within clusters 366 with significant between group differences of seed-367 to-voxel connectivity is represented in Fig. 2. suggesting partly distinct features between complex VH and other manifestations described as minor VH.

However, in that study, visual illusions were not distinguished from minor VH. Thus far, only two studies have specifically investigated the morphometric characteristics of PD-I compared to PD-H [START_REF] Marques | Visual hallucinations and illusions in Parkinson's disease: the role of ocular pathology[END_REF][START_REF] Barrell | Valida-779 tion of a Psycho-Sensory hAllucinations Scale (PSAS) in 780 schizophrenia and Parkinson's disease[END_REF], and reported a decrease in total gray matter volume [START_REF] Marques | Visual hallucinations and illusions in Parkinson's disease: the role of ocular pathology[END_REF][START_REF] Barrell | Valida-779 tion of a Psycho-Sensory hAllucinations Scale (PSAS) in 780 schizophrenia and Parkinson's disease[END_REF] and left accumbens [START_REF] Marques | Visual hallucinations and illusions in Parkinson's disease: the role of ocular pathology[END_REF] on brain MRI associated with the presence of VH compared to the presence of VI in PD, independent of age [START_REF] Barrell | Valida-779 tion of a Psycho-Sensory hAllucinations Scale (PSAS) in 780 schizophrenia and Parkinson's disease[END_REF], or disease duration [START_REF] Marques | Visual hallucinations and illusions in Parkinson's disease: the role of ocular pathology[END_REF]. In those studies, including one conducted by our group on the same population, a different tool was used to perform whole brain structural analyses (FreeSurfer), which could explain the apparently conflicting results compared to the cur-

416

rent study [START_REF] Blankstein | The complex minds of teenagers: neuroanatomy of personality differs between sexes[END_REF][START_REF] Voets | Evidence for abnormalities of cortical development in adolescent-onset schizophrenia[END_REF]. Indeed, in the current study we into multiple anatomic regions and quantifies tissue The cingulate cortex has a crucial role in the emergence of VH in PD [START_REF] Vogt | Cingulate cortex in Parkinson's disease[END_REF], and it was recently reported that access to consciousness in PD with VH was associated with hypoactivation in the cingulate, suggesting impaired involvement of attentional processing [START_REF] Lefebvre | Hallucinations and conscious access to visual inputs in Parkinson's disease[END_REF]. In particular, the anterior cingulate has a role in focusing attention on behaviorally relevant stimuli [START_REF] Weissman | Dorsal anterior cingulate cortex resolves conflict from distracting stimuli by boosting attention toward relevant events[END_REF], whereas the posterior cingulate cortex has a central role in supporting internally-directed cognition, such as autobiographical memories retrieval and conscious awareness [START_REF] Leech | The role of the posterior cingulate cortex in cognition and disease[END_REF],

and could be differentially involved in VI and VH.

The lateral occipital cortex is a well-known processing center for object recognition [START_REF] Palejwala | Anatomy and white matter connections of the lateral occipital cortex[END_REF], and has been reported to activate in fMRI studies in response to pictures of objects, independently of image feature or familiarity [START_REF] Malach | Object-related activity revealed by functional magnetic resonance imaging in human occipital cortex[END_REF]. Interestingly, previous neuroimaging studies using regional cerebral blood flow concluded that the inferior lateral temporal cortex (particularly fusiform gyrus), was the region most likely responsible for the complex visual hallucinations reported in Charles Bonnet Syndrome [START_REF] Kazui | Neuroimaging studies in 893 patients with Charles Bonnet Syndrome[END_REF].

As

174 68 PD

 68 patients (n = 23 PD-C; n = 22 PD-I; n = 3 PD-175 H), we excluded from our analysis six patients who 176 reported coexisting VH and VI (n = 3 PD-H with 177 mild VI, and n = 3 PD-I with mild VH), in order 178 to avoid any confounding effect on the brain struc-179 tural and functional characteristics related to VI or 180 VH (Supplementary Material Flow Diagram). We 181 excluded patients with neurological diseases other 182 than PD or psychiatric conditions characterized by VH, patients with cognitive impairment (Montreal Cognitive Assessment (MoCA) [38] score < 21/30) or with conditions incompatible with MRI scanning (e.g., claustrophobic, severe camptocormia, weight restrictions, congestive cardiac failure, severe dyskinesia or tremor). Treatment with antipsychotics represented a further exclusion, as well as modifications of anti-parkinsonian treatments within the month before inclusion.

60 /

 60 200, software: XVA 11-A) (n = 24 patients; 6 PD-C, 8 PD-H, 10 PD-I), due to change of scanner at our site during the study. The precise technical characteristics of each sequence have been harmonized between the two scanners in order to achieve the most homogeneous imaging data possible. A particular effort has been made on the definition of TR, TE, flip angle (␣), bandwidth, acceleration, temporal and spatial resolution, and field of view. This step was carried out using a NIST test object associated 241 with a quality control procedure to quantify various 242 geometric and signal parameters [48]. MRI scanners, 243 scans protocol and parameters were qualified after 244 images quality analysis by the CATI platform (Centre 245 d'Acquisition et de Traitement d'Images, Multicen-246 ter Neuroimaging Platform, Paris, France) to confirm 247 the acquisition reproducibility. Acquisition parame-248 ters are detailed in the Supplementary Material.

348Fig. 1 )

 1 Fig. 1). PD-I had functional hyper-connectivity between left IFG and left postcentral gyrus both compared to PD-H and PD-C. PD-I had hyper-connectivity between SFG and IFG compared to PD-H. Moreover, we observed in PD-I hypo-connectivity between SFG and posterior cingulate compared to PD-C, whereas PD-H had

Fig. 1 .

 1 Fig. 1. Seed-based resting state functional connectivity in PD-I, PD-H, and PD-C. For each seed (orange text), differences of functional connectivity between groups are represented; yellow clusters are regions presenting hyper-connectivity with the seed and blue clusters are regions presenting hypo-connectivity with the seed. PD-I and PD-H present a common functional hypo-connectivity pattern between lingual gyrus and parahippocampal region compared to PD-C. Yet, distinct functional patterns are also revealed between VH and VI in PD. PD-I have hyper-connectivity between inferior frontal gyrus (IFG) and postcentral gyrus compared to PD-C and PD-H, and hyper-connectivity between insula and occipital fusiform cortex, and from amygdala, hippocampus towards IFG, compared to PD-H. PD-I, Parkinson's disease patients with visual illusions; PD-H, Parkinson's disease patients with visual hallucinations; PD-C, Parkinson's disease patients without visual hallucinations or illusions.

368 DISCUSSION 369

 369 In this study, we compared the volumetric and 370 seed-based functional connectivity characteristics 371 respectively associated with VH and VI in PD. We 372 found no significant structural differences between 373 PD-I and PD-H. However, PD-I and PD-H showed distinct connectivity patterns, although partially overlapping; suggesting that whilst these symptoms share some partly common neural mechanisms, they are also underpinned by specific functional brain differences.Volumetric characteristics associated with VH and VI in PDUsing voxel-based morphometry (VBM), we observed no volumetric difference between PD-I and PD-H, nor between PD-I compared to PD-C, but found grey matter atrophy in PD-H compared to

Fig. 2 .

 2 Fig. 2. Average effect sizes in each group within clusters with significant seed-to-voxel differences between PD-I and PD-H. PD-I and PD-H present a common functional hypo-connectivity pattern between lingual gyrus and parahippocampal region compared to PD-. Yet, distinct functional patterns are also revealed between VH and VI in Parkinson's disease. PD-I have hyper-connectivity between insula and occipital fusiform cortex, and from amygdala and hippocampus to IFG compared to PD-H. PD-I, Parkinson's disease patients with visual illusions; PD-H, Parkinson's disease patients with visual hallucinations; PD-C, Parkinson's disease patients without visual hallucinations or illusions.

422

  volume in these predefined regions on an individual 423 case basis. Thus, even if more widespread brain atro-424 phy has been suggested in PD-H compared to PD-I 425[START_REF] Marques | Visual hallucinations and illusions in Parkinson's disease: the role of ocular pathology[END_REF], as of yet, no data suggests the existence of dis-426 tinct structural changes associated respectively with 427 VH and VI. Hereby, we are currently unable to distin-428 guish between VH and VI according to morphometric 429 characteristics. 430 Functional connectivity signatures of VH and VI 431 in PD 432 While we found no VBM difference between PD-I 433 and PD-H in this study, we identified partly overlap-434 ping and distinct resting state functional connectivity 435 patterns associated with the presence of VH and 436 VI in PD, independent of disease duration. These 437 connectivity changes observed in PD-H and PD-I 438 could reflect differential deposits of alpha-synuclein, 439 as it has been reported that patients with synucle-440 inopathies frequently experience hallucinations and 441 illusionary perceptions, related to dysfunction of both 442 associative visual areas and changes of limbic areas 443 or the ventral striatum [55]. 444 Common functional connectivity patterns to PD-I 445 and PD-H from lingual gyrus 446 Both PD-I and PD-H presented functional hypo-447 connectivity between the lingual gyrus and the 448 parahippocampal region, in line with occipital func-449 tional hypo-connectivity generally reported in PD 450 with VH [7, 11, 56]. Hippocampal functional con-451 nectivity with the visual cortices has been previously 452 reported to be lower in parkinsonian patients with VH 453 [57], and as suggested by the authors, this could dis-454 rupt visuospatial memory and constitute a common functional impairment to VH and VI in PD. 456 Contrasting functional patterns in PD-I 457 compared to PD-H: hyper-connectivity from 458 IFG, amygdala, hippocampus, and insula 459 Some contrasting functional differences were re-460 vealed between PD-I and PD-H. PD-I compared to 461 PD-H and PD-C had hyper-connectivity between the inferior frontal gyrus (involved in retrieval of episodic memories) [31] and postcentral gyrus. PD-I also had increased connectivity compared to PD-H from the amygdala and hippocampus to the IFG, and from the insula to the occipital fusiform cortex, as well as between SFG and IFG.Hyper-connectivity between IFG and amygdala, encoding positive and negative emotional memories[START_REF] Hamann | Cognitive and neural mechanisms of emotional memory[END_REF][START_REF] Wright | Novelty responses and differential effects of order in the amygdala, substantia innominata, and inferior temporal cortex[END_REF], suggests that emotional memory could influence and modify reality monitoring of an existing visual stimulus in PD with VI. Supporting this hypothesis, pronounced functional connectivity between the amygdala, hippocampus, and right IFG has been reported during autobiographical retrieval in the healthy population[31,[START_REF] Greenberg | Co-activation of the amygdala, hippocampus and inferior frontal gyrus during autobiographical memory retrieval[END_REF]. Our findings suggest that VI in PD could be associated with a strong influence of anxiety and fears, and prior experiences on interpretation of visual stimuli, as well as hyperacute reality monitoring, consistently with hyper-connectivity between the insula (involved in the detection of salient stimuli)[START_REF] Menon | Saliency, switching, attention and control: a network model of insula function[END_REF] and the occipital cortex.Contrasting functional connectivity towards cingulate regions in PD-I and PD-HDivergent patterns of connectivity between towards the cingulate were also observed in PD-I to PD-C: PD-H had hyper-connectivity from ILO with the anterior cingulate and the anterior paracingulate region compared to PD-C, whilst PD-I had hypoconnectivity from SFG with the posterior cingulate compared to PD-C.

601

  This advances our current comprehension of illu-602 sions and hallucinations in PD, which have to date 603 been limited to considering these phenomena as rep-604 resenting a clinical spectrum from simple to complex 605 hallucinations. Understanding the plurality of the 606 structural and functional characteristics underlying 607 VI and VH in PD may enlighten how these two forms 608 of perceptual experiences overlap and are distinct in their specific neural dysfunction, which may inform 610 clinical prognosis. Hence longitudinal studies focus-611 ing on structural and functional alterations in PD with 612 VI and VH should provide greater insight into the 613 natural history of these phenomena.

Table 1 and

 1 Supplementary Figure1).

	Mapping software (SPM12, Welcome Trust Centre
	for neuroimaging, London, UK, http://www.fil.ion.
	ucl.ac.uk/spm/) implemented in Matlab 2018a. A
	voxel-wise one-way ANOVA was performed to
	compare grey matter intensity across groups, with
	PD duration (years since symptoms onset), scanner
	manufacturer (Siemens or GE) and total intracra-
	nial volume as regressors of non-interest. Age and
	sex were not included as covariates since groups
	were matched for these variables. Statistical anal-
	ysis combined a p uncorrected < 0.001 threshold
	at the voxel level and an extent threshold of 100

adjacent voxels. Clusters which survived a multiple comparison correction (Family Wise Error rate) p FWE < 0.05 at the cluster level were explicitly labelled in the tables. Neuroanatomical regions of significance were visualized and identified using xjview toolbox (https://www.alivelearn.net/xjview).

Resting state fMRI analysis

Preprocessing (Supplementary Material) and seedbased connectivity analysis were performed using Conn functional connectivity toolbox (19.b) (http:// www.nitrc.org/projects/conn, RRID:SCR 009550)

[START_REF] Whitfield-Gabrieli | Conn: A functional connectivity toolbox for correlated and anticorrelated brain networks[END_REF]

, which streamlines functions from SPM12.

A seed-based connectivity analysis calculates the functional connectivity between a selected region of interest (ROI), the seed, with all the voxels in the brain. The analysis estimates the Fisher-transformed bivariate correlation coefficient between the seed BOLD signal and all the voxels BOLD signal across time. The following ROIs were selected (FSL Harvard-Oxford cortical/subcortical atlas, maximum probability threshold 25%, 1 mm, https://identifiers. controlling for age, duration of disease, benzodiazepine and antidepressant treatments and MRI scanner type. We performed parametric statistical testing, for comparisons between the three groups using two-sample t-tests for between two group differences and one-way ANOVA. We calculated the uncorrected p-value and false-discovery rate, with statistical significance specified at p < 0.05.

301

Statistical analysis of clinical data

312 was analyzed using the Bartlett test. When appro-313 priate (omnibus p-value < 0.05), post-hoc tests for 314 two by to multiple comparisons were applied, respec-315 tively Tukey-Kramer after ANOVA, Dunn's test after 316 Kruskal-Wallis and Marascuilo procedure after Chi-317 squared and Fisher exact tests. All statistical tests 318 were two-sided and a p-value < 0.05 was considered 319 statistically significant. 320 RESULTS 321 Clinical characteristics 322 Sixty-two PD patients were analyzed for clini-323 cal and brain volumetric data (n = 23 PD-C, n = 19 324 PD-I, n = 20 PD-H), and 58 patients analyzed for rest-325 ing state functional data (n = 21 PD-C, n = 19 PD-I, 326 n = 18 PD-H) after exclusion of corrupted files (Sup-327 plementary Material Flow Diagram). Demographic 328 and clinical characteristics of patients together with 329 group effects are shown in Supplementary Table

1

). 334 PD-I and PD-H or between PD-I and PD-C groups. 341 Seed-based functional connectivity at rest 342 Overlapping features between PD-I and PD-343 H. PD-I and PD-H both had functional hypo-344 connectivity between left lingual gyrus and left

Table 2

 2 Seed-based functional connectivity in parkinsonian patients with visual hallucinations, with illusions and controls

			P FDR -K E	P	BETA	T	X	Y	Z
			CORR		UNCORR			(MM) (MM) (MM)
	PD-I > PD-H
	R AMYGDALA	< 0.001 < 0.001 408	< 0.001	0.19	6.54	66	-38	-02 R MTG
		0.003	0.003 268	< 0.001	0.19	5.05	8	70	-6	R Frontal pole
		0.01	0.006 223	< 0.001	0.18	5.08	-58	24	20	L IFG
	L AMYGDALA	0.01	0.02	205	< 0.001	0.20	5.45	-58	12	24	L IFG
	R HIPPOCAMPUS	< 0.001 < 0.001 615	< 0.001	0.19	5.38	-60	8	18	L IFG (triang.)
		0.001	0.001 394	< 0.001	0.17	5.28	-14	20	14	L caudate
	L INSULA	0.001	0.002 358	< 0.001	0.19	5.07	26	-44	-14 R temporal occipital fusiform
											cortex
	R SFG	< 0.001 < 0.001 518	< 0.001	0.34	5.29	52	18	22	R IFG
	L IFG (TRIANG.)	0.01	0.02	239	< 0.001	0.18	5.25	-66	-6	18	L post central gyrus
		0.08	0.04	145	0.005	0.12	4.53	14	-46	-4	R lingual gyrus
		0.05	0.03	165	0.03	0.19	5.36	24	-28	-4	R hippocampus
	L CAUDATE	0.01	0.03	215	0.001	0.16	5.30	68	-34	-10 R MTG
	PD-I > PD-C
	R SFG	< 0.001 < 0.001 475	< 0.001	-0.08 4.23	-2	-54	30	Posterior cingulate gyrus,
											precuneus
		0.05	0.04	184	0.003	+0.20 5.57	+12	-22	+42 Posterior cingulate gyrus, r
											precentral gyrus
	R LINGUAL GYRUS	0.10	0.04	135	0.006	-0.11 5.65	-40	+4	-38 L temporal pole, L temporal
											fusiform gyrus
	L LINGUAL GYRUS	0.06	0.03	160	0.003	-0.07 -5.64 -36	-16	-30 L parahippocampal gyrus,
											temporal fusiform gyrus
	L IFG (TRIANG)	0.03	0.03	181	0.002	0.12	5.35	-66	-10	34	L postcentral gyrus
	PD-H > PD-C
	L INFERIOR LATERAL	0.009	0.01	258	< 0.001	-0.13 5.05	-16	38	10	Anterior cingulate gyrus, L
	OCCIPITAL CORTEX										paracingulate
	R LINGUAL	0.01	0.01	245	< 0.001	-0.14 5.55	36	2	-44 R parahippocampal gyrus, R
											temporal fusiform gyrus
		0.005	0.01	281	< 0.001	-0.16 6.58	-42	4	36	L parahippocampal gyrus, L
											temporal fusiform gyrus
	L LINGUAL	< 0.001 < 0.001 462	< 0.001	-0.17 5.85	-40	-4	2	L insular cortex
		0.01	0.009 250	< 0.001	-0.22 5.97	-30	-8	-42 L temporal fusiform gyrus
		0.006	0.004 268	< 0.001	-0.20 5.30	36	6	-46 R temporal fusiform gyrus

IFG, inferior frontal gyrus; ITG, inferior temporal gyrus; K E , cluster size; L, left; PD-C, Parkinson's disease without illusions or hallucinations; MTG, middle temporal gyrus; PD-H, Parkinson's disease with visual hallucinations; PD-I, Parkinson's disease with visual illusions; pFDR, p corrected for multiple comparisons using the false discovery rate; pFWE, p corrected for multiple comparisons using the family wise error; puncorr, p uncorrected; R, right; SFG, superior frontal gyrus; STG, superior temporal gyrus; Triang, triangularis; Beta values, represent

  of a common clinical spectrum related to the 562 functional alterations reported in our results.

			563
		Limitations	564
	proposed by the authors, visual loss due to eye
	pathology in Charles Bonnet Syndrome patients, pro-	Our study has several limitations that should be	565
	duces a state of sensory deprivation that releases the	considered when interpreting the results. First, our	566
	visual cortex from regulation by external stimuli that	study is ancillary, and analyses were conducted in	567
	normally has an inhibitory effect on the endogenous	a sample whose size was calculated for a previous	568
	activation of the visual cortex. Such a cortical release	analysis focusing on retinal thickness. Thus, the sam-	569
	phenomenon could also occur in PD-H patients sec-	ple size may have been too small to identify VBM	570
	ondary to retinal impairment [22], and result in	differences between PD-I and PD-C. However, 62	571
	visual hallucinations. Furthermore, reduced activity	parkinsonian patients were included in our functional	572
	of the superior frontal gyrus has been demonstrated in	data analysis, which is higher than the majority of	573
	healthy individuals who are prone to psychosis [33],	previous functional neuroimaging studies on parkin-	574
	and in patients with schizophrenia [69]. Contrasting	sonian patients with VH [7]. Another limitation of	575
	connectivity between cingulate, SFG and ILO could	this study is the difficulty to distinguish VI from	576
	represent distinct neural mechanisms leading to VH	minor hallucinations such as passage hallucinations	577
	or to VI in PD.	and sensation of presence. Presence hallucinations	578
		could be identified, during patients and caregiver
	Visual and non-visual hallucinations
	A significant proportion of PD patients with visual
	hallucinations also had olfactory hallucinations, and
	one might argue that it may have play a role in the spe-
	cific pattern of functional alterations observed in the
	PD-H group. Yet, olfactory, tactile and gustatory hal-
	lucinations usually coexist with VH in PD [70] and in
	other conditions such as schizophrenia [71, 72], and
	probably involve the alteration of common pathways
	responsible for reality monitoring. Indeed, a previous
	longitudinal study showed that whilst visual halluci-
	nations in isolation are classic in early PD, nonvisual
	hallucinations emerge over time, and the combination
	of visual with nonvisual hallucinations predominates
	in later stages of PD [73]. Moreover, the presence
	of visual/auditory hallucinations and sex has been		596
	reported recently to be the main variables predicting	neural activation patterns occurring during an actual	597
	the presence of olfactory hallucinations [74], with-	hallucinatory/illusory event.	598
	out any association with olfactory impairment [74,
	75]. Thus, hallucinations in PD may occur in one or	CONCLUSION	599
	multiple sensory modalities, and a common brain net-
	work responsible for hallucinations independent of	We show both overlapping and distinct functional	600
	the sensory modality has been suggested by previous	signatures related to VI versus VH in PD patients.
	studies [76]. In line with this hypothesis, a previous
	study compared PD patients with one versus multi-

ple hallucinatory modalities, and did not reveal any difference regarding demographic, clinical and medication parameters

[START_REF] Papapetropoulos | A questionnaire-908 based (UM-PDHQ) study of hallucinations in Parkinson's 909 disease[END_REF]

. Interestingly, we found that other sensory modalities of hallucinations are only reported in PD patients with visual hallucinations and not in PD patients with illusions, thus appearing to be part 579 interview, as they are closer to a delusional idea 580 or a social hallucination and defined by the belief 581 that a person is present behind oneself generally, but 582 without associated visual perception. Though, dis-583 tinguishing VI from passage hallucination can be 584 challenging since it depends on the patient's ability 585 to determine whether there was or not a real visual 586 external stimulation. More precise clinical criteria 587 need to be developed in the future in order to bet-588 ter classify these minor manifestations and illusions. 589 It should also be acknowledged that VI and VH are 590 dynamic phenomena that were not occurring during 591 the scans, even if patients had frequent and recent 592 VI and VH at the moment of inclusion in this study. 593 Thus, functional resting state analysis shows brain 594 patterns at rest that could predispose to the occur-595 rence of VI or VH in PD but does not allow to capture

Psychiatry 26, 1299-1309.

ACKNOWLEDGMENTS

CONFLICT OF INTEREST 618

The authors have no conflict of interest to report.

SUPPLEMENTARY MATERIAL 620

The supplementary material is available in the 621 electronic version of this article: https://dx.doi.org/ 622 10.3233/JPD212838. ease: A subtle symptom with major clinical implications.

627

Neurology 93, 259-266.


Holger Döring
email: haw-doering@t-online.de

On minimal Quasi Inertial Systems in Einstein Lifts

Keywords: local inertial-system, QUIS, quasi inertial-system, minimal reference frames, plancklength, bending of Einstein-Lift

In real Einstein-Lifts above resp. in radial gravitational-fields over planetary surfaces there appear tidal forces, which causes small deviations from real inertial systems because the Ricci-Tensor in the Lift is not equal to zero, caused by the non-parallelity of the gravity field lines. For these quasiinertial-systems (defined now as QUIS) there can be calculated a relative minimal-seize, which only depends from heigth over and radius of the planetary mass.

Introduction:

In the previous paper [1.] there was marginally noted, that real QUIS has to have a minimal seize because of measuring the movement of a real particle like an electron, proton, neutrino etc. In real universe, even in flat Minkowksi-space there can't be an "infinitesimal IS or QUIS", because this concept is only a mathematical construction without any sense for physical reality. If an Einstein-Lift falls in a real radial gravitational-field, there can be calculated this minimal seize of the QUIS in an explicite way. This minimal QUIS can be interpreted as the " fundamental quantum" of other, greater QUIS in radial gravitational-fields involving Einstein-lifts in the same heigth over same surface of the same planetary mass. This definition doesn't mean a real quantization because there are no gravity forces quantized but only local geometrical coordinate-systems over fields with very weak strength and very low Ricci-Tensors. But this model can be advanced to more genuine parameters. In addition, a real Einstein-lift in radial fields is not a cube or a cuboid with plane planes above and below but possesses a form of bending in ground and upper plane of the lift (further discussion of this problem see Appendix B and picture 1.).The equivalence principle is only involved in the usual manner ([2.]-[8.]).

2.Calculation:

From elementary geometrical laws (with proportional lengths by equal angles) there can be derived following equation for minimal size of Einstein-Lifts, which is here also defined as minimal, relative size of the local used QUIS. The size of this QUIS

x depends only from heigth over planet radial gravitational field and from planetary mass radius. Therefore a QUIS is never an absolute system but exists without changes only for equipotential "surfaces" of the local gravitational radius-field of the causing mass. Nevertheless there can be defined a minimal QUIS for every of this equipotential surface. The size of the QUIS changes with heigth over planet, because of the existence of radial field-lines.

The equation for this QUIS is given by: x=b+ √ b ²+2⋅b⋅(h+ r ) (1a.) concept-data for minimal QUISes: b -Planck-length r -radius of planetary or stellar mass (body) h -heigth over defined body-surface x -heigth, width and length of a relative, minimal, cubic Einstein-Lift (QUIS).

Therefore can be formulated a little advanced equation for QUISes of greater size:

x=n⋅b+√(n⋅b) ²+2⋅n⋅b⋅(r +h) ; n∈ℕ (1b.)
This formulates the "quantization"-process of QUISes.

Some remarks on the measuring process:

Agreed was in this process (see paper [START_REF] Döring | Formal Binary Logic, Tidal Forces and the Equivalence Principle[END_REF]) that there are two or more downwards pointing lasers resp. light-beams in the edges of the middle-line of the upper plane of the Einstein-Lift or in the middle-line itself. In the middle-line of the lower ground plane there are photometres installed, which measure the deviation from parallelity. If technical laser-problems of focusing are primarily first of all ignored, there can be determined the difference of the lift-systems QUIS relative to a real IS of special relativity theory with measuring the deviations of the real gravity field over planetary surface relative to parallel field lines of a homogenious, ideal field, which doesn't occur in reality of nature. If the possible minimal deviations b are defined as planck-length r Pl , then there can be derived the minimal seize of the QUIS in the condition of the given equipotential "surface" over planetary distance r +h .In this way all the local weak radial field of gravitation can be plastered with QUIS of minimal seize. There ergo is existing a form of "quantizing" local space-time through its local minimal coordinate-systems, but this process isn't a real quantizing of forces. But the minimal QUIS can be interpreted as the "quantum" of local geometrical space-time and they can be summed over to get greater QUISes because of b=r PL .

Example given:

For conditions of r=6,378⋅10⁶m , h=1⋅10⁵m , b=1,616255⋅10 ⁻³⁵ m=r Pl there is a minimal seize of the QUIS of: x=1,4470729⋅10 ⁻¹⁴ m For more exact information on the variation of minimal QUIS with height, see tables 1.-4. below. Of course, with the existence of QUIS, there are no longer any concepts of "potential planes " or "potential surfaces" in gravity fields of planets and stars but there is to speak by QUIS of "minimal potential volume-shells". Summing over all volume shells with increasing radius then leads to the whole gravity-field of the cosmic body to the limiting distance to which should be calculated. Disturbations from other cosmic bodies are not included yet but must have to be calculated to be more exactly.(Eg.:Earth-Moon-system or Sun-Jupiter).

Data of QUIS , belonging to planet earth:

Heigth h in m over planetary radius r (earth):

Minimal seize of QUIS x in 10 ¹ m: ⁻ ⁴ 0 1,

5.Conclusion and summary:

There can be constructed a system of minimal quasi-inertial-systems (QUIS) over every planetary or stellar surface, as a function depending from distance h over surface r . These QUIS vary with this distance in their heigth resp. minimal length of coordinate-axes. The Ricci-Tensor isn't equal to zero in this QUIS , because there are deviations of a real gravitational field with its radial field lines from parallelity. A QUIS is defined as including a deviation existing of minimal distance of planck-length between two gravity field lines at the inner bottom of Einstein-Lift, which contains the QUIS. Therefore are minmal defined QUISes existent.These minimal coordinatesystems can be developed to greater QUISes by multiplying the minimal distance b with an integer number n to get greater distances and greater, discrete QUISes. Therefore are minimal potential volume shells defined but no longer any "potential-surfaces". Summing over all QUISes gives the description of the whole gravity-field of the stellar body in defined distances r +h over surface. Minimal QUISes can be measured resp. defined by orthogonal light signals along the gravity-field lines between the upper and lower planes of the Einstein-Lift, which contains the QUIS. A QUIS is a real minimal reference-frame for local space-time to describe the moving of particles. There are no "infinitesimal IS" to get rid of the radial field-line deviations, because this imagination is not an allowed physical description.The variation of a QUIS with its heigth over planet-surface is a discrete process not a steady one.Therefore they have to be summarized, not integrated.

Appendix A:

QUIS-data for Earth, Jupiter and Sun.

Shown is the minmal length of QUIS in dependence of heigth over planetary gravity field-lines with increasing distance from planet in different intervalls.

Appendix B:

Tidal-force caused material bending through planetary gravity-fields in Einstein-Lift

1.The forces:

Because the gravity-field is weak, there can be calculated over Newton, Ricci-tensors aren't necessary. In picture 2 is seen, that there is bending in Einstein-Lift, because the middle-Force F 1 < F ,which works at the orthogonal middle-line center of the lift, is shorter than the field-lines, which leads to the upper or lower corners of the lift. In the corners there works the force F . So there are the following relations:

F 1 = m⋅M⋅G (r +h+ x )² = m⋅M⋅G R 1 ² (2a.) F= m⋅M⋅G R ² ( 2b.
)

F 2 = m⋅M⋅G ( r+ h)² (2c.) R= √ (r +h+ x )²+( x 2 )² ( 3a.
)

R 1 =r + h+ x (3b.) k =R-R 1 ( 3c.
)

F 1 F = ( R 1 )² R ² = (R-k)² R ² (3d.)
This leads to: k =1,92 * 10 ⁻⁶ m

This value of k is the length-difference between the gravitational field-lines in the upper corners of the lift (see picture 2) and the field-line in the middle of the lift.In analogy there can be constructed a value of k for the lower plane of the lift in the same clear method, which isn't done here now but the bending in both planes of the lift is calculated below. There follows for the force in the middle-line of the upper plane:

F 1 = F 1+5,957400266 * 10⁻ ¹³ (3e.)

2.Bending of lift planes:

The material volume of cubic supposed Einstein-lift is (numerical data below):

V =x 1 ³-x 2 ³ (4.) V =2,997001 m ³ .
The mass is with m=V⋅ϱ , adding:

(5.) m 2 =100 kg for the lift-passenger with space-suit and the material density for steel of the lift (iron,Fe): ϱ=7,86⋅10³ kg m ³ This leads to whole lift-mass of m=2,365642786⋅10⁴ kg .

Therefore the gravity force in the middle of the upper-deck of Einstein-lift is:

F 1 =223750,8375 N .
There follows the bending of upper deck with:

Δ u 1 = F 1 ⋅x 1 ³ 48⋅E⋅I y ( 6.
)

where E is the modulus of elasticity (Young-modul) for Iron/Steel (Fe) and I y the areamoment of inertia:

E=2,1⋅10¹¹ N m² , I y = 1 12 ⋅( x 1 ⁴-x 2 ⁴) ( 7.
)

and the bending for the lower deck with:

Δ u 2 = F 2 ⋅x 1 ³ 48⋅E⋅I y ( 8.
)

Therefore are the bendings in Einstein-Lift (calculated with whole lift-mass, not only with the upper and lower plane) of:

Δ u 1 =6,669756012⋅10⁻ ⁶m for the upper plane of lift and Δ u 2 =6,69744662⋅10⁻ ⁶ m for the lower one.

The difference between the two bendings in the middle of the lift from upper and lower deck is:

Δ (Δ u 1 -Δ u 2 )=2,0678⋅10 ⁻¹¹ m ,
This value is, in principle, measurable by the passengers two light beams along the field-lines of gravity field between the edge and the middle of the lift by measuring the running-time of his laserbeams because it occurs in the middle but not at the edges of the lift. This leads to:

Δ t= Δ( Δ u 1 -Δ u 2 ) c ; Δ t=6.797379516 * 10 ⁻²⁰ s .

This difference of time is in principle a measurable size for the lift passenger to make a difference between a gravitational radial field and a parallel, homogenious acceleration field, if he doesn't want to measure his position like described in [1.].

For calculation used numerical data:

M =M E =5,9742⋅10²⁴

Table 1 :

 1 Listed

are some sizes of minimal QUIS over earth-surface dependend from h between 0-100 km in distances of 10 km.

Table 2 :

 2 Shown is the minimal seize of QUIS over earth-surface in interval of 100km-1000km.

	2.Jupiter:
	distance from surface in m	Minimal size/heigth of QUIS in 10 ¹ m ⁻ ⁴
	0	4,807271627 (minimal defined QUIS for
		Jupiter on hypothetical surface of 71492 km
		equator-radius).
	1.000.000	4,840775918
	2.000.000	4,874049907
	3.000.000	4,907098276

Table 3 :

 3 Shown is the minimal seize of QUIS over jupiter-surface in interval of 1000 km -10.000km.

	3.Sun:
	distance from surface in m	Minimal size/heigth of QUIS in 10 ¹³ m ⁻
	0	1,499942319 (minimal defined QUIS for Sun
		on hypothetical photosphere-surface of 696000
		km equator-radius).
	10.000.000	1,510679337
	20.000.000	1,52134058
	30.000.000	1,531927629
	40.000.000	1,542442012
	50.000.000	1,552885205
	60.000.000	1,563258635
	70.000.000	1,573563682
	80.000.000	1,58380168
	90.000.000	1,593973921
	100.000.000	1,604081656

Table 4 :

 4 Shown is the minimal seize of QUIS over sun-surface in interval of 10.000 km -100.000km.

  kg , earth-mass r=r E =6,378⋅10 ⁶ m , earth-radius,equator h=1⋅10⁵ m , heigth over earth surface x 1 =10 m , outer length of Einstein-Lift x 2 =9,99 m , inner length of Einstein-Lift Picture 1: tidal-forced material bending in real Einstein-Lift meaning bending of Einstein-lift planes in real gravity-field. Drawed are only the bendings of the lower planes but the upper bends either.

	G=6,672041⋅10 ⁻¹¹	m³ kg⋅s ²	, Newton-gravity-constant
	c=2,99792⋅10⁸	m s	; local invariance-velocity of flat space-time.

Picture 2: effective gravity forces in real

Einstein-Lift.