prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What do dendritic spines contain?

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Context 1: JoVE | Peer Reviewed Scientific Video Journal - Methods and Protocols
A role for thrombospondin-1 deficits in astrocyte-mediated spine and synaptic pathology in Downs syndrome  Octavio Garcia, Maria Torres, Pablo Helguera, Pinar Coskun, Jorge Busciglio 
PUBLISHED: 07-02-2010	Downs syndrome (DS) is the most common genetic cause of mental retardation  Reduced number and aberrant architecture of dendritic spines are common features of DS neuropathology  However, the mechanisms involved in DS spine alterations are not known  In addition to a relevant role in synapse formation and maintenance, astrocytes can regulate spine dynamics by releasing soluble factors or by physical contact with neurons  We have previously shown impaired mitochondrial function in DS astrocytes leading to metabolic alterations in protein processing and secretion  In this study, we investigated whether deficits in astrocyte function contribute to DS spine pathology 
Analysis of Dendritic Spine Morphology in Cultured CNS Neurons Authors: Deepak P  Srivastava, Kevin M  Woolfrey, Peter Penzes  Published: 07-13-2011 JoVE Neuroscience
Dendritic spines are the sites of the majority of excitatory connections within the brain, and form the post-synaptic compartment of synapses  These structures are rich in actin and have been shown to be highly dynamic  In response to classical Hebbian plasticity as well as neuromodulatory signals, dendritic spines can change shape and number, which is thought to be critical for the refinement of neural circuits and the processing and storage of information within the brain  Within dendritic spines, a complex network of proteins link extracellular signals with the actin cyctoskeleton allowing for control of dendritic spine morphology and number  Neuropathological studies have demonstrated that a number of disease states, ranging from schizophrenia to autism spectrum disorders, display abnormal dendritic spine morphology or numbers  Moreover, recent genetic studies have identified mutations in numerous genes that encode synaptic proteins, leading to suggestions that these proteins may contribute to aberrant spine plasticity that, in part, underlie the pathophysiology of these disorders  In order to study the potential role of these proteins in controlling dendritic spine morphologies/number, the use of cultured cortical neurons offers several advantages  Firstly, this system allows for high-resolution imaging of dendritic spines in fixed cells as well as time-lapse imaging of live cells  Secondly, this in vitro system allows for easy manipulation of protein function by expression of mutant proteins, knockdown by shRNA constructs, or pharmacological treatments  These techniques allow researchers to begin to dissect the role of disease-associated proteins and to predict how mutations of these proteins may function in vivo 
Play ButtonIsolation and Culture of Mouse Cortical AstrocytesAuthors: Sebastian Schildge, Christian Bohrer, Kristina Beck, Christian Schachtrup  Institutions: University of Freiburg , University of Freiburg  Astrocytes are an abundant cell type in the mammalian brain, yet much remains to be learned about their molecular and functional characteristics  In vitro astrocyte cell culture systems can be used to study the biological functions of these glial cells in detail  This video protocol shows how to obtain pure astrocytes by isolation and culture of mixed cortical cells of mouse pups  The method is based on the absence of viable neurons and the separation of astrocytes, oligodendrocytes and microglia, the three main glial cell populations of the central nervous system, in culture  Representative images during the first days of culture demonstrate the presence of a mixed cell population and indicate the timepoint, when astrocytes become confluent and should be separated from microglia and oligodendrocytes  Moreover, we demonstrate purity and astrocytic morphology of cultured astrocytes using immunocytochemical stainings for well established and newly described astrocyte markers  This culture system can be easily used to obtain pure mouse astrocytes and astrocyte-conditioned medium for studying various aspects of astrocyte biology Neuroscience, Issue 71, Neurobiology, Cellular Biology, Medicine, Molecular Biology, Anatomy, Physiology, brain, mouse, astrocyte culture, astrocyte, fibroblast, fibrinogen, chondroitin sulfate proteoglycan, neuronal regeneration, cell culture, animal model50079Play ButtonImaging Dendritic Spines of Rat Primary Hippocampal Neurons using Structured Illumination MicroscopyAuthors: Marijn Schouten, Giulia M  R  De Luca, Diana K  Alatriste González, Babette E  de Jong, Wendy Timmermans, Hui Xiong, Harm Krugers, Erik M  M  Manders, Carlos P  Fitzsimons  Institutions: University of Amsterdam, University of Amsterdam Dendritic spines are protrusions emerging from the dendrite of a neuron and represent the primary postsynaptic targets of excitatory inputs in the brain  Technological advances have identified these structures as key elements in neuron connectivity and synaptic plasticity  The quantitative analysis of spine morphology using light microscopy remains an essential problem due to technical limitations associated with light's intrinsic refraction limit  Dendritic spines can be readily identified by confocal laser-scanning fluorescence microscopy  However, measuring subtle changes in the shape and size of spines is difficult because spine dimensions other than length are usually smaller than conventional optical resolution fixed by light microscopy's theoretical resolution limit of 200 nm 
Several recently developed super resolution techniques have been used to image cellular structures smaller than the 200 nm, including dendritic spines  These techniques are based on classical far-field operations and therefore allow the use of existing sample preparation methods and to image beyond the surface of a specimen  Described here is a working protocol to apply super resolution structured illumination microscopy (SIM) to the imaging of dendritic spines in primary hippocampal neuron cultures  Possible applications of SIM overlap with those of confocal microscopy  However, the two techniques present different applicability  SIM offers higher effective lateral resolution, while confocal microscopy, due to the usage of a physical pinhole, achieves resolution improvement at the expense of removal of out of focus light  In this protocol, primary neurons are cultured on glass coverslips using a standard protocol, transfected with DNA plasmids encoding fluorescent proteins and imaged using SIM  The whole protocol described herein takes approximately 2 weeks, because dendritic spines are imaged after 16-17 days in vitro, when dendritic development is optimal  After completion of the protocol, dendritic spines can be reconstructed in 3D from series of SIM image stacks using specialized software Neuroscience, Issue 87, Dendritic Spine, Microscopy, Confocal, Fluorescence, Neurosciences, hippocampus, primary neuron, super resolution microscopy, structured illumination microscopy (SIM), neuroscience, dendrite51276Play ButtonSetting-up an In Vitro Model of Rat Blood-brain Barrier (BBB): A Focus on BBB Impermeability and Receptor-mediated TransportAuthors: Yves Molino, Françoise Jabès, Emmanuelle Lacassagne, Nicolas Gaudin, Michel Khrestchatisky  Institutions: VECT-HORUS SAS, CNRS, NICN UMR 7259 The blood brain barrier (BBB) specifically regulates molecular and cellular flux between the blood and the nervous tissue  Our aim was to develop and characterize a highly reproducible rat syngeneic in vitro model of the BBB using co-cultures of primary rat brain endothelial cells (RBEC) and astrocytes to study receptors involved in transcytosis across the endothelial cell monolayer  Astrocytes were isolated by mechanical dissection following trypsin digestion and were frozen for later co-culture  RBEC were isolated from 5-week-old rat cortices  The brains were cleaned of meninges and white matter, and mechanically dissociated following enzymatic digestion  Thereafter, the tissue homogenate was centrifuged in bovine serum albumin to separate vessel fragments from nervous tissue  The vessel fragments underwent a second enzymatic digestion to free endothelial cells from their extracellular matrix  The remaining contaminating cells such as pericytes were further eliminated by plating the microvessel fragments in puromycin-containing medium  They were then passaged onto filters for co-culture with astrocytes grown on the bottom of the wells  RBEC expressed high levels of tight junction (TJ) proteins such as occludin, claudin-5 and ZO-1 with a typical localization at the cell borders  The transendothelial electrical resistance (TEER) of brain endothelial monolayers, indicating the tightness of TJs reached 300 ohm·cm2 on average  The endothelial permeability coefficients (Pe) for lucifer yellow (LY) was highly reproducible with an average of 0 26 ± 0 11 x 10-3 cm/min  Brain endothelial cells organized in monolayers expressed the efflux transporter P-glycoprotein (P-gp), showed a polarized transport of rhodamine 123, a ligand for P-gp, and showed specific transport of transferrin-Cy3 and DiILDL across the endothelial cell monolayer  In conclusion, we provide a protocol for setting up an in vitro BBB model that is highly reproducible due to the quality assurance methods, and that is suitable for research on BBB transporters and receptors Medicine, Issue 88, rat brain endothelial cells (RBEC), mouse, spinal cord, tight junction (TJ), receptor-mediated transport (RMT), low density lipoprotein (LDL), LDLR, transferrin, TfR, P-glycoprotein (P-gp), transendothelial electrical resistance (TEER),51278Play ButtonInducing Plasticity of Astrocytic Receptors by Manipulation of Neuronal Firing RatesAuthors: Alison X  Xie, Kelli Lauderdale, Thomas Murphy, Timothy L  Myers, Todd A  Fiacco  Institutions: University of California Riverside, University of California Riverside, University of California Riverside Close to two decades of research has established that astrocytes in situ and in vivo express numerous G protein-coupled receptors (GPCRs) that can be stimulated by neuronally-released transmitter  However, the ability of astrocytic receptors to exhibit plasticity in response to changes in neuronal activity has received little attention  Here we describe a model system that can be used to globally scale up or down astrocytic group I metabotropic glutamate receptors (mGluRs) in acute brain slices  Included are methods on how to prepare parasagittal hippocampal slices, construct chambers suitable for long-term slice incubation, bidirectionally manipulate neuronal action potential frequency, load astrocytes and astrocyte processes with fluorescent Ca2+ indicator, and measure changes in astrocytic Gq GPCR activity by recording spontaneous and evoked astrocyte Ca2+ events using confocal microscopy  In essence, a “calcium roadmap” is provided for how to measure plasticity of astrocytic Gq GPCRs  Applications of the technique for study of astrocytes are discussed  Having an understanding of how astrocytic receptor signaling is affected by changes in neuronal activity has important implications for both normal synaptic function as well as processes underlying neurological disorders and neurodegenerative disease Neuroscience, Issue 85, astrocyte, plasticity, mGluRs, neuronal Firing, electrophysiology, Gq GPCRs, Bolus-loading, calcium, microdomains, acute slices, Hippocampus, mouse51458Play ButtonInhibitory Synapse Formation in a Co-culture Model Incorporating GABAergic Medium Spiny Neurons and HEK293 Cells Stably Expressing GABAA ReceptorsAuthors: Laura E  Brown, Celine Fuchs, Martin W  Nicholson, F  Anne Stephenson, Alex M  Thomson, Jasmina N  Jovanovic  Institutions: University College London Inhibitory neurons act in the central nervous system to regulate the dynamics and spatio-temporal co-ordination of neuronal networks  GABA (γ-aminobutyric acid) is the predominant inhibitory neurotransmitter in the brain  It is released from the presynaptic terminals of inhibitory neurons within highly specialized intercellular junctions known as synapses, where it binds to GABAA receptors (GABAARs) present at the plasma membrane of the synapse-receiving, postsynaptic neurons  Activation of these GABA-gated ion channels leads to influx of chloride resulting in postsynaptic potential changes that decrease the probability that these neurons will generate action potentials  During development, diverse types of inhibitory neurons with distinct morphological, electrophysiological and neurochemical characteristics have the ability to recognize their target neurons and form synapses which incorporate specific GABAARs subtypes  This principle of selective innervation of neuronal targets raises the question as to how the appropriate synaptic partners identify each other  To elucidate the underlying molecular mechanisms, a novel in vitro co-culture model system was established, in which medium spiny GABAergic neurons, a highly homogenous population of neurons isolated from the embryonic striatum, were cultured with stably transfected HEK293 cell lines that express different GABAAR subtypes  Synapses form rapidly, efficiently and selectively in this system, and are easily accessible for quantification  Our results indicate that various GABAAR subtypes differ in their ability to promote synapse formation, suggesting that this reduced in vitro model system can be used to reproduce, at least in part, the in vivo conditions required for the recognition of the appropriate synaptic partners and formation of specific synapses  Here the protocols for culturing the medium spiny neurons and generating HEK293 cells lines expressing GABAARs are first described, followed by detailed instructions on how to combine these two cell types in co-culture and analyze the formation of synaptic contacts  Neuroscience, Issue 93, Developmental neuroscience, synaptogenesis, synaptic inhibition, co-culture, stable cell lines, GABAergic, medium spiny neurons, HEK 293 cell line52115Play ButtonTwo-Photon in vivo Imaging of Dendritic Spines in the Mouse Cortex Using a Thinned-skull PreparationAuthors: Xinzhu Yu, Yi Zuo  Institutions: University of California, Santa Cruz In the mammalian cortex, neurons form extremely complicated networks and exchange information at synapses  Changes in synaptic strength, as well as addition/removal of synapses, occur in an experience-dependent manner, providing the structural foundation of neuronal plasticity  As postsynaptic components of the most excitatory synapses in the cortex, dendritic spines are considered to be a good proxy of synapses  Taking advantages of mouse genetics and fluorescent labeling techniques, individual neurons and their synaptic structures can be labeled in the intact brain  Here we introduce a transcranial imaging protocol using two-photon laser scanning microscopy to follow fluorescently labeled postsynaptic dendritic spines over time in vivo  This protocol utilizes a thinned-skull preparation, which keeps the skull intact and avoids inflammatory effects caused by exposure of the meninges and the cortex  Therefore, images can be acquired immediately after surgery is performed  The experimental procedure can be performed repetitively over various time intervals ranging from hours to years  The application of this preparation can also be expanded to investigate different cortical regions and layers, as well as other cell types, under physiological and pathological conditions Neuroscience, Issue 87, dendritic spine, mouse cortex, in vivo, two-photon microscopy, thinned-skull, imaging51520Play ButtonModeling Astrocytoma Pathogenesis In Vitro and In Vivo Using Cortical Astrocytes or Neural Stem Cells from Conditional, Genetically Engineered MiceAuthors: Robert S  McNeill, Ralf S  Schmid, Ryan E  Bash, Mark Vitucci, Kristen K  White, Andrea M  Werneke, Brian H  Constance, Byron Huff, C  Ryan Miller  Institutions: University of North Carolina School of Medicine, University of North Carolina School of Medicine, University of North Carolina School of Medicine, University of North Carolina School of Medicine, University of North Carolina School of Medicine, Emory University School of Medicine, University of North Carolina School of Medicine Current astrocytoma models are limited in their ability to define the roles of oncogenic mutations in specific brain cell types during disease pathogenesis and their utility for preclinical drug development  In order to design a better model system for these applications, phenotypically wild-type cortical astrocytes and neural stem cells (NSC) from conditional, genetically engineered mice (GEM) that harbor various combinations of floxed oncogenic alleles were harvested and grown in culture  Genetic recombination was induced in vitro using adenoviral Cre-mediated recombination, resulting in expression of mutated oncogenes and deletion of tumor suppressor genes  The phenotypic consequences of these mutations were defined by measuring proliferation, transformation, and drug response in vitro  Orthotopic allograft models, whereby transformed cells are stereotactically injected into the brains of immune-competent, syngeneic littermates, were developed to define the role of oncogenic mutations and cell type on tumorigenesis in vivo  Unlike most established human glioblastoma cell line xenografts, injection of transformed GEM-derived cortical astrocytes into the brains of immune-competent littermates produced astrocytomas, including the most aggressive subtype, glioblastoma, that recapitulated the histopathological hallmarks of human astrocytomas, including diffuse invasion of normal brain parenchyma  Bioluminescence imaging of orthotopic allografts from transformed astrocytes engineered to express luciferase was utilized to monitor in vivo tumor growth over time  Thus, astrocytoma models using astrocytes and NSC harvested from GEM with conditional oncogenic alleles provide an integrated system to study the genetics and cell biology of astrocytoma pathogenesis in vitro and in vivo and may be useful in preclinical drug development for these devastating diseases Neuroscience, Issue 90, astrocytoma, cortical astrocytes, genetically engineered mice, glioblastoma, neural stem cells, orthotopic allograft51763Play ButtonPaired Whole Cell Recordings in Organotypic Hippocampal SlicesAuthors: Chantelle Fourie, Marianna Kiraly, Daniel V  Madison, Johanna M  Montgomery  Institutions: University of Auckland, Stanford University Pair recordings involve simultaneous whole cell patch clamp recordings from two synaptically connected neurons, enabling not only direct electrophysiological characterization of the synaptic connections between individual neurons, but also pharmacological manipulation of either the presynaptic or the postsynaptic neuron  When carried out in organotypic hippocampal slice cultures, the probability that two neurons are synaptically connected is significantly increased  This preparation readily enables identification of cell types, and the neurons maintain their morphology and properties of synaptic function similar to that in native brain tissue  A major advantage of paired whole cell recordings is the highly precise information it can provide on the properties of synaptic transmission and plasticity that are not possible with other more crude techniques utilizing extracellular axonal stimulation  Paired whole cell recordings are often perceived as too challenging to perform  While there are challenging aspects to this technique, paired recordings can be performed by anyone trained in whole cell patch clamping provided specific hardware and methodological criteria are followed  The probability of attaining synaptically connected paired recordings significantly increases with healthy organotypic slices and stable micromanipulation allowing independent attainment of pre- and postsynaptic whole cell recordings  While CA3-CA3 pyramidal cell pairs are most widely used in the organotypic slice hippocampal preparation, this technique has also been successful in CA3-CA1 pairs and can be adapted to any neurons that are synaptically connected in the same slice preparation  In this manuscript we provide the detailed methodology and requirements for establishing this technique in any laboratory equipped for electrophysiology Neuroscience, Issue 91, hippocampus, paired recording, whole cell recording, organotypic slice, synapse, synaptic transmission, synaptic plasticity51958Play ButtonImaging Intracellular Ca2+ Signals in Striatal Astrocytes from Adult Mice Using Genetically-encoded Calcium IndicatorsAuthors: Ruotian Jiang, Martin D  Haustein, Michael V  Sofroniew, Baljit S  Khakh  Institutions: University of California Los Angeles, University of California Los Angeles Astrocytes display spontaneous intracellular Ca2+ concentration fluctuations ([Ca2+]i) and in several settings respond to neuronal excitation with enhanced [Ca2+]i signals  It has been proposed that astrocytes in turn regulate neurons and blood vessels through calcium-dependent mechanisms, such as the release of signaling molecules  However, [Ca2+]i imaging in entire astrocytes has only recently become feasible with genetically encoded calcium indicators (GECIs) such as the GCaMP series  The use of GECIs in astrocytes now provides opportunities to study astrocyte [Ca2+]i signals in detail within model microcircuits such as the striatum, which is the largest nucleus of the basal ganglia  In the present report, detailed surgical methods to express GECIs in astrocytes in vivo, and confocal imaging approaches to record [Ca2+]i signals in striatal astrocytes in situ, are described  We highlight precautions, necessary controls and tests to determine if GECI expression is selective for astrocytes and to evaluate signs of overt astrocyte reactivity  We also describe brain slice and imaging conditions in detail that permit reliable [Ca2+]i imaging in striatal astrocytes in situ  The use of these approaches revealed the entire territories of single striatal astrocytes and spontaneous [Ca2+]i signals within their somata, branches and branchlets  The further use and expansion of these approaches in the striatum will allow for the detailed study of astrocyte [Ca2+]i signals in the striatal microcircuitry Neuroscience, Issue 93, astrocyte, calcium, striatum, GECI, GCaMP3, AAV2/5, stereotaxic injection, brain slice, imaging51972Play ButtonMethods to Assess Subcellular Compartments of Muscle in C  elegansAuthors: Christopher J  Gaffney, Joseph J  Bass, Thomas F  Barratt, Nathaniel J  Szewczyk  Institutions: University of Nottingham Muscle is a dynamic tissue that responds to changes in nutrition, exercise, and disease state  The loss of muscle mass and function with disease and age are significant public health burdens  We currently understand little about the genetic regulation of muscle health with disease or age  The nematode C  elegans is an established model for understanding the genomic regulation of biological processes of interest  This worm’s body wall muscles display a large degree of homology with the muscles of higher metazoan species  Since C  elegans is a transparent organism, the localization of GFP to mitochondria and sarcomeres allows visualization of these structures in vivo  Similarly, feeding animals cationic dyes, which accumulate based on the existence of a mitochondrial membrane potential, allows the assessment of mitochondrial function in vivo  These methods, as well as assessment of muscle protein homeostasis, are combined with assessment of whole animal muscle function, in the form of movement assays, to allow correlation of sub-cellular defects with functional measures of muscle performance  Thus, C  elegans provides a powerful platform with which to assess the impact of mutations, gene knockdown, and/or chemical compounds upon muscle structure and function  Lastly, as GFP, cationic dyes, and movement assays are assessed non-invasively, prospective studies of muscle structure and function can be conducted across the whole life course and this at present cannot be easily investigated in vivo in any other organism Developmental Biology, Issue 93, Physiology, C  elegans, muscle, mitochondria, sarcomeres, ageing52043Play ButtonImproved Preparation and Preservation of Hippocampal Mouse Slices for a Very Stable and Reproducible Recording of Long-term PotentiationAuthors: Agnès Villers, Laurence Ris  Institutions: University of Mons Long-term potentiation (LTP) is a type of synaptic plasticity characterized by an increase in synaptic strength and believed to be involved in memory encoding  LTP elicited in the CA1 region of acute hippocampal slices has been extensively studied  However the molecular mechanisms underlying the maintenance phase of this phenomenon are still poorly understood  This could be partly due to the various experimental conditions used by different laboratories  Indeed, the maintenance phase of LTP is strongly dependent on external parameters like oxygenation, temperature and humidity  It is also dependent on internal parameters like orientation of the slicing plane and slice viability after dissection 
The optimization of all these parameters enables the induction of a very reproducible and very stable long-term potentiation  This methodology offers the possibility to further explore the molecular mechanisms involved in the stable increase in synaptic strength in hippocampal slices  It also highlights the importance of experimental conditions in in vitro investigation of neurophysiological phenomena Neuroscience, Issue 76, Neurobiology, Anatomy, Physiology, Biomedical Engineering, Surgery, Memory Disorders, Learning, Memory, Neurosciences, Neurophysiology, hippocampus, long-term potentiation, mice, acute slices, synaptic plasticity, in vitro, electrophysiology, animal model50483Play ButtonIn Vivo Modeling of the Morbid Human Genome using Danio rerioAuthors: Adrienne R  Niederriter, Erica E  Davis, Christelle Golzio, Edwin C  Oh, I-Chun Tsai, Nicholas Katsanis  Institutions: Duke University Medical Center, Duke University, Duke University Medical Center Here, we present methods for the development of assays to query potentially clinically significant nonsynonymous changes using in vivo complementation in zebrafish  Zebrafish (Danio rerio) are a useful animal system due to their experimental tractability; embryos are transparent to enable facile viewing, undergo rapid development ex vivo, and can be genetically manipulated 1 These aspects have allowed for significant advances in the analysis of embryogenesis, molecular processes, and morphogenetic signaling  Taken together, the advantages of this vertebrate model make zebrafish highly amenable to modeling the developmental defects in pediatric disease, and in some cases, adult-onset disorders  Because the zebrafish genome is highly conserved with that of humans (~70% orthologous), it is possible to recapitulate human disease states in zebrafish  This is accomplished either through the injection of mutant human mRNA to induce dominant negative or gain of function alleles, or utilization of morpholino (MO) antisense oligonucleotides to suppress genes to mimic loss of function variants  Through complementation of MO-induced phenotypes with capped human mRNA, our approach enables the interpretation of the deleterious effect of mutations on human protein sequence based on the ability of mutant mRNA to rescue a measurable, physiologically relevant phenotype  Modeling of the human disease alleles occurs through microinjection of zebrafish embryos with MO and/or human mRNA at the 1-4 cell stage, and phenotyping up to seven days post fertilization (dpf)  This general strategy can be extended to a wide range of disease phenotypes, as demonstrated in the following protocol  We present our established models for morphogenetic signaling, craniofacial, cardiac, vascular integrity, renal function, and skeletal muscle disorder phenotypes, as well as others  Molecular Biology, Issue 78, Genetics, Biomedical Engineering, Medicine, Developmental Biology, Biochemistry, Anatomy, Physiology, Bioengineering, Genomics, Medical, zebrafish, in vivo, morpholino, human disease modeling, transcription, PCR, mRNA, DNA, Danio rerio, animal model50338Play ButtonDirect Imaging of ER Calcium with Targeted-Esterase Induced Dye Loading (TED)Authors: Samira Samtleben, Juliane Jaepel, Caroline Fecher, Thomas Andreska, Markus Rehberg, Robert Blum  Institutions: University of Wuerzburg, Max Planck Institute of Neurobiology, Martinsried, Ludwig-Maximilians University of Munich Visualization of calcium dynamics is important to understand the role of calcium in cell physiology  To examine calcium dynamics, synthetic fluorescent Ca2+ indictors have become popular  Here we demonstrate TED (= targeted-esterase induced dye loading), a method to improve the release of Ca2+ indicator dyes in the ER lumen of different cell types  To date, TED was used in cell lines, glial cells, and neurons in vitro  TED bases on efficient, recombinant targeting of a high carboxylesterase activity to the ER lumen using vector-constructs that express Carboxylesterases (CES)  The latest TED vectors contain a core element of CES2 fused to a red fluorescent protein, thus enabling simultaneous two-color imaging  The dynamics of free calcium in the ER are imaged in one color, while the corresponding ER structure appears in red  At the beginning of the procedure, cells are transduced with a lentivirus  Subsequently, the infected cells are seeded on coverslips to finally enable live cell imaging  Then, living cells are incubated with the acetoxymethyl ester (AM-ester) form of low-affinity Ca2+ indicators, for instance Fluo5N-AM, Mag-Fluo4-AM, or Mag-Fura2-AM  The esterase activity in the ER cleaves off hydrophobic side chains from the AM form of the Ca2+ indicator and a hydrophilic fluorescent dye/Ca2+ complex is formed and trapped in the ER lumen  After dye loading, the cells are analyzed at an inverted confocal laser scanning microscope  Cells are continuously perfused with Ringer-like solutions and the ER calcium dynamics are directly visualized by time-lapse imaging  Calcium release from the ER is identified by a decrease in fluorescence intensity in regions of interest, whereas the refilling of the ER calcium store produces an increase in fluorescence intensity  Finally, the change in fluorescent intensity over time is determined by calculation of ΔF/F0 Cellular Biology, Issue 75, Neurobiology, Neuroscience, Molecular Biology, Biochemistry, Biomedical Engineering, Bioengineering, Virology, Medicine, Anatomy, Physiology, Surgery, Endoplasmic Reticulum, ER, Calcium Signaling, calcium store, calcium imaging, calcium indicator, metabotropic signaling, Ca2+, neurons, cells, mouse, animal model, cell culture, targeted esterase induced dye loading, imaging50317Play ButtonPreparation of Dissociated Mouse Cortical Neuron CulturesAuthors: Lutz G  W  Hilgenberg, Martin A  Smith  Institutions: University of California, Irvine (UCI) This video will guide you through the process for generating cortical neuronal cultures from late embryo and early postnatal mouse brain  These cultures can be used for a variety of applications including immunocytochemistry, biochemistry, electrophysiology, calcium and sodium imaging, protein and/or RNA isolation  These cultures also provide a platform to study the neuronal development of transgenic animals that carry a late embryonic or postnatal lethal gene mutation  The procedure is relatively straight forward, requires some experience in tissue culture technique and should not take longer than two to three hours if you are properly prepared  Careful separation of the cortical rind from the thalamo-cortical fiber tract will reduce the number of unwanted non-neuronal cells  To increase yields of neuronal cells triturate the pieces of the cortical tissue gently after the enzyme incubation step  This is imperative as it prevents unnecessary injury to cells and premature neuronal cell death  Since these cultures are maintained in the absence of glia feeder cells, they also offer an added advantage of growing cultures enriched in neurons Neuroscience, Issue 10, cellular, molecular, neurobiology, neuron, calcium/sodium imaging, primary cultures, mouse562Play ButtonAnalysis of Schwann-astrocyte Interactions Using In Vitro AssaysAuthors: Fardad T  Afshari, Jessica C  Kwok, James W  Fawcett  Institutions: University of Cambridge Schwann cells are one of the commonly used cells in repair strategies following spinal cord injuries  Schwann cells are capable of supporting axonal regeneration and sprouting by secreting growth factors 1,2 and providing growth promoting adhesion molecules 3 and extracellular matrix molecules 4  In addition they myelinate the demyelinated axons at the site of injury 5 
However following transplantation, Schwann cells do not migrate from the site of implant and do not intermingle with the host astrocytes 6,7  This results in formation of a sharp boundary between the Schwann cells and astrocytes, creating an obstacle for growing axons trying to exit the graft back into the host tissue proximally and distally  Astrocytes in contact with Schwann cells also undergo hypertrophy and up-regulate the inhibitory molecules 8-13 
In vitro assays have been used to model Schwann cell-astrocyte interactions and have been important in understanding the mechanism underlying the cellular behaviour 
These in vitro assays include boundary assay, where a co-culture is made using two different cells with each cell type occupying different territories with only a small gap separating the two cell fronts  As the cells divide and migrate, the two cellular fronts get closer to each other and finally collide  This allows the behaviour of the two cellular populations to be analyzed at the boundary  Another variation of the same technique is to mix the two cellular populations in culture and over time the two cell types segregate with Schwann cells clumped together as islands in between astrocytes together creating multiple Schwann-astrocyte boundaries 
The second assay used in studying the interaction of two cell types is the migration assay where cellular movement can be tracked on the surface of the other cell type monolayer 14,15  This assay is commonly known as inverted coverslip assay  Schwann cells are cultured on small glass fragments and they are inverted face down onto the surface of astrocyte monolayers and migration is assessed from the edge of coverslip 
Both assays have been instrumental in studying the underlying mechanisms involved in the cellular exclusion and boundary formation  Some of the molecules identified using these techniques include N-Cadherins 15, Chondroitin Sulphate proteoglycans(CSPGs) 16,17, FGF/Heparin 18, Eph/Ephrins19 
This article intends to describe boundary assay and migration assay in stepwise fashion and elucidate the possible technical problems that might occur Cellular Biology, Issue 47, Schwann cell, astrocyte, boundary, migration, repulsion2214Play ButtonQuantifying Synapses: an Immunocytochemistry-based Assay to Quantify Synapse NumberAuthors: Dominic M  Ippolito, Cagla Eroglu  Institutions: Duke University, Duke University One of the most important goals in neuroscience is to understand the molecular cues that instruct early stages of synapse formation  As such it has become imperative to develop objective approaches to quantify changes in synaptic connectivity  Starting from sample fixation, this protocol details how to quantify synapse number both in dissociated neuronal culture and in brain sections using immunocytochemistry  Using compartment-specific antibodies, we label presynaptic terminals as well as sites of postsynaptic specialization  We define synapses as points of colocalization between the signals generated by these markers  The number of these colocalizations is quantified using a plug in Puncta Analyzer (written by Bary Wark, available upon request, c eroglu@cellbio duke edu) under the ImageJ analysis software platform  The synapse assay described in this protocol can be applied to any neural tissue or culture preparation for which you have selective pre- and postsynaptic markers  This synapse assay is a valuable tool that can be widely utilized in the study of synaptic development Neuroscience, Issue 45, synapse, immunocytochemistry, brain, neuron, astrocyte2270Play ButtonPreparation of Acute Hippocampal Slices from Rats and Transgenic Mice for the Study of Synaptic Alterations during Aging and Amyloid PathologyAuthors: Diana M  Mathis, Jennifer L  Furman, Christopher M  Norris  Institutions: University of Kentucky College of Public Health, University of Kentucky College of Medicine, University of Kentucky College of Medicine The rodent hippocampal slice preparation is perhaps the most broadly used tool for investigating mammalian synaptic function and plasticity  The hippocampus can be extracted quickly and easily from rats and mice and slices remain viable for hours in oxygenated artificial cerebrospinal fluid  Moreover, basic electrophysisologic techniques are easily applied to the investigation of synaptic function in hippocampal slices and have provided some of the best biomarkers for cognitive impairments  The hippocampal slice is especially popular for the study of synaptic plasticity mechanisms involved in learning and memory  Changes in the induction of long-term potentiation and depression (LTP and LTD) of synaptic efficacy in hippocampal slices (or lack thereof) are frequently used to describe the neurologic phenotype of cognitively-impaired animals and/or to evaluate the mechanism of action of nootropic compounds  This article outlines the procedures we use for preparing hippocampal slices from rats and transgenic mice for the study of synaptic alterations associated with brain aging and Alzheimer's disease (AD)1-3  Use of aged rats and AD model mice can present a unique set of challenges to researchers accustomed to using younger rats and/or mice in their research  Aged rats have thicker skulls and tougher connective tissue than younger rats and mice, which can delay brain extraction and/or dissection and consequently negate or exaggerate real age-differences in synaptic function and plasticity  Aging and amyloid pathology may also exacerbate hippocampal damage sustained during the dissection procedure, again complicating any inferences drawn from physiologic assessment  Here, we discuss the steps taken during the dissection procedure to minimize these problems  Examples of synaptic responses acquired in "healthy" and "unhealthy" slices from rats and mice are provided, as well as representative synaptic plasticity experiments  The possible impact of other methodological factors on synaptic function in these animal models (e g  recording solution components, stimulation parameters) are also discussed  While the focus of this article is on the use of aged rats and transgenic mice, novices to slice physiology should find enough detail here to get started on their own studies, using a variety of rodent models Neuroscience, Issue 49, aging, amyloid, hippocampal slice, synaptic plasticity, Ca2+, CA1, electrophysiology2330Play ButtonMesenteric Artery Contraction and Relaxation Studies Using Automated Wire MyographyAuthors: Lakeesha E  Bridges, Cicely L  Williams, Mildred A  Pointer, Emmanuel M  Awumey  Institutions: North Carolina Central University, Durham, North Carolina Central University, Durham, Wake Forest University School of Medicine Proximal resistance vessels, such as the mesenteric arteries, contribute substantially to the peripheral resistance  These small vessels of between 100-400 μm in diameter function primarily in directing blood flow to various organs according to the overall requirements of the body  The rat mesenteric artery has a diameter greater than 100 μm  The myography technique, first described by Mulvay and Halpern1, was based on the method proposed by Bevan and Osher2  The technique provides information about small vessels under isometric conditions, where substantial shortening of the muscle preparation is prevented  Since force production and sensitivity of vessels to different agonists is dependent on the extent of stretch, according to active tension-length relation, it is essential to conduct contraction studies under isometric conditions to prevent compliance of the mounting wires  Stainless steel wires are preferred to tungsten wires because of oxidation of the latter, which affects recorded responses3 The technique allows for the comparison of agonist-induced contractions of mounted vessels to obtain evidence for normal function of vascular smooth muscle cell receptors 
Medicine, Issue 55, cardiovascular, resistant arteries, contraction, relaxation, myography3119Play ButtonVisualization and Genetic Manipulation of Dendrites and Spines in the Mouse Cerebral Cortex and Hippocampus using In utero ElectroporationAuthors: Emilie Pacary, Matilda A  Haas, Hendrik Wildner, Roberta Azzarelli, Donald M  Bell, Djoher Nora Abrous, François Guillemot  Institutions: MRC National Institute for Medical Research, National Institute for Medical Research, Université de Bordeaux In utero electroporation (IUE) has become a powerful technique to study the development of different regions of the embryonic nervous system 1-5  To date this tool has been widely used to study the regulation of cellular proliferation, differentiation and neuronal migration especially in the developing cerebral cortex 6-8  Here we detail our protocol to electroporate in utero the cerebral cortex and the hippocampus and provide evidence that this approach can be used to study dendrites and spines in these two cerebral regions 
Finally, IUE provides a useful tool to identify functional interactions between genes involved in dendrite, spine and/or synapse development  Indeed, in contrast to other gene transfer methods such as virus, it is straightforward to combine multiple RNAi or transgenes in the same population of cells  In summary, IUE is a powerful method that has already contributed to the characterization of molecular mechanisms underlying brain function and disease and it should also be useful in the study of dendrites and spines Neuroscience, Issue 65, Developmental Biology, Molecular Biology, Neuronal development, In utero electroporation, dendrite, spines, hippocampus, cerebral cortex, gain and loss of function4163Play ButtonImaging Analysis of Neuron to Glia Interaction in Microfluidic Culture Platform (MCP)-based Neuronal Axon and Glia Co-culture SystemAuthors: Haruki Higashimori, Yongjie Yang  Institutions: Tufts University, Tufts Sackler School of Graduate Biomedical Sciences Proper neuron to glia interaction is critical to physiological function of the central nervous system (CNS)  This bidirectional communication is sophisticatedly mediated by specific signaling pathways between neuron and glia1,2   Identification and characterization of these signaling pathways is essential to the understanding of how neuron to glia interaction shapes CNS physiology  Previously, neuron and glia mixed cultures have been widely utilized for testing and characterizing signaling pathways between neuron and glia  What we have learned from these preparations and other in vivo tools, however, has suggested that mutual signaling between neuron and glia often occurred in specific compartments within neurons (i e , axon, dendrite, or soma)3  This makes it important to develop a new culture system that allows separation of neuronal compartments and specifically examines the interaction between glia and neuronal axons/dendrites  In addition, the conventional mixed culture system is not capable of differentiating the soluble factors and direct membrane contact signals between neuron and glia  Furthermore, the large quantity of neurons and glial cells in the conventional co-culture system lacks the resolution necessary to observe the interaction between a single axon and a glial cell 
In this study, we describe a novel axon and glia co-culture system with the use of a microfluidic culture platform (MCP)  In this co-culture system, neurons and glial cells are cultured in two separate chambers that are connected through multiple central channels  In this microfluidic culture platform, only neuronal processes (especially axons) can enter the glial side through the central channels  In combination with powerful fluorescent protein labeling, this system allows direct examination of signaling pathways between axonal/dendritic and glial interactions, such as axon-mediated transcriptional regulation in glia, glia-mediated receptor trafficking in neuronal terminals, and glia-mediated axon growth  The narrow diameter of the chamber also significantly prohibits the flow of the neuron-enriched medium into the glial chamber, facilitating probing of the direct membrane-protein interaction between axons/dendrites and glial surfaces Neuroscience, Issue 68, Molecular Biology, Cellular Biology, Biophysics, Microfluidics, Microfluidic culture platform, Compartmented culture, Neuron to glia signaling, neurons, glia, cell culture4448Play ButtonFluorescence Recovery After Photobleaching (FRAP) of Fluorescence Tagged Proteins in Dendritic Spines of Cultured Hippocampal NeuronsAuthors: Chan-Ying Zheng, Ronald S  Petralia, Ya-Xian Wang, Bechara Kachar  Institutions: National Institutes of Health, Bethesda FRAP has been used to quantify the mobility of GFP-tagged proteins  Using a strong excitation laser, the fluorescence of a GFP-tagged protein is bleached in the region of interest  The fluorescence of the region recovers when the unbleached GFP-tagged protein from outside of the region diffuses into the region of interest  The mobility of the protein is then analyzed by measuring the fluorescence recovery rate  This technique could be used to characterize protein mobility and turnover rate 
This FRAP protocol shows how to perform a basic FRAP experiment as well as how to analyze the data Neuroscience, Issue 50, Spine, FRAP, hippocampal neurons, live cell imaging, protein mobility2568Play ButtonPrimary Neuronal Cultures from the Brains of Late Stage Drosophila PupaeAuthors: Beatriz Sicaeros, Jorge M  Campusano, Diane K  O'Dowd  Institutions: University of California, Irvine (UCI) In this video, we demonstrate the preparation of primary neuronal cultures from the brains of late stage Drosophila pupae  The procedure begins with the removal of brains from animals at 70-78 hrs after puparium formation  The isolated brains are shown after brief incubation in papain followed by several washes in serum-free growth medium  The process of mechanical dissociation of each brain in a 5 ul drop of media on a coverslip is illustrated  The axons and dendrites of the post-mitotic neurons are sheered off near the soma during dissociation but the neurons begin to regenerate processes within a few hours of plating  Images show live cultures at 2 days  Neurons continue to elaborate processes during the first week in culture  Specific neuronal populations can be identified in culture using GAL4 lines to drive tissue specific expression of fluorescent markers such as GFP or RFP  Whole cell recordings have demonstrated the cultured neurons form functional, spontaneously active cholinergic and GABAergic synapses  A short video segment illustrates calcium dynamics in the cultured neurons using Fura-2 as a calcium indicator dye to monitor spontaneous calcium transients and nicotine evoked calcium responses in a dish of cultured neurons  These pupal brain cultures are a useful model system in which genetic and pharmacological tools can be used to identify intrinsic and extrinsic factors that influence formation and function of central synapses 
Context 2: \section{Introduction}
In recent years, vehicular technology has attracted significant attention from the automotive and telecommunication industries, leading to the emergence of vehicle-to-everything (V2X) communications for improving road safety, traffic management services and driving comfort 
V2X supported by the sixth generation (6G) is envisioned to be a key enabler of future connected autonomous vehicles \cite{9779322}  Although its transformative benefits for leveraging intelligent transportation systems, V2X still face several technical issues mainly related to performance and security 

The integration of sensing and communication (ISAC) has emerged very recently as a revolutionary element of 6G that could potentially help enabling adaptive learning and intelligent decision-making in future V2X applications 
The combination of sensing and communication allows vehicles to perceive their surroundings better, predict manoeuvres from nearby users and make intelligent decisions, thus paving the way toward a safer transportation system \cite{9665433} 
Modernized vehicles are augmented with various types of sensors divided into exteroceptive to observe their surrounding environment and proprioceptive to observe their internal states 
The former like GPS, Lidar, and Cameras are conveyed to improve situational awareness, while latter sensors, such as steering, pedal, and wheel speed, convey to improve self-awareness  

While sensing the environment, vehicles can exchange messages that assist in improving situational- and self-awareness and in coordinating maneuvers with other vehicles 
Those messages like the basic safety (BSMs) and cooperative awareness messages (CAMs) are composed of transmitting vehicle's states such as position and velocity and other vehicles' states in the vicinity  Vehicles might use their sensors, such as cameras and Lidar, to detect road users (e g , pedestrians), which can be communicated with other road users via the V2X messages to improve the overall performance  However, V2X communication links carrying those messages are inherently vulnerable to malicious attacks due to the open and shared nature of the wireless spectrum among vehicles and other cellular users \cite{8336901}  For instance, a jammer in the vicinity might alter the information to be communicated to nearby vehicles/users or can intentionally disrupt communication between a platoon of vehicles making the legitimate signals unrecognizable for on-board units (OBUs) and/or road side units (RSUs) that endanger vehicular safety 
\cite{8553649} 

In addition, the integrity of GPS signals and the correct acquisition of navigation data to compute position, velocity and time information is critical in V2X applications for their safe operation  However, since civil GPS receivers rely on unencrypted satellite signals, spoofers can easily replicate them by deceiving the GPS receiver to compute falsified positions \cite{9226611} 
Also, the long distance between satellites and terrestrial GPS receivers leads to an extremely weak signal that can be easily drowned out by a spoofer  
Thus, GPS sensors' vulnerability to spoofing attacks poses a severe threat that might be causing vehicles to be out of control or even hijacked and endanger human life \cite{9881548} 
Therefore, GPS spoofing attacks and jamming interference needs to be controlled and detected in real-time to reach secured vehicular communications allowing vehicles to securely talk to each other and interact with the infrastructure (e g , roadside terminals, base stations) \cite{9860410} 

Existing methods for GPS spoofing detection include GPS signal analysis methods and GPS message encryption methods \cite{9845684}  However, the former requires the ground truth source during the detection process, which is not always possible to collect  In contrast, the latter involves support from a secured infrastructure and advanced computing resources on GPS receivers, which hinders their adoption in V2X applications  On the other hand, existing methods for jammer detection in vehicular networks are based on analysing the packet drop rate as in \cite{9484071}, making it difficult to detect an advanced jammer manipulating the legitimate signal instead of disrupting it 
In this work, we propose a method to jointly detect GPS spoofing and jamming attacks in the V2X network  A coupled generalized dynamic Bayesian network (C-GDBN) is employed to learn the interaction between RF signals received by the RSU from multiple vehicles and their corresponding trajectories  This integration of vehicles' positional information with vehicle-to-infrastructure (V2I) communications allows semantic learning while mapping RF signals with vehicles' trajectories and enables the RSU to jointly predict the RF signals it expects to receive from the vehicles from which it can anticipate the expected trajectories 

The main contributions of this paper can be summarized as follows: \textit{i)} A joint GPS spoofing and jamming detection method is proposed for the V2X scenario, which is based on learning a generative interactive model as the C-GDBN  Such a model encodes the cross-correlation between the RF signals transmitted by multiple vehicles and their trajectories, where their semantic meaning is coupled stochastically at a high abstraction level  \textit{ii)} A cognitive RSU equipped with the acquired C-GDBN can predict and estimate vehicle positions based on real-time RF signals  This allows RSU to evaluate whether both RF signals and vehicles' trajectories are evolving according to the dynamic rules encoded in the C-GDBN and, consequently, to identify the cause (i e , a jammer attacking the V2I or a spoofer attacking the satellite link) of the abnormal behaviour that occurred in the V2X environment  \textit{iii)} Extensive simulation results demonstrate that the proposed method accurately estimates the vehicles' trajectories from the predicted RF signals, effectively detect any abnormal behaviour and identify the type of abnormality occurring with high detection probabilities 
To our best knowledge, this is the first work that studies the joint detection of jamming and spoofing in V2X systems 

\section{System model and problem formulation}
The system model depicted in Fig ~\ref{fig_SystemModel}, includes a single cell vehicular network consisting of a road side unit (RSU) located at $\mathrm{p}_{R}=[{x}_{R},{y}_{R}]$, a road side jammer (RSJ) located at $\mathrm{p}_{J}=[{x}_{J},{y}_{J}]$, a road side spoofer (RSS) located at $\mathrm{p}_{s}=[{x}_{s},{y}_{s}]$ and $N$ vehicles moving along multi-lane road in an urban area  The time-varying positions of the $n$-th vehicle is given by $\mathrm{p}_{n,t}=[{x}_{n,t},{y}_{n,t}]$ where $n \in N$  Among the $K$ orthogonal subchannels available for the Vehicle-to-Infrastructure (V2I) communications, RSU assigns one V2I link to each vehicle  Each vehicle exchanges messages composed of the vehicle's state (i e , position and velocity) with RSU through the $k$-th V2I link by transmitting a signal $\textrm{x}_{t,k}$ carrying those messages at each time instant $t$ where $k \in K$  We consider a reactive RSJ that aims to attack the V2I link by injecting intentional interference to the communication link between vehicles and RSU to alter the transmitted signals by the vehicles  In contrast, the RSS purposes to mislead the vehicles by spoofing the GPS signal and so registering wrong GPS positions  RSU aims to detect both the spoofer on the satellite link and the jammer on multiple V2I links in order to take effective actions and protect the vehicular network  
The joint GPS spoofing and jamming detection problem can be formulated as the following ternary hypothesis test:
\begin{equation}
    \begin{cases}
        \mathcal{H}_{0}: \mathrm{z}_{t,k} = \mathrm{g}_{t,k}^{nR} \mathrm{x}_{t,k} + \mathrm{v}_{t,k}, \\
        \mathcal{H}_{1}: \mathrm{z}_{t,k} = \mathrm{g}_{t,k}^{nR} \mathrm{x}_{t,k} + \mathrm{g}_{t,k}^{JR} \mathrm{x}_{t,k}^{j} + \mathrm{v}_{t,k}, \\
        \mathcal{H}_{2}: \mathrm{z}_{t,k} = \mathrm{g}_{t,k}^{nR} \mathrm{x}_{t,k}^{*} + \mathrm{v}_{t,k},
    \end{cases}
\end{equation}
where $\mathcal{H}_{0}$, $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ denote three hypotheses corresponding to the absence of both jammer and spoofer, the presence of the jammer, and the presence of the spoofer, respectively  $\textrm{z}_{t,k}$ is the received signal at the RSU at $t$ over the $k$-th V2I link, $\textrm{g}_{t,k}^{nR}$ is the channel power gain from vehicle $n$ to the RSU formulated as: $\textrm{g}_{t,k}^{nR} = \alpha_{t,k}^{nR} \mathrm{h}_{t,k}^{nR}$, where $\alpha_{t,k}^{nR}$ is the large-scale fading including path-loss and shadowing modeled as \cite{8723178}: $\alpha_{t,k}^{nR}=G\beta d_{t,nR}^{-\gamma}$ 
\begin{figure}[t!]
    \centering
    \includegraphics[height=5 3cm]{Figures/SystemModel_V1 pdf}
    \caption{An illustration of the system model }
    \label{fig_SystemModel}
\end{figure}
$G$ is the pathloss constant, $\beta$ is a log normal shadow fading random variable, $d_{t,nR}=\sqrt{({x}_{n,t}-x_{R})^{2}+({y}_{n,t}-y_{R})^{2}}$ is the distance between the $n$-th vehicle and the RSU  $\gamma$ is the power decay exponent and
$\mathrm{h}_{t,k}$ is the small-scale fading component distributed according to $\mathcal{CN}(0,1)$  In addition, $\mathrm{x}_{t,k}$ is the desired signal transmitted by the $n$-th vehicle, and $\mathrm{v}_{t,k}$ is an additive white Gaussian noise with variance $\sigma_{n}^{2}$  $\mathrm{x}_{t,k}^{J}$ is the jamming signal, $\mathrm{x}_{t,k}^{*}$ is the spoofed signal (i e , the signal that carries the bits related to the wrong GPS positions), $\mathrm{g}_{t,k}^{JR} = \alpha_{t,k}^{JR} \mathrm{h}_{t,k}^{JR}$ is the channel power gain from RSJ to RSU where $\alpha_{t,k}^{JR}=G\beta d_{t,JR}^{-\gamma}$ such that $d_{t,JR}=\sqrt{({x}_{J}-x_{R})^{2}+({y}_{J}-y_{R})^{2}}$ 
We assume that the channel state information (CSI) of V2I links is known and can be estimated at the RSU as in \cite{8345717}  
The RSU is equipped with an RF antenna which can track the vehicles' trajectories after decoding the received RF signals  RSU aims to learn the interaction between the RF signals received from multiple vehicles and their corresponding trajectories 

\section{Proposed method for joint detection of GPS spoofing and jamming}

\subsection{Environment Representation}
The RSU is receiving RF signals from each vehicle and tracking its trajectory (which we refer to as GPS signal) by decoding and demodulating the received RF signals  
The Generalized state-space model describing the $i$-th signal evolvement at multiple levels embodies the following equations: 
\begin{equation} \label{eq_discreteLevel}
    \mathrm{\Tilde{S}_{t}}^{(i)} = \mathrm{f}(\mathrm{\Tilde{S}_{t-1}}^{(i)}) + \mathrm{\tilde{w}}_{t},
\end{equation}
\begin{equation} \label{eq_continuousLevel}
    \mathrm{\Tilde{X}_{t}}^{(i)} = \mathrm{A} \mathrm{\Tilde{X}_{t-1}}^{(i)} + \mathrm{B} \mathrm{U}_{\mathrm{\Tilde{S}_{t}}^{(i)}} + \mathrm{\tilde{w}}_{t},
\end{equation}
\begin{equation} \label{eq_observationLevel}
    \mathrm{\Tilde{Z}_{t}}^{(i)} = \mathrm{H} \mathrm{\Tilde{X}_{t}}^{(i)} + \mathrm{\tilde{v}}_{t},
\end{equation}
where $i \in \{$RF, GPS$\}$ indicates the type of signal received by the RSU  The transition system model defined in \eqref{eq_discreteLevel} explains the evolution of the discrete random variables $\mathrm{\Tilde{S}_{t}}^{(i)}$ representing the clusters of the RF (or GPS) signal dynamics, $\mathrm{f}( )$ is a non linear function of its argument and the additive term $\mathrm{\tilde{w}}_{t}$ denotes the process noise  The dynamic model defined in \eqref{eq_continuousLevel} explains the RF signal dynamics evolution or the motion dynamics evolution of the $n$-th vehicle, where $\mathrm{\Tilde{X}_{t}}^{(i)}$ are hidden continuous variables generating sensory signals, $\mathrm{A} \in \mathbb{R}^{2d}$ and $\mathrm{B} \in \mathbb{R}^{2d}$ are the dynamic and control matrices, respectively, and $\mathrm{U}_{\mathrm{\Tilde{S}_{t}}^{(i)}}$ is the control vector representing the dynamic rules of how the signals evolve with time  The measurement model defined in \eqref{eq_observationLevel} describes dependence of the sensory signals $\mathrm{\Tilde{Z}_{t}}^{(i)}$ on the hidden states $\mathrm{\Tilde{X}_{t}}^{(i)}$ that is parametrized by the measurement matrix $\mathrm{B} \in \mathbb{R}^{2d}$ where $d$ stands for the data dimensionality and $\mathrm{\tilde{v}}_{t}$ is a random noise  

\subsection{Learning GDBN}
The hierarchical dynamic models defined in \eqref{eq_discreteLevel}, \eqref{eq_continuousLevel} and \eqref{eq_observationLevel} are structured in a Generalized Dynamic Bayesian Network (GDBN) \cite{9858012} as shown in Fig ~\ref{fig_GDBN_CGDBN}-(a) that provides a probabilistic graphical model expressing the conditional dependencies among random hidden variables and observable states  The generative process explaining how sensory signals have been generated can be factorized as:
\begin{equation} \label{eq_generative_process}
\begin{split}
    \mathrm{P}(\mathrm{\tilde{Z}}_{t}^{(i)}, \mathrm{\tilde{X}}_{t}^{(i)}, \mathrm{\tilde{S}}_{t}^{(i)}) = \mathrm{P}(\mathrm{\tilde{S}}_{0}^{(i)}) \mathrm{P}(\mathrm{\tilde{X}}_{0}^{(i)}) \\ \bigg[ \prod_{t=1}^{\mathrm{T}} \mathrm{P}(\mathrm{\tilde{Z}}_{t}^{(i)}|\mathrm{\tilde{X}}_{t}^{(i)}) \mathrm{P}(\mathrm{\tilde{X}}_{t}^{(i)}|\mathrm{\tilde{X}}_{t-1}^{(i)}, \mathrm{\tilde{S}}_{t}^{(i)}) \mathrm{P}(\mathrm{\tilde{S}}_{t}^{(i)}|\mathrm{\tilde{S}}_{t-1}^{(i)}) \bigg],
\end{split}
\end{equation}
where $\mathrm{P}(\mathrm{\tilde{S}}_{0}^{(i)})$ and $\mathrm{P}(\mathrm{\tilde{X}}_{0}^{(i)})$ are initial prior distributions, $\mathrm{P}(\mathrm{\tilde{Z}}_{t}^{(i)}|\mathrm{\tilde{X}}_{t}^{(i)})$ is the likelihood, $\mathrm{P}(\mathrm{\tilde{X}}_{t}^{(i)}|\mathrm{\tilde{X}}_{t-1}^{(i)}, \mathrm{\tilde{S}}_{t}^{(i)})$ and $\mathrm{P}(\mathrm{\tilde{S}}_{t}^{(i)}|\mathrm{\tilde{S}}_{t-1}^{(i)})$ are the transition densities describing the temporal and hierarchical dynamics of the generalized state-space model 
The generative process defined in \eqref{eq_generative_process} indicates the cause-effect relationships the model impose on the random variables $\mathrm{\tilde{S}}_{t}^{(i)}$, $\mathrm{\tilde{X}}_{t}^{(i)}$ and $\mathrm{\tilde{Z}}_{t}^{(i)}$ forming a chain of causality describing how one state contributes to the production of another state which is represented by the link $\mathrm{\tilde{S}}_{t}^{(i)} \rightarrow \mathrm{\tilde{X}}_{t}^{(i)} \rightarrow \mathrm{\tilde{Z}}_{t}^{(i)}$ 

The RSU starts perceiving the environment using a static assumption about the environmental states evolution by assuming that sensory signals are only subject to random noise  Hence, RSU predicts the RF signal (or vehciles trajectory) using the following simplified model:
$\mathrm{\tilde{X}}_{t}^{(i)} = \mathrm{A} \mathrm{\tilde{X}}_{t-1}^{(i)} + \mathrm{\tilde{w}}_{t}$, 
that differs from \eqref{eq_continuousLevel} in the control vector $\mathrm{U}_{\mathrm{\Tilde{S}_{t}}^{(i)}}$ which is supposed to be null, i e , $\mathrm{U}_{\mathrm{\Tilde{S}_{t}}^{(i)}} = 0$ as the dynamic rules explaining how the environmental states evolve with time are not discovered yet 
Those rules can be discovered by exploiting the generalized errors (GEs), i e , the difference between predictions and observations  The GEs projected into the measurement space are calculated as:
$\tilde{\varepsilon}_{\mathrm{\tilde{Z}}_{t}^{(i)}}^{} = \mathrm{\tilde{Z}}_{t}^{(i)} - \mathrm{H} \mathrm{\tilde{X}}_{t}^{(i)}$ 
Projecting $\tilde{\varepsilon}_{\mathrm{\tilde{Z}}_t}^{}$ back into the generalized state space can be done as follows:
\begin{equation}\label{GE_continuousLevel_initialModel}
    \tilde{\varepsilon}_{\mathrm{\tilde{X}}_t}^{(i)} = \mathrm{H}^{-1}\tilde{\varepsilon}_{\mathrm{\tilde{Z}}_{t}^{(i)}}^{}=\mathrm{H}^{-1}(\mathrm{\tilde{Z}}_{t}^{(i)}-\mathrm{H}\mathrm{\tilde{X}}_{t}^{(i)}) = \mathrm{H}^{-1}\mathrm{\tilde{Z}}_{t}^{(i)} - \mathrm{\tilde{X}}_{t}^{(i)} 
\end{equation}
The GEs defined in \eqref{GE_continuousLevel_initialModel} can be grouped into discrete clusters in an unsupervised manner by employing the Growing Neural Gas (GNG)  The latter produces a set of discrete variables (clusters) denoted by:
$\mathbf{\tilde{S}^{(i)}}=\{\mathrm{\tilde{S}}_{1}^{(i)},\mathrm{\tilde{S}}_{2}^{(i)},\dots,\mathrm{\tilde{S}}_{M_{i}}^{(i)}\}$,
where $M_{i}$ is the total number of clusters and each cluster $\mathrm{\tilde{S}}_{m}^{(i)} \in \mathbf{\tilde{S}^{(i)}}$ follows a Gaussian distribution composed of GEs with homogeneous properties, such that $\mathrm{\tilde{S}}_{m}^{(i)} \sim \mathcal{N}(\tilde{\mu}_{\mathrm{\tilde{S}}_{m}^{(i)}}=[\mu_{\tilde{S}_{m}^{(i)}}, \Dot{\mu}_{\tilde{S}_{m}^{(i)}}], \Sigma_{\mathrm{\tilde{S}}_{m}^{(i)}})$ 
\begin{figure}[t!]
    \begin{center}
        \begin{minipage}[b]{ 40\linewidth}
        \centering
           \includegraphics[width=2 5cm]{Figures/GDBN pdf}
        \\[-1 0mm]
        {\scriptsize (a)}
        \end{minipage}
        \begin{minipage}[b]{ 50\linewidth}
            \centering
            \includegraphics[width=5 0cm]{Figures/C_GDBN pdf}
           
            {\scriptsize (b)}
        \end{minipage}
        \caption{(a) The GDBN  (b) The coupled GDBN (C-GDBN) composed of two GDBNs representing the two signals received at the RSU where their discrete hidden variables are stochastically coupled }
        \label{fig_GDBN_CGDBN}
    \end{center}
\end{figure}
The dynamic transitions of the sensory signals among the available clusters can be captured in a time-varying transition matrix ($\Pi_{\tau}$) by estimating the time-varying transition probabilities $\pi_{ij}=\mathrm{P}(\mathrm{\tilde{S}}_{t}^{(i)}=i|\mathrm{\tilde{S}}_{t-1}^{(i)}=j, \tau)$ where $\tau$ is the time spent in $\mathrm{\tilde{S}}_{t-1}^{(i)}=j$ before transition to $\mathrm{\tilde{S}}_{t}^{(i)}=i$ 

\subsection{Learning Coupled GDBN (C-GDBN)}
The learning procedure described in the previous section can be executed for each signal type, i e , RF and GPS  After learning a separated GDBN model for each signal type, we analyse the interaction behaviour between RF signal and GPS signal received at the RSU by tracking the cluster firing among $\mathbf{\tilde{S}^{(1)}}$ and $\mathbf{\tilde{S}^{(2)}}$ during a certain experience  Such an interaction can be encoded in a Coupled GDBN (C-GDBN) as shown in Fig \ref{fig_GDBN_CGDBN}-(b) composed of the two GDBNs representing the two signals where their hidden variables at the discrete level are stochastically coupled (in $\mathrm{\tilde{C}}_{t}{=}[\mathrm{\tilde{S}}_{t}^{(1)},\mathrm{\tilde{S}}_{t}^{(2)}]$) as those variables are uncorrelated but have coupled means 
The interactive matrix $\Phi \in \mathbb{R}^{M_{1},M_{2}}$ encodes the firing cluster pattern allowing to predict the GPS signal from RF signal is defined as follows:
\begin{equation} \label{interactiveTM_fromRFtoGPS}
\Phi = 
        \begin{bmatrix} 
            \mathrm{P}(\mathrm{\Tilde{S}_{1}}^{(2)}|\mathrm{\Tilde{S}_{1}}^{(1)}) & \mathrm{P}(\mathrm{\Tilde{S}_{2}}^{(2)}|\mathrm{\Tilde{S}_{1}}^{(1)}) & \dots & \mathrm{P}(\mathrm{\Tilde{S}_{M_{2}}}^{(2)}|\mathrm{\Tilde{S}_{1}}^{(1)}) \\
            \mathrm{P}(\mathrm{\Tilde{S}_{1}}^{(2)}|\mathrm{\Tilde{S}_{2}}^{(1)}) & \mathrm{P}(\mathrm{\Tilde{S}_{2}}^{(2)}|\mathrm{\Tilde{S}_{2}}^{(1)}) & \dots & \mathrm{P}(\mathrm{\Tilde{S}_{M_{2}}}^{(2)}|\mathrm{\Tilde{S}_{2}}^{(1)}) \\
            \vdots & \vdots & \ddots & \vdots \\
            \mathrm{P}(\mathrm{\Tilde{S}_{1}}^{(2)}|\mathrm{\Tilde{S}_{M_{1}}}^{(1)}) & \mathrm{P}(\mathrm{\Tilde{S}_{2}}^{(2)}|\mathrm{\Tilde{S}_{M_{1}}}^{(1)}) & \dots & \mathrm{P}(\mathrm{\Tilde{S}_{M_{2}}}^{(2)}|\mathrm{\Tilde{S}_{M_{1}}}^{(1)}) 
        \end{bmatrix} 
\end{equation}

\subsection{Joint Prediction and Perception}
RSU starts predicting the RF signals it expects to receive from each vehicle based on a Modified Markov Jump Particle Filter (M-MJPF) \cite{9858012} that combines Particle filter (PF) and Kalman filter (KF) to perform temporal and hierarchical predictions  Since the acquired C-GDBN allows predicting a certain signal's dynamic evolution based on another's evolution, it requires an interactive Bayesian filter capable of dealing with more complicated predictions  To this purpose, we propose to employ an Interactive M-MJPF (IM-MJPF) on the C-GDBN  The IM-MJPF consists of a PF that propagates a set of $L$ particles equally weighted, such that $\{\mathrm{\tilde{S}}_{t,l}^{(1)}, \mathrm{W}_{t,l}^{(1)}\}{\sim}\{\pi(\mathrm{\tilde{S}}_{t}^{(1)}), \frac{1}{L}\}$, where $\mathrm{\tilde{S}}_{t,l}^{(1)}$, $l \in L$ and $( ^{(1)})$ is the RF signal type  In addition, RSU relies on $\Phi$ defined in \eqref{interactiveTM_fromRFtoGPS} to predict $\mathrm{\tilde{S}}_{t}^{(2)}$ realizing the discrete cluster of vehicle's trajectory starting from the predicted RF signal according to: $\{\mathrm{\tilde{S}}_{t}^{(2)},\mathrm{W}_{t,l}^{(2)}\}{\sim} \{\Phi(\mathrm{\tilde{S}}_{t,l}^{(1)}){=}\mathrm{P}( |\mathrm{\tilde{S}}_{t,l}^{(1)}), \mathrm{W}_{t,l}^{(2)}\}$  For each predicted discrete variable $\mathrm{\tilde{S}}_{t,l}^{(i)}$, a multiple KF is employed to predict multiple continuous variables which guided by the predictions at the higher level as declared in \eqref{eq_continuousLevel} that can be represented probabilistically as $\mathrm{P}(\mathrm{\tilde{X}}_{t}^{(i)}|\mathrm{\tilde{X}}_{t-1}^{(i)}, \mathrm{\tilde{S}}_{t}^{(i)})$  The posterior probability that is used to evaluate expectations is given by:
\begin{multline} \label{piX}
    \pi(\mathrm{\tilde{X}}_{t}^{(i)})=\mathrm{P}(\mathrm{\tilde{X}}_{t}^{(i)},\mathrm{\tilde{S}}_{t}^{(i)}|\mathrm{\tilde{Z}}_{t-1}^{(i)})= \\ \int \mathrm{P}(\mathrm{\tilde{X}}_{t}^{(i)}|\mathrm{\tilde{X}}_{t-1}^{(i)}, \mathrm{\tilde{S}}_{t}^{(i)}) \lambda(\mathrm{\tilde{X}}_{t-1}^{(i)})d\mathrm{\tilde{X}}_{t-1}^{(i)},
\end{multline}
where $\lambda(\mathrm{\tilde{X}}_{t-1}^{(i)}){=}\mathrm{P}(\mathrm{\tilde{Z}}_{t-1}^{(i)}|\mathrm{\tilde{X}}_{t-1}^{(i)})$  
The posterior distribution can be updated (and so representing the updated belief) after having seen the new evidence $\mathrm{\tilde{Z}}_{t}^{(i)}$ by exploiting the diagnostic message $\lambda(\mathrm{\tilde{X}}_{t}^{(i)})$ in the following form: $\mathrm{P}(\mathrm{\tilde{X}}_{t}^{(i)}, \mathrm{\tilde{S}}_{t}^{(i)}|\mathrm{\tilde{Z}}_{t}^{(i)}) {=} \pi(\mathrm{\tilde{X}}_{t}^{(i)})\lambda(\mathrm{\tilde{X}}_{t}^{(i)})$  Likewise, belief in discrete hidden variables can be updated according to: $\mathrm{W}_{t,l}^{(i)}{=}\mathrm{W}_{t,l}^{(i)}\lambda (\mathrm{\tilde{S}}_{t}^{(i)})$ where:
$\lambda (\mathrm{\tilde{S}}_{t}^{(i)}) {=} \lambda (\mathrm{\Tilde{X}}_{t}^{(i)})\mathrm{P}(\mathrm{\Tilde{X}}_{t}^{(i)}|\mathrm{\tilde{S}}_{t}^{(i)}) {=} \mathrm{P}(\mathrm{\tilde{Z}}_{t}^{(i)}|\mathrm{\Tilde{X}}_{t}^{(i)})\mathrm{P}(\mathrm{\Tilde{X}}_{t}^{(i)}|\mathrm{\tilde{S}}_{t}^{(i)})$ 

\subsection{Joint GPS spoofing and jamming detection}
RSU can evaluate the current situation and identify if V2I is under attack, or the satellite link is under spoofing based on a multiple abnormality indicator produced by the IM-MJPF  The first indicator calculates the similarity between the predicted RF signal and the observed one, which is defined as:
\begin{equation}\label{eq_CLA1}
    \Upsilon_{\mathrm{\tilde{X}}_{t}^{(1)}} = -ln \bigg( \mathcal{BC} \big(\pi(\mathrm{\tilde{X}}_{t}^{(1)}),\lambda(\mathrm{\tilde{X}}_{t}^{(1)}) \big) \bigg),
\end{equation}
where $\mathcal{BC}( ){=}\int \sqrt{\pi(\mathrm{\tilde{X}}_{t}^{(1)}),\lambda(\mathrm{\tilde{X}}_{t}^{(1)}})d\mathrm{\tilde{X}}_{t}^{(1)}$ is the Bhattacharyya coefficient 
The second indicator calculates the similarity between the predicted GPS signal (from the RF signal) and the observed one after decoding the RF signal which is defined as:
\begin{equation}\label{eq_CLA2}
    \Upsilon_{\mathrm{\tilde{X}}_{t}^{(2)}} = -ln \bigg( \mathcal{BC} \big(\pi(\mathrm{\tilde{X}}_{t}^{(2)}),\lambda(\mathrm{\tilde{X}}_{t}^{(2)}) \big) \bigg),
\end{equation}
where $\mathcal{BC}( ){=}\int \sqrt{\pi(\mathrm{\tilde{X}}_{t}^{(2)}),\lambda(\mathrm{\tilde{X}}_{t}^{(2)}})d\mathrm{\tilde{X}}_{t}^{(2)}$ 
Different hypotheses can be identified by the RSU to understand the current situation whether there is: a jammer attacking the V2I link, or a spoofer attacking the link between the satellite and the vehicle or both jammer and spoofer are absent according to:
\begin{equation}
    \begin{cases}
        \mathcal{H}_{0}: \text{if} \ \ \Upsilon_{\mathrm{\tilde{X}}_{t}^{(1)}} < \xi_{1} \ \text{and} \ \Upsilon_{\mathrm{\tilde{X}}_{t}^{(2)}} < \xi_{2}, \\
        \mathcal{H}_{1}: \text{if} \ \ \Upsilon_{\mathrm{\tilde{X}}_{t}^{(1)}} \geq \xi_{1} \ \text{and} \ \Upsilon_{\mathrm{\tilde{X}}_{t}^{(2)}} \geq \xi_{2}, \\
        \mathcal{H}_{2}: \text{if} \ \ \Upsilon_{\mathrm{\tilde{X}}_{t}^{(1)}} < \xi_{1} \ \text{and} \ \Upsilon_{\mathrm{\tilde{X}}_{t}^{(2)}} \geq \xi_{2},
    \end{cases}
\end{equation}
where $\xi_{1} = \mathbb{E}[\Bar{\Upsilon}_{\mathrm{\tilde{X}}_{t}^{(1)}}] + 3\sqrt{\mathbb{V}[\Bar{\Upsilon}_{\mathrm{\tilde{X}}_{t}^{(1)}}]}$, and $\xi_{2} = \mathbb{E}[\Bar{\Upsilon}_{\mathrm{\tilde{X}}_{t}^{(2)}}] + 3\sqrt{\mathbb{V}[\Bar{\Upsilon}_{\mathrm{\tilde{X}}_{t}^{(2)}}]}$  In $\xi_{1}$ and $\xi_{2}$, $\Bar{\Upsilon}_{\mathrm{\tilde{X}}_{t}^{(1)}}$ and $\Bar{\Upsilon}_{\mathrm{\tilde{X}}_{t}^{(2)}}$ stand for the abnormality signals during training (i e , normal situation when jammer and spoofer are absent) 

\subsection{Evaluation metrics}
In order to evaluate the performance of the proposed method to jointly detect jammer and GPS spoofer, we adopt the jammer detection probability ($\mathrm{P}_{d}^{j}$) and the spoofer detection probability ($\mathrm{P}_{d}^{s}$), respectively, which are defined as:
\begin{equation}
    \mathrm{P}_{d}^{j} = \mathrm{Pr}(\Upsilon_{\mathrm{\tilde{X}}_{t}^{(1)}}\geq \xi_{1}, \Upsilon_{\mathrm{\tilde{X}}_{t}^{(2)}} \geq \xi_{2}|\mathcal{H}_{1}),
\end{equation}
\begin{equation}
    \mathrm{P}_{d}^{s} = \mathrm{Pr}(\Upsilon_{\mathrm{\tilde{X}}_{t}^{(1)}}< \xi_{1}, \Upsilon_{\mathrm{\tilde{X}}_{t}^{(2)}} \geq \xi_{2}|\mathcal{H}_{2}) 
\end{equation}
Also, we evaluate the accuracy of the proposed method in predicting and estimating the vehicles' trajectories and the expected RF signals by adopting the root mean square error (RMSE) defined as:
\begin{equation}
    RMSE = \sqrt{ \frac{1}{T} \sum_{t=1}^{T}\bigg( \mathrm{\tilde{Z}}_{t}^{(i)}-\mathrm{\tilde{X}}_{t}^{(i)} \bigg)^{2} },
\end{equation}
where $T$ is the total number of predictions 

\section{Simulation Results}
In this section, we evaluate the performance of the proposed method to jointly detect the jammer and the spoofer using extensive simulations  We consider $\mathrm{N}=2$ vehicles interacting inside the environment and exchanging their states (i e , position and velocity) with the RSU  The vehicles move along predefined trajectories performing various maneuvers which are picked from the \textit{Lankershim} dataset proposed by \cite{5206559}  The dataset depicts a four way intersection and includes about $19$ intersection maneuvers  RSU assigns one subchannel realizing the V2I link for each vehicle over which the vehicles' states are transmitted  The transmitted signal carrying the vehicle's state and the jamming signal are both QPSK modulated  
The simulation settings are: carrier frequency of $2$GHz, BW${=}1 4$MHz, cell radius of $500$m, RSU antenna height and gain is $25$m and $8$ dBi, receiver noise figure of $5$dB, vehicle antenna height and gain is $1 5$m and $3$dBi, vehicle speed is $40$Km/h, V2I transmit power is $23$dBm, jammer transmit power ranging from $20$dBm to $40$dBm, SNR of $20$dB, path loss model ($128 1{+}37 6log d$), Log-normal shadowing with $8$dB standard deviation and a fast fading channel following the Rayleigh distribution 
\begin{figure}[ht!]
    \begin{center}
        \begin{minipage}[b]{ 55\linewidth}
        \centering
            \includegraphics[width=5 0cm]{Results/ObservedTrajectories_reference}
        \\[-1 5mm]
        {\scriptsize (a)}
        \end{minipage}
        \begin{minipage}[b]{ 49\linewidth}
            \centering
            \includegraphics[width=4 9cm]{Results/ObservedRFsignal_Veh1_reference}
            \\[-1 5mm]
            {\scriptsize (b)}
        \end{minipage}
        \begin{minipage}[b]{
Context 3: Paper Info

Title: Incorporating Human Path Preferences in Robot Navigation with Minimal Interventions
Publish Date: 16 Mar 2023
Author List: Oriana Peltzer, Dylan Asmar, Mac Schwager, Mykel Kochenderfer

Figure

Hyperplane arrangement of a twodimensional space containing two obstacles (colored in gray) The robot is located inside the pink polytope, surrounded by three adjacent obstacle-free polytopes Each hyperplane on the boundary of the robot's polytope corresponds to one of the nonredundant constraints in eq (4) (b)Graph derived from the hyperplane arrangement The nodes on the graph designate polytopes, and edges designate transitions to adjacent polytopes To estimate the human's preference, the robot updates a posterior over the goal and over which of the graph transitions φ 1 , φ 2 and φ 3 is preferred by the human (c)Example preference defined over the graph The location of the goal is indicated in yellow in the lower right polytope For each node, the outgoing pink arrow designates the edge on the graph corresponding to the preferred transition between polytopes 
Simple, 10 × 10, 8 polytopes (b) Map 2: Office, 10 × 10, 56 polytopes (c) Map 3: Classroom, 20 × 20, 73 polytopes (d) Sampled observations and robot's executed trajectories 
Fig 5: Maps used for simulating the robot navigation problem with path preferences In (d), the heading angles observed are indicated with arrows The goal is indicated with a pink circle, and the orange robot corresponds to the starting location The blue robot follows a policy that accounts for path preference, while the green robot does not The opacity of the robots increases with time 
Map 1 problem setup and example realizations for goal-only (green) and path preference (blue) solution methods The robot starts at the lower left corner of the environment, and the goal of the task (pink circle) is in the upper left area The robot does not know which goal, among 10 options (shown in light blue squares), is the correct goal The human provides noisy observations, indicated by arrows, at each iteration The green robot selects actions according to the goal-only baseline, and the blue robot uses our proposed method to infer path preferences The polytopes composing G are drawn in blue Probability of correct goal WLPHVWHS +J (c) Entropy of goal distribution g 
Fig  6: Probability of the correct goal, fig 6b, and entropy of the goal belief distribution P (g), fig 6c, for the same problem setup, fig 6a In this problem instance, the human's preference is to go to the goal by passing on the right side of the obstacle Results are averaged over 50 runs and the area filled represents one standard deviation above and below the mean value The goal-only baseline shows an over-confident prediction (shown by the strong reduction in belief entropy) that the correct goal is less likely, making it more difficult to reach the correct goal compared to a method that accounts for path preference 
Success rates in the simple environment (Map 1) The results are averaged over 6 randomly sampled problem instances (start location, goal location, and goal possibilities), and over 50 runs per problem instance ∆T is the number of time steps separating two consecutive human inputs The robot's mission time is Tmax = 30 time steps We selected γ h = 1 5, corresponding to relatively noisy human inputs and making the problem more difficult to solve for the robot 
Computation times for Goal Only and Path Preference methods on Map 1 (fig 5a),Map 2 (fig 5b), and Map 3 (fig 5c),averaged over 100 runs with randomly sampled problem instances The 95 % confidence interval is provided with the mean We evaluate computation time at the first iteration of each run (where the search depth takes on its highest value Tmax) 

abstract

Robots that can effectively understand human intentions from actions are crucial for successful human-robot collaboration  In this work, we address the challenge of a robot navigating towards an unknown goal while also accounting for a human's preference for a particular path in the presence of obstacles 
This problem is particularly challenging when both the goal and path preference are unknown a priori  To overcome this challenge, we propose a method for encoding and inferring path preference online using a partitioning of the space into polytopes  Our approach enables joint inference over the goal and path preference using a stochastic observation model for the human 
We evaluate our method on an unknown-goal navigation problem with sparse human interventions, and find that it outperforms baseline approaches as the human's inputs become increasingly sparse  We find that the time required to update the robot's belief does not increase with the complexity of the environment, which makes our method suitable for online applications 

INTRODUCTION

Collaboration between humans and robots has become increasingly important and one key aspect of this collaboration is the ability for robots to adapt to human decisions  In many scenarios, such as a robot navigating through a busy room to deliver an item, it is important for the robot to take into account human preferences 
For instance, humans may prefer a specific path that would allow their colleagues to notice the item being delivered, but this preference may change dynamically based on various factors such as changes in the environment or unforeseen circumstances  While some preferences can be incorporated into the path-planning process, accommodating dynamic user preferences in real-time remains challenging 
In this paper, we propose a way to enable robots to adapt to human preferences dynamically by leveraging real-time feedback to inform decision-making  In this work, we tackle the problem of robot navigation in which the robot cannot observe the goal or the preferred path to the goal, but must make navigation decisions that are influenced by humans through recommended actions 
Prior work has explored how to adapt to a human's preference through feedback, but such approaches often require a high level of intervention, which can be time-consuming and impractical in real-world scenarios  To optimize the use of human input and quickly infer the human's preference, Fig  : An autonomous robot navigates in a simulated classroom towards a goal location (pink circle) 
At the start of its mission, it receives direction indications (arrows) from a human that indicate which path it should take to get to the goal  In this scenario, the human wants the robot to go around the desks on the right side of the classroom  A robot that does not reason over path preferences (green) will take the shortest path to the goal regardless of the human's input 
Our method (blue) infers the human's path preference from these indications and adapts to their recommendations  we propose an approach that leverages probabilistic representations of human preference and incorporates real-time feedback  Previous research by Bajcsy et al  considered an online adaptation problem in a manipulation task, where the person can apply forces to the robot to indicate their preferences 
By allowing the robot to continue its task while taking into account a probabilistic representation of human preference, their approach does not require frequent inputs  Building on this idea, we adopt a similar approach to adapt to a human's preference in the context of a robot autonomously navigating through a known environment, such as a cluttered office space 
Specifically, we focus on allowing the human to influence the robot's trajectory with respect to obstacles, by providing guidance on preferred routes or paths, while the robot continues to execute its task  Paths can be represented using homotopy classes   However, homotopies can pose computational challenges when used to encode and infer human preferences 
When the robot maintains a belief over homotopy classes, the inference problem can become exponentially complex with the number of obstacles in the space  Additionally, when the goal is unknown, the number of variables increases with the number of candidate destinations  This complexity can render the decision-making problem intractable 
Our solution is to encode path preference based on a partitioning of the environment into polytopes   This representation allows path preferences to be expressed as sets of preferred transitions between adjacent polytopes  Paths belonging to different homotopy classes correspond to different sequences of transitions 
By leveraging conditional independence assumptions, we can make the Bayesian inference problem tractable  These assumptions exploit the fact that human actions provide information about the path in a piece-wise manner  For example, indicating a preference for navigating around a particular obstacle only provides information about the local area and not the entire path 
Finally, after updating its belief representation over the human's preference, the robot can adapt to indications by replanning online  Our contributions are as follows  • We formulate the human-robot collaboration problem as a Partially Observable Markov Decision Process (POMDP) where both the goal of the task and the human's path preference are unknown random variables 
• We propose an encoding of a human's path preference using a partitioning of the environment into polytopes, along with conditional independence assumptions that make the Bayesian inference problem tractable to infer the task goal and path preference online  • Through simulations in two environments of different sizes and complexity, we show that our method is effective for solving problems where the robot must reach a goal that is unknown a-priori while simultaneously adapting to a human's indications 
Our method shows higher success rates compared to baseline approaches when the human inputs are sparse  Our approach enables a robot to make effective navigation decisions in collaboration with a human, even when the goal and path preference are not known in advance, and with minimal human input  In recent years, there has been a growing interest in shared autonomy and interactive systems, where humans and robots work together to accomplish tasks 
Several approaches have been proposed to address the challenge of enabling effective collaboration between human and robot agents while still achieving high task performance  Losey et al  and Jeon, Losey, and Sadigh propose a framework where a human operator is given control of a task-relevant latent action space while an autonomous system handles the rest 
Dragan and Srinivasa present a formalism for arbitrating between a user's input and a robot's policy when both human and robot share control of the same action space  Cognetti et al  [7] provide a method for real-time modifications of a path,   



ground_truth: They are rich in actin and have been shown to be highly dynamic.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the name of the generative interactive model used in the method?

Read the following text and answer briefly.

Context 2: 49\linewidth}
            \centering
            \includegraphics[width=4 9cm]{Results/ObservedRFsignal_Veh2_reference}
            \\[-1 5mm]
            {\scriptsize (c)}
        \end{minipage}
        \caption{An example visualizing the received RF signals from the two vehicles and the corresponding trajectories: (a) Vehicles' trajectories, (b) received RF signal from vehicle 1, (c) received RF signal from vehicle 2 }
        \label{fig_receivedRFsignalandTrajectory}
    \end{center}
\end{figure}
\begin{figure}[t!]
    \begin{center}
        \begin{minipage}[b]{ 49\linewidth}
            \centering
                \includegraphics[width=4 5cm]{Results/clusters_trajectory_veh1}
            \\[-1 5mm]
            {\scriptsize (a)}
        \end{minipage}
        \begin{minipage}[b]{ 49\linewidth}
            \centering
                \includegraphics[width=4 5cm]{Results/clusters_trajectory_veh2}
            \\[-1 5mm]
            {\scriptsize (b)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=4 5cm]{Results/clusters_RFsignal_veh1}
            \\[-1 5mm]
            {\scriptsize (c)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=4 5cm]{Results/clusters_RFsignal_veh2}
            \\[-1 5mm]
            {\scriptsize (d)}
        \end{minipage}
       
        \caption{GNG output after clustering the generalized errors obtained from different experiences: (a) clustered trajectory of vehicle 1, (b) clustered trajectory of vehicle 2, (c) clustered RF signal received from vehicle 1, (d) clustered RF signal received from vehicle 2 }
        \label{fig_GNG_of_receivedRFsignalandTrajectory}
    \end{center}
\end{figure}

The RSU aims to learn multiple interactive models (i e , C-GDBN models) encoding the cross relationship between the received RF signal from each vehicle and its corresponding trajectory  These models allow the RSU to predict the trajectory the vehicle will follow based on the received RF signal and evaluate whether the V2I is under jamming attacks or the satellite link is under spoofing  It is to note that the RSU is receiving only the RF signals from the two vehicles and obtaining their positions after decoding the RF signals  Thus, the RSU should be able to evaluate if the received RF signals are evolving according to the dynamic rules learned so far and if the vehicles are following the expected (right) trajectories to decide whether the V2I links are really under attack or whether the satellite link is under spoofing 

Fig ~\ref{fig_receivedRFsignalandTrajectory}-(a) illustrates an example of the interaction between the two vehicles performing a particular manoeuvre, and Fig ~\ref{fig_receivedRFsignalandTrajectory}-(b) shows the received RF signals by the RSU from the two vehicles  At the beginning of the learning process, RSU performs predictions according to the simplified model defined in \eqref{eq_continuousLevel} where $\mathrm{U}_{\mathrm{\Tilde{S}_{t}}^{(i)}} {=} 0$ 
After obtaining the generalized errors as pointed out in \eqref{GE_continuousLevel_initialModel}, RUS clusters those errors using GNG to learn two GDBN models encoding the dynamic rules of how the RF signal and the GPS signal evolve with time, respectively, as showed in Fig ~\ref{fig_GNG_of_receivedRFsignalandTrajectory} and Fig ~\ref{fig_graphicalRep_transitionMatrices}  RSU can couple the two GDBNs by learning the interactive transition matrix that is encoded in a C-GDBN as shown in Fig ~\ref{fig_interactiveMatrices} 
\begin{figure}[t!]
    \begin{center}
        \begin{minipage}[b]{ 49\linewidth}
            \centering
                \includegraphics[width=4 5cm]{Results/graphTransition_Trajectory_veh1}
            \\[-1 5mm]
            {\scriptsize (a)}
        \end{minipage}
        \begin{minipage}[b]{ 49\linewidth}
            \centering
                \includegraphics[width=4 5cm]{Results/graphTransition_Trajectory_veh2}
            \\[-1 5mm]
            {\scriptsize (b)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=4 5cm]{Results/graphTransition_RFsignal_veh1}
            \\[-1 5mm]
            {\scriptsize (c)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=4 5cm]{Results/graphTransition_RFsignal_veh2}
            \\[-1 5mm]
            {\scriptsize (d)}
        \end{minipage}
        \caption{Graphical representation of the transition matrices (TM): (a) TM related to the trajectory of vehicle 1, (b) TM related to the trajectory of vehicle 2, (c) TM related to the RF signal received from vehicle 1, (d) TM related to the RF signal received from vehicle 2 }
        \label{fig_graphicalRep_transitionMatrices}
    \end{center}
\end{figure}
\begin{figure}[t!]
    \begin{center}
        \begin{minipage}[b]{ 49\linewidth}
            \centering
                \includegraphics[width=3 8cm]{Results/interactiveMatrix_RFtoGPS_Neu5_veh1}
            \\[-1 0mm]
            {\scriptsize (a)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=3 8cm]{Results/interactiveMatrix_RFtoGPS_Neu25_veh1}
            \\[-1 0mm]
            {\scriptsize (d)}
        \end{minipage}
       
        \caption{Interactive transition matrix defined in \eqref{interactiveTM_fromRFtoGPS} using different configurations: (a) $\mathrm{M_{1}}=5$, $\mathrm{M_{2}}=5$, (b) $\mathrm{M_{1}}=25$, $\mathrm{M_{2}}=25$ }
        \label{fig_interactiveMatrices}
    \end{center}
\end{figure}
\begin{figure}[t!]
    \begin{center}
        \begin{minipage}[b]{ 49\linewidth}
        \centering
            \includegraphics[width=4 9cm]{Results/RF_situation1_best_veh1}
        \\[-1 5mm]
        {\scriptsize (a)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=4 9cm]{Results/RF_situation1_worst_veh1}
            \\[-1 5mm]
            {\scriptsize (b)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=4 9cm]{Results/RF_situation1_best_veh2}
            \\[-1 5mm]
            {\scriptsize (c)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=4 9cm]{Results/RF_situation1_worst_veh2}
            \\[-1 5mm]
            {\scriptsize (d)}
        \end{minipage}
        \caption{An example visualizing the predicted and observed RF signals transmitted by the 2 vehicles using different configurations  Predicted RF signal from: (a) vehicle 1 using $\mathrm{M_{1}}{=}5$, $\mathrm{M_{2}}{=}5$, (b) vehicle 1 using $\mathrm{M_{1}}{=}25$, $\mathrm{M_{2}}{=}25$, (c) vehicle 2 using $\mathrm{M_{1}}{=}5$, $\mathrm{M_{2}}{=}5$, (d) vehicle 2 using $\mathrm{M_{1}}{=}25$, $\mathrm{M_{2}}{=}25$ }
            \label{fig_situation1_PredictedRF}
    \end{center}
\end{figure}

\begin{figure}[t!]
    \begin{center}
        \begin{minipage}[b]{ 49\linewidth}
        \centering
            \includegraphics[width=4 8cm]{Results/GPSfromRF_situation1_best}
        \\[-1 0mm]
        {\scriptsize (a)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=4 8cm]{Results/GPSfromRF_situation1_worst}
            \\[-1 0mm]
            {\scriptsize (b)}
        \end{minipage}
        %
        \caption{An example visualizing the predicted and observed trajectories of two vehicles interacting in the environment  (a) $\mathrm{M_{1}}{=}5$, $\mathrm{M_{2}}{=}5$, (b) $\mathrm{M_{1}}{=}25$, $\mathrm{M_{2}}{=}25$ }
            \label{fig_situation1_VehiclesTrajectories}
    \end{center}
\end{figure}

\begin{figure}[ht!]
    \begin{center}
        \begin{minipage}[b]{ 49\linewidth}
        \centering
            \includegraphics[width=4 8cm]{Results/rmse_on_trajectory}
        \\[-1 0mm]
        {\scriptsize (a)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=4 8cm]{Results/rmse_on_RFSignal}
            \\[-1 0mm]
            {\scriptsize (b)}
        \end{minipage}
        \caption{The average RMSE after testing different experiences and examples of: (a) trajectories and (b) RF signals }
            \label{fig_rmse_onTraj_onSig}
    \end{center}
\end{figure}

Fig ~\ref{fig_situation1_PredictedRF} illustrates an example comparing between predicted RF signals and observed ones based on two different configurations in learning the interactive matrix (as shown in Fig ~\ref{fig_interactiveMatrices})  Also, Fig ~\ref{fig_situation1_VehiclesTrajectories} illustrates an example comparing between the predicted and observed trajectories of the two vehicles using the two interactive matrices depicted in Fig ~\ref{fig_interactiveMatrices}  From Fig ~\ref{fig_situation1_PredictedRF} and Fig ~\ref{fig_situation1_VehiclesTrajectories} we can see that using an interactive matrix with less clusters allows to perform better predictions compared to that with more clusters  This can be validated by observing Fig ~\ref{fig_rmse_onTraj_onSig} that illustrates the RMSE values versus different number of clusters related to the two models representing the dynamics of the received RF signals and the vehicles' trajectories  It can be seen that as the number of clusters increases the RMSE error increases, since adding more clusters decreases the firing probability that explains the possibility to be in one of the $M_{2}$ clusters of the second model conditioned in being in a certain cluster of the first model 

Fig ~\ref{fig_exNormal_Spoofed_JammedTrajectories} illustrates an example of vehicle's trajectory under normal situation (i e , jammer and spoofer are absent), under jamming attacks and under spoofing attacks  Also the figure shows the predicted trajectory which should follow the same dynamic rules learned during a normal situation  After that, we implemented the IM-MJPF on the learned C-GDBN to perform multiple predictions, i e , to predict the RF signal that the RSU is expecting to receive from a certain vehicle and the corresponding trajectory that the vehicle is supposed to follow  IM-MJPF through the comparison between multiple predictions and observations, produces multiple abnormality signals as defined in \eqref{eq_CLA1} and \eqref{eq_CLA2} which are used to detect the jammer and the spoofer 

Fig ~\ref{fig_abnormalitySignals_JammerSpoofer} illustrates the multiple abnormality signals related to the example shown in Fig ~\ref{fig_exNormal_Spoofed_JammedTrajectories}  We can observe that the abnormal signals related to both RF signal (Fig ~\ref{fig_abnormalitySignals_JammerSpoofer}-(a)) and trajectory (Fig ~\ref{fig_abnormalitySignals_JammerSpoofer}-(b)) are below the threshold under normal situations  This proves that RSU learned the correct dynamic rules of how RF signals and trajectories evolve when the jammer and spoofer are absent (i e , under normal situations)  Also, we can see that the RSU can notice a high deviation on both the RF signal and the corresponding trajectory due to a jamming interference from what it has learned so far by relying on the abnormality signals  In contrast, we can see that under spoofing attacks, RSU notice a deviation only on the trajectory and not on the RF signal since the spoofer has affected only the positions without manipulating the RF signal  In addition, it is obvious how the proposed method allows the RSU to identify the type of abnormality occurring and to explain the cause of the detected abnormality (i e , understanding if it was because of a jammer attacking the V2I link or a spoofer attacking the satellite link) 
\begin{figure}[t!]
    \centering
    \includegraphics[width=6 5cm]{Results/trajectories_underJamming_andSpoofing}
   
    \caption{Vehicle's trajectory under: normal situation, jamming and spoofing }
    \label{fig_exNormal_Spoofed_JammedTrajectories}
\end{figure}
\begin{figure}[t!]
    \begin{center}
        \begin{minipage}[b]{ 92\linewidth}
        \centering
            \includegraphics[height=2 6cm]{Results/abnSignal_onRF}
        \\[-1 5mm]
        {\scriptsize (a)}
        \end{minipage}
        \begin{minipage}[b]{ 92\linewidth}
            \centering
            \includegraphics[height=2 6cm]{Results/abnSignal_onGPS}
            \\[-1 5mm]
            {\scriptsize (b)}
        \end{minipage}
        %
        \caption{Abnormality Signals related to the example shown in Fig \ref{fig_exNormal_Spoofed_JammedTrajectories}: (a) abnormality indicators related to the RF signal, (b) abnormality indicators related to the trajectory }
            \label{fig_abnormalitySignals_JammerSpoofer}
    \end{center}
\end{figure}
\begin{figure}[t!]
    \centering
    \includegraphics[height=3 2cm]{Results/Detection_Probability_RFfromGPS_versusPj}
    \caption{Detection probability ($\mathrm{P_{d}}$) versus jammer's power ($\mathrm{P_{J}}$) using different number of clusters $\mathrm{M}_{2}$ }
    \label{fig_jammerDetectionProb}
\end{figure}
\begin{figure}[t!]
    \centering
    \includegraphics[height=3 2cm]{Results/spoofingDetectionProbability_falseAlarm_versusM2}
    \caption{Spoofing detection probability ($\mathrm{P}_{d}^{s}$) and spoofing false alarm ($\mathrm{P}_{f}^{s}$) versus the number of clusters $\mathrm{M}_{2}$ }
    \label{fig_spooferDetectionProb}
\end{figure}

Fig ~\ref{fig_jammerDetectionProb} shows the overall performance of the proposed method in detecting the jammer by testing many situations and examples and by considering different jamming powers which ranges from $20$dBm to $40$dBm  It can be seen that the proposed method is able to detect the jammer with high probabilities (near $1$) and by considering low and high jamming powers  Also, the figure compares the performance in detecting the jammer by varying the number of clusters ($M_{2}$) 
Fig ~\ref{fig_spooferDetectionProb} shows the overall performance of the proposed method in detecting the spoofer by testing different different examples of driving maneuvers  It can be seen that the RSU is able to detect the spoofer with high detection probability and null false alarm versus different number of clusters 

\section{Conclusion}
A joint detection method of GPS spoofing and jamming attacks is proposed  The method is based on learning a dynamic interactive model encoding the cross-correlation between the received RF signals from multiple vehicles and their corresponding trajectories  Simulation results show the high effectiveness of the proposed approach in jointly detecting the GPS spoofer and jammer attacks  
Subsequent work will extend the system model to consider more than two vehicles with different channel conditions and various modulation schemes to evaluate the effectiveness of the proposed method 

\bibliographystyle{IEEEtran}

Context 3:    Fig  : We model the intent inference problem with the above diagram 
At each step in time, the robot receives an observation ot from the human conditioned on its current location st, the intended goal g, and the human's path preference θ  The robot updates its belief over g and θ and transitions to a next location st+1  while Hagenow et al  present a method that allows an outside agent to modify key robot state variables and blends the changes with the original control 
However, a common challenge of these approaches is the high level of intervention required from humans  Best and Fitch propose a method for predicting an agent's intended trajectory from observations  Rather than maintaining a belief over the agent's future path, they infer the agent's intended goal among a set of candidate locations at the boundary of the space 
This approach provides information on where the agent is heading and generates a distribution of candidate future trajectories for the agent  Inferring the goal of the task among a discrete set of candidates is also relevant to the area of shared autonomy  Javdani, Srinivasa, and Bagnell propose a formalism for shared control of a robotic arm, where the robot must assist the human in picking up an object but needs to infer which object the human has chosen from joystick inputs 
Planning with homotopy class constraints is useful in problems where the robot's requirements are given with respect to obstacles, and Yi, Goodrich, and Seppi consider topological constraints provided by human operators  Bhattacharya propose an efficient algorithm for solving pathplanning problems under homotopic constraints 
However, the number of homotopy classes for a given problem can be infinite, and as the robot changes location and updates its representation of the world, carrying out inference over homotopy classes in a dynamic environment requires recomputing the set of homotopies at every iteration, making the belief update challenging 
Prior work has addressed the challenge of shared autonomy by considering how robots can infer a human's intended goal, or how they can infer the preferred path to a goal  However, we argue that inferring the goal and the path as separate problems can lead to over-confidence in incorrect beliefs about the user's preferences 
To illustrate this point, consider the following scenario: a robot and a human are collaborating to move an object from one end of a room to Fig  : Using the hyperplanes composing the H-representation of each obstacle, we construct a hyperplane arrangement of the obstacle-free space (a)  We define the human's preference for the robot's one step action choices as the posterior distribution (given all human input up to that point) over transitions from the current to the neighboring polytopes, i e  edges on the graph 
Each time the robot transitions to a new polytope, the set of neighbor polytopes and the distribution over human preferences are updated  another, but there is an obstacle in the way  The human would like the robot to take a path around the obstacle on the left, even though the goal is on the right  If the robot only infers the goal from the human's inputs, it may incorrectly assume that the goal is on the right, and become over-confident in this belief 
On the other hand, if the robot only infers the preferred path, it may mistakenly assume that the goal is on the left, leading to a failure in completing the task  To overcome these challenges, our work proposes a joint inference approach that considers both the human's intended goal and their preferred path to that goal 
Specifically, we model the human's preference over different homotopy classes and leverage a conditional independence assumption to provide a tractable solution  In our approach, we assume that the human's inputs are noisily rational conditioned on both the goal and the preference  By jointly inferring the goal and path preference, we can avoid over-confidence in incorrect beliefs about the user's preferences, leading to improved system performance 
We consider the problem of robot navigation in a known environment to an unknown destination, where a human can intervene and provide a heading direction to the robot using a joystick or force cues  The human also has a preference on which path the robot should take with respect to obstacles, and our objective is for the robot to understand the human's intentions and execute the task with minimal interventions 
Let g be a discrete random variable denoting the goal of the task, belonging to a set of candidates Ω g , and let θ be a discrete-valued random variable representing the human's path preference, belonging to a set of possible preferences Θ  The physical location of the robot at time index t is denoted by s t ∈ R 2 , and the robot's action at time index t, belonging to some action space A, is denoted by a t  
The transition model T (s t+1 | s t , a t ) is deterministic, meaning the robot has full control over its future location  At any time step, the human may provide an observation to the robot  When the human intervenes, the robot receives a direction (heading angle) that can be mapped to a future location in space 
More specifically, we map the direction to an intended location, which is the resulting robot location after advancing in the indicated direction for one time step  For simplicity, we consider that the robot directly makes an observation o t of the location indicated by the human  We assume that the robot has a stochastic observation model for the human P (o t | s t , g, θ) that is conditioned on both the goal of the task g and the human's preferred path θ 
We further assume that having chosen a goal and path preference, the human takes actions to noisily minimize a cost function C g,θ that measures the cost of moving from the robot's current location to the goal along the preferred path  For example, C g,θ (s t , o t ) can be the length of the shortest path from location s t to the goal g after taking a first step to o t , and constrained by path preference θ 
We use C g,θ to induce a probability distribution over observations, given by: where γ h is a hyperparameter that designates the rationality coefficient  This model assumes the human will pick the lowest cost action with the highest probability and the likelihood of an action decreases exponentially with the increase in cost  
Our inclusion of the path preference θ sets our approach apart from   The model is shown in fig  represented as a Bayesian Network 

Inference

At each time step where the human provides an observation, the posterior P (g, θ) is given through the Bayesian update We note that the number of Bayesian updates required at each iteration to update the belief is equal to the cardinality of Ω g × Θ  In addition, each Bayesian update involves computing C g,θ (  
,   ) in eq  ( ), which involves solving an optimization problem (such as a shortest path problem)  In section IV, we propose a specific encoding of preference θ for resolving eq  ( ), while ensuring the number of computations of the cost C g,θ ( ,  ) per update does not grow exponentially with the number of obstacles 

Decision Making

We consider a navigation problem where the robot receives reward according to the model R(s t , g, θ, a t )  We wish to find the optimal policy π that maximizes the expected discounted sum of future rewards, with discount factor γ  The above problem is a Partially Observable Markov Decision Process (POMDP)  
In this section, we propose an encoding of human's path preference θ for computing the posterior in eq  ( )  Devifrom the concept of homotopy classes, we define the preference according to a partitioning of the environment into polytopes, as shown in fig  , creating a hyperplane arrangement of the space 
Hyperplane arrangements have been used by Vincent and Schwager in the context of Neural Network verification  In our setting, we leverage this representation to define path preferences as preferred transitions between adjacent regions of the space 

Hyperplane Arrangement

We assume a two-dimensional environment composed of m polytopic obstacles, each defined by their half-space representation (H-representation) where A i ∈ R di×2 and b i ∈ R di , and where d i is the number of edges (hyperplanes) composing polytope i  Let n = i d i be the total number of hyperplanes  We leverage each obstacle's H-representation to construct a hyperplane arrangement of the environment as shown in fig 
 e  a partitioning of the space into polytopes  More specifically, each location in space belongs to a polytope j for which we can write an H-representation of the form where α j i ∈ {−1, 1} di is a vector specific to polytope j and obstacle i corresponding to the relative position of any point in the set with respect to each hyperplane in O i  
Fig  : Intent inference model in a hyperplane arrangement of the obstacle free space  We spatially decompose the preference θ into a set of preferred neighboring polytopes per region of the space  Within each polytope j, the human preference pj is a discrete distribution over the preferred neighbor in N (j) 
We assume that for a location st belonging to polytope j, and given goal g and preference pj, the observation ot and any other preference p i,i =j are conditionally independent  Concatenating elements from each obstacle's Hrepresentation, we can write polytope j's H-representation as where Some of the constraints in eq  ( ) (corresponding to rows of A, b and α j ) are redundant, i e  the set P j does not change upon their removal 
We can further reduce the Hrepresentation of a polytope to include only non-redundant constraints  By removing the rows corresponding to redundant constraints, we obtain new matrices A j e , b j e and α j e such that we can write the polytope's reduced H-representation as The non-redundant constraints correspond to edges of the polytope 
In other words, as the robot continually moves in space, the first hyperplane that it will cross upon exiting the polytope will correspond to one of the polytope's nonredundant constraints  Vincent and Schwager outline an iterative method for removing redundant constraints by solving n linear programs 
We use this method in practice for computing α j e for each polytope  We can now characterize each polytope by a vector α j e ∈ {−1, 1} n j e , where n j e ≤ n is the number of essential constraints of the polytope  The polytopes P j partition the environment into a hyperplane arrangement 

Path Preference

In this section, we provide a definition of preference θ according to a graphical representation of the environment based on the hyperplane arrangement  Under this representation, a path preference corresponds to a set of preferred transitions  In other words, for each polytope in the space, the human will have a preference to which neighboring polytope they wish to transition 
Let G := (V, E) be an undirected graph, where vertices are obstacle-free polytopes, and edges connect two adjacent polytopes  Each polytope is described by a unique vector α j as defined in eq  ( )  Two polytopes are adjacent if they share non-redundant constraints (rows in eq  ( )) corresponding to the same hyperplane (i e  they are on opposite sides of the hyperplane)



ground_truth: The generative interactive model used in the method is called the Coupled Generalized Dynamic Bayesian Network (C-GDBN).

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the effect of accounting for path preference on the robot's belief update?

Read the following text and answer briefly.

Context 3: 
Let N (v) be the set of neighbors of a vertex v  For each vertex, we denote p v the discrete-valued random variable describing which edge in N (v) the human intends to transition to  Using this formalism, we define a path preference as the set of preferred transitions over all nodes in the graph, Let m θ = v∈V |N (v)| be the cardinality of Θ, and m g = |Ω g | the number of possible goals 
A priori, the number of Bayesian updates required to update the belief at every iteration should be m θ × m g   Now, let us assume the conditional independence relationships described by the new problem diagram in fig    More specifically, we introduce the assumption that conditioned on a robot location s t , the goal g, and the preference for the corresponding vertex p v in the graph, the observation o t and the preference for any other vertex are conditionally independent 
In other words, the observations the human provides can be defined conditioned only on the robot location, the goal, and the human's preference for its current vertex p v   By introducing this assumption, each update step only requires updating the joint (p v , g), reducing the number of cost computations to |N (v)| × m g  
We can notice that by introducing this assumption, we removed the direct relationship between the number of polytopes in the environment and the complexity of the Bayesian update in eq  ( )  In practice, components of θ are not mutually independent  For example, if the human preference at a vertex v 1 is
, it is unlikely that the human will also prefer p v2 = (v 2 , v 1 ) (turning back)  We can improve our model by assuming a dependent relationship between preferences for adjacent edges, which does not significantly increase the complexity of the inference problem  An interesting property of our encoding is that any two paths that belong to different homotopy classes will cross different sequences of polytopes, i e  they correspond to a different sequence of edges on G 
This can be proved by contradiction  Let us suppose that two continuous trajectories ξ 1 and ξ 2 , with the same start and end points and that do not intersect any obstacle, traverse the same regions in G in the same order  From the construction of the hyperplane arrangement, each polytope that the paths traverse through is obstacle-free 
Therefore, within each polytope, there is no obstacle in the area located in between the portions of ξ 1 and ξ 2 that belong to the region  A smooth transformation of ξ 1 into ξ 2 can be obtained by transforming each portion of ξ 1 belonging to the polytopes it intersects into the corresponding portion of ξ 2 for the same polytopes, where the extremities of the trajectory portions are connected to one another along the polytope's edges (where the same edge is crossed by both paths) 
Along this transformation, the paths do not intersect any obstacle, and therefore ξ 1 and ξ 2 belong to the same homotopy class 

EXPERIMENTS

We evaluate our model on a simulated navigation task where the robot must reach a goal that is unknown a priori while respecting the path preferences indicated by a human  The robot navigates in a grid world containing obstacles  The transition model is deterministic: the robot selects an adjacent location on the grid to reach at the next time step 
The robot is also allowed to take diagonal actions  Each location s t in the map can be mapped to a vertex v t ∈ G  Therefore, the actions leading to locations mapped to different vertices correspond to edges on the graph  We note f (s t , a t ) the edge crossed by taking action a t from location s t  
The robot is given a mission time limit T max for reaching the goal  In this problem, we assume that the human selects actions to noisily minimize a cost function C g,θ , where θ is defined as per eq  ( ), corresponding to the length of the shortest path to the goal constrained by the preference (where the robot is only allowed to make transitions on G along preferred edges) 
More specifically, where δ(s t , g | o t , p vt ) designates the length of the shortest path from s t to g passing by o t and constrained by preference p vt   This is a slight variant of the cost function proposed by Best and Fitch , where we add in a conditioning on the path preference  We compute costs by running the A path planning algorithm on the environment maps (grid worlds with diagonal actions) and impose preference constraints by pruning invalid transitions from the search tree 
Reward model  At each step in time, the robot receives a reward which is a sum of three components: a goal-specific reward a preference-specific reward or penalty We compute solutions to the POMDP defined in section III-B with the online solver POMCP , and with the particularity that within the rollouts, the robot does not expect to collect human inputs 
Each time a solution is computed, the robot takes an action and may receive an observation  If it does, it updates its belief distribution over the unknown problem variables and resolves the POMDP over a receding horizon 

Baselines

• Goal only  The robot solves the POMDP while ignoring the effects of path preference  Similarly to , we assume the human is taking action to minimize a goaldependent cost C g (s t , o t ) = δ(s t , g | o t ), where the conditioning on the preference is removed  We also omit the path preference's contribution to the reward R pref  
• Compliant  The robot complies with the human input, but does not take an initiative  If the user stops providing information, the robot continues in the last direction indicated for 5 time steps (conserving its momentum), then stops  • Blended  We designed an arbitration function to decide between our proposed policy (accounting for path preferences) and the user's recommendation when the robot receives inputs 
Our metric to evaluate confidence in the robot's prediction for the purpose of arbitration is the entropy of the intention distribution H(g, p i ), where p i denotes the preferred neighbor for the current region  Because our representation of the world is discrete, the arbitration is given by a step function 
Denoted by U , the action corresponding to the human's input, and P , the robot's prediction for the optimal action, we write the policy where we chose h = 1 6 as the confidence threshold 

Results

When evaluating the algorithm, we consider that a run is successful if the robot reached the goal within its allocated mission time T max and only made transitions between graph vertices corresponding to the human's preferences  We vary the time delay between human inputs, from constant guidance (∆ T = 1) to only a single observation (∆ T ≥ T max ) 
Success rates  Table I reports the success rates for experiments conducted over six randomly sampled problem instances and 50 runs per instance in Map 1 (fig  )  When the human provides inputs at every iteration, the compliant policy shows the highest success rates  However, as ∆ T increases, the compliant robot is not able to accomplish the task within the allotted time as it does not receive sufficient inputs to do so, and performance decreases compared to the autonomous baselines 
We find that in these runs, accounting for path preference consistently improves performance compared with the goal-only baseline  Results also show that blending the user's input with the robot's policy (Path Preference + Blend) when the human provides information leads to improved performance  Belief entropy 
Figure shows a challenging problem instance where the directions the human provides do not align directly with the shortest path to the goal  By ignoring the effects of preferences in the problem model (goal only), the robot quickly infers from observations that the upper left goal is less likely than others (P (g) drops) 
The strong decrease in entropy shows that the robot becomes overconfident in this prediction  Overconfidence in an incorrect goal will prevent the agent from finding the correct goal once the human's indications directly align with it, as it needs to correct for the wrong predictions, as shown in the path realization (fig 
)  In this realization, the goal-only method (green robot) fails to search the upper left area within the allotted time  By accounting for path preferences in its model, the blue robot's entropy over the goal distribution decreases more steadily, allowing for it to leverage the human's latest observations and reach the goal successfully 
shows an over-confident prediction (shown by the strong reduction in belief entropy) that the correct goal is less likely, making it more difficult to reach the correct goal compared to a method that accounts for path preference  Computation time  In table II we provide the time required to solve the POMDP, and the time required to update the robot's belief as it receives new observations 
We compute solutions on three maps: a simple 10 × 10 grid world with 8 polytopes (fig  ), a 10 × 10 grid world with 56 polytopes (fig  ), and a 20×20 grid world with 73 polytopes (fig  )  The latter environment being larger, we increase the mission time and the depth of the search tree in POMCP from T max = 30 (Map 1 and Map 2) to T max = 60 (Map 3) 
We do not notice an increase in the time required to update the robot's belief with an increase in problem complexity, which is consistent with our observation that the complexity of the Bayesian update should not increase with the number of obstacles or polytopes  On the contrary, the belief update time on Map 2 and Map 3, containing more obstacles, is reduced compared to the first map 
More obstacles result in fewer iterations when solving the constrained shortest path problem with A   Adding constraints due to the obstacles and polytopes reduces the size of the A search tree  C  Limitations Simulation environments  In our simulations, we hardcoded the preference policy over the maps (e g  in Map 1, go around the table counter-clockwise) 
We randomly sampled problem instances (start and goal locations, and goal options) to reduce the bias introduced by these preference choices  To best evaluate and compare the different approaches, it would be best to sample preferences among a distribution of preferences chosen by a human (for example, from benchmarks resulting from a collection of data) 
Creating such a benchmark is an interesting direction for future work  Hyperplane arrangement construction  The main limitation of our approach is that the size and geometry of each polytope depends strongly on the geometry of the obstacles, as seen in fig    Because of this, the robot can make predictions over preferences that are too refined compared with the topology of the environment 
A direct consequence is that when the size of the polytopes is small, the information provided by the human can be incorrectly interpreted as a preference on the robot's immediate action  Our method can be improved by changing the structure of the hyperplane arrangement so that it relies on the topology of the environment, but does not vary strongly with the geometry of the features in the environment 
For this purpose, topometric maps and region construction algorithms are promising directions  We presented an approach for encoding and inferring a human's path preference in an environment with obstacles  By leveraging a partitioning of the space into polytopes and a stochastic observation model, our method allows for joint inference over the goal and path preference even when both are unknown a-priori 
Our experiments on an unknown-goal navigation problem with sparse human interventions demonstrate the effectiveness of our approach and its suitability for online applications  The time required to update the robot's belief does not increase with the complexity of the environment, which further highlights the practicality of our method 



ground_truth: The belief entropy decreases more steadily.