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?

Read the following text and answer briefly.

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: \section{Introduction}\label{S1}

The multiple access interferences (MAI) is the root of user
limitation in CDMA systems \cite{R1,R3}  The parallel least mean
square-partial parallel interference cancelation (PLMS-PPIC) method
is a multiuser detector for code division multiple access (CDMA)
receivers which reduces the effect of MAI in bit detection  In this
method and similar to its former versions like LMS-PPIC \cite{R5}
(see also \cite{RR5}), a weighted value of the MAI of other users is
subtracted before making the decision for a specific user in
different stages \cite{cohpaper}  In both of these methods, the
normalized least mean square (NLMS) algorithm is engaged
\cite{Haykin96}  The $m^{\rm th}$ element of the weight vector in
each stage is the true transmitted binary value of the $m^{\rm th}$
user divided by its hard estimate value from the previous stage  The
magnitude of all weight elements in all stages are equal to unity 
Unlike the LMS-PPIC, the PLMS-PPIC method tries to keep this
property in each iteration by using a set of NLMS algorithms with
different step-sizes instead of one NLMS algorithm used in LMS-PPIC 
In each iteration, the parameter estimate of the NLMS algorithm is
chosen whose element magnitudes of cancelation weight estimate have
the best match with unity  In PLMS-PPIC implementation it is assumed
that the receiver knows the phases of all user channels  However in
practice, these phases are not known and should be estimated  In
this paper we improve the PLMS-PPIC procedure \cite{cohpaper} in
such a way that when there is only a partial information of the
channel phases, this modified version simultaneously estimates the
phases and the cancelation weights  The partial information is the
quarter of each channel phase in $(0,2\pi)$ 

The rest of the paper is organized as follows: In section \ref{S4}
the modified version of PLMS-PPIC with capability of channel phase
estimation is introduced  In section \ref{S5} some simulation
examples illustrate the results of the proposed method  Finally the
paper is concluded in section \ref{S6} 

\section{Multistage Parallel Interference Cancelation: Modified PLMS-PPIC Method}\label{S4}

We assume $M$ users synchronously send their symbols
$\alpha_1,\alpha_2,\cdots,\alpha_M$ via a base-band CDMA
transmission system where $\alpha_m\in\{-1,1\}$  The $m^{th}$ user
has its own code $p_m( )$ of length $N$, where $p_m(n)\in \{-1,1\}$,
for all $n$  It means that for each symbol $N$ bits are transmitted
by each user and the processing gain is equal to $N$  At the
receiver we assume that perfect power control scheme is applied 
Without loss of generality, we also assume that the power gains of
all channels are equal to unity and users' channels do not change
during each symbol transmission (it can change from one symbol
transmission to the next one) and the channel phase $\phi_m$ of
$m^{th}$ user is unknown for all $m=1,2,\cdots,M$ (see
\cite{cohpaper} for coherent transmission)  According to the above
assumptions the received signal is
\begin{equation}
\label{e1} r(n)=\sum\limits_{m=1}^{M}\alpha_m
e^{j\phi_m}p_m(n)+v(n),~~~~n=1,2,\cdots,N,
\end{equation}
where $v(n)$ is the additive white Gaussian noise with zero mean and
variance $\sigma^2$  Multistage parallel interference cancelation
method uses $\alpha^{s-1}_1,\alpha^{s-1}_2,\cdots,\alpha^{s-1}_M$,
the bit estimates outputs of the previous stage, $s-1$, to estimate
the related MAI of each user  It then subtracts it from the received
signal $r(n)$ and makes a new decision on each user variable
individually to make a new variable set
$\alpha^{s}_1,\alpha^{s}_2,\cdots,\alpha^{s}_M$ for the current
stage $s$  Usually the variable set of the first stage (stage $0$)
is the output of a conventional detector  The output of the last
stage is considered as the final estimate of transmitted bits  In
the following we explain the structure of a modified version of the
PLMS-PIC method \cite{cohpaper} with simultaneous capability of
estimating the cancelation weights and the channel phases 

Assume $\alpha_m^{(s-1)}\in\{-1,1\}$ is a given estimate of
$\alpha_m$ from stage $s-1$  Define
\begin{equation}
\label{e6} w^s_{m}=\frac{\alpha_m}{\alpha_m^{(s-1)}}e^{j\phi_m} 
\end{equation}
From (\ref{e1}) and (\ref{e6}) we have
\begin{equation}
\label{e7} r(n)=\sum\limits_{m=1}^{M}w^s_m\alpha^{(s-1)}_m
p_m(n)+v(n) 
\end{equation}
Define
\begin{subequations}
\begin{eqnarray}
\label{e8} W^s&=&[w^s_{1},w^s_{2},\cdots,w^s_{M}]^T,\\
\label{e9}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!X^{s}(n)\!\!\!&=&\!\!\![\alpha^{(s-1)}_1p_1(n),\alpha^{(s-1)}_2p_2(n),\cdots,\alpha^{(s-1)}_Mp_M(n)]^T 
\end{eqnarray}
\end{subequations}
where $T$ stands for transposition  From equations (\ref{e7}),
(\ref{e8}) and (\ref{e9}), we have
\begin{equation}
\label{e10} r(n)=W^{s^T}X^{s}(n)+v(n) 
\end{equation}
Given the observations $\{r(n),X^{s}(n)\}^{N}_{n=1}$, in modified
PLMS-PPIC, like the PLMS-PPIC \cite{cohpaper}, a set of NLMS
adaptive algorithm are used to compute
\begin{equation}
\label{te1} W^{s}(N)=[w^{s}_1(N),w^{s}_2(N),\cdots,w^{s}_M(N)]^T,
\end{equation}
which is an estimate of $W^s$ after iteration $N$  To do so, from
(\ref{e6}), we have
\begin{equation}
\label{e13} |w^s_{m}|=1 ~~~m=1,2,\cdots,M,
\end{equation}
which is equivalent to
\begin{equation}
\label{e14} \sum\limits_{m=1}^{M}||w^s_{m}|-1|=0 
\end{equation}
We divide $\Psi=\left(0,1-\sqrt{\frac{M-1}{M}}\right]$, a sharp
range for $\mu$ (the step-size of the NLMS algorithm) given in
\cite{sg2005}, into $L$ subintervals and consider $L$ individual
step-sizes $\Theta=\{\mu_1,\mu_2,\cdots,\mu_L\}$, where
$\mu_1=\frac{1-\sqrt{\frac{M-1}{M}}}{L}, \mu_2=2\mu_1,\cdots$, and
$\mu_L=L\mu_1$  In each stage, $L$ individual NLMS algorithms are
executed ($\mu_l$ is the step-size of the $l^{th}$ algorithm)  In
stage $s$ and at iteration $n$, if
$W^{s}_k(n)=[w^s_{1,k},\cdots,w^s_{M,k}]^T$, the parameter estimate
of the $k^{\rm th}$ algorithm, minimizes our criteria, then it is
considered as the parameter estimate at time iteration $n$  In other
words if the next equation holds
\begin{equation}
\label{e17} W^s_k(n)=\arg\min\limits_{W^s_l(n)\in I_{W^s}
}\left\{\sum\limits_{m=1}^{M}||w^s_{m,l}(n)|-1|\right\},
\end{equation}
where $W^{s}_l(n)=W^{s}(n-1)+\mu_l \frac{X^s(n)}{\|X^s(n)\|^2}e(n),
~~~ l=1,2,\cdots,k,\cdots,L-1,L$ and
$I_{W^s}=\{W^s_1(n),\cdots,W^s_L(n)\}$, then we have
$W^s(n)=W^s_k(n)$, and therefore all other algorithms replace their
weight estimate by $W^{s}_k(n)$  At time instant $n=N$, this
procedure gives $W^s(N)$, the final estimate of $W^s$, as the true
parameter of stage $s$ 

Now consider $R=(0,2\pi)$ and divide it into four equal parts
$R_1=(0,\frac{\pi}{2})$, $R_2=(\frac{\pi}{2},\pi)$,
$R_3=(\pi,\frac{3\pi}{2})$ and $R_4=(\frac{3\pi}{2},2\pi)$



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:  The
partial information of channel phases (given by the receiver) is in
a way that it shows each $\phi_m$ ($m=1,2,\cdots,M$) belongs to
which one of the four quarters $R_i,~i=1,2,3,4$  Assume
$W^{s}(N)=[w^{s}_1(N),w^{s}_2(N),\cdots,w^{s}_M(N)]^T$ is the weight
estimate of the modified algorithm PLMS-PPIC at time instant $N$ of
the stage $s$  From equation (\ref{e6}) we have
\begin{equation}
\label{tt3}
\phi_m=\angle({\frac{\alpha^{(s-1)}_m}{\alpha_m}w^s_m}) 
\end{equation}
We estimate $\phi_m$ by $\hat{\phi}^s_m$, where
\begin{equation}
\label{ee3}
\hat{\phi}^s_m=\angle{(\frac{\alpha^{(s-1)}_m}{\alpha_m}w^s_m(N))} 
\end{equation}
Because $\frac{\alpha^{(s-1)}_m}{\alpha_m}=1$ or $-1$, we have
\begin{eqnarray}
\hat{\phi}^s_m=\left\{\begin{array}{ll} \angle{w^s_m(N)} &
\mbox{if}~
\frac{\alpha^{(s-1)}_m}{\alpha_m}=1\\
\pm\pi+\angle{w^s_m(N)} & \mbox{if}~
\frac{\alpha^{(s-1)}_m}{\alpha_m}=-1\end{array}\right 
\end{eqnarray}
Hence $\hat{\phi}^s_m\in P^s=\{\angle{w^s_m(N)},
\angle{w^s_m(N)+\pi, \angle{w^s_m(N)}-\pi}\}$  If $w^s_m(N)$
sufficiently converges to its true value $w^s_m$, the same region
for $\hat{\phi}^s_m$ and $\phi_m$ is expected  In this case only one
of the three members of $P^s$ has the same region as $\phi_m$  For
example if $\phi_m \in (0,\frac{\pi}{2})$, then $\hat{\phi}^s_m \in
(0,\frac{\pi}{2})$ and therefore only $\angle{w^s_m(N)}$ or
$\angle{w^s_m(N)}+\pi$ or $\angle{w^s_m(N)}-\pi$ belongs to
$(0,\frac{\pi}{2})$  If, for example, $\angle{w^s_m(N)}+\pi$ is such
a member between all three members of $P^s$, it is the best
candidate for phase estimation  In other words,
\[\phi_m\approx\hat{\phi}^s_m=\angle{w^s_m(N)}+\pi \]
We admit that when there is a member of $P^s$ in the quarter of
$\phi_m$, then $w^s_m(N)$ converges  What would happen when non of
the members of $P^s$ has the same quarter as $\phi_m$? This
situation will happen when the absolute difference between $\angle
w^s_m(N)$ and $\phi_m$ is greater than $\pi$  It means that
$w^s_m(N)$ has not converged yet  In this case where we can not
count on $w^s_m(N)$, the expected value is the optimum choice for
the channel phase estimation, e g  if $\phi_m \in (0,\frac{\pi}{2})$
then $\frac{\pi}{4}$ is the estimation of the channel phase
$\phi_m$, or if $\phi_m \in (\frac{\pi}{2},\pi)$ then
$\frac{3\pi}{4}$ is the estimation of the channel phase $\phi_m$ 
The results of the above discussion are summarized in the next
equation
\begin{eqnarray}
\nonumber \hat{\phi}^s_m = \left\{\begin{array}{llll} \angle
{w^s_m(N)} & \mbox{if}~
\angle{w^s_m(N)}, \phi_m\in R_i,~~i=1,2,3,4\\
\angle{w^s_m(N)}+\pi & \mbox{if}~ \angle{w^s_m(N)}+\pi, \phi_m\in
R_i,~~i=1,2,3,4\\
\angle{w^n_m(N)}-\pi & \mbox{if}~ \angle{w^s_m(N)}-\pi, \phi_m\in
R_i,~~i=1,2,3,4\\
\frac{(i-1)\pi+i\pi}{4} & \mbox{if}~ \phi_m\in
R_i,~~\angle{w^s_m(N)},\angle
{w^s_m(N)}\pm\pi\notin R_i,~~i=1,2,3,4 \\
\end{array}\right 
\end{eqnarray}
Having an estimation of the channel phases, the rest of the proposed
method is given by estimating $\alpha^{s}_m$ as follows:
\begin{equation}
\label{tt4}
\alpha^{s}_m=\mbox{sign}\left\{\mbox{real}\left\{\sum\limits_{n=1}^{N}
q^s_m(n)e^{-j\hat{\phi}^s_m}p_m(n)\right\}\right\},
\end{equation}
where
\begin{equation} \label{tt5}
q^{s}_{m}(n)=r(n)-\sum\limits_{m^{'}=1,m^{'}\ne
m}^{M}w^{s}_{m^{'}}(N)\alpha^{(s-1)}_{m^{'}} p_{m^{'}}(n) 
\end{equation}
The inputs of the first stage $\{\alpha^{0}_m\}_{m=1}^M$ (needed for
computing $X^1(n)$) are given by
\begin{equation}
\label{qte5}
\alpha^{0}_m=\mbox{sign}\left\{\mbox{real}\left\{\sum\limits_{n=1}^{N}
r(n)e^{-j\hat{\phi}^0_m}p_m(n)\right\}\right\} 
\end{equation}
Assuming $\phi_m\in R_i$, then
\begin{equation}
\label{qqpp} \hat{\phi}^0_m =\frac{(i-1)\pi+i\pi}{4} 
\end{equation}
Table \ref{tab4} shows the structure of the modified PLMS-PPIC
method  It is to be notified that
\begin{itemize}
\item Equation (\ref{qte5}) shows the conventional bit detection
method when the receiver only knows the quarter of channel phase in
$(0,2\pi)$  \item With $L=1$ (i e  only one NLMS algorithm), the
modified PLMS-PPIC can be thought as a modified version of the
LMS-PPIC method 
\end{itemize}

In the following section some examples are given to illustrate the
effectiveness of the proposed method 

\section{Simulations}\label{S5}

In this section we have considered some simulation examples 
Examples \ref{ex2}-\ref{ex4} compare the conventional, the modified
LMS-PPIC and the modified PLMS-PPIC methods in three cases: balanced
channels, unbalanced channels and time varying channels  In all
examples, the receivers have only the quarter of each channel phase 
Example \ref{ex2} is given to compare the modified LMS-PPIC and the
PLMS-PPIC in the case of balanced channels 

\begin{example}{\it Balanced channels}:
\label{ex2}
\begin{table}
\caption{Channel phase estimate of the first user (example
\ref{ex2})} \label{tabex5} \centerline{{
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{6}{*}{\rotatebox{90}{$\phi_m=\frac{3\pi}{8},M=15~~$}} & N(Iteration) & Stage Number& NLMS & PNLMS  \\
&&&&\\
\cline{2-5} & \multirow{2}{*}{64}& s = 2 &  $\hat{\phi}^s_m=\frac{3 24\pi}{8}$ & $\hat{\phi}^s_m=\frac{3 18\pi}{8}$ \\
\cline{3-5} & & s = 3 & $\hat{\phi}^s_m=\frac{3 24\pi}{8}$ & $\hat{\phi}^s_m=\frac{3 18\pi}{8}$ \\
\cline{2-5} & \multirow{2}{*}{256}& s = 2 &  $\hat{\phi}^s_m=\frac{2 85\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 88\pi}{8}$ \\
\cline{3-5} & & s = 3 & $\hat{\phi}^s_m=\frac{2 85\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 88\pi}{8}$ \\
\cline{2-5} \hline
\end{tabular} }}
\end{table}
Consider the system model (\ref{e7}) in which $M$ users
synchronously send their bits to the receiver through their
channels



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 algorithm is engaged in the PLMS-PPIC method?

Read the following text and answer briefly.

Context 3:  It is assumed that each user's information consists of
codes of length $N$  It is also assumd that the signal to noise
ratio (SNR) is 0dB  In this example there is no power-unbalanced or
channel loss is assumed  The step-size of the NLMS algorithm in
modified LMS-PPIC method is $\mu=0 1(1-\sqrt{\frac{M-1}{M}})$ and
the set of step-sizes of the parallel NLMS algorithms in modified
PLMS-PPIC method are
$\Theta=\{0 01,0 05,0 1,0 2,\cdots,1\}(1-\sqrt{\frac{M-1}{M}})$,
i e  $\mu_1=0 01(1-\sqrt{\frac{M-1}{M}}),\cdots,
\mu_4=0 2(1-\sqrt{\frac{M-1}{M}}),\cdots,
\mu_{12}=(1-\sqrt{\frac{M-1}{M}})$  Figure~\ref{Figexp1NonCoh}
illustrates the bit error rate (BER) for the case of two stages and
for $N=64$ and $N=256$  Simulations also show that there is no
remarkable difference between results in two stage and three stage
scenarios  Table~\ref{tabex5} compares the average channel phase
estimate of the first user in each stage and over $10$ runs of
modified LMS-PPIC and PLMS-PPIC, when the the number of users is
$M=15$ 
\end{example}

Although LMS-PPIC and PLMS-PPIC, as well as their modified versions,
are structured based on the assumption of no near-far problem
(examples \ref{ex3} and \ref{ex4}), these methods and especially the
second one have remarkable performance in the cases of unbalanced
and/or time varying channels 

\begin{example}{\it Unbalanced channels}:
\label{ex3}
\begin{table}
\caption{Channel phase estimate of the first user (example
\ref{ex3})} \label{tabex6} \centerline{{
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{6}{*}{\rotatebox{90}{$\phi_m=\frac{3\pi}{8},M=15~~$}} & N(Iteration) & Stage Number& NLMS & PNLMS  \\
&&&&\\
\cline{2-5} & \multirow{2}{*}{64}& s=2 &  $\hat{\phi}^s_m=\frac{2 45\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 36\pi}{8}$ \\
\cline{3-5} & & s=3 & $\hat{\phi}^s_m=\frac{2 71\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 80\pi}{8}$ \\
\cline{2-5} & \multirow{2}{*}{256}& s=2 &  $\hat{\phi}^s_m=\frac{3 09\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 86\pi}{8}$ \\
\cline{3-5} & & s=3 & $\hat{\phi}^s_m=\frac{2 93\pi}{8}$ & $\hat{\phi}^s_m=\frac{3 01\pi}{8}$ \\
\cline{2-5} \hline
\end{tabular} }}
\end{table}
Consider example \ref{ex2} with power unbalanced and/or channel loss
in transmission system, i e  the true model at stage $s$ is
\begin{equation}
\label{ve7} r(n)=\sum\limits_{m=1}^{M}\beta_m
w^s_m\alpha^{(s-1)}_m c_m(n)+v(n),
\end{equation}
where $0<\beta_m\leq 1$ for all $1\leq m \leq M$  Both the LMS-PPIC
and the PLMS-PPIC methods assume the model (\ref{e7}), and their
estimations are based on observations $\{r(n),X^s(n)\}$, instead of
$\{r(n),\mathbf{G}X^s(n)\}$, where the channel gain matrix is
$\mathbf{G}=\mbox{diag}(\beta_1,\beta_2,\cdots,\beta_m)$  In this
case we repeat example \ref{ex2}  We randomly get each element of
$G$ from $[0,0 3]$  Figure~\ref{Figexp2NonCoh} illustrates the BER
versus the number of users  Table~\ref{tabex6} compares the channel
phase estimate of the first user in each stage and over $10$ runs of
modified LMS-PPIC and modified PLMS-PPIC for $M=15$ 
\end{example}

\begin{example}
\label{ex4} {\it Time varying channels}: Consider example \ref{ex2}
with time varying Rayleigh fading channels  In this case we assume
the maximum Doppler shift of $40$HZ, the three-tap
frequency-selective channel with delay vector of $\{2\times
10^{-6},2 5\times 10^{-6},3\times 10^{-6}\}$sec and gain vector of
$\{-5,-3,-10\}$dB  Figure~\ref{Figexp3NonCoh} shows the average BER
over all users versus $M$ and using two stages 
\end{example}


\section{Conclusion}\label{S6}

In this paper, parallel interference cancelation using adaptive
multistage structure and employing a set of NLMS algorithms with
different step-sizes is proposed, when just the quarter of the
channel phase of each user is known  In fact, the algorithm has been
proposed for coherent transmission with full information on channel
phases in \cite{cohpaper}  This paper is a modification on the
previously proposed algorithm  Simulation results show that the new
method has a remarkable performance for different scenarios
including Rayleigh fading channels even if the channel is
unbalanced 





ground_truth: The normalized least mean square (NLMS) algorithm.