it is well known that the brain has a highly developed and complex self - generated dynamical neural activity .we are therefore confronted with a dichotomy when attempting to understand the overall functioning of the brain or when designing an artificial cognitive system : a highly developed cognitive system , such as the brain , is influenced by sensory input but it is not driven directly by the input signals .the cognitive system needs however this sensory information vitally for adapting to a changing environment and survival .  in this contextwe then want to discuss two mutually interrelated questions :  * can we formulate a meaningful paradigm for the self - sustained internal dynamics of an autonomous cognitive system ? * how is the internal activity influenced by sensory signals , _viz _ which are the principles for the respective learning processes ?we believe that these topics represent important challenges for research in the field of recurrent neural networks and the modeling of neural processes . from an experimental point of viewwe note that an increasing flux of results from neurobiology supports the notion of quasi - stationary spontaneous neural activity in the cortex .it is therefore reasonable to investigate the two questions formulated above with the help of neural architectures centrally based on the notion of spontaneously generated transient states , as we will do in the present investigation using appropriate recurrent neural networks .standard classification schemes of dynamical systems are based on their long - time behavior , which may be characterized , e.g. , by periodic or chaotic trajectories .the term ` transient - state dynamics ' refers , on the other hand , to the type of activity occurring on intermediate time scales , as illustrated in fig .[ figure_trans_activity ] .a time series of semi - stable activity patterns , also denoted transient attractors , is characterized by two time scales .the typical duration of the activity plateaus and the typical time needed to perform the transition from one semi - stable state to the subsequent one .the transient attractors turn into stable attractors in the limit .transient state dynamics is intrinsically competitive in nature .when the current transient attractor turns unstable the subsequent transient state is selected by a competitive process .transient - state dynamics is a form of ` multi - winners - take - all ' process , with the winning coalition of dynamical variables suppressing all other competing activities .humans can discern about 10 - 12 objects per second and it is therefore tempting to identify the cognitive time scale of about 80 - 100ms with the duration of the transient - state dynamics illustrated in fig .[ figure_trans_activity ] .interestingly , this time scale also coincides with the typical duration of the transiently active neural activity patterns observed in the cortex .several high - level functionalities have been proposed for the spontaneous neural brain dynamics .edelman and tononi argue that ` critical reentrant events ' constitute transient conscious states in the human brain .these ` states - of - mind ' are in their view semi - stable global activity states of a continuously changing ensemble of neurons , the ` dynamic core ' .this activity takes place in what dehaene and naccache denote the ` global workspace ' .the global workspace serves , in the view of baars and franklin , as an exchange platform for conscious experience and working memory .crick and koch and koch have suggested that the global workspace is made - up of ` essential nodes ' , i.e. ensembles of neurons responsible for the explicit representation of particular aspects of visual scenes or other sensory information .   and for the length of activity - plateau and of the transient period respectively.,scaledwidth=85.0% ]   traditional neural network architectures are not continuously active on their own .feedforward setups are explicitly driven by external input and hopfield - type recurrent nets settle into a given attractor after an initial period of transient activities .the possibilities of performing cognitive computation with autonomously active neural networks , the route chosen by nature , are however investigated increasingly . in this contextthe time encoding of neural information , one of the possible neural codes , has been studied in various contexts .two network architectures , the echo state network suitable for rate - encoding neurons , and the liquid state machine suitable for spiking neurons , have been proposed to transiently encode in time a given input for further linear analysis by a subsequent perceptron .both architectures , the echo - state network and the liquid - state machine , are examples of reservoir architectures with fading memories , which however remain inactive in the absence of sensory input .an example of a continuously active recurrent network architecture is the winnerless competition based on stable heteroclinic cycles . in this casethe trajectory moves along heteroclines from one saddle point to the next approaching a complex limiting cycle .close to the saddle points the dynamics slows down leading to well defined transiently active neural activity patterns .in order to study the issues raised in the introduction , the notion of autonomous neural activity and its relation to the sensory input , we will consider a specific model based on clique - encoding recurrent nets .the emphasis will be on the discussion of the general properties and of the underlying challenges .we will therefore present here an overview of the algorithmic implementation , referring in part to the literature for further details .experimental evidence indicates that sparse neural coding is an important operating principle in the brain , as it minimizes energy consumption , maximizes storage capacity and contributes to make information encoding spatially explicit .a powerful form of sparse coding is multi - winners - take - all encoding in the form of cliques .the term cliques stems from network theory and denotes subgraphs which are fully interconnected , a few examples are given in fig .[ figure_networks_cliques ] .cliques are fully interconnected subgraphs of maximal size , in the sense that they are not part of another fully interconnected subgraph containing a larger number of vertices .clique encoding is an instance of sparse coding with spatially overlapping memory states .the use of clique encoding is in fact motivated by experimental findings indicating a hierarchical organization of overlapping neural clique assemblies for the real - time memory representation in the hippocampus . in the framework of a straightforward auto - associative neural networkthe cliques are defined by the network of the excitatory connections , which are shown as lines in fig .[ figure_networks_cliques ] , in the presence of an inhibitory background . in this setting all cliques correspond to attractors of the network , _ viz _ to spatially explicit and overlapping memory representations .one can transform the attractor network with clique encoding into a continuously active transient - state network , by introducing a reservoir variable for every neuron . in this setting the reservoir of a neuron is depleted whenever the neuron is active and refilled whenever the neuron is inactive . via a suitable local coupling between the individual neural activity and reservoir variables a well defined and stable transient state dynamics is obtained . when a given clique becomes a winning coalition ,the reservoirs of its constituting sites are depleted over time . when fully depleted the winning coalition becomes unstable and the subsequent winning coalition is activated through a competitive associative process , leading to an ever ongoing associative thought process .the resulting network architecture is a dense and homogeneous associative network ( dhan ) .an illustrative result of a numerical simulation is given in fig . [ figure_ai7 ] .  for the isolated system , not coupled to any sensory input , this associative thought process has no semantic content , as the transient attractors , the cliques , have none .the semantic content can be acquired only by coupling to a sensory input and by the generation of correlation between the transient attractors and patterns extracted from the input data stream .  .shown are , vertically displaced , the time developments of the respective neural activities ( solid lines ) and of the neural reservoirs ( dashed lines ) .the time series of spontaneously generated transient states are the cliques ( from the left to the right ) ., scaledwidth=55.0% ]   to be definite , we utilize a continuous - time formulation for the dhan architecture , with rate - encoding neurons , characterized by normalized activity levels  ] indicates that the transition is driven by the autonomous internal dynamics and that the series of winning coalitions take the form \atop\longrightarrow}\ ( 0,1 )    \ { [ s]\atop\longrightarrow}\ ( 3,6)~,\ ] ] in the presence of a sensory signal  ] the activity - levels of the neurons in the input layer . for subsequent usewe have defined in eq .( [ eq_delta_r_v_pq ] ) an auxiliary variable , which quantifies the influence of inactive input - neurons . the task is now to find a suitable learning algorithm which extracts relevant information from the input - data stream by mapping distinct input - patterns onto selected winning coalitions of the dhan layer .this setup is typical for an independent component analysis .the multi - winners - take - all dynamics in the dhan module implies individual neural activities to be close to 0/1 during the transient states and we can therefore define three types of inter - layer links ( see fig . [ fig_twolayers ] ) :  * ( _ ` act ' _ ) + links connecting active input neurons with the winning coalition of the dhan module . *( _ ` orth ' _ ) + links connecting inactive input neurons with the winning coalition of the dhan module .* ( _ ` ina ' _ ) + links connecting active input neurons with inactive neurons of the dhan module .the orthogonal links take their name from the circumstance that the receptive fields of the winning coalition of the target layer need to orthogonalize to all input - patters differing from the present one .note that it is not the receptive field of individual dhan neurons which is relevant , but rather the cumulative receptive field of a given winning coalition .we can then formulate three simple rules for the respective link - plasticity .whenever the new winning coalition in the dhan layer is activated by the input layer , _ viz _ whenever there is a substantial diffusive learning signal , i.e. when exceeds a certain threshold , the following optimization procedures should take place :  *  + the sum over active afferent links should take a large but finite value , *  + the sum over orthogonal afferent links should take a small value , *  + the sum over inactive links should take a small but non - vanishing value ,   the , and are the target values for the respective optimization processes . in order to implement these three rules we define three corresponding contributions to the link plasticities : where the inputs and to the dhan layer are defined by eq .( [ eq_delta_r_v_pq ] ) . for the sign - function is valid , for and respectively , denotes the heaviside - step function . in eq .( [ eq_back_contributions ] ) the , and are suitable optimization rates and the and the activity levels defining active and inactive dhan neurons respectively . a suitable set of parameters , which has been used for the numerical simulations ,is given in table [ tab_par_back ] ..the set of parameters entering the time - evolution equations of the links connecting the input to the dhan layer , with and , used in the actual simulations . [ cols="^,^,^,^",options="header " , ]   [ tab_par_back ]  using these definitions , the link plasticity may be written as ~ ,  \nonumber\ ] ] where is an appropriate threshold for the diffusive learning signal .the inter - layer links cease to be modified whenever the total input is optimal , _ viz _ when no more mistakes are made .we note , that a given interlayer - link is in general subject to competitive optimization from the three processes ( act / orth / ina ) .averaging would occur if the respective learning rates // would be of the same order of magnitude .it is therefore necessary , that and .bars problem with a probability for the occurrence of the individual horizontal or vertical bars .the problem is non - linear since the pattern intensity is not enhanced when an elementary horizontal and vertical bar overlap each other.,scaledwidth=85.0% ]   it is desirable that the interlayer connections neither grow unbounded with time ( runaway - effect ) nor disappear into irrelevance .suitable normalization procedures are therefore normally included explicitly into the respective neural learning rules and are present implicitly in eqs .( [ eq_back_contributions ] ) and eq .( [ eq_vdot ] ) .the strength of the input - signal is optimized by eq .( [ eq_vdot ] ) both for active as well as for inactive dhan neurons , a property referred to as fan - in normalization .( [ eq_back_contributions ] ) and ( [ eq_vdot ] ) also regulate the overall strength of inter - layer links emanating from a given input layer neuron , a property called fan - out normalization .next we note , that the time scales for the intrinsic autonomous dynamics in the dhan layer and for the input signal could in principle differ substantially .potential interference problems can be avoided when learning is switched - on very fast . in this casethe activation and decay rates for the diffusive learning signal are large and the corresponding characteristic time scales are smaller than both the typical time scales of the input and of the self - sustained dhan dynamics .a cognitive system needs to extract autonomously meaningful information about its environment from its sensory input data stream via signal separation and feature extraction .the identification of recurrently appearing patterns , i.e. of objects , in the background of fluctuation and of combinations of distinct and noisy patterns , constitutes a core demand in this context .this is the domain of the independent component analysis and blind source separation , which seeks to find distinct representations of statistically independent input patterns .  in order to test our system made - up by an input layer coupled to a dhan layer , as illustrated in fig .[ fig_twolayers ] , we have selected the bars problem .the bars problem constitutes a standard non - linear reference task for the feature extraction via a non - linear independent component analysis for an input layer .basic patterns are the vertical and horizontal bars . the individual input patterns are made - up of a non - linear superposition of the basic bars , containing with probability any one of them , as illustrated in fig .[ figure_bars_pattern ] .bars problem the response , as defined by eq .( [ eq_clique_rec_patt ] ) , for the 10 winning coalitions in the dhan layer ( compare fig . [ figure_networks_cliques ] and fig . [ fig_twolayers ] ) to the ten reference patterns , _ viz _ the 5 horizontal bars and the 5 vertical bars of the input field . in the top rowthe numbering of the cliques c - i, .. ,c - x having the maximal response to the respective reference patterns is given .in the bottom row , below each of the 10 black / white reference patterns , the receptive fields , eq .( [ eq_clique_recp_fields ] ) , for the winning coalitions c - i, .. ,c - x given in the top row are given color - coded , with black / blue / red / yellow coding synaptic strengths of increasing intensities ., scaledwidth=95.0% ]   for the simulations we presented to the system about randomly generated input patterns of the type shown in fig .[ figure_bars_pattern ] .the bars pattern are black / white with the for active / inactive sites , irrespectively of possible overlaps of vertical and horizontal bars .the individual patterns lasted with about for the time between two successive input signals .these time scales are to be compared with the time scale of the autonomous dhan dynamics illustrated in the figs .[ figure_sensitive_periods ] and [ figure_arb_7 ] , for which the typical stability - period for a transient state is about .we also note that there is no active training for the system .the associative thought process continuous in the dhan layer , at no time are the neural activities reset and the system restarted .all that happens is that the ongoing associative thought process is influenced from time to time by the input layer and that then the synaptic strengths connecting the input layer to the dhan layer are modified via eq .( [ eq_vdot ] ) .the results for the simulation are presented in fig .[ figure_crp_graph ] . for the geometry of the dhan network we used a regular 20-site star containing 10 cliques , with every clique being composed of four neurons ,see fig .[ figure_networks_cliques ] . in fig .[ figure_crp_graph ] we present the response of the 10 cliques in the dhan layer to the 10 basic input patterns , the isolated bars . herethe denotes the set of sites of the winning - coalition and its size , here .the response is equivalent to the clique averaged afferent synaptic signals , compare eq .( [ eq_delta_r_v_pq ] ) , in the presence of an elementary bar in the sensory input field .the individual potential winning coalitions , _ viz _ the cliques , have acquired in the course of the simulation , via the learning rule eq .( [ eq_vdot ] ) , distinct susceptibilities to the 10 bars , compare fig . [ figure_crp_graph ] . at the start of the simulationthe winning coalitions were just given by properties of the network typology , _ viz _ by the cliques , having no explicit semantic significance .the susceptibilities to the individual bars , which the cliques have acquired via the competition of the internal dhan dynamics with the sensory data input stream , can then be interpreted as a semantic assignment .the internal associative thought process of the dhan layer therefore becomes semantically meaningful via the coupling to the environment , corresponding to a sequence of horizontal and vertical bars .this learning paradigm is compatible with multi - electrode array studies of the visual cortex of developing ferrets , which indicate that the ongoing cortical dynamics is void of semantic content immediately after birth , acquiring semantic content however during the adolescence .the winning coalitions of the dhan layer are overlapping and every link targets in general more than one potential winning coalition in the dhan layer .this feature contrasts with the ` single - winner - takes - all ' setup , normally used for standard neural algorithms performing an independent component analysis , for which the target neurons are physically separated . for the regular 20-site network used in the simulationevery dhan neuron appertains to exactly two cliques , compare fig .[ figure_networks_cliques ] . the unsupervised learning procedure , eq .( [ eq_vdot ] ) , involves therefore a competition between the contribution , and , as given by eq .( [ eq_back_contributions ] ) . for the simulations we used a set of parameters ,see table [ tab_par_back ] , for which the contribution to is adapted at a much higher rate than the contributions to and .the responses of the winning coalitions are therefore close to , but somewhat below , the optimal value used for the simulations , compare fig .[ figure_crp_graph ] .the target value will not be reached even for extended simulations , due to the competition with the other optimization procedures , namely and , compare eq .( [ eq_back_contributions ] ) .bars problem the response , as defined by eq .( [ eq_clique_rec_patt ] ) , for the 20 winning coalitions c - i, .. ,c - xx in the dhan layer ( compare fig .[ figure_networks_cliques ] and fig . [ fig_twolayers ] ) to the twenty reference patterns , _ viz _ the 10 horizontal bars and the 10 vertical bars of the input field ., scaledwidth=85.0% ]   the averaged receptive fields of the cliques in the dhan layer with respect to the input neurons are also presented in fig .[ figure_crp_graph ] .the inter - layer synaptic weights can be both positive and negative and the orthogonalization procedure , eq .( [ eq_back_contributions ] ) , results in complex receptive fields .the time evolution equations for the inter - layer synaptic strengths ( [ eq_vdot ] ) are optimizing , but not maximizing , the response of the winning coalition to a given input signal .the receptive fields retain consequently a certain scatter , since the optimization via eq .( [ eq_vdot ] ) ceases whenever a satisfactory signal separation has been obtained . this behavior is consistent with the ` learning by mistakes ' paradigm , which states that a cognitive system needs to learn in general only when committing a mistake .bars problem the color - coded receptive fields , eq .( [ eq_clique_recp_fields ] ) , for the 20 cliques c - i, .. ,c - xx of the dhan layer , compare fig .[ figure_crp_100 ] , with black / blue / red / yellow coding synaptic strengths of increasing intensities ., scaledwidth=85.0% ]   the simulation results for the bars problem presented in fig .[ figure_crp_graph ] may be generalized to larger systems . for comparisonwe discuss now the results for the bars problem , for which there are ten horizontal and ten vertical elementary bars . for the dhan network we used a regular 40-site star with 20 cliques , a straightforward generalization of the regular star illustrated in fig .[ figure_networks_cliques ] .we used otherwise exactly the same set of parameters as previously for the bars problem , in particular also the same number of input training patterns .no optimization of parameters has been performed .the respective responses and receptive fields ( compare eqs .( [ eq_clique_rec_patt ] ) and ( [ eq_clique_recp_fields ] ) ) are presented in fig .[ figure_crp_100 ] and [ figure_crf_100 ] .the probability for any of the 20 bars to occur in a given input pattern , like the ones for the bars problem illustrated in fig .[ figure_bars_pattern ] , is and any individual input patterns contains on the average bars superposed non - linearly .the separation of the 20 statistically independent components in the input data stream is therefore a non - trivial task .the results presented in fig . [ figure_crp_100 ] indicate that the system performs the source separation surprisingly well , but not perfectly .the respective receptive fields , shown in fig . [ figure_crf_100 ] , are only in part self - evident .this is , again , due to the competitive nature of the unsupervised and local learning process , which has the task to optimize the input rates to the dhan layer and not to maximize the signal - to - noise ratio .we note in this context that the system contains no prior knowledge about the nature and statistics of the input signals .in fact , the system has not been constructed in the first place to tackle the non - linear independent component task .the setup used here has been motivated by two simple guiding principles , the occurrence of self - sustained internal neural activity and the principle of competitive neural dynamics .these principles have been used in our study to examine the interplay of the self - sustained internal neural dynamics with the inflow of external information via a sensory data stream .one can therefore interpret , to a certain extend , the capability of the system to perform a non - linear independent component analysis as an example of an ` emergent cognitive capability ' .this information processing capability emerges from general construction principles and does not result from the implementation of a specific neural algorithm .a standard approach in the field of neural networks is to optimize the design of a network such that a given cognitive or computational task can be tackled efficiently .this strategy has been very successful in the past with respect to technical applications like handwriting recognition and regarding the modeling of initial feed - forward sensory information processing in cortical areas like the primary optical cortex .task - driven network design standardly results in input - driven neural networks , with cognitive computation coming to a standstill in the absence of sensory inputs .real - world cognitive systems like the human brain are however driven by their own internal dynamics and it constitutes a challenge to present and to future research in the field of neural networks to combine models of this self - sustained brain activity with the processing of sensory data .this challenge regards especially recurrent neural networks , since recurrency is an essential ingredient for the occurrence of spontaneous internal neural activities .  in this work we studied the interplay of self - generated neural states , the time - series of winning coalitions , with the sensory input for the purpose of unsupervised feature extraction .we proposed learning to be autonomously activated during the transition from one winning coalition to the subsequent one .this general principle may be implemented algorithmically in various fashions .here we used a generalized neural net ( dhan - dense homogeneous associative net ) for the autonomous generation of a time series of associatively connected winning coalitions and controlled the unsupervised extraction of input - features by an autonomously generated diffusive learning signal .we tested the algorithm for the bars problem and found good and fast learning and that the initially semantically void transient states acquired , through interaction with the data input stream , a semantic significance .further preliminary results indicate that the learning algorithm retains functionality under a wide range of conditions and for various sets of parameters .we plan to extend the simulations to various forms of temporal inputs , especially to quasi - continuous input and to natural scene analysis , and to study the embedding of the here proposed concept within the framework of a full - fledged and autonomously active cognitive system .there is a growing research effort trying to develop universal operating principles for biologically inspired cognitive systems , the rational being , that the number of genes in the human genome is by far too small for the detailed encoding of the fast array of neural algorithms the brain is capable off .there is therefore a growing consensus , that universal operating principles may be potentially of key importance also for synthetic cognitive and complex systems .the present work is motivated by this line of approach .universal operating principles for a cognitive system remain functionally operative for a wide range of environmental conditions .examples are , universal time prediction tasks for the unsupervised extraction of abstract concepts and intrinsic generalized grammars from the sensory data input stream and the optimization of complexity and information theoretical measures for closed - loop sensorimotor behavioral studies of simulated robots .the present study is motivated by a similar line of thinking , investigating the consequences of a self - sustained internal neural activity in recurrent networks , being based on the notion of transient - state and competitive neural dynamics .the long - term goal of an autonomous cognitive system is pursued in this approach via a modular approach , with each module being based on one of the above mentioned general architectural and operational principles .m. rabinovich , a. volkovskii , p. lecanda , r. huerta , h.d.i .abarbanel and g. laurent , _ dynamical encoding by networks of competing neuron groups : winnerless competition _ , physical review letters 87 ( 2001 ) 68102 .    c. gros ,_ self - sustained thought processes in a dense associative network _, in ki 2005 , u. furbach ( ed . ), springer lecture notes in artificial intelligence 3698 ( 2005 ) 366 - 379 ; also available as http://arxiv.org/abs/q-bio.nc/0508032 .             j.l .elman , _ an alternative view of the mental lexicon _, trends in cognitive sciences 8 ( 2004 ) 301 - 306 .seth and g.m .edelman , _ environment and behavior influence the complexity of evolved neural networks _ , adaptive behavior 12 ( 2004 ) 5 - 20 .

Abstract: the human brain is autonomously active , being characterized by a self - sustained neural activity which would be present even in the absence of external sensory stimuli . here we study the interrelation between the self - sustained activity in autonomously active recurrent neural nets and external sensory stimuli .   there is no a priori semantical relation between the influx of external stimuli and the patterns generated internally by the autonomous and ongoing brain dynamics . the question then arises when and how are semantic correlations between internal and external dynamical processes learned and built up ?   we study this problem within the paradigm of transient state dynamics for the neural activity in recurrent neural nets , i.e. for an autonomous neural activity characterized by an infinite time - series of transiently stable attractor states . we propose that external stimuli will be relevant during the sensitive periods , _ viz _ the transition period between one transient state and the subsequent semi - stable attractor . a diffusive learning signal is generated unsupervised whenever the stimulus influences the internal dynamics qualitatively .  for testing we have presented to the model system stimuli corresponding to the bars and stripes problem . we found that the system performs a non - linear independent component analysis on its own , being continuously and autonomously active . this emergent cognitive capability results here from a general principle for the neural dynamics , the competition between neural ensembles .   recurrent neural networks , autonomous neural dynamics , transient state dynamics , emergent cognitive capabilities


in  a cellular automaton model for traffic flow is proposed that is based on the nagel and schreckenberg ( nasch ) model  and that is capable to reproduce all of the empirically observed traffic states  , i.e. , free flow , wide moving jams and especially synchronized traffic ( see for recent reviews on the modeling of traffic flow ) . moreover, the accordance of the model with the empirics can be found even on a microscopic level of description . in  this degree of realismis underscored by considering a more realistic simulation setup , i.e. , a simulation of a two - lane highway segment with an onramp .empirical observations suggest  that lane changes are responsible for the occurrence of strong correlations of the velocity and flow measurements between neighboring lanes in the synchronized states and in wide moving jams . however , it was concluded  that lane changes do only select rather than generate traffic states .although the results of  do not depend on the special choice of the lane change update , the application of the model in more sophisticated topologies should be provided with realistic lane changing rules .much progress has been made in the simulation of large traffic networks like the inner - city of duisburg  , the dallas / forth - worth area  and the highway network of north - rhine - westfalia  .these simulations were based on the nasch model and use a detailed representation of the systems infrastructure like multi - lane traffic , on- and offramps and highway intersections .however , a realistic microscopic description of the dynamics of each individual vehicle in a network requires to take into account the peculiarities of the complex road structures .it is therefore necessary to formulate simple rules in the language of cellular automata which mimic the behavior of the drivers as simple as possible and yield reasonable results compared with empirical findings .unfortunately it turned out that the incorporation of two - lane traffic even in the nasch model is a formidable task and the formulation of realistic lane changing rules is very difficult .only a few empirical results for two - lane traffic exist that help to specify lane changing rules  .the ability to change lanes should increase with the density , shows a maximum in the vicinity of the flow maximum and then decreases with increasing density  .nevertheless , lane changes are still possible at large densities .a special feature of german highway traffic is the empirically observed lane usage inversion .although there is a right lane preference , the distribution of the flow becomes asymmetric and the flow is larger on the left than on the right lane .in contrast , in urban traffic or on highways without the right lane constraints the traffic flow is evenly distributed on both lanes  .several two - lane extensions  of the nasch model were proposed which are able to reproduce the empirically observed lane usage inversion .however , they fail to model a realistic density dependence of the number of lane changes .moreover , it was shown that some problems of the cellular automaton approach exist if one introduces different kinds of vehicles , but these shortcomings can be minimized by the consideration of anticipation effects  .  here , we extend the recently introduced traffic model  to two - lane traffic and answer the question which mechanism gives rise to the empirically observed lane usage inversion .it is clarified that this extension can be made without changing the basic realistic properties of the single - lane model .moreover , it is demonstrated that lane changes do lead to strong correlations between neighboring lanes .the simulations are ( if not stated otherwise ) performed on a highway of length cells with periodic boundary conditions .in addition to the local measurements of  traffic data are also recorded by averaging over the whole highway segment .this allows , in particular , to observe lane changes which can not be measured by local detectors .the averaging process , however , does not distinguish between the various traffic states on different parts of the highway  .  in the next sectionwe briefly summarize the definition of the single - lane model and present a simple two - lane extension . because it is not possible to generate a lane usage inversion by the introduction of different kinds of cars , we present in section [ section : asym ] an asymmetric lane changing rule set .in section [ section : local ] we show that the vehicle dynamics of the single - lane model is not influenced by the lane - changes , but the car dynamics is determined by the single - lane update rules only .the advantages of two - lane traffic in comparison to single - lane traffic become obvious in section [ section : comparison ] .finally , we conclude with section [ section : conclusion ] .before we will present our lane changing rule set , we briefly define the single - lane model ( see  for a detailed explanation ) . the model used hereis based on the nagel - schreckenberg cellular automaton model for traffic flow  .the road is divided into cells and a vehicle has a length of five cells .the movement of the vehicles is determined by a set of simple update rules and the system update is performed in parallel .the model of is based on the desire of the drivers for smooth and comfortable driving whereas the dynamics of the nasch model only assures the avoidance of crashes .this more sophisticated behavior is incorporated in a driving strategy which comprises three aspects : ( i ) at large distances the movement of a vehicle is determined by the maximum possible velocity .vehicles at rest have a smaller acceleration capability than moving vehicles .additionally , the velocity of the leading car is anticipated which allows for smaller gaps and larger velocities .( ii ) at intermediate distances vehicles anticipate last minute braking by means of brake lights .( iii ) at small distances the drivers adjust their velocity such that safe driving is possible ..summary of the variables and of the parameters of the model used in the model definition . [ cols="<,<,<,<",options="header " , ]   the two times and give the time a vehicle needs to reach the position of its predecessor and its predecessor on the destination lane .since the time - headways take the velocity of the vehicles into account , slow vehicles are allowed to change the lane even at small distances ) in order to retard the change from the left to the right lane .unfortunately , this has major drawbacks since for larger densities this condition is never be fulfilled  . ] .  with the two parameters and it is possible to trigger the lane usage inversion .if the movement of a vehicle is undisturbed ( at small densities ) it changes the lane if its time - headway to the preceding vehicle on its own and on the destination lane is larger than and , respectively .otherwise , at large densities the condition ensures an incentive to change to the right lane .figure [ asym : fundi ] illustrates the asymmetry of the lane changing rules with the fundamental diagram : at small densities , the flow on the left lane is larger than on the right lane . as a consequence , the flow breakdown occurs first on the left lane .once the maximum flow on the left lane is reached more and more cars swerve on the right lane until the flow is distributed evenly on both lanes .therefore , the system breakdown is mainly triggered by a single - lane breakdown on the left lane  .however , the mean velocity on the right lane starts to decrease before the breakdown on the left lane occurs , so that the velocity is always larger on the left rather than on the right lane .thus , the system separates into a fast and a slow lane .the two competing mechanisms of the lane changing rules , namely the right lane overtaking ban and the right lane preference , lead to a lane usage inversion ( fig .[ asym : laneusage ] ) . at small densitiesthe right lane preference dominates so that most of the vehicles are driving on the right lane .however , as a consequence of the right lane overtaking ban , with increasing density more vehicles avoid driving on the right lane .the position of the inversion point is mainly controlled by while the extent of the lane usage inversion is controlled by . with increasing total densitythe lane usage inversion decreases ( fig .[ asym : laneusage ] ) .  due to the retarded lane change from left toright the number of lane changes is decreased significantly compared to the symmetric model . like in the symmetric model ,the maximum of lane changes is reached in the vicinity of the maximum flow . further increasing the density the lane change number drops to a plateau value until lane changes are no longer possible at very large densities ( fig .[ asym : lanechangenumber ] ) .in contrast to the symmetric model , the ping - pong changes are strongly suppressed due to the asymmetric rule set .next , we consider an inhomogeneous model with trucks .in contrast to the symmetric model trucks are now allowed to change from the right to the left lane . in fig .[ asym : inhomo : sharenoban ] the distribution of the trucks on both lanes is depicted .although at small densities considerably more trucks are on the right than on the left lane , the trucks do not condensate completely on the right lane .therefore , the flow is dominated by trucks on both lanes even for small densities and a small fractions of trucks . as a result of the lane usage inversion and a small number of trucks on the left lane, vehicles can go faster on the right rather than on the left lane .however , at densities larger than the maximum velocity of the vehicles are smaller than the mean velocity so that the trucks are evenly distributed on both lanes .the stability of the plugs is mainly determined by velocity fluctuations since the distance between two trucks driving side by side in free flow can only be increased by a repeated deceleration of one truck in the randomization step of the velocity update .we therefore distributed the maximum velocity of the cars according to a gaussian profile with different variances and a mean . despite the possibility that now small differences of the maximum velocity of the trucks are allowed , the plugs remain still very robust .even different gaussian distributions of the maximum velocities of cars and trucks or special lane changing rules for trucks do not decrease the stability of the plugs .plugs are therefore no artifact of the discretization but are formed by the dynamics . because of the asymmetry of the lane changing rules , once a truck changed to the left lane , there has to be a large gap in front on the left and on the right lane to change back .this increases the time trucks are on the left lane and supports the formation of plugs .in contrast to the symmetric model , even one truck on the left lane is able to dominate the flow of the left lane completely .as long as the truck stays on the left lane passing is not possible : due to the right lane overtaking ban , like in reality , the fast cars pile up behind the truck waiting for a possibility to pass .  in order to avoid the formation of plugs we reintroduce the lane changing ban for the trucks , so that the trucks now are lane stationary on the right lane . like in the symmetric model ,the flow of the right lane is dominated by trucks only in the vicinity of ( fig .[ asym : inhomo : ban : flow ] ) which is a consequence of the increased robustness of the model regarding slow vehicles . as a result of the moving hindrances on the right lane ,the number of lane changes is increased compared to the homogeneous model ( fig .[ asym : inhomo : ban : lanechanges ] ) .now , lane changes are necessary even at very small densities to avoid being trapped behind a truck . therefore, the lane usage inversion is increased significantly in the vicinity of ( fig .[ asym : inhomo : ban : laneusage ] ) .  like in the homogeneous model the traffic breakdownis mainly controlled by the capacity of the left lane ( fig .[ asym : inhomo : breakdown ] ) . at small densitiesthe flow on the left lane is larger than on the right lane since the density is distributed asymmetrically . further increasing the density shifts the excess flow on the right lane up to its capacity . at large densitiesmore vehicles are on the left than on the right lane because of the lane usage inversion , which results in a larger flow on the right lane .in the last sections we focused our analysis on two - lane properties induced by lane changing maneuvers like the number of lane changes , the flow enhancement and the lane usage inversion .it turned out that it is possible to produce a realistic lane changing behavior which can reproduce empirical data already by means of a simple asymmetric rule set .nevertheless , it is an open question whether the vehicle dynamics of the single - lane model is modified by the introduction of lane changing rules .  in order to compare the two - lane model with the corresponding single - lane model, we evaluated the simulation data by a virtual inductive loop , i.e. , we measured the speed and the time - headway of the vehicles at a given link of the lattice .the density is calculated via the relation where and are the mean flow and the mean velocity of cars passing the detector in a time interval of min .in addition to the aggregated data , the single - vehicle data of each car passing the detector are also analyzed .  a detailed analysis of the locally measured data of the homogeneous model shows no difference of the two - lane model to the single - lane model on a macroscopic ( i.e. , data averaged over an interval of min ) as well as on a microscopic ( i.e. , single - vehicle data ) level in the congested regimemoreover , there is no difference of the locally measured data between the two - lanes in the congested regime .in contrast , in the free flow regime the time - headway distribution of the left lane shows an increased number of small headways compared to the right lane .the lane usage inversion at small densities increases the car - car interactions on the left lane which leads to clusters of cars with small gaps and large velocities . in the inhomogeneous modelthe inclusion of disorder dissolves these clusters so that the difference between the time - headway distributions to the single - lane model vanishes again .moreover , trucks on the right lane reduce the mean velocity in the free flow regime and shift the velocity - headway curve to smaller velocities at small distances compared to the left lane .nevertheless , these results show , that the vehicle dynamics is not influenced by the lane changing rules but by the traffic state .empirical observations suggest a strong coupling of the lanes in the synchronized state  . in order to quantify the interaction of the lanes ,we calculated the crosscorrelation of the time - series of measurements of the density , the flow and the velocity on different lanes :    here denotes either the flow , the density or the velocity in lane .  in the free flow state the weak interaction of the lanes results in a small correlation of all quantities ( fig .[ asym : crosscorr1 ] ) .since the flow is mainly controlled by density fluctuations the density and the flow measurements show the same correlations .in contrast , at larger densities the velocity and the flow on both lanes is strongly correlated ( at ) , i.e. , indicating that the system is in the synchronized regime .however , due to the large variance of the density measurements in the synchronized regime the density on both lanes is not correlated ( fig .[ asym : crosscorr2 ] ) .lane changes are responsible for the synchronization of the velocities of different lanes . in case of a velocity difference between the lanes, vehicles are changing to the faster lanes , thus decreasing the velocity difference .this leads to a strong coupling of the lanes in the congested regime .in particular , in the vicinity of on- and offramps cars enter or leave only the right lane , but both lanes are disturbed at large in- or outflows . in order to verify this , we simulated a system with open boundary conditions and generated a jam on the right lane by a large input of an onramp .as a result , the jam spreads to the left lane , so that the time - series of the velocity measurements of both , the left and the right lane are highly correlated .moreover , the traffic breakdown occurs at the same time ( fig .[ asym : open_jam ] ) leading to a synchronization of the lanes .a detailed comparison of the single - lane model with the corresponding two - lane models shows that it is not possible to increase the flow of a two - lane model to more than twice the flow of a single - lane model by means of an asymmetric lane interaction .in contrast to symmetric models , the flow of the asymmetric model on the right lane is decreased whereas the flow on the left lane is increased compared with the single - lane model ( fig . [ asym : fundi : compare ] ) . the right lane overtaking ban and the right lane preference lead to a suboptimal gap usage and , therefore , to a reduction of the flow on the right lane . as a result ,the mean flow per lane is smaller than the flow of a homogeneous single - lane model .however , the benefits of the two - lane model become visible if different types of cars with different maximum velocities are considered .( note , that the trucks are not allowed to change the lane . ) due to the possibility of overtaking , the system is more robust regarding disorder . while the flow of a single - lane model is dominated already by just one slow car  ( since passingis not possible ) , the flow on the right lane of the two - lane model is only decreased in the vicinity of compared with the homogeneous model .furthermore , the travel - time of a vehicle is reduced considerably by the lane interaction . the travel - time is measured as the time a vehicle needs to pass a given fixed link of the lattice twice .thus the travel - time on the right lane is given by vehicles that passed the measurement section on the right lane .here it is not distinguished whether the car started on the left or the right lane and lane changes during the trip through the system are allowed .compared to the single - lane model , the width of the travel - time distribution in the congested regime of the two - lane model is larger ( fig .[ asym : traveltime1 ] ) but the maximum has moved to smaller times . on one hand , vehicles which can not change the lane due to an insufficient safety gap lead to larger travel - times . on the other hand , the possibility to change the lane reduces the travel - time .moreover , the asymmetric distribution of the density leads to a smaller travel - time on the right lane than on the left lane again , the results in the congested regime are independent on the fraction of trucks since driving with maximum velocity is no longer possible .  in the free flow regime ,the inclusion of trucks on the right lane is responsible for a larger mean travel - time compared with the homogeneous two - lane model ( fig .[ asym : traveltime2 ] ) .the trucks on the right lane enforce clustering so that the travel - time of cars trapped behind a truck increases which leads to a second small peak at large times in the travel - time distribution .therefore , the more trucks are in the system , the larger is the variance of the travel - time distribution .we have presented an asymmetric two - lane model with a simple and local lane changing rule set that shows the empirically observed lane usage inversion and reproduces the density dependence of the number of lane changes . in order to observe lane usage inversionit is not sufficient to implement asymmetry into the model by means of disorder , i.e. , different types of cars .moreover , the introduction of the right lane overtaking ban or the right lane preference alone fails to reproduce the lane usage inversion , too .thus , realistic behavior is only obtained by incorporating both a right lane preference and a right lane overtaking ban simultaneously into the model s rules .the benefits of the two - lane model compared with the single - lane model become visible by introducing disorder : there is no flow enhancement of the two - lane model with one type of cars , but the robustness of the flow regarding slow vehicles can be increased significantly by means of lane interaction .this lane interaction leads to a strong coupling of both lanes in the synchronized regime , which results in strong correlations of the time - series of the velocity .  as a consequence, a jam generated on the right lane by an onramp can branch to the left lane leading to a congested region on both lanes . in order to improve and optimize the throughput of the highway it is possible to use ramp metering systems  . here, such systems will help to reduce perturbations of the right lane and , thus , stabilize the flow even on the left lane .finally , a detailed analysis of local measurements shows that the car dynamics remains unchanged by the introduction of the lane changing rules .therefore , the lane change extension of the single - lane model can be used for the propagation of traffic state information to all lanes , e.g. , in order to synchronize the lanes , or to trigger distinct traffic states ( e.g. , in the vicinity of on- and offramps ) . moreover, the independence of the velocity update and the lane change update allows for further improvements of the car dynamics and/or developments of more sophisticated lane changing rules ( e.g. , for bottlenecks ) .  *acknowledgments * : the authors are grateful to the ministry of economics and midsize businesses , technology and transport for the financial support .l. s. acknowledges support from the deutsche forschungsgemeinschaft under grant no .sa864/1 - 2 .99        j. esser and m. schreckenberg , int .j. mod .c 8 , 1025 ( 1997 ) .m. rickert and k. nagel , int .j. mod .c 8 , 483 ( 1997 ) .o. kaumann , k. froese , r. chrobok , j. wahle , l. neubert , and m. schreckenberg , in _ traffic and granular flow 99 _ , p. 351, d. helbing , h. j. herrmann , m. schreckenberg , d. e. wolf ( eds . ) ( springer , 2000 ) .gang - len chang and yang - ming kao , trans .a * 25 * , 375 ( 1991 ) .f. l. hall and t. n. lam , trans .a * 22 * , 45 ( 1988 ) .m. brackstone , and m. mcdonald , in _ traffic and granular flow _ , p. 151, d. e. wolf , m. schreckenberg and a. bachem ( eds . ) , ( world scientific , 1996 ) w. leutzbach and f. busch , _ spurwechselvorgnge auf dreispurigen bab - richtungsfahrbahnen _ , institut fr verkehrswesen , universitt karlsruhe ( 1984 ) ( in german ) .t. nagatani , in _ traffic and granular flow _ , p. 57, d. e. wolf , m. schreckenberg and a. bachem ( eds . ) ( world scientific , 1996 ) .t. nagatani , j. phys .a * 29 * , 6531 ( 1996 ) w. knospe , l. santen , a. schadschneider , and m. schreckenberg , physica a * 265 * , 614 ( 1999 ) .r. barlovic , l. santen , a. schadschneider , and m. schreckenberg , eur . phys .j * 5 * , 793 ( 1998 ) .d. chowdhury , d.e .wolf , and m. schreckenberg , physica a * 235 * , 417 ( 1997 ) .

Abstract: a two - lane extension of a recently proposed cellular automaton model for traffic flow is discussed . the analysis focuses on the reproduction of the lane usage inversion and the density dependence of the number of lane changes . it is shown that the single - lane dynamics can be extended to the two - lane case without changing the basic properties of the model which are known to be in good agreement with empirical single - vehicle data . therefore it is possible to reproduce various empirically observed two - lane phenomena , like the synchronization of the lanes , without fine - tuning of the model parameters .


a langmuir monolayer is a single molecular layer of insoluble surfactant molecules spread on the air / water interface .the surfactants are typically amphiphilic molecules with a hydrophilic headgroup and a hydrophobic tail .each of the individual molecules has internal degrees of freedom , namely the tilt and the tilt azimuth .such a system exhibits complex phase structure .the `` tilted '' phases have uniform tilt and possess mesoscopic ordering in the tilt azimuth .the structure of the tilt azimuth is typically observed as a variation in the light intensity under a polarized light microscope .the tilt azimuth organization is referred to as the texture .various classes of the texture have been observed , such as the stripes in the bulk , the star configuration and `` boojums'' in the domains of the tilted phase , when it coexists with an isotropic phase .the term boojum refers to a class of textures that has a tilt azimuth distribution which resembles the structure of the orbital angular momentum in a superfluid droplet .the `` boojum '' texture , in which the tilt azimuth is distributed continuously without singularity , will be the subject of this report .domains observed to contain a boojum texture are not circular in shape . in addition , micron - size bubbles , which are regions of isotropic phase surrounded by a `` tilted '' phase , have been found to have non - circular shapes .the local tilt azimuth in the `` tilted '' phase around the bubble exhibits a non - trivial structure , which has been termed an `` inverse boojum '' .the relationship between experimentally observed textures and the underlying structure of the ordered phase has attracted attention in the literature recently . in particular , the `` boojum '' texture was first discussed by mermin in the context of orbital angular momental distribution in superfluid droplet .similar texture of has been found and discussed in the liquid crystal films .an extensive discussion on the various classes of textures in the langmuir monolayers can be found in ref . .the problem of the equilibrium shape of , and the texture contained in domains in a langmuir monolayer has been investigated by rudnick and bruinsma who varied both the texture and the boundary analytically in a perturbative manner .it was discovered that a non - circular boundary represents the equilibrium shape of a domain only when there is bulk or line - tension anisotropy .the equilibrium domain boundary was derived as a function of line - tension anisotropy .galatola and fournier obtained numerically , in a fixed background texture , the equilibrium boundary when both elastic and line - tension anisotropies are present .rivire and meunier have attempted to explain the observed non - circular domains in terms of elastic anisotropy .evidence of bubbles with a non - circular boundary and an `` inverse boojum '' has been reported , and a qualitative theoretical discussion of the equilibrium shape and texture associated with the bubbles can be found in ref .  . in the spirit of ref . , the authors have analyzed in ref . the equilibrium texture and boundary shape combinations perturbatively to the first order in both the bulk elastic and line - tension anisotropies .the approach describes the infinitesimal response of the texture and the boundary to anisotropic parameters .however , when the correction is large enough to be observed , the validity of first order perturbative calculations becomes questionable .the extension of the perturbative approach to include higher order corrections is algebraically formidable . if one is to go beyond first order effects , the use of numerical techniques in this problem is inevitable .the major challenges in this problem are , first , the evaluation of a 2d texture with a boundary condition on the boundary , which is , itself , variable .secondly , not only must the texture be evaluated with high accuracy , but a precise determinations of the derivatives of the texture on the boundary is also crucial to the computation of the boundary shape .the authors have developed a numerical algorithm based on the finite element method ( fem ) with adaptive mesh refinement for the evaluation of a 2d texture and its derivatives .the boundary corrections can then be computed using the runge - kutta methods .implementation of the numerical method reveals various classes of domain shapes ranging from those with indentations to those with protruding features and , additionally , of cigar - shaped domains .the effects of bulk elastic anisotropy have also been examined .these studies lead us to the conclusion that , as least for those domain shapes observed to date , it is more likely that the line - tension anisotropy is responsible for non - circular domains .a brief account of the study described above has appeared in an earlier publication .the numerical results also confirm that the qualitative conclusion to be drawn from the perturbative treatment are preserved up to large anisotropic parameter .  in this report , we describe in detail the implementation of the numerical methods that lead us to the results reported in ref  .the extension of the algorithm to allow for computation in the case of bubble has also been examined .it is verified numerically that bubbles acquire a non - trivial boundary shape when only the first term in the fourier expansion of the line tension is present .this result contrasts with what is known to be true in the case of domain , which remains circular in the presence of this low - order line - tension anisotropy . with the use of our numerical algorithm , we are able to examine the effects of the bulk elastic anisotropy on the shape of the bubble and on the texture that surrounds it . we find that bulk elastic anisotropy significantly affects the texture in the condensed phase around the bubble while leaving the boundary nearly unmodified .the organization of this paper is as follows .section [ sec : numerics ] contains the details of the computational scheme for the evaluation of the equilibrium textural and boundary configuration for domains .the discussion covers the derivation of the simplest variational formulation of the finite element method in our specific application , the runge - kutta method and the combined algorithm . in sec .[ sec : domains ] , results for the domain are examined .section [ sec : bubbles ] describes the extension of the numerical algorithm to the problem of bubbles .an examination of the results of the perturbative treatment follows .new results on the effect of the bulk elastic anisotropy on the textures around the bubbles are discussed .finally , sec .[ sec : conclusions ] contains concluding remarks and discusses possible future extensions of the numerical methods discussed in this report .the model that we adopt for the langmuir monolayer is a simple elastic model of an ordered media associated with -like order parameter a 2 dimensional unit vector , which can be parameterized as  .the quantities and are unit vectors in a cartesian coordinate system , and is the angle between and the x - axis .the energy of the system contains contributions from the boundary , , in addition to the bulk , .the most general form of the elastic energy for such a system with in - plane reflection symmetry ( an achiral system ) can be written as =\int_\omega { \cal h}_b da+\oint_\gamma\sigma(\vartheta-\theta)ds , \label{sysenr}\end{aligned}\ ] ] where  , are respectively the splay and bend elastic moduli , and is the angle between the outward normal of and the x - axis .the setup of the computations is shown in fig .[ setup ] . in terms of the average frank modulus and the coefficient of elastic anisotropy , where and , the extrema of the elastic energy eq .( [ sysenr ] ) occurs when and the bounding curve satisfy their respective equilibrium conditions .the extremum equations for are &&= 0 , \label{mbulk}\end{aligned}\ ] ] in and +&&\nonumber\\ \kappa b\theta_t\sin 2(\vartheta-\theta)- \sigma^{\prime}(\vartheta-\theta)&&=0 , \label{mbc}\end{aligned}\ ] ] along , where , , being the tangential vector .the extremum equation for the bounding curve , in terms of , and , is \frac{d\vartheta}{ds } + \lambda&&=0,\label{mgamma}\end{aligned}\ ] ] where is the length element of traversing in the positive direction of and is a lagrange multiplier that enforces the condition of constant enclosed area.  the equations for both and are complex and highly non - linear .closed form analytic solution of the extremum equations is almost impossible .attempts have been made to solve the simultaneous equation perturbatively to first order in the elastic and line - tension anistropies  .when the corrections to the boundaries are large enough to be observable , it is not expected that the results are accurate and high order corrections have to be taken into account .however , these attempts provide us with insight with regard to the infinitesimal response of the boundary to the anisotropies under investigation . in the work to be described below ,we analyze the equations numerically in order to further explore the implications of the simple model eq .( [ sysenr ] ) for a larger range of the anisotropic parameters .we retain coefficients up to in the expansion of the line tension in our analysis , i.e. , where the quantity is defined for convenience .we remark that the analysis will be based on the exact `` boojum '' texture with circular domain when .the boundary will be computed in terms of the corrections to the circular boundary .the discussions will be restricted to those domains with boundaries for which the distance from each points on the curves to the origin is a single - valued function of the polar angle .the numerical algorithm consists of two parts : in the first part , one evaluates the texture using an assumed boundary , and , in the second part , one computes using a fixed .simultaneous equilibrium conditions for and are achieved when a set of predefined self - consistent criteria is met .it is evident from the form of eq .( [ mgamma ] ) that accurate determinations of and its derivatives are the key factors in the solution of the problem . the requirement that the assumed in the first part of the algorithm be an arbitrary curve rules out finite difference methods , and militates in favor of finite element methods ( fem ) .a key feature of the fem is flexibility in the choice the set of points at which the functional values are to be evaluated , including those on the boundary of the region of interest .this feature is exactly what is needed in our problem , because of the non - trivial geometry of the boundary .one of the simplest constructions of the fem in 2 dimension is described as follows .we first approximate by a polygonal curve , then subdivide into a set of non - overlapping triangles .no vertex of one triangle lies on the edge of another in the set .the edges of the set of triangle forms a mesh that covers .the process of creating this set of triangles is called mesh generation .the resulting set of triangles is referred to as the triangulation of .functions are defined by their values on the vertices of the triangles in the triangulation .the value of a function within a triangle is obtained by interpolation using the values on the vertices .integration over is the sum of integrations over the triangles which can generally be trivially evaluated .we have now projected our problem , originally on an infinite dimensional space onto a n dimensional space , where n is the number of vertices in the triangulation of .we may write , and where is a set of basis functions of the n dimensional space .these s should not be confused with the polar angle which is denoted by the symbol without a subscript . the discrete version of the elastic energy functional eq .( [ sysenr ] ) is a function of n variables and it can be rewritten in terms of and as \right\}\;da+ \oint_\gamma\sigma(\vartheta-\theta)\;ds,\end{aligned}\ ] ] where , , and .the equilibrium condition becomes which is a discretized version of eqs .( [ mbulk ] ) and ( [ mbc ] ) .the set of above equations is not linear .however , if we write them in the form of where is an matrix , and are column matrix as shown below , \label{matrixa}\\ b_i(\bf \theta)&=&\kappa b\int_\omega [ ( -\theta_x^2+\theta_y^2 ) \sin 2\theta+\nonumber\\ & & 2\theta_x\theta_y\cos 2\theta]\varphi_ida + \nonumber\\ & & \oint_\gamma\sigma^\prime(\vartheta-\theta)\varphi_ids\label{vectorb}\end{aligned}\ ] ] we are able to solve for iteratively using a standard numerical algorithm for the solution of systems of linear equations .we have adopted the method of lu decomposition for solving .the mesh generation algorithm plays an important role in the efficiency of the fem .an adaptive mesh generation algorithm is used in our program to determine .we start with a mesh that is nearly regular throughout with a predefined grid size . after obtaining a first estimate of ,a refined mesh is generated .the refined mesh has variable grid sizes over depending on the variation of .figure [ mesht ] depicts the process of mesh generation with adaptive refinement .we are able to determine not only , but also the derivatives and , which are necessary for the evaluation the bounding curve , accurately and efficiently with the adaptive mesh generation algorithm .the next part of the algorithm is the determination of the bounding curve .we assume the order parameter field is fixed in eq .( [ mgamma ] ) so as to simplify the problem .we then pick an origin in and parameterize the bounding curve as , where is the distance between the origin and on , and is the polar angle . in this parameterization ,( [ mgamma ] ) is a second order non - linear differential equation in .if we rewrite eq .( [ mgamma ] ) as where and ( 1+k^{\prime2})+\nonumber\\ & & \frac{{\mbox e}^k{\cal h}_b+\lambda}{\sigma+\sigma^{\prime\prime}}(1+k^{\prime2})^{3/2}. \label{geq2}\end{aligned}\ ] ] again , it is possible to integrate the equation for iteratively using standard method for the solution of ordinary differential equation .the runge - kutta method is chosen for our application .  the problem of solving eqs . ( [ mbulk ] ) , ( [ mbc ] ) and ( [ mgamma ] ) for and reformulated in terms of the solution of eqs .( [ fem ] ) and ( [ rk ] ) iteratively for and .we begin by assuming an initial boundary and texture , from which the texture can be computed using fem . then iterated texture is in turn used to evaluate a new accepted boundary , where is obtained using the runge - kutta ordinary differential equation integrator on eq .( [ mgamma ] ) .the process is repeated until both and are less then a preset tolerance of the order .the factors and with initial magnitude in the order of is introduced to avoid numerical instability in these iterative processes .these factors , and , are adjusted depending on   2 and , and at the final iterations take on values close to unity .it is well established that the boundary is strictly circular for a domain with a boojum texture when elastic anisotropy and line - tension anisotropy are not present , or  .this texture - boundary combination is indeed a local minimum of eq .( [ sysenr]) .thus , in order for there to be non - circular domains , it is necessary to retain terms in the expansion eq .( [ linetension ] ) up to at least terms going as . using the numerical algorithmdescribed above , we have performed systematic studies of the domain textures and shapes in terms of the elastic anisotropy , the line - tension anisotropy as well as the domain size .before we describe our observation , we note that when , the exact result is given by a circular boundary of radius together with a `` boojum '' texture with a defect located a distance from the center of the domain , where is the normalized domain radius  . an exact equilibrium texture - boundary combination is shown in fig . [ exact ] .the simulated image obtained using the brewster angle microscopy ( bam ) is also displayed in the background .the signature of a `` boojum '' in a bam image is a set of straight constant - intensity lines emerging from a virtual defect slightly outside the domain .the light intensity in a bam image depends on the exact experimental setup and the properties of the monolayer  . in the case of all simulated bam images presented in fig .[ exact ] and elsewhere in this report , the brewster angle is taken to be that of water , the angle of the analyzer is equal to , the thickness of the monolayer is assumed to be , the tilt is , the dielectric constants of the monolayer are , and it is assumed that the wavelength of the light .we first concentrate on the effects of and keep .when , the texture is altered in such a way that the virtual defect appears to move closer to the boundary .this is observed as accelerated convergence of the constant - intensity lines to a point on the boundary . on the other hand ,when , the texture relaxes as if the virtual defect has moved away from the boundary .the deviation of the texture from the boojum texture is as large as when .the textural response is qualitatively in accord with that reported in ref .  .although there are significant textural corrections due to the presence of bulk elastic anisotropy , the resultant textures very much resemble a `` boojum '' as seen in a bam image as shown in fig .[ bdepd ] .this means that it is difficult to identify elastic anisotropy based on observation of the textures in the domains .the response of the boundary to elastic anisotropy that we obtain contrasts to that reported in ref .the domain acquires an indentation when .the indentation remains observable for a large range of domain sizes .the boundary protrudes slightly for , as depicted in fig .[ bdepd ] .the protrusion when is subtle and does not resemble the sharp feature observed experimentally .thus , elastic anisotropy alone is not capable of accounting for the shapes of the domains observed experimentally .figure [ brdep ] shows domains of various sizes when ( a ) and ( b ) .we now proceed to discuss the role of the line - tension anisotropy , parameterized by , in the textures and the boundaries of the domains .elastic anisotropy will be eliminated ( ) for simplicity .we first investigate situations when .for very small domains where , the texture is almost constant and the dominant contribution to the boundary deformation comes from the contribution .the domain is elongated at both ends along the axis connecting its center and the virtual defect when and is flattened at both ends along the same axis when .domain shapes exhibit a 2-fold symmetry . when , the texture closely resembles the boojum texture and contributes significantly to the boundary distortion through the influence of in the line - tension .the domain nolonger displays 2-fold symmetry and acquires a protrusion when , or an indentation when .figure [ adepn ] shows domains of with ranging from to .the numerical algorithm also allows us to examine domain shape and texture when and . in this case, the domain acquires a `` cigar - shape '' and the texture is associated with two virtual defects  .the progressive changes of the texture and the shape from a system with and to one for which and are shown in fig .[ cgshpdn ] .when both and are nonzero , the texture can be thought of a superposition of pure and pure textures . typically at ,the effect of the set of two defects become observable when .domains with indentation , protrusions , and the `` cigar - shaped '' domains , have all been observed experimentally  .we have already briefly discussed the issue of size dependence in the previous paragraph . to look into this matter in detail, we will examine the particular set of data reported in ref . in which the domains investigated possess protruding features sharp enough that `` excluded angles '' , , characterizing the boundaries can be identified . the definition of and the experimental data are depicted in fig . [ andenf ] .the key features of this set of experimental data are : ( 1 ) goes through a maximum as varies ; ( 2 ) there is an abrupt onset of in the small region ; and ( 3 ) the intercept at the -axis when the curve is extrapolated implies .  before we make comparisons between theoretical results and the experimental data , we comment on the extraction of from computed domain boundaries .it has been shown that within the parameter regime of our discussions , the domain boundaries are smooth and continuous .there is no cusp - like singularity on the boundary .this can be seen in the domains of various sizes shown in fig .[ nurdp ] .nevertheless , can be unambiguously measured for some of these domains .the values of the parameters utilized here are , and .to determine for these domains , we adopt a systematic scheme that utilizes the function  ] given in eq .( [ sysenr ] ) now becomes \right\}\;da \oint_\gamma\sigma(\vartheta-\theta)\;ds\label{femhamil}\end{aligned}\ ] ] where we denote , and .we differentiate eq .( [ femhamil ] ) with respect to a yields \theta_j \nonumber\\ & & -\kappa b\int_\omega [ ( -\theta_x^2+\theta_y^2)\sin 2\theta+\nonumber\\ & & 2\theta_x\theta_y\cos 2\theta]\varphi_ida -\oint_\gamma\sigma^\prime(\vartheta-\theta)\varphi_ids\end{aligned}\ ] ] the equilibrium condition gives with and provided in eqs .( [ matrixa ] ) and ( [ vectorb ] ) .the integrals in eqs .( [ matrixa ] ) and ( [ vectorb ] ) over are broken up into sums of integration over the triangles in the triangulation of .integration over the interior individual triangle can usually be carried out analytically depending on the specific forms of the basis functions and the matrix elements and .we have chosen to be a continuous , piecewise linear function in and within a triangle .the line integral in will must be evaluated numerically , because integrand depends on the polar angle , which is not linear in or .this does not degrade the efficiency of the computation because first of all , only triangles whose perimeters coincide with contribute to the line integral and secondly it is a line integral over a short distance .  in eq .( [ thetatria ] ) , we express a function for in terms of its values at the nodes of the triangulation and the corresponding basis functions . within an individual triangle , we can write where is the index of the vertex of the triangle , , are respectively the functional value of and the coordinates of the vertex . is the restriction of in .the actual index of the vertex is .it is , however , awkward to carry the in the symbol throughout the discussion .we will use to identity the vertex for simplicity from now on , i.e. is simplified as .we introduce a set of natural coordinates and such that where  ] and . transformation between variable sets and can be obtained from eq .( [ universal ] ) by substituting with and .we then have the followings relations ,\\ v=&&\frac{1}{\delta}\left[\left(y_1-y_2\right)x-\left(x_1-x_2\right)y+ x_1y_2-x_2y_1\right],\end{aligned}\ ] ] where is the jacobian determinant given by   identifying eqs . ( [ restriction ] ) and ( [ universal ] ) ,we find , and . in terms of , we have ,\\ \varphi_2(x , y)=&&\frac{1}{\delta}\left[\left(y_3-y_1\right)x- \left(x_3-x_1\right)y+x_3y_1-x_1y_3\right],\\ \varphi_3(x , y)=&&\frac{1}{\delta}\left[\left(y_1-y_2\right)x- \left(x_1-x_2\right)y+x_1y_2-x_2y_1\right].\end{aligned}\ ] ]  evaluation of the matrix element involves of the following area integrals which can be computed analytically .the trivial one is the area of , which is and ,\\ \int_k \sin 2\theta \;da = & & \frac{|\delta|}{4}\left[\frac{\sin 2\theta_1}{(\theta_1-\theta_2 ) ( \theta_2- \theta_3)}+\frac{\sin 2\theta_2}{(\theta_2-\theta_3)(\theta_3-\theta_1)}+\frac{\sin 2\theta_3}{(\theta_3-\theta_1)(\theta_1-\theta_2)}\right],\end{aligned}\ ] ] as and are constants .evaluation of involves ,\\ \int_k\varphi_2\cos 2\theta\;da = & & \frac{|\delta|}{4}\left[\frac{\cos 2\theta_2}{(\theta_1-\theta_2)(\theta_2- \theta_3)}+\frac{\sin 2\theta_2-\sin 2\theta_3}{2(\theta_2-\theta_3)^2(\theta_3- \theta_1)}-\frac{\sin 2\theta_2-\sin 2\theta_1}{2(\theta_1-\theta_2)^2(\theta_3- \theta_1)}\right],\\\int_k\varphi_3\cos 2\theta\;da = & & \frac{|\delta|}{4}\left[\frac{\cos 2\theta_3}{(\theta_2-\theta_3)(\theta_3- \theta_1)}+\frac{\sin 2\theta_3-\sin 2\theta_2}{2(\theta_2-\theta_3)^2(\theta_2- \theta_1)}-\frac{\sin 2\theta_3-\sin 2\theta_1}{2(\theta_1-\theta_3)^2(\theta_3- \theta_1)}\right],\\ \int_k\varphi_1\sin 2\theta\;da = & & \frac{|\delta|}{4}\left[\frac{\sin 2\theta_1}{(\theta_3-\theta_1)(\theta_1- \theta_2)}-\frac{\cos 2\theta_1-\cos 2\theta_3}{2(\theta_1-\theta_3)^2(\theta_3- \theta_2)}+\frac{\cos 2\theta_1-\cos 2\theta_2}{2(\theta_1-\theta_2)^2(\theta_3- \theta_2)}\right],\\ \int_k\varphi_2\sin 2\theta\;da = & & \frac{|\delta|}{4}\left[\frac{\sin 2\theta_2}{(\theta_1-\theta_2)(\theta_2- \theta_3)}-\frac{\cos 2\theta_2-\cos 2\theta_3}{2(\theta_2-\theta_3)^2(\theta_3- \theta_1)}+\frac{\cos 2\theta_2-\cos 2\theta_1}{2(\theta_1-\theta_2)^2(\theta_3- \theta_1)}\right],\\ \int_k\varphi_3\sin 2\theta\;da = & & \frac{|\delta|}{4}\left[\frac{\sin 2\theta_3}{(\theta_2-\theta_3)(\theta_3- \theta_1)}-\frac{\cos 2\theta_3-\cos 2\theta_2}{2(\theta_2-\theta_3)^2(\theta_2- \theta_1)}+\frac{\cos 2\theta_3-\cos 2\theta_1}{2(\theta_1-\theta_3)^2(\theta_3- \theta_1)}\right].\end{aligned}\ ] ] the above formulae will work only if .let us consider cases where the values of at 2 vertices of triangle and at the other vertex .we obtain the following for the integrals in \\ \int_k\sin 2\theta\;da&=&\frac{|\delta|}{4}\left[\frac{\sin 2\theta_e-\sin 2\theta_o } { ( \theta_e-\theta_o)^2}-\frac{2\cos 2\theta_e}{\theta_e-\theta_o}\right]\end{aligned}\ ] ] we denote the restrictions of the basis functions at the nodes that have , and the restriction of the basis function at the node that has . we arrive at \\ \int_k\varphi_o\cos 2\theta\;da&=&\frac{|\delta|}{4}\left[-\frac{\cos 2\theta_e+ \cos 2\theta_o}{(\theta_e-\theta_o)^2 } + \frac{\sin 2\theta_e-\sin 2\theta_o}{(\theta_e-\theta_o)^3}\right]\\ \int_k\varphi_e\sin 2\theta\;da&=&\frac{|\delta|}{4}\left[-\frac{\cos 2\theta_e } { \theta_e-\theta_o}+\frac{\sin 2\theta_e}{(\theta_e-\theta_o)^2}+\frac{\cos 2 \theta_e-\cos 2\theta_o}{2(\theta_e-\theta_o)^3}\right]\\ \int_k\varphi_o\sin 2\theta\;da&=&\frac{|\delta|}{4}\left[- \frac{\sin 2\theta_e+\sin 2\theta_o}{(\theta_e-\theta_o)^2}-\frac{\cos 2\theta_e- \cos 2\theta_o}{(\theta_e-\theta_o)^3}\right]\end{aligned}\ ] ] for the integrals required to evaluate . finally , when for all s , one will need .  2one of the biggest benefits of the fem is that it enables straightforward determination of the derivatives of on the boundary .the tangential derivative at node as \end{aligned}\ ] ] the normal derivative of is given by eq .( [ mbc ] ) which reads }.\end{aligned}\ ] ]   j. ruiz - garcia , x. qiu , m .- w .tsao , g. marshall , c.m .knobler , g. a. overbeck and d. mbius , j. phys .chem * 97 * , 6955 ( 1993 ) ; d.k .schwartz , j.ruiz-garcia , x. qiu , j.v .seligner and c.m .knobler , physica a * 204 * , 606 ( 1994);j.v .selinger , z .- g . wang , r.f .bruinsma and c.m .knobler , phys .* 70 * , 1139 ( 1993 ) .

Abstract: a numerical algorithm based on the finite element methods has been developed to accurately determine the shape of the boundary of a domain containing `` boojum '' textures . within the context of the simple model we adopt , the effects of both bulk elastic anisotropy and line - tension anisotropy on the domain boundary can be examined . it is found that line - tension anisotropy must be present in order to account for domains with protruding features . both elastic anisotropy and anisotropic line - tension can result in domains with indentations . the numerical algorithm has been extended to investigate the problem of a bubble in extended region ordered phase .   # 1  # 1  ( 290,80)(80,500 ) ( 65,500 ) ( 1 , 0)255(320,500 ) ( 0 , 1 ) 5  # 1  # 1  ( 290,80)(80,500 ) ( 330,500 ) ( 1 , 0)255(330,500 ) ( 0 , -1 ) 5  2