while the existence and characteristic distributions of empty regions are already implicit in early self - similar models of structure hierarchy in the universe ( cf . , ) , it was the observational discovery of large , approximately spherical regions almost devoid of visible galaxies known as voids and supervoids ( e.g. , , ) that sparked further theoretical work on underdense regions in the universe .this included analytical studies of the dynamical evolution of individual voids and shell - crossing ( ) as well as statistical properties of the void distribution . applying a simple sign - reversal argument , pointed out that the sphericity of underdense regions increases due to gravitational dynamics , so that it is in fact natural to expect approximately spherical voids . early theoretical approaches to void statistics used poisson statistics of empty regions , as developed by , voronoi tesselations ( ) , and structure formation theory in analogy to the statistics of overdensity peaks ( ) in gaussian primordial density fluctuations ( e.g. , ) .self - similar features in the void size distribution were also noted ( e.g. , ) . of course , much of this earlier work concentrated on einstein - de - sitter cosmology with , as was favoured then . especially after the discovery of the near - isotropic cmb , it was realized that the largest observable voids were difficult to explain theoretically and may therefore provide important cosmological constraints ( e.g. , ) .attention was also drawn to galaxy properties within voids ( e.g. , the void phenomenon of ) .now , with the advent of the paradigm , detailed numerical studies of void statistics ( e.g. , for a comparison ) and analytical studies of void shape evolution in redshift space ( e.g. , ) have been conducted .moreover , the void formation theory based on the excursion set formalism of primordial density fluctuations has been refined to include the notion of hierarchy ( ) . this formalism has been extended more recently to investigate effects of non - gaussianity ( e.g. , ) and modified theories of gravity ( e.g. , ) on voids .the possibility of using voids for precision cosmology has also been explored ( e.g. , ) . on the observational side ,a void catalogue extracted from the sdss data release 7 has recently been published ( ) , which illustrates the strong dependence of void statistics on the underlying void definition .the effect of survey masks and boundaries has been considered in detail by , and void sphericity properties have been noted by . shown how weak lensing observations may be used to measure the mass distribution of voids . in this paper , we propose a new geometrical approach to void statistics based on the idea that voids can be modelled well by spheres in the euclidean 3-space .each such sphere is represented by a point in the 4-dimensional configuration space of spheres , with radius and centre position as coordinates . since the spatial contact between these spheres is physically important ( e.g. , in the evolution of voids ) , it is interesting to consider transformations of the configuration space that preserve such contact relations , namely conformal transformations , in order to define a distance measure on the configuration space .given these assumptions , it turns out that the configuration space has a de - sitter geometry , with timelike radii and spacelike centre positions of the spheres , which emerges from classical sphere geometry and is different from cosmological de - sitter spacetime .one can now study the size distribution of spherical voids in the universe in terms of their distribution over this de - sitter configuration space .the approach presented here has been inspired by other notions of configuration space measures in cosmology ( e.g. , ) , and another application of this de - sitter configuration space of spheres in a different astronomical context has been proposed recently ( ) .the outline of this work is as follows . in section [ sec : desitter ] , we give a brief review of the de - sitter configuration space for the general case of -spheres in -dimensional euclidean space and indicate its conformal structure .this is followed by a discussion of a uniform distribution over the 4-dimensional de - sitter configuration space used to model voids , that is , in section [ sec : uniform ] .we shall derive , firstly , the corresponding void size distribution for voids in the whole ( unrestricted ) 3-space and find a self - similar power - law .secondly , the effect of restricting void positions in 3-space due to survey geometry is studied , which naturally breaks the self - similarity of the size distribution .this is carried out analytically for a survey volume shaped as a spherical sector . then in section [ sec : comparison ] , we investigate how the void size distributions predicted by the unrestricted and restricted uniform de - sitter distributions compare with actual void size distributions .we consider , firstly , theoretical models of self - similarity and the excursion set formalism , to find that the power - law from the unrestricted uniform distribution is , to an extent , consistent with the expectation from structure formation theory .to illustrate this and the effect of survey geometry restriction in a data set , we apply an n - body simulation using the gadget2 ( ) code and extract spherical voids .a comparison with the fitting formula of is given .we also discuss how the scaling dimension of the large scale structure can be estimated , using a fractal model of random spherical cutouts which is reviewed in the appendix .section 5 contains our conclusions and comments on possible further applications of this idea .we begin by briefly recapitulating the notion of a configuration space of spheres in euclidean space and its de - sitter geometry , as discussed in some more detail in ; see also , pp .646647 . consider an -dimensional unoriented sphere in -dimensional euclidean space given in terms of its radius and its centre at .it turns out that , to each such sphere , one can uniquely assign a point , in -dimensional minkowski space endowed with the usual metric , with on the -dimensional hypersurface which is a de - sitter quadric .in other words , each sphere with a given centre and radius corresponds to a point on an -dimensional de - sitter space which can therefore be interpreted as a configuration space of spheres . since for unoriented spheresone may set , we have .therefore , the space of unoriented spheres is in fact given by half of the full de - sitter quadric , sometimes referred to as de - sitter space modulo the antipodal map .geometrically , the coordinates of a sphere can be regarded as a form of lie cycle coordinates for the laguerre cycle representing the sphere ( see , e.g. , the appendix of for more mathematical details ) . taking , as coordinates of the de - sitter configuration space ,its metric induced by the ambient minkowski metric in the usual way can be read off from the line element so that , with respect to the coordinate - induced basis ,   -spheres in , here illustrated for the case and , in projection , , can be characterized by the euclidean angle given in eq . ( [ angle ] ) , which is an invariant of the isometries of the de - sitter configuration space . ]  hence , this metric measures the distance between spheres in terms of their configuration , that is , their radii and positions in space .isometries of this de - sitter space can be identified with conformal transformations of spheres , for the following reason .consider two intersecting spheres and given by and , respectively .then so that isometries leave the euclidean angle between the two spheres invariant which , as illustrated in fig .[ fig : angle ] , measures their contact .indeed , it provides a geometrical interpretation for the line element ( [ lineelement ] ) , since for two spheres that are infinitesimally close in the configuration space so that they are nearly identical in their radii and positions , one finds , finally , a measure of the number of spheres having radii within  ] and with centres in the volume element at becomes from eq .( [ ds - volume ] ) . also , the infinitesimal number of voids with radii in  ] of the volume parameter can then be expressed as so that by eq .( [ f1 ] ) in spherical polar coordinates , using the limits of for voids with volume parameter given by ( [ limits ] ) and the uniform distribution ( [ f2 ] ) , whose constant of proportionality is kept , we can recast ( [ density1 ] ) to obtain \label{density2}.\end{aligned}\ ] ] this expression can in turn be integrated to yield the corresponding cumulative number density , and fig .[ fig : cumul ] shows it for three choices of the opening angle of the survey geometry .these results for the number density ( [ density2 ] ) and cumulative number density of voids illustrate how the intrinsic distribution and the observed distribution can differ substantially due to the restrictions imposed by the survey geometry .small voids , of course , tend to be affected less by the survey geometry and therefore approach asymptotically the power - law ( [ cumul ] ) of the intrinsic distribution .as shown in section [ sec : unrestricted ] , the uniform de - sitter distribution without survey geometry restriction yields a power - law size distribution given by for the differential number density of voids , from eq .( [ density ] ) .this is obviously self - similar in the sense that a change in volume scale , , also implies .interpreted as a probability distribution , we can identify this as a pareto distribution of index one .considering , on the other hand , a sufficiently large but finite set of voids in a fixed volume of , then the integer rank of a void of volume in a decreasing ordered sequence of volumes is proportional to .hence , one can see that zipf s law is satisfied , self - similarity and power - laws are a characteristic feature of fractals , and their applicability to cosmic voids has been investigated early on ( e.g. , ) .more recently , zipf s law in this context has been studied by , followed by more general considerations of the fractal properties of void distributions ( e.g. , ) including the notion of multifractals , which admits a range rather than a single fractal dimension , and lacunarity , which characterizes the void distribution at a given fractal dimension .this is , of course , also related to the question whether and to what extent the large scale structure itself can be described by fractals ( see , e.g. , the review by ) . denoting the ( comoving ) distances of galaxies in by , one can define a local scaling index in terms of the cumulative number count of galaxies ( c.f . , pp .12441245 ) , this can be interpreted as a ( single ) fractal dimension if it is constant and independent of the scale , for instance the hausdorff dimension of a cantor set if ( cf .appendix a ) , in which case the corresponding number count is simply a power - law of the form .if the homogeneity postulate of the cosmological principle holds , one expects as .the detailed interpretation of , however , depends on the homogeneity scale , the correlation length and sample size ( c.f . , pp .236240 ) , as well as on the finiteness of the sample number ( e.g. , ) .it is interesting to note that a simple fractal model , mandelbrot s _ random tremas _ interpreted as voids , provides another geometrical perspective on the cumulative number density of eq .( [ cumul ] ) implied by the uniform de - sitter distribution , which also connects its constant of proportionality with the fractal dimension of the large scale structure mentioned above .consider with a homogeneous mass density and introduce a sequence of spherical cutouts ( tremas ) to model voids .these spheres are placed randomly so that arbitrary overlap is allowed . for a particular size distribution of these spheres , it turns out that an infinite sequence will not empty the space completely but leave in general a non - empty complementary set of some hausdorff dimension .the cumulative number density of the spheres in this case , counting all regardless of overlap , is ( cf .appendix a ) which has the same scaling behaviour as the unrestricted uniform de - sitter distribution , for which arbitrary overlap of spheres is also allowed as discussed in section [ sec : unrestricted ] . interpreting the unrestricted uniform de - sitter distribution in terms of the random trema model, we can therefore identify from eq .( [ cumul ] ) . in the case of a finite data set, one should require that the sample size is much greater than the correlation length , , for eq .( [ trema2 ] ) to apply , as will be discussed further in section [ sec : discussion ] below .finally , it may be emphasized that it is the overlap of the cutouts allowed in this model which causes the fixed exponent in the power - law and relegates information about to the constant of proportionality .this is different from the disjoint cutouts considered more usually , e.g. by , which contain in the exponent .   in their pioneering paper , out that , starting from gaussian density perturbations in a friedmann cosmology and considering the linear quasi - newtonian perturbation analysis of the growing mode , the mass distribution at late times does not depend on the initial mass distribution .they identify a simple physical reason for this self - similarity in the existence of two dimensionless quantities governing the gravitational collapse , which remain approximately _ constant _ during the matter - dominated phase on subhorizon scales .the press - schechter argument yielding a power - law size distribution can be summarized as follows .consider massive particles distributed in some comoving volume .then the mass density function on this volume is obtained by applying some smoothing window function with scale length at each point . at sufficiently early times, this density function will be approximately constant , however , if the density within a window is greater than some ( fixed dimensionless , say ) critical density so that around overdensity peaks due to fluctuations in the density function , then the mass within the window will gravitationally collapse and ultimately form a bound object .again , for sufficiently early times , one may take these fluctuations to be gaussian , such that the volume fraction of points with collapsing windows is given by these windows will contain slightly different masses , but basically . also , the variance may depend on the smoothing scale and hence also on the mass , , for some spectral index .thus , is a function as well , and so the differential number density of ultimately bound objects obeys and hence ,\ ] ] where is the mass scale of the exponential term , which gives rise to a cutoff at large masses ( or volumes ) . for and a scale - invariant with ,this yields the inverse square power - law it turns out that this power - law emerges also in the modified press - schechter approach proposed by , which uses an adaptive window scale .now in the context of voids , one can consider underdensity troughs rather than overdensity peaks ( as in ) and , since gaussian fluctuations are symmetric about the average density , the same power - law applies , as pointed out by .then since , we recover ( [ square ] ) from ( [ pslimit ] ) .of course , in addition to the large volume cutoff , this simple excursion set argument ignores effects of void hierarchy which affect small volumes in particular ( i.e. , by the void - in - void and void - in - cloud processes , cf . ) , so that the power - law will only apply within a range of volumes .hence , at least within a range of volumes , the power - law ( [ square ] ) has a physical basis in the excursion set formalism : the gaussian fluctuations producing voids give rise to a self - similar power - law size distribution which can be described geometrically as a uniform distribution over the de - sitter configuration space of spheres .now in order to compare the void size distributions predicted by the unrestricted and restricted uniform de - sitter distributions from section [ sec : uniform ] with data , we perform an n - body simulation with the parallel tree - particle mesh code , gadget2 ( ) , using cosmological parameters consistent with the wmap seven - year results ( ) , namely spatial curvature , dark energy density , hubble parameter and spectral index . the initial condition for the simulation is generated at redshift using the code developed by , who employ second order lagrangian perturbation theory .the simulation is run in a cubic box with side length in comoving coordinates with periodic boundary conditions , and dark matter particles whose individual mass is scaled to match the mass density of the universe . at redshift , voids are extracted according to the following algorithm .firstly , the discrete simulation data are smoothed to construct the matter density function , using a gaussian window with adaptive length scale .this smoothing length is chosen to be the distance to the 20th nearest particle at each cell .secondly , a set of spherical voids is extracted by defining density minima as void centres and determining the radius for each void as the largest radius centered at the density minimum for which the average density of the enclosed sphere is less than a critical value relative to the average density of the box .hence , for our present purposes , we allow arbitrary overlap of the spherical voids and do not apply an additional merging algorithm to these voids ( cf . , also called _ protovoids _ ) , which would give rise to non - spherical voids .this should allow for are meaningful comparison with the uniform distribution over the de - sitter configuration space of spheres , which allows void overlap as noted in section [ sec : unrestricted ] .the resulting cumulative number density of voids in the simulation box as a function of void volume is shown in fig .[ fig : plot ] .the largest void has a volume of , which can be interpreted in terms of the exponential cutoff in the excursion set formalism mentioned in section [ sec : excursion ] , and whose value is comparable with the simulation results reported in fig . 7 of .until the curve flattens at very small volumes here , of course , rather due to simulation resolution and smoothing scale than effects of void hierarchy , the cumulative number density for can indeed be approximated by the power - law of eq .( [ cumul ] ) derived from the unrestricted uniform de - sitter distribution .it is also instructive to compare this result with the fitting formula of , which was shown to model well the cumulative volume filling factor of voids derived from magnitude - limited samples of galaxies in the 2df galaxy redshift survey , , \label{fitting}\ ] ] where is the effective spherical radius of a void of volume and any shape , denotes the mean separation between galaxies , and are parameters . in order to convert this to a cumulative number density as a function of volume, one needs to rewrite eq .( [ fitting ] ) in terms of . introducing a factor to correct for the fact that our voids are not merged and therefore have a higher number density , we let where denotes the differential number density as before , to obtain by differentiating so that can be computed from ( [ fitting ] ) .while a detailed comparison is beyond the scope of this paper , we note that the parameters , , , and , which seem appropriate choices given the data in tables 1 and 2 of , yield a curve in reasonable agreement with our simulation data , as seen in fig .( [ fig : plot ] ) , using a factor of which is compatible with the void number densities shown in fig .6 and fig . 7 of .we also note that the constant of proportionality in eq .( [ cumul ] ) which fits the power - law range is .thus , as described in section [ sec : self - similarity ] , one would expect the fractal dimension of the large scale structure to be by eq .( [ trema2 ] ) , which seems reasonable given that real observational data can be approximated by at small scales transitioning to at large scales , with a homogeneity scale of and a correlation length of ( e.g. , pp .12311235 , table 1 , and references therein ) , so that our box size is indeed much larger than the correlation length . given , then , that our simulation appears to produce a realistic size distribution of spherical voids without merging , which does indeed agree with the uniform de - sitter distribution for voids in the range , we shall now turn to the effect of restricting the survey geometry . to this end, we select a survey region in the simulation box shaped like a spherical sector as in fig .[ fig : survey ] , with its apex at a corner of the box and its axis oriented along the diagonal .consider such a spherical sector with an opening angle of and a radius of half the maximum radius within the box , .although such a survey volume has only of the total box volume , the volume of the largest void within this survey region is , from eq .( [ largest ] ) , which is greater than . the corresponding cumulative number density is also shown in fig .[ fig : plot ] , together with the theoretical prediction of eq .( [ cumul2 ] ) from the restricted uniform de - sitter distribution .since , for voids in the whole box , the unrestricted uniform de - sitter distribution is a good approximation in the range where is of the order , we expect from fig .[ fig : cumul ] that the survey geometry restriction causes a significant deviation in this range from the power - law asymptote .furthermore , since , we expect a small effect of void overcounting in the restricted uniform distribution due to the large volume cutoff .both expectations are indeed borne out by the data , as can be seen in the figure ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ in parallel with efforts to explain , i think it indispensable to describe clustering , and to mimic reality by purely geometric means . " _ , p. 84_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  in this article , we have considered an application of the 4-dimensional de - sitter configuration space of 2-spheres in euclidean 3-space to cosmology .modelling cosmic voids as spheres and allowing any overlap , it was shown that a uniform distribution over this configuration space gives rise to a self - similar power - law size distribution of voids .it appears to agree well with data in an intermediate range of void volumes , and this can be understood physically from the excursion set formalism as long as the large volume cutoff and void hierarchy effects at small volumes may be ignored .moreover , we pointed out that this power - law may also be reinterpreted in terms of mandelbrot s random trema model , allowing one to estimate the fractal dimension of the large scale structure in a different way compared to other approaches .we have also seen how restrictions of void positions in 3-space due to survey geometry significantly affect the size distribution . now in order to refine this approach , one might implement void merging conditions using this configuration space language . furthermore , since the uniform distribution appears to provide a reasonably good match with actual void distributions of intermediate volume , it may be worthwhile to treat the uniform distribution as a null hypothesis and study the physical interpretation of non - uniform distributions .then deviations from uniformity will , as we have seen , encode physically interesting effects such as void hierarchy processes or the large volume cutoff , which may be particularly sensitive to the underlying cosmology . finally ,while de - sitter geometry plays an important and well - known rle in cosmological spacetimes , we would like to emphasize the interesting , and perhaps rather surprising , fact that the present application of de - sitter geometry to cosmology arises in a physically entirely different context . at this stage ,our approach is primarily a descriptive device which also offers a novel geometrical interpretation of an aspect of self - similarity in cosmology , in the spirit of the quotation above .mcw gratefully acknowledges support from the world premier international research center initiative ( wpi initiative ) , mext , japan , and would like to thank masahiro takada and jean - philippe uzan for useful discussions .ny acknowledges financial support from the japan society for the promotion of science ( jsps ) grant - 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beckmann a. m. , mller v. , 2008 , mnras , 384 , 1189  zee a. , 2013 , einstein gravity in a nutshell , princeton university press , princetonhere we give a simple heuristic argument for the cumulative number density of the random trema model of section [ sec : self - similarity ] , as stated in eq .[ trema1 ] .this idea was introduced by mandelbrot and is sketched in , pp .281283 and pp .301302 . see also , pp .372373 . a rigorous exposition of the present argument can be found in , pp .136142 , where the relevant mathematical literature is also cited .the notion of disjoint cutouts as used , e.g. , by and discussed , e.g. , in , pp .5156 , is different from the one reviewed here .consider , in for simplicity , the closed interval  ] whose endpoints are identified . if the sum of subinterval lengths counted regardless of overlap diverges , it is plausible that the overlapping subintervals will almost certainly remove the entire interval  ] is in fact non - empty with hausdorff dimension ( cf .proposition 8.8 , p. 140, in ) .while the rigorous argument is technical , some insight into the origin of this result can be gained by estimating the hausdorff dimension with a box - counting algorithm in an early stage with negligible overlap .thus , at the stage of the sequence , the remaining length of the interval will be thinking of the sequence as dividing the interval successively into intervals of length at stage , the corresponding number of intervals is and then the hausdorff dimension can be computed by interval - counting , as required . nowthe cumulative number density of these one - dimensional voids , counting all subintervals regardless of overlap , is so that for sufficiently large or small voids of length , we obtain a self - similar power - law as a limiting case , this can be generalized to and a random trema model of -spheres as voids. then the cumulative number density of overlapping voids in terms of their -volume becomes ( cf .p. 302 in ) , with in eq .( [ trema1 ] ) .in addition to the hausdorff dimension of the remaining set , which is a real number and a metric concept , one can also define the lebesgue covering dimension , which is an integer and a topological concept , and in general .now if exceeds , the set is called _fractal_. for a random set , it can be shown using an intersection argument that is a sufficient condition for ( cf . , pp .215216 ) , which is called a cantor set .thus , we expect that a random trema model in 3-dimensional euclidean space gives rise to a fractal cantor set if , as seems to be indicated by the data , c.f .section [ sec : discussion ] .[ lastpage ]

Abstract: starting from the geometrical concept of a 4-dimensional de - sitter configuration of spheres in euclidean 3-space and modelling voids in the universe as spheres , we show that a uniform distribution over this configuration space implies a power - law for the void number density which is consistent with results from the excursion set formalism and with data , for an intermediate range of void volumes . the scaling dimension of the large scale structure can be estimated as well . we also discuss the effect of restricting the survey geometry on the void statistics . this work is a new application of de - sitter geometry to cosmology and also provides a new geometrical perspective on self - similarity in cosmology .   [ firstpage ]  cosmology : theory large - scale structure of universe


synchronization phenomena are ubiquitous in nature and play an important role in biology , ecology , climatology , sociology , technology , and even fine arts . in the study of nonlinear dynamical systems , synchronization is recurrently being placed in the focus of attention , and recently insightful findings regarding the synchronization phenomena on complex networks were reported and comprehensively reviewed .it is interesting to see that one can literary infer topological scales of complex networks based solely on synchronization , thus making a closed loop of dependence between the synchronizability and the structure of underlying interactions of network elements . in neural systems ,in particular , the interplay between the network structure and the dynamics taking place on it is closely interrelated .the function - follow - form paradigm , for example , is central to attest to this observation .it is also well known that the cerebral cortex features properties that are characteristic for complex networks . as a results ,the firing activities of individual neurons are often related to the synchronization of the underlying network , and accordingly synchronized firings can be observed at virtually all processing levels , including the retina , the lateral geniculate nucleus , and the cortex .synchronization on complex networks , therefore , has actually become a focal topic in theoretical neurosciences , as evidenced by several recent studies devoted to the explorations of this subject .important works have been elaborating on general aspects of synchronization on scale - free as well as complex - gradient networks , among many other models .it is now clear that synchronization is key to the efficient processing and transmission of information across a nervous system such as the brain .the handling of information transmission over a neuronal network , however , is still an open avenue for research .since information transmission delays are inherent to neuronal systems because of the finite speeds at which action potentials propagate across neuron axons , and due to the time lapses occurring in both dendritic and synaptic processing , studies are in need of catching up with the most recent advances in synchronization research on complex networks .  indeed , delays have been found responsible for several interesting phenomena in coupled dynamical systems .for example , ernst __ have identified mechanisms of synchronization among pulse - coupled oscillators in the presence of time delay .moreover , it has been shown that coupled oscillators undergo a transition towards amplitude death faster if the time delays in coupling are distributed over an interval rather than being uniform throughout the system .the role of delays and connection topologies for the synchronization of coupled chaotic maps has been studied in , where it is reported that on scale - free and random networks sufficiently large coupling strengths can offset the delayed flow of information .it has also been shown that networks with delays can sometimes synchronize more easily than in their absence , and it has been argued that this may be particularly relevant for neuronal networks for establishing a concept of collective information processing in the presence of delayed information transmission . in the present paper we give further support to the latter assumption by considering the impact of delays in scale - free neuronal networks .more recently , the role of delays by the formation of the so - called chimera , _ i.e. _ coexistent coherent and incoherent states in a system of nonlocally coupled phase oscillators has also been examined , and it has been shown that time - delays can induce a transition towards phase clustering , giving rise to clustered chimera states that have spatially distributed phase coherence separated by incoherence with adjacent coherent regions in antiphase . herewe support the theory of delay - induced dynamical transitions in terms of the synchrony of neuronal noise - induced excitations on a scale - free network .notably , previous studies have already considered particularly the neuronal dynamics on large networks in conjunction with information transmission delay , but the focus was primarily on the bifurcation structure of transitions between different delay - induced states , including oscillatory bumps , aperiodic regimes , traveling , lurching and standing waves , as well as regimes of multistability . the impact of information transmission delay on neuronal synchronization , on the other hand , has been studied in , where the emergence of zigzag fronts , clustering antiphase synchronization and in - phase synchronization on regular and small - world neuronal networks has been discussed .  at present, we aim to extend the scope of the above - mentioned investigations by studying front propagation and synchronization transitions in dependence on the coupling strength and information transmission delays over scale - free neuronal networks with different average degrees and scaling exponents .notably , it has been reported that , by using functional magnetic resonance imaging , power - law distributions can be obtained upon linking correlated fmri voxels , and that the robustness against simulated lesions of anatomic cortical networks relies mostly on the scale - free structure .this study thus addresses a relevant system setup that is still widely open for new research .more specifically , we report several non - trivial effects induced by finite delay lengths , and the ability of its fine - tuning towards highly synchronized fronts of excitations .these findings are compared to the impacts of different coupling strengths and average degrees , and their robustness is examined at different levels of additive noise , variations in system size and different scaling exponents of the underlying scale - free topology .remarkably , we found that , irrespective of the system size and the scaling exponent , properly adjusted information transmission delays play a pivotal role in warranting synchronized fronts of excitations on noisy scale - free neuronal networks , which can be further enhanced via larger coupling strengths or higher average degrees of the constitutive nodes .we argue that this is primarily attributed to the emergence of locking between the delay and the inherent oscillation frequency of individual neurons of the scale - free network .the remainder of this paper is organized as follows . in the next section ,we describe the rulkov map , which will be employed to obtain an efficient setup for simulating neuronal dynamics on scale - free networks . in sectionii we also present the coupling scheme and the measure for synchronization of excitatory fronts , as well as other mathematical methods to be used . in sectioniii we present the main results , and in the last section we summarize our findings and discuss their potential implications .for simulating the neuronal dynamics on a scale - free network effectively , the rulkov map is employed , which succinctly captures all the major dynamical features of the complex continuous - time models .the spatial - temporal evolution of the studied network , corrupted with additive gaussian noise and experienced with information transmission delays , is described by the following iteration equations +y^{(i)}(n)+w \xi^{(i)}(n)\nonumber\\ * & & + d\sum_{j}\varepsilon^{{i , j}}\left[x^{j}(n-\tau)-x^{i}(n)\right ] , \\y^{(i)}(n+1)&= & y^{(i)}(n)-\beta x^{(i)}(n)-\gamma , \i=1,\ldots , n \nonumber\end{aligned}\ ] ] where is the discrete time index , is the membrane potential and the variation of ion concentration of the -th neuron , representing the fast and the slow variable , respectively .the slow temporal evolution of is due to the small values of the positive parameters and , which within this study equal unless stated otherwise .moreover , is the main parameter determining the dynamics of individual neurons on the scale - free network . according to ,if all neurons are situated in excitable steady states $ ] , whereas if complex firing and bursting patterns of temporal activity emerge via a hopf bifurcation . herewe set and initiate each neuron from steady state initial conditions , so that the additive spatiotemporal gaussian noise , having mean and autocorrelation , acts as the only source of large - amplitude excitations . moreover , in eq . ( 1 ) is a nonlinear function warranting the essential ingredient of neuronal dynamics , parameter determines the noise intensity , is the coupling strength , and is the information transmission delay .the latter two parameters will be in the focus of attention within this work , whereas , and will be varied only occasionally .  as the interaction base between neurons we primarily use the scale - free network generated via growth and preferential attachment as proposed by barabsi and albert , consisting of nodes unless stated otherwise .each node corresponds to one neuron , whose dynamics is governed by the noise - driven rulkov map . in eq .( 1 ) if neuron is coupled to neuron and otherwise .following ref . , the preferential attachment is introduced via the probability , which states that a new node will be connected to node depending on its connectivity according to , as demonstrated schematically in fig .this growth and preferential attachment scheme yields a network with an average degree , and a power - law degree distribution with the slope of the line equaling on a double - logarithmic graph .notably , analytical estimations predict the slope of the line to equal  .we will use barabsi - albert scale - free networks having throughout this work ( see fig .[ fig1 ] ) , unless stated otherwise .in order to quantitatively study the degree of spatiotemporal synchronization in the network , and thus support below presented visual assessments of front propagation via space - time plots , we introduce , by means of the standard deviation , a synchronization parameter ( see _ e.g. _ ) , which can be calculated effectively according to : ^ 2-[\frac{1}{n}\sum\limits_{i=1}^{n}x^{i}(n)]^2\ ] ] it turns out that is an excellent indicator for numerically measuring the spatiotemporal synchronization of excitations , hence revealing different synchronization levels and related transitions . from eq .( 2 ) , it is evident that the smaller the synchronization parameter , the more synchronous the neuronal network .accordingly , when the network reaches complete synchrony .final results shown below were averaged over independent runs for each set of parameter values to warrant appropriate statistical accuracy with respect to the scale - free network generation and numerical simulations .we start by presenting space - time plots obtained with a fixed information transmission delay and noise intensity , but different values of the coupling strength .results shown in the left three panels of fig . [ fig2 ] illustrate the spatiotemporal dynamics of neurons on the scale - free neuronal network having .evidently , for small coupling strengths [ see panel ( a ) , left ] the excitatory fronts are quite nicely ordered in both time and space . however , both the temporal and spatial regularity increase further and substantially as is increased [ see panels ( b ) and ( c ) , left ] . interestingly , by setting the information transmission delay to and keeping the same noise intensity , the order in both time and space deteriorates substantially , as depicted by the right three panels of fig .[ fig2 ] . nevertheless , increasing the coupling strength can still improve the overall regularity of the excitatory fronts [ comparing panels ( a ) , ( b ) and ( c ) , right ] .it is thus revealed that different information transmission delays have a profound impact on the spatiotemporal regularity of excitatory fronts , whereas increasing the coupling strength always leads to an improvement of temporal and spatial synchronization .    to investigate the impact of different information transmission delays , outlined in fig .[ fig2 ] more precisely , we show in fig .[ fig3 ] space - time plots obtained with fixed coupling strength and noise intensity , but different values of .it can be observed that the spatiotemporal dynamics is ordered nicely if [ see panel ( a ) ] .when , this deteriorates drastically [ see panel ( b ) ] , but is again revived at [ see panel ( c ) ] .in fact , by closely examining space - time plots obtained with and , respectively , one can observe that the non - zero yet appropriately tuned information transmission delay can further enhance the regularity of excitatory fronts as compared to the case of .quite remarkably , when the regularity of excitatory fronts is again heavily impaired [ see panel ( d ) ] , yet with the order is restored anew [ see panel ( e ) ] .indeed , the information transmission delay induced transitions to superbly synchronized neuronal activities on scale - free networks seem to appear intermittently , at roughly integer multiples of a given value of , which equals approximately in fig .[ fig3 ] . in accordance with this preliminary assessment ,it is expectable that with disorder in the temporal as well as the spatial domain sets in again [ see panel ( f ) ] .visual investigations of fig .[ fig3 ] thus reveal that regular and irregular front propagation appears intermittently as the delay is increased .hence , it can be stated that finite ( non - zero ) information transmission delays play a pivotal role in the generation of spatiotemporal patterns of neuronal activity on scale - free networks .  inwhat follows , the degree of spatiotemporal synchronization will be studied quantitatively via ( see eq .2 ) , so as to support and validate the above visual assessments .furthermore , it remains of interest to examine the impact of different and . in fig .[ fig4](a ) , we first plot in dependence on for three different values of . as visually interpreted by space - time plots presented in fig .[ fig2 ] , larger coupling strengths indeed facilitate spatiotemporal synchronization in an monotonous manner .that is , as increases , decreases ( irrespective of ) , which is in agreement with previous studies examining synchronization phenomena in neuronal as well as many other nonlinear systems . more eventful are results presented in fig . [fig4](b ) , where is presented in dependence on for three different coupling strengths .it can be observed clearly that certain values of significantly facilitate spatiotemporal synchronization of excitatory fronts on scale - free neuronal networks .the two minima of appear at and , respectively , and are largely independent of .this confirms the above claim that the information transmission delay induced transitions to spatiotemporally synchronized neuronal activity appear intermittently , at integer multiples of the given value of . on the other hand ,values of outside these regions significantly impair synchronization , as can be inferred from the rather sharp ascends towards larger beyond the optimal delays .the delay - induced synchronization transitions , as well as the impact of increasing , can be presented succinctly also via contour plots of in dependence on the two parameters .figure [ fig5 ] features two such graphs obtained for [ panel ( a ) ] and [ panel ( b ) ] .transitions to highly synchronized states in dependence on the information transition delay are clearly visible as extensive white regions ( denoting smallest values of ) occurring at and , corresponding to the two minima depicted previously in fig .[ fig4](b ) .moreover , results presented in fig . [ fig5 ] clearly convey the impact of increasing the coupling strength .in fact , not only do increasing decrease , as outlined above when interpreting fig . [fig4](a ) , but also they broaden the span of within which synchronous spatiotemporal neuronal activity is warranted .notably , the optimal values of shift insignificantly during the broadening .these features can be inferred also from fig .[ fig4](b ) , yet the contour plots in fig .[ fig5 ] convey them more clearly . finally , it is interesting to note that different values of do not evoke significantly different results , as can be appreciated by comparing panels ( a ) and ( b ) of fig .[ fig5 ] . from this, we conclude that the delay - induced transitions to synchronous neuronal activity on scale - free networks are largely independent of ( apart from the broadening of the interval of warranting optimal synchronization ) and robust against reasonable variations of the noise intensity .thus far , we have considered only scale - free networks having . since the average degree determines the density of interneuronal links , and is thus arguably an important parameter , we present effects of different in fig .panel ( a ) features space - time plots obtained with and for increasing average degree from left to right .visual inspection reveals that the impact of increasing is comparable to the impact of increasing ( comparing with the space - time plots presented in fig .[ fig2 ] ) in that the excitatory fronts propagate increasingly ordered in both time and space as the average degree increases .this can be confirmed quantitatively via , as shown in fig .[ fig6](b ) .indeed , larger shift lower the whole outlay of in dependence on , thus indicating improvement in the quality of spatiotemporal synchronization of neuronal activity on scale - free networks .importantly , however , the oscillating outlay of , along with the optimal value of , is preserved and does not vary in dependence on .therefore we conclude that the delay - induced synchronization transitions are a robust phenomenon of neuronal dynamics on scale - free networks , which an important role in achieving synchronized information transmission among neighboring as well as distant neurons .  in order to further test the generality of our findings, we examine the impact of different as well as different scaling exponents characterizing the scale - free degree distribution .while the growth and preferential attachment algorithm proposed by barabsi and albert yields a power - law degree distribution with the slope of the line on a double - logarithmic graph and serves as the most commonly used structure for testing theoretical models , also relevant for neuronal networks are slopes around , as reported in .accordingly , we employ an alternative algorithm for scale - free network generation based on assigning a quenched fitness value to every node , and drawing links among them with a probability depending on the fitness of the two involved sites . using exponentially distributed fitness and a threshold rule for linking nodes, we obtain a scale - free network with the scaling exponent , as presented in fig . 3 of .results presented in figs .[ fig7](a ) and ( b ) show clearly that variations of the system size do not notably influence the outcome of our simulations . in fact , the minima of remain located at the same values of and are of the same depth ( with a reasonable error margin ) irrespective of .likewise , by changing the scaling exponent from ( as given by the barabsi - albert algorithm ) to ( as given by the algorithm described in ) the results also do not change significantly in that the minima of appear by roughly the same values of [ compare panels ( a ) and ( b ) of fig .[ fig7 ] ] .notable as a qualitative difference between the two scaling exponents characterizing the underlying scale - free topology is the very first local minimum of by , occurring if the slope of the degree distribution equals , as depicted in fig .[ fig7](b ) .we conjecture that this minimum may be related to the subharmonic of the optimal ( of the first minimum at ) .nevertheless , results presented in fig . [ fig7 ] attest to the fact that reported synchronization transitions on scale - free neuronal networks due to finite information transmission delays are largely robust to variations of the system size and of the scaling exponent characterizing the scale - free degree distribution .finally in this section , we provide an explanation for the emergence of the newly reported delay - induced synchronization transitions .up to now we have shown that the optimal value of , resulting in the occurrence of the first minimum of , as well as its reappearance at integer multiples , does not vary significantly in dependence on [ see fig .[ fig4](b ) ] , ( see fig .[ fig5 ] ) , [ see fig .[ fig6](b ) ] , [ see fig .[ fig7](a ) ] or the scaling exponent characterizing the underlying scale - free network [ see fig .[ fig7](b ) ] .this leads to the conclusion that parameters determining the global neuronal dynamics do not play a significant role .hence , one may adjust the local dynamics of each neuron by varying and ( thus far we have not varied them ) .these two parameters affect the speed of the temporal evolution of , and consequently the predominant oscillation period of excitations .results presented in fig . [ fig8 ] clearly show that the locations of minima of shift to different values of as and are varied .we thus determined the predominant oscillation period of individual neurons within the scale - free network , and remarkably found that for it is , for it is , and for it is .these data agree very well with the occurrence of the first minimum of in dependence on as presented in fig .accordingly , we also conclude that the information transmission delay induced transitions to spatiotemporal synchronization of neuronal activity are due to the locking between and the predominant oscillation period of individual neurons on the scale - free network .in sum , we have studied front propagation and synchronization transitions on scale - free neuronal networks in dependence on the information transmission delay , coupling strength and the average degree .we found that an appropriately adjusted delay length can significantly enhance synchronization of excitatory fronts in an intermittent fashion in dependence on .the intermittent outlay emerges due to the locking between the delay and the inherent oscillation frequency of individual neurons on the scale - free network .thus , approximately at every multiple of the inherent oscillation period of each neuron , the information transmission delay between coupled neurons results in supremely ordered and synchronized fronts of excitations .the widths of these dynamical regimes in dependence on can be broadened , and the synchronization further improved , if the coupling strength or the average degree of the network is enlarged . in addition , we have examined the robustness of these findings to different levels of additive noise , as well as to different system sizes and scaling exponents characterizing the scale - free topology , finding that all the conclusions prevail .we have shown that fine - tuned information transmission delays can effectively supplement some recently identified mechanisms for the enhancement of synchronization , as well as weak signals in general , on scale - free networks .these conclusions seem to be supported by real biological data stating that conduction velocities along axons connecting neurons vary from 20 to 60 m / s .real - life transmission delays are within the range of milliseconds , suggesting that substantially lower or higher values may be preclusive for optimal functioning of neuronal tissue . in future studies, it would be interesting to examine the impact of synaptic noise and different conductance states on synchronization transitions in delayed complex networks , as well as to pinpoint the precise role of different aspects of structure and functioning of active neuronal networks .we hope that our present study will be a useful source of information when striving towards these goals .this work was supported by the national science foundation of china ( funds nos .10702023 and 10832006 ) and china s post - doctoral science foundation ( funds nos . 200801020 and 20070410022 )matja perc additionally acknowledges support from the slovenian research agency ( grant z1 - 2032 - 2547 ) .

Abstract: we investigate front propagation and synchronization transitions in dependence on the information transmission delay and coupling strength over scale - free neuronal networks with different average degrees and scaling exponents . as the underlying model of neuronal dynamics , we use the efficient rulkov map with additive noise . we show that increasing the coupling strength enhances synchronization monotonously , whereas delay plays a more subtle role . in particular , we found that depending on the inherent oscillation frequency of individual neurons , regions of irregular and regular propagating excitatory fronts appear intermittently as the delay increases . these delay - induced synchronization transitions manifest as well - expressed minima in the measure for spatial synchrony , appearing at every multiple of the oscillation frequency . larger coupling strengths or average degrees can broaden the region of regular propagating fronts by a given information transmission delay and further improve synchronization . these results are robust against variations in system size , intensity of additive noise and the scaling exponent of the underlying scale - free topology . we argue that fine - tuned information transmission delays are vital for assuring optimally synchronized excitatory fronts on complex neuronal networks , and indeed , they should be seen as important as the coupling strength or the overall density of interneuronal connections . we finally discuss some biological implications of the presented results .


traffic dynamics has naturally attracted much attention from engineers since the volume of vehicular traffic outstripped road capacity . especially for the last decades , it has attracted a lot of interest , in addition , from physicists and mathematicians as a typical example of non - equilibrium statistical mechanics of self - driven many particle systems .since self - driven particles do not obey newton s laws of motion , their collective phenomena are far from predictable and are strongly dependent on their density .recently , a number of different approaches to the subject have been made from various viewpoints such as microscopic and macroscopic , continuous and discrete , deterministic and stochastic .the burgers equation , is known as an elementary model of traffic flow .it formulates the evolution of density distribution of vehicles , i.e. , it is a macroscopic , continuous model .recent studies show that the burgers equation is directly connected with other basic models of traffic flow .the burgers equation is , at first , transformed into a cellular automaton ( ca ) model ( a microscopic discrete one ) through _ ultra - discretization _note that ca models are microscopic ones because they explicitly define each particle s motion .it provides a method to reveal another profile of models , and enables us to return from macroscopic to microscopic . from the burgers equation ,we thereby obtain the so - called _ burgers cellular automata _ ( bca ) , which is an extension of the rule-184 ca .this ca has a very simple update rule , i.e. , if the adjacent site is not occupied , the vehicle move ahead with a given probability , but otherwise it does not .each site contains one vehicle at most , and the rule is generally called _ the hard - core exclusion rule_. in the case of bca , each site can contain more than one vehicle .moreover , bca can be transformed into another basic model , i.e. , the _ optimal velocity ( ov ) model _ , through the _ discrete euler - lagrange ( e - l ) transformation_. the discrete e - l transformation is made on fully discrete variables , and then field variables change to particle variables . the ov model is a continuous , microscopic model , and is expressed as ,\label{ov}\ ] ] where denotes the position of the -th vehicle at time and the function is called the _ optimal velocity ( ov ) function _ . the ov function gives the optimal velocity of a vehicle in terms of the headway , where the -th vehicle follows the -th in the same lane , and then the ov function , in general , is monotonically increasing . in particular , the ov model obtained from bca has ov function which is a step function .this suggests that the ov model with a step function is essential as well as elementary .cellular automaton models are efficient and flexible compared to those described by differential equations , and they have been used to model complex traffic systems such as ramps and crossings . the nagel - schreckenberg ( n - s ) model , a well - known ca model , successfully reproduces typical properties of real traffic .what makes it sophisticated is a _ random braking rule _ , which is a plausible mechanism to simulate the motion of vehicles in a single lane .nevertheless , the n - s model does not succeed in reproducing the so - called _ metastable state _, i.e. , an unstable state which breaks down to the lower - flux stable state under some perturbations.  extensive study of traffic flow has revealed that the metastable property , appearing in the medium density region , is universal in real traffic flows , and accordingly that property is essential in modelling .moreover , it is quite distinct among non - equilibrium statistical systems . in other words ,this metastable property plays a critical role in characterizing traffic flow from the viewpoint of dynamics , and it is hence required for traffic models to exhibit this property .thus far , one needs the _ slow - start rule _ to reproduce a metastable state in existing ca models .it introduces a delay for vehicles to respond to the changing traffic situation , i.e. , if a vehicle stops due to the hard - core exclusion rule , the slow - start rule forces it to stop again at the next time step .in contrast , due to the second - order derivative , the ov model naturally includes a similar mechanism to the slow - start rule .the intrinsic parameter in the ov model ( [ ov ] ) corresponds to the driver s sensitivity to a traffic situation ( e.g. the distances relatively to the vehicles ahead ) , and plays an important role in the stability of a traffic flow .note that the reciprocal of represents a driver s response time , connecting the ov model and the newell model .  in the next section, we introduce a stochastic ca model following , and then we show that the model inherits the sophisticated features from the ov model .most stochastic models incorporate noise , taking into account uncertain effects such as different driver characteristics , driver error false operation , and external influences . in general, randomness disturbs metastable states and thus stochastic models do not show any metastable state in the fundamental diagram .in contrast , as will be seen in the latter part of this paper , the probability of the sov model plays an essential role in producing metastable states , and the metastable states emerge by extracting a deterministic mechanism .note that a similar trick was considered in .first of all , we explain the general framework of our stochastic ca model for one - lane traffic .the roadway , being divided into cells , is regarded as a one - dimensional array of sites , and each site contains one vehicle at most .let be a stochastic variable which denotes the number of sites through which the -th vehicle moves at time , and be the probability that .then , we assume a principle of motion that the probability depends on the probability distribution , and the positions of vehicles at the previous time .the updating procedure is as follows :  * calculate the next intention from the present , intention and positions ; * determine the number of sites through which a vehicle moves ( i.e. the velocity ) probabilistically according to the intention .* the new position of each vehicle is where denotes the headway .( headway is defined to be the clear space in front of the vehicle , and thus in a ca model we need to subtract 1 to take account of the site occupied by the vehicle itself . )the hard - core exclusion rule is incorporated through the second term of the right hand side of ( [ genx ] ) .we call the probability distribution _ the intention _ because it is an intrinsic variable of the vehicle and drives themselves .it brings uncertainty of operation into the traffic model and has no physical counterpart .   inwhat follows , we assume for . it is notable that by definition and the expectation value , and hence , setting , we have and .from ( [ gen ] ) we have and we therefore express the intention by in stead of . as long as vehicles move separately ( i.e. ) , the positions are updated according to the simple form and consequently we have in the sense of expectation value .this equation expresses the fact that the intention can be regarded as the average velocity at time .let us take an evolution equation in ( [ gen ] ) , where is a parameter and the function takes the value in  ] .equation ( [ sov ] ) consists of two terms , i.e. , a term turning over the intention into the next , and an effect of the situation ( the headway ) .the intrinsic parameter indicates the sensitivity of vehicles to the traffic situation , and the larger is , the less time a vehicle takes to change its intention .a discrete version of the ov model is expressed as \label{dovv}\delta t,\end{aligned}\ ] ] where , and is a time interval . due to the formal correspondence between ( [ sov ] ) and ( [ dovv ] ), we call a stochastic ca model defined by ( [ sov ] ) the _ stochastic optimal velocity ( sov ) model _ , hereafter .  as we noted in the preceding work , the sov model reduces to two exactly solvable stochastic models , the asymmetric simple exclusion process ( asep ) and the zero range process ( zrp ) , when the parameter takes the values of and respectively .the sov model has a fundamental diagram similar to that of zrp as approaches 1 ( the zrp limit ) .however , when takes a small value ( the asep limit ) , the sov model shows quite different properties from those of asep . since we already discussed this point in detail , we do not devote any space to the limiting case .it should be noted here that a model including zrp and asep was proposed in .however , this model does not satisfy the essential standard that a traffic model should reproduce the metastable state observed universally in empirical traffic data .in this section , we take a step function as the ov function .then , we impose a periodic boundary condition and adopt the parallel updating as usual .   ) with a step function ( ) .we observe a hysteresis loop ( thick gray line ) around the discontinuous point of the step ov function ( thin black line ) , where the phase of a vehicle goes round in the direction of arrows . in comparison with fig . 3 of ,two corners of the parallelogram are folded by an effect of the hard - core exclusion rule . ]  ) with a step function ( ) simulated with the number of site and the number of vehicles , where periodic boundary condition is imposed on the roadway .there appear a lot of stable clusters ( small jams ) propagating backward . ]  in preceding works , the original ov model with the step ov function ( [ step ] ) has been studied in detail . in an ideal case, they gave an exact solution for cluster formation ( i.e. the traveling cluster solution ) .they use figures to illustrate the behaviour of vehicles forming a cluster , and find a closed loop or _ hysteresis loop _ in the velocity - headway diagram ( phase space ) .moreover , from the microscopic viewpoint , they evaluate the velocity of a jam transmitting backward , and two specific headways ; the distance with which vehicles stop in a jam , and with which vehicles move in a free - flow region .( see [ criticalpoint ] . )figure [ hv ] shows the phase space of our model .note that the vertical axis of figure [ hv ] means the expectation value of velocity .we find that there appears a hysteresis loop corresponding to figure 3 in .since our model incorporates the hard - core exclusion rule as well as a ca model with maximum allowed velocity 1 , it takes a different shape from the parallelogram of figure 3 in .note that , in contrast with the original ov model , our model contains the element of randomness and hence the vehicles do not always move in accordance with the hysteresis loop . in figure [ stp ], we show the spatio - temporal pattern of our model , which should be compared with figure 4 in . due to the randomness, the clusters do not hold their sizes constant and the number of them fluctuates with time .these two figures suggest that the sov model inherits , from the original ov model , the mechanism of vehicles clustering and then separating .however , from the viewpoint of many - particle systems , the sov model shows apparently different collective phenomena from that of the ov model . in the following part of this section, we estimate the fundamental diagram of the sov model with a step function as well as comparing the estimated specific headways and velocity of our model with those given in .we denote the density of vehicles to sites by ( is the number of sites , and the number of vehicles ) , which is a macroscopic variable , and a conserved quantity of motion under the periodic boundary condition , no entrances or exits . another macroscopic variable flux , ,is defined using the average velocity in a steady state ; where time should be taken large enough for the system to reach a steady state .  a _ fundamental diagram _ , a plot of the flux versus the density , illustrates how traffic conditions depend on density .it represents the characteristics of a traffic model , and hence traffic models are required to reproduce a fundamental diagram observed in real traffic flow .as far as the above - mentioned exactly solvable models ( asep and zrp ) are concerned , we can make an exact calculation of the fundamental diagram .we only exhibit an explicit formula of the fundamental diagram of zrp ( i.e. ) especially when a step function is adopted as the ov function : where is the discontinuous point of the step ov function ( [ step ] ) .  since the flux of r184-ca ( asep with the hopping probability ) is obtained in the simple form of , considering that each vehicle occupies sites , we directly obtain the formula ( [ fluxofzrp ] ) .   ) plotted at each value of sensitivity parameter , where the discontinuous point is , the initial value of the intention is , and the system size is .theoretical curve ( gray ) has a complete agreement with simulated data ( dots ) in these cases ., title="fig : " ] ) plotted at each value of sensitivity parameter , where the discontinuous point is , the initial value of the intention is , and the system size is .theoretical curve ( gray ) has a complete agreement with simulated data ( dots ) in these cases ., title="fig : " ] + ) plotted at each value of sensitivity parameter , where the discontinuous point is , the initial value of the intention is , and the system size is .theoretical curve ( gray ) has a complete agreement with simulated data ( dots ) in these cases ., title="fig : " ] ) plotted at each value of sensitivity parameter , where the discontinuous point is , the initial value of the intention is , and the system size is .theoretical curve ( gray ) has a complete agreement with simulated data ( dots ) in these cases ., title="fig : " ]  figure [ fd ] shows the fundamental diagram of the sov model with the step ov function ( [ step ] ) .we find that the diagram consists of two lines corresponding respectively to _ free - flow _ phase ( positive slope ) and _ jam _ phase ( negative slope ) .it is remarkable that there is a region of density where two states ( a free - flow state and a jam state ) coexist .the second - order difference allows the model to show this property , so - called _ hysteresis _ , as pointed out in section 1 .the free - flow line in the fundamental diagram has a slope of 1 , i.e. , all the vehicles are moving deterministically ( i.e. ) without jamming .this kind of state can be implemented in the uniform state ; equal spacing of vehicles and the initial velocity for all . as can be seen from ( [ sov ] ), the intention changes and consequently the uniform states constitute a line segment in the fundamental diagram .the maximum - flux density , at which the vehicles take the minimum value of equal spacing , is given as follows : in figure [ fd ] , we see that the formula ( [ h ] ) is in the complete agreement with the simulated results in the case of .  , the number of vehicles , and the sensitivity parameter takes the value of ( a ) and ( b ) .it imply that , in contrast with our preceding paper , the metastable state has no lifetime ., title="fig : " ] , the number of vehicles , and the sensitivity parameter takes the value of ( a ) and ( b ) .it imply that , in contrast with our preceding paper , the metastable state has no lifetime ., title="fig : " ]  the uniform states divide into two classes especially under an external perturbation which makes a vehicle accidentally slow down .one is a class of the states recovering their uniform configurations , and the other is that of the states never recovering them . in this paper , we call such a state unstable against external perturbations a _ metastable state _ in accordance with customary practice . figure [ flux - time ] shows that , under perturbation , the flux of a metastable state is decreasing and tends to that of a jam state .( note that it also implies that there is not such a long - lived metastable state as observed in . )consequently , in the region of density where two states coexist , the uniform states appear as a metastable , higher - flux branch .   the flux of traffic flow increases in proportion to the density of vehicles while the density is small .however , as the density becomes bigger , close - range interaction between vehicles makes a wide , strong correlation over them , and consequently gives rise to a jam .then , there appears a turning point at which the flux declines for the first time . around that point ( the so - called _ critical point _ ) , the states of traffic flow bifurcates into a stable branch and a metastable branch , and moreover phase transition occurs between them .in order to estimate the critical point , we consider that the jam line should be expressed by where and denote respectively the density at which flux vanishes and that of the critical point . from ( [ jamline ] ) and , we have equation ( [ headways ] ) suggests that the spatial pattern divides into two kinds , i.e. , clustering ( ) and free flow ( ) .then , since the vehicles move at velocity 1 or 0 in the present model , just indicates the ratio of those in free flow , and moreover total average headway is calculated from the average headway of the clustered vehicles and that of the vehicles in free flow .these two values reflect a macroscopic property of the sov model with the ov function ( [ step ] ) , but we should estimate them from a microscopic viewpoint . in what follows , we consider the case of , following .   approaching a cluster , andfinally comes to be in the cluster .the vehicle stops with a headway to the cluster ahead , which is estimated from the ov function . ]first , we think of as the limit density at which all free - flow domains of the roadway close up and no vehicle can move then .let us consider the situation illustrated in figure [ in ] that a free vehicle with its intention 1 is going into a cluster .then , taking the time when the headway firstly gets equal to , the intention decreases as . since it gives the probability of the vehicle moving at , the average headway , while in cluster , amounts to ^{1/6},\end{aligned}\ ] ] where are the elliptic theta functions .consequently , we obtain the density of clustering vehicles as a function of the sensitivity parameter .note that since , and is equivalent to the probability of taking the value of 1 .( thick line ) at which the flux vanishes with the corresponding numerical results ( cross ) .they have perfect agreement .we also show a corresponding curve of the original ov model ( thin line ) by use of ( [ sugi1 ] ) . ]figure [ jam ] shows the graph of , and it has a perfect agreement with the numerical results read off from figure [ fd ] . in ,they give the explicit formula of , the uniform headway with which vehicles stop in a jam .it reads where and in the present case , and . by use of ( [ sugi1 ] ), we illustrate the corresponding graph in fig . [ jam ] as well .since the original ov model does not incorporate a hard - core exclusion rule , the sensitivity parameter is limited in the scope of so as to avoid any collision .in contrast , the maximum density of the sov model is retained , due to that rule , not to diverge within .  next , in order to estimate we consider that two vehicles in the front of a cluster are getting out of it as illustrated in figure [ out ] .then , as described above , there occur two cases since ; with probability and with probability , where we again take the time when becomes . corresponding to and , we describe respectively as and , and accordingly our main result is expressed as follows : where   let denote the interval of time for the front vehicle to get out of the cluster . provided that clusters are large enough to take approximately and that the second vehicle leaving the cluster maintains a headway of at least ,we conclude that , , , and from ( [ v1 ] ) .note that is a stochastic valuable , and hence and are dependent on .  in the case of : since indicates the probability of moving ahead at one site , the probability of amounts to and moreover , the distance which the two make in free flow amounts to =\frac{1-a}av^0_{i+1}(\tau).\ ] ] consequently , we have in a convenient form for computation :     ( thick line ) at which the flux bifurcates into a metastable branch and a stable jam branch ( i.e. hysteresis ) , with the corresponding numerical results ( cross ) .they also have a perfect agreement . the corresponding curve ( thin line ) of the original ov model is illustrated by use of ( [ sugi2 ] ) . ]  in the case of : the probability of amounts to \ ] ] consequently , from ( [ v1 ] ) we have ,\\\ ] ] and finally is formulated as a function of the sensitivity parameter .the formula of in the corresponding case of the ov model , given in , reads where , , and .figure [ critical ] shows the critical density versus the sensitivity parameter . it also has a perfect agreement with the numerical results read out from fig .[ fd ] . by use of ( [ sugi2 ] ), we illustrate the corresponding graph in fig .[ critical ] as well .we find that the two theoretical curves present a qualitative agreement , while there are some quantitative differences due to the choice of unit car size .the stochastic optimal velocity model was introduced in the preceding paper as a stochastic cellular automaton model extending two exactly solvable models ( the asymmetric simple exclusion process and the zero range process ) . moreover , since it has the same formulation as the optimal velocity model , the sov model can be regarded as a stochastic extension of the ov model .  in the present paper , we take a step function as the ov function in order to investigate an elementary property of the sov model . in previous papers , the original ov model with this ov functionare studied in detail and some analytical results are given .accordingly , we first make a qualitative comparison between them in terms of the motion of each vehicle , and then we see a similar hysteresis loop in the velocity - headway diagram ( phase space ) as long as the density is low .that is , as far as each vehicle s motion is concerned , the vehicles of the two models show a similar motion within a low - density region .however , as the density of vehicles grows large , the structural difference between ordinary differential equations and cellular automata , as well as the presence or absence of randomness , causes crucial differences especially in the fundamental diagram . as discussed in ,the metastability observed in the ov model is derived from the instability of a uniform / homogeneous flow against perturbations .in general , randomness introduced in traffic models tends to eliminate metastable states as well as unstable ones . nevertheless , the metastable states appearing in the fundamental diagram of the sov model fend off the disturbance of randomness with a trick , i.e. , the special choice of the ov function and the configuration .since that choice allows vehicles to move with probability 1 , the uniform flow continues to be stable while the uniform headway takes the value not less than the discontinuous point of the ov function . that simple consideration leads to the formula of the highest - flux density , i.e. , the end - point of a metastable branch. however , all the uniform flows are not entirely stable , but higher - flux states of them are metastable , i.e. , unstable against external perturbations as shown in fig .[ flux - time ] .it is one of the most important things to evaluate the critical point of density where traffic flow becomes unstable and clustering of vehicles sets in .  in , they gave the fundamental diagram of the ov model with a practical ov function and discussed the stability of uniform flow .then , they showed the region of density where uniform flows become unstable and gave the flux of jammed vehicles formulated analytically in the fundamental diagram .in contrast with the original ov model , the sov model incorporates a hard - core exclusion rule , and is thus collision - free under any configuration of vehicles .moreover , as the intention ( or average velocity ) is suppressed since density is high , the vehicle s motion is controlled by randomness as well as the hard - core exclusion rule .these aspects give the sov model a fundamental diagram which is obvious different from that of the ov model .we point out some apparent differences from the ov model attributed to the effect of hard - core exclusion rule and randomness . in the fundamental diagram of the sov model, there is a metastable branch with positive slope ( equal to the maximum velocity 1 ) , and the jam line vanishes at a density less than 1 .the simulated results reveal that the fundamental diagram of the sov model with a step ov function consists of two line segments with which it has exactly the shape of inversed - lambda . from the diagrams , we conclude that the vehicles move approximately with either headway of free - flow or that of jammed vehicles , and our issue is thereby reduced to estimation of two specific headways .it can be done entirely by probabilistic calculations , and consequently the whole fundamental diagram ( obviously including the critical point ) is successfully formulated as a function of the sensitivity parameter .these analytical results are attributed to the simplicity of the sov model and the choice of ov function .further studies on the sov model under open boundary condition and on the multi - velocity version of sov model will be given in subsequent publications .the authors appreciate nimmo j j c for critical reading and helpful comments .this work is supported in part by grant - in - aid for scientific research from the japan society for the promotion of science ( no .15760047 ) .99 helbing d 2001 _ rev .phys . _ * 73 * 1067 chowdhury d , santen l and schadschneider a 2000 _ phys . rep . _ * 329 * 199 nagatani t 2002 _ repphys . _ * 65 * 1331 matsukidaira j and nishinari k 2003 _ phys .lett . _ * 90 * 088701 tokihiro t , takahashi d , matsukidaira j and satsuma j 1996 _ phys . rev. lett . _* 76 * 3247 bando m , hasebe k , nakayama a , shibata a and sugiyama y 1995 _ phys .e _ * 51 * 1035 bando m , hasebe k , nakanishi k , nakayama a , shibata a and sugiyama y 1995 _j. phys .i france _ * 5 * 1389 nagel k and schreckenberg m 1992 _ j. phys .i france _ * 2 * 2221 kerner b s 2004 _ the physics of traffic _ ( berlin , new york : springer - verlag ) nishinari k , fukui m and schadschneider a 2004 _ j. phys .a _ * 37 * 3101 takayasu m and takayasu h 1993 _ fractals _ * 1 * 860 schadschneider a and schreckenberg m 1997 _ ann . physik _* 6 * 541 nishinari k 2001 _ j. phys .a _ * 34 * 10727 newell g f 1961 _ oper .res . _ * 9 * 209 kanai m , nishinari k and tokihiro t 2005 _ phys .e _ * 72 * 035102(r ) rajewsky n , santen l , schadschneider a and schreckenberg m 1998 _ j. stat .phys . _ * 92 * 151 sch " utz g m 2003 _ j. phys .2003 * 36 * r339 spitzer f 1970 _ adv ._ * 5 * 246 evans m r and hanney t 2005 _ j. phys .a _ * 38 * r195 klauck k and schadschneider a 1999 _ physica a _ * 271 * 102 appert c and santen l 2001 _ phys .lett . _ * 86 * 2498 sugiyama y and yamada h 1997 _ phys .e _ * 55 * 7749 nakanishi k , itoh k and igarashi y 1997 _ phys .* 55 * 6519 kanai m , nishinari k and tokihiro t ( to be published ) schreckenberg m , schadschneider a , nagel k and ito n 1995 _ phys .e _ * 51 * 2939 oloan o j , evans m r and cates m e 1998 _ phys .e _ * 58 * 1404

Abstract: in recent works , we have proposed a stochastic cellular automaton model of traffic flow connecting two exactly solvable stochastic processes , i.e. , the asymmetric simple exclusion process and the zero range process , with an additional parameter . it is also regarded as an extended version of the optimal velocity model , and moreover it shows particularly notable properties . in this paper , we report that when taking optimal velocity function to be a step function , all of the flux - density graph ( i.e. the fundamental diagram ) can be estimated . we first find that the fundamental diagram consists of two line segments resembling an _ inversed- _ form , and next identify their end - points from a microscopic behaviour of vehicles . it is notable that by using a microscopic parameter which indicates a driver s sensitivity to the traffic situation , we give an explicit formula for the critical point at which a traffic jam phase arises . we also compare these analytical results with those of the optimal velocity model , and point out the crucial differences between them .


the idea of thermoacoustic tomography ( tct , sometimes also called tat )  can be briefly described as follows ( see figure [ fig : tctsketch ] ) .a short pulse of radiofrequency ( rf ) electromagnetic waves is sent through a biological object heating up the tissue .it is known that the cancerous cells absorb several times more rf energy than the healthy ones  . as a resulta significant increase of temperature occurs at the tumor locations causing a thermal expansion of cancerous masses pressing on the neighboring healthy tissue .the created pressure wave is registered by the transducers located on the edge of the object .assuming speed of propagation of these acoustic waves constant inside the object ( an assumption which is not always correct , but is satisfactory , e.g. for mammography ) , the signals registered at any transducer location are generated by the inclusions laying on a sphere centered at that location .in fact the measured data are the integrals of rf absorption coefficient over those spheres or , in other words , the spherical radon transform of the rf absorption coefficient .hence , to reconstruct the image one needs to invert the spherical radon transform .the spherical radon transform of is defined as where is the surface area on the sphere centered at .  in the definitionwe allow arbitrary set of centers and radii .however , from the dimensional consideration it is clear , that this mapping is overdetermined .the tomographic motivation suggests to restrict the set of centers to a surface ( the set of transducers locations ) , while not imposing any restrictions on the radii . in this paperwe will deal only with the case of spherical acquisition ( i.e. the transducers are located on a unit sphere ) and from now on we will suppose .two different approaches have been used to derive exact inversion formulae for this case .fourier - bessel and spherical harmonic expansions result in solutions written as an infinite series for two and three dimensions respectively  .the tct analog of -filtered backprojection inversion is derived in  as well as a filtered backprojection type version both formulas can be generalized to higher odd dimensions  .notice that , as one can expect for a codimension 1 radon transform in 3d , the formulas are local .  in section 2 we describe the numerical simulation of the data acquisition . the reconstruction algorithms based on the -filtered backprojection formula ( [ eq : rhofilt ] ) and the filtered backprojection one ( [ eq : fbp ] )are discussed in section 3 .the region of reconstruction is the unit ball centered at the origin ( see fig . [ fig1 ] ) .all phantoms considered in the paper are sums of indicator functions supported in ellipsoids completely contained inside the unit ball .the transducers are located on the surface of the unit sphere .we parameterize the transducer location by two angles , where is the the azimuthal angle in the -plane and  ] and use the trapezoidal rule to compute the integral . for , numerical integration usingthe trapezoidal rule is accurate to  .if , however , is only piecewise smooth on $ ] , then we locate the pieces of where it is smooth and use gaussian quadrature to integrate over each piece .once we have generated the projection data , we reconstruct the original indicator functions of the phantoms . the reconstruction algorithmsare based on the -filtered backprojection ( [ eq : rhofilt ] ) or the filtered backprojection ( [ eq : fbp ] ) .the integrals over the unit sphere in ( [ eq : rhofilt ] ) and ( [ eq : fbp ] ) are computed as double integrals with respect to the azimuthal angle and the polar angle .the function to be integrated is periodic with respect to making the trapezoidal rule an appealing quadrature choice .integration with respect to is done by gaussian quadrature .the laplace operator is implemented through the matlab built - in function del2 " .the reconstructions were generated using matlab 5.0 .  in the results belowthe resolution is over a volume resulting in isotropic pixel dimension of .the algorithm is tested on the defrise phantom which consists of five thin ellipsoids symmetrically centered along the -axis ( see fig . [fig : def2d ] ) .we numerate them from to starting with the lowest .  [ cols="^,^,^",options="header " , ]   the data was acquired from the transducers located discretely over the sphere in the following way .the azimuthal angles of the transducer locations were uniformly discretized to points between 0 and .the polar angles of the transducer locations corresponded to gaussian nodes on the interval form 0 to , as described in the previous section .the radii of the integration spheres were uniformly discretized to points from 0 to 2 .the reconstruction was done by both methods : filtered backprojection ( fbp ) and -filtered backprojection .+  the obtained results validate reconstruction formulas ( [ eq : rhofilt ] ) and ( [ eq : fbp ] ) ( see fig . [ defrise ] ) . in both casesthe defrise phantom has a good reconstruction everywhere except along the -axis ( ) , where some noise is present which , while not always noticeable on reconstructions , is visible on the graphs .the reason for appearance of that noise is the correlation of numerical errors along that axis of phantom s symmetry and is discussed in section [ subsec : errors ] .half - scan reconstructions were done using data from only the eastern hemisphere ( , ) or the southern hemisphere ( , ) .these hemispheres are highlighted in figure [ fig : defpartialscan ] .the rest of the data has been zero - filled .+  -0.3 in  it is known  that in case of incomplete data one can expect to recover stably only certain parts of the image the rest of it being blurred out .namely some parts of the wavefront set of the image will be lost . for our phantomthe singularities are jump discontinuities ( edges ) of imaged value across an interface ( a surface of an ellipsoid in 3d ) .the wavefront of in this situation is the set of pairs , where is a point on , and is a vector normal to at .as it was shown in  using microlocal analysis , a point can be stably detected from the radon data , if and only if includes data obtained from a sphere passing through and normal to . in other words , one can see only those parts of an interface , that can be tangentially touched by spheres of integration centered at available transducer locations .the rest of the interface will be blurred .edges in the defrise phantom were reconstructed in figure [ fig : defpartialscan ] as expected . when the data is collected from the eastern hemispherethere are enough spheres to touch tangentially all edges in the eastern hemisphere ( see fig .[ fig : def2d ] ) but none to do it in the western hemisphere .that is why the locations of the edges in the eastern hemisphere were correctly reconstructed while those in the western part were blurred .when the data is collected from the southern hemisphere there are enough spheres to touch tangentially all edges in the defrise phantom , hence all of them were resolved .from the geometric description above it is not hard to see that there may exist certain regions of reconstruction ( locations of , sometimes called audible zones ) where any possible pair belonging to is recognizable from . in our examples ,when the data is collected from the eastern or southern hemisphere , these regions are the eastern and southern half of the unit ball correspondingly .notice that the image values were not reconstructed correctly , since part of the data was missing . however , certain iterative techniques allow one to improve substantially the image values in the audible zone  .   in early experimental work on thermoacoustic tomography ,an approximate backprojection formula was used .it was written in analogy with the backprojection of regular radon transform and looked similar to equation ( [ eq : rhofilt ] ) , except the missing weight factor .the composition of this operator with the direct radon transform is an elliptic pseudo - differential operator of order zero ( see , e.g. ,  . )thus the locations and strengths " of image singularities are recovered correctly .however the values of the image function will not be recovered correctly .the obtained reconstructions validate the predictions correctly recovering locations of edges .the values of image functions are accurate near the center where but degrade slowly with distance from the origin , as expected .+  -0.3 in   as it was mentioned before , the reconstructions of defrise phantom have some noise along the axis of phantom s symmetry ( see figs .[ defrise ] , [ fig : defpartialscan ] ) . to discuss the reasons of appearance of that noise we consider reconstructions of some simpler phantoms consisting of indicator functions of a perfect ball .this allows us to compute the radon transform analytically , hence to exclude the errors in the data simulation . for every fixed , is a third order polynomial with respect to for and is zero for every other . filtered backprojection requires differentiating with respect to the radial variable .we used centered finite differences to estimate which is exact on the third degree polynomials .therefore , we compute exactly for all radii , , at least away from and . hence the only errors in numerical differentiation that spread into the backprojection come with the data from spheres close to the ones touching tangentially the phantom ball .none of these spheres passes inside the phantom ball , hence backprojection at those points is free of errors from numerical differentiation ( see fig .[ errors ] ) .now let us consider a point on the axis of symmetry of the ball phantoms ( the line connecting the center of the phantom ball and the origin ) .there are two sets of spheres that pass through that point and touch the phantom ball tangentially .the spheres in the first set contain the phantom ball , while the spheres in the second set do not .a 2d slice of this scenario is presented in figure [ circles ] .notice that all spheres in the same set have the same radius .so the errors from the numerical differentiation that they will bring into the backprojection algorithm are absolutely the same .the axis of symmetry is the only location in the reconstruction region where these errors are perfectly correlated .this resonance increases the magnitude of errors resulting in the noise along the symmetry axis on reconstructed images ( see fig .[ errors ] ) .  in case of ellipsoids in the defrisephantom everything said above holds .in fact magnitude of errors is five times bigger since there are five ellipsoids with the same axis of symmetry there . at the same time, the reconstruction of an ellipsoidal phantom without any rotational symmetry has no axis of emphasized errors ( see fig .[ ellipsoid ] ) .we have implemented a straightforward numerical validation of both fbp and -filtered inversion formulae for tct data on the high - frequency defrise phantom .fbp and -filtered have virtually identical performance with noise - free simulated data from this high - contrast object .artifacts due to numerical errors are more severe in fbp than -filtered images and might be reduced by mollification techniques  . comparing fbp and -filtered performance in the presence of noise and for low - contrast detectability will appear in future publications .the authors would like to thank mark anastasio , david finch , peter kuchment , leonid kunyansky and rakesh for information about their work and discussions on the subject .the first author would also like to thank his advisor peter kuchment for constant support .the first author was supported in part by the nsf grants dms 9971674 and 0002195 and thanks the nsf for that support .any opinions , findings , conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the nsf .m. haltmeier , t. schuster , o. scherzer , filtered backprojection for thermoacoustic computed tomography in spherical geometry " , _ mathematical methods in the applied sciences , accepted for publication ._     r.a .kruger , k.d .miller , h.e .reynolds , w.l .kiser jr . ,reinecke , g.a .kruger , `` contrast enhancement of breast cancer in vivo using thermoacoustic ct at 434 mhz , '' _ radiology _ 2000 ; * 216 * : 279 - 283 .louis , e.t .quinto , local tomographic methods in sonar , " in _ surveys on solution methods for inverse problems _ , d colton , h engl , ak louis , jr mclaughlin , & w rundell , eds . , springer - verlag 2000 ; 147 - 154 .   s.j .norton , m. linzer , `` ultrasonic reflectivity imaging in three dimensions : exact inverse scattering solutions for plane , cylindrical , and spherical apertures , '' _ ieee trans . biomed .eng . _ , 1981 ; bme-*28* : 200 - 202 .

Abstract: the problem of image reconstruction in thermoacoustic tomography requires inversion of a generalized radon transform , which integrates the unknown function over circles in 2d or spheres in 3d . the paper investigates implementation of the recently discovered backprojection type inversion formulas for the case of spherical acquisition in 3d . a numerical simulation of the data acquisition with subsequent reconstructions are made for the defrise phantom as well as for some other phantoms . both full and partial scan situations are considered . the results are compared with the implementation of a previously used approximate inversion formula .