an inverse problem for circle packing and conformal mapping figure 1 1 illustrates the fact that if a region is almost packed with circles of radius e in the hexagonal pattern and if the unit disk is packed in an isomorphic pattern with circles of varying radii then after suitable normalization the correspondence of circles converges to the riemann mapping function as e 0 see 15 in the present paper an inverse of this result is obtained as illustrated by figure 1 2 namely if the unit disk is almost packed with e circles there is an isomorphic circle packing almost filling the region such that after suitable normalization the circle correspondence converges to the conformal map of the disk onto the region as e 0

the schwarz lemma at the boundary of the symmetrized bidisc abstract the symmetrized bidisc g 2 is defined by g 2 z 1 z 2 z 1 z 2 c 2 z 1 1 z 2 1 z 1 z 2 c it is a bounded inhomogeneous pseudoconvex domain without c 1 boundary and especially the symmetrized bidisc hasn t any strongly pseudoconvex boundary point and the boundary behavior of both caratheodory and kobayashi metrics over the symmetrized bidisc is hard to describe precisely in this paper we study the boundary schwarz lemma for holomorphic self mappings of the symmetrized bidisc g 2 and our boundary schwarz lemma in the paper differs greatly from the earlier related results

fixed points koebe uniformization and circle packings a domain in the riemann sphere hat mathbb c is called a circle domain if every connected component of its boundary is either a circle or a point in 1908 p koebe ko1 posed the following conjecture known as koebe s kreisnormierungsproblem any plane domain is conformally homeomorphic to a circle domain in hat mathbb c when the domain is simply connected this is the content of the riemann mapping theorem

fokker planck description of conductance based integrate and fire neuronal networks steady dynamics of coupled conductance based integrate and fire neuronal networks in the limit of small fluctuations is studied via the equilibrium states of a fokker planck equation an asymptotic approximation for the membrane potential probability density function is derived and the corresponding gain curves are found validity conditions are discussed for the fokker planck description and verified via direct numerical simulations

computational design and additive manufacturing of periodic conformal metasurfaces by synthesizing topology optimization with conformal mapping abstract in this paper we present a computational framework for computational design and additive manufacturing of free form periodic metasurfaces the proposed scheme rests on the level set based topology approach and the conformal mapping theory a metamaterial with pre specified performance is created using a level set based topology optimization method the achieved unit cell is further mapped to the 3d quad meshes on a free form surface by applying the conformal mapping method which can preserve the local shape and angle during the mapping with embedded geometric information the proposed level set based optimization methods not only can act as a motivator for design synthesis but also can be seamlessly hooked with additive manufacturing without the need of cad reconstructions the current computational framework provides a solution to increasing applications involving innovative metamaterial designs on free form surfaces for different fields of interests the performance of the proposed scheme is illustrated through two benchmark examples where a negative poisson s ratio unit cell pattern and a stiff and light inner structure are mapped to 3d free form surfaces and fabricated through additive manufacturing

schwarz lemma at the boundary of strongly pseudoconvex domain in mathbb c n in this paper we establish a new type of the classical boundary schwarz lemma for holomorphic self mappings of strongly pseudoconvex domain in mathbb c n

hexagonal circle patterns and integrable systems patterns with constant angles hexagonal circle patterns with constant intersection angles are introduced and studied it is shown that they are described by discrete integrable systems of toda type conformally symmetric patterns are classified circle pattern analogs of holomorphic mappings z sp c and log z are constructed as special isomonodromic solutions circle patterns studied in the paper include schramm s circle patterns with the combinatorics of the square grid as a special case

uniformization of sierpinski carpets in the plane let si i i be a countable collection of jordan curves in the extended complex plane widehat mathbb c that bound pairwise disjoint closed jordan regions if the jordan curves are uniform quasicircles and are uniformly relatively separated then there exists a quasiconformal map f colon widehat mathbb c rightarrow widehat mathbb c such that f si is a round circle for all i i this implies that every sierpinski carpet in widehat mathbb c whose peripheral circles are uniformly relatively separated uniform quasicircles can be mapped to a round sierpinski carpet by a quasisymmetric map

square sierpinski carpets and lattes maps we prove that every quasisymmetric homeomorphism of a standard square sierpinski carpet s p p ge 3 odd is an isometry this strengthens and completes earlier work by the authors bonk and merenkov in ann math 2 177 591 643 2013 theorem 1 2 we also show that a similar conclusion holds for quasisymmetries of the double of s p across the outer peripheral circle finally as an application of the techniques developed in this paper we prove that no standard square carpet s p is quasisymmetrically equivalent to the julia set of a postcritically finite rational map

conformal welding and koebe s theorem it is well known that not every orientation preserving homeomorphism of the circle to itself is a conformal welding but in this paper we prove several results which state that every homeomorphism is almost a welding in a precise way the proofs are based on koebe s circle domain theorem we also give a new proof of the well known fact that quasisymmetric maps are conformal weldings

population rate codes carried by mean fluctuation and synchrony of neuronal firings abstract a population of firing neurons is expected to carry information not only by the mean firing rate but also by fluctuation and synchrony among neurons in order to examine this possibility we have studied responses of neuronal ensembles to three kinds of inputs mean fluctuation and synchrony driven inputs the generalized rate code model including additive and multiplicative noise h hasegawa phys rev e 75 2007 051904 has been studied by direct simulations dss and the augmented moment method amm in which equations of motion for mean firing rate fluctuation and synchrony are derived results calculated by the amm are in good agreement with those by dss the independent component analysis ica of our results has shown that mean firing rate fluctuation or variability and synchrony may carry independent information in the population rate code model the input output relation of mean firing rates is shown to have higher sensitivity for larger multiplicative noise as recently observed in prefrontal cortex a comparison is made between results obtained by the integrate and fire if model and our rate code model

on the teichmuller theory of circle patterns given a circle pattern on the riemann sphere in this paper we prove that its quasiconformal deformation space can be naturally identified with the product of the teichmuller spaces of its interstices

the generalised zakharov shabat system and the gauge group action the generalized zakharov shabat systems with complex valued non regular cartan elements and the systems studied by caudrey beals and coifman cbc systems and their gauge equivalent are studied this study includes the properties of fundamental analytical solutions for the gauge equivalent to cbc systems and the minimal set of scattering data the description of the class of nonlinear evolutionary equations solvable by the inverse scattering method and the recursion operator related to such systems the hierarchies of hamiltonian structures the results are illustrated in the example of the multi component nonlinear schrodinger equations and the corresponding gauge equivalent multi component heisenberg ferromagnetic type models related to so 5 c algebra

reductions of integrable equations dihedral group we discuss the algebraic and analytic structure of rational lax operators with algebraic reductions of lax equations we associate a reduction group a group of automorphisms of the corresponding infinite dimensional lie algebra we present a complete study of dihedral reductions for sl 2 c lax operators with simple poles and corresponding integrable equations in the last section we give three examples of dihedral reductions for sl n c lax operators

rigidity theorems for circle domains a circle domain omega in the riemann sphere is conformally rigid if every conformal map from omega onto another circle domain is the restriction of a mobius transformation we show that circle domains satisfying a certain quasihyperbolic condition which was considered by jones and smirnov ark mat 38 263 279 2000 are conformally rigid in particular holder circle domains and john circle domains are all conformally rigid this provides new evidence for a conjecture of he and schramm relating rigidity and conformal removability

two dimensional generalized toda lattice the zero curvature representation is obtained for the two dimensional generalized toda lattices connected with semisimple lie algebras the reduction group and conservation laws are found and the mass spectrum is calculated

on the use of analytical expressions for the voltage distribution to analyze intracellular recordings different analytical expressions for the membrane potential distribution of membranes subject to synaptic noise have been proposed and can be very helpful in analyzing experimental data however all of these expressions are either approximations or limit cases and it is not clear how they compare and which expression should be used in a given situation in this note we provide a comparison of the different approximations available with an aim of delineating which expression is most suitable for analyzing experimental data

constructing conformal maps of triangulated surfaces abstract we describe a means of computing the uniformizing conformal map from a triangulated surface whose triangles are realized as euclidean triangles in r 3 onto a fundamental domain in the unit disc d plane c or sphere s 2 mapping such triangulated surfaces arises in a number of applications such as conformal brain flattening we use the circle packing technique of bowers hurdal stephenson et al to first create a quasiconformal approximation to the conformal map then we apply a discrete form of conformal welding to reduce the distortion and converge to the conformal map in the limit

quasisymmetric structures on surfaces we show that a locally ahlfors 2 regular and locally linearly locally contractible metric surtace is locally quasisymmetrically equivalent to tne disk we also discuss an application of this result to the problem of characterizing surfaces embedded in some euclidean spaces that are locally bi lipschitz equivalent to a ball in the plane

characterization of subthreshold voltage fluctuations in neuronal membranes synaptic noise due to intense network activity can have a significant impact on the electrophysiological properties of individual neurons this is the case for the cerebral cortex where ongoing activity leads to strong barrages of synaptic inputs which act as the main source of synaptic noise affecting on neuronal dynamics here we characterize the subthreshold behavior of neuronal models in which synaptic noise is represented by either additive or multiplicative noise described by ornstein uhlenbeck processes we derive and solve the fokker planck equation for this system which describes the time evolution of the probability density function for the membrane potential we obtain an analytic expression for the membrane potential distribution at steady state and compare this expression with the subthreshold activity obtained in hodgkin huxley type models with stochastic synaptic inputs the differences between multiplicative and additive noise models suggest that multiplicative noise is adequate to describe the high conductance states similar to in vivo conditions because the steady state membrane potential distribution is easily obtained experimentally this approach provides a possible method to estimate the mean and variance of synaptic conductances in real neurons

the c convergence of circle packings of bounded degree to the riemann mapping abstract thurston conjectured that hexagonal circle packings can be used to approximate the riemann mapping the corresponding convergence was proven by rodin and sullivan he and schramm showed that for hexagonal circle packings the convergence is c here the c convergence is generalized to the case of non hexagonal circle packings with bounded degree furthermore the estimation of the convergence rate is obtained for arbitrary order derivatives

hexagonal circle patterns with constant intersection angles and discrete painleve and riccati equations hexagonal circle patterns with constant intersection angles mimicking holomorphic maps zc and log z are studied it is shown that the corresponding circle patterns are immersed and described by special separatrix solutions of discrete painleve and riccati equations the general solution of the riccati equation is expressed in terms of the hypergeometric function global properties of these solutions as well as of the discrete zc and log z are established

texture analysis using graphs generated by deterministic partially self avoiding walks texture is one of the most important visual attributes for image analysis it has been widely used in image analysis and pattern recognition a partially self avoiding deterministic walk has recently been proposed as an approach for texture analysis with promising results this approach uses walkers called tourists to exploit the gray scale image contexts in several levels here we present an approach to generate graphs out of the trajectories produced by the tourist walks the generated graphs embody important characteristics related to tourist transitivity in the image computed from these graphs the statistical position degree mean and dispersion entropy of two vertices with the same degree measures are used as texture descriptors a comparison with traditional texture analysis methods is performed to illustrate the high performance of this novel approach

minimal surfaces from infinitesimal deformations of circle packings abstract we study circle packings with the combinatorics of a triangulated disk in the plane and parametrize deformations of circle packings in terms of vertex rotation and cross ratios we show that there is a weierstrass representation formula relating infinitesimal deformations of circle packings to discrete minimal surfaces of koebe type furthermore every minimal surface of koebe type can be extended naturally to a discrete minimal surface of general type in this way discrete minimal surfaces via steiner s formula are unified

boundary schwarz lemma for solutions to poisson s equation abstract let d be the closure of the unit disk d in c and g a continuous function of d in this paper we establish the schwarz lemma at the boundary for solutions to the poisson s equation f g

synaptic shot noise and conductance fluctuations affect the membrane voltage with equal significance the subthreshold membrane voltage of a neuron in active cortical tissue is a fluctuating quantity with a distribution that reflects the firing statistics of the presynaptic population it was recently found that conductancebased synaptic drive can lead to distributions with a significant skew here it is demonstrated that the underlying shot noise caused by poissonian spike arrival also skews the membrane distribution but in the opposite sense using a perturbative method we analyze the effects of shot noise on the distribution of synaptic conductances and calculate the consequent voltage distribution to first order in the perturbation theory the voltage distribution is a gaussian modulated by a prefactor that captures the skew the gaussian component is identical to distributions derived using current based models with an effective membrane time constant the well known effective time constant approximation can therefore be identified as the leading order solution to the full conductance based model the higher order modulatory prefactor containing the skew comprises terms due to both shot noise and conductance fluctuations the diffusion approximation misses these shot noise effects implying that analytical approaches such as the fokker planck equation or simulation with filtered white noise cannot be used to improve on the gaussian approximation it is further demonstrated that quantities used for fitting theory to experiment such as the voltage mean and variance are robust against these non gaussian effects the effective time constant approximation is therefore relevant to experiment and provides a simple analytic base on which other pertinent biological details may be added

removability rigidity of circle domains and koebe s conjecture abstract a circle domain in the riemann sphere is conformally rigid if every conformal map of onto another circle domain is the restriction of a mobius transformation we show that two rigidity conjectures of he and schramm are in fact equivalent at least for a large family of circle domains the proof follows from a result on the removability of countable unions of certain conformally removable sets we also introduce trans quasiconformal deformation of schottky groups to prove that a circle domain is conformally rigid if and only if it is quasiconformally rigid thereby providing new evidence for the aforementioned conjectures

effects of synaptic conductance on the voltage distribution and firing rate of spiking neurons a neuron in an active cortical circuit is subject to a fluctuating synaptic drive mediated by conductance changes it was recently demonstrated that synaptic conductance effects in vivo significantly alter the integrative properties of neurons these effects are missed in models that approximate the synaptic drive as a fluctuating current here the membrane potential distribution and firing rate are derived for the integrate andfire neuron withd correlated conductance based synaptic input using the fokker planck formalism a number of different input scenarios are examined including balanced drive and fluctuation changes at constant conductance the latter of which corresponds to shifts in synchrony in the presynaptic population this minimal model captures many experimentally observed conductance related effects such as reduced membranepotential fluctuations in response to increasing synaptic noise the solvability of the model allows for a direct comparison with current based approaches providing a basis for assessing the validity of existing approximation schemes that have dealt with conductance change in particular a commonly used heuristic approach whereby the passive membrane time constant is replaced by a drive dependent effective time constant is examined it is demonstrated that this approximation is valid in the same limit that the underlying diffusion approximation holds both for d correlated as well as filtered synaptic drive

minimal surfaces from circle patterns geometry from combinatorics the theory of polyhedral surfaces and more generally the field of discrete differential geometry are presently emerging on the border of differential and discrete geometry whereas classical differential geometry investigates smooth geometric shapes such as surfaces and discrete geometry studies geometric shapes with a finite number of elements polyhedra the theory of polyhedral surfaces aims at a development of discrete equivalents of the geometric notions and methods of surface theory the latter appears then as a limit of the refinement of the discretization current progress in this field is to a large extent stimulated by its relevance for computer graphics and visualization one of the central problems of discrete differential geometry is to find proper discrete analogues of special classes of surfaces such as minimal con stant mean curvature isothermic surfaces etc usually one can suggest vari ous discretizations with the same continuous limit which have quite different geometric properties the goal of discrete differential geometry is to find a dis cretization which inherits as many essential properties of the smooth geometry as possible our discretizations are based on quadrilateral meshes i e we discretize parametrized surfaces for the discretization of a special class of surfaces it is natural to choose an adapted parametrization in this paper we investigate conformai discretizations of surfaces i e discretizations in terms of circles and spheres and introduce a new discrete model for minimal surfaces see figures 1 and 2 in comparison with direct methods see in particular 23 leading

quasiconformal mappings on the heisenberg group on etudie les applications conformes et quasi conformes sur le groupe de heisenberg la metrique invariante a gauche naturelle sur cet espace est non isotrope sous des dilatations et ne peut se decrire par une metrique de riemann ceci engendre de nouveaux phenomenes

twisted hierarchies associated with the generalized sine gordon equation twisted u and twisted u k hierarchies are soliton hierarchies introduced by terng to find higher flows of the generalized sine gordon equation twisted o j j o j o j hierarchies are among the most important classes of twisted hierarchies in this paper we derive explicit interesting first and higher flows of twisted o j j o j o j hierarchies justify that the one dimensional systems of twisted o j j o j o j hierarchies for j iq n q 1 q n 1 called the generalized sinh gordon equations are the gauss codazzi equations for n dimensional timelike submanifolds with constant sectional curvature 1 and index q in pseudo euclidean 2n 1 dimensional space r2q 12n 1 with index 2q 1 furthermore a unified treatment of the inverse scattering theory for twisted o j j o j o j hierarchies is provided

topological invariants and lipschitz equivalence of fractal squares abstract fractal sets typically have very complex geometric structures and a fundamental problem in fractal geometry is to characterize how similar different fractal sets are the lipschitz equivalence of fractal sets is often used to classify fractal sets that are geometrically similar interesting links between lipschitz equivalence and algebraic properties of contraction ratios for self similar sets have been uncovered and widely analyzed however with the exception of very few papers the study of lipschitz equivalence has largely focused on totally disconnected self similar sets for connected self similar sets this problem becomes rather challenging even for well known fractal models such as fractal squares in this paper we introduce geometric and topological methods to study the lipschitz equivalence of connected fractal squares in particular we completely characterize the lipschitz equivalence of fractal squares of order 3 in which one or two squares are removed we also discuss the lipschitz equivalence of fractal squares of more general orders our paper is the first study of lipschitz equivalence for nontrivial connected self similar sets and it raises also some interesting questions for the more general setting

the c convergence of hexagonal disk packings to the riemann map let omega mathop subset limits ne c be a simply connected domain the rodin sullivan theorem states that a sequence of disk packings in the unit disk u converges in a well defined sense to a conformal map from to u moreover it is known that the first and second derivatives converge as well here it is proven that for hexagonal disk packings the convergence is c this is done by studying mobius invariants of the disk packings that are discrete analogs of the schwarzian derivative

mean variance and autocorrelation of subthreshold potential fluctuations driven by filtered conductance shot noise we study the subthreshold voltage fluctuations of a conductance based passive point neuron stimulated by filtered poissonian shot noise we give exact analytical expressions in terms of quadratures for the first two time dependent moments and the autocorrelation function of the membrane voltage we also derive simplified expressions for the moments in terms of elementary functions that hold true in the limit case of short filter time small spike amplitude and a single synaptic reversal potential by means of these expressions we show that for an ensemble of equilibrated conductances but sharp initial voltage corresponding to a short voltage clamp at the initial time the mean and the standard deviation can display nonmonotonic time courses in particular transient changes in the standard deviation disagree strongly with the predictions of the commonly used effective time constant approximation over a large parameter range we also study the dependence of the correlation time of the voltage on the synaptic spike amplitude and the synaptic input rate all results are confirmed by extensive stochastic simulations

rigidity of infinite disk patterns let p be a locally flnite disk pattern on the complex plane c whose combinatorics are described by the one skeleton g of a triangulation of the open topological disk and whose dihedral angles are equal to a function e 0 2 on the set of edges let p be a combinatorially equivalent disk pattern on the plane with the same dihedral angle function we show that p and p difier only by a euclidean similarity in particular when the dihedral angle function is identically zero this yields the rigidity theorems of b rodin and d sullivan and of o schramm whose arguments rely essentially on the pairwise disjointness of the interiors of the disks the approach here is analytical and uses the maximum principle the concept of vertex extremal length and the recurrency of a family of electrical networks obtained by placing resistors on the edges in the contact graph of the pattern a similar rigidity property holds for locally flnite disk patterns in the hyperbolic plane where the proof follows by a simple use of the maximum principle also we have a uniformization result for disk patterns in a future paper the techniques of this paper will be extended to the case when 0 in particular we will show a rigidity property for a class of inflnite convex polyhedra in the 3 dimensional hyperbolic space

definitions of quasiconformality summary we establish that the infinitesimal h definition for quasiconforreal mappings on carnot groups implies global quasisymmetry and hence the absolute continuity on almost all lines our method is new even in r where we obtain that the limsup condition in the h definition can be replaced by a liminf condition this leads to a new removability result for quasi conformal mappings in euclidean spaces an application to parametrizations of chord arc surfaces is also given

characterizations of circle patterns and finite convex polyhedra in hyperbolic 3 space the aim of this paper is to study finite convex polyhedra in three dimensional hyperbolic space mathbb h 3 we characterize the quasiconformal deformation space of each finite convex polyhedron as a corollary we obtain some results on finite circle patterns in the riemann sphere with dihedral angle 0 le theta pi that is for any circle pattern on hat mathbb c its quasiconformal deformation space can be naturally identified with the product of the teichmuller spaces of its interstices

quasisymmetric rigidity of square sierpinski carpets we prove that every quasisymmetric self homeomorphism of the standard 1 3 sierpinski carpet s3 is a euclidean isometry for carpets in a more general family the standard 1 p sierpinski carpets sp p 3 odd we show that the groups of quasisymmetric self maps are finite dihedral we also establish that sp and sq are quasisymmetrically equivalent only if p q the main tool in the proof for these facts is a new invariant a certain discrete modulus of a path family that is preserved under quasisymmetric

on the convergence of circle packings to the riemann map rodin and sullivan 1987 proved thurston s conjecture that a scheme based on the circle packing theorem converges to the riemann mapping thereby providing a refreshing geometric view of riemann s mapping theorem we now present a new proof of the rodin sullivan theorem this proof is based on the argument principle and has the following virtues 1 it applies to more general packings the rodin sullivan paper deals with packings based on the hexagonal combinatorics later quantitative estimates were found which also worked for bounded valence packings here the bounded valence assumption is unnecessary and irrelevant 2 our method is rather elementary and accessible to non experts in particular quasiconformal maps are not needed consequently this gives an independent proof of riemann s conformal mapping theorem the rodin sullivan proof uses results that rely on riemann s mapping theorem 3 our approach gives the convergence of the first and second derivatives without significant additional difficulties while previous work has established the convergence of the first two derivatives for bounded valence packings now the bounded valence assumption is unnecessary

planar relative schottky sets and quasisymmetric maps a relative schottky set in a planar domain is a subset of obtained by removing from open geometric discs whose closures are in and are pairwise disjoint in this paper we study quasisymmetric and related maps between relative schottky sets of measure zero we prove in particular that quasisymmetric maps between such sets in jordan domains are conformal locally bi lipschitz and that their first derivatives are locally lipschitz we also provide a locally bi lipschitz uniformization result for relative schottky sets in jordan domains and establish rigidity with respect to local quasisymmetric maps for relative schottky sets in the unit disc

hyperbolic groups with planar boundaries we prove that the class of convex cocompact kleinian groups is quasi isometrically rigid we also establish that a word hyperbolic group with a planar boundary different from the sphere is virtually a convex cocompact kleinian group provided that its boundary has ahlfors regular conformal dimension strictly less than 2 or if it acts geometrically on a cat 0 cube complex

reductions of integrable equations on a iii type symmetric spaces we study a class of integrable nonlinear differential equations related to the a iii type symmetric spaces these spaces are realized as factor groups of the form su n s u n k u k we use the cartan involution corresponding to this symmetric space as an element of the reduction group and restrict generic lax operators to this symmetric space the symmetries of the lax operator are inherited by the fundamental analytic solutions and give a characterization of the corresponding riemann hilbert data

from local to global in quasiconformal structures abstract we exhibit a large class of metric spaces whose infinitesimal quasiconformal structure is strong enough to capture the global quasiconformal structure a sufficient condition for this to happen is described in terms of a poincare type inequality

quasisymmetries of sierpinski carpet julia sets abstract we prove that if is a quasisymmetric homeomorphism between sierpinski carpets that are julia sets of postcritically finite rational maps then is the restriction of a mobius transformation this implies that the group of quasisymmetric homeomorphisms of a sierpinski carpet julia set of a postcritically finite rational map is finite

a twisted integrable hierarchy with d2 symmetry a loop algebra approach to the gerdjikov mikhailov valchev gmv equation is provided to exploit the associated twisted integrable structure and a new twisted integrable hierarchy is discovered using the twisted loop algebra structure we obtain a transparent treatment of the associated scattering and inverse scattering theory and solve the initial value problem for the gmv equation

schwarz s lemma for circle packings burt rodin department of mathematics university of california la jolla ca 92093 usa 1 introduction conformal mappings can be approximated by circle packing isomorphisms this is proved in 10 roughly speaking see sect 5 for a more detailed description a bounded region f2 is almost filled by e circles 7 from the regular hexagonal e circle packing of the plane denote this approximate circle pack ing of q by q by results of andreev and thurston there is a combinatorially isomorphic circle packing d of the unit disk d denote this isomorphism suitably normalized by 7 7 f2 d let f be the piecewise linear quasicon formal mapping determined by the associated triangulations as e 0 f con verges uniformly on compacta to the riemann mapping function f f2 d let

an extended analytic expression for the membrane potential distribution of conductance based synaptic noise synaptically generated subthreshold membrane potential vm fluctuations can be characterized within the framework of stochastic calculus it is possible to obtain analytic expressions for the steady state vm distribution even in the case of conductance based synaptic currents however as we show here the analytic expressions obtained may substantially deviate from numerical solutions if the stochastic membrane equations are solved exclusively based on expectation values of differentials of the stochastic variables hence neglecting the spectral properties of the underlying stochastic processes we suggest a simple solution that corrects these deviations leading to extended analytic expressions of the vm distribution valid for a parameter regime that covers several orders of magnitude around physiologically realistic values these extended expressions should enable finer characterization of the stochasticity of synaptic currents by analyzing experimentally recorded vm distributions and may be applicable to other classes of stochastic processes as well

branched circle packings and discrete blaschke products in this paper we introduce the notion of discrete blaschke products via circle packing we first establish necessary and sufficient conditions for the existence of finite branched circle packings next discrete blaschke products are defined as circle packing maps from univalent circle packings that properly fill d z izi 1 to the corresponding branched circle packings that properly cover d it is verified that such maps have all geometric properties of their classical counterparts finally we show that any classical finite blaschke product can be approximated uniformly on compacta of d by discrete ones

rigidity of schottky sets we call the complement of a union of at least three disjoint round open balls in the unit sphere bbb s n a schottky set we prove that every quasisymmetric homeomorphism of a schottky set of spherical measure zero to another schottky set is the restriction of a m obius transformation on bbb s n in the other direction we show that every schottky set in bbb s 2 of positive measure admits nontrivial quasisymmetric maps to other schottky sets these results are applied to establish rigidity statements for convex subsets of hyperbolic space that have totally geodesic boundaries

reduction groups and automorphic lie algebras we study a new class of infinite dimensional lie algebras which has important applications to the theory of integrable equations the construction of these algebras is very similar to the one for automorphic functions and this motivates the name automorphic lie algebras for automorphic lie algebras we present bases in which they are quasigraded and all structure constants can be written out explicitly these algebras have useful factorisations on two subalgebras similar to the factorisation of the current algebra on the positive and negative parts

method to calculate the moments of the membrane voltage in a model neuron driven by multiplicative filtered shot noise neurons are subject to synaptic inputs from many other cells these inputs consist of spikes changing the conductivity of the target cell i e they enter the neural dynamics as multiplicative shot noise up to now only for simplified models like current based additive noise point neurons or models with gaussian white noise input exact solutions are available we present a method to calculate the exact time dependent moments for the voltage of a point neuron with conductance based shot noise and a passive membrane the exact solutions show features for instance maxima of the moments vs time which are also confirmed by numerical simulations the theoretical analysis of subthreshold membrane fluctuations may contribute to a better comprehension of neural noise in general we also discuss how the analytical results may provide additional conditions for estimating parameters from experimental data

recognizing constant curvature discrete groups in dimension 3 we characterize those discrete groups g which can act properly discontinuously isometrically and cocompactly on hyperbolic 3 space ihi3 in terms of the combinatorics of the action of g on its space at infinity the major ingredients in the proof are the properties of groups that are negatively curved in the large that is gromov hyperbolic the combinatorial riemann mapping theorem and the sullivan tukia theorem on groups which act uniformly quasiconformally on the 2 sphere

quasiconformal maps in metric spaces with controlled geometry this paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry the principal message is that such a theory is both relevant and viable the first main issue is the problem of definition which we next describe quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount this requirement makes sense in every metric space given a homeomorphism f from a metric space x to a metric space y then for x x and r 0 set

on the recursion operators for the gerdjikov mikhailov and valchev system we consider the recursion operator approach to the soliton equations related to an auxiliary linear system introduced recently by gerdjikov mikhailov and valchev gmv system we discuss the recursion operators obtained for the gmv system and show that they are particular cases of the generating operators related to a zakharov shabat type linear system on the algebra sl 3 c in pole gauge

estimation of time dependent input from neuronal membrane potential the set of firing rates of the presynaptic excitatory and inhibitory neurons constitutes the input signal to the postsynaptic neuron estimation of the time varying input rates from intracellularly recorded membrane potential is investigated here for that purpose the membrane potential dynamics must be specified we consider the ornstein uhlenbeck stochastic process one of the most common single neuron models with time dependent mean and variance assuming the slow variation of these two moments it is possible to formulate the estimation problem by using a state space model we develop an algorithm that estimates the paths of the mean and variance of the input current by using the empirical bayes approach then the input firing rates are directly available from the moments the proposed method is applied to three simulated data examples constant signal sinusoidally modulated signal and constant signal with a jump for the constant signal the estimation performance of the method is comparable to that of the traditionally applied maximum likelihood method further the proposed method accurately estimates both continuous and discontinuous time variable signals in the case of the signal with a jump which does not satisfy the assumption of slow variability the robustness of the method is verified it can be concluded that the method provides reliable estimates of the total input firing rates which are not experimentally measurable

dynamics of the langevin model subjected to colored noise functional integral method abstract we have discussed the dynamics of langevin model subjected to colored noise by using the functional integral method fim combined with equations of motion for mean and variance of the state variable two sets of colored noise have been investigated a one additive and one multiplicative colored noise and b one additive and two multiplicative colored noise the case b is examined with relevance to a recent controversy on the stationary subthreshold voltage distribution of an integrate and fire model including stochastic excitatory and inhibitory synapses and a noisy input we have studied the stationary probability distribution and dynamical responses to time dependent pulse and sinusoidal inputs of the linear langevin model model calculations have shown that results of the fim are in good agreement with those of direct simulations dss a comparison is made among various approximate analytic solutions such as the universal colored noise approximation ucna it has been pointed out that dynamical responses to pulse and sinusoidal inputs calculated by the ucna are rather different from those of ds and the fim although they yield the same stationary distribution

rigidity of infinite circle packings the nerve of a packing is a graph that encodes its combinatorics the vertices of the nerve correspond to the packed sets and an edge occurs between two vertices in the nerve precisely when the corresponding sets of the packing intersect

rigidity of circle domains whose boundary hasσ finite linear measure let be a circle domain in the riemann sphere whose boundary has finite linear measure we show that is rigid in the sense that any conformal homeomorphism of onto any other circle domain is equal to the restriction of a mobius transformation previously kaufman and bishop have independently found examples of non rigid circle domains whose boundary is a cantor set of hausdorff dimension one

the c convergence of sg circle patterns to the riemann mapping abstract thurston conjectured that the riemann mapping function from a simply connected region onto the unit disk can be approximated by regular hexagonal packings schramm introduced circle patterns with combinatorics of the square grid sg and showed that sg circle patterns converge to meromorphic functions he and schramm proved that hexagonal disk packings converge in c to the riemann mapping in this paper we show a similar c convergence for sg circle patterns

comment on characterization of subthreshold voltage fluctuations in neuronal membranes by m rudolph and a destexhe in two recent articles rudolph and destexhe 2003 2005 studied a leaky integrator model an rc circuit driven by correlated colored gaussian conductance noise and gaussian current noise in the first article they derived an expression for the stationary probability density of the membrane voltage in the second they modified this expression to cover a larger parameter regime here we show by standard analysis of solvable limit cases white noise limit of additive and multiplicative noise sources only slow multiplicative noise only additive noise and by numerical simulations that their first result does not hold for the general colored noise case and uncover the errors made in the derivation of a fokker planck equation for the probability density furthermore we demonstrate analytically including an exact integral expression for the time dependent mean value of the voltage and by comparison to simulation results that the extended expression for the probability density works much better but still does not exactly solve the full colored noise problem we also show that at stronger synaptic input the stationary mean value of the linear voltage model may diverge and give an exact condition relating the system parameters for which this takes place