the helmholtz machine discovering the structure inherent in a set of patterns is a fundamental aim of statistical inference or learning one fruitful approach is to build a parameterized stochastic generative model independent draws from which are likely to produce the patterns for all but the simplest generative models each pattern can be generated in exponentially many ways it is thus intractable to adjust the parameters to maximize the probability of the observed patterns we describe a way of finessing this combinatorial explosion by maximizing an easily computed lower bound on the probability of the observations our method can be viewed as a form of hierarchical self supervised learning that may relate to the function of bottom up and top down cortical processing pathways

insights from a simple expression for linear fisher information in a recurrently connected population of spiking neurons a simple expression for a lower bound of fisher information is derived for a network of recurrently connected spiking neurons that have been driven to a noise perturbed steady state we call this lower bound linear fisher information as it corresponds to the fisher information that can be recovered by a locally optimal linear estimator unlike recent similar calculations the approach used here includes the effects of nonlinear gain functions and correlated input noise and yields a surprisingly simple and intuitive expression that offers substantial insight into the sources of information degradation across successive layers of a neural network here this expression is used to 1 compute the optimal i e information maximizing firing rate of a neuron 2 demonstrate why sharpening tuning curves by either thresholding or the action of recurrent connectivity is generally a bad idea 3 show how a single cortical expansion is sufficient to instantiate a redundant population code that can propagate across multiple cortical layers with minimal information loss and 4 show that optimal recurrent connectivity strongly depends on the covariance structure of the inputs to the network

continuous attractors of lotka volterra recurrent neural networks with infinite neurons continuous attractors of lotka volterra recurrent neural networks lv rnns with infinite neurons are studied in this brief a continuous attractor is a collection of connected equilibria and it has been recognized as a suitable model for describing the encoding of continuous stimuli in neural networks the existence of the continuous attractors depends on many factors such as the connectivity and the external inputs of the network a continuous attractor can be stable or unstable it is shown in this brief that a lv rnn can possess multiple continuous attractors if the synaptic connections and the external inputs are gussian like in shape moreover both stable and unstable continuous attractors can coexist in a network explicit expressions of the continuous attractors are calculated simulations are employed to illustrate the theory

analytic solutions for a stefan problem with gibbs thomson correction we provide existence of a unique smooth solution for a class of oneand two phase stefan problems with gibbs thomson correction in arbitrary space dimensions in addition it is shown that the moving interface depends analytically on the temporal and spatial variables of crucial importance for the analysis is the property of maximal lpregularity for the linearized problem which is fully developed in this paper as well

on convergence of solutions to equilibria for quasilinear parabolic problems abstract we show convergence of solutions to equilibria for quasilinear parabolic evolution equations in situations where the set of equilibria is non discrete but forms a finite dimensional c 1 manifold which is normally hyperbolic our results do not depend on the presence of an appropriate lyapunov functional as in the ojasiewicz simon approach but are of local nature

orthogonality conditions and asymptotic stability in the stefan problem with surface tension we prove nonlinear asymptotic stability of steady spheres in the two phase stefan problem with surface tension our method relies on the introduction of appropriate orthogonality conditions in conjunction with a high order energy method

continuous attractors of a class of recurrent neural networks recurrent neural networks rnns may possess continuous attractors a property that many brain theories have implicated in learning and memory there is good evidence for continuous stimuli such as orientation moving direction and the spatial location of objects could be encoded as continuous attractors in neural networks the dynamical behaviors of continuous attractors are interesting properties of rnns this paper proposes studying the continuous attractors for a class of rnns in this network the inhibition among neurons is realized through a kind of subtractive mechanism it shows that if the synaptic connections are in gaussian shape and other parameters are appropriately selected the network can exactly realize continuous attractor dynamics conditions are derived to guarantee the validity of the selected parameters simulations are employed for illustration

optimal l p l q estimates for parabolic boundary value problems with inhomogeneous data in this paper we investigate vector valued parabolic initial boundary value problems mathcal a t x d mathcal b j t x d subject to general boundary conditions in domains g in mathbb r n with compact c 2m boundary the top order coefficients of mathcal a are assumed to be continuous we characterize optimal l p l q regularity for the solution of such problems in terms of the data we also prove that the normal ellipticity condition on mathcal a and the lopatinskii shapiro condition on mathcal a mathcal b 1 dots mathcal b m are necessary for these l p l q estimates as a byproduct of the techniques being introduced we obtain new trace and extension results for sobolev spaces of mixed order and a characterization of triebel lizorkin spaces by boundary data

difficulty of singularity in population coding fisher information has been used to analyze the accuracy of neural population coding this works well when the fisher information does not degenerate but when two stimuli are presented to a population of neurons a singular structure emerges by their mutual interactions in this case the fisher information matrix degenerates and the regularity condition ensuring the cramer rao paradigm of statistics is violated an animal shows pathological behavior in such a situation we present a novel method of statistical analysis to understand information in population coding in which algebraic singularity plays a major role the method elucidates the nature of the pathological case by calculating the fisher information we then suggest that synchronous firing can resolve singularity and show a method of analyzing the binding problem in terms of the fisher information our method integrates a variety of disciplines in population coding such as nonregular statistics bayesian statistics singularity in algebraic geometry and synchronous firing under the theme of fisher information

qualitative behaviour of solutions for the two phase navier stokes equations with surface tension the two phase free boundary value problem for the isothermal navier stokes system is studied for general bounded geometries in absence of phase transitions external forces and boundary contacts it is shown that the problem is well posed in an l p setting and that it generates a local semiflow on the induced state manifold if the phases are connected the set of equilibria of the system forms a n 1 dimensional manifold each equilibrium is stable and it is shown that global solutions which do not develop singularities converge to an equilibrium as time goes to infinity the latter is proved by means of the energy functional combined with the generalized principle of linearized stability

critical spaces for quasilinear parabolic evolution equations and applications abstract we present a comprehensive theory of critical spaces for the broad class of quasilinear parabolic evolution equations the approach is based on maximal l p regularity in time weighted function spaces it is shown that our notion of critical spaces coincides with the concept of scaling invariant spaces in case that the underlying partial differential equation enjoys a scaling invariance applications to the vorticity equations for the navier stokes problem convection diffusion equations the nernst planck poisson equations in electro chemistry chemotaxis equations the mhd equations and some other well known parabolic equations are given

a general approach to stability in free boundary problems abstract every solution of a linear elliptic equation on a bounded domain may be considered as an equilibrium of a free boundary problem the free boundary problem consists of the corresponding parabolic equation on a variable unknown domain with free boundary conditions prescribing both dirichlet and neumann data we establish a rigorous stability analysis of such equilibria including the construction of stable and unstable manifolds for this purpose we transform the free boundary problem to a fully nonlinear and nonlocal parabolic problem on a fixed domain with fully nonlinear lateral boundary conditions and we develop the general theory for such problems as an illustration we give two examples the second being the focussing flame problem in combustion theory

nonlinear analytic semiflows in this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems x t f x t and quasilinear initial value problems x t a x t x t f x t this theory extends the theory of daprato and grisvard of 1979 and shows how various properties like analyticity of solutions can be derived as a direct corollary of the existence theorem

correlation and independence in the neural code the decoding scheme of a stimulus can be different from the stochastic encoding scheme in the neural population coding the stochastic fluctuations are not independent in general but an independent version could be used for the ease of decoding how much information is lost by using this unfaithful model for decoding there are discussions concerning loss of information nirenberg latham 2003 schneidman bialek berry 2003 we elucidate the nirenberg latham loss from the point of view of information geometry

free boundary problems with surface tension conditions abstract in this paper we consider free boundary problems with a surface tension condition at the free boundary the simplest of these problems is the classical hele shaw problem which describes viscous flow between two parallel plates with small spacing we shall consider models which describe tumor growth and are expressed by systems of pdes we shall also consider the hele shaw problem constrained in a half space

on a coupled bulk surface allen cahn system with an affine linear transmission condition and its approximation by a robin boundary condition abstract we study a coupled bulk surface allen cahn system with an affine linear transmission condition that is the trace values of the bulk variable and the values of the surface variable are connected via an affine relation and this serves to generalize the usual dynamic boundary conditions we tackle the problem of well posedness via a penalization method using robin boundary conditions in particular for the relaxation problem the strong well posedness and long time behaviour of solutions can be shown for more general and possibly nonlinear relations new difficulties arise since the surface variable is no longer the trace of the bulk variable and uniform estimates in the relaxation parameter are scarce nevertheless weak convergence to the original problem with affine linear relations can be shown using the approach of colli and fukao 2015 we show strong existence to the original problem with affine linear relations and derive an error estimate between solutions to the relaxed and original problems

singularities affect dynamics of learning in neuromanifolds the parameter spaces of hierarchical systems such as multilayer perceptrons include singularities due to the symmetry and degeneration of hidden units a parameter space forms a geometrical manifold called the neuromanifold in the case of neural networks such a model is identified with a statistical model and a riemannian metric is given by the fisher information matrix however the matrix degenerates at singularities such a singular structure is ubiquitous not only in multilayer perceptrons but also in the gaussian mixture probability densities arma time series model and many other cases the standard statistical paradigm of the cramer rao theorem does not hold and the singularity gives rise to strange behaviors in parameter estimation hypothesis testing bayesian inference model selection and in particular the dynamics of learning from examples prevailing theories so far have not paid much attention to the problem caused by singularity relying only on ordinary statistical theories developed for regular nonsingular models only recently have researchers remarked on the effects of singularity and theories are now being developed this article gives an overview of the phenomena caused by the singularities of statistical manifolds related to multilayer perceptrons and gaussian mixtures we demonstrate our recent results on these problems simple toy models are also used to show explicit solutions we explain that the maximum likelihood estimator is no longer subject to the gaussian distribution even asymptotically because the fisher information matrix degenerates that the model selection criteria such as aic bic and mdl fail to hold in these models that a smooth bayesian prior becomes singular in such models and that the trajectories of dynamics of learning are strongly affected by the singularity causing plateaus or slow manifolds in the parameter space the natural gradient method is shown to perform well because it takes the singular geometrical structure into account the generalization error and the training error are studied in some examples

a center manifold analysis for the mullins sekerka model abstract the mullins sekerka model is a nonlocal evolution model for hypersurfaces which arises as a singular limit for the cahn hilliard equation we show that classical solutions exist globally and tend to spheres exponentially fast provided that they are close to a sphere initially our analysis is based on center manifold theory and on maximal regularity

continuous attractors of a class of neural networks with a large number of neurons a class of simplified background neural networks model with a large number of neurons is proposed continuous attractors of the simplified model are studied in this paper it contains 1 when the background inputs are set to zero and the excitatory connections are in gaussian shape continuous attractors of the new network are obtained under some condition 2 when the background inputs are nonzero and the excitatory connections are still in gaussian shape continuous attractors are achieved under some appropriately selected condition 3 discussions and examples are used to illustrate the theories developed

dynamics of the ericksen leslie equations with general leslie stress ii the compressible isotropic case in this article the non isothermal compressible ericksen leslie system for nematic liquid crystals subject to general leslie stress is considered it is shown that this system is locally well posed within the l q setting and that for initial data close to equilibria points which are identical with the ones for the incompressible situation the solution exists globally moreover any global solution which does not develop singularities converges to an equilibrium in the topology of the natural state manifold note that no structural assumptions on the leslie coefficients are imposed and in particular parodi s relation is not being assumed the results can be viewed as an extension of the studies in hieber and pruss math ann 369 977 996 2017 for the incompressible case to the compressible situation

change based inference in attractor nets linear analysis one standard interpretation of networks of cortical neurons is that they form dynamical attractors computations such as stimulus estimation are performed by mapping inputs to points on the networks attractive manifolds these points represent population codes for the stimulus values however this standard interpretation is hard to reconcile with the observation that the firing rates of such neurons constantly change following presentation of stimuli we have recently suggested an alternative interpretation according to which computations are realized by systematic changes in the states of such networks over time this way of performing computations is fast accurate readily learnable and robust to various forms of noise here we analyze the computation of stimulus discrimination in this change based setting relating it directly to the computation of stimulus estimation in the conventional attractor based view we use a common linear approximation to compare the two methods and show that perfect performance at estimation implies chance performance at discrimination

on the flow of non axisymmetric perturbations of cylinders via surface diffusion we study the surface diusion ow acting on a class of general non axisymmetric perturbations of cylinders cr in ir 3 using tools from parabolic theory on uniformly regular manifolds and maximal regularity we establish existence and uniqueness of solutions to surface diusion ow starting from spatially unbounded surfaces dened over cr via scalar height functions which are uniformly bounded away from the central cylindrical axis addition ally we show that cr is normally stable with respect to 2 axially periodic perturbations if the radius r 1 and unstable if 0 r 1 stability is also shown to hold in settings with axial neumann boundary conditions

population coding with correlation and an unfaithful model this study investigates a population decoding paradigm in which the maximum likelihood inference is based on an unfaithful decoding model umli this is usually the case for neural population decoding because the encoding process of the brain is not exactly known or because a simplified decoding model is preferred for saving computational cost we consider an unfaithful decoding model that neglects the pair wise correlation between neuronal activities and prove that umli is asymptotically efficient when the neuronal correlation is uniform or of limited range the performance of umli is compared with that of the maximum likelihood inference based on the faithful model and that of the center of mass decoding method it turns out that umli has advantages of decreasing the computational complexity remarkably and maintaining high level decoding accuracy moreover it can be implemented by a biologically feasible recurrent network pouget zhang deneve latham 1998 the effect of correlation on the decoding accuracy is also discussed

a moving bump in a continuous manifold a comprehensive study of the tracking dynamics of continuous attractor neural networks understanding how the dynamics of a neural network is shaped by the network structure and consequently how the network structure facilitates the functions implemented by the neural system is at the core of using mathematical models to elucidate brain functions this study investigates the tracking dynamics of continuous attractor neural networks canns due to the translational invariance of neuronal recurrent interactions canns can hold a continuous family of stationary states they form a continuous manifold in which the neural system is neutrally stable we systematically explore how this property facilitates the tracking performance of a cann which is believed to have clear correspondence with brain functions by using the wave functions of the quantum harmonic oscillator as the basis we demonstrate how the dynamics of a cann is decomposed into different motion modes corresponding to distortions in the amplitude position width or skewness of the network state we then develop a perturbation approach that utilizes the dominating movement of the network s stationary states in the state space this method allows us to approximate the network dynamics up to an arbitrary accuracy depending on the order of perturbation used we quantify the distortions of a gaussian bump during tracking and study their effects on tracking performance results are obtained on the maximum speed for a moving stimulus to be trackable and the reaction time for the network to catch up with an abrupt change in the stimulus

ostwald ripening for dilute systems under quasistationary dynamics we consider a dilute mixture in 3d of a finite number of particles initially close to spherical but of varying size evolving according to quasistationary dynamics under the scaling hypotheses that 1 size distance 2 deviation from sphericity size are both initially small we show robustness of the almost spherical shape by associating centers and radii to each particle and then derive rigorously a set of o d e s for the radii which we relate to the lifschitz slyozov wagner theory of coarsening and also establish an estimate for the speed of the centers

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

quasi static motion of a capillary drop ii the three dimensional case abstract a theory is presented to analyze the nonlinear stability of a drop of incompressible viscous fluid with negligible inertia the theory is developed here on the three dimensional version of the relevant free boundary model for stokes equations within this context we show that if the free boundary initiates close to a sphere r 1 e 0 e small then there exists a global in time solution with free boundary r 1 t e 1 n 1 n t e n which approaches a sphere exponentially fast as t moreover we prove that if 0 is analytic resp c in then the velocity u x t e the pressure p x t e and the free boundary are all jointly analytic resp c in x e in an earlier paper we considered the analogous problem for a two dimensional drop although the three dimensional problem proceeds along similar lines the analysis is more complicated due to the fact that we work here with spherical harmonics and vector spherical harmonics

stability of spherical caps under the volume preserving mean curvature flow with line tension abstract we show stability of spherical caps scs lying on a flat surface where the motion is governed by the volume preserving mean curvature flow mcf moreover we introduce a dynamic boundary condition that models a line tension effect on the boundary the proof is based on the generalized principle of linearized stability

invariant manifolds and bifurcation for quasilinear reaction diffusion systems in this paper we consider parameter dependent quasilinear reaction diffusion equations on a bounded domain of r it is our purpose to study the dynamical behavior of solutions to these equations in particular we are interested in the behavior of the solution set if the parameter is changed to be more precise let us assume that sz c r is a bounded domain having a smooth boundary then we consider the parameter dependent equation

on fourier multiplier transformations of banach valued functions we obtain analogues of the mihlin multiplier theorem and littlewood paley inequalities for functions with values in a suitable banach space b the requirement on b is that it have the unconditionality property for martingale difference sequences

well posedness and uniform bounds for a nonlocal third order evolution operator on an infinite wedge we investigate regularity and well posedness for a fluid evolution model in the presence of a three phase contact point we consider a fluid evolution governed by darcy s law after linearization we obtain a nonlocal third order operator which contains the dirichlet neumann operator on the wedge with opening angle epsilon 0 we show well posedness and regularity for this linear evolution equation in the limit of vanishing opening angle we show the convergence of solutions to a fourth order degenerate parabolic operator related to the thin film equation in the course of the analysis we introduce and characterize a new type of sum of weighted sobolev spaces which are suitable to capture the singular limit as epsilon to 0 in particular the nature of the problem requires the use of techniques that are adapted to the problem in the singular domain as well as the degenerate limit problem

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

maximal lp regularity of parabolic problems with boundary dynamics of relaxation type in this paper we investigate vector valued parabolic initial boundary value problems of relaxation type typical examples for such boundary conditions are dynamic boundary conditions or linearized free boundary value problems like in the stefan problem we present a complete lp theory for such problems which is based on maximal regularity of certain model problems

traces and embeddings of anisotropic function spaces in this paper we characterize the trace spaces of a class of weighted function spaces of intersection type with mixed regularities to a large extent we can overcome the difficulty of mixed scales by employing a microscopic improvement in sobolev and mixed derivative embeddings with fixed integrability we apply the general results to prove maximal l p l q regularity for the linearized fully inhomogeneous two phase stefan problem with gibbs thomson correction

lie group action and stability analysis of stationary solutions for a free boundary problem modelling tumor growth abstract in this paper we study asymptotic behavior of solutions for a free boundary problem modelling tumor growth we first establish a general result for differential equations in banach spaces possessing a lie group action which maps a solution into new solutions we prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either d a or d a type by using this general result and making delicate analysis of the spectrum of the linearization of the stationary free boundary problem we prove that if the surface tension coefficient is larger than a threshold value then the unique stationary solution is asymptotically stable modulo translations provided the constant c is sufficiently small whereas if then this stationary solution is unstable

classical solutions for diffusion induced grain boundary motion we prove existence and uniqueness of classical solutions for the motion of hypersurfaces driven by mean curvature and diffusion of a solute along the surface this free boundary problem involves solving a coupled system of nonlinear partial differential equations

computing with continuous attractors stability and online aspects two issues concerning the application of continuous attractors in neural systems are investigated the computational robustness of continuous attractors with respect to input noises and the implementation of bayesian online decoding in a perfect mathematical model for continuous attractors decoding results for stimuli are highly sensitive to input noises and this sensitivity is the inevitable consequence of the system s neutral stability to overcome this shortcoming we modify the conventional network model by including extra dynamical interactions between neurons these interactions vary according to the biologically plausible hebbian learning rule and have the computational role of memorizing and propagating stimulus information accumulated with time as a result the new network model responds to the history of external inputs over a period of time and hence becomes insensitive to short term fluctuations also since dynamical interactions provide a mechanism to convey the prior knowledge of stimulus that is the information of the stimulus presented previously the network effectively implements online bayesian inference this study also reveals some interesting behavior in neural population coding such as the trade off between decoding stability and the speed of tracking time varying stimuli and the relationship between neural tuning width and the tracking speed

population coding and decoding in a neural field a computational study this study uses a neural field model to investigate computational aspects of population coding and decoding when the stimulus is a single variable a general prototype model for the encoding process is proposed in which neural responses are correlated with strength specified by a gaussian function of their difference in preferred stimuli based on the model we study the effect of correlation on the fisher information compare the performances of three decoding methods that differ in the amount of encoding information being used and investigate the implementation of the three methods by using a recurrent network this study not only rediscovers main results in existing literatures in a unified way but also reveals important new features especially when the neural correlation is strong as the neural correlation of firing becomes larger the fisher information decreases drastically we confirm that as the width of correlation increases the fisher information saturates and no longer increases in proportion to the number of neurons however we prove that as the width increases further wider than 2 times the effective width of the turning function the fisher information increases again and it increases without limit in proportion to the number of neurons furthermore we clarify the asymptotic efficiency of the maximum likelihood inference mli type of decoding methods for correlated neural signals it shows that when the correlation covers a nonlocal range of population excepting the uniform correlation and when the noise is extremely small the mli type of method whose decoding error satisfies the cauchy type distribution is not asymptotically efficient this implies that the variance is no longer adequate to measure decoding accuracy

dynamics and computation of continuous attractors continuous attractor is a promising model for describing the encoding of continuous stimuli in neural systems in a continuous attractor the stationary states of the neural system form a continuous parameter space on which the system is neutrally stable this property enables the neutral system to track time varying stimuli smoothly but it also degrades the accuracy of information retrieval since these stationary states are easily disturbed by external noise in this work based on a simple model we systematically investigate the dynamics and the computational properties of continuous attractors in order to analyze the dynamics of a large size network which is otherwise extremely complicated we develop a strategy to reduce its dimensionality by utilizing the fact that a continuous attractor can eliminate the noise components perpendicular to the attractor space very quickly we therefore project the network dynamics onto the tangent of the attractor space and simplify it successfully as a one dimensional ornstein uhlenbeck process based on this simplified model we investigate 1 the decoding error of a continuous attractor under the driving of external noisy inputs 2 the tracking speed of a continuous attractor when external stimulus experiences abrupt changes 3 the neural correlation structure associated with the specific dynamics of a continuous attractor and 4 the consequence of asymmetric neural correlation on statistical population decoding the potential implications of these results on our understanding of neural information processing are also discussed

stimulus dependent correlations and population codes the magnitude of correlations between stimulus driven responses of pairs of neurons can itself be stimulus dependent we examine how this dependence affects the information carried by neural populations about the stimuli that drive them stimulus dependent changes in correlations can both carry information directly and modulate the information separately carried by the firing rates and variances we use fisher information to quantify these effects and show that although stimulus dependent correlations often carry little information directly their modulatory effects on the overall information can be large in particular if the stimulus dependence is such that correlations increase with stimulus induced firing rates this can significantly enhance the information of the population when the structure of correlations is determined solely by the stimulus however in the presence of additional strong spatial decay of correlations such stimulus dependence may have a negative impact opposite relationships hold when correlations decrease with firing rates

well posedness of the muskat problem with h2 initial data abstract we study the dynamics of the interface between two incompressible fluids in a two dimensional porous medium whose flow is modeled by the muskat equations for the two phase muskat problem we establish global well posedness and decay to equilibrium for small h 2 perturbations of the rest state for the one phase muskat problem we prove local well posedness for h 2 initial data of arbitrary size finally we show that solutions to the muskat equations instantaneously become infinitely smooth

a causal perspective on the analysis of signal and noise correlations and their role in population coding the role of correlations between neuronal responses is crucial to understanding the neural code a framework used to study this role comprises a breakdown of the mutual information between stimuli and responses into terms that aim to account for different coding modalities and the distinction between different notions of independence here we complete the list of types of independence and distinguish activity independence related to total correlations conditional independence related to noise correlations signal independence related to signal correlations coding independence related to information transmission and information independence related to redundancy for each type we identify the probabilistic criterion that defines it indicate the information theoretic measure used as statistic to test for it and provide a graphical criterion to recognize the causal configurations of stimuli and responses that lead to its existence using this causal analysis we first provide sufficiency conditions relating these types second we differentiate the use of the measures as statistics to test for the existence of independence from their use for quantification we indicate that signal and noise correlation cannot be quantified separately third we explicitly define alternative system configurations used to construct the measures in which noise correlations or noise and signal correlations are eliminated accordingly we examine which measures are meaningful only as a comparison across configurations and which ones provide a characterization of the actually observed responses without resorting to other configurations fourth we compare the commonly used nonparametric approach to eliminate noise correlations with a functional model based approach showing that the former approach does not remove those effects of noise correlations captured by the tuning properties of the individual neurons and implies nonlocal causal structure manipulations these results improve the interpretation of the measures on the framework and help in understanding how to apply it to analyze the role of correlations

dual decompositions of 4 manifolds this paper concerns decompositions of smooth 4 manifolds as the union of two handlebodies each with handles of index 5 results of smale trivial 1 and wall general 1 describe analogous decompositions up to diffeomorphism in terms of homotopy type of skeleta or chain complexes in dimension 4 we show the same data determines decompositions up to 2 deformation of their spines in higher dimensions spine 2 deformation implies diffeomorphism but in dimension 4 the fundamental group of the boundary is not determined sample results 1 5 two 2 complexes are up to 2 deformation spines of a dual decomposition of the 4 sphere if and only if they satisfy the conclusions of alexander lefshetz duality h 1 k h 2 l and h 2 k h 1 l 3 3 if n n is 1 connected then there is a pseudo handle decomposition without 1 handles in the sense that there is a pseudo collar m n a relative 2 handlebody with spine that 2 deforms to n and n is obtained from this by attaching handles of index 2

dynamics of the ericksen leslie equations with general leslie stress i the incompressible isotropic case the ericksen leslie model for nematic liquid crystals in a bounded domain with general leslie and isotropic ericksen stress tensor is studied in the case of a non isothermal and incompressible fluid this system is shown to be locally strongly well posed in the l p setting and a dynamical theory is developed the equilibria are identified and shown to be normally stable in particular a local solution extends to a unique global strong solution provided the initial data are close to an equilibrium or the solution is eventually bounded in the topology of the natural state manifold in this case the solution converges exponentially to an equilibrium in the topology of the state manifold the above results are proven without any structural assumptions on the leslie coefficients and in particular without assuming parodi s relation

bounded h_ infty calculus for pseudodifferential operators and applications to the dirichlet neumann operator operators of the form a a x d k with a pseudodifferential symbol x belonging to the hormander class s m 1 m 0 0 1 and certain perturbations k are shown to possess a bounded h calculus in besov triebel lizorkin and certain subspaces of holder spaces provided a is suitably elliptic applications concern pseudodifferential operators with mildly regular symbols and operators on manifolds of low regularity an example is the dirichlet neumann operator for a compact domain with c 1 r bounda r y

analysis of a tumor model free boundary problem with a nonlinear boundary condition abstract in this paper we study a free boundary problem modeling the growth of nonnecrotic tumors with angiogenesis this model differs from the other tumor models studied in existing literatures at the point that it has a nonlinear boundary value condition for the nutrient concentration we first study spherically symmetric version of this model we prove that there exists a unique spherically symmetric stationary solution which is asymptotically stable under spherically symmetric perturbation next we make rigorous analysis to the spherically asymmetric version of this model by using some abstract theory of parabolic differential equations in banach manifold we prove that this free boundary problem is locally well posed in little holder spaces and the radial stationary solution is asymptotically stable in case the surface tension coefficient is larger than a threshold value whereas unstable in case is less than this threshold value

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

well posedness and stability of a free boundary problem modeling the growth of multi layer tumors abstract in this paper we study well posedness and stability of a free boundary problem modeling the growth of multi layer tumors under the action of external inhibitors an important feature of this problem is that the surface tension of the free boundary is taken into account we first reduce this free boundary problem into an evolution equation in little holder space and use the well posedness theory for differential equations in banach spaces of parabolic type i e equations which are treatable by using the analytic semi group theory to prove that this free boundary problem is locally well posed for initial data belonging to a little holder space next we study flat solutions of this problem we obtain all flat stationary solutions and give a precise description of asymptotic stability of these stationary solutions under flat perturbations finally we investigate asymptotic stability of flat stationary solutions under non flat perturbations by carefully analyzing the spectrum of the linearized stationary problem and employing the theory of linearized stability for differential equations in banach spaces of parabolic type we give a complete analysis of stability and instability of all flat stationary solutions under small non flat perturbations

existence of analytic solutions for the classical stefan problem we prove that under mild regularity assumptions on the initial data the two phase classical stefan problem admits a unique solution that is analytic in space and time

dual decompositions of 4 manifolds ii linear invariants this paper continues the study of decompositions of a smooth 4 manifold into two handlebodies with handles of index 2 part i trans amer math soc 354 2002 1373 1392 gave existence results in terms of spines and chain complexes over the fundamental group of the ambient manifold here we assume that one side of a decomposition has larger fundamental group and use this to define algebraic topological invariants these reveal a basic asymmetry in these decompositions subtle changes on one side can force algebraic topologically detectable changes on the other a solvable iteration of the basic invariant gives an obstruction theory using lower commutator quotients by thinking of a 2 handlebody as essentially determined by the links used as attaching maps for its 2 handles this theory can be thought of as giving ambient link invariants the moves used are related to the grope cobordism of links developed by conant teichner and the cochran orr teichner filtration of the link concordance groups the invariants give algebraically sophisticated finite type invariants in the sense of vassilaev

front migration in the nonlinear cahn hilliard equation the method of matched asymptotic expansions is used to describe solutions of the nonlinear cahn hilliard equation for phase separation in n 1 space dimensions the expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature on the dominant slowest timescale the interface velocity is determined by the mean curvature of the interface by a non local relation which is identical to that in a well known quasi static model of solidification which exhibits a shape instability discovered by mullins sekerka j appl phys 34 323 329 1963 on a faster timescale the cahn hilliard equation regularizes a classic two phase stefan problem similarity solutions of the two phase stefan problem should describe boundary layers existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

on the blow up scenario for a class of parabolic moving boundary problems abstract we consider maximally continued classical solutions of a large class of parabolic moving boundary problems if the maximal existence time is finite we describe the blow up mechanism either a suitable norm of the bulk density blows up or the geometry of the interface collapses this can also be seen as a sufficient condition for global in time existence of classical solutions moreover we prove a representation theorem saying that any closed compact connected hypersurface of holder regularity class c k can be regarded as a graph over an analytic hypersurface provided k 2

reaction diffusion systems of maxwell stefan type with reversible mass action kinetics abstract the mass based maxwell stefan approach to one phase multicomponent reactive mixtures is mathematically analyzed it is shown that the resulting quasilinear strongly coupled reaction diffusion system is locally well posed in an l p setting and generates a local semiflow on its natural state space solutions regularize instantly and become strictly positive if their initial components are all nonnegative and nontrivial for a class of reversible mass action kinetics the positive equilibria are identified these are precisely the constant chemical equilibria of the system which may form a manifold here the total free energy of the system is employed which serves as a lyapunov function for the system by the generalized principle of linearized stability positive equilibria are proved to be normally stable

ostwald ripening in two dimensions the rigorous derivation of the equations from the mullins sekerka dynamics we consider a dilute mixture of a finite number of particles and we are interested in the coarsening of the spatial distribution in two space dimensions under mullins sekerka dynamics under the appropriate scaling hypotheses we associate radii and centers to each particle and derive equations for the whole evolution

bayesian computation in recurrent neural circuits a large number of human psychophysical results have been successfully explained in recent years using bayesian models however the neural implementation of such models remains largely unclear in this article we show that a network architecture commonly used to model the cerebral cortex can implement bayesian inference for an arbitrary hidden markov model we illustrate the approach using an orientation discrimination task and a visual motion detection task in the case of orientation discrimination we show that the model network can infer the posterior distribution over orientations and correctly estimate stimulus orientation in the presence of significant noise in the case of motion detection we show that the resulting model network exhibits direction selectivity and correctly computes the posterior probabilities over motion direction and position when used to solve the well known random dots motion discrimination task the model generates responses that mimic the activities of evidence accumulating neurons in cortical areas lip and fef the framework we introduce posits a new interpretation of cortical activities in terms of log posterior probabilities of stimuli occurring in the natural world

the hele shaw problem and area preserving curve shortening motions we prove existence locally in time of a solution of the following hele shaw problem given a simply connected curve contained in a smooth bounded domain find the motion of the curve such that its normal velocity equals the jump of the normal derivatives of a function which is harmonic in the complement of the curve in and whose boundary value on the curve equals its curvature we show that this motion is a curve shortening motion which does not change the area of the region enclosed by the curve in case is the whole plane 2 we also show that if the initial curve is close to an equilibrium curve i e to a circle then there exists a global solution and the global solution tends to some circle exponentially fast as time tends to infinity

parameter extraction from population codes a critical assessment in perceptual systems a stimulus parameter can be extracted by determining the center of gravity of the response profile of a population of neural sensors likewise at the motor end of a neural system center of gravity decoding also known as vector decoding generates a movement direction from the neural activation profile we evaluate these schemes from a statistical perspective by comparing their statistical variance with the minimum variance possible for an unbiased parameter extraction from the noisy neuronal ensemble activation profile center of gravity decoding can be statistically optimal this is the case for regular arrays of sensors with gaussian tuning profiles that have an output described by poisson statistics and for arrays of sensors with a sinusoidal tuning profile for the angular parameter estimated however there are also many cases in which center of gravity decoding is highly inefficient this includes the important case where sensor positions are very irregular finally we study the robustness of center of gravity decoding against response nonlinearities at different stages of an information processing hierarchy we conclude that in neural systems instead of representing a parameter explicitly it is safer to leave the parameter coded implicitly in a neuronal ensemble activation profile

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

analysis of a free boundary problem modeling the growth of multicell spheroids with angiogenesis abstract in this paper we study a free boundary problem modeling the growth of vascularized tumors the model is a modification to the byrne chaplain tumor model that has been intensively studied during the past two decades the modification is made by replacing the dirichlet boundary value condition with the robin condition which causes some new difficulties in making rigorous analysis of the model particularly on existence and uniqueness of a radial stationary solution in this paper we successfully solve this problem we prove that this free boundary problem has a unique radial stationary solution which is asymptotically stable for large surface tension coefficient whereas unstable for small surface tension coefficient tools used in this analysis are the geometric theory of abstract parabolic differential equations in banach spaces and spectral analysis of the linearized operator

global classical solution of quasi stationary stefan free boundary problem we consider an one phase quasi stationary stefan problem hele shaw problem in multidimensional case under some reasonable conditions we prove that the problem has a classical solution globally in time the method can be used in two phase problem as well we also discuss asymptotic behavior of solution as t the method developed here can be extended to a general class of free boundary problems

neuronal tuning to sharpen or broaden sensory and motor variables are typically represented by a population of broadly tuned neurons a coarser representation with broader tuning can often improve coding accuracy but sometimes the accuracy may also improve with sharper tuning the theoretical analysis here shows that the relationship between tuning width and accuracy depends crucially on the dimension of the encoded variable a general rule is derived for how the fisher information scales with the tuning width regardless of the exact shape of the tuning function the probability distribution of spikes and allowing some correlated noise between neurons these results demonstrate a universal dimensionality effect in neural population coding

the second periodic eigenvalue and the alikakos fusco conjecture abstract we investigate the maximality properties of the second periodic eigenvalue of the hill s operator the potential function is normalized so that its average over a period is zero apart from its own significance this question is related to the study of the motion by curvature equation

well posedness and qualitative behaviour of the mullins sekerka problem with ninety degree angle boundary contact we show local well posedness for a mullins sekerka system with ninety degree angle boundary contact we will describe the motion of the moving interface by a height function over a fixed reference surface using the theory of maximal regularity together with a linearization of the equations and a localization argument we will prove well posedness of the full nonlinear problem via the contraction mapping principle here one difficulty lies in choosing the right space for the neumann trace of the height function and showing maximal l p l q regularity for the linear problem in the second part we show that solutions starting close to certain equilibria exist globally in time are stable and converge to an equilibrium solution at an exponential rate

optimal short term population coding when fisher information fails efficient coding has been proposed as a first principle explaining neuronal response properties in the central nervous system the shape of optimal codes however strongly depends on the natural limitations of the particular physical system here we investigate how optimal neuronal encoding strategies are influenced by the finite number of neurons n place constraint the limited decoding time window length t time constraint the maximum neuronal firing rate fmax power constraint and the maximal average rate f max energy constraint while fisher information provides a general lower bound for the mean squared error of unbiased signal reconstruction its use to characterize the coding precision is limited analyzing simple examples we illustrate some typical pitfalls and thereby show that fisher information provides a valid measure for the precision of a code only if the dynamic range fmint fmaxt is sufficiently large in particular we demonstrate that the optimal width of gaussian tuning curves depends on the available decoding time t within the broader class of unimodal tuning functions it turns out that the shape of a fisher optimal coding scheme is not unique we solve this ambiguity by taking the minimum mean square error into account which leads to fiat tuning curves the tuning width however remains to be determined by energy constraints rather than by the principle of efficient coding

population coding with motion energy filters the impact of correlations the codes obtained from the responses of large populations of neurons are known as population codes several studies have shown that the amount of information conveyed by such codes and the format of this information is highly dependent on the pattern of correlations however very little is known about the impact of response correlations as found in actual cortical circuits on neural coding to address this problem we investigated the properties of population codes obtained from motion energy filters which provide one of the best models for motion selectivity in early visual areas it is therefore likely that the correlations that arise among energy filters also arise among motion selective neurons we adopted an ideal observer approach to analyze filter responses to three sets of images noisy sine gratings random dots kinematograms and images of natural scenes we report that in our model the structure of the population code varies with the type of image we also show that for all sets of images correlations convey a large fraction of the information 40 to 90 of the total information moreover ignoring those correlations when decoding leads to considerable information loss from 50 to 93 depending on the image type finally we show that it is important to consider a large population of motion energy filters in order to see the impact of correlations study of pairs of neurons as is often done experimentally can underestimate the effect of correlations

dual decompositions of 4 manifolds iii s cobordisms the main result is that an s cobordism topological or smooth of 4 manifolds has a product structure outside a core sub s cobordism these cores are arranged to have quite a bit of structure for example they are smooth and abstractly forgetting boundary structure diffeomorphic to a standard neighborhood of a 1 complex the decomposition is highly nonunique so cannot be used to define an invariant but it shows that the topological s cobordism question reduces to the core case the simply connected version of the decomposition with 1 complex a point is due to curtis freedman hsiang and stong controlled surgery is used to reduce topological triviality of core s cobordisms to a question about controlled homotopy equivalence of 4 manifolds there are speculations about further reductions the decompositions on the ends of the s cobordism are dual decompositions with homotopically controlled handle structures and the main result is an application of earlier papers in the series