analytical model for the effects of learning on spike count distributions the spike count distribution observed when recording from a variety of neurons in many different conditions has a fairly stereotypical shape with a single mode at zero or close to a low average count and a long quasi exponential tail to high counts such a distribution has been suggested to be the direct result of three simple facts the firing frequency of a typical cortical neuron is close to linear in the summed input current entering the soma above a threshold the input current varies on several timescales both faster and slower than the window used to count spikes and the input distribution at any timescale can be taken to be approximately normal the third assumption is violated by associative learning which generates correlations between the synaptic weight vector on the dendritic tree of a neuron and the input activity vectors it is repeatedly subject to we show analytically that for a simple feed forward model the normal distribution of the slow components of the input current becomes the sum of two quasi normal terms the term important below threshold shifts with learning while the term important above threshold does not shift but grows in width these deviations from the standard distribution may be observable in appropriate recording experiments

c1 weierstrass for compact sets in hilbert space abstract the c 1 weierstrass approximation theorem is proved for any compact subset x of a hilbert space h the same theorem is also proved for whitney 1 jets on x when x satisfies the following further condition there exist finite dimensional linear subspaces h 1 h 2 h such that n 1 h n is dense in span x and n x x h n for each n 1 here n h h n is the orthogonal projection it is also shown that when x is compact convex with span x h and satisfies the above condition then c 1 x is complete if and only if the c 1 whitney extension theorem holds for x finally for compact subsets of h an extension of the c 1 weierstrass approximation theorem is proved for c 1 maps h h with compact derivatives

self similar groups and the zig zag and replacement products of graphs abstract every finitely generated self similar group naturally produces an infinite sequence of finite d regular graphs n we construct self similar groups whose graphs n can be represented as an iterated zig zag product and graph powering n 1 n k z k 1 we also construct self similar groups whose graphs n can be represented as an iterated replacement product and graph powering n 1 n k r k 1 this gives simple explicit examples of self similar groups whose graphs n form an expanding family and examples of automaton groups whose graphs n have linear diameters diam n o n and bounded girth

volumes of balls in large riemannian manifolds we prove two lower bounds for the volumes of balls in a riemannian manifold if m n g is a complete riemannian manifold with lling radius at least r then it contains a ball of radius r and volume at least n r n if m n hyp is a closed hyperbolic manifold and if g is another metric on m with volume no greater than n vol m hyp then the universal cover of m g contains a unit ball with volume greater than the volume of a unit ball in hyperbolic n space

free abelian covers short loops stable length and systolic inequalities we explore the geometry of the abel jacobi map f from a closed orientable riemannian manifold x to its jacobi torus h 1 x mathbb r h 1 x mathbb z mathbb r applying m gromov s filling inequality to the typical fiber of f we prove an interpolating inequality for two flavors of shortest length invariants of loops the procedure works provided the lift of the fiber is non trivial in the homology of the maximal free abelian cover overline x classified by f we show that the finite dimensionality of the rational homology of overline x is a sufficient condition for the homological non triviality of the fiber when applied to nilmanifolds our fiberwise inequality typically gives stronger information than the filling inequality for x itself in dimension 3 we present a sufficient non vanishing condition in terms of massey products this condition holds for certain manifolds that do not fiber over their jacobi torus such as 0 framed surgeries on suitable links our systolic inequality applies to surface bundles over the circle provided the algebraic monodromy has 1 dimensional coinvariants even though the massey product invariant vanishes for some of these bundles

firing rate distributions and efficiency of information transmission of inferior temporal cortex neurons to natural visual stimuli the distribution of responses of sensory neurons to ecological stimulation has been proposed to be designed to maximize information transmission which according to a simple model would imply an exponential distribution of spike counts in a given time window we have used recordings from inferior temporal cortex neurons responding to quasi natural visual stimulation presented using a video of everyday lab scenes and a large number of static images of faces and natural scenes to assess the validity of this exponential model and to develop an alternative simple model of spike count distributions we find that the exponential model has to be rejected in 84 of cases at the p 0 01 level a new model which accounts for the firing rate distribution found in terms of slow and fast variability in the inputs that produce neuronal activation is rejected statistically in only 16 of cases finally we show that the neurons are moderately efficient at transmitting information but not optimally efficient

mean instantaneous firing frequency is always higher than the firing rate frequency coding is considered one of the most common coding strategies employed by neural systems this fact leads in experiments as well as in theoretical studies to construction of so called transfer functions where the output firing frequency is plotted against the input intensity the term firing frequency can be understood differently in different contexts basically it means that the number of spikes over an interval of preselected length is counted and then divided by the length of the interval but due to the obvious limitations the length of observation cannot be arbitrarily long then firing frequency is defined as reciprocal to the mean interspike interval in parallel an instantaneous firing frequency can be defined as reciprocal to the length of current interspike interval and by taking a mean of these the definition can be extended to introduce the mean instantaneous firing frequency all of these definitions of firing frequency are compared in an effort to contribute to a better understanding of the input output properties of a neuron

isometries rigidity and universal covers the goal of this paper is to describe all closed aspherical riemannian manifolds m whose universal covers m have have a nontrivial amount of symmetry by this we mean that isom m is not discrete by the well known theorem of myers steenrod ms this condition is equivalent to isom m 1 m also note that if any cover of m has a nondiscrete isometry group then so does its universal cover m our description of such m is given in theorem 1 2 below the proof of this theorem uses methods from lie theory harmonic maps large scale geometry and the homological theory of transformation groups the condition that m have nondiscrete isometry group appears in a wide variety of problems in geometry since theorem 1 2 provides a taxonomy of such m it can be used to reduce many general problems to verifications of specific examples actually it is not always theorem 1 2 which is applied directly but the main subresults from its proof after explaining in 1 1 the statement of theorem 1 2 we give in 1 2 a number of such applications these range from new characterizations of locally symmetric manifolds to the classification of contractible manifolds covering both compact and finite volume manifolds to a new proof of the nadel frankel theorem in complex geometry

local boundary rigidity of a compact riemannian manifold with curvature bounded above this paper considers the boundary rigidity problem for a compact convex riemannian manifold m g with boundary o9m whose curvature satisfies a general upper bound condition this includes all nonpositively curved manifolds and all sufficiently small convex domains on any given riemannian manifold it is shown that in the space of metrics g on m there is a c3 neighborhood of g such that g is the unique metric with the given boundary distance function i e the function that assigns to any pair of boundary points their distance as measured in m more precisely given any metric g in this neighborhood with the same boundary distance function there is diffeomorphism o which is the identity on o9m such that g g there is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distance function 1 statement of the result the general boundary rigidity problem reads to which extent is a riemannian metric on a compact manifold with boundary determined from the distances between boundary points more precisely it can be formulated as follows let m g be a compact riemannian manifold with boundary om let g be another riemannian metric on m we say that g and g have the same boundary distance function if d1 x y d9 x y for arbitrary boundary points x y e om where d1 resp d9 represents distance in m with respect to g resp g it is easy to give examples of pairs of metrics with the same boundary distance function indeed if m m is an arbitrary diffeomorphism of m onto itself which is the identity on the boundary then the metrics g and g g have the same boundary distance function here g og is the pull back of g under i e for arbitrary vectors 1 e txm we have r1 71 x i where p txm t x m is the differential of at x and resp is the inner product with respect to the metric g resp g we say that a compact riemannian manifold is boundary rigid if this is the only type of nonuniqueness more precisely m g is boundary rigid if for every received by the editors august 10 1998 2000 mathematics subject classification primary 53c65 53c22 53c20 secondary 58c25 58c40 the first and third authors were supported by crdf grant rm2 143 the first author was also supported by nsf grants mds 9505175 and mds 9626232 the third author was also supported by intas rfbr grant 95 0763 2000 american matlhematical society 3937 this content downloaded from 207 46 13 167 on thu 21 jul 2016 04 20 00 utc all use subject to http about jstor org terms 3938 c b croke n s dairbekov and v a sharafnjtdinov riemannian metric g on m1with the same boundary distance function as g there is a diffeomorphism o m m which is the identity on the boundary and for which g 0 of in ward and outward vectors here v is the unit outward norn al to om1 given x c qmvi we denote by k x the maximum of the sectional curvatures of all two planes oc t1 m such that c a for m g a cdrm we define the following invariant

strong convergence rate in averaging principle for stochastic fitzhugh nagumo system with two time scales abstract this article deals with averaging principle for stochastic fitzhugh nagumo system with different time scales under suitable conditions the existence of an averaging equation eliminating the fast variable for this coupled system is proved and as a consequence the system can be reduced to a single stochastic ordinary equation with a modified coefficient moreover the rate of convergence for the slow component towards the solution of the averaging equation is of order 1 2

logarithmic growth of systole of arithmetic riemann surfaces along congruence subgroups we apply a study of orders in quaternion algebras to the differential geometry of riemann surfaces the least length of a closed geodesic on a hyperbolic surface is called its systole and denoted sys 1 p buser and p sarnak constructed riemann surfaces x whose systole behaves logarithmically in the genus g x the fuchsian groups in their examples are principal congruence subgroups of a fixed arithmetic group with rational trace field we generalize their construction to principal congruence subgroups of arbitrary arithmetic surfaces the key tool is a new trace estimate valid for an arbitrary ideal in a quaternion algebra we obtain a particularly sharp bound for a principal congruence tower of hurwitz surfaces pch namely the 4 3bound sys 1 xpch 4 log g xpch similar results are obtained for the systole of hyperbolic 3 manifolds relative to their simplicial volume

stability for the lens rigidity problem let g be a riemannian metric for mathbb r d d geqq 3 which differs from the euclidean metric only in a smooth and strictly convex bounded domain m the lens rigidity problem is concerned with recovering the metric g inside m from the corresponding lens relation on the boundary partial m in this paper the stability of the lens rigidity problem is investigated for metrics which are a priori close to a given non trapping metric satisfying the strong fold regular condition a metric g is called strong fold regular if for each point x in m there exists a set of geodesics passing through x whose conormal bundle covers t x m moreover these geodesics contain either no conjugate points or only fold conjugate points with a non degeneracy condition examples of strong fold regular metrics are constructed our main result gives the first stability result for the lens rigidity problem in the case of anisotropic metrics which allow conjugate points the approach is based on the study of the linearized inverse problem of recovering a metric from its induced geodesic flow which is a weighted geodesic x ray transform problem for symmetric 2 tensor fields a key ingredient is to show that the kernel of the x ray transform on symmetric solenoidal 2 tensor fields is of finite dimension it remains open whether the kernel space is trivial or not

differentiable functions defined in closed sets a problem of whitney in 1934 whitney raised the question of how to recognize whether a function f defined on a closed subset x of n is the restriction of a function of class p a necessary and sufficient criterion was given in the case n 1 by whitney using limits of finite differences and in the case p 1 by glaeser 1958 using limits of secants we introduce a necessary geometric criterion for general n and p involving limits of finite differences that we conjecture is sufficient at least if x has a tame topology we prove that if x is a compact subanalytic set then there exists q q x p such that the criterion of order q implies that f is p the result gives a new approach to higher order tangent bundles or bundles of differential operators on singular spaces

volume of minimal hypersurfaces in manifolds with nonnegative ricci curvature we establish a min max estimate on the volume width of a closed riemannian manifold with nonnegative ricci curvature more precisely we show that every closed riemannian manifold with nonnegative ricci curvature admits a pl morse function whose level set volume is bounded in terms of the volume of the manifold as a consequence of this sweep out estimate there exists an embedded closed possibly singular minimal hypersurface whose volume is bounded in terms of the volume of the manifold

bounding the distance of quantum surface codes homological quantum codes also called topological codes are low density parity check error correcting codes that come from surfaces and higher dimension manifolds homological codes from surfaces i e surface codes have also been suggested as a possible way to construct stable quantum memory and fault tolerant computation it has been conjectured that all homological codes have a square root bound on there distance and therefore cannot produce good codes this claim has been disputed in dimension four using the geometric property of systolic freedom we will show in this paper that the conjecture holds in dimension two due to the negation of systolic freedom i e systolic rigidity

geodesic loops and periodic geodesics on a riemannian manifold diffeomorphic to s 3 let m n be a closed 2 connected riemannian manifold such that 3 m n 0 in this paper we prove that either there exists a periodic geodesic on m n of length 6d where d is the diameter of m n or at each point p m n there exists a geodesic loop of length 2d

fundamental groups of manifolds with positive isotropic curvature a central theme in riemannian geometry is understanding the relationships between the curvature and the topology of a riemannian manifold positive isotropic curvature pic is a natural and much studied curvature condition which includes manifolds with pointwise quarter pinched sectional curvatures and manifolds with positive curvature operator by the results of micallef and moore there is only one topological type of compact simply connected manifold with pic namely any such manifold must be homeomorphic to the sphere on the other hand there is a large class of nonsimply connected manifolds with pic an important open problem has been to understand the fundamental groups of manifolds with pic in this paper we prove a new result in this direction we show that the fundamental group of a compact manifold m n with pic n 5 does not contain a subgroup isomorphic to the techniques used involve minimal surfaces

upper bounds on the length of the shortest closed geodesic on simply connected manifolds the subject of this paper is upper bounds on the length of the shortest closed geodesic on simply connected manifolds with non trivial second homology group we will give three estimates the first estimate will explicitly depend on volume and the upper bound for the sectional curvature the second estimate will depend on diameter a positive lower bound for the volume and on the possibly negative lower bound on sectional curvature the third estimate will depend on diameter on a possibly negative lower bound for the sectional curvature and on a lower bound for the simply connectedness radius the technique that we develop in order to obtain the last result will also enable us to estimate the homotopy distance between any two closed curves on compact simply connected manifolds of sectional curvature bounded from below and diameter bounded from above more precisely let c be a constant such that any metric ball of radius leq c is simply connected there exists a homotopy connecting any two closed curves such that the length of the trajectory of the points during this homotopy has an upper bound in terms of the lower bound of the curvature the upper bound of diameter and c

volume growth of some uniformly contractible riemannian manifolds it is shown that if a uniformly contractible riemannian n manifold m g is k quasi isometric to an n dimensional normed space v n cdot k 1 then liminf r rightarrow infty frac vol g ball r r n omega n geq frac 1 k 2n where n is the volume of the unit euclidean ball in particular if m is uniformly contractible and d gh m d g v n cdot infty then m has at least euclidean volume growth this corollary covers an earlier result by burago and ivanov our results are motivated by a volume growth theorem contained in gromov s book gromov in progress in mathematics vol 152 birkhauser boston 1999 p 256 we give a detailed proof of this theorem using the same argument we also derive a generalization of the theorem which is pointed out by gromov

systoles des surfaces plates singulieres de genre deux the extremal metrics for the isosystolic problem on surfaces of genus two are studied in this paper we show that contrary to e calabi s conjecture in genus three no flat metric with conical singularities is extremal for this problem in genus two

optimal control of convective fitzhughnagumo equation we investigate smooth and sparse optimal control problems for convective fitzhughnagumo equation with traveling wave solutions in moving excitable media the cost function includes distributed spacetime and terminal observations or targets the state and adjoint equations are discretized in space by symmetric interior point galerkin sipg method and by backward euler method in time several numerical results are presented for the control of the traveling waves we also show numerically the validity of the second order optimality conditions for the local solutions of the sparse optimal control problem for vanishing tikhonov regularization parameter further we estimate the distance between the discrete control and associated local optima numerically by the help of the perturbation method and the smallest eigenvalue of the reduced hessian

riccati based feedback control of the monodomain equations with the fitzhugh nagumo model feedback control for the monodomain equations is studied the dynamics of interest are governed by a coupled pde ode reaction diffusion system with nonmonotone nonlinearity of fitzhugh nagumo type a localized distributed control is used to locally stabilize the nonlinear system this is achieved by a riccati based feedback law determined by the linearized system it is shown that the riccati equation corresponding to the pde variable suffices for exponential stabilization of the linearized pde ode system the theoretical findings are underlined by several numerical examples

ends of finite volume nonpositively curved manifolds we study complete finite volume n manifolds m of bounded nonpositive sectional curvature a classical theorem of m gromov says that if such m has negative curvature then it is homeomorphic to the interior of a compact manifold with boundary and we denote this boundary partial m if n ge 3 we prove that the universal cover of the boundary widetilde partial m and also the pi 1m cover of the boundary partial widetilde m have vanishing n 2 dimensional homology for n 4 the first of these recovers a result of nguy tilde hat mathrm e n phan saying that each component of the boundary partial m is aspherical for any n ge 3 the second of these implies the vanishing of the first group cohomology group with group ring coefficients h 1 b pi 1m mathbb z pi 1m 0 a consequence is that pi 1m is freely indecomposable these results extend to manifolds m of bounded nonpositive curvature if we assume that m is homeomorphic to the interior of a compact manifold with boundary our approach is a form of homological collapse for ends of finite volume manifolds of bounded nonpositive curvature this paper is very much influenced by earlier yet still unpublished work of nguy tilde hat mathrm e n phan

boundary rigidity with partial data we study the boundary rigidity problem with partial data consisting of determining locally the riemannian metric of a riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary we show that one can recover uniquely and in a stable way a conformal factor near a strictly convex point where we have the information in particular this implies that we can determine locally the isotropic sound speed of a medium by measuring the travel times of waves joining points close to a convex point on the boundary the local results lead to a global lens rigidity uniqueness and stability result assuming that the manifold is foliated by strictly convex hypersurfaces

homological and homotopical dehn functions are different the homological and homotopical dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space the homological dehn function measures fillings of cycles by chains whereas the homotopical dehn function measures fillings of curves by disks because the two definitions involve different sorts of boundaries and fillings there is no a priori relationship between the two functions however before this work there were no known examples of finitely presented groups for which the two functions differ this paper gives such examples constructed by amalgamating a free by cyclic group with several bestvina brady groups

on the spread of positively curved alexandrov spaces it was proved by f wilhelm that gromov s filling radius of closed positively curved manifolds with a uniform lower bound on sectional curvature attains the maximum with the round sphere recently the author proved that this is also the case for closed finite dimensional alexandrov spaces with a positive lower curvature bound these were proved as a corollary of a comparison theorem for the invariant called spread of those spaces in this paper we extend the latter result to infinite dimensional alexandrov spaces

error estimates for the semidiscrete galerkin approximations of the fitzhugh nagumo equations l 2 error estimates are obtained for the semidiscrete approximation of the fitzhugh nagumo equations with nonsmooth initial data the equations consist of a nonlinear parabolic equation coupled with an ordinary differential equation smoothing due to the parabolic equation is used to show increased accuracy for the approximation for positive time