a stochastic method to predict the consequence of arbitrary forms of spike timing dependent plasticity synapses in various neural preparations exhibit spike timing dependent plasticity stdp with a variety of learning window functions the window functions determine the magnitude and the polarity of synaptic change according to the time difference of pre and postsynaptic spikes numerical experiments revealed that stdp learning with a single expo nential window function resulted in a bimodal distribution of synaptic conductances as a consequence of competition between synapses a slightly modified window function however resulted in a unimodal distribution rather than a bimodal distribution since various window functions have been observed in neural preparations we develop a rigorous mathematical method to calculate the conductance distribution for any given window function our method is based on the fokker planck equation to determine the conductance distribution and on the ornstein uhlenbeck process to characterize the membrane potential fluctuations demonstrating that our method reproduces the known quantitative resuits of stdp learning we apply the method to the type of stdp learning found recently in the ca1 region of the rat hippocampus we find that this learning can result in nearly optimized competition between synapses meanwhile we find that the type of stdp learning found in the cerebellum like structure of electric fish can result in all or none synapses either all the synaptic conductances are maximized or none of them becomes significantly large our method also determines the window function that optimizes synaptic competition

the proximal normal formula in banach space approximation by proximal normals to the clarke generalized subdifferential for a distance function generated by a nonempty closed set and the normal cone to the set generated by the proximal normals are important tools in nonsmooth analysis we give simple general versions of such formulae in infinite dimensional banach spaces which satisfy different geometrical conditions our first class of spaces with uniformly g teaux differentiable norm includes the hilbert space case and the formulae is attained through dense subsets our second class of reflexive kadec smooth spaces is the most general for which such formulae can be obtained for all nonempty closed sets in the space our technique also allows us to establish the existence of solutions for a class of optimization problems substantially extending similar work of ekeland and lebourg

optimal factorization of muckenhoupt weights peter jones theorem on the factorization of ap weights is sharpened for weights with bounds near 1 allowing the factorization to be performed continuously near the limiting unweighted case when 1 p oo and w is an ap weight with bound ap w 1 e it is shown that there exist a1 weights u v such that both the formula w uv1 p and the estimates al u al v 1 0 v hold the square root in these estimates is also proven to be the correct asymptotic power as e 0

extremal approximately convex functions and estimating the size of convex hulls abstract a real valued function f defined on a convex set k is an approximately convex function iff it satisfies f x y 2 f x f y 2 1 a thorough study of approximately convex functions is made the principal results are a sharp universal upper bound for lower semi continuous approximately convex functions that vanish on the vertices of a simplex and an explicit description of the unique largest bounded approximately convex function e vanishing on the vertices of a simplex a set a in a normed space is an approximately convex set iff for all a b a the distance of the midpoint a b 2 to a is 1 the bounds on approximately convex functions are used to show that in rn with the euclidean norm for any approximately convex set a any point z of the convex hull of a is at a distance of at most log2 n 1 1 n 1 2 log2 n 1 from a examples are given to show this is the sharp bound bounds for general norms on rn are also given

approximate convexity and submonotonicity abstract it is shown that a locally lipschitz function is approximately convex if and only if its clarke subdifferential is a submonotone operator consequently in finite dimensions the class of locally lipschitz approximately convex functions coincides with the class of lower c 1 functions directional approximate convexity is introduced and shown to be a natural extension of the class of lower c 1 functions in infinite dimensions the following characterization is established a multivalued operator is maximal cyclically submonotone if and only if it coincides with the clarke subdifferential of a locally lipschitz directionally approximately convex function which is unique up to a constant furthermore it is shown that in asplund spaces every regular function is generically approximately convex

robustness analysis of identification and adaptive control for stochastic systems abstract this paper gives a preliminary robustness analysis for stochastic systems with unmodelled dynamics the extended least squares algorithm and stochastic adaptive control for tracking are defined on the basis of the modelled part of the system and it is shown that the parameter estimate is closed to the true one and the tracking error is small if the unmodelled dynamics is small

a robust stochastic adaptive controller the adaptive tracking problem is considered for discrete time stochastic systems consisting of a modeled part being a stable armax process and unmodeled dynamics dominated by a small constant epsilon multiplied by a quantity independent of epsilon but tending to infinity as the past input output and noise grow the adaptive control law proposed is switched at stopping times and is disturbed by a sequence of random vectors bounded by an arbitrary small but fixed constant sigma it is shown that the closed loop system is globally stable the estimation errors for parameters contained in the modeled part of the order epsilon and the tracking error differs from the minimum tracking error for systems without unmodeled dynamics by value of o epsilon sup 2 o sigma sup 2

subdifferential representation formula and subdifferential criteria for the behavior of nonsmooth functions abstract several kinds of behaviors of extended real valued lower semicontinuous functions are known to be equivalent to certain appropriate conditions in terms of the clarke subdifferential the paper provides a systematic study showing that any such condition with the clarke subdifferential is valid if and only if it holds with any operator representing the clarke subdifferential like in the subdifferential proximal formula

subdifferential characterization of approximate convexity the lower semicontinuous case it is known that a locally lipschitz function is approximately convex if and only if its clarke subdifferential is a submonotone operator the main object of this work is to extend the above characterization to the class of lower semicontinuous functions to this end we establish a new approximate mean value inequality involving three points we also show that an analogue of the rockafellar maximal monotonicity theorem holds for this class of functions and we discuss the case of arbitrary subdifferentials

integration of subdifferentials of lower semicontinuous functions on banach spaces abstract this paper studies the lower semicontinuous functions on banach spaces whose subdifferentials are not disassociated from several conditions are given to ensure that two lower semicontinuous functions with the same subdifferential are equal up to an additive constant

locally optimal adaptive control without persistent excitation it is shown that without a persistent excitation condition the least squares ls estimates will converge to some manifold when a ls estimator is combined with a controller computed from the estimated model an equation of this manifold depends upon a type of the control law applied it is proven that there exists the unique control law which is invariable in this manifold this analysis leads to designing adaptive locally optimal control without any excitation signals and modifications of the algorithm

integration of subdifferentials of nonconvex functions in nonsmooth analysis and optimization subgradients come in many different flavors e g approximate dini proximal clarke generalized see 2 5 6 12 14 191 these subgradients are important and valuable tools however many questions remain unsolved concerning the exact link between the function and its subgradients for instance can two functions not differing by an additive constant have the same subgradients in this paper we study the fundamental problem of determining functions that can be recovered up to an additive constant from the knowledge of their subgradients this integration problem is not very well understood and very few functions or classes of functions are known to be recoverable from their subgradients in section 4 we show that if the basic constraint qualification holds at r then the composition of a closed i e lowersemicontinuous proper convex function with a twice continuously differentiable mapping is determined up to an additive constant by its generalized subgradients actually in this case all above mentioned subgradients are the same beside the obvious theoretical interest of this integration problem it is our hope or perhaps our long term goal that once this problem is better understood we can then tackle the question of uniqueness of solutions to generalized differential equations involving subgradients in place of partial derivatives an example of such an equation that well deserves study is the extended hamilton jacobi equation used in optimal control see clarke 2 let us also mention that a problem similar to the integration problem is the one of determining the set valued mappings that are in fact subgradient set valued mappings uniqueness is not mandatory for a contribution to this problem see janin 8 before we look at some of the known cases where the function can be recovered from its subgradients let us look at some negative examples it is clear that not every function can be recovered up to an additive constant from its subgradients we only need to look at the following two functions 0 x50 0 x10 f x 1 x o g x 2 x o

adaptive stabilization of unstable and nonminimum phase stochastic systems abstract the system under consideration is the discrete time stochastic system a z yn zb z un wn driven by a martingale difference sequence wn where a z are polynomilas in backward shift operator z with unknown coefficients and both a z and b z may be unstable with the purpose of demonstrating theoretical possibility rather than designing a practically applicable control law this paper constructs an adaptive control that stabilizes the system and simultaneously guarantees strong consistency of the least squares estimates for unknown coefficients

consistent order estimation for linear stochastic feedback control systems carma model abstract a new criterion cic is introduced to estimate orders p0 q0 r0 of the linear stochastic feedback control system with correlated noise described by a carma model it is proved that the estimate is strongly consistent when the upper bounds for p0 q0 and r0 are available but neither the stability condition nor the ergodicity of the input and output are imposed on the system

subsmooth sets functional characterizations and related concepts prox regularity of a set poliquin rockafellar thibault 2000 or its global version proximal smoothness clarke stern wolenski 1995 plays an important role in variational analysis not only because it is associated with some fundamental properties as the local continuous differentiability of the function dist c or the local uniqueness of the projection mapping but also because in the case where c is the epigraph of a locally lipschitz function it is equivalent to the weak convexity lower c 2 property of the function in this paper we provide an adapted geometrical concept called sub smoothness which permits an epigraphic characterization of the approximate convex functions or lower c 1 property subsmooth sets turn out to be naturally situated between the classes of prox regular and of nearly radial sets this latter class has been recently introduced by lewis in 2002 we hereby relate it to the mifflin semismooth functions

stochastic perturbation methods for spike timing dependent plasticity online machine learning rules and many biological spike timing dependent plasticity stdp learning rules generate jump process markov chains for the synaptic weights we give a perturbation expansion for the dynamics that unlike the usual approximation by a fokker planck equation fpe is well justified our approach extends the related system size expansion by giving an expansion for the probability density as well as its moments we apply the approach to two observed stdp learning rules and show that in regimes where the fpe breaks down the new perturbation expansion agrees well with monte carlo simulations the methods are also applicable to the dynamics of stochastic neural activity like previous ensemble analyses of stdp we focus on equilibrium solutions although the methods can in principle be applied to transients as well

adaptive linear quadratic gaussian control the cost biased approach revisited in adaptive control a standard approach is to resort to the so called certainty equivalence principle which consists of generating some standard parameter estimate and then using it in the control law as if it were the true parameter as a consequence of this philosophy the estimation problem is decoupled from the control problem and this substantially simplifies the corresponding adaptive control scheme on the other hand the complete absence of dual properties makes certainty equivalent controllers run into an identifiability problem which generally leads to a strictly suboptimal performance in this paper we introduce a cost biased parameter estimator to overcome this difficulty this estimator is applied to a linear quadratic gaussian controller the corresponding adaptive scheme is proven to be stable and optimal when the unknown system parameter lies in an infinite yet compact parameter set

stochastic adaptive one step ahead optimal controllers based on input matching optimal adaptive controller based on the els algorithm is established using the input matching technique the control signal is reduced to a constant weighted sum of the measurable information state vector components using a one step ahead quadratic cost function to govern the behavior of the stochastic linear systems the control effort can be estimated globally the algorithm also predicts the convergence rate with no excitation condition the closed loop system is globally stable and the input converges to the one step ahead optimal input

integration of multivalued operators and cyclic submonotonicity we introduce a notion of cyclic submonotonicity for multivalued operators from a banach space x to its dual we show that if the clarke subdifferential of a locally lipschitz function is strictly submonotone on an open subset u of x then it is also maximal cyclically submonotone on u and conversely that every maximal cyclically submonotone operator on u is the clarke subdifferential of a locally lipschitz function which is unique up to a constant if u is connected in finite dimensions these functions are exactly the lower c 1 functions considered by spingarn and rockafellar

identification and adaptive control for systems with unknown orders delay and coefficients an analysis that gives recursive estimates for time delay system orders and coefficients of single input single output linear discrete time deterministic and stochastic systems with uncorrelated noise is presented it is assumed that a lower bound for the time delay and upper bounds for system order are known the optimal adaptive control is designed for both tracking and linear quadratic regulation when the system parameters including time delay orders and coefficients are unknown the rates of convergence are derived the coefficient estimates to their true values and the loss functions to their minima

dynamics of temporal learning rules the changes of synaptic strength are analyzed on two time scales the fast local field dynamics and the slow synaptic modification dynamics the fast dynamics are determined by the synaptic strengths and background noise in the system the slow dynamics are determined by the functional form of a temporal learning rule temporal learning rules are defined to be functions yielding state dependent changes in synaptic strengths depending on the timing of pre and postsynaptic states in the network the evolution of local field dynamics that result from various learning rules are analyzed for a stochastic discrete time neural model with no relative refractory period that receives a series of delayed adaptive inputs a fixed point is found in the learning dynamics and conditions for two types of instabilities are analyzed four universality classes of dynamics are found that are independent of the details of the temporal learning rules examples are given of biological systems in which these temporal learning rules have been identified and their functional consequences are discussed

lipschitz behavior of the robust regularization to minimize or upper bound the value of a function robustly we might instead minimize or upper bound the epsilon robust regularization defined as the map from a point to the maximum value of the function within an epsilon radius this regularization may be easy to compute convex quadratics lead to semidefinite representable regularizations for example and the spectral radius of a matrix leads to pseudospectral computations for favorable classes of functions we show that the robust regularization is lipschitz around any given point for all small epsilon 0 even if the original function is non lipschitz like the spectral radius one such favorable class consists of the semi algebraic functions such functions have graphs that are finite unions of sets defined by finitely many polynomial inequalities and are commonly encountered in applications

essentially smooth lipschitz functions in this paper we address some of the most fundamental questions regarding the differentiability structure of locally lipschitz functions defined on separable banach spaces for example we examine the relationship between integrability d representability and strict differentiability in addition to this we show that on any separable banach space there is a significant family of locally lipschitz functions that are integrable d representable and possess desirable differentiability properties we also present some striking applications of our results to distance functions

random walks for spike timing dependent plasticity random walk methods are used to calculate the moments of negative image equilibrium distributions in synaptic weight dynamics governed by spike timing dependent plasticity stdp the neural architecture of the model is based on the electrosensory lateral line lobe ell of mormyrid electric fish which forms a negative image of the reafferent signal from the fish s own electric discharge to optimize detection of sensory electric fields of particular behavioral importance to the fish is the variance of the equilibrium postsynaptic potential in the presence of noise which is determined by the variance of the equilibrium weight distribution recurrence relations are derived for the moments of the equilibrium weight distribution for arbitrary postsynaptic potential functions and arbitrary learning rules for the case of homogeneous network parameters explicit closed form solutions are developed for the covariances of the synaptic weight and postsynaptic potential distributions

hebbian learning and spiking neurons a correlation based hebbian learning rule at a spike level with millisecond resolution is formulated mathematically analyzed and compared with learning in a firing rate description the relative timing of presynaptic and postsynaptic spikes influences synaptic weights via an asymmetric learning window a differential equation for the learning dynamics is derived under the assumption that the time scales of learning and neuronal spike dynamics can be separated the differential equation is solved for a poissonian neuron model with stochastic spike arrival it is shown that correlations between input and output spikes tend to stabilize structure formation with an appropriate choice of parameters learning leads to an intrinsic normalization of the average weight and the output firing rate noise generates diffusion like spreading of synaptic weights s1063 651x 99 02804 4

nonstationary time series identification abstract in this paper strongly consistent estimates are given for both unknown parameters and orders of the nonstationary time series where the nonstationarity arises because 1 at any time n a feedback control is added to the usual arma process which is successfully applied to modelling economic systems and 2 det a z may have zeros on the unit circle in addition to those located outside the unit disk where a z is the matrix polynomial corresponding to the ar part of the arma process

optimal adaptive control and consistent parameter estimates for armax model withquadratic cost we consider the multidimensional armax model a z y n b z u n c z w n with loss function j u mathop overline lim limits n to infty frac 1 n sum limits i 1 n left y i tau q 1 y i u i tau q 2 u i right where the coefficients in the matrix polynomials a z b z and c z are unknown conditions used here are 1 stability of a z and full rank of a p 2 strictly positive realness of c z frac 1 2 i and 3 controllability and observability of a matrix triple consisting of coefficients in a z b z and q 1 on the basis of the estimates given by the stochastic gradient algorithm for unknown parameters an adaptive control law is recursively defined it is proved that the parameter estimates are strongly consistent and the quadratic loss function reaches its minimum this paper also includes some general theorems on parameter estimation on which the results about adaptive control are essentially based

time structure of the activity in neural network models several neural network models in continuous time are reconsidered in the framework of a general mean field theory which is exact in the limit of a large and fully connected network the theory assumes pointlike spikes which are generated by a renewal process the effect of spikes on a receiving neuron is described by a linear response kernel which is the dominant term in a weak coupling expansion it is shown that the resulting spike response model is the most general renewal model with linear inputs the standard integrate and fire model forms a special case in a network structure with several pools of identical spiking neurons the global states and the dynamic evolution are determined by a nonlinear integral equation which describes the effective interaction within and between different pools we derive explicit stability criteria for stationary incoherent and oscillatory coherent solutions it is shown that the stationary state of noiseless systems is almost always unstable noise suppresses fast oscillations and stabilizes the system furthermore collective oscillations are stable only if the firing occurs while the synaptic potential is increasing in particular collective oscillations in a network with delayless excitatory interaction are at most semistable inhibitory interactions with short delays or excitatory interactions with long delays lead to stable oscillations our general results allow a straightforward application to different network models with spiking neurons furthermore the theory allows an estimation of the errors introduced in firing rate or graded response models

least squares identification for armax models without the positive real condition recursive identification problems of linear stochastic feedback control systems described by armax models are studied without imposing the strictly positive real condition on the noise model an increasing lag least squares is used in the first step to estimate the noise process while the parameter estimate is formed in the second step by an extended least squares in the algorithm there is an increase in computational cost in comparison to traditional algorithms however the results of this note do not need any a priori information or conditions on the noise model except that of stability and they are applicable to the identification of general feedback control systems

softness sleekness and regularity properties in nonsmooth analysis abstract we study continuity properties of tangent and normal cones to a closed subset of a banach space such a study can be seen as a nonsmooth set theoretic analogue of the study of continuously differentiable functions we give particular attention to the case where different concepts coincide a desirable feature such properties are able to offer we identify a wide class of sets and functions for which this property occurs

existence and differentiability of metric projections in hilbert spaces this paper considers metric projections onto a closed subset s of a hilbert space if the set s is convex then it is well known that the corresponding metric projections always exist unique and directionally differentiable at boundary points of s these properties of metric projections are considered for possibly nonconvex sets s in particular existence and directional differentiability of metric projections for certain classes of sets are established and will be referred to as nearly convex sets

strong and weak convexity of sets and functions in this paper we study two classes of sets strongly and weakly convex sets for each class we derive a series of properties which involve either the concept of supporting ball an obvious extension of the concept of supporting hyperplane or the normal cone to the set we also study a class of functions denoted convex which satisfy for arbitrary points x1 and x2 and any value 0 1 the classical inequality of convex functions up to a term 1 x1 x2 2 depending on the sign of the constant the function is said to be strongly or weakly convex we provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets finally we give three applications a separation theorem a sufficient condition for global optimum of a nonconvex programming problem and a sufficient geometrical condition for a set to be a manifold

adaptive lqg control of input output systems a cost biased approach in this paper we consider linear systems in input output form and introduce a new adaptive linear quadratic gaussian lqg control scheme which is shown to be self optimizing the identification algorithm incorporates a cost biasing term which favors the parameters with smaller lqg optimal cost and a second term that aims at moderating the time variability of the estimate the corresponding closed loop scheme is proven to be stable and to achieve an asymptotic lqg cost equal to the one obtained under complete knowledge of the true system self optimization the results of this paper extend in a nontrivial way previous results established along the cost biased approach in other settings

approximation and decomposition properties of some classes of locally d c function we study the connexion between local and global decompositions of some important subclasses of locally d c functions functions which locally split as a difference of two convex functions then we tackle the problem of regularizing such functions by the moreau yosida process and prove in particular that the class of lower c2 functions fits well this approximation procedure

local differentiability of distance functions recently clarke stern and wolenski characterized in a hilbert space the closed subsets c for which the distance function dc is continuously differentiable everywhere on an open tube of uniform thickness around c here a corresponding local theory is developed for the property of dc being continuously differentiable outside of c on some neighborhood of a point x c this is shown to be equivalent to the prox regularity of c at x which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation additional characterizations are provided in terms of dc being locally of class c 1 or such that dc 2 is convex around x for some 0 prox regularity of c at x corresponds further to the normal cone mapping nc having a hypomonotone truncation around x and leads to a formula for pc by way of nc the local theory also yields new insights on the global level of the clarke stern wolenski results and on a property of sets introduced by shapiro as well as on the concept of sets with positive reach considered by federer in the finite dimensional setting

stability of negative image equilibria in spike timing dependent plasticity we investigate the stability of negative image equilibria in mean synaptic weight dynamics governed by spike timing dependent plasticity stdp the model architecture closely follows the anatomy and physiology of the electrosensory lateral line lobe ell of mormyrid electric fish the ell uses a spike timing dependent learning rule to form a negative image of the reafferent signal from the fish s own electric discharge thus improving detectability of external electric fields we derive sufficient conditions for existence of the negative image and necessary and sufficient conditions for stability for arbitrary postsynaptic potential functions and arbitrary learning rules this significantly generalizes earlier investigations we then apply the general result to several examples of biological interest including a class of learning rules consistent with the rule observed experimentally in the mormyrid ell

an extension of attouch s theorem and its application to second order epi differentiation of convexly composite functions in 1977 hedy attouch established that a sequence of closed proper convex functions epi converges to a convex function if and only if the graphs of the subdierentials converge in the mosco sense to the subdifferential of the limiting function and roughly speaking there is a condition that fixes the constant of integration we show that the theorem is valid if instead one considers functions that are the composition of a closed proper convex function with a twice continuously differentiable mapping in addition a constraint qualification is imposed

approximately convex functions and approximately monotonic operators abstract we present characterizations of some generalized convexity properties of functions with the help of a general subdifferential we stress the case of lower semicontinuous functions we also study the important case of marginal functions and we provide representation results

characterizations of weak sharp minima for lower c1 functions abstract we derive characterizations of weak sharp minimizers of order one for the class of lower c 1 functions the characterizations of such minimizers are obtained for a nonlinear programming problem with an abstract set constraint the first characterization is formulated in terms of the proximal normals to a given set relative to the abstract set constraint and the directional derivative of the objective function two examples are given to illustrate this characterization the other characterizations are extensions of the main characterizations of weak sharp minima for convex functions recently proved by burke and deng 2002 13 theorems 2 2 and 2 3 to the class of lower c 1 functions