spike timing dependent plasticity the relationship to rate based learning for models with weight dynamics determined by a stable fixed point experimental evidence indicates that synaptic modification depends on the timing relationship between the presynaptic inputs and the output spikes that they generate in this letter results are presented for models of spike timing dependent plasticity stdp whose weight dynamics is determined by a stable fixed point four classes of stdp are identified on the basis of the time extent of their input output interactions the effect on the potentiation of synapses with different rates of input is investigated to elucidate the relationship of stdp with classical studies of long term potentiation and depression and rate based hebbian learning the selective potentiation of higher rate synaptic inputs is found only for models where the time extent of the input output interactions is input restricted i e restricted to time domains delimited by adjacent synaptic inputs and that have a time asymmetric learning window with a longer time constant for depression than for potentiation the analysis provides an account of learning dynamics determined by an input selective stable fixed point the effect of suppressive interspike interactions on stdp is also analyzed and shown to modify the synaptic dynamics

linear free resolutions and minimal multiplicity let s k x x be a polynomial ring over a field and let a4 a m be a finitely generated graded module in the most interesting case a4 is an ideal of s for a given natural number p there is a great interest in the question can m be generated by homogeneous elements of degree p no simple answer say in terms of the local cohomology of m is known but somewhat surprisingly the stronger question can the jth syzygy of m be generated by elements of degree qp j for all j 0 l n does admit a simple response the following is neither new nor difficult to prove though it seems not well known

on the noetherianity of some associative finitely presented algebras abstract we consider finitely generated associative algebras over a fixed field k of arbitrary characteristic for such an algebra a we impose some structural restrictions we call a strictly ordered we are interested in the implication of strict order on a for its noetherian properties in particular we prove that if a is a graded standard finitely presented strictly ordered algebra then a is left noetherian if and only if it is almost commutative in this case a has polynomial growth

bifurcation from infinity for reaction diffusion equations under nonlinear boundary conditions we consider reaction diffusion equations under nonlinear boundary conditions where the nonlinearities are asymptotically linear at infinity and depend on a parameter we prove that as the parameter crosses some critical values a resonance type phenomenon provides solutions that bifurcate from infinity we characterize the bifurcated branches when they are sub or supercritical we obtain both landesman lazer type conditions that guarantee the existence of solutions in the resonant case and an anti maximum principle

fixed point attractor analysis for a class of neurodynamics nearly all models in neural networks start from the assumption that the input output characteristic is a sigmoidal function on parameter space we present a systematic and feasible method for analyzing the whole spectrum of attractors all saturated all but one saturated all but twosaturated and so on of a neurodynamical system with a saturated sigmoidal function as its input output characteristic we present an argument that claims under a mild condition that only all saturated or all but one saturated attractors are observable for the neurodynamics for any given all saturated configuration all but one saturated configuration the article shows how to construct an exact parameter region r such that if and only if the parameters fall within r then is an attractor a fixed point of the dynamics the parameter region for an all saturated fixed point attractor is independent of the specific choice of a saturated sigmoidal function whereas for an all but one saturated fixed point it

generalized castelnuovo mumford regularity for affine kac moody algebras abstract the graded modules over noncommutative algebras often have minimal free resolutions of infinite length resulting in infinite castelnuovo mumford regularity in kang et al 2010 6 we introduced a generalized notion of castelnuovo mumford regularity to overcome this difficulty in this paper we compute the generalized castelnuovo mumford regularity for integrable highest weight representations of all affine kac moody algebras it is shown that the generalized regularity depends only on the type and rank of algebras and the level of representations

non commutative graded algebras and their hilbert series abstract this paper begins to develop a theory of non commutative graded algebras and their hilbert series which parallels the extensive already existing theory on commutative graded algebras we begin with a discussion of order ideals of monomials which give us vector space bases for these algebras we develop next the concept of weak summand which is a special relationship a subalgebra may have to an algebra and the concept of strongly free sets strongly free sets have many properties which are analogous to the properties of regular sequences in commutative algebras we discuss combinatorially free sets which are useful for constructing examples of many of the preceding ideas we conclude with an application of our results to algebraic topology by showing that there is a finite cw complex with only nine positive dimensional cells whose loop space has an irrational poincare series

the information encoded in initial ideals we consider homogeneous ideals i and the initial ideal in i for the revlex order first we give a sequence of invariants computed from i giving better and better approximations to the initial ideal and ending in an equivalent description then we apply this to different settings in algebraic geometry to understand what information is encoded in the generic initial ideal of the ideal of a projective scheme we also consider the higher initial ideals as defined in a paper by fl0ystad in particular we show that giving the generic higher initial ideal of a space curve is equivalent to giving the generic initial ideal of a linked curve

a non markovian random walk underlies a stochastic model of spike timing dependent plasticity a stochastic model of spike timing dependent plasticity stdp proposes that spike timing influences the probability but not the amplitude of synaptic strength change at single synapses the classic biphasic stdp profile emerges as a spatial average over many synapses presented with a single spike pair or as a temporal average over a single synapse presented with many spike pairs we have previously shown that the model accounts for a variety of experimental data including spike triplet results and has a number of desirable theoretical properties including being entirely self stabilizing in all regions of parameter space our earlier analyses of the model have employed cumbersome spike to spike averaging arguments to derive results here we show that the model can be reformulated as a non markovian random walk in synaptic strength the step sizes being fixed as postulated this change of perspective greatly simplifies earlier calculations by integrating out the proposed switch mechanism by which changes in strength are driven and instead concentrating on the changes in strength themselves moreover this change of viewpoint is generative facilitating further calculations that would be intractable if not impossible with earlier approaches we prepare the machinery here for these later calculations but also briefly indicate how this machinery may be used by considering two particular applications

an analysis of synaptic normalization in a general class of hebbian models in standard hebbian models of developmental synaptic plasticity synaptic normalization must be introduced in order to constrain synaptic growth and ensure the presence of activity dependent competitive dynamics in such models multiplicative normalization cannot segregate afferents whose patterns of electrical activity are positively correlated while subtractive normalization can it is now widely believed that multiplicative normalization cannot segregate positively correlated afferents in any hebbian model however we recently provided a counterexample to this belief by demonstrating that our own neurotrophic model of synaptic plasticity which can segregate positively correlated afferents can be reformulated as a nonlinear hebbian model with competition implemented through multiplicative normalization we now perform an analysis of a general class of hebbian models under general forms of synaptic normalization in particular we extract conditions on the forms of these rules that guarantee that such models possess a fixed point structure permitting the segregation of all but perfectly correlated afferents we find that the failure of multiplicative normalization to segregate positively correlated afferents in a standard hebbian model is quite atypical

hebbian learning of recurrent connections a geometrical perspective we show how a hopfield network with modifiable recurrent connections undergoing slow hebbian learning can extract the underlying geometry of an input space first we use a slow and fast analysis to derive an averaged system whose dynamics derives from an energy function and therefore always converges to equilibrium points the equilibria reflect the correlation structure of the inputs a global object extracted through local recurrent interactions only second we use numerical methods to illustrate how learning extracts the hidden geometrical structure of the inputs indeed multidimensional scaling methods make it possible to project the final connectivity matrix onto a euclidean distance matrix in a high dimensional space with the neurons labeled by spatial position within this space the resulting network structure turns out to be roughly convolutional the residual of the projection defines the nonconvolutional part of the connectivity which is minimized in the process finally we show how restricting the dimension of the space where the neurons live gives rise to patterns similar to cortical maps we motivate this using an energy efficiency argument based on wire length minimization finally we show how this approach leads to the emergence of ocular dominance or orientation columns in primary visual cortex via the self organization of recurrent rather than feedforward connections in addition we establish that the nonconvolutional or long range connectivity is patchy and is co aligned in the case of orientation learning

taming fluctuations in a stochastic model of spike timing dependent plasticity a stochastic model of spike timing dependent plasticity proposes that single synapses express fixed amplitude jumps in strength the amplitudes being independent of the spike time difference however the probability that a jump in strength occurs does depend on spike timing although the model has a number of desirable features the stochasticity of response of a synapse introduces potentially large fluctuations into changes in synaptic strength these can destabilize the segregated patterns of afferent connectivity characteristic of neuronal development previously we have taken these jumps to be small relative to overall synaptic strengths to control fluctuations but doing so increases developmental times cales unacceptably here we explore three alternative ways of taming fluctuations first a calculation of the variance for the change in synaptic strength shows that the mean change eventually dominates fluctuations but on timescales that are too long second it is possible that fluctuations in strength may cancel between synapses but we show that correlations between synapses emasculate the law of large numbers finally by separating plasticity induction and expression we introduce a temporal window during which induction signals are low pass filtered before expression in this way fluctuations in strength are tamed stabilizing segregated states of afferent connectivity

stability against fluctuations scaling bifurcations and spontaneous symmetry breaking in stochastic models of synaptic plasticity in stochastic models of synaptic plasticity based on a random walk the control of fluctuations is imperative we have argued that synapses could act as low pass filters filtering plasticity induction steps before expressing a step change in synaptic strength earlier work showed in simulation that such a synaptic filter tames fluctuations very well leading to patterns of synaptic connectivity that are stable for long periods of time here we approach this problem analytically we explicitly calculate the lifetime of meta stable states of synaptic connectivity using a fokker planck formalism in order to understand the dependence of this lifetime on both the plasticity step size and the filtering mechanism we find that our analytical results agree very well with simulation results despite having to make two approximations our analysis reveals however a deeper significance to the filtering mechanism and the plasticity step size we show that a filter scales the step size into a smaller effective step size this scaling suggests that the step size may itself play the role of a temperature parameter so that a filter cools the dynamics thereby reducing the influence of fluctuations using the master equation we explicitly demonstrate a bifurcation at a critical step size confirming this interpretation at this critical point spontaneous symmetry breaking occurs in the class of stochastic models of synaptic plasticity that we consider

discrete states of synaptic strength in a stochastic model of spike timing dependent plasticity a stochastic model of spike timing dependent plasticity stdp postulates that single synapses presented with a single spike pair exhibit all or none quantal jumps in synaptic strength the amplitudes of the jumps are independent of spiking timing but their probabilities do depend on spiking timing by making the amplitudes of both upward and downward transitions equal synapses then occupy only a discrete set of states of synaptic strength we explore the impact of a finite discrete set of strength states on our model finding three principal results first a finite set of strength states limits the capacity of a single synapse to express the standard exponential stdp curve we derive the expression for the expected change in synaptic strength in response to a standard experimental spike pair protocol finding a deviation from exponential behavior we fit our prediction to recent data from single dendritic spine heads finding results that are somewhat better than exponential fits second we show that the fixed point dynamics of our model regulate the upward and downward transition probabilities so that these are on average equal leading to a uniform distribution of synaptic strength states however third under long term potentiation ltp and long term depression ltd protocols these probabilities are unequal skewing the distribution away from uniformity if the number of states of strength is at least of order 10 then we find that three effective states of synaptic strength appear consistent with some experimental data on ternary strength synapses on this view ltp and ltd protocols may therefore be saturating protocols

down up algebras abstract the algebra generated by the down and up operators on a differential or uniform partially ordered set poset encodes essential enumerative and structural properties of the poset motivated by the algebras generated by the down and up operators on posets we introduce here a family of infinite dimensional associative algebras called down up algebras we show that down up algebras exhibit many of the important features of the universal enveloping algebrau s l2 of the lie algebra s l 2including a poincare birkhoff witt type basis and a well behaved representation theory we investigate the structure and representations of down up algebras and focus especially on verma modules highest weight representations and category o modules for them we calculate the exact expressions for all the weights since that information has proven to be particularly useful in determining structural results about posets

on neurodynamics with limiter function and linsker s developmental model the limiter function is used in many learning and retrieval models as the constraint controlling the magnitude of the weight or state vectors in this paper we developed a new method to relate the set of saturated fixed points to the set of system parameters of the models that use the limiter function and then as a case study applied this method to linsker s hebbian learning network we derived a necessary and sufficient condition to test whether a given saturated weight or state vector is stable or not for any given set of system parameters and used this condition to determine the whole regime in the parameter space over which the given state is stable this approach allows us to investigate the relative stability of the major receptive fields reported in linsker s simulations and to demonstrate the crucial role played by the synaptic density functions

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

hebb type dynamics is sufficient to account for the inverse magnification rule in cortical somatotopy the inverse magnification rule in cortical somatotopy is the experimentally derived inverse relationship between cortical magnification area of somatotopic map representing a unit area of skin surface and receptive field size area of restricted skin surface driving a cortical neuron we show by computer simulation of a simple multilayer model that hebb type synaptic modification subject to competitive constraints is sufficient to account for the inverse magnification rule

simple modification of oja rule limits l1 norm of weight vector and leads to sparse connectivity this letter describes a simple modification of the oja learning rule which asymptotically constrains the l1 norm of an input weight vector instead of the l2 norm as in the original rule this constraining is local as opposed to commonly used instant normalizations which require the knowledge of all input weights of a neuron to update each one of them individually the proposed rule converges to a weight vector that is sparser has more zero weights than the vector learned by the original oja rule with or without the zero bound which could explain the developmental synaptic pruning

convergence of stochastic learning in perceptrons with binary synapses the efficacy of a biological synapse is naturally bounded and at some resolution and is discrete at the latest level of single vesicles the finite number of synaptic states dramatically reduce the storage capacity of a network when online learning is considered i e the synapses are immediately modified by each pattern the trace of old memories decays exponentially with the number of new memories palimpsest property moreover finding the discrete synaptic strengths which enable the classification of linearly separable patterns is a combinatorially hard problem known to be np complete in this paper we show that learning with discrete binary synapses is nevertheless possible with high probability if a randomly selected fraction of synapses is modified following each stimulus presentation slow stochastic learning as an additional constraint the synapses are only changed if the output neuron does not give the desired response as in the case of classical perceptron learning we prove that for linearly separable classes of patterns the stochastic learning algorithm converges with arbitrary high probability in a finite number of presentations provided that the number of neurons encoding the patterns is large enough the stochastic learning algorithm is successfully applied to a standard classification problem of nonlinearly separable patterns by using multiple stochastically independent output units with an achieved performance which is comparable to the maximal ones reached for the task

a generalization of castelnuovo mumford regularity for representations of noncommutative algebras abstract we introduce and generalize the notion of castelnuovo mumford regularity for representations of noncommutative algebras effectively establishing a measure of complexity for such objects the grobner shirshov basis theory for modules over noncommutative algebras is developed by which a noncommutative analogue of schreyer s theorem is proved for computing syzygies by a repeated application of this theorem we construct free resolutions for representations of noncommutative algebras some interesting examples are included in which graded free resolutions and regularities are computed for representations of various algebras in particular using the bernstein gelfand gelfand resolutions for integrable highest weight modules over kac moody algebras we compute the projective dimensions and regularities explicitly for the cases of finite type and affine type a n 1

memory nearly on a spring a mean first passage time approach to memory lifetimes we study memory lifetimes in a perceptron based framework with binary synapses using the mean first passage time for the perceptron s total input to fall below firing threshold to define memory lifetimes working with the simplest memory related model of synaptic plasticity we may obtain exact results for memory lifetimes or working in the continuum limit good analytical approximations that afford either much qualitative insight or extremely good quantitative agreement in one particular limit we find that memory dynamics reduce to the well understood ornstein uhlenbeck process we show that asymptotically the lifetimes of memories grow logarithmically in the number of synapses when the perceptron s firing threshold is zero reproducing standard results from signal to noise ratio analyses however this is only an asymptotically valid result and we show that extending its application outside the range of its validity leads to a massive overestimate of the minimum number of synapses required for suc

axonal processes and neural plasticity a reply we examine the claim that a class of sprouting and retraction models is mathematically equivalent to a fixed anatomy model we accept subject to important caveats a narrow mathematical equivalence of the energy functions in both classes of model we argue that this narrow equivalence of energy functions does not however entail equivalence of the models indeed the claim of complete model equivalence hides significant dynamical differences between the approaches which we discuss we also disagree that our work demonstrates that subtractive constraint enforcement is natural in fixed anatomy models

reduced representation by neural networks with restricted receptive fields model neural networks can perform dimensional reductions of input data sets using correlation based learning rules to adjust their weights simple hebbian learning rules lead to an optimal reduction at the single unit level but result in highly redundant network representations more complex rules designed to reduce or remove this redundancy can develop optimal principal component representations but they are not very compelling from a biological perspective neurons in biological networks have restricted receptive fields limiting their access to the input data space we find that within this restricted receptive field architecture simple correlation based learning rules can produce surprisingly efficient reduced representations when noise is present the size of the receptive fields can be optimally tuned to maximize the accuracy of reconstructions of input data from a reduced representation

the role of weight normalization in competitive learning the effect of different kinds of weight normalization on the outcome of a simple competitive learning rule is analyzed it is shown that there are important differences in the representation formed depending on whether the constraint is enforced by dividing each weight by the same amount divisive enforcement or subtracting a fixed amount from each weight subtractive enforcement for the divisive cases weight vectors spread out over the space so as to evenly represent typical inputs whereas for the subtractive cases the weight vectors tend to the axes of the space so as to represent extreme inputs the consequences of these differences are examined

the role of constraints in hebbian learning models of unsupervised correlation based hebbian synaptic plasticity are typically unstable either all synapses grow until each reaches the maximum allowed strength or all synapses decay to zero strength a common method of avoiding these outcomes is to use a constraint that conserves or limits the total synaptic strength over a cell we study the dynamic effects of such constraints two methods of enforcing a constraint are distinguished multiplicative and subtractive for otherwise linear learning rules multiplicative enforcement of a constraint results in dynamics that converge to the principal eigenvector of the operator determining unconstrained synaptic development subtractive enforcement in contrast typically leads to a final state in which almost all synaptic strengths reach either the maximum or minimum allowed value this final state is often dominated by weight configurations other than the principal eigenvector of the unconstrained operator multiplicative enforcement yields a graded receptive field in which most mutually correlated inputs are represented whereas subtractive enforcement yields a receptive field that is sharpened to a subset of maximally correlated inputs if two equivalent input populations e g two eyes innervate a common target multiplicative enforcement prevents their segregation ocular dominance segregation when the two populations are weakly correlated whereas subtractive enforcement allows segregation under these circumstances these results may be used to understand constraints both over output cells and over input cells a variety of rules that can implement constrained dynamics are discussed

grobner shirshov bases for irreducible sln 1 modules abstract we determine the grobner shirshov bases for finite dimensional irreducible representations of the special linear lie algebra sl n 1 and construct explicit monomial bases for these representations we also show that each of these monomial bases is in 1 1 correspondence with the set of semistandard young tableaux of a given shape

removing time variation with the anti hebbian differential synapse i describe a local synaptic learning rule that can be used to remove the effects of certain types of systematic temporal variation in the inputs to a unit according to this rule changes in synaptic weight result from a conjunction of short term temporal changes in the inputs and the output formally this is like the differential rule proposed by klopf 1986 and kosko 1986 except for a change of sign which gives it an anti hebbian character by itself this rule is insufficient a weight conservation condition is needed to prevent the weights from collapsing to zero and some further constraint implemented here by a biasing term to select particular sets of weights from the subspace of those which give minimal variation as an example i show that this rule will generate center surround receptive fields that remove temporally varying linear gradients from the inputs

linear algebraic approach to grobner shirshov basis theory we construct a new efficient algorithm for finding grobner shirshov bases for noncommutative algebras and their representations this algorithm uses the macaulay matrix f s macaulay on some formula in elimination proc london math soc 33 1 1902 3 27 and can be viewed as a representation theoretic analogue of the f4 algorithm developed by j c faugere we work out some examples of universal enveloping algebras of lie algebras and of their representations to illustrate the algorithm

constrained optimization for neural map formation a unifying framework for weight growth and normalization computational models of neural map formation can be considered on at least three different levels of abstraction detailed models including neural activity dynamics weight dynamics that abstract from the neural activity dynamics by an adiabatic approximation and constrained optimization from which equations governing weight dynamics can be derived constrained optimization uses an objective function from which a weight growth rule can be derived as a gradient flow and some constraints from which normalization rules are derived in this article we present an example of how an optimization problem can be derived from detailed nonlinear neural dynamics a systematic investigation reveals how different weight dynamics introduced previously can be derived from two types of objective function terms and two types of constraints this includes dynamic link matching as a special case of neural map formation we focus in particular on the role of coordinate transformations to derive different weight dynamics from the same optimization problem several examples illustrate how the constrained optimization framework can help in understanding generating and comparing different models of neural map formation the techniques used in this analysis may also be useful in investigating other types of neural dynamics