dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions we establish two conditions that ensure the nondivergence of additive recurrent networks with unsaturating piecewise linear transfer functions also called linear threshold or semilinear transfer functions as hahnloser sarpeshkar mahowald douglas and seung 2000 showed networks of this type can be efficiently built in silicon and exhibit the coexistence of digital selection and analog amplification in a single circuit to obtain this behavior the network must be multistable and nondivergent and our conditions allow determining the regimes where this can be achieved with maximal recurrent amplification the first condition can be applied to nonsymmetric networks and has a simple interpretation of requiring that the strength of local inhibition match the sum over excitatory weights converging onto a neuron the second condition is restricted to symmetric networks but can also take into account the stabilizing effect of nonlocal inhibitory interactions we demonstrate the application of the conditions on a simple example and the orientation selectivity model of ben yishai lev bar or and sompolinsky 1995 we show that the conditions can be used to identify in their model regions of maximal orientation selective amplification and symmetry breaking

transient turing patterns in a neural field model we investigate turing bifurcations in a neural field model with one spatial dimension for some parameter values the resulting turing patterns are stable while for others the patterns appear transiently we show that this difference is due to the relative position in parameter space of the saddle node bifurcation of a spatially periodic pattern and the turing bifurcation point by varying parameters we are able to observe transient patterns whose duration scales in the same way as type i intermittency similar behavior occurs in two spatial dimensions

multistability of second order competitive neural networks with nondecreasing saturated activation functions in this paper second order interactions are introduced into competitive neural networks nns and the multistability is discussed for second order competitive nns socnns with nondecreasing saturated activation functions firstly based on decomposition of state space cauchy convergence principle and inequality technique some sufficient conditions ensuring the local exponential stability of 2n equilibrium points are derived secondly some conditions are obtained for ascertaining equilibrium points to be locally exponentially stable and to be located in any designated region thirdly the theory is extended to more general saturated activation functions with 2r corner points and a sufficient criterion is given under which the socnns can have r 1 n locally exponentially stable equilibrium points even if there is no second order interactions the obtained results are less restrictive than those in some recent works finally three examples with their simulations are presented to verify the theoretical analysis

waves and bumps in neuronal networks with axo dendritic synaptic interactions we consider a firing rate model of a neuronal network continuum that incorporates axo dendritic synaptic processing and the finite conduction velocities of action potentials the model equation is an integral one defined on a spatially extended domain apart from a spatial integral mixing the network connectivity function with space dependent delays arising from non instantaneous axonal communication the integral model also includes a temporal integration over some appropriately identified distributed delay kernel these distributed delay kernels are biologically motivated and represent the response of biological synapses to spiking inputs they are interpreted as green s functions of some linear dierential operator exploiting this green s function description we discuss formal reductions of this non local system to equivalent partial dierential equation pde models we distinguish between those spatial connectivity functions that give rise to local pde models and those that give rise to pde models with delayed non local terms for cases in which local pdes are derived we investigate traveling wave solutions in a comoving frame by numerically computing global heteroclinic connections for sigmoidal firing rate functions we also calculate exact solutions parameterized by axonal conduction velocity for the heaviside firing rate function the sigmoidal firing rate function in the limit of infinte gain the inclusion of synaptic adaptation is shown to alter traveling wave fronts to traveling pulses which we study analytically and numerically in terms of a global homoclinic orbit finally we consider the impact of dendritic interactions on waves and on static spatially localized solutions exact analysis for

multiperiodicity and exponential attractivity evoked by periodic external inputs in delayed cellular neural networks we show that an n neuron cellular neural network with time varying delay can have 2n periodic orbits located in saturation regions and these periodic orbits are locally exponentially attractive in addition we give some conditions for ascertaining periodic orbits to be locally or globally exponentially attractive and allow them to locate in any designated region as a special case of exponential periodicity exponential stability of delayed cellular neural networks is also characterized these conditions improve and extend the existing results in the literature to illustrate and compare the results simulation results are discussed in three numerical examples

permitted and forbidden sets in discrete time linear threshold recurrent neural networks the concepts of permitted and forbidden sets enable a new perspective of the memory in neural networks such concepts exhibit interesting dynamics in recurrent neural networks this paper studies the basic theories of permitted and forbidden sets of the linear threshold discrete time recurrent neural networks the linear threshold transfer function has been regarded as an adequate transfer function for recurrent neural networks networks with this transfer function form a class of hybrid analog and digital networks which are especially useful for perceptual computations networks in discrete time can directly provide algorithms for efficient implementation in digital hardware the main contribution of this paper is to establish foundations of permitted and forbidden sets necessary and sufficient conditions for the linear threshold discrete time recurrent neural networks are obtained for complete convergence existence of permitted and forbidden sets as well as conditionally multiattractivity respectively simulation studies explore some possible interesting practical applications

turing instability and pattern formation in a two population neuronal network model abstract a two population firing rate model describing the dynamics of excitatory and inhibitory neural activity in one spatial dimension is investigated with respect to formation of patterns in particular stationary periodic patterns and spatiotemporal oscillations conditions for existence of spatially homogeneous equilibrium states are first determined and the stability properties of these equilibria are investigated it is shown that the nonlocal synaptic interactions may promote a finite bandwidth instability in a way analogous to diffusion effects in the classical turing instability for reaction diffusion equations and modulational instability in the theory of nonlinear waves in nonlocal defocusing kerr media our analysis relies on the wave number dependent invariants of the 2 2 matrix representing the spatially fourier transformed linearized evolution equations the generic picture which emerges is an instability consisting of a finite set of well separated unstable bands in wave number space gain bands the case with symmetrical exponentially decaying connectivity functions is investigated in detail allowing for a more comprehensive analysis of the gain band structure and in particular conditions for the excitation of a single gain band through a turing hopf bifurcation with the relative inhibition time constant as control parameter two typical situations emerge depending on the thresholds and inclinations of the sigmoidal firing rate functions i a single gain band is excited through a turing hopf bifurcation and the resulting state is a spatiotemporally oscillating pattern or ii the instability develops into a stationary periodic pattern i e a set of equidistant bumps the dependence of instability type on the inclinations of the firing rate function and the time constant are comprehensively investigated demonstrating for example that only stationary patterns can be generated for sufficiently small inhibitory time constants the nonlinear development of the gain band instabilities is further elucidated by direct numerical simulations

an improved global asymptotic stability criterion for delayed cellular neural networks a new lyapunov krasovskii functional is constructed for delayed cellular neural networks and the s procedure is employed to handle the nonlinearities an improved global asymptotic stability criterion is also derived that is a generalization of and an improvement over previous results numerical examples demonstrate the effectiveness of the criterion

two bump solutions of amari type models of neuronal pattern formation abstract we study a partial integro differential equation defined on a spatially extended domain that arises in the modeling of pattern formation in neuronal networks for a one dimensional domain we develop criteria for the existence and stability of equal width 2 bump solutions under the assumption that the firing rate function is the heaviside function we apply these criteria to an example for which the connectivity is of lateral inhibition type i e the coupling function has one positive zero and find that families of 2 bump solutions exist but none of the solutions are stable extensive numerical searches suggest that this is true for all coupling functions of this form however for a large class of coupling functions which have three positive zeros we find the coexistence of both stable and unstable 2 bump solutions we also extend our investigation to two spatial dimensions and give numerical evidence for the coexistence of 1 bump and 2 bump solutions our results imply that lateral inhibition type coupling is not sufficient to produce stable patterns that are more complex than single isolated patches of high activity

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

multistability and instability of neural networks with discontinuous nonmonotonic piecewise linear activation functions in this paper we discuss the coexistence and dynamical behaviors of multiple equilibrium points for recurrent neural networks with a class of discontinuous nonmonotonic piecewise linear activation functions it is proved that under some conditions such n neuron neural networks can have at least 5 n equilibrium points 3 n of which are locally stable and the others are unstable based on the contraction mapping theorem and the theory of strict diagonal dominance matrix the investigation shows that the neural networks with the discontinuous activation functions introduced in this paper can have both more total equilibrium points and more locally stable equilibrium points than the ones with continuous mexican hat type activation function or discontinuous two level activation functions an illustrative example with computer simulations is presented to verify the theoretical analysis

stability and bifurcations in neural fields with finite propagation speed and general connectivity a stability analysis is presented for neural field equations in the presence of finite propagation speed along axons and for a general class of connectivity kernels and synaptic proper ties sufficient conditions are given for the stability of equilibrium solutions it is shown that the propagation delays play a significant role in nonstationary bifurcations of equilibria whereas the stationary bifurcations depend only on the connectivity kernel in the case of nonstationary bifurca tions bounds are determined on the frequencies of the resulting oscillatory solutions a perturbative scheme is used to calculate the types of bifurcations leading to spatial patterns oscillations and traveling waves for high propagation speeds a simple method is derived that allows the determi nation of the bifurcation type by visual inspection of the fourier transforms of the kernel and its first moment results are numerically illustrated on a class of neurologically plausible systems with combinations of gaussian excitatory and inhibitory connections

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

generalization errors of the simple perceptron to find an exact form for the generalization error of a learning machine is an open problem even in the simplest case simple perceptron learning we introduce a new approach to tackle the problem the generalization error of the simple perceptron is expressed as a linear combination of extreme values of inputs with the help of extreme value theory in statistics we then obtain an exact form of the generalization error of the simple perceptron in the case of the worst learning generalization errors of the higher order perceptron taking the form of an inverse power law in the number of examples are also considered

incremental rough set approach for hierarchical multicriteria classification abstract multicriteria classification refers to classify objects evaluated by a set of criteria to preference ordered decision classes dominance based rough set approach has been successfully introduced to express and reason inconsistencies with a dominance principle in multicriteria classification problems hierarchical attribute values exist extensively within many real world applications which provide a hierarchical form to organize view and analyze data from different perspectives for accommodating the preference variability in this study we consider an extension of dominance based rough set approach by applying an incremental learning technique for hierarchical multicriteria classification while attribute values dynamically vary across different levels of granulations we formalize the dynamic characteristics of knowledge granules with the cut refinement and coarsening through attribute value taxonomies in the hierarchical multicriteria decision systems in consequence incremental algorithms for computing dominance based rough approximations of preference ordered decision classes are developed by applying the resulted prior knowledge as the input and only recomputing those outputs which depend on the changed attribute values this paper presents the theoretical foundation of the proposed approach example analysis and experimental evaluation are also provided for illustration of the feasibility and efficiency

new algebraic criteria for synchronization stability of chaotic memristive neural networks with time varying delays in this brief we consider the exponential synchronization of chaotic memristive neural networks with time varying delays using the lyapunov functional method and inequality technique the dynamic analysis here employs the theory of differential equations with discontinuous right hand side as introduced by filippov the designing laws in the synchronization of neural networks are proposed via state or output coupling in addition the new proposed algebraic criteria are very easy to verify and they also enrich and improve the earlier publications finally an example is given to show the effectiveness of the obtained results

the weakly nonlocal limit of a one population wilson cowan model abstract we derive the weakly nonlocal limit of a one population neuronal field model of the wilson cowan type in one spatial dimension by transforming this equation to an equation in the firing rate variable it is shown that stationary periodic solutions exist by appealing to a pseudo potential analysis the solutions of the full nonlocal equation obey a uniform bound and the stationary periodic solutions in the weakly nonlocal limit satisfying the same uniform bound are characterized by finite ranges of pseudo energy constants the time dependent version of the model is reformulated as a ginzburg landau khalatnikov type of equation in the firing rate variable where the maximum minimum points correspond stable unstable homogeneous solutions of the weakly nonlocal limit based on this formulation it is also conjectured that the stationary periodic solutions are unstable we implement a numerical method for the weakly nonlocal limit of the wilson cowan type of model based on the wavelet galerkin approach we perform some numerical tests to illustrate the stability of homogeneous solutions and the evolution of the bumps

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

multiple bumps in a neuronal model of working memory we study a partial integro differential equation defined on a spatially extended domain that arises from the modeling of working or short term memory in a neuronal network the equation is capable of supporting spatially localized regions of high activity which can be switched on and off by transient external stimuli we analyze the effects of coupling between units in the network showing that if the connection strengths decay monotonically with distance then no more thanon e regionof high activity canpersist whereas if they decay inanoscillatory fashion then multiple regions can persist

delay induced periodicity in a neural netlet of excitation and inhibition abstract the dynamical behaviour of a two neuron netlet of excitation and inhibition with a transmission delay is investigated it is shown that in the absence of delay the netlet relaxes to the trivial resting state if the delay is of sufficient magnitude the network is excited to a temporally periodic cyclic behaviour the analytical mechanism for the onset of cyclic behaviour is through a hopf type bifurcation approximate solusions to the periodic output of the netlet is calculated stability of the temporally periodic cycle is investigated it is shown that the bifurcation is supercritical a related discrete version of the continuous time system is formulated it is found that the discrete system also displays a cyclic behaviour results of a number of computer simulations are displayed graphically the article concludes with a brief neurobiological discussion

spatially structured activity in synaptically coupled neuronal networks ii lateral inhibition and standing pulses we consider the existence and stability of standing pulse solutions to a system of integro differential equations used to describe the activity of synaptically coupled networks of excitatory and inhibitory neurons in a single spatial domain assuming an arrangement of synaptic connections described by lateral inhibition previous formal arguments have demonstrated the existence of both stable and unstable standing pulses s amari biol cybern 27 1977 pp 77 87 these results have formed the basis for several recent hypotheses regarding the generation of sustained activity patterns in prefrontal cortex and other brain regions implicit in the lateral inhibition arrangement however is the assumption that the dynamics of inhibition are instantaneous here we present two arguments demonstrating the loss of stability of standing pulse solutions through a hopf bifurcation when more realistic inhibitory dynamics are considered the first argument parallels amari s formal presentation while the seco

multistability of two kinds of recurrent neural networks with activation functions symmetrical about the origin on the phase plane in this paper we investigate multistability of two kinds of recurrent neural networks with time varying delays and activation functions symmetrical about the origin on the phase plane one kind of activation function is with zero slope at the origin on the phase plane while the other is with nonzero slope at the origin on the phase plane we derive sufficient conditions under which these two kinds of n dimensional recurrent neural networks are guaranteed to have 2m 1 n equilibrium points with m 1 n of them being locally exponentially stable these new conditions improve and extend the existing multistability results for recurrent neural networks finally the validity and performance of the theoretical results are demonstrated through two numerical examples

on attracting basins of multiple equilibria of a class of cellular neural networks in this paper we study the distribution of attraction basins of multiple equilibrium points of cellular neural networks cnns under several conditions the boundaries of the attracting basins of the stable equilibria of a completely stable cnn system are composed of the closures of the stable manifolds of unstable equilibria of n 1 dimensions as demonstrations of this idea under the conditions proposed in the literature which depicts stable and unstable equilibria we identify the attraction basin of each stable equilibrium of which the boundary is composed of the stable manifolds of the unstable equilibria precisely we also investigate the attracting basins of a simple class of symmetric 1 d cnns via identifying the unstable equilibria of which the stable manifold is n 1 dimensional and the completely stable asymmetric cnns with stable equilibria less than 2n

a fast and robust cluster update algorithm for image segmentation in spin lattice models without annealing visual latencies revisited image segmentation in spin lattice models relies on the fast and reliable assignment of correct labels to those groups of spins that represent the same object commonly used local spin update algorithms are slow because in each iteration only a single spin is flipped and a careful annealing schedule has to be designed in order to avoid local minima and correctly label larger areas updating of complete spin clusters is more efficient but often clusters that should represent different objects will be conjoined in this study we propose a cluster update algorithm that similar to most local update algorithms calculates an energy function and determines the probability for flipping a whole cluster of spins by the energy gain calculated for a neighborhood of the regarded cluster the novel algorithm called energy based cluster update ecu algorithm is compared to its predecessors a convergence proof is derived and it is shown that the algorithm outperforms local update algorithms by far in speed and re

boundary detection by constrained optimization a statistical framework is used for finding boundaries and for partitioning scenes into homogeneous regions the model is a joint probability distribution for the array of pixel gray levels and an array of labels in boundary finding the labels are binary zero or one representing the absence or presence of boundary elements in partitioning the label values are generic two labels are the same when the corresponding scene locations are considered to belong to the same region the distribution incorporates a measure of disparity between certain spatial features of block pairs of pixel gray levels using the kolmogorov smirnov nonparametric measures of difference between the distributions of these features the number of model parameters is minimized by forbidding label configurations which are assigned probability zero the maximum a posteriori estimator of boundary placements and partitionings is examined the forbidden states introduce constraints into the calculation of these configurations stochastic relaxation methods are extended to accommodate constrained optimization

activity invariant sets and exponentially stable attractors of linear threshold discrete time recurrent neural networks this technical note proposes to study the activity invariant sets and exponentially stable attractors of linear threshold discrete time recurrent neural networks the concept of activity invariant sets deeply describes the property of an invariant set by that the activity of some neurons keeps invariant all the time conditions are obtained for locating activity invariant sets under some conditions it shows that an activity invariant set can have one equilibrium point which attracts exponentially all trajectories starting in the set since the attractors are located in activity invariant sets each attractor has binary pattern and also carries analog information such results can provide new perspective to apply attractor networks for applications such as group winner take all associative memory etc

localized activity patterns in two population neuronal networks abstract we investigate a two population neuronal network model of the wilson cowan type with respect to existence of localized stationary solutions bumps and focus on the situation where two separate bump solutions one narrow pair and one broad pair exist the stability of the bumps is investigated by means of two different approaches the first generalizes the amari approach while the second is based on a direct linearization procedure a classification scheme for the stability problem is formulated and it is shown that the two approaches yield the same predictions except for one notable exception the narrow pair is generically unstable while the broad pair is stable for small and moderate values of the relative inhibition time at a critical relative inhibition time the broad pair is typically converted to stable breathers through a hopf bifurcation in our numerical example the broad pulse pair remains stable even when the inhibition time constant is three times longer than the excitation time constant thus our model results do not support the claim that slow excitation mediated by e g nmda receptors is needed to allow stable bumps

qualitative behaviour of some simple networks some classical neural network systems including the hartline ratliff system the linsker system and the general sigmoid dynamics are reconsidered within a more general class of dynamical systems for synchronous dynamics the existence uniqueness local and global stability of stationary points is investigated for asynchronous dynamics a convergence theorem is proved the application of the theory of quasimonotone flows leads to some insights so far not widespread in network theory

monostability and multistability for almost periodic solutions of fractional order neural networks with unsaturating piecewise linear activation functions since the unsaturating activation function is unbounded more complex dynamics may exist in neural networks with this kind of activation function in this article monostability and multistability results of almost periodic solutions are developed for fractional order neural networks with unsaturating piecewise linear activation functions some globally mittag leffler attractive sets are given and the existence of globally mittag leffler stable almost periodic solution is demonstrated by using ascoli arzela theorem in particular some sufficient conditions are provided to ascertain the multistability of almost periodic solutions based on locally positively invariant set it shows that there exists an almost periodic solution in each positively invariant set and all trajectories converge to this periodic trajectory in that rectangular area two illustrative examples are provided to demonstrate the effectiveness of the proposed sufficient criteria

on the pdf of the sum of random vectors there are various cases in physics and engineering sciences especially communications where one requires the envelope probability density function pdf of the sum of several random sinusoidal signals according to the correspondence between a random sinusoidal signal and a random vector the sum of random vectors can be considered as an abstract mathematical model for the above sum now it is desired to obtain the pdf of the length of the resulting vector considering the common and reasonable assumption of uniform distributions for the angles of the vectors many researchers have obtained the pdf of the length of the resulting vector only for special cases however in this paper the pdf is obtained for the most general case in which the lengths of vectors are arbitrary dependent random variables this pdf is in the form of a definite integral which may be inappropriate for analytic manipulations and numerical computations so an appropriate infinite laguerre expansion is also derived finally the results are applied to solve a typical example in computing the scattering cross section of random scatterers

capacity of the hopfield model for a given if neurons deviating from the memorized patterns are allowed we constructively show that if and only if all stored patterns are fixed points of the hopfield model if neurons are allowed with then where is the distribution function of the normal distribution the result obtained by amit and co workers only formally coincides with the latter case which indicates that the replica trick approach to the capacity of the hopfield model is only valid in the case

an analysis on neural dynamics with saturated sigmoidal functions we propose a unified approach to study the relation between the set of saturated attractors and the set of system parameters of the hopfield model linsker s model and the dynamic link network dln which use saturated sigmoidal functions in its dynamics of the state or weight the key point for this approach is to rigorously derive a necessary and sufficient condition to test whether a given saturated state in the hopfield model or weight vector in linsker s model and the dln is stable or not for any given set of system parameters and used this to determine the complete regime in the parameter space over which the given state or weight is stable our approach allows us to give an exact characterization between the parameters and the capacity in the hopfield model to generalize our previous results on linsker s network and the dln to have a better understanding of the underlying mechanism among these models the method reported here could be adopted to analyze a variety of models in the field of the neural networks

subspace clustering using a low rank constrained autoencoder the performance of subspace clustering is affected by data representation data representation for subspace clustering maps data from the original space into another space with the property of better separability many data representation methods have been developed in recent years typical among them are low rank representation lrr and an autoencoder lrr is a linear representation method that captures the global structure of data with low rank constraint alternatively an autoencoder nonlinearly maps data into a latent space using a neural network by minimizing the difference between the reconstruction and input to combine the advantages of an lrr globality and autoencoder self supervision based locality we propose a novel data representation method for subspace clustering the proposed method called low rank constrained autoencoder lrae forces the latent representation of the neural network to be of low rank and the low rank constraint is computed as a prior from the input space one major advantage of the lrae is that the learned data representation not only maintains the local features of the data but also preserves the underlying low rank global structure extensive experiments on several datasets for subspace clustering were conducted they demonstrated that the proposed lrae substantially outperformed state of the art subspace clustering methods

lyapunov functions for neural nets with nondifferentiable input output characteristics i construct lyapunov functions for asynchronous dynamics and synch ronous dynamics of neural networks with nondifferentiable input output characteristics

on the stationarity of signal scattering by two dimensional slightly rough random surfaces we present statistical properties of signal scattering by two dimensional slightly rough random surface the work concerns the intermediate field zone calculations are carried out within the framework of the first order small perturbation method the surface is assumed to be ergodic and stationary and the height distribution to be gaussian under an oblique incidence we demonstrate that the first order scattered field is not wide sense stationary for a given altitude under the normal incidence the scattered field is a strictly stationary random process by using the stationary phase method we show that the scattered field becomes asymptotically a strictly stationary random process when increasing the altitude of the observation point we also define the condition that must be satisfied by an antenna transfer function so that the measured scattered field becomes stationary and ergodic

multistability and new attraction basins of almost periodic solutions of delayed neural networks in this paper we investigate multistability of almost periodic solutions of recurrently connected neural networks with delays simply called delayed neural networks we will reveal that under some conditions the space rn can be divided into 2n subsets and in each subset the delayed n neuron neural network has a locally stable almost periodic solution furthermore we also investigate the attraction basins of these almost periodic solutions we reveal that the attraction basin of almost periodic trajectory is larger than the subset where the corresponding almost periodic trajectory is located in addition several numerical simulations are presented to corroborate the theoretical results

multistability of recurrent neural networks with time varying delays and the piecewise linear activation function in this brief stability of multiple equilibria of recurrent neural networks with time varying delays and the piecewise linear activation function is studied a sufficient condition is obtained to ensure that n neuron recurrent neural networks can have 4k 1 n equilibrium points and 2k n of them are locally exponentially stable this condition improves and extends the existing stability results in the literature simulation results are also discussed in one illustrative example

boundedness and complete stability of complex valued neural networks with time delay in this paper the boundedness and complete stability of complex valued neural networks cvnns with time delay are studied some conditions to guarantee the boundedness of the cvnns are derived using local inhibition moreover under the boundedness conditions a compact set that globally attracts all the trajectories of the network is also given additionally several conditions in terms of real valued linear matrix inequalities lmis for complete stability of the cvnns are established via the energy minimization method and the approach that converts the complex valued lmis to real valued ones examples with simulation results are given to show the effectiveness of the theoretical analysis

closed form expressions for the moments of the binomial probability distribution this work develops closed form expressions for the raw and central moments of the binomial probability distribution for this i first derive a recursive formula for the raw moments from the moment generating function then it is shown that the recursion involved is essentially the same as for the stirling numbers of the second kind from this fact it is then possible to derive the closed formulae finally i discuss an application of these formulae to the analysis of neural associative memory

energy approach to rivalry dynamics soliton stability and pattern formation in neuronal networks hopfield s lyapunov function is used to view the stability and topology of equilibria in neuronal networks for visual rivalry and pattern formation for two neural populations with reciprocal inhibition and slow adaptation the dynamics of neural activity is found to include a pair of limit cycles one for oscillations between states where one population has high activity and the other has low activity as in rivalry and one for oscillations between states where both populations have the same activity hopfield s lyapunov function is used to find the dynamical mechanism for oscillations and the basin of attraction of each limit cycle for a spatially continuous population with lateral inhibition stable equilibria are found for local regions of high activity solitons and for bound states of two or more solitons bound states become stable when moving two solitons together minimizes the lyapunov function a result of decreasing activity in regions between peaks of high activity when the firing rate is described by a sigmoid function lowering the barrier to soliton formation leads to a pattern forming instability and a nonlinear solution to the dynamical equations is found to be given by a soliton lattice which is completely characterized by the soliton width and the spacing between neighboring solitons fluctuations due to noise create lattice vacancies analogous to point defects in crystals leading to activity which is spatially inhomogeneous

on the co polarized phase difference of rough layered surfaces formulae derived from the small perturbation method we determine the statistical distribution of the co polarized phase difference of fields scattered from a stack of two two dimensional rough interfaces in the incidence plane the electromagnetic fields are represented by rayleigh expansions and a perturbation method is used to solve the boundary value problem and to determine the first order scattering amplitudes for slightly rough interfaces with infinite length and gaussian height distributions we show that the probability density function is only a function of two parameters for a sand layer on a granite surface in backscattering configurations we study the influence of the incidence angle the layer thickness the cross spectral density and the wave frequency upon both parameters of the probability law

multistability of discrete time recurrent neural networks with unsaturating piecewise linear activation functions this paper studies the multistability of a class of discrete time recurrent neural networks with unsaturating piecewise linear activation functions it addresses the nondivergence global attractivity and complete stability of the networks using the local inhibition conditions for nondivergence are derived which not only guarantee nondivergence but also allow for the existence of multiequilibrium points under these nondivergence conditions global attractive compact sets are obtained complete stability is studied via constructing novel energy functions and using the well known cauchy convergence principle examples and simulation results are used to illustrate the theory

on the co polarized scattered intensity ratio of rough layered surfaces the probability law derived from the small perturbation method we determine the probability law of the ratio between the co polarized intensities scattered from a stack of two two dimensional rough interfaces in the incidence plane calculations are carried out within the framework of the first order small perturbation method for slightly rough interfaces with infinite length and gaussian height distributions we show that the probability density function is only a function of two parameters and has an infinite average and an infinite variance for a sand layer on a granite surface in backscattering configurations we study the influence of the incidence angle and the cross spectral density upon this probability law

dynamical behaviors analysis of memristor based fractional order complex valued neural networks with time delay abstract the robust stability for memristor based fractional order complex valued neural networks fcvnns with time delay is investigated here in complex plane by using the lyapunov method and under the sense of filippov solutions the existence of unique equilibrium and globally asymptotic stability for such neural networks nns have been obtained when the nonlinear complex valued activation functions could be split into two real and imaginary parts moreover locally asymptotic stability for such neural nns have been proposed when the nonlinear complex activation functions are bounded lastly three numerical examples are given to confirm the efficiency of theorems

linear threshold discrete time recurrent neural networks stability and globally attractive sets the stability of linear threshold dynamic neural networks is studied and a series of methods to obtain globally attractive sets is proposed a sufficient condition to judge whether an invariant set is a globally attractive set is also proposed this method requires only the solution to a class of linear matrix inequalities also two direct methods to obtain globally attractive sets are given the stability criteria presented are based on the proposed globally attractive sets some numerical examples are given to illustrate the effectiveness of the obtained results

existence and global exponential stability of almost periodic solution for cellular neural networks with variable coefficients and time varying delays in this paper we study cellular neural networks with almost periodic variable coefficients and time varying delays by using the existence theorem of almost periodic solution for general functional differential equations introducing many real parameters and applying the lyapunov functional method and the technique of young inequality we obtain some sufficient conditions to ensure the existence uniqueness and global exponential stability of almost periodic solution the results obtained in this paper are new useful and extend and improve the existing ones in previous literature

some statistical and spatial properties of signal scattering by 2 d slightly rough random surfaces within the framework of the first order small perturbation method spm we establish some statistical and spatial properties of signal scattering by 2 d slightly rough random surfaces the work concerns the total field within the intermediate field zone for an ergodic and stationary gaussian surface under illumination by a monochromatic plane wave for infinite extension surfaces we obtain the probability density functions pdfs for the modulus and phase of the total field components the modulus pdf is expressed as an infinite sum of modified bessel functions while the phase pdf is expressed in terms of the error function under oblique illumination the total field is not wide sense stationary from a spatial point of view for a given elevation and under all incidences we show that the total field is ergodic to the second order under oblique incidence the spatial distribution of the total field modulus is a rayleigh law and the phase is uniform under normal incidence we establish that the spatial and statistical distributions are interchangeable

on the foundations of relaxation labeling processes a large class of problems can be formulated in terms of the assignment of labels to objects frequently processes are needed which reduce ambiguity and noise and select the best label among several possible choices relaxation labeling processes are just such a class of algorithms they are based on the parallel use of local constraints between labels this paper develops a theory to characterize the goal of relaxation labeling the theory is founded on a definition of con sistency in labelings extending the notion of constraint satisfaction in certain restricted circumstances an explicit functional exists that can be maximized to guide the search for consistent labelings this functional is used to derive a new relaxation labeling operator when the restrictions are not satisfied the theory relies on variational cal culus it is shown that the problem of finding consistent labelings is equivalent to solving a variational inequality a procedure nearly identical to the relaxation operator derived under restricted circum stances serves in the more general setting further a local convergence result is established for this operator the standard relaxation labeling formulas are shown to approximate our new operator which leads us to conjecture that successful applications of the standard methods are explainable by the theory developed here observations about con vergence and generalizations to higher order compatibility relations are described

selectable and unselectable sets of neurons in recurrent neural networks with saturated piecewise linear transfer function the concepts of selectable and unselectable sets are proposed to describe some interesting dynamical properties of a class of recurrent neural networks rnns with saturated piecewise linear transfer function a set of neurons is said to be selectable if it can be co unsaturated at a stable equilibrium point by some external input a set of neurons is said to be unselectable if it is not selectable i e such set of neurons can never be co unsaturated at any stable equilibrium point regardless of what the input is the importance of such concepts is that they enable a new perspective of the memory in rnns necessary and sufficient conditions for the existence of selectable and unselectable sets of neurons are obtained as an application the problem of group selection is discussed by using such concepts it shows that under some conditions each group is a selectable set and each selectable set is contained in some group thus groups are indicated by selectable sets of the rnns and can be selected by external inputs simulations are carried out to further illustrate the theory

analysis of cyclic dynamics for networks of linear threshold neurons the network of neurons with linear threshold lt transfer functions is a prominent model to emulate the behavior of cortical neurons the analysis of dynamic properties for lt networks has attracted growing interest such as multistability and boundedness however not much is known about how the connection strength and external inputs are related to oscillatory behaviors periodic oscillation is an important characteristic that relates to nondivergence which shows that the network is still bounded although unstable modes exist by concentrating on a general parameterized two cell network theoretical results for geometrical properties and existence of periodic orbits are presented although it is restricted to two dimensional systems the analysis can provide a useful contribution to analyze cyclic dynamics of some specific lt networks of high dimension as an application it is extended to an important class of biologically motivated networks of large scale the winner take all model using local excitation and global inhibition

global asymptotic stability of delayed cellular neural networks a new criterion for the global asymptotic stability of the equilibrium point of cellular neural networks with multiple time delays is presented the obtained result possesses the structure of a linear matrix inequality and can be solved efficiently using the recently developed interior point algorithm a numerical example is used to show the effectiveness of the obtained result

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

convergence analysis of a discrete time recurrent neural network to perform quadratic real optimization with bound constraints presents a model of a discrete time recurrent neural network designed to perform quadratic real optimization with bound constraints the network iteratively improves the estimate of the solution always maintaining it inside of the feasible region several neuron updating rules which assure global convergence of the net to the desired minimum have been obtained some of them also assure exponential convergence and maximize a lower bound for the convergence degree simulation results are presented to show the net performance

piecewise affine bijections of rn and the equation sx tx y we prove two criteria for bijectivity of piecewise affine maps f of rn it is assumed that det f has the same nonzero sign everywhere the first criterion is based solely on the branching properties of the partition of rn into domains of linearity of f the second one concerns the map itself and depends on the images under f of certain two dimensional angles this generalizes and improves results of r schramm the methods are topological our results imply k g murty s solution of the linear complementarity problem and some other partly known facts which we show are equivalent to it for example the map x sx tx is bijective iff all principal minors of s 1t are strictly positive

projective synchronization of fractional order memristive neural networks with switching jumps mismatch abstract this paper investigates the projective synchronization of fractional order memristor based neural networks fmnns with switching jumps mismatch and time varying delays according to the theory of discontinuous differential system filippov and differential inclusions a fractional order inequality was introduced based on notoriously barbalat s lemma and razumikhin type stability theorem some projective synchronization criteria of fmnns are derived these results extend the previous publications to illustrate the effectiveness of the proposed results two numerical examples are presented

dynamic stability conditions for lotka volterra recurrent neural networks with delays the lotka volterra model of neural networks derived from the membrane dynamics of competing neurons have found successful applications in many winner take all types of problems this paper studies the dynamic stability properties of general lotka volterra recurrent neural networks with delays conditions for nondivergence of the neural networks are derived these conditions are based on local inhibition of networks thereby allowing these networks to possess a multistability property multistability is a necessary property of a network that will enable important neural computations such as those governing the decision making process under these nondivergence conditions a compact set that globally attracts all the trajectories of a network can be computed explicitly if the connection weight matrix of a network is symmetric in some sense and the delays of the network are in l2 space we can prove that the network will have the property of complete stability

multistability of neural networks with mexican hat type activation functions in this paper we are concerned with a class of neural networks with mexican hat type activation functions due to the different structure from neural networks with saturated activation functions a set of new sufficient conditions are presented to study the multistability including the total number of equilibrium points their locations and stability furthermore the attraction basins of stable equilibrium points are investigated for two neuron neural networks the investigation shows that the stable manifolds of unstable equilibrium points constitute the boundaries of attraction basins of stable equilibrium points several illustrative examples are given to verify the effectiveness of our results

equilibria and their bifurcations in a recurrent neural network involving iterates of a transcendental function some practical models contain so complicated mathematical expressions that it is hard to determine the number and distribution of all equilibria not mentioning the qualitative properties and bifurcations of those equilibria the three node recurrent neural network system with two free weight parameters originally introduced by ruiz owens and townley in 1997 is such a system for which the equation of equilibria involves transcendental function and its iterates not computing coordinates of its equilibria in this paper we display an effective technique to determine the number and distribution of its equilibria without full information about equilibria our method enables to further study qualitative properties of those equilibria and discuss their saddle node pitchfork and hopf bifurcations by approximating center manifolds

on the properties of nonlinear nonlocal operators arising in neural field models abstract we study the existence and continuous dependence of stationary solutions of the one population wilson cowan model on the steepness of the firing rate functions we investigate the properties of the nonlinear nonlocal operators which arise when formulating the stationary one population wilson cowan model as a fixed point problem the theory is used to study the existence and continuous dependence of localized stationary solutions of this model on the steepness of the firing rate functions the present work generalizes and complements previously obtained results as we relax on the assumptions that the firing rate functions are given by smoothed heaviside functions

eigenspectrum bounds for semirandom matrices with modular and spatial structure for neural networks the eigenvalue spectrum of the matrix of directed weights defining a neural network model is informative of several stability and dynamical properties of network activity existing results for eigenspectra of sparse asymmetric random matrices neglect spatial or other constraints in determining entries in these matrices and so are of partial applicability to cortical like architectures here we examine a parameterized class of networks that are defined by sparse connectivity with connection weighting modulated by physical proximity i e asymmetric euclidean random matrices modular network partitioning and functional specificity within the excitatory population we present a set of analytical constraints that apply to the eigenvalue spectra of associated weight matrices highlighting the relationship between connectivity rules and classes of network dynamics

global exponential stability of multitime scale competitive neural networks with nonsmooth functions in this paper we study the global exponential stability of a multitime scale competitive neural network model with nonsmooth functions which models a literally inhibited neural network with unsupervised hebbian learning the network has two types of state variables one corresponds to the fast neural activity and another to the slow unsupervised modification of connection weights based on the nonsmooth analysis techniques we prove the existence and uniqueness of equilibrium for the system and establish some new theoretical conditions ensuring global exponential stability of the unique equilibrium of the neural network numerical simulations are conducted to illustrate the effectiveness of the derived conditions in characterizing stability regions of the neural network

absolute stability conditions for discrete time recurrent neural networks an analysis of the absolute stability for a general class of discrete time recurrent neural networks rnn s is presented a discrete time model of rnn s is represented by a set of nonlinear difference equations some sufficient conditions for the absolute stability are derived using ostrowski s theorem and the similarity transformation approach for a given rnn model these conditions are determined by the synaptic weight matrix of the network the results reported in this paper need fewer constraints on the weight matrix and the model than in previously published studies

spatially structured activity in synaptically coupled neuronal networks i traveling fronts and pulses we consider traveling front and pulse solutions to a system of integro differential equations used to describe the activity of synaptically coupled neuronal networks in a single spatial dimension our first goal is to establish a series of direct links between the abstract nature of the equations and their interpretation in terms of experimental findings in the cortex and other brain regions this is accomplished first by presenting a biophysically motivated derivation of the system and then by establishing a framework for comparison between numerical and experimental measures of activity propagation speed our second goal is to establish the existence of traveling pulse solutions using more rigorous methods two techniques are presented the first a shooting argument reduces the problem from finding a specific solution to an integro differential equation system to finding any solution to an ode system the second a singular perturbation argument provides a construction of traveling pulse solutions un

stability analysis of discrete time recurrent neural networks with stochastic delay this paper is concerned with the stability analysis of discrete time recurrent neural networks rnns with time delays as random variables drawn from some probability distribution by introducing the variation probability of the time delay a common delayed discrete time rnn system is transformed into one with stochastic parameters improved conditions for the mean square stability of these systems are obtained by employing new lyapunov functions and novel techniques are used to achieve delay dependence the merit of the proposed conditions lies in its reduced conservatism which is made possible by considering not only the range of the time delays but also the variation probability distribution a numerical example is provided to show the advantages of the proposed conditions

modulational instability in the nonlocal χ 2 model abstract we investigate in detail the linear regime of the modulational instability mi properties of the plane waves of the nonlocal model for 2 media formulated in nikolov et al n i nikolov d neshev o bang w z krolikowski quadratic solitons as nonlocal solitons phys rev e 68 2003 036614 i v shadrivov a a zharov dynamics of optical spatial solitons near the interface between two quadratically nonlinear media j opt soc amer b 19 2002 596 602 it is shown that the mi is of the oscillatory type and of finite bandwidth moreover it is possible to identify regions in the parameter space for which a fundamental gain band exists and regions for which higher order gain bands and modulational stability exist we also show that the mi analysis for the nonlocal model is applicable in the finite walk off case finally we show that the plane waves of the nonlocal 2 model are recovered as the asymptotic limit of one of the branches of the plane waves i e the adiabatic branch or the acoustic branch of the full 2 model by means of a singular perturbational approach it is also proven that the stability results for the adiabatic branch continuously approach those of the nonlocal 2 model by using the singular perturbational approach the other branch of the plane waves i e the nonadiabatic branch or the optical branch is always modulationally unstable we compare the mi results for the adiabatic branch with the predictions obtained from the full 2 model in the non walk off limit it is concluded that for most physical relevant parameter regimes it suffices to use the nonlocal model in order to determine the mi properties

multiperiodicity and attractivity of delayed recurrent neural networks with unsaturating piecewise linear transfer functions this paper studies multiperiodicity and attractivity for a class of recurrent neural networks rnns with unsaturating piecewise linear transfer functions and variable delays using local inhibition conditions for boundedness and global attractivity are established these conditions allow coexistence of stable and unstable trajectories moreover multiperiodicity of the network is investigated by using local invariant sets it shows that under some interesting conditions there exists one periodic trajectory in each invariant set which exponentially attracts all trajectories in that region correspondingly simulations are carried out to illustrate the theories

exponential stabilization of memristive neural networks with time delays in this paper a general class of memristive neural networks with time delays is formulated and studied some sufficient conditions in terms of linear matrix inequalities are obtained in order to achieve exponential stabilization the result can be applied to the closed loop control of memristive systems in particular several succinct criteria are given to ascertain the exponential stabilization of memristive cellular neural networks in addition a simplified and effective algorithm is considered for design of the optimal controller these conditions are the improvement and extension of the existing results in the literature two numerical examples are given to illustrate the theoretical results via computer simulations

a comprehensive review of stability analysis of continuous time recurrent neural networks stability problems of continuous time recurrent neural networks have been extensively studied and many papers have been published in the literature the purpose of this paper is to provide a comprehensive review of the research on stability of continuous time recurrent neural networks including hopfield neural networks cohen grossberg neural networks and related models since time delay is inevitable in practice stability results of recurrent neural networks with different classes of time delays are reviewed in detail for the case of delay dependent stability the results on how to deal with the constant variable delay in recurrent neural networks are summarized the relationship among stability results in different forms such as algebraic inequality forms m matrix forms linear matrix inequality forms and lyapunov diagonal stability forms is discussed and compared some necessary and sufficient stability conditions for recurrent neural networks without time delays are also discussed concluding remarks and future directions of stability analysis of recurrent neural networks are given

selectively grouping neurons in recurrent networks of lateral inhibition winner take all networks have been proposed to underlie many of the brain s fundamental computational abilities however not much is known about how to extend the grouping of potential winners in these networks beyond single neuron or uniformly arranged groups of neurons we show that competition between arbitrary groups of neurons can be realized by organizing lateral inhibition in linear threshold networks given a collection of potentially overlapping groups with the exception of some degenerate cases the lateral inhibition results in network dynamics such that any permitted set of neurons that can be coactivated by some input at a stable steady state is contained in one of the groups the information about the input is preserved in this operation the activity level of a neuron in a permitted set corresponds to its stimulus strength amplified by some constant sets of neurons that are not part of a group cannot be coactivated by any input at a stable steady state we analyze the storage capacity of such a network for random groups the number of random groups the network can store as permitted sets without creating too many spurious ones in this framework we calculate the optimal sparsity of the groups maximizing group entropy we find that for dense inputs the optimal sparsity is unphysiologically small however when the inputs and the groups are equally sparse we derive a more plausible optimal sparsity we believe our results are the first steps toward attractor theories in hybrid analog digital networks

frustration stability and delay induced oscillations in a neural network model the effect of time delays on the linear stability of equilibria in an artificial neural network of hopfield type is analyzed the possibility of delay induced oscillations occurring is characterized in terms of properties of the not necessarily symmetric connection matrix of the network such oscillations are possible exactly when the network is frustrated equivalently when the signed digraph of the matrix does not require the perron property nonlinear analysis centre manifold computation of a three unit frustrated network is presented giving the nature of the bifurcations taking place a supercritical hopf bifurcation is shown to occur and a codimension two bifurcation is unfolded

analysis of nonlocal neural fields for both general and gamma distributed connectivities abstract this work studies the stability of equilibria in spatially extended neuronal ensembles we first derive the model equation from statistical properties of the neuron population the obtained integro differential equation includes synaptic and space dependent transmission delay for both general and gamma distributed synaptic connectivities the latter connectivity type reveals infinite finite and vanishing self connectivities the work derives conditions for stationary and nonstationary instabilities for both kernel types in addition a nonlinear analysis for general kernels yields the order parameter equation of the turing instability to compare the results to findings for partial differential equations pdes two typical pde types are derived from the examined model equation namely the general reaction diffusion equation and the swift hohenberg equation hence the discussed integro differential equation generalizes these pdes in the case of the gamma distributed kernels the stability conditions are formulated in terms of the mean excitatory and inhibitory interaction ranges as a novel finding we obtain turing instabilities in fields with local inhibition lateral excitation while wave instabilities occur in fields with local excitation and lateral inhibition numerical simulations support the analytical results

multistability analysis for recurrent neural networks with unsaturating piecewise linear transfer functions multistability is a property necessary in neural networks in order to enable certain applications e g decision making where monostable networks can be computationally restrictive this article focuses on the analysis of multistability for a class of recurrent neural networks with unsaturating piecewise linear transfer functions it deals fully with the three basic properties of a multistable network boundedness global attractivity and complete convergence this article makes the following contributions conditions based on local inhibition are derived that guarantee boundedness of some multistable networks conditions are established for global attractivity bounds on global attractive sets are obtained complete convergence conditions for the network are developed using novel energy like functions and simulation examples are employed to illustrate the theory thus developed

foundations of implementing the competitive layer model by lotka volterra recurrent neural networks the competitive layer model clm can be described by an optimization problem the problem can be further formulated by an energy function called the clm energy function in the subspace of nonnegative orthant the set of minimum points of the clm energy function forms the set of solutions of the clm problem solving the clm problem means to find out such solutions recurrent neural networks rnns can be used to implement the clm to solve the clm problem the key point is to make the set of minimum points of the clm energy function just correspond to the set of stable attractors of the recurrent neural networks this paper proposes to use lotka volterra rnns lv rnns to implement the clm the contribution of this paper is to establish foundations of implementing the clm by lv rnns the contribution mainly contains three parts the first part is on the clm energy function necessary and sufficient conditions for minimum points of the clm energy function are established by detailed study the second part is on the convergence of the proposed model of the lv rnns it is proven that interesting trajectories are convergent the third part is the most important it proves that the set of stable attractors of the proposed lv rnn just equals the set of minimum points of the clm energy function in the nonnegative orthant thus the lv rnns can be used to solve the problem of the clm it is believed that by establishing such basic rigorous theories more and interesting applications of the clm can be found

lagrange exponential stability for fuzzy cohen grossberg neural networks with time varying delays this paper focuses on the globally exponential stability in lagrange sense for takagi sugeno t s fuzzy cohen grossberg neural networks with time varying delays by employing lyapunov method and delay inequality technique we analyze two different types of activation functions which include both lipschitz function and general activation functions several easily verifiable sufficient criteria about linear matrix inequality form are obtained to guarantee the lagrange exponential stability of cohen grossberg neural networks with time varying delays which are represented by t s fuzzy models meanwhile the estimations of the globally exponentially attractive sets are given here the existence and uniqueness of the equilibrium points need not be considered finally two numerical examples with simulations are given to illustrate the effectiveness of the theoretical results

nontrivial global attractors in 2 d multistable attractor neural networks attractor dynamics is a crucial problem for attractor neural networks as it is the underling computational mechanism for memory storage and retrieval in neural systems this brief studies a class of attractor network consisting of linearized threshold neurons and analyzes global attractors based on a parameterized 2 d model on the basis of previous results on nondegenerate and degenerate equilibria in mathematics we further elucidate all possible nontrivial global attractors our theoretical result provides precise descriptions on how the changes of network parameters affect the attractors distribution and landscape and it may give a feasible solution towards specifying attractors by specifying weights simulations are presented to illustrate the theoretical results