main overlap dynamics for multistate neural networks the authors present explicit formulae for one step dynamics of retrieval pattern errors main overlap for the q state potts and q or 4 clock neural networks solution of the fixed point equations and critical values of the saturation parameters alpha c q in the one step approximation are considered in the potts case

stability properties of potts neural networks with biased patterns and low loading the q state potts glass model of neural networks is extended to include biased patterns for a finite number of such patterns the existence and stability properties of the mattis states and symmetric states are discussed in detail as a function of the bias analytic results are presented for all q at zero temperature for finite temperatures numerical results are obtained for q 3 and two classes of representative bias parameters a comparison is made with the hopfield model

convergence properties of the softassign quadratic assignment algorithm the softassign quadratic assignment algorithm is a discrete time continuous state synchronous updating optimizing neural network while its effectiveness has been shown in the traveling salesman problem graph matching and graph partitioning in thousands of simulations its convergence properties have not been studied here we construct discrete time lyapunov functions for the cases of exact and approximate doubly stochastic constraint satisfaction which show convergence to a fixed point the combination of good convergence properties and experimental success makes the softassign algorithm an excellent choice for neural quadratic assignment optimization

classical open systems with nonlinear nonlocal dissipation and state dependent diffusion dynamical responses and the jarzynski equality we have studied dynamical responses and the jarzynski equality je of classical open systems described by the generalized caldeira leggett model with the nonlocal system bath coupling in the derived non markovian langevin equation the nonlinear nonlocal dissipative term and state dependent diffusion term yielding multiplicative colored noise satisfy the fluctuation dissipation relation simulation results for harmonic oscillator systems have shown the following a averaged responses of the system x to applied sinusoidal and step forces significantly depend on model parameters of magnitudes of additive and multiplicative noises and the relaxation time of colored noise although stationary marginal probability distribution functions are independent of them b a combined effect of nonlinear dissipation and multiplicative colored noise induces enhanced fluctuations x x 2 for an applied sinusoidal force and c the je holds for an applied ramp force independently of the model parameters with a work distribution function which is symmetric gaussian and asymmetric non gaussian for additive and multiplicative noises respectively it has been shown that the non markovian langevin equation in the local and overdamped limits is quite different from the widely adopted phenomenological markovian langevin equation subjected to additive and multiplicative noises

thermodynamic properties of fully connected q ising neural networks the thermodynamic and retrieval properties of fully connected q ising networks are studied in the replica symmetric mean field approximation in particular capacity gain parameter and capacity temperature phase diagrams are derived for q 3 4 and q infinity and different distributions of the stored patterns furthermore the optimal gain function is determined in order to obtain the best performance where appropriate the results are compared with the diluted and layered versions of these models

cccp algorithms to minimize the bethe and kikuchi free energies convergent alternatives to belief propagation this article introduces a class of discrete iterative algorithms that are provably convergent alternatives to belief propagation bp and generalized belief propagation gbp our work builds on recent results by yedidia freeman and weiss 2000 who showed that the fixed points of bp and gbp algorithms correspond to extrema of the bethe and kikuchi free energies respectively we obtain two algorithms by applying cccp to the bethe and kikuchi free energies respectively cccp is a procedure introduced here for obtaining discrete iterative algorithms by decomposing a cost function into a concave and a convex part we implement our cccp algorithms on two and three dimensional spin glasses and compare their results to bp and gbp our simulations show that the cccp algorithms are stable and converge very quickly the speed of cccp is similar to that of bp and gbp unlike cccp bp will often not converge for these problems gbp usually but not always converges the results found by cccp applied to the bethe or kikuchi free energies are equivalent or slightly better than those found by bp or gbp respectively when bp and gbp converge note that for these and other problems bp and gbp give very accurate results see yedidia et al 2000 and failure to converge is their major error mode finally we point out that our algorithms have a large range of inference and learning applications

the mechanism of stochastic resonance it is shown that a dynamical system subject to both periodic forcing and random perturbation may show a resonance peak in the power spectrum which is absent when either the forcing or the perturbation is absent

invariant recognition in potts glass neural networks the authors show that a potts glass neural network can be used for invariant recognition each neuron is modelled as a q state potts variable and p potts configurations are stored in the network the learning rule is a generalized hebb rule so that all pq permutations of the patterns are stabilized simultaneously they analyse the model for an extensive number of patterns and present results of numerical simulations to confirm the analytical results and to illustrate the relaxation to equilibrium the model is applied to the recognition of isomorphic graphs finally they show how to generalize the model such that its stable states are patterns were the permutational symmetry is broken in a hierarchical manner

numerical integration of stochastic differential equations several numerical methods for treating stochastic differential equations are considered both the convergence in the mean square limit and the convergence of the moments is discussed and the generation of appropriate random numbers is treated the necessity of simulations at various time steps with an extrapolation to time step zero is emphasized and demonstrated by a simple example

two state representations of three state neural networks the authors investigate to what extent the dynamical behaviour of nets consisting of three state neurons can be realised in nets using two state neurons they find that the generalisation from a two state neuron to a three state neuron cannot introduce any new behaviour in the deterministic case but can do so for noisy nets even in the zero noise limit for those noisy three state nets which have a hamiltonian description and a two state representation they investigate the relationship between the three state hamiltonian and the two state hamiltonian and find an interesting level splitting phenomenon

on the overlap dynamics of multi state neural networks with a finite number of patterns neural networks with multi state neurons are studied in the case of low loading for symmetric couplings satisfying a certain positivity condition a lyapunov function is shown to exist in the space of overlaps between the instantaneous microscopic state of the system and the learned patterns furthermore an algorithm is derived for zero temperature to determine all the fixed points as an illustration the three state model is worked out explicitly for hebbian couplings for finite temperature the time evolution of the overlap is studied for couplings which need not be symmetric the stability properties are discussed in detail for the three state model for asymmetric couplings limit cycle behaviour is shown to be possible

stationary and dynamical properties of finite n unit langevin models subjected to multiplicative noises abstract we have studied the finite n unit langevin model subjected to multiplicative noises by using the augmented moment method amm as a continuation of our previous paper h hasegawa j phys soc japan 75 2006 033001 effects of couplings on stationary and dynamical properties of the model have been investigated the difference and similarity between the results of diffusive and sigmoid couplings are studied in details time dependences of average and fluctuations in local and global variables calculated by the amm are in good agreement with those of direct simulations dss we also discuss stationary distributions of local and global variables with the use of the fokker planck equation fpe method and dss it is demonstrated that stationary distributions show much variety when multiplicative noise and external inputs are taken into account

learning grey toned patterns in neural networks the problem of learning multi state patterns in neural networks is investigated an analysis of the space of couplings gardner approach yields the distribution of local fields the critical storage capacity alpha c and the minimum number of errors for an overloaded network for noisy local fields the classification error is minimized if the local fields of the patterns are allowed to lie in intervals of finite width a fast converging adaptive learning algorithm is presented which finds the coupling matrix of optimal stability

stochastic resonance in bistable systems with nonlinear dissipation and multiplicative noise a microscopic approach abstract we have studied the stochastic resonance sr of bistable systems coupled to a bath with a nonlinear system bath interaction by using the microscopic generalized caldeira leggett cl model the adopted cl model yields the non markovian langevin equation with nonlinear dissipation and state dependent multiplicative diffusion which preserve the fluctuation dissipation relation fdr results of our simulations are given as follows 1 the spectral power amplification spa exhibits sr not only for a and b but also for while the stationary probability distribution function is independent of them where a and b denote magnitudes of multiplicative and additive noises respectively and expresses the relaxation time of ornstein uhlenbeck ou colored noise 2 the spa for coexisting additive and multiplicative noises has a single peak but two peak structure as functions of a b and or results 1 and 2 are qualitatively different from previous ones obtained by phenomenological langevin models where the fdr is not held or indefinite these show an importance of the fdr in a study on sr of open bistable systems

a lower bound for the memory capacity in the potts hopfield model we consider potts hopfield networks of sizen we prove the result c 0 such that for all 0 0 in such a way that whenn we can store n patterns all of them being sorrounded by energy barriers at distance

replica symmetric theory of nonlinear analogue neural networks on the basis of the theory of the naive mean field model of spin glasses analogue neural networks of the hopfield type are investigated in the saturation limit the saddle point equations for the order parameters describing retrieval and spin glass phases of the networks are obtained by means of a statistical mechanical analysis within the framework of the replica symmetric theory and are shown to undergo a modification to that of ags theory due to the absence of the onsager reaction term in the tap equation based on the equations the memory storage capacity of the networks is analysed as a function of the analogue gain beta a small increase in the critical storage capacity is found for finite values of beta compared with that of the ising model networks with corresponding inverse temperature although the qualitative nature of the phase diagram is unchanged

the mean field theory of a q state neural network model the mean field equations of a q state clock neural network model are derived in the replica symmetric approximation using the replica method these equations are studied for the cases q 2 3 4 and q to infinity it is shown that an infinite number of patterns can be stored in the network even in the limit q to infinity the phase diagram and storage capacity of the network are calculated and the information content considered although the overlap achieved with a nominated pattern is shown to decrease as q increases the information stored in the network is shown to increase

storing an extensive number of grey toned patterns in a neural network using multistate neurons grey toned patterns are pictures composed of pixels of several shades of grey the ability of neural networks using multistate neurons to store such patterns is systematically investigated if conventional generalizations of hopfield networks using analogue or soft neurons are considered it is impossible to stabilize these grey tones nevertheless it is shown that it can be done with networks that use neurons which have only a discrete set of possible activities this is demonstrated for the pseudo inverse rule for the synaptic couplings where only the stability of the patterns shrinks with increasing number q of grey tones one wants to store if the patterns are uncorrelated one can use the hebb rule and in this case the mean field theory is presented applying this rule the storage capacity decreases as q 2 with the number of grey tones

on the parallel dynamics of the q state potts and q ising neural networks using a probabilistic approach the parallel dynamics of theq state potts andq ising neural networks are studied at zero and at nonzero temperatures evolution equations are derived for the first time step and arbitraryq these formulas constitute recursion relations for the exact parallel dynamics of the extremely diluted asymmetric versions of these networks an explicit analysis including dynamical capacity temperature diagrams and the temperature dependence of the overlap is carried out forq 3 both types of models are compared

neural network with hierarchical clustering near saturation the authors perform a detailed investigation of the storage properties of a model for neural networks that exhibits the same organization into clusters as dyson s hierarchical model 1969 for ferromagnetism combined with hebb s learning algorithm for an extensive number of stored patterns p alpha n where n is the size of the network the authors first perform a signal to noise analysis obtaining a succession of critical storage capacities for the ancestor and its descendants that are below the hopfield value afterwards they apply the statistical mechanics formulation of amit gutfreund and sompolinsky 1987 to obtain also in this case a succession of critical storage capacities that are below the corresponding value for hopfield s model in both cases they consider the ratio of the critical storage capacity for the ancestor to the same quantity as evaluated in hopfield s model and they prove rigorously that the signal to noise method provides a lower bound for this ratio that is bounded from above by unity

information processing in three state neural networks we introduce networks of three state neurons where the additional state embodies the absence of information their dynamical behavior is studied from the standpoint of information processing these networks display strong pattern completion capabilities moreover inference naturally occurs between coherent patterns

a moment approach to non gaussian colored noise abstract the langevin system subjected to non gaussian colored noise has been discussed by using the second order moment approach with two kinds of models for generating the noise we have derived the effective differential equation de for a variable x from which the stationary probability distribution p x has been calculated with the use of the fokker planck equation the result of p x calculated by the moment method is compared to several expressions obtained by different methods such as the universal colored noise approximation ucna jung and hanggi phys rev a 35 1987 4464 and the functional integral method it has been shown that our p x is in good agreement with that of direct simulations dss we have also discussed dynamical properties of the model with an external input solving des in the moment method

global nonexistence for the cauchy problem of some nonlinear reaction diffusion systems we first consider the parabolic equationut 2um a x uq f t x u p w t x t 0 x rn where 2 is the 2 fractional power of the laplacian which for 0 2 stands for impurities and f t x and w t x are given nonnegative functions and find its critical exponent then we consider the criticality for five systems of strongly coupled parabolic equations two of them with nonlinear convective terms our results answer positively some open problems raised recently by bandle deng levine and zhang

diffusion terms in systems of reaction diffusion equations can lead to blow up we present a system of reaction diffusion equations posed in r in which the diffusion terms are responsible for the finite time blow up of its solutions

the concave convex procedure the concave convex procedure cccp is a way to construct discrete time iterative dynamical systems that are guaranteed to decrease global optimization and energy functions monotonically this procedure can be applied to almost any optimization problem and many existing algorithms can be interpreted in terms of it in particular we prove that all expectation maximization algorithms and classes of legendre minimization and variational bounding algorithms can be reexpressed in terms of cccp we show that many existing neural network and mean field theory algorithms are also examples of cccp the generalized iterative scaling algorithm and sinkhorn s algorithm can also be expressed as cccp by changing variables cccp can be used both as a new way to understand and prove the convergence of existing optimization algorithms and as a procedure for generating new algorithms

adiabatic elimination for systems of brownian particles with nonconstant damping coefficients we discuss the problem of eliminating the momentum variable in the phase space langevin equations for a system of brownian particles in two related situations i position dependent damping and ii existence of hydrodynamic interactions we discuss the problems associated with the conventional elimination and we develop an alternative elimination procedure in the lagevin framework which leads to the correct smoluchowski equation we give a heuristic argument on the basis of stochastic differential equations for the smoluchowski limit and establish rigorously the limit for the general case of position dependent friction and diffusion coefficents

a lagrange multiplier and hopfield type barrier function method for the traveling salesman problem a lagrange multiplier and hopfield type barrier function method is proposed for approximating a solution of the traveling salesman problem the method is derived from applications of lagrange multipliers and a hopfield type barrier function and attempts to produce a solution of high quality by generating a minimum point of a barrier problem for a sequence of descending values of the barrier parameter for any given value of the barrier parameter the method searches for a minimum point of the barrier problem in a feasible descent direction which has a desired property that lower and upper bounds on variables are always satisfied automatically if the step length is a number between zero and one at each iteration the feasible descent direction is found by updating lagrange multipliers with a globally convergent iterative procedure for any given value of the barrier parameter the method converges to a stationary point of the barrier problem without any condition on the objective function theoretical and numerical results show that the method seems more effective and efficient than the softassign algorithm

stochastic resonance in a bistable system subject to multiplicative and additive noise the stochastic resonance sr phenomenon in a bistable system under the simultaneous action of multiplicative and additive noise and periodic signal is studied by using the theory of signal to noise ratio snr in the adiabatic limit two cases have been considered the case of no correlations between multiplicative and additive noise and the case of correlations between two noises the expressions of the snr for both cases are obtained the effects of intensity of multiplicative and additive noise and the intensity of the correlations between noises on the snr are discussed for both cases respectively it is found that the existence of a maximum in the snr is the identifying characteristic of the sr phenomenon in the case of no correlations between multiplicative and additive noise the snr is independent of the initial condition of the system however the snr is not only dependent on the intensity of correlations between noises but also on the initial condition of the system in the presence of correlations between two noises

stability and synchronization of discrete time markovian jumping neural networks with mixed mode dependent time delays in this paper we introduce a new class of discrete time neural networks dnns with markovian jumping parameters as well as mode dependent mixed time delays both discrete and distributed time delays specifically the parameters of the dnns are subject to the switching from one to another at different times according to a markov chain and the mixed time delays consist of both discrete and distributed delays that are dependent on the markovian jumping mode we first deal with the stability analysis problem of the addressed neural networks a special inequality is developed to account for the mixed time delays in the discrete time setting and a novel lyapunov krasovskii functional is put forward to reflect the mode dependent time delays sufficient conditions are established in terms of linear matrix inequalities lmis that guarantee the stochastic stability we then turn to the synchronization problem among an array of identical coupled markovian jumping neural networks with mixed mode dependent time delays by utilizing the lyapunov stability theory and the kronecker product it is shown that the addressed synchronization problem is solvable if several lmis are feasible hence different from the commonly used matrix norm theories such as the m matrix method a unified lmi approach is developed to solve the stability analysis and synchronization problems of the class of neural networks under investigation where the lmis can be easily solved by using the available matlab lmi toolbox two numerical examples are presented to illustrate the usefulness and effectiveness of the main results obtained

neural networks that use three state neurons the dynamics of a neural network which uses three state neurons 1 0 and 1 is solved exactly in the limit of non symmetric and highly dilute synapses recursion relations for the activity the fraction of non zero neurons and overlap of the network with a given pattern are derived which have three generic kinds of fixed points a retrieval fixed point a chaotic fixed point which corresponds to non zero activity but no overlap and a zero fixed point where all the neurons go to the 0 state as the non retrieval fixed points both have activities different from the retrieval fixed point one can easily tell whether a pattern has been recovered an analysis of which fixed points occur as a function of the thresholds and the storage ratio of the system yields remarkably rich phase diagrams optimising the threshold level can be very important especially when low activity patterns are stored a similar analysis can be applied to biased networks using two state 1 0 neurons finally one finds that mixture states which have an overlap with two patterns can be stabilised by a threshold in the networks using three state neurons this property allows larger higher activity memories to be naturally constructed out of smaller ones

optimization via intermittency with a self organizing neural network one of the major obstacles in using neural networks to solve combinatorial optimization problems is the convergence toward one of the many local minima instead of the global minima in this letter we propose a technique that enables a self organizing neural network to escape from local minima by virtue of the intermittency phenomenon it gives rise to novel search dynamics that allow the system to visit multiple global minima as meta stable states numerical experiments performed suggest that the phenomenon is a combined effect of kohonen type competitive learning and the iterated softmax function operating near bifurcation the resultant intermittent search exhibits fractal characteristics when the optimization performance is at its peak in the form of 1 f signals in the time evolution of the cost as well as power law distributions in the meta stable solution states then queens problem is used as an example to illustrate the meta stable convergence process that sequentially generates in a single run 92 solutions to the 8 queens problem and 4024 solutions to the 17 queens problem

on systems of reaction diffusion equations with a balance law the sequel abstract this paper is a sequel of the review paper on reaction diffusion systems with a balance law by martin and pierre 1990 here we present the results obtained after martin and pierre 1990 not only for classical systems but also for systems with either time fractional derivatives or space fractional derivatives accounting for anomalous diffusions

numerical treatment of stochastic differential equations we define general runge kutta approximations for the solution of stochastic differential equations sde these approximations are proved to converge in quadratic mean to the solution of an sde with a corrected drift the explicit form of the correction term is given concerning the order of convergence we show that in general it is impossible for the quadratic mean of the one step error to be of an order greater than o h 3 this order is attained e g by the stochastic analogue of heun s method in the n dimensional case the highest order of convergence is in general only o h 2 attained by euler s method the order o h 3 can only be reached if nabla x sigma r sigma s nabla x sigma s sigma r for the diffusion matrix

finite state neural networks a step toward the simulation of very large systems neural networks composed of neurons withq n states and synapses withq states are studied analytically and numerically analytically it is shown that these finite state networks are much more efficient at information storage than networks with continuous synapses in order to take the utmost advantage of networks with finite state elements a multineuron and multisynapse coding scheme is introduced which allows the simulation of networks having 1 0 109 couplings at a speed of 7 1 109 coupling evaluations per second on asingle processor of the cray ymp a local learning algorithm is also introduced which allows for the efficient training of large networks with finite state elements

convex potentials and their conjugates in analog mean field optimization this paper deals with the problem of mapping hybrid i e both discrete and continuous constrained optimization problems onto analog networks the saddle point paradigm of mean field methods in statistical physics provides a systematic procedure for finding such a mapping via the notion of effective energy specifically it is shown that within this paradigm to each closed bounded constraint set is associated a smooth convex potential function using the conjugate or the legendre fenchel transform of the convex potential the effective energy can be transformed to yield a cost function that is a natural generalization of the analog hopfield energy descent dynamics and deterministic annealing can then be used to find the global minimum of the original minimization problem when the conjugate is hard to compute explicitly it is shown that a minimax dynamics similar to that of arrow and hurwicz in lagrangian optimization can be used to find the saddle points of the effective energy as an illustration of its wide applicability the effective energy framework is used to derive hopfield like energy functions and descent dynamics for two classes of networks previously considered in the literature winner take all networks and rotor networks even when the cost function of the original optimization problem is not quadratic

on the dynamics of analogue neurons with nonsigmoidal gain functions abstract the characteristic properties of the macroscopic retrieval dynamics of analogue neurons with hebbian coupling strengths are studied usign the shape of the gain function as a modeling parameter already at low loading a rich diversity in dynamical behaviour is observed covering the full range from point attractors to chaotic dynamics the attractors which are not of the fixed point type are interpreted as an intermediate phase between the well known retrieval states and the zero state enabling a waiting mode in the system using a probabilistic approach it is shown that these features persist in extremely and asymmetrically diluted systems at sufficiently low loading a number of generic examples are worked out in detail illustrating some properties of networks governed by nonmonotonic piecewise linear gain functions the critical storage level above which the chaotic behaviour is absent is numerically determined

on the phase diagram of the q state potts glass neural network the q state potts model of neural networks extended to include biased patterns is studied for extensive loading a mean field equations are written down for general q and arbitrary temperature t within the replica symmetric approximation the complete t phase diagram is discussed especially for q 3 a study of the bifurcation diagrams for the overlap and the free energy clarifies the results obtained

a novel optimizing network architecture with applications we present a novel optimizing network architecture with applications in vision learning pattern recognition and combinatorial optimization this architecture is constructed by combining the following techniques 1 deterministic annealing 2 self amplification 3 algebraic transformations 4 clocked objectives and 5 softassign deterministic annealing in conjunction with self amplification avoids poor local minima and ensures that a vertex of the hypercube is reached algebraic transformations and clocked objectives help partition the relaxation into distinct phases the problems considered have doubly stochastic matrix constraints or minor variations thereof we introduce a new technique softassign which is used to satisfy this constraint experimental results on different problems are presented and discussed

layered neural networks exact solutions for the dynamics of layered feedforward neural networks are presented these networks are expected to respond to an input by going through a sequence of preassigned states on the various layers the family of networks considered has a variety of interlayer couplings linear and non linear hebbian hebbian with gaussian synaptic noise and with various kinds of dilution in addition the authors also solve the problem of layered networks with the pseudoinverse projector matrix of couplings in all cases the solutions take the form of layer to layer recursions for the mean overlap with a random key pattern and for the width of the embedding field distribution the dynamics is governed by the fixed points of these recursions for all cases non trivial domains of attraction of the memory states are found and graphically displayed

stochastic resonance driven by two different kinds of colored noise in a bistable system the phenomenon of stochastic resonance in a bistable nonlinear system is investigated when both the multiplicative noise and the coupling between additive and multiplicative noise are colored with different values of noise correlation time ensuremath tau 1 and ensuremath tau 2 combining the functional analysis and unified colored noise approximation the two different kinds of colored noise in the nonlinear system can be simplified the signal to noise ratio is calculated when a weakly periodic signal is added to the system it is found that there appears a transition between one peak and two peaks in the curve of the signal to noise ratio when either the noise correlation time ensuremath tau 1 and ensuremath tau 2 or the coupling strength ensuremath lambda between additive and multiplicative noise is increased the transition between one and two peaks depending on ensuremath tau 1 and ensuremath lambda is more complex than that depending on ensuremath tau 2

image evolution in potts glass neural networks neural networks of the potts type where neurons can occupy q or 2 discrete states are considered for couplings which need not be symmetric the time evolution of the macroscopic overlap between the instantaneous microscopic state of the system and the learned patterns is studied for small values of q and for a finite small number of patterns retrieval and limit cycle behaviour in this type of model is discussed in some detail

effect of correlation between additive and multiplicative noises on the activation from a double well we study the thermal activation problem of a bistable system driven by correlated additive and multiplicative noises by means of numerical simulation we find that in the colored noise case the suppression effect of the positive correlation decreases and finally is released as the autocorrelation time tau of the colored noise grows when tau is large enough negative correlation becomes more suppressive than the positive correlation such phenomena are ascribed to the collaboration between noises as well as the memory effect of the multiplicative colored noise

simple central pattern generator model using phasic analog neurons many biological neurons called phasic or adapting neurons display neural adaptation their response to a constant input diminishes with time a simple method of adding adaptive firing thresholds to existing analog or graded response neural models is described a half center central pattern generator is modeled using two mutually inhibitory phasic analog neurons hopf bifurcation analysis shows that oscillatory solutions will arise if the mutual inhibition is sufficiently strong and allows us to characterize the stability of the cycles which arise

temporal sequence storage capacity of time summating neural networks the maximum capacity for storage of temporal sequences in a time summating neural network is analysed within a statistical mechanical framework each neuron of such a network maintains an activity trace consisting of a decaying sum of all previous inputs to that neuron the maximum storage capacity is expressed as a function of the temporal correlations of the local fields of the neurons and evaluated using a perturbation expansion in powers of the decay rate

macroscopic lyapunov functions for separable stochastic neural networks with detailed balance we derive macroscopic lyapunov functions for large long range ising spin neural networks with separable symmetric interactions which evolve in time according to local field alignment we generalize existing constructions which correspond todeterministic zero temperature evolution and to specific choices of the interaction structure to the case ofstochastic evolution and arbitrary separable interaction matrices for both parallel and sequential spin updating we find a direct relation between the form of the lyapunov functions which describe dynamical processes and the saddle point integration that results from performing equilibrium statistical mechanical studies of the present type of model

stochastic langevin equations markovian and non markovian dynamics non markovian stochastic langevin like equations of motion are compared to their corresponding markovian local approximations the validity of the local approximation for these equations when contrasted with the fully nonlocal ones is analyzed in detail the conditions for when the equation in a local form can be considered a good approximation are then explicitly specified we study both the cases of additive and multiplicative noises including system dependent dissipation terms according to the fluctuation dissipation theorem

optimising threshold levels for information transmission in binary threshold networks independent multiplicative noise on each threshold abstract the problem of optimising the threshold levels in multilevel threshold system subject to multiplicative gaussian and uniform noise is considered similar to previous results for additive noise we find a bifurcation phenomenon in the optimal threshold values as the noise intensity changes this occurs when the number of threshold units is greater than one we also study the optimal thresholds for combined additive and multiplicative gaussian noise and find that all threshold levels need to be identical to optimise the system when the additive noise intensity is a constant however this identical value is not equal to the signal mean unlike the case of additive noise when the multiplicative noise intensity is instead held constant the optimal threshold levels are not all identical for small additive noise intensity but are all equal to zero for large additive noise intensity the model and our results are potentially relevant for sensor network design and understanding neurobiological sensory neurons such as in the peripheral auditory system

low firing rates an effective hamiltonian for excitatory neurons the authors analyse the behaviour of an attractor neural network which exhibits low mean temporal activity levels despite the fact that the intrinsic neuronal cycle time is very short 2 3 ms information and computation are represented on the excitatory neurons only the influence of inhibitory neurons which are assumed to react on a shorter timescale than the excitatory ones is expressed as an effective interaction of the excitatory neurons this leads to an effective model which describes the interplay of excitation and inhibition acting on excitatory neurons in terms of the excitatory neural variables alone the network operates in the presence of fast noise which is large relative to the frozen randomness induced by the stored patterns the overall fraction of active neurons is controlled by a single free parameter which expresses the relative strength of the effective inhibition associative retrieval is identified as usual with the breakdown of ergodicity in the dynamics of the network in particular with the presence of dynamical attractors corresponding to the retrieval of a given pattern in such an attractor the activity of neurons corresponding to active sites in the stored patterns increases at the expense of other neurons yet only a small fraction of the neurons active in the pattern are in the active state in each elementary time cycle and they vary from cycle to cycle in an uncorrelated fashion due to the noise hence the observed mean activity rate of any individual neuron is kept low this scenario is demonstrated by an analytical study based on the replica method and the results are tested by numerical simulations