dynamics of moments of fitzhugh nagumo neuronal models and stochastic bifurcations for the study of the behavior of noisy neuronal models rodriguez and tuckwell have introduced an elegant and systematic method which consists of replacing the system of stochastic differential equations with a system of deterministic equations representing the dynamics of the means variances and covariance of the state variables r rodriguez and h c tuckwell phys rev e 54 5585 1996 in this work we first report a modification of their method in the case of the fitzhugh nagumo model which enhances the accuracy of the approximation without including higher order moments this method is then combined with a self consistency argument in order to better characterize the behavior of the underlying stochastic processes through the computation of approximate auto and cross correlation functions of the state variables finally we argue that the moments equations can also reveal the existence of stochastic bifurcations i e qualitative changes in the dynamics of stochastic systems

noise in integrate and fire neurons from stochastic input to escape rates we analyze the effect of noise in integrate and fire neurons driven by time dependent input and compare the diffusion approximation for the membrane potential to escape noise it is shown that for time dependent subthreshold input diffusive noise can be replaced by escape noise with a hazard function that has a gaussian dependence on the distance between the noise free membrane voltage and threshold the approximation is improved if we add to the hazard function a probability current proportional to the derivative of the voltage stochastic resonance in response to periodic input occurs in both noise models and exhibits similar characteristics

sandpiles on a sierpinski gasket abstract we perform extensive simulations of the sandpile model on a sierpinski gasket critical exponents for waves and avalanches are determined we extend the existing theory of waves to the present case this leads to an exact value for the exponent w which describes the distribution of wave sizes w ln 9 5 ln 3 numerically it is found that the number of waves in an avalanche is proportional to the number of distinct sites toppled in the avalanche this leads to a conjecture for the exponent that determines the distribution of avalanche sizes 1 w ln 27 5 ln 3 our predictions are in good agreement with the numerical results

dimer monomer model on the sierpinski gasket abstract we present the numbers of dimer monomers m d n on the sierpinski gasket s g d n at stage n with dimension d equal to two three and four the upper and lower bounds for the asymptotic growth constant defined as z s g d lim v ln m d n v where v is the number of vertices on s g d n are derived in terms of the results at a certain stage as the difference between these bounds converges quickly to zero as the calculated stage increases the numerical value of z s g d can be evaluated with more than a hundred significant figures accurate from the results for d 2 3 4 we conjecture the upper and lower bounds of z s g d for general dimension the corresponding results on the generalized sierpinski gasket s g d b n with d 2 and b 3 4 are also obtained

multilevel monte carlo for stochastic differential equations with small noise we consider the problem of numerically estimating expectations of solutions to stochastic differential equations driven by brownian motions in the commonly occurring small noise regime we consider i standard monte carlo methods combined with numerical discretization algorithms tailored to the small noise setting and ii a multilevel monte carlo method combined with a standard euler maruyama implementation under the assumptions we make on the underlying model the multilevel method combined with euler maruyama is often found to be the most efficient option moreover under a wide range of scalings the multilevel method is found to give the same asymptotic complexity that would arise in the idealized case where we have access to exact samples of the required distribution at a cost of o 1 per sample a key step in our analysis is to analyze the variance between two coupled paths directly as opposed to their l 2 distance careful simulations are provided to illustrate the asymptotic results

distribution of sizes of erased loops for loop erased random walks the loop erased random walk lerw is a simpler variant of the well known self avoiding walk saw problem which retains the no self intersection property of saws but is closer to the ordinary random walk problem in this paper we study the distribution of sizes of erased loops for lerws on arbitrary graphs and relate it to the distribution of sizes of loops formed when a bond is added to a random spanning tree on the same graph in the themodynamic limit this distribution has a power law tail and we express the exponent in terms of the fractal dimension of the chemical paths on spanning trees we show also how this relation is modified for deterministic fractals in spite of the fact that the lerw model is somewhat more tractable analytically than the saw problem the number of papers devoted to this problem has remained rather small the model was defined by lawler 1 who called it the loop erased self avoiding walk lesaw this terminology is somewhat inappropriate and

reconstructing the transition rate function of a broadwell random walk from exit times in this paper we utilize the layer stripping method a method used in seismology by geophysicists to study a stochastic inverse problem arising from a one dimensional broadwell process the broadwell process can be described as a random walk of a particle that transitions with a spatially dependent flip rate between two states associated with the same speed but with different signs of velocities our goal is to reconstruct the spatially dependent flip rate function of a one dimensional broadwell process from exit time distributions out of a finite interval in principle we are able to reconstruct flip rate functions with error within o 10 2 from o 10 5 exit times when the particle speed is unity for smaller particle speeds the noise increases for a fixed number of exit times this method is less time consuming compared with traditional projection methods which involve optimization

packing dimers on 2p 1 x 2q 1 lattices we use computational method to investigate the number of ways to pack dimers on m x n lattices where both m and n are odd in this case there is always a single vacancy in the lattices we show that the dimer configuration numbers on 2k 1 x 2k 1 odd square lattices have some remarkable number theoretical properties in parallel to those of close packed dimers on 2k x 2k even square lattices for which an exact solution exists furthermore we demonstrate that there is an unambiguous logarithmic term in the finite size correction of free energy of odd by odd lattice strips with any width n or 1 this logarithmic term determines the distinct behavior of the free energy of odd square lattices these findings reveal a deep and previously unexplored connection between statistical physics models and number theory and indicate the possibility that the monomer dimer problem might be solvable

corrigendum to conical plurisubharmonic measure and new cross theorems j math anal appl 365 2010 429 434

monomer dimer model in two dimensional rectangular lattices with fixed dimer density the classical monomer dimer model in two dimensional lattices has been shown to belong to the p complete class which indicates the problem is computationally intractable we use exact computational method to investigate the number of ways to arrange dimers on mxn two dimensional rectangular lattice strips with fixed dimer density rho for any dimer density 0 rho 1 we find a logarithmic correction term in the finite size correction of the free energy per lattice site the coefficient of the logarithmic correction term is exactly 12 this logarithmic correction term is explained by the newly developed asymptotic theory of pemantle and wilson the sequence of the free energy of lattice strips with cylinder boundary condition converges so fast that very accurate free energy f 2 rho for large lattices can be obtained for example for a half filled lattice f 2 12 0 633195588930 while f 2 14 0 4413453753046 and f 2 34 0 64039026 for rho 0 65 f 2 rho is accurate at least to ten decimal digits the function f 2 rho reaches the maximum value f 2 rho 0 662798972834 at rho 0 6381231 with 11 correct digits this is also the monomer dimer constant for two dimensional rectangular lattices the asymptotic expressions of free energy near close packing are investigated for finite and infinite lattice widths for lattices with finite width dependence on the parity of the lattice width is found for infinite lattices the data support the functional form obtained previously through series expansions

spanning trees on the sierpinski gasket we present the numbers of spanning trees on the sierpinski gasket sg d n at stage n with dimension d equal to two three and four the general expression for the number of spanning trees on sg d n with arbitrary d is conjectured the numbers of spanning trees on the generalized sierpinski gasket sg d b n with d 2 and b 3 4 are also obtained

conical plurisubharmonic measure and new cross theorems we study the boundary behavior of the conical plurisubharmonic measure as an application we establish some new extension theorems for separately holomorphic mappings defined on cross sets

the exponent for the mean square displacement of self avoiding random walk on the sierpinski gasket the authors rigorously prove that the exponent for the mean square displacement of self avoiding random walk on the sierpinski gasket is log 2 log left frac 7 sqrt 5 2 right 0 79862 0 5 log 2 log 5

self avoiding paths on the pre sierpinski gasket we study a statistical mechanics of self avoiding paths on the pre sierpinski gasket we first show the existence of the thermodynamic limit of the appropriately scaled free energy then we show that there are two domains in the weight parameters i e two phases between which the scaling differs i e there is a certain kind of phase transition in our model and we find the critical exponents of the free energy at the phase transition point we also show the convergence of the distribution of the scaled length of the paths at thermodynamic limit

determination of firing times for the stochastic fitzhugh nagumo neuronal model we present for the first time an analytical approach for determining the time of firing of multicomponent nonlinear stochastic neuronal models we apply the theory of first exit times for markov processes to the fitzhugh nagumo system with a constant mean gaussian white noise input representing stochastic excitation and inhibition partial differential equations are obtained for the moments of the time to first spike the observation that the recovery variable barely changes in the prespike trajectory leads to an accurate one dimensional approximation for the moments of the time to reach threshold this leads to ordinary differential equations that may be easily solved several analytical approaches are explored that involve perturbation expansions for large and small values of the noise parameter for ranges of the parameters appropriate for these asymptotic methods the perturbation solutions are used to establish the validity of the one dimensional approximation for both small and large values of the noise parameter additional verification is obtained with the excellent agreement between the mean and variance of the firing time found by numerical solution of the differential equations for the one dimensional approximation and those obtained by simulation of the solutions of the model stochastic differential equations such agreement extends to intermediate values of the noise parameter for the mean time to threshold we find maxima at small noise values that constitute a form of stochastic resonance we also investigate the dependence of the mean firing time on the initial values of the voltage and recovery variables when the input current has zero mean

exact finite size corrections of the free energy for the square lattice dimer model under different boundary conditions we express the partition functions of the dimer model on finite square lattices under five different boundary conditions free cylindrical toroidal mobius strip and klein bottle obtained by others kasteleyn temperley and fisher mccoy and wu brankov and priezzhev and lu and wu in terms of the partition functions with twisted boundary conditions z alpha beta with alpha beta 1 2 0 0 1 2 and 1 2 1 2 based on such expressions we then extend the algorithm of ivashkevich izmailian and hu j phys a 35 5543 2002 to derive the exact asymptotic expansion of the logarithm of the partition function for all boundary conditions mentioned above we find that the aspect ratio dependence of finite size corrections is sensitive to boundary conditions and the parity of the number of lattice sites along the lattice axis we have also established several groups of identities relating dimer partition functions for the different boundary conditions

dimers on a simple quartic net with a vacancy a seminal milestone in lattice statistics is the exact solution of the enumeration of dimers on a simple quartic net obtained by fisher kasteleyn and temperley fkt in 1961 an outstanding related and yet unsolved problem is the enumeration of dimers on a net with vacant sites here we consider this vacant site problem with a single vacancy occurring at certain specific sites on the boundary of a simple quartic net first using a bijection between dimer and spanning tree configurations due to temperley kenyon propp and wilson we establish that the dimer generating function is independent of the location of the vacancy and deduce a closed form expression for the generating function we next carry out finite size analyses of this solution as well as that of the fkt solution our analyses lead to a logarithmic correction term in the large size expansion for the vacancy problem with free boundary conditions a concrete example exhibiting this difference is given we also find the central charge c 2 in the language of conformal field theory for the vacancy problem as versus the value c 1 when there is no vacancy

two dimensional monomer dimer systems are computationally intractable the classic problem of counting monomer dimer arrangements on a two dimensional lattice is analyzed using techniques from theoretical computer science under a certain assumption made precise in the text it can be shown that the general problem is computationally intractable this negative result contrasts with the special case of a system with monomer density zero for which efficient solutions have been known for some time a second much easier result obtained under the same assumption is that the partition function of a three dimensional ising system is computationally intractable again the negative result contrasts with known efficient techniques for evaluating the partition function of a two dimensional system

reconstruction of potential energy profiles from multiple rupture time distributions we explore the mathematical and numerical aspects of reconstructing a potential energy profile of a molecular bond from its rupture time distribution while reliable reconstruction of gross attributes such as the height and the width of an energy barrier can be easily extracted from a single first passage time fpt distribution the reconstruction of finer structure is ill conditioned more careful analysis shows the existence of optimal bond potential amplitudes represented by an effective peclet number and initial bond configurations that yield the most efficient numerical reconstruction of simple potentials furthermore we show that reconstruction of more complex potentials containing multiple minima can be achieved by simultaneously using two or more measured fpt distributions obtained under different physical conditions for example by changing the effective potential energy surface by known amounts additional measured fpt distributions improve the reconstruction we demonstrate the possibility of reconstructing potentials with multiple minima motivate heuristic rules ofthumb for optimizing the reconstruction and discuss further applications and extensions

phase transitions on fractals i quasi linear lattices magnetic spin models and resistor networks are studied on certain self similar fractal lattices which are described as quasi linear because they share a significant property of the line finite portions can be isolated from the rest by removal of two points sites in all cases there is no long range order at finite temperature the transition at zero temperature has a discontinuity in the magnetisation and the associated magnetic exponent is equal to the fractal dimensionality d when the lattice reduces to a non branching curve the thermal exponent v 1 y is equal to d when the lattice is a branching curve y is related respectively to the dimensionality of the single channel segments of the curve for the ising model or to the exponent describing the resistivity for models with continuous spin symmetry

time to first spike in stochastic hodgkin huxley systems abstract the time to first spike is an experimentally observed quantity in laboratory experiments in the auditory somatic and visual sensory modalities the times of first spikes in the corresponding cortical neurons have been implicated as coding much of the information about stimulus properties we describe an analytical approach for determining the time to first spike from a given initial state which may be applied to a general nonlinear stochastic model neuron we illustrate with a standard hodgkin huxley model with a gaussian white noise input current whose drift parameter is and whose variance parameter is partial differential equations pdes of second order are obtained for the first two moments of the time taken for the depolarization to reach a threshold value from rest state as functions of the initial values simulation confirms that for small noise amplitudes a 2 component model is reasonably accurate for small values of the noise parameter including the deterministic case 0 perturbation methods are used to find the moments of the firing time and the results compare favorably with those from simulation the approach is accurate for almost all when is above threshold for action potentials in the absence of noise and over a considerable range of values of when is as small as 2 the same methods may be applied to models similar to hodgkin huxley which involve channels for additional or different ionic currents