on the continuous differentiability of inter spike intervals of synaptically connected cortical spiking neurons in a neuronal network we derive conditions for continuous differentiability of inter spike intervals isis of spiking neurons with respect to parameters decision variables of an external stimulating input current that drives a recurrent network of synaptically connected neurons the dynamical behavior of individual neurons is represented by a class of discontinuous single neuron models we report here that isis of neurons in the network are continuously differentiable with respect to decision variables if 1 a continuously differentiable trajectory of the membrane potential exists between consecutive action potentials with respect to time and decision variables and 2 the partial derivative of the membrane potential of spiking neurons with respect to time is not equal to the partial derivative of their firing threshold with respect to time at the time of action potentials our theoretical results are supported by showing fulfillment of these conditions for a class of known bidimensional spiking neuron models

the novikov and entropy conditions of multidimensional diffusion processes with singular drift we consider multidimensional stochastic differential equations of the form dx t b t x t dt dw t 0 t r r 1 with arbitrary initial probability distribution on d d 1 the first aim of this paper is to give handy to verify analytic i e nonstochastic conditions for the existence of a weak solution of 1 where the drift b will be allowed to have singularities these investigations are illustrated by various examples we first concentrate on a uniform form of the novikov condition e exp1 2 0 r b s x s 2 ds 2 and then investigate further sufficient conditions for the applicability of the girsanov maruyama theorem which are not covered by 2 the outcoming results improve some of those of h j engelbert and w schmidt math nachr 119 97 115 for time independent drifts b x and n i portenko theory probab appl 20 27 37 1975 translation from teor veroyatn primen 20 29 39 for time dependent drifts b t x one of the examples involves a drift which is singular on a dense set in d but nevertheless satisfies 2 the second aim of this paper is to discuss some general properties and applications of 2 for instance we investigate whether the factor 1 2 in the novikov condition 2 can be replaced by 1 2 e e 0 furthermore we give several equivalence characterizations of 2 being connected to the well known r z khas minskij lemma ibid 4 309 318 1960 resp ibid 4 332 341 finally it is shown that under the novikov condition 2 the diffusion process with drift b has finite relative entropy with respect to wiener measure and thus finite energy

strong uniqueness for sdes in hilbert spaces with nonregular drift we prove pathwise uniqueness for a class of stochastic dierential equations sde on a hilbert space with cylindrical wiener noise whose non linear drift parts are sums of the subdierential of a convex function and a bounded part this generalizes a classical result by one of the authors to innite dimensions our results also generalize

toward the fourier law for a weakly interacting anharmonic crystal for a system of weakly interacting anharmonic oscillators perturbed by an energy preserving stochastic dynamics we prove an autonomous stochastic evolution for the energies at large time scale with respect to the coupling parameter it turn out that this macroscopic evolution is given by the so called conservative non gradient ginzburg landau system of stochastic differential equations the proof exploits hypocoercivity and hypoellipticity properties of the uncoupled dynamics

nonequilibrium statistical mechanics of hamiltonian rotators with alternated spins we consider a finite region of a d dimensional lattice of nonlinear hamiltonian rotators where neighbouring rotators have opposite alternated spins and are coupled by a small potential of size varepsilon a a ge 1 2 we weakly stochastically perturb the system in such a way that each rotator interacts with its own stochastic thermostat with a force of order varepsilon then we introduce action angle variables for the system of uncoupled rotators varepsilon 0 and note that the sum of actions over all nodes is conserved by the purely hamiltonian dynamics of the system with varepsilon 0 we investigate the limiting as varepsilon rightarrow 0 dynamics of actions for solutions of the varepsilon perturbed system on time intervals of order varepsilon 1 it turns out that the limiting dynamics is governed by a certain autonomous stochastic equation for the vector of actions this equation has a completely non hamiltonian nature this is a consequence of the fact that the system of rotators with alternated spins do not have resonances of the first order the varepsilon perturbed system has a unique stationary measure widetilde mu varepsilon and is mixing any limiting point of the family widetilde mu varepsilon of stationary measures as varepsilon rightarrow 0 is an invariant measure of the system of uncoupled integrable rotators there are plenty of such measures however it turns out that only one of them describes the limiting dynamics of the varepsilon perturbed system we prove that a limiting point of widetilde mu varepsilon is unique its projection to the space of actions is the unique stationary measure of the autonomous equation above which turns out to be mixing and its projection to the space of angles is the normalized lebesque measure on the torus mathbb t n the results and convergences which concern the behaviour of actions on long time intervals are uniform in the number n of rotators those concerning the stationary measures are uniform in n in some natural cases

critical strong feller regularity for markov solutions to the navier stokes equations abstract the main purpose of this paper is to show that markov solutions to the 3d navier stokes equations driven by gaussian noise have the strong feller property up to the critical topology given by the domain of the stokes operator to the power one fourth

onsager s conjecture on the energy conservation for solutions of euler s equation we give a simple proof of a result conjectured by onsager 1 on energy conservation for weak solutions of euler s equation

green kubo formula for weakly coupled systems with noise we study the green kubo formula kappa varepsilon varsigma for the heat conductivity of an infinite chain of d dimensional finite systems cells coupled by a smooth nearest neighbor potential varepsilon v the uncoupled systems evolve according to hamiltonian dynamics perturbed stochastically by an energy conserving noise of strength varsigma noting that kappa varepsilon varsigma exists and is finite whenever varsigma 0 we are interested in what happens when the strength of the noise varsigma to 0 for this we start in this work by formally expanding kappa varepsilon varsigma in a power series in varepsilon kappa varepsilon varsigma varepsilon 2 sum n geq 2 varepsilon n 2 kappa n varsigma and investigating the formal equations satisfied by kappa n varsigma we show in particular that kappa 2 varsigma is well defined when no pinning potential is present and coincides formally with the heat conductivity obtained in the weak coupling van hove limit where time is rescaled as varepsilon 2 t for the cases where the latter has been established liverani and olla in jams 25 555 583 2012 dolgopyat and liverani in commun math phys 308 201 225 2011 for one dimensional systems we investigate kappa 2 varsigma as varsigma to 0 in three cases the disordered harmonic chain the rotor chain and a chain of strongly an harmonic oscillators moreover we formally identify kappa 2 varsigma with the conductivity obtained by having the chain between two reservoirs at temperature t and t delta t in the limit delta t to 0 n to infty varepsilon to 0

bifurcation analysis of a general class of nonlinear integrate and fire neurons in this paper we define a class of formal neuron models being computationally efficient and biologically plausible i e able to reproduce a wide range of behaviors observed in in vivo or in vitro recordings of cortical neurons this class includes for instance two models widely used in computational neuroscience the izhikevich and the brette gerstner models these models consist of a 4 parameter dynamical system we provide the full local bifurcation diagram of the members of this class and show that they all present the same bifurcations an andronov hopf bifurcation manifold a saddle node bifurcation manifold a bogdanov takens bifurcation and possibly a bautin bifurcation i e all codimension two local bifurcations in a two dimensional phase space except the cusp among other global bifurcations this system shows a saddle homoclinic bifurcation curve we show how this bifurcation diagram generates the most prominent cortical neuron behaviors this study leads us to introduce a new neuron mode

nonsmooth dynamics in spiking neuron models large scale studies of spiking neural networks are a key part of modern approaches to understanding the dynamics of biological neural tissue one approach in computational neuroscience has been to consider the detailed electrophysiological properties of neurons and build vast computational compartmental models an alternative has been to develop minimal models of spiking neurons with a reduction in the dimensionality of both parameter and variable space that facilitates more effective simulation studies in this latter case the single neuron model of choice is often a variant of the classic integrate and fire model which is described by a nonsmooth dynamical system in this paper we review some of the more popular spiking models of this class and describe the types of spiking pattern that they can generate ranging from tonic to burst firing we show that a number of techniques originally developed for the study of impact oscillators are directly relevant to their analysis particularly those for treating grazing bifurcations importantly we highlight one particular single neuron model capable of generating realistic spike trains that is both computationally cheap and analytically tractable this is a planar nonlinear integrate and fire model with a piecewise linear vector field and a state dependent reset upon spiking we call this the pwl if model and analyse it at both the single neuron and network level the techniques and terminology of nonsmooth dynamical systems are used to flesh out the bifurcation structure of the single neuron model as well as to develop the notion of liapunov exponents we also show how to construct the phase response curve for this system emphasising that techniques in mathematical neuroscience may also translate back to the field of nonsmooth dynamical systems the stability of periodic spiking orbits is assessed using a linear stability analysis of spiking times at the network level we consider linear coupling between voltage variables as would occur in neurobiological networks with gap junction coupling and show how to analyse the properties existence and stability of both the asynchronous and synchronous states in the former case we use a phase density technique that is valid for any large system of globally coupled limit cycle oscillators whilst in the latter we develop a novel technique that can handle the nonsmooth reset of the model upon spiking finally we discuss other aspects of neuroscience modelling that may benefit from further translation of ideas from the growing body of knowledge on nonsmooth dynamics

total variation flow and sign fast diffusion in one dimension abstract we consider the dynamics of the total variation flow tvf u t div d u d u and of the sign fast diffusion equation sfde u t sign u in one spatial dimension we find the explicit dynamic and sharp asymptotic behaviour for the tvf and we deduce the one for the sfde by an explicit correspondence between the two equations

existence of densities for the 3d navier stokes equations driven by gaussian noise we prove three results on the existence of densities for the laws of finite dimensional functionals of the solutions of the stochastic navier stokes equations in dimension 3 in particular under very mild assumptions on the noise we prove that finite dimensional projections of the solutions have densities with respect to the lebesgue measure which have some smoothness when measured in a besov space this is proved thanks to a new argument inspired by an idea introduced in fournier and printems 2010

gradient estimates and applications for sdes in hilbert space with multiplicative noise and dini continuous drift abstract consider the stochastic evolution equation in a separable hilbert space h with a nice multiplicative noise and a locally dini continuous drift we prove that for any initial data the equation has a unique possibly explosive mild solution under a reasonable condition ensuring the non explosion of the solution the strong feller property of the associated markov semigroup is proved gradient estimates and log harnack inequalities are derived for the associated semigroup under certain global conditions which are new even in finite dimensions

unique ergodicity for a class of stochastic hyperbolic equations with additive space time white noise in this paper we consider a certain class of second order nonlinear pdes with damping and space time white noise forcing posed on the d dimensional torus this class includes the wave equation for d 1 and the beam equation for d le 3 we show that the gibbs measure is the unique invariant measure for this system since the flow does not satisfy the strong feller property we introduce a new technique for showing unique ergodicity this approach may be also useful in situations in which finite time blowup is possible

well posedness of semilinear stochastic wave equations with holder continuous coefficients abstract we prove that semilinear stochastic abstract wave equations including wave and plate equations are well posed in the strong sense with an holder continuous drift coefficient if 2 3 1 the uniqueness may fail for the corresponding deterministic pde and well posedness is restored by adding an external random forcing of white noise type this shows a kind of regularization by noise for the semilinear wave equation to prove the result we introduce an approach based on backward stochastic differential equations we also establish regularizing properties of the transition semigroup associated to the stochastic wave equation by using control theoretic results

on the asymptotic behaviour of solutions of a stochastic energy balance climate model abstract we prove the existence of a random global attractor for the multivalued random dynamical system associated to a nonlinear multivalued parabolic equation with a stochastic term of amplitude of the order of e the equation was initially suggested by north and cahalan following a previous deterministic model proposed by m i budyko for the modeling of some non deterministic variability as for instance the cyclones which can be treated as a fast varying component and are represented as a white noise process in the context of energy balance climate models we also prove the convergence in some sense of the global attractors when e 0 i e the convergence to the global attractor for the associated deterministic case e 0

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

energy transport in stochastically perturbed lattice dynamics we consider lattice dynamics with a small stochastic perturbation of order varepsilon and prove that for a space time scale of order varepsilon 1 the local spectral density wigner function evolves according to a linear transport equation describing inelastic collisions for an energy and momentum conserving chain the transport equation predicts a slow decay as 1 sqrt t for the energy current correlation in equilibrium this is in agreement with previous studies using a different method

pathwise uniqueness for a class of sde in hilbert spaces and applications abstract an abstract evolution equation in hilbert spaces is considered in the deterministic case it includes several examples of non uniqueness it is proved that pathwise uniqueness is restored by means of a suitably non degenerate additive noise the proof is based on the associated infinite dimensional kolmogorov equation

local times for solutions of the complex ginzburg landau equation and the inviscid limit abstract we consider the behaviour of the distribution for stationary solutions of the complex ginzburg landau equation perturbed by a random force it was proved in s kuksin and a shirikyan 2004 4 that if the random force is proportional to the square root of the viscosity 0 then the family of stationary measures possesses an accumulation point as 0 we show that if is such a point then the distributions of the l 2 norm and of the energy possess a density with respect to the lebesgue measure the proofs are based on ito s formula and some properties of local time for semimartingales

well posedness of the transport equation by stochastic perturbation we consider the linear transport equation with a globally holder continuous and bounded vector field with an integrability condition on the divergence while uniqueness may fail for the deterministic pde we prove that a multiplicative stochastic perturbation of brownian type is enough to render the equation well posed this seems to be the first explicit example of a pde of fluid dynamics that becomes well posed under the influence of a multiplicative noise the key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of ito tanaka type

stabilization of linear systems by noise it is proved that the biggest lyapunov number lambda max of the system dot x a f t x where a is a fixed d times d matrix and f t is a zero mean strictly stationary matrix valued stochastic process satisfies 1 d trace a leqq lambda max on the other hand for each varepsilon 0 there is a process f t for which lambda max leqq 1 d trace a varepsilon in particular the system dot x ax can be stabilized by zero mean stationary parameter noise if and only if trace a 0 the stabilization can be accomplished by a one dimensional noise source the results carry over to the case where a is a stationary process they are also true for f t white noise

irreducibility of the 3 d stochastic navier stokes equation abstract a 3 dimensional navier stokes equation with random force is investigated a form of irreducibility of interest in ergodic theory is proved under a full noise assumption the basic tool is the fact that even if the equation is a priori non well posed the solutions depend continuously on the noise around regular solutions

the stampacchia maximum principle for stochastic partial differential equations and applications abstract stochastic partial differential equations spdes are considered linear and nonlinear for which we establish comparison theorems for the solutions or positivity results a e and a s for suitable data comparison theorems for spdes are available in the literature the originality of our approach is that it is based on the use of truncations following the stampacchia approach to maximum principle we believe that our method which does not rely too much on probability considerations is simpler than the existing approaches and to a certain extent more directly applicable to concrete situations among the applications boundedness results and positivity results are respectively proved for the solutions of a stochastic boussinesq temperature equation and of reaction diffusion equations perturbed by a non lipschitz nonlinear noise stabilization results to a chafee infante equation perturbed by a nonlinear noise are also derived

a generalized linear integrate and fire neural model produces diverse spiking behaviors for simulations of neural networks there is a trade off between the size of the network that can be simulated and the complexity of the model used for individual neurons in this study we describe a generalization of the leaky integrate and fire model that produces a wide variety of spiking behaviors while still being analytically solvable between firings for different parameter values the model produces spiking or bursting tonic phasic or adapting responses depolarizing or hyperpolarizing after potentials and so forth the model consists of a diagonalizable set of linear differential equations describing the time evolution of membrane potential a variable threshold and an arbitrary number of firing induced currents each of these variables is modified by an update rule when the potential reaches threshold the variables used are intuitive and have biological significance the model s rich behavior does not come from the differential equations which are linear but rather from complex update rules this single neuron model can be implemented using algorithms similar to the standard integrate and fire model it is a natural match with event driven algorithms for which the firing times are obtained as a solution of a polynomial equation

partial regularity for the stochastic navier stokes equations the effects of random forces on the emergence of singularities in the navier stokes equations are investigated in spite of the presence of white noise the paths of a martingale suitable weak solution have a set of singular points of one dimensional hausdorff measure zero furthermore statistically stationary solutions with finite mean dissipation rate are analysed for these stationary solutions it is proved that at any time t the set of singular points is empty the same result holds true for every martingale solution starting from 0 a e initial condition u 0 where 0 is the law at time zero of a stationary solution finally the previous result is non trivial when the noise is sufficiently non degenerate since for any stationary solution the measure 0 is supported on the whole space h of initial conditions

long time behavior of weakly coupled oscillators we consider small perturbations of a simple completely integrable system with many degrees of freedom a collection of independent one degree of freedom oscillators in the perturbed system the individual oscillators are no longer independent we show that the long time behavior of such a system even in the case of purely deterministic perturbations should in general be described as a stochastic process the limiting stochastic process is a markov process on an open book space corresponding to the collection of first integrals of the non perturbed system

noise prevents singularities in linear transport equations a stochastic linear transport equation with multiplicative noise is considered and the question of no blow up is investigated the drift is assumed only integrable to a certain power opposite to the deterministic case where smooth initial conditions may develop discontinuities we prove that a certain sobolev degree of regularity is maintained which implies holder continuity of solutions the proof is based on a careful analysis of the associated stochastic flow of characteristics

speed of propagation for hamilton jacobi equations with multiplicative rough time dependence and convex hamiltonians we show that the initial value problem for hamilton jacobi equations with multiplicative rough time dependence typically stochastic and convex hamiltonians satisfies finite speed of propagation we prove that in general the range of dependence is bounded by a multiple of the length of the skeleton of the path that is a piecewise linear path obtained by connecting the successive extrema of the original one when the driving path is a brownian motion we prove that its skeleton has almost surely finite length we also discuss the optimality of the estimate

martingale solutions and markov selection of stochastic 3d navier stokes equations with jump abstract in this paper we study the existence of martingale solutions of stochastic 3d navier stokes equations with jump and following flandoli and romito 2008 7 and goldys et al 2009 8 we prove the existence of markov selections for the martingale solutions

neural dynamics bifurcations and firing rates in a quadratic integrate and fire model with a recovery variable i deterministic behavior we study the dynamics of a quadratic integrate and fire model of a single compartment neuron with a slow recovery variable as input current and parameters describing timescales recovery variable and postspike reset change analysis of a codimension 2 bifurcation reveals that the domain of attraction of a stable hyperpolarized rest state interacts subtly with reset parameters which reposition the system state after spiking we obtain explicit approximations of instantaneous firing rates for fixed values of the recovery variable and use the averaging theorem to obtain asymptotic firing rates as a function of current and reset parameters along with the different phase plane geometries these computations provide explicit tools for the interpretation of different spiking patterns and guide parameter selection in modeling different cortical cell types

synchronization by noise we provide sufficient conditions for synchronization by noise i e under these conditions we prove that weak random attractors for random dynamical systems consist of single random points in the case of sde with additive noise these conditions are also essentially necessary in addition we provide sufficient conditions for the existence of a minimal weak point random attractor consisting of a single random point as a result synchronization by noise is proven for a large class of sde with additive noise in particular we prove that the random attractor for an sde with drift given by a multidimensional double well potential and additive noise consists of a single random point all examples treated in tearne probab theory relat fields 141 1 2 1 18 2008 are also included

exponential convergence to non equilibrium stationary states in classical statistical mechanics we continue the study of a model for heat conduction 6 consisting of a chain of non linear oscillators coupled to two hamiltonian heat reservoirs at different temperatures we establish existence of a liapunov function for the chain dynamics and use it to show exponentially fast convergence of the dynamics to a unique stationary state ingredients of the proof are the reduction of the infinite dimensional dynamics to a finite dimensional stochastic process as well as a bound on the propagation of energy in chains of anharmonic oscillators

time space white noise eliminates global solutions in reaction diffusion equations abstract we prove that perturbing the reaction diffusion equation u t u x x u p p 1 with time space white noise produces that solutions explodes with probability one for every initial datum opposite to the deterministic model where a positive stationary solution exists

stochastic reaction diffusion systems with multiplicative noise and non lipschitz reaction term we study existence and uniqueness of a mild solution in the space of continuous functions and existence of an invariant measure for a class of reaction diffusion systems on bounded domains of d perturbed by a multiplicative noise the reaction term is assumed to have polynomial growth and to be locally lipschitz continuous and monotone the noise is white in space and time if d 1 and coloured in space if d 1 in any case the covariance operator is never assumed to be hilbert schmidt the multiplication term in front of the noise is assumed to be lipschitz continuous and no restrictions are given either on its linear growth or on its degenaracy our results apply in particular to systems of stochastic ginzburg landau equations with multiplicative noise

regularization by noise for stochastic hamilton jacobi equations we study regularizing effects of nonlinear stochastic perturbations for fully nonlinear pde more precisely path by path l infty bounds for the second derivative of solutions to such pde are shown these bounds are expressed as solutions to reflected sde and are shown to be optimal

on the regularization effect of space time white noise on quasi linear parabolic partial differential equations we show that an existence and uniqueness and a comparison theorem hold if we add a space time white noise to a quasi linear parabolic equation in one space dimension even if the nonlinearity is only measurable and not even locally bounded

on quasi linear stochastic partial differential equations we prove existence and uniqueness of the solution of a parabolic spde in one space dimension driven by space time white noise in the case of a measurable drift and a constant diffusion coefficient as well as a comparison theorem

ergodicity of the 2 d navier stokes equation under random perturbations a 2 dimensional navier stokes equation perturbed by a sufficiently distributed white noise is considered existence of invariant measures is known from previous works the aim is to prove uniqueness of the invariant measures strong law of large numbers and convergence to equilibrium

non equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures we study the statistical mechanics of a finite dimensional non linear hamiltonian system a chain of anharmonic oscillators coupled to two heat baths described by wave equations assuming that the initial conditions of the heat baths are distributed according to the gibbs measures at two different temperatures we study the dynamics of the oscillators under suitable assumptions on the potential and on the coupling between the chain and the heat baths we prove the existence of an invariant measure for any temperature difference i e we prove the existence of steady states furthermore if the temperature difference is sufficiently small we prove that the invariant measure is unique and mixing in particular we develop new techniques for proving the existence of invariant measures for random processes on a non compact phase space these techniques are based on an extension of the commutator method of hormander used in the study of hypoelliptic differential operators

importance of the cutoff value in the quadratic adaptive integrate and fire model the quadratic adaptive integrate and fire model izhikevich 2003 2007 is able to reproduce various firing patterns of cortical neurons and is widely used in large scale simulations of neural networks this model describes the dynamics of the membrane potential by a differential equation that is quadratic in the voltage coupled to a second equation for adaptation integration is stopped during the rise phase of a spike at a voltage cutoff value vc or when it blows up subsequently the membrane potential is reset and the adaptation variable is increased by a fixed amount we show in this note that in the absence of a cutoff value not only the voltage but also the adaptation variable diverges in finite time during spike generation in the quadratic model the divergence of the adaptation variable makes the system very sensitive to the cutoff changing vc can dramatically alter the spike patterns furthermore from a computational viewpoint the divergence of the adaptation variable implies that the time steps for numerical simulation need to be small and adaptive however divergence of the adaptation variable does not occur for the quartic model touboul 2008 and the adaptive exponential integrate and fire model brette gerstner 2005 hence these models are robust to changes in the cutoff value

martingale and stationary solutions for stochastic navier stokes equations we prove the existence of martingale solutions and of stationary solutions of stochastic navier stokes equations under very general hypotheses on the diffusion term the stationary martingale solutions yield the existence of invariant measures when the transition semigroup is well defined the results are obtained by a new method of compactness

markov selections for the 3d stochastic navier stokes equations we investigate the markov property and the continuity with respect to the initial conditions strong feller property for the solutions to the navier stokes equations forced by an additive noise first we prove by means of an abstract selection principle that there are markov solutions to the navier stokes equations due to the lack of continuity of solutions in the space of finite energy the markov property holds almost everywhere in time then depending on the regularity of the noise we prove that any markov solution has the strong feller property for regular initial conditions we give also a few consequences of these facts together with a new sufficient condition for well posedness

on the simulation of nonlinear bidimensional spiking neuron models bidimensional spiking models are garnering a lot of attention for their simplicity and their ability to reproduce various spiking patterns of cortical neurons and are used particularly for large network simulations these models describe the dynamics of the membrane potential by a nonlinear differential equation that blows up in finite time coupled to a second equation for adaptation spikes are emitted when the membrane potential blows up or reaches a cutoff i the precise simulation of the spike times and of the adaptation variable is critical for it governs the spike pattern produced and is hard to compute accurately because of the exploding nature of the system at the spike times we thoroughly study the precision of fixed time step integration schemes for this type of model and demonstrate that these methods produce systematic errors that are unbounded as the cutoff value is increased in the evaluation of the two crucial quantities the spike time and the value of the adaptation variable at this time precise evaluation of these quantities therefore involves very small time steps and long simulation times in order to achieve a fixed absolute precision in a reasonable computational time we propose here a new algorithm to simulate these systems based on a variable integration step method that either integrates the original ordinary differential equation or the equation of the orbits in the phase plane and compare this algorithm with fixed time step euler scheme and other more accurate simulation algorithms