an eigenvalue problem for hemivariational inequalities with combined nonlinearities on an infinite strip in this paper a class ofeigenvalue problems f hemivariational inequalities is studied which is defined on domains ofthe type r is a bounded open subset of r m m 1 and it involves

semilinear elliptic equations on unbounded domains on presente une nouvelle approche variationnelle a l existence des solutions pour le probleme aux valeurs limites u u r x f u x x x h 0 1 sur des domaines non bornes

quasilinear elliptic boundary value problems on unbounded cylinders and a related mountain pass lemma sufficient conditions for the existence of a weak solution to quasilinear elliptic boundary value problems on unbounded cylinders are obtained

comparison of langevin and markov channel noise models for neuronal signal generation the stochastic opening and closing of voltage gated ion channels produce noise in neurons the effect of this noise on the neuronal performance has been modeled using either an approximate or langevin model based on stochastic differential equations or an exact model based on a markov process model of channel gating yet whether the langevin model accurately reproduces the channel noise produced by the markov model remains unclear here we present a comparison between langevin and markov models of channel noise in neurons using single compartment hodgkin huxley models containing either text na and text k or only text k voltage gated ion channels the performance of the langevin and markov models was quantified over a range of stimulus statistics membrane areas and channel numbers we find that in comparison to the markov model the langevin model underestimates the noise contributed by voltage gated ion channels overestimating information rates for both spiking and nonspiking membranes even with increasing numbers of channels the difference between the two models persists this suggests that the langevin model may not be suitable for accurately simulating channel noise in neurons even in simulations with large numbers of ion channels

quantum markov semigroups constructed from quantum bernoulli noises quantum bernoulli noises qbns are the family of annihilation and creation operators acting on bernoulli functionals which can describe a two level quantum system with infinitely many sites in this paper we consider the problem to construct quantum markov semigroups qmss directly from qbns we first establish several new theorems concerning qbns in particular we define the number operator acting on bernoulli functionals by using the canonical orthonormal basis prove its self adjoint property and describe precisely its connections with qbn in a mathematically rigorous way we then show the possibility to construct qms directly from qbn this is done by combining the general results on qms with our new results on qbn obtained here finally we examine some properties of qms constructed from qbn

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

sufficient conditions for conservativity of minimal quantum dynamical semigroups abstract the conservativity of a minimal quantum dynamical semigroup is proved whenever there exists a generalized subharmonic operator bounded from below by the dissipative part of the infinitesimal generator we discuss applications of this criteria in mathematical physics and quantum probability

interspike interval statistics in the stochastic hodgkin huxley model coexistence of gamma frequency bursts and highly irregular firing when the classical hodgkin huxley equations are simulated with na and k channel noise and constant applied current the distribution of interspike intervals is bimodal one part is an exponential tail as often assumed while the other is a narrow gaussian peak centered at a short interspike interval value the gaussian arises from bursts of spikes in the gamma frequency range the tail from the interburst intervals giving overall an extraordinarily high coefficient of variation up to 2 5 for 180 000 na channels when i 7 acm2 since neurons with a bimodal isi distribution are common it may be a useful model for any neuron with class 2 firing the underlying mechanism is due to a subcritical hopf bifurcation together with a switching region in phase space where a fixed point is very close to a system limit cycle this mechanism may be present in many different classes of neurons and may contribute to widely observed highly irregular neural spiking

quantum bernoulli noises approach to stochastic schrodinger equation of exclusion type stochastic schrodinger equations are a special type of stochastic evolution equations in complex hilbert spaces which arise in the study of open quantum systems quantum bernoulli noises refer to annihilation and creation operators acting on bernoulli functionals which satisfy a canonical anti commutation relation in equal time in this paper we investigate a linear stochastic schrodinger equation of exclusion type in terms of quantum bernoulli noises among others we prove the well posedness of the equation illustrate the results with examples and discuss the consequences our main work extends that of chen and wang j math phys 58 5 053510 2017

linear stochastic schrodinger equations in terms of quantum bernoulli noises quantum bernoulli noises qbn are the family of annihilation and creation operators acting on bernoulli functionals which satisfy a canonical anti commutation relation in this paper we study linear stochastic schrodinger equations lsses associated with qbn in the space of square integrable complex valued bernoulli functionals we first rigorously prove a formula concerning the number operator n on bernoulli functionals and then by using this formula as well as mora and rebolledo s results on a general lsse c m mora and r rebolledo infinite dimens anal quantum probab relat top 10 237 259 2007 we obtain an easily checking condition for a lsse associated with qbn to have a unique nr strong solution of mean square norm conservation for given r 0 finally as an application of this condition we examine a special class of lsses associated with qbn and some further results are proven

entropy production for quantum markov semigroups an invariant state of a quantum markov semigroup is an equilibrium state if it satisfies a quantum detailed balance condition in this paper we introduce a notion of entropy production for faithful normal invariant states of a quantum markov semigroup on mathcal b mathsf h as a numerical index measuring how far they are from equilibrium the entropy production rate is defined as the derivative of the relative entropy of the one step forward and backward evolution in analogy with the classical probabilistic concept we prove an explicit trace formula expressing the entropy production rate in terms of the completely positive part of the generator of a norm continuous quantum markov semigroup showing that it turns out to be zero if and only if a standard quantum detailed balance condition holds

existence and bifurcation theorems for nonlinear elliptic eigenvalue problems on unbounded domains abstract we consider nonlinear elliptic eigenvalue problems on unbounded domains g r n using an extended ljusternik schnirelman theory we prove the existence of infinitely many eigenfunctions on every sphere in l 2 g moreover we establish that the infimum of the spectrum of the linearized problem l is always a bifurcation point in addition there is an infinity of branches emanating at from the trivial line of solutions if belongs to the essential spectrum of l

stochastic differential equation models for ion channel noise in hodgkin huxley neurons the random transitions of ion channels between conducting and nonconducting states generate a source of internal fluctuations in a neuron known as channel noise the standard method for modeling the states of ion channels nonlinearly couples continuous time markov chains to a differential equation for voltage beginning with the work of r f fox and y n lu phys rev e 49 3421 1994 there have been attempts to generate simpler models that use stochastic differential equation sdes to approximate the stochastic spiking activity produced by markov chain models recent numerical investigations however have raised doubts that sde models can capture the stochastic dynamics of markov chain models we analyze three sde models that have been proposed as approximations to the markov chain model one that describes the states of the ion channels and two that describe the states of the ion channel subunits we show that the former channel based approach can capture the distribution of channel noise and its effects on spiking in a hodgkin huxley neuron model to a degree not previously demonstrated but the latter two subunit based approaches cannot our analysis provides intuitive and mathematical explanations for why this is the case the temporal correlation in the channel noise is determined by the combinatorics of bundling subunits into channels but the subunit based approaches do not correctly account for this structure our study confirms and elucidates the findings of previous numerical investigations of subunit based sde models moreover it presents evidence that markov chain models of the nonlinear stochastic dynamics of neural membranes can be accurately approximated by sdes this finding opens a door to future modeling work using sde techniques to further illuminate the effects of ion channel fluctuations on electrically active cells

perturbation analysis of spontaneous action potential initiation by stochastic ion channels a stochastic interpretation of spontaneous action potential initiation is developed for the morris lecar equations initiation of a spontaneous action potential can be interpreted as the escape from one of the wells of a double well potential and we develop an asymptotic approximation of the mean exit time using a recently developed quasistationary perturbation method using the fact that the activating ionic channel s random openings and closings are fast relative to other processes we derive an accurate estimate for the mean time to fire an action potential mft which is valid for a below threshold applied current previous studies have found that for above threshold applied current where there is only a single stable fixed point a diffusion approximation can be used we also explore why different diffusion approximation techniques fail to estimate the mft

analysis of inverse stochastic resonance and the long term firing of hodgkin huxley neurons with gaussian white noise abstract in order to explain the occurrence of a minimum in firing rate which occurs for certain mean input levels as noise level increases inverse stochastic resonance isr in hodgkin huxley hh systems we analyze the underlying transitions from a stable equilibrium point to limit cycle and vice versa for a value of at which isr is pronounced properties of the corresponding stable equilibrium point are found a linearized approximation around this point has oscillatory solutions from whose maxima spikes tend to occur a one dimensional diffusion is also constructed for small noise properties of the basin of attraction of the limit cycle spike are investigated heuristically long term trials of duration 500000 ms are carried out for values of from 0 to 2 0 the graph of mean spike count versus is divided into 4 regions r 1 r 4 where r 3 contains the minimum associated with isr in r 1 transitions to the basin of attraction of the rest point are not observed until a small critical value of c 1 is reached at the beginning of r 2 the sudden decline in firing rate when is just greater than c 1 implies that there is only a small range of noise levels 0 c 1 where repetitive spiking is safe from annihilation by noise the firing rate remains small throughout r 3 at a larger critical value c 2 which signals the beginning of r 4 the probability of transitions from the basin of attraction of the equilibrium point to that of the limit cycle apparently becomes greater than zero and the spike rate thereafter increases with increasing the quantitative scheme underlying the isr curve is outlined in terms of the properties of exit time random variables in the final subsection several statistical properties of the main random variables associated with long term spiking activity are given including distributions of exit times from the two relevant basins of attraction and the interspike interval

discrete time quantum bernoulli noises in this paper we show that annihilation and creation operators on the space of bernoulli functionals in discrete time can be naturally interpreted as discrete time quantum bernoulli noises qbns we first examine their algebraic properties and prove among other things that the family of annihilation creation sum operators generates a commutative group of unitary operators additionally then by using qbn we prove an operator version of the discrete time clark formula we also obtain a qbn based necessary and sufficient condition for an operator process to be adapted we define integrals of adapted operator processes with respect to qbn and show their operator martingale property finally we give a representation formula for operators on the space of bernoulli functionals

weighted number operators on bernoulli functionals and quantum exclusion semigroups quantum bernoulli noises qbn for short are the family of annihilation and creation operators acting on bernoulli functionals which satisfy a canonical anticommutation relation in equal time in this paper by using qbn we first introduce a class of self adjoint operators acting on bernoulli functionals which we call the weighted number operators we then make clear spectral decompositions of these operators and establish their commutation relations with the annihilation as well as the creation operators we also obtain a necessary and sufficient condition for a weighted number operator to be bounded finally as an application of the above results we construct a class of quantum markov semigroups associated with the weighted number operators which belong to the category of quantum exclusion semigroups some basic properties of these quantum markov semigroups are shown and examples are given

linking and multiplicity results for the p laplacian on unbounded cylinders abstract we consider the p laplacian problem formula on unbounded cylinders rn m rn n m 2 where pu div u p 2 u is a constant in a certain range and a ln p l is nonnegative a 0 using the principle of symmetric criticality existence and multiplicity are proved under suitable conditions on a and f

gradient flow and entropy inequalities for quantum markov semigroups with detailed balance abstract we study a class of ergodic quantum markov semigroups on finite dimensional unital c algebras these semigroups have a unique stationary state and we are concerned with those that satisfy a quantum detailed balance condition with respect to we show that the evolution on the set of states that is given by such a quantum markov semigroup is gradient flow for the relative entropy with respect to in a particular riemannian metric on the set of states this metric is a non commutative analog of the 2 wasserstein metric and in several interesting cases we are able to show in analogy with work of otto on gradient flows with respect to the classical 2 wasserstein metric that the relative entropy is strictly and uniformly convex with respect to the riemannian metric introduced here as a consequence we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum markov semigroups with detailed balance

an investigation of the stochastic hodgkin huxley models under noisy rate functions the effects of ion channel fluctuations on the transmembrane voltage activity are potentially profound in small size excitable membrane patches different groups have extended hodgkin huxley equations into stochastic differential equations to capture the effects of ion channel noise analytically fox lu 1994 linaro storace giugliano 2011 guler 2013 studies have shown that the accuracy of spiking statistics by fox and lu s model does not match well with the corresponding statistics from the exact microscopic simulations the models of both linaro et al and guler however were found to produce highly accurate statistics here we extend the examination of these models to the case in which the rate functions for the opening and closing of gates are under the influence of noise for that purpose the usual rate functions are accompanied additively by ornstein uhlenbeck type stochastic angular variables moreover we argue that the existence of such noise in the rate functions is a plausible physiological phenomenon for finite size membranes it is observed that the presence of noise in the rates is not effective on the degree of inaccuracies within the fox and lu model guler model s accuracy is found to remain high as in the case of noise free rates but the performance of linaro et al s model is seen to degrade seriously with the increasing strength of the introduced rate function noise we attribute this failure of linaro et al s model to the use of the covariance function of open channels at the steady state in its derivation

metabolically efficient information processing energy efficient information transmission may be relevant to biological sensory signal processing as well as to low power electronic devices we explore its consequences in two different regimes in an immediate regime we argue that the information rate should be maximized subject to a power constraint and in an exploratory regime the transmission rate per power cost should be maximized in the absence of noise discrete inputs are optimally encoded into boltzmann distributed output symbols in the exploratory regime the partition function of this distribution is numerically equal to 1 the structure of the optimal code is strongly affected by noise in the transmission channel the arimoto blahut algorithm generalized for cost constraints can be used to derive and interpret the distribution of symbols for optimal energy efficient coding in the presence of noise we outline the possibilities and problems in extending our results to information coding and transmission in neurobiological systems

a boundary preserving numerical scheme for the wright fisher model abstract we are interested in the numerical approximation of non linear stochastic differential equations sdes with solution in a certain domain our goal is to construct explicit numerical schemes that preserve that structure we generalize the semi discrete method halidias and stamatiou 2016 and propose a numerical scheme for which we prove a strong convergence result to a class of sdes that appears in population dynamics and ion channel dynamics within cardiac and neuronal cells we furthermore extend our scheme to a multidimensional case

stochastic hodgkin huxley equations with colored noise terms in the conductances the excitability of cells is facilitated by voltage gated ion channels these channels accommodate a multiple number of gates individually the possible impact of that gate multiplicity on the cell s function specifically when the membrane area is of limited size was investigated in the author s prior work guler 2011 there it was found that a nontrivially persistent correlation takes place between the transmembrane voltage fluctuations also between the fluctuations in the gating variables and the component of open channel fluctuations attributed to the gate multiplicity this nontrivial phenomenon was found to be playing a major augmentative role for the elevation of excitability and spontaneous firing in small cells in addition the same phenomenon was found to be enhancing spike coherence significantly here we extend fox and lu s 1994 stochastic hodgkin huxley equations by incorporating colored noise terms into the conductances there to obtain a formalism capable of capturing the addressed cross correlations statistics of spike generation spike coherence firing efficiency latency and jitter from the articulated set of equations are found to be highly accurate in comparison with the corresponding statistics from the exact microscopic markov simulations this way it is demonstrated vividly that our formulation overcomes the inherent inadequacy of the fox and lu equations finally a recently proposed diffusion approximation method linaro storace giugliano 2011 is taken into consideration and a discussion on its character is pursued

subharmonic projections for a quantum markov semigroup this article introduces a concept of subharmonic projections for a quantum markov semigroup in view of characterizing the support projection of a stationary state in terms of the semigroup generator these results together with those of our previous article j math phys 42 1296 2001 lead to a method for proving the existence of faithful stationary states this is often crucial in the analysis of ergodic properties of quantum markov semigroups the method is illustrated by applications to physical models

quasicycles in the stochastic hybrid morris lecar neural model intrinsic noise arising from the stochastic opening and closing of voltage gated ion channels has been shown experimentally and mathematically to have important effects on a neuron s function study of classical neuron models with stochastic ion channels is becoming increasingly important especially in understanding a cell s ability to produce subthreshold oscillations and to respond to weak periodic stimuli while it is known that stochastic models can produce oscillations quasicycles in parameter regimes where the corresponding deterministic model has only a stable fixed point little analytical work has been done to explore these connections within the context of channel noise using a stochastic hybrid morris lecar ml model we combine a system size expansion in k and a quasi steady state qss approximation in persistent na in order to derive an effective langevin equation that preserves the low dimensional planar structure of the underlying deterministic ml model the qss analysis exploits the fact that persistent na channels are fast by calculating the corresponding power spectrum we determine analytically how noise significantly extends the parameter regime in which subthreshold oscillations occur