gap junctions destroy persistent states in excitatory networks gap junctions between excitatory neurons are shown to disrupt the persistent state the asynchronous state of the network loses stability via a hopf bifurcation and then the active state is destroyed via a homoclinic bifurcation with a stationary state a partial differential equation pde is developed to analyze the hopf and the homoclinic bifurcations the simplified dynamics are compared to a biophysical model where similar behavior is observed in the low noise case the dynamics of the pde is shown to be very complicated and includes possible chaotic behavior the onset of synchrony is studied by the application of averaging to obtain a simple criterion for destabilization of the asynchronous persistent state

existence positivity and contractivity for quantum stochastic flows with infinite dimensional noise quantum stochastic differential equations of the form govern stochastic flows on a c algebra we analyse this class of equation in which the matrix of fundamental quantum stochastic integrators is infinite dimensional and the coefficient matrix consists of bounded linear operators on weak and strong forms of solution are distinguished and a range of regularity conditions on the mapping matrix are considered for investigating existence and uniqueness of solutions necessary and sufficient conditions on are determined for any sufficiently regular weak solution k to be completely positive the further conditions on for k to also be a contraction process are found and when is a von neumann algebra and the components of are normal these in turn imply sufficient regularity for the equation to have a strong solution weakly multiplicative and homomorphic solutions and their generators are also investigated we then consider the right and left hudson parthasarathy equations in which f is a matrix of bounded hilbert space operators their solutions are interchanged by a time reversal operation on processes the analysis of quantum stochastic flows is applied to obtain characterisations of the generators f of contraction isometry and coisometry processes in particular weak solutions that are contraction processes are shown to have bounded generators and to be necessarily strong solutions

variability in noise driven integrator neurons neural variability in the presence of noise has been studied mainly in resonator neurons such as hodgkinhuxley or fitzhugh nagumo models here we investigate this variability for integrator neurons whose excitability is due to a saddle node bifurcation of the rest state instead of a hopf bifurcation using simple theoretical expressions for the interspike times distributions we obtain coefficients of variation in good agreement with numerical calculations in realistic neuron models the main features of this coefficient as a function of noise depend on the refractory period and on the presence of bistability the bistability is responsible for the existence of two different time scales in the spiking behavior giving an antiresonance effect

the moser trudinger inequality on kahler einstein manifolds we prove the conjecture of tian on the strong form of the moser trudinger inequality for k ahler einstein manifolds with positive first chern class when there are no holomorphic vector fields and more generally when the setting is invariant under a maximal compact subgroup of the automorphism group

effects of noise near a homoclinic bifurcation in an electrochemical system the interaction of external noise with an electrochemical oscillator anodic dissolution of iron has been studied experimentally on both sides of the homoclinic bifurcation in the oscillatory regime the regularity of the limit cycle behavior was destroyed with increasing noise amplitude until no periodicity was observable any more in contrast the response of the excitable state fixed point regime became more regular at intermediate noise levels

time scales of spike train correlation for neural oscillators with common drive we examine the effect of the phase resetting curve on the transfer of correlated input signals into correlated output spikes in a class of neural models receiving noisy superthreshold stimulation we use linear response theory to approximate the spike correlation coefficient in terms of moments of the associated exit time problem and contrast the results for type i vs type ii models and across the different time scales over which spike correlations can be assessed we find that on long time scales type i oscillators transfer correlations much more efficiently than type ii oscillators on short time scales this trend reverses with the relative efficiency switching at a time scale that depends on the mean and standard deviation of input currents this switch occurs over time scales that could be exploited by downstream circuits

firing variability is higher than deduced from the empirical coefficient of variation a convenient and often used summary measure to quantify the firing variability in neurons is the coefficient of variation cv defined as the standard deviation divided by the mean it is therefore important to find an estimator that gives reliable results from experimental data that is the estimator should be unbiased and have low estimation variance when the cv is evaluated in the standard way empirical standard deviation of interspike intervals divided by their average then the estimator is biased underestimating the true cv especially if the distribution of the interspike intervals is positively skewed moreover the estimator has a large variance for commonly used distributions the aim of this letter is to quantify the bias and propose alternative estimation methods if the distribution is assumed known or can be determined from data parametric estimators are proposed which not only remove the bias but also decrease the estimation errors if no distribution is assumed and the data are very positively skewed we propose to correct the standard estimator when defining the corrected estimator we simply use that it is more stable to work on the log scale for positively skewed distributions the estimators are evaluated through simulations and applied to experimental data from olfactory receptor neurons in rats

stochastic integration on the cuntz algebra o abstract we develop a theory of non commutative stochastic integration with respect to the creation and annihilation process on the full fock space over l2 r our theory largely parallels the theories of non commutative stochastic ito integration on boson and fermion fock space as developed by r hudson and k r parthasarathy it provides the first example of a non commutative stochastic calculus which does not depend on the quantum mechanical commutation or anticommutation relations but it is based on the theory of reduced free products of c algebras by d voiculescu this theory shows that the creation and annihilation processes on the full fock space over l2 r which generate the cuntz algebra o can be interpreted as a generalized brownian motion we should stress the fact that in contrast to the other theories of stochastic integration our integrals converge in the c norm on o i e uniformly rather than in some state dependent strong operator topology or non commutative l2 norm

population rate codes carried by mean fluctuation and synchrony of neuronal firings abstract a population of firing neurons is expected to carry information not only by the mean firing rate but also by fluctuation and synchrony among neurons in order to examine this possibility we have studied responses of neuronal ensembles to three kinds of inputs mean fluctuation and synchrony driven inputs the generalized rate code model including additive and multiplicative noise h hasegawa phys rev e 75 2007 051904 has been studied by direct simulations dss and the augmented moment method amm in which equations of motion for mean firing rate fluctuation and synchrony are derived results calculated by the amm are in good agreement with those by dss the independent component analysis ica of our results has shown that mean firing rate fluctuation or variability and synchrony may carry independent information in the population rate code model the input output relation of mean firing rates is shown to have higher sensitivity for larger multiplicative noise as recently observed in prefrontal cortex a comparison is made between results obtained by the integrate and fire if model and our rate code model

type i membranes phase resetting curves and synchrony type i membrane oscillators such as the connor model connor et al 1977 and the morris lecar model morris and lecar 1981 admit very low frequency oscillations near the critical applied current hansel et al 1995 have numerically shown that synchrony is difficult to achieve with these models and that the phase resetting curve is strictly positive we use singular perturbation methods and averaging to show that this is a general property of type i membrane models we show in a limited sense that so called type ii resetting occurs with models that obtain rhythmicity via a hopf bifurcation we also show the differences between synapses that act rapidly and those that act slowly and derive a canonical form for the phase interactions

inhibition and modulation of rhythmic neuronal spiking by noise we investigated the effects of noise on periodic firing in the hodgkin huxley nonlinear system with mean input current mu as a bifurcation parameter a bifurcation to repetitive spiking occurs at a critical value microc approximately 6 44 the firing behavior was studied as a function of the mean and variance of the input current firstly with initial resting conditions noise of a small amplitude can turn off the spiking for values of micro close to microc and the number of spikes undergoes a minimum as a function of the noise level the robustness of these phenomena was confirmed by simulations with random initial conditions and with random time of commencement of the noise furthermore their generality was indicated by their occurrence when additive noise was replaced by conductance based noise for long periods of observation many frequent transitions may occur from spiking to nonspiking activity when the noise is sufficiently strong explanations of the above phenomena are sought in terms of the underlying bifurcation structure and the probabilities that noise shifts the process from the basin of attraction of a stable limit cycle to that of a stable rest state the waiting times for such transitions depend strongly on the values of mu and sigma and on the forms of the basins of attraction the observed effects of noise are expected to occur in diverse fields in systems with the same underlying dynamical structure

coding stimulus amplitude by correlated neural activity while correlated activity is observed ubiquitously in the brain its role in neural coding has remained controversial recent experimental results have demonstrated that correlated but not single neuron activity can encode the detailed time course of the instantaneous amplitude i e envelope of a stimulus these have furthermore demonstrated that such coding required and was optimal for a nonzero level of neural variability however a theoretical understanding of these results is still lacking here we provide a comprehensive theoretical framework explaining these experimental findings specifically we use linear response theory to derive an expression relating the correlation coefficient to the instantaneous stimulus amplitude which takes into account key single neuron properties such as firing rate and variability as quantified by the coefficient of variation the theoretical prediction was in excellent agreement with numerical simulations of various integrate and fire type neuron models for various parameter values further we demonstrate a form of stochastic resonance as optimal coding of stimulus variance by correlated activity occurs for a nonzero value of noise intensity thus our results provide a theoretical explanation of the phenomenon by which correlated but not single neuron activity can code for stimulus amplitude and how key single neuron properties such as firing rate and variability influence such coding correlation coding by correlated but not single neuron activity is thus predicted to be a ubiquitous feature of sensory processing for neurons responding to weak input

feature selection in simple neurons how coding depends on spiking dynamics the relationship between a neuron s complex inputs and its spiking output defines the neuron s coding strategy this is frequently and effectively modeled phenomenologically by one or more linear filters that extract the components of the stimulus that are relevant for triggering spikes and a nonlinear function that relates stimulus to firing probability in many sensory systems these two components of the coding strategy are found to adapt to changes in the statistics of the inputs in such a way as to improve information transmission here we show for two simple neuron models how feature selectivity as captured by the spike triggered average depends on both the parameters of the model and the statistical characteristics of the input

the α invariant on certain surfaces with symmetry groups the global holomorphic invariant introduced by tian is closely related to the existence of kahler einstein metrics we apply the result of tian yau and zelditch on polarized kahler metrics to approximate plurisubharmonic functions and compute the invariant on cp 2 ncp 2 for n 1 2 3

models of stochastic biperiodic oscillations and extended serial correlations in electroreceptors of paddlefish two types of minimal models were used to study stochastic oscillations in sensory receptors composed of two coupled oscillators as in the electroreceptors of paddlefish they have populations of cells in sensory epithelia undergoing approximately 26 hz oscillations these are coupled unidirectionally via synaptic excitation to a few afferent neurons each of whose terminal contains a 30 70 hz oscillator expressed as a dominant peak in the power spectra of spontaneous afferent firing corresponding to the mean firing rate the two distinct types of internal noisy oscillators result in stochastic biperiodic firing patterns of the primary afferent sensory neurons however the functions of the oscillations have remained elusive motivating this study the models we used here are based on the circle map or on the ermentraut koppell canonical phase theta neuron model parameters were chosen according to experimental data we used the models to demonstrate that the presence of epithelial oscillations leads to extended negative correlations of afferent interspike intervals and to show that the correlation structure depends crucially on the ratio of the afferent to epithelial oscillation frequencies being most pronounced when this ratio is close to 2 as observed in experiments our studies of stochastic versions of these models are of general interest for a wide range of coupled excitable systems especially for understanding the functional roles of noisy oscillations in auditory and other types of hair cell primary afferent sensory receptors

stochastic resonance of electrochemical aperiodic spike trains aperiodic stochastic resonance in an electrochemical system with excitable dynamics is characterized in experiments and simulations two different spike trains one with stochastic and the other with chaotic interspike intervals are imposed on the system as subthreshold aperiodic signals information transmission is quantified by the cross correlation between the subthreshold input signal and the noise induced system response a maximum is exhibited in the input output correlation as a function of the noise amplitude numerical simulations with an electrochemical model are in excellent agreement with the experimental observations

stochastic calculus with respect to free brownian motion and analysis on wigner space we define stochastic integrals with respect to free brownian motion and show that they satisfy burkholder gundy type inequalities in operator norm we prove also a version of ito s predictable representation theorem as well as product form and functional form of ito s formula finally we develop stochastic analysis on the free fock space in analogy with stochastic analysis on the wiener space

critical phenomena in globally coupled excitable elements critical phenomena in globally coupled excitable elements are studied by focusing on a saddle node bifurcation at the collective level critical exponents that characterize divergent fluctuations of interspike intervals near the bifurcation are calculated theoretically the calculated values appear to be in good agreement with those determined by numerical experiments the relevance of our results to jamming transitions is also mentioned

stochastic resonance in continuous and spiking neuron models with levy noise levy noise can help neurons detect faint or subthreshold signals levy noise extends standard brownian noise to many types of impulsive jump noise processes found in real and model neurons as well as in models of finance and other random phenomena two new theorems and the ito calculus show that white levy noise will benefit subthreshold neuronal signal detection if the noise process s scaled drift velocity falls inside an interval that depends on the threshold values these results generalize earlier forbidden interval theorems of neuronal stochastic resonance sr or noise injection benefits global and local lipschitz conditions imply that additive white levy noise can increase the mutual information or bit count of several feedback neuron models that obey a general stochastic differential equation sde simulation results show that the same noise benefits still occur for some infinite variance stable levy noise processes even though the theorems themselves apply only to finite variance levy noise the appendix proves the two ito theoretic lemmas that underlie the new levy noise benefit theorems

theito formula for quantum semimartingales we construct a sequence of concordance invariants for classical links which depend on the peripheral isomorphism type of the nilpotent quotients of the link fundamental group the terminology stems from the fact that we replace the magnus expansion in the definition of milnor s bar mu invariants by the similar campbell hausdorff expansion the main point is that we introduce a new universal indeterminacy which depends only on the number of components of the link the campbell hausdorff invariants are new effectively computable and can efficiently distinguish unordered and unoriented isotopy types of links as we indicate on several families of closed braid examples they also satisfy certain natural dependence relations which generalize well known symmetries of the bar mu invariants 1991 mathematics subject classification 81s25 46l10 46l50 47a60

stochastic resonance in an autonomous system with a nonuniform limit cycle in a recent numerical study gang et al phys rev lett 71 807 1993 presented the first example of stochastic resonance in an autonomous system they considered a two variable model in which a limit cycle is born as some parameter is varied their numerical experiments revealed various noise induced effects including a noise induced shift in the frequency of the limit cycle oscillations and noise induced oscillations in the absence of a deterministic limit cycle we show that both of these effects are simple consequences of the nonuniformity of the motion along the limit cycle

pseudo regular oscillations induced by external noise an examination of the effect of noise on a general system at a saddle node bifurcation has revealed that in the limit of weak noise the probability density of the time to pass through the saddle node has a universal shape the specific kinetics of the particular system serving only to set the time scale this probability density is displayed and its salient features are explicated in the case of a saddle node bifurcation leading to relaxation oscillations this analysis leads to the prediction of the existence of noise induced oscillations which appear much less random than might at first be expected the period of these oscillations has a well defined nonzero most probable value the inverse of which is a noise induced frequency this frequency can be detected as a peak in power spectra from numerical simulations of such a system this is the first case of the prediction and detection of a noise induced frequency of which the authors are aware

fermion ito s formula and stochastic evolutions an ito product formula is proved for stochastic integrals against fermion brownian motion and used to construct unitary processes satisfying stochastic differential equations as in the corresponding boson theory 10 11 these give rise to stochastic dilations of completely positive semigroups

physiological gain leads to high isi variability in a simple model of a cortical regular spiking cell to understand the interspike interval isi variability displayed by visual cortical neurons softky koch 1993 it is critical to examine the dynamics of their neuronal integration as well as

analytic approximations of statistical quantities and response of noisy oscillators abstract nonlinear oscillators have been utilized in many contexts because they encompass a large class of phenomena for a reduced phase oscillator model with weak noise forcing that is necessarily multiplicative we provide analytic formulas for the stationary statistical quantities of the random period this is an important quantity which we term response i e the spike times instantaneous frequency in neuroscience the cycle time in chemical reactions etc that is often analytically intractable in noisy oscillator systems the analytic formulas are accurate in the weak noise limit so that one does not have to numerically solve a time varying fokker planck equation the steady state and dynamic responses are also analyzed with deterministic forcing a second order analytic formula is derived for the steady state response whereas the dynamic response with time varying forcing is more complicated we focus on the specific case where the forcing is sinusoidal and accurately capture the frequency response with an analytic approximation that is obtained with a rescaling of the equation by utilizing various techniques in the weak noise regime this work leads to a better understanding of how the random period of nonlinear oscillators are affected by multiplicative noise and external forcing comparisons of the asymptotic formulas with a full oscillator system confirm the qualitative accurateness of the theory

firing rate response of linear and nonlinear integrate and fire neurons to modulated current based and conductance based synaptic drive integrate and fire models are mainstays of the study of single neuron response properties and emergent states of recurrent networks of spiking neurons they also provide an analytical base for perturbative approaches that treat important biological details such as synaptic filtering synaptic conductance increase and voltage activated currents steady state firing rates of both linear and nonlinear integrate and fire models receiving fluctuating synaptic drive can be calculated from the time independent fokker planck equation the dynamic firing rate response is less easy to extract even at the first order level of a weak modulation of the model parameters but is an important determinant of neuronal response and network stability for the linear integrate and fire model the response to modulations of current based synaptic drive can be written in terms of hypergeometric functions for the nonlinear exponential and quadratic models no such analytical forms for the response are available here it is demonstrated that a rather simple numerical method can be used to obtain the steady state and dynamic response for both linear and nonlinear models to parameter modulation in the presence of current based or conductance based synaptic fluctuations to complement the full numerical solution generalized analytical forms for the high frequency response are provided a special case is also identified time constant modulation for which the response to an arbitrarily strong modulation can be calculated exactly

a quantum nonadapted ito formula and stochastic analysis in fock scale abstract a generalized definition of quantum stochastic qs integrals and differentials is given in the free of adaptiveness and dimensionality form in terms of malliavin derivative on a projective fock space and their uniform continuity with respect to the inductive limite convergence is proved a new form of qs calculus based on an inductive algebraic structure in an indefinite space is developed and a nonadaptive generalization of the qs ito formula for its representation in fock space is derived 1 the problem of solution of general qs evolution equations in a hilbert space is solved in terms of the constructed operator representation of chronological products defined in the indefinite space and the isometry and homomorphism property respectively for operators and maps of these solutions corresponding to the peseudounitary and homomorphism property of the qs integrable generators is proved

cohomology of power sets with applications in quantum probability square integrable wiener functionals may be represented as sums of multiple ito integrals this leads to an identification of such functionals with square integrable functions on the symmetric measure space of the lebesgue spacer when the pointwise product of wiener functionals is thus carried over the product takes a pleasing form cf wick s theorem and various non commutative perturbations of this wiener product have been considered here we employ cohomological arguments to analyse deformations of an abstract wiener product this leads to the construction of levy fields which are neither bosonic nor fermionic and also gives rise to homotopies between quasi free boson and fermion fields finally we unify existence and uniqueness results for quantum stochastic differential equations by treating mixed noise differential equations

theoretical analysis for critical fluctuations of relaxation trajectory near a saddle node bifurcation a langevin equation whose deterministic part undergoes a saddle node bifurcation is investigated theoretically it is found that statistical properties of relaxation trajectories in this system exhibit divergent behaviors near a saddle node bifurcation point in the weak noise limit while the final value of the deterministic solution changes discontinuously at the point a systematic formulation for analyzing a path probability measure is constructed on the basis of a singular perturbation method in this formulation the critical nature turns out to originate from the neutrality of exiting time from a saddle point the theoretical calculation explains results of numerical simulations

analytic expressions for rate and cv of a type i neuron driven by white gaussian noise we study the one dimensional normal form of a saddle node system under the influence of additive gaussian white noise and a static bias current input parameter a model that can be looked upon as the simplest version of a type i neuron with stochastic input this is in contrast with the numerous studies devoted to the noise driven leaky integrate and fire neuron we focus on the firing rate and coefficient of variation cv of the interspike interval density for which scaling relations with respect to the input parameter and noise intensity are derived quadrature formulas for rate and cv are numerically evaluated and compared to numerical simulations of the system and to various approximation formulas obtained in different limiting cases of the model we also show that caution must be used to extend these results to the neuron model with multiplicative gaussian white noise the correspondence between the first passage time statistics for the saddle node model and the neuron model is obtained only in the stratonovich interpretation of the stochastic neuron model while previous results have focused only on the ito interpretation the correct stratonovich interpretation yields cvs that are still relatively high although smaller than in the ito interpretation it also produces certain qualitative differences especially at larger noise intensities our analysis provides useful relations for assessing the distance to threshold and the level of synaptic noise in real type i neurons from their firing statistics we also briefly discuss the effect of finite boundaries finite values of threshold and reset on the firing statistics

existence of quantum diffusions a quantum diffusion a a j comprises of unital algebras a and a and a family of identity preserving homomorphisms j j t t 0 from a into a also j satisfies a system of quantum stochastic differential equations dj t x 0 j t x 0 dm j 0 x 0 x 0 i for all x 0 a where 1 i j n are maps from a to itself and are known as the structure maps in this paper an existence proof is given for a class of quantum diffusions for which the structure maps are bounded in the operator norm sense a solution to the system of quantum stochastic differential equations is first produced using a variation of the picard iteration method another application of this method shows that the solution is a quantum diffusion

quantum ito s formula and stochastic evolutions using only the boson canonical commutation relations and the riemann lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes this leads to a noncommutative ito product formula a realisation of the classical poisson process in fock space which gives a noncommutative central limit theorem the construction of solutions of certain noncommutative stochastic differential equations and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps the classical ito product formula for stochastic differentials with respect to brownian motion and the poisson process is a special case

noncommutative stochastic integration through decoupling abstract we present an abstract approach to noncommutative stochastic integration in the context of a finite von neumann algebra equipped with a normal faithful tracial state with respect to processes with tensor or freely independent increments satisfying a stationarity condition using a decoupling technique we obtain necessary and sufficient conditions for stochastic integrability of l p processes with respect to such integrators we apply the theory to stochastic integration with respect to boson and free brownian motion

quantum stochastic calculus with maximal operator domains quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains operator adaptedness conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of fock space together with sections of the adapted gradient operator free from exponential vector domains our stochastic integrals may be satisfactorily composed yielding quantum ito formulas for operator products as sums of stochastic integrals the calculus has seen two reformulations since its discovery one closely related to classical ito calculus the other to noncausal stochastic analysis and malliavin calculus our theory extends both of these approaches and may be viewed as a synthesis of the two the main application given here is existence and uniqueness for the attal meyer equations for implicit definition of quantum stochastic integrals

an introduction to quantum filtering this paper provides an introduction to quantum filtering theory an introduction to quantum probability theory is given focusing on the spectral theorem and the conditional expectation as a least squares estimate and culminating in the construction of wiener and poisson processes on the fock space we describe the quantum ito calculus and its use in the modeling of physical systems we use both reference probability and innovations methods to obtain quantum filtering equations for system probe models from quantum optics

hilbert modules and stochastic dilation of a quantum dynamical semigroup on a von neumann algebra a general theory for constructing a weak markov dilation of a uniformly continuous quantum dynamical semigroup tt on a von neumann algebra with respect to the fock filtration is developed with the aid of a coordinate free quantum stochastic calculus starting with the structure of the generator of tt existence of canonical structure maps in the sense of evans and hudson is deduced and a quantum stochastic dilation of tt is obtained through solving a canonical flow equation for maps on the right fock module l2 k0 where k0 is some hilbert space arising from a representation of this gives rise to a homomorphism jt of moreover it is shown that every such flow is implemented by a partial isometry valued process this leads to a natural construction of a weak markov process in the sense of b p with respect to fock filtration

dynamics of membrane excitability determine interspike interval variability a link between spike generation mechanisms and cortical spike train statistics we propose a biophysical mechanism for the high interspike interval variability observed in cortical spike trains the key lies in the nonlinear dynamics of cortical spike generation which are consistent with type i membranes where saddle node dynamics underlie excitability rinzel ermentrout 1989 we present a canonical model for type i membranes the neuron the neuron is a phase model whose dynamics reflect salient features of type i membranes this model generates spike trains with coefficient of variation cv above 0 6 when brought to firing by noisy inputs this happens because the timing of spikes for a type i excitable cell is exquisitely sensitive to the amplitude of the suprathreshold stimulus pulses a noisy input current giving random amplitude kicks to the cell evokes highly irregular firing across a wide range of firing rates an intrinsically oscillating cell gives regular spike trains we corroborate the results with simulations of the morris lecar m l neural model with random synaptic inputs type i m l yields high cvs when this model is modified to have type ii dynamics periodicity arises via a hopf bifurcation however it gives regular spike trains cv below 0 3 our results suggest that the high cv values such as those observed in cortical spike trains are an intrinsic characteristic of type i membranes driven to firing by random inputs in contrast neural oscillators or neurons exhibiting type ii excitability should produce regular spike trains

stochastic integral representation of bounded quantum martingales in fock space it is a classical theorem of kunita and watanabe 6 that every square integrable martingale adapted to the standard brownian motion can be uniquely expressed as the stochastic integral of a nonanticipating square integrable brownian functional with respect to the same brownian motion the central aim of this paper is to establish a similar integral represen tation of a quantum martingale with respect to the annihilation creation and gauge processes in the context of quantum stochastic calculus in fock space as developed in 11 in the case of non fock quantum brownian motion such an integral representation was achieved by hudson and lindsay 2 33 with much less difficulty owing to the non existence of the so called gauge processes as special cases of our main result we obtain the differentials of hilbert schmidt and unitary martingales 4 as an application the uniqueness of fermion martingales in boson fock space is established the first author wishes to thank hudson and lindsay for several useful conversations on this subject 2

quantum and non causal stochastic calculus the quantum stochastic calculus initiated by hudson and parthasarathy and the non causal stochastic calculus originating with the papers of hitsuda and skorohod are two potent extensions of the ito calculus currently enjoying intensive development the former provides a quantum probabilistic extension of schrodinger s equation enabling the construction of a markov process for a quantum dynamical semigroup the latter allows the treatment of stochastic differential equations which involve terms which anticipate the future in this paper the close relationship between these theories is displayed and a noncausal quantum stochastic calculus already in demand from physics is described

evaluation of entrainment of a nonlinear neural oscillator to white noise the lyapunov exponent for a one dimensional neural oscillator model the theta neuron is computed for white noise forcing using the steady state solution to the associated fokker planck equation the latter is mildly singular due to the nature of the multiplicative input in agreement with previous results with similar models the exponent is negative for all forcing amplitudes but here it is shown to be small relative to that for periodic drive in a range of forcing strengths thus the synchronization of an ensemble of independent neurons receiving common but random input can be slow moreover this implies that aperiodic input may be suboptimal in some contexts for preserving the reliability of fine spike timing a potentially important component of the neural code