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

from spiking neurons to rate models a cascade model as an approximation to spiking neuron models with refractoriness a neuron that is stimulated repeatedly by the same time dependent stimulus exhibits slightly different spike timing at each trial we compared the exact solution of the time dependent firing rate for a stochastically spiking neuron model with refractoriness spike response model with that of an inhomogeneous poisson process subject to the same stimulus to arrive at a mapping between the two models we used alternatively i a systematic parameter free volterra expansion of the exact solution or ii a linear filter combined with nonlinear poisson rate model linear nonlinear poisson cascade model with a single free parameter both the cascade model and the second order volterra model showed excellent agreement with the exact rate dynamics of the spiking neuron model with refractoriness even for strong and rapidly changing input cascade rate models are widely used in systems neuroscience our method could help to connect experimental rate measurements to the theory of spiking neurons

low dimensional firing rate dynamics for populations of renewal type spiking neurons the macroscopic dynamics of large populations of neurons can be mathematically analyzed using low dimensional firing rate or neural mass models however these models fail to capture spike synchronization effects and nonstationary responses of the population activity to rapidly changing stimuli here we derive low dimensional firing rate models for homogeneous populations of neurons modeled as time dependent renewal processes the class of renewal neurons includes integrate and fire models driven by white noise and has been frequently used to model neuronal refractoriness and spike synchronization dynamics the derivation is based on an eigenmode expansion of the associated refractory density equation which generalizes previous spectral methods for fokker planck equations to arbitrary renewal models we find a simple relation between the eigenvalues characterizing the timescales of the firing rate dynamics and the laplace transform of the interspike interval density for which explicit expressions are available for many renewal models retaining only the first eigenmode already yields a reliable low dimensional approximation of the firing rate dynamics that captures spike synchronization effects and fast transient dynamics at stimulus onset we explicitly demonstrate the validity of our model for a large homogeneous population of poisson neurons with absolute refractoriness and other renewal models that admit an explicit analytical calculation of the eigenvalues the eigenmode expansion presented here provides a systematic framework for alternative firing rate models in computational neuroscience based on spiking neuron dynamics with refractoriness

fast global oscillations in networks of integrate and fire neurons with low firing rates we study analytically the dynamics of a network of sparsely connected inhibitory integrate and fire neurons in a regime where individual neurons emit spikes irregularly and at a low rate in the limit when the number of neurons n 1 the network exhibits a sharp transition between a stationary and an oscillatory global activity regime where neurons are weakly synchronized the activity becomes oscillatory when the inhibitory feedback is strong enough the period of the global oscillation is found to be mainly controlled by synaptic times but depends also on the characteristics of the external input in large but finite networks the analysis shows that global oscillations of finite coherence time generically exist both above and below the critical inhibition threshold their characteristics are determined as functions of systems parameters in these two different regimes the results are found to be in good agreement with numerical simulations

the number of synaptic inputs and the synchrony of large sparse neuronal networks the prevalence of coherent oscillations in various frequency ranges in the central nervous system raises the question of the mechanisms that synchronize large populations of neurons we study synchronization in models of large networks of spiking neurons with random sparse connectivity synchrony occurs only when the average number of synapses m that a cell receives is larger than a critical value mc below mc the system is in an asynchronous state in the limit of weak coupling assuming identical neurons we reduce the model to a system of phase oscillators that are coupled via an effective interaction in this framework we develop an approximate theory for sparse networks of identical neurons to estimate mc analytically from the fourier coefficients of our approach relies on the assumption that the dynamics of a neuron depend mainly on the number of cells that are presynaptic to it we apply this theory to compute mc for a model of inhibitory networks of integrate and fire i f neurons as a function of the intrinsic neuronal properties e g the refractory period tr the synaptic time constants and the strength of the external stimulus iext the number mc is found to be nonmonotonous with the strength of iext for tr 0 we estimate the minimum value of mc over all the parameters of the model to be 363 8 above mc the neurons tend to fire in smeared one cluster states at high firing rates and smeared two or more cluster states at low firing rates refractoriness decreases mc at intermediate and high firing rates these results are compared to numerical simulations we show numerically that systems with different sizes n behave in the same way provided the connectivity m is such that 1 meff 1 m 1 n remains constant when n varies this allows extrapolating the large n behavior of a network from numerical simulations of networks of relatively small sizes n 800 in our case we find that our theory predicts with remarkable accuracy the value of mc and the patterns of synchrony above mc provided the synaptic coupling is not too large we also study the strong coupling regime of inhibitory sparse networks all of our simulations demonstrate that increasing the coupling strength reduces the level of synchrony of the neuronal activity above a critical coupling strength the network activity is asynchronous we point out a fundamental limitation for the mechanisms of synchrony relying on inhibition alone if heterogeneities in the intrinsic properties of the neurons and spatial fluctuations in the external input are also taken into account

a population density approach that facilitates large scale modeling of neural networks extension to slow inhibitory synapses a previously developed method for efficiently simulating complex networks of integrate and fire neurons was specialized to the case in which the neurons have fast unitary postsynaptic conductances however inhibitory synaptic conductances are often slower than excitatory ones for cortical neurons and this difference can have a profound effect on network dynamics that cannot be captured with neurons that have only fast synapses we thus extend the model to include slow inhibitory synapses in this model neurons are grouped into large populations of similar neurons for each population we calculate the evolution of a probability density function pdf which describes the distribution of neurons over state space the population firing rate is given by the flux of probability across the threshold voltage for firing an action potential in the case of fast synaptic conductances the pdf was one dimensional as the state of a neuron was completely determined by its transmembrane voltage an exact extension to slow inhibitory synapses increases the dimension of the pdf to two or three as the state of a neuron now includes the state of its inhibitory synaptic conductance however by assuming that the expected value of a neuron s inhibitory conductance is independent of its voltage we derive a reduction to a one dimensional pdf and avoid increasing the computational complexity of the problem we demonstrate that although this assumption is not strictly valid the results of the reduced model are surprisingly accurate

spike frequency adapting neural ensembles beyond mean adaptation and renewal theories we propose a markov process model for spike frequency adapting neural ensembles that synthesizes existing mean adaptation approaches population density methods and inhomogeneous renewal theory resulting in a unified and tractable framework that goes beyond renewal and mean adaptation theories by accounting for correlations between subsequent interspike intervals a method for efficiently generating inhomogeneous realizations of the proposed markov process is given numerical methods for solving the population equation are presented and an expression for the first order interspike interval correlation is derived further we show that the full five dimensional master equation for a conductance based integrate and fire neuron with spike frequency adaptation and a relative refractory mechanism driven by poisson spike trains can be reduced to a two dimensional generalization of the proposed markov process by an adiabatic elimination of fast variables for static and dynamic stimulation negative serial interspike interval correlations and transient population responses respectively of monte carlo simulations of the full five dimensional system can be accurately described by the proposed two dimensional markov process

dynamics of encoding in neuron populations some general mathematical features the use of a population dynamics approach promises efficient simulation of large assemblages of neurons depending on the issues addressed and the degree of realism incorporated in the simulated neurons a wide range of different population dynamics formulations can be appropriate here we present a common mathematical structure that these various formulations share and that implies dynamical behaviors that they have in common this underlying structure serves as a guide toward efficient means of simulation as an example we derive the general population firing rate frequency response and show how it may be used effectively to address a broad range of interacting population response and stability problems a few specific cases will be worked out a summary of this work appears at the end before the appendix

alternating oscillatory and stochastic dynamics in a model for a neuronal assembly abstract in recent neurophysiological experiments stimulus related neuronal oscillations were discovered in various species the oscillations are not persistent during the whole time of stimulation but instead seem to be restricted to rather short periods interrupted by stochastic periods in this contribution we argue that these observations can be explained by a bistability in the ensemble dynamics of coupled integrate and fire neurons this dynamics can be cast in terms of a high dimensional map for the time evolution of a phase density which represents the ensemble state a numerical analysis of this map reveals the coexistence of two stationary states in a broad parameter regime when the synaptic transmission is nonlinear the one state corresponds to a stochastic firing of individual neurons the other state describes a periodic activation we demonstrate that under the influence of additional external noise the system can switch between these states in this way reproducing the experimentally activity we also investigate the connection between the nonlinearity of the synaptic transmission function and the bistability of the dynamics to this purpose we heuristically reduce the high dimensional assembly dynamics to a one dimensional map which in turn yields a simple explanation for the relation between nonlinearity and bistability in our system

stable propagation of activity pulses in populations of spiking neurons we investigate the propagation of pulses of spike activity in a neuronal network with feedforward couplings the neurons are of the spike response type with a firing probability that depends linearly on the membrane potential after firing neurons enter a phase of refractoriness spike packets are described in terms of the moments of the firing time distribution so as to allow for an analytical treatment of the evolution of the spike packet as it propagates from one layer to the next analytical results and simulations show that depending on the synaptic coupling strength a stable propagation of the packet with constant waveform is possible crucial for this observation is neither the existence of a firing threshold nor a sigmoidal gain function both are absent in our model but the refractory behavior of the neurons

stochastic dynamics of a finite size spiking neural network we present a simple markov model of spiking neural dynamics that can be analytically solved to characterize the stochastic dynamics of a finite size spiking neural network we give closed form estimates for the equilibrium distribution mean rate variance and autocorrelation function of the network activity the model is applicable to any network where the probability of firing of a neuron in the network depends on only the number of neurons that fired in a previous temporal epoch networks with statistically homogeneous connectivity and membrane and synaptic time constants that are not excessively long could satisfy these conditions our model completely accounts for the size of the network and correlations in the firing activity it also allows us to examine how the network dynamics can deviate from mean field theory we show that the model and solutions are applicable to spiking neural networks in biophysically plausible parameter regimes

theory of oscillatory firing induced by spatially correlated noise and delayed inhibitory feedback a network of leaky integrate and fire neurons with global inhibitory feedback and under the influence of spatially correlated noise is studied we calculate the spectral statistics of the network power spectrum of the population activity cross spectrum between spike trains of different neurons as well as of a single neuron power spectrum of spike train cross spectrum between external noise and spike train within the network as shown by comparison with numerical simulations our theory works well for arbitrary network size if the feedback is weak and the amount of external noise does not exceed that of the internal noise by means of our analytical results we discuss the quality of the correlation induced oscillation in a large network as a function of the transmission delay and the internal noise intensity it is shown that the strongest oscillation is obtained in a system with zero internal noise and adiabatically long delay i e the delay period is longer than any other time scale in the system for a neuron with a strong intrinsic frequency the oscillation becomes strongly anharmonic in the case of a long delay time we also discuss briefly the kind of synchrony introduced by the feedback induced oscillation

on the phase reduction and response dynamics of neural oscillator populations weundertake a probabilistic analysis of the response of repetitively firing neural populations to simple pulselike stimuli recalling and extending results from the literature we compute phase response curves prcs valid near bifurcations to periodic firing for hindmarsh rose hodgkin huxley fitzhugh nagumo and morris lecar models encompassing the four generic codimension one bifurcations phase density equations are then used to analyze the role of the bifurcation and the resulting prc in responses to stimuli in particular we explore the interplay among stimulus duration baseline firing frequency and population level response patterns we interpret the results in terms of the signal processing measure of gain and discuss further applications and experimentally testable predictions

stationary bumps in networks of spiking neurons we examine the existence and stability of spatially localized bumps of neuronal activity in a network of spiking neurons bumps have been proposed in mechanisms of visual orientation tuning the rat head direction system and working memory we show that a bump solution can exist in a spiking network provided the neurons fire asynchronously within the bump we consider a parameter regime where the bump solution is bistable with an all off state and can be initiated with a transient excitatory stimulus we show that the activity profile matches that of a corresponding population rate model the bump in a spiking network can lose stability through partial synchronization to either a traveling wave or the all off state this can occur if the synaptic timescale is too fast through a dynamical effect or if a transient excitatory pulse is applied to the network a bump can thus be activated and deactivated with excitatory inputs that may have physiological relevance

integrate and fire neurons with threshold noise a tractable model of how interspike interval correlations affect neuronal signal transmission many neurons exhibit interval correlations in the absence of input signals we study the influence of these intrinsic interval correlations of model neurons on their signal transmission properties for this purpose we employ two simple firing models one of which generates a renewal process while the other leads to a nonrenewal process with negative interval correlations different methods to solve for spectral statistics in the presence of a weak stimulus spike train power spectra cross spectra and coherence functions are presented and their range of validity is discussed using these analytical results we explore a lower bound on the mutual information rate between output spike train and input stimulus as a function of the system s parameters we demonstrate that negative correlations in the baseline activity can lead to enhanced information transfer of a weak signal by means of noise shaping of the background noise spectrum we also show that an enhancement is not compulsory char22 for a stimulus with power exclusively at high frequencies the renewal model can transfer more information than the nonrenewal model does we discuss the application of our analytical results to other problems in neuroscience our results are also relevant to the general problem of how a signal affects the power spectrum of a nonlinear stochastic system

self organizing dual coding based on spike time dependent plasticity it has been a matter of debate how firing rates or spatiotemporal spike patterns carry information in the brain recent experimental and theoretical work in part showed that these codes especially a population rate code and a synchronous code can be dually used in a single architecture however we are not yet able to relate the role of firing rates and synchrony to the spatiotemporal structure of inputs and the architecture of neural networks in this article we examine how feedforward neural networks encode multiple input sources in the firing patterns we apply spiketime dependent plasticity as a fundamental mechanism to yield synaptic competition and the associated input filtering we use the fokker planck formalism to analyze the mechanism for synaptic competition in the case of multiple inputs which underlies the formation of functional clusters in downstream layers in a self organizing manner depending on the types of feedback coupling and shared connectivity clusters are independently engaged in population rate coding or synchronous coding or they interact to serve as input filters classes of dual codings and functional roles of spike time dependent plasticity are also discussed

a master equation formalism for macroscopic modeling of asynchronous irregular activity states many efforts have been devoted to modeling asynchronous irregular ai activity states which resemble the complex activity states seen in the cerebral cortex of awake animals most of models have considered balanced networks of excitatory and inhibitory spiking neurons in which ai states are sustained through recurrent sparse connectivity with or without external input in this letter we propose a mesoscopic description of such ai states using master equation formalism we derive a second order mean field set of ordinary differential equations describing the temporal evolution of randomly connected balanced networks this formalism takes into account finite size effects and is applicable to any neuron model as long as its transfer function can be characterized we compare the predictions of this approach with numerical simulations for different network configurations and parameter spaces considering the randomly connected network as a unit this approach could be used to build large scale networks of such connected units with an aim to model activity states constrained by macroscopic measurements such as voltage sensitive dye imaging

asynchronous states and the emergence of synchrony in large networks of interacting excitatory and inhibitory neurons we investigate theoretically the conditions for the emergence of synchronous activity in large networks consisting of two populations of extensively connected neurons one excitatory and one inhibitory the neurons are modeled with quadratic integrate and fire dynamics which provide a very good approximation for the subthreshold behavior of a large class of neurons in addition to their synaptic recurrent inputs the neurons receive a tonic external input that varies from neuron to neuron because of its relative simplicity this model can be studied analytically we investigate the stability of the asynchronous state as of the network with given average firing rates of the two populations first we show that the as can remain stable even if the synaptic couplings are strong then we investigate the conditions under which this state can be destabilized we show that this can happen in four generic ways the first is a saddle node bifurcation which leads to another state with different average firing rates this bifurcation which occurs for strong enough recurrent excitation does not correspond to the emergence of synchrony in contrast in the three other instability mechanisms hopf bifurcations which correspond to the emergence of oscillatory synchronous activity occur we show that these mechanisms can be differentiated by the firing patterns they generate and their dependence on the mutual interactions of the inhibitory neurons and cross talk between the two populations we also show that besides these codimension 1 bifurcations the system can display several codimension 2 bifurcations takens bogdanov gavrielov guckenheimer and double hopf bifurcations

stability of incoherence in a population of coupled oscillators we analyze a mean field model of coupled oscillators with randomly distributed frequencies this system is known to exhibit a transition to collective oscillations for small coupling the system is incoherent with all the oscillators running at their natural frequencies but when the coupling exceeds a certain threshold the system spontaneously synchronizes we obtain the first rigorous stability results for this model by linearizing the fokker planck equation about the incoherent state an unexpected result is that the system has pathological stability properties the incoherent state is unstable above threshold butneutrally stable below threshold we also show that the system is singular in the sense that its stability properties are radically altered by infinitesimal noise

dynamics and computation of continuous attractors continuous attractor is a promising model for describing the encoding of continuous stimuli in neural systems in a continuous attractor the stationary states of the neural system form a continuous parameter space on which the system is neutrally stable this property enables the neutral system to track time varying stimuli smoothly but it also degrades the accuracy of information retrieval since these stationary states are easily disturbed by external noise in this work based on a simple model we systematically investigate the dynamics and the computational properties of continuous attractors in order to analyze the dynamics of a large size network which is otherwise extremely complicated we develop a strategy to reduce its dimensionality by utilizing the fact that a continuous attractor can eliminate the noise components perpendicular to the attractor space very quickly we therefore project the network dynamics onto the tangent of the attractor space and simplify it successfully as a one dimensional ornstein uhlenbeck process based on this simplified model we investigate 1 the decoding error of a continuous attractor under the driving of external noisy inputs 2 the tracking speed of a continuous attractor when external stimulus experiences abrupt changes 3 the neural correlation structure associated with the specific dynamics of a continuous attractor and 4 the consequence of asymmetric neural correlation on statistical population decoding the potential implications of these results on our understanding of neural information processing are also discussed

reduction of the hodgkin huxley equations to a single variable threshold model it is generally believed that a neuron is a threshold element that fires when some variable u reaches a threshold here we pursue the question of whether this picture can be justified and study the four dimensional neuron model of hodgkin and huxley as a concrete example the model is approximated by a response kernel expansion in terms of a single variable the membrane voltage the first order term is linear in the input and its kernel has the typical form of an elementary postsynaptic potential higher order kernels take care of nonlinear interactions between input spikes in contrast to the standard volterra expansion the kernels depend on the firing time of the most recent output spike in particular a zero order kernel that describes the shape of the spike and the typical after potential is included our model neuron fires if the membrane voltage given by the truncated response kernel expansion crosses a threshold the threshold model is tested on a spike train generated by the hodgkin huxley model with a stochastic input current we find that the threshold model predicts 90 percent of the spikes correctly our results show that to good approximation the description of a neuron as a threshold element can indeed be justified

period focusing induced by network feedback in populations of noisy integrate and fire neurons the population dynamics of an ensemble of nonleaky integrate and fire stochastic neurons is studied the model selected allows for a detailed analysis of situations where noise plays a dominant role simulations in a regime with weak to moderate interactions show that a mechanism of excitatory message interchange among the neurons leads to a decrease in the firing period dispersion of the individual units the dispersion reduction observed is larger than what would be expected from the decrease in the period this period focusing is explained using a mean field model it is a dynamical effect that arises from the progressive decrease of the effective firing threshold as a result of the messages received by each unit from the rest of the population a back of the envelope formula to calculate this nontrivial dispersion reduction and a simple geometrical description of the effect are also provided

an adaptive least squares fem for the stokes equations with optimal convergence rates this paper introduces the first adaptive least squares finite element method ls fem for the stokes equations with optimal convergence rates based on the newest vertex bisection with lowest order raviart thomas and conforming p 1 p1 discrete spaces for the divergence least squares formulation in 2d although the least squares functional is a reliable and efficient error estimator the novel refinement indicator stems from an alternative explicit residual based a posteriori error control with exact solve particular interest is on the treatment of the data approximation error which requires a separate marking strategy the paper proves linear convergence in terms of the levels and optimal convergence rates in terms of the number of unknowns relative to the notion of a non linear approximation class it extends and generalizes the approach of carstensen and park siam j numer anal 53 43 62 2015 from the poisson model problem to the stokes equations

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

critical analysis of dimension reduction by a moment closure method in a population density approach to neural network modeling computational techniques within the population density function pdf framework have provided time saving alternatives to classical monte carlo simulations of neural network activity efficiency of the pdf method is lost as the underlying neuron model is made more realistic and the number of state variables increases in a detailed theoretical and computational study we elucidate strengths and weaknesses of dimension reduction by a particular moment closure method cai tao shelley mclaughlin 2004 cai tao rangan mclaughlin 2006 as applied to integrate and fire neurons that receive excitatory synaptic input only when the unitary postsynaptic conductance event has a single exponential time course the evolution equation for the pdf is a partial differential integral equation in two state variables voltage and excitatory conductance in the moment closure method one approximates the conditional kth centered moment of excitatory conductance given voltage by the corresponding unconditioned moment the result is a system of k coupled partial differential equations with one state variable voltage and k coupled ordinary differential equations moment closure at k 2 works well and at k 3 works even better in the regime of high dynamically varying synaptic input rates both closures break down at lower synaptic input rates phase plane analysis of the k 2 problem with typical parameters proves and reveals why no steady state solutions exist below a synaptic input rate that gives a firing rate of 59 s 1 in the full 2d problem closure at k 3 fails for similar reasons low firing rate solutions can be obtained only with parameters for the amplitude or kinetics or both of the unitary postsynaptic conductance event that are on the edge of the physiological range we conclude that this dimension reduction method gives ill posed problems for a wide range of physiological parameters and we suggest future directions

clustering in globally coupled inhibitory neurons abstract a model of a large population of identical excitable neurons with a global slowly decaying inhibitory coupling is studied and its patterns of synchrony are examined in addition to converging to a homogeneous fixed point and a homogeneous limit cycle the system exhibits cluster states in which it breaks spontaneously into a few macroscopically big clusters each of which is fully synchronized a method for calculating the stability of cluster states is described and used for investigating the dynamical behavior of the network versus the parameters that describe the neurons and synapses effects of stochastic noise on the network dynamics are discussed at large enough noise the system goes to a globally stationary state low levels of noise preserve the cluster like neuron trajectories in the regime where a noiseless system converges to a fully synchronized periodic state relatively low noise levels cause neurons to burst only every two or more consecutive time periods

finite size dynamics of inhibitory and excitatory interacting spiking neurons the dynamic mean field approach we recently developed is extended to study the dynamics of population emission rates nstd for a finite network of coupled excitatory sed and inhibitory sid integrate and fire if neurons the power spectrum of nstd in an asynchronous state is computed and compared to simulations we calculate the interpopulations transfer functions and show how synaptic interaction modulates the otherwise low pass filter with resonances which go well beyond the filter s cut sv nd allowing efficient information transmission on very short time scales determined by spike transmission delays the saddle node instability of the asynchronous state is studied and a simple exact dependence of the stability condition on the current to rate gain functions is derived by which self couplings ee and ii decrease stability while mutual interaction ei and ie favor stability realistic models of interacting spiking neurons must ultimately encompass multiple interacting modules for which the intermodule and intramodule connectivity pattern and synaptic couplings define the architecture of interest in a specific setting even at the crude modeling level of integrateand fire if current driven neurons with instantaneous synaptic input predicting the dynamical properties of multiple interacting neural populations is a difficult task it has been increasingly recognized that noisy regimes in which the collective firing of a population of neurons exhibits important fluctuations provide a key for the description of biologically relevant dynamic phenomena e g fast collective oscillations with frequencies largely exceeding the single neuron firing rate 1 3 the diffusion approximation has been widely adopted for the description of the noisy dynamics of a single neuron or a homogeneous population of if neurons a few attempts have dealt so far with multiple populations of interacting if neurons in this brief report by extending an approach introduced in 4 to two self and mutually interacting populations of excitatory and inhibitory neurons we attempt a further step towards the analytical and quantitative description of biologically interesting architectures besides this generic interest the approach taken here has some implications on the problem of information transmission in complex neural systems 5 6 it is known that a network of spiking neurons firing asynchronously can encode signals covering a frequency band extending well beyond the emission rate of the individual neurons in the second part of the present paper we briefly focus on the input output properties of the neural populations as linear systems characterized by suitable transfer functions both the intramodule and intermodule synaptic interactions produce resonances in the frequency response which select optimal bands for the information transmission the present work illustrates how one can analytically put in relatively simple terms the expected role of synaptic couplings in shaping the signal to noise transmission properties before describing the results we summarize the formalism referring for details to 4 where we introduced a dynamic mean field emission rate equation describing the time dependent firing activity nstd of a finite number of interacting if neurons in noisy regimes with a distribution of spike transmission delays and instantaneous synaptic currents in the diffusion approximation 7 the stochastic dynamics of an if neuron s membrane potential is described by the langevin equation v fsvd msv td ssv tdgstd where fsvd is the leakage term and msv td and s 2 sv td are the contributions to the infinitesimal mean and variance ofv due to the afferent current gstd is a white noise with zero mean and unit variance the associated fokker planck fp equation 8 for the probability psv tddv of having vstd p fv v dvg is

time structure of the activity in neural network models several neural network models in continuous time are reconsidered in the framework of a general mean field theory which is exact in the limit of a large and fully connected network the theory assumes pointlike spikes which are generated by a renewal process the effect of spikes on a receiving neuron is described by a linear response kernel which is the dominant term in a weak coupling expansion it is shown that the resulting spike response model is the most general renewal model with linear inputs the standard integrate and fire model forms a special case in a network structure with several pools of identical spiking neurons the global states and the dynamic evolution are determined by a nonlinear integral equation which describes the effective interaction within and between different pools we derive explicit stability criteria for stationary incoherent and oscillatory coherent solutions it is shown that the stationary state of noiseless systems is almost always unstable noise suppresses fast oscillations and stabilizes the system furthermore collective oscillations are stable only if the firing occurs while the synaptic potential is increasing in particular collective oscillations in a network with delayless excitatory interaction are at most semistable inhibitory interactions with short delays or excitatory interactions with long delays lead to stable oscillations our general results allow a straightforward application to different network models with spiking neurons furthermore the theory allows an estimation of the errors introduced in firing rate or graded response models

systematic fluctuation expansion for neural network activity equations population rate or activity equations are the foundation of a common approach to modeling for neural networks these equations provide mean field dynamics for the firing rate or activity of neurons within a network given some connectivity the shortcoming of these equations is that they take into account only the average firing rate while leaving out higher order statistics like correlations between firing a stochastic theory of neural networks that includes statistics at all orders was recently formulated we describe how this theory yields a systematic extension to population rate equations by introducing equations for correlations and appropriate coupling terms each level of the approximation yields closed equations they depend only on the mean and specific correlations of interest without an ad hoc criterion for doing so we show in an example of an all to all connected network how our system of generalized activity equations captures phenomena missed by the mean field rate equations alone

robust h sub infinity control for linear systems with norm bounded time varying uncertainty robust h sub infinity control design for linear systems with uncertainty in both the state and input matrices is treated a state feedback control design which stabilizes the plant and guarantees an h sub infinity norm bound constraint on disturbance attenuation for all admissible uncertainties is presented the robust h sub infinity control problem is solved via the notion of quadratic stabilization with an h sub infinity norm bound necessary and sufficient conditions for quadratic stabilization with an h sub infinity norm bound are derived the results can be regarded as extensions of existing results on h sub infinity control and robust stabilization of uncertain linear systems

temporal correlations in stochastic networks of spiking neurons the determination of temporal and spatial correlations in neuronal activity is one of the most important neurophysiological tools to gain insight into the mechanisms of information processing in the brain its interpretation is complicated by the difficulty of disambiguating the effects of architecture single neuron properties and network dynamics we present a theory that describes the contribution of the network dynamics in a network of spiking neurons for a simple neuron model including refractory properties we calculate the temporal cross correlations in a completely homogeneous excitatory fully connected network in a stable stationary state for stochastic dynamics in both discrete and continuous time we show that even for this simple network architecture the crosscorrelations exhibit a large variety of qualitatively different properties strongly dependent on the level of noise the decay constant of the refractory function and the network activity at the critical point the crosscorrelations oscillate with a frequency that depends on the refractory properties or decay exponentially with a diverging damping constant for weak refractory properties we also investigate the effect of the synaptic time constants it is shown that these time constants may apart from their influence on the asymmetric peak arising from the direct synaptic connection also affect the long term properties of the cross correlations

spike train statistics and dynamics with synaptic input from any renewal process a population density approach in the probability density function pdf approach to neural network modeling a common simplifying assumption is that the arrival times of elementary postsynaptic events are governed by a poisson process this assumption ignores temporal correlations in the input that sometimes have important physiological consequences we extend pdf methods to models with synaptic event times governed by any modulated renewal process we focus on the integrate and fire neuron with instantaneous synaptic kinetics and a random elementary excitatory postsynaptic potential epsp a between presynaptic events the membrane voltage v decays exponentially toward rest while s the time since the last synaptic input event evolves with unit velocity when a synaptic event arrives v jumps by a and s is reset to zero if v crosses the threshold voltage an action potential occurs and v is reset to vreset the probability per unit time of a synaptic event at time t given the elapsed time s since the last event h s t depends on specifics of the renewal process we study how regularity of the train of synaptic input events affects output spike rate pdf and coefficient of variation cv of the interspike interval and the autocorrelation function of the output spike train in the limit of a deterministic clocklike train of input events the pdf of the interspike interval converges to a sum of delta functions with coefficients determined by the pdf for a the limiting autocorrelation function of the output spike train is a sum of delta functions whose coefficients fall under a damped oscillatory envelope when the epsp cv a a is equal to 0 45 a cv for the intersynaptic event interval t t 0 35 is functionally equivalent to a deterministic periodic train of synaptic input events cv 0 with respect to spike statistics we discuss the relevance to neural network simulations

asynchronous states in networks of pulse coupled oscillators we use a mean field approach to analyze the stability of the asynchronous state in a population of all to all pulse coupled nonlinear oscillators we determine the conditions that must be satisfied by the time constants and phase dependence characterizing the coupling between the oscillators in order for the asynchronous state to be stable we also consider the effects of noise this work complements results on synchronous states in similar models and allows us to study the validity of firing rate models commonly used for neural networks

increase of coherence in excitable systems by delayed feedback the control of coherence and spectral properties of noise induced oscillations by time delayed feedback is studied in a fitzhugh nagumo system which serves as a paradigmatic model of excitable systems a semianalytical approach based on a discrete model with waiting time densities is developed which allows one to predict quantitatively the increase of coherence measured by the correlation time and the modulation of the main frequencies of the stochastic dynamics in dependence on the delay time the analytical mean field approximation is in good agreement with numerical results for the full nonlinear model

what matters in neuronal locking exploiting local stability we show what neuronal characteristics are essential to ensure that coherent oscillations are asymptotically stable in a spatially homogeneous network of spiking neurons under standard conditions a necessary and in the limit of a large number of interacting neighbors also sufficient condition is that the postsynaptic potential is increasing in time as the neurons fire if the postsynaptic potential is decreasing oscillations are bound to be unstable this is a kind of locking theorem and boils down to a subtle interplay of axonal delays postsynaptic potentials and refractory behavior the theorem also allows for mixtures of excitatory and inhibitory interactions on the basis of the locking theorem we present a simple geometric method to verify the existence and local stability of a coherent oscillation

collective synchronization of pulse coupled oscillators and excitable units abstract a phase model is proposed and studied for a population of integrate and fire type oscillators and excitable units with pulsatile coupling the units are subject to external noise and their mutual connection is all to all an evolution equation is derived for the phase distribution in a closed form its steady state analysis in the limit of weak noise shows the possibility of collective bistability the evolution equation is simplified drastically when the coupling as well as noise is sufficiently weak this reduction of equation enables analytical calculation of various quantities among others the stability of the steady distribution is analyzed explicity and the onset of collective oscillations is described in terms of the hopf bifurcation it is also shown that the present model is improved in some respects by introducing absolute refractoriness

global competition and local cooperation in a network of neural oscillators abstract an architecture of locally excitatory globally inhibitory oscillator networks is proposed and investigated both analytically and by computer simulation the model for each oscillator corresponds to a standard relaxation oscillator with two time scales oscillators are locally coupled by a scheme that resembles excitatory synaptic coupling and each oscillator also inhibits other oscillators through a common inhibitor oscillators are driven to be oscillatory by external stimulation the network exhibits a mechanism of selective gating whereby an oscillator jumping up to its active phase rapidly recruits the oscillators stimulated by the same pattern while preventing the other oscillators from jumping up we show analytically that with the selective gating mechanism the network rapidly achieves both synchronization within blocks of oscillators that are stimulated by connected regions and desynchronization between different blocks computer simulations demonstrate the model s promising ability for segmenting multiple input patterns in real time this model lays a physical foundation for the oscillatory correlation theory of feature binding and may provide an effective computational framework for scene segmentation and figure ground segregation

aperiodic stochastic resonance stochastic resonance sr is a phenomenon wherein the response of a nonlinear system to a weak periodic input signal is optimized by the presence of a particular level of noise recently we presented a method and theory for characterizing sr type behavior in excitable systems with aperiodic i e broadband input signals phys rev e 52 r3321 1995 we coined the term aperiodic stochastic resonance asr to describe this general type of behavior in that earlier study we demonstrated asr in the fitzhugh nagumo neuronal model here we demonstrate asr in three additional systems a bistable well system an integrate and fire neuronal model and the hodgkin huxley hh neuronal model we present computational and theoretical results for each system in the context of the hh model we develop a general theory for asr in excitable membranes this work clearly shows that sr type behavior is not limited to systems with periodic inputs thus in general noise can serve to enhance the response of a nonlinear system to a weak input signal regardless of whether the signal is periodic or aperiodic textcopyright 1996 the american physical society

phase locking in weakly heterogeneous neuronal networks abstract we examine analytically the existence and stability of phase locked states in a weakly heterogeneous neuronal network we consider a model of n neurons with all to all synaptic coupling where the heterogeneity is in the intrinsic firing frequency of the individual neurons we consider both inhibitory and excitatory coupling we derive the conditions under which stable phase locking is possible in homogeneous networks many different periodic phase locked states are possible their stability depends on the dynamics of the neuron and the coupling for weak heterogeneity the phase locked states are perturbed from the homogeneous states and can remain stable if their homogeneous counterparts are stable for enough heterogeneity phase locked solutions either lose stability or are destroyed completely we analyze the possible states the network can take when phase locking is broken

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

population dynamics of spiking neurons fast transients asynchronous states and locking an integral equation describing the time evolution of the population activity in a homogeneous pool of spiking neurons of the integrate and fire type is discussed it is analytically shown that transients from a state of incoherent firing can be immediate the stability of incoherent firing is analyzed in terms of the noise level and transmission delay and a bifurcation diagram is derived the response of a population of noisy integrate and fire neurons to an input current of small amplitude is calculated and characterized by a linear filter l the stability of perfectly synchronized locked solutions is analyzed

analysis of nonlocal neural fields for both general and gamma distributed connectivities abstract this work studies the stability of equilibria in spatially extended neuronal ensembles we first derive the model equation from statistical properties of the neuron population the obtained integro differential equation includes synaptic and space dependent transmission delay for both general and gamma distributed synaptic connectivities the latter connectivity type reveals infinite finite and vanishing self connectivities the work derives conditions for stationary and nonstationary instabilities for both kernel types in addition a nonlinear analysis for general kernels yields the order parameter equation of the turing instability to compare the results to findings for partial differential equations pdes two typical pde types are derived from the examined model equation namely the general reaction diffusion equation and the swift hohenberg equation hence the discussed integro differential equation generalizes these pdes in the case of the gamma distributed kernels the stability conditions are formulated in terms of the mean excitatory and inhibitory interaction ranges as a novel finding we obtain turing instabilities in fields with local inhibition lateral excitation while wave instabilities occur in fields with local excitation and lateral inhibition numerical simulations support the analytical results

gaussian process approach to spiking neurons for inhomogeneous poisson inputs this article presents a new theoretical framework to consider the dynamics of a stochastic spiking neuron model with general membrane response to input spike we assume that the input spikes obey an inhomogeneous poisson process the stochastic process of the membrane potential then becomes a gaussian process when a general type of the membrane response is assumed the stochastic process becomes a markov gaussian process we present a calculation method for the membrane potential density and the firing probability density our new formulation is the extension of the existing formulation based on diffusion approximation although the single markov assumption of the diffusion approximation simplifies the stochastic process analysis the calculation is inaccurate when the stochastic process involves a multiple markov property we find that the variation of the shape of the membrane response which has often been ignored in existing stochastic process studies significantly affects the firing probability our approach can consider the reset effect which has been difficult to deal with by analysis based on the first passage time density

decoding spikes in a spiking neuronal network we investigate how to reliably decode the input information from the output of a spiking neuronal network a maximum likelihood estimator of the input signal together with its fisher information is rigorously calculated the advantage of the maximum likelihood estimation over the brute force rate coding estimate is clearly demonstrated it is pointed out that the ergodic assumption in neuroscience i e a temporal average is equivalent to an ensemble average is in general not true averaging over an ensemble of neurons usually gives a biased estimate of the input information a method on how to compensate for the bias is proposed reconstruction of dynamical input signals with a group of spiking neurons is extensively studied and our results show that less than a spike is sufficient to accurately decode dynamical inputs