insights from a simple expression for linear fisher information in a recurrently connected population of spiking neurons a simple expression for a lower bound of fisher information is derived for a network of recurrently connected spiking neurons that have been driven to a noise perturbed steady state we call this lower bound linear fisher information as it corresponds to the fisher information that can be recovered by a locally optimal linear estimator unlike recent similar calculations the approach used here includes the effects of nonlinear gain functions and correlated input noise and yields a surprisingly simple and intuitive expression that offers substantial insight into the sources of information degradation across successive layers of a neural network here this expression is used to 1 compute the optimal i e information maximizing firing rate of a neuron 2 demonstrate why sharpening tuning curves by either thresholding or the action of recurrent connectivity is generally a bad idea 3 show how a single cortical expansion is sufficient to instantiate a redundant population code that can propagate across multiple cortical layers with minimal information loss and 4 show that optimal recurrent connectivity strongly depends on the covariance structure of the inputs to the network

output of a neuronal population code in the population coding framework we consider how the response distributions affect output distribution a general theory for the output of neuronal population code is presented when the spike train is a renewal process under a given condition on the response distribution the most probable value of the output distribution is the center of input preferred values whereas in the other cases the most improbable value of the output distribution is the center of input preferred values or there are no most probable states depending on the exact form of the response distributions the variance of the output distributions can either enlarge or reduce the tuning width of the tuning curves

on the generation of mono implicit runge kutta nystro m methods by mono implicit runge kutta methods mono implicit runge kutta methods can be used to generate implicit runge kutta nystrom irkn methods for the numerical solution of systems of second order differential equations the paper is concerned with the investigation of the conditions to be fulfilled by the mono implicit runge kutta mirk method in order to generate a mono implicit runge kutta nystrom method mirkn that is p stable one of the main theoretical results is the property that mirk methods in standard form cannot generate mirkn methods in standard form of order greater than 4 many examples of mirkn methods generated by mirk methods are presented

neural population densities shape network correlations the way sensory microcircuits manage cellular response correlations is a crucial question in understanding how such systems integrate external stimuli and encode information most sensory systems exhibit heterogeneities in terms of population sizes and features which all impact their dynamics this work addresses how correlations between the dynamics of neural ensembles depend on the relative size or density of excitatory and inhibitory populations to do so we study an apparently symmetric system of coupled stochastic differential equations that model the evolution of the populations activities excitatory and inhibitory populations are connected by reciprocal recurrent connections and both receive different stimuli exhibiting a certain level of correlation with each other a stability analysis is performed which reveals an intrinsic asymmetry in the distribution of the fixed points with respect to the threshold of the nonlinearities based on this we show how the cross correlation between the population responses depends on the density of the inhibitory population and that a specific ratio between both population sizes leads to a state of zero correlation we show that this so called asynchronous state subsists despite the presence of stimulus correlation and most importantly that it occurs only in asymmetrical systems where one population outnumbers the other using linear approximations we derive analytical expressions for the root of the cross correlation function and study how the asynchronous state is impacted by the model s parameters this work suggests a possible explanation for why inhibitory cells outnumber excitatory cells in the visual system

fisher information and stochastic complexity by taking into account the fisher information and removing an inherent redundancy in earlier two part codes a sharper code length as the stochastic complexity and the associated universal process are derived for a class of parametric processes the main condition required is that the maximum likelihood estimates satisfy the central limit theorem the same code length is also obtained from the so called maximum likelihood code

multiscale codes in the nervous system the problem of noise correlations and the ambiguity of periodic scales encoding information about continuous variables using noisy computational units is a challenge nonetheless asymptotic theory shows that combining multiple periodic scales for coding can be highly precise despite the corrupting influence of noise mathis herz and stemmler phys rev lett 109 018103 2012 indeed the cortex seems to use periodic multiscale grid codes to represent position accurately here we show how such codes can be read out without taking the long term limit even on short time scales the precision of such codes scales exponentially in the number n of neurons does this finding also hold for neurons that are not firing in a statistically independent fashion to assess the extent to which biological grid codes are subject to statistical dependences we first analyze the noise correlations between pairs of grid code neurons in behaving rodents we find that if the grids of two neurons align and have the same length scale the noise correlations between the neurons can reach values as high as 0 8 for increasing mismatches between the grids of the two neurons the noise correlations fall rapidly incorporating such correlations into a population coding model reveals that the correlations lessen the resolution but the exponential scaling of resolution with n is unaffected

computation in a single neuron hodgkin and huxley revisited a spiking neuron computes by transforming a complex dynamical input into a train of action potentials or spikes the computation performed by the neuron can be formulated as dimensional reduction or feature detection followed by a nonlinear decision function over the low dimensional space generalizations of the reverse correlation technique with white noise input provide a numerical strategy for extracting the relevant low dimensional features from experimental data and information theory can be used to evaluate the quality of the low dimensional approximation we apply these methods to analyze the simplest biophysically realistic model neuron the hodgkin huxley hh model using this system to illustrate the general methodological issues we focus on the features in the stimulus that trigger a spike explicitly eliminating the effects of interactions between spikes one can approximate this triggering feature space as a two dimensional linear subspace in the high dimensional space of input histories capturing in this way a substantial fraction of the mutual information between inputs and spike time we find that an even better approximation however is to describe the relevant subspace as two dimensional but curved in this way we can capture 90 of the mutual information even at high time resolution our analysis provides a new understanding of the computational properties of the hh model while it is common to approximate neural behavior as integrate and fire the hh model is not an integrator nor is it well described by a single threshold

fisher information and temporal correlations for spiking neurons with stochastic dynamics population coding accuracy can be studied using fisher information here the fisher information and correlation functions are determined analytically for a network of coupled spiking neurons with a more general than poisson stochastic dynamics it is shown that stimulus driven temporal correlations between neurons always increase the fisher information whereas stimulus independent correlations need not do so additionally we find that for subthreshold stimuli there is some nonzero level of noise for which network coding is optimal we also find that the fisher information is larger for purely excitatory than for purely inhibitory networks but only in a limited range of values of synaptic coupling strengths in most cases the dependence of the fisher information on time is linear except for excitatory networks with strong synaptic couplings and for strong stimuli in the latter case this dependence shows two distinct regimes fast and slow for excitatory networks short term synaptic depression can improve the coding accuracy significantly whereas short term facilitation can lower the coding accuracy for inhibitory networks coding accuracy is insensitive to short term synaptic dynamics

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

representational accuracy of stochastic neural populations fisher information is used to analyze the accuracy with which a neural population encodes d stimulus features it turns out that the form of response variability has a major impact on the encoding capacity and therefore plays an important role in the selection of an appropriate neural model in particular in the presence of baseline firing the reconstruction error rapidly increases with d in the case of poissonian noise but not for additive noise the existence of limited range correlations of the type found in cortical tissue yields a saturation of the fisher information content as a function of the population size only for an additive noise model we also show that random variability in the correlation coefficient within a neural population as found empirically considerably improves the average encoding quality finally the representational accuracy of populations with inhomogeneous tuning properties either with variability in the tuning widths or fragmented into specialized subpopulations is superior to the case of identical and radially symmetric tuning curves usually considered in the literature

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

high orderp stable formulae for the numerical integration of periodic initial value problems recently there has been considerable interest in the approximate numerical integration of the special initial value problemy f x y for cases where it is known in advance that the required solution is periodic the well known class of stormer cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems an appropriate stability requirement for the numerical integration of periodic problems is that ofp stability however lambert and watson have shown that ap stable linear multistep method cannot have an order of accuracy greater than 2 in the present paper a class of 2 step methods of runge kutta type is discussed for the numerical solution of periodic initial value problems p stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods

mutual information of population codes and distance measures in probability space we studied the mutual information between a stimulus and a system consisting of stochastic statistically independent elements that respond to a stimulus using statistical mechanical methods the properties of the mutual information mi in the limit of a large system size n are calculated for continuous valued stimuli the mi increases logarithmically with n and is related to the log of the fisher information of the system for discrete stimuli the mi saturates exponentially with n we find that the exponent of saturation of the mi is the chernoff distance between response probabilities that are induced by different stimuli

neuronal discrimination capacity we explore neuronal mechanisms of discriminating between masked signals it is found that when the correlation between input signals is zero the output signals are separable if and only if input signals are separable with positively negatively correlated signals the output signals are separable mixed even when input signals are mixed separable exact values of discrimination capacity are obtained for two most interesting cases the exactly balanced inhibitory and excitatory input case and the uncorrelated input case interestingly the discrimination capacity obtained in these cases is independent of model parameters input distribution and is universal our results also suggest a functional role of inhibitory inputs and correlated inputs or more generally the large variability of efferent spike trains observed in in vivo experiments the larger the variability of efferent spike trains the easier it is to discriminate between masked input signals

fundamental limits on persistent activity in networks of noisy neurons neural noise limits the fidelity of representations in the brain this limitation has been extensively analyzed for sensory coding however in short term memory and integrator networks where noise accumulates and can play an even more prominent role much less is known about how neural noise interacts with neural and network parameters to determine the accuracy of the computation here we analytically derive how the stored memory in continuous attractor networks of probabilistically spiking neurons will degrade over time through diffusion by combining statistical and dynamical approaches we establish a fundamental limit on the network s ability to maintain a persistent state the noise induced drift of the memory state over time within the network is strictly lower bounded by the accuracy of estimation of the network s instantaneous memory state by an ideal external observer this result takes the form of an information diffusion inequality we derive some unexpected consequences despite the persistence time of short term memory networks it does not pay to accumulate spikes for longer than the cellular time constant to read out their contents for certain neural transfer functions the conditions for optimal sensory coding coincide with those for optimal storage implying that short term memory may be co localized with sensory representation

response properties of an integrate and fire model that receives subthreshold inputs a computational technique is described for calculation of the interspike interval and poststimulus time histograms for the responses of an integrate and fire model to arbitrary inputs the effects of the model parameters on the response statistics were studied systematically specifically the probability distribution of the membrane potential was calculated as a function of time and the mean interspike interval and pst histogram were calculated for arbitrary inputs for stationary inputs the regularity of the output was studied in detail for various model parameters for nonstationary inputs the effects of the model parameters on the output synchronization index were explored the results show that enhanced synchronization in response to low frequency stimuli required a large number n 25 of weak inputs irregular responses and a linear input output rate relationship required strong but subthreshold inputs with a small time constant a model cell with mixed amplitude synaptic inputs can respond to stationary inputs irregularly and have enhanced synchronization to nonstationary inputs that are phase locked to low frequency inputs both of these response properties have been reported for some cells in the ventral cochlear nucleus in the auditory brainstem

estimates for operator norms on weighted spaces and reverse jensen inequalities we examine the dependence on the a p norm of w of the operator norms of singular integrals maximal functions and other operators in l p w we also examine connections between some fairly general reverse jensen inequalities and the a p and rh p weight conditions

analytical investigation of the effects of lateral connections on the accuracy of population coding we studied how lateral connections affect the accuracy of a population code by using a model of orientation selectivity in the primary visual cortex investigating the effects of lateral connections on population coding is a complex problem because these connections simultaneously change the shape of tuning curves and correlations between neurons both of these changes caused by lateral connections have to be taken into consideration to correctly evaluate their effects we propose a theoretical framework for analytically computing the fisher information which measures the accuracy of a population code in stochastic spiking neuron models with refractory periods within our framework we accurately evaluated both the changes in tuning curves and correlations caused by lateral connections and their effects on the fisher information we found that their effects conflicted with each other and the answer to whether or not the lateral connections increased the fisher information strongly depended on the intrinsic properties of the model neuron by systematically changing the coupling strengths of excitations and inhibitions we found the parameter regions of lateral connectivities where sharpening of tuning curves through mexican hat connectivities led to an increase in information which is in contrast to some previous findings

estimating parameters of generalized integrate and fire neurons from the maximum likelihood of spike trains when a neuronal spike train is observed what can we deduce from it about the properties of the neuron that generated it a natural way to answer this question is to make an assumption about the type of neuron select an appropriate model for this type and then choose the model parameters as those that are most likely to generate the observed spike train this is the maximum likelihood method if the neuron obeys simple integrate and fire dynamics paninski pillow and simoncelli 2004 showed that its negative log likelihood function is convex and that at least in principle its unique global minimum can thus be found by gradient descent techniques many biological neurons are however known to generate a richer repertoire of spiking behaviors than can be explained in a simple integrate and fire model for instance such a model retains only an implicit through spike induced currents not an explicit memory of its input an example of a physiological situation that cannot be explained is the absence of firing if the input current is increased very slowly therefore we use an expanded model mihalas niebur 2009 which is capable of generating a large number of complex firing patterns while still being linear linearity is important because it maintains the distribution of the random variables and still allows maximum likelihood methods to be used in this study we show that although convexity of the negative log likelihood function is not guaranteed for this model the minimum of this function yields a good estimate for the model parameters in particular if the noise level is treated as a free parameter furthermore we show that a nonlinear function minimization method r algorithm with space dilation usually reaches the global minimum

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

what causes a neuron to spike the computation performed by a neuron can be formulated as a combination of dimensional reduction in stimulus space and the nonlinearity inherent in a spiking output white noise stimulus and reverse correlation the spike triggered average and spike triggered covariance are often used in experimental neuroscience to ask neurons which dimensions in stimulus space they are sensitive to and to characterize the nonlinearity of the response in this article we apply reverse correlation to the simplest model neuron with temporal dynamics the leaky integrate and fire model and find that for even this simple case standard techniques do not recover the known neural computation to overcome this we develop novel reverse correlation techniques by selectively analyzing only isolated spikes and taking explicit account of the extended silences that precede these isolated spikes we discuss the implications of our methods to the characterization of neural adaptation although these methods are developed in the context of the leaky integrate and fire model our findings are relevant for the analysis of spike trains from real neurons

population coding and the labeling problem extrinsic versus intrinsic representations current population coding methods including weighted averaging and bayesian estimation are based on extrinsic representations these require that neurons be labeled with response parameters such as tuning curve peaks or noise distributions which are tied to some external world based metric scale firing rates alone without this external labeling are insufficient to represent a variable however the extrinsic approach does not explain how such neural labeling is implemented a radically different and perhaps more physiological approach is based on intrinsic representations which have access only to firing rates because neurons are unlabeled intrinsic coding represents relative rather than absolute values of a variable we show that intrinsic coding has representational advantages including invariance categorization and discrimination and in certain situations it may also recover absolute stimulus values

decorrelation of spiking variability and improved information transfer through feedforward divisive normalization response variability is often positively correlated in pairs of similarly tuned neurons in the visual cortex many authors have considered correlated variability to prevent postsynaptic neurons from averaging across large groups of inputs to obtain reliable stimulus estimates however a simple average of variability ignores nonlinearities in cortical signal integration this study shows that feedforward divisive normalization of a neuron s inputs effectively decorrelates their variability furthermore we show that optimal linear estimates of a stimulus parameter that are based on normalized inputs are more accurate than those based on nonnormalized inputs due partly to reduced correlations and that these estimates improve with increasing population size up to several thousand neurons this suggests that neurons may possess a simple mechanism for substantially decorrelating noise in their inputs further work is needed to reconcile this conclusion with past evidence that correlated noise impairs visual perception

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

comment on a two stage fourth order almost p stable method for oscillatory problems abstract in chawla and al zanaidi j comput appl math 89 1997 115 118 a fourth order almost p stable method for y f x y is proposed we claim that it is possible to retrieve this combination of multistep methods by means of the theory of parameterized runge kutta nystrom rkn methods and moreover to generalize the method discussed by chawla and al zanaidi j comput appl math 89 1997 115 118

estimation of the input parameters in the feller neuronal model the stochastic feller neuronal model is studied and estimators of the model input parameters depending on the firing regime of the process are derived closed expressions for the first two moments of functionals of the first passage time ftp through a constant boundary in the suprathreshold regime are derived which are used to calculate moment estimators in the subthreshold regime the exponentiality of the ftp is utilized to characterize the input parameters the methods are illustrated on simulated data finally approximations of the first passage time moments are suggested and biological interpretations and comparisons of the parameters in the feller and the ornstein uhlenbeck models are discussed

generalized fisher information matrix in nonextensive systems with spatial correlation by using the q gaussian distribution derived by the maximum entropy method for spatially correlated n unit nonextensive systems we have calculated the generalized fisher information matrix of g ensuremath theta n ensuremath theta m for ensuremath theta 1 ensuremath theta 2 ensuremath theta 3 ensuremath mu q ensuremath sigma q 2 s where ensuremath mu q ensuremath sigma q 2 and s denote the mean variance and degree of spatial correlation respectively for a given entropic index q it has been shown from the cram er rao theorem that 1 an accuracy of an unbiased estimate of ensuremath mu q is improved degraded by a negative positive correlation s 2 that of ensuremath sigma q 2 is worsen with increasing s and 3 that of s is much improved for s ensuremath simeq ensuremath 1 n ensuremath 1 or s ensuremath simeq 1 0 though it is worst at s n ensuremath 2 2 n ensuremath 1 our calculation provides a clear insight to the long standing controversy whether the spatial correlation is beneficial or detrimental to decoding in neuronal ensembles we discuss also a calculation of the q gaussian distribution applying the superstatistics to the langevin model subjected to spatially correlated inputs

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

optimal population codes for space grid cells outperform place cells rodents use two distinct neuronal coordinate systems to estimate their position place fields in the hippocampus and grid fields in the entorhinal cortex whereas place cells spike at only one particular spatial location grid cells fire at multiple sites that correspond to the points of an imaginary hexagonal lattice we study how to best construct place and grid codes taking the probabilistic nature of neural spiking into account which spatial encoding properties of individual neurons confer the highest resolution when decoding the animal s position from the neuronal population response a priori estimating a spatial position from a grid code could be ambiguous as regular periodic lattices possess translational symmetry the solution to this problem requires lattices for grid cells with different spacings the spatial resolution crucially depends on choosing the right ratios of these spacings across the population we compute the expected error in estimating the position in both the asymptotic limit using fisher information and for low spike counts using maximum likelihood estimation achieving high spatial resolution and covering a large range of space in a grid code leads to a trade off the best grid code for spatial resolution is built of nested modules with different spatial periods one inside the other whereas maximizing the spatial range requires distinct spatial periods that are pairwisely incommensurate optimizing the spatial resolution predicts two grid cell properties that have been experimentally observed first short lattice spacings should outnumber long lattice spacings second the grid code should be self similar across different lattice spacings so that the grid field always covers a fixed fraction of the lattice period if these conditions are satisfied and the spatial tuning curves for each neuron span the same range of firing rates then the resolution of the grid code easily exceeds that of the best possible place code with the same number of neurons

multidimensional encoding strategy of spiking neurons neural responses in sensory systems are typically triggered by a multitude of stimulus features using information theory we study the encoding accuracy of a population of stochastically spiking neurons characterized by different tuning widths for the different features the optimal encoding strategy for representing one feature most accurately consists of narrow tuning in the dimension to be encoded to increase the single neuron fisher information and broad tuning in all other dimensions to increase the number of active neurons extremely narrow tuning without sufficient receptive field overlap will severely worsen the coding this implies the existence of an optimal tuning width for the feature to be encoded empirically only a subset of all stimulus features will normally be accessible in this case relative encoding errors can be calculated that yield a criterion for the function of a neural population based on the measured tuning curves

computing with continuous attractors stability and online aspects two issues concerning the application of continuous attractors in neural systems are investigated the computational robustness of continuous attractors with respect to input noises and the implementation of bayesian online decoding in a perfect mathematical model for continuous attractors decoding results for stimuli are highly sensitive to input noises and this sensitivity is the inevitable consequence of the system s neutral stability to overcome this shortcoming we modify the conventional network model by including extra dynamical interactions between neurons these interactions vary according to the biologically plausible hebbian learning rule and have the computational role of memorizing and propagating stimulus information accumulated with time as a result the new network model responds to the history of external inputs over a period of time and hence becomes insensitive to short term fluctuations also since dynamical interactions provide a mechanism to convey the prior knowledge of stimulus that is the information of the stimulus presented previously the network effectively implements online bayesian inference this study also reveals some interesting behavior in neural population coding such as the trade off between decoding stability and the speed of tracking time varying stimuli and the relationship between neural tuning width and the tracking speed

optimal signal estimation in neuronal models we study optimal estimation of a signal in parametric neuronal models on the basis of interspike interval data fisher information is the inverse asymptotic variance of the best estimator its dependence on the parameter value indicates accuracy of estimation our models assume that the input signal is estimated from neuronal output interspike interval data where the frequency transfer function is sigmoidal if the coefficient of variation of the interspike interval is constant with respect to the signal the fisher information is unimodal and its maximum for the most estimable signal can be found we obtain a general result and compare the signal producing maximal fisher information with the inflection point of the sigmoidal transfer function in several basic neuronal models

efficient neural codes that minimize lp reconstruction error the efficient coding hypothesis assumes that biological sensory systems use neural codes that are optimized to best possibly represent the stimuli that occur in their environment most common models use information theoretic measures whereas alternative formulations propose incorporating downstream decoding performance here we provide a systematic evaluation of different optimality criteria using a parametric formulation of the efficient coding problem based on the reconstruction error of the maximum likelihood decoder this parametric family includes both the information maximization criterion and squared decoding error as special cases we analytically derived the optimal tuning curve of a single neuron encoding a one dimensional stimulus with an arbitrary input distribution we show how the result can be generalized to a class of neural populations by introducing the concept of a meta tuning curve the predictions of our framework are tested against previously measured characteristics of some early visual systems found in biology we find solutions that correspond to low values of suggesting that across different animal models neural representations in the early visual pathways optimize similar criteria about natural stimuli that are relatively close to the information maximization criterion

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

population coding and decoding in a neural field a computational study this study uses a neural field model to investigate computational aspects of population coding and decoding when the stimulus is a single variable a general prototype model for the encoding process is proposed in which neural responses are correlated with strength specified by a gaussian function of their difference in preferred stimuli based on the model we study the effect of correlation on the fisher information compare the performances of three decoding methods that differ in the amount of encoding information being used and investigate the implementation of the three methods by using a recurrent network this study not only rediscovers main results in existing literatures in a unified way but also reveals important new features especially when the neural correlation is strong as the neural correlation of firing becomes larger the fisher information decreases drastically we confirm that as the width of correlation increases the fisher information saturates and no longer increases in proportion to the number of neurons however we prove that as the width increases further wider than 2 times the effective width of the turning function the fisher information increases again and it increases without limit in proportion to the number of neurons furthermore we clarify the asymptotic efficiency of the maximum likelihood inference mli type of decoding methods for correlated neural signals it shows that when the correlation covers a nonlocal range of population excepting the uniform correlation and when the noise is extremely small the mli type of method whose decoding error satisfies the cauchy type distribution is not asymptotically efficient this implies that the variance is no longer adequate to measure decoding accuracy

stimulus dependent correlations and population codes the magnitude of correlations between stimulus driven responses of pairs of neurons can itself be stimulus dependent we examine how this dependence affects the information carried by neural populations about the stimuli that drive them stimulus dependent changes in correlations can both carry information directly and modulate the information separately carried by the firing rates and variances we use fisher information to quantify these effects and show that although stimulus dependent correlations often carry little information directly their modulatory effects on the overall information can be large in particular if the stimulus dependence is such that correlations increase with stimulus induced firing rates this can significantly enhance the information of the population when the structure of correlations is determined solely by the stimulus however in the presence of additional strong spatial decay of correlations such stimulus dependence may have a negative impact opposite relationships hold when correlations decrease with firing rates

switch of encoding characteristics in single neurons by subthreshold and suprathreshold stimuli spike triggered analysis is a statistical method used to elucidate encoding properties in neural systems by estimating the statistical structure of input stimulus preceding spikes a recent numerical study suggested that the profile of the spike triggered average sta changes depending on whether the mean input stimuli are subthreshold or suprathreshold here we analytically verify the difference between subthreshold sta and suprathreshold sta by using the spike response model srm we show by moment expansion that the suprathreshold sta is proportional to the first derivative of the response kernel and that the subthreshold sta is expressed by a linear combination of the response kernel and its first derivative we verify whether the analytical results obtained from the srm can be applied to a multicompartment model with hodgkin huxley type dynamics

a mono implicit runge kutta nystrom modification of the numerov method we present two two parameter families of fourth order mono implicit runge kutta nystrom methods each member of these families can be considered as a modification of the numerov method we analyze the stability and periodicity properties of these methods it is shown that i within one of these families there exist a stable even l stable and p stable methods and ii in both families there exist methods with a phase lag of order six

a causal perspective on the analysis of signal and noise correlations and their role in population coding the role of correlations between neuronal responses is crucial to understanding the neural code a framework used to study this role comprises a breakdown of the mutual information between stimuli and responses into terms that aim to account for different coding modalities and the distinction between different notions of independence here we complete the list of types of independence and distinguish activity independence related to total correlations conditional independence related to noise correlations signal independence related to signal correlations coding independence related to information transmission and information independence related to redundancy for each type we identify the probabilistic criterion that defines it indicate the information theoretic measure used as statistic to test for it and provide a graphical criterion to recognize the causal configurations of stimuli and responses that lead to its existence using this causal analysis we first provide sufficiency conditions relating these types second we differentiate the use of the measures as statistics to test for the existence of independence from their use for quantification we indicate that signal and noise correlation cannot be quantified separately third we explicitly define alternative system configurations used to construct the measures in which noise correlations or noise and signal correlations are eliminated accordingly we examine which measures are meaningful only as a comparison across configurations and which ones provide a characterization of the actually observed responses without resorting to other configurations fourth we compare the commonly used nonparametric approach to eliminate noise correlations with a functional model based approach showing that the former approach does not remove those effects of noise correlations captured by the tuning properties of the individual neurons and implies nonlocal causal structure manipulations these results improve the interpretation of the measures on the framework and help in understanding how to apply it to analyze the role of correlations

dynamics of moment neuronal networks a theoretical framework is developed for moment neuronal networks mnns within this framework the behavior of the system of spiking neurons is specified in terms of the first and second order statistics of their interspike intervals i e the mean the variance and the cross correlations of spike activity since neurons emit and receive spike trains which can be described by renewal but generally non poisson processes we first derive a suitable diffusion type approximation of such processes two approximation schemes are introduced the usual approximation scheme uas and the ornstein uhlenbeck scheme it is found that both schemes approximate well the input output characteristics of spiking models such as the if and the hodgkin huxley models the mnn framework is then developed according to the uas scheme and its predictions are tested on a few examples

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

optimal control of neuronal activity we investigate the optimal control of neuronal spiking activity for neurons receiving a class of random synaptic inputs characterized by a positive parameter alpha optimal control signals and optimal variances are found exactly for the diffusion process approximating an integrate and fire model when synaptic inputs are sub poisson alpha 0 5 inputs the optimal control is smooth and unique the optimal variance obtained in the current paper sets the lowest possible bound in controlling the stochasticity of neuronal activity we also discuss how to implement the optimal control signal for certain model neurons

analysis of integrate and fire neurons synchronization of synaptic input and spike output a new technique for analyzing the probability distribution of output spikes for the integrate and fire model is presented this technique enables us to investigate models with arbitrary synaptic response functions that incorporate both leakage across the membrane and a rise time of the postsynaptic potential the results which are compared with numerical simulations are exact in the limit of a large number of small amplitude inputs this method is applied to the synchronization problem in which we examine the relationship between the spread in arrival times of the inputs the temporal jitter of the synaptic input and the resultant spread in the times at which the output spikes are generated output jitter the results of previous studies which indicated that the ratio of the output jitter to the input jitter is consistently less than one and that it decreases for increasing numbers of inputs are confirmed for three classes of the integrate and fire model in addition to the previously identified factors of axonal propagation times and synaptic jitter we identify the variation in the spike generating thresholds of the neurons and the variation in the number of active inputs as being important factors that determine the timing jitter in layered networks previously observed phase differences between optimally and suboptimally stimulated neurons may be understood in terms of the relative time taken to reach threshold

population coding in neuronal systems with correlated noise neuronal representations of external events are often distributed across large populations of cells we study the effect of correlated noise on the accuracy of these neuronal population codes our main question is whether the inherent error in the population code can be suppressed by increasing the size of the population n in the presence of correlated noise we address this issue using a model of a population of neurons that are broadly tuned to an angular variable in two dimensions the fluctuations in the neuronal activities are modeled as gaussian noises with pairwise correlations that decay exponentially with the difference between the preferred angles of the correlated cells we assume that the system is broadly tuned which means that both the correlation length and the width of the tuning curves of the mean responses span a substantial fraction of the entire system length the performance of the system is measured by the fisher information fi which bounds its estimation error by calculating the fi in the limit of a large n we show that positive correlations decrease the estimation capability of the network relative to the uncorrelated population the information capacity saturates to a finite value as the number of cells in the population grows in contrast negative correlations substantially increase the information capacity of the neuronal population these results are supplemented by the effect of correlations on the mutual information of the system our analysis provides an estimate of the effective number of statistically independent degrees of freedom denoted n mathrm eff that a large correlated system can have according to our theory n mathrm eff remains finite in the limit of a large n estimating the parameters of the correlations and tuning curves from experimental data in some cortical areas that code for angles we predict that the number of effective degrees of freedom embedded in localized populations in these areas is less than or of the order of ensuremath approx 10 2

the theory of weights and the dirichlet problem for elliptic equations in this work we consider questions from the area of partial differential equations and the related questions from harmonic analysis specifically we will study the dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients roughly speaking we consider the following question suppose that we are given two operators lo and li and that we have good estimates for the dirichlet problem for lo good in this context could mean several different things for instance it might mean that there exists p 1 p oo so that the dirichlet problem is solvable in a suitable sense with lp data what are the optimal conditions on the difference between the coefficients of lo and l1 which guarantee that the dirichlet problem for li also has good estimates this question has been extensively studied recently cf f j k r f and d4 our work uses in a crucial way the results in r f the main results in d4 and r f are proved using an ingenious differential inequality for a family of harmonic measures introduced in d4 another of our goals in this article is to give new direct proofs of the results in d4 and r f without the use of this differential inequality theorems 2 4 2 5 and 2 18 these direct approaches have already had important applications k p2 an interesting class of examples for the theorems quoted above arises when one pulls back the laplacian via a quasiconformal mapping of the plane into itself specializing our theorem to this case suggests a new characterization of the class of am weights introduced in connection with several problems in harmonic analysis by muckenhoupt mul and coifman c fefferman co f we shall present in theorem 2 3 a criterion in terms of carleson measures which is very close to log w e bmo and is necessary and sufficient for w e ad

explicit runge kutta nystro m methods with reduced phase errors for computing oscillating solutions we construct explicit runge kutta nystrom methods for the integration of first and second order differential equations having an oscillatory solution special attention is paid to the phase errors or dispersion of the dominant components in the numerical oscillations when these methods are applied to a linear homogeneous test model rk n methods are constructed which are dispersive of orders up to 10 whereas the algebraic order of accuracy is only 2 or 3 application of these methods to equations describing free and weakly forced oscillations and to semidiscretized hyperbolic equations reveals that the phase errors can significantly be reduced

piecewise linear differential equations and integrate and fire neurons insights from two dimensional membrane models we derive and study two dimensional generalizations of integrate and fire models which can be found from a piecewise linear idealization of the fitzhugh nagumo or morris lecar model these models give rise to new properties not present in one dimensional integrate and fire models a detailed analytical study of the models is presented in particular i for the piecewise linear fitzhugh nagumo model we determine analytically the bistability regime between stationary solutions and oscillations that is typical for class ii models ii in the piecewise morris lecar model we find a noncanonical class i transition from a stationary state to oscillations with logarithmic dependence similar to that found for leaky integrate and fire models iii furthermore we establish a relation to the recently proposed resonate and fire model and show that a short input current pulse can trigger several spikes

theory of correlations in stochastic neural networks one of the main experimental tools in probing the interactions between neurons has been the measurement of the correlations in their activity in general however the interpretation of the observed correlations is difficult since the correlation between a pair of neurons is influenced not only by the direct interaction between them but also by the dynamic state of the entire network to which they belong thus a comparison between the observed correlations and the predictions from specific model networks is needed in this paper we develop a theory of neuronal correlation functions in large networks comprising several highly connected subpopulations and obeying stochastic dynamic rules when the networks are in asynchronous states the cross correlations are relatively weak i e their amplitude relative to that of the autocorrelations is of order of 1 n n being the size of the interacting populations using the weakness of the cross correlations general equations that express the matrix of cross correlations in terms of the mean neuronal activities and the effective interaction matrix are presented the effective interactions are the synaptic efficacies multiplied by the gain of the postsynaptic neurons the time delayed cross correlation matrix can be expressed as a sum of exponentially decaying modes that correspond to the nonorthogonal eigenvectors of the effective interaction matrix the theory is extended to networks with random connectivity such as randomly dilute networks this allows for a comparison between the contribution from the internal common input and that from the direct interactions to the correlations of monosynaptically coupled pairs a closely related quantity is the linear response of the neurons to external time dependent perturbations we derive the form of the dynamic linear response function of neurons in the above architecture in terms of the eigenmodes of the effective interaction matrix the behavior of the correlations and the linear response when the system is near a bifurcation point is analyzed near a saddle node bifurcation the correlation matrix is dominated by a single slowly decaying critical mode near a hopf bifurcation the correlations exhibit weakly damped sinusoidal oscillations the general theory is applied to the case of a randomly dilute network consisting of excitatory and inhibitory subpopulations using parameters that mimic the local circuit of 1 mathrm mm 3 of the rat neocortex both the effect of dilution as well as the influence of a nearby bifurcation to an oscillatory state are demonstrated

optimal short term population coding when fisher information fails efficient coding has been proposed as a first principle explaining neuronal response properties in the central nervous system the shape of optimal codes however strongly depends on the natural limitations of the particular physical system here we investigate how optimal neuronal encoding strategies are influenced by the finite number of neurons n place constraint the limited decoding time window length t time constraint the maximum neuronal firing rate fmax power constraint and the maximal average rate f max energy constraint while fisher information provides a general lower bound for the mean squared error of unbiased signal reconstruction its use to characterize the coding precision is limited analyzing simple examples we illustrate some typical pitfalls and thereby show that fisher information provides a valid measure for the precision of a code only if the dynamic range fmint fmaxt is sufficiently large in particular we demonstrate that the optimal width of gaussian tuning curves depends on the available decoding time t within the broader class of unimodal tuning functions it turns out that the shape of a fisher optimal coding scheme is not unique we solve this ambiguity by taking the minimum mean square error into account which leads to fiat tuning curves the tuning width however remains to be determined by energy constraints rather than by the principle of efficient coding

sample skewness as a statistical measurement of neuronal tuning sharpness we propose using the statistical measurement of the sample skewness of the distribution of mean firing rates of a tuning curve to quantify sharpness of tuning for some features like binocular disparity tuning curves are best described by relatively complex and sometimes diverse functions making it difficult to quantify sharpness with a single function and parameter skewness provides a robust nonparametric measure of tuning curve sharpness that is invariant with respect to the mean and variance of the tuning curve and is straightforward to apply to a wide range of tuning including simple orientation tuning curves and complex object tuning curves that often cannot even be described parametrically because skewness does not depend on a specific model or function of tuning it is especially appealing to cases of sharpening where recurrent interactions among neurons produce sharper tuning curves that deviate in a complex manner from the feedforward function of tuning since tuning curves for all neurons are not typically well described by a single parametric function this model independence additionally allows skewness to be applied to all recorded neurons maximizing the statistical power of a set of data we also compare skewness with other nonparametric measures of tuning curve sharpness and selectivity compared to these other nonparametric measures tested skewness is best used for capturing the sharpness of multimodal tuning curves defined by narrow peaks maximum and broad valleys minima finally we provide a more formal definition of sharpness using a shape based information gain measure and derive and show that skewness is correlated with this definition

a mixture model for population codes of gabor filters population coding is a coding scheme which is ubiquitous in neural systems and is also of more general use in coding stimuli for example in vision problems a population of responses to a stimulus can be used to represent not only the value of some variable in the environment but a full probability distribution for that variable the information is held in a distributed and encoded form which may in some situations be more robust to noise and failures than conventional representations gabor filters are a popular choice for detecting edges in the visual field for several reasons they are easily tuned for a variety of edge widths and orientations and are considered a close model of the edge filters in the human visual system in this paper we consider population codes of gabor filters with different orientations a probabilistic model of gabor filter responses is presented based on the analytically derived orientation tuning function and a parametric mixture model of the filter responses in the presence of local edge structure with single or multiple orientations a probability density function pdf of the local orientation in any point x y can be extracted through a parameter estimation procedure the resulting pdf of the local contour orientation captures not only angular information at edges corners or t junctions but also describes the certainty of the measurement which can be characterized in terms of the entropy of the individual mixture components

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

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

information transfer for small amplitude signals we study the optimality conditions of information transfer in systems with memory in the low signal to noise ratio regime of vanishing input amplitude we find that the optimal mutual information is represented by a maximum variance of the signal time course with correlation structure determined by the fisher information matrix we provide illustration of the method on a simple biologically inspired model of electrosensory neuron our general results apply also to the study of information transfer in single neurons subject to weak stimulation with implications to the problem of coding efficiency in biological systems

replica free evaluation of the neuronal population information with mixed continuous and discrete stimuli from the linear to the asymptotic regime recent studies have explored theoretically the ability of populations of neurons to carry information about a set of stimuli both in the case of purely discrete or purely continuous stimuli and in the case of multidimensional continuous angular and discrete correlates in the presence of additional quenched disorder in the distribution an analytical expression for the mutual information has been obtained in the limit of large noise by means of the replica trick here we show that the same results can actually be obtained in most cases without the use of replicas by means of a much simpler expansion of the logarithm fitting the theoretical model to real neuronal data we show that the introduction of correlations in the quenched disorder improves the fit suggesting a possible role of signal correlations actually detected in real data in a redundant code we show that even in the more difficult analysis of the asymptotic regime an explicit expression for the mutual information can be obtained without resorting to the replica trick despite the presence of quenched disorder both with a gaussian and with a more realistic thresholded gaussian model when the stimuli are mixed continuous and discrete we find that with both models the information seem to grow logarithmically to infinity with the number of neurons and with the inverse of the noise even though the exact general dependence cannot be derived explicitly for the thresholded gaussian model in the large noise limit lower values of information were obtained with the thresholded gaussian model for a fixed value of the noise and of the population size on the contrary in the asymptotic regime with very low values of the noise a lower information value is obtained with the gaussian model

correlated firing in a feedforward network with mexican hat type connectivity we report on deterministic and stochastic evolutions of firing states through a feedforward neural network with mexican hat type connectivity the prevalence of columnar structures in a cortex implies spatially localized connectivity between neural pools although feedforward neural network models with homogeneous connectivity have been intensively studied within the context of the synfire chain the effect of local connectivity has not yet been studied so thoroughly when a neuron fires independently the dynamics of macroscopic state variables a firing rate and spatial eccentricity of a firing pattern is deterministic from the law of large numbers possible stable firing states which are derived from deterministic evolution equations are uniform localized and nonfiring the multistability of these three states is obtained where the excitatory and inhibitory interactions among neurons are balanced when the presynapse dependent variance in connection efficacies is incorporated into the network the variance generates common noise then the evolution of the macroscopic state variables becomes stochastic and neurons begin to fire in a correlated manner due to the common noise the correlation structure that is generated by common noise exhibits a nontrivial bimodal distribution the development of a firing state through neural layers does not converge to a certain fixed point but keeps on fluctuating

parameter extraction from population codes a critical assessment in perceptual systems a stimulus parameter can be extracted by determining the center of gravity of the response profile of a population of neural sensors likewise at the motor end of a neural system center of gravity decoding also known as vector decoding generates a movement direction from the neural activation profile we evaluate these schemes from a statistical perspective by comparing their statistical variance with the minimum variance possible for an unbiased parameter extraction from the noisy neuronal ensemble activation profile center of gravity decoding can be statistically optimal this is the case for regular arrays of sensors with gaussian tuning profiles that have an output described by poisson statistics and for arrays of sensors with a sinusoidal tuning profile for the angular parameter estimated however there are also many cases in which center of gravity decoding is highly inefficient this includes the important case where sensor positions are very irregular finally we study the robustness of center of gravity decoding against response nonlinearities at different stages of an information processing hierarchy we conclude that in neural systems instead of representing a parameter explicitly it is safer to leave the parameter coded implicitly in a neuronal ensemble activation profile

an explicit hybrid method of numerov type for second order periodic initial value problems abstract we consider a two parameter family of explicit hybrid methods of numerov type for the numerical integration of second order intial value problems when these methods are applied to the linear equation y 2 y 0 0 we determine the parameters so that the phase lag frequency distortion of the method is minimal the resulting method has algebraic order 4 and a small frequency distortion of size 1 3628800 v 8 v h h being the step size and in addition it possesses an interval of periodicity of size 4 63 which is larger than the interval of periodicity corresponding to the explicit method of chawla and rao 1986 the application of this method to equations describing free and weakly forced oscillations reveals its superiority over other methods

nonlinear population codes theoretical and experimental studies of distributed neuronal representations of sensory and behavioral variables usually assume that the tuning of the mean firing rates is the main source of information however recent theoretical studies have investigated the effect of crosscorrelations in the trial to trial fluctuations of the neuronal responses on the accuracy of the representation assuming that only the first order statistics of the neuronal responses are tuned to the stimulus these studies have shown that in the presence of correlations similar to those observed experimentally in cortical ensembles of neurons the amount of information in the population is limited yielding nonzero error levels even in the limit of infinitely large populations of neurons in this letter we study correlated neuronal populations whose higher order statistics and in particular response variances are also modulated by the stimulus we ask two questions does the correlated noise limit the accuracy of the neuronal representation of the stimulus and how can a biological mechanism extract most of the information embedded in the higher order statistics of the neuronal responses specifically we address these questions in the context of a population of neurons coding an angular variable we show that the information embedded in the variances grows linearly with the population size despite the presence of strong correlated noise this information cannot be extracted by linear readout schemes including the linear population vector instead we propose a bilinear readout scheme that involves spatial decorrelation quadratic non linearity and population vector summation we show that this nonlinear population vector scheme yields accurate estimates of stimulus parameters with an efficiency that grows linearly with the population size this code can be implemented using biologically plausible neurons

cosine tuning minimizes motor errors cosine tuning is ubiquitous in the motor system yet a satisfying explanation of its origin is lacking here we argue that cosine tuning minimizes expected errors in force production which makes it a natural choice for activating muscles and neurons in the final stages of motor processing our results are based on the empirically observed scaling of neuromotor noise whose standard deviation is a linear function of the mean such scaling predicts a reduction of net force errors when redundant actuators pull in the same direction we confirm this prediction by comparing forces produced with one versus two hands and generalize it across directions under the resulting neuromotor noise model we prove that the optimal activation profile is a possibly truncated cosine for arbitrary dimensionality of the workspace distribution of force directions correlated or uncorrelated noise with or without a separate cocontraction command the model predicts a negative force bias truncated cosine tuning at low muscle cocontraction levels and misalignment of preferred directions and lines of action for nonuniform muscle distributions all predictions are supported by experimental data

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

representational capacity of a set of independent neurons the capacity with which a system of independent neuron like units represents a given set of stimuli is studied by calculating the mutual information between the stimuli and the neural responses both discrete noiseless and continuous noisy neurons are analyzed in both cases the information grows monotonically with the number of neurons considered under the assumption that neurons are independent the mutual information rises linearly from zero and approaches exponentially its maximum value we find the dependence of the initial slope on the number of stimuli and on the sparseness of the representation

efficiency characterization of a large neuronal network a causal information approach abstract when inhibitory neurons constitute about 40 of neurons they could have an important antinociceptive role as they would easily regulate the level of activity of other neurons we consider a simple network of cortical spiking neurons with axonal conduction delays and spike timing dependent plasticity representative of a cortical column or hypercolumn with a large proportion of inhibitory neurons each neuron fires following a hodgkin huxley like dynamics and it is interconnected randomly to other neurons the network dynamics is investigated estimating bandt and pompe probability distribution function associated to the interspike intervals and taking different degrees of interconnectivity across neurons more specifically we take into account the fine temporal structures of the complex neuronal signals not just by using the probability distributions associated to the interspike intervals but instead considering much more subtle measures accounting for their causal information the shannon permutation entropy fisher permutation information and permutation statistical complexity this allows us to investigate how the information of the system might saturate to a finite value as the degree of interconnectivity across neurons grows inferring the emergent dynamical properties of the system

neuronal tuning to sharpen or broaden sensory and motor variables are typically represented by a population of broadly tuned neurons a coarser representation with broader tuning can often improve coding accuracy but sometimes the accuracy may also improve with sharper tuning the theoretical analysis here shows that the relationship between tuning width and accuracy depends crucially on the dimension of the encoded variable a general rule is derived for how the fisher information scales with the tuning width regardless of the exact shape of the tuning function the probability distribution of spikes and allowing some correlated noise between neurons these results demonstrate a universal dimensionality effect in neural population coding

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

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

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

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

extracting oscillations neuronal coincidence detection with noisy periodic spike input how does a neuron vary its mean output firing rate if the input changes from random to oscillatory coherent but noisy activity what are the critical parameters of the neuronal dynamics and input statistics to answer these questions we investigate the coincidence detection properties of an integrate and fire neuron we derive an expression indicating how coincidence detection depends on neuronal parameters specifically we show how coincidence detection depends on the shape of the postsynaptic response function the number of synapses and the input statistics and we demonstrate that there is an optimal threshold our considerations can be used to predict from neuronal parameters whether and to what extent a neuron can act as a coincidence detector and thus can convert a temporal code into a rate code

a noumerov type method with minimal phase lag for the integration of second order periodic initial value abstract in a recent paper 2 we gave a noumerov type method with minimal phase lag for the integration of second order initial value problems y f t y y t0 y0 y t0 y 0 however the method given there is implicit we show here the interesting result that if the noumerov type methods of 2 are made explicit with the help of the classical second order method then there exists a selection of the free parameter for which the resulting method has a considerably small frequency distortion of size 1 40320 h 6 and also a slightly larger interval of periodicity of size 2 75 than the phase lag of size 1 12096 h 6 and interval of periodicity of size 2 71 for the implicit method 2 more interestingly it turns out that noumerov made explicit of chawla 3 also has less frequency distortion than the implicit noumerov method we shall assume a familiarity with the notation and discussion given in 2

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

difficulty of singularity in population coding fisher information has been used to analyze the accuracy of neural population coding this works well when the fisher information does not degenerate but when two stimuli are presented to a population of neurons a singular structure emerges by their mutual interactions in this case the fisher information matrix degenerates and the regularity condition ensuring the cramer rao paradigm of statistics is violated an animal shows pathological behavior in such a situation we present a novel method of statistical analysis to understand information in population coding in which algebraic singularity plays a major role the method elucidates the nature of the pathological case by calculating the fisher information we then suggest that synchronous firing can resolve singularity and show a method of analyzing the binding problem in terms of the fisher information our method integrates a variety of disciplines in population coding such as nonregular statistics bayesian statistics singularity in algebraic geometry and synchronous firing under the theme of fisher information

theory of correlation in a network with synaptic depression synaptic depression affects not only the mean responses of neurons but also the correlation of response variability in neural populations although previous studies have constructed a theory of correlation in a spiking neuron model by using the mean field theory framework synaptic depression has not been taken into consideration we expanded the previous theoretical framework in this study to spiking neuron models with short term synaptic depression on the basis of this theory we analytically calculated neural correlations in a ring attractor network with mexican hat type connectivity which was used as a model of the primary visual cortex the results revealed that synaptic depression reduces neural correlation which could be beneficial for sensory coding furthermore our study opens the way for theoretical studies on the effect of interaction change on the linear response function in large stochastic networks

efficient sensory encoding and bayesian inference with heterogeneous neural populations the efficient coding hypothesis posits that sensory systems maximize information transmitted to the brain about the environment we develop a precise and testable form of this hypothesis in the context of encoding a sensory variable with a population of noisy neurons each characterized by a tuning curve we parameterize the population with two continuous functions that control the density and amplitude of the tuning curves assuming that the tuning widths vary inversely with the cell density this parameterization allows us to solve in closed form for the information maximizing allocation of tuning curves as a function of the prior probability distribution of sensory variables for the optimal population the cell density is proportional to the prior such that more cells with narrower tuning are allocated to encode higher probability stimuli and that each cell transmits an equal portion of the stimulus probability mass we also compute the stimulus discrimination capabilities of a perceptual system that relies on this neural representation and find that the best achievable discrimination thresholds are inversely proportional to the sensory prior we examine how the prior information that is implicitly encoded in the tuning curves of the optimal population may be used for perceptual inference and derive a novel decoder the bayesian population vector that closely approximates a bayesian least squares estimator that has explicit access to the prior finally we generalize these results to sigmoidal tuning curves correlated neural variability and a broader class of objective functions these results provide a principled embedding of sensory prior information in neural populations and yield predictions that are readily testable with environmental physiological and perceptual data

markov analysis of stochastic resonance in a periodically driven integrate and fire neuron we model the dynamics of the leaky integrate and fire neuron under periodic stimulation as a markov process with respect to the stimulus phase this avoids the unrealistic assumption of a stimulus reset after each spike made in earlier papers and thus solves the long standing reset problem the neuron exhibits stochastic resonance both with respect to input noise intensity and stimulus frequency the latter resonance arises by matching the stimulus frequency to the refractory time of the neuron the markov approach can be generalized to other periodically driven stochastic processes containing a reset mechanism