fast sweeping algorithms for a class of hamilton jacobi equations we derive a godunov type numerical flux for the class of strictly convex homogeneous hamiltonians that includes h p q sqrt ap 2 bq 2 2cpq c 2 ab we combine our godunov numerical fluxes with simple gauss seidel type iterations for solving the corresponding hamilton jacobi hj equations the resulting algorithm is fast since it does not require a sorting strategy as found e g in the fast marching method in addition it providesa way to compute solutions to a class of hj equations for which the conventional fast marching method is not applicable our experiments indicate convergence after a few iterations even in rather difficult cases

a parallel two scale method for eikonal equations numerous applications of eikonal equations prompted the development of many efficient numerical algorithms the heap cell method hcm is a recent serial two scale technique that has been shown to have advantages over other serial state of the art solvers for a wide range of problems a chacon and a vladimirsky siam j sci comput 34 2012 pp a547 a578 this paper presents a parallelization of hcm for a shared memory architecture the numerical experiments in r 3 show that the parallel hcm exhibits good algorithmic behavior and scales well resulting in a very fast and practical solver

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

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

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

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

an approximation scheme for the minimum time function this paper presents an approximation scheme for the nonlinear minimum time problem with compact target the scheme is derived from a discrete dynamic programming principle and the main convergence result is obtained by applying techniques related to discontinuous viscosity solutions for hamilton jacobi equations the convergence is proved under general controllability assumptions on both the continuous time and the discrete time systems an explicit sufficient condition on the system and the target ensuring the desired controllability is given this condition is shown to be necessary and sufficient for the lipschitz continuity of the minimum time function if the target is smooth an extension to the case of a point shaped target is given

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

an adaptive high order discontinuous galerkin method with error control for the hamilton jacobi equations part i the one dimensional steady state case we propose and study an adaptive version of the discontinuous galerkin method for hamilton jacobi equations it works as follows given the tolerance and the degree of the polynomial of the approximate solution the adaptive algorithm finds a mesh on which the approximate solution has an l 1 distance to the viscosity solution no bigger than the prescribed tolerance the algorithm uses three main tools the first is an iterative solver combining the explicit runge kutta discontinuous galerkin method and the implicit newton s method that enables us to solve the hamilton jacobi equations efficiently the second is a new a posteriori error estimate based on the approximate resolution of an approximate problem for the actual error the third is a method that allows us to find a new mesh as a function of the old mesh and the ratio of the a posteriori error estimate to the tolerance we display extensive numerical evidence that indicates that for any given polynomial degree the method achieves its goal with optimal complexity independently of the tolerance this is done in the framework of one dimensional steady state model problems with periodic boundary conditions

accuracy of rate coding when shorter time window and higher spontaneous activity help it is widely accepted that neuronal firing rates contain a significant amount of information about the stimulus intensity nevertheless theoretical studies on the coding accuracy inferred from the exact spike counting distributions are rare we present an analysis based on the number of observed spikes assuming the stochastic perfect integrate and fire model with a change point representing the stimulus onset for which we calculate the corresponding fisher information to investigate the accuracy of rate coding we analyze the effect of changing the duration of the time window and the influence of several parameters of the model in particular the level of the presynaptic spontaneous activity and the level of random fluctuation of the membrane potential which can be interpreted as noise of the system the results show that the fisher information is nonmonotonic with respect to the length of the observation period this counterintuitive result is caused by the discrete nature of the count of spikes we observe also that the signal can be enhanced by noise since the fisher information is nonmonotonic with respect to the level of spontaneous activity and in some cases also with respect to the level of fluctuation of the membrane potential

impact of correlated inputs on the output of the integrate and fire model for the integrate and fire model with or without reversal potentials we consider how correlated inputs affect the variability of cellular output for both models the variability of efferent spike trains measured by coefficient of variation cv of the interspike interval is a nondecreasing function of input correlation when the correlation coefficient is greater than 0 09 the cv of the integrate and fire model without reversal potentials is always above 0 5 no matter how strong the inhibitory inputs when the correlation coefficient is greater than 0 05 cv for the integrateand fire model with reversal potentials is always above 0 5 independent of the strength of the inhibitory inputs under a given condition on correlation coefficients we find that correlated poisson processes can be decomposed into independent poisson processes we also develop a novel method to estimate the distribution density of the first passage time of the integrate and fire model

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

a fast sweeping method for eikonal equations in this paper a fast sweeping method for computing the numerical solution of eikonal equations on a rectangular grid is presented the method is an iterative method which uses upwind difference for discretization and uses gauss seidel iterations with alternating sweeping ordering to solve the discretized system the crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding eikonal equation in a certain direction simultaneously the method has an optimal complexity of o n for n grid points and is extremely simple to implement in any number of dimensions monotonicity and stability properties of the fast sweeping algorithm are proven convergence and error estimates of the algorithm for computing the distance function is studied in detail it is shown that 2 n gauss seidel iterations is enough for the distance function in n dimensions an estimation of the number of iterations for general eikonal equations is also studied numerical examples are used to verify the analysis

a patchy dynamic programming scheme for a class of hamilton jacobi bellman equations in this paper we present a new algorithm for the solution of hamilton jacobi bellman equations related to optimal control problems the key idea is to divide the domain of computation into subdomains which are shaped by the optimal dynamics of the underlying control problem this can result in a rather complex geometrical subdivision but it has the advantage that every subdomain is invariant with respect to the optimal dynamics and then the solution can be computed independently in each subdomain the features of this dynamics dependent domain decomposition can be exploited to speed up the computation and for an efficient parallelization since the classical transmission conditions at the boundaries of the subdomains can be avoided for their properties the subdomains are patches in the sense introduced by ancona and bressan esaim control optim calc var 4 1999 pp 445 471 several examples in two and three dimensions illustrate the properties of the new method

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

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

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

random projections for classification a recovery approach random projection has been widely used in data classification it maps high dimensional data into a low dimensional subspace in order to reduce the computational cost in solving the related optimization problem while previous studies are focused on analyzing the classification performance in the low dimensional space in this paper we consider the recovery problem i e how to accurately recover the optimal solution to the original high dimensional optimization problem based on the solution learned after random projection we present a simple algorithm termed dual random projection which uses the dual solution of the low dimensional optimization problem to recover the optimal solution to the original problem our theoretical analysis shows that with a high probability the proposed algorithm is able to accurately recover the optimal solution to the original problem provided that the data matrix is approximately low rank and or optimal solution is approximately sparse we further show that the proposed algorithm can be applied iteratively to reducing the recovery error exponentially

a discontinuous galerkin finite element method for directly solving the hamilton jacobi equations in this paper we propose a new discontinuous galerkin finite element method to solve the hamilton jacobi equations unlike the discontinuous galerkin method of c hu c w shu a discontinuous galerkin finite element method for hamilton jacobi equations siam journal on scientific computing 21 1999 666 690 which applies the discontinuous galerkin framework on the conservation law system satisfied by the derivatives of the solution the method in this paper applies directly to the solution of the hamilton jacobi equations for the linear case this method is equivalent to the traditional discontinuous galerkin method for conservation laws with source terms thus stability and error estimates are straightforward for the nonlinear convex hamiltonians numerical experiments demonstrate that the method is stable and provides the optimal k 1 th order of accuracy for smooth solutions when using piecewise kth degree polynomials singularities in derivatives can also be resolved sharply if the entropy condition is not violated special treatment is needed for the entropy violating cases both one and two dimensional numerical results are provided to demonstrate the good qualities of the scheme

ordered upwind methods for static hamilton jacobi equations theory and algorithms we develop a family of fast methods for approximating the solutions to a wide class of static hamilton jacobi pdes these fast methods include both semi lagrangian and fully eulerian versions numerical solutions to these problems are typically obtained by solving large systems of coupled nonlinear discretized equations our techniques which we refer to as ordered upwind methods oums use partial information about the characteristic directions to decouple these nonlinear systems greatly reducing the computational labor our techniques are considered in the context of control theoretic and front propagation problems we begin by discussing existing oums focusing on those designed for isotropic problems we then introduce a new class of oums which decouple systems for general anisotropic problems we prove convergence of one such scheme to the viscosity solution of the corresponding hamilton jacobi pde next we introduce a set of finite differences methods based on an analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems the performance of the methods is analyzed and computational experiments are performed using test problems from computational geometry and seismology

an adaptive domain decomposition technique for parallelization of the fast marching method abstract the fast marching method fmm is an efficient technique to solve numerically the eikonal equation the parallelization of the fmm is not easy because of its intrinsic sequential nature in this paper we propose a novel approach to parallelize the fmm it leads to an equation dependent domain decomposition and it turns out to be particularly suitable for machines with two or four cores that are in common use today compared to other techniques in the field the proposed method is much simpler to implement and it gives a slightly better computational speed up in order to test the new method on a real world application we solve the shape from shading problem based on a hamilton jacobi equation on a standard four core machine the method confirms the good properties it shows a reasonable speedup factor of about 2 5 and it reveals its potential to good performance if the arithmetic density of the problem is high

flow stability of patchy vector fields and robust feedback stabilization the paper is concerned with patchy vector fields a class of discontinuous piecewise smooth vector fields that were introduced by the authors to study feedback stabilization problems we prove the stability of the corresponding solution set w r t a wide class of impulsive perturbations these results yield the robustness of patchy feedback controls in the presence of measurement errors and external disturbances

pulse number distribution for the neural spike train in the cat s auditory nerve pulse number distributions pnds were recorded from primary afferent fibers in the auditory nerve of the cat using standard extracellular microelectrode recording techniques pure tone and broadband noise stimuli were used the number of neural spikes pulses n was measured in a set of contiguous intervals each of duration t seconds the quantity n varies from one interval to another these data were then used to determine the pnd which is the probability p n t of occurrence of n spikes in the time t versus the number n the estimated mean and variance of p n t were obtained two different values of t were used an unexpected observation was that the count mean to variance ratio r is relatively constant and independent of the stimulus intensity use of the pnd as a statistical measure of the underlying neural point process has a number of virtues for example the pnd readily exhibits the existence of spike clusters e g pairs for some units the pnd is essentially unaffected by time jitter and time quantization and provides a statistically significant measure for units firing at low rates a study of the scaled and unscaled pulse interval distributions pids under conditions of spontaneous firing demonstrates that the occurrences of neural events are generally not describable by a renewal process our investigation shows that none of the point processes customarily used to model the auditory neural spike train is consistent with all of the data it appears that the encoding of acoustic information into nerve spikes in the peripheral auditory system takes the form of a cluster point process similar to the neyman scott type for pure tone excitation the pnd will be well represented as a multinomial distribution in this case

efficient algorithms for globally optimal trajectories we present serial and parallel algorithms for solving a system of equations that arises from the discretization of the hamilton jacobi equation associated to a trajectory optimization problem of the following type a vehicle starts at a prespecified point x sub o and follows a unit speed trajectory x t inside a region in spl rscr sup m until an unspecified time t that the region is exited a trajectory minimizing a cost function of the form spl int sub 0 sup t r x t dt q x t is sought the discretized hamilton jacobi equation corresponding to this problem is usually solved using iterative methods nevertheless assuming that the function r is positive we are able to exploit the problem structure and develop one pass algorithms for the discretized problem the first algorithm resembles dijkstra s shortest path algorithm and runs in time o n log n where n is the number of grid points the second algorithm uses a somewhat different discretization and borrows some ideas from a variation of dial s shortest path algorithm 1969 that we develop here it runs in time o n which is the best possible under some fairly mild assumptions finally we show that the latter algorithm can be efficiently parallelized for two dimensional problems and with p processors its running time becomes o n p provided that p o spl radic n log n

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

fast semi lagrangian schemes for the eikonal equation and applications we introduce and analyze a fast version of the semi lagrangian algorithm for front propagation originally proposed in m falcone the minimum time problem and its applications to front propagation in motion by mean curvature and related topics a visintin and g buttazzo eds de gruyter berlin 1994 pp 70 88 the new algorithm is obtained using the local definition of the approximate solution typical of semi lagrangian schemes and redefining the set of neighboring nodes necessary for fast marching schemes a new proof of convergence is needed since that definition produces a new narrow band centered at the interphase which is larger than the one used in fast marching methods based on finite differences we show that the new algorithm converges to the viscosity solution of the problem and that its complexity is o n log n nb as it is for the fast marching method based on finite difference n and n nb being respectively the total number of nodes and the number of nodes in the narrow band a new sufficient condition for the convergence of the standard finite difference fast marching method is also given we present several tests comparing the two algorithms and other fast methods e g fast sweeping on a series of benchmarks which include the minimum time problem and the shape from shading problem

control and stabilization of nonholonomic dynamic systems a class of inherently nonlinear control problems has been identified the nonlinear features arising directly from physical assumptions about constraints on the motion of a mechanical system models are presented for mechanical systems with nonholonomic constraints represented both by differential algebraic equations and by reduced state equations control issues for this class of systems are studied and a number of fundamental results are derived although a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback a general procedure for constructing a piecewise analytic state feedback which achieves the desired result is suggested

fisher and shannon information in finite neural populations the precision of the neural code is commonly investigated using two families of statistical measures shannon mutual information and derived quantities when investigating very small populations of neurons and fisher information when studying large populations these statistical tools are no longer the preserve of theorists and are being applied by experimental research groups in the analysis of empirical data although the relationship between information theoretic and fisher based measures in the limit of infinite populations is relatively well understood how these measures compare in finite size populations has not yet been systematically explored we aim to close this gap we are particularly interested in understanding which stimuli are best encoded by a given neuron within a population and how this depends on the chosen measure we use a novel monte carlo approach to compute a stimulus specific decomposition of the mutual information the ssi for populations of up to 256 neurons and show that fisher information can be used to accurately estimate both mutual information and ssi for populations of the order of 100 neurons even in the presence of biologically realistic variability noise correlations and experimentally relevant integration times according to both measures the stimuli that are best encoded are those falling at the flanks of the neuron s tuning curve in populations of fewer than around 50 neurons however fisher information can be misleading

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

convergence of a generalized fast marching method for an eikonal equation with a velocity changing sign we present a new fast marching algorithm for an eikonal equation with a velocity changing sign this first order equation models a front propagation in the normal direction the algorithm is an extension of the fast marching method in two respects the first is that the new scheme can deal with a time dependent velocity and the second is that there is no restriction on its change in sign we analyze the properties of the algorithm and we prove its convergence in the class of discontinuous viscosity solutions finally we present some numerical simulations of fronts propagating in mathbb r 2

local minimization algorithms for dynamic programming equations the numerical realization of the dynamic programming principle for continuous time optimal control leads to nonlinear hamilton jacobi bellman equations which require the minimization of a nonlinear mapping over the set of admissible controls this minimization is often performed by comparison over a finite number of elements of the control set in this paper we demonstrate the importance of an accurate realization of these minimization problems and propose algorithms by which this can be achieved effectively the considered class of equations includes nonsmooth control problems with ell 1 penalization which lead to sparse controls

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

an efficient policy iteration algorithm for dynamic programming equations we present an accelerated algorithm for the solution of static hamilton jacobi bellman equations related to optimal control problems our scheme is based on a classic policy iteration procedure which is known to have superlinear convergence in many relevant cases provided the initial guess is sufficiently close to the solution this limitation often degenerates into a behavior similar to a value iteration method with an increased computation time the new scheme circumvents this problem by combining the advantages of both algorithms with an efficient coupling the method starts with a coarse mesh value iteration phase and then switches to a fine mesh policy iteration procedure when a certain error threshold is reached a delicate point is to determine this threshold in order to avoid cumbersome computations with the value iteration and at the same time to ensure the convergence of the policy iteration method to the optimal solution we analyze the methods and efficient coupling in a number of examples