on a system of nonlinear boltzmann equations of semiconductor physics a system of kinetic equations of semiconductors is studied these are vlasov poisson equations with additional nonlinear boltzmann operators first h theorems for these integral operators are given and their null space is determined then the existence and uniqueness of smooth solutions is proven the proof relies on the construction of a sequence of solutions for a suitable linearized system which converges strongly

network amplification of local fluctuations causes high spike rate variability fractal firing patterns and oscillatory local field potentials we investigate a model for neural activity in a two dimensional sheet of leaky integrate and fire neurons with feedback connectivity consisting of local excitation and surround inhibition each neuron receives stochastic input from an external source independent in space and time as recently suggested by softky and koch 1992 1993 independent stochastic input alone cannot explain the high interspike interval variability exhibited by cortical neurons in behaving monkeys we show that high variability can be obtained due to the amplification of correlated fluctuations in a recurrent network furthermore the cross correlation functions have a dual structure with a sharp peak on top of a much broader hill this is due to the inhibitory and excitatory feedback connections which cause hotspots of neural activity to form within the network these localized patterns of excitation appear as clusters or stripes that coalesce disintegrate or fluctuate in size while simultaneously moving in a random walk constrained by the interaction with other clusters the synaptic current impinging upon a single neuron shows large fluctuations at many time scales leading to a large coefficient of variation cv for the interspike interval statistics the power spectrum associated with single units shows a 1 f decay for small frequencies and is flat at higher frequencies while the power spectrum of the spiking activity averaged over many cells equivalent to the local field potential shows no 1 f decay but a prominent peak around 40 hz in agreement with data recorded from cat and monkey cortex gray et al 1990 eckhorn et al 1993 firing rates exhibit self similarity between 20 and 800 msec resulting in 1 f like noise consistent with the fractal nature of neural spike trains teich 1992

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

spike train statistics for consonant and dissonant musical accords in a simple auditory sensory model the phenomena of dissonance and consonance in a simple auditory sensory model composed of three neurons are considered two of them here so called sensory neurons are driven by noise and subthreshold periodic signals with different ratio of frequencies and its outputs plus noise are applied synaptically to a third neuron so called interneuron we present a theoretical analysis with a probabilistic approach to investigate the interspike intervals statistics of the spike train generated by the interneuron we find that tones with frequency ratios that are considered consonant by musicians produce at the third neuron inter firing intervals statistics densities that are very distinctive from densities obtained using tones with ratios that are known to be dissonant in other words at the output of the interneuron inharmonious signals give rise to blurry spike trains while the harmonious signals produce more regular less noisy spike trains theoretical results are compared with numerical simulations

polygonal figures and their form spaces ii angle permutations in equilateral polygons abstract in a closed nonregular equilateral polygonal curve angles cannot be permuted arbitrarily without opening the curve here we investigate the possible angle permutations that leave a polygonal curve closed to this end we introduce angle sequences i e symbol sequences describing the number and arrangement of like angle values in classes of forms we find that there exist universal angle sequences giving closed polygons for arbitrary angle values provided that the angle sum is kept constant at one of certain permissible values the universal angle sequences are periodic they consist of at least two identical series of symbols within a series the symbols can be permuted freely on condition that in all other series the very same permutation is applied all these correlated permutations when applied to the angles of a corresponding polygonal curve leave this curve closed polygons belonging to a given angle sequence form class constitute a submanifold of the manifold m i of equilateral polygons within the form space these submanifolds structure m i in a hierarchical manner so that the form class concept gives a valuable complementary classification scheme for the forms of m i furthermore we find that polygons with constant angle composition the term composition is used here in its chemical sense show a wide variety of possible shapes from quasi circular to wormlike and even self intersecting ones corresponding to different degrees of segregation of like angles within the polygonal curve

propagation of moments and regularity for the 3 dimensional vlasov poisson system we prove that if the initial data has moments inv higher than three then the solution of vlasov poisson has also moments inv higher than three we deduce from this different regularity results on the local density the force field or the solution itself also we give a new uniqueness result and new regularity results for solutions satisfying only the energy andl bounds our proofs are based on a new representation formula and logarithmic estimates for the force field

crisp and fuzzy motif and arrangement symmetries in composite geometric figures abstract the notions of motif and arrangement symmetries within composite geometric figures are defined the relationships between these types of symmetry and the symmetry of the whole figure are clarified by making use of the crystallographic concepts of site symmetry and direction symmetry from this it has been deduced that a figure with arbitrary symmetry can be assembled from motifs of likewise arbitrary symmetries if a motif with symmetry g m is placed on a site having the site symmetry g s g m its contribution to the figure symmetry g is only a subgroup g mo of its direction symmetry g mo where g s g mo g mo g m supernumerary symmetry elements of the motif give rise to intermediate or latent symmetries of the figure a consequent decomposition of a geometric figure into its constituent points reveals that a large part of the o n symmetry of every single point is lost when assembling these points to build up the figure all lost symmetries can however be detected as intermediate symmetries of this figure they can be displayed as fuzzy symmetry landscapes and symmetry profiles for a given figure showing all crisp and intermediate symmetries of interest

spike output jitter mean firing time and coefficient of variation to understand how a single neurone processes information it is critical to examine the relationship between input and output marsalek koch and maunsell s study focused on output jitter standard deviation of output interpike interval found that for the integrate and fire if whereas an input distribution with a more slowly decreasing tail induces divergence of output jitter exact formulae for mean firing time are also obtained which enable us to estimate the coefficient of variation the i f model with leakage is also briefly considered

on the cauchy problem of the vlasov poisson bgk system global existence of weak solutions global in time existence of weak solutions to the cauchy problem of the three dimensional vlasov poisson bgk system is shown for initial data belonging to the space lp 3 3 with p 9 and having finite second order velocity moments this result solves partially the well posed problem for the vlasov poisson bgk system proposed by b perthame higher moments for kinetic equations the vlasov poisson and fokker planck cases math meth appl sci 13 441 452 1990

bistability and resonance in the periodically stimulated hodgkin huxley model with noise we describe general characteristics of the hodgkin huxley neuron s response to a periodic train of short current pulses with gaussian noise the deterministic neuron is bistable for antiresonant frequencies when the stimuli arrive at the resonant frequency the firing rate is a continuous function of the current amplitude i 0 and scales as i 0 i th 1 2 characteristic of a saddle node bifurcation at the threshold i th intervals of continuous irregular response alternate with integer mode locked regions with bistable excitation edge there is an even all multimodal transition between the 2 1 and 3 1 states in the vicinity of the main resonance which is analogous to the odd all transition discovered earlier in the high frequency regime for i 0 i th and small noise the firing rate has a maximum at the resonant frequency for larger noise and subthreshold stimulation the maximum firing rate initially shifts toward lower frequencies then returns to higher frequencies in the limit of large noise the stochastic coherence antiresonance defined as a simultaneous occurrence of i the maximum of the coefficient of variation and ii the minimum of the firing rate vs the noise intensity occurs over a wide range of parameter values including monostable regions results of this work can be verified experimentally

responses of a hodgkin huxley neuron to various types of spike train inputs numerical investigations have been made of responses of a hodgkin huxley hh neuron to spike train inputs whose interspike interval isi is modulated by deterministic semi deterministic chaotic and stochastic signals as deterministic one we adopt inputs with the time independent isi and with time dependent isi modulated by sinusoidal signal the rossler and lorentz models are adopted for chaotic modulations of isi stochastic isi inputs with the gamma distribution are employed it is shown that distribution of output isi data depends not only on the mean of isis of spike train inputs but also on their fluctuations the distinction of responses to the three kinds of inputs can be made by return maps of input and output isis but not by their histograms the relation between the variations of input and output isis is shown to be different from that of the integrate and fire if model because of the refractory period in the hh neuron

deformations isosymmetric manifolds and higher dimensional form space symmetries for point ensembles polygonal forms under o 2 symmetry i two and three points abstract the analysis of polygonal forms and their form transitions using normal deformations 1 has been extended to a global analysis of form spaces for polygonal forms specified by n points in a plane coincidences of the points are explicitly allowed and the origin of the form space is taken to correspond to a coincidence of all n points this choice is natural since for every form there exists a totally symmetric deformation leading to exactly this configuration the form space f for n points in a plane is a 2n dimensional euclidean space whose points represent all possible n vertex polygons simple and self intersecting ones by fixing the centres of gravity of the forms in the origin of the x y plane the dimension of the form space reduces to 2n 2 to give a reduced form space f within the form spaces isosymmetric manifolds geometric loci of all forms having the same symmetry g are determined these manifolds inscribed in a form space f define the corresponding symmetry space s the symmetry g s of the reduced symmetry space for three points has been determined in part the four dimensional symmetry group 21 04 has been found to be a subgroup of the full noncrystallographic symmetry of s for some cases the relation g f g s has been exemplified

global weak solutions to the cometary flow equation with a self generated electric field abstract we study the cauchy problem of a cometary flow equation with a self generated electric field this kinetic model originates from the theory of astrophysical plasmas and can be viewed as a perturbation by a wave particle collision operator of the classical vlasov poisson system by asymptotic methods in kinetic theory global existence of nonnegative weak solutions to the cauchy problem in three space variables is established for bounded initial data having finite second order velocity moments

dynamical mean field theory of noisy spiking neuron ensembles application to the hodgkin huxley model a dynamical mean field approximation dma previously proposed by the present author h hasegawa phys rev e 67 041903 2003 has been extended to ensembles described by a general noisy spiking neuron model ensembles of n unit neurons each of which is expressed by coupled k dimensional differential equations des are assumed to be subject to spatially correlated white noises the original mathrm kn dimensional stochastic des have been replaced by k k 2 dimensional deterministic des expressed in terms of means and the second order moments of local and global variables the fourth order contributions are taken into account by the gaussian decoupling approximation our dma has been applied to an ensemble of hodgkin huxley hh neurons k 4 for which effects of the noise the coupling strength and the ensemble size on the response to a single spike input have been investigated numerical results calculated by the dma theory are in good agreement with those obtained by direct simulations although the former computation is about a thousand times faster than the latter for a typical hh neuron ensemble with n 100

dynamical response of the hodgkin huxley model in the high input regime the response of the hodgkin huxley neuronal model subjected to stochastic uncorrelated spike trains originating from a large number of inhibitory and excitatory post synaptic potentials is analyzed in detail the model is examined in its three fundamental dynamical regimes silence bistability and repetitive firing its response is characterized in terms of statistical indicators interspike interval distributions and their first moments as well as of dynamical indicators autocorrelation functions and conditional entropies in the silent regime the coexistence of two different coherence resonances is revealed one occurs at quite low noise and is related to the stimulation of subthreshold oscillations around the rest state the second one at intermediate noise variance is associated with the regularization of the sequence of spikes emitted by the neuron bistability in the low noise limit can be interpreted in terms of jumping processes across barriers activated by stochastic fluctuations in the repetitive firing regime a maximization of incoherence is observed at finite noise variance finally the mechanisms responsible for the different features appearing in the interspike interval distributions like multimodality and exponential tails are clearly identified in the various regimes

dynamical entropies applied to stochastic resonance we will show that sr is accompanied by an extremal behavior of these quantities when sweeping noise intensity over the resonance region we will prove that an increase of applied noise at first leads to an increase of temporal order in the output sequence simultaneously the predictability of the output sequence will improve in the resonance region from the kullback entropy analysis we will learn that for a certain value of noise intensity the input and output distributions of binary subsequences maximally match moreover input and output subsequences registered at equal times exhibit maximum correlation finally we relate these results to the syn

multimodal transition and stochastic antiresonance in squid giant axons the experimental data of takahashi et al physica d 43 318 1990 on the response of squid giant axons stimulated by periodic sequence of short current pulses is interpreted within the hodgkin huxley model the minimum of the firing rate as a function of the stimulus amplitude i 0 in the high frequency regime is due to the multimodal transition below this singular point only odd multiples of the driving period remain and the system is sensitive to noise the coefficient of variation has a maximum and the firing rate has a minimum as a function of the noise intensity which is an indication of the stochastic coherence antiresonance the model calculations reproduce the frequency of occurrence of the most common modes in the vicinity of the transition a linear relation of output frequency vs i 0 above the transition is also confirmed

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

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

response of a hodgkin huxley neuron to a high frequency input we study the response of a hodgkin huxley neuron stimulated by a periodic sequence of conductance pulses arriving through the synapse in the high frequency regime in addition to the usual excitation threshold there is a smooth crossover from the firing to the silent regime for increasing pulse amplitude g syn the amplitude of the voltage spikes decreases approximately linearly with g syn in some regions of parameter space the response is irregular probably chaotic in the chaotic regime between the mode locked regions 3 1 and 2 1 near the lower excitation threshold the output interspike interval histogram isih undergoes a sharp transition if the driving period is below the critical value t i l t ensuremath ast the output histogram contains only odd multiples of t i for t i g t ensuremath ast even multiples of t i also appear in the histogram starting from the largest values near t ensuremath ast the isih scales logarithmically on both sides of the transition the coefficient of variation of isih has a cusp singularity at t ensuremath ast the average response period has a maximum slightly above t ensuremath ast near the excitation threshold in the chaotic regime the average firing rate rises sublinearly from frequencies of order 1 hz

interspike interval attractors from chaotically driven neuron models abstract sequences of intervals between firing times interspike intervals isis from single neuron models with chaotic forcing are investigated we analyze how the dynamical properties of the chaotic input determine those of the output isi sequence and assess how various biophysical parameters affect this input output relationship the attractors constructed from delay embeddings of isis and of the chaotic input are compared from the points of view of geometry and nonlinear forecastability for the three integrate and fire if models investigated the similarity between these attractors is high only when the mean firing rate is high and when firings occur over a large range of the input signal when these conditions are satisfied isis are related to input values at which firings occur by a simple map and their distribution can be derived from that of input signal values attractor reconstruction is found to be more sensitive to mean firing rate than to isi distribution at low firing rates for which the input is under sampled and isis become larger than the short term prediction horizon or when the dynamics do not allow a smooth invertible mapping between signal and isis information is lost our results relevant to all dynamical systems generating events show that information about continuous time dynamics is difficult to retrieve at low event rates and that information about inputs can be isolated only under restricted conditions

isosymmetric manifolds in form spaces and the normal deformations of polygonal forms abstract form variations are described in an appropriately constructed form space f typically an r n where every point of f represents a different form regarding the symmetries of the forms f can be divided into disjunct isosymmetric manifolds i e points lines surfaces and volumes whose points correspond to forms with equal symmetries these manifolds are derived from a symmetry analysis of possible deformations of the forms this analysis is comparable to the construction of symmetry coordinates in a normal coordinate analysis of molecules and results in normal modes of deformation normal deformations of these forms from the symmetry species of a normal deformation the symmetry of the resulting form can be inferred transformation of the form space coordinates into normal coordinates the differentials of which are the normal deformations facilitates the description of the high dimensional form spaces and can be made the basis of an easy symmetry diagnosis of forms furthermore the problem of an ascent in symmetry by deformation is discussed

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

microscopic mechanism for self organized quasiperiodicity in random networks of nonlinear oscillators self organized quasiperiodicity is one of the most puzzling dynamical phases observed in systems of nonlinear coupled oscillators the single dynamical units are not locked to the periodic mean field they produce but they still feature a coherent behavior through an unexplained complex form of correlation we consider a class of leaky integrate and fire oscillators on random sparse and massive networks with dynamical synapses featuring self organized quasiperiodicity and we show how complex collective oscillations arise from constructive interference of microscopic dynamics in particular we find a simple quantitative relationship between two relevant microscopic dynamical time scales and the macroscopic time scale of the global signal we show that the proposed relation is a general property of collective oscillations common to all the partially synchronous dynamical phases analyzed we argue that an analogous mechanism could be at the origin of similar network dynamics

polygonal figures and their form spaces i the structure of isodiastemic manifolds manifolds of equilateral polygons abstract the geometric structure of isodiastemic manifolds i e manifolds on which the equilaterality of polygons is preserved within a form space f spanned by n 2 consecutive vertex angles i internal coordinates of plane n gons is reported for n 5 and n 6 the curved isodiastemic manifolds are almost everywhere locally n 3 dimensional exceptions are singular points linear forms for even n where the local dimension is n 2 the isodiastemic manifolds for n 5 and n 6 are subdivided into five n 3 dimensional submanifolds of nondegenerate forms forms without coincident points each comprising forms with the same angle sum called a form family the submanifolds are bounded by n 4 dimensional dividing manifolds of degenerate forms which are structured hierarchically according to the types of degeneracy of their forms for n 6 the boundary polyhedra for two submanifolds are described in detail the manifold of spearhead shaped forms is equivalent to an octahedron whereas the hexagon manifold is equivalent to a special icosihexahedron 1

a fuzzy symmetry concept for forms with imperfect symmetries we develop a new symmetry concept for forms whose symmetries are only approximate the concept is based on fuzzy set theory the crisp set s g of objects having a certain symmetry g is replaced by a fuzzy set s g with a membership function 0 1 corresponding to the degree of fulfilment of the symmetry requirements of g degree of symmetry the degree of symmetry s g of a given form s relative to a symmetry group g is evaluated on the basis of a harmonic analysis of the corresponding form function r depending on the symmetry of the form certain coefficients of the fourier series vanish the extinction rules for the fourier coefficients are deduced from the symmetries of the individual cosine terms cos we define the degree of symmetry as the relative weight of all terms which are compatible with g thus s is a continous measure of symmetry it may quantitatively characterise continuous transitions of forms between different symmetries and also the degree of membership of s in s g depending on the values of s we propose a gradation of fuzzy symmetries actual apparent approximate distant hidden missing symmetry

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

existence and uniqueness of a global smooth solution for the vlasov poisson fokker planck system in three dimensions abstract studying precisely the regularity of the force field created by the solution of the linear vlasov fokker planck equation we prove existence and uniqueness of a smooth solution to the vlasov poisson fokker planck system in three dimensions the attractive and repulsive cases are treated the method involves an analysis of the effect of the fokker planck kernel without any force field on non linear expressions coming from the vlasov term

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