Journal of Mathematical Neuroscience最新文献

{"title":"Path integral methods for stochastic differential equations.","authors":"Carson C Chow,&nbsp;Michael A Buice","doi":"10.1186/s13408-015-0018-5","DOIUrl":"https://doi.org/10.1186/s13408-015-0018-5","url":null,"abstract":"<p><p>Stochastic differential equations (SDEs) have multiple applications in mathematical neuroscience and are notoriously difficult. Here, we give a self-contained pedagogical review of perturbative field theoretic and path integral methods to calculate moments of the probability density function of SDEs. The methods can be extended to high dimensional systems such as networks of coupled neurons and even deterministic systems with quenched disorder. </p>","PeriodicalId":54271,"journal":{"name":"Journal of Mathematical Neuroscience","volume":"5 ","pages":"8"},"PeriodicalIF":2.3,"publicationDate":"2015-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/s13408-015-0018-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"33079377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A formalism for evaluating analytically the cross-correlation structure of a firing-rate network model.","authors":"Diego Fasoli,&nbsp;Olivier Faugeras,&nbsp;Stefano Panzeri","doi":"10.1186/s13408-015-0020-y","DOIUrl":"https://doi.org/10.1186/s13408-015-0020-y","url":null,"abstract":"<p><p>We introduce a new formalism for evaluating analytically the cross-correlation structure of a finite-size firing-rate network with recurrent connections. The analysis performs a first-order perturbative expansion of neural activity equations that include three different sources of randomness: the background noise of the membrane potentials, their initial conditions, and the distribution of the recurrent synaptic weights. This allows the analytical quantification of the relationship between anatomical and functional connectivity, i.e. of how the synaptic connections determine the statistical dependencies at any order among different neurons. The technique we develop is general, but for simplicity and clarity we demonstrate its efficacy by applying it to the case of synaptic connections described by regular graphs. The analytical equations so obtained reveal previously unknown behaviors of recurrent firing-rate networks, especially on how correlations are modified by the external input, by the finite size of the network, by the density of the anatomical connections and by correlation in sources of randomness. In particular, we show that a strong input can make the neurons almost independent, suggesting that functional connectivity does not depend only on the static anatomical connectivity, but also on the external inputs. Moreover we prove that in general it is not possible to find a mean-field description à la Sznitman of the network, if the anatomical connections are too sparse or our three sources of variability are correlated. To conclude, we show a very counterintuitive phenomenon, which we call stochastic synchronization, through which neurons become almost perfectly correlated even if the sources of randomness are independent. Due to its ability to quantify how activity of individual neurons and the correlation among them depends upon external inputs, the formalism introduced here can serve as a basis for exploring analytically the computational capability of population codes expressed by recurrent neural networks. </p>","PeriodicalId":54271,"journal":{"name":"Journal of Mathematical Neuroscience","volume":"5 ","pages":"6"},"PeriodicalIF":2.3,"publicationDate":"2015-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/s13408-015-0020-y","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"33073730","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Path-integral methods for analyzing the effects of fluctuations in stochastic hybrid neural networks.","authors":"Paul C Bressloff","doi":"10.1186/s13408-014-0016-z","DOIUrl":"10.1186/s13408-014-0016-z","url":null,"abstract":"<p><p>We consider applications of path-integral methods to the analysis of a stochastic hybrid model representing a network of synaptically coupled spiking neuronal populations. The state of each local population is described in terms of two stochastic variables, a continuous synaptic variable and a discrete activity variable. The synaptic variables evolve according to piecewise-deterministic dynamics describing, at the population level, synapses driven by spiking activity. The dynamical equations for the synaptic currents are only valid between jumps in spiking activity, and the latter are described by a jump Markov process whose transition rates depend on the synaptic variables. We assume a separation of time scales between fast spiking dynamics with time constant [Formula: see text] and slower synaptic dynamics with time constant τ. This naturally introduces a small positive parameter [Formula: see text], which can be used to develop various asymptotic expansions of the corresponding path-integral representation of the stochastic dynamics. First, we derive a variational principle for maximum-likelihood paths of escape from a metastable state (large deviations in the small noise limit [Formula: see text]). We then show how the path integral provides an efficient method for obtaining a diffusion approximation of the hybrid system for small ϵ. The resulting Langevin equation can be used to analyze the effects of fluctuations within the basin of attraction of a metastable state, that is, ignoring the effects of large deviations. We illustrate this by using the Langevin approximation to analyze the effects of intrinsic noise on pattern formation in a spatially structured hybrid network. In particular, we show how noise enlarges the parameter regime over which patterns occur, in an analogous fashion to PDEs. Finally, we carry out a [Formula: see text]-loop expansion of the path integral, and use this to derive corrections to voltage-based mean-field equations, analogous to the modified activity-based equations generated from a neural master equation. </p>","PeriodicalId":54271,"journal":{"name":"Journal of Mathematical Neuroscience","volume":"5 ","pages":"4"},"PeriodicalIF":2.3,"publicationDate":"2015-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4385107/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"33073727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Approximate, not Perfect Synchrony Maximizes the Downstream Effectiveness of Excitatory Neuronal Ensembles.","authors":"Christoph Börgers,&nbsp;Jie Li,&nbsp;Nancy Kopell","doi":"10.1186/2190-8567-4-10","DOIUrl":"https://doi.org/10.1186/2190-8567-4-10","url":null,"abstract":"<p><p>The most basic functional role commonly ascribed to synchrony in the brain is that of amplifying excitatory neuronal signals. The reasoning is straightforward: When positive charge is injected into a leaky target neuron over a time window of positive duration, some of it will have time to leak back out before an action potential is triggered in the target, and it will in that sense be wasted. If the goal is to elicit a firing response in the target using as little charge as possible, it seems best to deliver the charge all at once, i.e., in perfect synchrony. In this article, we show that this reasoning is correct only if one assumes that the input ceases when the target crosses the firing threshold, but before it actually fires. If the input ceases later-for instance, in response to a feedback signal triggered by the firing of the target-the \"most economical\" way of delivering input (the way that requires the least total amount of input) is no longer precisely synchronous, but merely approximately so. If the target is a heterogeneous network, as it always is in the brain, then ceasing the input \"when the target crosses the firing threshold\" is not an option, because there is no single moment when the firing threshold is crossed. In this sense, precise synchrony is never optimal in the brain. </p>","PeriodicalId":54271,"journal":{"name":"Journal of Mathematical Neuroscience","volume":"4 1","pages":"10"},"PeriodicalIF":2.3,"publicationDate":"2014-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/2190-8567-4-10","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"34492411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Numerical Bifurcation Theory for High-Dimensional Neural Models.","authors":"Carlo R Laing","doi":"10.1186/2190-8567-4-13","DOIUrl":"https://doi.org/10.1186/2190-8567-4-13","url":null,"abstract":"<p><p>Numerical bifurcation theory involves finding and then following certain types of solutions of differential equations as parameters are varied, and determining whether they undergo any bifurcations (qualitative changes in behaviour). The primary technique for doing this is numerical continuation, where the solution of interest satisfies a parametrised set of algebraic equations, and branches of solutions are followed as the parameter is varied. An effective way to do this is with pseudo-arclength continuation. We give an introduction to pseudo-arclength continuation and then demonstrate its use in investigating the behaviour of a number of models from the field of computational neuroscience. The models we consider are high dimensional, as they result from the discretisation of neural field models-nonlocal differential equations used to model macroscopic pattern formation in the cortex. We consider both stationary and moving patterns in one spatial dimension, and then translating patterns in two spatial dimensions. A variety of results from the literature are discussed, and a number of extensions of the technique are given. </p>","PeriodicalId":54271,"journal":{"name":"Journal of Mathematical Neuroscience","volume":"4 1","pages":"13"},"PeriodicalIF":2.3,"publicationDate":"2014-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/2190-8567-4-13","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"34505122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation.","authors":"Khashayar Pakdaman,&nbsp;Benoît Perthame,&nbsp;Delphine Salort","doi":"10.1186/2190-8567-4-14","DOIUrl":"https://doi.org/10.1186/2190-8567-4-14","url":null,"abstract":"<p><p>Motivated by a model for neural networks with adaptation and fatigue, we study a conservative fragmentation equation that describes the density probability of neurons with an elapsed time s after its last discharge. In the linear setting, we extend an argument by Laurençot and Perthame to prove exponential decay to the steady state. This extension allows us to handle coefficients that have a large variation rather than constant coefficients. In another extension of the argument, we treat a weakly nonlinear case and prove total desynchronization in the network. For greater nonlinearities, we present a numerical study of the impact of the fragmentation term on the appearance of synchronization of neurons in the network using two \"extreme\" cases. Mathematics Subject Classification (2000)2010: 35B40, 35F20, 35R09, 92B20. </p>","PeriodicalId":54271,"journal":{"name":"Journal of Mathematical Neuroscience","volume":"4 ","pages":"14"},"PeriodicalIF":2.3,"publicationDate":"2014-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/2190-8567-4-14","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"32578327","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Frequency Preference Response to Oscillatory Inputs in Two-dimensional Neural Models: A Geometric Approach to Subthreshold Amplitude and Phase Resonance.","authors":"Horacio G Rotstein","doi":"10.1186/2190-8567-4-11","DOIUrl":"https://doi.org/10.1186/2190-8567-4-11","url":null,"abstract":"<p><p>We investigate the dynamic mechanisms of generation of subthreshold and phase resonance in two-dimensional linear and linearized biophysical (conductance-based) models, and we extend our analysis to account for the effect of simple, but not necessarily weak, types of nonlinearities. Subthreshold resonance refers to the ability of neurons to exhibit a peak in their voltage amplitude response to oscillatory input currents at a preferred non-zero (resonant) frequency. Phase-resonance refers to the ability of neurons to exhibit a zero-phase (or zero-phase-shift) response to oscillatory input currents at a non-zero (phase-resonant) frequency. We adapt the classical phase-plane analysis approach to account for the dynamic effects of oscillatory inputs and develop a tool, the envelope-plane diagrams, that captures the role that conductances and time scales play in amplifying the voltage response at the resonant frequency band as compared to smaller and larger frequencies. We use envelope-plane diagrams in our analysis. We explain why the resonance phenomena do not necessarily arise from the presence of imaginary eigenvalues at rest, but rather they emerge from the interplay of the intrinsic and input time scales. We further explain why an increase in the time-scale separation causes an amplification of the voltage response in addition to shifting the resonant and phase-resonant frequencies. This is of fundamental importance for neural models since neurons typically exhibit a strong separation of time scales. We extend this approach to explain the effects of nonlinearities on both resonance and phase-resonance. We demonstrate that nonlinearities in the voltage equation cause amplifications of the voltage response and shifts in the resonant and phase-resonant frequencies that are not predicted by the corresponding linearized model. The differences between the nonlinear response and the linear prediction increase with increasing levels of the time scale separation between the voltage and the gating variable, and they almost disappear when both equations evolve at comparable rates. In contrast, voltage responses are almost insensitive to nonlinearities located in the gating variable equation. The method we develop provides a framework for the investigation of the preferred frequency responses in three-dimensional and nonlinear neuronal models as well as simple models of coupled neurons. </p>","PeriodicalId":54271,"journal":{"name":"Journal of Mathematical Neuroscience","volume":"4 ","pages":"11"},"PeriodicalIF":2.3,"publicationDate":"2014-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/2190-8567-4-11","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"32376767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Effects of synaptic plasticity on phase and period locking in a network of two oscillatory neurons.","authors":"Zeynep Akcay,&nbsp;Amitabha Bose,&nbsp;Farzan Nadim","doi":"10.1186/2190-8567-4-8","DOIUrl":"https://doi.org/10.1186/2190-8567-4-8","url":null,"abstract":"<p><p>We study the effects of synaptic plasticity on the determination of firing period and relative phases in a network of two oscillatory neurons coupled with reciprocal inhibition. We combine the phase response curves of the neurons with the short-term synaptic plasticity properties of the synapses to define Poincaré maps for the activity of an oscillatory network. Fixed points of these maps correspond to the phase-locked modes of the network. These maps allow us to analyze the dependence of the resulting network activity on the properties of network components. Using a combination of analysis and simulations, we show how various parameters of the model affect the existence and stability of phase-locked solutions. We find conditions on the synaptic plasticity profiles and the phase response curves of the neurons for the network to be able to maintain a constant firing period, while varying the phase of locking between the neurons or vice versa. A generalization to cobwebbing for two-dimensional maps is also discussed. </p>","PeriodicalId":54271,"journal":{"name":"Journal of Mathematical Neuroscience","volume":"4 ","pages":"8"},"PeriodicalIF":2.3,"publicationDate":"2014-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/2190-8567-4-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"32312318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Well-posedness of a density model for a population of theta neurons.","authors":"Grégory Dumont, Jacques Henry, Carmen Oana Tarniceriu","doi":"10.1186/2190-8567-4-2","DOIUrl":"10.1186/2190-8567-4-2","url":null,"abstract":"<p><p>Population density models used to describe the evolution of neural populations in a phase space are closely related to the single neuron model that describes the individual trajectories of the neurons of the population and which gives in particular the phase-space where the computations are made. Based on a transformation of the quadratic integrate and fire single neuron model, the so called theta-neuron model is obtained and we shall introduce in this paper a corresponding population density model for it. Existence and uniqueness of a solution will be proved and some numerical simulations are presented.</p>","PeriodicalId":54271,"journal":{"name":"Journal of Mathematical Neuroscience","volume":"4 1","pages":"2"},"PeriodicalIF":2.3,"publicationDate":"2014-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"32272083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Large deviations for nonlocal stochastic neural fields.","authors":"Christian Kuehn, Martin G Riedler","doi":"10.1186/2190-8567-4-1","DOIUrl":"10.1186/2190-8567-4-1","url":null,"abstract":"<p><p>We study the effect of additive noise on integro-differential neural field equations. In particular, we analyze an Amari-type model driven by a Q-Wiener process, and focus on noise-induced transitions and escape. We argue that proving a sharp Kramers' law for neural fields poses substantial difficulties, but that one may transfer techniques from stochastic partial differential equations to establish a large deviation principle (LDP). Then we demonstrate that an efficient finite-dimensional approximation of the stochastic neural field equation can be achieved using a Galerkin method and that the resulting finite-dimensional rate function for the LDP can have a multiscale structure in certain cases. These results form the starting point for an efficient practical computation of the LDP. Our approach also provides the technical basis for further rigorous study of noise-induced transitions in neural fields based on Galerkin approximations.Mathematics Subject Classification (2000): 60F10, 60H15, 65M60, 92C20. </p>","PeriodicalId":54271,"journal":{"name":"Journal of Mathematical Neuroscience","volume":"4 1","pages":"1"},"PeriodicalIF":2.3,"publicationDate":"2014-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3991906/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"32271224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}