Dynamics of stochastic integrate-and-fire networks
Gabriel Koch Ocker
备注：
A previous version of this article, now retracted (https://journals.aps.org/prx/abstract/10.1103/ PhysRevX.12.041007), contained errors described in the retraction notice (https://journals.aps.org/prx/ abstract/10.1103/PhysRevX.13.029904). These errors have been corrected in the current article, which was re-reviewed and accepted back at Phys. Rev. X
The neural dynamics generating sensory, motor, and cognitive functions are commonly understood through field theories for neural population activity. Classic neural field theories are derived from highly simplified models of individual neurons, while biological neurons are highly complex cells. Integrate-and-fire neuron models balance biophysical detail and analytical tractability. Here, we develop a statistical field theory for networks of integrate-and-fire neurons with stochastic spike emission. This reveals an exact mapping to a self-consistent renewal process and a new mean field theory for the activity in these networks. The mean field theory has a rate-dependent leak, approximating the spike-driven resets of the membrane voltage. This gives rise to bistability between quiescent and active states in homogenous and excitatory-inhibitory pulse-coupled networks. The field-theoretic framework also exposes fluctuation corrections to the mean field theory. We find that due to the spike reset, fluctuations suppress activity. We then examine the roles of spike resets and recurrent inhibition in stabilizing network activity. We calculate the phase diagram for inhibitory stabilization and find that an inhibition-stabilized regime occurs in wide regions of parameter space, consistent with experimental reports of inhibitory stabilization in diverse brain regions. Fluctuations narrow the region of inhibitory stabilization, consistent with their role in suppressing activity through spike resets.
We consider a single Leaky integrate-and-fire neuron stimulated with Poisson process. We develop a method, which allows one to obtain the first passage time probability density function without any additional approximations.
One of the fundamental characteristics of a nonlinear system is how it transfers correlations in its inputs to correlations in its outputs. This is particularly important in the nervous system, where correlations between spiking neurons are prominent. Using linear response and asymptotic methods for pairs of unconnected integrate-and-fire (IF) neurons receiving white noise inputs, we show that this correlation transfer depends on the output spike firing rate in a strong, stereotyped manner, and is, surprisingly, almost independent of the interspike variance. For cells receiving heterogeneous i…
We compute the firing rate of a leaky integrate-and-fire (LIF) neuron with stochastic conductance-based inputs in the limit when synaptic decay times are much shorter than the membrane time constant. A comparison of our analytical results to numeric simulations is presented for a range of biophysically-realistic parameters.
We derive analytical formulae for the firing rate of integrate-and-fire neurons endowed with realistic synaptic dynamics. In particular we include the possibility of multiple synaptic inputs as well as the effect of an absolute refractory period into the description.
Spike-train responses of single Hodgkin-Huxley (HH) and integrate-and-fire (IF) neurons with and without the refractory period, are calculated and compared. The HH and IF neurons are assumed to receive spike-train inputs with the constant interspike intervals (ISIs) and stochastic ISIs given by the Gamma distribution, through excitatory and inhibitory synaptic couplings: for both the couplings the HH neuron can fire while the IF neuron can only for the excitatory one. It is shown that the response to the constant-ISI inputs of the IF neuron strongly depends on the refractory period and the syn…
This paper proposes a methodology to extract a low-dimensional integrate-and-fire model from an arbitrarily detailed single-compartment biophysical model. The method aims at relating the modulation of maximal conductance parameters in the biophysical model to the modulation of parameters in the proposed integrate-and-fire model. The approach is illustrated on two well-documented examples of cellular neuromodulation: the transition between Type I and Type II excitability and the transition between spiking and bursting.
We study the synchronization of fully-connected and totally excitatory integrate and fire neural networks in presence of Gaussian white noises. Using a large deviation principle, we prove the stability of the synchronized state under stochastic perturbations. Then, we give a lower bound on the probability of synchronization for networks which are not initially synchronized. This bound shows the robustness of the emergence of synchronization in presence of small stochastic perturbations.
Probability density function of output interspike intervals is found in exact form for leaky integrate and fire neuron stimulated with Poisson stream. The diffusion approximation is not exploited.
We propose another integrate-and-fire model as a single neuron model. We study a globally coupled noisy integrate-and-fire model with inhibitory interaction using the Fokker-Planck equation and the Langevin equation, and find a reentrant transition of oscillatory states. Intermittent time evolutions of neuron firing are found in strongly inhibited systems. We propose another integrate-and-fire-or-burst model including the dynamics of the low-threshold Ca$^{2+}$ current based on the new integrate-and-fire model. We study a globally coupled noisy integrate-and-fire-or-burst model with inhibitory…
We explore the dynamics of an integrate-and-fire neuron with an oscillatory stimulus. The frustration due to the competition between the neuron's natural firing period and that of the oscillatory rhythm, leads to a rich structure of asymptotic phase locking patterns and ordering dynamics. The phase transitions between these states can be classified as either tangent or discontinuous bifurcations, each with its own characteristic scaling laws. The discontinuous bifurcations exhibit a new kind of phase transition that may be viewed as intermediate between continuous and first order, while tangen…
By controlling the state of neuronal populations, neuromodulators ultimately affect behaviour. A key neuromodulation mechanism is the alteration of neuronal excitability via the modulation of ion channel expression. This type of neuromodulation is normally studied via conductance-based models, but those models are computationally challenging for large-scale network simulations needed in population studies. This paper studies the modulation properties of the Multi-Quadratic Integrate-and-Fire (MQIF) model, a generalisation of the classical Quadratic Integrate-and-Fire (QIF) model. The model is …
The quadratic integrate-and-fire (QIF) model captures the normal form bifurcation dynamics of Type-I neurons found in cortex. Remarkably, this model is known to have a dual equivalent representation in terms of phase, called the $θ$-model, which has advantages for numerical simulations and analysis over the QIF model. Here, I investigate the nature of the dual representation and derive the general phase model expression for all integrate-and-fire models. Moreover, I show the condition for which the phase models exhibit discontinuous onset firing rate, the hallmark of Type-II spiking neurons.
This work reports a transfer function-based approach to characterizing the operation of single neuronal cells in terms of the instantaneous frequency of the input and output signals. We adopt the leaky integrate-and-fire model. The transfer relationship is obtained by performing successive numeric-computational simulations and statistical regressions. Several interesting results are reported, including the identification and characterization of linearity in the transfer relationship, as well as the identification of regions in the parameter space characterized by sharper transfer functions. Th…
We develop a method to investigate the effect of noise timescales on the first-passage time of nonlinear oscillators. Using Fredholm theory, we derive an exact integral equation for the mean event rate of a leaky-integrate-and-fire oscillator that receives constant input and temporally correlated noise. Furthermore, we show that Fredholm theory provides a unified framework to determine system scaling behavior for small and large noise timescales. In this framework, the leading order and higher-order asymptotic corrections for slow and fast noise are naturally emerging. We show the scaling beha…
We consider a noise driven network of integrate-and-fire neurons. The network evolves as result of the activities of the neurons following spike-timing-dependent plasticity rules. We apply a self-consistent mean-field theory to the system to obtain the mean activity level for the system as a function of the mean synaptic weight, which predicts a first-order transition and hysteresis between a noise-dominated regime and a regime of persistent neural activity. Assuming Poisson firing statistics for the neurons, the plasticity dynamics of a synapse under the influence of the mean-field environmen…
We extensively explore networks of weakly unbalanced, leaky integrate-and-fire (LIF) neurons for different coupling strength, connectivity, and by varying the degree of refractoriness, as well as the delay in the spike transmission. We find that the neural network does not only exhibit a microscopic (single-neuron) stochastic-like evolution, but also a collective irregular dynamics (CID). Our analysis is based on the computation of a suitable order parameter, typically used to characterize synchronization phenomena and on a detailed scaling analysis (i.e. simulations of different network sizes…
We study the dynamics of identical leaky integrate-and-fire neurons with symmetric non-local coupling. Upon varying control parameters (coupling strength, coupling range, refractory period) we investigate the system's behaviour and highlight the formation of chimera states. We show that the introduction of a refractory period enlarges the parameter region where chimera states appear and affects the chimera multiplicity.
Reduced models of neuronal activity such as Integrate-and-Fire models allow a description of neuronal dynamics in simple, intuitive terms and are easy to simulate numerically. We present a method to fit an Integrate-and-Fire-type model of neuronal activity, namely a modified version of the Spike Response Model, to a detailed Hodgkin-Huxley-type neuron model driven by stochastic spike arrival. In the Hogkin-Huxley model, spike arrival at the synapse is modeled by a change of synaptic conductance. For such conductance spike input, more than 70% of the postsynaptic action potentials can be predic…
Nonlinear Noisy Leaky Integrate and Fire (NNLIF) models for neurons networks can be written as Fokker-Planck-Kolmogorov equations on the probability density of neurons, the main parameters in the model being the connectivity of the network and the noise. We analyse several aspects of the NNLIF model: the number of steady states, a priori estimates, blow-up issues and convergence toward equilibrium in the linear case. In particular, for excitatory networks, blow-up always occurs for initial data concentrated close to the firing potential. These results show how critical is the balance between n…
We have studied neuronal synchronisation in a random network of adaptive exponential integrate-and-fire neurons. We study how spiking or bursting synchronous behaviour appears as a function of the coupling strength and the probability of connections, by constructing parameter spaces that identify these synchronous behaviours from measurements of the inter-spike interval and the calculation of the order parameter. Moreover, we verify the robustness of synchronisaton by applying an external perturbation to each neuron. The simulations show that bursting synchronisation is more robust than spike …
We study a class of planar integrate and fire (IF) models called adaptive integrate and fire (AIF) models, which possesses an adaptation variable on top of membrane potential, and whose subthreshold dynamics is piecewise linear (PWL). These AIF models therefore have two reset conditions, which enable bursting dynamics to emerge for suitable parameter values. Such models can be thought of as hybrid dynamical systems. We consider a particular slow dynamics within AIF models and prove the existence of bursting cycles with $N$ resets, for any integer $N$. Furthermore, we study the transition betwe…
We show that the stochastic Morris-Lecar neuron, in a neighborhood of its stable point, can be approximated by a two-dimensional Ornstein-Uhlenbeck (OU) modulation of a constant circular motion. The associated radial OU process is an example of a leaky integrate-and-fire (LIF) model prior to firing. A new model constructed from a radial OU process together with a simple firing mechanism based on detailed Morris-Lecar firing statistics reproduces the Morris-Lecar Interspike Interval (ISI) distribution, and has the computational advantages of a LIF. The result justifies the large amount of atten…
Artificial neural networks (ANNs) have been extensively used for the description of problems arising from biological systems and for constructing neuromorphic computing models. The third generation of ANNs, namely, spiking neural networks (SNNs), inspired by biological neurons enable a more realistic mimicry of the human brain. A large class of the problems from these domains is characterized by the necessity to deal with the combination of neurons, spikes and synapses via integrate-and-fire neuron models. Motivated by important applications of the integrate-and-fire of neurons in neuromorphic…
Mathematical models are an important tool for neuroscientists. During the last thirty years many papers have appeared on single neuron description and specifically on stochastic Integrate and Fire models. Analytical results have been proved and numerical and simulation methods have been developed for their study. Reviews appeared recently collect the main features of these models but do not focus on the methodologies employed to obtain them. Aim of this paper is to fill this gap by upgrading old reviews on this topic. The idea is to collect the existing methods and the available analytical res…
The collective dynamics of neural populations are often characterized in terms of correlations in the spike activity of different neurons. Open questions surround the basic nature of these correlations. In particular, what leads to higher-order correlations -- correlations in the population activity that extend beyond those expected from cell pairs? Here, we examine this question for a simple, but ubiquitous, circuit feature: common fluctuating input arriving to spiking neurons of integrate-and-fire type. We show that leads to strong higher-order correlations, as for earlier work with discrete…
We consider the criticality for firing structures of a simplified integrate-and-fire neural model on the regular network, small-world network, and random networks. We simplify an integrate-and-fire model suggested by Levina, Herrmann and Geisel (LHG). In our model we set up the synaptic strength as a constant value. We observed the power law behaviors of the probability distribution of the avalanche size and the life time of the avalanche. The critical exponents in the small-world network and the random network were the same as those in the fully connected network. However, in the regular one-…
Providing an analytical treatment to the stochastic feature of neurons' dynamics is one of the current biggest challenges in mathematical biology. The noisy leaky integrate-and-fire model and its associated Fokker-Planck equation are probably the most popular way to deal with neural variability. Another well-known formalism is the escape-rate model: a model giving the probability that a neuron fires at a certain time knowing the time elapsed since its last action potential. This model leads to a so-called age-structured system, a partial differential equation with non-local boundary condition …
Qualitatively, some real networks in the brain could be characterized as 'small worlds', in the sense that the structure of their connections is intermediate between the extremes of an orderly geometric arrangement and of a geometry-independent random mesh. Small worlds can be defined more precisely in terms of their mean path length and clustering coefficient; but is such a precise description useful to better understand how the type of connectivity affects memory retrieval? We have simulated an autoassociative memory network of integrate-and-fire units, positioned on a ring, with the network…
We analytically derive mean-field models for all-to-all coupled networks of heterogeneous, adapting, two-dimensional integrate and fire neurons. The class of models we consider includes the Izhikevich, adaptive exponential, and quartic integrate and fire models. The heterogeneity in the parameters leads to different moment closure assumptions that can be made in the derivation of the mean-field model from the population density equation for the large network. Three different moment closure assumptions lead to three different mean-field systems. These systems can be used for distinct purposes s…
The leaky integrate and fire (LIF) neuron represents standard neuronal model used for numerical simulations. The leakage is implemented in the model as exponential decay of trans-membrane voltage towards its resting value. This makes inevitable the usage of machine floating point numbers in the course of simulation. It is known that machine floating point arithmetic is subjected to small inaccuracies, which prevent from exact comparison of floating point quantities. In particular, it is incorrect to decide whether two separate in time states of a simulated system composed of LIF neurons are ex…
The statistics of the output activity of a neuron during its stimulation by the stream of input impulses that forms the stochastic Poisson process is studied. The leaky integrate-and-fire neuron is considered as a neuron model. A new representation of the probability distribution function of the output interspike interval durations is found. Based on it, the moment-generating function of the probability distribution is calculated explicitly. The latter, according to Curtiss theorem, completely determines the distribution itself. In particular, explicit expressions are derived from the moment-g…
An event-based integration scheme for an integrate-and-fire neuron model with exponentially decaying excitatory synaptic currents and double exponential inhibitory synaptic currents has recently been introduced by Carnevale and Hines. This integration scheme imposes non-physiological constraints on the time constants of the synaptic currents it attempts to model which hamper the general applicability. This paper addresses this problem in two ways. First, we provide physical arguments to show why these constraints on the time constants can be relaxed. Second, we give a formal proof showing whic…
The Nonlinear Noisy Leaky Integrate and Fire neuronal models are mathematical models that describe the activity of neural networks. These models have been studied at a microscopic level, using Stochastic Differential Equations, and at a mesoscopic/macroscopic level, through the mean field limits using Fokker-Planck type equations. The aim of this paper is to improve their understanding, using a numerical study of their particle systems. We analyse in depth the behaviour of the classical and physical solutions of the Stochastic Differential Equations and, we compare it with what is already know…
The voltage-conductance kinetic equation for integrate and fire neurons has been used in neurosciences since a decade and describes the probability density of neurons in a network. It is used when slow conductance receptors are activated and noticeable applications to the visual cortex have been worked-out. In the simplest case, the derivation also uses the assumption of fully excitatory and moderately all-to-all coupled networks; this is the situation we consider here. We study properties of solutions of the kinetic equation for steady states and time evolution and we prove several global a p…
Tactile afferents such as (RA), and Pacinian (PC) afferents that respond to external stimuli enable complicated actions such as grasping, stroking and identifying an object. To understand the tactile sensation induced by these actions deeply, the activities of the tactile afferents need to be revealed. For this purpose, we develop a computational model for each tactile afferent for vibration stimuli, combining finite element analysis finite element method (FEM) analysis and a leaky integrate-and-fire model that represents the neural characteristics. This computational model can easily estimate…
The response properties of excitable systems driven by colored noise are of great interest, but are usually mathematically only accessible via approximations. For this reason, dichotomous noise, a rare example of a colored noise leading often to analytically tractable problems, has been extensively used in the study of stochastic systems. Here, we calculate exact expressions for the power spectrum and the susceptibility of a leaky integrate-and-fire neuron driven by asymmetric dichotomous noise. While our results are in excellent agreement with simulations, they also highlight a limitation of …
We derive the Kuramoto model (KM) corresponding to a population of weakly coupled, nearly identical quadratic integrate-and-fire (QIF) neurons with both electrical and chemical coupling. The ratio of chemical to electrical coupling determines the phase lag of the characteristic sine coupling function of the KM and critically determines the synchronization properties of the network. We apply our results to investigate chimera states in two coupled populations of identical QIF neurons. We find that the presence of both electrical and chemical coupling is a necessary condition for chimera states …
Although there is increasing evidence of criticality in the brain, the processes that guide neuronal networks to reach or maintain criticality remain unclear. The present research examines the role of neuronal gain plasticity in time-series of simulated neuronal networks composed of integrate-and-fire stochastic spiking neurons, and the utility of fractal methods in assessing network criticality. Simulated time-series were derived from a network model of fully connected discrete-time stochastic excitable neurons. Monofractal and multifractal analyses were applied to neuronal gain time-series. …
One of the most important challenges in mathematical neuroscience is to properly illustrate the stochastic nature of neurons. Among different approaches, the noisy leaky integrate-and-fire and the escape rate models are probably the most popular. These two models are usually chosen to express different noise action over the neural cell. In this paper we investigate the link between the two formalisms in the case of a neuron subject to a time dependent input. To this aim, we introduce a new general stochastic framework. As we shall prove, our general framework entails the two already existing o…
Chimera states are spatiotemporal patterns in which coherence and incoherence coexist. We observe the coexistence of synchronous (coherent) and desynchronous (incoherent) domains in a neuronal network. The network is composed of coupled adaptive exponential integrate-and-fire neurons that are connected by means of chemical synapses. In our neuronal network, the chimera states exhibit spatial structures both with spikes and bursts activities. Furthermore, those desynchronised domains not only have either spike or burst activity, but we show that the structures switch between spikes and bursts a…
We present a biophysical approach for the coupling of neural network activity as resulting from proper dipole currents of cortical pyramidal neurons to the electric field in extracellular fluid. Starting from a reduced threecompartment model of a single pyramidal neuron, we derive an observation model for dendritic dipole currents in extracellular space and thereby for the dendritic field potential that contributes to the local field potential of a neural population. This work aligns and satisfies the widespread dipole assumption that is motivated by the "open-field" configuration of the dendr…
We study the collective dynamics of a Leaky Integrate and Fire network in which precise relative phase relationship of spikes among neurons are stored, as attractors of the dynamics, and selectively replayed at differentctime scales. Using an STDP-based learning process, we store in the connectivity several phase-coded spike patterns, and we find that, depending on the excitability of the network, different working regimes are possible, with transient or persistent replay activity induced by a brief signal. We introduce an order parameter to evaluate the similarity between stored and recalled …
We present two Bayesian procedures to infer the interactions and external currents in an assembly of stochastic integrate-and-fire neurons from the recording of their spiking activity. The first procedure is based on the exact calculation of the most likely time courses of the neuron membrane potentials conditioned by the recorded spikes, and is exact for a vanishing noise variance and for an instantaneous synaptic integration. The second procedure takes into account the presence of fluctuations around the most likely time courses of the potentials, and can deal with moderate noise levels. The…
The spiking activity of single neurons can be well described by a nonlinear integrate-and-fire model that includes somatic adaptation. When exposed to fluctuating inputs sparsely coupled populations of these model neurons exhibit stochastic collective dynamics that can be effectively characterized using the Fokker-Planck equation. [...] Here we derive from that description four simple models for the spike rate dynamics in terms of low-dimensional ordinary differential equations using two different reduction techniques: one uses the spectral decomposition of the Fokker-Planck operator, the othe…
Statistical properties of spike trains as well as other neurophysiological data suggest a number of mathematical models of neurons. These models range from entirely descriptive ones to those deduced from the properties of the real neurons. One of them, the diffusion leaky integrate-and-fire neuronal model, which is based on the Ornstein-Uhlenbeck stochastic process that is restricted by an absorbing barrier, can describe a wide range of neuronal activity in terms of its parameters. These parameters are readily associated with known physiological mechanisms. The other model is descriptive, Gamm…
We investigate numerically the collective dynamical behavior of pulse-coupled non-leaky integrate-and-fire-neurons that are arranged on a two-dimensional small-world network. To ensure ongoing activity, we impose a probability for spontaneous firing for each neuron. We study network dynamics evolving from different sets of initial conditions in dependence on coupling strength and rewiring probability. Beside a homogeneous equilibrium state for low coupling strength, we observe different local patterns including cyclic waves, spiral waves, and turbulent-like patterns, which -- depending on netw…
Long-range dependence (LRD) has been observed in a variety of phenomena in nature, and for several years also in the spiking activity of neurons. Often, this is interpreted as originating from a non-Markovian system. Here we show that a purely Markovian integrate-and-fire (IF) model, with a noisy slow adaptation term, can generate interspike intervals (ISIs) that appear as having LRD. However a proper analysis shows that this is not the case asymptotically. For comparison, we also consider a new model of individual IF neuron with fractional (non-Markovian) noise. The correlations of its spike …
Author summary: Synchronization of neuronal spiking in the brain is related to cognitive functions, such as perception, attention, and memory. It is therefore important to determine which properties of neurons influence their collective behavior in a network and to understand how. A prominent feature of many cortical neurons is spike frequency adaptation, which is caused by slow transmembrane currents. We investigated how these adaptation currents affect the synchronization tendency of coupled model neurons. Using the efficient adaptive exponential integrate-and-fire (aEIF) model and a biophys…
Neural network dynamics emerge from the interaction of spiking cells. One way to formulate the problem is through a theoretical framework inspired by ideas coming from statistical physics, the so-called mean-field theory. In this document, we investigate different issues related to the mean-field description of an excitatory network made up of leaky integrate-and-fire neurons. The description is written in the form a nonlinear partial differential equation which is known to blow up in finite time when the network is strongly connected. We prove that in a moderate coupling regime the equation i…
We consider a leaky integrate-and-fire neuron with deterministic subthreshold dynamics and a firing threshold that evolves as an Ornstein-Uhlenbeck process. The formulation of this minimal model is motivated by the experimentally observed widespread variation of neural firing thresholds. We show numerically that the mean first passage time can depend non-monotonically on the noise amplitude. For sufficiently large values of the correlation time of the stochastic threshold the mean first passage time is maximal for non-vanishing noise. We provide an explanation for this effect by analytically t…
How extracellular electric fields, as generated endogenously or through transcranial brain stimulation, affect the dynamics of large neuronal populations is of great interest but not well understood. To study the collective dynamics of large populations single-compartment (point) model neurons have been proven very successful. These models, however, lack the dendritic morphology to biophysically account for the effects of electric fields, and for changes in synaptic integration due to morphology alone. Here we (i) characterize the response of a canonical spatial (ball-and-stick) model neuron t…
We investigate the effect of electric synapses (gap junctions) on collective neuronal dynamics and spike statistics in a conductance-based Integrate-and-Fire neural network, driven by a Brownian noise, where conductances depend upon spike history. We compute explicitly the time evolution operator and show that, given the spike-history of the network and the membrane potentials at a given time, the further dynamical evolution can be written in a closed form. We show that spike train statistics is described by a Gibbs distribution whose potential can be approximated with an explicit formula, whe…
