Firing patterns in the adaptive exponential integrate-and-fire model

For simulations of large spiking neuron networks, an accurate, simple and versatile single-neuron modeling framework is required. Here we explore the versatility of a simple two-equation model: the adaptive exponential integrate-and-fire neuron. We show that this model generates multiple firing patterns depending on the choice of parameter values, and present a phase diagram describing the transition from one firing type to another. We give an analytical criterion to distinguish between continuous adaption, initial bursting, regular bursting and two types of tonic spiking. Also, we report that the deterministic model is capable of producing irregular spiking when stimulated with constant current, indicating low-dimensional chaos. Lastly, the simple model is fitted to real experiments of cortical neurons under step current stimulation. The results provide support for the suitability of simple models such as the adaptive exponential integrate-and-fire neuron for large network simulations.     

Exact analytical results for integrate-and-fire neurons driven by excitatory shot noise

A neuron receives input from other neurons via electrical pulses, so-called spikes. The pulse-like nature of the input is frequently neglected in analytical studies; instead, the input is usually approximated to be Gaussian. Recent experimental studies have shown, however, that an assumption underlying this approximation is often not met: Individual presynaptic spikes can have a significant effect on a neuron’s dynamics. It is thus desirable to explicitly account for the pulse-like nature of neural input, i.e. consider neurons driven by a shot noise – a long-standing problem that is mathematically challenging. In this work, we exploit the fact that excitatory shot noise with exponentially distributed weights can be obtained as a limit case of dichotomous noise, a Markovian two-state process. This allows us to obtain novel exact expressions for the stationary voltage density and the moments of the interspike-interval density of general integrate-and-fire neurons driven by such an input. For the special case of leaky integrate-and-fire neurons, we also give expressions for the power spectrum and the linear response to a signal. We verify and illustrate our expressions by comparison to simulations of leaky-, quadratic- and exponential integrate-and-fire neurons.     

Lapicque’s 1907 paper: from frogs to integrate-and-fire

Exactly 100 years ago, Louis Lapicque published a paper on the excitability of nerves that is often cited in the context of integrate-and-fire neurons. We discuss Lapicque’s contributions along with a translation of the original publication.     

Sensitive dependence of the coefficient of variation of interspike intervals on the lower boundary of membrane potential for the leaky integrate-and-fire neuron model

After the report of Softky and Koch [Softky, W.R., Koch, C., 1993. The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs. J. Neurosci. 13, 334-350], leaky integrate-and-fire models have been investigated to explain high coefficient of variation (CV) of interspike intervals (ISIs) at high firing rates observed in the cortex. The purpose of this paper is to study the effect of the position of a lower boundary of membrane potential on the possible value of CV of ISIs based on the diffusional leaky integrate-and-fire models with and without reversal potentials. Our result shows that the irregularity of ISIs for the diffusional leaky integrate-and-fire neuron significantly changes by imposing a lower boundary of membrane potential, which suggests the importance of the position of the lower boundary as well as that of the firing threshold when we study the statistical properties of leaky integrate-and-fire neuron models. It is worth pointing out that the mean-CV plot of ISIs for the diffusional leaky integrate-and-fire neuron with reversal potentials shows a close similarity to the experimental result obtained in Softky and Koch [Softky, W.R., Koch, C., 1993. The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs. J. Neurosci. 13, 334-350].     

An extended model for a spiking neuron class

This paper proposes an extension to the model of a spiking neuron for information processing in artificial neural networks, developing a new approach for the dynamic threshold of the integrate-and-fire neuron. This new approach invokes characteristics of biological neurons such as the behavior of chemical synapses and the receptor field. We demonstrate how such a digital model of spiking neurons can solve complex nonlinear classification with a single neuron, performing experiments for the classical XOR problem. Compared with rate-coded networks and the classical integrate-and-fire model, the trained network demonstrated faster information processing, requiring fewer neurons and shorter learning periods. The extended model validates all the logic functions of biological neurons when such functions are necessary for the proper flow of binary codes through a neural network.     

Integrate-and-fire models of insolation-driven entrainment of broadcast spawning in corals

The circa-annual cycle of gametogenesis produces mature gametes at the spawning “season” for successful mass spawning of broadcast corals. We develop a bioenergetic integrate-and-fire model that reveals how annual insolation rhythms can entrain the gametogenetic cycles in tropical hermatypic corals to the appropriate spawning season, since photosynthate is their primary source of energy. In the presence of short-term fluctuations in the energy input, a feedback regulatory mechanism is likely required to achieve coherence of spawning times to within one lunar cycle, in order for subsequent signals such as lunar and diurnal light cycles to unambiguously determine the “correct” night of spawning. The feedback mechanism can also provide robustness against population heterogeneity that may arise due to genetic and environmental effects. We solve the integrate-and-fire bioenergetic model numerically using the Fokker–Planck equation and use analytical tools such as rotation number to study entrainment.     

The response of cortical neurons to in vivo-like input current: theory and experiment: II. Time-varying and spatially distributed inputs

The response of a population of neurons to time-varying synaptic inputs can show a rich phenomenology, hardly predictable from the dynamical properties of the membrane’s inherent time constants. For example, a network of neurons in a state of spontaneous activity can respond significantly more rapidly than each single neuron taken individually. Under the assumption that the statistics of the synaptic input is the same for a population of similarly behaving neurons (mean field approximation), it is possible to greatly simplify the study of neural circuits, both in the case in which the statistics of the input are stationary (reviewed in La Camera et al. in Biol Cybern, 2008) and in the case in which they are time varying and unevenly distributed over the dendritic tree. Here, we review theoretical and experimental results on the single-neuron properties that are relevant for the dynamical collective behavior of a population of neurons. We focus on the response of integrate-and-fire neurons and real cortical neurons to long-lasting, noisy, in vivo-like stationary inputs and show how the theory can predict the observed rhythmic activity of cultures of neurons. We then show how cortical neurons adapt on multiple time scales in response to input with stationary statistics in vitro. Next, we review how it is possible to study the general response properties of a neural circuit to time-varying inputs by estimating the response of single neurons to noisy sinusoidal currents. Finally, we address the dendrite–soma interactions in cortical neurons leading to gain modulation and spike bursts, and show how these effects can be captured by a two-compartment integrate-and-fire neuron. Most of the experimental results reviewed in this article have been successfully reproduced by simple integrate-and-fire model neurons.     

Orientation Selectivity in Inhibition-Dominated Networks of Spiking Neurons: Effect of Single Neuron Properties and Network Dynamics

The neuronal mechanisms underlying the emergence of orientation selectivity in the primary visual cortex of mammals are still elusive. In rodents, visual neurons show highly selective responses to oriented stimuli, but neighboring neurons do not necessarily have similar preferences. Instead of a smooth map, one observes a salt-and-pepper organization of orientation selectivity. Modeling studies have recently confirmed that balanced random networks are indeed capable of amplifying weakly tuned inputs and generating highly selective output responses, even in absence of feature-selective recurrent connectivity. Here we seek to elucidate the neuronal mechanisms underlying this phenomenon by resorting to networks of integrate-and-fire neurons, which are amenable to analytic treatment. Specifically, in networks of perfect integrate-and-fire neurons, we observe that highly selective and contrast invariant output responses emerge, very similar to networks of leaky integrate-and-fire neurons. We then demonstrate that a theory based on mean firing rates and the detailed network topology predicts the output responses, and explains the mechanisms underlying the suppression of the common-mode, amplification of modulation, and contrast invariance. Increasing inhibition dominance in our networks makes the rectifying nonlinearity more prominent, which in turn adds some distortions to the otherwise essentially linear prediction. An extension of the linear theory can account for all the distortions, enabling us to compute the exact shape of every individual tuning curve in our networks. We show that this simple form of nonlinearity adds two important properties to orientation selectivity in the network, namely sharpening of tuning curves and extra suppression of the modulation. The theory can be further extended to account for the nonlinearity of the leaky model by replacing the rectifier by the appropriate smooth input-output transfer function. These results are robust and do not depend on the state of network dynamics, and hold equally well for mean-driven and fluctuation-driven regimes of activity.     

神经元网络动力学的统计物理方法

本文回顾了利用统计物理的方法研究神经元网络动力学的数学降维描述.以一个全兴奋性的“整合-发放”神经元网络为出发点,导出了描写神经元群体活动的概率分布函数的(2+1)-维对流-扩散方程.在没有引入任何新参数的情况下,讨论了如何利用moment closure scheme得到(1+1)-维的动力学方程.我们将此方程的预测...     

The membrane potential process of a single neuron seen as a cumulative damage process

A simple integrate-and-fire mechanism of a single neuron can be compared with a cumulative damage process, where the spiking process is analogous to rupture sequences of a material under cycles of stress. Although in some cases lognormal-like patterns can be recognized in the inter-spike times under a simple integrate-and-fire mechanism, fatigue life models as the inverse Gaussian distribution and the Birnbaum–Saunders distribution (which was recently introduced in the neural activity framework) provide theoretical arguments that make them more suitable for the modeling of the resulting inter-spike times.     

Decoding Input Signals in Time Domain—A Model Approach

From an observation of efferent interspike intervals of a neuron, we consider how to decode the input temporal information. It is found that the integrate-and-fire model is blind in the temporal domain due to the fact that its efferent firing rate is independent of the input temporal frequency. The conclusion is then confirmed for the integrate-and-fire model with correlated inputs, with reversal potentials, with a nonlinear leakage and with a subthreshold oscillation. For the Hodgkin-Huxley model, however, in terms of efferent firing rates alone, it is possible to read out the input temporal information.     

Robustness and regularity of oscillations in neuronal populations

We study a biologically plausible but computationally simplified integrate-and-fire neuronal model. Oscillatory activity is analyzed in the networks with and without self-connections. We perform a detailed scan of four major parameters that represent the properties of neurons and synapses: connection ratio, connection strengths, post-synaptic potential decay rate and soma's potential decay rate. It is observed that networks with different properties exhibit different periods and different patterns of synchrony. We find that generally these oscillations are robust against changes of parameters, meanwhile we also locate the parametric boundaries where oscillations break down.     

A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties

The integrate-and-fire neuron model describes the state of a neuron in terms of its membrane potential, which is determined by the synaptic inputs and the injected current that the neuron receives. When the membrane potential reaches a threshold, an action potential (spike) is generated. This review considers the model in which the synaptic input varies periodically and is described by an inhomogeneous Poisson process, with both current and conductance synapses. The focus is on the mathematical methods that allow the output spike distribution to be analyzed, including first passage time methods and the Fokker–Planck equation. Recent interest in the response of neurons to periodic input has in part arisen from the study of stochastic resonance, which is the noise-induced enhancement of the signal-to-noise ratio. Networks of integrate-and-fire neurons behave in a wide variety of ways and have been used to model a variety of neural, physiological, and psychological phenomena. The properties of the integrate-and-fire neuron model with synaptic input described as a temporally homogeneous Poisson process are reviewed in an accompanying paper (Burkitt in Biol Cybern, 2006).     

The response of cortical neurons to in vivo-like input current: theory and experiment

The study of several aspects of the collective dynamics of interacting neurons can be highly simplified if one assumes that the statistics of the synaptic input is the same for a large population of similarly behaving neurons (mean field approach). In particular, under such an assumption, it is possible to determine and study all the equilibrium points of the network dynamics when the neuronal response to noisy, in vivo-like, synaptic currents is known. The response function can be computed analytically for simple integrate-and-fire neuron models and it can be measured directly in experiments in vitro. Here we review theoretical and experimental results about the neural response to noisy inputs with stationary statistics. These response functions are important to characterize the collective neural dynamics that are proposed to be the neural substrate of working memory, decision making and other cognitive functions. Applications to the case of time-varying inputs are reviewed in a companion paper (Giugliano et al. in Biol Cybern, 2008). We conclude that modified integrate-and-fire neuron models are good enough to reproduce faithfully many of the relevant dynamical aspects of the neuronal response measured in experiments on real neurons in vitro.     

When inhibition not excitation synchronizes neural firing

Excitatory and inhibitory synaptic coupling can have counter-intuitive effects on the synchronization of neuronal firing. While it might appear that excitatory coupling would lead to synchronization, we show that frequently inhibition rather than excitation synchronizes firing. We study two identical neurons described by integrate-and-fire models, general phase-coupled models or the Hodgkin-Huxley model with mutual, non-instantaneous excitatory or inhibitory synapses between them. We find that if the rise time of the synapse is longer than the duration of an action potential, inhibition not excitation leads to synchronized firing.     

Resonantlike Synchronization and Bursting in a Model of Pulse-Coupled Neurons with Active Dendrites

We analyze the dynamical effects of active, linearized dendritic membranes on the synchronization properties of neuronal interactions. We show that a pair of pulse-coupled integrate-and-fire neurons interacting via active dendritic cables can exhibit resonantlike synchronization when the frequency of the oscillators is approximately matched to the resonant frequency of the membrane impedance. For weak coupling the neurons are phase-locked with constant interspike intervals whereas for strong coupling periodic bursting patterns are observed. This bursting behavior is reflected by the occurrence of a Hopf bifurcation in the firingrates of a corresponding rate-coded model.     

The relationship between neuronal calcium concentration and firing rate during stochastic synaptic inputs

We propose a novel, nonlinear theory about reading neuronal information using intracellular calcium concentrations, which includes the linear theory already developed in the literature as a special case. The theory is numerically confirmed using the Pinsky-Rinzel and integrate-and-fire models with constant rate Poisson inputs. Applying the theory to models with non-constant inputs, we find that there is a time lag equal to the calcium buffering time constant between the instantaneous firing rate and the firing rate estimated using calcium concentrations.     

How synapses in the auditory system wax and wane: theoretical perspectives

Spike-timing-dependent synaptic plasticity has recently provided an account of both the acuity of sound localization and the development of temporal-feature maps in the avian auditory system. The dynamics of the resulting learning equation, which describes the evolution of the synaptic weights, is governed by an unstable fixed point. We outline the derivation of the learning equation for both the Poisson neuron model and the leaky integrate-and-fire neuron with conductance synapses. The asymptotic solutions of the learning equation can be described by a spectral representation based on a biorthogonal expansion.