Higher-order statistics of input ensembles and the response of simple model neurons

Pairwise correlations among spike trains recorded in vivo have been frequently reported. It has been argued that correlated activity could play an important role in the brain, because it efficiently modulates the response of a postsynaptic neuron. We show here that a neuron's output firing rate critically depends on the higher-order statistics of the input ensemble. We constructed two statistical models of populations of spiking neurons that fired with the same rates and had identical pairwise correlations, but differed with regard to the higher-order interactions within the population. The first ensemble was characterized by clusters of spikes synchronized over the whole population. In the second ensemble, the size of spike clusters was, on average, proportional to the pairwise correlation. For both input models, we assessed the role of the size of the population, the firing rate, and the pairwise correlation on the output rate of two simple model neurons: a continuous firing-rate model and a conductance-based leaky integrate-and-fire neuron. An approximation to the mean output rate of the firing-rate neuron could be derived analytically with the help of shot noise theory. Interestingly, the essential features of the mean response of the two neuron models were similar. For both neuron models, the three input parameters played radically different roles with respect to the postsynaptic firing rate, depending on the interaction structure of the input. For instance, in the case of an ensemble with small and distributed spike clusters, the output firing rate was efficiently controlled by the size of the input population. In addition to the interaction structure, the ratio of inhibition to excitation was found to strongly modulate the effect of correlation on the postsynaptic firing rate.     

Correlation between uncoupled conductance-based integrate-and-fire neurons due to common and synchronous presynaptic firing

We investigate the firing characteristics of conductance-based integrate-and-fire neurons and the correlation of firing for uncoupled pairs of neurons as a result of common input and synchronous firing of multiple synaptic inputs. Analytical approximations are derived for the moments of the steady state potential and the effective time constant. We show that postsynaptic firing barely depends on the correlation between inhibitory inputs; only the inhibitory firing rate matters. In contrast, both the degree of synchrony and the firing rate of excitatory inputs are relevant. A coefficient of variation CV > 1 can be attained with low inhibitory firing rates and (Poisson-modulated) synchronized excitatory synaptic input, where both the number of presynaptic neurons in synchronous firing assemblies and the synchronous firing rate should be sufficiently large. The correlation in firing of a pair of uncoupled neurons due to common excitatory input is initially increased for increasing firing rates of independent inhibitory inputs but decreases for large inhibitory firing rates. Common inhibitory input to a pair of uncoupled neurons barely induces correlated firing, but amplifies the effect of common excitation. Synchronous firing assemblies in the common input further enhance the correlation and are essential to attain experimentally observed correlation values. Since uncorrelated common input (i.e., common input by neurons, which do not fire in synchrony) cannot induce sufficient postsynaptic correlation, we conclude that lateral couplings are essential to establish clusters of synchronously firing neurons.     

Synchronous firing and higher-order interactions in neuron pool

The stochastic mechanism of synchronous firing in a population of neurons is studied from the point of view of information geometry. Higher-order interactions of neurons, which cannot be reduced to pairwise correlations, are proved to exist in synchronous firing. In a neuron pool where each neuron fires stochastically, the probability distribution q(r) of the activity r, which is the fraction of firing neurons in the pool, is studied. When q(r) has a widespread distribution, in particular, when q(r) has two peaks, the neurons fire synchronously at one time and are quiescent at other times. The mechanism of generating such a probability distribution is interesting because the activity r is concentrated on its mean value when each neuron fires independently, because of the law of large numbers. Even when pairwise interactions, or third-order interactions, exist, the concentration is not resolved. This shows that higher-order interactions are necessary to generate widespread activity distributions. We analyze a simple model in which neurons receive common overlapping inputs and prove that such a model can have a widespread distribution of activity, generating higher-order stochastic interactions.     

Optimality model of unsupervised spike-timing-dependent plasticity: synaptic memory and weight distribution

We studied the hypothesis that synaptic dynamics is controlled by three basic principles: (1) synapses adapt their weights so that neurons can effectively transmit information, (2) homeostatic processes stabilize the mean firing rate of the postsynaptic neuron, and (3) weak synapses adapt more slowly than strong ones, while maintenance of strong synapses is costly. Our results show that a synaptic update rule derived from these principles shares features, with spike-timing-dependent plasticity, is sensitive to correlations in the input and is useful for synaptic memory. Moreover, input selectivity (sharply tuned receptive fields) of postsynaptic neurons develops only if stimuli with strong features are presented. Sharply tuned neurons can coexist with unselective ones, and the distribution of synaptic weights can be unimodal or bimodal. The formulation of synaptic dynamics through an optimality criterion provides a simple graphical argument for the stability of synapses, necessary for synaptic memory.     

Correlated firing improves stimulus discrimination in a retinal model

Synchronous firing limits the amount of information that can be extracted by averaging the firing rates of similarly tuned neurons. Here, we show that the loss of such rate-coded information due to synchronous oscillations between retinal ganglion cells can be overcome by exploiting the information encoded by the correlations themselves. Two very different models, one based on axon-mediated inhibitory feedback and the other on oscillatory common input, were used to generate artificial spike trains whose synchronous oscillations were similar to those measured experimentally. Pooled spike trains were summed into a threshold detector whose output was classified using Bayesian discrimination. For a threshold detector with short summation times, realistic oscillatory input yielded superior discrimination of stimulus intensity compared to rate-matched Poisson controls. Even for summation times too long to resolve synchronous inputs, gamma band oscillations still contributed to improved discrimination by reducing the total spike count variability, or Fano factor. In separate experiments in which neurons were synchronized in a stimulus-dependent manner without attendant oscillations, the Fano factor increased markedly with stimulus intensity, implying that stimulus-dependent oscillations can offset the increased variability due to synchrony alone.     

Spike-timing-dependent plasticity in small-world networks

Biologically plausible excitatory neural networks develop a persistent synchronized pattern of activity depending on spontaneous activity and synaptic refractoriness (short term depression). By fixed synaptic weights synchronous bursts of oscillatory activity are stable and involve the whole network. In our modeling study we investigate the effect of a dynamic Hebbian-like learning mechanism, spike-timing-dependent plasticity (STDP), on the changes of synaptic weights depending on synchronous activity and network connection strategies (small-world topology). We show that STDP modifies the weights of synaptic connections in such a way that synchronization of neuronal activity is considerably weakened. Networks with a higher proportion of long connections can sustain a higher level of synchronization in spite of STDP influence. The resulting distribution of the synaptic weights in single neurons depends both on the global statistics of firing dynamics and on the number of incoming and outgoing connections.     

Stochastic synchrony of chaos in a pulse-coupled neural network with both chemical and electrical synapses among inhibitory neurons

The synchronous firing of neurons in a pulse-coupled neural network composed of excitatory and inhibitory neurons is analyzed. The neurons are connected by both chemical synapses and electrical synapses among the inhibitory neurons. When electrical synapses are introduced, periodically synchronized firing as well as chaotically synchronized firing is widely observed. Moreover, we find stochastic synchrony where the ensemble-averaged dynamics shows synchronization in the network but each neuron has a low firing rate and the firing of the neurons seems to be stochastic. Stochastic synchrony of chaos corresponding to a chaotic attractor is also found.     

Distinguishing the causes of firing with the membrane potential slope

In this letter, we aim to measure the relative contribution of coincidence detection and temporal integration to the firing of spikes of a simple neuron model. To this end, we develop a method to infer the degree of synchrony in an ensemble of neurons whose firing drives a single postsynaptic cell. This is accomplished by studying the effects of synchronous inputs on the membrane potential slope of the neuron and estimating the degree of response-relevant input synchrony, which determines the neuron's operational mode. The measure is calculated using the normalized slope of the membrane potential prior to the spikes fired by a neuron, and we demonstrate that it is able to distinguish between the two operational modes. By applying this measure to the membrane potential time course of a leaky integrate-and-fire neuron with the partial somatic reset mechanism, which has been shown to be the most likely candidate to reflect the mechanism used in the brain for reproducing the highly irregular firing at high rates, we show that the partial reset model operates as a temporal integrator of incoming excitatory postsynaptic potentials and that coincidence detection is not necessary for producing such high irregular firing.     

Intrinsic stabilization of output rates by spike-based Hebbian learning.

We study analytically a model of long-term synaptic plasticity where synaptic changes are triggered by presynaptic spikes, postsynaptic spikes, and the time differences between presynaptic and postsynaptic spikes. The changes due to correlated input and output spikes are quantified by means of a learning window. We show that plasticity can lead to an intrinsic stabilization of the mean firing rate of the postsynaptic neuron. Subtractive normalization of the synaptic weights (summed over all presynaptic inputs converging on a postsynaptic neuron) follows if, in addition, the mean input rates and the mean input correlations are identical at all synapses. If the integral over the learning window is positive, firing-rate stabilization requires a non-Hebbian component, whereas such a component is not needed if the integral of the learning window is negative. A negative integral corresponds to anti-Hebbian learning in a model with slowly varying firing rates. For spike-based learning, a strict distinction between Hebbian and anti-Hebbian rules is questionable since learning is driven by correlations on the timescale of the learning window. The correlations between presynaptic and postsynaptic firing are evaluated for a piecewise-linear Poisson model and for a noisy spiking neuron model with refractoriness. While a negative integral over the learning window leads to intrinsic rate stabilization, the positive part of the learning window picks up spatial and temporal correlations in the input.     

How noise affects the synchronization properties of recurrent networks of inhibitory neurons

GABAergic interneurons play a major role in the emergence of various types of synchronous oscillatory patterns of activity in the central nervous system. Motivated by these experimental facts, modeling studies have investigated mechanisms for the emergence of coherent activity in networks of inhibitory neurons. However, most of these studies have focused either when the noise in the network is absent or weak or in the opposite situation when it is strong. Hence, a full picture of how noise affects the dynamics of such systems is still lacking. The aim of this letter is to provide a more comprehensive understanding of the mechanisms by which the asynchronous states in large, fully connected networks of inhibitory neurons are destabilized as a function of the noise level. Three types of single neuron models are considered: the leaky integrate-and-fire (LIF) model, the exponential integrate-and-fire (EIF), model and conductance-based models involving sodium and potassium Hodgkin-Huxley (HH) currents. We show that in all models, the instabilities of the asynchronous state can be classified in two classes. The first one consists of clustering instabilities, which exist in a restricted range of noise. These instabilities lead to synchronous patterns in which the population of neurons is broken into clusters of synchronously firing neurons. The irregularity of the firing patterns of the neurons is weak. The second class of instabilities, termed oscillatory firing rate instabilities, exists at any value of noise. They lead to cluster state at low noise. As the noise is increased, the instability occurs at larger coupling, and the pattern of firing that emerges becomes more irregular. In the regime of high noise and strong coupling, these instabilities lead to stochastic oscillations in which neurons fire in an approximately Poisson way with a common instantaneous probability of firing that oscillates in time.     

The relationship between synchronization among neuronal populations and their mean activity levels.

In the past decade the importance of synchronized dynamics in the brain has emerged from both empirical and theoretical perspectives. Fast dynamic synchronous interactions of an oscillatory or nonoscillatory nature may constitute a form of temporal coding that underlies feature binding and perceptual synthesis. The relationship between synchronization among neuronal populations and the population firing rates addresses two important issues: the distinction between rate coding and synchronization coding models of neuronal interactions and the degree to which empirical measurements of population activity, such as those employed by neuroimaging, are sensitive to changes in synchronization. We examined the relationship between mean population activity and synchronization using biologically plausible simulations. In this article, we focus on continuous stationary dynamics. (In a companion article, Chawla (forthcoming), we address the same issue using stimulus-evoked transients.) By manipulation parameters such as extrinsic input, intrinsic noise, synaptic efficacy, density of extrinsic connections, the voltage-sensitive nature of postsynaptic mechanisms, the number of neurons, and the laminar structure within the populations, we were able to introduce variations in both mean activity and synchronization under a variety of simulated neuronal architectures. Analyses of the simulated spike trains and local field potentials showed that in nearly every domain of the model's parameter space, mean activity and synchronization were tightly coupled. This coupling appears to be mediated by an increase in synchronous gain when effective membrane time constants are lowered by increased activity. These observations show that under the assumptions implicit in our models, rate coding and synchrony coding in neural systems with reciprocal interconnections are two perspectives on the same underlying dynamic. This suggests that in the absence of specific mechanisms decoupling changes in synchronization from firing levels, indexes of brain activity that are based purely on synaptic activity (e.g., functional magnetic resonance imaging) may also be sensitive to changes in synchronous coupling.     

Short-term synaptic plasticity can enhance weak signal detectability in nonrenewal spike trains

We study the encoding of weak signals in spike trains with interspike interval (ISI) correlations and the signals' subsequent detection in sensory neurons. Motivated by the observation of negative ISI correlations in auditory and electrosensory afferents, we assess the theoretical performance limits of an individual detector neuron receiving a weak signal distributed across multiple afferent inputs. We assess the functional role of ISI correlations in the detection process using statistical detection theory and derive two sequential likelihood ratio detector models: one for afferents with renewal statistics; the other for afferents with negatively correlated ISIs. We suggest a mechanism that might enable sensory neurons to implicitly compute conditional probabilities of presynaptic spikes by means of short-term synaptic plasticity. We demonstrate how this mechanism can enhance a postsynaptic neuron's sensitivity to weak signals by exploiting the correlation structure of the input spike trains. Our model not only captures fundamental aspects of early electrosensory signal processing in weakly electric fish, but may also bear relevance to the mammalian auditory system and other sensory modalities.     

Learning Temporally Encoded Patterns in Networks of Spiking Neurons

Networks of spiking neurons are very powerful and versatile models for biological and artificial information processing systems. Especially for modelling pattern analysis tasks in a biologically plausible way that require short response times with high precision they seem to be more appropriate than networks of threshold gates or models that encode analog values in average firing rates. We investigate the question how neurons can learn on the basis of time differences between firing times. In particular, we provide learning rules of the Hebbian type in terms of single spiking events of the pre- and postsynaptic neuron and show that the weights approach some value given by the difference between pre- and postsynaptic firing times with arbitrary high precision.     

Theory of input spike auto- and cross-correlations and their effect on the response of spiking neurons

Spike correlations between neurons are ubiquitous in the cortex, but their role is not understood. Here we describe the firing response of a leaky integrate-and-fire neuron (LIF) when it receives a temporarily correlated input generated by presynaptic correlated neuronal populations. Input correlations are characterized in terms of the firing rates, Fano factors, correlation coefficients, and correlation timescale of the neurons driving the target neuron. We show that the sum of the presynaptic spike trains cannot be well described by a Poisson process. In fact, the total input current has a nontrivial two-point correlation function described by two main parameters: the correlation timescale (how precise the input correlations are in time) and the correlation magnitude (how strong they are). Therefore, the total current generated by the input spike trains is not well described by a white noise gaussian process. Instead, we model the total current as a colored gaussian process with the same mean and two-point correlation function, leading to the formulation of the problem in terms of a Fokker-Planck equation. Solutions of the output firing rate are found in the limit of short and long correlation timescales. The solutions described here expand and improve on our previous results (Moreno, de la Rocha, Renart, & Parga, 2002) by presenting new analytical expressions for the output firing rate for general IF neurons, extending the validity of the results for arbitrarily large correlation magnitude, and by describing the differential effect of correlations on the mean-driven or noise-dominated firing regimes. Also the details of this novel formalism are given here for the first time. We employ numerical simulations to confirm the analytical solutions and study the firing response to sudden changes in the input correlations. We expect this formalism to be useful for the study of correlations in neuronal networks and their role in neural processing and information transmission.     

Population dynamics of spiking neurons: fast transients, asynchronous states, and locking

An integral equation describing the time evolution of the population activity in a homogeneous pool of spiking neurons of the integrate-and-fire type is discussed. It is analytically shown that transients from a state of incoherent firing can be immediate. The stability of incoherent firing is analyzed in terms of the noise level and transmission delay, and a bifurcation diagram is derived. The response of a population of noisy integrate-and-fire neurons to an input current of small amplitude is calculated and characterized by a linear filter L. The stability of perfectly synchronized"locked"solutions is analyzed.     

Computational properties of networks of synchronous groups of spiking neurons

We demonstrate a model in which synchronously firing ensembles of neurons are networked to produce computational results. Each ensemble is a group of biological integrate-and-fire spiking neurons, with probabilistic interconnections between groups. An analogy is drawn in which each individual processing unit of an artificial neural network corresponds to a neuronal group in a biological model. The activation value of a unit in the artificial neural network corresponds to the fraction of active neurons, synchronously firing, in a biological neuronal group. Weights of the artificial neural network correspond to the product of the interconnection density between groups, the group size of the presynaptic group, and the postsynaptic potential heights in the synchronous group model. All three of these parameters can modulate connection strengths between neuronal groups in the synchronous group models. We give an example of nonlinear classification (XOR) and a function approximation example in which the capability of the artificial neural network can be captured by a neural network model with biological integrate-and-fire neurons configured as a network of synchronously firing ensembles of such neurons. We point out that the general function approximation capability proven for feedforward artificial neural networks appears to be approximated by networks of neuronal groups that fire in synchrony, where the groups comprise integrate-and-fire neurons. We discuss the advantages of this type of model for biological systems, its possible learning mechanisms, and the associated timing relationships.     

Influences of synaptic location on the synchronization of rhythmic bursting neurons

We study how the location of synaptic input influences the stablex firing states in coupled model neurons bursting rhythmically at the gamma frequencies (20-70 Hz). The model neuron consists of two compartments and generates one, two, three or four spikes in each burst depending on the intensity of input current and the maximum conductance of M-type potassium current. If the somata are connected by reciprocal excitatory synapses, we find strong correlations between the changes in the bursting mode and those in the stable phase-locked states of the coupled neurons. The stability of the in-phase phase-locked state (synchronously firing state) tends to change when the individual neurons change their bursting patterns. If, however, the synaptic connections are terminated on the dendritic compartments, no such correlated changes occur. In this case, the coupled bursting neurons do not show the in-phase phase-locked state in any bursting mode. These results indicate that synchronization behaviour of bursting neurons significantly depends on the synaptic location, unlike a coupled system of regular spiking neurons.     

Estimation of time-dependent input from neuronal membrane potential

The set of firing rates of the presynaptic excitatory and inhibitory neurons constitutes the input signal to the postsynaptic neuron. Estimation of the time-varying input rates from intracellularly recorded membrane potential is investigated here. For that purpose, the membrane potential dynamics must be specified. We consider the Ornstein-Uhlenbeck stochastic process, one of the most common single-neuron models, with time-dependent mean and variance. Assuming the slow variation of these two moments, it is possible to formulate the estimation problem by using a state-space model. We develop an algorithm that estimates the paths of the mean and variance of the input current by using the empirical Bayes approach. Then the input firing rates are directly available from the moments. The proposed method is applied to three simulated data examples: constant signal, sinusoidally modulated signal, and constant signal with a jump. For the constant signal, the estimation performance of the method is comparable to that of the traditionally applied maximum likelihood method. Further, the proposed method accurately estimates both continuous and discontinuous time-variable signals. In the case of the signal with a jump, which does not satisfy the assumption of slow variability, the robustness of the method is verified. It can be concluded that the method provides reliable estimates of the total input firing rates, which are not experimentally measurable.