Despite their biological plausibility, neural network models with asymmetric weights are rarely solved analytically, and closed-form solutions are available only in some limiting cases or in some mean-field approximations. We found exact analytical solutions of an asymmetric spin model of neural networks with arbitrary size without resorting to any approximation, and we comprehensively studied its dynamical and statistical properties. The network had discrete time evolution equations and binary firing rates, and it could be driven by noise with any distribution. We found analytical expressions of the conditional and stationary joint probability distributions of the membrane potentials and the firing rates. By manipulating the conditional probability distribution of the firing rates, we extend to stochastic networks the associating learning rule previously introduced by Personnaz and coworkers. The new learning rule allowed the safe storage, under the presence of noise, of point and cyclic attractors, with useful implications for content-addressable memories. Furthermore, we studied the bifurcation structure of the network dynamics in the zero-noise limit. We analytically derived examples of the codimension 1 and codimension 2 bifurcation diagrams of the network, which describe how the neuronal dynamics changes with the external stimuli. This showed that the network may undergo transitions among multistable regimes, oscillatory behavior elicited by asymmetric synaptic connections, and various forms of spontaneous symmetry breaking. We also calculated analytically groupwise correlations of neural activity in the network in the stationary regime. This revealed neuronal regimes where, statistically, the membrane potentials and the firing rates are either synchronous or asynchronous. Our results are valid for networks with any number of neurons, although our equations can be realistically solved only for small networks. For completeness, we also derived the network equations in the thermodynamic limit of infinite network size and we analytically studied their local bifurcations. All the analytical results were extensively validated by numerical simulations.

This letter introduces a study to precisely measure what an increase in spike timing precision can add to spike-driven pattern recognition algorithms. The concept of generating spikes from images by converting gray levels into spike timings is currently at the basis of almost every spike-based modeling of biological visual systems. The use of images naturally leads to generating incorrect artificial and redundant spike timings and, more important, also contradicts biological findings indicating that visual processing is massively parallel, asynchronous with high temporal resolution. A new concept for acquiring visual information through pixel-individual asynchronous level-crossing sampling has been proposed in a recent generation of asynchronous neuromorphic visual sensors. Unlike conventional cameras, these sensors acquire data not at fixed points in time for the entire array but at fixed amplitude changes of their input, resulting optimally sparse in space and time—pixel individually and precisely timed only if new, (previously unknown) information is available (event based). This letter uses the high temporal resolution spiking output of neuromorphic event-based visual sensors to show that lowering time precision degrades performance on several recognition tasks specifically when reaching the conventional range of machine vision acquisition frequencies (30–60 Hz). The use of information theory to characterize separability between classes for each temporal resolution shows that high temporal acquisition provides up to 70% more information that conventional spikes generated from frame-based acquisition as used in standard artificial vision, thus drastically increasing the separability between classes of objects. Experiments on real data show that the amount of information loss is correlated with temporal precision. Our information-theoretic study highlights the potentials of neuromorphic asynchronous visual sensors for both practical applications and theoretical investigations. Moreover, it suggests that representing visual information as a precise sequence of spike times as reported in the retina offers considerable advantages for neuro-inspired visual computations.

We study the relationship between the accuracy of a large neuronal population in encoding periodic sensory stimuli and the width of the tuning curves of individual neurons in the population. By using general simple models of population activity, we show that when considering one or two periodic stimulus features, a narrow tuning width provides better population encoding accuracy. When encoding more than two periodic stimulus features, the information conveyed by the population is instead maximal for finite values of the tuning width. These optimal values are only weakly dependent on model parameters and are similar to the width of tuning to orientation ormotion direction of real visual cortical neurons. A very large tuning width leads to poor encoding accuracy, whatever the number of stimulus features encoded. Thus, optimal coding of periodic stimuli is different from that of nonperiodic stimuli, which, as shown in previous studies, would require infinitely large tuning widths when coding more than two stimulus features.

We demonstrate that the information contained in the spike occurrence times of a population of neurons can be broken up into a series of terms, each reflecting something about potential coding mechanisms. This is possible in the coding regime in which few spikes are emitted in the relevant time window. This approach allows us to study the additional information contributed by spike timing beyond that present in the spike counts and to examine the contributions to the whole information of different statistical properties of spike trains, such as firing rates and correlation functions. It thus forms the basis for a new quantitative procedure for analyzing simultaneous multiple neuron recordings and provides theoretical constraints on neural coding strategies. We find a transition between two coding regimes, depending on the size of the relevant observation timescale. For time windows shorter than the timescale of the stimulus-induced response fluctuations, there exists a spike count coding phase, in which the purely temporal information is of third order in time. For time windows much longer than the characteristic timescale, there can be additional timing information of first order, leading to a temporal coding phase in which timing information may affect the instantaneous information rate. In this new framework, we study the relative contributions of the dynamic firing rate and correlation variables to the full temporal information, the interaction of signal and noise correlations in temporal coding, synergy between spikes and between cells, and the effect of refractoriness. We illustrate the utility of the technique by analyzing a few cells from the rat barrel cortex.

The effectiveness of various stimulus identification (decoding) procedures for extracting the information carried by the responses of a population of neurons to a set of repeatedly presented stimuli is studied analytically, in the limit of short time windows. It is shown that in this limit, the entire information content of the responses can sometimes be decoded, and when this is not the case, the lost information is quantified. In particular, the mutual information extracted by taking into account only the most likely stimulus in each trial turns out to be, if not equal, much closer to the true value than that calculated from all the probabilities that each of the possible stimuli in the set was the actual one. The relation between the mutual information extracted by decoding and the percentage of correct stimulus decodings is also derived analytically in the same limit, showing that the metric content index can be estimated reliably from a few cells recorded from brief periods. Computer simulations as well as the activity of real neurons recorded in the primate hippocampus serve to confirm these results and illustrate the utility and limitations of the approach.

The distribution of responses of sensory neurons to ecological stimulation has been proposed to be designed to maximize information transmission, which according to a simple model would imply an exponential distribution of spike counts in a given time window. We have used recordings from inferior temporal cortex neurons responding to quasi-natural visual stimulation (presented using a video of everyday lab scenes and a large number of static images of faces and natural scenes) to assess the validity of this exponential model and to develop an alternative simple model of spike count distributions. We find that the exponential model has to be rejected in 84% of cases (at the p < 0.01 level). A new model, which accounts for the firing rate distribution found in terms of slow and fast variability in the inputs that produce neuronal activation, is rejected statistically in only 16% of cases. Finally, we show that the neurons are moderately efficient at transmitting information but not optimally efficient.

It is difficult to extract the information carried by neuronal responses about a set of stimuli because limited data samples result in biased es timates. Recently two improved procedures have been developed to calculate information from experimental results: a binning-and-correcting procedure and a neural network procedure. We have used data produced from a model of the spatiotemporal receptive fields of parvocellular and magnocellular lateral geniculate neurons to study the performance of these methods as a function of the number of trials used. Both procedures yield accurate results for one-dimensional neuronal codes. They can also be used to produce a reasonable estimate of the extra information in a three-dimensional code, in this instance, within 0.05-0.1 bit of the asymptotically calculated value—about 10% of the total transmitted information. We believe that this performance is much more accurate than previous procedures.

Extracting information measures from limited experimental samples, such as those normally available when using data recorded in vivo from mammalian cortical neurons, is known to be plagued by a systematic error, which tends to bias the estimate upward. We calculate here the average of the bias, under certain conditions, as an asymptotic expansion in the inverse of the size of the data sample. The result agrees with numerical simulations, and is applicable, as an additive correction term, to measurements obtained under such conditions. Moreover, we discuss the implications for measurements obtained through other usual procedures.