Self-organizing dual coding based on spike-time-dependent plasticity

It has been a matter of debate how firing rates or spatiotemporal spike patterns carry information in the brain. Recent experimental and theoretical work in part showed that these codes, especially a population rate code and a synchronous code, can be dually used in a single architecture. However, we are not yet able to relate the role of firing rates and synchrony to the spatiotemporal structure of inputs and the architecture of neural networks. In this article, we examine how feedforward neural networks encode multiple input sources in the firing patterns. We apply spike-time-dependent plasticity as a fundamental mechanism to yield synaptic competition and the associated input filtering. We use the Fokker-Planck formalism to analyze the mechanism for synaptic competition in the case of multiple inputs, which underlies the formation of functional clusters in downstream layers in a self-organizing manner. Depending on the types of feedback coupling and shared connectivity, clusters are independently engaged in population rate coding or synchronous coding, or they interact to serve as input filters. Classes of dual codings and functional roles of spike-time-dependent plasticity are also discussed.     

Dynamics of Learning in Multilayer Perceptrons Near Singularities

The dynamical behavior of learning is known to be very slow for the multilayer perceptron, being often trapped in the “plateau.” It has been recently understood that this is due to the singularity in the parameter space of perceptrons, in which trajectories of learning are drawn. The space is Riemannian from the point of view of information geometry and contains singular regions where the Riemannian metric or the Fisher information matrix degenerates. This paper analyzes the dynamics of learning in a neighborhood of the singular regions when the true teacher machine lies at the singularity. We give explicit asymptotic analytical solutions (trajectories) both for the standard gradient (SGD) and natural gradient (NGD) methods. It is clearly shown, in the case of the SGD method, that the plateau phenomenon appears in a neighborhood of the critical regions, where the dynamical behavior is extremely slow. The analysis of the NGD method is much more difficult, because the inverse of the Fisher information matrix diverges. We conquer the difficulty by introducing the “blow-down” technique used in algebraic geometry. The NGD method works efficiently, and the state converges directly to the true parameters very quickly while it staggers in the case of the SGD method. The analytical results are compared with computer simulations, showing good agreement. The effects of singularities on learning are thus qualitatively clarified for both standard and NGD methods.      

Population coding and decoding in a neural field: a computational study

This study uses a neural field model to investigate computational aspects of population coding and decoding when the stimulus is a single variable. A general prototype model for the encoding process is proposed, in which neural responses are correlated, with strength specified by a gaussian function of their difference in preferred stimuli. Based on the model, we study the effect of correlation on the Fisher information, compare the performances of three decoding methods that differ in the amount of encoding information being used, and investigate the implementation of the three methods by using a recurrent network. This study not only rediscovers main results in existing literatures in a unified way, but also reveals important new features, especially when the neural correlation is strong. As the neural correlation of firing becomes larger, the Fisher information decreases drastically. We confirm that as the width of correlation increases, the Fisher information saturates and no longer increases in proportion to the number of neurons. However, we prove that as the width increases further--wider than (sqrt)2 times the effective width of the turning function--the Fisher information increases again, and it increases without limit in proportion to the number of neurons. Furthermore, we clarify the asymptotic efficiency of the maximum likelihood inference (MLI) type of decoding methods for correlated neural signals. It shows that when the correlation covers a nonlocal range of population (excepting the uniform correlation and when the noise is extremely small), the MLI type of method, whose decoding error satisfies the Cauchy-type distribution, is not asymptotically efficient. This implies that the variance is no longer adequate to measure decoding accuracy.     

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.     

An exact method to quantify the information transmitted by different mechanisms of correlational coding

We derive a new method to quantify the impact of correlated firing on the information transmitted by neuronal populations. This new method considers, in an exact way, the effects of high order spike train statistics, with no approximation involved, and it generalizes our previous work that was valid for short time windows and small populations. The new technique permits one to quantify the information transmitted if each cell were to convey fully independent information separately from the information available in the presence of synergy-redundancy effects. Synergy-redundancy effects are shown to arise from three possible contributions: a redundant contribution due to similarities in the mean response profiles of different cells; a synergistic stimulus-dependent correlational contribution quantifying the information content of changes of correlations with stimulus, and a stimulus-independent correlational contribution term that reflects interactions between the distribution of rates of individual cells and the average level of cross-correlation. We apply the new method to simultaneously recorded data from somatosensory and visual cortices. We demonstrate that it constitutes a reliable tool to determine the role of cross-correlated activity in stimulus coding even when high firing rate data (such as multi-unit recordings) are considered.     

Ergodicity of spike trains: when does trial averaging make sense?

Neuronal information processing is often studied on the basis of spiking patterns. The relevant statistics such as firing rates calculated with the peri-stimulus time histogram are obtained by averaging spiking patterns over many experimental runs. However, animals should respond to one experimental stimulation in real situations, and what is available to the brain is not the trial statistics but the population statistics. Consequently, physiological ergodicity, namely, the consistency between trial averaging and population averaging, is implicitly assumed in the data analyses, although it does not trivially hold true. In this letter, we investigate how characteristics of noisy neural network models, such as single neuron properties, external stimuli, and synaptic inputs, affect the statistics of firing patterns. In particular, we show that how high membrane potential sensitivity to input fluctuations, inability of neurons to remember past inputs, external stimuli with large variability and temporally separated peaks, and relatively few contributions of synaptic inputs result in spike trains that are reproducible over many trials. The reproducibility of spike trains and synchronous firing are contrasted and related to the ergodicity issue. Several numerical calculations with neural network examples are carried out to support the theoretical results.     

The effects of pair-wise and higher order correlations on the firing rate of a post-synaptic neuron

Coincident firing of neurons projecting to a common target cell is likely to raise the probability of firing of this postsynaptic cell. Therefore, synchronized firing constitutes a significant event for postsynaptic neurons and is likely to play a role in neuronal information processing. Physiological data on synchronized firing in cortical networks are based primarily on paired recordings and cross-correlation analysis. However, pair-wise correlations among all inputs onto a postsynaptic neuron do not uniquely determine the distribution of simultaneous postsynaptic events. We develop a framework in order to calculate the amount of synchronous firing that, based on maximum entropy, should exist in a homogeneous neural network in which the neurons have known pair-wise correlations and higher-order structure is absent. According to the distribution of maximal entropy, synchronous events in which a large proportion of the neurons participates should exist even in the case of weak pair-wise correlations. Network simulations also exhibit these highly synchronous events in the case of weak pair-wise correlations. If such a group of neurons provides input to a common postsynaptic target, these network bursts may enhance the impact of this input, especially in the case of a high postsynaptic threshold. The proportion of neurons participating in synchronous bursts can be approximated by our method under restricted conditions. When these conditions are not fulfilled, the spike trains have less than maximal entropy, which is indicative of the presence of higher-order structure. In this situation, the degree of synchronicity cannot be derived from the pair-wise correlations.     

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.     

Representational accuracy of stochastic neural populations.

Fisher information is used to analyze the accuracy with which a neural population encodes D stimulus features. It turns out that the form of response variability has a major impact on the encoding capacity and therefore plays an important role in the selection of an appropriate neural model. In particular, in the presence of baseline firing, the reconstruction error rapidly increases with D in the case of Poissonian noise but not for additive noise. The existence of limited-range correlations of the type found in cortical tissue yields a saturation of the Fisher information content as a function of the population size only for an additive noise model. We also show that random variability in the correlation coefficient within a neural population, as found empirically, considerably improves the average encoding quality. Finally, the representational accuracy of populations with inhomogeneous tuning properties, either with variability in the tuning widths or fragmented into specialized subpopulations, is superior to the case of identical and radially symmetric tuning curves usually considered in the literature.     

Singularities affect dynamics of learning in neuromanifolds

The parameter spaces of hierarchical systems such as multilayer perceptrons include singularities due to the symmetry and degeneration of hidden units. A parameter space forms a geometrical manifold, called the neuromanifold in the case of neural networks. Such a model is identified with a statistical model, and a Riemannian metric is given by the Fisher information matrix. However, the matrix degenerates at singularities. Such a singular structure is ubiquitous not only in multilayer perceptrons but also in the gaussian mixture probability densities, ARMA time-series model, and many other cases. The standard statistical paradigm of the Cramér-Rao theorem does not hold, and the singularity gives rise to strange behaviors in parameter estimation, hypothesis testing, Bayesian inference, model selection, and in particular, the dynamics of learning from examples. Prevailing theories so far have not paid much attention to the problem caused by singularity, relying only on ordinary statistical theories developed for regular (nonsingular) models. Only recently have researchers remarked on the effects of singularity, and theories are now being developed.This article gives an overview of the phenomena caused by the singularities of statistical manifolds related to multilayer perceptrons and gaussian mixtures. We demonstrate our recent results on these problems. Simple toy models are also used to show explicit solutions. We explain that the maximum likelihood estimator is no longer subject to the gaussian distribution even asymptotically, because the Fisher information matrix degenerates, that the model selection criteria such as AIC, BIC, and MDL fail to hold in these models, that a smooth Bayesian prior becomes singular in such models, and that the trajectories of dynamics of learning are strongly affected by the singularity, causing plateaus or slow manifolds in the parameter space. The natural gradient method is shown to perform well because it takes the singular geometrical structure into account. The generalization error and the training error are studied in some examples.     

Random neural networks with synchronized interactions

Large-scale distributed systems, such as natural neuronal and artificial systems, have many local interconnections, but they often also have the ability to propagate information very fast over relatively large distances. Mechanisms that enable such behavior include very long physical signaling paths and possibly saccades of synchronous behavior that may propagate across a network. This letter studies the modeling of such behaviors in neuronal networks and develops a related learning algorithm. This is done in the context of the random neural network (RNN), a probabilistic model with a well-developed mathematical theory, which was inspired by the apparently stochastic spiking behavior of certain natural neuronal systems. Thus, we develop an extension of the RNN to the case when synchronous interactions can occur, leading to synchronous firing by large ensembles of cells. We also present an O(N3) gradient descent learning algorithm for an N-cell recurrent network having both conventional excitatory-inhibitory interactions and synchronous interactions. Finally, the model and its learning algorithm are applied to a resource allocation problem that is NP-hard and requires fast approximate decisions.     

Duality of rate coding and temporal coding in multilayered feedforward networks

A functional role for precise spike timing has been proposed as an alternative hypothesis to rate coding. We show in this article that both the synchronous firing code and the population rate code can be used dually in a common framework of a single neural network model. Furthermore, these two coding mechanisms are bridged continuously by several modulatable model parameters, including shared connectivity, feedback strength, membrane leak rate, and neuron heterogeneity. The rates of change of these parameters are closely related to the response time and the timescale of learning.     

Spatiotemporal coding in the cortex: information flow-based learning in spiking neural networks

We introduce a learning paradigm for networks of integrate-and-fire spiking neurons that is based on an information-theoretic criterion. This criterion can be viewed as a first principle that demonstrates the experimentally observed fact that cortical neurons display synchronous firing for some stimuli and not for others. The principle can be regarded as the postulation of a nonparametric reconstruction method as optimization criteria for learning the required functional connectivity that justifies and explains synchronous firing for binding of features as a mechanism for spatiotemporal coding. This can be expressed in an information-theoretic way by maximizing the discrimination ability between different sensory inputs in minimal time.     

A unified approach to the study of temporal, correlational, and rate coding

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.     

Nonlinear population codes

Theoretical and experimental studies of distributed neuronal representations of sensory and behavioral variables usually assume that the tuning of the mean firing rates is the main source of information. However, recent theoretical studies have investigated the effect of cross-correlations in the trial-to-trial fluctuations of the neuronal responses on the accuracy of the representation. Assuming that only the first-order statistics of the neuronal responses are tuned to the stimulus, these studies have shown that in the presence of correlations, similar to those observed experimentally in cortical ensembles of neurons, the amount of information in the population is limited, yielding nonzero error levels even in the limit of infinitely large populations of neurons. In this letter, we study correlated neuronal populations whose higher-order statistics, and in particular response variances, are also modulated by the stimulus. Weask two questions: Does the correlated noise limit the accuracy of the neuronal representation of the stimulus? and, How can a biological mechanism extract most of the information embedded in the higher-order statistics of the neuronal responses? Specifically, we address these questions in the context of a population of neurons coding an angular variable. We show that the information embedded in the variances grows linearly with the population size despite the presence of strong correlated noise. This information cannot be extracted by linear readout schemes, including the linear population vector. Instead, we propose a bilinear readout scheme that involves spatial decorrelation, quadratic nonlinearity, and population vector summation. We show that this nonlinear population vector scheme yields accurate estimates of stimulus parameters, with an efficiency that grows linearly with the population size. This code can be implemented using biologically plausible neurons.     

An invariance principle for maintaining the operating point of a neuron

Sensory neurons adapt to changes in the natural statistics of their environments through processes such as gain control and firing threshold adjustment. It has been argued that neurons early in sensory pathways adapt according to information-theoretic criteria, perhaps maximising their coding efficiency or information rate. Here, we draw a distinction between how a neuron's preferred operating point is determined and how its preferred operating point is maintained through adaptation. We propose that a neuron's preferred operating point can be characterised by the probability density function (PDF) of its output spike rate, and that adaptation maintains an invariant output PDF, regardless of how this output PDF is initially set. Considering a sigmoidal transfer function for simplicity, we derive simple adaptation rules for a neuron with one sensory input that permit adaptation to the lower-order statistics of the input, independent of how the preferred operating point of the neuron is set. Thus, if the preferred operating point is, in fact, set according to information-theoretic criteria, then these rules nonetheless maintain a neuron at that point. Our approach generalises from the unimodal case to the multimodal case, for a neuron with inputs from distinct sensory channels, and we briefly consider this case too.     

Optimal short-term population coding: when Fisher information fails

Efficient coding has been proposed as a first principle explaining neuronal response properties in the central nervous system. The shape of optimal codes, however, strongly depends on the natural limitations of the particular physical system. Here we investigate how optimal neuronal encoding strategies are influenced by the finite number of neurons N (place constraint), the limited decoding time window length T (time constraint), the maximum neuronal firing rate f(max) (power constraint), and the maximal average rate (f)(max) (energy constraint). While Fisher information provides a general lower bound for the mean squared error of unbiased signal reconstruction, its use to characterize the coding precision is limited. Analyzing simple examples, we illustrate some typical pitfalls and thereby show that Fisher information provides a valid measure for the precision of a code only if the dynamic range (f(min)T, f(max)T) is sufficiently large. In particular, we demonstrate that the optimal width of gaussian tuning curves depends on the available decoding time T. Within the broader class of unimodal tuning functions, it turns out that the shape of a Fisher-optimal coding scheme is not unique. We solve this ambiguity by taking the minimum mean square error into account, which leads to flat tuning curves. The tuning width, however, remains to be determined by energy constraints rather than by the principle of efficient coding.     

Implications of noise and neural heterogeneity for vestibulo-ocular reflex fidelity

The vestibulo-ocular reflex (VOR) is characterized by a short-latency, high-fidelity eye movement response to head rotations at frequencies up to 20 Hz. Electrophysiological studies of medial vestibular nucleus (MVN) neurons, however, show that their response to sinusoidal currents above 10 to 12 Hz is highly nonlinear and distorted by aliasing for all but very small current amplitudes. How can this system function in vivo when single cell response cannot explain its operation? Here we show that the necessary wide VOR frequency response may be achieved not by firing rate encoding of head velocity in single neurons, but in the integrated population response of asynchronously firing, intrinsically active neurons. Diffusive synaptic noise and the pacemaker-driven, intrinsic firing of MVN cells synergistically maintain asynchronous, spontaneous spiking in a population of model MVN neurons over a wide range of input signal amplitudes and frequencies. Response fidelity is further improved by a reciprocal inhibitory link between two MVN populations, mimicking the vestibular commissural system in vivo, but only if asynchrony is maintained by noise and pacemaker inputs. These results provide a previously missing explanation for the full range of VOR function and a novel account of the role of the intrinsic pacemaker conductances in MVN cells. The values of diffusive noise and pacemaker currents that give optimal response fidelity yield firing statistics similar to those in vivo, suggesting that the in vivo network is tuned to optimal performance. While theoretical studies have argued that noise and population heterogeneity can improve coding, to our knowledge this is the first evidence indicating that these parameters are indeed tuned to optimize coding fidelity in a neural control system in vivo.     

Insights from a simple expression for linear fisher information in a recurrently connected population of spiking neurons

A simple expression for a lower bound of Fisher information is derived for a network of recurrently connected spiking neurons that have been driven to a noise-perturbed steady state. We call this lower bound linear Fisher information, as it corresponds to the Fisher information that can be recovered by a locally optimal linear estimator. Unlike recent similar calculations, the approach used here includes the effects of nonlinear gain functions and correlated input noise and yields a surprisingly simple and intuitive expression that offers substantial insight into the sources of information degradation across successive layers of a neural network. Here, this expression is used to (1) compute the optimal (i.e., information-maximizing) firing rate of a neuron, (2) demonstrate why sharpening tuning curves by either thresholding or the action of recurrent connectivity is generally a bad idea, (3) show how a single cortical expansion is sufficient to instantiate a redundant population code that can propagate across multiple cortical layers with minimal information loss, and (4) show that optimal recurrent connectivity strongly depends on the covariance structure of the inputs to the network.     

Optimal neural rate coding leads to bimodal firing rate distributions

Many experimental studies concerning the neuronal code are based on graded responses of neurons, given by the emitted number of spikes measured in a certain time window. Correspondingly, a large body of neural network theory deals with analogue neuron models and discusses their potential use for computation or function approximation. All physical signals, however, are of limited precision, and neuronal firing rates in cortex are relatively low. Here, we investigate the relevance of analogue signal processing with spikes in terms of optimal stimulus reconstruction and information theory. In particular, we derive optimal tuning functions taking the biological constraint of limited firing rates into account. It turns out that depending on the available decoding time T, optimal encoding undergoes a phase transition from discrete binary coding for small T towards analogue or quasi-analogue encoding for large T. The corresponding firing rate distributions are bimodal for all relevant T, in particular in the case of population coding.