Serial correlation in neural spike trains: experimental evidence, stochastic modeling, and single neuron variability

A dynamic spike threshold with correlated noise predicts observed patterns of negative interval correlations in neuronal spike trains

Negative correlations in the sequential evolution of interspike intervals (ISIs) are a signature of memory in neuronal spike-trains. They provide coding benefits including firing-rate stabilization, improved detectability of weak sensory signals, and enhanced transmission of information by improving signal-to-noise ratio. Primary electrosensory afferent spike-trains in weakly electric fish fall into two categories based on the pattern of ISI correlations: non-bursting units have negative correlations which remain negative but decay to zero with increasing lags (Type I ISI correlations), and bursting units have oscillatory (alternating sign) correlation which damp to zero with increasing lags (Type II ISI correlations). Here, we predict and match observed ISI correlations in these afferents using a stochastic dynamic threshold model. We determine the ISI correlation function as a function of an arbitrary discrete noise correlation function [Formula: see text], where k is a multiple of the mean ISI. The function permits forward and inverse calculations of the correlation function. Both types of correlation functions can be generated by adding colored noise to the spike threshold with Type I correlations generated with slow noise and Type II correlations generated with fast noise. A first-order autoregressive (AR) process with a single parameter is sufficient to predict and accurately match both types of afferent ISI correlation functions, with the type being determined by the sign of the AR parameter. The predicted and experimentally observed correlations are in geometric progression. The theory predicts that the limiting sum of ISI correlations is [Formula: see text] yielding a perfect DC-block in the power spectrum of the spike train. Observed ISI correlations from afferents have a limiting sum that is slightly larger at [Formula: see text] ([Formula: see text]). We conclude that the underlying process for generating ISIs may be a simple combination of low-order AR and moving average processes and discuss the results from the perspective of optimal coding.

Mean-return-time phase of a stochastic oscillator provides an approximate renewal description for the associated point process

Stochastic oscillations can be characterized by a corresponding point process; this is a common practice in computational neuroscience, where oscillations of the membrane voltage under the influence of noise are often analyzed in terms of the interspike interval statistics, specifically the distribution and correlation of intervals between subsequent threshold-crossing times. More generally, crossing times and the corresponding interval sequences can be introduced for different kinds of stochastic oscillators that have been used to model variability of rhythmic activity in biological systems. In this paper we show that if we use the so-called mean-return-time (MRT) phase isochrons (introduced by Schwabedal and Pikovsky) to count the cycles of a stochastic oscillator with Markovian dynamics, the interphase interval sequence does not show any linear correlations, i.e., the corresponding sequence of passage times forms approximately a renewal point process. We first outline the general mathematical argument for this finding and illustrate it numerically for three models of increasing complexity: (i) the isotropic Guckenheimer-Schwabedal-Pikovsky oscillator that displays positive interspike interval (ISI) correlations if rotations are counted by passing the spoke of a wheel; (ii) the adaptive leaky integrate-and-fire model with white Gaussian noise that shows negative interspike interval correlations when spikes are counted in the usual way by the passage of a voltage threshold; (iii) a Hodgkin-Huxley model with channel noise (in the diffusion approximation represented by Gaussian noise) that exhibits weak but statistically significant interspike interval correlations, again for spikes counted when passing a voltage threshold. For all these models, linear correlations between intervals vanish when we count rotations by the passage of an MRT isochron. We finally discuss that the removal of interval correlations does not change the long-term variability and its effect on information transmission, especially in the neural context.

Analytical approach to the mean-return-time phase of isotropic stochastic oscillators

One notion of phase for stochastic oscillators is based on the mean return-time (MRT): a set of points represents a certain phase if the mean time to return from any point in this set to this set after one rotation is equal to the mean rotation period of the oscillator (irrespective of the starting point). For this so far only algorithmically defined phase, we derive here analytical expressions for the important class of isotropic stochastic oscillators. This allows us to evaluate cases from the literature explicitly and to study the behavior of the MRT phase in the limits of strong noise. We also use the same formalism to show that lines of constant return time variance (instead of constant mean return time) can be defined, and that they in general differ from the MRT isochrons.

Stochastic Spiking Behavior in Neuromorphic Networks Enables True Random Number Generation

There is currently a great deal of interest in the use of nanoscale devices to emulate the behaviors of neurons and synapses and to facilitate brain-inspired computation. Here, it is shown that percolating networks of nanoparticles exhibit stochastic spiking behavior that is strikingly similar to that observed in biological neurons. The spiking rate can be controlled by the input stimulus, similar to "rate coding" in biology, and the distributions of times between events are log-normal, providing insights into the atomic-scale spiking mechanism. The stochasticity of the spiking behavior is then used for true random number generation, and the high quality of the generated random bit-streams is demonstrated, opening up promising routes toward integration of neuromorphic computing with secure information processing.

Interspike interval correlations in neuron models with adaptation and correlated noise

The generation of neural action potentials (spikes) is random but nevertheless may result in a rich statistical structure of the spike sequence. In particular, contrary to the popular renewal assumption of theoreticians, the intervals between adjacent spikes are often correlated. Experimentally, different patterns of interspike-interval correlations have been observed and computational studies have identified spike-frequency adaptation and correlated noise as the two main mechanisms that can lead to such correlations. Analytical studies have focused on the single cases of either correlated (colored) noise or adaptation currents in combination with uncorrelated (white) noise. For low-pass filtered noise or adaptation, the serial correlation coefficient can be approximated as a single geometric sequence of the lag between the intervals, providing an explanation for some of the experimentally observed patterns. Here we address the problem of interval correlations for a widely used class of models, multidimensional integrate-and-fire neurons subject to a combination of colored and white noise sources and a spike-triggered adaptation current. Assuming weak noise, we derive a simple formula for the serial correlation coefficient, a sum of two geometric sequences, which accounts for a large class of correlation patterns. The theory is confirmed by means of numerical simulations in a number of special cases including the leaky, quadratic, and generalized integrate-and-fire models with colored noise and spike-frequency adaptation. Furthermore we study the case in which the adaptation current and the colored noise share the same time scale, corresponding to a slow stochastic population of adaptation channels; we demonstrate that our theory can account for a nonmonotonic dependence of the correlation coefficient on the channel’s time scale. Another application of the theory is a neuron driven by network-noise-like fluctuations (green noise). We also discuss the range of validity of our weak-noise theory and show that by changing the relative strength of white and colored noise sources, we can change the sign of the correlation coefficient. Finally, we apply our theory to a conductance-based model which demonstrates its broad applicability.

The elementary processing units in the central nervous system are neurons that transmit information by short electrical pulses, so called action potentials or spikes. The generation of the action potential is a random process that can be shaped by correlated fluctuations (colored noise) and by adaptation. A consequence of these two ubiquitous features is that the successive time intervals between spikes, the interspike intervals, are not independent but correlated. As these correlations can significantly improve information transmission and weak-signal detection, it is an important task to develop analytical approaches to these statistics for well-established computational models. Here we present a theory of interval correlations for a widely used class of integrate-and-fire models endowed with an adaptation mechanism and subject to correlated fluctuations. We demonstrate which patterns of interval correlations can be expected from the interplay of colored noise, adaptation and intrinsic nonlinear dynamics.

Statistical moments of the interspike intervals for a neuron model driven by trichotomous noise

The influence of a colored three-level input noise (trichotomous noise) on the spike generation of a perfect integrate-and-fire (PIF) model of neurons is studied. Using a first-passage-time formulation, exact expressions for the Laplace transform of the output interspike interval (ISI) density and for the statistical moments of the ISIs (such as the coefficient of variation, the skewness, the serial correlation coefficient, and the Fano factor) are derived. To model the anomalous subdiffusion that can arise from, e.g., the trapping properties of dendritic spines, the model is extended by including a random operational time in the form of an inverse strictly increasing Lévy-type subordinator, and exact formulas for ISI statistics are given for this case as well. Particularly, it is shown that at some parameter regimes, the ISI density exhibits a three-modal structure. The results for the extended model show that the ISI serial correlation coefficient and the Fano factor are nonmonotonic with respect to the input current, which indicates that at an intermediate value of the input current the variability of the output spike trains is minimal. Similarities and differences between the behavior of the presented models and the previously investigated PIF models driven by dichotomous noise are also discussed.

Fano Factor: A Potentially Useful Information

The Fano factor, defined as the variance-to-mean ratio of spike counts in a time window, is often used to measure the variability of neuronal spike trains. However, despite its transparent definition, careless use of the Fano factor can easily lead to distorted or even wrong results. One of the problems is the unclear dependence of the Fano factor on the spiking rate, which is often neglected or handled insufficiently. In this paper we aim to explore this problem in more detail and to study the possible solution, which is to evaluate the Fano factor in the operational time. We use equilibrium renewal and Markov renewal processes as spike train models to describe the method in detail, and we provide an illustration on experimental data.

Enhanced Signal Detection by Adaptive Decorrelation of Interspike Intervals

Spike trains with negative interspike interval (ISI) correlations, in which long/short ISIs are more likely followed by short/long ISIs, are common in many neurons. They can be described by stochastic models with a spike-triggered adaptation variable. We analyze a phenomenon in these models where such statistically dependent ISI sequences arise in tandem with quasi-statistically independent and identically distributed (quasi-IID) adaptation variable sequences. The sequences of adaptation states and resulting ISIs are linked by a nonlinear decorrelating transformation. We establish general conditions on a family of stochastic spiking models that guarantee this quasi-IID property and establish bounds on the resulting baseline ISI correlations. Inputs that elicit weak firing rate changes in samples with many spikes are known to be more detectible when negative ISI correlations are present because they reduce spike count variance; this defines a variance-reduced firing rate coding benchmark. We performed a Fisher information analysis on these adapting models exhibiting ISI correlations to show that a spike pattern code based on the quasi-IID property achieves the upper bound of detection performance, surpassing rate codes with the same mean rate-including the variance-reduced rate code benchmark-by 20% to 30%. The information loss in rate codes arises because the benefits of reduced spike count variance cannot compensate for the lower firing rate gain due to adaptation. Since adaptation states have similar dynamics to synaptic responses, the quasi-IID decorrelation transformation of the spike train is plausibly implemented by downstream neurons through matched postsynaptic kinetics. This provides an explanation for observed coding performance in sensory systems that cannot be accounted for by rate coding, for example, at the detection threshold where rate changes can be insignificant.

Information filtering by coincidence detection of synchronous population output: analytical approaches to the coherence function of a two-stage neural system

Information about time-dependent sensory stimuli is encoded in the activity of neural populations; distinct aspects of the stimulus are read out by different types of neurons: while overall information is perceived by integrator cells, so-called coincidence detector cells are driven mainly by the synchronous activity in the population that encodes predominantly high-frequency content of the input signal (high-pass information filtering). Previously, an analytically accessible statistic called the partial synchronous output was introduced as a proxy for the coincidence detector cell's output in order to approximate its information transmission. In the first part of the current paper, we compare the information filtering properties (specifically, the coherence function) of this proxy to those of a simple coincidence detector neuron. We show that the latter's coherence function can indeed be well-approximated by the partial synchronous output with a time scale and threshold criterion that are related approximately linearly to the membrane time constant and firing threshold of the coincidence detector cell. In the second part of the paper, we propose an alternative theory for the spectral measures (including the coherence) of the coincidence detector cell that combines linear-response theory for shot-noise driven integrate-and-fire neurons with a novel perturbation ansatz for the spectra of spike-trains driven by colored noise. We demonstrate how the variability of the synaptic weights for connections from the population to the coincidence detector can shape the information transmission of the entire two-stage system.

Synaptic Information Transmission in a Two-State Model of Short-Term Facilitation

Action potentials (spikes) can trigger the release of a neurotransmitter at chemical synapses between neurons. Such release is uncertain, as it occurs only with a certain probability. Moreover, synaptic release can occur independently of an action potential (asynchronous release) and depends on the history of synaptic activity. We focus here on short-term synaptic facilitation, in which a sequence of action potentials can temporarily increase the release probability of the synapse. In contrast to the phenomenon of short-term depression, quantifying the information transmission in facilitating synapses remains to be done. We find rigorous lower and upper bounds for the rate of information transmission in a model of synaptic facilitation. We treat the synapse as a two-state binary asymmetric channel, in which the arrival of an action potential shifts the synapse to a facilitated state, while in the absence of a spike, the synapse returns to its baseline state. The information bounds are functions of both the asynchronous and synchronous release parameters. If synchronous release facilitates more than asynchronous release, the mutual information rate increases. In contrast, short-term facilitation degrades information transmission when the synchronous release probability is intrinsically high. As synaptic release is energetically expensive, we exploit the information bounds to determine the energy-information trade-off in facilitating synapses. We show that unlike information rate, the energy-normalized information rate is robust with respect to variations in the strength of facilitation.

Surrogate Data Methods Based on a Shuffling of the Trials for Synchrony Detection: The Centering Issue

We investigate several distribution-free dependence detection procedures, all based on a shuffling of the trials, from a statistical point of view. The mathematical justification of such procedures lies in the bootstrap principle and its approximation properties. In particular, we show that such a shuffling has mainly to be done on centered quantities-that is, quantities with zero mean under independence-to construct correct p-values, meaning that the corresponding tests control their false positive (FP) rate. Thanks to this study, we introduce a method, named permutation UE, which consists of a multiple testing procedure based on permutation of experimental trials and delayed coincidence count. Each involved single test of this procedure achieves the prescribed level, so that the corresponding multiple testing procedure controls the false discovery rate (FDR), and this with as few assumptions as possible on the underneath distribution, except independence and identical distribution across trials. The mathematical meaning of this assumption is discussed, and it is in particular argued that it does not mean what is commonly referred in neuroscience to as cross-trials stationarity. Some simulations show, moreover, that permutation UE outperforms the trial-shuffling of Pipa and Grün ( 2003 ) and the MTGAUE method of Tuleau-Malot et al. ( 2014 ) in terms of single levels and FDR, for a comparable amount of false negatives. Application to real data is also provided.

Slot-like capacity and resource-like coding in a neural model of multiple-item working memory

For the past decade, research on the storage limitations of working memory has been dominated by two fundamentally different hypotheses. On the one hand, the contents of working memory may be stored in a limited number of "slots," each with a fixed resolution. On the other hand, any number of items may be stored but with decreasing resolution. These two hypotheses have been invaluable in characterizing the computational structure of working memory, but neither provides a complete account of the available experimental data or speaks to the neural basis of the limitations it characterizes. To address these shortcomings, we simulated a multiple-item working memory task with a cortical network model, the cellular resolution of which allowed us to quantify the coding fidelity of memoranda as a function of memory load, as measured by the discriminability, regularity, and reliability of simulated neural spiking. Our simulations account for a wealth of neural and behavioral data from human and nonhuman primate studies, and they demonstrate that feedback inhibition lowers both capacity and coding fidelity. Because the strength of inhibition scales with the number of items stored by the network, increasing this number progressively lowers fidelity until capacity is reached. Crucially, the model makes specific, testable predictions for neural activity on multiple-item working memory tasks. NEW & NOTEWORTHY Working memory is the ability to keep information in mind and is fundamental to cognition. It is actively debated whether the storage limitations of working memory reflect a small number of storage units (slots) or a decrease in coding resolution as a limited resource is allocated to more items. In a cortical model, we found that slot-like capacity and resource-like neural coding resulted from the same mechanism, offering an integrated explanation for storage limitations.

Sub-threshold signal encoding in coupled FitzHugh-Nagumo neurons

Despite intensive research, the mechanisms underlying the neural code remain poorly understood. Recent work has focused on the response of a single neuron to a weak, sub-threshold periodic signal. By simulating the stochastic FitzHugh-Nagumo (FHN) model and then using a symbolic method to analyze the firing activity, preferred and infrequent spike patterns (defined by the relative timing of the spikes) were detected, whose probabilities encode information about the signal. As not individual neurons but neuronal populations are responsible for sensory coding and information transfer, a relevant question is how a second neuron, which does not perceive the signal, affects the detection and the encoding of the signal, done by the first neuron. Through simulations of two stochastic FHN neurons we show that the encoding of a sub-threshold signal in symbolic spike patterns is a plausible mechanism. The neuron that perceives the signal fires a spike train that, despite having an almost random temporal structure, has preferred and infrequent patterns which carry information about the signal. Our findings could be relevant for sensory systems composed by two noisy neurons, when only one detects a weak external input.

Multimodal integration and stimulus categorization in putative mushroom body output neurons of the honeybee

Flowers attract pollinating insects like honeybees by sophisticated compositions of olfactory and visual cues. Using honeybees as a model to study olfactory-visual integration at the neuronal level, we focused on mushroom body (MB) output neurons (MBON). From a neuronal circuit perspective, MBONs represent a prominent level of sensory-modality convergence in the insect brain. We established an experimental design allowing electrophysiological characterization of olfactory, visual, as well as olfactory-visual induced activation of individual MBONs. Despite the obvious convergence of olfactory and visual pathways in the MB, we found numerous unimodal MBONs. However, a substantial proportion of MBONs (32%) responded to both modalities and thus integrated olfactory-visual information across MB input layers. In these neurons, representation of the olfactory-visual compound was significantly increased compared with that of single components, suggesting an additive, but nonlinear integration. Population analyses of olfactory-visual MBONs revealed three categories: (i) olfactory, (ii) visual and (iii) olfactory-visual compound stimuli. Interestingly, no significant differentiation was apparent regarding different stimulus qualities within these categories. We conclude that encoding of stimulus quality within a modality is largely completed at the level of MB input, and information at the MB output is integrated across modalities to efficiently categorize sensory information for downstream behavioural decision processing.

Neural correlates of side-specific odour memory in mushroom body output neurons

Humans and other mammals as well as honeybees learn a unilateral association between an olfactory stimulus presented to one side and a reward. In all of them, the learned association can be behaviourally retrieved via contralateral stimulation, suggesting inter-hemispheric communication. However, the underlying neuronal circuits are largely unknown and neural correlates of across-brain-side plasticity have yet not been demonstrated. We report neural plasticity that reflects lateral integration after side-specific odour reward conditioning. Mushroom body output neurons that did not respond initially to contralateral olfactory stimulation developed a unique and stable representation of the rewarded compound stimulus (side and odour) predicting its value during memory retention. The encoding of the reward-associated compound stimulus is delayed by about 40 ms compared with unrewarded neural activity, indicating an increased computation time for the read-out after lateral integration.

Evoking prescribed spike times in stochastic neurons

Single cell stimulation in vivo is a powerful tool to investigate the properties of single neurons and their functionality in neural networks. We present a method to determine a cell-specific stimulus that reliably evokes a prescribed spike train with high temporal precision of action potentials. We test the performance of this stimulus in simulations for two different stochastic neuron models. For a broad range of parameters and a neuron firing with intermediate firing rates (20-40 Hz) the reliability in evoking the prescribed spike train is close to its theoretical maximum that is mainly determined by the level of intrinsic noise.

Similarity in Neuronal Firing Regimes across Mammalian Species

The architectonic subdivisions of the brain are believed to be functional modules, each processing parts of global functions. Previously, we showed that neurons in different regions operate in different firing regimes in monkeys. It is possible that firing regimes reflect differences in underlying information processing, and consequently the firing regimes in homologous regions across animal species might be similar. We analyzed neuronal spike trains recorded from behaving mice, rats, cats, and monkeys. The firing regularity differed systematically, with differences across regions in one species being greater than the differences in similar areas across species. Neuronal firing was consistently most regular in motor areas, nearly random in visual and prefrontal/medial prefrontal cortical areas, and bursting in the hippocampus in all animals examined. This suggests that firing regularity (or irregularity) plays a key role in neural computation in each functional subdivision, depending on the types of information being carried.

SIGNIFICANCE STATEMENT

By analyzing neuronal spike trains recorded from mice, rats, cats, and monkeys, we found that different brain regions have intrinsically different firing regimes that are more similar in homologous areas across species than across areas in one species. Because different regions in the brain are specialized for different functions, the present finding suggests that the different activity regimes of neurons are important for supporting different functions, so that appropriate neuronal codes can be used for different modalities.

Serial Spike Time Correlations Affect Probability Distribution of Joint Spike Events

Detecting the existence of temporally coordinated spiking activity, and its role in information processing in the cortex, has remained a major challenge for neuroscience research. Different methods and approaches have been suggested to test whether the observed synchronized events are significantly different from those expected by chance. To analyze the simultaneous spike trains for precise spike correlation, these methods typically model the spike trains as a Poisson process implying that the generation of each spike is independent of all the other spikes. However, studies have shown that neural spike trains exhibit dependence among spike sequences, such as the absolute and relative refractory periods which govern the spike probability of the oncoming action potential based on the time of the last spike, or the bursting behavior, which is characterized by short epochs of rapid action potentials, followed by longer episodes of silence. Here we investigate non-renewal processes with the inter-spike interval distribution model that incorporates spike-history dependence of individual neurons. For that, we use the Monte Carlo method to estimate the full shape of the coincidence count distribution and to generate false positives for coincidence detection. The results show that compared to the distributions based on homogeneous Poisson processes, and also non-Poisson processes, the width of the distribution of joint spike events changes. Non-renewal processes can lead to both heavy tailed or narrow coincidence distribution. We conclude that small differences in the exact autostructure of the point process can cause large differences in the width of a coincidence distribution. Therefore, manipulations of the autostructure for the estimation of significance of joint spike events seem to be inadequate.

Multi-scale detection of rate changes in spike trains with weak dependencies

The statistical analysis of neuronal spike trains by models of point processes often relies on the assumption of constant process parameters. However, it is a well-known problem that the parameters of empirical spike trains can be highly variable, such as for example the firing rate. In order to test the null hypothesis of a constant rate and to estimate the change points, a Multiple Filter Test (MFT) and a corresponding algorithm (MFA) have been proposed that can be applied under the assumption of independent inter spike intervals (ISIs). As empirical spike trains often show weak dependencies in the correlation structure of ISIs, we extend the MFT here to point processes associated with short range dependencies. By specifically estimating serial dependencies in the test statistic, we show that the new MFT can be applied to a variety of empirical firing patterns, including positive and negative serial correlations as well as tonic and bursty firing. The new MFT is applied to a data set of empirical spike trains with serial correlations, and simulations show improved performance against methods that assume independence. In case of positive correlations, our new MFT is necessary to reduce the number of false positives, which can be highly enhanced when falsely assuming independence. For the frequent case of negative correlations, the new MFT shows an improved detection probability of change points and thus, also a higher potential of signal extraction from noisy spike trains.

Drifting States and Synchronization Induced Chaos in Autonomous Networks of Excitable Neurons

The study of balanced networks of excitatory and inhibitory neurons has led to several open questions. On the one hand it is yet unclear whether the asynchronous state observed in the brain is autonomously generated, or if it results from the interplay between external drivings and internal dynamics. It is also not known, which kind of network variabilities will lead to irregular spiking and which to synchronous firing states. Here we show how isolated networks of purely excitatory neurons generically show asynchronous firing whenever a minimal level of structural variability is present together with a refractory period. Our autonomous networks are composed of excitable units, in the form of leaky integrators spiking only in response to driving currents, remaining otherwise quiet. For a non-uniform network, composed exclusively of excitatory neurons, we find a rich repertoire of self-induced dynamical states. We show in particular that asynchronous drifting states may be stabilized in purely excitatory networks whenever a refractory period is present. Other states found are either fully synchronized or mixed, containing both drifting and synchronized components. The individual neurons considered are excitable and hence do not dispose of intrinsic natural firing frequencies. An effective network-wide distribution of natural frequencies is however generated autonomously through self-consistent feedback loops. The asynchronous drifting state is, additionally, amenable to an analytic solution. We find two types of asynchronous activity, with the individual neurons spiking regularly in the pure drifting state, albeit with a continuous distribution of firing frequencies. The activity of the drifting component, however, becomes irregular in the mixed state, due to the periodic driving of the synchronized component. We propose a new tool for the study of chaos in spiking neural networks, which consists of an analysis of the time series of pairs of consecutive interspike intervals. In this space, we show that a strange attractor with a fractal dimension of about 1.8 is formed in the mentioned mixed state.

Emergence of spike correlations in periodically forced excitable systems

In sensory neurons the presence of noise can facilitate the detection of weak information-carrying signals, which are encoded and transmitted via correlated sequences of spikes. Here we investigate the relative temporal order in spike sequences induced by a subthreshold periodic input in the presence of white Gaussian noise. To simulate the spikes, we use the FitzHugh-Nagumo model and to investigate the output sequence of interspike intervals (ISIs), we use the symbolic method of ordinal analysis. We find different types of relative temporal order in the form of preferred ordinal patterns that depend on both the strength of the noise and the period of the input signal. We also demonstrate a resonancelike behavior, as certain periods and noise levels enhance temporal ordering in the ISI sequence, maximizing the probability of the preferred patterns. Our findings could be relevant for understanding the mechanisms underlying temporal coding, by which single sensory neurons represent in spike sequences the information about weak periodic stimuli.

ASSET: Analysis of Sequences of Synchronous Events in Massively Parallel Spike Trains

With the ability to observe the activity from large numbers of neurons simultaneously using modern recording technologies, the chance to identify sub-networks involved in coordinated processing increases. Sequences of synchronous spike events (SSEs) constitute one type of such coordinated spiking that propagates activity in a temporally precise manner. The synfire chain was proposed as one potential model for such network processing. Previous work introduced a method for visualization of SSEs in massively parallel spike trains, based on an intersection matrix that contains in each entry the degree of overlap of active neurons in two corresponding time bins. Repeated SSEs are reflected in the matrix as diagonal structures of high overlap values. The method as such, however, leaves the task of identifying these diagonal structures to visual inspection rather than to a quantitative analysis. Here we present ASSET (Analysis of Sequences of Synchronous EvenTs), an improved, fully automated method which determines diagonal structures in the intersection matrix by a robust mathematical procedure. The method consists of a sequence of steps that i) assess which entries in the matrix potentially belong to a diagonal structure, ii) cluster these entries into individual diagonal structures and iii) determine the neurons composing the associated SSEs. We employ parallel point processes generated by stochastic simulations as test data to demonstrate the performance of the method under a wide range of realistic scenarios, including different types of non-stationarity of the spiking activity and different correlation structures. Finally, the ability of the method to discover SSEs is demonstrated on complex data from large network simulations with embedded synfire chains. Thus, ASSET represents an effective and efficient tool to analyze massively parallel spike data for temporal sequences of synchronous activity.

A Model for Single Neuron Activity With Refractory Effects and Spike Rate Estimation Techniques

The use of random point processes as models for neural spike trains allows the derivation of powerful statistical estimation techniques for time varying firing rates. Frequently, however, such estimators are based on the assumption that spike sequences follow a Poisson point process. Because of the bio-physical properties of neuronal action potentials, spike trains are affected by the refractory phenomenon that induces history dependency, and hence contradicts the Poisson assumption. In this work we present a neural spiking model, and a Maximum Likelihood (ML) estimation framework for time varying firing rates, that account for history dependencies in spike trains. Our framework is based on an exponential of polynomial model for the excitation function (stimulus), that generates a self exciting point process representing spike trains with absolute as well as relative refractory effects. Using this framework we employ techniques based on non-convex optimization and model order selection to derive ML estimators for neuronal firing rates. Results on simulated data with a refractory period show an improvement in accuracy when our estimation technique, that accounts for the complete refractory phenomenon, is used. Employing this estimation method for measured neuronal data shows an improvement in goodness of fit over estimators that do not account for the refractory effect, and also over other commonly used techniques.

The Gamma renewal process as an output of the diffusion leaky integrate-and-fire neuronal model

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 (OU) 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, Gamma renewal process, and its parameters only reflect the observed experimental data or assumed theoretical properties. Both of these commonly used models are related here. We show under which conditions the Gamma model is an output from the diffusion OU model. In some cases, we can see that the Gamma distribution is unrealistic to be achieved for the employed parameters of the OU process.

Impact of slow K(+) currents on spike generation can be described by an adaptive threshold model

A neuron that is stimulated by rectangular current injections initially responds with a high firing rate, followed by a decrease in the firing rate. This phenomenon is called spike-frequency adaptation and is usually mediated by slow K(+) currents, such as the M-type K(+) current (I M ) or the Ca(2+)-activated K(+) current (I AHP ). It is not clear how the detailed biophysical mechanisms regulate spike generation in a cortical neuron. In this study, we investigated the impact of slow K(+) currents on spike generation mechanism by reducing a detailed conductance-based neuron model. We showed that the detailed model can be reduced to a multi-timescale adaptive threshold model, and derived the formulae that describe the relationship between slow K(+) current parameters and reduced model parameters. Our analysis of the reduced model suggests that slow K(+) currents have a differential effect on the noise tolerance in neural coding.

A minimum-error, energy-constrained neural code is an instantaneous-rate code

Sensory neurons code information about stimuli in their sequence of action potentials (spikes). Intuitively, the spikes should represent stimuli with high fidelity. However, generating and propagating spikes is a metabolically expensive process. It is therefore likely that neural codes have been selected to balance energy expenditure against encoding error. Our recently proposed optimal, energy-constrained neural coder (Jones et al. Frontiers in Computational Neuroscience, 9, 61 2015) postulates that neurons time spikes to minimize the trade-off between stimulus reconstruction error and expended energy by adjusting the spike threshold using a simple dynamic threshold. Here, we show that this proposed coding scheme is related to existing coding schemes, such as rate and temporal codes. We derive an instantaneous rate coder and show that the spike-rate depends on the signal and its derivative. In the limit of high spike rates the spike train maximizes fidelity given an energy constraint (average spike-rate), and the predicted interspike intervals are identical to those generated by our existing optimal coding neuron. The instantaneous rate coder is shown to closely match the spike-rates recorded from P-type primary afferents in weakly electric fish. In particular, the coder is a predictor of the peristimulus time histogram (PSTH). When tested against in vitro cortical pyramidal neuron recordings, the instantaneous spike-rate approximates DC step inputs, matching both the average spike-rate and the time-to-first-spike (a simple temporal code). Overall, the instantaneous rate coder relates optimal, energy-constrained encoding to the concepts of rate-coding and temporal-coding, suggesting a possible unifying principle of neural encoding of sensory signals.

Shot-noise Fano factor

A variability measure of the times of uniform events based on a shot-noise process is proposed and studied. The measure is inspired by the Fano factor, which we generalize by considering the time-weighted influence of the events given by a shot-noise response function. The sequence of events is assumed to be an equilibrium renewal process, and based on this assumption we present formulas describing the behavior of the variability measure. The formulas are derived for a general response function, restricted only by some natural conditions, but the main focus is given to the shot noise with exponential decrease. The proposed measure is analyzed and compared with the Fano factor.

Cooperativity leads to temporally-correlated fluctuations in the bacteriophage lambda genetic switch

Cooperative interactions are widespread in biochemical networks, providing the nonlinear response that underlies behavior such as ultrasensitivity and robust switching. We introduce a temporal correlation function-the conditional activity-to study the behavior of these phenomena. Applying it to the bistable genetic switch in bacteriophage lambda, we find that cooperative binding between binding sites on the prophage DNA lead to non-Markovian behavior, as quantified by the conditional activity. Previously, the conditional activity has been used to predict allosteric pathways in proteins; here, we show that it identifies the rare unbinding events which underlie induction from lysogeny to lysis.

On Firing Rate Estimation for Dependent Interspike Intervals

If interspike intervals are dependent, the instantaneous firing rate does not catch important features of spike trains. In this case, the conditional instantaneous rate plays the role of the instantaneous firing rate for the case of samples of independent interspike intervals. If the conditional distribution of the interspikes intervals (ISIs) is unknown, it becomes difficult to evaluate the conditional firing rate. We propose a nonparametric estimator for the conditional instantaneous firing rate for Markov, stationary, and ergodic ISIs. An algorithm to check the reliability of the proposed estimator is introduced, and its consistency properties are proved. The method is applied to data obtained from a stochastic two-compartment model and to in vitro experimental data.

Estimating latency from inhibitory input

Stimulus response latency is the time period between the presentation of a stimulus and the occurrence of a change in the neural firing evoked by the stimulation. The response latency has been explored and estimation methods proposed mostly for excitatory stimuli, which means that the neuron reacts to the stimulus by an increase in the firing rate. We focus on the estimation of the response latency in the case of inhibitory stimuli. Models used in this paper represent two different descriptions of response latency. We consider either the latency to be constant across trials or to be a random variable. In the case of random latency, special attention is given to models with selective interaction. The aim is to propose methods for estimation of the latency or the parameters of its distribution. Parameters are estimated by four different methods: method of moments, maximum-likelihood method, a method comparing an empirical and a theoretical cumulative distribution function and a method based on the Laplace transform of a probability density function. All four methods are applied on simulated data and compared.

Event-triggered feedback in noise-driven phase oscillators

Using a stochastic nonlinear phase oscillator model, we study the effect of event-triggered feedback on the statistics of interevent intervals. Events are associated with the entering of a new cycle. The feedback is modeled by an instantaneous increase (positive feedback) or decrease (negative feedback) of the oscillator frequency whenever an event occurs followed by an exponential decay on a slow time scale. In addition to the known excitable and oscillatory regimes, which are separated by a saddle node on invariant circle bifurcation, positive feedback can lead to bistable dynamics and a change of the system's excitability. The feedback has also a strong effect on noise-induced phenomena like coherence resonance or anticoherence resonance. Both positive and negative feedback can lead to more regular output for particular noise strengths. Finally, we investigate serial correlations in the sequence of interevent intervals that occur due to the additional slow dynamics. We derive approximations for the serial correlation coefficient and show that positive feedback results in extended positive interval correlations, whereas negative feedback yields short-ranging negative correlations. Investigating the interplay of feedback and the nonlinear phase dynamics close to the bifurcation, we find that correlations are most pronounced for optimal feedback strengths.

Estimating nonstationary inputs from a single spike train based on a neuron model with adaptation

Because every spike of a neuron is determined by input signals, a train of spikes may contain information about the dynamics of unobserved neurons. A state-space method based on the leaky integrate-and-fire model, describing neuronal transformation from input signals to a spike train has been proposed for tracking input parameters represented by their mean and fluctuation [11]. In the present paper, we propose to make the estimation more realistic by adopting an LIF model augmented with an adaptive moving threshold. Moreover, because the direct state-space method is computationally infeasible for a data set comprising thousands of spikes, we further develop a practical method for transforming instantaneous firing characteristics back to input parameters. The instantaneous firing characteristics, represented by the firing rate and non-Poisson irregularity, can be estimated using a computationally feasible algorithm. We applied our proposed methods to synthetic data to clarify that they perform well.

Fano factor estimation

Fano factor is one of the most widely used measures of variability of spike trains. Its standard estimator is the ratio of sample variance to sample mean of spike counts observed in a time window and the quality of the estimator strongly depends on the length of the window. We investigate this dependence under the assumption that the spike train behaves as an equilibrium renewal process. It is shown what characteristics of the spike train have large effect on the estimator bias. Namely, the effect of refractory period is analytically evaluated. Next, we create an approximate asymptotic formula for the mean square error of the estimator, which can also be used to find minimum of the error in estimation from single spike trains. The accuracy of the Fano factor estimator is compared with the accuracy of the estimator based on the squared coefficient of variation. All the results are illustrated for spike trains with gamma and inverse Gaussian probability distributions of interspike intervals. Finally, we discuss possibilities of how to select a suitable observation window for the Fano factor estimation.

Patterns of interval correlations in neural oscillators with adaptation

Neural firing is often subject to negative feedback by adaptation currents. These currents can induce strong correlations among the time intervals between spikes. Here we study analytically the interval correlations of a broad class of noisy neural oscillators with spike-triggered adaptation of arbitrary strength and time scale. Our weak-noise theory provides a general relation between the correlations and the phase-response curve (PRC) of the oscillator, proves anti-correlations between neighboring intervals for adapting neurons with type I PRC and identifies a single order parameter that determines the qualitative pattern of correlations. Monotonically decaying or oscillating correlation structures can be related to qualitatively different voltage traces after spiking, which can be explained by the phase plane geometry. At high firing rates, the long-term variability of the spike train associated with the cumulative interval correlations becomes small, independent of model details. Our results are verified by comparison with stochastic simulations of the exponential, leaky, and generalized integrate-and-fire models with adaptation.

Natural image sequences constrain dynamic receptive fields and imply a sparse code

In their natural environment, animals experience a complex and dynamic visual scenery. Under such natural stimulus conditions, neurons in the visual cortex employ a spatially and temporally sparse code. For the input scenario of natural still images, previous work demonstrated that unsupervised feature learning combined with the constraint of sparse coding can predict physiologically measured receptive fields of simple cells in the primary visual cortex. This convincingly indicated that the mammalian visual system is adapted to the natural spatial input statistics. Here, we extend this approach to the time domain in order to predict dynamic receptive fields that can account for both spatial and temporal sparse activation in biological neurons. We rely on temporal restricted Boltzmann machines and suggest a novel temporal autoencoding training procedure. When tested on a dynamic multi-variate benchmark dataset this method outperformed existing models of this class. Learning features on a large dataset of natural movies allowed us to model spatio-temporal receptive fields for single neurons. They resemble temporally smooth transformations of previously obtained static receptive fields and are thus consistent with existing theories. A neuronal spike response model demonstrates how the dynamic receptive field facilitates temporal and population sparseness. We discuss the potential mechanisms and benefits of a spatially and temporally sparse representation of natural visual input.

Impact of spike train autostructure on probability distribution of joint spike events

The discussion whether temporally coordinated spiking activity really exists and whether it is relevant has been heated over the past few years. To investigate this issue, several approaches have been taken to determine whether synchronized events occur significantly above chance, that is, whether they occur more often than expected if the neurons fire independently. Most investigations ignore or destroy the autostructure of the spiking activity of individual cells or assume Poissonian spiking as a model. Such methods that ignore the autostructure can significantly bias the coincidence statistics. Here, we study the influence of the autostructure on the probability distribution of coincident spiking events between tuples of mutually independent non-Poisson renewal processes. In particular, we consider two types of renewal processes that were suggested as appropriate models of experimental spike trains: a gamma and a log-normal process. For a gamma process, we characterize the shape of the distribution analytically with the Fano factor (FFc). In addition, we perform Monte Carlo estimations to derive the full shape of the distribution and the probability for false positives if a different process type is assumed as was actually present. We also determine how manipulations of such spike trains, here dithering, used for the generation of surrogate data change the distribution of coincident events and influence the significance estimation. We find, first, that the width of the coincidence count distribution and its FFc depend critically and in a nontrivial way on the detailed properties of the structure of the spike trains as characterized by the coefficient of variation CV. Second, the dependence of the FFc on the CV is complex and mostly nonmonotonic. Third, spike dithering, even if as small as a fraction of the interspike interval, can falsify the inference on coordinated firing.

Firing statistics of inhibitory neuron with delayed feedback. II: Non-Markovian behavior

The instantaneous state of a neural network consists of both the degree of excitation of each neuron the network is composed of and positions of impulses in communication lines between the neurons. In neurophysiological experiments, the neuronal firing moments are registered, but not the state of communication lines. But future spiking moments depend essentially on the past positions of impulses in the lines. This suggests, that the sequence of intervals between firing moments (inter-spike intervals, ISIs) in the network could be non-Markovian. In this paper, we address this question for a simplest possible neural "net", namely, a single inhibitory neuron with delayed feedback. The neuron receives excitatory input from the driving Poisson stream and inhibitory impulses from its own output through the feedback line. We obtain analytic expressions for conditional probability density P(tn+1|tn, …, t1, t0), which gives the probability to get an output ISI of duration tn+1 provided the previous (n+1) output ISIs had durations tn, …, t1, t0. It is proven exactly, that P(tn+1|tn, …, t1, t0) does not reduce to P(tn+1|tn, …, t1) for any n≥0. This means that the output ISIs stream cannot be represented as a Markov chain of any finite order.

Channel noise from both slow adaptation currents and fast currents is required to explain spike-response variability in a sensory neuron

Spike-timing variability has a large effect on neural information processing. However, for many systems little is known about the noise sources causing the spike-response variability. Here we investigate potential sources of spike-response variability in auditory receptor neurons of locusts, a classic insect model system. At low-spike frequencies, our data show negative interspike-interval (ISI) correlations and ISI distributions that match the inverse Gaussian distribution. These findings can be explained by a white-noise source that interacts with an adaptation current. At higher spike frequencies, more strongly peaked distributions and positive ISI correlations appear, as expected from a canonical model of suprathreshold firing driven by temporally correlated (i.e., colored) noise. Simulations of a minimal conductance-based model of the auditory receptor neuron with stochastic ion channels exclude the delayed rectifier as a possible noise source. Our analysis suggests channel noise from an adaptation current and the receptor or sodium current as main sources for the colored and white noise, respectively. By comparing the ISI statistics with generic models, we find strong evidence for two distinct noise sources. Our approach does not involve any dendritic or somatic recordings that may harm the delicate workings of many sensory systems. It could be applied to various other types of neurons, in which channel noise dominates the fluctuations that shape the neuron's spike statistics.

Enhancing the resolution of a sensor via negative correlation: a biologically inspired approach

We demonstrate that a neuronal system, underpinned by "fire-then-reset" dynamics, can display an enhanced resolution R~T(ob)(-1) where T(ob) is the observation time of the measurement; this occurs when the interspike intervals are negatively correlated and T(ob)<Δ/ε, where ε is a parameter characterizing the level of correlation between interspike intervals and Δ is the average interspike interval. We also show that by introducing negative correlations into the time domain response of a nonlinear dynamical sensor it is possible to replicate this enhanced scaling of the resolution. Thus, we demonstrate the potential for designing a novel class of biomimetic sensors that afford improved signal resolution by functionally utilizing negative correlations.

Estimating individual firing frequencies in a multiple spike train record

Neuronal activity of several neurons is commonly recorded by a single electrode and then the individual spike trains are separated. If the separation is difficult or fails, then as a minimal result of the experiment, the individual firing rates are of interest. The proposed method solves the problem of their identification. This is possible under the condition that the recorded neurons are independent in their activities. The number of the neurons in the multi-unit record needs to be given (known or assumed) prior the calculation. The proposed method is based on the presence of the refractory period in neuronal firing, however, its precise value is not required. In addition to the determination of the individual firing rates the method can be used for an inference about the refractory period itself.

Population-wide distributions of neural activity during perceptual decision-making

Cortical activity involves large populations of neurons, even when it is limited to functionally coherent areas. Electrophysiological recordings, on the other hand, involve comparatively small neural ensembles, even when modern-day techniques are used. Here we review results which have started to fill the gap between these two scales of inquiry, by shedding light on the statistical distributions of activity in large populations of cells. We put our main focus on data recorded in awake animals that perform simple decision-making tasks and consider statistical distributions of activity throughout cortex, across sensory, associative, and motor areas. We transversally review the complexity of these distributions, from distributions of firing rates and metrics of spike-train structure, through distributions of tuning to stimuli or actions and of choice signals, and finally the dynamical evolution of neural population activity and the distributions of (pairwise) neural interactions. This approach reveals shared patterns of statistical organization across cortex, including: (i) long-tailed distributions of activity, where quasi-silence seems to be the rule for a majority of neurons; that are barely distinguishable between spontaneous and active states; (ii) distributions of tuning parameters for sensory (and motor) variables, which show an extensive extrapolation and fragmentation of their representations in the periphery; and (iii) population-wide dynamics that reveal rotations of internal representations over time, whose traces can be found both in stimulus-driven and internally generated activity. We discuss how these insights are leading us away from the notion of discrete classes of cells, and are acting as powerful constraints on theories and models of cortical organization and population coding.

Analysis and modelling of variability and covariability of population spike trains across multiple time scales

As multi-electrode and imaging technology begin to provide us with simultaneous recordings of large neuronal populations, new methods for modelling such data must also be developed. We present a model of responses to repeated trials of a sensory stimulus based on thresholded Gaussian processes that allows for analysis and modelling of variability and covariability of population spike trains across multiple time scales. The model framework can be used to specify the values of many different variability measures including spike timing precision across trials, coefficient of variation of the interspike interval distribution, and Fano factor of spike counts for individual neurons, as well as signal and noise correlations and correlations of spike counts across multiple neurons. Using both simulated data and data from different stages of the mammalian auditory pathway, we demonstrate the range of possible independent manipulations of different variability measures, and explore how this range depends on the sensory stimulus. The model provides a powerful framework for the study of experimental and surrogate data and for analyzing dependencies between different statistical properties of neuronal populations.

Beyond statistical significance: implications of network structure on neuronal activity

It is a common and good practice in experimental sciences to assess the statistical significance of measured outcomes. For this, the probability of obtaining the actual results is estimated under the assumption of an appropriately chosen null-hypothesis. If this probability is smaller than some threshold, the results are deemed statistically significant and the researchers are content in having revealed, within their own experimental domain, a “surprising” anomaly, possibly indicative of a hitherto hidden fragment of the underlying “ground-truth”. What is often neglected, though, is the actual importance of these experimental outcomes for understanding the system under investigation. We illustrate this point by giving practical and intuitive examples from the field of systems neuroscience. Specifically, we use the notion of embeddedness to quantify the impact of a neuron's activity on its downstream neurons in the network. We show that the network response strongly depends on the embeddedness of stimulated neurons and that embeddedness is a key determinant of the importance of neuronal activity on local and downstream processing. We extrapolate these results to other fields in which networks are used as a theoretical framework.

Spike train auto-structure impacts post-synaptic firing and timing-based plasticity

Cortical neurons are typically driven by several thousand synapses. The precise spatiotemporal pattern formed by these inputs can modulate the response of a post-synaptic cell. In this work, we explore how the temporal structure of pre-synaptic inhibitory and excitatory inputs impact the post-synaptic firing of a conductance-based integrate and fire neuron. Both the excitatory and inhibitory input was modeled by renewal gamma processes with varying shape factors for modeling regular and temporally random Poisson activity. We demonstrate that the temporal structure of mutually independent inputs affects the post-synaptic firing, while the strength of the effect depends on the firing rates of both the excitatory and inhibitory inputs. In a second step, we explore the effect of temporal structure of mutually independent inputs on a simple version of Hebbian learning, i.e., hard bound spike-timing-dependent plasticity. We explore both the equilibrium weight distribution and the speed of the transient weight dynamics for different mutually independent gamma processes. We find that both the equilibrium distribution of the synaptic weights and the speed of synaptic changes are modulated by the temporal structure of the input. Finally, we highlight that the sensitivity of both the post-synaptic firing as well as the spike-timing-dependent plasticity on the auto-structure of the input of a neuron could be used to modulate the learning rate of synaptic modification.

Epileptic seizures from abnormal networks: why some seizures defy predictability

Seizure prediction has proven to be difficult in clinically realistic environments. Is it possible that fluctuations in cortical firing could influence the onset of seizures in an ictal zone? To test this, we have now used neural network simulations in a computational model of cortex having a total of 65,536 neurons with intercellular wiring patterned after histological data. A spatially distributed Poisson driven background input representing the activity of neighboring cortex affected 1% of the neurons. Gamma distributions were fit to the interbursting phase intervals, a non-parametric test for randomness was applied, and a dynamical systems analysis was performed to search for period-1 orbits in the intervals. The non-parametric analysis suggests that intervals are being drawn at random from their underlying joint distribution and the dynamical systems analysis is consistent with a nondeterministic dynamical interpretation of the generation of bursting phases. These results imply that in a region of cortex with abnormal connectivity analogous to a seizure focus, it is possible to initiate seizure activity with fluctuations of input from the surrounding cortical regions. These findings suggest one possibility for ictal generation from abnormal focal epileptic networks. This mechanism additionally could help explain the difficulty in predicting partial seizures in some patients.

Identifying temporal codes in spontaneously active sensory neurons

The manner in which information is encoded in neural signals is a major issue in Neuroscience. A common distinction is between rate codes, where information in neural responses is encoded as the number of spikes within a specified time frame (encoding window), and temporal codes, where the position of spikes within the encoding window carries some or all of the information about the stimulus. One test for the existence of a temporal code in neural responses is to add artificial time jitter to each spike in the response, and then assess whether or not information in the response has been degraded. If so, temporal encoding might be inferred, on the assumption that the jitter is small enough to alter the position, but not the number, of spikes within the encoding window. Here, the effects of artificial jitter on various spike train and information metrics were derived analytically, and this theory was validated using data from afferent neurons of the turtle vestibular and paddlefish electrosensory systems, and from model neurons. We demonstrate that the jitter procedure will degrade information content even when coding is known to be entirely by rate. For this and additional reasons, we conclude that the jitter procedure by itself is not sufficient to establish the presence of a temporal code.