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1.  Coding and Decoding with Adapting Neurons: A Population Approach to the Peri-Stimulus Time Histogram 
PLoS Computational Biology  2012;8(10):e1002711.
The response of a neuron to a time-dependent stimulus, as measured in a Peri-Stimulus-Time-Histogram (PSTH), exhibits an intricate temporal structure that reflects potential temporal coding principles. Here we analyze the encoding and decoding of PSTHs for spiking neurons with arbitrary refractoriness and adaptation. As a modeling framework, we use the spike response model, also known as the generalized linear neuron model. Because of refractoriness, the effect of the most recent spike on the spiking probability a few milliseconds later is very strong. The influence of the last spike needs therefore to be described with high precision, while the rest of the neuronal spiking history merely introduces an average self-inhibition or adaptation that depends on the expected number of past spikes but not on the exact spike timings. Based on these insights, we derive a ‘quasi-renewal equation’ which is shown to yield an excellent description of the firing rate of adapting neurons. We explore the domain of validity of the quasi-renewal equation and compare it with other rate equations for populations of spiking neurons. The problem of decoding the stimulus from the population response (or PSTH) is addressed analogously. We find that for small levels of activity and weak adaptation, a simple accumulator of the past activity is sufficient to decode the original input, but when refractory effects become large decoding becomes a non-linear function of the past activity. The results presented here can be applied to the mean-field analysis of coupled neuron networks, but also to arbitrary point processes with negative self-interaction.
Author Summary
How can information be encoded and decoded in populations of adapting neurons? A quantitative answer to this question requires a mathematical expression relating neuronal activity to the external stimulus, and, conversely, stimulus to neuronal activity. Although widely used equations and models exist for the special problem of relating external stimulus to the action potentials of a single neuron, the analogous problem of relating the external stimulus to the activity of a population has proven more difficult. There is a bothersome gap between the dynamics of single adapting neurons and the dynamics of populations. Moreover, if we ignore the single neurons and describe directly the population dynamics, we are faced with the ambiguity of the adapting neural code. The neural code of adapting populations is ambiguous because it is possible to observe a range of population activities in response to a given instantaneous input. Somehow the ambiguity is resolved by the knowledge of the population history, but how precisely? In this article we use approximation methods to provide mathematical expressions that describe the encoding and decoding of external stimuli in adapting populations. The theory presented here helps to bridge the gap between the dynamics of single neurons and that of populations.
doi:10.1371/journal.pcbi.1002711
PMCID: PMC3464223  PMID: 23055914
2.  Reconstructing Stimuli from the Spike Times of Leaky Integrate and Fire Neurons 
Reconstructing stimuli from the spike trains of neurons is an important approach for understanding the neural code. One of the difficulties associated with this task is that signals which are varying continuously in time are encoded into sequences of discrete events or spikes. An important problem is to determine how much information about the continuously varying stimulus can be extracted from the time-points at which spikes were observed, especially if these time-points are subject to some sort of randomness. For the special case of spike trains generated by leaky integrate and fire neurons, noise can be introduced by allowing variations in the threshold every time a spike is released. A simple decoding algorithm previously derived for the noiseless case can be extended to the stochastic case, but turns out to be biased. Here, we review a solution to this problem, by presenting a simple yet efficient algorithm which greatly reduces the bias, and therefore leads to better decoding performance in the stochastic case.
doi:10.3389/fnins.2011.00001
PMCID: PMC3046364  PMID: 21390287
decoding; spiking neurons; Bayesian inference; population coding; leaky integrate and fire
3.  Neuronal Spike Timing Adaptation Described with a Fractional Leaky Integrate-and-Fire Model 
PLoS Computational Biology  2014;10(3):e1003526.
The voltage trace of neuronal activities can follow multiple timescale dynamics that arise from correlated membrane conductances. Such processes can result in power-law behavior in which the membrane voltage cannot be characterized with a single time constant. The emergent effect of these membrane correlations is a non-Markovian process that can be modeled with a fractional derivative. A fractional derivative is a non-local process in which the value of the variable is determined by integrating a temporal weighted voltage trace, also called the memory trace. Here we developed and analyzed a fractional leaky integrate-and-fire model in which the exponent of the fractional derivative can vary from 0 to 1, with 1 representing the normal derivative. As the exponent of the fractional derivative decreases, the weights of the voltage trace increase. Thus, the value of the voltage is increasingly correlated with the trajectory of the voltage in the past. By varying only the fractional exponent, our model can reproduce upward and downward spike adaptations found experimentally in neocortical pyramidal cells and tectal neurons in vitro. The model also produces spikes with longer first-spike latency and high inter-spike variability with power-law distribution. We further analyze spike adaptation and the responses to noisy and oscillatory input. The fractional model generates reliable spike patterns in response to noisy input. Overall, the spiking activity of the fractional leaky integrate-and-fire model deviates from the spiking activity of the Markovian model and reflects the temporal accumulated intrinsic membrane dynamics that affect the response of the neuron to external stimulation.
Author Summary
Spike adaptation is a property of most neurons. When spike time adaptation occurs over multiple time scales, the dynamics can be described by a power-law. We study the computational properties of a leaky integrate-and-fire model with power-law adaptation. Instead of explicitly modeling the adaptation process by the contribution of slowly changing conductances, we use a fractional temporal derivative framework. The exponent of the fractional derivative represents the degree of adaptation of the membrane voltage, where 1 is the normal leaky integrator while values less than 1 produce increasing correlations in the voltage trace. The temporal correlation is interpreted as a memory trace that depends on the value of the fractional derivative. We identify the memory trace in the fractional model as the sum of the instantaneous differentiation weighted by a function that depends on the fractional exponent, and it provides non-local information to the incoming stimulus. The spiking dynamics of the fractional leaky integrate-and-fire model show memory dependence that can result in downward or upward spike adaptation. Our model provides a framework for understanding how long-range membrane voltage correlations affect spiking dynamics and information integration in neurons.
doi:10.1371/journal.pcbi.1003526
PMCID: PMC3967934  PMID: 24675903
4.  STDP Allows Fast Rate-Modulated Coding with Poisson-Like Spike Trains 
PLoS Computational Biology  2011;7(10):e1002231.
Spike timing-dependent plasticity (STDP) has been shown to enable single neurons to detect repeatedly presented spatiotemporal spike patterns. This holds even when such patterns are embedded in equally dense random spiking activity, that is, in the absence of external reference times such as a stimulus onset. Here we demonstrate, both analytically and numerically, that STDP can also learn repeating rate-modulated patterns, which have received more experimental evidence, for example, through post-stimulus time histograms (PSTHs). Each input spike train is generated from a rate function using a stochastic sampling mechanism, chosen to be an inhomogeneous Poisson process here. Learning is feasible provided significant covarying rate modulations occur within the typical timescale of STDP (∼10–20 ms) for sufficiently many inputs (∼100 among 1000 in our simulations), a condition that is met by many experimental PSTHs. Repeated pattern presentations induce spike-time correlations that are captured by STDP. Despite imprecise input spike times and even variable spike counts, a single trained neuron robustly detects the pattern just a few milliseconds after its presentation. Therefore, temporal imprecision and Poisson-like firing variability are not an obstacle to fast temporal coding. STDP provides an appealing mechanism to learn such rate patterns, which, beyond sensory processing, may also be involved in many cognitive tasks.
Author Summary
In vivo neural responses to stimuli are known to have a lot of variability across trials. If the same number of spikes is emitted from trial to trial, the neuron is said to be reliable. If the timing of such spikes is roughly preserved across trials, the neuron is said to be precise. Here we demonstrate both analytically and numerically that the well-established Hebbian learning rule of spike-timing-dependent plasticity (STDP) can learn response patterns despite relatively low reliability (Poisson-like variability) and low temporal precision (10–20 ms). These features are in line with many experimental observations, in which a poststimulus time histogram (PSTH) is evaluated over multiple trials. In our model, however, information is extracted from the relative spike times between afferents without the need of an absolute reference time, such as a stimulus onset. Relevantly, recent experiments show that relative timing is often more informative than the absolute timing. Furthermore, the scope of application for our study is not restricted to sensory systems. Taken together, our results suggest a fine temporal resolution for the neural code, and that STDP is an appropriate candidate for encoding and decoding such activity.
doi:10.1371/journal.pcbi.1002231
PMCID: PMC3203056  PMID: 22046113
5.  Analysis of Slow (Theta) Oscillations as a Potential Temporal Reference Frame for Information Coding in Sensory Cortices 
PLoS Computational Biology  2012;8(10):e1002717.
While sensory neurons carry behaviorally relevant information in responses that often extend over hundreds of milliseconds, the key units of neural information likely consist of much shorter and temporally precise spike patterns. The mechanisms and temporal reference frames by which sensory networks partition responses into these shorter units of information remain unknown. One hypothesis holds that slow oscillations provide a network-intrinsic reference to temporally partitioned spike trains without exploiting the millisecond-precise alignment of spikes to sensory stimuli. We tested this hypothesis on neural responses recorded in visual and auditory cortices of macaque monkeys in response to natural stimuli. Comparing different schemes for response partitioning revealed that theta band oscillations provide a temporal reference that permits extracting significantly more information than can be obtained from spike counts, and sometimes almost as much information as obtained by partitioning spike trains using precisely stimulus-locked time bins. We further tested the robustness of these partitioning schemes to temporal uncertainty in the decoding process and to noise in the sensory input. This revealed that partitioning using an oscillatory reference provides greater robustness than partitioning using precisely stimulus-locked time bins. Overall, these results provide a computational proof of concept for the hypothesis that slow rhythmic network activity may serve as internal reference frame for information coding in sensory cortices and they foster the notion that slow oscillations serve as key elements for the computations underlying perception.
Author Summary
Neurons in sensory cortices encode objects in our sensory environment by varying the timing and number of action potentials that they emit. Brain networks that ‘decode’ this information need to partition those spike trains into their individual informative units. Experimenters achieve such partitioning by exploiting their knowledge about the millisecond precise timing of individual spikes relative to externally presented sensory stimuli. The brain, however, does not have access to this information and has to partition and decode spike trains using intrinsically available temporal reference frames. We show that slow (4–8 Hz) oscillatory network activity can provide such an intrinsic temporal reference. Specifically, we analyzed neural responses recorded in primary auditory and visual cortices. This revealed that the oscillatory reference frame performs nearly as well as the precise stimulus-locked reference frame and renders neural encoding robust to sensory noise and temporal uncertainty that naturally occurs during decoding. These findings provide a computational proof-of-concept that slow oscillatory network activity may serve the crucial function as temporal reference frame for sensory coding.
doi:10.1371/journal.pcbi.1002717
PMCID: PMC3469413  PMID: 23071429
6.  Spike-Based Population Coding and Working Memory 
PLoS Computational Biology  2011;7(2):e1001080.
Compelling behavioral evidence suggests that humans can make optimal decisions despite the uncertainty inherent in perceptual or motor tasks. A key question in neuroscience is how populations of spiking neurons can implement such probabilistic computations. In this article, we develop a comprehensive framework for optimal, spike-based sensory integration and working memory in a dynamic environment. We propose that probability distributions are inferred spike-per-spike in recurrently connected networks of integrate-and-fire neurons. As a result, these networks can combine sensory cues optimally, track the state of a time-varying stimulus and memorize accumulated evidence over periods much longer than the time constant of single neurons. Importantly, we propose that population responses and persistent working memory states represent entire probability distributions and not only single stimulus values. These memories are reflected by sustained, asynchronous patterns of activity which make relevant information available to downstream neurons within their short time window of integration. Model neurons act as predictive encoders, only firing spikes which account for new information that has not yet been signaled. Thus, spike times signal deterministically a prediction error, contrary to rate codes in which spike times are considered to be random samples of an underlying firing rate. As a consequence of this coding scheme, a multitude of spike patterns can reliably encode the same information. This results in weakly correlated, Poisson-like spike trains that are sensitive to initial conditions but robust to even high levels of external neural noise. This spike train variability reproduces the one observed in cortical sensory spike trains, but cannot be equated to noise. On the contrary, it is a consequence of optimal spike-based inference. In contrast, we show that rate-based models perform poorly when implemented with stochastically spiking neurons.
Author Summary
Most of our daily actions are subject to uncertainty. Behavioral studies have confirmed that humans handle this uncertainty in a statistically optimal manner. A key question then is what neural mechanisms underlie this optimality, i.e. how can neurons represent and compute with probability distributions. Previous approaches have proposed that probabilities are encoded in the firing rates of neural populations. However, such rate codes appear poorly suited to understand perception in a constantly changing environment. In particular, it is unclear how probabilistic computations could be implemented by biologically plausible spiking neurons. Here, we propose a network of spiking neurons that can optimally combine uncertain information from different sensory modalities and keep this information available for a long time. This implies that neural memories not only represent the most likely value of a stimulus but rather a whole probability distribution over it. Furthermore, our model suggests that each spike conveys new, essential information. Consequently, the observed variability of neural responses cannot simply be understood as noise but rather as a necessary consequence of optimal sensory integration. Our results therefore question strongly held beliefs about the nature of neural “signal” and “noise”.
doi:10.1371/journal.pcbi.1001080
PMCID: PMC3040643  PMID: 21379319
7.  Model-Based Decoding, Information Estimation, and Change-Point Detection Techniques for Multineuron Spike Trains 
Neural computation  2010;23(1):1-45.
One of the central problems in systems neuroscience is to understand how neural spike trains convey sensory information. Decoding methods, which provide an explicit means for reading out the information contained in neural spike responses, offer a powerful set of tools for studying the neural coding problem. Here we develop several decoding methods based on point-process neural encoding models, or forward models that predict spike responses to stimuli. These models have concave log-likelihood functions, which allow efficient maximum-likelihood model fitting and stimulus decoding. We present several applications of the encoding model framework to the problem of decoding stimulus information from population spike responses: (1) a tractable algorithm for computing the maximum a posteriori (MAP) estimate of the stimulus, the most probable stimulus to have generated an observed single- or multiple-neuron spike train response, given some prior distribution over the stimulus; (2) a gaussian approximation to the posterior stimulus distribution that can be used to quantify the fidelity with which various stimulus features are encoded; (3) an efficient method for estimating the mutual information between the stimulus and the spike trains emitted by a neural population; and (4) a framework for the detection of change-point times (the time at which the stimulus undergoes a change in mean or variance) by marginalizing over the posterior stimulus distribution. We provide several examples illustrating the performance of these estimators with simulated and real neural data.
doi:10.1162/NECO_a_00058
PMCID: PMC3625985  PMID: 20964538
8.  Information Carried by Population Spike Times in the Whisker Sensory Cortex can be Decoded Without Knowledge of Stimulus Time 
Computational analyses have revealed that precisely timed spikes emitted by somatosensory cortical neuronal populations encode basic stimulus features in the rat's whisker sensory system. Efficient spike time based decoding schemes both for the spatial location of a stimulus and for the kinetic features of complex whisker movements have been defined. To date, these decoding schemes have been based upon spike times referenced to an external temporal frame – the time of the stimulus itself. Such schemes are limited by the requirement of precise knowledge of the stimulus time signal, and it is not clear whether stimulus times are known to rats making sensory judgments. Here, we first review studies of the information obtained from spike timing referenced to the stimulus time. Then we explore new methods for extracting spike train information independently of any external temporal reference frame. These proposed methods are based on the detection of stimulus-dependent differences in the firing time within a neuronal population. We apply them to a data set using single-whisker stimulation in anesthetized rats and find that stimulus site can be decoded based on the millisecond-range relative differences in spike times even without knowledge of stimulus time. If spike counts alone are measured over tens or hundreds of milliseconds rather than milliseconds, such decoders are much less effective. These results suggest that decoding schemes based on millisecond-precise spike times are likely to subserve robust and information-rich transmission of information in the somatosensory system.
doi:10.3389/fnsyn.2010.00017
PMCID: PMC3059688  PMID: 21423503
information theory; somatosensation; neural coding; decoding; spike patterns; population coding
9.  Modeling the impact of common noise inputs on the network activity of retinal ganglion cells 
Synchronized spontaneous firing among retinal ganglion cells (RGCs), on timescales faster than visual responses, has been reported in many studies. Two candidate mechanisms of synchronized firing include direct coupling and shared noisy inputs. In neighboring parasol cells of primate retina, which exhibit rapid synchronized firing that has been studied extensively, recent experimental work indicates that direct electrical or synaptic coupling is weak, but shared synaptic input in the absence of modulated stimuli is strong. However, previous modeling efforts have not accounted for this aspect of firing in the parasol cell population. Here we develop a new model that incorporates the effects of common noise, and apply it to analyze the light responses and synchronized firing of a large, densely-sampled network of over 250 simultaneously recorded parasol cells. We use a generalized linear model in which the spike rate in each cell is determined by the linear combination of the spatio-temporally filtered visual input, the temporally filtered prior spikes of that cell, and unobserved sources representing common noise. The model accurately captures the statistical structure of the spike trains and the encoding of the visual stimulus, without the direct coupling assumption present in previous modeling work. Finally, we examined the problem of decoding the visual stimulus from the spike train given the estimated parameters. The common-noise model produces Bayesian decoding performance as accurate as that of a model with direct coupling, but with significantly more robustness to spike timing perturbations.
doi:10.1007/s10827-011-0376-2
PMCID: PMC3560841  PMID: 22203465
Retina; Generalized linear model; State-space model; Multielectrode; Recording; Random-effects model
10.  Decorrelation of Neural-Network Activity by Inhibitory Feedback 
PLoS Computational Biology  2012;8(8):e1002596.
Correlations in spike-train ensembles can seriously impair the encoding of information by their spatio-temporal structure. An inevitable source of correlation in finite neural networks is common presynaptic input to pairs of neurons. Recent studies demonstrate that spike correlations in recurrent neural networks are considerably smaller than expected based on the amount of shared presynaptic input. Here, we explain this observation by means of a linear network model and simulations of networks of leaky integrate-and-fire neurons. We show that inhibitory feedback efficiently suppresses pairwise correlations and, hence, population-rate fluctuations, thereby assigning inhibitory neurons the new role of active decorrelation. We quantify this decorrelation by comparing the responses of the intact recurrent network (feedback system) and systems where the statistics of the feedback channel is perturbed (feedforward system). Manipulations of the feedback statistics can lead to a significant increase in the power and coherence of the population response. In particular, neglecting correlations within the ensemble of feedback channels or between the external stimulus and the feedback amplifies population-rate fluctuations by orders of magnitude. The fluctuation suppression in homogeneous inhibitory networks is explained by a negative feedback loop in the one-dimensional dynamics of the compound activity. Similarly, a change of coordinates exposes an effective negative feedback loop in the compound dynamics of stable excitatory-inhibitory networks. The suppression of input correlations in finite networks is explained by the population averaged correlations in the linear network model: In purely inhibitory networks, shared-input correlations are canceled by negative spike-train correlations. In excitatory-inhibitory networks, spike-train correlations are typically positive. Here, the suppression of input correlations is not a result of the mere existence of correlations between excitatory (E) and inhibitory (I) neurons, but a consequence of a particular structure of correlations among the three possible pairings (EE, EI, II).
Author Summary
The spatio-temporal activity pattern generated by a recurrent neuronal network can provide a rich dynamical basis which allows readout neurons to generate a variety of responses by tuning the synaptic weights of their inputs. The repertoire of possible responses and the response reliability become maximal if the spike trains of individual neurons are uncorrelated. Spike-train correlations in cortical networks can indeed be very small, even for neighboring neurons. This seems to be at odds with the finding that neighboring neurons receive a considerable fraction of inputs from identical presynaptic sources constituting an inevitable source of correlation. In this article, we show that inhibitory feedback, abundant in biological neuronal networks, actively suppresses correlations. The mechanism is generic: It does not depend on the details of the network nodes and decorrelates networks composed of excitatory and inhibitory neurons as well as purely inhibitory networks. For the case of the leaky integrate-and-fire model, we derive the correlation structure analytically. The new toolbox of formal linearization and a basis transformation exposing the feedback component is applicable to a range of biological systems. We confirm our analytical results by direct simulations.
doi:10.1371/journal.pcbi.1002596
PMCID: PMC3487539  PMID: 23133368
11.  The Chronotron: A Neuron That Learns to Fire Temporally Precise Spike Patterns 
PLoS ONE  2012;7(8):e40233.
In many cases, neurons process information carried by the precise timings of spikes. Here we show how neurons can learn to generate specific temporally precise output spikes in response to input patterns of spikes having precise timings, thus processing and memorizing information that is entirely temporally coded, both as input and as output. We introduce two new supervised learning rules for spiking neurons with temporal coding of information (chronotrons), one that provides high memory capacity (E-learning), and one that has a higher biological plausibility (I-learning). With I-learning, the neuron learns to fire the target spike trains through synaptic changes that are proportional to the synaptic currents at the timings of real and target output spikes. We study these learning rules in computer simulations where we train integrate-and-fire neurons. Both learning rules allow neurons to fire at the desired timings, with sub-millisecond precision. We show how chronotrons can learn to classify their inputs, by firing identical, temporally precise spike trains for different inputs belonging to the same class. When the input is noisy, the classification also leads to noise reduction. We compute lower bounds for the memory capacity of chronotrons and explore the influence of various parameters on chronotrons' performance. The chronotrons can model neurons that encode information in the time of the first spike relative to the onset of salient stimuli or neurons in oscillatory networks that encode information in the phases of spikes relative to the background oscillation. Our results show that firing one spike per cycle optimizes memory capacity in neurons encoding information in the phase of firing relative to a background rhythm.
doi:10.1371/journal.pone.0040233
PMCID: PMC3412872  PMID: 22879876
12.  Recurrent Network Models for Perfect Temporal Integration of Fluctuating Correlated Inputs 
PLoS Computational Biology  2009;5(6):e1000404.
Temporal integration of input is essential to the accumulation of information in various cognitive and behavioral processes, and gradually increasing neuronal activity, typically occurring within a range of seconds, is considered to reflect such computation by the brain. Some psychological evidence suggests that temporal integration by the brain is nearly perfect, that is, the integration is non-leaky, and the output of a neural integrator is accurately proportional to the strength of input. Neural mechanisms of perfect temporal integration, however, remain largely unknown. Here, we propose a recurrent network model of cortical neurons that perfectly integrates partially correlated, irregular input spike trains. We demonstrate that the rate of this temporal integration changes proportionately to the probability of spike coincidences in synaptic inputs. We analytically prove that this highly accurate integration of synaptic inputs emerges from integration of the variance of the fluctuating synaptic inputs, when their mean component is kept constant. Highly irregular neuronal firing and spike coincidences are the major features of cortical activity, but they have been separately addressed so far. Our results suggest that the efficient protocol of information integration by cortical networks essentially requires both features and hence is heterotic.
Author Summary
Spikes are the words that neurons use for communicating with one another through their networks. While individual cortical neurons generate highly irregular spike trains, coincidently arriving spikes are considered to exert a strong impact on postsynaptic-cell firing and hence to play an active role in neural information processing. However, little is known about whether computations by the brain benefit from such coincident spikes. Here, we show in a recurrent network model that coincident spikes embedded in random spike trains provide a neural code useful for highly accurate temporal integration of external input. In fact, the proposed neural integration is almost perfectly accurate in the mathematical sense. A wide range of cognitive behavior relies on temporal integration. For instance, it is a central player in sensory discrimination tasks and interval timing perception. Our model provides the neural basis for quantitative understanding of animal's decision behavior. In addition, it may account for why cortical activity shows a heterotic feature with irregular firing and synchronous spikes.
doi:10.1371/journal.pcbi.1000404
PMCID: PMC2685482  PMID: 19503816
13.  Rate Dynamics of Leaky Integrate-and-Fire Neurons with Strong Synapses 
Firing-rate models provide a practical tool for studying the dynamics of trial- or population-averaged neuronal signals. A wealth of theoretical and experimental studies has been dedicated to the derivation or extraction of such models by investigating the firing-rate response characteristics of ensembles of neurons. The majority of these studies assumes that neurons receive input spikes at a high rate through weak synapses (diffusion approximation). For many biological neural systems, however, this assumption cannot be justified. So far, it is unclear how time-varying presynaptic firing rates are transmitted by a population of neurons if the diffusion assumption is dropped. Here, we numerically investigate the stationary and non-stationary firing-rate response properties of leaky integrate-and-fire neurons receiving input spikes through excitatory synapses with alpha-function shaped postsynaptic currents for strong synaptic weights. Input spike trains are modeled by inhomogeneous Poisson point processes with sinusoidal rate. Average rates, modulation amplitudes, and phases of the period-averaged spike responses are measured for a broad range of stimulus, synapse, and neuron parameters. Across wide parameter regions, the resulting transfer functions can be approximated by a linear first-order low-pass filter. Below a critical synaptic weight, the cutoff frequencies are approximately constant and determined by the synaptic time constants. Only for synapses with unrealistically strong weights are the cutoff frequencies significantly increased. To account for stimuli with larger modulation depths, we combine the measured linear transfer function with the nonlinear response characteristics obtained for stationary inputs. The resulting linear–nonlinear model accurately predicts the population response for a variety of non-sinusoidal stimuli.
doi:10.3389/fncom.2010.00149
PMCID: PMC3014599  PMID: 21212832
leaky integrate-and-fire neuron; spiking neuron model; firing-rate model; linear response; transfer function; diffusion limit; finite synaptic weights; linear–nonlinear model
14.  Action potential timing precision in dorsal cochlear nucleus pyramidal cells 
Journal of neurophysiology  2007;97(6):4162-4172.
Many studies of the dorsal cochlear nucleus (DCN) have focused on the representation of acoustic stimuli in terms of average firing rate. However, recent studies have emphasized the role of spike timing in information encoding. We sought to ascertain whether DCN pyramidal cells might employ similar strategies and to what extent intrinsic excitability regulates spike timing. Gaussian distributed low-pass noise current was injected into pyramidal cells in a brain slice preparation. The shuffled auto-correlation-based analysis was used to compute a correlation index of spike times across trials. The noise causes the cells to fire with temporal precision (standard deviation ≅ 1-2 msec) and high reproducibility. Increasing the coefficient of variation of the noise improved the reproducibility of the spike trains, whereas increasing the firing rate of the neuron decreased the neurons' ability to respond with predictable patterns of spikes. Simulated IPSPs superimposed on the noise stimulus enhanced spike timing for > 300 msec, although the enhancement was greatest during the first 100 msec. We also found that populations of pyramidal neurons respond to the same noise stimuli with correlated spike trains, suggesting that ensembles of neurons in the DCN receiving shared input can fire with similar timing. These results support the hypothesis that spike timing can be an important aspect of information coding in the DCN.
doi:10.1152/jn.00469.2006
PMCID: PMC2365897  PMID: 17442767
auditory system; hearing; cochlear nucleus; spike timing; inhibition; synchrony; potassium channels
15.  The relationship between optimal and biologically plausible decoding of stimulus velocity in the retina 
A major open problem in systems neuroscience is to understand the relationship between behavior and the detailed spiking properties of neural populations. We assess how faithfully velocity information can be decoded from a population of spiking model retinal neurons whose spatiotemporal receptive fields and ensemble spike train dynamics are closely matched to real data. We describe how to compute the optimal Bayesian estimate of image velocity given the population spike train response and show that, in the case of global translation of an image with known intensity profile, on average the spike train ensemble signals speed with a fractional standard deviation of about 2% across a specific set of stimulus conditions. We further show how to compute the Bayesian velocity estimate in the case where we only have some a priori information about the (naturalistic) spatial correlation structure of the image but do not know the image explicitly. As expected, the performance of the Bayesian decoder is shown to be less accurate with decreasing prior image information. There turns out to be a close mathematical connection between a biologically plausible “motion energy” method for decoding the velocity and the Bayesian decoder in the case that the image is not known. Simulations using the motion energy method and the Bayesian decoder with unknown image reveal that they result in fractional standard deviations of 10% and 6%, respectively, across the same set of stimulus conditions. Estimation performance is rather insensitive to the details of the precise receptive field location, correlated activity between cells, and spike timing.
PMCID: PMC3612545  PMID: 19884914
16.  From Spiking Neuron Models to Linear-Nonlinear Models 
PLoS Computational Biology  2011;7(1):e1001056.
Neurons transform time-varying inputs into action potentials emitted stochastically at a time dependent rate. The mapping from current input to output firing rate is often represented with the help of phenomenological models such as the linear-nonlinear (LN) cascade, in which the output firing rate is estimated by applying to the input successively a linear temporal filter and a static non-linear transformation. These simplified models leave out the biophysical details of action potential generation. It is not a priori clear to which extent the input-output mapping of biophysically more realistic, spiking neuron models can be reduced to a simple linear-nonlinear cascade. Here we investigate this question for the leaky integrate-and-fire (LIF), exponential integrate-and-fire (EIF) and conductance-based Wang-Buzsáki models in presence of background synaptic activity. We exploit available analytic results for these models to determine the corresponding linear filter and static non-linearity in a parameter-free form. We show that the obtained functions are identical to the linear filter and static non-linearity determined using standard reverse correlation analysis. We then quantitatively compare the output of the corresponding linear-nonlinear cascade with numerical simulations of spiking neurons, systematically varying the parameters of input signal and background noise. We find that the LN cascade provides accurate estimates of the firing rates of spiking neurons in most of parameter space. For the EIF and Wang-Buzsáki models, we show that the LN cascade can be reduced to a firing rate model, the timescale of which we determine analytically. Finally we introduce an adaptive timescale rate model in which the timescale of the linear filter depends on the instantaneous firing rate. This model leads to highly accurate estimates of instantaneous firing rates.
Author Summary
Deciphering the encoding of information in the brain implies understanding how individual neurons emit action potentials (APs) in response to time-varying stimuli. This task is made difficult by two facts: (i) although the biophysics of AP generation are well understood, the dynamics of the membrane potential in response to a time-varying input are highly complex; (ii) the firing of APs in response to a given stimulus is inherently stochastic as only a fraction of the inputs to a neuron are directly controlled by the stimulus, the remaining being due to the fluctuating activity of the surrounding network. As a result, the input-output transform of individual neurons is often represented with the help of simplified phenomenological models that do not take into account the biophysical details. In this study, we directly relate a class of such phenomenological models, the so called linear-nonlinear models, with more biophysically detailed spiking neuron models. We provide a quantitative mapping between the two classes of models, and show that the linear-nonlinear models provide a good approximation of the input-output transform of spiking neurons, as long as the fluctuating inputs from the surrounding network are not exceedingly weak.
doi:10.1371/journal.pcbi.1001056
PMCID: PMC3024256  PMID: 21283777
17.  Predictive Coding of Dynamical Variables in Balanced Spiking Networks 
PLoS Computational Biology  2013;9(11):e1003258.
Two observations about the cortex have puzzled neuroscientists for a long time. First, neural responses are highly variable. Second, the level of excitation and inhibition received by each neuron is tightly balanced at all times. Here, we demonstrate that both properties are necessary consequences of neural networks that represent information efficiently in their spikes. We illustrate this insight with spiking networks that represent dynamical variables. Our approach is based on two assumptions: We assume that information about dynamical variables can be read out linearly from neural spike trains, and we assume that neurons only fire a spike if that improves the representation of the dynamical variables. Based on these assumptions, we derive a network of leaky integrate-and-fire neurons that is able to implement arbitrary linear dynamical systems. We show that the membrane voltage of the neurons is equivalent to a prediction error about a common population-level signal. Among other things, our approach allows us to construct an integrator network of spiking neurons that is robust against many perturbations. Most importantly, neural variability in our networks cannot be equated to noise. Despite exhibiting the same single unit properties as widely used population code models (e.g. tuning curves, Poisson distributed spike trains), balanced networks are orders of magnitudes more reliable. Our approach suggests that spikes do matter when considering how the brain computes, and that the reliability of cortical representations could have been strongly underestimated.
Author Summary
Two observations about the cortex have puzzled and fascinated neuroscientists for a long time. First, neural responses are highly variable. Second, the level of excitation and inhibition received by each neuron is tightly balanced at all times. Here, we demonstrate that both properties are necessary consequences of neural networks representing information reliably and with a small number of spikes. To achieve such efficiency, spikes of individual neurons must communicate prediction errors about a common population-level signal, automatically resulting in balanced excitation and inhibition and highly variable neural responses. We illustrate our approach by focusing on the implementation of linear dynamical systems. Among other things, this allows us to construct a network of spiking neurons that can integrate input signals, yet is robust against many perturbations. Most importantly, our approach shows that neural variability cannot be equated to noise. Despite exhibiting the same single unit properties as other widely used network models, our balanced networks are orders of magnitudes more reliable. Our results suggest that the precision of cortical representations has been strongly underestimated.
doi:10.1371/journal.pcbi.1003258
PMCID: PMC3828152  PMID: 24244113
18.  Timing Precision in Population Coding of Natural Scenes in the Early Visual System 
PLoS Biology  2008;6(12):e324.
The timing of spiking activity across neurons is a fundamental aspect of the neural population code. Individual neurons in the retina, thalamus, and cortex can have very precise and repeatable responses but exhibit degraded temporal precision in response to suboptimal stimuli. To investigate the functional implications for neural populations in natural conditions, we recorded in vivo the simultaneous responses, to movies of natural scenes, of multiple thalamic neurons likely converging to a common neuronal target in primary visual cortex. We show that the response of individual neurons is less precise at lower contrast, but that spike timing precision across neurons is relatively insensitive to global changes in visual contrast. Overall, spike timing precision within and across cells is on the order of 10 ms. Since closely timed spikes are more efficient in inducing a spike in downstream cortical neurons, and since fine temporal precision is necessary to represent the more slowly varying natural environment, we argue that preserving relative spike timing at a ∼10-ms resolution is a crucial property of the neural code entering cortex.
Author Summary
Neurons convey information about the world in the form of trains of action potentials (spikes). These trains are highly repeatable when the same stimulus is presented multiple times, and this temporal precision across repetitions can be as fine as a few milliseconds. It is usually assumed that this time scale also corresponds to the timing precision of several neighboring neurons firing in concert. However, the relative timing of spikes emitted by different neurons in a local population is not necessarily as fine as the temporal precision across repetitions within a single neuron. In the visual system of the brain, the level of contrast in the image entering the retina can affect single-neuron temporal precision, but the effects of contrast on the neural population code are unknown. Here we show that the temporal scale of the population code entering visual cortex is on the order of 10 ms and is largely insensitive to changes in visual contrast. Since closely timed spikes are more efficient in inducing a spike in downstream cortical neurons, and since fine temporal precision is necessary in representing the more slowly varying natural environment, preserving relative spike timing at a ∼10-ms resolution may be a crucial property of the neural code entering cortex.
Early neural representation of visual scenes occurs with a temporal precision on the order of 10 ms, which is precise enough to strongly drive downstream neurons in the visual pathway. Unlike individual neurons, the neural population code is largely insensitive to pronounced changes in visual contrast.
doi:10.1371/journal.pbio.0060324
PMCID: PMC2602720  PMID: 19090624
19.  Predicting the Responses of Repetitively Firing Neurons to Current Noise 
PLoS Computational Biology  2014;10(5):e1003612.
We used phase resetting methods to predict firing patterns of rat subthalamic nucleus (STN) neurons when their rhythmic firing was densely perturbed by noise. We applied sequences of contiguous brief (0.5–2 ms) current pulses with amplitudes drawn from a Gaussian distribution (10–100 pA standard deviation) to autonomously firing STN neurons in slices. Current noise sequences increased the variability of spike times with little or no effect on the average firing rate. We measured the infinitesimal phase resetting curve (PRC) for each neuron using a noise-based method. A phase model consisting of only a firing rate and PRC was very accurate at predicting spike timing, accounting for more than 80% of spike time variance and reliably reproducing the spike-to-spike pattern of irregular firing. An approximation for the evolution of phase was used to predict the effect of firing rate and noise parameters on spike timing variability. It quantitatively predicted changes in variability of interspike intervals with variation in noise amplitude, pulse duration and firing rate over the normal range of STN spontaneous rates. When constant current was used to drive the cells to higher rates, the PRC was altered in size and shape and accurate predictions of the effects of noise relied on incorporating these changes into the prediction. Application of rate-neutral changes in conductance showed that changes in PRC shape arise from conductance changes known to accompany rate increases in STN neurons, rather than the rate increases themselves. Our results show that firing patterns of densely perturbed oscillators cannot readily be distinguished from those of neurons randomly excited to fire from the rest state. The spike timing of repetitively firing neurons may be quantitatively predicted from the input and their PRCs, even when they are so densely perturbed that they no longer fire rhythmically.
Author Summary
Most neurons receive thousands of synaptic inputs per second. Each of these may be individually weak but collectively they shape the temporal pattern of firing by the postsynaptic neuron. If the postsynaptic neuron fires repetitively, its synaptic inputs need not directly trigger action potentials, but may instead control the timing of action potentials that would occur anyway. The phase resetting curve encapsulates the influence of an input on the timing of the next action potential, depending on its time of arrival. We measured the phase resetting curves of neurons in the subthalamic nucleus and used them to accurately predict the timing of action potentials in a phase model subjected to complex input patterns. A simple approximation to the phase model accurately predicted the changes in firing pattern evoked by dense patterns of noise pulses varying in amplitude and pulse duration, and by changes in firing rate. We also showed that the phase resetting curve changes systematically with changes in total neuron conductance, and doing so predicts corresponding changes in firing pattern. Our results indicate that the phase model may accurately represent the temporal integration of complex patterns of input to repetitively firing neurons.
doi:10.1371/journal.pcbi.1003612
PMCID: PMC4014400  PMID: 24809636
20.  A neurally-efficient implementation of sensory population decoding 
A sensory stimulus evokes activity in many neurons, creating a population response that must be “decoded” by the brain to estimate the parameters of that stimulus. Most decoding models have suggested complex neural circuits that compute optimal estimates of sensory parameters on the basis of responses in many sensory neurons. We propose a slightly suboptimal but practically simpler decoder. Decoding neurons integrate their inputs across 100 ms; incoming spikes are weighted by the preferred stimulus of the neuron of origin; and a local, cellular non-linearity approximates divisive normalization without dividing explicitly. The suboptimal decoder includes two simplifying approximations. It uses estimates of firing rate across the population rather than computing the total population response, and it implements divisive normalization with local cellular mechanisms of single neurons rather than more complicated neural circuit mechanisms. When applied to the practical problem of estimating target speed from a realistic simulation of the population response in extrastriate visual area MT, the suboptimal decoder has almost the same accuracy and precision as traditional decoding models. It succeeds in predicting the precision and imprecision of motor behavior using a suboptimal decoding computation because it adds only a small amount of imprecision to the code for target speed in MT, which is itself imprecise.
doi:10.1523/JNEUROSCI.6776-10.2011
PMCID: PMC3085943  PMID: 21451025
population decoding; divisive normalization; spike timing; MT; vector averaging
21.  Noise Suppression and Surplus Synchrony by Coincidence Detection 
PLoS Computational Biology  2013;9(4):e1002904.
The functional significance of correlations between action potentials of neurons is still a matter of vivid debate. In particular, it is presently unclear how much synchrony is caused by afferent synchronized events and how much is intrinsic due to the connectivity structure of cortex. The available analytical approaches based on the diffusion approximation do not allow to model spike synchrony, preventing a thorough analysis. Here we theoretically investigate to what extent common synaptic afferents and synchronized inputs each contribute to correlated spiking on a fine temporal scale between pairs of neurons. We employ direct simulation and extend earlier analytical methods based on the diffusion approximation to pulse-coupling, allowing us to introduce precisely timed correlations in the spiking activity of the synaptic afferents. We investigate the transmission of correlated synaptic input currents by pairs of integrate-and-fire model neurons, so that the same input covariance can be realized by common inputs or by spiking synchrony. We identify two distinct regimes: In the limit of low correlation linear perturbation theory accurately determines the correlation transmission coefficient, which is typically smaller than unity, but increases sensitively even for weakly synchronous inputs. In the limit of high input correlation, in the presence of synchrony, a qualitatively new picture arises. As the non-linear neuronal response becomes dominant, the output correlation becomes higher than the total correlation in the input. This transmission coefficient larger unity is a direct consequence of non-linear neural processing in the presence of noise, elucidating how synchrony-coded signals benefit from these generic properties present in cortical networks.
Author Summary
Whether spike timing conveys information in cortical networks or whether the firing rate alone is sufficient is a matter of controversial debate, touching the fundamental question of how the brain processes, stores, and conveys information. If the firing rate alone is the decisive signal used in the brain, correlations between action potentials are just an epiphenomenon of cortical connectivity, where pairs of neurons share a considerable fraction of common afferents. Due to membrane leakage, small synaptic amplitudes and the non-linear threshold, nerve cells exhibit lossy transmission of correlation originating from shared synaptic inputs. However, the membrane potential of cortical neurons often displays non-Gaussian fluctuations, caused by synchronized synaptic inputs. Moreover, synchronously active neurons have been found to reflect behavior in primates. In this work we therefore contrast the transmission of correlation due to shared afferents and due to synchronously arriving synaptic impulses for leaky neuron models. We not only find that neurons are highly sensitive to synchronous afferents, but that they can suppress noise on signals transmitted by synchrony, a computational advantage over rate signals.
doi:10.1371/journal.pcbi.1002904
PMCID: PMC3617020  PMID: 23592953
22.  Synaptic and nonsynaptic plasticity approximating probabilistic inference 
Learning and memory operations in neural circuits are believed to involve molecular cascades of synaptic and nonsynaptic changes that lead to a diverse repertoire of dynamical phenomena at higher levels of processing. Hebbian and homeostatic plasticity, neuromodulation, and intrinsic excitability all conspire to form and maintain memories. But it is still unclear how these seemingly redundant mechanisms could jointly orchestrate learning in a more unified system. To this end, a Hebbian learning rule for spiking neurons inspired by Bayesian statistics is proposed. In this model, synaptic weights and intrinsic currents are adapted on-line upon arrival of single spikes, which initiate a cascade of temporally interacting memory traces that locally estimate probabilities associated with relative neuronal activation levels. Trace dynamics enable synaptic learning to readily demonstrate a spike-timing dependence, stably return to a set-point over long time scales, and remain competitive despite this stability. Beyond unsupervised learning, linking the traces with an external plasticity-modulating signal enables spike-based reinforcement learning. At the postsynaptic neuron, the traces are represented by an activity-dependent ion channel that is shown to regulate the input received by a postsynaptic cell and generate intrinsic graded persistent firing levels. We show how spike-based Hebbian-Bayesian learning can be performed in a simulated inference task using integrate-and-fire (IAF) neurons that are Poisson-firing and background-driven, similar to the preferred regime of cortical neurons. Our results support the view that neurons can represent information in the form of probability distributions, and that probabilistic inference could be a functional by-product of coupled synaptic and nonsynaptic mechanisms operating over several timescales. The model provides a biophysical realization of Bayesian computation by reconciling several observed neural phenomena whose functional effects are only partially understood in concert.
doi:10.3389/fnsyn.2014.00008
PMCID: PMC3986567  PMID: 24782758
Bayes' rule; synaptic plasticity and memory modeling; intrinsic excitability; naïve Bayes classifier; spiking neural networks; Hebbian learning
23.  Temporal Correlation Mechanisms and Their Role in Feature Selection: A Single-Unit Study in Primate Somatosensory Cortex 
PLoS Biology  2014;12(11):e1002004.
How neurons pay attention Top-down selective attention mediates feature selection by reducing the noise correlations in neural populations and enhancing the synchronized activity across subpopulations that encode the relevant features of sensory stimuli.
Studies in vision show that attention enhances the firing rates of cells when it is directed towards their preferred stimulus feature. However, it is unknown whether other sensory systems employ this mechanism to mediate feature selection within their modalities. Moreover, whether feature-based attention modulates the correlated activity of a population is unclear. Indeed, temporal correlation codes such as spike-synchrony and spike-count correlations (rsc) are believed to play a role in stimulus selection by increasing the signal and reducing the noise in a population, respectively. Here, we investigate (1) whether feature-based attention biases the correlated activity between neurons when attention is directed towards their common preferred feature, (2) the interplay between spike-synchrony and rsc during feature selection, and (3) whether feature attention effects are common across the visual and tactile systems. Single-unit recordings were made in secondary somatosensory cortex of three non-human primates while animals engaged in tactile feature (orientation and frequency) and visual discrimination tasks. We found that both firing rate and spike-synchrony between neurons with similar feature selectivity were enhanced when attention was directed towards their preferred feature. However, attention effects on spike-synchrony were twice as large as those on firing rate, and had a tighter relationship with behavioral performance. Further, we observed increased rsc when attention was directed towards the visual modality (i.e., away from touch). These data suggest that similar feature selection mechanisms are employed in vision and touch, and that temporal correlation codes such as spike-synchrony play a role in mediating feature selection. We posit that feature-based selection operates by implementing multiple mechanisms that reduce the overall noise levels in the neural population and synchronize activity across subpopulations that encode the relevant features of sensory stimuli.
Author Summary
Attention can select stimuli in space based on the stimulus features most relevant for a task. Attention effects have been linked to several important phenomena such as modulations in neuronal spiking rate (i.e., the average number of spikes per unit time) and spike-spike synchrony between neurons. Attention has also been associated with spike count correlations, a measure that is thought to reflect correlated noise in the population of neurons. Here, we studied whether feature-based attention biases the correlated activity between neurons when attention is directed towards their common preferred feature. Simultaneous single-unit recordings were obtained from multiple neurons in secondary somatosensory cortex in non-human primates performing feature-attention tasks. Both firing rate and spike-synchrony were enhanced when attention was directed towards the preferred feature of cells. However, attention effects on spike-synchrony had a tighter relationship with behavior. Further, attention decreased spike-count correlations when it was directed towards the receptive field of cells. Our data indicate that temporal correlation codes play a role in mediating feature selection, and are consistent with a feature-based selection model that operates by reducing the overall noise in a population and synchronizing activity across subpopulations that encode the relevant features of sensory stimuli.
doi:10.1371/journal.pbio.1002004
PMCID: PMC4244037  PMID: 25423284
24.  Statistical Analysis and Decoding of Neural Activity in the Rodent Geniculate Ganglion Using a Metric-Based Inference System 
PLoS ONE  2013;8(5):e65439.
We analyzed the spike discharge patterns of two types of neurons in the rodent peripheral gustatory system, Na specialists (NS) and acid generalists (AG) to lingual stimulation with NaCl, acetic acid, and mixtures of the two stimuli. Previous computational investigations found that both spike rate and spike timing contribute to taste quality coding. These studies used commonly accepted computational methods, but they do not provide a consistent statistical evaluation of spike trains. In this paper, we adopted a new computational framework that treated each spike train as an individual data point for computing summary statistics such as mean and variance in the spike train space. We found that these statistical summaries properly characterized the firing patterns (e. g. template and variability) and quantified the differences between NS and AG neurons. The same framework was also used to assess the discrimination performance of NS and AG neurons and to remove spontaneous background activity or “noise” from the spike train responses. The results indicated that the new metric system provided the desired decoding performance and noise-removal improved stimulus classification accuracy, especially of neurons with high spontaneous rates. In summary, this new method naturally conducts statistical analysis and neural decoding under one consistent framework, and the results demonstrated that individual peripheral-gustatory neurons generate a unique and reliable firing pattern during sensory stimulation and that this pattern can be reliably decoded.
doi:10.1371/journal.pone.0065439
PMCID: PMC3667800  PMID: 23738016
25.  Faithful Representation of Stimuli with a Population of Integrate-and-Fire Neurons 
Neural computation  2008;20(11):10.1162/neco.2008.06-07-559.
We consider a formal model of stimulus encoding with a circuit consisting of a bank of filters and an ensemble of integrate-and-fire neurons. Such models arise in olfactory systems, vision, and hearing. We demonstrate that bandlimited stimuli can be faithfully represented with spike trains generated by the ensemble of neurons. We provide a stimulus reconstruction scheme based on the spike times of the ensemble of neurons and derive conditions for perfect recovery. The key result calls for the spike density of the neural population to be above the Nyquist rate. We also show that recovery is perfect if the number of neurons in the population is larger than a threshold value. Increasing the number of neurons to achieve a faithful representation of the sensory world is consistent with basic neurobiological thought. Finally we demonstrate that in general, the problem of faithful recovery of stimuli from the spike train of single neurons is ill posed. The stimulus can be recovered, however, from the information contained in the spike train of a population of neurons.
doi:10.1162/neco.2008.06-07-559
PMCID: PMC3816090  PMID: 18533815

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