Modeling the variability of firing rate of retinal ganglion cells.

Impulse trains simulating the maintained discharges of retinal ganglion cells were generated by digital realizations of the integrate-and-fire model. If the mean rate were set by a "bias" level added to "noise," the variability of firing would be related to the mean firing rate as an inverse square root law; the maintained discharges of retinal ganglion cells deviate systematically from such a relationship. A more realistic relationship can be obtained if the integrate-and-fire mechanism is "leaky"; with this refinement, the integrate-and-fire model captures the essential features of the data. However, the model shows that the distribution of intervals is insensitive to that of the underlying variability. The leakage time constant, threshold, and distribution of the noise are confounded, rendering the model unspecifiable. Another aspect of variability is presented by the variance of responses to repeated discrete stimuli. The variance of response rate increases with the mean response amplitude; the nature of that relationship depends on the duration of the periods in which the response is sampled. These results have defied explanation. But if it is assumed that variability depends on mean rate in the way observed for maintained discharges, the variability of responses to abrupt changes in lighting can be predicted from the observed mean responses. The parameters that provide the best fits for the variability of responses also provide a reasonable fit to the variability of maintained discharges.     

Amplification of trial-to-trial response variability by neurons in visual cortex

The visual cortex responds to repeated presentations of the same stimulus with high variability. Because the firing mechanism is remarkably noiseless, the source of this variability is thought to lie in the membrane potential fluctuations that result from summated synaptic input. Here this hypothesis is tested through measurements of membrane potential during visual stimulation. Surprisingly, trial-to-trial variability of membrane potential is found to be low. The ratio of variance to mean is much lower for membrane potential than for firing rate. The high variability of firing rate is explained by the threshold present in the function that converts inputs into firing rates. Given an input with small, constant noise, this function produces a firing rate with a large variance that grows with the mean. This model is validated on responses recorded both intracellularly and extracellularly. In neurons of visual cortex, thus, a simple deterministic mechanism amplifies the low variability of summated synaptic inputs into the large variability of firing rate. The computational advantages provided by this amplification are not known.     

Tuned normalization explains the size of attention modulations

The effect of attention on firing rates varies considerably within a single cortical area. The firing rate of some neurons is greatly modulated by attention while others are hardly affected. The reason for this variability across neurons is unknown. We found that the variability in attention modulation across neurons in area MT of macaques can be well explained by variability in the strength of tuned normalization across neurons. The presence of tuned normalization also explains a striking asymmetry in attention effects within neurons: when two stimuli are in a neuron's receptive field, directing attention to the preferred stimulus modulates firing rates more than directing attention to the nonpreferred stimulus. These findings show that much of the neuron-to-neuron variability in modulation of responses by attention depends on variability in the way the neurons process multiple stimuli, rather than differences in the influence of top-down signals related to attention.     

A model for the variability of maintained discharges and responses to flashes of light

Previous studies of the variability of firing of retinal ganglion cells have led to apparently contradictory conclusions. To a first approximation, maintained discharges derive their variability from a noise source that is linearly added to the signal setting the mean firing rate. On the other hand, the variability of responses to abrupt changes in lighting seems to result from a nonlinear interaction between signal and noise. In both the cat and the goldfish retinae, the variance of rate is a fractional power function of the mean response amplitude (impulses/s). The exponent of that power function depends on the duration of the period in which the response is sampled after each transition in luminance; longer durations have a larger exponent. These results are difficult to explain with any simple model. The variability of the maintained discharges also deviates from the predictions of simple additivity. We propose a model for the variability of responses to abrupt changes in lighting that incorporates variability of the form observed for maintained discharges. The parameters of our model that provide the best fits to the variability of responses also provide a reasonable fit to the variability of maintained discharges. Thus, a single explanation can account for the variability of maintained discharges and responses of ganglion cells.     

Encoding odorant identity by spiking packets of rate-invariant neurons in awake mice

Background

How do neural networks encode sensory information? Following sensory stimulation, neural coding is commonly assumed to be based on neurons changing their firing rate. In contrast, both theoretical works and experiments in several sensory systems showed that neurons could encode information as coordinated cell assemblies by adjusting their spike timing and without changing their firing rate. Nevertheless, in the olfactory system, there is little experimental evidence supporting such model.

Methodology/Principal Findings

To study these issues, we implanted tetrodes in the olfactory bulb of awake mice to record the odorant-evoked activity of mitral/tufted (M/T) cells. We showed that following odorant presentation, most M/T neurons do not significantly change their firing rate over a breathing cycle but rather respond to odorant stimulation by redistributing their firing activity within respiratory cycles. In addition, we showed that sensory information can be encoded by cell assemblies composed of such neurons, thus supporting the idea that coordinated populations of globally rate-invariant neurons could be efficiently used to convey information about the odorant identity. We showed that different coding schemes can convey high amount of odorant information for specific read-out time window. Finally we showed that the optimal readout time window corresponds to the duration of gamma oscillations cycles.

Conclusion

We propose that odorant can be encoded by population of cells that exhibit fine temporal tuning of spiking activity while displaying weak or no firing rate change. These cell assemblies may transfer sensory information in spiking packets sequence using the gamma oscillations as a clock. This would allow the system to reach a tradeoff between rapid and accurate odorant discrimination.     

Predicting the Responses of Repetitively Firing Neurons to Current Noise

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.     

Modeling peripheral olfactory coding in Drosophila larvae

The Drosophila larva possesses just 21 unique and identifiable pairs of olfactory sensory neurons (OSNs), enabling investigation of the contribution of individual OSN classes to the peripheral olfactory code. We combined electrophysiological and computational modeling to explore the nature of the peripheral olfactory code in situ. We recorded firing responses of 19/21 OSNs to a panel of 19 odors. This was achieved by creating larvae expressing just one functioning class of odorant receptor, and hence OSN. Odor response profiles of each OSN class were highly specific and unique. However many OSN-odor pairs yielded variable responses, some of which were statistically indistinguishable from background activity. We used these electrophysiological data, incorporating both responses and spontaneous firing activity, to develop a bayesian decoding model of olfactory processing. The model was able to accurately predict odor identity from raw OSN responses; prediction accuracy ranged from 12%-77% (mean for all odors 45.2%) but was always significantly above chance (5.6%). However, there was no correlation between prediction accuracy for a given odor and the strength of responses of wild-type larvae to the same odor in a behavioral assay. We also used the model to predict the ability of the code to discriminate between pairs of odors. Some of these predictions were supported in a behavioral discrimination (masking) assay but others were not. We conclude that our model of the peripheral code represents basic features of odor detection and discrimination, yielding insights into the information available to higher processing structures in the brain.     

A New Approach for Determining Phase Response Curves Reveals that Purkinje Cells Can Act as Perfect Integrators

Cerebellar Purkinje cells display complex intrinsic dynamics. They fire spontaneously, exhibit bistability, and via mutual network interactions are involved in the generation of high frequency oscillations and travelling waves of activity. To probe the dynamical properties of Purkinje cells we measured their phase response curves (PRCs). PRCs quantify the change in spike phase caused by a stimulus as a function of its temporal position within the interspike interval, and are widely used to predict neuronal responses to more complex stimulus patterns. Significant variability in the interspike interval during spontaneous firing can lead to PRCs with a low signal-to-noise ratio, requiring averaging over thousands of trials. We show using electrophysiological experiments and simulations that the PRC calculated in the traditional way by sampling the interspike interval with brief current pulses is biased. We introduce a corrected approach for calculating PRCs which eliminates this bias. Using our new approach, we show that Purkinje cell PRCs change qualitatively depending on the firing frequency of the cell. At high firing rates, Purkinje cells exhibit single-peaked, or monophasic PRCs. Surprisingly, at low firing rates, Purkinje cell PRCs are largely independent of phase, resembling PRCs of ideal non-leaky integrate-and-fire neurons. These results indicate that Purkinje cells can act as perfect integrators at low firing rates, and that the integration mode of Purkinje cells depends on their firing rate.     

Modulation of synchrony without changes in firing rates

It was often reported and suggested that the synchronization of spikes can occur without changes in the firing rate. However, few theoretical studies have tested its mechanistic validity. In the present study, we investigate whether changes in synaptic weights can induce an independent modulation of synchrony while the firing rate remains constant. We study this question at the level of both single neurons and neuronal populations using network simulations of conductance based integrate-and-fire neurons. The network consists of a single layer that includes local excitatory and inhibitory recurrent connections, as well as long-range excitatory projections targeting both classes of neurons. Each neuron in the network receives external input consisting of uncorrelated Poisson spike trains. We find that increasing this external input leads to a linear increase of activity in the network, as well␣as an increase in the peak frequency of oscillation. In␣contrast, balanced changes of the synaptic weight of␣excitatory long-range projections for both classes of postsynaptic neurons modulate the degree of synchronization without altering the firing rate. These results demonstrate that, in a simple network, synchronization and firing rate can be modulated independently, and thus, may be used as independent coding dimensions. Electronic supplementary material  The online version of this article (doi: ) contains supplementary material, which is available to authorized users.     

Working Memory Cells' Behavior May Be Explained by Cross-Regional Networks with Synaptic Facilitation

Neurons in the cortex exhibit a number of patterns that correlate with working memory. Specifically, averaged across trials of working memory tasks, neurons exhibit different firing rate patterns during the delay of those tasks. These patterns include: 1) persistent fixed-frequency elevated rates above baseline, 2) elevated rates that decay throughout the tasks memory period, 3) rates that accelerate throughout the delay, and 4) patterns of inhibited firing (below baseline) analogous to each of the preceding excitatory patterns. Persistent elevated rate patterns are believed to be the neural correlate of working memory retention and preparation for execution of behavioral/motor responses as required in working memory tasks. Models have proposed that such activity corresponds to stable attractors in cortical neural networks with fixed synaptic weights. However, the variability in patterned behavior and the firing statistics of real neurons across the entire range of those behaviors across and within trials of working memory tasks are typical not reproduced. Here we examine the effect of dynamic synapses and network architectures with multiple cortical areas on the states and dynamics of working memory networks. The analysis indicates that the multiple pattern types exhibited by cells in working memory networks are inherent in networks with dynamic synapses, and that the variability and firing statistics in such networks with distributed architectures agree with that observed in the cortex.     

Modeling the spontaneous activity in suprachiasmatic nucleus neurons: Role of cation single channels

A population of interconnected neurons of the mammalian suprachiasmatic nuclei (SCN) controls circadian rhythms in physiological functions. In turn, a circadian rhythm of individual neurons is driven by intracellular processes, which via activation of specific membrane channels, produce circadian modulation of electrical firing rate. Yet the membrane target(s) of the cellular clock have remained enigmatic. Previously, subthreshold voltage-dependent cation (SVC) channels have been proposed as the membrane target of the cellular clock responsible for circadian modulation of the firing rate in SCN neurons. We tested this hypothesis with computational modeling based on experimental results from on-cell recording of SVC channel openings in acutely isolated SCN neurons and long-term continuous recording of activity from dispersed SCN neurons in a multielectrode array dish (MED). The model reproduced the circadian behavior if the number of SVC channels or their kinetics were modulated in accordance with protein concentration in a model of the intracellular clock (Scheper et al., 1999. J. Neurosci. 19, 40-47). Such modulation changed the average firing rate of the model neuron from zero (“subjective-night” silence) up to 18 Hz (“subjective-day” peak). Furthermore, the variability of interspike intervals (ISI) and the circadian pattern of firing rate (i.e. silence-to-activity ratio and shape of circadian peaks) are in reasonable agreement with experimental data obtained in dispersed SCN neurons in MED. These results suggest that the variability of ISI in intact SCN neurons is mostly due to stochastic single-channel openings, and that the circadian pattern of the firing rate is specified by threshold properties of dependence of the spontaneous firing rate on the number of single channels (R-N relationship). This plausible mathematical modeling supports the hypothesis that SVC channels could be a critical element in circadian modulation of firing rate in SCN neurons.     

Time course of perceptual discrimination and single neuron reliability

The reliability of identification of a visual target increases with time available for inspection of the stimulus. We suggest that the neural basis of this improvement is the existence of a mechanism for integrating a noisy firing rate over some period, leading to a reduction in mean firing rate variance with available processing time. We have determined the experimental time course of the improvement in reliability in a parallel search task where the available inspection time is limited by the presentation of a mask at various times after a brief stimulus. We compare the resulting psychometric functions with the predictions of a model based on Signal Detection Theory. The model is based on the assumption that the reliability of the observer's response is limited by the variability of the responses of individual neurons. The reliability of the discrimination between two stimuli at the neuronal level is then directly related to the ratio of the difference between their integrated mean responses (over many trials) to the response standard deviation. This reliability increases with inspection time. To demonstrate application of the model to electrophysiological data, neurometric functions are derived from the firing rates of a monkeyV1 cortical neuron. The data were obtained while the animal was active in a discrimination task. The results correspond qualitatively to our observed human psychometric functions.     

Effects of postural changes and vestibular lesions on genioglossal muscle activity in conscious cats.

Previous studies in humans showed that genioglossal muscle activity is higher when individuals are supine than when they are upright, and prior experiments in anesthetized or decerebrate animals suggested that vestibular inputs might participate in triggering these alterations in muscle firing. The present study determined the effects of whole body tilts in the pitch (nose-up) plane on genioglossal activity in a conscious feline model and compared these responses with those generated by roll (ear-down) tilts. We also ascertained the effects of a bilateral vestibular neurectomy on the alterations in genioglossal activity elicited by changes in body position. Both pitch and roll body tilts produced modifications in muscle firing that were dependent on the amplitude of the rotation; however, the relative effects of ear-down and nose-up tilts on genioglossal activity were variable from animal to animal. The response variability observed might reflect the fact that genioglossus has a complex organization and participates in a variety of tongue movements; in each animal, electromyographic recordings presumably sampled the firing of different proportions of fibers in the various compartments and subcompartments of the muscle. Furthermore, removal of labyrinthine inputs resulted in alterations in genioglossal responses to postural changes that persisted until recordings were discontinued approximately 1 mo later, demonstrating that the vestibular system participates in regulating the muscle's activity. Peripheral vestibular lesions were subsequently demonstrated to be complete through the postmortem inspection of temporal bone sections or by observing that vestibular nucleus neurons did not respond to rotations in vertical planes.     

Stochastic emergence of repeating cortical motifs in spontaneous membrane potential fluctuations in vivo

It was recently discovered that subthreshold membrane potential fluctuations of cortical neurons can precisely repeat during spontaneous activity, seconds to minutes apart, both in brain slices and in anesthetized animals. These repeats, also called cortical motifs, were suggested to reflect a replay of sequential neuronal firing patterns. We searched for motifs in spontaneous activity, recorded from the rat barrel cortex and from the cat striate cortex of anesthetized animals, and found numerous repeating patterns of high similarity and repetition rates. To test their significance, various statistics were compared between physiological data and three different types of stochastic surrogate data that preserve dynamical characteristics of the recorded data. We found no evidence for the existence of deterministically generated cortical motifs. Rather, the stochastic properties of cortical motifs suggest that they appear by chance, as a result of the constraints imposed by the coarse dynamics of subthreshold ongoing activity.     

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.     

Variability and randomness in stationary neuronal activity

The patterns of neuronal activity can be different even if the mean firing rate is fixed. Investigating the variability of the firing may not be sufficient and we suggest to take into account the notion of randomness. The randomness is related to the entropy of the firing, which is bounded from above by the entropy of the Poisson process (given the mean interspike interval). Thus, we propose the Kullback-Leibler distance with respect to the Poisson process as a measure of randomness in a stationary neuronal activity. Under the condition of equal mean values the KL distance does not depend on the time scale and therefore can be compared to the coefficient of variation employed to measure the variability. Furthermore, this measure can be extended to account for correlated neuronal firing. Finally, we analyze the variability and randomness for three common ISI distributions in detail: gamma, lognormal and inverse Gaussian.     

Neural Resolution of Formant Frequencies in the Primary Auditory Cortex of Rats

Pulse-resonance sounds play an important role in animal communication and auditory object recognition, yet very little is known about the cortical representation of this class of sounds. In this study we shine light on one simple aspect: how well does the firing rate of cortical neurons resolve resonant (“formant”) frequencies of vowel-like pulse-resonance sounds. We recorded neural responses in the primary auditory cortex (A1) of anesthetized rats to two-formant pulse-resonance sounds, and estimated their formant resolving power using a statistical kernel smoothing method which takes into account the natural variability of cortical responses. While formant-tuning functions were diverse in structure across different penetrations, most were sensitive to changes in formant frequency, with a frequency resolution comparable to that reported for rat cochlear filters.     

A generalized linear model for estimating spectrotemporal receptive fields from responses to natural sounds

In the auditory system, the stimulus-response properties of single neurons are often described in terms of the spectrotemporal receptive field (STRF), a linear kernel relating the spectrogram of the sound stimulus to the instantaneous firing rate of the neuron. Several algorithms have been used to estimate STRFs from responses to natural stimuli; these algorithms differ in their functional models, cost functions, and regularization methods. Here, we characterize the stimulus-response function of auditory neurons using a generalized linear model (GLM). In this model, each cell's input is described by: 1) a stimulus filter (STRF); and 2) a post-spike filter, which captures dependencies on the neuron's spiking history. The output of the model is given by a series of spike trains rather than instantaneous firing rate, allowing the prediction of spike train responses to novel stimuli. We fit the model by maximum penalized likelihood to the spiking activity of zebra finch auditory midbrain neurons in response to conspecific vocalizations (songs) and modulation limited (ml) noise. We compare this model to normalized reverse correlation (NRC), the traditional method for STRF estimation, in terms of predictive power and the basic tuning properties of the estimated STRFs. We find that a GLM with a sparse prior predicts novel responses to both stimulus classes significantly better than NRC. Importantly, we find that STRFs from the two models derived from the same responses can differ substantially and that GLM STRFs are more consistent between stimulus classes than NRC STRFs. These results suggest that a GLM with a sparse prior provides a more accurate characterization of spectrotemporal tuning than does the NRC method when responses to complex sounds are studied in these neurons.     

Effect of an exponentially decaying threshold on the firing statistics of a stochastic integrate-and-fire neuron

We study a white-noise driven integrate-and-fire (IF) neuron with a time-dependent threshold. We give analytical expressions for mean and variance of the interspike interval assuming that the modification of the threshold value is small. It is shown that the variability of the interval can become both smaller or larger than in the case of constant threshold depending on the decay rate of threshold. We also show that the relative variability is minimal for a certain finite decay rate of the threshold. Furthermore, for slow threshold decay the leaky IF model shows a minimum in the coefficient of variation whenever the firing rate of the neuron matches the decay rate of the threshold. This novel effect can be seen if the firing rate is changed by varying the noise intensity or the mean input current.     

Neuronal activity in rat barrel cortex underlying texture discrimination

Rats and mice palpate objects with their whiskers to generate tactile sensations. This form of active sensing endows the animals with the capacity for fast and accurate texture discrimination. The present work is aimed at understanding the nature of the underlying cortical signals. We recorded neuronal activity from barrel cortex while rats used their whiskers to discriminate between rough and smooth textures. On whisker contact with either texture, firing rate increased by a factor of two to ten. Average firing rate was significantly higher for rough than for smooth textures, and we therefore propose firing rate as the fundamental coding mechanism. The rat, however, cannot take an average across trials, but must make an immediate decision using the signals generated on each trial. To estimate single-trial signals, we calculated the mutual information between stimulus and firing rate in the time window leading to the rat's observed choice. Activity during the last 75 ms before choice transmitted the most informative signal; in this window, neuronal clusters carried, on average, 0.03 bits of information about the stimulus on trials in which the rat's behavioral response was correct. To understand how cortical activity guides behavior, we examined responses in incorrect trials and found that, in contrast to correct trials, neuronal firing rate was higher for smooth than for rough textures. Analysis of high-speed films suggested that the inappropriate signal on incorrect trials was due, at least in part, to nonoptimal whisker contact. In conclusion, these data suggest that barrel cortex firing rate on each trial leads directly to the animal's judgment of texture.