Instantaneous Firing Rate Research Articles

Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons

With the aim to describe the interaction between a couple of neurons a stochastic model is proposed and formalized. In such a model, maintaining statements of the Leaky Integrate-and-Fire framework, we include a random component in the synaptic current, whose role is to modify the equilibrium point of the membrane potential of one of the two neurons and when a spike of the other one occurs it is turned on. The initial and after spike reset positions do not allow to identify the inter-spike intervals with the corresponding first passage times. However, we are able to apply some well-known results for the first passage time problem for the Ornstein-Uhlenbeck process in order to obtain (i) an approximation of the probability density function of the inter-spike intervals in one-way-type interaction and (ii) an approximation of the tail of the probability density function of the inter-spike intervals in the mutual interaction. Such an approximation is admissible for small instantaneous firing rates of both neurons.

Hemodynamic and electrophysiological spontaneous low-frequency oscillations in the cortex: Directional influences revealed by Granger causality

We used a combined electrophysiological/hemodynamic system to examine low-frequency oscillations (LFOs) in spontaneous neuronal activities (spike trains and local field potentials) and hemodynamic signals (cerebral blood flow) recorded from the anesthetized rat somatosensory and visual cortices. The laser Doppler flowmetry (LDF) probe was tilted slightly to approach the area in which a microelectrode array (MEA) was implanted for simultaneous recordings. Spike trains (STs) were converted into continuous-time rate functions (CRFs) using the ST instantaneous firing rates. LFOs were detected for all three of the components using the multi-taper method (MTM). The frequencies of these LFOs ranged from 0.052 to 0.167Hz (mean±SD, 0.10±0.026Hz) for cerebral blood flow (CBF), from 0.027 to 0.26Hz (mean±SD, 0.12±0.041Hz) for the CRFs of the STs and from 0.04 to 0.19Hz (mean±SD, 0.11±0.035Hz) for local field potentials (LFPs). We evaluated the Granger causal relationships of spontaneous LFOs among CBF, LFPs and CRFs using Granger causality (GC) analysis. Significant Granger causal relationships were observed from LFPs to CBF, from STs to CBF and from LFPs to STs at approximately 0.1Hz. The present results indicate that spontaneous LFOs exist not only in hemodynamic components but also in neuronal activities of the rat cortex. To the best of our knowledge, the present study is the first to identify Granger causal influences among CBF, LFPs and STs and show that spontaneous LFOs carry important Granger causal influences from neural activities to hemodynamic signals.

Firing-rate models for neurons with a broad repertoire of spiking behaviors

Instantaneous firing rates are commonly used to describe either the compound spiking activity of neuron ensembles (population rate) or the trial-averaged response of individual neurons to multiple repetitions of the same stimulus. The dynamics of firing rates is often studied by means of population or firing-rate models. The main motivation for using such models rather than spiking neuron models is to reduce the dimensionality and complexity of the microscopic dynamics to allow analytical tractability, efficient simulation, and intuitive understanding. In most cases, population models have to be treated as purely phenomenological descriptions. Only under simplifying assumptions can they be derived or extracted from the single-neuron dynamics. Previous studies in this field were mainly restricted to homogeneous populations of simple integrate-and-fire neurons which can, to some extent, mimic the rather stereotypical behaviour of regularly firing neurons (e.g., pyramidal cells in the neocortex). Electrophysiological studies have revealed a large diversity of neuron types with different dynamical behaviours, ranging from regular spiking to bursting cells, from accommodating to non-accommodating cells, from integrators to resonators, from depolarization- to hyperpolarization-activated cells, etc. So far, the effect of this neuron-type diversity on the dynamics of neural populations is unclear. Here, we investigate the rate-response characteristics for a variety of neuron types in simulation experiments. To this end, we extend previous work on simple integrate-and-fire neurons [1,2] to the multi-timescale adaptive threshold (MAT) [3] and the Izhikevich model [4]. The model neurons are probed by spike trains modeled as realizations of inhomogeneous Poisson or Gamma point processes with sinusoidally modulated intensity. Average rates, modulation amplitudes and phases of the period-averaged spike responses are measured for a broad range of stimulus, neuron and synapse parameters. The stationary and nonstationary response properties are parameterized by means of nonlinear activation functions and linear filter kernels, respectively, serving as ingredients for simple linear-nonlinear firing rate models. We show that linear-nonlinear rate models can accurately predict the population responses to novel test stimuli for a broad range of parameters and neuron types.

Influence of biophysical properties on temporal filters in a sensory neuron

Sensory pathways implement filters that extract relevant information from the environment [1]. These filtering properties depend in intricate ways on the biophysical parameters of the underlying neuronal architecture. Understanding the link between computational aspects, such as response properties or spike statistics, and the underlying biophysics is a question that can be addressed with theoretical methods. Many primary sensory neurons operate in a mean-driven regime, where mean intensity is represented by the average firing rate, while the temporal structure of the stimulus around its mean is encoded into the particular structure of the spike train in analogy to an irregular sampler [2]. Examples of neurons applying this strategy include the paddlefish's electrosensory system [3], grasshopper auditory receptors [4], or the vestibular system of the turtle [5]. Spikes are generated from the dynamics of voltage-dependent ion channels. Different sensory neurons, however, may use different compositions of spike-generating channel types. These do not only render the cell excitable in the first place, they also have a large influence on a neuron's transfer function. In addition, some ion channels shape the subthreshold dynamics of neurons, but it is less clear, how they affect the temporal structure of spike trains when the neurons are driven tonically. Several channel types, including HCN channels, are known to cause subthreshold resonance. This property results in largest subthreshold response amplitudes of a neuron at a particular frequency, hence termed the resonance frequency. The resulting band-pass like filter properties have a strong effect on fluctuation-driven neurons. Their influence on tonic spiking, however, has been less explored. Accordingly, the influence of channels that cause subthreshold resonance, such as HCN, on sensory systems like the hair cells of the mouse inner ear is still under debate [6]. Here, we use phase-response-curve-based methods within the framework of numerical continuation to resolve whether and how the effects of these channels carry over into the suprathreshold, mean-driven dynamics. To this end, we use computational models of hair-cell mechanoreceptors [7]. Our approach is to, first, derive the phase oscillator that is input-output equivalent to the full biophysical model. Then, applying linear response theory, the transfer function can be estimated in the regime of weak-amplitude stimulation [8]. This transfer function maps time-dependent stimuli to the instantaneous firing rate. We show that the poles and zeros of the transfer spectrum are related to the Fourier components of the systems phase response curve (PRC) as well as the average intrinsic noise level, which is due to a finite ensemble of stochastic ion channels. We find that both properties (1) the PRC - a consequence of the deterministic part of the dynamical system - and (2) the intrinsic stochasticity, are affected by the presence of HCN channels. Further, we use an overall measure of information transmission, the stimulus-response information, to approximate the contribution of HCN channels to information transfer in given frequency bands.

Sustained firing of model central auditory neurons yields a discriminative spectro-temporal representation for natural sounds.

The processing characteristics of neurons in the central auditory system are directly shaped by and reflect the statistics of natural acoustic environments, but the principles that govern the relationship between natural sound ensembles and observed responses in neurophysiological studies remain unclear. In particular, accumulating evidence suggests the presence of a code based on sustained neural firing rates, where central auditory neurons exhibit strong, persistent responses to their preferred stimuli. Such a strategy can indicate the presence of ongoing sounds, is involved in parsing complex auditory scenes, and may play a role in matching neural dynamics to varying time scales in acoustic signals. In this paper, we describe a computational framework for exploring the influence of a code based on sustained firing rates on the shape of the spectro-temporal receptive field (STRF), a linear kernel that maps a spectro-temporal acoustic stimulus to the instantaneous firing rate of a central auditory neuron. We demonstrate the emergence of richly structured STRFs that capture the structure of natural sounds over a wide range of timescales, and show how the emergent ensembles resemble those commonly reported in physiological studies. Furthermore, we compare ensembles that optimize a sustained firing code with one that optimizes a sparse code, another widely considered coding strategy, and suggest how the resulting population responses are not mutually exclusive. Finally, we demonstrate how the emergent ensembles contour the high-energy spectro-temporal modulations of natural sounds, forming a discriminative representation that captures the full range of modulation statistics that characterize natural sound ensembles. These findings have direct implications for our understanding of how sensory systems encode the informative components of natural stimuli and potentially facilitate multi-sensory integration.

Local and Global Contrast Adaptation in Retinal Ganglion Cells

Retinal ganglion cells react to changes in visual contrast by adjusting their sensitivity and temporal filtering characteristics. This contrast adaptation has primarily been studied under spatially homogeneous stimulation. Yet, ganglion cell receptive fields are often characterized by spatial subfields, providing a substrate for nonlinear spatial processing. This raises the question whether contrast adaptation follows a similar subfield structure or whether it occurs globally over the receptive field even for local stimulation. We therefore recorded ganglion cell activity in isolated salamander retinas while locally changing visual contrast. Ganglion cells showed primarily global adaptation characteristics, with notable exceptions in certain aspects of temporal filtering. Surprisingly, some changes in filtering were most pronounced for locations where contrast did not change. This seemingly paradoxical effect can be explained by a simple computational model, which emphasizes the importance of local nonlinearities in the retina and suggests a reevaluation of previously reported local contrast adaptation.

A Spike-Based Model of Neuronal Intrinsic Plasticity

The discovery of neuronal intrinsic plasticity (IP) processes which persistently modify a neuron's excitability necessitates a new concept of the neuronal plasticity mechanism and may profoundly influence our ideas on learning and memory. In this paper, we propose a spike-based IP model/adaptation rule for an integrate-and-fire (IF) neuron to model this biological phenomenon. By utilizing spikes denoted by Dirac delta functions rather than computing instantaneous firing rates for the time-dependent stimulus, this simple adaptation rule adjusts two parameters of an individual IF neuron to modify its excitability. As a result, this adaptation rule helps an IF neuron to keep its firing activity in a relatively “low but not too low” level and makes the spike-count distributions computed with adjusted window sizes similar to the experimental results.

REM Sleep Reorganizes Hippocampal Excitability

Sleep is composed of an alternating sequence of REM and non-REM episodes, but their respective roles are not known. We found that the overall firing rates of hippocampal CA1 neurons decreased across sleep concurrent with an increase in the recruitment of neuronal spiking to brief "ripple" episodes, resulting in a net increase in neural synchrony. Unexpectedly, within non-REM episodes, overall firing rates gradually increased together with a decrease in the recruitment of spiking to ripples. The rate increase within non-REM episodes was counteracted bya larger and more rapid decrease of discharge frequency within the interleaved REM episodes. Both the decrease in firing rates and the increase in synchrony during the course of sleep were correlated with the power of theta activity during REM episodes. These findings assign a prominent role of REM sleep in sleep-related neuronal plasticity.

Discharge patterns in the lateral superior olive of decerebrate cats

Anatomical and pharmacological studies have shown that the lateral superior olive (LSO) receives inputs from a number of sources and that LSO cells can alter the balance of their own excitatory and inhibitory drive. It is thus likely that the ongoing sound-evoked responses of LSO cells reflect a complex interplay of excitatory and inhibitory events, which may be affected by anesthesia. The goal of this study was to characterize the temporal discharge patterns of single units in the LSO of unanesthetized, decerebrate cats in response to long-duration ipsilateral best-frequency tone bursts. A decision tree is presented to partition LSO units on the basis of poststimulus time histogram shape, adaptation of instantaneous firing rate as a function of time, and sustained discharge rate. The results suggest that LSO discharge patterns form a continuum with four archetypes: sustained choppers that show two or more peaks of activity at stimulus onset and little adaptation of rate throughout the response, transient choppers that undergo a decrease in rate that eventually stabilizes with time, primary-like units that display an initial peak of activity followed by a monotonic decline in rate to a steady-state value, and onset-sustained units that exhibit an initial peak of activity at stimulus onset followed by a low sustained activity. Compared with the chopper units, the nonchopper units tend to show longer first-spike latencies, lower peak firing rates, and more irregular sustained discharge patterns. Modeling studies show that the full range of LSO response types can be obtained from an underlying sustained chopper by varying the strength and latency of a sound-driven ipsilateral inhibition relative to that of excitation. Together, these results suggest that inhibition plays a major role in shaping the temporal discharge patterns of units in unanesthetized preparations.

Inflammatory pain disrupts the orbitofrontal neuronal activity and risk-assessment performance in a rodent decision-making task

It has been recently described that disruption of the neural mechanisms of emotion-based decision making occurs in both chronic pain patients and in animal models of pain; moreover, it also has been shown that chronic pain causes morphological and functional changes in the prefrontal cortex that may be crucial for this decision-making dysfunction. However, it is not known whether pain alone is capable of altering the neuronal encoding of decision exhibited by prefrontal neurons. We have previously shown that naïve animals have risk-averse performance in the rodent gambling task, whereas chronic pain animals reverse their choice preference and become risk prone. Using this paradigm, we chronically implanted arrays of multielectrodes and recorded from neuronal ensembles in the orbitofrontal cortex of freely moving animals performing 4 sessions of the rodent gambling task: 2 in control conditions and 2 after the onset of inflammatory pain induced by complete Freund’s adjuvant injection. Our results show that the instantaneous neuronal firing rate was correlated with the probability of choosing a specific lever in 62.5% of the neurons; however, although in the control sessions 61% of the neurons encoded the reward magnitude, after the pain onset only 16% of the neurons differentiated small from large rewards. Moreover, we found that the fraction of risk-sensitive neurons recorded in each session predicted the overall risk bias of the animal. Our data suggest that orbitofrontal cortex encoding of risk preference is compromised in chronic pain animals.

Neural Dynamics, Bifurcations, and Firing Rates in a Quadratic Integrate-and-Fire Model with a Recovery Variable. I: Deterministic Behavior

We study the dynamics of a quadratic integrate-and-fire model of a single-compartment neuron with a slow recovery variable, as input current and parameters describing timescales, recovery variable, and postspike reset change. Analysis of a codimension 2 bifurcation reveals that the domain of attraction of a stable hyperpolarized rest state interacts subtly with reset parameters, which reposition the system state after spiking. We obtain explicit approximations of instantaneous firing rates for fixed values of the recovery variable, and use the averaging theorem to obtain asymptotic firing rates as a function of current and reset parameters. Along with the different phase-plane geometries, these computations provide explicit tools for the interpretation of different spiking patterns and guide parameter selection in modeling different cortical cell types.

Comparative study on song behavior, and ultra-structural, electrophysiological and immunoreactive properties in RA among deafened, untutored and normal-hearing Bengalese finches

To gain additional insight into how a birdsong is learned, we compared the songs of Bengalese finch males that were deafened early in development or raised without tutors to control finches that learned songs from adult models. Fewer note types and a more variable number of notes per bout were observed in untutored male songs, and no audible songs were detected in deafened males. We then investigated the ultrastructural, immunohistological, and electrophysiological correlates of the outcomes of song learning within the robust nucleus of the archopallium (RA), a forebrain nucleus for song production. In comparison to control birds, untutored and deafened birds had more synapses per unit volume, fewer vesicles per synapse, longer postsynaptic densities, and a lower proportion of perforated synapses, which suggest lower activity or decreased efficiency of synaptic transmission within the RA of the treated birds. For anesthetized birds, neurons within the RA of untutored and deafened males had lower spontaneous firing rates, fewer and shorter bursts, and higher coefficient of variation of the instantaneous firing rate than the normally reared males. Compared with controls, the untutored and deafened males had higher staining intensities within the RA of GABA and the GABAA receptor, less staining of tyrosine hydroxylase and no difference in the staining of NMDA receptors. Thus, both the ultrastructural and immunohistochemical results could explain for the stronger electrophysiological activities in normally reared birds. Because RA is involved in generating the motor commands, these data might account for the deficits in birds with abnormal song learning.

Temporal Encoding of Spatial Information during Active Visual Fixation

Humans and other species continually perform microscopic eye movements, even when attending to a single point. These movements, which include drifts and microsaccades, are under oculomotor control, elicit strong neural responses, and have been thought to serve important functions. The influence of these fixational eye movements on the acquisition and neural processing of visual information remains unclear. Here, we show that during viewing of natural scenes, microscopic eye movements carry out a crucial information-processing step: they remove predictable correlations in natural scenes by equalizing the spatial power of the retinal image within the frequency range of ganglion cells' peak sensitivity. This transformation, which had been attributed to center-surround receptive field organization, occurs prior to any neural processing and reveals a form of matching between the statistics of natural images and those of normal eye movements. We further show that the combined effect of microscopic eye movements and retinal receptive field organization is to convert spatial luminance discontinuities into synchronous firing events, beginning the process of edge detection. Thus, microscopic eye movements are fundamental to two goals of early visual processing: redundancy reduction and feature extraction.

The ionic mechanism of gamma resonance in rat striatal fast-spiking neurons

Striatal fast-spiking (FS) cells in slices fire in the gamma frequency range and in vivo are often phase-locked to gamma oscillations in the field potential. We studied the firing patterns of these cells in slices from rats ages 16-23 days to determine the mechanism of their gamma resonance. The resonance of striatal FS cells was manifested as a minimum frequency for repetitive firing. At rheobase, cells fired a doublet of action potentials or doublets separated by pauses, with an instantaneous firing rate averaging 44 spikes/s. The minimum rate for sustained firing was also responsible for the stuttering firing pattern. Firing rate adapted during each episode of firing, and bursts were terminated when firing was reduced to the minimum sustainable rate. Resonance and stuttering continued after blockade of Kv3 current using tetraethylammonium (0.1-1 mM). Both gamma resonance and stuttering were strongly dependent on Kv1 current. Blockade of Kv1 channels with dendrotoxin-I (100 nM) completely abolished the stuttering firing pattern, greatly lowered the minimum firing rate, abolished gamma-band subthreshold oscillations, and slowed spike frequency adaptation. The loss of resonance could be accounted for by a reduction in potassium current near spike threshold and the emergence of a fixed spike threshold. Inactivation of the Kv1 channel combined with the minimum firing rate could account for the stuttering firing pattern. The resonant properties conferred by this channel were shown to be adequate to account for their phase-locking to gamma-frequency inputs as seen in vivo.

Attracting states in frontal cortex networks associated with working memory and decision making

A frequent hypothesis in theoretical neuroscience is that cognitive entities are represented and processed by attracting states of the underlying neural system [1]. For instance, different attractor-like states may represent different spatial locations or cognitive entities, and transitions between these attracting sets could be associated with the recall of a memory sequence or the execution of a motor plan. Attractor states underlying cognition were previously proposed in the context of working memory [1] and decision making tasks. However, although theoretically suggested, experimental evidence is still sparse for the hypothesis that higher cognitive processes proceed by moving between attracting states in higher cortical areas. Using state space reconstruction theorems [3] and statistical learning techniques, we were able to reveal dynamical properties, not easily accessible in previous studies, of anterior cingulate cortex (ACC) multiple single-unit activity (MSUA) during a cognitive task. The approach worked by constructing high-dimensional state spaces from delays of the original single-unit instantaneous firing-rates and all possible products (multinomials) among them up to some specific order. The dynamics within these sparse and high-dimensional spaces of neural activity interactions were then statistically accessed using strongly regularized kernel methods [4,5]. Results showed cognitive-epoch-specific neural ensemble states (dependent on behavioral performance [3]) in ACC while the rats performed an ecologically valid eight-arm radial arm-maze task. More interestingly, these cognitively defined ensemble states showed some hallmarks of attracting behavior which became apparent in high-dimensional expansions of the MSU spaces due to a proper unfolding of the neural activity flow. It turned out that optimal unfolding of neural trajectories was achieved in an embedding space characterized by a specific maximum order of neural interactions, common across different animals. In further analyses, ensemble states as a function of the animal’s spatial position i.e. the arm visited were analyzed. From these analyses the intrinsic dimensionality which is relevant to the animal’s arm choices could be computed [4]. Results showed that cognitively relevant network states were restricted to a low-dimensional nonlinear manifold within the high-dimensional space. Finally, preliminary analyses indicate that it might be possible to directly infer from this reduced space an idealized neural ensemble model with predictive properties across future trials with respect to the animal’s arm choices. An analysis of such a model would permit a full characterization of the putative neural attractors underlying working memory. To summarize, results suggest that ACC networks may process different subcomponents of higher cognitive tasks by transitioning between different attracting states. Moreover it may be possible to use these analyses to characterize the dynamical states underlying the animal’s choices without further assumptions about the biophysical variables.

Motoneuron afterhyperpolarisation duration in amyotrophic lateral sclerosis

Motor unit (MU) potentials were registered from 20 ALS patients and 13 age-matched control individuals during isometric constant force contractions of brachial biceps (BB). The registered signals were decomposed into single MU potential trains. The estimates of duration of the afterhyperpolarisation (AHP) in MNs, derived from the interspike interval variability, was compared between ALS patients (124 MNs) and control subjects (111 MNs) and no significant differences were encountered. However, the relationship between TI and age for patients appeared to be qualitatively different from that of the control group. The dependence of patients' AHPs on relative force deficit (RFD), which quantified muscle involvement, was more specific. For RFDs below 30%, the AHP estimate was significantly lower than control values and then increased thereafter with increasing RFDs. Moreover, firing rates of patients with the smallest RFDs were significantly higher while firing rates of patients with the greatest RFDs were significantly lower than control values. The AHP shortening in the early stages of muscle impairment is consistent with the decrease in firing threshold of ‘fast' MNs found in spinal cord slices from neonatal SOD1 mice. The later elongation of the AHP may be caused by the higher vulnerability of ‘fast' MNs to degeneration and by the influence of reinnervation. Our results are comparable to what has been observed in acute experiments in animal models, providing a bridge between animal and clinical research that may be relevant for identification of mechanism(s) underlying neurodegeneration in ALS.

History dependence of human muscle-fiber conduction velocity during voluntary isometric contractions

The conduction velocity (CV) of a muscle fiber is affected by the fiber's discharge history going back ∼1 s. We investigated this dependence by measuring CV fluctuations during voluntary isometric contractions of the human brachioradialis muscle. We recorded electromyogram (EMG) signals simultaneously from multiple intramuscular electrodes, identified potentials belonging to the same motor unit using EMG decomposition, and estimated the CV of each discharge from the interpotential interval. In 12 of 14 subjects, CV increased by ∼10% during the first second after recruitment and then fluctuated by about ±2% in a way that mirrored the fluctuations in the instantaneous firing rate. The CV profile could be precisely described in terms of the discharge history by a simple mathematical model. In the other two subjects, and one subject retested after cooling the arm, the CV fluctuations were inversely correlated with instantaneous firing rate. In all subjects, CV was additionally affected by very short interdischarge intervals (<25 ms): it was increased in doublets at recruitment, but decreased in doublets during continuous firing and after short interdischarge intervals in doubly innervated fibers. CV also exhibited a slow trend of about -0.05%/s that did not depend on the immediate discharge history. We suggest that measurements of CV fluctuations during voluntary contractions, or during stimulation protocols that involve longer and more complex stimulation patterns than are currently being used, may provide a sensitive approach for estimating the dynamic characteristics of ion channels in the human muscle-fiber membrane.

On nonlinearity in neural encoding models applied to the primary visual cortex

Within the regression framework, we show how different levels of nonlinearity influence the instantaneous firing rate prediction of single neurons. Nonlinearity can be achieved in several ways. In particular, we can enrich the predictor set with basis expansions of the input variables (enlarging the number of inputs) or train a simple but different model for each area of the data domain. Spline-based models are popular within the first category. Kernel smoothing methods fall into the second category. Whereas the first choice is useful for globally characterizing complex functions, the second is very handy for temporal data and is able to include inner-state subject variations. Also, interactions among stimuli are considered. We compare state-of-the-art firing rate prediction methods with some more sophisticated spline-based nonlinear methods: multivariate adaptive regression splines and sparse additive models. We also study the impact of kernel smoothing. Finally, we explore the combination of various local models in an incremental learning procedure. Our goal is to demonstrate that appropriate nonlinearity treatment can greatly improve the results. We test our hypothesis on both synthetic data and real neuronal recordings in cat primary visual cortex, giving a plausible explanation of the results from a biological perspective.

From spiking neuron models to linear-nonlinear models.

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.

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.