Firing Rate Estimation Research Articles

Estimation of firing rate from instantaneous interspike intervals

The rate coding hypothesis is the oldest and still one of the most accepted hypotheses of neural coding. Consequently, many approaches have been devised for the firing rate estimation, ranging from simple binning of the time axis to advanced statistical methods. Nonetheless the concept of firing rate, while informally understood, can be mathematically defined in several distinct ways. These definitions may yield mutually incompatible results unless implemented properly. Recently it has been shown that the notions of the instantaneous and the classical firing rates can be made compatible, at least in terms of their averages, by carefully discerning the time instant at which the neuronal activity is observed. In this paper we revisit the properties of instantaneous interspike intervals in order to derive several novel firing rate estimators, which are free of additional assumptions or parameters and their temporal resolution is ’locally self-adaptive’. The estimators are simple to implement and are numerically efficient even for very large sets of data.

Spatiotemporal dynamics of human high gamma discriminate naturalistic behavioral states

In analyzing the neural correlates of naturalistic and unstructured behaviors, features of neural activity that are ignored in a trial-based experimental paradigm can be more fully studied and investigated. Here, we analyze neural activity from two patients using electrocorticography (ECoG) and stereo-electroencephalography (sEEG) recordings, and reveal that multiple neural signal characteristics exist that discriminate between unstructured and naturalistic behavioral states such as "engaging in dialogue" and "using electronics". Using the high gamma amplitude as an estimate of neuronal firing rate, we demonstrate that behavioral states in a naturalistic setting are discriminable based on long-term mean shifts, variance shifts, and differences in the specific neural activity's covariance structure. Both the rapid and slow changes in high gamma band activity separate unstructured behavioral states. We also use Gaussian process factor analysis (GPFA) to show the existence of salient spatiotemporal features with variable smoothness in time. Further, we demonstrate that both temporally smooth and stochastic spatiotemporal activity can be used to differentiate unstructured behavioral states. This is the first attempt to elucidate how different neural signal features contain information about behavioral states collected outside the conventional experimental paradigm.

Improved Spike-Based Brain-Machine Interface Using Bayesian Adaptive Kernel Smoother and Deep Learning

Multiunit activity (MUA) has been proposed to mitigate the robustness issue faced by single-unit activity (SUA)-based brain-machine interfaces (BMIs). Most MUA-based BMIs still employ a binning method for estimating firing rates and linear decoder for decoding behavioural parameters. The limitations of binning and linear decoder lead to suboptimal performance of MUA-based BMIs. To address this issue, we propose a method which consists of Bayesian adaptive kernel smoother (BAKS) as the firing rate estimation algorithm and deep learning, particularly quasi-recurrent neural network (QRNN), as the decoding algorithm. We evaluated the proposed method for reconstructing (offline) hand kinematics from intracortical neural data chronically recorded from the primary motor cortex of two non-human primates. Extensive empirical results across recording sessions and subjects showed that the proposed method consistently outperforms other combinations of firing rate estimation algorithm and decoding algorithm. Overall results suggest the effectiveness of the proposed method for improving the decoding performance of MUA-based BMIs.

On modeling of a recurrent neural network from neural spiking data.

We present a theoretical and computational work, aiming at the estimation of firing rate based excitatory and inhibitory neural network from realistic stimulus-response data. The stimulus and response recordings are taken from a previous study which performs a measurement on the H1 neurons of the order Diptera flies. The parameter estimation is performed by maximum likelihood method. As the stimulus-response data is a single recording of 20 minutes, it is segmented and individual segments are superimposed on each other to increase the statistical content of information. The true values of the model parameters are unknown as we are not using synthetic data. Because of this fact, two sample Kolmogorov-Smirnov test is applied to compare the interspiking intervals of the recorded and model responses. Estimation and analysis results are presented in tabular and graphical forms. In addition, a comparison with previous research employing a modified Fitzhugh-Nagumo model is made.

Estimation of Motor Unit Global Firing Rate by Maximum Power Amplitude.

Motor unit (MU) global firing rate is widely applied in physiological and clinical investigation. Currently it still remains difficult to measure the MU global firing rate from sEMG. In this study, we propose a new feature of maximum power amplitude (MPA) from sEMG power spectrum. Based on an analysis of mathematical model and simulated signals, MPA was demonstrated to be highly correlated with the MU global firing rate. The performance of MPA was comparable with features based on sEMG amplitude in the time domain. Moreover, the simulation results showed that the square of MPA changed accordingly with the output force, indicating potential application estimating force using MPA2.

The Firing Rate Speed Code of Entorhinal Speed Cells Differs across Behaviorally Relevant Time Scales and Does Not Depend on Medial Septum Inputs.

The firing rate of speed cells, a dedicated subpopulation of neurons in the medial entorhinal cortex (MEC), is correlated with running speed. This correlation has been interpreted as a speed code used in various computational models for path integration. These models consider firing rate to be linearly tuned by running speed in real-time. However, estimation of firing rates requires integration of spiking events over time, setting constraints on the temporal accuracy of the proposed speed code. We therefore tested whether the proposed speed code by firing rate is accurate at short time scales using data obtained from open-field recordings in male rats and mice. We applied a novel filtering approach differentiating between speed codes at multiple time scales ranging from deciseconds to minutes. In addition, we determined the optimal integration time window for firing-rate estimation using a general likelihood framework and calculated the integration time window that maximizes the correlation between firing rate and running speed. Data show that these time windows are on the order of seconds, setting constraints on real-time speed coding by firing rate. We further show that optogenetic inhibition of either cholinergic, GABAergic, or glutamatergic neurons in the medial septum/diagonal band of Broca does not affect modulation of firing rates by running speed at each time scale tested. These results are relevant for models of path integration and for our understanding of how behavioral activity states may modulate firing rates and likely information processing in the MEC.SIGNIFICANCE STATEMENT Path integration is the most basic form of navigation relying on self-motion cues. Models of path integration use medial septum/diagonal band of Broca (MSDB)-dependent MEC grid-cell firing patterns as the neurophysiological substrate of path integration. These models use a linear speed code by firing rate, but do not consider temporal constraints of integration over time for firing-rate estimation. We show that firing-rate estimation for speed cells requires integration over seconds. Using optogenetics, we show that modulation of firing rates by running speed is independent of MSDB inputs. These results enhance our understanding of path integration mechanisms and the role of the MSDB for information processing in the MEC.

Firing rate estimation using infinite mixture models and its application to neural decoding.

Neural decoding is a framework for reconstructing external stimuli from spike trains recorded by various neural recordings. Kloosterman et al. proposed a new decoding method using marked point processes (Kloosterman F, Layton SP, Chen Z, Wilson MA. J Neurophysiol 111: 217-227, 2014). This method does not require spike sorting and thereby improves decoding accuracy dramatically. In this method, they used kernel density estimation to estimate intensity functions of marked point processes. However, the use of kernel density estimation causes problems such as low decoding accuracy and high computational costs. To overcome these problems, we propose a new decoding method using infinite mixture models to estimate intensity. The proposed method improves decoding performance in terms of accuracy and computational speed. We apply the proposed method to simulation and experimental data to verify its performance.NEW & NOTEWORTHY We propose a new neural decoding method using infinite mixture models and nonparametric Bayesian statistics. The proposed method improves decoding performance in terms of accuracy and computation speed. We have successfully applied the proposed method to position decoding from spike trains recorded in a rat hippocampus.

Estimation of the synaptic input firing rates and characterization of the stimulation effects in an auditory neuron.

To understand information processing in neuronal circuits, it is important to infer how a sensory stimulus impacts on the synaptic input to a neuron. An increase in neuronal firing during the stimulation results from pure excitation or from a combination of excitation and inhibition. Here, we develop a method for estimating the rates of the excitatory and inhibitory synaptic inputs from a membrane voltage trace of a neuron. The method is based on a modified Ornstein-Uhlenbeck neuronal model, which aims to describe the stimulation effects on the synaptic input. The method is tested using a single-compartment neuron model with a realistic description of synaptic inputs, and it is applied to an intracellular voltage trace recorded from an auditory neuron in vivo. We find that the excitatory and inhibitory inputs increase during stimulation, suggesting that the acoustic stimuli are encoded by a combination of excitation and inhibition.

On firing rate estimation for dependent interspike intervals.

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

Real-time estimation and biofeedback of single-neuron firing rates using local field potentials.

The long-term stability and low-frequency composition of local field potentials (LFPs) offer important advantages for robust and efficient neuroprostheses. However, cortical LFPs recorded by multi-electrode arrays are often assumed to contain only redundant information arising from the activity of large neuronal populations. Here we show that multichannel LFPs in monkey motor cortex each contain a slightly different mixture of distinctive slow potentials that accompany neuronal firing. As a result, the firing rates of individual neurons can be estimated with surprising accuracy. We implemented this method in a real-time biofeedback brain–machine interface, and found that monkeys could learn to modulate the activity of arbitrary neurons using feedback derived solely from LFPs. These findings provide a principled method for monitoring individual neurons without long-term recording of action potentials.

Signal Processing With Direct Computations on Compressively Sensed Data

Sparsity is characteristic of a signal that potentially allows us to represent information efficiently. We present an approach that enables efficient representations based on sparsity to be utilized throughout a signal processing system, with the aim of reducing the energy and/or resources required for computation, communication, and storage. The representation we focus on is compressive sensing. Its benefit is that compression is achieved with minimal computational cost through the use of random projections; however, a key drawback is that reconstruction is expensive. We focus on inference frameworks for signal analysis. We show that reconstruction can be avoided entirely by transforming signal processing operations (e.g., wavelet transforms, finite impulse response filters, etc.) such that they can be applied directly to the compressed representations. We present a methodology and a mathematical framework that achieve this goal and also enable significant computational-energy savings through operations over fewer input samples. This enables explicit energy-versus-accuracy tradeoffs that are under the control of the designer. We demonstrate the approach through two case studies. First, we consider a system for neural prosthesis that extracts wavelet features directly from compressively sensed spikes. Through simulations, we show that spike sorting can be achieved with 54× fewer samples, providing an accuracy of 98.63% in spike count, 98.56% in firing-rate estimation, and 96.51% in determining the coefficient of variation; this compares with a baseline Nyquist-domain detector with corresponding performance of 98.97%, 99.69%, and 97.09%, respectively. Second, we consider a system for detecting epileptic seizures by extracting spectral-energy features directly from compressively sensed electroencephalogram. Through simulations of the end-to-end algorithm, we show that detection can be achieved with 21× fewer samples, providing a sensitivity of 94.43%, false alarm rate of 0.1543/h, and latency of 4.70 s; this compares with a baseline Nyquist-domain detector with corresponding performance of 96.03%, 0.1471/h, and 4.59 s, respectively.

Strong Correlations between Sensitivity and Variability Give Rise to Constant Discrimination Thresholds across the Otolith Afferent Population

The vestibular system is vital for our sense of linear self-motion. At the earliest processing stages, the otolith afferents of the vestibular nerve encode linear motion. Their resting discharge regularity has long been known to span a wide range, suggesting an important role in sensory coding, yet to date, the question of how this regularity alters the coding of translational motion is not fully understood. Here, we recorded from single otolith afferents in macaque monkeys during linear motion along the preferred directional axis of each afferent over a wide range of frequencies (0.5-16 Hz) corresponding to physiologically relevant stimulation. We used signal-detection theory to directly measure neuronal thresholds and found that values for single afferents were substantially higher than those observed for human perception even when a Kaiser filter was used to provide an estimate of firing rate. Surprisingly, we further found that neuronal thresholds were independent of both stimulus frequency and resting discharge regularity. This was because increases in trial-to-trial variability were matched by increases in sensitivity such that their ratio remains constant: a coding strategy that markedly differs from that used by semicircular canal vestibular afferents to encode rotations. Finally, using Fisher information, we show that pooling the activities of multiple otolith afferents gives rise to neural thresholds comparable with those measured for perception. Together, our results strongly suggest that higher-order structures integrate inputs across afferent populations to provide our sense of linear motion and provide unexpected insight into the influence of variability on sensory encoding.

Estimation of neural firing rate: the wavelet density estimation approach

The computation of neural firing rates based on spike sequences has been introduced as a useful tool for extraction of an animal's behavior. Different estimating methods of such neural firing rates have been developed by neuroscientists, and among these methods, time histogram and kernel estimators have been used more than other approaches. In this paper, the problem in the estimation of firing rates using wavelet density estimators has been considered. The results of simulation study in estimation of underlying rates based on spike sequences sampled from two different variable firing rates show that the proposed wavelet density method provides better and more accurate estimation of firing rates with smooth results compared to two other classical approaches. Furthermore, the performance of a different family of wavelet density estimators in the estimation of the underlying firing rate of biological data have been compared with results of both time histogram and kernel estimators. All in all, the results show that the proposed method can be useful in the estimation of firing rate of neural spike trains.

A General Likelihood Framework for Characterizing the Time Course of Neural Activity

We develop a general likelihood-based framework for use in the estimation of neural firing rates, which is designed to choose the temporal smoothing parameters that maximize the likelihood of missing data. This general framework is algorithm-independent and thus can be applied to a multitude of established methods for firing rate or conditional intensity estimation. As a simple example of the use of the general framework, we apply it to the peristimulus time histogram and kernel smoother, the methods most widely used for firing rate estimation in the electrophysiological literature and practice. In doing so, we illustrate how the use of the framework can employ the general point process likelihood as a principled cost function and can provide substantial improvements in estimation accuracy for even the most basic of rate estimation algorithms. In particular, the resultant kernel smoother is simple to implement, efficient to compute, and can accurately determine the bandwidth of a given rate process from individual spike trains. We perform a simulation study to illustrate how the likelihood framework enables the kernel smoother to pick the bandwidth parameter that best predicts missing data, and we show applications to real experimental spike train data. Additionally, we discuss how the general likelihood framework may be used in conjunction with more sophisticated methods for firing rate and conditional intensity estimation and suggest possible applications.

Efficient, adaptive estimation of two-dimensional firing rate surfaces via Gaussian process methods

Estimating two-dimensional firing rate maps is a common problem, arising in a number of contexts: the estimation of place fields in hippocampus, the analysis of temporally nonstationary tuning curves in sensory and motor areas, the estimation of firing rates following spike-triggered covariance analyses, etc. Here we introduce methods based on Gaussian process nonparametric Bayesian techniques for estimating these two-dimensional rate maps. These techniques offer a number of advantages: the estimates may be computed efficiently, come equipped with natural errorbars, adapt their smoothness automatically to the local density and informativeness of the observed data, and permit direct fitting of the model hyperparameters (e.g., the prior smoothness of the rate map) via maximum marginal likelihood. We illustrate the method's flexibility and performance on a variety of simulated and real data.

Hidden structures detection in nonstationary spike trains

Event Abstract Back to Event Hidden structures detection in nonstationary spike trains Ken Takiyama1* and Masato Okada2 1 The University of Tokyo, Japan 2 RIKEN ,The University of Tokyo, Japan Elucidating neural encoding from irregular neural activities is one of the most important issues in neuroscience. Firing rates abundantly include information encoded by neurons and are often estimated averaging neural activities across trials. We however need to estimate firing rates using only one or the least spike trains especially in the case that neural activities represent internal processes such as decision making and motor planning because the process differs from trial to trial. Firing rate estimation using only one spike train hence plays a crucial role to elucidate neural encoding. Many studies have been proving probabilistic models can estimate firing rates using one spike train in recent years. These studies assume stationarity of mean and temporal correlation within a trial. Stationarity means time series has temporally uniform statistical property. We call what does not have stationarity as nonstationary. Many studies, using hidden markov model (HMM), have been showing neural network activity transit among neural states within a trial. Since statistical properties of firing rates differ drastically in all neural states, firing rates include nonstationarity with in a trial in many cases. Recent studies have been indicating that neural states reflect the properties of inputs encoded by neurons. Because the studies also suggest the transitions show trial-by-trial variation, neural state estimation using only one spike train plays a crucial role for elucidating neural encoding. We construct the algorithm that can simultaneously estimate nonstationary firing rates, neural state transition timings and neural state numbers using only one spike train. Our algorithm consists of Switching State Space Model(SSSM). SSSM defines more than one prior distributions one of which generates observation data. SSSM can also estimate neural state transition timings where a prior distribution that generates observation switches. The HMMs assume firing rates are constant in each neural state. Firing rates however naturally showing temporal variation, Excluding the time variation probably obscures the neural state information. SSSM enables to estimate neural state transitions with considering the temporal variation. Learning and estimation algorithm are constituted of Variational Bayes method. Automatic relevance determination induced by Variational Bayes enables to estimate neural state number at a time. Synthetic data analysis shows our algorithm can simultaneously estimate nonstationary firing rates, neural state transition timings and neural state numbers using one spike train with high accuracy. Applying to real data, which is available on Neural Signal Archive, reveals our algorithm can estimate neural state information from area MT data. These neural states probably correspond to transient and sustained states which have been detected heuristically in area MT data. We confirm the HMM can not detect these neural states. These results suggest our algorithm has the versatility in neural state estimation and the effectiveness on neural encoding elucidation. Conference: Computational and Systems Neuroscience 2010, Salt Lake City, UT, United States, 25 Feb - 2 Mar, 2010. Presentation Type: Poster Presentation Topic: Poster session III Citation: Takiyama K and Okada M (2010). Hidden structures detection in nonstationary spike trains. Front. Neurosci. Conference Abstract: Computational and Systems Neuroscience 2010. doi: 10.3389/conf.fnins.2010.03.00168 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 02 Mar 2010; Published Online: 02 Mar 2010. * Correspondence: Ken Takiyama, The University of Tokyo, Chiba, Japan, takiyama@mns.k.u-tokyo.ac.jp Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Ken Takiyama Masato Okada Google Ken Takiyama Masato Okada Google Scholar Ken Takiyama Masato Okada PubMed Ken Takiyama Masato Okada Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Modelling spike trains and extracting response latency with Bayesian binning

The peristimulus time histogram (PSTH) and the spike density function (SDF) are commonly used in the analysis of neurophysiological data. The PSTH is usually obtained by binning spike trains, the SDF being a (Gaussian) kernel smoothed version of the PSTH. While selection of the bin width or kernel size is often relatively arbitrary there have been recent attempts to remedy this situation ( Shimazaki and Shinomoto, 2007c,b,a). We further develop an exact Bayesian generative model approach to estimating PSTHs ( Endres et al., 2008) and demonstate its superiority to competing methods using data from early (LGN) and late (STSa) visual areas. We also highlight the advantages of our scheme’s automatic complexity control and generation of error bars. Additionally, our approach allows extraction of excitatory and inhibitory response latency from spike trains in a principled way, both on repeated and single trial data. We show that the method can be applied to data with high background firing rates and inhibitory responses (LGN) as well as to data with low firing rate and excitatory responses (STSa). Furthermore, we demonstrate on simulated data that our latency extraction method works for a range of signal-to-noise ratios and background firing rates. While further studies are needed to examine the sensitivity of our method to, for example, gradual changes in firing rate and adaptation, the current results suggest that Bayesian binning is a powerful method for the estimation of firing rate and the extraction response latency from neuronal spike trains.

A new look at state-space models for neural data.

State space methods have proven indispensable in neural data analysis. However, common methods for performing inference in state-space models with non-Gaussian observations rely on certain approximations which are not always accurate. Here we review direct optimization methods that avoid these approximations, but that nonetheless retain the computational efficiency of the approximate methods. We discuss a variety of examples, applying these direct optimization techniques to problems in spike train smoothing, stimulus decoding, parameter estimation, and inference of synaptic properties. Along the way, we point out connections to some related standard statistical methods, including spline smoothing and isotonic regression. Finally, we note that the computational methods reviewed here do not in fact depend on the state-space setting at all; instead, the key property we are exploiting involves the bandedness of certain matrices. We close by discussing some applications of this more general point of view, including Markov chain Monte Carlo methods for neural decoding and efficient estimation of spatially-varying firing rates.

Exact Inference in Discontinuous Firing Rate Estimation Using Belief Propagation

We apply image restoration methods to estimate the firing rate of neurons that include discontinuity according to the input stimulus. Image restoration methods are effective in estimating firing rate because images and firing rates can be respectively modeled by two-dimensional and one-dimensional Markov random field (MRF) models. Our method uses the line process, which was developed to detect edges in images, and to estimate discontinuous firing rates and entirely-unknown stimulus timings. We construct the firing rate estimation algorithm for our model using belief propagation (BP). BP gives the exact inference in the firing rate estimation because there is no loop in the first-order MRF. BP simultaneously estimates the discontinuous firing rate, unknown stimulus timings, and calculates a marginal likelihood value that enables us to estimate the hyperparameter based on an empirical Bayes method. By applying our method to synthetic spike trains, whose firing rates include discontinuity, we show that our m...

Methods for estimating neural firing rates, and their application to brain-machine interfaces.

Neural spike trains present analytical challenges due to their noisy, spiking nature. Many studies of neuroscientific and neural prosthetic importance rely on a smoothed, denoised estimate of a spike train's underlying firing rate. Numerous methods for estimating neural firing rates have been developed in recent years, but to date no systematic comparison has been made between them. In this study, we review both classic and current firing rate estimation techniques. We compare the advantages and drawbacks of these methods. Then, in an effort to understand their relevance to the field of neural prostheses, we also apply these estimators to experimentally gathered neural data from a prosthetic arm-reaching paradigm. Using these estimates of firing rate, we apply standard prosthetic decoding algorithms to compare the performance of the different firing rate estimators, and, perhaps surprisingly, we find minimal differences. This study serves as a review of available spike train smoothers and a first quantitative comparison of their performance for brain-machine interfaces.