Single-trial estimation of neuronal firing rates: From single-neuron spike trains to population activity

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Abstract

We present a method to estimate the neuronal firing rate from single-trial spike trains. The method, based on convolution of the spike train with a fixed kernel function, is calibrated by means of simulated spike trains for a representative selection of realistic dynamic rate functions. We derive rules for the optimized use and performance of the kernel method, specifically with respect to an effective choice of the shape and width of the kernel functions. An application of our technique to the on-line, single-trial reconstruction of arm movement trajectories from multiple single-unit spike trains using dynamic population vectors illustrates a possible use of the proposed method.

Introduction

Most prevailing models of neural coding rely heavily on neuronal firing rates. It has been demonstrated in many physiological studies that the firing rate reveals relevant aspects of a neuron’s involvement in information processing and computation. The availability of methods to measure firing rates from neuronal spike trains is therefore essential. The conventional strategy is to average the spike responses over repeated trials in the form of a peri-stimulus time histogram (PSTH; Gerstein and Kiang, 1960), and to interpret the outcome as an estimate of the time-varying rate function (Fig. 1). Using this technique, important insights into the neural mechanisms of sensory and motor processing have been gained.

There are, however, a number of problems with this approach: (1) Not all interesting experiments can be forced into a repeated-trial design; (2) averaging across trials requires stationarity across trials, which is not always guaranteed; (3) potentially relevant dynamic effects which are not strictly time-locked to the trigger event do not survive trial-averaging. For these various reasons it is becoming increasingly important to eliminate the need for trial-averaging and to consider, instead, the alternative of estimating spike rates on the basis of single-trial responses. In addition, (4) the issue of trial-by-trial variability of neuronal responses is recently receiving increasing interest (Arieli et al., 1996, Azouz and Gray, 1999). Moreover, (5) certain multiple-electrode recording experiments (e.g. Chapin et al., 1999) create the need for a reliable on-line estimate of neuronal firing rates.

The firing rate is a fundamental concept for the description of a spiking neuron (and a point process in general; Cox and Isham, 1980). The underlying firing rate ρ(t), also termed intensity function, is a non-negative deterministic function of time, such that the integraltatbρtdtrepresents the expected number of spikes encountered in an observation of the neuron during the observation interval [ta, tb).

In the context of the current paper, the underlying rate function is assumed to be invariant across trials. In reality, this rate function may change as a result of circumstances beyond the control of the experimenter. However, even if the rate function is the same over trials, individual spike trains in repeated observations may vary considerably, as a result of random fluctuations. The underlying rate is nevertheless reflected by the instantaneous density of spikes.

The rate function underlying the spiking of a real neuron, however, cannot be observed directly, it must be reconstructed from the recorded spike trains. Unfortunately, the theory of stochastic point processes does not currently provide a substantial apparatus for the direct inference of underlying dynamics from experimental data. Here, we describe a method to estimate the neuronal firing rate from single-trial spike trains by convolution with a fixed kernel function. The method is calibrated on the basis of simulated spike trains for a selected set of typical dynamic underlying rate functions. From this we derive rules for the optimized use and performance of the kernel method. Finally, we present an application of our technique to the on-line, single-trial reconstruction of arm movement trajectories from multiple single-unit spike trains using dynamic population vectors. Preliminary results have been presented in abstract form (Nawrot et al., 1997, Nawrot et al., 1999).

Section snippets

Estimation

Consider a single spike train, comprised of a finite number of discrete spike events at times t1, . . .,tn. We define the estimation of the time-varying rate function asλti=1nKt−tiwhere K(t) is called kernel function. Thus, the desired underlying ‘true’ rate function is estimated from a single-trial spike train by taking the sum over kernel functions K(tti), centered at spike occurrence times ti (Fig. 2).

We require K(t) to be non-negative to avoid negative rates. Moreover, the kernel should be

Results

The results of our calibration procedure, expressed in terms of the mean integrated square error MISE, show how the performance of the rate estimation depends on shape and width of the kernel function. Based on a systematic variation of the parameters (b, A, w), we derived rules for the construction of optimized kernels when applying the method to experimental data.

Discussion

We described a method to estimate the neuronal firing rate from single-trial spike train data. The method, basically a convolution of the spike train with a fixed kernel function, was calibrated on the basis of simulated spike trains. Our findings demonstrate that estimation of neuronal firing rates from single-trial spike trains is feasible for a representative selection of physiologically realistic dynamic spike responses, including difficult cases with weak responses against a relatively

Acknowledgements

We gratefully acknowledge the comments by M. Diesmann and D. Heck on an earlier version of the manuscript. This research was supported by the Deutsche Forschungsgemeinschaft (DFG), the German-Israeli Foundation for Scientific Research and Development (GIF) and the Human Frontier Science Program (HFSP).

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