Statistics of inverse interspike intervals: The instantaneous firing rate revisited

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.

Variability and Randomness of the Instantaneous Firing Rate

The apparent stochastic nature of neuronal activity significantly affects the reliability of neuronal coding. To quantify the encountered fluctuations, both in neural data and simulations, the notions of variability and randomness of inter-spike intervals have been proposed and studied. In this article we focus on the concept of the instantaneous firing rate, which is also based on the spike timing. We use several classical statistical models of neuronal activity and we study the corresponding probability distributions of the instantaneous firing rate. To characterize the firing rate variability and randomness under different spiking regimes, we use different indices of statistical dispersion. We find that the relationship between the variability of interspike intervals and the instantaneous firing rate is not straightforward in general. Counter-intuitively, an increase in the randomness (based on entropy) of spike times may either decrease or increase the randomness of instantaneous firing rate, in dependence on the neuronal firing model. Finally, we apply our methods to experimental data, establishing that instantaneous rate analysis can indeed provide additional information about the spiking activity.

Fano Factor: A Potentially Useful Information

The Fano factor, defined as the variance-to-mean ratio of spike counts in a time window, is often used to measure the variability of neuronal spike trains. However, despite its transparent definition, careless use of the Fano factor can easily lead to distorted or even wrong results. One of the problems is the unclear dependence of the Fano factor on the spiking rate, which is often neglected or handled insufficiently. In this paper we aim to explore this problem in more detail and to study the possible solution, which is to evaluate the Fano factor in the operational time. We use equilibrium renewal and Markov renewal processes as spike train models to describe the method in detail, and we provide an illustration on experimental data.

Revealing Spectrum Features of Stochastic Neuron Spike Trains

Power spectra of spike trains reveal important properties of neuronal behavior. They exhibit several peaks, whose shape and position depend on applied stimuli and intrinsic biophysical properties, such as input current density and channel noise. The position of the spectral peaks in the frequency domain is not straightforwardly predictable from statistical averages of the interspike intervals, especially when stochastic behavior prevails. In this work, we provide a model for the neuronal power spectrum, obtained from Discrete Fourier Transform and expressed as a series of expected value of sinusoidal terms. The first term of the series allows us to estimate the frequencies of the spectral peaks to a maximum error of a few Hz, and to interpret why they are not harmonics of the first peak frequency. Thus, the simple expression of the proposed power spectral density (PSD) model makes it a powerful interpretative tool of PSD shape, and also useful for neurophysiological studies aimed at extracting information on neuronal behavior from spike train spectra.

Deviance detection in physiologically identified cell types in the rat auditory cortex

Auditory deviance detection is a function of the auditory system that allows reduction of the processing demand for repetitive stimuli while stressing unpredictable ones, which are potentially more informative. Deviance detection has been extensively studied in humans using the oddball paradigm, which evokes an event-related potential known as mismatch negativity (MMN). The same stimulation paradigms are used in animal studies that aim to elucidate the neuronal mechanisms underlying deviance detection. In order to understand the circuitry responsible for deviance detection in the auditory cortex (AC), it is necessary to determine the properties of excitatory and inhibitory neurons separately. Measuring the spike widths of neurons recorded extracellularly from the anaesthetized rat AC, we classified them as fast spiking or regular spiking units. These two neuron types are generally considered as putative inhibitory or excitatory, respectively. In response to an oddball paradigm, we found that both types of units showed similar amounts of deviance detection overall. When considering each AC field separately, we found that only in A1 fast spiking neurons showed higher deviance detection levels than regular spiking neurons, while in the rest of the fields there was no such distinction. Interpreting these responses in the context of the predictive coding framework, we found that the responses of both types of units reflect mainly prediction error signaling (i.e., genuine deviance detection) rather than repetition suppression.

Online Detection of Multiple Stimulus Changes Based on Single Neuron Interspike Intervals

Nervous systems need to detect stimulus changes based on their neuronal responses without using any additional information on the number, times, and types of stimulus changes. Here, two relatively simple, biologically realistic change point detection methods are compared with two common analysis methods. The four methods are applied to intra- and extracellularly recorded responses of a single cricket interneuron (AN2) to acoustic simulation. Solely based on these recorded responses, the methods should detect an unknown number of different types of sound intensity in- and decreases shortly after their occurrences. For this task, the methods rely on calculating an adjusting interspike interval (ISI). Both simple methods try to separate responses to intensity in- or decreases from activity during constant stimulation. The Pure-ISI method performs this task based on the distribution of the ISI, while the ISI-Ratio method uses the ratio of actual and previous ISI. These methods are compared to the frequently used Moving-Average method, which calculates mean and standard deviation of the instantaneous spike rate in a moving interval. Additionally, a classification method provides the upper limit of the change point detection performance that can be expected for the cricket interneuron responses. The classification learns the statistical properties of the actual and previous ISI during stimulus changes and constant stimulation from a training data set. The main results are: (1) The Moving-Average method requires a stable activity in a long interval to estimate the previous activity, which was not always given in our data set. (2) The Pure-ISI method can reliably detect stimulus intensity increases when the neuron bursts, but it fails to identify intensity decreases. (3) The ISI-Ratio method detects stimulus in- and decreases well, if the spike train is not too noisy. (4) The classification method shows good performance for the detection of stimulus in- and decreases. But due to the statistical learning, this method tends to confuse responses to constant stimulation with responses triggered by a stimulus change. Our results suggest that stimulus change detection does not require computationally costly mechanisms. Simple nervous systems like the cricket's could effectively apply ISI-Ratios to solve this fundamental task.

Review: Methods of firing rate estimation

Neuronal firing rate is traditionally defined as the number of spikes per time window. The concept is essential for the rate coding hypothesis, which is still the most commonly investigated scenario in neuronal activity analysis. The estimation of dynamically changing firing rate from neural data can be challenging due to the variability of spike times, even under identical external conditions; hence a wide range of statistical measures have been employed to solve this particular problem. In this paper, we review established firing rate estimation methods, briefly summarize the technical aspects of each approach and discuss their practical applications.