Evolution of moments and correlations in nonrenewal escape-time processes

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.

Non-additive activity modulation during a decision making task involving tactic selection

Human brain imaging has revealed that stimulus-induced activity does generally not simply add to the pre-stimulus activity, but rather builds in a non-additive way on this activity. Here we investigate this subject at the single neuron level and address the question whether and to what extent a strong form of non-additivity where activity drops post-cue is present in different areas of monkey cortex, including prefrontal and agranular frontal areas, during a perceptual decision making task involving action and tactic selection. Specifically we analyze spike train data recorded in vivo from the posterior dorsomedial prefrontal cortex (pmPFC), the supplementary motor area (SMA) and the presupplementary motor area (pre-SMA). For each neuron, we compute the ratio of the trial-averaged pre-stimulus spike count to the trial-averaged post-stimulus count. We also perform the ratio and averaging procedures in reverse order. We find that the statistics of these quantities behave differently across areas. pmPFC involved in tactic selection shows stronger non-additivity compared to the two other areas which more generically just increase their firing rate pos-stimulus. pmPFC behaved more similarly to pre-SMA, a likely consequence of the reciprocal connections between these areas. The trial-averaged ratio statistic was reproduced by a surrogate inhomogeneous Poisson process in which the measured trial-averaged firing rate for a given neuron is used as its time-dependent rate. Principal component analysis (PCA) of the trial-averaged firing rates of neuronal ensembles further reveals area-specific time courses of response to the stimulus, including latency to peak neural response, for the typical population activity. Our work demonstrates subtle forms of area-specific non-additivity based on the fine variability structure of pre- and post-stimulus spiking activity on the single neuron level. It also reveals significant differences between areas for PCA and surrogate analysis, complementing previous observations of regional differences based solely on post-stimulus responses. Moreover, we observe regional differences in non-additivity which are related to the monkey's successful tactic selection and decision making.

Supplementary Information

The online version contains supplementary material available at 10.1007/s11571-021-09702-0.

Effects of transient subordinators on the firing statistics of a neuron model driven by dichotomous noise

The behavior of a stochastic perfect integrate-and-fire (PIF) model of neurons is considered. The effect of temporally correlated random activity of synaptic inputs is modeled as a combination of an asymmetric dichotomous noise and a random operation time in the form of an inverse strictly increasing Lévy-type subordinator. Using a first-passage-time formulation, we find exact expressions for the output interspike interval (ISI) statistics. Particularly, it is shown that at some parameter regimes the ISI density exhibits a multimodal structure. Moreover, it is demonstrated that the coefficient of variation, the serial correlation coefficient, and the Fano factor display a nonmonotonic dependence on the mean input current μ, i.e., the ISI's regularity is maximized at an intermediate value of μ. The features of spike statistics, analytically revealed in our study, are compared with previously obtained results for a perfect integrate-and-fire neuron model driven by dichotomous noise (without subordination).

Receiver operating characteristic curves for a simple stochastic process that carries a static signal

The detection of a weak signal in the presence of noise is an important problem that is often studied in terms of the receiver operating characteristic (ROC) curve, in which the probability of correct detection is plotted against the probability for a false positive. This kind of analysis is typically applied to the situation in which signal and noise are stochastic variables; the detection problem emerges, however, also often in a context in which both signal and noise are stochastic processes and the (correct or false) detection is said to take place when the process crosses a threshold in a given time window. Here we consider the problem for a combination of a static signal which has to be detected against a dynamic noise process, the well-known Ornstein-Uhlenbeck process. We give exact (but difficult to evaluate) quadrature expressions for the detection rates for false positives and correct detections, investigate systematically a simple sampling approximation suggested earlier, compare to an approximation by Stratonovich for the limit of high threshold, and briefly explore the case of multiplicative signal; all theoretical results are compared to extensive numerical simulations of the corresponding Langevin equation. Our results demonstrate that the sampling approximation provides a reasonable description of the ROC curve for this system, and it clarifies limit cases for the ROC curve.

Interspike interval correlations in networks of inhibitory integrate-and-fire neurons

We study temporal correlations of interspike intervals, quantified by the network-averaged serial correlation coefficient (SCC), in networks of both current- and conductance-based purely inhibitory integrate-and-fire neurons. Numerical simulations reveal transitions to negative SCCs at intermediate values of bias current drive and network size. As bias drive and network size are increased past these values, the SCC returns to zero. The SCC is maximally negative at an intermediate value of the network oscillation strength. The dependence of the SCC on two canonical schemes for synaptic connectivity is studied, and it is shown that the results occur robustly in both schemes. For conductance-based synapses, the SCC becomes negative at the onset of both a fast and slow coherent network oscillation. We then show by means of offline simulations using prerecorded network activity that a neuron's SCC is highly sensitive to its number of presynaptic inputs. Finally, we devise a noise-reduced diffusion approximation for current-based networks that accounts for the observed temporal correlation transitions.

Statistics of inverse interspike intervals: The instantaneous firing rate revisited

The rate coding hypothesis is the oldest and still one of the most accepted and investigated scenarios in neuronal activity analyses. However, the actual neuronal firing rate, while informally understood, can be mathematically defined in several different ways. These definitions yield distinct results; even their average values may differ dramatically for the simplest neuronal models. Such an inconsistency, together with the importance of "firing rate," motivates us to revisit the classical concept of the instantaneous firing rate. We confirm that different notions of firing rate can in fact be compatible, at least in terms of their averages, by carefully discerning the time instant at which the neuronal activity is observed. Two general cases are distinguished: either the inspection time is synchronised with a reference time or with the neuronal spiking. The statistical properties of the instantaneous firing rate, including parameter estimation, are analyzed, and compatibility with the intuitively understood concept is demonstrated.

Sub-threshold signal encoding in coupled FitzHugh-Nagumo neurons

Despite intensive research, the mechanisms underlying the neural code remain poorly understood. Recent work has focused on the response of a single neuron to a weak, sub-threshold periodic signal. By simulating the stochastic FitzHugh-Nagumo (FHN) model and then using a symbolic method to analyze the firing activity, preferred and infrequent spike patterns (defined by the relative timing of the spikes) were detected, whose probabilities encode information about the signal. As not individual neurons but neuronal populations are responsible for sensory coding and information transfer, a relevant question is how a second neuron, which does not perceive the signal, affects the detection and the encoding of the signal, done by the first neuron. Through simulations of two stochastic FHN neurons we show that the encoding of a sub-threshold signal in symbolic spike patterns is a plausible mechanism. The neuron that perceives the signal fires a spike train that, despite having an almost random temporal structure, has preferred and infrequent patterns which carry information about the signal. Our findings could be relevant for sensory systems composed by two noisy neurons, when only one detects a weak external input.