Approaches based on point processes provide a principled statistical framework for modeling neural spiking activity.

From the conditional intensity function (CIF) defined in Eq. 1, we can obtain the survival function The terms over-or underdispersion describe empirical quantile distributions that do not match the point process model. A.3 Renewal processes A.3.1 Firing rates and ISIs The law of large numbers for renewal processes [24] shows that for a Markov renewal process lim Below we give the parametric densities for renewal processes used in the paper. To evaluate the CIF for renewal processes, we need to compute the hazard function as discussed above. Gamma The cumulative density function is C ( τ) = 1 Γ(α) γ ( α,τ) (32) where γ (,) denotes the lower incomplete Gamma function. B.1 Sparse variational Gaussian processes B.1.1 Gaussian processes as priors over functions In addition, closed-form inference and prediction are not possible for non-Gaussian likelihoods as used in this paper.

Statistical features of neuronal spike trains are known to be non-Poisson. Here, we investigate the extent to which the non-Poissonian feature affects the efficiency of transmitting information on fluctuating firing rates. For this purpose, we introduce the Kullback-Leibler (KL) divergence as a measure of the efficiency of information encoding, and assume that spike trains are generated by time-rescaled renewal processes. We show that the KL divergence determines the lower bound of the degree of rate fluctuations below which the temporal variation of the firing rates is undetectable from sparse data. We also show that the KL divergence, as well as the lower bound, depends not only on the variability of spikes in terms of the coefficient of variation, but also significantly on the higher-order moments of interspike interval (ISI) distributions. We examine three specific models that are commonly used for describing the stochastic nature of spikes (the gamma, inverse Gaussian (IG) and lognormal ISI distributions), and find that the time-rescaled renewal process with the IG distribution achieves the largest KL divergence, followed by the lognormal and gamma distributions.

Statistical features of neuronal spike trains are known to be non-Poisson. Here, we investigate the extent to which the non-Poissonian feature affects the efficiency of transmitting information on fluctuating firing rates. For this purpose, we introduce the Kullbuck-Leibler (KL) divergence as a measure of the efficiency of information encoding, and assume that spike trains are generated by time-rescaled renewal processes. We show that the KL divergence determines the lower bound of the degree of rate fluctuations below which the temporal variation of the firing rates is undetectable from sparse data. We also show that the KL divergence, as well as the lower bound, depends not only on the variability of spikes in terms of the coefficient of variation, but also significantly on the higher-order moments of interspike interval (ISI) distributions. We examine three specific models that are commonly used for describing the stochastic nature of spikes (the gamma, inverse Gaussian (IG) and lognormal ISI distributions), and find that the time-rescaled renewal process with the IG distribution achieves the largest KL divergence, followed by the lognormal and gamma distributions.

Cell assemblies exhibiting episodes of recurrent coherent activity have been observed in several brain regions including the striatum and hippocampus CA3. Here we address the question of how coherent dynamically switching assemblies appear in large networks of biologically realistic spiking neurons interacting deterministically. We show by numerical simulations of large asymmetric inhibitory networks with fixed external excitatory drive that if the network has intermediate to sparse connectivity, the individual cells are in the vicinity of a bifurcation between a quiescent and firing state and the network inhibition varies slowly on the spiking timescale, then cells form assemblies whose members show strong positive correlation, while members of different assemblies show strong negative correlation. We show that cells and assemblies switch between firing and quiescent states with time durations consistent with a power-law. Our results are in good qualitative agreement with the experimental studies. The deterministic dynamical behaviour is related to winner-less competition shown in small closed loop inhibitory networks with heteroclinic cycles connecting saddle-points.

Neurons perform computations, and convey the results of those computations through the statistical structure of their output spike trains. Here we present a practical method, grounded in the information-theoretic analysis of prediction, for inferring a minimal representation of that structure and for characterizing its complexity. Starting from spike trains, our approach finds their causal state models (CSMs), the minimal hidden Markov models or stochastic automata capable of generating statistically identical time series. We then use these CSMs to objectively quantify both the generalizable structure and the idiosyncratic randomness of the spike train. Specifically, we show that the expected algorithmic information content (the information needed to describe the spike train exactly) can be split into three parts describing (1) the time-invariant structure (complexity) of the minimal spike-generating process, which describes the spike train statistically; (2) the randomness (internal entropy rate) of the minimal spike-generating process; and (3) a residual pure noise term not described by the minimal spike-generating process. We use CSMs to approximate each of these quantities. The CSMs are inferred nonparametrically from the data, making only mild regularity assumptions, via the causal state splitting reconstruction algorithm. The methods presented here complement more traditional spike train analyses by describing not only spiking probability and spike train entropy, but also the complexity of a spike train's structure. We demonstrate our approach using both simulated spike trains and experimental data recorded in rat barrel cortex during vibrissa stimulation.