Revealing Spectrum Features of Stochastic Neuron Spike Trains

Detecting rhythmic spiking through the power spectra of point process model residuals

Objective

Oscillations figure prominently as neurological disease hallmarks and neuromodulation targets. To detect oscillations in a neuron's spiking, one might attempt to seek peaks in the spike train's power spectral density (PSD) which exceed a flat baseline. Yet for a non-oscillating neuron, the PSD is not flat: The recovery period ("RP", the post-spike drop in spike probability, starting with the refractory period) introduces global spectral distortion. An established "shuffling" procedure corrects for RP distortion by removing the spectral component explained by the inter-spike interval (ISI) distribution. However, this procedure sacrifices oscillation-related information present in the ISIs, and therefore in the PSD. We asked whether point process models (PPMs) might achieve more selective RP distortion removal, thereby enabling improved oscillation detection.

Approach

In a novel "residuals" method, we first estimate the RP duration (nr) from the ISI distribution. We then fit the spike train with a PPM that predicts spike likelihood based on the time elapsed since the most recent of any spikes falling within the preceding nr milliseconds. Finally, we compute the PSD of the model's residuals.

Main results

We compared the residuals and shuffling methods' ability to enable accurate oscillation detection with flat baseline-assuming tests. Over synthetic data, the residuals method generally outperformed the shuffling method in classification of true- versus false-positive oscillatory power, principally due to enhanced sensitivity in sparse spike trains. In single-unit data from the internal globus pallidus (GPi) and ventrolateral anterior thalamus (VLa) of a parkinsonian monkey -- in which alpha-beta oscillations (8-30 Hz) were anticipated -- the residuals method reported the greatest incidence of significant alpha-beta power, with low firing rates predicting residuals-selective oscillation detection.

Significance

These results encourage continued development of the residuals approach, to support more accurate oscillation detection. Improved identification of oscillations could promote improved disease models and therapeutic technologies.

Spike Spectra for Recurrences

In recurrence analysis, the τ-recurrence rate encodes the periods of the cycles of the underlying high-dimensional time series. It, thus, plays a similar role to the autocorrelation for scalar time-series in encoding temporal correlations. However, its Fourier decomposition does not have a clean interpretation. Thus, there is no satisfactory analogue to the power spectrum in recurrence analysis. We introduce a novel method to decompose the τ-recurrence rate using an over-complete basis of Dirac combs together with sparsity regularization. We show that this decomposition, the inter-spike spectrum, naturally provides an analogue to the power spectrum for recurrence analysis in the sense that it reveals the dominant periodicities of the underlying time series. We show that the inter-spike spectrum correctly identifies patterns and transitions in the underlying system in a wide variety of examples and is robust to measurement noise.

Stochastic Bifurcations and Excitement in the ZS-Model of a Thermochemical Reaction

The Zeldovich–Semenov model of the continuous stirred tank reactor with parametric random disturbances in temperature is considered. We study a phenomenon of noise-induced transformation of the equilibrium mode into the mixed-mode oscillatory stochastic regime with alternations between small and large amplitudes. In the parametric analysis of the stochastic excitement, we use the analytical method of confidence domains based on the stochastic sensitivity technique. Analyzing a mutual arrangement of the confidence ellipses and separatrices, we estimate the critical intensity of the noise that causes the excitation. The phenomena of stochastic P-bifurcations and coherence resonances are discovered and studied by probability density functions and the statistics of interspike intervals.

Analysis of Stochastic Generation and Shifts of Phantom Attractors in a Climate–Vegetation Dynamical Model

A problem of the noise-induced generation and shifts of phantom attractors in nonlinear dynamical systems is considered. On the basis of the model describing interaction of the climate and vegetation we study the probabilistic mechanisms of noise-induced systematic shifts in global temperature both upward (“warming”) and downward (“freezing”). These shifts are associated with changes in the area of Earth covered by vegetation. The mathematical study of these noise-induced phenomena is performed within the framework of the stochastic theory of phantom attractors in slow-fast systems. We give a theoretical description of stochastic generation and shifts of phantom attractors based on the method of freezing a slow variable and averaging a fast one. The probabilistic mechanisms of oppositely directed shifts caused by additive and multiplicative noise are discussed.