Chaotic dynamics from interspike intervals

Characterization of the chaos-hyperchaos transition based on return times

We discuss the problem of the detection of hyperchaotic oscillations in coupled nonlinear systems when the available information about this complex dynamical regime is very limited. We demonstrate the ability of diagnosing the chaos-hyperchaos transition from return times into a Poincaré section and show that an appropriate selection of the secant plane allows a correct estimation of two positive Lyapunov exponents (LEs) from even a single sequence of return times. We propose a generalized approach for extracting dynamics from point processes that allows avoiding spurious identification of the dynamical regime caused by artifacts. The estimated LEs are nearly close to their expected values if the second positive LE is essentially different from the largest one. If both exponents become nearly close, an underestimation of the second LE may be obtained. Nevertheless, distinctions between chaotic and hyperchaotic regimes are clearly possible.

Quantifying chaotic dynamics from integrate-and-fire processes

Characterizing chaotic dynamics from integrate-and-fire (IF) interspike intervals (ISIs) is relatively easy performed at high firing rates. When the firing rate is low, a correct estimation of Lyapunov exponents (LEs) describing dynamical features of complex oscillations reflected in the IF ISI sequences becomes more complicated. In this work we discuss peculiarities and limitations of quantifying chaotic dynamics from IF point processes. We consider main factors leading to underestimated LEs and demonstrate a way of improving numerical determining of LEs from IF ISI sequences. We show that estimations of the two largest LEs can be performed using around 400 mean periods of chaotic oscillations in the regime of phase-coherent chaos. Application to real data is discussed.

A method for determining neural connectivity and inferring the underlying network dynamics using extracellular spike recordings

In the present paper we propose a novel method for the identification and modeling of neural networks using extracellular spike recordings. We create a deterministic model of the effective network, whose dynamic behavior fits experimental data. The network obtained by our method includes explicit mathematical models of each of the spiking neurons and a description of the effective connectivity between them. Such a model allows us to study the properties of the neuron ensemble independently from the original data. It also permits to infer properties of the ensemble that cannot be directly obtained from the observed spike trains. The performance of the method is tested with spike trains artificially generated by a number of different neural networks.

Analysis of the noise-induced bursting-spiking transition in a pancreatic beta-cell model

A stochastic model of the electrophysiological behavior of the pancreatic beta cell is studied, as a paradigmatic example of a bursting biological cell embedded in a noisy environment. The analysis is focused on the distortion that a growing noise causes to the basic properties of the membrane potential signals, such as their periodic or chaotic nature, and their bursting or spiking behavior. We present effective computational tools to obtain as much information as possible from these signals, and we suggest that the methods could be applied to real time series. Finally, a universal dependence of the main characteristics of the membrane potential on the size of the considered cell cluster is presented.