How Noisy Adaptation of Neurons Shapes Interspike Interval Histograms and Correlations

Channel noise is the dominant intrinsic noise source of neurons causing variability in the timing of action potentials and interspike intervals (ISI). Slow adaptation currents are observed in many cells and strongly shape response properties of neurons. These currents are mediated by finite populations of ionic channels and may thus carry a substantial noise component. Here we study the effect of such adaptation noise on the ISI statistics of an integrate-and-fire model neuron by means of analytical techniques and extensive numerical simulations. We contrast this stochastic adaptation with the commonly studied case of a fast fluctuating current noise and a deterministic adaptation current (corresponding to an infinite population of adaptation channels). We derive analytical approximations for the ISI density and ISI serial correlation coefficient for both cases. For fast fluctuations and deterministic adaptation, the ISI density is well approximated by an inverse Gaussian (IG) and the ISI correlations are negative. In marked contrast, for stochastic adaptation, the density is more peaked and has a heavier tail than an IG density and the serial correlations are positive. A numerical study of the mixed case where both fast fluctuations and adaptation channel noise are present reveals a smooth transition between the analytically tractable limiting cases. Our conclusions are furthermore supported by numerical simulations of a biophysically more realistic Hodgkin-Huxley type model. Our results could be used to infer the dominant source of noise in neurons from their ISI statistics

Active Brownian Particles. From Individual to Collective Stochastic Dynamics

We review theoretical models of individual motility as well as collective dynamics and pattern formation of active particles. We focus on simple models of active dynamics with a particular emphasis on nonlinear and stochastic dynamics of such self-propelled entities in the framework of statistical mechanics. Examples of such active units in complex physico-chemical and biological systems are chemically powered nano-rods, localized patterns in reaction-diffusion system, motile cells or macroscopic animals. Based on the description of individual motion of point-like active particles by stochastic differential equations, we discuss different velocity-dependent friction functions, the impact of various types of fluctuations and calculate characteristic observables such as stationary velocity distributions or diffusion coefficients. Finally, we consider not only the free and confined individual active dynamics but also different types of interaction between active particles. The resulting collective dynamical behavior of large assemblies and aggregates of active units is discussed and an overview over some recent results on spatiotemporal pattern formation in such systems is given.Comment: 161 pages, Review, Eur Phys J Special-Topics, accepte

Theory for serial correlations of interevent intervals

We consider stochastic systems with m internal states in which discrete events (e.g. hopping events between metastable states or firing events of neurons) occur at a state-dependent rate. Transitions between states are possible with certain fixed rates. Because the state immediately after an event depends in general on the history of the process, the intervals between two consecutive events (“residence times”) are correlated among each other, i.e. the residence time sequence constitutes a nonrenewal process. We construct a general kinetic scheme that accounts for the number of events at a given time. The count statistics is used to derive a general expression for the correlation coefficient of residence times with a certain lag. We apply the theoretical result to a simple neuron model with discrete threshold states leading to negative interspike interval correlations

Theory for serial correlations of interevent intervals

We consider stochastic systems with m internal states in which discrete events (e.g. hopping events between metastable states or firing events of neurons) occur at a state-dependent rate. Transitions between states are possible with certain fixed rates. Because the state immediately after an event depends in general on the history of the process, the intervals between two consecutive events (“residence times”) are correlated among each other, i.e. the residence time sequence constitutes a nonrenewal process. We construct a general kinetic scheme that accounts for the number of events at a given time. The count statistics is used to derive a general expression for the correlation coefficient of residence times with a certain lag. We apply the theoretical result to a simple neuron model with discrete threshold states leading to negative interspike interval correlations

When the leak is weak – how the first-passage statistics of a biased random walk can approximate the ISI statistics of an adapting neuron

Sequences of first-passage times can describe the interspike intervals (ISI) between subsequent action potentials of sensory neurons. Here, we consider the ISI statistics of a stochastic neuron model, a leaky integrate-and-fire neuron, which is driven by a strong mean input current, white Gaussian current noise, and a spike-frequency adaptation current. In previous studies, it has been shown that without a leak current, i.e. for a so-called perfect integrate-and-fire (PIF) neuron, the ISI density can be well approximated by an inverse Gaussian corresponding to the first-passage-time density of a biased random walk. Furthermore, the serial correlations between ISIs, which are induced by the adaptation current, can be described by a geometric series. By means of stochastic simulations, we inspect whether these results hold true in the presence of a modest leak current. Specifically, we measure mean and variance of the ISI in the full model with leak and use the analytical results for the perfect IF model to relate these cumulants of the ISI to effective values of the mean input and noise intensity of an equivalent perfect IF model. This renormalization procedure yields semi-analytical approximations for the ISI density and the ISI serial correlation coeffcient in the full model with leak. We find that both in the absence and the presence of an adaptation current, the ISI density can be well approximated in this way if the leak current constitutes only a weak modification of the dynamics. Moreover, also the serial correlations of the model with leak are well reproduced by the expressions for a PIF model with renormalized parameters. Our results explain, why expressions derived for the rather special perfect integrate-and-fire model can nevertheless be often well fit to experimental data