A study of dependency features of spike trains through copulas.

Despite the progresses of statistical and machine learning techniques, simultaneous recordings from many neurons hide important information and the connections characterizing the network remain generally undiscovered. Discerning the presence of direct links between neurons from data is still a not completely solved problem. We propose the use of copulas, to enlarge the number of tools for detecting the network structure, pursuing on a research direction we started in Sacerdote et al.

The Gamma renewal process as an output of the diffusion leaky integrate-and-fire neuronal model.

Statistical properties of spike trains as well as other neurophysiological data suggest a number of mathematical models of neurons. These models range from entirely descriptive ones to those deduced from the properties of the real neurons. One of them, the diffusion leaky integrate-and-fire neuronal model, which is based on the Ornstein-Uhlenbeck (OU) stochastic process that is restricted by an absorbing barrier, can describe a wide range of neuronal activity in terms of its parameters.

On firing rate estimation for dependent interspike intervals.

If interspike intervals are dependent, the instantaneous firing rate does not catch important features of spike trains. In this case, the conditional instantaneous rate plays the role of the instantaneous firing rate for the case of samples of independent interspike intervals. If the conditional distribution of the interspikes intervals (ISIs) is unknown, it becomes difficult to evaluate the conditional firing rate.

Cooperative behavior in a jump diffusion model for a simple network of spiking neurons.

The distribution of time intervals between successive spikes generated by a neuronal cell --the interspike intervals (ISI)-- may reveal interesting features of the underlying dynamics. In this study we analyze the ISI sequence --the spike train-- generated by a simple network of neurons whose output activity is modeled by a jump-diffusion process. We prove that, when specific ranges of the involved parameters are chosen, it is possible to observe multimodal ISI distributions which reveal that the modeled network fires with more than one single preferred time interval.

Joint distribution of first exit times of a two dimensional Wiener process with jumps with application to a pair of coupled neurons.

Motivated by a neuronal modeling problem, a bivariate Wiener process with two independent components is considered. Each component evolves independently until one of them reaches a threshold value. If the first component crosses the threshold value, it is reset while the dynamics of the other component remains unchanged.

On dependency properties of the ISIs generated by a two-compartmental neuronal model.

One-dimensional leaky integrate and fire neuronal models describe interspike intervals (ISIs) of a neuron as a renewal process and disregarding the neuron geometry. Many multi-compartment models account for the geometrical features of the neuron but are too complex for their mathematical tractability. Leaky integrate and fire two-compartment models seem a good compromise between mathematical tractability and an improved realism.

Detecting dependencies between spike trains of pairs of neurons through copulas.

The dynamics of a neuron are influenced by the connections with the network where it lies. Recorded spike trains exhibit patterns due to the interactions between neurons. However, the structure of the network is not known.

Leaky integrate and fire models coupled through copulas: association properties of the interspike intervals.

We propose a model able to describe the Interspike Intervals of two or more neurons subject to common inputs from the network. The single neuron dynamic is described through a classical Leaky Integrate and Fire model, but the model also catches the joint behavior of two neurons resorting to the use of copulas. Copulas are mathematical objects largely used to describe dependencies laws.

How sample paths of leaky integrate-and-fire models are influenced by the presence of a firing threshold.

Neural membrane potential data are necessarily conditional on observation being prior to a firing time. In a stochastic leaky integrate-and-fire model, this corresponds to conditioning the process on not crossing a boundary. In the literature, simulation and estimation have almost always been done using unconditioned processes.

A copulas approach to neuronal networks models.

Simultaneous recordings from groups of neurons request to improve models. To switch from single unit description to multivariate models describing the coding activity of two or more neurons, we propose to use the copula notion. This mathematical object catches the coupling properties and allows a mathematical description of the dependencies between two or more random variables.

Errors in estimation of the input signal for integrate-and-fire neuronal models.

Estimation of the input parameters of stochastic (leaky) integrate-and-fire neuronal models is studied. It is shown that the presence of a firing threshold brings a systematic error to the estimation procedure. Analytical formulas for the bias are given for two models, the randomized random walk and the perfect integrator.

Optimum signal in a diffusion leaky integrate-and-fire neuronal model.

An optimum signal in the Ornstein-Uhlenbeck neuronal model is determined on the basis of interspike interval data. Two criteria are proposed for this purpose. The first, the classical one, is based on searching for maxima of the slope of the frequency transfer function.

Effect of periodic stimulus on a neuronal diffusion model with signal-dependent noise.

To relate the noise intensity with a periodically modulated input signal in a single neuron stochastic model we introduce a diffusion model with both time modulated drift and diffusion coefficient. Such a model is the continuous version of a Stein model with time oscillating frequencies for the Poisson processes describing the inputs impinging on the neuron. We focus here on some aspects of the resonance phenomenon for such a model.

Mean instantaneous firing frequency is always higher than the firing rate.

Frequency coding is considered one of the most common coding strategies employed by neural systems. This fact leads, in experiments as well as theoretical studies, to construction of so-called transfer functions, where the output firing frequency is plotted against the input intensity. The term "firing frequency" can be understood differently in different contexts.

Interspike interval statistics in the Ornstein-Uhlenbeck neuronal model with signal-dependent noise.

The stochastic leaky integrate-and-fire (LIF) continuous model is studied under the condition that the amplitude of noise is a function of the input signal. The coefficient of variation (CV) of interspike intervals (ISIs) is investigated for different types of dependencies between the noise and the signal. Finally, we present the CV and the ISI density resulting from the special choice of parameters of the input that gave rise to a contra-intuitive behavior of the transfer function in Lánský and Sacerdote [Phys.

Effects of random jumps on a very simple neuronal diffusion model.

The effects of taking into account in a perfect integrate and fire model of neuronal activity the spatial localization of the synapses are studied by superposing to the diffusion a simple discrete jump component. Different criteria are employed to assess the role of excitatory and inhibitory discrete contributions. Comparisons are performed with respect to the case where contributions coming from synapses more distal from the trigger zone are summed up in a continuous model.