Estimating input parameters from intracellular recordings in the Feller neuronal model

We study the estimation of the input parameters in a Feller neuronal model from a trajectory of the membrane potential sampled at discrete times. These input parameters are identified with the drift and the infinitesimal variance of the underlying stochastic diffusion process with multiplicative noise. The state space of the process is restricted from below by an inaccessible boundary. Further, the model is characterized by the presence of an absorbing threshold, the first hitting of which determines the length of each trajectory and which constrains the state space from above. We compare, both in the presence and in the absence of the absorbing threshold, the efficiency of different known estimators. In addition, we propose an estimator for the drift term, which is proved to be more efficient than the others, at least in the explored range of the parameters. The presence of the threshold makes the estimates of the drift term biased, and two methods to correct it are proposed

Stationary distributions of systems with discreteness-induced transitions

We provide a theoretical analysis of some autocatalytic reaction networks exhibiting the phenomenon of discreteness-induced transitions. The family of networks that we address includes the celebrated Togashi and Kaneko model. We prove positive recurrence, finiteness of all moments and geometric ergodicity of the models in the family. For some parameter values, we find the analytic expression for the stationary distribution and discuss the effect of volume scaling on the stationary behaviour of the chain. We find the exact critical value of the volume for which discreteness-induced transitions disappear

Motoneuron membrane potentials follow a time inhomogeneous jump diffusion process

Stochastic leaky integrate-and-fire models are popular due to their simplicity and statistical tractability. They have been widely applied to gain understanding of the underlying mechanisms for spike timing in neurons, and have served as building blocks for more elaborate models. Especially the Ornstein–Uhlenbeck process is popular to describe the stochastic fluctuations in the membrane potential of a neuron, but also other models like the square-root model or models with a non-linear drift are sometimes applied. Data that can be described by such models have to be stationary and thus, the simple models can only be applied over short time windows. However, experimental data show varying time constants, state dependent noise, a graded firing threshold and time-inhomogeneous input. In the present study we build a jump diffusion model that incorporates these features, and introduce a firing mechanism with a state dependent intensity. In addition, we suggest statistical methods to estimate all unknown quantities and apply these to analyze turtle motoneuron membrane potentials. Finally, simulated and real data are compared and discussed. We find that a square-root diffusion describes the data much better than an Ornstein–Uhlenbeck process with constant diffusion coefficient. Further, the membrane time constant decreases with increasing depolarization, as expected from the increase in synaptic conductance. The network activity, which the neuron is exposed to, can be reasonably estimated to be a threshold version of the nerve output from the network. Moreover, the spiking characteristics are well described by a Poisson spike train with an intensity depending exponentially on the membrane potential

Cogarch models: A statistical application

One of the reason that suggests to use COGARCH models to fit financial log-return data is due to the fact that they are able to capture the so called stylized facts observed in real data: uncorrelated log-returns but correlated absolute log-return, time varying volatility, conditional heteroscedasticity, cluster in volatility, heavy tailed and asymmetric unconditional distributions, leverage effects. The aims of this paper is to fit the COGARCH models to a real financial data set, estimate the parameters of the models via the prediction based estimating functions and to look at the performance of these estimates

The Ornstein-Uhlenbeck process as a model for filtered white noise

The Ornstein–Uhlenbeck process is presented with its main mathematical properties and with original results on the first crossing times in the case of two threshold barriers. The interpretation of filtered white noise, its stationary spectrum and Allan variance are also presented for ease of use in the time and frequency metrology field. An improved simulation scheme for the evaluation of first passage times between two barriers is also introduced