Stochastic processes and their rst passage times have been widely used to describe the membrane potential dynamics of single neurons and to reproduce neuronal spikes, respectively.However, cerebral cortex in human brains is estimated to contain 10-20 billions of neurons and each of them is connected to thousands of other neurons. The rst question is: how to model neural networks through stochastic processes? A multivariate Ornstein-Uhlenbeck process, obtained as a diffusion approximation of a jump process, is the proposed answer. Obviously, dependencies between neurons imply dependencies between their spike times. Therefore, the second question is: how to detect neural network connectivity from simultaneously recorded spike trains? Answering this question corresponds to investigate the joint distribution of sequences of rst passage times. A non-parametric method based on copulas is proposed. As a first step toward a theoretical analysis, a simplified framework with two neurons and their first spikes is considered. For computing the joint distribution of the passage times, theoretical and numerical results are provided.
Now imagine to observe neurons characterized by spontaneous generation of pikes. When a stimulus is applied to the network, the spontaneous rings may prevail and hamper detection of the effects of the stimulus. Therefore, the spontaneous rings cannot be ignored and the response latency has to be detected on top of a background signal. Everything becomes more dicult if the latencies are expressed as a sum of deterministic (absolute response latency) and stochastic (relative response latency) parts. The third question is:what is the response latency to the stimulus? Non-parametric and parametric estimators of the two components are proposed in a single neuron framework.