Time series: theory and methods
Time series: theory and methods
On the evaluation of first passage time densities for Gaussian processes
Signal Processing
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
Diffusion models of neuron activity
The handbook of brain theory and neural networks
Spiking Neuron Models: An Introduction
Spiking Neuron Models: An Introduction
Vectorized Simulations of Normal Processes for First Crossing-Time Problems
EUROCAST '97 Proceedings of the A Selection of Papers from the 6th International Workshop on Computer Aided Systems Theory
Simulation of Gaussian Processes and First Passage Time Densities Evaluation
EUROCAST '99 Proceedings on Computer Aided Systems Theory
Upcrossing first passage times for correlated gaussian processes
EUROCAST'05 Proceedings of the 10th international conference on Computer Aided Systems Theory
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This work is a contribution towards the understanding of certain features of mathematical models of single neurons. Emphasis is set on neuronal firing, for which the first passage time (FPT) problem bears a fundamental relevance. We focus the attention on modeling the change of the neuron membrane potential between two consecutive spikes by Gaussian stochastic processes, both of Markov and of non-Markov types. Methods to solve the FPT problems, both of a theoretical and of a computational nature, are sketched, including the case of random initial values. Significant similarities or diversities between computational and theoretical results are pointed out, disclosing the role played by the correlation time that has been used to characterize the neuronal activity. It is highlighted that any conclusion on this matter is strongly model-dependent. In conclusion, an outline of the asymptotic behavior of FPT densities is provided, which is particularly useful to discuss neuronal firing under certain slow activity conditions.