Una introducció a les matemàtiques de l'activitat neuronal

In this paper, we want to provide the reader a first approximation of the role
of mathematics in neuroscience, with a quick overview of the most classical
models and showing recent applications. We start presenting the biophysics
of the most known models of activity of a single neuron, generally using differential
equations, pointing out the implications of the main physiological properties
on the dynamics. While we emphasize on conductance-based models,
we mention as well the usefulness of more simplified models like the integrate
& fire or Morris-Lecar, and the existence of diffusion models like the cable
equation. The modeling of populations of neurons, highly demanding computationally
speaking, is illustrated in this paper through the description of two
models of primary visual cortex. The last part of the paper is devoted, as an
example of the use of mathematical tools different from differential equations,
to a short description of visual receptive fields by means of minimizing
certain functionals.