Neural dependency structures: mathematical models and statistical methods

Neural information processing is a challenging topic. Mathematicians, physicists, biologists and computer scientists have devoted important eorts to the study of this subject since the second half of the last century. However, despite important improvements in our knowledge, we are still far from a complete comprehension of the problem. Many experimental data show that one of the primary ingredients of neural information processing is the dependency structure between the involved variables. However many classical mathematical neural models and the associated statistical tools for their analysis are typically based on independence assumptions. Actually the independence hypothesis often makes a model simpler and mathematically tractable, but also farther from the real nature of the problem. To improve our knowledge of the features related to dependency properties, new models should be proposed. Furthermore specic methods for the study of dependency between the variables involved should be developed. The aim of this thesis is to give a contribution to this subject. In particular we consider a two-compartment neural model. It accounts for the interaction between dierent parts of the nerve cell and seems to be a good compromise between mathematical tractability and an improved realism. Then we develop suitable mathematical methods for the statistical analysis of this model as well as a method to estimate the neural ring rate in the presence of dependence.