Estimation of local dependence graphs via Hawkes processes to unravel functional neuronal connectivity (2/2)

Sensory perception emerges from the coordinated activity of many cortical neurons that individually respond to selected features of a stimulus. While their discharge is primarily driven by feedforward inputs from the periphery, each cortical neuron might contribute to the responsiveness of the surrounding neurons of the network and thus to the formation of a perception. Indeed, feedforward inputs comprise only 10-15% of the anatomical inputs onto a given cortical neuron. What are the other 90% of the inputs doing? How do they shape the neuronal response to sensory stimulation? Answering these questions not only implies to simultaneously record multiple neuronal ensemble activities but also to develop new analysis methods to quantify the dependencies among the recorded neurons.

The first part of the seminar aims to review some of the current techniques that allow a time-resolved analysis of the correlation dynamics between pair of neurons and the few available methods, which might deal with multiple neurons.

In a second step, we will present an adaptation of the Least Absolute Shrinkage and Selection Operator “LASSO” method to the analysis of correlation dynamics of small neuronal populations. Indeed, due to its low computational cost, Lasso is an attractive regularization method for high-dimensional statistical settings. Within our framework, we consider multivariate counting processes depending on an unknown function parameter to be estimated by linear combinations of a fixed dictionary. To select coefficients, we propose an adaptive l-penalization methodology, where data-driven weights of the penalty are derived from new Bernstein type inequalities for martingales. Oracle inequalities are established under assumptions on the Gram matrix of the dictionary. This method is then applied to Hawkes processes as model for spike train analysis. The estimation allows us to recover the functional underlying connectivity as the local dependence graph that has been estimated. Simulations and real data analysis show the excellent performances of our method in practice.