Inference of functional connectivity in Neurosciences via Hawkes processes

Abstract

We use Hawkes processes as models for spike trains analysis. A new Lasso method designed for general multivariate counting processes [1] enables us to estimate the functional connectivity graph between the different recorded neurons.

Goodness-of-Fit Tests and Nonparametric Adaptive Estimation for Spike Train Analysis

“…On simulated data (Fig. 8), ${\stackrel{falseˆ}{f}}^{\mathrm{B}}$ and ${\stackrel{falseˆ}{f}}^{\mathrm{BO}}$ recover the support of the interaction functions and also find that $h_1^{false(2false)}=0$, which could not have been possible with a classical least-square estimate (see also more comments on the functional connectivity graph in [58]). Moreover, ${\stackrel{falseˆ}{f}}^{\mathrm{BO}}$ is less biased than ${\stackrel{falseˆ}{f}}^{\mathrm{B}}$ as expected.…”

“…The matrix $false({\sum }_{i=1}^n{\mathbf{G}}^{false(ifalse)}false)$ cumulates all these features (and also the fixed effect due to the spontaneous parameter, which needs to be subtracted) and inverting it enables us to find an estimate of the true interactions. See also [58] for a more visual transcription. …”

“…Therefore, it is difficult to interpret the resulting estimate in terms of functional connectivity graph [58]. Moreover, if we wish to capture all the features, it is preferable to take A large and δ small.…”

Semi-Analytic Nonparametric Bayesian Inference for Spike-Spike Neuronal Connectivity

“…While most of the existing literature based on Hawkes processes assumes a simple parametrization for the response functions, several new studies introduced the use of nonparametric methods [21][22][23]. Hawkes processes have been successfully used in neuroscience settings in order to infer spike-spike causal connectivity [24,25]. In a Hawkes process the spike density of the j-th unit is a linear functional of the spike sequences of the other units:…”

Locally stationary Hawkes processes

“…Intuitively similar to a Poisson process, the conditional intensity function of a Hawkes process is however stochastic as it depends on its own past events. Whereas Hawkes' model was introduced to reproduce the ripple effects generated after the occurrence of an earthquake, applications of this model have become since then really numerous in many and diverse fields such as seismology (see e.g., [5], for a recent review), biology ([6] on genome analysis) or neuroscience ( [7] on brain data analysis), to name but a few. Recently, this model is also being widely used in finance where self-exciting processes led to many applications such as microstructure dynamics ( [8]), order arrival rate modelling and high-frequency data ( [9], [10], [11]), financial price modelling across scales ( [12]), and many others.…”