EFFICIENT INFERENCE OF FLEXIBLE INTERACTION IN SPIKING-NEURON NETWORKS

Abstract

Hawkes process provides an effective statistical framework for analyzing the timedependent interaction of neuronal spiking activities. Although utilized in many real applications, the classic Hawkes process is incapable of modelling inhibitory interactions among neurons. Instead, the nonlinear Hawkes process allows for a more flexible influence pattern with excitatory or inhibitory interactions. In this paper, three sets of auxiliary latent variables (Pólya-Gamma variables, latent marked Poisson processes and sparsity variables) are augmented to make functional connection weights in a Gaussian form, which allows for a simple iterative algorithm with analytical updates. As a result, an efficient expectationmaximization (EM) algorithm is derived to obtain the maximum a posteriori (MAP) estimate. We demonstrate the accuracy and efficiency performance of our algorithm on synthetic and real data. For real neural recordings, we show our algorithm can estimate the temporal dynamics of interaction and reveal the interpretable functional connectivity underlying neural spike trains.

1. INTRODUCTION

One of the most important tracks in neuroscience is to examine the neuronal activity in the cerebral cortex under varying experimental conditions. Recordings of neuronal activity are represented through a series of action potentials or spike trains. The transmitted information and functional connection between neurons are considered to be primarily represented by spike trains (Kass et al., 2014; Kass & Ventura, 2001; Brown et al., 2004; 2002) . A spike train is a sequence of recorded times at which a neuron fires an action potential and each spike may be considered to be a timestamp. Spikes occur irregularly both within and across multiple trials, so it is reasonable to consider a spike train as a point process with the instantaneous firing rate being the intensity function of point processes (Perkel et al., 1967; Paninski, 2004; Eden et al., 2004 ). An example of spike trains for multiple neurons is shown in Fig. 2a in the real data experiment. Despite many existing applications, the classic point process models, e.g., Poisson processes, neglect the time-dependent interaction within one neuron and between multiple neurons, so fail to capture the complex temporal dynamics of a neural population. In contrast, Hawkes process is one type of point processes which is able to model the self-exciting interaction between past and future events. Existing applications cover a wide range of domains including seismology (Ogata, 1998; 1999 ), criminology (Mohler et al., 2011;; Lewis et al., 2012 ), financial engineering (Bacry et al., 2015; Filimonov & Sornette, 2015) and epidemics (Saichev & Sornette, 2011; Rizoiu et al., 2018) . Unfortunately, due to the linearly additive intensity, the vanilla Hawkes process can only represent the purely excitatory interaction because a negative firing rate may exist with inhibitory interaction. This makes the vanilla version inappropriate in the neuroscience domain where the influence between neurons is a mixture of excitation and inhibition (Maffei et al., 2004; Mongillo et al., 2018) . In order to reconcile Hawkes process with inhibition, various nonlinear Hawkes process variants are proposed to allow for both excitatory and inhibitory interactions. The core point of nonlinear Hawkes process is a nonlinearity which maps the convolution of the spike train with a causal influential kernel to a nonnegative conditional intensity, such as rectifier (Reynaud-Bouret et al., 2013), 

