On the stability and dynamics of stochastic spiking neuron models: Nonlinear Hawkes process and point process GLMs

Point process generalized linear models (PP-GLMs) provide an important statistical framework for modeling spiking activity in single-neurons and neuronal networks. Stochastic stability is essential when sampling from these models, as done in computational neuroscience to analyze statistical properties of neuronal dynamics and in neuro-engineering to implement closed-loop applications. Here we show, however, that despite passing common goodness-of-fit tests, PP-GLMs estimated from data are often unstable, leading to divergent firing rates. The inclusion of absolute refractory periods is not a satisfactory solution since the activity then typically settles into unphysiological rates. To address these issues, we derive a framework for determining the existence and stability of fixed points of the expected conditional intensity function (CIF) for general PP-GLMs. Specifically, in nonlinear Hawkes PP-GLMs, the CIF is expressed as a function of the previous spike history and exogenous inputs. We use a mean-field quasi-renewal (QR) approximation that decomposes spike history effects into the contribution of the last spike and an average of the CIF over all spike histories prior to the last spike. Fixed points for stationary rates are derived as self-consistent solutions of integral equations. Bifurcation analysis and the number of fixed points predict that the original models can show stable, divergent, and metastable (fragile) dynamics. For fragile models, fluctuations of the single-neuron dynamics predict expected divergence times after which rates approach unphysiologically high values. This metric can be used to estimate the probability of rates to remain physiological for given time periods, e.g., for simulation purposes. We demonstrate the use of the stability framework using simulated single-neuron examples and neurophysiological recordings. Finally, we show how to adapt PP-GLM estimation procedures to guarantee model stability. Overall, our results provide a stability framework for data-driven PP-GLMs and shed new light on the stochastic dynamics of state-of-the-art statistical models of neuronal spiking activity.Author summary: Earthquakes, gene regulatory elements, financial transactions, and action potentials produced by nerve cells are examples of sequences of discrete events in space or time. In many cases, such events do not appear independently of each other. Instead, the occurrence of one event changes the rate of upcoming events (e.g, aftershocks following an earthquake). The nonlinear Hawkes process is a statistical model that captures these complex dependencies. Unfortunately, for a given model, it is hard to predict whether stochastic samples will produce an event pattern consistent with observations. In particular, with positive feedback loops, the process might diverge and yield unrealistically high event rates. Here, we show that an approximation to the mathematical model predicts dynamical properties, in particular, whether the model will exhibit stable and finite rates. In the context of neurophysiology, we find that models estimated from experimental data often tend to show metastability or even unstable dynamics. Our framework can be used to add constraints to data-driven estimation procedures to find the optimal model with realistic event rates and help to build more robust models of single-cell spiking dynamics. It is a first step towards studying the stability of large-scale nonlinear spiking neural network models estimated from data.