title: On the stability and dynamics of stochastic spiking neuron models: Nonlinear Hawkes process and point process GLMs.
abstract: 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.
authors.affiliation: Department of Neuroscience, Brown University, Providence, Rhode Island, United States of America.
authors.forename: Felipe
authors.identifier: 0000-0002-1733-3229
authors.initials: F
authors.lastname: Gerhard
authors.affiliation: School of Computer and Communication Sciences and School of Life Sciences, Brain Mind Institute, École polytechnique fédérale de Lausanne (EPFL), Lausanne, Switzerland.|Institute for Zoology, Faculty of Mathematics and Natural Sciences, University of Cologne, Cologne, Germany.
authors.forename: Moritz
authors.identifier: 0000-0002-2775-2611
authors.initials: M
authors.lastname: Deger
authors.affiliation: Department of Neuroscience, Brown University, Providence, Rhode Island, United States of America.|Institute for Brain Science, Brown University, Providence, Rhode Island, United States of America.|Center for Neurorestoration & Neurotechnology, U. S. Department of Veterans Affairs, Providence, Rhode Island, United States of America.
authors.forename: Wilson
authors.initials: W
authors.lastname: Truccolo
mesh_terms: D000200:Action Potentials / Q000502:physiology*; D000818:Animals; D001683:Biological Clocks / Q000502:physiology*; D003198:Computer Simulation; D006801:Humans; D016014:Linear Models; D008959:Models, Neurological; D015999:Multivariate Analysis; D009474:Neurons / Q000502:physiology*; D017711:Nonlinear Dynamics; D012815:Signal Processing, Computer-Assisted; D013269:Stochastic Processes