Universität zu Köln

Gerhard, Felipe, Deger, Moritz ORCID: 0000-0002-2775-2611 and Truccolo, Wilson (2017). On the stability and dynamics of stochastic spiking neuron models: Nonlinear Hawkes process and point process GLMs. PLoS Comput. Biol., 13 (2). SAN FRANCISCO: PUBLIC LIBRARY SCIENCE. ISSN 1553-7358

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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 PPGLM 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.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Gerhard, Felipe UNSPECIFIED UNSPECIFIED UNSPECIFIED Deger, Moritz UNSPECIFIED orcid.org/0000-0002-2775-2611 UNSPECIFIED Truccolo, Wilson UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-241919
DOI:	10.1371/journal.pcbi.1005390
Journal or Publication Title:	PLoS Comput. Biol.
Volume:	13
Number:	2
Date:	2017
Publisher:	PUBLIC LIBRARY SCIENCE
Place of Publication:	SAN FRANCISCO
ISSN:	1553-7358
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language TIME-RESCALING THEOREM; CONNECTIVITY; VARIABILITY; ADAPTATION; INFERENCE; ENSEMBLE; SPECTRA Multiple languages Biochemical Research Methods; Mathematical & Computational Biology Multiple languages
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/24191