On the Stability of GLMs: Alteration of the Nonlinearity Governing Spike History Inputs

Abstract

Establishing a functional relationship between stimuli and neural response is a fundamental goal of sensory systems neuroscience. A standard method for determining this relationship is to build a probabilistic generative model and estimate its parameters by maximizing the likelihood of the data under the model. However, models that consider the spike history may lead to instability. This instability arises from the production of an unstable and high firing rate from positive feedback through a self-excitatory history filter. To address this issue, I present a method in which I alter the nonlinearity governing the spike history inputs, such that this allows for a model with realistic event rates. I then demonstrate the effectiveness of this altered nonlinearity in producing a stable spike-history dependent model by creating methods to simulate spiking models, obtaining model parameters through the maximization of the likelihood of the data under the model, and then simulating data with these fitted parameters. These methods allow for the investigation of model stability through examination of the simulated data and firing rates. Through this analysis I demonstrate the instability arising from the exponential nonlinearity as well as the stability arising from the modified softplus function as an altered nonlinearity. Model stability is important for spike train generation and for simulation of large-scale models of brain function. This project aids in the development of more robust models of spiking dynamics.