Mutual Information and Parameter Estimation in the Generalized Inverse Gaussian Diffusion Model of Cortical Neurons

We propose a novel model of postsynaptic potential buildup and action potential upswing of cortical neurons using the generalized inverse Gaussian (GIG) diffusion of Barndorff-Nielsen, Blaesild, and Halgreen. A certain subset of GIG diffusions exhibit attraction to a threshold, a phenomenon widely observed in biological neurons. The parameters in this model control the variance, components of the drift, and the height of threshold. These parameters can be estimated from a sample path of the GIG diffusion using a modification of the maximum likelihood estimator from the Euler method approximation. The input intensity of the cortical neuron is also estimated using similar methods. We use Monte Carlo simulations to evaluate the effectiveness of the estimators. Of interest to information theorists and neuroscientists is the mutual information between the input intensity of a cortical neuron and the time of firing of its output spike in response to that input intensity. We determine said mutual information for the GIG diffusion model of the neuron.