Building Logistic Spiking Neuron Models Using Analytical Approach

Spiking neuron models are inspired by biological neurons. They can simulate the neuronal activities of the mammalian brains, such as spiking (integrator) and periodic oscillation (resonator). A spiking neural network consisting of a cluster of spiking neurons can be used to simulate the collective dynamic behaviors of a brain neural network. This paper presents step-by-step analyses for the non-linear dynamics of mathematical spiking neuron models and sets forth a novel spiking model based on logistic function using an analytical approach. The logistic function is a well-known one-dimensional dynamical system and can generate spiking or periodic oscillation based on the system parameter. The novel spiking neural model is a combination of the integrate-and-fire and the quadratic integrate-and-fire neuron models with an added parameter to control the neural dynamics in order to generate stable, periodic, or chaotic neural behavior with flexibility. The analytical approach presented in this paper can be applied extensively to the design and analyses of multi-dimensional neuron models. The goal of this research project is to understand the dynamical behaviors of biological neurons in order to design biologically inspired spiking neuron model for building artificial intelligence, treating cognitive disorders, and advancing the scientific frontiers of brain research.