Modeling and Simulation Methods of Neuronal Populations and Neuronal Networks

Abstract

This thesis presents numerical methods and modeling related to simulating neurons. Two approaches to the simulation are taken: a population density approach and a neuronal network approach.

The first two chapters present the results from the population density approach and its applications. The population density approach assumes that each neuron can be identified by its states (e.g., membrane potential, conductance of ion channels). Additionally, it assumes the population is large such that it can be approximated by a continuous population density distribution in the state space. By updating this population density, we can learn the macroscopic behavior of the population, such as the average firing rate and average membrane potential. The Population density approach avoids the need to simulate every single neuron when the population is large. While many previous population-density methods, such as the mean-field method, make further simplifications to the models, we developed the Asymmetric Particle Population Density (APPD) method to simulate the population density directly without the need to simplify the dynamics of the model. This enables us to simulate the macroscopic properties of coupled neuronal populations as accurately as a direct simulation.

The APPD method tracks multiple asymmetric Gaussians as they advance in time due to a convection-diffusion equation, and our main theoretical innovation is deriving this update algorithm by tracking a level set. Tracking a single Gaussian is also applicable to the Bayesian filtering for continuous-discrete systems. By adding a measurement-update step, we reformulated our tracking method as the Level Set Kalman Filter(LSKF) method and find that it offers greater accuracy than state-of-the-art methods.

Chapter IV presents the methods for direct simulation of a neuronal network. For this approach, the aim is to build a high-performance and expandable framework that can be used to simulate various neuronal networks. The implementation is done on GPUs using CUDA, and this framework enables simulation for millions of neurons on a high-performance desktop computer. Additionally, real-time visualization of neuron activities is implemented. Pairing with the simulation framework, a detailed mouse cortex model with experiment-determined morphology using the CUBIC-Atlas, and neuron connectome information from Allen's brain atlas is generated.