Population density methods for large-scale modeling of neuronal networks with realistic synaptic kinetics

Population density methods provide promising time-saving alternatives to direct Monte-Carlo simulations of neuronal network activity, in which one tracks the state of thousands of individual neurons and synapses. Previously Nykamp and Tranchina (2000a) implemented a new population density method introduced by Knight and colleagues (1996, 2000a, 2000b, 2000c). They found the approach to be roughly a hundred times faster than direct simulation for various test networks of integrate-and-fire model neurons with instantaneous excitatory and inhibitory post-synaptic conductances (Nykamp and Tranchina, 2000a). In this method, neurons are grouped into large populations of similar neurons. For each population, one calculates the evolution of a probability density function (PDF) which describes the distribution of neurons over state space. The population firing rate is then given by the total flux of probability across the threshold voltage for firing an action potential. Extending the method beyond instantaneous synapses is necessary for obtaining accurate results, since synaptic kinetics play an important role in network dynamics. Embellishments incorporating more realistic synaptic kinetics for the underlying neuron model increase the dimension of the PDF, which was one-dimensional in the instantaneous synapse case. This increase in dimension causes a substantial increase in computation time to find the exact PDF, decreasing the computational speed advantage of the population density method over direct Monte-Carlo simulation. In this thesis, I develop a one-dimensional model of the PDF for neurons with realistic synaptic kinetics. I then compare the results of the model to those from Monte-Carlo simulation of populations of individual integrate-and-fire neurons. Since the model is based on integrate-and-fire neuron dynamics, averaging these direct Monte-Carlo simulations over many trials with large population sizes is considered to provide correct, results. I also compare with a mean-field method and find the model to be more accurate than the mean-field method in the steady state, where the mean-field approximation works best, and also under dynamic-stimulus conditions. Limitations of the method and computational efficiencies are demonstrated, and possible improvements are discussed.