Population density methods for large-scale modelling of neuronal networks with realistic synaptic kinetics: cutting the dimension down to size

Population density methods provide promising time-saving alternatives to di rect Monte Carlo simulations of neuronal network activity, in which one tra cks the state of thousands of individual neurons and synapses. A population density method has been found to be roughly a hundred times faster than di rect simulation for various test networks of integrate-and-fire model neuro ns with instantaneous excitatory and inhibitory post-synaptic conductances. In this method, neurons are grouped into large populations of similar neur ons. For each population, one calculates the evolution of a probability den sity function (PDF) which describes the distribution of neurons over state space. The population firing rate is then given by the total flux of probab ility across the threshold voltage for firing an action potential. Extendin g the method beyond instantaneous synapses is necessary for obtaining accur ate results, because synaptic kinetics play an important role in network dy namics. Embellishments incorporating more realistic synaptic kinetics for t he 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, de creasing the computational speed advantage of the population density method over direct Monte Carlo simulation. We report here on a one-dimensional mo del of the PDF for neurons with arbitrary synaptic kinetics. The method is more accurate than the mean-held method in the steady state, where the mean -field approximation works best, and also under dynamic-stimulus conditions . The method is much faster than direct simulations. Limitations of the met hod are demonstrated, and possible improvements are discussed.