Fast algorithms for simulation of neuronal dynamics based on the bilinear dendritic integration rule

We aim to develop fast algorithms for neuronal simulations to capture the dynamics of a neuron with realistic dendritic morphology. To achieve this, we perform the asymptotic analysis on a cable neuron model with branched dendrites. Using the second-order asymptotic solutions, we derive a bilinear dendritic integration rule to characterize the voltage response at the soma when receiving multiple spatiotemporal synaptic inputs from dendrites, with a dependency on the voltage state of the neuron at input arrival times. Based on the derived bilinear rule, we finally propose two fast algorithms and demonstrate numerically that, in comparison with solving the original cable neuron model numerically, the algorithms can reduce the computational cost of simulation for neuronal dynamics enormously while retaining relatively high accuracy in terms of both sub-threshold dynamics and firing statistics.