Numerical methods for stochastic processes
Cao Y, Li H and Petzold, L Efficient formulation of the stochastic simulation algorithm for chemically reacting systems. J Chem Phys, Vol. 121, No. 9. (1 September 2004), pp. 4059-4067.
Summary of various stochastic simulation algorithms for chemical systems and a discussion of their efficiency.
Kim T. Blackwell An efficient stochastic diffusion algorithm for modeling second messengers in dendrites and spines Journal of Neuroscience Methods Volume 157, Issue 1, 15 October 2006, Pages 142-153
Reaction-diffusion systems using a tau-leap style method and lookup tables.
ABSTRACT: Intracellular signaling pathways, which encompass both biochemical reactions and second messenger diffusion, interact non-linearly with neuronal membrane properties in their role as essential intermediaries for synaptic plasticity and neuromodulation. Computational modeling is a productive approach for investigating these phenomena; however, most current strategies for modeling neurons exclude signaling pathways. To overcome this deficiency, a new algorithm is presented to simulate stochastic diffusion in a highly efficient manner. The gain in speed is obtained by considering collections of molecules, instead of tracking the movement of individual molecules. The probability of a molecule leaving a spatially discrete compartment is used to create a lookup table that stores the probability of km molecules leaving the compartment as a function of the total number of molecules in the compartment. During the simulation, the number of molecules leaving the compartment is determined using a uniform random number as an index into the lookup table. Simulations illustrate the accuracy of this algorithm by comparing it with the theoretical solution for deterministic diffusion. Additional simulations show how spines on a dendritic branch compartmentalize diffusible molecules. The efficiency of the algorithm is sufficient to allow simulation of second messenger pathways in a multitude of spines on an entire neuron.
Voltage propagation on trees
Hines, M. Efficient computation of branched nerve equations. (PDF) Int. J. Bio-Med. Comput. 15:69-76, 1984
Describes the differenceing scheme used in Neuron, and in particular, the efficient voltage propagation method.