TIME-DEPENDENT SOLUTIONS FOR A CABLE MODEL OF AN OLFACTORY RECEPTOR NEURON

A mathematical model for an olfactory receptor neuron is investigated. The physiological and anatomical background required for the construc tion of a mathematical model are explained. The model, which has been described previously, has three components, including the sensory dend rite on which are found the receptor proteins themselves, and others c onsisting of a passive cable leading to a trigger zone and axon. In th e present paper, we pursue an analytical approach for determining the change in time of the receptor potential in the important case of a su bthreshold square pulse of odorant stimulation delivered uniformly at the sensory dendrite. Then, the input current increases in time to its asymptotic value. This latter condition means that we can use a Green 's function approach in order to obtain accurate representations for t he solution for the entire length of the nerve cell. In the case of fi nite cables the solution is obtained as an infinite series which is sh own to converge and can be easily used to find the depolarization at a ll space and time points of interest. A steady-state result is obtaine d directly by solving the relevant ordinary differential equation. For a semi-infinite cable an explicit expression is found for the voltage as a function of time and space variables involving a single integral . However, the exact expression follows from this for the steady-state result. The analytical results obtained are compared to numerical sol utions and employed to investigate the effect of varying the position of the trigger zone and the electronic length of the neuron. (C) 1996 Academic Press Limited