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 construction of a mathematical model are explained. The model, which has been described previously, has three components, including the sensory dendrite on which are found the receptor proteins themselves, and others consisting of a passive cable leading to a trigger zone and axon. In the present paper, we pursue an analytical approach for determining the change in time of the receptor potential in the important case of a subthreshold 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 the solution for the entire length of the nerve cell. In the case of finite cables the solution is obtained as an infinite series which is shown to converge and can be easily used to find the depolorization at all space and time points of interest. A steady-state result is obtained directly by solving the relevant ordinary differential equation. For a semi-infinite cable an explicit expression is found for the voltage as 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 solutions and employed to investigate the effect of varying the position of the trigger zone and the electronic length of the neuron.