Chapter 2. Introduction to a random walk simulation for amperometric and cyclic voltammetry recordings of evoked dopamine overflow in brain slices

In this second lesson, we add two new concepts that help adapt the previous quantal release model to experiments in the brain, where there is release from many sites. One is the action of uptake transporters, in this example the dopamine uptake transporter, which removes dopamine from the extracellular milieu to repackage in the neuron. Thus, we have to worry about a decrease in the signal from consuming electrodes (in amperometry) that oxidize the transmitter, diffusion, and reuptake. CP caudate putamen, cc corpus callosum

Schematic drawing: Dopamine overflow is evoked in striatal slices by a bipolar stimulation electrode. The recording carbon fiber electrode is positioned at a distance of ~ 50-100 µm from both poles of the stimulation electrode. Dopamine overflow is measured with either cyclic voltammetry, for which a triangular voltage wave (-400 mV to +1000 mV) is applied to the electrode at 10 Hz, or with continuous amperometry, for which a constant voltage of +400 mV is applied. The two methods yield recordings of dopamine overflow with very different kinetics. However, both can be simulated with the model described below.

The essentials of diffusion can be characterized by a one-dimensional random walk (Berg, 1983). A spreadsheet modeling diffusion in one dimension over time can be made with columns representing distance and rows representing time intervals. Since the vectors of diffusion are random, for a population of molecules diffusing in one dimension, one half will move to the right and one half will move to the left. The bins in a row t (x+1) equal therefore the average of the two neighboring bins in row t (x).The time steps are calculated (t=x^2/2D) for the diffusion coefficient D for dopamine in brain tissue: 2.7*10-6 cm^2/s (Nicholson, 1996). For 0.5 µm bins a time step is 0.463 ms. (See also chapter 1.)

The first row in the spreadsheets below represents the initial dopamine concentration in µM after stimulation. The middle column represents the electrode surface (located in the center of the stimulated field, represented by the orange doughnut in the schematic drawing above). An important parameter that determines the kinetics of the recorded signals is a dead space around the electrode where no dopamine release occurs (represented by the inner blue circle in the figure above). For cyclic voltammetry, a reflecting electrode surface is modeled, assuming that dopamine molecules are regenerated during the reducing voltage scan. The columns next to the electrode (in blue) are determined using the same formula as the other columns in the spreadsheet.

For amperometry, a consuming surface is modeled, assuming that dopamine molecules are oxidized at the electrode surface and that the oxidation product is not reduced back to dopamine. The bin in the column next to the electrode (in red) in row t (x+1) receives only half of the molecules from its other neighboring bin in row t (x), and no molecules from the bin in the 'electrode column'.

(Edges: We chose to model a diffusing edge: of the initial molecules 50% fall off the edge, but 50% of those return during the next iteration, and so on. This process reaches a limit, so that 71% of the molecules that reach the edge eventually return. However, it should be noted that if the release area is sufficiently large, the way the edge is modeled does not play an important role.)

Dopamine uptake

To include dopamine uptake in the simulation, the dopamine concentration is corrected after each iteration by subtracting the amount of dopamine that is taken up during a time step according to the Michaelis-Menten equation:

d[DA]/dt=Vmax*[DA]/(Km+[DA]

with the maximal uptake rate Vmax (µM/s), the apparent affinity Km (µM), and the dopamine concentration [DA] (µM) 