Physica A: Statistical Mechanics and its Applications
Simulation of miniature endplate potentials in neuromuscular junctions by using a cellular automaton☆
Introduction
The transmission of synaptic signals due to the diffusion of neurotransmitter molecules is one of the most important ways of communication between adjacent neurons in the nervous system. When one nerve impulse reaches the presynaptic membrane, it induces the release and diffusion of neurotransmitter molecules, initially contained inside synaptic vesicles. These molecules cross the cleft and bind receptor proteins on the postsynaptic membrane. While the binding lasts, each receptor opens an ionic channel which allows the flow of ions like Na+, K+, Cl−, Ca+, Mg+, etc, (depending on the channel). This flux increases or diminishes the postsynaptic potential, thus producing an action potential. Miniature potentials at neuromuscular junctions are produced by acetylcholine (ACh), which binds nicotinic receptors (NAChR). Around 10 000 ACh molecules are released by each vesicle [1].
Two ACh molecules bind an NAChR, open a channel and depolarize the membrane; its potential becomes more positive. After around 1 ms, the receptor releases both molecules and the channel closes. Then K+, Na+ and Cl− ions flow through non-gated (leakage) channels until the rest potential is completely restored [20]. In addition, acetylcholinesterase (AChE) clusters hydrolyze ACh molecules.
Sometimes, however, one or a few synaptic vesicles release their contents in the absence of presynaptic action potentials, generating small voltage spikes in the postsynaptic membrane called miniature endplate potentials (MEPP); for the case of the frog neuromuscular junction (NMJ) these are little depolarizations of around 0.7 mV and last between 3 and 5 ms [21]. Since their discovery by Fatt and Katz in 1952 [2], [3], MEPPs have shown the quantized nature of the neural transmission in chemical synapses [14], [23], [25]. In addition, they can show the effect of activation of a single kind of receptor (which is not always possible when looking at whole action potentials). Therefore, the understanding of how one MEPP is produced gives deep insight into the mechanism of the electrical signaling in the nervous system, i.e. the structure of presynaptic spontaneous activity in the neuromuscular junction. Despite the fact that they have been worked on for many years, some aspects of the generation mechanisms are still not completely understood [15], [24], [25], [26], [29]. Although some of these aspects can be analytically computed, many others, like the effect of inhomogeneities in the distribution of receptors or the consequences of the shape of the cleft, deserve numerical simulation.
The simulation of a process started by a reaction–diffusion process includes two elements: the diffusion equation should be properly reproduced; and the interactions (which may be non-linear and strongly correlated) must be included. For this last point it is very useful to be able to track the history of each single particle. The most relevant numerical approaches are those based on finite elements [5], [6] and finite differences [8], on one side; and Brownian Monte Carlo dynamics [9], on the other. The former are very successful in solving differential equations, but they ask for large computational resources and the tracking of each single particle is impossible. The latter allow the tracking of single particles (like in the MCell project [16]), but at the price of enormous computational effort. In contrast, cellular automata are excellent models for reproducing reaction–diffusion processes, in the absence of other external force [10]. In fact, they are very simple models that include both the diffusion equation and the tracking of each single particle. Actually, it has been proved [12] that they obey the exact diffusion–reaction differential equation in the macroscopic limit, and their results are quantitative in both space and time. Therefore, they are good candidates for simulating synaptic transmission.
This paper explores this possibility by constructing a cellular automaton to simulate a MEPP in a neuromuscular synapse. The model is based on a two-dimensional diffusion automaton that moves the ACh molecules, plus one NAChR receptor per cell that binds and releases those molecules. In addition, the action of AChE clusters is included as a probability of destroying ACh molecules. The model allows one to compute the number of open receptors against the time. Once this number is known, it is possible to compute the currents and voltages of the MEPP. In Section 2 we present a brief description of the CA model, including the rules of interaction between ACh molecules and NAChR receptors. This section also describes the spatial initial distribution that we assumed and the way to compute currents and voltages. Section 3 shows the results that we obtained for the time evolution of the number of open receptors, the current and the voltage signals for a single MEPP by using parameter values taken from experimental data. We present the conclusions in the last section.
Section snippets
The model
A cellular automaton (CA) is used for a type of simulation where both space and time are discrete. Our model is based on a 2D diffusion CA [10]; it consists of a flat square grid of square boxes each containing a single NAChR and with space for up to four ACh molecules (one in each direction: north, south, east, west). Since, in the cellular automata theory, the spatial grid is called a “lattice”, we will refer to it as a lattice from now on. The lattice constant, , i.e. the mean distance
Results
Fig. 2 shows the number of open receptors as a function of the time for several bound rate constants. The probability of ACh destruction by AChE has been fixed at 0.03456, which corresponds to the reported value of [5], [6]. With this probability, it was necessary to use bound rate constants that are ten times larger than the reported value, but the reason for this will be discussed in the conclusions.
It can be observed that the maximal number of open receptors occurs around
Conclusions
In order to explore the use of cellular automata in the simulation of synaptic transmission, we have developed a two-dimensional cellular automaton model to simulate the generation of a MEPP in the neuromuscular junction. Based on a diffusion cellular automaton, our model reproduces both the diffusion equation for ACh and the kinetics of the reaction between ACh molecules and nAChR receptors, and includes to some extent the effect of AChE clusters onto ACh.
When experimental data are used to
Acknowledgments
We are grateful to Dr. Leonardo de María and Prof. María Marcela Camacho for useful discussions and help in finding some experimental parameters. We also thank an anonymous referee for many useful suggestions and corrections. This work was supported by the National Research Division of the National University of Colombia (Grant No. 20101006180).
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This work was supported by the National Research Division of the National University of Colombia (Grant No. 2010100).