A stochastic simulator for axons, neurons, and networks
Ion channels are transmembrane proteins which selectively allow ions to cross the lipid bilayer. As is the case for almost any molecular process in biology, ion channels opening and closing are stochastic processes (Faisal, Selen, and Wolpert, 2008). When modelling neurons, it is often implicitly assumed that averaging large numbers of ion channels allows the elimination of the effects of individual channels’ randomness at higher levels (such as spike timing). The standard deviation of variability over the mean activity of a set of signalling molecules is inversely proportional to the square root of the number of involved molecules. In large cells, such as the squid giant axon which has been key in understanding neural mechanisms, the variability averages out quickly. In comparison, neuronal connections in our brains can be over 10000 times smaller. This implies that even a small number of channels randomly opening can have a systematic effect on the cell’s behaviour (Faisal, White, and Laughlin, 2005; Faisal and Laughlin, 2007; Faisal, Selen, and Wolpert, 2008). Moreover, neurons perform highly non- linear operations involving high gain amplification and positive feedback. Therefore, small random fluctuations can therefore significantly change whole cell responses.
In order to capture the effects of the variability of ion channels, we need to use the stochastic partial differential equations describing the evolution of the membrane potential in space and time inside neurons. We use Modigliani to perform Monte-Carlo integration on these equations (Faisal, Laughlin, and White, 2002). Modigliani is a fast, accurate stochastic partial differential equation solver (Faisal, White, and Laughlin, 2005). It is completely object oriented, and its classes are mapped onto relevant entities in neuroscience (see Figure 2). Using Modigliani, we can build complex neurons from the ground up by plugging objects, such as compartments or ion channels together and change them at during runtime.
Modigliani simulates each individual ion channel as a Markov process switching between finite states. A typical NaV1.6 channel, for instance, is simulated as an 8-state markov process, where transition probabilities depend on the membrane potential, and the transition probabilities are given by the well known kinetic rate functions that govern gating particles in the deterministic Hodgkin-Huxley type model (Hodgkin and Huxley, 1952). While simulating neurons stochastically, the choice of integration algorithm can have a crucial effect on the behaviour of the cell. We typically use the binomial algorithm (Faisal, 2010), where the channels of each compartment are treated as a binomial population. At each step, binomial random numbers are generated to estimate the number of channels switching states based on the individual transition probabilities of each ion channel. This algorithm is chosen because of its accuracy as compared to the Langevin algorithm, and speed when compared to the Gillespie algorithm (For details about algorithms and their performance, see Faisal, 2010).