Mathematical analysis and algorithms for efficiently and accurately implementing stochastic simulations of short-term synaptic depression and facilitation

Abstract

The release of neurotransmitter vesicles after arrival of a pre-synaptic action potential (AP) at cortical synapses is known to be a stochastic process, as is the availability of vesicles for release. These processes are known to also depend on the recent history of AP arrivals, and this can be described in terms of time-varying probabilities of vesicle release. Mathematical models of such synaptic dynamics frequently are based only on the mean number of vesicles released by each pre-synaptic AP, since if it is assumed there are sufficiently many vesicle sites, then variance is small. However, it has been shown recently that variance across sites can be significant for neuron and network dynamics, and this suggests the potential importance of studying short-term plasticity using simulations that do generate trial-to-trial variability. Therefore, in this paper we study several well-known conceptual models for stochastic availability and release. We state explicitly the random variables that these models describe and propose efficient algorithms for accurately implementing stochastic simulations of these random variables in software or hardware. Our results are complemented by mathematical analysis and statement of pseudo-code algorithms.

Keywords: facilitation; short term depression; short term plasticity; short term synaptic dynamics; stochastic simulation; stochastic synapse; synaptic plasticity models; vesicle site model.

Figure 1

Absolute errors for incorrect implementation 1 (A) and incorrect implementation 2 (B), for Availability Model 1, with exponentially distributed availability times. The data was obtained by empirically estimating the probability of release after i spikes, as a function of the frequency of periodically arriving pre-synaptic action potentials, by stochastically simulating Z = 100,000 trials for each condition. The absolute error can be as high as 0.1, and higher errors occur at low frequencies.

Figure 2

Relative errors for incorrect implementation 1 (A) and incorrect implementation 2 (B) for Availability Model 1, with exponentially distributed availability times. The data was obtained by empirically estimating the probability of release after i spikes, as a function of the frequency of periodically arriving pre-synaptic action potentials, by stochastically simulating Z = 100,000 trials for each condition. The largest relative errors occur for higher frequencies.

Figure 3

Absolute error between correct and incorrect implementation for Availability Model 2, with exponentially distributed availability times. The data was obtained by empirically estimating the probability of release after i spikes, as a function of the frequency of periodically arriving pre-synaptic action potentials, by stochastically simulating Z = 100,000 trials for each condition. The largest error occurs for low frequencies, but is much smaller than for Availability Model 1.

Figure 4

Mean (solid traces) number of vesicles released in total after 100 periodic pre-synaptic action potential arrivals for Availability Model 1 (A) and Availability Model 2 (B). The minimum and maximum number of vesicles released are shown with (dashed traces). For each frequency f, all statistics are derived from 10,000 stochastic simulations. Clearly the incorrect implementations can over or under estimate the correct number of vesicles released.

Figure 5

Fraction of 1000 trials in which vesicles are released, for each of a sequence of 20 periodic spikes (A), and 50 Poisson spikes (B), and vesicles with exponentially distributed availability times. The frequency in both cases is 10 Hz. The traces for Deterministic, and Steady state were obtained using Equations (9) and (10). This data shows that the incorrect implementations give markedly different outcomes to the correct stochastic simulation implementations, and to the deterministic expression for the mean number of trials in which vesicles are released.

Figure 6

Fraction of 1000 trials in which vesicles are released, for each of a sequence of 20 periodic spikes (A), and 50 Poisson spikes (B), and vesicles with Rayleigh distributed availability times. The frequency in both cases is 10 Hz. This data shows that Availability Models 1 and 2 provide markedly different outcomes for Rayleigh distributed availability times, unlike the identical outcomes for exponentially distributed times. The data also shows that extension of the binomial approach matches the data where each trial is individually simulated.