Binomial Quantal Release Simulation

The binomial model

Set n, the number of independent release sites; p, the probability that each site releases on a trial; and q, the EPSC amplitude contributed by one release (pA). Each site releases either zero or one quantum. All sites have the same p and q, their responses add linearly, and these properties stay fixed across trials.

The number of released quanta k follows a binomial distribution: P(k) = C(n,k) p^k(1 − p)^n−k, and the amplitude is kq. Without measurement noise, mean = npq, variance = np(1 − p)q², and the failure probability is (1 − p)ⁿ.

Read the plots

The upper plot shows the theoretical distribution of EPSC amplitudes, displayed as positive magnitudes in pA. These are calculated probabilities, not a finite set of randomly generated trials. With Noise = 0, the point at zero represents release failures. The cumulative plot gives the predicted percentage of responses at or below each amplitude.

Show as histogram groups probability into amplitude bins. With noise, the ordinary curve shows probability assigned to small numerical intervals; histogram bars combine these intervals. Their heights therefore depend on bin width: do not compare curve heights directly with histogram heights. White markers indicate the underlying noiseless response amplitudes.

Noise and summary statistics

Noise is the standard deviation of additive, zero-mean Gaussian measurement noise, in pA. It broadens each response peak, can obscure failures, and can produce apparent negative amplitudes around zero. It does not change the release probabilities.

Show more displays the theoretical mean, standard deviation (sDev), and coefficient of variation (CV = SD / mean). These statistics describe release variability only; they exclude the added measurement noise. For independent measurement noise, the measured variance would be the release variance plus Noise squared. CV is undefined when the mean is zero. Total p is the numerically summed probability, normally close to 1; finite plotting limits and numerical approximation can cause small deviations.

Controls and figures

Use the sliders or edit the number fields; plots update automatically. The max fields set control limits, not additional biological parameters. Text size adjusts graph and parameter text. Hover over controls for explanations and graphs for values; use a graph’s camera button to save a PNG. CSV data saves parameters, exact quantal probabilities, and the displayed distribution. Reset restores the model parameters.

Try it

1. Start with Noise = 0 and n = 5. Increase p: how do failures, mean amplitude, and CV change?
2. Double q while holding n and p fixed. The peak spacing, mean, and SD double, but CV stays the same.
3. Compare n = 5, p = 0.6 with n = 10, p = 0.3 at the same q. Both have the same mean, but do they have the same variability or failure probability?
4. Increase Noise until individual peaks merge. Could you still infer q from this distribution?

This model does not include different properties across sites, correlated release, multivesicular release at an individual site, or changes in release properties during a train. Use the Heterogeneous Release simulation to explore differences between sites.

References

The quantal interpretation of synaptic transmission was established by del Castillo & Katz (1954), Quantal components of the end-plate potential. For a modern treatment of what can and cannot be inferred from a binomial synapse, see Gontier & Pfister (2020), Identifiability of a Binomial Synapse.