Improved signaling as a result of randomness in synaptic vesicle release
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Edited by Charles F. Stevens, The Salk Institute for Biological Studies, La Jolla, CA, and approved October 21, 2015 (received for review July 4, 2015)

Significance
Noise is not only a source of disturbance, but it also can be beneficial for neuronal information processing. The release of neurotransmitter vesicles in synapses is an unreliable process, especially in the central nervous system. Here we show that the probabilistic nature of neurotransmitter release directly influences the functional role of a synapse, and that a small probability of release per docked vesicle helps reduce the error in the reconstruction of desired signals from the time series of vesicle release events.
Abstract
The probabilistic nature of neurotransmitter release in synapses is believed to be one of the most significant sources of noise in the central nervous system. We show how p0, the probability of release per docked vesicle when an action potential arrives, affects the dynamics of the rate of vesicle release in response to changes in the rate of arrival of action potentials. Furthermore, we examine the theoretical capability of a synapse in the estimation of desired signals using information from the stochastic vesicle release events under the framework of optimal linear filter theory. We find that a small p0, such as 0.1, reduces the error in the reconstruction of the input, or in the reconstruction of the time derivative of the input, from the time series of vesicle release events. Our results imply that the probabilistic nature of synaptic vesicle release plays a direct functional role in synaptic transmission.
Randomness is present in almost all levels of all nervous systems, such as in ionic channels of individual neurons, in synapses between neurons, and in environmental stimuli. The probabilistic nature of the synaptic vesicle release process is believed to be one of the most significant sources of randomness. Stochastic vesicle release affects information transfer from a presynaptic neuron to a postsynaptic neuron, and hence may play not only an important role in synaptic plasticity (1, 2) but also a significant role in determining the functionality of certain synapses, a point we will argue in this article. The functional role of stochastic vesicle release in synaptic transmission is likely to be more significant in those synapses in the central nervous system, where the size of the readily releasable vesicle pool is usually smaller than those in the periphery.
Synaptic vesicle release has only recently been studied in a probabilistic or information–theoretic manner (3⇓–5). A systematic perspective on how stochastic vesicle release affects neural code processing at synapses is still lacking. On the experimental front, a comprehensive, quantitative knowledge about the nature of vesicle docking, priming, fusing, undocking, replenishing, and recycling is far from complete, and hence a biophysically detailed model of the entire vesicle release process is not yet possible.
Despite current limited understanding of the synaptic vesicle release process, a probabilistic model that captures the essential elements of that process can still be built to study the effect of various sources of randomness on synaptic transmission. Rosenbaum et al. (3) recently constructed such a model to study how variability in vesicle dynamics affects information transfer from one neuron to another. They found that the depletion of docked vesicles at higher rates of arrival of action potentials makes the synapse act as a high-pass filter from a signal processing point of view. Motivated by Rosenbaum et al.’s (3) model of the presynaptic vesicle release process, we here build a probabilistic model that includes four stochastic subprocesses involved in synaptic vesicle release: the stochastic process that generates the presynaptic spike density
More generally, we provide a systematic framework, based on a cascade of stochastic models of the presynaptic vesicle docking and release processes coupled with optimal linear filter theory, to address how various sources of randomness in synaptic vesicle release affect synaptic transmission between two neurons. We believe that our model framework can be further extended or modified to incorporate biophysical details of the synaptic vesicle release process, or by changing the objective function used in the optimal filter theory, to study various issues in synaptic transmission.
The Presynaptic Model of Synaptic Vesicle Release
We first build a model of stochastic vesicle release in the presynaptic neuron.
A key assumption of our model is that we allow for an unlimited number of docking sites. Let vesicle docking occur by a homogeneous Poisson process with mean rate
Let presynaptic action potentials occur by a stochastic process with mean rate
When an action potential occurs in the presynaptic neuron, each vesicle that is docked has a probability
The Expected Rate of Synaptic Vesicle Release
Using the above model of the presynaptic vesicle docking and release processes, we first study how
Let
If we differentiate Eq. 1 with respect to time on both sides and make use of
To our knowledge, Eq. 4 is new. In its linearized form, however, it is closely related to the theory of Rosenbaum et al. (3). Eq. 4 shows that during any time interval in which the spike density
Plotted in Fig. 1 are the probability per unit time of an action potential in the presynaptic neuron [blue curve, also referred to as the presynaptic spike density
Effect of probability of release per docked vesicle (
Effect of probability of release per docked vesicle (
Note that in our model, any steady level
SI Materials and Methods
Synaptic Vesicle Release with Finite Docking Sites.
Suppose there are m docking sites, and let
Recall from the main text that
Then Eq. 1 in the main text implies that the expected rate of vesicle release
Let
To obtain the case of an unlimited number of docking sites, let
Optimal Filter.
Plotted in Figs. S1 and S2 are the optimal filters
The optimal filters in the estimation of
The optimal filters in the estimation of
Synapse as an Optimal Filter
So far we have only discussed the effect of randomness on the expected rate of vesicle release. In this section, we address the theoretical capability of a postsynaptic neuron in using the time series of vesicle release events to estimate the instantaneous rate of action potentials in the presynaptic spike train
We consider the optimal filtering of the rate of synaptic vesicle release
Let
Note also that we do not impose any causality constraint on the optimal filter
The actual postsynaptic neuron, of course, does not have this luxury. In future work we shall consider the causal case in which
The point of view here is that we are given
Two remarks are of importance here. First, the optimization problem is not fully specified until we define the stochastic processes that generate the presynaptic spike density
The mean square error
Therefore, to find the optimal filter
Once we obtain the optimal filter
The perspective we adopt here is to provide a theoretical upper bound on how well the synapse can possibly perform in the estimation of a desired signal in the mean square sense under optimal linear filter theory. Using this method, we can address how various sources of randomness in synaptic vesicle release affect the fidelity of synaptic transmission.
A Small p 0 Enhances Synaptic Transmission
In this section, we show that a small value of
We assume that the presynaptic spike density
We assume that the presynaptic spike train
We simulate numerically the stochastic synaptic vesicle release process defined above at different values of
Effect of probability of release per docked vesicle (
The filtered outputs in Fig. 2 clearly show the gradual improvement of the estimation of the presynaptic spike density
Fig. 4 plots the mean square error
Effect of probability of release per docked vesicle (
The perspective adopted in this paper is that we do not presume the biophysical details of a synapse’s postsynaptic response to the presynaptic vesicle release events, but instead we use optimal linear filter theory to address the ability of our idealized postsynaptic neuron to estimate a desired signal derived from the presynaptic rate of arrival of action potentials as a function of time, given that we know the statistical properties of the ensemble from which the presynaptic rate of arrival of action potentials as a function of time has been drawn. We have shown in this paper how the optimal filter is calculated, and that a lower probability of release per docked vesicle upon arrival of a presynaptic spike leads to a more accurate estimation of two kinds of desired signals; we believe that this result goes a long way to explain why synaptic vesicle release is a random process.
Materials and Methods
Derivation of the Optimal Linear Filter.
Our goal is to find an optimal linear filter
If
Because
Note that
Now take the limit
Then Eq. 15 becomes
It is easy to check that Eq. 18 holds for all
Simulation of the Vesicle Docking and Release Processes.
Under the assumption that vesicle docking occurs by a homogeneous Poisson process, we show that
To see this fact, let us first define
The process that generates the numbers
Let
For any integers
Noting that
we have
where
Note that the random variables
Hence, given the presynaptic spike times
Acknowledgments
C.Z. is a Courant Instructor. C.S.P. is partially supported by the Systems Biology Center New York under NIH Grant P50-GM071558.
Footnotes
- ↵1To whom correspondence should be addressed. Email: calvinz{at}cims.nyu.edu.
Author contributions: C.Z. and C.S.P. designed research, performed research, contributed analytic tools, analyzed data, and wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1513160112/-/DCSupplemental.
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