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# Improved signaling as a result of randomness in synaptic vesicle release

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 *p*_{0}, 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 *p*_{0}, 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 *k*th presynaptic action potential by *t*. The vesicle docking process and the process that generates action potentials are independent of each other, regardless of whether we condition on *S*; this is a consequence of the assumption of an unlimited number of docking sites. Otherwise, vesicle release by action potentials would increase the rate of docking by making more sites available.

When an action potential occurs in the presynaptic neuron, each vesicle that is docked has a probability *n* is the number of docked vesicles when the action potential occurs.

## The Expected Rate of Synaptic Vesicle Release

Using the above model of the presynaptic vesicle docking and release processes, we first study how

Let *t*, and the expected rate of vesicle release (**1**, the product of *t*. Further multiplying this product by the spike density *t*.

If we differentiate Eq. **1** with respect to time on both sides and make use of **2** becomes**3** can be written simply as**4** is rigorously correct given our assumptions, including the assumption of a Poisson spike train; it can be derived, for example, from the master equation for the process by taking first moments to get expected values.

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 **4** shows that the variable

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

Note that in our model, any steady level *SI Materials and Methods*) (3).

## 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 *t*.

Then Eq. **1** in the main text implies that the expected rate of vesicle release *ε* is a small number, *ε*, and *ω* is the frequency variable and *t* is the time variable.

Let

To obtain the case of an unlimited number of docking sites, let

## 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 *f* is actually a sum, because

Note also that we do not impose any causality constraint on the optimal filter *t* because we are idealizing the postsynaptic neuron as an observer who can record the vesicle release events and process them at leisure to reconstruct the desired signal. It is well known in optimal filter theory that such an ideal observer can be approximated by a real observer whose task is to produce the desired signal after some specified delay. The approximation improves as the delay increases (8).

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 *t* at which we choose to minimize the error. This difficulty will disappear, however, if we take the limit

The mean square error *Materials and Methods*):**8**, we get

Therefore, to find the optimal filter **9** and **10**; this can be done semianalytically for some specific choices of the stochastic processes for vesicle docking and release by first finding the right-hand sides of Eqs. **9** and **10** conditioned on

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 *k*th action potential, is generated using the following two steps: first, generate a provisional spike train *t* is the new (physical) time variable. Then our desired spike train *t*, is given by

We simulate numerically the stochastic synaptic vesicle release process defined above at different values of *Materials and Methods*). For each

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

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 **8**.

### Simulation of the Vesicle Docking and Release Processes.

Under the assumption that vesicle docking occurs by a homogeneous Poisson process, we show that *k*th action potential, are independent and Poisson distributed, conditioned on the presynaptic spike times

To see this fact, let us first define *k*th action potential that docked during the time interval *j*th action potential, presumed to be given. Then

The process that generates the numbers *j* can be described as follows: first, choose *D* from Poisson distribution with mean *D* vesicles independently undergo a branching process such that each unreleased vesicle has a probability of *K* possible outcomes; there are *K* possible outcomes are release on any one of those, and nonrelease.

Let *k*th outcome, and *k*th action potential (*K*). Then,

For any integers

Noting that

we have

where *j*,

Note that the random variables *j*, because different *j* refers to different cohort of vesicles, and both docking and release are independent for different vesicles. Because the sum of independent, Poisson-distributed random variables is a Poisson-distributed random variable with its mean equal to the sum of the means of the original Poisson-distributed random variables, we conclude that

Hence, given the presynaptic spike times *k*th action potential can be simulated in an efficient manner by drawing a number from Poisson distribution with its mean defined by the above Eq. **24**.

## 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

- ↵
^{1}To 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.

## References

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- Han EB,
- Stevens CF

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- Hall JE

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- Lee YW

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- Knight BW

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- Knight BW

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