Abstract
In the nervous system, communication occurs via synaptic transmission where signaling molecules (neurotransmitters) are released by the presynaptic neuron, and they influence electrical activity of another neuron (postsynaptic neuron). The inherent probabilistic release of neurotransmitters is a significant source of noise that critically impacts the timing of spikes (action potential) in the postsynaptic neuron. We develop a stochastic model that incorporates noise mechanisms in synaptic transmission, such as, random docking of neurotransmitter-filled vesicle to a finite number of docking sites, with each site having a probability of vesicle release upon arrival of an action potential. This random, burst-like release of neurotransmitters serves as an input to an integrate-and-fire model, where spikes in the postsynaptic neuron are triggered when its membrane potential reaches a critical threshold for the first time. We derive novel analytical results for the probability distribution function of spike timing, and systematically investigate how underlying model parameters and noise processes regulate variability in the inter-spike times. Interestingly, in some parameter regimes, independent arrivals of action potentials in the presynaptic neuron generate strong dependencies in the spike timing of the postsynaptic neuron. Finally, we argue that probabilistic release of neurotransmitters is not only a source of disturbance, but plays a beneficial role in synaptic information processing.
I. INTRODUCTION
Communication between neurons occurs through chemical synapses, where signals in the form of neurotransmitters are passed from a presynaptic neuron to a postsynaptic neuron. Synaptic structures are generally present at the axon terminals of the presynaptic neuron, and consist of neurotransmitter-filled vesicles that are loaded on to a finite number of docking sites. An action potential reaching the axon terminal triggers opening of voltage-gated calcium channels. Calcium influx into the axon terminal causes fusion of docked synaptic vesicles to the cell membrane, and release of neurotransmitters. The depletion of docked synaptic vesicles leads to the system becoming less responsive to the next action potential, creating some sort of memory that is often referred to as synaptic depression [1]. Released neurotransmitters bind to receptors on the postsynaptic neuron, and open ion channels that drive short-term changes in the membrane potential of the postsynaptic neuron. While individual neurons are known to receive synaptic contacts (both inhibitory and excitatory) from thousand of other neurons, we here consider the simplest case of a single presynaptic neuron forming an excitatory synapse with a single postsynaptic neuron
Much prior work treats neurotransmitter release as a deterministic processes [2–6], but these models fail to capture the variability introduced at each trial by the inherent probabilistic nature of vesicle release and other noise mechanisms [7–10]. Moreover, several experimental and computational works have argued that these stochastic effects are functionally important for understanding information flow across a synapse [11–16]. As part of this contribution we consider a stochastic model of synaptic transmission that incorporates three different noise mechanisms:
Random arrival of action potentials at the axon terminal, with inter-arrival times drawn from an arbitrary probability distribution.
When the action potential arrives, each docking site loaded with a vesicle has a probability of releasing the vesicle.
Empty sites recover probabilistically, and over time they become occupied by synaptic vesicles.
Before formally introducing the stochastic model and our main analysis, we discuss some mathematical preliminaries.
II. MATHEMATICAL PRELIMINARIES
We start by recalling some basic probability concepts related to the binomial distribution, as this distribution arises ubiquitously in the modeling of noisy synaptic transmission. If the random variable X follows the binomial distribution with parameters M ∈ {1, 2,…} (the number of trials) and p ∈ [0-1] (success probability in each trial), then we write
and its probability mass function is given by
k ∈ {0,1, 2,…}. The mean and variance of X is respectively, where 〈 〉 denotes the expected value operator. Moreover, denotes the moment generating function (MGF) of X. A key result that will be used later relates to conditional binomials. If , and we define a new random variable Y such that conditioned on (i.e., the number of trials itself is binomially distributed) then
This result can be can be straightforwardly proven using the MGF in (4) and we omit the proof due to space constraints. Next, we describe a stochastic model for neurotransmitter release from the presynaptic neuron, and show how binomial concepts introduced here aid the model analysis.
III. STOCHASTIC MODELING OF NEUROTRANSMITTER RELEASE
Let action potentials arrive at the presynaptic neuron at discrete times ti, i ∈ {1, 2,…}, and the intervals
be independent and identically distributed (i.i.d.) random variables drawn from a given probability density function. We denote by and , the time just before and after the ith action potential, respectively. Upon arrival of an action potential, neurotransmitter molecules are released based on a stochastic model with the following three ingredients:
There are M docking sites on the presynaptic neuron, and each site can either be empty, or occupied by a synaptic vesicle that is filled with neurotransmitters.
When the action potential arrives, each occupied site has a probability pr ∈ [0-1] of releasing the vesicle and becoming empty. The release process is assumed to occur instantaneously in time.
In the time interval between two successive action potentials, an occupied sites remains occupied, and an empty sites become occupied with a rate k.
As a consequence of the last point, if a site is empty after the ith action potential, then the probability pi ∈ [0-1] of a vesicle docking there before the arrival of the i + 1th action potential is
Intuitively, if the next action potential takes much longer to arrive by random chance, then there is a higher probability of an empty site getting occupied by a synaptic vesicle. Note that the probabilities pi are themselves i.i.d random variables since Ti are i.i.d. As an example, if the action potential arrives based on a Poisson process and Ti is exponentially distributed with mean 〈Ti〉, then as per (7) the probability density function of pi is given by with the following mean and variance
As expected, 〈pi〉 increases with increasing gap between successive action potentials, and in the limit 〈Ti〉 →∞, pi = 1 with probability one. Another scenario that we will discuss later on is deterministic arrival of action potentials, for which Ti = 〈Ti〉 with probability one, and with probability one. A sample path of the number of synaptic vesicles released over time is shown in Fig. 1. For convenience, notations used in the model and subsequent analysis are summarized in Table I. The following Theorem quantifies the amount of neurotransmitter released.
Let c denote the number of neurotransmitters in each synaptic vesicle. Then, upon arrival of the ith action potential, the number of neurotransmitter molecules released is cZi, where is the number of synaptic vesicles released. The probability in (12) follows the random discrete-time system where pi are i.i.d random variables defined in (7).
Proof Let denote the number of occupied sites at time (just before the arrival of the ith action potential). Given the probability of vesicle release, pr, the number of occupied sites releasing vesicles is and the number of occupied sites at time is given by
Now the number of empty sites at is , and each empty site has a probability pi of getting occupied before the arrival of i + 1th action potential. Then, the number of occupied sites just before the arrival of the i + 1th action potential is
To solve this stochastic map we assume that follows a binomial distribution and apply the concept of conditional binomials from (5) to (15) and (17)
In the Appendix A, we show using (18) that the right-hand-side of (16) follows a binomial random variable
From (17), the left-hand-side of (16) is also binomially distributed
Equation (13) in the theorem results from matching the success probability parameter in (19) and (20). In summary, solving (13) based on some initial condition, i.e., if all sites are occupied when the first action potential comes, provides the distribution for the number occupied sites at time via (17). Using (14), (17), and the property of conditional binomials, the number of synaptic vesicles released is
Two special cases resulting from Theorem 1 are
pr = 1 (all occupied sites release vesicles), in which case
pi = 1 (all empty sites get occupied before the next action potential), in which case
Next, we focus on computing the statistical moments of Zi.
IV. MOMENTS OF THE NUMBER OF NEUROTRANSMITTERS RELEASED
Applying the formulas for the moments of a binomially distributed random variable in (3) to yields the following conditional moments
To uncondition (24), we first obtain the steady-state moments of from (13). Taking the expected values on both sides of (13), and using the fact that pi are i.i.d. random variables drawn independently of ,
Similarly, squaring both sides in (13), taking the expected value and the limit i →∞ yields
Now unconditioning (24) using the moments of , the average number of synaptic vesicles released per action potential at equilibrium is
and using a similar approach,
We now use these moments of Zi to investigate how stochasticity in the amount of neurotransmitter released can be regulated. In particular,
How does noise in Zi (quantified by its coefficient of variation squared) depend on noise in Ti (action potential arrival times)?
Is there an optimal way to choose M (number of docking sites) and pr (vesicle release probability) so as to minimize ?
To explore the first question, recall from (7) that pi is a monotonically increasing function of Ti. Hence, increasing noise in the action potential timing can be modeled as increasing noise in pi. Fig. 2 plots as a function of the coefficient of variation of pi for a fixed mean 〈pi〉. Our results reveal an interesting trade off: when the release probability is small, then starts high but remains flat (Fig. 2). In contrast, when the release probability is close to one, then starts low but increases sharply.
Fig. 3 plots as a function of pr for deterministic arrivals (Ti = 〈Ti〉 with probability one), and Poisson arrivals (Ti is exponentially distributed with mean 〈Ti〉) of action potentials. The number of docking sites is correspondingly changes as per (28) to keep limi→∞〈Zi〉 fixed. For deterministic arrivals, is always minimized by choosing pr = 1 (Fig. 3; top). Intriguingly, for Poisson arrivals, if the recovery probability 〈pi〉 is not large, then is minimized at an intermediate value of pr (Fig. 3; bottom). Note the qualitative shift in behavior when stochasticity is incorporated in Ti (compare Fig. 3 bottom and top), and in some cases is minimized by choosing pr significantly smaller than one (see 〈pi〉 = 0.5 line in Fig. 3 bottom).
V. ACTION POTENTIAL TIMING IN THE POSTSYNAPTIC NEURON
Having quantified the release of vesicles from the presynaptic neuron, we next focus on the stochastic dynamics of neurotransmitter counts in the synaptic gap (space between the cell membranes of the presynaptic and the postsynaptic neuron). Let z(t) denote the level of neurotransmitters in the synaptic gap, and its time evolution is characterized by production in bursts followed by exponential decay (Fig. 4). Whenever an action potential arrives at the presynaptic neuron, multiple vesicles are released causing a jump in z(t) where random variable Zi, is the number of vesicles released and c is the amount of neurotransmitter per vesicle. In between action potentials, z(t) decreases as per a first-order decay process where γz is the degradation rate of individual molecules. Given the average time between two action potentials (or burst events) 〈Ti〉, the average number of neurotransmitters released per burst c〈Zi〉, and each molecule lives for an average time 1/γz in the synaptic gap, we obtain the following steady-state mean levels from (28)1
Furthermore, recall that for deterministic and Poisson arrivals of action potentials, respectively. Substituting (34) in (33), one can see that the mean neurotransmitter level increases with increasing frequency of arrivals (1/〈Ti〉), and at high frequencies
the system saturates to the same limit irrespective of deterministic or Poisson arrivals.
The stochastic process z(t) drives the linear dynamical system that determines when action potentials fire in the postsynaptic neuron. Model (36) is commonly referred to as the integrate-and-fire model [18], and here v(t) is the membrane potential of the postsynaptic neuron, τυ is the membrane time constant and kv is a positive constant. An action potential in the postsynaptic neuron is triggered when v(t) hits a threshold vth for the first time. At this time, v(t) is reset to zero and the process starts anew. We use v = 0 to represent the membrane resting potential that is typically −70 mV. In essence, the time to the next action potential in the postsynaptic neuron is the first-passage time
Note that if fluctuations in z(t) are long lived (for example, low turnover rate of neurotransmitters in the synaptic gap), then the timing of successive action potentials in the postsynaptic neuron will be correlated.
Since fluctuations in z(t) are at steady-state, the mean membrane potential increases as starting from υ(0) = 0. Assuming and small fluctuations in T*, the mean first-passage time can be approximated using
Using (33)-(34) in (40), Fig. 5 plots the average frequency of action potentials in the postsynaptic neuron (1 /T*) as a function of the input frequency (1 /〈Ti〉) and shows a linear increase followed by saturation. The saturation limit is obtained by substituting (35) in (40), and interestingly, this limit is dependent on M, but completely independent of pr. Thus, at sufficiently high input frequencies of action potentials, changing pr will not affect the action potential timing in the postsynaptic neuron.
VI. NOISE IN ACTION POTENTIAL TIMING
Having determined the mean time to the next action potential, the focus now is on the noise properties of T*. We assume that the noise in the first-passage time T* is sufficiently small, that it is simply proportional (or monotonically related) to the noise in υ(t) computed at the mean first-passage time t = 〈T*〉. This assumption naturally motives the question: Can we derive closed-form expressions for temporal dynamics of v(t) noise levels?
Recall from the previou section that random processes z(t) and υ(t) are defined through the stochastic hybrid system (31), (33) and (36). To aid moment computations described below, we assume Poisson arrivals of action potentials and
in (13) which makes an i.i.d random variable. Referring interested readers to [19–21] for details on moment dynamics for stochastic hybrid systems, the time evolution of all the first and second-order moment of z(t), υ(t) is obtained as
Based on the first two equation in (42), the steady-state moments of z(t) are
where is the noise in the neurotransmitter release process defined in (30) 2. Since v(0) = 0 just after an action potential firing, and fluctuations in z(t) are at steady-state, solving the linear dynamical system (42) with the following initial conditions
yields the following noise levels in υ(t) (quantified by its coefficient of variation squared)
where and cosh (sinh) are hyperbolic cosine (sine) functions. Based on our earlier assumption, and given (47), noise in the first-passage time T* is
As one would expect, random fluctuations in T* are connected to noise in the neurotransmitter counts , which in turn is related to noise in Zi via (43). Thus, mechanisms to buffer noise in Zi illustrated in Figs. 2 & 3 are critical for achieving precise action potential firing times. Fig. 6 plots 〈T*〉 and as a function of the firing threshold υth - while the average time to the next action potential increases, the noise in T* decreases with increasing threshold. Furthermore, it can be seen from (48) that and hence the decrease in noise levels in Fig. 6 is sharper for higher membrane time constant τν, or faster turnover rate of neurotransmitters γz.
VII. DISCUSSION
We have performed a systematic analysis of noise mechanisms in synaptic transmission between two neuron pairs. Our key mathematical contributions are
Quantifying the statistics of neurotransmitter release, each time the presynaptic neuron is stimulated by an action potential (Theorem 1).
Characterizing fluctuations in the number of neurotransmitter molecules in the synaptic gap.
Derivation of approximate formulas for the timing of action potentials in the postsynaptic neuron based on the integrate-and-fire modeling framework.
These results have led to some intriguing biological insights that are summarized below. Counter-intuitively, when action potentials randomly arrive at the presynaptic neuron, having an optimal probability of releases pr minimizes the noise in the number of neurotransmitters released (Figs. 2 & 3). Furthermore, in some cases deterministically releasing vesicles based on pr = 1 would considerably amplify noise (Fig. 3; bottom). Our results further show that high frequencies of action potentials in the presynaptic neuron lead to saturation in the neurotransmitter count, and the action potential frequency in the postsynaptic neuron (Fig. 5). Interestingly, this saturation limit depends on the number of docking sites, but completely independent of pr. Finally, our model analysis reveals that increasing the firing threshold in integrate-and-fire models can buffer noise and enhance precision in action potential timing (Fig. 6).
In recent work, we have developed exact analytical results for the first-passage time in stochastic models of bursty gene expression [24–26]. Motivated by this literature, an important direction of future research is to perform a mathematically rigorous derivation of the first-passage time T*, and explore how the results in Fig. 6 change when the magnitude of fluctuations in T* are large, or when threshold crossings are noise induced. Other directions of future work include:
Expand the current model to consider multiple synapses (both inhibitory and excitatory).
Investigate how negative feedback in the form of an autapse (synapse from a neuron onto itself) can enhance noise buffering in synaptic transmission [27, 28].
While the current work assumes biophysical parameters, such as, the number of docking sites and release probability to be constant, experimental work points to dynamic regulation of these parameters [29–31]. It will be interesting to investigate the effects of such dynamic regulation through local feedbacks on synaptic transmission.
ACKNOWLEDGMENT
AS is supported by the National Science Foundation Grant DMS-1312926, University of Delaware Research Foundation (UDRF) and Oak Ridge Associated Universities (ORAU).
Appendix A
Here we prove that given (18), the right-hand-side of (16) is binomially distributed as
The MGF of the right-hand-side of (16) conditioned on is given by
Now unconditioning (A3) with respect to yields
Since is also a Binomial random variable, we have from (4) and (18)
Simplifying (A4) using (A6) where is given by (13) and (A7) is the MGF of a binomial random variable .
Footnotes
↵* absingh{at}udel.edu
1 If we think molecules in the synaptic gap as customers in a queue, this result, can be derived based on Little’s law for a G/M/∞ queue, where customers arrives in batches based on some general inter-burst, time distribution Ti [17].
2 Results in (44) are analogous to protein noise levels in bursty models of gene expression [19, 22, 23], with the only exception that Zi is binomially distributed, while protein burst sizes typically follow geometric distributions