Abstract
Action potential-triggered release of neurotransmitters at chemical synapses forms the key basis of communication between two neurons. To quantify the stochastic dynamics of the number of neurotransmitters released, we investigate a model where neurotransmitter-filled vesicles attach to a finite number of docking sites in the axon terminal, and are subsequently released when the action potential arrives. We formulate the model as a Stochastic Hybrid System (SHS) that combines three key noise mechanisms: random arrival of action potentials, stochastic refilling of docking sites, and probabilistic release of docked vesicles. This SHS representation is used to derive exact analytical formulas for the mean and noise (as quantified by Fano factor) in the number of vesicles released per action potential. Interestingly, results show that in relevant parameter regimes, noise in the number of vesicles released is sub-Poissonian at low frequencies, super-Poissonian at intermediate frequencies, and approaches a Poisson limit at high frequencies. In contrast, noise in the number of neurotransmitters in the synaptic cleft is always super-Poissonian, but is lowest at intermediate frequencies. We further investigate changes in these noise properties for non-Poissonian arrival of action potentials, and when the probability of release is frequency dependent. In summary, these results provide the first glimpse into synaptic parameters not only determining the mean synaptic strength, but also shaping its stochastic dynamics that is critical for information transfer between neurons.
1 INTRODUCTION
We present a mechanistic stochastic model to characterize the statistics of the number of neurotransmitters released at a neuronal synapse. While there is rich literature regarding synaptic transmission as a deterministic process [1–5], increasing evidence points to diverse noise mechanisms at play during this process [6–10]. Counterintuitively, noise has been shown to sometimes enhance signaling and information transfer between neurons [11–16].
Stochastic Hybrid Systems (SHS) constitute an important class of mathematical models that integrate discrete stochastic events with continuous dynamics. Given their generality and scope, SHS have been successfully used for modeling stochastic phenomena in a variety of biological processes [17–33], including neuronal dynamics [34, 35]. Building up on our previous work [36], we consider the SHS framework for modeling the neurotransmitter dynamics. More specifically, the model consists of M ∈ {1, 2,…} docking site at the axon terminal (Fig. 1). Neurotransmitter-filled vesicles attach to these docking sites with a given probabilistic rate that is proportional to M −n, where n is the number of already docked vesicles. Action Potentials (APs) arrive at a given frequency, such that, the time between two successive APs follows an arbitrary positively-valued probability density function g. Each docked vesicle has a certain probability of release, and based on this probability, APs cause a fraction of the docked vesicles to release their neurotransmitter content into the synaptic cleft (gap between neurons). Finally, released neurotransmitters decay (or are removed) at a constant rate. Details of the model along with sample stochastic realizations are shown in Fig.1.
The level of neurotransmitters in the synaptic cleft ultimately determines the downstream biological activity through binding and opening of ion channels in the postsynaptic neuron. How does noise in the neurotransmitter level depend on the frequency of AP arrivals, the probability of vesicular release, and other synaptic parameters? To address this question, we derive exact analytical results for the statistical moments of the SHS state space. While solving moment dynamics for SHS often suffers from the problem of moment closure [37–39], here we find exact analytical formulas for the steady-state mean and noise levels for both the number of neurotransmitters released per AP, and the neurotransmitter level in the synaptic cleft.
Analysis of these formulas reveals that depending on the parameter regime, the noise in the number of vesicles released can vary monotonically or non-monotonically with the arrival frequency. In the latter case, noise is sub-Poissonian at low frequencies, super-Poissonian at intermediate frequencies, and approaches a Poisson limit at high frequencies. In contrast, noise in the number of neurotransmitters in the synaptic cleft is always super-Poissonian, but is lowest at intermediate frequencies. The paper is organized as follows: in Section II we recast synaptic transmission models in the SHS framework, and systematically analyze this class of stochastic models in Section III to derive exact moment dynamics. Based on these results, we present our key findings in Section IV, followed by the Conclusion section.
2 Model Formulation
Fig.1(a) shows the structure of a (chemical) synapse that permits communication between neurons via released neurotransmitters. Briefly, this process is triggered when an electrical stimulus or Action Potential (AP) reaches the axon terminal of the presynaptic neuron. Inside the axon terminal, vesicles filled with neurotransmitters are synthesized, and they become release-ready by docking close to the cell membrane. Arrival of an AP leads to opening of calcium ion channels, and the influx of calcium causes these docked vesicles to release their neurotransmitter content in the synaptic cleft (extracellular space between the two neurons). The vesicular release process is known to be probabilistic, with each docked vesicle having a certain probability of release. Released neurotransmitters bind to receptors on the cell membrane, triggering an AP in the postsynaptic neuron.
We model the stochastic dynamics of synaptic transmission by a SHS shown in Fig.1(b). The SHS state-space x = [n z]T, where n(t) is the number of docked vesicles, and z(t) is the number of neurotransmitters in the cleft at time t. Two families of resets occur randomly over time that impact the time evolution of x. The first reset is the arrival of an AP that results in the release of vesicles, and the second reset is the replenishment of the vesicle pool from newly synthesized vesicles. We describe these resets in further detail below, starting with the AP arrival process.
2.1 Timer-based arrival of Action Potentials
We assume that APs arrive at times ts, s ∈ {1, 2, 3,…}, such that the time intervals τs≡ ts −ts−1 are independent and identically distributed random variables following an arbitrary positively-valued continuous probability density function (pdf) g. To model the timing of APs, we introduce a timer τ that linearly increases over time and resets to zero whenever an AP arrives. The arrival of the next AP depends on the state of the timer, and this introduces memory in the arrival process. More specifically, the probability that the AP arrives in the next infinitesimal time interval (t, t + dt] is given by h(τ)dt, where the hazard rate
[40–42]. Defining the arrival of events as per (3) ensures that τs follows the pdf g and the corresponding pdf of τ is given by
For example, if g is exponentially-distributed with mean ⟨τs,⟩ then h(τ) = 1/⟨τs⟩ would be a constant corresponding to a Poisson arrival of events.
Having modeled the timing of the AP arrival process, we next describe its impact on the SHS state space. We assume that each vesicle has a certain probability of release (pr). Thus, an AP causes the SHS state space to jump as per the following reset where b is the number of vesicles released and follows a Binomial distribution and c is the number of neurotransmitters in each vesicle. The reset (6) can be written in terms of x as where random variable x+ is the state of system immediately after an AP. Using (7) and (8), the mean and covariance matrix of x+ is given by and respectively. Note that the stochastic jumps in x+ are state-dependent due to the binomial nature of the release process [43, 44].
2.2 Stochastic hybrid system model
In addition to timer-based arrival of APs, we also have a second family of resets corresponding to replenishment of vesicles. We assume that the docked-vesicle pool replenishes with a rate k(M −n), where M is the total number of docking sites (i.e., maximum possible number of vesicles), and k represents the refilling rate per empty site. We rewrite this rate in terms of x as where . The replenishment process is assumed to occur independently of APs. In the stochastic sense, the probability that a replenishment event occurs in the next time interval (t, t + dt] is given by k(M − n)dt, and whenever this event occurs where and corresponds to n increasing by one. In between the two families of resets, the state evolves continuously via a linear system where γ is the constant rate of removal or decay of released neurotransmitters.
In summary, we have defined a SHS model for synaptic transmission with continuous dynamics (13), two family of rests that occur with rates (3) and (11), and when they occur, the state space resets via (8) and (12), respectively. It is important to point out that this model with just the timer-dependent reset has been referred to in literature as time-triggered SHS [45–48]. Allowing for a second family of resets that occurs at a rate linearly dependent on x is a novelty of this work.
3 Solving moment dynamics
Next, we derive exact analytical formulas for the first- and second-order moments of x = [n z]T. In this sections we use ⟨ ⟩ to denote the expected value, and to show the expected value at steady-state.
3.1 First-order moment
As shown in Appendix, in between two successive AP arrivals, the conditional moment ⟨x|τ⟩ evolves as
[49]. By solving (14) we have where ⟨x|τ = 0⟩ is the expected value just after AP arrival. Given that the time interval between two APs is τs and the reset in mean levels (9), then at steady-state
Using (15) for τ= τs together with (16) solves , and then using (5) to make ⟨x|τ⟩ unconditioned with respect to τ, yields where I2 is an identity matrix of size 2. Additionally, the system is stable if eighenvalues of are inside the unit circle. Using all the constant vector and matrices defined in (9)-(12), we can solve for the mean number of docked vesicles and the mean neurotransmitters level
3.2 Second order moment
To obtain the second-order moments we define a new vector where vec(xxT)∈ℝn×1 is a vector representation of the matrix xxT∈ℝn×n. The new vector µ =[n z n2nz z n z2]T contains all the first and second order moments. It can be shown after some algebraic steps (see also Appendix), where ⊗ is Kronecker product. Solving (21) gives
Similar to (16), based on the resets (9) and (10) we can write and then using an approach analogous to the first order moment results in where I6 is an identity matrix of size 6. The eighenvalues of should be inside the unit circle in order for the system to be stable.Using (26) and the first order moments (18) and (19), we obtain the second order moments (43) and (44). Due to the size of these formulas they are presented at the end on the last page.
Throughout the paper we use the steady-state Fano factor (variance over mean) defined by as a metric to quantify noise. Here, FFb and FFz denote the Fano factor for the number of vesicles released, and the Fano factor for the number of neurotransmitters, respectively. While FFz can be calculated using the first and second order moments in (19) and (44), to obtain FFb we recall that given n, b ∼ β(n, pr) follows a binomial distribution. Hence,
Given formulas for and in (18) and (43), respectively, FFb is computed by applying (30) and (31) in (29). Interestingly, with these formulas one can show the following limits for low (f → 0) and high frequency (f → ∞) stimulation regardless of τs:
This result assumes that the probability of release pr is constant. As Fano factor is one for a Poisson distribution, the number of vesicles released follows Poissonian statistics at high frequencies, and is sub-Poissonian at low frequencies. Assuming c ≫ 1, the noise in neurotransmitter levels is super-Poissonian at both low and high frequencies.
4 Noise Analysis
In this section, we consider specific distributions for the arrival of APs.
4.1 Poisson arrival of APs
Let τs be an exponentially-distributed random variable with mean ⟨τs⟩ = 1/f, where f is the frequency of AP arrival. Then, as per the moment generating function,
Using (18), (19), (30) and (34), we obtain the following simplified steady-state means
The number of vesicles released per AP decreases with increasing frequency of stimulation due to vesicular depletion in the axon terminal. In contrast, neurotransmitter levels increase monotonically with f. Similarly, using (43) and (44), the respective Fano factors can be calculated as
Further analysis reveals an intriguing find: if Mpr < 2, then FFb monotonically increases with frequency approaching the Poisson limit FFb → 1 as f → ∞ (Fig. 2). Note in this case FFb < 1 and is always sub-Poissonian. However, if Mpr > 2, then FFb has a non-monotonic profile, where FFb > 1, i.e., super-Poissonian at an intermediate frequency. Our results further show that FFz also varies non-monotonically with f, but exhibits a minima (Fig. 2).
4.2 Deterministic arrival of APs
Next consider a deterministic arrival process where τs = 1/f with probability one. In this case, the Fano factors are given by
As in the Poisson case, depending on the values of M and pr, FFb can be a monotonic/non-monotonic function of frequency f (Fig. 2). More specifically, if , then FFb is non-monotonic, and monotonic for .
4.3 Frequency-dependent probability of release
Up till now, we have assumed that the probability of release is a constant. We relax this assumption by allowing for a frequency-dependent probability of release. Mechanistically, an increased frequency of APs cause a higher buildup of calcium in the axon terminal which enhances pr. This frequency-dependence is modeled as with positive constants h, pmax and F. As per this formulation, pr → 0 for f → 0, and pr approaches a maximum value of pmax as f. → ∞ In the case, the Fano factor limits in (32) and (33) are modified to and are the same for low and high-frequency stimulation. While the statistic of vesicle release is Poissonian (FFb = 1) in the extreme frequency limits, it shows dramatic changes for in-between frequencies with first having a minimum, and then a maximum (Fig. 3). The Fano factor of the neurotransmitter level mirrors these fluctuations with opposite effects – a maximum first followed by a minimum (Fig. 3).
5 Conclusion
We have applied SHS-based modeling and analysis to systematically characterize the stochastic dynamics of neuronal synaptic transmission. It is important to mention that this contribution also advances the theory for SHS. For example, prior work was focused on either a single timer-dependent reset [33, 46], or two resets where the other occurred at a constant rate [49]. Here we allow the second family of reset to occur at a rate for a constant vector and scalar k1. Hence, the results on moment derivations presented in Section III are by themselves novel.
Applying the results of Section III to the SHS model in Fig. 1b yields exact analytical formulas for the mean synaptic strength (quantified by (18) and (19)), and the extent of random fluctuations around it (quantified by (43) and (44)). While the noise formulas are quite complicated, they have provided valuable insights in different limits. For example, the limits (32), (33), (41) and (42) show the expected Fano factor at low and high-frequency stimulation for either a constant pr or frequency-dependent pr. Apart from predicting noise levels, these limits may prove useful for actually inferring parameters by experimentally measuring the corresponding Fano factor, thus providing a unique method that exploits stochastic fluctuations for determining synaptic parameters.
Another interesting finding in the non-monotonicity of noise levels shown in Fig. 2 and Fig.3. For the case of Poisson and deterministic arrivals, we identified parameter regimes that exhibit such behaviors, and these results can be expanded to Gamma or lognormally-distributed AP arrivals. The predictions made in Fig. 2 and Fig.3 can be tested experimentally and, they present an exciting direction of future research in collaboration with neuroscientists.
ACKNOWLEDGMENT
This work is supported by the National Science Foundation Grant ECCS-1711548.
APPENDIX
In order to prove (14) we use forward Kolmogrov equation as we need to gain the joint probability density function of states x and timer τ where ∂/∂x defined as partial derivative vector. Now, having joint probability distribution from (45), conditional mean is
By taking derivative with respect to t in (46) we have
Substituting (5) and (45) into mean dynamics, we have
By solving for (48) we get
For simplicity, and and so we get (14) Similar to (46), conditional second order moment is
After taking the same steps as (47) and (48) and also also using vectorization [50], we have