Estimating synaptic parameters from mean, variance, and covariance in trains of synaptic responses

Fluctuation analysis of synaptic transmission using the variance-mean approach has been restricted in the past to steady-state responses. Here we extend this method to short repetitive trains of synaptic responses, during which the response amplitudes are not stationary. We consider intervals between trains, long enough so that the system is in the same average state at the beginning of each train. This allows analysis of ensemble means and variances for each response in a train separately. Thus, modifications in synaptic efficacy during short-term plasticity can be attributed to changes in synaptic parameters. In addition, we provide practical guidelines for the analysis of the covariance between successive responses in trains. Explicit algorithms to estimate synaptic parameters are derived and tested by Monte Carlo simulations on the basis of a binomial model of synaptic transmission, allowing for quantal variability, heterogeneity in the release probability, and postsynaptic receptor saturation and desensitization. We find that the combined analysis of variance and covariance is advantageous in yielding an estimate for the number of release sites, which is independent of heterogeneity in the release probability under certain conditions. Furthermore, it allows one to calculate the apparent quantal size for each response in a sequence of stimuli.     

A Monte Carlo model reveals independent signaling at central glutamatergic synapses

We have developed a biophysically realistic model of receptor activation at an idealized central glutamatergic synapse that uses Monte Carlo techniques to simulate the stochastic nature of transmission following release of a single synaptic vesicle. For the a synapse with 80 AMPA and 20 NMDA receptors, a single quantum, with 3000 glutamate molecules, opened approximately 3 NMDARs and 20 AMPARs. The number of open receptors varied directly with the total number of receptors, and the fraction of open receptors did not depend on the ratio of co-localized AMPARs and NMDARs. Variability decreased with increases in either total receptor number or quantal size, and differences between the variability of AMPAR and NMDAR responses were due solely to unequal numbers of receptors at the synapse. Despite NMDARs having a much higher affinity for glutamate than AMPARs, quantal release resulted in similar occupancy levels in both receptor types. Receptor activation increased with number of transmitter molecules released or total receptor number, whereas occupancy levels were only dependent on quantal size. Tortuous diffusion spaces reduced the extent of spillover and the activation of extrasynaptic receptors. These results support the conclusion that signaling is spatially independent within and between central glutamatergic synapses.     

The probability of quantal secretion within an array of calcium channels of an active zone

A Monte Carlo analysis has been made of calcium dynamics in submembranous domains of active zones in which the calcium contributed by the opening of many channels is pooled. The kinetics of calcium ions in these domains has been determined using simulations for channels arranged in different geometries, according to the active zone under consideration: rectangular grids for varicosities and boutons and lines for motor-nerve terminals. The effects of endogenous fixed and mobile buffers on the two-dimensional distribution of free calcium ions at these active zones are then given, together with the extent to which these are perturbed and can be detected with different affinity calcium indicators when the calcium channels open stochastically under an action potential. A Monte Carlo analysis of how the dynamics of calcium ions in the submembranous domains determines the probability of exocytosis from docked vesicles is also presented. The spatial distribution of exocytosis from rectangular arrays of secretory units is such that exocytosis is largely excluded from the edges of the array, due to the effects of endogenous buffers. There is a steeper than linear increase in quantal release with an increase in the number of secretory units in the array, indicating that there is not just a local interaction between secretory units. Conditioning action potentials promote an increase in quantal release by a subsequent action potential primarily by depleting the fixed and mobile buffers in the center of the array. In the case of two parallel lines of secretory units exocytosis is random, and diffusion, together with the endogenous calcium buffers, ensures that the secretory units only interact over relatively short distances. As a consequence of this and in contrast to the case of the rectangular array, there is a linear relationship between the extent of quantal secretion from these zones and their length, for lengths greater than a critical value. This Monte Carlo analysis successfully predicts the relationship between the size and geometry of active zones and the probability of quantal secretion at these, the existence of quantal versus multiquantal release at different active zones, and the origins of the F1 phase of facilitation in synapses possessing different active zone geometries.     

Building a bilayer model of the neuromuscular synapse

Progress over the past 10 years has made it possible to construct a simple model of neurotransmitter release. Currently, some models use artificially formed vesicles to represent synaptic vesicles and a planar lipid bilayer as a presynaptic membrane. Fusion of vesicles with the bilayer is via channel proteins in the vesicle membrane and an osmotic gradient. In this paper, a framework is presented for the successful construction of a more complete model of synaptic transmission. This model includes real synaptic vesicles that fuse with a planar bilayer. The bilayer contains acetylcholine receptor (AChR) channels which function as autoreceptors in the membrane. Vesicle fusion is initiated following a Ca²⁺ flux through voltage-gated Ca²⁺ channels. Key steps in the plan are validated by mathematical modeling. Specifically, the probability that a reconstituted AChR channel opens following the release of ACh from a fusing vesicle, is calculated as a function of time, quantal content, and number of reconstituted AChRs. Experimentally obtainable parameters for construction of a working synapse are given. The inevitable construction of a full working model will mean that the minimal structures necessary for synaptic transmission are identified. This will open the door in determining regulatory and modulatory factors of transmitter release.     

Modulatory role of purines in neuromuscular transmission

The data on purine modulation of myoneural transmission are reviewed. A particular attention is paid to adenosine-5′-triphosphoric acid (ATP), the co-transmitter of the principal mediator (acetylcholine), and adenosine, the final ATP metabolite in the synaptic cleft. The effects of these endogenous modulators on pre- and postsynaptic current are discussed. The contributions of purines to the process of quantal and non-quantal acetylcholine release into the synaptic cleft and the effects of ATP and adenosine on cholinoceptor function have been assessed. It is concluded that the role of endogenous purines is mainly confined to enhancement of the efficiency of neuromuscular transmission and synaptic adjustment of a motor unit at different modes of function.     

Modern Concepts of Cholinergic Neurotransmission at the Motor Synapse

Cholinergic synaptic contact between motor neuron and skeletal muscle fiber is perhaps one of the core objects for investigations of molecular mechanisms underlying the communication between neurons and innervated cells. In the studies conducted on this object in the past few decades, a large amount of experimental data was obtained that substantially complemented a traditional view on synaptic transmission. In particular, it was established that (i) acetylcholine is released from the nerve ending in both quantal and nonquantal ways; (ii) molecular mechanisms of the processes of the quantal acetylcholine release—spontaneous and evoked by electrical stimuli—have unique features and can be regulated independently; (iii) acetylcholine release from the nerve ending is accompanied by a release of a number of synaptically active molecules modulating the processes of secretion or reception of the main mediator; (iv) signal molecules affecting the process of cholinergic neurotransmission can be released not only from the nerve ending but also from glial cells and muscle fiber; (v) molecular mechanisms of the regulation of synaptic transmission are highly diverse and go beyond the alteration of the number of the released acetylcholine quanta. Thus, the neuromuscular junction shall be deemed currently as complicated and adaptive synapse characterized by a wide range of multiloop intercellular signaling pathways between presynaptic motor neuron ending, muscle fiber, and glial cells ensuring a high safety factor of synaptic transmission and the possibility of its fine tuning.     

Synaptic vesicle recruitment for release explored by Monte Carlo stimulation at the crayfish neuromuscular junction

Neurotransmission at chemically transmitting synapses requires calcium-mediated fusion of synaptic vesicles with the presynaptic membrane. Utilizing ultrastructural information available for the crustacean excitatory neuromuscular junction, we developed a model that employs the Monte Carlo simulation technique to follow the entry and movement of Ca2+ ions at a presynaptic active zone, where synaptic vesicles are preferentially docked for release. The model includes interaction of Ca2+ with an intracellular buffer, and variable separation between calcium channels and vesicle-associated Ca(2+)-binding targets that react with Ca2+ to trigger vesicle fusion. The end point for vesicle recruitment for release was binding of four Ca2+ ions to the target controlling release. The results of the modeling experiments showed that intracellular structures that interfere with Ca2+ diffusion (in particular synaptic vesicles) influence recruitment or priming of vesicles for release. Vesicular recruitment is strongly influenced by the separation distance between an opened calcium channel and the target controlling release, and by the concentration and binding properties of the intracellular buffers, as in previous models. When a single opened calcium channel is very close to the target, a single synaptic vesicle can be recruited. However, many of the single-channel openings actuated by a nerve impulse are likely to be ineffective for release, although they contribute to the buildup of total intracellular Ca2+. Thus, the overall effectiveness of single calcium channels in causing vesicles to undergo exocytosis is likely quite low.     

Localization effects of acetylcholine release from a synaptic vesicle at the neuromuscular junction.

A two-dimensional compartment model devised for the appropriate representation of the transient process of the spontaneous generation of miniature endplate current (MEPC) at the neuromuscular junction is applied for clarifying the biochemical significance of the quantal release mechanism of acetylcholine (ACh), a typical neurotransmitter, in the synaptic chemical transmission process. The simulation analysis with the model demonstrates that the localization of the ACh release due to the fusion of a synaptic vesicle with the presynaptic membrane has significant effects on the amplitude of MEPC and that the stronger effects are caused with the smaller diffusion coefficients of ACh in the cleft. The sharpest and highest response of MEPC is achieved when the release area is about 4 times to the natural release through the narrow pore. On the other hand, the actual localization corresponding to the natural release of ACh makes the amplitude of MEPC higher by a factor about 2.5 compared with that in the most extended release of ACh examined, implying that the natural release mechanism works as an amplifier of the MEPC with the fixed amount of ACh available.     

Quantal transmission at purinergic junctions: stochastic interaction between ATP and its receptors.

The time course of most quantal currents recorded with a small diameter electrode placed over visualized varicosities of sympathetic nerve terminals that secrete ATP was determined: these had a time to reach 90% of peak of 1.3-1.8 ms and a time constant of decay of 12-18 ms; they were unaffected by blocking ectoenzymes or the uptake of adenosine. Monte Carlo methods were used to analyze the stochastic interaction between ATP, released in a packet from a varicosity, and the underlying patch of purinoceptors, to reconstitute the time course of the quantal current. This leads to certain restrictions on the possible number of ATP molecules in a quantum (about 1000) and the density of purinoceptors at the junctions (about 1000 microns-1), given the known geometry of the junction and the kinetics of ATP action. The observed quantal current has a relatively small variability (coefficient of variation < 0.1), and this stochastic property is reproduced for a given quantum of ATP. Potentiation effects (of about 12%) occur if two quanta are released from the same varicosity because the receptor patch is not saturated even by the release of two quanta. The simulations show that quantal currents have a characteristically distinct shape for varicosities with different junctional cleft widths (50-200 nm). Finally, incorporation of an ectoenzyme with the known kinetics of ATPase into the junctional cleft allows for a quantal current of the observed time course, provided the number of ATP molecules in a quantum is increased over the number in the absence of the ATPase.     

A Machine Learning Method for the Prediction of Receptor Activation in the Simulation of Synapses

Chemical synaptic transmission involves the release of a neurotransmitter that diffuses in the extracellular space and interacts with specific receptors located on the postsynaptic membrane. Computer simulation approaches provide fundamental tools for exploring various aspects of the synaptic transmission under different conditions. In particular, Monte Carlo methods can track the stochastic movements of neurotransmitter molecules and their interactions with other discrete molecules, the receptors. However, these methods are computationally expensive, even when used with simplified models, preventing their use in large-scale and multi-scale simulations of complex neuronal systems that may involve large numbers of synaptic connections. We have developed a machine-learning based method that can accurately predict relevant aspects of the behavior of synapses, such as the percentage of open synaptic receptors as a function of time since the release of the neurotransmitter, with considerably lower computational cost compared with the conventional Monte Carlo alternative. The method is designed to learn patterns and general principles from a corpus of previously generated Monte Carlo simulations of synapses covering a wide range of structural and functional characteristics. These patterns are later used as a predictive model of the behavior of synapses under different conditions without the need for additional computationally expensive Monte Carlo simulations. This is performed in five stages: data sampling, fold creation, machine learning, validation and curve fitting. The resulting procedure is accurate, automatic, and it is general enough to predict synapse behavior under experimental conditions that are different to the ones it has been trained on. Since our method efficiently reproduces the results that can be obtained with Monte Carlo simulations at a considerably lower computational cost, it is suitable for the simulation of high numbers of synapses and it is therefore an excellent tool for multi-scale simulations.     

Subunits in quantal transmission at the mouse neuromuscular junction: tests of peak intervals in amplitude distributions

The regular spacing of peaks throughout the amplitude distribution of miniature end-plate potentials, quantal evoked end-plate potentials and quantal currents was demonstrated using autocorrelations and power density spectra calculated from the number of events in the successive bins of the histograms built by Matteson et al. (1979), Kriebel & Florey (1983) and Erxleben & Kriebel (1984). At the same mouse neuromuscular junction, the calculated interpeak was constant for evoked and spontaneous quantal releases, throughout sequential sampling and after change of bin size. The presence of regular peak intervals supports the hypothesis that quantal potentials are composed of potential subunits the size of the smallest subminiature potential. Challenging the hypothesis of an acetylcholine quantum composed of acetylcholine subunits, a postsynaptic origin of the subunit is proposed on the basis of the spatial arrangement in rows of the ACh receptors. The ACh-saturating patch evoked by a quantum release (Land et al., 1980, 1981) activates 10-20 rows of receptors, which is roughly the number of subunits composing a quantal event. Therefore the position of the ACh patch or the continuous variations in its size might cause stepwise variations in the total number of ACh receptors activated by an ACh quantum.     

A Brownian simulation model of glutamate synaptic diffusion in the femtosecond time scale

 To gain a better understanding of the elementary unit of synaptic communication between hippocampal neurons, we simulated the release of glutamate from a single pre-synaptic vesicle and its diffusion into the synaptic cleft. Diffusion of glutamate was simulated by a Brownian model based on Langevin equations. The model was implemented for parallel computer simulation and tested under different conditions of glutamate release and different geometrical and physical characteristics of the synaptic cleft. All the tested parameters have shown to be important for the synaptic responses. The results show that the synaptic transmission efficacy is influenced by many different geometrical parameters and, as a consequence, the quality of the excitatory post-synaptic response can be very different in the same synapse. The variability in the quantal response found by several authors can also be explained by physical parameters other than by variations in the quantal content of the synaptic vesicle as proposed by these authors. Received: 6 October 1999 / Accepted: 29 February 2000     

Stochastic model of central synapses: slow diffusion of transmitter interacting with spatially distributed receptors and transporters.

A detailed mathematical analysis of the diffusion process of neurotransmitter inside the synaptic cleft is presented and the spatio-temporal concentration profile is calculated. Using information about the experimentally observed time course of glutamate in the cleft the effective diffusion coefficient Dnet is estimated as Dnet approximately 20-50 nm(2) microseconds(-1), implying a strong reduction compared with free diffusion in aqueous solution. The tortuosity of the cleft and interactions with transporter molecules are assumed to affect the transmitter motion. We estimate the transporter density to be 5170 to 8900 micrometer(-2) in the synaptic cleft and its vicinity, using the experimentally observed time constant of glutamate. Furthermore a theoretical model of synaptic transmission is presented, taking the spatial distribution of post-synaptic (AMPA-) receptors into account. The transmitter diffusion and receptor dynamics are modeled by Monte Carlo simulations preserving the typically observed noisy character of post-synaptic responses. Distributions of amplitudes, rise and decay times are calculated and shown to agree well with experiments. Average open probabilities are computed from a novel kinetic model and are shown to agree with averages over many Monte Carlo runs. Our results suggest that post-synaptic currents are only weakly potentiated by clustering of post-synaptic receptors, but increase linearly with the total number of receptors. Distributions of amplitudes and rise times are used to discriminate between different morphologies, e.g. simple and perforated synapses. A skew in the miniature amplitude distribution can be caused by multiple release of pre-synaptic vesicles at perforated synapses.     

The effects of geometrical parameters on synaptic transmission: a Monte Carlo simulation study

Monte Carlo simulations of transmitter diffusion and its interactions with postsynaptic receptors have been used to study properties of quantal responses at central synapses. Fast synaptic responses characteristic of those recorded at glycinergic junctions on the teleost Mauthner cell (time to peak approximately 0.3-0.4 ms and decay time constant approximately 3-6 ms) served as the initial reference, and smaller contacts with fewer postsynaptic receptors were also modeled. Consistent with experimental findings, diffusion, simulated using a random walk algorithm and assuming a diffusion coefficient of 0.5-1.0 x 10(-5) cm2 s(-1), was sufficiently fast to account for transmitter removal from the synaptic cleft. Transmitter-receptor interactions were modeled as a two-step binding process, with the double-bound state having opened and closed conformations. Addition of a third binding step only slightly decreased response amplitude but significantly slowed both its rising and decay phases. The model allowed us to assess the sources of response variability and the likelihood of postsynaptic saturation as functions of multiple kinetic and spatial parameters. The method of nonstationary fluctuation analysis, typically used to estimate the number of functional channels at a synapse and single channel current, proved unreliable, presumably because the receptors in the postsynaptic matrix are not uniformly exposed to the same profile of transmitter concentration. Thus, the time course of the probability of channel opening most likely varies among receptors. Finally, possible substrates for phenomena of synaptic plasticity, such as long-term potentiation, were explored, including the diameter of the contact zone, defined by the region of pre- and postsynaptic apposition, the number and distribution of the receptors, and the degree of vesicle filling. Surprisingly, response amplitude is quite sensitive to the size of the receptor-free annulus surrounding the receptor cluster, such that expansion of the contact zone could produce an appreciable increase in quantal size, normally attributed to either the presence of more receptors or the release of more transmitter molecules.     

Testing the fit of a quantal model of neurotransmission

Many studies of synaptic transmission have assumed a parametric model to estimate the mean quantal content and size or the effect upon them of manipulations such as the induction of long-term potentiation. Classical tests of fit usually assume that model parameters have been selected independently of the data. Therefore, their use is problematic after parameters have been estimated. We hypothesized that Monte Carlo (MC) simulations of a quantal model could provide a table of parameter-independent critical values with which to test the fit after parameter estimation, emulating Lilliefors's tests. However, when we tested this hypothesis within a conventional quantal model, the empirical distributions of two conventional goodness-of-fit statistics were affected by the values of the quantal parameters, falsifying the hypothesis. Notably, the tests' critical values increased when the combined variances of the noise and quantal-size distributions were reduced, increasing the distinctness of quantal peaks. Our results support two conclusions. First, tests that use a predetermined critical value to assess the fit of a quantal model after parameter estimation may operate at a differing unknown level of significance for each experiment. Second, a MC test enables a valid assessment of the fit of a quantal model after parameter estimation.     

The Influence of Synaptic Size on AMPA Receptor Activation: A Monte Carlo Model

Physiological and electron microscope studies have shown that synapses are functionally and morphologically heterogeneous and that variations in size of synaptic junctions are related to characteristics such as release probability and density of postsynaptic AMPA receptors. The present article focuses on how these morphological variations impact synaptic transmission. We based our study on Monte Carlo computational simulations of simplified model synapses whose morphological features have been extracted from hundreds of actual synaptic junctions reconstructed by three-dimensional electron microscopy. We have examined the effects that parameters such as synaptic size or density of AMPA receptors have on the number of receptors that open after release of a single synaptic vesicle. Our results indicate that the maximum number of receptors that will open after the release of a single synaptic vesicle may show a ten-fold variation in the whole population of synapses. When individual synapses are considered, there is also a stochastical variability that is maximal in small synapses with low numbers of receptors. The number of postsynaptic receptors and the size of the synaptic junction are the most influential parameters, while the packing density of receptors or the concentration of extrasynaptic transporters have little or no influence on the opening of AMPA receptors.     

Analytical description of the activation of multi-state receptors by continuous neurotransmitter signals at brain synapses.

Chemical synaptic transmission is a fundamental component of interneuronal communications in the central nervous system (CNS). Discharge of a presynaptic vesicle containing a few thousand molecules (a quantum) of neurotransmitter into the synaptic cleft generates a transmitter concentration signal that drives postsynaptic ion-channel receptors. These receptors exhibit multiple states, with state transition kinetics dependent on neurotransmitter concentration. Here, a novel and simple analytical approach for describing gating of multi-state receptors by signals with complex continuous time courses is used to describe the generation of glutamate-mediated quantal postsynaptic responses at brain synapses. The neurotransmitter signal, experienced by multi-state N-methyl-D-aspartate (NMDA)- and L-alpha-amino-3-hydroxy-5-methyl-4-isoxazolepropionate (AMPA)-type glutamate receptors at specific points in a synaptic cleft, is approximated by a series of step functions of different intensity and duration and used to drive a Markovian, multi-state kinetic scheme that describes receptor gating. Occupancy vectors at any point in time can be computed interatively from the occupancy vectors at the times of steps in transmitter concentration. Multi-state kinetic schemes for both the low-affinity AMPA subtype of glutamate receptor and for the high-affinity NMDA subtype are considered, and expected NMDA and AMPA components of synaptic currents are calculated. The amplitude of quantal responses mediated by postsynaptic receptor clusters having specific spatial distributions relative to foci of quantal neurotransmitter release is then calculated and related to the displacement between the center of the postsynaptic receptor cluster and the focus of synaptic vesicle discharge. Using this approach we show that the spatial relation between the focus of release and the center of the postsynaptic receptor cluster affects synaptic efficacy. We also show how variation in this relation contributes to variation in synaptic current amplitudes.     

Modelling Vesicular Release at Hippocampal Synapses

We study local calcium dynamics leading to a vesicle fusion in a stochastic, and spatially explicit, biophysical model of the CA3-CA1 presynaptic bouton. The kinetic model for vesicle release has two calcium sensors, a sensor for fast synchronous release that lasts a few tens of milliseconds and a separate sensor for slow asynchronous release that lasts a few hundred milliseconds. A wide range of data can be accounted for consistently only when a refractory period lasting a few milliseconds between releases is included. The inclusion of a second sensor for asynchronous release with a slow unbinding site, and thereby a long memory, affects short-term plasticity by facilitating release. Our simulations also reveal a third time scale of vesicle release that is correlated with the stimulus and is distinct from the fast and the slow releases. In these detailed Monte Carlo simulations all three time scales of vesicle release are insensitive to the spatial details of the synaptic ultrastructure. Furthermore, our simulations allow us to identify features of synaptic transmission that are universal and those that are modulated by structure.