Chapter Thirteen - Data-Driven Modeling of Synaptic Transmission and Integration

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Abstract

In this chapter, we describe how to create mathematical models of synaptic transmission and integration. We start with a brief synopsis of the experimental evidence underlying our current understanding of synaptic transmission. We then describe synaptic transmission at a particular glutamatergic synapse in the mammalian cerebellum, the mossy fiber to granule cell synapse, since data from this well-characterized synapse can provide a benchmark comparison for how well synaptic properties are captured by different mathematical models. This chapter is structured by first presenting the simplest mathematical description of an average synaptic conductance waveform and then introducing methods for incorporating more complex synaptic properties such as nonlinear voltage dependence of ionotropic receptors, short-term plasticity, and stochastic fluctuations. We restrict our focus to excitatory synaptic transmission, but most of the modeling approaches discussed here can be equally applied to inhibitory synapses. Our data-driven approach will be of interest to those wishing to model synaptic transmission and network behavior in health and disease.

Introduction

Some of the first intracellular voltage recordings from the neuromuscular junction (NMJ) revealed the presence of spontaneous miniature end plate potentials with fast rise and slower decay kinetics.1 The similarity of these “mini” events to the smallest events evoked by nerve stimulation, together with the discrete nature of the fluctuations in the amplitude of the end plate potentials,2 lead to the hypothesis that transmitter was released probabilistically in discrete all-or-none units called “quanta,”3 units that were subsequently shown to be vesicles containing neurotransmitter. The quantum hypothesis is an elegantly simple yet extremely powerful statistical model of transmitter release: the average number of quanta released at a synapse per stimulus (quantal content, m) is simply the product of the total number of quanta available for release (NT) and their release probability (P):m=NTPQuantitative comparison of the predictions of the quantum hypothesis against experimental measurements confirmed the hypothesis,3 albeit under nonphysiological conditions of low release probabilities. Subsequent electron micrograph studies revealed presynaptic vesicles clustered at active zones,4, 5, 6, 7 providing compelling morphological equivalents for the quanta and their specialized release sites. Other work around the same time revealed the dynamic nature of synaptic transmission at the NMJ, providing the first concepts for activity-dependent short-term changes in synaptic strength.8, 9 Further work by Katz and colleagues lead to the concept of Ca2 +-dependent vesicular release and the refinement of ideas regarding the activation of postsynaptic receptors.3 Together, this early body of work on the NMJ provided the basis for our current understanding of the intricate signaling cascade underlying synaptic transmission. The basic mechanisms underlying synaptic transmission are summarized in Fig. 13.1: an action potential, propagating down the axon of the presynaptic neuron, invades synaptic terminals. The brief depolarization of the terminals causes voltage-gated Ca2 + channels (VGCCs) to open, leading to Ca2 + influx and a transient increase in the intracellular Ca2 + concentration ([Ca]i) in the vicinity of the VGCCs. For those vesicles docked at a release site near one or more VGCCs, the local increase in [Ca]i triggers the vesicles to fuse with the terminal membrane and release their content of neurotransmitter into the synaptic cleft. The released neurotransmitter diffuses across the narrow synaptic cleft and binds to postsynaptic ionotropic receptors, transiently increasing their open probability. The resulting flow of Na+ and K+ through the receptors’ ion channels results in an excitatory postsynaptic potential (EPSP) or excitatory postsynaptic current (EPSC) depending on whether the intracellular recording is made under a current- or voltage-clamp configuration.

Some 20 years after the early work on the NMJ, development of the patch-clamp method increased the signal-to-noise ratio of electrophysiological recordings by several orders of magnitude over traditional sharp-electrode recordings.10 The patch-clamp method not only confirmed the existence of individual ion channels but also enabled resolution of significantly smaller EPSCs, thereby paving the way for studies of synaptic transmission in the central nervous system (CNS). Although these studies revealed the basic mechanisms underlying synaptic transmission are largely similar at the NMJ and in the brain (Fig. 13.1), there are a number of key differences. For example, whereas synaptic transmission in the NMJ is mediated by the release of 100–1000 vesicles2 at highly elongated active zones,11 synaptic transmission between neurons in the brain is typically mediated by the release of just a few vesicles at a handful of small active zones.12, 13 The number of postsynaptic receptors is also quite different: vesicle release activates thousands of postsynaptic receptors in the NMJ14 but only a few (~ 10–100) at central excitatory synapses.15, 16 These differences in scale link directly to synaptic function: the large potentials generated at the NMJ ensure a reliable relay of motor command signals from presynaptic neuron to postsynaptic muscle. In contrast, the much smaller potentials generated by central synapses require spatiotemporal summation in order to trigger action potentials.

Another important distinction between the NMJ and central synapses is the difference in neurotransmitter (acetylcholine at the NMJ vs. glutamate, GABA, glycine, etc., in the CNS) and the diversity in postsynaptic receptors and their function. Here we focus on excitatory central synapses, where two major classes of ionotropic glutamate receptors, AMPA and NMDA receptors (AMPARs and NMDARs), are colocalized.17, 18 These two receptor types have different gating kinetics and current–voltage relations and therefore play distinct roles in synaptic transmission. The majority of AMPARs, for example, have relatively fast kinetics and a linear (ohmic) current–voltage relation, often expressed as:IAMPAR=GAMPARVEAMPARwhere V is the membrane potential and EAMPAR is the reversal potential of the AMPAR conductance (GAMPAR), which is typically 0 mV. Both of these properties, that is, fast kinetics and a linear current–voltage relation, make AMPARs well suited for mediating temporally precise signaling and setting synaptic weight. NMDARs, in contrast, have slower kinetics and a nonlinear current–voltage relation, the latter caused by Mg2 + block at hyperpolarized potentials.19 These properties make NMDARs well suited for coincidence detection and plasticity, since presynaptic glutamate release and postsynaptic depolarization are required for NMDAR activation.20 Certain subtypes of NMDARs, however, show a weaker Mg2 + block (i.e., those containing the GluN2C and GluN2D subunits) and therefore create substantial synaptic current at hyperpolarized potentials.21, 22 These types of NMDARs are thought to enhance synaptic transmission by enabling temporal integration of low-frequency inputs.22 Of course, numerous other differences exist between the NMJ and central synapses, including those pertaining to stochasticity- and time-dependent plasticity. These are discussed further in the next section where we introduce the MF-to-GC synapse, our synapse of choice for providing accurate data for the synaptic models presented in this chapter.

The input layer of the cerebellum receives sensory and motor signals via MFs23 which form large en passant synapses, each of which contacts several GCs (Fig. 13.2A). Although GCs are the smallest neuron in the vertebrate brain, they account for more than half of all neurons. Each GC receives excitatory synaptic input from 2 to 7 MFs, and each synaptic connection consists of a handful of active zones.27, 28 The small number of synaptic inputs, along with a small soma and electrically compact morphology, makes GCs particularly suitable for studying synaptic transmission.15, 18 In Fig. 13.2B, we show representative examples of EPSCs recorded at a single MF–GC synaptic connection under resting basal conditions (gray traces). Here, fluctuations in the peak amplitude of the EPSCs highlight the stochastic behavior of synaptic transmission introduced above. Analysis of such fluctuations using multiple-probability fluctuation analysis (MPFA), a technique based on a multinomial statistical model, has provided estimates for NT, P and the postsynaptic response to a quantum of transmitter (Q), for single MF–GC connections. MPFA indicates that at low frequencies synaptic transmission is meditated by 5–10 readily releasable vesicles (or, equivalently the number of functional release sites NT), with each vesicle or site having a vesicular release probability (P) of ~ 0.5.26, 29 Experiments with rapidly equilibrating AMPAR antagonists suggest that release is predominantly univesicular at this synapse (one vesicle released per synaptic contact), an interpretation that is supported by the finding that at some weak MF–GC connections a maximum of only one vesicle is released even when P is increased to high levels.15, 26

Synaptic responses to low-frequency presynaptic stimuli (e.g., those in Fig. 13.2B) provide useful information about NT, P, and Q under resting conditions. To explore how these quantal synaptic parameters change in an activity-dependent manner, however, paired-pulse stimulation protocols or high-frequency trains of stimuli are required. Figure 13.2C shows an example of the latter, where responses of a single MF–GC connection to the same 100 Hz train of stimuli are superimposed (gray traces). Here, fluctuations in the peak amplitude of the EPSCs can still be seen (see inset), but successive peaks between stimuli also show clear signs of depression. The average of all responses (black trace) reveals the depression more clearly. Although by eye, signs of facilitation are not apparent in Fig. 13.2C, facilitation at this synapse most likely exists. We know this since lowering P at this synapse, by lowering the extracellular Ca2+ concentration, has revealed the presence of both depression and facilitation; however, because depression predominates under normal conditions, facilitation is not always apparent.29 As described in detail later in this chapter, mathematical models have been developed to simulate synaptic depression and facilitation. If used appropriately, these models can provide useful insights into the underlying mechanisms of synaptic transmission. Such models have revealed, for example, a rapid rate of vesicle reloading at the MF–GC synapse (k1 = 60–80 ms 1) as well as a large pool of vesicles that can be recruited rapidly at each release site (~ 30029, 30, 31). These findings offer an explanation as to how the MF–GC synapse can sustain high-frequency signaling for prolonged periods of time.

The MF–GC synapse forms part of a glomerular-type synapse, which also occur in the thalamus and dorsal spinocerebellar tract. While the purpose of the glomerulus has not been determined definitively, experimental evidence from the MF–GC synapse indicates this glial-ensheathed structure promotes transmitter spillover between excitatory synaptic connections24, 32 and between excitatory and inhibitory synaptic connections.33 AMPAR-mediated EPSCs recorded from a MF–GC connection, therefore, exhibit both a fast “direct” component arising from quantal release at the MF–GC connection under investigation (Fig. 13.2B, green trace) and a slower component mediated by glutamate spillover from neighboring MF–GC connections (blue trace). While direct quantal release is estimated to activate about 50% of postsynaptic AMPARs at the peak of the EPSC,34 spillover is estimated to activate a significantly smaller fraction. However, because spillover produces a prolonged presence of glutamate in the synaptic cleft, activation of AMPARs by spillover can contribute as much as 50% of the AMPAR-mediated charge delivered to GCs.24

Glutamate spillover also activates NMDARs, but mostly at mature MF–GC synapses when the NMDARs occupy a perisynaptic location.35 At a more mature time of development, MF–GC synapses also exhibit a weak Mg2 + block due to the expression of GluN2C and/or GluN2D subunits.22, 36, 37 The weak Mg2 + block allows NMDARs to pass a significant amount of charge at subthreshold potentials, thereby creating a spillover current comparable in size to the AMPAR-mediated spillover current. Using several of the modeling techniques discussed in this chapter, we were able to show the summed contribution from both AMPAR and NMDAR spillover currents enables GCs to integrate over comparatively long periods of time, thereby enabling transmission of low-frequency MF signals through the input layer of the cerebellum.22

In the following sections, we describe how to capture the various properties of synaptic transmission recorded at the MF–GC synapse in mathematical forms that can be used in computer simulations. We start with the most basic features of the synapse, the postsynaptic conductance waveform, and the resulting postsynaptic current, and add biological detail from there. However, several aspects of synaptic transmission are beyond the scope of this chapter. These include long-term plasticity (i.e., Hebbian learning) and presynaptic Ca2 + dynamics. Mathematical models of these synaptic processes can be found elsewhere.38, 39, 40, 41, 42

Section snippets

Constructing Synaptic Conductance Waveforms from Voltage-Clamp Recordings

The time course of a synaptic conductance, denoted Gsyn(t), can be computed from the synaptic current, Isyn(t), measured at a particular holding potential (Vhold) using the whole-cell voltage-clamp technique. If the synapse under investigation is electrotonically close to the somatic patch pipette, as is the case with the MF–GC synapse, then adequate voltage clamp can be achieved and the measured Isyn(t) will have relatively small distortions due to poor space clamp. On the other hand, if the

Empirical Models of Voltage-Dependent Mg2 + Block of the NMDA Receptor

The voltage dependence of the synaptic AMPAR component can usually be modeled with the simple linear current–voltage relation described in Eq. (13.2). In contrast, the synaptic NMDAR component exhibits strong voltage dependence due to Mg2 + binding inside the receptor's ion channel.19 The block is strongest near the neuronal resting potential and becomes weaker as the membrane potential becomes more depolarized. This unique characteristic of NMDARs allows them to behave like logical AND gates:

Construction of Presynaptic Spike Trains with Refractoriness and Pseudo-Random Timing

To simulate the temporal patterns of activation that a synapse is likely to experience in vivo, it is necessary to construct trains of discrete events that can be used to activate model synaptic conductance events, Gsyn(t), as described in Eqs. (13.4), (13.5), (13.6), (13.7), at specific times (i.e., tj). These trains can then be used to mimic the timing of presynaptic action potentials as they reach the synaptic terminals. Real presynaptic spike trains can exhibit a wide range of statistics.

Synaptic Integration in a Simple Conductance-Based Integrate-and-Fire Neuron

Once we have built a train of presynaptic spike times (tj) and synapses with realistic conductance waveforms (GAMPAR and GNMDAR) and current–voltage relations (IAMPAR and INMDAR), we are well on our way to simulating synaptic integration in a simple point neuron like the GC, which is essentially a single RC circuit with a battery. The simplest neuronal integrator is the integrate-and-fire (IAF) model.72 Most modern versions of the IAF model act as a leaky integrator with a voltage threshold and

Short-Term Synaptic Depression and Facilitation

So far, we have only considered the simulation of fixed amplitude synaptic conductances recorded under basal conditions. At synapses with a relatively high release probability, repetitive stimulation at short time intervals often results in depression of the postsynaptic response (see, e.g., Fig. 13.2C). This kind of synaptic depression was first described by Eccles et al.8 for endplate potentials at the NMJ and has since been described for synapses in the CNS. Because recovery from synaptic

Simulating Trial-to-Trial Stochasticity

Up until now, the synaptic models we have presented are deterministic. However, as mentioned in Section 1, synapses exhibit considerable variability in their trial-to-trial response (see, e.g., Fig. 13.2B) due to the probabilistic nature of the mechanisms underlying synaptic transmission, from the release of quanta to the binding and opening of postsynaptic ionotropic receptors (Fig. 13.1). Here, we discuss the simulation of three sources of stochastic variation that account for the bulk of the

Going Microscopic

The models discussed in this chapter are intended to capture the basic macroscopic features of synaptic transmission, mainly the time and voltage dependence of the transfer of charge into the postsynaptic neuron. These types of models are useful for investigating signal processing at the cellular and network level but are generally not as useful for investigating the mechanism underlying signal transmission at a microscopic level. Hence, other modeling approaches are usually adopted when

Simulators and Standardized Model Descriptions

A number of options exist for creating computer simulations of the synaptic models presented in this chapter. Generic simulation and analysis packages like MATLAB (http://www.mathworks.co.uk/products/matlab/) and Igor Pro (http://www.wavemetrics.com) are commonly used for simulating relatively simple models. These packages have the advantage that the user is completely in control of the model structure and can perform analysis in the same scripting language as that used in the model

Summary

In this chapter, we discussed how mathematical models can capture various aspects of synaptic transmission. At their most basic level, the models are simple empirical descriptions of the average conductance waveform and current–voltage relation of postsynaptic receptors. Above this basic level, the models can be extended to capture more and more of the behavior of real synapses, including stochasticity and short-term plasticity. Throughout the chapter, we examined how well the different models

Acknowledgments

We thank Bóris Marin, Eugenio Piasini, Arnd Roth, and Stefan Hallermann for their comments on the manuscript, and Padraig Gleeson for his contribution to the section on simulator packages. This work was funded by the Wellcome Trust (086699) and ERC. R. A. S. holds a Wellcome Trust Principal Research Fellowship (095667) and an ERC Advanced Grant (294667).

References (103)

  • J.W. Johnson et al.

    Voltage-dependent block by intracellular Mg2 + of N-methyl-D-aspartate-activated channels

    Biophys J

    (1990)
  • D.H. Johnson et al.

    Application of a point process model to responses of cat lateral superior olive units to ipsilateral tones

    Hear Res

    (1986)
  • R.B. Stein

    A theoretical analysis of neuronal variability

    Biophys J

    (1965)
  • N. Lipstein et al.

    Dynamic control of synaptic vesicle replenishment and short-term plasticity by Ca2 +-calmodulin-Munc13-1 signaling

    Neuron

    (2013)
  • T. Sakaba et al.

    Calmodulin mediates rapid recruitment of fast-releasing synaptic vesicles at a calyx-type synapse

    Neuron

    (2001)
  • C.F. Stevens et al.

    Activity-dependent modulation of the rate at which synaptic vesicles become available to undergo exocytosis

    Neuron

    (1998)
  • L.G. Wu et al.

    The reduced release probability of releasable vesicles during recovery from short-term synaptic depression

    Neuron

    (1999)
  • J.I. Wadiche et al.

    Multivesicular release at climbing fiber-Purkinje cell synapses

    Neuron

    (2001)
  • J.D. Clements et al.

    Unveiling synaptic plasticity: a new graphical and analytical approach

    Trends Neurosci

    (2000)
  • R.A. Silver

    Estimation of nonuniform quantal parameters with multiple-probability fluctuation analysis: theory, application and limitations

    J Neurosci Methods

    (2003)
  • F. Minneci et al.

    Estimation of the time course of neurotransmitter release at central synapses from the first latency of postsynaptic currents

    J Neurosci Methods

    (2012)
  • A.L. Fogelson et al.

    Presynaptic calcium diffusion from various arrays of single channels. Implications for transmitter release and synaptic facilitation

    Biophys J

    (1985)
  • I. Bucurenciu et al.

    Nanodomain coupling between Ca2 + channels and Ca2 + sensors promotes fast and efficient transmitter release at a cortical GABAergic synapse

    Neuron

    (2008)
  • P. Fatt et al.

    Spontaneous subthreshold activity at motor nerve endings

    J Physiol

    (1952)
  • J. del Castillo et al.

    Quantal components of the end-plate potential

    J Physiol

    (1954)
  • B. Katz

    The Release of Neural Transmitter Substances

    (1969)
  • R. Couteaux et al.

    Synaptic vesicles and pouches at the level of “active zones” of the neuromuscular junction

    C R Acad Sci Hebd Seances Acad Sci D

    (1970)
  • E.D. De Robertis et al.

    Some features of the submicroscopic morphology of synapses in frog and earthworm

    J Biophys Biochem Cytol

    (1955)
  • G.E. Palade et al.

    Electron microscope observations of interneuronal and neuromuscular synapses

    Anat Rec

    (1954)
  • S.L. Palay

    Synapses in the central nervous system

    J Biophys Biochem Cytol

    (1956)
  • J.C. Eccles et al.

    Nature of the ‘end-plate potential’ in curarized muscle

    J Neurophysiol

    (1941)
  • A.W. Liley et al.

    An electrical investigation of effects of repetitive stimulation on mammalian neuromuscular junction

    J Neurophysiol

    (1953)
  • E. Neher et al.

    The extracellular patch clamp: a method for resolving currents through individual open channels in biological membranes

    Pflugers Arch

    (1978)
  • M.L. Harlow et al.

    The architecture of active zone material at the frog's neuromuscular junction

    Nature

    (2001)
  • A.A. Biró et al.

    Quantal size is independent of the release probability at hippocampal excitatory synapses

    J Neurosci

    (2005)
  • R.A. Silver et al.

    High-probability uniquantal transmission at excitatory synapses in barrel cortex

    Science

    (2003)
  • R.A. Silver et al.

    Non-NMDA glutamate receptor occupancy and open probability at a rat cerebellar synapse with single and multiple release sites

    J Physiol

    (1996)
  • J.M. Bekkers et al.

    NMDA and non-NMDA receptors are co-localized at individual excitatory synapses in cultured rat hippocampus

    Nature

    (1989)
  • R.A. Silver et al.

    Rapid-time-course miniature and evoked excitatory currents at cerebellar synapses in situ

    Nature

    (1992)
  • P. Ascher et al.

    The role of divalent cations in the N-methyl-D-aspartate responses of mouse central neurones in culture

    J Physiol

    (1988)
  • T.V. Bliss et al.

    A synaptic model of memory: long-term potentiation in the hippocampus

    Nature

    (1993)
  • E.J. Schwartz et al.

    NMDA receptors with incomplete Mg2 + block enable low-frequency transmission through the cerebellar cortex

    J Neurosci

    (2012)
  • J.C. Eccles et al.

    The Cerebellum as a Neuronal Machine

    (1967)
  • S.L. Palay et al.

    Cerebellar Cortex: Cortex and Organization

    (1974)
  • P.B. Sargent et al.

    Rapid vesicular release, quantal variability, and spillover contribute to the precision and reliability of transmission at a glomerular synapse

    J Neurosci

    (2005)
  • L. Cathala et al.

    Changes in synaptic structure underlie the developmental speeding of AMPA receptor-mediated EPSCs

    Nat Neurosci

    (2005)
  • R.L. Jakab et al.

    Quantitative morphology and synaptology of cerebellar glomeruli in the rat

    Anat Embryol (Berl)

    (1988)
  • C. Saviane et al.

    Fast vesicle reloading and a large pool sustain high bandwidth transmission at a central synapse

    Nature

    (2006)
  • S.J. Mitchell et al.

    GABA spillover from single inhibitory axons suppresses low-frequency excitatory transmission at the cerebellar glomerulus

    J Neurosci

    (2000)
  • D.A. DiGregorio et al.

    Desensitization properties of AMPA receptors at the cerebellar mossy fiber granule cell synapse

    J Neurosci

    (2007)
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