Data-driven modeling of synaptic transmission and integration

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.

Keywords: AMPA receptor; Chemical synapses; Conductance waveforms; Depletion models; Integrate-and-fire models; Mathematical models; NMDA receptor; Poisson spike trains; Quantal release; Short-term plasticity; Synaptic depression; Synaptic integration; Synaptic transmission; Vesicular release.

Figure 13.1. Cartoon illustrating the basic sequence of events underlying synaptic transmission

The sequence starts with an action potential (AP) invading a presynaptic terminal, leading to the opening of voltage-gated Ca²⁺ channels (VGCCs), some of which are located near vesicle release sites within one or more active zones. For those release sites containing a readily releasable vesicle, the local rise in [Ca²⁺]_i causes the fusion of the vesicle with the terminal’s membrane, resulting in the release of neurotransmitter packed inside the vesicle. The neurotransmitter diffuses across the synaptic cleft to reach the postsynaptic membrane where it binds to ionotropic receptors, causing the channels to open and pass Na⁺ and K⁺. The permeation of these ions through the ionotropic receptors leads to a local injection of current, known as the EPSC. The EPSC often contains fast and slow components due to the fast activation of receptors immediately opposite to the vesicle release site and the slower activation of receptors further away (i.e., extrasynaptic). Kinetics of the EPSC will also depend on the receptor’s affinity for the neurotransmitter and the receptor’s gating properties, which may include blocked and desensitization states.

Figure 13.2. Synaptic transmission at the cerebellar MF–GC synapse

(A) Electron micrograph of a cerebellar MF terminal filled with thousands of synaptic vesicles and a few large mitochondria. Synaptic contacts with GC dendrites appear along the contours of the MF membrane at several locations, evident by the wider and darker appearance of the membrane due to clustering of proteins within the presynaptic active zone and postsynaptic density. (B) Superimposed AMPAR-mediated EPSCs (gray) recorded from a single MF–GC connection, showing considerable variability in amplitude and time course from trial to trial. On some trials, failure of direct release revealed a spillover current with slow rise time. Such trials were separated using the rise time criteria of Ref. . The average direct-release component (green) was computed by subtracting the average spillover current (blue) from the average total EPSC (black). Arrow denotes time of extracellular MF stimulation, which occurred at a slow frequency of 2 Hz; most of the stimulus artifact has been blanked for display purposes. (C) Superimposed AMPAR-mediated EPSCs (gray) recorded from a single MF–GC connection and their average (black). The MF was stimulated at 100 Hz with an external electrode (arrows at top). Successive EPSCs show clear signs of depression. Inset shows EPSC responses to fourth stimulus on expanded timescale, showing the variation in peak amplitude. Stimulus artifacts have been blanked. (D) Average direct-release AMPAR conductance waveform (gray) fit with G_syn(t) defined by the following functions: alpha (Eq. 13.5), one-exponential (Eq. 13.4), two-exponential (Eq. 13.6), multiexponential (4-Exp, Eq. 13.7). Most functions gave a good fit except the one-exponential function (blue). The conductance waveform was computed from the average current waveform in (B) via Eq. (13.3). (E) Same as (D) but for the average spillover component in (B). Most functions gave a good fit except the alpha function (green). (F) Same as (D) but for an average NMDAR-mediated conductance waveform computed from four different MF–GC connections. Again, most functions gave a good fit except the alpha function (green). Dashed lines denote 0. (A) Image from Palay and Chan-Palay with permission. (B) Data from Sargent et al. with permission.

Figure 13.3. Weak Mg²⁺ block in GluN2C-containing NMDARs

(A) Current–voltage relation of an NMDAR current from a mature GC (black) fit to Eq. (13.8) (E_NMDAR = 0 mV) where φ(V) was defined by either a two-state kinetic model (blue; Eq. 13.11) or a three-state kinetic model that includes Mg²⁺ permeation (red; Eq. 13.12). The latter kinetic model produced the better fit. Kinetic models are shown at top. (B) Percent of unblocked NMDARs, φ(V), from the three-state kinetic model fit in (A) (red), compared to φ(V) derived from fits to the same model for another data set of mature GCs (purple; data from Ref. 46) and immature GCs (black; data from Ref. 50). At nearly all potentials, NMDARs from mature GCs show weaker Mg²⁺ block than those from immature GCs. This difference is presumably due to the developmental maturation switch in GCs from GluN2A/B-containing receptors to GluN2C-containing receptors, discussed in text. (C) IAF simulations (Eq. 13.20) of a GC with immature (top, +GluN2A/B) and mature (bottom, +GluN2C) NMDARs, using φ(V) functions in (B) (black and red, respectively), demonstrating the enhanced depolarization and spiking under mature NMDAR conditions. Identical simulations were repeated with G_NMDAR = 0 (yellow) and G_AMPAR = 0 (green) to compare the contribution of AMPARs and NMDARs to depolarizing the membrane. G_AMPAR consisted of a simulated direct and spillover component, both with depression, as described in Fig. 13.5F. G_NMDAR was simulated with both depression and facilitation, as described in Fig. 13.5G. The peak value of the G_NMDAR waveform equaled that of the G_AMPAR waveform, giving an amplitude ratio of unity, which is in the physiological range for GCs. The total synaptic current consisted of the sum of four independent I_syn, each representing a different MF input. Spike times for each MF input were generated for a constant mean rate of 60 Hz (Eq. 13.16), producing a total MF input of 240 Hz. Total I_syn also contained the following tonic GABA-receptor current not discussed in this chapter: I_GABAR = 0.438(V + 75). IAF membrane parameters matched the average values computed from a population of 242 GCs: C_m = 3.0 pF, R_m = 0.92 GΩ, V_rest = −80 mV. Action potential parameters were: V_thresh = −40 mV (gray dashed line), V_peak = 32 mV, V_reset = −63 mV, τ_AR = 2 ms. Action potentials were truncated to −15 mV for display purposes. (D) Average output spike rate of the IAF GC model as shown in (C) as a function of total MF input rate for immature (bottom left) and mature (bottom right) NMDARs, again demonstrating the enhanced spiking caused by GluN2C subunits. A total of 242 simulations were computed using C_m, R_m, V_rest values derived from a data base of 242 real GCs (top distributions, red lines denote average population values), with the average output spike rate plotted as black circles. Red line denotes one GC simulation whose C_m, R_m, V_rest matched the average population values shown at top, which are the same parameters used in (C). Note, the output spike rate of this “average GC” simulation is twice as large as the average of all 242 GC simulations due to the nonlinear behavior of the IAF model. Data in this figure is from Schwartz et al. with permission.

Figure 13.4. Simulated spike trains with refractoriness and pseudorandom timing

(A1) Trains of spike event times (top) computed for an instantaneous rate function λ(t) with exponential decay time constant of 150 ms (bottom, solid red line) and absolute refractory period (τ_AR) of 1 ms. To compute the trains, a refractory-corrected rate function Λ(t) (dashed red line) was first derived from Eq. (13.18) and then used in the integral of Eq. (13.15) to compute the spike intervals in sequence. The PSTH (black, 2-ms bins) computed from 200 such trains closely matches λ(t). (A2) Interspike interval histogram (ISIH) computed from the same 200 trains in (A1), showing the 1 ms absolute refractory period. The overall exponential decay of the ISIH is a hallmark sign of a random Poisson process. (B1) and (B2) Same as (A1) and (A2) except λ(t) was a half-wave rectified sinusoid with 250 ms period, and refractoriness was both absolute and relative: τ_AR = 0.5 ms and τ_RR = 0.5 ms. Intervals were computed via Eq. (13.19).

Figure 13.5. Modeling short-term depression and facilitation

(A) Original depletion model of Liley and North describing release of freely diffusing transmitter (N). N is in equilibrium with a large store of precursor molecules (N_s), governed by forward and backward rate constants k₁ and k₋₁. The arrival of an action potential causes a rise in [Ca²⁺]_i, triggering a fraction (P) of N to be released (NP) into the synaptic cleft (red), disrupting the balance between N and N_s. N recovers back to its steady-state value (N_∞) with an exponential time course (τ_r), where N_∞ and τ_r are set by k₁ and/or k₋₁. (B) A modern version of the depletion model with a large store of synaptic vesicles (N_s, gray circles) and a fixed number of vesicle release sites (N_T, blue), where N now represents the number of vesicles docked at a release site and are therefore readily releasable (orange circle). The arrival of an action potential now triggers a certain fraction of the readily releasable vesicles to be released (NP), freeing release sites. The number of free release sites at any given time is equal to N_T – N. Variations of this model include a k₁ that is dependent on residual [Ca²⁺]_i (red star), in which case [Ca²⁺]_i is explicitly simulated, and the inclusion of a backwards rate constant k₋₁ (gray arrow) representing the undocking of a vesicle, that is, the return of N to N_s. (C) A more recent version of the depletion model, similar to that in (B), has two pools of readily releasable vesicles (N₁ and N₂) with low- and high-release probabilities, respectively (P₁ and P₂). The difference in probabilities is related to the distance vesicles in pools N₁ and N₂ are from VGCCs, where vesicles in pool N₂ are more distant. Here, the model includes a maturation process where N₂ emerges from N₁ at a rate set by k₂, but some models have N₂ emerging from N_s in parallel with N₁. In some models, k₂ is dependent on residual [Ca²⁺]_i (red star), in which case [Ca²⁺]_i is explicitly simulated. Only the second pool has a fixed number of vesicle release sites (N_T2). (D) Synaptic model with depression implemented using RP recursive algorithm described in Eqs. (13.28)–(13.32) (τ_r = 20 ms; R = N/N_T). The time evolution of R and P are shown at top (blue and red), where P_∞ = 0.4. Since there is no facilitation (Δ_P = 0), P is constant. At the arrival of an action potential at t_j, the fraction of vesicles released (R_e) is computed: R_e = RP (gray circles). R_e is then used to scale a synaptic conductance waveform G_syn(t = t_j) (Eq. 13.30) and also subtracted from R (Eq. 13.31). The time evolution of the sum of all G_syn(t = t_j) is shown at the bottom (gray). (E) The same simulation in (D), except the synaptic model includes facilitation (Δ_P = 0.5, τ_f = 30 ms). For comparison, R_e and the sum of all G_syn(t = t_j) are plotted in black (+F) along with their values in (D) (−F, gray). (F) Fit of a synaptic model with depression (yellow) to a 30 Hz MF–GC AMPAR conductance train (black). The fit consisted of the sum of two separate components, the direct and spillover components, where each component had its own depression parameters. Parameters for the fit can be found in Schwartz et al. (G) Same as (F) but for a corresponding 30 Hz MF–GC NMDAR conductance train. This time the fit (green) consisted of a single component that had depression and facilitation. Scale bars are for (F) and (G), with two different y-scale values denoted on the left and right, respectively. Data in (F) and (G) is from Schwartz et al. with permission.

Figure 13.6. Simulating trial-to-trial variability using a binomial model with quantal variability and asynchronous release

(A1) Simulations from a binomial model of a typical MF–GC connection with five release sites (N_T) each with 0.5 release probability (P) and 0.2 nS peak conductance response (Q). Q was used to scale a G_AMPAR waveform with only a direct component. A total of 1000 trials were computed, 20 of which are displayed (blue). Inset shows σ²–μ relation computed from the peak amplitudes of all 1000 trials (blue circle), matching the theoretical expected value computed from Eq. (13.36) (dashed line). Repeating the simulations using a low P (0.1, red) and high P (0.9, green) confirmed the parabolic σ²–μ relation of the binomial model. (A2) Frequency distribution (bottom, blue) of the 1000 peak amplitudes computed in (A1), which closely matched the expected distribution computed via Eq. (13.37) (not shown). Circles on x-axis denote μ. Distributions for low and high P are also shown (red and green). Top graph shows Q which lacked variation. (B1) Same as (A1) except Q included intrasynaptic variation (CV_QS = 0.26) and intersynaptic variation (CV_QII = 0.31), creating a larger combination of peak amplitudes and therefore larger variance. Inset shows theoretical σ²–μ relation with (solid line) and without (dashed line) variation in Q, the former computed using Eq. 11 of Silver. (B2) Same as (A2) but for the simulations in (B1). Top graph shows the distribution of the average Q at each site i (i.e., Q_i, colored circles) with μ ± σ = 0.20 ± 0.62 nS (black circle), as defined by CV_QII. Gaussian curves show distribution of Q at each site, defined by Q_i and CV_QS. (C1) Same as (A1) except a delay, or release time (t_release), was added to each quantal release event. Values for t_release were randomly sampled from the release time course shown in the inset, which is typical for a single release site at a MF–GC connection. (C2) Same as (A2) but for the simulations in (C1). Peaks were measured over the entire simulation window.

Figure 13.7. A binomial model with short-term depression and facilitation

(A1) Simulation of a binomial synapse with N_T = 5, P = 0.5, and Q = 0.2 nS using the RP recursive algorithm described in Eqs. (13.28)–(13.32) (τ_r = 50 ms, Δ_P = 0.5, τ_f = 12 ms). The stimulus was a 100 Hz train of action potentials (top). Bottom graphs show time evolution of R (blue) and P (red) for each release sight during the train. Note, action potentials always caused facilitation, but only caused depression when there was success of vesicle release. At each release site, vesicle release was a success if a random number drawn from [0, 1] was less than the product RP. Gray trace (bottom) shows the sum of resulting quantal waveforms from all five sites. The quantal waveform was a G_AMPAR waveform with only a direct component, the same used in Fig. 13.6. (A2) Conductance trains of the same simulation in (A1) for 100 trials (gray). Black trace shows average of the 100 trials, which closely matches the time course of the same simulation computed for a deterministic RP model (green dashed line). (B1 and B2) Same as (A1) and (A2) but for a 300 Hz train of action potentials.