Table 1.
Parameters and their values used for figures.
Parameters are grouped together as: those pertaining to occupancy and release statistics; those required for calculation of the post-synaptic response; and those used to generated ISI statistics. Certain neuronal parameters, such as τ and μ, are used for both pre and post-synaptic integrate-fire models.
Fig 1.
Vesicle occupancy and neurotransmitter release rate are a function of the ISI distribution.
(A) Presynaptic spike train, vesicle occupancy and neurotransmitter release time-course for gamma-distributed ISIs with α as marked (in panel B) for the same release probabilty p = 0.6, presynaptic rate r = 5Hz and postsynaptic rate λ = 2Hz. (B) Corresponding ISI distributions for the three α values. (C) The cumulative distribution. (D) The mean 〈x〉 (grey) and the pre-spike 〈x〉∞ (black) release-site occupancies (Eqs 9 and 10). Note that 〈x〉∞ increases with increasing regularity, larger α. (E) Variance of the two occupancies exhibiting non-monotonic behaviour. (F) The mean neurotransmitter release rate 〈χ〉 is directly proportional to 〈x〉∞ and so shares its qualitative dependence on α. The code used to generate this figure is provided in the Supporting Material.
Fig 2.
Temporal and spectral statistics of the presynaptic spike-train and neurotransmitter release required for post-synaptic voltage mean and variance.
(A) The spike-train (Ai) and neurotransmitter release (Aii) event-triggered rate for three cases ranging from bursty to regular with α as marked (see Eq 22 and surrounding text). (B) The equivalent autocovariances of the spike train (Bi) and synaptic release events (Bii). For bursty presynaptic trains (small α = 0.4) the autocovariance is positive; however all cases have negative autocovariances for the release events themselves (Eq 18) because of the filtering by the depressing synapses. (C) The corresponding power spectra (Eq 21) of the spike train (Ci) and synaptic release events (Cii). (D) The postsynaptic voltage mean, standard deviation (std) and coefficient of variation std/mean (Eqs 24 and 29). Though the mean and std both increase with increasing presynaptic regularity, mirroring the behaviour of 〈x〉∞ (Fig 1D), the voltage CV itself decreases with increasing regularity. For panels A-C parameters and color are same as Fig 1, and in panel D parameter α is varied with other parameters provided in Table 1. The code used to generate this figure is provided in the Supporting Material.
Fig 3.
Correlated occupancy, release and post-synaptic voltage statistics when there are multiple contacts per presynaptic cell.
(A) Illustration of a case with one presynaptic cell N = 1 making three contacts n = 3. Shown in descending order: presynaptic spike train (α = 0.4, r = 5Hz), occupancy of the three release sites (restock λ = 2Hz), the corresponding release-event time courses (p = 0.6), and the voltage time course (see Table 1 for parameters). (B) The occupancy correlation 〈xz〉∞ for a pair of sites receiving the same presynaptic spike train and the covariance (see Eqs 34 and 35). (C) The cross-covariance of release events from a pair of sites sharing the same presynaptic cell (Eq 39). (D) Post-synaptic voltage standard deviation (top: from Eq 42) and CV (bottom) as a function of α for four combinations of N, n as marked. With each of these choices, where Nn = 1000, the mean voltage (Eq 32) is the same as in Fig 2D. The code used to generate this figure is provided in the Supporting Material.
Fig 4.
Occupancy and post-synaptic voltage statistics from a presynaptic pool of LIF neurons.
(A) Presynaptic rate (Eq 45) as a function of drive μ for three pairs of (vth − vre, σ) as marked leading to bursty (blue), intermediate (green) and regular (red) firing statistics. Lines of constant rate are at 5, 10 and 20Hz are marked (dotted lines). (B) Example presynaptic voltage time courses (Eq 44) with spikes marked (black) for the parameters marked with symbols in panel A (μ is varied so they are all at rate 5Hz). (C) The parameters vre and σ were co-varied linearly (with μ compensating so that rates were constant at 5, 10 and 20Hz: see dotted lines in panel A, with same color coding used) and the occupancy (Eq 9), post-synaptic voltage mean (Eq 24), standard deviation (from Eq 29) and CV plotted (against vth − vre). The behaviour seen is qualitatively the same as for the gamma-generated ISIs. In this figure N = 1000, n = 1 and vth = 10mV, with other parameters given in Table 1. The code used to generate this figure is provided in the Supporting Material.
Fig 5.
Approximate rate of a postsynaptic EIF neuron driven by a pool of presynaptic EIF neurons, using the matched-variance approximation.
(A) Firing rate of presynaptic neurons as a function of μ for three pairs (vT − vre, σ) representing bursting (−3mV, 2.0mV; blue), intermediate (1.0mV, 1.45mV; green) and regular (10mV, 0.2mV; red) firing. (B) Example presynaptic-voltage time courses (Eq 49) corresponding to symbols in panel A. In panels C-F parameters vre and σ were simultaneously varied linearly between the values of the blue and red curves in panel A (μ adjusted so the presynaptic rate remained 10Hz). Hence, vT − vre ranges from bursting −3mV to regular 10mV. (C) Shows the mean pre-spike vesicle occupancy (Eq 9), (D) the mean post-synaptic voltage (Eq 32) and (E) its standard deviation (From Eq 42). In (F) the post-synaptic rate was approximated using the matched-variance approximation, in which the voltage mean and variance (exactly calculated for this filtered drive) are used in the white-noise EIF firing rate calculation. For comparison, the symbols are simulations of the true firing rate. In Panels D-F Nn = 1000 with (N, n) taking the values (1000, 1: grey), (100, 10: yellow), (50, 20: orange) and (25, 40: purple). The reset for the post-synaptic neuron was held constant at vre = 5mV and other parameters are given in Table 1. The code used to generate this figure is provided in the Supporting Material.