Figure 1.
(A) The axon of a presynaptic neuron (orange) makes synaptic contacts onto a postsynaptic neuron (green). (B) Synaptic vesicles in the synaptic terminal of the presynaptic neuron contain neurotransmitter molecules. A presynaptic action potential releases these neurotransmitter molecules with some probability,
. Once released, these molecules bind to the postsynaptic neuron's membrane and cause a transient change in membrane conductance. (C,D) After a vesicle is released, the synapse enters a refractory state where it is unavailable to release additional neurotransmitter until it recovers by replacing the released vesicle.
Figure 2.
Stochastic versus deterministic models of short term depression.
(A) An example presynaptic spike train, . Each vertical bar represents an action potential. (B) The number of synaptic vesicles,
, available for release and the conductance,
, induced in the postsynaptic cell for one realization of the stochastic model. Filled circles in (B) represent vesicle recovery events. (C) A second realization of the stochastic model with the same input. Observe in (B) and (C) that the number of vesicles released by the stochastic model during one second is primarily determined by the number of recovery events during that second and does not reflect the number of presynaptic spikes. (D) The number of synaptic vesicles and the conductance induced by the deterministic model with the input from (A). Parameters in (A–D) were chosen for illustrative purposes as
,
,
, and
. (E) The steady state mean conductance,
, as a function of the presynaptic firing rate,
. The inset shows the gain,
. (F) The steady state variance of
as a function of
for the deterministic (solid blue) and stochastic (dashed red) models of vesicle dynamics with Poisson inputs. Variability in the deterministic model is introduced only by variability in the input,
. Synaptic parameters for (E–F) and for all subsequent figures are given in Table 1.
Figure 3.
Synaptic filtering of a single Poisson presynaptic spike train.
(A)–(B) The low-pass filter, , and the high-pass filter,
, are multiplied with the presynaptic rate (cf. Eq. (2)) to determine the band-pass cross-spectrum,
, between a Poisson presynaptic spike train,
, and postsynaptic conductance,
. The cross-spectrum is identical for the stochastic (solid blue) and deterministic (dashed red) models. (C)–(D) The power spectrum,
, of the conductance is larger for the stochastic model than the deterministic model due to the additive terms,
and
, that quantify the increase in variability due to stochastic vesicle release and recovery (see Eq. (3)). For this and all subsequent figures, solid blue lines and dashed red lines show plots obtained from closed form expressions for the stochastic and deterministic models, respectively. Light blue and light red lines indicate simulations of the stochastic and deterministic models, respectively.
Figure 4.
Low frequency signal transfer in a variety of parameter regimes.
Low frequency cross-spectrum (), auto-spectrum (
), and coherence (
) between a Poisson presynaptic spike train,
, and postsynaptic conductance,
, plotted as a function of the presynaptic rate,
(Ai–iii), the vesicle recovery timescale,
(Bi–iii), the number of synaptic contacts,
(Ci–iii), and presynaptic population size,
(Di–iii). Columns A–C are for a single presynaptic spike train (
). The zero-frequency coherence in Diii is shown for three values of the presynaptic correlation coefficient:
,
, and
. The power spectrum and coherence predicted by the stochastic model (solid blue) and the deterministic model (dashed red) disagree by orders of magnitude unless
is small,
is large,
is small, or
is large with
.
Table 1.
Table of synaptic parameters.
Figure 5.
Coherence between a single presynaptic spike train and the postsynaptic conductance it induces.
The coherence, , between a Poisson presynaptic spike train,
, and the resulting postsynaptic conductance,
. The stochastic model (solid blue) yields a high pass coherence that is dramatically smaller than the flat coherence predicted by the deterministic model (dashed red).
Figure 6.
Signal transfer at high and low frequencies.
The firing rate of a single presynaptic spike train () is modulated by the signal,
, producing a postsynaptic conductance,
. The coherence between the signal and conductance for (A) a slowly varying signal with peak frequency
and (B) a quickly varying signal with
. The stochastic model (solid blue) transmits the higher frequency signal more reliably than the lower frequency signal. The deterministic model (dashed red) transmits the signal with equal fidelity in both cases.
Figure 7.
Linear information transfer rate as a function of signal frequency.
The linear mutual information rate, , between a rate-coded signal,
, and the total conductance,
, produced by (A)
, (B) n = 100, and (C)
presynaptic spike trains, each encoding
. The information rate is plotted as a function of the central frequency,
, at which
is encoded. The stochastic model (solid blue) transmits quickly varying signals more reliable than slowly varying signals. The deterministic model (dashed red) transmits information encoded at any frequency equally well.
Figure 8.
Synaptic filtering at the population level.
A population, , of
Poisson presynaptic spike trains with pairwise correlation
drive a postsynaptic neuron to produce postsynaptic conductances,
. (A) The cross-spectrum between the total presynaptic input and the total conductance. (B) The power spectrum of the total conductance has maximal power within the beta frequency band for both the deterministic (dashed red) and stochastic (solid blue) models. (C) The coherence between the total presynaptic input and the total conductance. Stochastic vesicle dynamics increase the power spectrum and therefore decrease the coherence, especially at low frequencies. All three plots are obtained in the absence of a rate-coded signal (
).