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Edited by: Tatjana Tchumatchenko, Max Planck Institute for Brain Research, Germany

Reviewed by: Jean-Pascal Pfister, Cambridge University, UK (in collaboration with Simone Surace); Michael Graupner, New York University, USA

*Correspondence: Alex D. Bird, Warwick Systems Biology Centre, Senate House, University of Warwick, CV4 7AL, Coventry, UK e-mail:

This article was submitted to the journal Frontiers in Computational Neuroscience.

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Synchrony in a presynaptic population leads to correlations in vesicle occupancy at the active sites for neurotransmitter release. The number of independent release sites per presynaptic neuron, a synaptic parameter recently shown to be modified during long-term plasticity, will modulate these correlations and therefore have a significant effect on the firing rate of the postsynaptic neuron. To understand how correlations from synaptic dynamics and from presynaptic synchrony shape the postsynaptic response, we study a model of multiple release site short-term plasticity and derive exact results for the crosscorrelation function of vesicle occupancy and neurotransmitter release, as well as the postsynaptic voltage variance. Using approximate forms for the postsynaptic firing rate in the limits of low and high correlations, we demonstrate that short-term depression leads to a maximum response for an intermediate number of presynaptic release sites, and that this leads to a tuning-curve response peaked at an optimal presynaptic synchrony set by the number of neurotransmitter release sites per presynaptic neuron. These effects arise because, above a certain level of correlation, activity in the presynaptic population is overly strong resulting in wastage of the pool of releasable neurotransmitter. As the nervous system operates under constraints of efficient metabolism it is likely that this phenomenon provides an activity-dependent constraint on network architecture.

Synapses play a key role in transmitting and processing information throughout the nervous system and long-term shifts in synaptic efficacy are believed to underpin learning and memory (Hebb,

The usage of vesicles due to presynaptic firing and stochastic replenishment means that the number of vesicles available for release is a highly dynamic quantity that is dependent on the history of afferent activity. In the immature cortex, the relatively high release probability and limited availability of vesicles causes a progressive reduction in synaptic efficacy during a period of sustained neuronal activity (Reyes and Sakmann,

Correlations in neurotransmitter release between different sites can arise from two sources: from multiple contacts onto a postsynaptic neuron from the same presynaptic cell and from synchronous activity across the presynaptic population. The number of sites between a pair of neurons is fixed over short timescales, unlike the number of vesicles ready to release from the sites, but can vary widely over longer periods (Loebel et al.,

A detailed stochastic model of neurotransmitter dynamics at the presynaptic terminal is required to analyze the effects of presynaptic synchrony, particularly when long-term plasticity varies the structure of synapses through altering the number of release sites. It can be noted that multiple contacts between cells and transient correlations within a presynaptic population are likely to introduce considerable redundancy in the usage of vesicles: correlated events may lead to EPSPs many times larger than that required to reach threshold. However, evidence points to the nervous system operating under constraints of efficient metabolism (Levy and Baxter,

Following the basic model definitions, we first derive exact forms for the crosscorrelations of vesicle occupancies and release at multiple contacts from the same and different presynaptic cells. These correlations were previously derived by Rosenbaum et al. (

We consider a population of _{a}. In between presynaptic action potentials, empty release sites are restocked independently at a constant Poissonain rate _{r}. Initially, we consider that the total number of release sites onto the postsynaptic cell is fixed at _{r} and _{a}. The binary random variable ϱ_{k}(_{k}(_{k}(

_{a}); an empty release site; and restock of an empty release site (Poissonian rate _{r}).

When a presynaptic neuron spikes, available vesicles at each of the

The population of neurons driving a common target often displays substantial synchrony in spiking activity (Salinas and Sejnowski, _{a}/_{a} as required and also that, given that one presynaptic neuron spikes, the probability that a particular other presynaptic neuron has a spike at the same time is _{j} following de la Rocha and Parga (

We treat the postsynaptic neuron as a leaky integrate-and-fire model with each neurotransmitter release event causing the voltage to jump by an amount _{th}. After a spike, _{r} to model the refractory period. If _{ij} is the occupancy variable for the _{ij}. Note that the spike times ^{i}_{k} are identical for all release sites with the same presynaptic neuron

Postsynaptic membrane voltage | Varies | |

Number of presynaptic cells that fire together | Varies | |

Number of release sites per presynaptic neuron | Varies | |

Number of presynaptic neurons | Varies | |

Total number of vesicle release sites ( |
5000 | |

_{r} |
Rate at which empty vesicles are replaced at release sites | 2Hz |

_{a} |
Rate of presynaptic spiking | 2Hz |

Probability of spike arrival inducing neurotransmitter release at a site with a vesicle present | 0.66 | |

τ_{j} |
Jitter standard deviation timescale | 2ms |

Resting membrane voltage | −70mV | |

_{th} |
Threshold at which action potentials are initiated | −55mV |

τ_{r} |
Refractory period of a neuron after a spike | 2ms |

τ | Membrane time constant | 10ms |

EPSP amplitude induced by neurotransmitter released from a single vesicle | 0.2mV |

We first derive exact forms for the crosscorrelations of vesicle-occupancy and of neurotransmitter-release time series. The latter can then be used to calculate the exact membrane voltage variance. Two approximations of the postsynaptic firing rate then lead us to the main result of the paper: that long-term synaptic plasticity—through its alternation of the synaptic parameter

For the calculation of the crosscorrelations of objects separated by a time _{x} and steady-state occupancy 〈

That the second quantity must be the steady-state occupancy 〈

The autocorrelation of release-site occupancy can be calculated by making use of the fact that for the binary variable ^{2} = ^{2}〉 = 〈

The terms on the left-hand side represent the total rate into the double occupancy state, whereas the terms on the right-hand side multiplying the expectation are the combined rates of individual vesicle release minus the coincidence term to prevent overcounting of events. We now combine terms to obtain the required expectation

Example plots of Equation (7), and Equation (10) for cases with γ = 1 and γ =

Though synchrony in the presynaptic population leads to positive correlations for release-site occupancy, we now show that the delayed restock following release leads to negative cross-correlations in the release events themselves. Let χ(_{a}〈_{a} given vesicle occupancy multiplied by the occupancy probability 〈_{a}〈_{a}〈^{2} because in the limit _{a} multiplied by the probability that each contact releases a vesicle ^{2}〈_{γ}. The prefactor of the exponential shares the same squared component −〈χ〉^{2} = −(_{a}〈^{2} as the autocorrelation, but also has a non-zero contribution from 〈χ(_{γ} multiplied by the probability of a release from site _{a}_{a}. This exact result is again identical to that derived previously using a diffusion and additive-noise approximation (Rosenbaum et al.,

The tonic component of the presynaptic drive can be characterized by the mean voltage, which is straightforward to calculate in the absence of a threshold. The dynamics of this quantity can be found by taking the expectation of Equation (2) to yield the steady-state result

Note that the mean voltage is independent of the synchrony

The effect of correlated synaptic fluctuations on the postsynaptic neuron can also be characterized by deriving the steady-state variance of the postsynaptic voltage (again in the absence of a threshold-reset mechanism). This quantity is derived in the Appendix using the auto and crosscorrelations of χ (Equations 12, 13) and takes the form

The first term arises from the δ-functions in Equations (12, 13) and the second term comes from the negative correlations in vesicle release due to short-term depression (the terms featuring exponentials in the same equations). For a related model (de la Rocha and Parga,

Though the voltage variance measures one aspect of presynaptic fluctuations, it misses its increasing shot-noise nature as the correlations increase. Shot noise causes a non-Gaussian component in the tails of the membrane voltage distribution that, because they extend to the region of action-potential initiation, can significantly affect the post-synaptic firing rate (Richardson and Swarbrick,

As correlations from increasing

As the analyses of the previous section and examples in Figure

For the low ^{2} equal to that of Equation (15). The firing rate of a leaky-integrate-and-fire neuron with these parameters is given (Brunel and Hakim, _{th} = (_{th} − _{re} = −μ/σ.

For sufficiently large _{th} − _{a}/

Therefore, increasing the presynaptic synchrony

Under conditions of a fixed number of release sites onto the postsynaptic cell

The post-synaptic firing rate is sensitive to correlations arising from multiple release sites, as discussed above, as well as to presynaptic synchrony (de la Rocha and Parga,

Analyses of long-term plasticity data (over a 12 h period) by Loebel et al. (

The effects of fluctuations in a synchronous presynaptic population can be modeled by adding a Gaussian-distributed jitter, of timescale τ_{j}, to the timing of each action potential. When the individual components of the synchronous MIP event are too dispersed temporally, i.e., when the jitter is greater than the membrane time constant τ_{j} > τ, the MIP event will fail to integrate in the postsynaptic cell. Under these circumstances the effect of correlations is diminished, as illustrated in Figure _{j} = 2 ms (Figure _{j} = τ (Figure

_{j}. _{j} = 0. _{j} = 2ms. _{j} = 10ms.

Throughout much of the above analysis we held the total number of release sites _{0} where _{0} is a constant. This corresponds to a scenario in which the entire presynaptic population has either potentiated or depressed their contacts, thereby changing the number of release sites _{κ}/^{κ} with κ = 3/4, 1/2, 1/4 a maximal postsynaptic rate again occurs at some intermediate

_{κ}/^{κ} for κ = 0, 1/4, 1/2, 3/4, 1, where κ = 1 corresponds to the homeostatic scenario principally considered in this paper for which there is a restriction _{κ} chosen so that they all pass through the point

We considered the effects of afferent correlations arising from multiple neurotransmitter release sites and a partially synchronized presynaptic population. We derived exact forms for the crosscorrelations of vesicle release site occupancy and vesicle release, and demonstrated that these are identical to those recently obtained from a diffusion and additive-noise approximation (Rosenbaum et al.,

Neurons respond maximally to specific stimuli when processing sensory input. A coordination of long-term plasticity, afferent synchrony and short-term depression therefore provides a potential tuning mechanism for cells to achieve this sensitivity. Efficient responsiveness would then depend on historical changes in synaptic connectivity (Taschenberger et al.,

To investigate maximal firing rate response to a defined excitatory drive, we have neglected the effects of synaptic inhibition. As

This research was supported by a Warwick Systems Biology Doctoral Training Centre fellowship to Alex D. Bird funded by the UK EPSRC and BBSRC funding agencies.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The voltage equation can be written in the form
_{ij} takes the form of Equation (11) for the ^{2} crosscorrelations of χ for release trains with different presynaptic neurons given by Equation (13) with γ =

Taking expectations of both side of Equation (19) in the steady state gives

We can now solve Equation (19) to give

As discussed above, the autocorrelation of ζ is the sum of the various crosscorrelations of χ so that it must take the form

On substituting the appropriate forms for α and β the result given in Equation (15) is obtained.