Synaptic dynamics plays a key role in neuronal communication. Due to its high dimensionality, the main fundamental mechanisms triggering different synaptic dynamics and their relation with the neurotransmitter release regimes (facilitation, biphasic, and depression) are still elusive. For a general set of parameters, and employing an approximated solution for a set of differential equations associated with a synaptic model, we obtain a discrete map that provides analytical solutions that shed light on the dynamics of synapses. Assuming that the presynaptic neuron perturbing the neuron whose synapse is being modelled is spiking periodically, we derive the stable equilibria and the maximal values for the release regimes as a function of the percentage of neurotransmitter released and the mean frequency of the presynaptic spiking neuron. Assuming that the presynaptic neuron is spiking stochastically following a Poisson distribution, we demonstrate that the equations for the time average of the trajectory are the same as the map under the periodic presynaptic stimulus, admitting the same equilibrium points. Thus, the synapses under stochastic presynaptic spikes, emulating the spiking behaviour produced by a complex neural network, wander around the equilibrium points of the synapses under periodic stimulus, which can be fully analytically calculated.

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短期塑性的分析解决方案

突触动力学在神经元通讯中起着关键作用。由于其高维性，触发不同突触动力学的主要基本机制及其与神经递质释放机制（促进、双相和抑制）的关系仍然难以捉摸。对于一组通用参数，并采用与突触模型相关的一组微分方程的近似解，我们获得了一个离散图，该图提供了阐明突触动力学的解析解。假设扰动正在建模的神经元的突触前神经元周期性地尖峰，我们推导出释放模式的稳定平衡和最大值，作为释放的神经递质百分比和突触前尖峰神经元的平均频率的函数。假设突触前神经元随机地遵循泊松分布，我们证明轨迹的时间平均值方程与周期性突触前刺激下的图相同，允许相同的平衡点。因此，随机突触前尖峰下的突触模拟复杂神经网络产生的尖峰行为，在周期性刺激下围绕突触的平衡点徘徊，这可以完全分析计算。