Analytical solutions for the short-term plasticity

Abstract

Synaptic dynamics plays a key role in neuronal communication. Due to its high-dimensionality, the main fundamental mechanisms triggering different synaptic dynamics and its relation with the neurotransmitters release regimes (facilitation, biphasic, and depression) are still elusive. For a general set of parameters, and by means of 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 into 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. Author summaryBased on the model proposed by Tsodyks et al., we obtained a map approximation to study analytically the dynamics of short-term synaptic plasticity. We identified the synaptic regimes named facilitation, depression, and biphasic in the parameters space, and determined the maximal and equilibrium points of active neurotransmitters for presynaptic neurons spiking periodically and stochastically following a Poisson process. Besides that, we verify that the time average of the variables for the synaptic dynamics driven by presynaptic neurons spiking following a Poisson distribution presents the equilibrium points obtained for the synaptic driven by periodic presynaptic neurons, spiking with a frequency that is the mean frequency of the Poisson distribution. These results shed analytical light into the understanding of synaptic dynamics.