Stochastic variability of synaptic responses

Excitatory synaptic response shows a large variability which is influenced by several factors both of pre and post-synaptic origin. The average number of synapses on a single hippocampal CA1 pyramidal neuron is $\sim 3 \times 10^4$ [2]. Each synapse on the dendritic tree is then influenced by the electrical activity of other synapses. We have simulated the single synaptic event in terms of a difference of two exponential as follows: \begin{equation} V(t)=-k(e^{-\frac{t}{\tau_1}}-e^{-\frac{t}{\tau_2}}) \end{equation} where $k$ modulates the Excitatory Post Synaptic Potential (EPSP) peak amplitude and $\tau_1$ and $\tau_2$ are the rising and decay time constants of the EPSP. Although $k$, $\tau_1$ and $\tau_2$ can be all random variables, we have used their mean values for a typical EPSP in order to study the effect of synaptic population and of electrical activity of the cell on the stochastic variability of the single synaptic event [1,3].\\ We have tested a sinusoid wave which can well simulate the effect of a filtered retrograde spike producing \begin{equation} V_s(t)=V(t)+\alpha \sin(\omega t+\phi) \end{equation} where $\alpha$ is the wave amplitude which depends on the distance of the synapse with respect to the soma, $\omega t$ is the frequency and $\phi$ is the phase of the sinusoid wave. We have analyzed the variability of the EPSP as function of the amplitude, frequency and phase of the background wave.\\ In addition we have tested the effect of a gaussian noise $G(\mu,\sigma$) both with respect to a fixed level of resting potential \begin{equation} V_w(t)=V(t)+G(\mu,\sigma) \end{equation} and in a cooperation with the sinusoid wave \begin{equation} V_{w,s}(t)=V(t)+\alpha \sin(\omega t+\phi)+G(\mu,\sigma) \end{equation} The variability of the simulated EPSPs has been analyzed as function of the stochastic processes occurring outside the synaptic space according to their frequency and phase of occurrence.