Activity-dependent synaptic plasticity is widely believed to play the major role in learning and memory. Moreover, the robustness of memories depends on the stability of synapses. Recent studies, however, have shown that synapses exhibit significant volatility which to some extent appears activity-independent [1]. This stochastic turnover therefore can put a challenge on encoding and preserving information in synaptic connectivity [2]. In this study, using in vivo STED nanoscopy, to assess the dynamics of morphological features of excitatory spines in mouse cortical circuits including the dynamics of head size, neck length, and neck width measured over short (hour) and long (days) intervals for up to 30 days, monitoring their changes in time [3,4]. We model two scenarios where the change of spine morphological features was either activity dependent or completely spontaneous. In the first scenario the synaptic changes solely depend on the timing of discrete ‘learning events’ following a Poisson distribution as the most generic case. In the second scenario we model the spine dynamics based on stochastic dynamics of actin filaments, with the synaptic changes are independent of pre and post synaptic activity.Comparing theoretical predictions with measured cross-correlation functions of spine features we distinguish distinct roles of activity (in)dependent plasticity in governing synaptic turnover over short and long time intervals. Our results indicate that quenched disorder – the heterogeneity in the stable component of synaptic measures – is necessary to capture the non-vanishing part of cross correlation functions that our data reveal.