Previous upper bounds on the variability of spine volume in the hippocampus, based on the whole spine volume (Sorra and Harris, 1993; O'Connor et al., 2005), underestimated the precision by including the spine neck volume (Figure 4—figure supplement 1A), which was not correlated between pairs of spines in our volume (Figure 4—figure supplement 1B). Our dense reconstruction included a complete inventory of every synapse in the reconstructed volume and in this respect was unbiased. Additional pairs of synapses from two other rats confirmed that this finding is not confined to a single brain. Of course, additional measurements in the hippocampus and other brain regions would be needed to confirm and extend this finding. The very high statistical significance of the finding (Figure 4C, KS test p=0.0002) despite a relatively small number of pairs in our sample implies a large effect magnitude, which would be much smaller if many more samples were needed to reach the same level of significance. To make this p value concrete, if 17 random pairs were chosen from all 287 synapses in the reconstructed volume, there is only a one in 5000 chance that the spine heads would be as precisely matched as the 17 axon-coupled pairs discovered here.

Previous studies have shown that there is a high correlation of the size of the spine head with the PSD area and numbers of docked vesicles (Harris and Stevens, 1989; Lisman and Harris, 1994; Harris and Sultan, 1995; Schikorski and Stevens, 1997; Murthy et al., 2001; Branco et al., 2008; Bourne et al., 2013). Since the correlations between the head sizes of axon-coupled pairs of spines is high, the high correlation between the PSD areas and numbers of docked vesicles observed in axon-coupled spines is not surprising (Figures 4E and 4F). However, it was unexpected to find that the spine neck diameters were also highly correlated between axon-coupled pairs of spines (Figure 4D r2=0.70), since the correlation between spine head volumes and spine neck diameters is not statistically significant (Figure 1D). Thus, there are at least two geometric aspects of the spine geometry that are under tight control of synaptic plasticity, which may reflect different aspects of synaptic function. The diameter of the spine neck may reflect the need for trafficking of materials between the spine shaft and spine head, which is known to be regulated by LTP and LTD (Araki et al., 2015).

Complementing our observations and analysis in the hippocampus, highly correlated p r at multiple contacts in the neocortex between the axon of a given layer 2/3 pyramidal neuron and the same target cell has been reported (Koester and Johnston, 2005). Our estimate of synaptic variability, based on spine head volume, is an order of magnitude lower. In a recent connectomic reconstruction of the mouse cortex, the similarity in the volumes of axon-coupled pairs of dendritic spines were statistically significant (Kasthuri et al., 2015). This observation is further evidence for the high precision of synaptic plasticity and suggests that the same may be true in other brain areas.

The axon-coupled pairs of synapses that we studied were within a few microns of each other on the same dendrite, which raises the question of how far apart the two synapses can be and still converge to the same size. Related to this question, two synapses from the same axon on two different dendrites of the same neuron might not share the same postsynaptic history. These questions cannot be answered with our current data due to the small dimensions and the fact that the position in the neuropil from which our reconstruction was taken makes it highly unlikely that any of the dendrites, other than the one branch point captured in the volume, belong to the same neuron (Ishizuka et al., 1995). Synaptic tagging and capture, in which inputs that are too weak to trigger LTP or LTD can be ‘rescued’ by a stronger input to neighboring synapses if it occurs within an hour (Frey and Morris, 1997; O’Donnell and Sejnowski, 2014), is much less effective when the synapses are on different branches (Govindarajan et al., 2011), which would tend to make two synapses from the same axon on different dendritic branches less similar. Probing these questions will require reconstructing a larger extent of hippocampus when a single axon can contact multiple dendritic branches of the same neuron (Sorra and Harris, 1993) or of other cells, such as layer 5 pyramidal cells, which can have 4–8 connections between pairs of neurons (Markram et al., 1997).

An unusual triple synapse from a single axon (Figure 4B, ‘k, l, m’) was excluded from the analysis because the presynaptic terminal was a single large varicosity filled with vesicles (i.e. an MSB) shared by three synapses, unlike the other pairs that had isolated presynaptic specializations (n=9), or an MSB shared by two synapses (n=8). It is possible that the large, central spine had an effectively larger pool of vesicles by virtue of proximity, whereas the two synapses on the outside had a more limited population to draw from, and the size of the postsynaptic spine was influenced by the size of the available pool. More examples are needed before we can reach any conclusions. Regardless of the explanation, our estimate of the variability would not be greatly affected by including these 3 additional pairs of synapses in the analysis.

How can the high precision in spine head volume be achieved despite the many sources of stochastic variability observed in synaptic responses? These include: 1) The low probability of release from the presynaptic axon in response to an action potential (Murthy et al., 2001); 2) Short-term plasticity of release of neurotransmitter (Dobrunz et al., 1997); 3) Stochastic fluctuations in the opening of postsynaptic NMDA receptors, with only a few of the 2–20 conducting at any time (Nimchinsky, 2004); 4) Location of release site relative to AMPA receptors (Franks et al., 2003; Ashby et al., 2006; Kusters et al., 2013) 5) Few voltage-dependent calcium channels (VDCCs) in spines that affect synaptic plasticity (smallest spines contain none) (Mills et al., 1994; Magee and Johnston, 1995); 6) VDCCs depress after back propagating action potentials (Yasuda et al., 2003); 7) Capacity for local ribosomal protein synthesis in some spines while others depend on transport of proteins from the dendrites (Ostroff et al., 2002; Sutton and Schuman, 2006; Bourne et al., 2007; Bourne and Harris, 2011); 8) Homeostatic mechanisms for synaptic scaling may vary (Turrigiano, 2008; Bourne and Harris, 2011); 9) Presence or absence of glia (Ventura and Harris, 1999; Witcher et al., 2007; Clarke and Barres, 2013); and 10) Frequency of axonal firing (Callaway and Ross, 1995).

One way that high precision can be achieved is through time averaging. Long-term changes in the structure of the synapse and the efficacy of synaptic transmission are triggered by the entry of calcium into the spine. A strategy for identifying the time-averaging mechanism is to follow the calcium. Phosphorylation of calcium/calmodulin-dependent protein kinase II (CaMKII), required for spike-timing dependent plasticity processes, integrates calcium signals over minutes to hours and is a critical step in enzyme cascades leading to structural changes induced by long-term potentiation (LTP) and long-term depression (LTD) (Kennedy et al., 2005), including rearrangements of the cytoskeleton (Kramár et al., 2012). The time window over which CaMKII integrates calcium signals is within the range of time windows we predict would be needed to achieve the observed precision (Table 1). Similar time windows occur in synaptic tagging and capture, which also requires CaMKII (Redondo and Morris, 2011; de Carvalho Myskiw et al., 2014). These observations suggest that biochemical pathways within the postsynaptic spine have the long time scales required to record and maintain the history of activity patterns leading to structural changes in the size of the spine heads.

The information stored at a single synapse is encoded in the form of the synaptic strength, which reflects the pre- and postsynaptic history experienced by the synapse. But due to the many sources of variability, this information cannot be read out with a single input spike. This apparent limitation may have several advantages. First, the stochastic variability might reflect a sampling strategy designed for energetic efficiency since it is the physical substrate that must be stable for long-term memory retention, not the read out of individual spikes (Laughlin and Sejnowski, 2003). Second, some algorithms depend on stochastic sampling, such as the Markov Chain Monte Carlo algorithm that achieves estimates by sampling from probability distributions, and can be used for Bayesian inference (Gamerman and Lopes, 2006). Each synapse in essence samples from a probability distribution with a highly accurate mean, which collectively produces a sample from the joint probability distribution across all synapses. A final advantage derives from the problem of overfitting, which occurs when the number of parameters in a model is very large. This problem can be ameliorated by ‘drop out’, a procedure in which only a random fraction of the elements in the model are used on any given trial (Wan et al., 2013; Srivastava et al., 2014). Drop out regularizes the learning since a different network is being used on every learning trial, which reduces co-adaptation and overfitting.

We are just beginning to appreciate the level of precision with which synapses are regulated and the wide range of time scales that govern the structural organization of synapses. The upper bound on the variability that we have found may be limited by errors in the reconstruction and could be even lower if a more accurate method could be devised to compute the volume of a spine head, neck diameter, PSD area, number of docked vesicles, or other salient features of dendritic spines. Much can be learned about the computational resources of synapses by exploring axon-coupled synaptic pairs in other brain regions and in other species.