What is the memory capacity of a human brain? The storage capacity in a computer memory is measured in bits, each of which can have a value of 0 or 1. In the brain, information is stored in the form of synaptic strength, a measure of how strongly activity in one neuron influences another neuron to which it is connected. The number of different strengths can be measured in bits. The total storage capacity of the brain therefore depends on both the number of synapses and the number of distinguishable synaptic strengths.

Structurally, neurons consist of a cell body that influences other neurons through a cable-like axon. The cell body bears numerous short branches called dendrites, which are covered in tiny protrusions, or “spines”. Most excitatory synapses are formed between the axon of one neuron and a dendritic spine on another. When two neurons on either side of a synapse are active simultaneously, that synapse becomes stronger, a form of memory. The dendritic spine also becomes larger to accommodate the extra molecular machinery needed to support a stronger synapse.

Some axons form two or more synapses with the same dendrite, but on different dendritic spines. These synapses should be the same strength because they will have experienced the same history of neural activity. Bartol et al. used a technique called serial section electron microscopy to create a 3D reconstruction of part of the brain that allowed the sizes of the dendritic spines these synapses form on to be compared. This revealed that the synaptic areas and volumes of the spine heads were nearly identical. This remarkable similarity can be used to estimate the number of bits of information that a single synapse can store, since the size of dendritic spines and their synapses can be used as proxies for synaptic strength.

Measurements in a small cube of brain tissue revealed 26 different dendritic spine sizes, each associated with a distinct synaptic strength. This number translates into a storage capacity of roughly 4.7 bits of information per synapse. This estimate is markedly higher than previous suggestions. It implies that the total memory capacity of the brain – with its many trillions of synapses – may have been underestimated by an order of magnitude. Additional measurements in the same and other brain regions are needed to confirm this possibility.

Synapses between neurons control the flow of information in the brain and their strengths are regulated by experience. Synapses in the hippocampus are involved in the formation of new declarative memories. Understanding how and why synaptic strengths undergo changes in the hippocampus is important for understanding how we remember facts about the world. A fundamental question is the degree of precision in the adjustment of synaptic strengths in view of the many sources of variability at synapses. In this study we provide an upper bound on the variability of synaptic plasticity and quantify a lower bound on the amount of information that can be stored at a single synapse.

Excitatory synapses on dendritic spines of hippocampal pyramidal neurons have a wide range of sizes. Anatomical measurements of the spine size, the area of the postsynaptic density (PSD), the number of AMPA receptors, the area of the presynaptic active zone and the number of docked vesicles in the presynaptic terminal are all highly correlated with each other and with physiological measurements of the release probability and the efficacy of the synapse (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). Thus, each of these individual characteristics is a correlate of synaptic strength. The sizes and strengths of these synapses can increase or decrease according to the history of relative timing of presynaptic inputs and postsynaptic spikes (Bi and Poo, 1998).

If experience regulates synaptic strength then one might expect that synapses having the same pre- and postsynaptic histories would be adjusted to have the same strength. But what would be the inherent variability, or conversely the precision, of this process? Due to the high failure rate and other sources of stochastic variability at synapses one might expect that the precision of changes in the strengths of these synapses in vivo to be low. The failure rate at synapses depends inversely on the strength, and therefore the size, of the synapse. On this basis the strengths of weaker, and therefore smaller and less reliable synapses, would be expected to be less precisely controlled than the larger and stronger synapses, which have a lower failure rate.

An ideal experiment to test for the precision of the changes in synaptic strength would be to stimulate in vivo the axonal inputs to two well-separated spines on the same dendrite to insure that they have the same presynaptic and postsynaptic history of stimulation. Nature has already done the experiment for us as pairs of spines on the same dendrite contacting the same axon satisfy this condition. Prior work suggests that such pairs of spines are more similar in size than those from the same axon on different dendrites (Sorra and Harris, 1993). Here we evaluated this axon-spine coupling in a complete nanoconnectomic three-dimensional reconstruction from serial electron microscopy (3DEM) (Harris et al., 2015) of hippocampal neuropil. We determined the similarity of synapses among pairs of spines and set an upper bound on the variability and the time window over which pre- and postsynaptic histories would need to be averaged to achieve the observed precision.

In a 6 x 6 x 5 μm³ complete 3DEM from the middle of stratum radiatum in hippocampal area CA1 (Mishchenko et al., 2010; Kinney et al., 2013; see Materials and Methods). We identified 449 synapses, 446 axons and 149 dendrites, which except for one identified branch point, are likely to originate from different neurons based on the size of the reconstructed volume and the obtuse branching angles of dendrites from these neurons (Ishizuka et al., 1995; Megı́as et al., 1997). We measured spine head volume and surface area, surface area of the postsynaptic density (PSD) adjacent to the presynaptic active zone, and spine neck volume, neck length and neck diameter at the 287 spines that were fully contained within the volume. We also quantified the number of vesicles at the 236 spines and presynaptic boutons that were fully contained within the volume. The strong correlations between these metrics, the skewed shape of the frequency histograms, and the number of synapses per unit of volume (Figures 1–3), are consistent with previous observations (Harris and Stevens, 1989; Schikorski and Stevens, 1997; Sorra et al., 2006; Bourne and Harris, 2011; Bourne et al., 2013; Bell et al., 2014). To reduce error, we averaged over multiple independent spine volume measurements for each spine (Figure 3—figure supplement 1). We determined that the relationship between PSD area and spine head volume did not differ significantly across different dendrites (Figure 1—figure supplement 1). The correlation between spine head area and spine head volume accounted for 99% of the variance, despite the wide range in spine head shapes and dimensions (Figure 1A), which suggests that the accuracy of our measurements matched the precision of the spine. We also measured spine neck length, diameter, and volume and found a weak trend between the neck diameter (Figure 1D) and spine head volume but no correlation between neck length (Figure 1E) and spine head volume, consistent with previous studies (Harris and Stevens, 1989; Tønnesen et al., 2014).

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Next, we analyzed spine volumes according to their axonal connectivity and dendrite origin. Pairs of spines on the same dendrite that received input from the same axon (‘axon-coupled’), were of the same size and had nearly identical head volumes (Figure 4). We compared this sample of 10 axon-coupled pairs on the same dendrite (Figure 4B, pairs a-j) to those identified on dendrites from the two additional animals (Bourne et al., 2013), for a total of 17 axon-coupled spine pairs. When plotted against one another, the paired head volumes were highly correlated with slope 0.91, and despite the small sample size, were highly significantly different from random pairings of spines (Figure 4C and Figure 4—figure supplement 1A, KS test p=0.0002). Similarly, there was a strong positive correlation between their paired neck diameters (Figure 4D), PSD areas (Figure 4E), and number of presynaptic docked vesicles (Figure 4F). These features of axon-coupled spines from the same dendrite spanned the distribution of the overall spine population (Figure 3). In contrast, the spine neck lengths (Figure 4G), and neck volumes (Figure 4—figure supplement 1B) of the pairs were not well-correlated indicating that regulation of neck length and neck volume are not important for synaptic strength.

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The coupled triplet of synapses (Figure 4C, gray points ‘k, l, m’) are on three different spines along a single dendrite and receive synaptic input from a single multi-synaptic bouton. A larger central spine between two similar in size (Figure 4B, ‘k, l, m’) produces one same size pair (‘k’ ) and two different size pairs (‘ l’, ‘m’). This unusual configuration is probably driven by processes, such as competition for available resources, that differ from the other pairs (Sorra and Harris, 1993; Sorra et al., 1998). As one possibility, perhaps the the size of the larger postsynaptic spine was influenced by the larger size of the available pool of presynaptic vesicles in close proximity to its active zone. Excluding this triple synapse, the median value of the coefficient of variation of volume differences between pairs was CV = 0.083 and was as precise for small synapses as it was for large ones (Figure 5). This precision (i.e. low CV) suggests that accurately maintaining the size of every synapse, regardless of size and strength, could be important for the function, flexibility and computational power of the hippocampus.

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This near-identical size relationship does not hold for axon-coupled spines on different dendrites (Figure 6B, CV = 0.39, n = 127, example Figure 6A), nor for non-axon-coupled spines on the same or different dendrites (Figures 6E,F, example Figure 6D) -- all cases which would have had different activation histories. The volumes of axon-coupled different-dendrite spines are no different from the volumes of random pairs when plotted against one another (KS test p=0.94, Figure 4—figure supplement 2A, and Figures 6B,C) and the distribution of their sizes was no different from the whole population (KS test p=0.41). The number of docked vesicles for pairs on different dendrites (Figure 6—figure supplement 1B) is not different from random pairings (KS test p=0.08), nor are the neck diameters (Figure 6—figure supplement 1C, KS test p=0.06), nor the neck lengths (Figure 6—figure supplement 1D, KS test p=0.75). The size difference of pairs of axon-coupled spines on the same or different dendrites shows a weak trend with separation distance along the axon or dendrite (Figure 6—figure supplement 2). The sizes of pairs of axon-coupled spines on the same or different dendrites is unaffected by proximity of glia processes to the synapses (Figure 7) (Ventura and Harris, 1999; Witcher et al., 2007), or location of mitochondria in the axon (Billups and Forsythe, 2002).

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We found that spine head volumes ranged in size over a factor of 60 from smallest to largest while the CV of any given size was 0.083 and was constant across the range of sizes. Measurements of these of 20 pairs allowed us to estimate the number of distinct spine sizes, and by extension synaptic strengths, that can be reliably distinguished across this range. Signal detection theory holds that at a Signal-to-Noise Ratio (SNR) of 1, a common detection threshold used in psychophysical experiments, an ideal observer can correctly detect whether a signal is higher or lower than some threshold 69% of the time (Green and Swets, 1966; Schultz, 2007). Put another way, if random samples are drawn from two Gaussian distributions whose areas overlap by 31%, an ideal observer will correctly assign a given sample to the correct distribution 69% of the time. Using this logic, we found that ~26 different mean synaptic strengths could span the entire range, assuming CV = 0.083 for each strength level, and a 69% discrimination threshold (Figure 8, see Materials and methods). These 26 distinct strength levels can be represented with 4.7 bits of information (i.e. 2^4.7 ≈ 26) which means 4.7 bits of information that can be stored at each synapse as synaptic strength. At a discrimination threshold of 76% (corresponding to SNR = 2) there would be ~23 distinct strengths and 4.5 bits of information.

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To explain the high precision observed in spine head volumes, we propose that time-window averaging smooths out fluctuations due to plasticity and other sources of variability including differences in the age of the synapses. To set a lower bound on averaging time, we chose to examine neurotransmitter release probability as a single source of variability. Let us first consider release caused by single action potentials, ignoring short-term plasticity. Release of this type can be analyzed using a binomial model in which n presynaptic action potentials, each with a probability p_r of releasing one or more vesicles, leads to a mean number of releases μ = n*p_r having variance σ² = n*p_r*(1−p_r). The coefficient of variation around the mean is CV = σ/μ = sqrt [(1−p_r)/(n*p_r)] and can be compared with the measured values. Therefore, the number of spikes that are needed to reduce the variability to achieve a given CV is n=(1−p_r)/(p_r*CV²). Table 1 gives averaging time windows T = n/R, where R is spiking rate of the presynaptic axon, for representative values of p_r and a range of spiking rates.

Accounting for other known sources of variability at dendritic spines would require even longer time windows. In particular, the impact of short-term plasticity during bursts of action potentials on the length of the time-window is complicated by the interplay of facilitation and depression. Synapses with a low initial p_r (and corresponding long time-window in Table 1) exhibit marked facilitation and slowly depress during bursts (Kandaswamy et al., 2010; Nadkarni et al., 2010) which would shorten the time-window. But synapses with a high initial p_r (and short time-window) only weakly facilitate, if at all, and quickly depress, which would lengthen the time-window.

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.

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Author details

1. Thomas M Bartol Jr

   Howard Hughes Medical Institute, Salk Institute for Biological Studies, La Jolla, United States

   Contribution

   TMB, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article

   For correspondence

   bartol@salk.edu

   Competing interests

   The authors declare that no competing interests exist.

2. Cailey Bromer

   Howard Hughes Medical Institute, Salk Institute for Biological Studies, La Jolla, United States

   Contribution

   CB, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

3. Justin Kinney

   1. Howard Hughes Medical Institute, Salk Institute for Biological Studies, La Jolla, United States
   2. McGovern Institute for Brain Research, Massachusetts Institute of Technology, Cambridge, United States

   Present address

   Massachusetts Institute of Technology, Cambridge, United States

   Contribution

   JK, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

4. Michael A Chirillo

   Center for Learning and Memory, Department of Neuroscience, The University of Texas at Austin, Austin, United States

   Contribution

   MAC, Acquisition of data; Analysis and interpretation of data; Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

5. Jennifer N Bourne

   Center for Learning and Memory, Department of Neuroscience, The University of Texas at Austin, Austin, United States

   Present address

   University of Colorado Denver, Denver, United States

   Contribution

   JNB, Acquisition of data; Analysis and interpretation of data; Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

6. Kristen M Harris

   Center for Learning and Memory, Department of Neuroscience, The University of Texas at Austin, Austin, United States

   Contribution

   KMH, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article

   For correspondence

   kmh2249@gmail.com

   Competing interests

   The authors declare that no competing interests exist.

7. Terrence J Sejnowski

   1. Howard Hughes Medical Institute, Salk Institute for Biological Studies, La Jolla, United States
   2. Division of Biological Sciences, University of California, San Diego, San Diego, United States

   Contribution

   TJS, Conception and design, Analysis and interpretation of data, Drafting or revising the article

   For correspondence

   terry@salk.edu

   Competing interests

   The authors declare that no competing interests exist.

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