## Storage Capacity of a Single Macrocolumn

This page has my original 1996 thesis results characterizing the storage capacity of a single macrocolumn, or "mac". An analysis of the storage capacity of an overall hierarchy of macs is a future project, but this page has my general argument that for naturalistic input domains, such a hierarchical system could operate indefinitely storing all novel, important inputs, and with high likelihood, never reaching capacity.

A Sparsey mac is a set of *Q* minicolumns each with *K* units. The minicolumns are WTA competitive modules and abbreviated, "CM". The *K* units are proposed as analogous to the L2/3 pyramidal population of a real cortical minicolumn. We plan to gradually enrich the model and its correspondence to the other parts of a minicolumn, e.g., the sub-populations in other layers, over time, but we consider the L2/3 population in particular to be the principal repository of memory, most importantly, because it is the largest sub-population of the minicolumn (and thus of the mac, and of the cortical region, and of the whole cortex) and therefore corresponds to the largest possible capacity given our coding scheme. Figure 1a shows a biologically realistically-sized mac (*Q=100*, *K=40*) (red outline) (actually, *Q*=70, *K*=20 is slightly more realistic). In the current Sparsey model, a mac code is defined as a set of active units, one per CM. Thus, in this case, there are *K ^{Q}* (in Figure 1, 40

^{100}) unique codes possible for the mac. [However note that is not the storage capacity of the mac. Indeed, the storage capacity will generally be a tiny fraction of that. The limiting factor is the number of afferent weights to the mac's units: because weights are shared across memory traces, interference (cross-talk) accrues as additional codes are stored, resulting in the actual number of codes that can be stored and

*reliably*retrieved being tiny compared to the overall size of the code space.]

However, note that in these experiments, we used a slightly different architecture than the current model. Specifically, instead of all input features projecting to all units of all CMs, each input feature projects only to the units of its associated CM (gray bundles). That is, there is a 1-to-1 association of input features and CMs here (in contrast to the current model described throughout the rest of this website). Therefore, rather than activate a winner in all *Q* CMs, we activate winners only in the CMs associated with the active features. So in these studies, the SDR codes consist of 20 active units. Note however that although the bottom-up (U) connectivity scheme is different, the horizontal connectivity scheme is the same as for the current model, i.e., all mac units project to all other mac units except those in the source unit's CM. Thus, in the depicted model, there are 4000 x 3960 = 15,840,000 H weights (a tiny sample of which suggested by the green arrow sprays). Figure 1b shows the nature of the random sequences used in these studies. Each input sequence has 10 frames and 20 out of 100 features are chosen randomly to be active on each frame. Despite this variant architecture, we believe that these capacity results are qualitatively similar--specifically, the number of inputs that can be stored increases faster-than-linearly in the units and linearly in the weights--to those that would result for the general architecture case (i.e., full connectivity from input units to all of the mac's units). Studies confirming this are on the short list for near future projects.

**Figure 1:** (a) The model has *Q*=100 CMs, each with *K*=40 WTA CMs. Horizantal weights suggested by green arrows. (b) Example of a training sequence used to test capacity. All sequences had 10 frames on which 20 binary features were randomly selected to be active. All sequences are presented once.

A further point to be aware of here is that when we test for recall accuracy, we actually prompt the model not with the input pattern, but with the mac code that was assigned to that input pattern during the learning trial. We then let the propogation of H signals from the code active at T reinstate the code at T+1. And the accuracy measure reported is in terms mac codes, not reinstated input codes. This is a slight "cheat", since we are assisting the model by telling it which unit to pick in each of the 20 active CMs on the first frame of each sequence. However, after the first frame, all code activations depend only on the learned signals propagating via the H matrix, which constitutes the vast majority of the signalling accounting for reported accuracies.

The Table and graphs below give the primary result which is that storage capacity (in terms of items, in this case, sequences) increases: 1) superlinearly in the number of units (*L*), as shown in Table column "*E*" and charts a and b (chart b zooms in on the bottom two curves of chart a); and 2) linearly in the number of weights (chart b). The table holds the data corresponding to the second lowest graph in chart a (labelled "UNC, STC=200", i.e., sequences were 10 frames long with 20 randomly chosen active features per frame; "UNC" means uncorrelated). Each row represents the maximum number of such sequences storable while retaining average recall accuracy ≥ 97%, for a model of a given size. Model size, *L* and total number of weights (*W*, no column) systematically increases with row. Note: "L1" refers to input level.

It is also very important that we also tested the case of *complex sequences* (here, called "correlated patterns", denoted "COR") corresponding to the dashed lines in charts (table not shown). For the correlated pattern experiments, we first generated a "lexicon" of 100 frames (each a random choice of 20 of 100 features). Then we generated the 10-frame training sequences by choosing randomly with replacement from the lexicon. Hence, each lexicon element occured approximately 10% of the time. This is important because the complex sequence case corresponds to natural lanquage lexicons. It is essential that any sequential/temporal memory be able to store massive numbers of sequences composed of frequently recurring items, as well as frequently ocurring sub-sequences of items.

In addition, chart c shows that capacity in bits increases superlinearly in the weights. The amount of information contained in each sequence (for the UNC case), was computed as shown below.

NOTE: This method overestimates the information contained in the whole set of sequences because it ignores the correlations (redundancies) amongst the set. We will attempt to find an approriate correction, but given that the sequences are completely random, we believe the overestimate may not be too severe. In any case, we believe that a bits/synapse value of this approximate value is quite tolerable for scaling to massive data sizes for at least two reasons: a) it comes in the context of a model with the more essential properties of being able to learn, *with single trial** s*, new sequences and retrieve the best-matching sequence in time that is

*for the life of the model; and b) we expect the overall model bits/synapse value to increase significantly when hierarchy is leveraged. This latter point will be one of our research focuses going forward.*

**fixed**