Glassman, Robert - Topology, Graph Theory & the Magic Number Four in Neuroscience

In a review of so many four-part models, one has to wonder what is so special about the number four. Do these models represent something in nature that is four-fold? Perhaps it is only some common analytical habit that predisposes us to see things in this particular four-fold manner. Whether either or both of these are true, it remains incumbent upon us to question what might be so special about four-fold models.

In a series of articles (Glassman, 2003; Glassman, 1999a; Glassman, 1999b; Glassman, 1999c; Glassman et al., 1998; Glassman, 1997; Glassman et al., 1994) Robert Glassman undertakes this investigation. His goal is to explain the capacity limits of working memory; i.e. the well-studied fact that both we and other species can keep about 7 ±2 items of active information in mind in any span of time. Since limited working memory seems to be a very stable and robust finding in living organisms, Glassman looks into the organizational and operational features that might explain why selection favored these particular limits.

A lower limit upon working memory is furnished by the mathematics of association. To be useful, working memory should sustain at least three chunks of active information. Two chunks may be logically inadequate for cognitive representation, i.e. for supporting decisions or constructing larger representations. Direct one-to-one associations leave no room for decision-making. At least three representations are needed to represent a contingency, with the third element providing a context or occasion for the paired association. If we were to give this decision-making function mathematical expression as a search or walk through an option-space, we would need at least three non-colinear points to define that space – a triangle. Glassman also notes that the syntax of natural language strongly features the action set of subject, verb and object, and that the emergence of three-word utterances in child development ushers in a period of rapid linguistic growth. Three is thus a representationally significant number. (Glassman, 2003)

Furthermore, at least three nodes are needed to explore associativity and represent groups, transitivity and other more complex relations. To determine that A, B and C are associative in mathematical terms, one has to be able to determine that A+(B+C)=(A+B)+C. "If a hypothetical mental buffer were able only to hold pairs of elements, then, if it begins with A and B, one of these must be dropped in order to pick up C. That leaves only the possibility of piecemeal chaining of pairs in long-term memory (LTM)." Glassman concedes that a two-node system could recursively chunk paired items (AB) to associate them as a unit with a third item (C). However, these associations finalize meaning, like a decision, commitment or rule. There is no more room for representing conditions and contingencies. Meaning is narrowed to a single pathway of definite associations, rather than an open space for context-sensitive choices.

While working memory must involve at least three-way associations, Glassman reviews a number of experimental findings that indicate that the upper limit on simultaneous associations in working memory is only four (Glassman et al., 1998; Glassman et al., 1994). We get to the magic number 7 by looping or continually refreshing working memory in time. Four items at time t can thus be chunked into one item at time t+1, providing a context for the three remaining working memory slots. I will not take the time to duplicate Glassman’s empirical argument here. Instead, I want to focus on a further observation that he offers. He argues that, aside from the empirical reasons to believe that working memory capacity is limited to four items, there are structural reasons why this must be so. Four represents a mathematical upper limit for simultaneous associative interrelations in the brain, because of the topology of the isocortex.

Many structures in the central nervous system are sheet-like or laminar in structure. That means that local interactions must take place on surfaces that are effectively two-dimensional. Isocortex is the most conspicuously sheet-like structure of all. This planar organization imposes certain mathematical constraints upon local associations. On any two-dimensional or sheet-like surface, if four sub-regions are defined, any one of them can grow an edge to contact any of the other three without ‘cutting across’ another patch, isolating or ‘trapping’ part of the invaded sub-region. With five or more sub-regions defined, some patches will be disconnected. “So long as there are no more than four planar regions, any of them has free access to grow an edge to any other in some way that does not split a subpatch nor divide any of the other subpatches from each other.” (Glassman, 2003) This mathematical limit flows from the “four-color theorum”, the proven fact that you can color any flat map in only four colors, and no region of it need border on another region of the same color, no matter how serpentine the regions. Four regions can maintain undisrupted contact with each other on a flat surface. Add a fifth and some isolation or disconnection in inevitable.

It might even be argued that we see this growth in complexity during conversations at cocktail parties. Robin Dunbar has observed that human conversation groups have a “decisive upper limit” of four individuals. The addition of a fifth listener will destabilize the group, resulting in side conversations and a division of the group in two. (Dunbar, 1996; p. 121) This four-unit threshold for all-way associativity may represent a universal limit on the unity of simple systems, with bifurcation and differentiation occurring above that threshold. Glassman argues that this may explain the upper limits of working memory.

The importance of the number four in the mathematics of planar surfaces is further explored in graph theory. Graph theory deals with systems of vertices and arcs, or the points in a network diagram and the lines that connect them. On a sheet-like surface, you can connect up to four vertices with arcs such that no arcs bisect or cross any other arcs. In other words, on a flat surface you can connect three points in a triangle, then add a fourth point that you can connect to the three triangle points without crossing any lines (forming a diamond, and then connecting the two far points with an arc or ‘handle’). From the fifth point on, if you want to connect every point to every other point, you have to cross or overlay lines. On a planar surface, a four-point graph is the largest complete graph (where every vertex is connected to every other) possible. (Glassman, 2003)

These combinatorial issues are important for Glassman because, on his account, working memory must involve some kind of dynamic allocation of cortical space over brief time intervals. He proposes that “…mental associations among the WM items are embodied neurally as topological associations of activated areas and subareas of cortex.” In other words, the active working memory ensemble would be topologically adjacent to each other, and in range of local cortical connectivity. Long cortico-cortical connections might participate in binding the features of items represented working memory subpatches, but for economy of time and energy, local and neighborhood connections would have to serve active duty as well. To support rapid and flexible working memory operations, certain cortical areas may sustain “pointers”, “proxies” or “surrogates” for otherwise more distributed representations, so that the speed of adjacent or overlapping cortical associations can be used for quick thinking and rapid responding. Glassman writes:

Glassman sees a role for neuromodulators in this hypothetical surrogacy function.
These organizational and topological considerations may help explain why the structure of concern seems so widespread. Fourfold arrangements are special in the mathematics of surface topology, and surfaces or boundaries of all kinds have great natural and biological significance. Guts, skins, gas exchange surfaces and other biological interfaces all exploit sheet-like structures, as does isocortex. Furthermore, topological arrangements of representations are largely conserved as information is processed throughout the nervous system. From the flat surface of the retina to the lateral geniculate nuclei to the visual cortex to the object and position analyzers of higher association cortices, the topological relationships between all the various bits of data in the visual image remain remarkably intact. The same can be said for other sensory systems: interoceptive and exteroceptive. These topological relationships are again largely preserved in projections to the basal ganglia. If local associative relationships are important for decision and evaluation processes, four-color constraints may well impose their structure upon cortical computations.

Fourfold structure of concern models may in part reflect a four-field local association zone in the cerebral cortex. It is also probable that four color considerations play a role in the intellectual and representational practices involved in the creation of these models. Investigators who draw and write on paper and other two-dimensional surfaces to clarify visual-gestalt types of ideas may well fall upon four-cell charts as a powerful way for describing different yet associated dynamics. Structure of concern models may represent in part the organizational necessities of their own representation. The similarities of content across these various models would not be explained by these representational issues, but perhaps a richer account of the phenomena described might be given using models framed using different modeling tactics.

1. Glassman, R. B. (1997). “Might harmonies in the gamma brain wave octave coordinate binding of multiple items in working memory?” Society for Neuroscience. Abstracts, 23. Part 2(#820.16), 21-11.
2. Glassman, R. B. (1998). “A ‘theory of relativity’ for paradoxical time span flexibility of limited item-capacity working memory. Involvement of harmonics among ultradian clocks?” Society for Neuroscience. Abstracts, 24((Part 1, #67.10)), 166.
3. Glassman, R. B. (1999a). “Hypothesized neural dynamics of working memory: several chunks might be marked simultaneously by harmonic frequencies within an octave band of brain waves.” Brain Research Bulletin, 50, 73-77.
4. Glassman, R. B. (1999b). “Working memory: planar graph theory suggests that a segmented, activated cortical domain can serve associativities of up to four chunk-features.” Society for Neuroscience. Abstracts, 25(#355.19).
5. Glassman, R. B. (1999c). “A working memory ‘theory of relativity’: elasticity over temporal, spatial, and modality ranges conserves 7 ±2 item capacity in radial maze, verbal tasks and other cognition.” Brain Research Bulletin, 48, 475-489.
6. Glassman, R. B. (2003). “Topology and graph theory applied to cortical anatomy may help explain working memory capacity for three or four simultaneous items.” Brain Research Bulletin, 60, 25-42.
7. Glassman, R. B., Garvey, K. J., Elkins, K. M., Kasal, K. L., & Couillard, N. (1994). “Spatial working memory score of humans in a large radial maze, similar to published score of rats, implies capacity close to the magical number 7 ±2.” Brain Research Bulletin, 34(151-159).
8. Glassman, R. B., Leniek, K. M., & Haegerich, T. M. (1998). “Human working memory capacity is 7±2 in a radial maze with distracting interruption: Possible implication for neural mechanisms of declarative and implicit long-term memory.” Brain Research Bulletin, 47, 249-256.
9. Dunbar, R. (1996). Grooming, Gossip and the Evolution of Language. Cambridge, Massachusetts: Harvard University Press.
10. Dunbar, R. (2003). “The Social Brain: Mind, Language, and Society in Evolutionary Perspective.” Annual Review of Anthropology, 32, 163-181.
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