William H. Calvin, "Error-correcting Codes" (1993)

Error-correcting Codes:

Coherent Hexagonal Copying from Fuzzy Neuroanatomy


Department of Psychiatry and Behavioral Sciences,

University of Washington

Seattle, Washington 98195-1800 USA


Calvin, William H. (1993). Error-correcting codes: Coherent hexagonal copying from fuzzy neuroanatomy. World Congress on Neural Networks 1:101-104 (1993).

Copyright 1993 by W. H. Calvin.


Fuzzy approximations to a point-to-star wiring principle for recurrent excitation can clone spatial patterns of activity. Because of temporary entrainment of otherwise active neurons, the homologous points form a triangular mosaic of synchronized neurons. The largest such collection of points that can be so copied without screen-wrap-like overwriting is a hexagon as wide as the point-to-star distance (about 0.5 mm in the case of pyramidal neurons of the superficial layers of primate neocortex). This hexagonal spatiotemporal neuron firing pattern becomes a candidate for a cerebral code that can represent a mental image in the same manner as a barcode represents the contents of a grocery package. This pattern may clone considerable territories, with conformity enforced as a variant point becomes surrounded by up to six identical homologous points. If the long corticocortical bundles approximate the same point-to-star principle as the local connectivity, long-distance copying becomes possible without the point-to-point mapping achieved by a coherent light pipe, as the parent pattern can seed a mosaic of the same pattern at a distant cortical site.

Since Hebb's 1949 cell-assembly proposal1, we have realized that evoking a memory could involve an ensemble of cortical neurons, each of which helps to implement other memories as well. Different schemas might be characterized by different firing patterns in time and space. The analogous problem in genetic memory, that of the genetic code, was solved by first identifying which physical patterns could be reliably copied. But copying has not been on the neurophysiological agenda; we usually assume transformations rather than cloning when information is moved.

Many movement command clones were, however, inferred2,3 from the need to reduce timing jitter during precision throwing. And the notion of convergence zones for associative memories4 raises the issue of maintaining the identity of a cerebral code during long-distance corticocortical transmission, such as through the corpus callosum. Because of the reciprocal connections between distant cortical regions, any distortions of the original spatiotemporal firing pattern during forwards transmission would need to be compensated during reverse transmission in order to maintain the pattern as the local code for a sensory or motor schema. Here I suggest an error-correction mechanism in both directions that avoids the need for an inverse transform. Convergence to 

Entrainment via Minor Recurrent Excitation

The average cortical neuron contacts fewer than 1 percent of the neurons within a 1 mm radius5. One wiring principle can be seen from the axons of pyramidal neurons in the superficial cortical layers6,7,8. An axon goes a characteristic lateral distance without any terminations, then it produces a tight cluster (it may continue for an identical distance and then produce another cluster, though this will not be addressed here). J. S. Lund (personal communication) estimates this gap as about 0.5 mm in visual areas, 0.7 mm in prefrontal cortex, and nearly 1.0 mm in motor cortex of monkeys. The synapses are excitatory and often onto cells of the same type9. Because of the standard axon length (hereafter, simply 0.5 mm), recurrent excitation becomes possible.

Even if the synaptic strengths were high, much longer axons are required for a reverberating loop. But even weak coupling between relaxation oscillators is known to produce entrainment10,11, and that is the feature utilized here. Were the cells otherwise active, they would soon tend to produce some spikes at about the same time (Fig. 1A). This intermittent synchrony12 suggests the entrainment of other superficial pyramidal neurons located at the two points which are equidistant from the first pair (Fig. 1B).

If sufficient axons traveled in approximately the right directions (up to six, at 60deg angles from a point), this recruitment could continue to eventually form a mosaic of synchronous activity at 0.5 mm spacings (Fig. 2A). While the most efficient anatomy would be six equally-spaced axon branches from the same cell (Fig. 1B, Fig. 3AB), a more general wiring principle would be enough neighboring work-alike neurons to collectively send axons in all directions and thereby create an annulus of excitation. Cells in cortical minicolumns are often interested in the same kinds of stimuli (orientation columns) and fire in synchrony (at least in development); since there are about 100 neurons in such a column, coverage may well be sufficient without individual axons being able to branch at 60deg angles. Because simultaneous arrivals at the outlying neuron are the actual criterion rather than equal length, the standard axon can vary so long as conduction velocity or synaptic delay is tuneable. Thus fuzzy anatomy, so long as the 60deg directions are not avoided, can yield a self-organizing triangular mosaic. The largest copyable pattern is a 0.5 mm hexagon (Fig. 2B). Concurrent nonhexagonal activities may well be compatible, especially in other cortical layers. Indeed, something needs to get the cells firing so that weak coupling can entrain them (this also suggests that mosaics can be erased with diffuse waves of inhibition). Note that hexagons are not an assumption: they are an emergent property of standard axon lengths and weak excitatory coupling that entrains.

triangular, hexagon 
mosaic relations

Error-correction tendencies

Like a crystal in formation, a neighborhood of hexagons will tend toward conformity (Fig. 3B) because any deviating point repeatedly receives synchronous inputs from six homologous points. The same fuzzy point-to-star wiring as used locally (Fig. 3B) could also operate over long corticocortical paths (Fig. 3A). While some inputs will go astray via topographic mapping errors or temporal dispersal, only several repeatedly synchronous inputs might be needed to entrain the activity of the distant neurons; these coherent inputs could come from any of seven (center plus six neighbors) distant points. While point-to-star will not copy spatially unorganized activity in the manner of a coherent light pipe, it does allow a chorus to recreate their common pattern among a few distant listeners, that then clone their own territory.

Two hexagonal patterns could be superimposed, via either local or long-distance copying, and so form a new category, an episodic memory, or an association between a sensory schema and a response schema. If the dot matrix is sparsely populated (and estimates suggest that a dozen minicolumns might be active, out of 300 in the hexagon), the composite could serve as a form of associative memory in the manner of a double exposure. I discuss elsewhere13 the implications of hexagonal copying competitions for implementing a msec-to-minute version of the same darwinian process familiar from the immune response and species evolution. Neurallike networks could also 1) use hexagonal dot-matrix patterns as item representations, could 2) form associations, and 3) so implement a novel computing architecture with wiring that is diffuse enough to perform multiple tasks.

point to star 


1. Hebb, D.O., The Organization of Behavior (Wiley, New York, 1949).

2. Calvin, W. H., J. Theor Biol. 104, 121-135 (1983).

3. Calvin, W. H., The Cerebral Symphony(Bantam, New York, 1989).

4. Damasio, A.R., Cognition 33, 25-62 (1989).

5. Stevens, C. F., Neural Computation 1, 473-479 (1989).

6. Katz, L.C., Callaway, E.M., Ann. Rev. Neurosci. 15, 31-56(1992).

7. Rockland, K.S., Lund, J.S., Science 215, 1532-1534 (1982).

8. Gilbert, C.D., Hubel, D.H., J. Neurosci. 3, 1116-1133 (1983).

9. White, E.L., Cortical Circuits: Synaptic Organization of the Cerebral Cortex Structure, Function, and Theory (Birkhauser, Boston, 1989).

10. Winfree, A. T., The Geometry of Biological Time (Springer Verlag, Berlin, 1980).

11. Somers, D., Kopell, N., Biol. Cybernetics (in press, 1992).

12. Engel, A.K., Koenig, P., Kreiter, A.K., Schillen, T.B., Singer, W., Trends Neurosci. 15, 218-226 (1992).

13. Calvin, W. H., Soc. Neurosci. Abstr. 18, 214.18 (1992)

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Mailing address: University of Washington, Seattle WA 98195-1800 USA.

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Revised 23 September 1995 WHC