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Last update 7 May 2006

The stochastic connectivity of the cortex

At the scale of organs, brains have a well-defined structure. Parts of the brain have a reasonable well-defined structure in smaller scale, in the region of 1mm. The connectivity at lower scale (low-level connectivity), however, is not well specified.

For example, When an axon from the Lateral Geniculate Nucleus enters the visual cortex, it is directed to some location in the cortex, to preserve the topographic mapping of the information. This is commonly given as an example of highly ordered connection (e.g. Shepherd (1990), p.395). However, in the cortex the axon branches to an 'axon tree' which span more than 1mm squared, and is made of hundreds of branches (Shepherd (1990), p.396). Within this region the neuron forms contacts with only part of the neurons, depending on the type of the target neuron and location of its dendrites (mostly layer 4, in this case). This still leaves a choice of several tens of thousands of neurons to choose from (or even more), and the axon forms connections with few thousands of these. The selection of these few thousands is essentially stochastic, by which I mean it is not related in a consistent way to the selection that other neurons do, in the same brain or in other brains.

The evidence for this is from comparison of the axon trees of different neurons, within the same brain and from brains of different animals of the same species. It is clear that the structure of the axon trees of individual neurons is not well specified. When it come to comparison between brains, or between the two hemispheres in the same brain, it is not even possible to match individual neurons between brains, because they are too different. Since the low-level connectivity is different between individuals, it cannot be specified during development (by the genes or otherwise), and hence must be stochastic.

This conclusion tells us more than just about differences between individual brains. It tells us that the set of neurons which will tend to become active as a result of activity of some specific neuron is stochastic, i.e. uncorrelated to the set that will tend to become active as a result of the activity of any other neuron, even in the same brain. It follows immediately that the set of neurons that will become active as a result of the activity of some specific set of neurons is stochastic. In other words, the relation between a some pattern of activity |X| and the pattern of activity |Y| that it will activate (the transformation |X|-> |Y|) is stochastic, i.e. uncorrelated to the relation between any other pattern of activity |X'| and the pattern of activity |Y'| that it will activate (the transformation |X'| -> |Y'|).

It should be noted that this lack of relations applies within the same brain, and this is what is meant by the term stochastic connectivity in this article. In particular, the term does not mean variations over time, and does not mean lack of correlation between relations between patterns of activity (|X| -> |Y|) and relations between entities in the outside world.

It can be argued is that even though the connectivity as defined by the axon trees are not well defined, some process reduce the strength of irrelevant synapses, so they become insignificant. The problem with this possibility, however, is that this process requires that the information about the correct connectivity be stored somewhere, and then affect the modification of the synapses. This information cannot be stored in the neurons themselves (because of their stochastic connectivity), and there is no other place in the brain, body or outside the body where this information (i.e. which synapses needed to be eliminated to get the right connectivity) can be stored.

Note what the argument above does not say:

The brain is completely unspecified.
This is not true, both because at the level of cell populations there is order in the brain, and at that level it is possible to match different brains, and because even at the individual neuron level there are various tendencies. The point is that these are tendencies, and are very very far from being complete specification.
The low-level connectivity cannot be related to anything.
This is not true because information about other things (in particular, about the outside world) can be used in modification of synapse strength. Thus the low-level connectivity can be related in some way to the outside world.

In the Peripheral Nervous System (PNS) the individual connections are less stochastic, but even there, in most of the cases, the low-level connectivity is not well specified. For example, normally each muscle fiber is innervated by a single axon. Initially the fiber is innervated by several axons, and then there is a process of selection, which causes all of these, except one, to retract. Which axon stays is a stochastic choice, a conclusion that is again based on comparison between individual animals.

The stochastic nature of the low-level connectivity is almost never mentioned explicitly in neurobiological textbooks, probably because they don't believe that this fact has any consequences. Instead, these books emphasize the order that exists in coarser resolution, many times in a confusing way.

For example, Nicholls, Martin & Wallace (1992, p. 341) ask: "What cellular mechanism enable one neuron to select another out of myriad of choices, to grow toward it, and to form synapses?". They later bring examples of specific connectivity. However, in all the examples that concern vertebrate Central Nervous System (CNS), the specificity is in the level of cell populations, rather than individual connections. Thus the answer to the question is that in the CNS a neuron does not "select another". Rather, it selects a region and cell types, which still leaves quiet a large spectrum for individual choices.

Maybe the worst example is in kandel et al (1991). On page 20 appears, as part of the 'principle of connectional specificity' which is supposed to be general property of neurons, this assertion: ".. (3) Each cell makes specific connections of precise and specialized points of synaptic contacts - with some postsynaptic target cells but not with others." The 'specific connections' is true in some invertebrate systems, but it is simply false when applied to the vertebrate brain. In chapter 58, 'Cell migration and axon guidance', the author tries to support this assertion, but all the examples of specific connectivity are from invertebrates. There are some examples from vertebrates, but they all show connectivity between cell populations, rather than individual cells. In addition, they are all about peripheral neural system, except one example from the spine of bullfrog. The vertebrate brain is not even mentioned in this chapter. It is obvious that this is because there are no example of specific connectivity there, but the text does not actually say this. The next chapter, 'Neural Survival and synapse formation', discusses only neuron-muscle junctions, and there is no further discussion on the question of specific connectivity.

Disappointingly, This is true even in books that are explicitly about the computational aspect of the brain, e.g. Churchland & Sejnowski (1992), Baron (1987), Gutnick & Mody (Eds.)(1995). For example, in Gutnick & Mody (Eds.)(1995), Section iii is about "The Cortical Neuron as Part of a Network". However, only the chapter about modeling this network (Bush & Sejnowski, 1995) mentions individual connections, by saying that they assume them to be random in their simulations (P. 187). Even they don't actually discuss the point, and none of the other chapters in this section, or in the rest of the book, touches the point.

Even though the stochastic nature is not explicitly stated, it is clear from the data that is presented in these books that this is the case. One of the 'distal' targets of this article is to show the significance of this fact, and hence to convince neurobiologists (and others) to pay attention to it.

newer additions

[8 Apr 2004] This review (Visual Object Understanding, Palmeri & Gauthier , Nature Reviews Neuroscience 5, 291-303 (2004); doi) is a typical example ignoring the stochastic connectivity. Most of it is non-sense, because the theories they describe are either obviously false or don't have any meaning in the cortex with its stochastic connectivity. The two authors come from psychology but also do "cognitive neuroscience", and it is not obvious whether they know that the connectivity inside the cortex is stochastic or not.

[ 11 Nov 2004 ] To add to the confusion, neuroscientists commonly confuse individual neuron specificity, i.e. which neuron connects with which neuron, with other types of specificity. Most commonly it is neuronal type specificity, i.e. what types of neurons connect with each other. Other are laminar specificity, i.e. in which layers are the neurons. The other specificties are much much less specific, because there are only few (as far as specificity is concerned, certainly less than 1000) neuronal types, compare to billions of individual neurons, out of which at least few millions are within the range to connect with each neuron. Thus neuronal type specificty is much much weaker specificity, and still leave the connectivity very very far from being 'precise' in any snese.

This confusion of specificities seems sometimes to be done intentionally. for example, in this summary/discussion article (Journal of Neurocytology 31 (3-5): 387-416, March - May - June, 2002 ) , Edward L. White says, in a respond to the question what "specific" means (end of page 391):

In this paper, "specific" is used to describe synaptic connections between "specific neurons or neuronal types".
He goes on to say that he does that to "be comprehensive", but that is nonsense. He clearly does that so he doesn't have to admit lack of specificity in the cortex. In the article itself (White, Journal of Neurocytology, 31 (3-5): 195-202, March - May - June, 2002) he does his best to defend specificity in the cortex, to a ridiculous extent. His first argument, "contrasting synaptic patterns on the surfaces of spiny vs. non-spiny neurons", is specially ridiculous, because these synaptic patterns give us exactly zero information about which neuron connects with which neuron, and are therefore totally irrelevant.

[22 Jan 2005] In this new article (Kalisman, Silberberg an Markram, PNAS | January 18, 2005 | vol. 102 | no. 3 | 880-885) they claim to "..provide the first direct experimental evidence for a tabula rasa-like structural matrix between neocortical pyramidal neurons..". For the senior author, this seems to be a change of heart from his position two years ago.

Another example of research that wouldn't have happened if people realize the issue of stochastic connectivity is in Machens et al, Science, V 307, p.1121 18 Feb 2005 (First author publication with full text). To get their model working, they need to get the inputs to their system in the right way, by specific input connectivity (see their figure 3, G and H). Note that they need to control the input in the same way to all of their units, which is clearly impossible with stochasic connectivity. But since they ignore this point (in this case it seems they are genuinely not aware of it), they don't see a problem.

The authors of the highly positive comentary in Nature Neuroscience Latham &, Dayan, Nature Neuroscience 8, 408 - 409 (2005) say about this circuitry : "Machens et al avail themselves of creative wiring to achieve this." I thought Maybe they are aware of the problem and this is supposed to point it in an ironic way, but Latham told me it is not. It seems they are not aware of the problem too.

[7 May 2006] In >this artcile in PLoS computational Biology (The Human Connectome: A Structural Description of the Human Brain, Olaf Sporns, Giulio Tononi and Rolf Kötter) they advocate creating a "Connectome", which at least shows that they realize that the pattern of connections is important. However, that is what they believe about variability across individuals:

The large-scale connectivity structure of the brain above the synaptic level represents a relatively invariant characteristic of our species.
That researchers in areas close to neuroscience can write such rubbish is not new. What is stunning in this case is that in the next paragraph they bring evidence against it, including evidence that even at the level of sulci and gyri there is significant variability across individuals. Thus it is not that they don't know the facts: they just ignore them.

Their justification is by analogy to the genome. That makes sense only to somebody that doesn't know the facts. In the case of the genome, the variability is minute compared to the total size of the genome. Between any par of individuals, the vast majority of the genome is identical (that is true even between a human and a chimpanzee). In the case of the brain, below the level of regions it is simply not possible to match "units" across individuals, even between identical twins. Therefore the analogy to the genome is completely broken.



Baron, Robert J. (1987), The Cereblar Computer: An introduction to the computational structure of the human brain. Hillsdale, NJ: Lawrence Erlbaum Associates.

Brodar, Per(1992) The Central Nervous System, structure and function. New York, NY: Oxford University Press.

Bush, Paul & Sejnowski, Terrence J. (1995), `Models of Cortical Networks', in Gutnick, Michael J. & Mody, Istvan (Eds.) The cortical neuron. New york, NY: Oxford University Press, pp.174-189.

Churchland, Patricia S. & Sejnowski, Terrence J. (1992). The Computational Brain. Cambridge, MA:MIT Press.

Gutnick, Michael J. & Mody, Istvan (Eds.)(1995), The cortical neuron. New York, NY: Oxford University Press.

Johnson-Laird, Philip (1993), The Computer And The Mind (second edition). London, UK: Fontana Press.

Kandel, Eric P., Schwartz, James H., Jessel, Thomas M. (Eds.) (1991), Principles of Neural Sciences (third edition). New York: Elsevier.

Matlin, Margaret W. (1994), Cognition (third edition). Fort Worth: Harcourt Brach Publishers.

Nicholls, John G., Martin, Robert A. & Wallace, Bruce G. (1992), From Neuron to Brain (third edition). Sunderland, MA: Snauer Associates Inc.

Shepherd, Gordon M. (Ed.)(1990), The synaptic organization of the brain (third edition). New York, NY: Oxford University Press.

Shepherd, Gordon M. (1994). Neurobiology (third edition). New York, NY: Oxford University Press.

Yehouda Harpaz
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