Valentino Braitenberg

Brain size and number of neurons: an exercise in synthetic neuroanatomy.

Certain remarkable invariances have long been known in comparative neuroanatomy, such as the proportionality between neuronal density and the inverse of the cubic root of brain volume or that between the square root of brain weight and the cubic root of body weight. Very likely these quantitative relations reflect some general principles of the architecture of neuronal networks. Under the assumption that most of brain volume is due to fibers, we propose four abstract models: I, constant fiber length per neuron; II, fiber length proportionate to brain diameter; III, complete set of connections between all neurons; IV, complete set of connections between compartments each containing the square root of the total number of neurons. Model I conforms well to the cerebellar cortex. Model II yields the observed comparative invariances between number of neurons and brain size. Model III is totally unrealistic, while Model IV is compatible with the volume of the hemispheric white substance in different mammalian species.

The cerebellar network: attempt at a formalization of its structure

The well defined histology of the cerebellar cortex makes it possible to translate its structure directly into statements about the transformation of cerebellar input into output. In the system of ‘parallel fibres’ the input activity is shifted in space and time in an anisotropic way. One consequence of this is that input patterns moving across the cerebellar cortex at a velocity corresponding to that of conduction in parallel fibres should elicit a much larger response than static input.

Information from structure: a sketch of neuroanatomy

After the discovery of electrical phenomena in brains, which suggested recording with electrodes as the obvious approach to the problem of cerebral function, the field of neuroanatomy underwent a long period of stagnation. It is now once more an important source of information, which model makers are tapping abundantly (those who are not would be well advised to consider doing so).

Density of Axons

The sum total of the length of all the nerve cell processes (dendrites and axons) in a given volume of brain substance provides, in a sense, a measure of the information-handling capacity of the nerve tissue. From a different point of view, we may take it as indicating the complexity of the neural operations performed there.

Density of Neurons

Contrary to the density of synapses which, as we shall see, does not vary much from animal to animal and from place to place in the same brain, the number of neurons per unit volume varies a great deal. As a rule the density of neurons is lower in larger brains, a fact which is explained in a qualitative way by the larger size of the neurons in the larger animal.

What Sort of Information Can Be Gained from Cytoarchitectonics on Nissl Pictures

What leaps to the eye in the Nissl picture viewed at low magnification is the alternation of “darker” and “lighter” layers. At a higher magnification it becomes obvious that the darkness is composed out of two parameters, number and size of neural cell bodies. (We neglect another possible source of variation, the density of Nissl bodies within the perikaryon.) How can we interpret these two variables in terms of connectivity?

Length of the Dendritic Trees

We measured dendrites with the same methods we had used for axons. Here again, besides the few specimens measured in Braitenberg (1978b), we relied on Staiger’s (1984) collection. The sample was larger than in the case of the axons, since satisfactory impregnation of the dendritic tree is much more common in Golgi preparations than a complete stain of both dendrites and axons. The stain of the initial segment of the axon, in conjunction with the shape of the dendritic tree, is sufficient in most cases to confidently classify the neurons as belonging to one of our three types.

Relative Density of Axons and Dendrites

We found two questions interesting in connexion with the shape of neuronal ramification in the cortex. One is, how wide is the territory in which the intracortical axonal branches of a certain neuron are distributed? The other, what proportion of the synapses present within that territory is served by that one axon? The two questions are related, and the answer to one can be deduced from the answer to the other once we know, as we already do, the total density of axons in the tissue and the axonal length of individual neurons.

Quantitative Aspects of the Three Types of Neurons. Methods

The reason why we first measured the length of axonal branches of individual neurons in the cortex, was our impression that the density of the axonal tree is very different in different neurons and may be used as one of the distinguishing characteristics of pyramidal and non-pyramidal cells. This was indeed confirmed, the pyramidal cells producing the least dense axonal ramification and some stellate cells, as well as some extracortical afferents (Fig. 31) the densest in the cortex.

Comparison Between the Densities of Neurons, Synapses and Axons

The fact that 4 km of axons belonging to 9.2 x 104 neurons allot more than 4 cm of axon to each neuron is surprising, especially since direct measurements on Golgi preparations have led to values less than half that figure. It must be noted, however, that the measurements in Braitenberg 1978b which refer to pyramidal cells (the classification of cell types will be introduced in Chap. 15) do not consider the reentrant axons of these cells. These may make considerable terminal arborizations which are only rarely stained by the Golgi procedure and if they are, are practically never seen in continuity with the other part of the axonal tree.