Waves, Pulses, and the Theory of Neural Masses

It is a truism that systems as complex as vertebrate nervous systems are more than the sum of their parts. What is meant is that the interconnection of numbers of neurons gives rise to collective properties belonging to the neural populations and not to the neurons taken one at a time. The purpose of this essay is to explore some facets of the nature of neural collective properties. Conventional wisdom holds that such properties emerge from the interconnection of finite numbers of neurons in discrete chains and networks, which are logical and anatomical counterparts of the Jacksonian-Sherringtonian heirarchy of reflex arcs. According to a popular analogy, neurons are like the electronic components of a television receiver which can be connected in a certain way or set of ways to give the properties of the receiver. The central thesis of this essay is the idea that, when neurons strongly interact in sufficiently large numbers (on the order of 10' or more), new collective properties emerge that demand a different kind or level of conceptualization. An analogy equivalent to that given above is the notion that temperature and pressure exist only for a mass, in contrast to the thermal kinetic energy of molecules in the mass. The suggestion is that certain interactive phenomena in vertebrate brains occur only as broadly distributed and continuous events or waves across masses of neurons, and that in some instances these cooperative phenomena may be essential aspects of normal brain function. The task is to describe some of these wave phenomena in terms of underlying collective properties, and to do so in such a way as to minimize confusion between observables and principles. Again by analogy, brain potentials (EEG waves) appear to have somewhat the relation to wave activity of neural masses that flow patterns have to temperature and pressure waves in atmospheric storms. They are observable side effects that are of interest mainly because they give access to the internal dynamics. The approach used is to review the historical interplay between ideas concerning neural networks and masses, to develop a set of rules for describing neural masses as dynamic entities, and then to discuss some of the implications of those rules for neurophysiology. Throughout the development the emphasis is placed on the idea of graded neural synaptic interaction, because it is interaction of neurons that gives rise to something more than the sum of parts. Neurons are connected to each other by structural synaptic linkages. For each neuron there is a certain density of these anatomical connections, referring to the number and size of contacts of each neuron with its neighbors within each unit volume of neural mass. But the significant quantity is the momentary functional or effective connection density, which denotes the level of transfer of influence across a given set of connections at a given time and place. If, for example, a volley arrives on an afferent path to a neuron that is in an absolute refractory state, the functional connection density is zero, even though the anatomical connection density is nonzero. Two kinds of massive connections are distinguished. The first is a one-way or forward connection from one neuron to neurons in another mass; the second is feedback connection of one neuron with many others in the same mass. Both types give rise to mass actions of many neurons, but only the second gives rise to the collective properties of interest in the present context. That is, neural interactions based on functional interconnection densities give rise to wave phenomena, and, as is shown for some of the neural masses in the mammalian olfactory system, the observable effects of wave patterns in turn provide the means for measuring the intensities of interactions