Journal of Neurophysiology, Vol 60, Issue 2 751-768, Copyright © 1988 by APS
Theory of dynamic similarity in neuronal systems
R. J. MacGregor and G. Tajchman
Aerospace Engineering Sciences, University of Colorado, Boulder 80309-0429.
1. The techniques of dynamic similarity from the engineering science of
fluid mechanics are applied to neuronal systems to suggest how to scale
down critical parameters (such as numbers of constituent cells and
synapses, synaptic strengths, thresholds, etc.) from naturally occurring
systems to computer models. 2. The interconnectivity of a prototypical
neuronal junction is defined in terms of the total number of projecting
fibers, receiving cells, synapses, and directly connected cell fiber pairs.
Critical derivative parameters are defined in terms of these, including: a
global convergence factor, alpha ij, which is the ratio of the numbers of
projecting fibers to receiving cells; and an interconnectivity completeness
parameter or microscopic convergence/divergence parameter, gamma ij, which
measures both the percentage of cells to which a given sending fiber
projects (and the percentage of fibers from which a given cell receives)
and the percentage of cell fiber combinations which are directly connected.
3. Analysis of the differential equations governing neuroelectric activity
in constituent neurons suggests the definition of a sensitivity parameter
complex, sigma ij (with components eta ij and mu ij) for each ij junction.
These numbers represent the ratio of synaptic drive to current leakage in
nonactive neurons. 4. A model for quasi-steady firing suggests the
definition of a parameter, rho *j, which may be used to characterize the
level of activity in a given neuronal population in terms of its synaptic
drive and system parameters. It may be considered as the neuronal analog of
the Reynolds number in fluid mechanics. 5. The analysis implies that
computer models of neuronal systems should be scaled so as to keep the
parameters alpha ij, gamma ij, and sigma ij for every junction at the same
values as in the corresponding junctions of naturally occurring system
being modeled. Equations for a scaling factor, chi, numbers of constituent
synapses, thresholds, etc., are provided. The scaling method is illustrated
by a computer simulation example and by application to the junction of the
perforant path fibers to the granule cells of the hippocampus. 6. The
analysis shows that there is a fundamental trade-off in scaled down
computer models between verisimilitude at the level of network
interconnectivity and verisimilitude at the level of individual neuronal
dynamics. 7. The approach of dynamic similarity is discussed with respect
to compression of free parameters and predictive comparison of naturally
occurring systems.
