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Characterization, scaling, and partial representation of neural junctions and coordinated firing patterns by dynamic similarity

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Abstract

This paper presents a dynamic-similarity-based system for mathematically characterizing the functional connectivity and information flow of neural junctions. This approach allows for quantitative comparison of operations of neural junctions across systems, and an interpretation of their connectivity parameters in terms of the flow of multiunit firing patterns. The paper further uses this characterization to show how to rationally construct reduced operational models of neural junctions. Both uniformly proportional scaling and partial fragmentary representations are developed. The uniformly scaled models are better adapted to overall capacities and broader theoretical conceptualizations; the partial representations are better adapted to direct comparison with microelectrode experimentation. The characterization of information flow is based on coordinated multiunit patterns such as synfire chains or sequential configurations. The system can be applied to component parts of large composite networks including junctions with topographical patchiness and other irregularities. The characterization should be of use to anatomists, physiologists, modelers, and theorists. The theory predicts that the necessity for cooperative confluence of synaptic potentials in sending and receiving sequential configurations across topographically constrained projection fields requires the existence of functional ‘pattern modules’ within the topographical synaptology of the junction.

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Abbreviations

N :

number of cells in the sending population

t :

average number of output terminals per sending cell

s :

average number of input synapses per receiving cell

S :

total number of synapses

M :

number of cells in the receiving population

(β):

synaptic redundancy factor

P :

number of distinct connected fiber-cell pairs

f :

average fraction of area of receiving population in the terminal field of a given sender cell, and fraction of area in sending population in receptive field of a given receiving cell

t p :

averge number of distinct receiver cells connected by single sending cells

s p :

average number of distinct sending cells received by single receiving cells alpha (α)-overall convergence ratio gamma (γ)-microscopic connectivity ratio: fraction of total possible fiber-cell connections which are in fact connected, and average fraction of receiving cells connected to by a given sender cell, and average fraction of sending cells received by a given receiver cell gammaf (γ f)-topographical microscopic connectivity factor: average fraction of receiving cells within a sending cells terminal field connected by that sending cell, and average fraction of sending cells within the receptive field of a given receiving cell, connected by that receiving cell

SQ:

sequential configuration; a particular, coordinated, multiunit, dynamic firing pattern

nf i :

average number of SQ cells which fire per unit time in the input population

nf o :

average number of SQ cells which fire per unit time in the output population

q :

average fraction of nf o cells connected to by a given nf i cell

n L :

average number of time units over which synaptic connections are made from SQi to SQo cells

m :

average fraction of output terminals from single sending cells which are involved in single SQs

r * :

threshold number of EPSPs which must be applied every time step for n L time steps in order to trigger firing in single receiving cells

a :

safety factor elevating input to single receiving cells to ar * for a given SQ active at level nf i

p i :

packing density of an active SQ in the input population = nf i/N

p o :

packing density of an active SQ in the population = nf o/M

p t :

packing transfer ratio of SQs: The ratio p i/p o

d :

average amplitude of single EPSPs (mv)

(g4):

average firing threshold of individual receiving cells (mV)

S m :

amplification factor for temporal summation of EPSPs presented every time unit for n L times

O c :

diminution factor for EPSPs from occlusion

(ω):

characteristic average firing rate in input population

J t :

average number of connected input cells which must fire in order to activate single receiving cells sigma (σ)-relative overall average input activation level in the receiving population

f p :

the total firing power of a junction = sd/

C 1 :

the ratio nf o/t

c 2 :

the ratio nf i/s

L i :

average number of time units required for a single SQi

L o :

average number of time units required for a single SQo

C i :

fraction of sending cells utilized by a single SQi

C o :

fraction of receiving cells utilized by a single SQo

C J :

fraction of total synapses utilized by a single SQ

C if :

fraction of sending cells within a sending module used by a single SQi

C of :

fraction of receiving cells within a receiving module used by a single SQo

C Jf :

fraction of total synapses within a sending module used by a single SQ

′:

denotes quantities referring to scaled or abstracted junctions

x :

primary scaling factor

u :

scaled ratio of overall convergence factor

v :

scaled ratio of microscopic connectivity

x′ :

scaled ratio of redundancy factor, β

y :

scaled ratio of topographical factor, f

(ξ):

primary scaling factor for SQs

y * :

scaled ratio of nf i/r *

v′ :

scaled ratio of packing transfer density, p t

q′/q :

scaling factor: ratio of scaled q to original q

x min :

minimal scaling possible with discrete junctions

f min :

minimum value possible for topographical factor

x min2 :

constraint on x min implied by constraint on f

F(x, f) :

hypergeometric-based function involved in topograhical constraints

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Authors and Affiliations

  1. Department of Aerospace Engineering Sciences, University of Colorado, Campus Box 429, 80309-0429, Boulder, Colorado, USA

    R. J. MacGregor, F. G. Ascarrunz & M. A. Kisley

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  1. R. J. MacGregor
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  2. F. G. Ascarrunz
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  3. M. A. Kisley
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MacGregor, R.J., Ascarrunz, F.G. & Kisley, M.A. Characterization, scaling, and partial representation of neural junctions and coordinated firing patterns by dynamic similarity. Biol. Cybern. 73, 155–166 (1995). https://doi.org/10.1007/BF00204054

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