ABSTRACT
I.INTRODUCTION The effect of neural-dendritic interactions haS so far been only weakly probed in the realm
.. ,However, the more frequent situation is that the body of a neuron has synapses with axons ofmany other neurons. It even appears that, occasionally, several axons from one neuron form synapses on another. Thus the possible stimulators are many, and the patterns of stimulation that may be effective have more complicated definitions than the simple "and" and "or" schemes described .,. It may wen be that certain nerve pulse combinations will
stimulate a given neuron not simply by virtue of their number but also by virtue of the spatial relations of the synapses to which they arrive ... [po 54]
Stein [1965,67] treated the neural activation as a continuous Marlcov process whose sampie paths liad discontinuities of the first kind. Solutions to this type of model are difficult due to the nature of the differential-difference dynamics. Also, this approach modeled the inputs to a neuron as discrete current pulses arriving according to a Poisson process. The value of "I(t) would change instantaneously upon arrival of a distinct excitatory or inhibitory event. and then decay
(lla)
(lle)
a7R-dc.' < I i>I, (12) ,
Hg 1. Effective nonlinearity parameter ß/a (computed from (llb»
to be a good approximation (and we are aIready in violation of (12». It is important to point out that, contrary to conventional mean field theories, the adiabatic elimination of fast variables utilized in this worlc does not necessarily yield better results as the number N of entities increases; rather it depends solelyon the separation of time-scales embodied in the inequality (2). Hence, repeating the simulations of figure 2 with larger N is unlikely to shed any fresh light on this aspect of the problem. Increasing N does, however, lead to other changes in the bifurcation properties of the reduced dynamics (9). These changes have been detailed elsewhere [Bulsara, Maren and Schmera 1992] and are not discussed here.
