A mathematical model, of general character for the dynamic description of coupled neural oscillators is presented. The population approach that is employed applies equally to coupled cells as to populations of such coupled cells. The formulation includes stochasticity and preserves details of precisely firing neurons. Based on the generally accepted view of cortical wiring, this formulation is applied to the retinal ganglion cell (RGC)/lateral geniculate nucleus (LGN) relay cell system, of the early mammalian visual system. The smallness of quantal voltage jumps at the retinal level permits a Fokker-Planck approximation for the RGC contribution; however, the LGN description requires the use of finite jumps, which for fast synaptic dynamics appears as finite jumps in the membrane potential.
Analyses of equilibrium spiking behavior for both the deterministic and stochastic cases are presented. Green's function methods form the basis for the asymptotic and exact results that are presented. This determines the spiking ratio (i.e., the number of RGC arrivals per LGN spike), which is the reciprocal of the transfer ratio, under wide circumstances. Criteria for spiking regimes, in terms of the relatively few parameters of the model, are presented.
Under reasonable hypotheses, it is shown that the transfer ratio is ≤1/2, in the absence of input from other areas. Thus, the model suggests that the LGN/RGC system may be a relatively unsophisticated spike editor. In the absence of other input, the system is designed to fire an LGN spike only when two or more RGC spikes appear in a relatively short time. Transfer ratios that briefly exceed 1/2 (but are less than 1) have been recorded in the laboratory. Inclusion of brain stem input has been shown to provide a signal that elevates the transfer ratio (Ozaki & Kaplan, 2006). A model that includes this contribution is also presented.