Testing a model of excitatory interactions between oscillators

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Abstract
The experimentally observed influence of regularly arriving tugs upon the AP discharge of the slowly-adapting stretch receptor organs (SAO) of crayfish was compared to a model of pacemaker excitatory synaptic interactions (Segundo and Kohn 1981). Criteria for compliance referred to facets as A) the excitation, B) the postulates, and C) the behavior. Excitation was implied primarily by the tug initially increasing the AP rate (it subsequently decreased it). The pacemaker AP discharges, and with more reason the electronically driven tugs, were considered acceptably regular sequence (postulate i). Tugs advanced the next AP (postulate ii); the “delay function” plost of delays vs. phases, i.e. interval shortenings vs. the time from the last AP to the tug, were close to theV of postulate iii (Fig. 2), even though the shortest phases tended to postopone the next AP and the longest ones did not trigger immediately but with an around 5 ms latency. These effects were displayed also as “old phase vs. new phase” plots (Figs.2 and 3). The interval following that with the tug tended to be lengthened, but the pre-tug timing was not recovered (Fig. 4). Behavior during a train of excitatory events, both in model and experiments, went through very similar initial settlings and eventual steady-states. The latter were characterized in the model by 1. an average excitatory vs. excited rate display formed by an endless number of segments with all positive rational slopes separated by negative-going discontinuities, 2. locking in the sense of preferential phases, and 3. periodic repetition of the same phases and inter-AP intervals. Experimental results were compatible with this (Figs. 5 and 7–9). Such behavior was absent when the tug sequence was highly irregular (Fig. 9). The initial settling, in the SAO as in the model, depended jointly on the first phaseΦ 1 and the intertug intervalE. If the former was under λ, it went through one or two monotonic phase-decreasing stages (one smaller, the other larger, than λ), or through a single increasing one, depending onE being smaller or greater than, respectively, an estimated but never actually observedE * leading to unstable lockings. If the initial phase was greater than λ, settling withE's underrN+λ involved jumps between larger than and smaller than λ phases; withE's overrn+λ, it involved an intermediate stable locking with Φ=E-rN. The model-embodiment correspondence in terms of postulates was judged acceptable in spite of discrepancies like the imperfect regularities, the dispersion and extra segment in the delay function (Fig. 2), and the post-tug interval lengthening (Fig. 4). The correspondence in terms of behavior turned out to be quite precise (see above) and at times surprising because of its counterintuitiveness. This reiterates the epistemologically interesting issue of why the generality of model applicability.