The nature of the coupling between segmental oscillators of the lamprey spinal generator for locomotion: A mathematical model

Abstract

We present a theoretical model which is used to explain the intersegmental coordination of the neural networks responsible for generating locomotion in the isolated spinal cord of lamprey. A simplified mathematical model of a limit cycle oscillator is presented which consists of only a single dependent variable, the phase theta(t). By coupling N such oscillators together we are able to generate stable phase locked motions which correspond to traveling waves in the spinal cord, thus simulating "fictive swimming". We are also able to generate irregular "drifting" motions which are compared to the experimental data obtained from cords with selective surgical lesions.

Modeling inter-human movement coordination: synchronization governs joint task dynamics

“…Schmidt et al [1993] employ the task paradigm of swinging pendulums originally introduced by Turvey et al [1986]. An extension of the HKB coupling function by a frequency detuning term similar to the coupling function proposed by Cohen et al [1982] is found to account for both the effects of different eigenfrequencies and external forcing frequencies.…”

“…Frequency detuning imposed through different eigenfrequencies and frequency levels are introduced as control parameters. Depending on the frequency level and the intended phase relation, the authors obtain the coupling strength of a local dynamical model similar to Cohen et al [1982]. The number of coordination breakdowns, the phase fluctuation and the coupling strength reveal interpersonal coordination to be weaker than intrapersonal coordination.…”

A Model of the Effects of Applied Electric Fields on Neuronal Synchronization

“…We now argue that the transition to synchrony in the network including the wedged region can be explained using a simple phase oscillator model (Winfree, 1980;Cohen et al, 1982;Kuramoto, 1984;Ermentrout and Kopell, 1984). Specifically, the boundary between the synchronous and the asynchronous states is highly correlated with the degree of natural frequency mismatch among the individual units within the network.…”

“…This periodic behavior can typically be reduced to a simple phase equation under a suitable parameterization and the assumption of weak coupling. Various authors (Rand and Holmes, 1980;Cohen et al, 1982;Kuramoto, 1984;Ermentrout and Kopell, 1984) have given detailed descriptions for similar phase reductions in different biological systems. For the following discussion, we assume that the reduced phase equation for the ith uncoupled neuron can be written as…”

“…(Kuramoto, 1984;Cohen et al, 1982;Ermentrout and Kopell, 1984) and further assume that the input from the presynaptic neuron is not so strong that it significantly alters the form of the spike. From our numerical simulations, this is a valid assumption for almost the entire range of parameters examined.…”