Period doubling bifurcation in a train of spontaneous activity

Abstract

Introduction: It has been shown that fibrillations (FIBs) and positive sharp waves (PSWs) occur during spontaneous oscillations of the membrane potential when the threshold is reached. According to the models of excitable membranes these oscillations should be nonlinear. We wondered whether the nonlinearity of the underlying process could also be demonstrated indirectly in the pattern of spontaneous discharges that can be recorded much easier extracellularly in routine electromyography. A simple example of nonlinear behaviour is a period doubling bifurcation in a periodic system. Methods: In our clinical neurophysiology laboratory discharges are saved on hard disc and the interval between successive discharges is analyzed if there is any doubt whether the discharges occur at regular intervals. In one recording out of 32 we found a phenomenon that we believe to represent a bifurcation. The recording is from the left anterior tibial muscle of a 67 year old patient. The patient showed widespread presence of spontaneous activity and fasciculations in muscles. Neurography was normal. Thus, a motor neuron disease was suspected. The recordings were made with the commercial Neurophysiology workstation „Keypoint“ from ALPINE Biomed with concentric needle electrodes. The samples were exported in ASCII format at a sampling frequency of 24kHz and a resolution of 1µV. Discharges were selected manually and the interdischarge interval (IDI) was calculated. Results: The upper trace in Fig.1 shows the signal as recorded with the concentric needle electrode. The peaks of the discharges that were used to calculate the interdischarge intervals are marked with little circles. The lower trace shows the interdischarge intervals as a function of time. Note that the discharge intervals decrease initially, and increase slowly afterward. At the moment indicated by the double arrow two different IDIs occur that alternate. This resembles behaviour resembles very closely period doubling bifurcations that can be observed in simple nonlinear systems such as the logistic map. Discussion: The recording presented here illustrates that irregularity at the end of trains of spontaneous activity can be the result of nonlinear deterministic behaviour. Further studies have to show whether more irregular behaviour that also can be observed at the end of these trains can also be described with methods of nonlinear dynamics.