Periodicity of Non-Expanding Piecewise Linear Maps
Presented in Dynamical Systems !

ResearchRobust Professional Abilities

Periodicity of Non-Expanding Piecewise Linear Maps <BR>Presented in Dynamical Systems !

Fumihiko Nakamura, a pilot student from the Hokkaido University Graduate School of Science, Department of Mathematics, precisely depicted sufficient conditions to show that the most simplified dynamical system (a model that describes the “firing” phenomenon of brain neurons) has periodic points and proved that those conditions have the same structure as the Farey sequence. These results suggest the possibility of the neuron-firing phenomenon being almost entirely periodic and that of the existence of arbitrary firing patterns with periodicity. Moreover, since this model has complex periodic structures, it also suggests the existence of the possibility of periodicity changes occurring due to slight changes in parameters contingent on the coupling coefficient between cells, external stimulation, and the threshold value for firing. These results were published as an essay in Dynamical Systems, and they are attracting attention as the first step toward understanding the complex periodic structures of this model.