Large-scale brain simulations require the investigation of large networks of realistic neuron models, usually represented by sets of differential equations. Here we report a detailed fine-scale study of the dynamical response over extended parameter ranges of a computationally inexpensive model, the two-dimensional Rulkov map, which reproduces well the spiking and spiking-bursting activity of real biological neurons. In addition, we provide evidence of the existence of nested arithmetic progressions among periodic pulsing and bursting phases of Rulkov’s neuron. We find that specific remarkably complex nested sequences of periodic neural oscillations can be expressed as simple linear combinations of pairs of certain basal periodicities. Moreover, such nested progressions are robust and can be observed abundantly in diverse control parameter planes which are described in detail. We believe such findings to add significantly to the knowledge of Rulkov neuron dynamics and to be potentially helpful in large-scale simulations of the brain and other complex neuron networks.
Bibliographical noteFunding Information:
This work was initiated during a visit of the authors to the Max-Planck Institute for the Physics of Complex Systems, Dresden, gratefully supported by an Advanced Study Group on Forecasting with Lyapunov vectors. GMRA was supported by the German Academic Exchange Service, Re-invitation Programme. IMJ was supported by the Hungarian National Research, Development and Innovation Office under Grant K-125171. JACG was supported by CNPq, Brazil, Grant PQ-305305/2020-4.
© 2022, The Author(s).