Distribution of spiking and bursting in Rulkov’s neuron model

Abstract

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.

New phenomena in Rulkov map based on Poincaré cross section

“…There are some different phenomena from the existing literatures such as Ref. [20]. When the initial value is (−1, −3), the split lines of the period of the original map are like "lightning-shaped", otherwise, when the initial value is (−1, 3), the split line of the period are smooth curve.…”

“…The y n is the slow dynamics variables, thus it can be used to judge the periodicity of the orbit with relatively high precision.In Ref. [20], it shows that the chaos also occurs in the region of bursts of spikes and spikes in Fig. 1, the chaos can be computed by Lyapunov exponents, besides the spikes and bursts of spikes with different period can be separated by smooth critical lines.…”

“…The general method is by means of time series of fast variable and slow variable directly to find the period and patterns of spikes of the neuron numerically. For example, the patterns of spikes of Rulkov's map was proposed in the parameter space by the method of computation mentioned above in [20], where the chaos was given Lyapunov exponents. Some others work also involved the computation of the Rulkov's map by improved the model or considering the model in the networks, such as memristive neuron for Rulkov model including magnetic effects [21][22][23]; the synchronization problem of chaotic oscillators [24][25][26][27]; and delayed feedback control of bursting synchronization [28][29][30][31]; phase synchronisation of bursting neurons in small-world networks [32][33][34][35].…”

A novel chaotic map with a shifting parameter and stair-like bifurcation diagram: dynamical analysis and multistability

“…For instance, a hyperchaotic map based on offset boosting [20], a memristive map with a cosine memristor and hidden multistable dynamics [21], and 2D rational memristive maps with hidden attractors and various solutions [22]. Additionally, a comprehensive fine-scale analysis of the dynamical response across wide parameter ranges of the 2D Rulkov map was presented in [23].…”