Controlling the bursting size in the two-dimensional Rulkov model

Abstract

We propose to control the orbits of the two-dimensional Rulkov model affected by bounded noise. For the correct parameter choice the phase space presents two chaotic regions separated by a transient chaotic region in between. One of the chaotic regions is the responsible to give birth to the neuronal bursting regime. Normally, an orbit in this chaotic region cannot pass through the transient chaotic one and reach the other chaotic region. As a consequence the burstings are short in time. Here, we propose a control technique to connect both chaotic regions and allow the neuron to exhibit very long burstings. This control method defines a region Q covering the transient chaotic region where it is possible to find an advantageous set S ⊂ Q through which the orbits can be driven with a minimal control. In addition we show how the set S changes depending on the noise intensity affecting the map, and how the set S can be used in different scenarios of control.