Rulkov Map

The Rulkov map is a two-dimensional iterated map used to model a biological neuron. It was proposed by Nikolai F. Rulkov in 2001."Modelling of spiking-bursting neural behavior using two dimensional map

/ref> The use of this map to study neural networks has computational advantages because the map is easier to iterate than a continuous dynamical system. This saves memory and simplifies the computation of large neural networks.

The model

The Rulkov map, with $n$ as discrete time, can be represented by the following dynamical equations:

:$x_=\frac+y_n$
:$y_=y_n-\mu(x_-\sigma)$

where $x$ represents the membrane potential of the neuron. The variable $y$ in the model is a slow variable due to a very small value of $\mu$ $(0 < \mu << 1)$. Unlike variable $x$, variable $y$ does not have explicit biological meaning, though some analogy to gating variables can be drawn. The parameter $\sigma$ can be thought of as an external dc current given to the neuron and $\alpha$ is a nonlinearity parameter of the map. Different combinations of parameters $\sigma$ and $\alpha$ give rise to different dynamical states of the neuron like resting, tonic spiking and chaotic bursts. The chaotic bursting is enabled above $\alpha > 4$

Analysis

The dynamics of the Rulkov map can be analyzed by analyzing the dynamics of its one dimensional fast submap. Since the variable $y$ evolves very slowly, for moderate amount of time it can be treated as a parameter with constant value in the $x$ variable's evolution equation (which we now call as one dimensional fast submap because as compared to $y$, $x$ is a fast variable). Depending on the value of $y$, this submap can have either one or three fixed points. One of these fixed points is stable, another is unstable and third may change the stability. As $y$ increases, two of these fixed points (stable one and unstable one) merge and disappear by saddle-node bifurcation.

See also

*Biological neuron model
*Hodgkin–Huxley model
*FitzHugh–Nagumo model
*Chialvo map

References

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Dynamical systems