Rulkov map

Time series of Rulkov map showing three different dynamical regimes

The Rulkov map is a two-dimensional iterated map used to model a biological neuron. It was proposed by Nikolai F. Rulkov in 2001.^[1] 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:

$x_{{n+1}}={\frac {\alpha }{1+x_{n}^{2}}}+y_{n}$

$y_{{n+1}}=y_{n}-\mu (x_{{n}}-\sigma )$

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.^[2] 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 paramter 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.^[3] 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

References

1. "Modelling of spiking-bursting neural behavior using two dimensional map",
2. Igor Franovic´; Vladimir Miljkovic´ (2011). "The effects of synaptic time delay on motifs of chemically coupled Rulkov model neurons". Commun Nonlinear Sci Numer Simulat. 16: 623–633. doi:10.1016/j.cnsns.2010.05.007.
3. N.F. Rulkov (2001). "Regularization of Synchronized Chaotic Bursts". Physical Review Letters. 86: 183–186. doi:10.1103/physrevlett.86.183.