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. 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 can be represented by 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 very small value of parameter $\mu$ $(0<\mu <<1)$. Unlike variable $x$, variable $y$ does not have explicit biological meaning though some analogy to gating variables can be drawn.

See also

• Biological neuron model
• Hodgkin–Huxley model
• FitzHugh–Nagumo model