Chialvo map

Introduction

The Chialvo map is a two-dimensional map that captures the excitable behavior of neurons. It was proposed by Dante R. Chialvo in 1995 . The model is used to simulate the activity of one neuron and by using few parameters is able to mimic generic neuronal dynamics.

The model

The model is an iterative map where at each time step, the behavior of one neuron is updated as the following equations:

$${\begin{aligned}x_{n+1}=&f(x_{n},y_{n})=x_{n}^{2}\exp {(y_{n}-x_{n})}+k\\y_{n+1}=&g(x_{n},y_{n})=ay_{n}-bx_{n}+c\\\end{aligned}}$$

in which, $x$ is called activation or actions potential variable, and $y$ is the recovery variable. The model has four parameters, $k$ is a time-dependent additive perturbation or a constant bias, $a$ is the time constant of recovery $(a<1)$, $b$ is the activation-dependence of the recovery process $(b<1)$ and $c$ is an offset constant. The model has a rich dynamics, presenting from oscillatory  to chaotic behavior.

Analysis

Bursting and chaos

The map is able to capture the aperiodic solutions and the bursting behavior which are remarkable in the context of neural systems. For example, for the values $a=0.89$, $c=0.28$ and $k=0.025$ and changing b from $0.6$ to $0.18$ the system passes from oscillations to aperiodic bursting solutions.

Activation variable as a function of time for the chaotic regime

Solution for the Chialvo map equations for the chaotic regime

Activation variable as a function of time for the excitable regime

Solution for the Chialvo map equations for the excitable regime

Fixed Points

Considering the case where $k=0$ and $b<<a$ the model mimics the lack of ‘voltage-dependence inactivation’ for real neurons and the evolution of the recovery variable is fixed at $y_{f0}$. Therefore, the dynamics of the activation variable is basically described by the iteration of the following equations

${\begin{aligned}x_{n+1}=&f(x_{n},y_{f0})=x_{n}^{2}exp(r-x_{n})\\r=&y_{f0}=c/(1-a)\\\end{aligned}}$

in which $f(x_{n},y_{f0})$ as a function of r has a period-doubling bifurcation structure.

Period-doubling bifurcation structure of the Chialvo Map

Examples

Example 1:

A practical implementation is the combination of $N$ neurons over a lattice, for that, it can be defined $0>d<1$ as a coupling constant for combining the neurons. For neurons in a single row, we can define the evolution of action potential on time by the diffusion of the local temperature $x$ in:

$x_{n+1}^{i}=(1-d)f(x_{n}^{i})+(d/2)[f(x_{n}^{i+1})+f(x_{n}^{i-1})]$

where $n$ is the time step and $i$ is the index of each neuron. For the values $a=0.89$, $b=0.6$, $c=0.28$ and $k=0.02$, in absence of perturbations they are at the resting state. If we introduce a stimulus over cell 1, it induces two propagated waves circulating in opposite directions that eventually collapse and die in the middle of the ring.

Example 2:

Analogous to the previous example, it’s possible create an set of coupling neurons over a 2-D lattice, in this case the evolution of actions potentials is given by:

$x_{n+1}^{i,j}=(1-d)f(x_{n}^{i,j})+(d/4)[f(x_{n}^{i+1,j})+f(x_{n}^{i-1,j})+f(x_{n}^{i,j+1})+f(x_{n}^{i,j-1})]$

where $i$, $j$, represent the index of each neuron in a square lattice of size $I$, $J$. With this example spiral waves can be represented for specific values of parameters.  We can see an example in the figure X for a $100x100$ lattice and $a=0.89$, $b=0.6$, $c=0.26$, and $k=0.02$. In order to visualize the spirals, we set the initial condition in a specific configuration $x^{ij}=i*0.0033$ and the recovery as $y^{ij}=y_{f}-(j*0.0066)$.

Example of spiral waves for the Two-dimensional Chialvo map in 100 x 100 lattice and parameters a=0.89, b=0.6, c= 0.26, and k=0.02.