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 ${\displaystyle a=0.89}$, ${\displaystyle c=0.28}$ and ${\displaystyle k=0.025}$ and changing b from ${\displaystyle 0.6}$ to ${\displaystyle 0.18}$ the system passes from oscillations to aperiodic bursting solutions.

Fixed points

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

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

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

Example 1

A practical implementation is the combination of ${\displaystyle N}$ neurons over a lattice, for that, it can be defined ${\displaystyle 0>d<1}$ 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 ${\displaystyle x}$ in:

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

where ${\displaystyle n}$ is the time step and ${\displaystyle i}$ is the index of each neuron. For the values ${\displaystyle a=0.89}$, ${\displaystyle b=0.6}$, ${\displaystyle c=0.28}$ and ${\displaystyle k=0.02}$, 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 a set of coupling neurons over a 2-D lattice, in this case the evolution of action potentials is given by:

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

where ${\displaystyle i}$, ${\displaystyle j}$, represent the index of each neuron in a square lattice of size ${\displaystyle I}$, ${\displaystyle J}$. With this example spiral waves can be represented for specific values of parameters. In order to visualize the spirals, we set the initial condition in a specific configuration ${\displaystyle x^{ij}=i*0.0033}$ and the recovery as ${\displaystyle y^{ij}=y_{f}-(j*0.0066)}$.

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.

The map can also present chaotic dynamics for certain parameter values. In the following figure we show the chaotic behavior of the variable ${\displaystyle x}$ on a square network of ${\displaystyle 500\times 500}$ for the parameters ${\displaystyle a=0.89}$, ${\displaystyle b=0.18}$, ${\displaystyle c=0.28}$ and ${\displaystyle k=0.026}$.

Evolution of Potential X as a function of time in a 500x500 lattice for a chaotic regime with parameters a=0.89, b=0.18, c= 0.28, and k=0.026.

The map can be used to simulated a nonquenched disordered lattice (as in Ref ^[8]), where each map connects with four nearest neighbors on a square lattice, and in addition each map has a probability ${\displaystyle p}$ of connecting to another one randomly chosen, multiple coexisting circular excitation waves will emerge at the beginning of the simulation until spirals takes over.

Example of spiral waves for the Two-dimensional Chialvo map in an annealed random network starting from a 128 x 128 lattice with probability of rewiring ${\displaystyle p=0.25}$ and parameters a=0.89, b=0.6, c= 0.26, and k=0.02.