Chialvo map

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

in which, $x$ is called activation or action 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 , $b$ is the activation-dependence of the recovery process and $c$ is an offset constant. The model has a rich dynamics, presenting from oscillatory  to chaotic behavior,{{Cite arXiv |last1=Pilarczyk |first1=Paweł |last2=Signerska-Rynkowska |first2=Justyna |last3=Graff |first3=Grzegorz |date=2022-09-07 |title=Topological-numerical analysis of a two-dimensional discrete neuron model |class=math.DS |eprint=2209.03443}}{{Cite journal |last1=Wang |first1=Fengjuan |last2=Cao |first2=Hongjun |date=2018-03-01 |title=Mode locking and quasiperiodicity in a discrete-time Chialvo neuron model |url=https://www.sciencedirect.com/science/article/pii/S1007570417303155 |journal=Communications in Nonlinear Science and Numerical Simulation |language=en |volume=56 |pages=481–489 |doi=10.1016/j.cnsns.2017.08.027 |bibcode=2018CNSNS..56..481W |issn=1007-5704|url-access=subscription }} as well as non trivial responses to small stochastic fluctuations.{{Cite journal |last1=Chialvo |first1=Dante R. |last2=Apkarian |first2=A. Vania |date=1993-01-01 |title=Modulated noisy biological dynamics: Three examples |url=https://doi.org/10.1007/BF01053974 |journal=Journal of Statistical Physics |language=en |volume=70 |issue=1 |pages=375–391 |doi=10.1007/BF01053974 |bibcode=1993JSP....70..375C |s2cid=121830779 |issn=1572-9613|url-access=subscription }}{{Cite journal |last1=Bashkirtseva |first1=Irina |last2=Ryashko |first2=Lev |last3=Used |first3=Javier |last4=Seoane |first4=Jesús M. |last5=Sanjuán |first5=Miguel A. F. |date=2023-01-01 |title=Noise-induced complex dynamics and synchronization in the map-based Chialvo neuron model |url=https://www.sciencedirect.com/science/article/pii/S1007570422003549 |journal=Communications in Nonlinear Science and Numerical Simulation |language=en |volume=116 |pages=106867 |doi=10.1016/j.cnsns.2022.106867 |bibcode=2023CNSNS.11606867B |s2cid=252140483 |issn=1007-5704|doi-access=free |hdl=10115/27394 |hdl-access=free }}

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.

Considering the case where $k=0$ and

\\ r = & y_{f0} = c/(1-a) \\ \end{align}

in which $f(x\_n,y\_\{f0\})$ as a function of $r$ has a period-doubling bifurcation structure.

A practical implementation is the combination of $N$ neurons over a lattice, for that, it can be defined 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.

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:

$x\_\{n+1\}^\{i,j\}\; =\; (1-d)f(x\_n^\{i,j\})\; +\; (d/4)[f(x\_n^\{i+1,j\})\; +\; f(x\_n^\{i-1,j\})+f(x\_n^\{i,j+1\})\; +\; f(x\_n^\{i,j-1\})]$

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. In order to visualize the spirals, we set the initial condition in a specific configuration $x^\{ij\}=i*0.0033$ and the recovery as $y^\{ij\}=y\_f-(j\; *\; 0.0066)$.

File:SpiralChialvomap.gif

The map can also present chaotic dynamics for certain parameter values. In the following figure we show the chaotic behavior of the variable $x$ on a square network of $500\backslash times500$ for the parameters $a=0.89$, $b=0.18$, $c=0.28$ and $k=0.026$.

File:Evolution of Potential X as a function of time in a 500x500 lattice for a chaotic regime.gif

The map can be used to simulated a nonquenched disordered lattice (as in Ref {{Cite journal |last1=Sinha |first1=Sitabhra |last2=Saramäki |first2=Jari |last3=Kaski |first3=Kimmo |date=2007-07-09 |title=Emergence of self-sustained patterns in small-world excitable media |url=https://link.aps.org/doi/10.1103/PhysRevE.76.015101 |journal=Physical Review E |language=en |volume=76 |issue=1 |pages=015101 |doi=10.1103/PhysRevE.76.015101 |pmid=17677522 |arxiv=cond-mat/0701121 |bibcode=2007PhRvE..76a5101S |s2cid=11714109 |issn=1539-3755}}), where each map connects with four nearest neighbors on a square lattice, and in addition each map has a probability $p$ of connecting to another one randomly chosen, multiple coexisting circular excitation waves will emerge at the beginning of the simulation until spirals takes over.

File:ESPIRALES2.gif

For a neuron, in the limit of $b=0$, the map becomes 1D, since $y$ converges to a constant. If the parameter $b$ is scanned in a range, different orbits will be seen, some periodic, others chaotic, that appear between two fixed points, one at $x=1$ ; $y=1$ and the other close to the value of $k$ (which would be the regime excitable).

File:Chialvo_map_for_an_Neuron.png

File:Chialvomal_aneuron.gif

Category:Chaotic maps

Category:Neuroscience