Chialvo map
![](http://upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Chaotic_act_variable.png/400px-Chaotic_act_variable.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Chaotic_regime.png/400px-Chaotic_regime.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Excitable_act_variable.png/400px-Excitable_act_variable.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/a/af/Excitable_regime.png/400px-Excitable_regime.png)
The Chialvo map is a two-dimensional map proposed by Dante R. Chialvo in 1995[1] to describe the generic dynamics of excitable systems. The model is inspired by Kunihiko Kaneko's Coupled map lattice numerical approach which considers time and space as discrete variables but state as a continuous one. Later on Rulkov popularized a similar approach.[2] By using only three parameters the model is able to efficiently mimic generic neuronal dynamics in computational simulations, as single elements or as parts of inter-connected networks.
The model
[edit]The model is an iterative map where at each time step, the behavior of one neuron is updated as the following equations:
in which, is called activation or action potential variable, and is the recovery variable. The model has four parameters, is a time-dependent additive perturbation or a constant bias, is the time constant of recovery , is the activation-dependence of the recovery process and is an offset constant. The model has a rich dynamics, presenting from oscillatory to chaotic behavior,[3][4] as well as non trivial responses to small stochastic fluctuations.[5][6]
Analysis
[edit]Bursting and chaos
[edit]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 , and and changing b from to the system passes from oscillations to aperiodic bursting solutions.
Fixed points
[edit]Considering the case where and the model mimics the lack of ‘voltage-dependence inactivation’ for real neurons and the evolution of the recovery variable is fixed at . Therefore, the dynamics of the activation variable is basically described by the iteration of the following equations
in which as a function of has a period-doubling bifurcation structure.
Examples
[edit]Example 1
[edit]A practical implementation is the combination of 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 in:
where is the time step and is the index of each neuron. For the values , , and , 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
[edit]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:
where , , represent the index of each neuron in a square lattice of size , . 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 and the recovery as .
![](http://upload.wikimedia.org/wikipedia/commons/thumb/4/4b/SpiralChialvomap.gif/400px-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 on a square network of for the parameters , , and .
![](http://upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Evolution_of_Potential_X_as_a_function_of_time_in_a_500x500_lattice_for_a_chaotic_regime.gif/459px-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 [7]), where each map connects with four nearest neighbors on a square lattice, and in addition each map has a probability of connecting to another one randomly chosen, multiple coexisting circular excitation waves will emerge at the beginning of the simulation until spirals takes over.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/4/4d/ESPIRALES2.gif/459px-ESPIRALES2.gif)
Chaotic and periodic behavior for a neuron
[edit]For a neuron, in the limit of , the map becomes 1D, since converges to a constant. If the parameter is scanned in a range, different orbits will be seen, some periodic, others chaotic, that appear between two fixed points, one at ; and the other close to the value of (which would be the regime excitable).
![](http://upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Chialvo_map_for_an_Neuron.png/400px-Chialvo_map_for_an_Neuron.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Chialvomal_aneuron.gif/400px-Chialvomal_aneuron.gif)
References
[edit]- ^ Chialvo, Dante R. (1995-03-01). "Generic excitable dynamics on a two-dimensional map". Chaos, Solitons & Fractals. Nonlinear Phenomena in Excitable Physiological Systems. 5 (3): 461–479. Bibcode:1995CSF.....5..461C. doi:10.1016/0960-0779(93)E0056-H. ISSN 0960-0779.
- ^ Rulkov, Nikolai F. (2002-04-10). "Modeling of spiking-bursting neural behavior using two-dimensional map". Physical Review E. 65 (4): 041922. arXiv:nlin/0201006. Bibcode:2002PhRvE..65d1922R. doi:10.1103/PhysRevE.65.041922. PMID 12005888. S2CID 1998912.
- ^ Pilarczyk, Paweł; Signerska-Rynkowska, Justyna; Graff, Grzegorz (2022-09-07). "Topological-numerical analysis of a two-dimensional discrete neuron model". arXiv:2209.03443 [math.DS].
- ^ Wang, Fengjuan; Cao, Hongjun (2018-03-01). "Mode locking and quasiperiodicity in a discrete-time Chialvo neuron model". Communications in Nonlinear Science and Numerical Simulation. 56: 481–489. Bibcode:2018CNSNS..56..481W. doi:10.1016/j.cnsns.2017.08.027. ISSN 1007-5704.
- ^ Chialvo, Dante R.; Apkarian, A. Vania (1993-01-01). "Modulated noisy biological dynamics: Three examples". Journal of Statistical Physics. 70 (1): 375–391. Bibcode:1993JSP....70..375C. doi:10.1007/BF01053974. ISSN 1572-9613. S2CID 121830779.
- ^ Bashkirtseva, Irina; Ryashko, Lev; Used, Javier; Seoane, Jesús M.; Sanjuán, Miguel A. F. (2023-01-01). "Noise-induced complex dynamics and synchronization in the map-based Chialvo neuron model". Communications in Nonlinear Science and Numerical Simulation. 116: 106867. Bibcode:2023CNSNS.11606867B. doi:10.1016/j.cnsns.2022.106867. ISSN 1007-5704. S2CID 252140483.
- ^ Sinha, Sitabhra; Saramäki, Jari; Kaski, Kimmo (2007-07-09). "Emergence of self-sustained patterns in small-world excitable media". Physical Review E. 76 (1): 015101. arXiv:cond-mat/0701121. Bibcode:2007PhRvE..76a5101S. doi:10.1103/PhysRevE.76.015101. ISSN 1539-3755. PMID 17677522. S2CID 11714109.