Volume 15 Supplement 1

## Abstracts from the Twenty Third Annual Computational Neuroscience Meeting: CNS*2014

- Poster presentation
- Open Access

# A map-based logistic neuron model: an efficient way to obtain many different neural behaviors

- Rafael V Stenzinger
^{1}, - Jheniffer J Gonsalves
^{1}, - Mauricio Girardi-Schappo
^{1}and - Marcelo HR Tragtenberg
^{1}Email author

**15 (Suppl 1)**:P24

https://doi.org/10.1186/1471-2202-15-S1-P24

© Stenzinger et al.; licensee BioMed Central Ltd. 2014

**Published:**21 July 2014

## Keywords

- Phase Diagram
- Sigmoidal Function
- Neuron Model
- Excitable Neuron
- Burst Size

where $\delta $ is related to the refractory period, ${x}_{R}$ is the reversal potential, $\lambda $ controls the burst sizes, ${I}_{t}$ is an external current and $K$ and $T$ are gain parameters for neuron self-interactions. The hyperbolic tangent is biologically plausible, since it is a sigmoidal function that saturates at large absolute inputs.

This model exhibits a rich repertoire of dynamical behaviors, especially those of excitable neurons: bursting, fast and regular spiking, spikes with plateau, type I and II excitability, chaotic firing, among others [1–4]. When associated in networks, they exhibit synchronization, power law avalanches, criticality, etc. [4, 5].

However, hyperbolic tangent is a computationally expensive sigmoidal function. Considering the need of a good trade-off between computational efficiency and biological relevance [6–8], we then propose the use of the logistic function $\text{f}\left(u\right)=u/\left(1+\left|u\right|\right)$, which displays a similar asymptotic behavior and is computationally faster. We call it the KTz logistic model. A similar model has been recently used to study the unlearning hypothesis in REM sleep [9].

We determine the complete phase diagram of the KTz logistic model with details for ${z}_{t}=\delta =\lambda =0$. This phase diagram is similar to the hyperbolic tangent KTz under the same conditions [5], although it is more analytically tractable. We also determine the phase diagram of the logistic KTz model with $\delta =\lambda =0$ and ${z}_{t}\ne 0$ and compare with the hyperbolic tangent KTz [1, 4].

The myriad of dynamical behaviors of the logistic KTz model ($\delta \ne 0$ and $\lambda \ne 0$) and its $T$ versus ${x}_{R}$ phase diagram are also described and compared with those of the hyperbolic tangent KTz [1–4], showing the regions of cardiac like behavior, bursting, fast and slow spiking and fixed point. A detailed analysis of the order of the transitions between these regimes is also made.

The preliminary results of networks of logistic KTz models connected by map chemical synapses [2, 4, 5] are displayed and compared with those of hyperbolic KTz model. Finally, we make qualitative and quantitative comparisons between computational efficiency of the hyperbolic tangent and the logistic KTz models, for many different behaviors of the studied neurons.

## Declarations

### Acknowledgements

RVS, JJG and MGS acknowledge financial support from Brazilian agency CNPq.

## Authors’ Affiliations

## References

- Kinouchi O, Tragtenberg MHR: Modeling neurons by simple maps. Int J Bifurcat Chaos. 1996, 6: 2353-2360.View ArticleGoogle Scholar
- Kuva SM, Lima GF, Kinouchi O, Tragtenberg MHR, Roque AC: A minimal model for excitable and bursting elements. Neurocomputing. 2001, 38-40: 255-261.View ArticleGoogle Scholar
- Copelli M, Tragtenberg MHR, Kinouchi O: Stability diagrams for bursting neurons modeled by three-variable maps. Physica A. 2004, 342 (1): 263-269.View ArticleGoogle Scholar
- Girardi-Schappo M, Kinouchi O, Tragtenberg MHR: A brief history of excitable map-based neurons and neural networks. J Neurosci Methods. 2013, 220 (2): 116-130. 10.1016/j.jneumeth.2013.07.014.View ArticlePubMedGoogle Scholar
- Girardi-Schappo M, Kinouchi O, Tragtenberg MHR: Critical avalanches and subsampling in map-based neural networks coupled with noisy synapses. Phys Rev E. 2013, 88: 024701-View ArticleGoogle Scholar
- Izhikevich EM, Edelman GM: Large-scale model of mammalian thalamocortical systems. Proc Natl Acad Sci USA. 2008, 105 (9): 3593-3598. 10.1073/pnas.0712231105.PubMed CentralView ArticlePubMedGoogle Scholar
- de Garis H, Shuo C, Goertzel B, Ruiting L: A world survey of artificial brain projects, Part I: Large-scale brain simulations. Neurocomputing. 2010, 74: 3-29. 10.1016/j.neucom.2010.08.004.View ArticleGoogle Scholar
- Eliasmith C, Stewart TC, Choo X, Bekolay T, DeWolf T, Tang Y, Rasmussen D: A Large-Scale Model of the Functioning Brain. Science. 2012, 338: 1202-1205. 10.1126/science.1225266.View ArticlePubMedGoogle Scholar
- Kinouchi O, Kinouchi RR: Dreams, endocannabinoids and itinerant dynamics in neural networks: re elaborating Crick-Mitchison unlearning hypothesis. arXiv:cond-mat/0208590Google Scholar

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