Topological-numerical analysis of a two-dimensional discrete neuron model

Data and software

Paweł Pilarczyk

This website contains the data and software described in the paper Topological-numerical analysis of a two-dimensional discrete neuron model by J. Signerska-Rynkowska, P. Pilarczyk, and G. Graff. A preprint of the paper is available from arXiv: https://doi.org/10.48550/arXiv.2209.03443.

Continuation diagrams

There are a few continuation diagrams available here for browsing. Please, click the small diagram on the left to launch an interactive browser.

Continuation Diagram	Description	Data Files
	The diagram shown in Figure 7 in the version 1 of the paper and in Figure 32 in version 2 of the paper. Parameter space a=0.89, b∊[0,1], c=0.28, k∊[0.0,0.2] sampled at 100×100. Phase space [−0.1,9] × [−5,3] sampled at 1024×1024.	neuron09php.zip – PHP code for the browser neuron09c.zip – Conley-Morse graphs neuron09p.zip – phase space portraits
	The diagram shown in Figure 8 in version 1 of the paper and in Figure 5 in version 2 of the paper. Parameter space a=0.89, b∊[0,1], c=0.28, k∊[0.015,0.030] sampled at 200×75. Phase space [−0.1,9] × [−5,3] sampled at 1024×1024.	neuron12php.zip – PHP code for the browser neuron12c.zip – Conley-Morse graphs neuron12p.zip – phase space portraits
	The diagram for the low-resolution computations described in Section 4.5 of the paper (both version 1 and 2). Parameter space a=0.89, b∊[0,0.5], c=0.28, k∊[0.017,0.027] sampled at 200×50. Phase space [−0.1,7.5] × [−1.3,2.7] sampled at 256×256.	neuron25php.zip – PHP code for the browser neuron25c.zip – Conley-Morse graphs neuron25p.zip – phase space portraits neuron25r.zip – recurrence diagrams neuron25h.zip – recurrence histograms neuron25t.zip – reduced recurrence histograms neuron25o.zip – raw recurrence data

The analysis was conducted for the Chialvo model:
x_n+1 = x_n² exp(y_n−x_n) + k
y_n+1 = ay_n − bx_n + c

Each continuation diagram shows b in the horizontal axis, and k in the vertical axis.

Each phase space shows x in the horizontal axis, and y in the vertical axis, with a color bar at the bottom.

You are welcome to browse the numerical Morse decompositions and the Conley-Morse diagrams to discover the information obtaind about the dynamics using this method.

Data files used for the browser are provided in 3 zipped files:

• PHP code files and PNG images for the browser provided with the source code (see below),
• the Conley-Morse graphs encoded in the text format compatible with the dot program from the Graphviz Graph Visualization Software package (https://graphviz.org/),
• the computed Morse decompositions encoded in terms of a PNG image in which a single pixel corresponds to a square in the phase space.

(Note: if your browser doesn't download some of the data files, the reason might be that you are browsing the current page using the secure https protocol, while the files are available through the unsecure http protocol. A solution to the problem might be to switch to the http protocol, e.g. by clicking here.)

Software

The computations were conducted using the Conley-Morse Graphs software configured for the Chialvo model. The original software was published with the first paper about the database approach to cataloging dynamics in 2009, and updated afterwards. The source code of the software that was actually used for the computations is available below for the benefit of the academic community. It is expected that those who are interested in doing similar computations for different systems or to verify our computations may benefit from this software.

• cmgraphsSrc.zip (July 18, 2022); please, modify the file config.h to choose computation no. 09, 12 or 25

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