Topologicalnumerical analysis of a twodimensional
discrete neuron model
Data and software
PaweÅ‚ Pilarczyk
This website contains the data and software described
in the paper Topologicalnumerical analysis
of a twodimensional discrete neuron model
by J. SignerskaRynkowska, 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 – ConleyMorse 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 – ConleyMorse graphs
neuron12p.zip – phase space portraits 

The diagram
for the lowresolution 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 – ConleyMorse 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_{n}^{2}
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 ConleyMorse 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 ConleyMorse 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.
The third browser also shows recurrence diagrams and recurrence histograms
(actual and reduced, the latter used for machine learning in the paper).
The raw recurrence data is provided in yet another file.
(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
ConleyMorse
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
This software is written in C++ and its compilation requires
the
CAPD software library,
version 4.2.153 (the limited version "capdDynSys" is sufficient),
and the
Original CHomP
library, as well as several other standard software libraries:
boost, zlib, libpng, libbz2, and LAPACK.
A relatively recent GNU C++ compiler is also necessary.
Specific instructions for downloading and compiling the software
in Ubuntu GNU/Linux 20.04 are provided
here.