Transverse-instabilities

This software provides the main tools to reproduce the results of the paper: Clusella et al. , Complex spatiotemporal oscillations emerge from transverse instabilities in large-scale brain networks (2023). Cite this work if you use this, or parts of this, repository.

Summary and dependencies

Mainly, this repository provides 3 tools that could be used independently:

• Integration of a network of coupled Jansen-Rit NMMs and computation of their largest Lyapunov exponents. This is provided by the src/netdynamics.c and src/system.c C codes.
• Scripts to compute 1-parameter bifurcation diagrams as those in Fig. 1(a-d) of the paper in auto-07p.
• Computation of the MSF of the coupled Jansen model, provided by the src/floquet.c code.

The C codes in this repository require the GNU Scientific Library (tested using version 2.7). Auto-07p can be obtained from the project github.

Integration of the Jansen-Rit network

The C-codes to simulate a network of Jansen-Rit elements can be compiled using gcc with the provided makefile. From the src directory just run:

make

If the compilation is successful, the netdyn executable will be created in the main workspace. This can be executed with: ./netdyn <output identifier> <file to network topology> <p> <epsilon> <number of Lyap. exp. to compute>

where:

• <output identifier> is a string to identify the output files (no file extension or path should be included).
• <network topology> is the path to the file containing the network topology. Notice that this needs to be in the appropiate format (see Network data format below).
• <p> is the parameter $p$ of the Jansen model (see paper).
• <epsilon> is the parameter $\epsilon$ of the Jansen model (see paper).
• <nLE> is the number of Lyapunov exponents to be computed, can be from 0 up to the number of system dimensions. Notice that computing Lyap. exp. increases the computation time.

An example call to the software could be:

./netdyn testing data/network.dat 230 50 2

The results can be plot, for instance, using gnuplot:

set xlabel 'Node index'
set ylabel 'time [s]'
set cblabel 'Activity [mV]'
plot 'outputs/pattern_testing.bin'  binary format='%double' array=(90,1000) with image

Network data format

The network files must adhere to the following format:

• First line of the file should be the number of nodes (N).
• The following lines must contain three columns separated by spaces: i j w, where A[i,j]=w is the structural connectivity matrix of the system. Indices should start from 0: i,j=0,...,N-1.

The file should then contain L+1 lines, where L are the number of non-zero entries in A. Notice that the simulation algorithm performs the row-normalization used in the paper by default.

Bifurcation diagrams with auto-07p

The directory auto-07p contains the necessary files to compute 1-dimensional bifurcation diagrams with auto-07p. The files bifur_epsXX.auto.py contain step-by-step explanations to obtain the diagrams using the Python interface. The also contain the instruction to obtain initial conditions to be used for computation of the Floquet exponents floquet.c (see Master Stability Function below).

Master Stability Function

The standalone C code floquet.c in src computes the largest Floquet exponent of the self-coupled Jansen model. The code can be compiled with:

gcc src/floquet.c -o floquet -lm -lgsl -lgslcblas -lblas -O3

The simulations are called with:

./floquet <p> <eps> <lambda> <ic file>

where

• p is the $p$ parameter of the Jansen model.
• eps is the coupling strength.
• lambda is the structural connectivity eigenvalue that you want to use. To obtain the entire dispersion relation of the system one needs to loop over all the eigenvalues.
• ic file is the path to the initial conditions of limit cycle, as provided by the auto scripts.

Upon success the algorithm provides, in standard output 4 numbers: <p> <eps> <lambda> <largest Floquet exponent>. For instance, a call to

./floquet 120 50 0.7 auto-07p/msf_initial_conditions.dat 

gives

230.000000 50.000000 0.700000 0.6125043913964573

Structural Connectivity Data

The connectome used in the article is provided in data/sc2017.dat. The data is given in the format specified in the Network data format section above. This structural connectivity corresponds to the average connectome of 16 healthy subjects obtained from Diffusion Tensor Imaging and parcelled with the AAL90 atlas.
For details about data collection and postprocessing users should refer to the original publication by Deco et al. (2017). Any works using this data must cite that paper as well as Clusella et. al (2023).