by Bram Alexander Siebert, University of Limerick
Modularity plays a significant role in the formation of patterns of activity on
brain networks due to the small spectral gap of modular networks. As a byproduct, the
modularity leads to the formation of macroscopic patterns, where the amount of activator
and inhibitor on nodes in the same modules is approximately equal. These are similar
to the patterns reported in the brain, and understanding the mathematical underpinnings
of pattern formations can assist researchers in understanding brain development
and neurological conditions.
This work was generously funded by the Irish research council and Science foundation Ireland, and is a collaboration between researchers in the University of Limerick and the University of Bristol.
The main findings of our research are that modular networks aid in the formation of patterns, the theory of which was laid down by Alan Turing in 1952. Furthermore, we found that the network topology is what drives the final shape of patterns, more so than the dynamical processes.
Patterns form due to an instability in a two species reaction-diffusion model. We begin with a
homogeneous steady state in the absence of diffusion. Then, when we allow for diffusion, if the
eigenvalues of the extended Jacobian are positive, patterns will form if the steady state is perturbed. These perturbations then grow according to the linear solutions, and the final shape of the patterns is thus influenced by the eigenvector of the dominant eigenvalue.
In the images, we compare the (non-existent) pattern on a Newman Watts network, on the left, to the pattern which has formed on the modular network on the right. Looking at the dispersion relation in the centre, we can see that the spectral gap of the modular network is much smaller than that of the Newman Watts network. Because one of the eigenvalues of the extended Jacobian of the modular networks is positive, a pattern has formed.
We can use a perturbative approach to see why a modular network has a small spectral gap. On the left, we have five disconnected ER networks, and thus, there are five zero eigenvalues of the Laplacian. When we add inter-connections between the modules, these zero eigenvalues are perturbed from the zero point, and help to close the spectral gap in the dispersion relation. Thus patterns form on the connected modular network.
Depending on which eigenvalues dominate, different patterns will form. The set of eigenvalues closest to zero we denote as “modular” eigenvalues, and when they dominate we obtain a pattern that is heterogeneous globally, but homogeneous within modules. Conversely, when the other eigenvalues dominate, here denoted as “non-modular” eigenvalues, then the pattern is heterogeneous both globally and locally. Finally, if the two sets of eigenvalues are similarly values, and thus are competing, then we find that come modules have a homogeneous pattern, and some are heterogeneous. In order to predict the final patterns, we can look at the eigenvector of the dominant eigenvalue of the Laplacian. When we normalize both the eigenvector and the pattern, we can see that the two match quite well, thus predicting the shape of the final pattern.
Great question! Turing patterns were in fact originally defined on a continuous space. Essentially all we do is substitute the graph Laplacian with the continuous Laplacian (the diffusion equation), and allow u and v to depend on space as well as time.
p.s. sorry if I commented twice, I didn’t see my reply appear the first time
Hi, How’s the continuous version defined?
Great question! Turing patterns were originally defined only for a continuous domain. The system is essentially the same, but instead of u_i and v_i depending only on time, they now also depend on space, that is, u_i(t) becomes u(x,y,t). Furthermore we substitute the discrete Laplacian with the continuous Laplacian.
Thanks. I think I didn’t ask clearly. I meant how’s the continuous space defined as the limit of the modular network? I’m only familiar with the PDE scenario. The network looks pretty discrete to me, not sure what continuum it is converging to.
Ah ok! The continuous curves are mostly added as a sort of visual aid. This is because the eigenvalues of the extended Jacobian will match those of the PDE formulation. So I’m not intentionally defining any limit of the modular network. Interestingly, if one were to be on a directed network, then the eigenvalues of the Laplacian are complex, and then the real component of the eigenvalues of the extended Jacobian no longer “sit” on the continuous dispersion.