Hodge–Kodaira decomposition of evolving neural networks

Abstract

Although it is very important to scrutinize recurrent structures of neural networks for elucidating brain functions, conventional methods often have difficulty in characterizing global loops within a network systematically. Here we applied the Hodge–Kodaira decomposition, a topological method, to an evolving neural network model in order to characterize its loop structure. By controlling a learning rule parametrically, we found that a model with an STDP-rule, which tends to form paths coincident with causal firing orders, had the most loops. Furthermore, by counting the number of global loops in the network, we detected the inhomogeneity inside the chaotic region, which is usually considered intractable.

Introduction

Although the network topology can be very important for communicating neurons, the conventional network analyses are often limited to locally defined variables such as degrees. Those intrinsically local variables cannot capture the global recurrent structures ubiquitously observed in neural networks. In fact, a network that alters its coupling strengths under STDP learning rule has tendency to make long paths of sequential firings (Aoki and Aoyagi, 2007, Aoki and Aoyagi, 2009, Aoki and Aoyagi, 2011, Aoki and Aoyagi, 2012, Buonomano, 2005, Edelman et al., 2004, Liu and Buonomano, 2009, Magnasco et al., 2009, Masuda et al., 2009, Morrison et al., 2007, Takahashi et al., 2009, Tsubo et al., 2007).

To characterize the global loop structures, the approaches based on algebraic topology are needed (Bossavit, 1997, Curto et al., 2013, Fulton, 1995, Hatcher, 2002). Recent advances in the field of computational topology made it possible to compute topological invariants, such as the number of “holes” in proteins (Gameiro et al., 2012), in an accessible way (Arai et al., 2009, Edelsbrunner and Harer, 2009, Kaczynski et al., 2010). For example, topological methods can count the number of “marbles” in an image irrespective of their shapes, which can be much more informative in detecting cancers than just using raw pixels. For discrete graphs, “graph invariants”, independent of labeling, are desired (Chandrasekaran, Parrilo, & Willsky, 2012) in the same vein, especially when, say, you randomize initial conditions as in this paper and, therefore, do not care about specific labels.

Here we apply the Hodge–Kodaira decomposition of graph flows (de Rham, 1984, Hodge, 1941, Jiang et al., 2011, Kodaira, 1949, Warner, 1983) to evolving neural network models (Aoki & Aoyagi, 2009) in order to count the number of global loops as a topological measure of network structures. Specifically, it is interesting to see if the measure reflects the bifurcation diagrams and even subdivides chaotic parameter regions, which are usually considered intractable.

In Section  2, we explain the evolving neural network model which we simulated. We also show the method of Hodge–Kodaira decomposition. In Section  3, we show the results of Hodge–Kodaira decomposition applied to the evolving neural networks. Finally, Section  4 presents a summary and discussions.