Symmetry breaker governs synchrony patterns in neuronal inspired networks

TL;DR

This work considers two layers with the top layer directly coupled to the bottom layer, and demonstrates that the symmetry breaker prevents complete synchronization in the bottom layer, a situation that would not be desirable in a normal functioning brain.

Abstract

Experiments in the human brain reveal switching between different activity patterns and functional network organization over time. Recently, multilayer modeling has been employed across multiple neurobiological levels (from spiking networks to brain regions) to unveil novel insights into the emergence and time evolution of synchrony patterns. We consider two layers with the top layer directly coupled to the bottom layer. When isolated, the bottom layer would remain in a specific stable pattern. However, in the presence of the top layer, the network exhibits spatiotemporal switching. The top layer in combination with the inter-layer coupling acts as a symmetry breaker, governing the bottom layer and restricting the number of allowed symmetry-induced patterns. This structure allows us to demonstrate the existence and stability of pattern states on the bottom layer, but most remarkably, it enables a simple mechanism for switching between patterns based on the unique symmetry-breaking role of the governing layer. We demonstrate that the symmetry breaker prevents complete synchronization in the bottom layer, a situation that would not be desirable in a normal functioning brain. We illustrate our findings using two layers of Hindmarsh-Rose (HR) oscillators, employing the Master Stability function approach in small networks to investigate the switching between patterns.

Figures (7)

• Figure 1: Symmetry breaker underlies switching pattern states in neuronal networks. a) Schematic diagram of the switching between brain network states. This represents our motivational toy example. b) Each layer contains the same number of nodes, representing different instances of coupled units. Our focus is to observe the change of pattern in the bottom layer by the presence of the top layer and inter-layer connections, the symmetry breaker. The left panel shows the bottom layer exhibits the pattern state 1 for an intra-coupling $\beta$. The right panel shows that once the symmetry breaker is coupled to the bottom layer, the bottom layer shifts its state, exhibiting pattern state 2. Pattern state 1 leaves to be flow-invariant, giving space to pattern state 2. The intra-layer $\alpha$ and inter-coupling $\sigma$ are tuned to pinpoint pattern state 2's stability.
• Figure 2: Symmetry-induced pattern states. Each orbit partition induced by the graph automorphism and its subgroups creates different patterns in the network with five nodes. Four patterns are illustrated where node color corresponds to a cluster.
• Figure 3: Eight symmetry-induced pattern states in a duplex. Node colors denote the clusters of each layer. The top and bottom layers are illustrated using blue and red edges, respectively. The inter-layer connections are denoted as dashed directed edges. The construction of these patterns can be performed by breaking different synchronous clusters lodi2021one. Pattern $\mathcal{P}$ represents the pattern state constructed from the orbital partition of the group in Equation \ref{['eq:group_symm_duplex']} while $\mathcal{P}^1-\mathcal{P}^7$ are patterns from breaking different synchronous clusters. $\mathcal{P}^1-\mathcal{P}^4$ are generated from breaking $\mathcal{K}^1_{\mathfrak{T}}=\{1,2,3\}$, $\mathcal{P}^5$ from $\mathcal{K}^2_{\mathfrak{T}}=\{4,5\}$, $\mathcal{P}^6$ from $\mathcal{K}^2_{\mathfrak{B}}=\{2,3\}$, and $\mathcal{P}^7$ from $\mathcal{K}^3_{\mathfrak{B}}=\{4,5\}$.
• Figure 4: Symmetry breaker governs patterns states in coupled HR oscillators. (a) Initially, when $\sigma=0$, bottom layer is in one pattern state $\mathcal{P}^1_{\mathfrak{B}}=\{(a, b, b, c, d)\}$. (b) The bottom layer switches to pattern $\mathcal{P}^2_{\mathfrak{B}}=\{(a, b, c, d, d)\}$ in the presence of the symmetry breaker with $\alpha = 0.225$, when $\sigma = 0.5$ at $t=1500$, represented by vertical dashed line. Node color corresponds to nodes in the same cluster, but the same color across layers has no meaning. The HR parameters are $I_\mathfrak{T}=3.2, r_\mathfrak{T}=0.01, I_\mathfrak{B}=3.27, r_\mathfrak{B}=0.01, \beta=0.3$. The intra and inter-layer coupling functions are as shown in Equations \ref{['eq:intra_coupling_fun']} and \ref{['eq:inter_coupling_fun']}.
• Figure 5: Bottom layer can attain other patterns by changing $\alpha$. (a) The bottom layer switches from $\mathcal{P}^1_{\mathfrak{B}}=\{(a, b, b, c, d)\}$ to an incoherent pattern $\mathcal{P}^0_{\mathfrak{B}} = \{(a, b, c, d, e)\}$ when $\alpha = 0.1$ and $\sigma$ is turned on at $t=1500$. (b) The bottom layer switches from $\mathcal{P}^1_{\mathfrak{B}}=\{(a, b, b, c, d)\}$ to $\mathcal{P}^3_{\mathfrak{B}}=\{(a, b, b, c, c)\}$ when $\alpha = 0.425$.

Theorems & Definitions (9)

• Proposition 4.1: Symmetry compatibility DellaRossa2020
• proof
• Definition 4.2: Duplex clusters
• Proposition 5.1: Block-diagonalization of coupling matrices.
• proof
• Remark 5.2: Laplacian coupling case
• Corollary A.1: Bottom layer's cluster: all or nothing.
• proof
• proof