Heteroclinic networks in an ensemble of generalized Lotka-Volterra elements

TL;DR

The paper investigates an ensemble of four generalized Lotka–Volterra elements and demonstrates that its phase space contains a heteroclinic network formed by connected heteroclinic cycles on the invariant hypercube $0 \leq \rho_i \leq 1$. It derives necessary and sufficient inequalities for the existence of heteroclinic cycles on the cube vertices, identifies 21 distinct cycles, and classifies them into six networks with symmetry relations. The authors partition the $(\alpha,\beta)$ parameter plane into regions where different heteroclinic networks exist and show how time-reversal and symmetry map cycles between networks. This work provides a framework for stable sequential switching dynamics (Winnerless Competition) in multi-element neural-like ensembles, with potential implications for understanding transient neural activity patterns.

Abstract

In this article the generalized Lotka-Volterra model of ensemble of four excitory or inhibitory coupled elements are studied. It is shown that in the phase space of the model there exist heteroclinic network: a connected union of two or more heteroclinic cycles. A partition of the plane of coupling parameters into sets of existence of various heteroclinic networks is constructed.

Figures (5)

• Figure 1: Graph of possible transitions between equilibria at the vertices of the invariant hypercube $0 \leq \rho_i \leq 1$.
• Figure 2: Partition of the set $\alpha + 1 < \gamma < \beta$ (the boundaries of the set are thick black lines) into sets of existence of different heteroclinic networks. The set marked in red is the existence set of the network $\frak{G_1}$, the set marked in green is the existence set of the network $\frak{G_2}$, the set marked in blue is the existence set of the network $\frak{G_3}$, the set marked in cyan is the existence set of the network $\frak{G_4}$, the set marked in yellow is the existence set of the network $\frak{G_5}$, the set marked in white is the existence set of the network $\frak{G_6}$.
• Figure 3: Graph showing a heteroclinic network consisting of cycles $\Gamma_1$ (is shown in red), $\Gamma_5$ (is shown in blue), $\Gamma_6$ (is shown in green), $\Gamma_{10}$ (is shown in light brown), $\Gamma_{11}$ (is shown in cyan), $\Gamma_{13}$ (is shown in brown), $\Gamma_{16}$ (is shown in violet). Each of the network cycles has a symmetric cycle in the same network with respect to the line passing through the vertices $O_{2, 3}$, $O_{1, 4}$ and $O_{1, 2, 3, 4}$. The cycle $\Gamma_1$ is symmetric to the cycle $\Gamma_{16}$, the cycle $\Gamma_6$ is symmetric to the cycle $\Gamma_{10}$, and the remaining cycles are symmetric to themselves.
• Figure 4: Time series (in the figure (a) new coordinates $x_i$ are used, in the figure (b) original coordinates $\rho_i$ are used). Initial conditions: (0.65, 0.81, 0.67, 0.97). Parameters: $\alpha = -2$, $\beta = 2.1$, $\gamma = 0.8$.
• Figure 5: A graph containing all heteroclinic cycles that do not contain the equilibria $O_1$, $O_2$, $O_3$ and $O_4$.

Theorems & Definitions (7)

• Theorem 1
• Proposition 1
• proof
• Proposition 2
• proof
• Corollary 1
• proof