Revisiting the Haken Lighthouse Model

TL;DR

The paper revisits the Haken Lighthouse model to provide a rigorous analytical treatment of synchrony, waves, and pattern formation in spiking neural networks. It develops linear stability conditions for phase-locked states on graphs with $\alpha$-function synapses and delays, and introduces a Master Stability Function framework to assess stability across arbitrary networks. By extending to a spatial continuum, it derives nonlocal Turing-type instabilities, periodic travelling waves, and analytically tractable localised bump solutions, connecting spike-based dynamics with rate-based reductions in the slow-synapse limit. Collectively, these results reinforce the Lighthouse model as a powerful, tractable platform for studying structured spiking interactions and self-organisation with potential applications in neuromorphic design and AI.

Abstract

Simple spiking neural network models, such as those built from interacting integrate-and-fire (IF) units, exhibit rich emergent behaviours but remain notoriously difficult to analyse, particularly in terms of their pattern-forming properties. In contrast, rate-based models and coupled phase oscillators offer greater mathematical tractability but fail to capture the full dynamical complexity of spiking networks. To bridge these modelling paradigms, Hermann Haken -- the pioneer of Synergetics -- introduced the Lighthouse model, a framework that provides insights into synchronisation, travelling waves, and pattern formation in neural systems. In this work, we revisit the Lighthouse model and develop new mathematical results that deepen our understanding of self-organisation in spiking neural networks. Specifically, we derive the linear stability conditions for phase-locked spiking states in Lighthouse networks structured on graphs with realistic synaptic interactions ($α$-function synapses) and axonal conduction delays. Extending the analysis on graphs to a spatially continuous (non-local) setting, we develop a variant of Turing instability analysis to explore emergent spiking patterns. Finally, we show how localised spiking bump solutions -- which are difficult to mathematically analyse in IF networks -- are far more tractable in the Lighthouse model and analyse their linear stability to wandering states. These results reaffirm the Lighthouse model as a valuable tool for studying structured neural interactions and self-organisation, further advancing the synergetic perspective on spiking neural dynamics.

Figures (12)

• Figure 1: A plot of the nonlinear function $S$ (in blue) given by equation (\ref{['S']}) and its mid-range linear approximation $S_L$ (in red). Parameters: $r=0.1$ and $h=1$.
• Figure 2: The emergent frequency $\Omega = 2 \pi/T$ of the (nonlinear) Lighthouse network model with a network structure such that $\sum_{j} w_{ij} = \Gamma$ for all $j$ (row sum constraint) as a function of the synaptic rate parameter $\alpha$. Parameters: $r=1$, $h=-1$, and $\tau=0$.
• Figure 3: A raster plot of spike times illustrating oscillator death (arising from the instability of a synchronous state) in a globally coupled network with $w_{ij} = (\Gamma+1) \delta_{ij} - N^{-1}$. Open circles denote times of firing. Parameters: $N=30$, $\Gamma=1$, $\gamma=\pi$, $\Theta=-1$, $\alpha = 5$, and $\tau=0$.
• Figure 4: The Master Stability Function $\text{MSF} (\beta)$ for a Haken Lighthouse network with $\alpha$-function synapses, shown as a colour plot. The parameters are $\Theta=-1$, and $\alpha = 0.1$. The synchronous solution is stable provided all of the eigenvalues of $\gamma w$ lie within the black closed curve.
• Figure 5: A novel bursting behaviour may arise as one leaves the negative region of the MSF along the imaginary axis. Here we show a simulation of a Haken Lighthouse model with an anti-symmetric circulant connectivity matrix generated by the row vector $(0, -\epsilon, \epsilon, -\epsilon, \epsilon, \ldots, -\epsilon, \epsilon)$ for $N=21$. The parameters are $\Theta=-1$, $\gamma=1$, $\alpha = 0.1$, and $\epsilon=1.5$ (for which $\beta \simeq 20 {\rm i}$).