Adaptive synapse-based neuron model with heterogeneous multistability and riddled basins.

Biological systems often display intricate firing patterns that depend heavily on their starting states. Prior research has shown that neurons maintain various stable states simultaneously. That uncertainty drove the need for better mathematical representations of these phenomena. No prior work had resolved how adaptive synaptic connections influence these diverse firing modes. This gap motivated the development of models incorporating time-varying equilibrium points. Previous studies often relied on static synaptic weights to describe neuronal activity. Such approaches frequently failed to capture the full range of observed multistability. The current investigation addresses these limitations by proposing a dynamic synaptic framework.

The aim of this study is to introduce a novel mathematical model for neurons that incorporates adaptive synaptic connections. This work addresses the challenge of representing complex, coexisting firing patterns in simplified neuronal systems. The authors seek to understand how time-varying equilibria influence the stability of these firing modes. They investigate the role of externally applied currents in triggering transitions between stable and unstable states. The research explores the emergence of heterogeneous multistability through specific bifurcation processes. By analyzing the distribution of attractive regions, the team intends to map the system's global dynamics. They also aim to demonstrate the existence of riddled-like basins of attraction within this framework. Finally, the study intends to validate these theoretical findings through experimental implementation on electronic hardware.

The researchers employed a comprehensive computational approach to analyze the proposed mathematical framework. They utilized bifurcation diagrams to track how equilibrium points shift under varying external currents. Phase portraits provided a visual method to map the complex regions of attraction. Dynamical distributions helped characterize the global behavior of the system. The team calculated basins of attraction to identify sensitivity to initial conditions. They performed numerical simulations to observe the emergence of multiple firing states. A simple microcontroller platform facilitated the physical implementation of the model. Experimental data collected from this hardware verified the accuracy of the theoretical findings.

The model reveals that a single neuron can support up to twelve coexisting heterogeneous attractors. This count exceeds previous reports for similar simplified neuronal architectures. The analysis identifies that fold and Hopf bifurcations regulate the stability of these points. These transitions create complex, riddled-like boundaries between different regions of attraction. Small variations in starting parameters lead to distinct firing patterns, as shown in the four illustrative examples. The numerical simulations align closely with the physical data obtained from the microcontroller. These findings confirm the presence of heterogeneous multistability across the system. The results establish a clear link between adaptive synaptic mechanisms and complex neuronal output.

The authors demonstrate that their model supports up to twelve distinct coexisting firing patterns. This finding suggests that adaptive synapses significantly increase the dynamical complexity of individual neurons. The researchers highlight that these heterogeneous attractors emerge through specific bifurcation mechanisms. Their analysis confirms that stable and unstable points transition via fold and Hopf bifurcations. The study provides evidence that these complex basins of attraction are sensitive to initial conditions. Experimental implementation on a microcontroller validates the theoretical predictions regarding these diverse states. The team concludes that their framework offers a robust way to simulate complex neuronal behaviors. These results imply that synaptic adaptation is a key driver of neuronal multistability.