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Vibrational Resonance and Delay-Induced Resonance in a FitzHugh-Nagumo Neuron Model with State-Dependent Time Delay


Core Concepts
Incorporating a state-dependent time delay, specifically a velocity component, into a FitzHugh-Nagumo neuron model significantly impacts the emergence and characteristics of both vibrational resonance and delay-induced resonance, offering potential for controlling these phenomena.
Abstract

Bibliographic Information:

Siewe Siewe, M., Rajasekar, S., Coccolo, M., & Sanjuán, M. A. F. (2024). Vibrational resonance in the FitzHugh-Nagumo neuron model under state-dependent time delay. arXiv preprint arXiv:2410.06861v1.

Research Objective:

This research paper investigates the influence of state-dependent time delay, particularly the velocity component, on the occurrence and behavior of vibrational resonance and delay-induced resonance in a FitzHugh-Nagumo neuron model.

Methodology:

The authors employ numerical analysis to study a modified FitzHugh-Nagumo neuron model incorporating an asymmetric double-well potential, state-dependent time delay (modeled using a sigmoid function), and a biharmonic force. They analyze the system's response amplitude at the low frequency to characterize vibrational resonance and delay-induced resonance under varying parameters of the time delay and forcing frequencies.

Key Findings:

  • The presence of a state-dependent time delay velocity component can induce vibrational resonance and multi-resonance in the neuron model, particularly for smaller values of the velocity component amplitude.
  • Increasing the amplitude of the state-dependent time delay velocity component generally leads to a decrease in the response amplitude of vibrational resonance.
  • Delay-induced resonance occurs for specific combinations of the time delay position component and the forcing frequency.
  • The time delay velocity component can broaden the range of forcing frequencies that trigger delay-induced resonance when the delay position component is already conducive to the phenomenon.

Main Conclusions:

The study demonstrates that the state-dependent time delay, specifically the velocity component, plays a crucial role in regulating the emergence and characteristics of both vibrational resonance and delay-induced resonance in the FitzHugh-Nagumo neuron model. This finding suggests potential avenues for controlling these resonance phenomena by manipulating the time delay parameters.

Significance:

This research enhances our understanding of how time delays, particularly state-dependent ones, influence the dynamics of neuronal systems. The findings have implications for comprehending information processing in neurons and potentially for developing treatments for neurological disorders characterized by abnormal neuronal firing patterns.

Limitations and Future Research:

The study primarily relies on numerical analysis of a specific neuron model. Further research could explore the generalizability of these findings to other neuron models and experimental validation in biological systems. Additionally, investigating the impact of noise and network interactions on these resonance phenomena in the presence of state-dependent time delays would be valuable.

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Stats
γ = 0.2, f = 0.1, ω = 1, Ω= 10 are fixed parameters for the model. The asymmetrical double-well potential uses fixed parameters ω0 = 0.81, λ = 1.60 and β = 0.62. Initial conditions for simulations are x0 = 1.80 and ˙x0 = −0.04. For the sigmoid function representing time delay, p = -1 is used in most analyses. When analyzing delay-induced resonance, g = 0. A threshold value of τ0 ≈ 3.5 is identified for significant effects on delay-induced resonance.
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Deeper Inquiries

How might the findings regarding state-dependent time delay in neuron models be applied to developing artificial neural networks with more biologically realistic dynamics?

The findings regarding state-dependent time delay could significantly advance the development of artificial neural networks (ANNs) that exhibit more biologically realistic dynamics. Here's how: Incorporating Realistic Time Delays: Current ANNs typically don't account for the time delays inherent in biological neuronal signaling. By integrating state-dependent time delays, ANNs can better mimic the way real neurons communicate, leading to more accurate models of brain function. Enhancing Temporal Processing: The brain relies heavily on the precise timing of neuronal spikes for information processing. Introducing state-dependent time delays in ANNs can improve their ability to process temporal information, making them more adept at tasks like speech recognition, natural language processing, and motor control. Modeling Complex Oscillatory Patterns: State-dependent time delays have been shown to contribute to the emergence of complex oscillatory patterns in neuronal systems. Incorporating these delays in ANNs could enable them to generate richer dynamics, potentially leading to breakthroughs in understanding consciousness, memory, and other cognitive functions. Developing Novel Learning Rules: The presence of state-dependent time delays necessitates the development of new learning rules for ANNs. These rules would need to account for the dynamic nature of the delays and how they influence the network's overall behavior. This could lead to more efficient and robust learning algorithms. Building Energy-Efficient ANNs: Biological neuronal networks are remarkably energy-efficient. By incorporating state-dependent time delays, which can regulate neuronal firing patterns, ANNs could potentially become more energy-efficient, reducing their computational cost. However, implementing state-dependent time delays in ANNs also presents challenges. Determining the appropriate delay functions, developing efficient computational methods, and understanding the impact of these delays on network stability and learning are all areas that require further research.

Could other factors, such as network topology or synaptic plasticity, mitigate or amplify the effects of state-dependent time delay on resonance phenomena in neuronal systems?

Absolutely, factors like network topology and synaptic plasticity can significantly interact with state-dependent time delays, either mitigating or amplifying resonance phenomena in neuronal systems. Network Topology: The architecture of neuronal connections plays a crucial role. Amplification: In networks with recurrent connections or specific motifs that favor signal propagation, state-dependent time delays could be amplified, leading to enhanced resonance. For instance, delays might synchronize neuronal firing, leading to stronger network oscillations. Mitigation: Conversely, in sparsely connected networks or those with architectures that dampen signal propagation, the effects of state-dependent time delays on resonance might be mitigated. Synaptic Plasticity: The ability of synapses to change their strength over time adds another layer of complexity. Amplification: Synaptic plasticity could amplify resonance effects by strengthening connections between neurons that fire synchronously due to state-dependent time delays. This could lead to the formation of strongly connected neuronal assemblies that exhibit robust oscillations. Mitigation: Conversely, plasticity mechanisms could also act to homeostatically regulate network activity and prevent runaway excitation caused by resonance. For example, synapses could weaken if they consistently contribute to overly strong oscillations. Furthermore, the interplay between these factors is likely to be highly non-linear and context-dependent. For instance, the impact of state-dependent time delay in a network with strong plasticity might differ depending on the specific learning rule governing synaptic changes. Investigating these interactions through computational modeling and experimental studies is crucial for a comprehensive understanding of how resonance phenomena arise and are modulated in complex neuronal networks.

What are the broader implications of understanding and controlling resonance in dynamical systems beyond the realm of neuroscience, such as in physics, engineering, or social systems?

The understanding and control of resonance phenomena in dynamical systems extend far beyond neuroscience, holding significant implications across diverse fields: Physics: Enhancing Energy Transfer: In areas like acoustics and optics, manipulating resonance is crucial for optimizing energy transfer and signal amplification. For example, designing resonant cavities for lasers or musical instruments relies heavily on controlling resonant frequencies. Developing Novel Sensors: Understanding resonance phenomena is essential for developing highly sensitive sensors. By designing systems that resonate at specific frequencies corresponding to target signals, minute changes can be detected, leading to advancements in fields like medical imaging and environmental monitoring. Engineering: Optimizing Structural Design: Resonance plays a critical role in structural engineering. By understanding and controlling resonant frequencies, engineers can design buildings, bridges, and other structures that are resistant to vibrations and external forces, preventing catastrophic failures. Improving Control Systems: In control theory, resonance can be both beneficial and detrimental. Understanding how delays and feedback loops influence resonance is crucial for designing robust and stable control systems for applications ranging from robotics to aircraft design. Social Systems: Understanding Opinion Dynamics: Social systems can exhibit emergent behavior analogous to resonance. The spread of information, opinions, and trends can be amplified or dampened depending on network structure and individual interactions. Understanding these dynamics can inform strategies for effective communication and social intervention. Predicting Market Fluctuations: Financial markets are complex dynamical systems where resonance-like phenomena can lead to booms and crashes. By analyzing market data and identifying potential resonant frequencies, it might be possible to develop early warning systems for financial instability. Overall, the study of resonance in dynamical systems transcends disciplinary boundaries. By unraveling the principles governing resonance, we gain valuable insights into the behavior of complex systems, enabling us to design more robust structures, develop innovative technologies, and potentially even influence the dynamics of social interactions.
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