Energy analysis of bursting Hindmarsh-Rose neurons with time-delayed coupling
Abstract
Mathematical modeling is an important tool to study the role of delay in neural systems and to evaluate its effects on the signaling activity of coupled neurons. Models for delayed neurons are often used to represent the dynamics of real neurons, but rarely to assess the energy required to maintain these dynamics. In this work, we address these questions from an energy perspective by considering a pair of Hindmarsh-Rose burst neurons coupled by reciprocal time-delayed coupling with electrical and chemical synapses. We examine the average energy consumption required to maintain cooperative behavior and quantify the contribution of synapses to total energy consumption. We show that unlike electrical coupling, where the time delay appears to reduce the instantaneous average relative weight of the synaptic contribution, in chemical coupling this average synaptic contribution appears to be much higher in delayed coupling than in instantaneous coupling, except at certain values of coupling strength where the instantaneous synaptic contribution is more important.
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Energy analysis of bursting Hindmarsh-Rose neurons with time-delayed coupling Abdelmalik Moujahid1, F. Vadillo1 1University of the Basque Country UPV/EHU, San Sebasti´an, Spain Published in: Chaos, Solitons & Fractals, Elseiver, 2022. DOI: 10.1016/j.chaos.2022.112071. Volume 158, May 2022, 112071. Published online 11 April 2022 Abstract Mathematical modeling is an important tool to study the role of delay in neural systems and to evaluate its effects on the signaling activity of coupled neurons. Models for delayed neurons are often used to represent the dynamics of real neurons, but rarely to assess the energy required to maintain these dynamics. In this work, we address these questions from an energy perspective by considering a pair of Hindmarsh-Rose burst neurons coupled by reciprocal time-delayed coupling with electrical and chemical synapses. We examine the average energy consumption required to maintain cooperative behavior and quantify the contribution of synapses to total energy consumption. We show that unlike electrical coupling, where the time delay appears to reduce the instantaneous average relative weight of the synaptic contribution, in chemical coupling this average synaptic contribution appears to be much higher in delayed coupling than in instantaneous coupling, except at certain values of coupling strength where the instantaneous synaptic contribution is more important. Main Contributions •Provides a mathematical and computational framework to study the energetic demands of neural synchronization with both electrical and chemical synapses, introducing explicit time delay in the coupling. •Quantifies the contribution of synapses to the energy balance required to maintain bursting and cooperative behaviors in coupled neuron systems. •Distinguishes clear differences in the impact of time delay depending on the type of coupling: time delay generally increases energy demand for electrical synapses but can reduce demand or enable new regimes for chemical synapses, thereby revealing optimal conditions for energetically efficient synchronization. •Offers a comprehensive analysis of how coupling parameters shape the metabolic cost of synchronous neural signaling, with implications for both neuroscience and the bio-inspired design of artificial neural and communication systems. 1
Impact of the Paper This research provides a foundational step in linking the fields of theoretical neuroscience, biophysics, and the design of energy-efficient artificial systems through a nuanced mathematical treatment of synchronization energetics in coupled neuron models. Key impacts are: •Neurobiology and physiology: The work clarifies how different modes of synaptic coupling, and especially time delays, critically modulate the energetic burden of synchronized neuronal signaling. By quantifying these energy flows, it advances our mechanistic understanding of neural resource allocation and may help explain metabolic trade-offs underlying neural coding, disease mechanisms (e.g., epilepsy, Parkinson’s, essential tremor), and adaptive processes in biological brains. •Modeling and computational neuroscience: By offering analytical tools for precise energy accounting in dynamic neural networks, the study enables benchmarking of bio-inspired algorithmic designs, both for simulating neural function and for synthetic neuromorphic circuits, toward optimal energy consumption, a key limitation in large-scale and mobile computational intelligence. •Bio-Inspired engineering and applications: The identification of optimal (low-energy) coupling regimes and parameter ranges, particularly for delayed chemical synapses, serves as a blueprint for building robust, synchronized, and power-conscious artificial neural networks, with applications in robotics, real-time pattern recognition, and biorobotics where resource constraints are severe. •General complex systems science: The explicit demonstration that synchronization does not always require high-energy regimes (and that delays can enhance stability/efficiency) is applicable beyond neuroscience: in areas like power grids, distributed computing, and communication networks, where synchrony must be energetically sustainable over long time frames. In essence, this work bridges theoretical modeling with practical, energetic considerations, offering deeper insight into both how the brain operates and how next-generation technologies can be designed for maximum efficiency and synchronization stability even in the presence of biologically realistic constraints such as communication delays. 2