From Sigmoid Power Control Algorithm to Hopfield-like Neural Networks: "SIR" ("Signal"-to-"Interference"-Ratio)-Balancing Sigmoid-Based Networks- Part I: Continuous Time

Continuous-time Hopfield network has been an important focus of research area since 1980s whose applications vary from image restoration to combinatorial optimization from control engineering to associative memory systems. On the other hand, in wireless communications systems literature, power control has been intensively studied as an essential mechanism for increasing the system performance. A fully distributed power control algorithm (DPCA), called Sigmoid DPCA, is presented by Uykan in [10] and [11], which is obtained by discretizing the continuous-time system. In this paper, we present a Sigmoid-based “Signal-to-Interference Ratio, (SIR)” balancing dynamic networks, called Sgm"SIR"NN, which includes both the Sigmoid power control algorithm (SgmDPCA) and the Hopfield neural networks, two different areas whose scope of interest, motivations and settings are completely different. It’s shown that the Sgm"SIR"N5C5C5C5C5CN exhibits features which are generally attributed to Hopfield Networks. Computer simulations show the effectiveness of the proposed network as compared to traditional Hopfield Network.

💡 Research Summary

The paper introduces a novel continuous‑time neural network called the Sigmoid “Signal‑to‑Interference‑Ratio” Neural Network (Sgm‑SIR‑NN) that unifies two historically separate research domains: Hopfield‑type recurrent neural networks and distributed power‑control algorithms used in wireless communications. The authors start by revisiting the Sigmoid Distributed Power Control Algorithm (SgmDPCA) originally proposed by Uykan, which adjusts each user’s transmit power so that its measured SIR approaches a target value. By keeping the algorithm in continuous time and expressing the power update as a differential equation that incorporates a sigmoid activation, the authors reveal a structural similarity to the dynamics of continuous‑time Hopfield networks, whose evolution is driven by the gradient of an energy (Lyapunov) function.

Mathematically, the state vector (v(t)) (interpreted either as neuron membrane potentials or as user transmit powers) evolves according to

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