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System stability is a fundamental concept in signal processing, often assessed using convolution. For a system to be considered bounded-input bounded-output (BIBO) stable, any bounded input signal must produce a bounded output signal. A bounded input signal is one where the modulus does not exceed a certain constant at any point in time.
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An RC circuit consists of resistance and capacitance, while in an RL circuit, capacitance is replaced by an inductor. RL and RC circuits are first-order differential circuits that store energy. An RC circuit stores energy in the electric field, while an RL circuit stores energy in the magnetic field. When connected to a battery, an RC circuit charges the capacitor, causing the current to decrease from maximum to zero upon being fully charged. This increases the voltage across the capacitor from...
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Frequency response analysis in electrical circuits provides vital insights into a circuit's behavior as the frequency of the input signal changes. The transfer function, a mathematical tool, is instrumental in understanding this behavior. It defines the relationship between phasor output and input and comes in four types: voltage gain, current gain, transfer impedance, and transfer admittance. The critical components of the transfer function are the poles and zeros.
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Unconditional stability of a recurrent neural circuit implementing divisive normalization.

Shivang Rawat1,2, David J Heeger3,4, Stefano Martiniani1,2,5

  • 1Courant Institute of Mathematical Sciences, NYU.

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|November 7, 2025
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Summary
This summary is machine-generated.

We introduce Oscillatory Recurrent Gated Neural Integrator Circuits (ORGaNICs), a biologically plausible model demonstrating unconditional stability. This breakthrough allows training without gradient issues, enhancing neurodynamical model development.

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Area of Science:

  • Computational Neuroscience
  • Machine Learning
  • Dynamical Systems

Background:

  • Recurrent neural models face stability challenges, hindering biologically plausible and trainable neurodynamical systems.
  • Traditional cortical models are difficult to train due to complex nonlinearities.
  • Standard Recurrent Neural Networks (RNNs) lack biological plausibility and interpretability.

Purpose of the Study:

  • To link dynamic divisive normalization (DN) to the stability of Oscillatory Recurrent Gated Neural Integrator Circuits (ORGaNICs).
  • To establish ORGaNICs as a biologically plausible, stable, and trainable recurrent cortical circuit model.
  • To provide a normative principle for circuit and neuronal function.

Main Methods:

  • Utilized the indirect method of Lyapunov to prove unconditional local stability for arbitrary-dimensional ORGaNICs with an identity recurrent weight matrix.
  • Connected ORGaNICs to coupled damped harmonic oscillators to derive an energy function.
  • Proved stability for the 2D ORGaNICs model with a generic recurrent weight matrix and empirically validated higher dimensions.

Main Results:

  • Proved unconditional local stability for ORGaNICs with an identity recurrent weight matrix.
  • Derived an energy function for ORGaNICs, offering a normative principle.
  • Demonstrated empirical stability for generic weight matrices in higher dimensions.
  • Showcased successful training of ORGaNICs via backpropagation through time without gradient clipping/scaling.

Conclusions:

  • ORGaNICs exhibit intrinsic stability, enabling training without common gradient issues (exploding, vanishing, oscillating).
  • ORGaNICs outperform other neurodynamical models in static image classification and match LSTMs in sequential tasks.
  • The model offers a promising direction for developing biologically plausible and efficiently trainable neural networks.