Transmission Neural Networks: Inhibitory and Excitatory Connections

This paper establishes a nonlinear separation principle based on contraction theory and derives sharp stability conditions for recurrent neural networks (RNNs). First, we introduce a nonlinear separation principle that guarantees global exponential stability for the interconnection of a contracting state-feedback controller and a contracting observer, alongside parametric extensions for robustness and equilibrium tracking. Second, we derive sharp linear matrix inequality (LMI) conditions that guarantee the contractivity of both firing rate and Hopfield neural network architectures. We establish structural relationships among these certificates-demonstrating that continuous-time models with monotone non-decreasing activations maximize the admissible weight space-and extend these stability guarantees to interconnected systems and Graph RNNs. Third, we combine our separation principle and LMI framework to solve the output reference tracking problem for RNN-modeled plants. We provide LMI synthesis methods for feedback controllers and observers, and rigorously design a low-gain integral controller to eliminate steady-state error. Finally, we derive an exact, unconstrained algebraic parameterization of our contraction LMIs to design highly expressive implicit neural networks, achieving competitive accuracy and parameter efficiency on standard image classification benchmarks.

Spiking neural networks (SNNs) promise low-power event-driven computation for temporally rich tasks, but commonly used neuron models often trade off gradient-based trainability, dynamical richness, and high activity sparsity. These limitations are acute in regression, where approximation error, noise and spike discretization can severely degrade continuous-valued outputs. Indeed, many state-of-the-art (SOTA) SNNs rely on simple phenomenological dynamics trained with surrogate gradients and offer limited control over spiking diversity and sparsity. To overcome such limitations, we introduce multi-timescale conductance spiking networks, a gradient-trainable framework in which neural dynamics emerge from shaping the current-voltage (I-V) curve by tuning fast, slow and ultra-slow conductances. This parametrization allows systematic control over excitability, can be implemented efficiently in analog circuits, and yields rich firing regimes including tonic, phasic and bursting responses within a single model. We derive a discrete-time formulation of these differentiable dynamics, enabling direct backpropagation through time without surrogate-gradient approximations. To probe both trainability and accuracy, we evaluate feedforward networks of these neurons at the predictability limit of Mackey-Glass time-series regression and compare them to baseline LIF and SOTA AdLIF networks. Our model outperforms LIF and AdLIF networks, while exhibiting substantially sparser activity from both communication and computational perspectives. These results highlight multi-timescale conductance spiking neurons as a promising building block for energy-aware temporal processing and neuromorphic implementation.

Artificial neural networks (ANNs), as core components of modern deep learning (DL) systems, lack the adaptive flexibility and long-term stability exhibited by biological systems. This limitation largely stems from the fact that conventional ANNs rely on uniform, local, and gradient-based parameter updates, while neglecting internal learning principles that are biological mechanisms such as neurotransmitters signalling or neuroplasticity. Consequently, many existing approaches focus on architectural expansion or mathematical fine-tuning techniques such as regularisation or parameter isolation. Inspired by the superior adaptability and plasticity of mammalian brains, we posit that neuromodulation with neurotransmitters constitutes a third axis of learning, complementary to neural activity and synaptic plasticity, and should be explicitly modelled in artificial neural networks. In this positional paper, we argue that incorporating neuromodulatory principles into ANN design represents a promising and underexplored research direction, and we advocate for greater attention to this perspective in the development of adaptive and continual learning systems.