Transmission Neural Networks: Inhibitory and Excitatory Connections

Shuang Gao; Peter E. Caines

cs.SIApr 5, 2026

Transmission Neural Networks: Inhibitory and Excitatory Connections

Authors: Shuang Gao, Peter E. Caines

Organizations: Department of Electrical Engineering, Polytechnique Montreal, GERAD (Group for Research in Decision Analysis), and UNIQUE (Unifying Neuroscience and Artificial Intelligence - Quebec), Montreal, QC, Canada · Department of Electrical and Computer Engineering, McGill University & GERAD, Montreal, QC, Canada

Abstract

This paper extends the Transmission Neural Network model proposed by Gao and Caines in [1]-[3] to incorporate inhibitory connections and neurotransmitter populations. The extended network model contains binary neuronal states, transmission dynamics, and inhibitory and excitatory connections. Under technical assumptions, we establish the characterization of the firing probabilities of neurons, and show that such a characterization considering inhibitions can be equivalently represented by a neural network where each neuron has a continuous state of dimension 2. Moreover, we incorporated neurotransmitter populations into the modeling and establish the limit network model when the number of neurotransmitters at all synaptic connections go to infinity. Finally, sufficient conditions for stability and contraction properties of the limit network model are established.

Figures & tables

Fig. 1: NOR gate with inputs

A

and

B

, and output

C

, created with inhibitory connections and a constant signal 1.

Neural Network Stability

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A Nonlinear Separation Principle via Contraction Theory: Applications to Neural Networks, Control, and Learning

Anand Gokhale, Anton V. Proskurnikov, Yu Kawano, Francesco Bullo

Center for Control, Dynamical Systems, and Computation, UC Santa Barbara, Santa Barbara, CA 93106 USA · Department of Electronics and Telecommunications, Politecnico di Torino, Italy, 10129 · Graduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan

Dynamical Systems Contraction Theory Recurrent Neural Networks Neural Network Stability

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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.

Abstract

Neural Network Stability

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