Modelling and Verifying Neuronal Archetypes in Coq

TL;DR

This paper models and verifies fundamental neuronal circuit archetypes in Coq, providing formal proofs of their dynamic properties to support understanding of larger neural networks.

Contribution

It introduces a formal Coq model of key neuronal archetypes and proves their core properties, advancing the formal verification of biological neural systems.

Findings

Proved properties for six of seven archetypes.

Developed a general model of neuronal circuits.

Created reusable Coq libraries for future neural network modeling.

Abstract

Formal verification has become increasingly important because of the kinds of guarantees that it can provide for software systems. Verification of models of biological and medical systems is a promising application of formal verification. Human neural networks have recently been emulated and studied as a biological system. In this paper, we provide a model of some crucial neuronal circuits, called "archetypes", in the Coq Proof Assistant and prove properties concerning their dynamic behavior. Understanding the behavior of these modules is crucial because they constitute the elementary building blocks of bigger neuronal circuits. We consider seven fundamental archetypes (simple series, series with multiple outputs, parallel composition, positive loop, negative loop, inhibition of a behavior, and contralateral inhibition), and prove an important representative property for six of them. In building up to our model of archetypes, we also provide a general model of "neuronal circuits", and prove a variety of general properties about neurons and circuits. In addition, we have defined our model with a longer term goal of modelling the composition of basic archetypes into larger networks, and structured our libraries with definitions and lemmas useful for proving the properties in this paper as well as those to be proved as future work.