Modelling and Verifying Neuronal Archetypes in Coq

TL;DR

This work presents a Coq-based formalization of Leaky Integrate-and-Fire neurons and seven neuronal archetypes, providing a general, parameter-agnostic framework for modeling neurons, circuits, and archetypes. It develops inductive proofs of key properties for multi-input and single-input neurons and extends these results to general circuits, establishing lemmas that ensure equivalence between per-neuron updates and circuit-level updates. The paper then proves representative properties for each archetype, including delayer behavior, oscillations, and winner-takes-all dynamics, all within Coq’s higher-order logic. This formalization enables reasoning about arbitrary input lengths, time horizons, and circuit sizes, offering a rigorous foundation for composing archetypes into larger networks with provable guarantees.

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.

Figures (22)

• Figure 1: The basic neuronal archetypes
• Figure 2: General form of a function definition in Coq
• Figure 3: Coq pattern matching forms for natural numbers and lists
• Figure 4: Example definition and variable declaration for a record type
• Figure 5: Coq definition of a neuron

Theorems & Definitions (42)

• Definition 2.1
• Lemma 3.3: Always Non-Negative Potential
• proof
• Lemma 3.4: Always Non-Positive Potential
• Lemma 3.6: Properties of AfterNstepsNeuron
• Lemma 3.7: Properties of $\mathop{\equiv_{len}}$
• Lemma 3.8: Properties of Circuits and of Initial Neurons and Circuits
• Lemma 3.9: Output and CurPot Unchanged with Empty Input
• Lemma 4.1: $\mathit{CurPot}_N$ Always Non-Negative for Multiple-Input Neuron
• proof