Modelling and Formal Verification of Neuronal Archetypes Coupling

Abstract

In the literature, neuronal networks are often represented as graphs where each node symbolizes a neuron and each arc stands for a synaptic connection. Some specific neuronal graphs have biologically relevant structures and behaviors and we call them archetypes. Six of them have already been characterized and validated using formal methods. In this work, we tackle the next logical step and proceed to the study of the properties of their couplings. For this purpose, we rely on Leaky Integrate and Fire neuron modeling and we use the synchronous programming language Lustre to implement the neuronal archetypes and to formalize their expected properties. Then, we exploit an associated model checker called kind2 to automatically validate these behaviors. We show that, when the archetypes are coupled, either these behaviors are slightly modulated or they give way to a brand new behavior. We can also observe that different archetype couplings can give rise to strictly identical behaviors. Our results show that time coding modeling is more suited than rate coding modeling for this kind of studies.

Modelling and Verifying Dynamic Properties of Biological Neural Networks in Coq

“…In the literature, this is not the first attempt to the formal investigation of neural networks. In [21,22], the synchronous paradigm has been exploited to model neurons and some small neuronal circuits with a relevant topological structure and behavior and to prove some properties concerning their dynamics. Our approach based on the use of the Coq proof assistant (which is, to the best of our knowledge, the first one), turned out to be much more general.…”

“…As an example, let us consider the simple series. In [22], the authors were able to write a function (more precisely, a Lustre node) which encodes the expected behavior of the circuit. Then, they could call a model checker to test whether the property at issue is valid for some input series with a fixed length.…”

“…As far as human neural networks are concerned, there is recent work that has focused on their formal verification. In [21,22], the authors consider the synchronous paradigm to model and verify some specific graphs composed of a few biological neurons. These graphs or mini-circuits, characterized by biologically relevant structures and behaviors, are referred to as archetypes and constitute the fundamental elements of neuronal information processing.…”

“…In the work proposed in [21,22], some model checkers such as Kind2 [24] are employed to automatically verify properties concerning the dynamics of six basic archetypes and their coupling. However, model checkers prove properties for some given parameter intervals, and do not handle inputs of arbitrary length.…”

“…Another recent application of formal methods in computer science to neuro-sciences is given in [21,22], where the authors model LI&F neurons and some basic small circuits using the synchronous language Lustre. Such a language is dedicated to the modelling of reactive systems, i.e., systems which constantly interact with the environment and which may have an infinite duration.…”