Computer-Aided Formal Proofs about Dendritic Integration within a Neuron

Description

This article is threefold: (i) we define the first formal framework able to model dendritic integration within biological neurons, (ii) we show how we can turn continuous time into discrete time consistently and (iii) we show how a Lustre model checker can automatically perform proofs about neuron input/output behavioursowing to our framework.Our innovative formal framework is a carefully defined trade-off between abstraction and biological relevance in order to facilitate proofs. This framework is hybrid: inputs entering the synapses as well as the soma output are discrete signals made of spikes but, inside the dendrites, we combine signals quantitatively using real numbers. The soma potential is inevitably specified as a differential equation to keep a biologically accurate modelling of signal accumulation. This prevents from performing simple formal proofs. This has been our motivation to discretize time. Owing to this discretization, we are able to encode our neuron models in Lustre. Lustre is a particularly well suited flow-based language for our purpose. We also encode in Lustre a property of input/output equivalence between neurons in such a way that the model checker Kind2 is able to automatically handle the proof.

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