Modelling and analysing neural networks using a hybrid process algebra

Research involving artificial neural networks has tended to be driven towards efficient computation, especially in the domain of pattern recognition, or towards elucidating biological processes in the brain. Models have become more detailed as our understanding of the biology of the brain has increased, incorporating real-time behaviour of individual neurons interacting within complex system structures and dynamics. There are few examples of abstract and fully formal models of biologically plausible neural networks: in the neural networks literature models are often presented as a mixture of mathematical equations and natural language, supported by simulation code and associated experimental results. The informality often hides or obscures important aspects of a particular model, and leaves a large conceptual gap between the model descriptions and the usually low-level programming code used to simulate them. The main contribution of this paper is formally modelling and analysing a biologically plausible neural network model from the literature that exhibits complex neuron-level behaviour and network-level structure. To achieve this a modelling language ‘Pann’ is developed, based on the process algebras CSP and Hybridχ. It is designed to be convenient for mixing the behaviour of discrete events (such as a neuron spike) with mutable continuous and discrete variables (representing chemical properties of a neuron, for instance). Its behaviour is defined using an operational semantics, from which a set of general properties of the language is proved. The groundwork for the biological model is laid by first formalising some well-known concepts from the artificial neural networks domain, such as feedforward behaviour, backpropagation, and recurrent neural networks. The Pann model of a feedforward network, comprising a set of communicating processes representing individual neurons, is proved equivalent to the standard one-line calculation of feedforward behaviour.