Representations of Neuronal Models Using Minimal and Bilinear Realisations

Abstract

Construction of large scale simulations of neuronal circuits is often limited by the intractability of implementing numerical solutions of large numbers of differential equations describing the neuronal elements of the circuit. To make modelling more tractable, simplified models are often used. The relationship between these simplified models and real neuronal circuits is often only qualitative.We demonstrate differential ge- ometric techniques that allow the formal construction of neuronal models in terms of their minimal realisation. A minimal model can be described in terms of a rational series with an associated formal language.These techniques preserve the fundamental behavior of the system. A Lie alge- bra approach is used to produce approximations of arbitrary order and of minimal dimension. It is shown that the dimension of the minimal representation of a neuronal model is determined by the order of ap- proximation and not the number of states in the original description. A bilinear realisation of Hodgkin Huxley models shows that in the critical region of behaviour below the threshold for firing an action potential, the system should not be described as a leaky, linear, integrator, but as a non-linear integrator.