Stochasticity and functionality of neural systems: Mathematical modelling of axon growth in the spinal cord of tadpole

In this paper we study a simple mathematical model of axon growth in the spinal cord of tadpole. Axon development is described by a system of three difference equations (the dorso-ventral and longitudinal coordinates of the growth cone and the growth angle) with stochastic components. We find optimal parameter values by fitting the model to experimentally measured characteristics of the axon and using the quadratic cost function. The fitted model generates axons for different neuron types in both ascending and descending directions which are similar to the experimentally measured axons. Studying the model of axon growth we have found the analytical solution for dynamics of the variance of the dorso-ventral coordinate and the variance of the growth angle. Formulas provide conditions for the case when the increase of the variance is limited and the analytical expression for the saturation level. It is remarkable that optimal parameters always satisfy the condition of limited variance increase. Taking into account experimental data on distribution of neuronal cell bodies along the spinal cord and dorso-ventral distribution of dendrites we generate a biologically realistic architecture of the whole tadpole spinal cord. Preliminary study of the electrophysiological properties of the model with Morris-Lecar neurons shows that the model can generate electrical activity corresponding to the experimentally observed swimming pattern activity of the tadpole in a broad range of parameter values.