One important application of the parameter identification method in neuroscience is to determine how voltage-dependent ion channels are spatially distributed through measurements of voltage along the dendrites. While in the past 50 years we have made tremendous strides experimentally to understand how neurons process information, there has also been a lot of supporting mathematical modeling work, initiated by the successful modeling work of Hodgkin and Huxley. As these models and experiments got more detailed, questions about the values of model parameters became more important, initiating parameter estimation investigations.

In this study, we used optimization techniques to solve a parameter identification problem in mathematical modelling of neuronal processing of information at the cellular level, which is governed by a set of nonlinear partial differential equations. We first applied this method to a simplified neuron equation system, which we successfully identified spine density functions. The method is then extended to a more realistic model. We tested our method on cases where the densities to be estimated were either continuous functions with very steep gradients, or piecewise constant functions with large jumps. The method correctly identified the location and size of the jumps, and closely approximated the main features of the unknown density.

We have successfully demonstrated the possibility and advantages of using a PDE-constrained optimization approach to recover the spatial distribution of dendritic spines from measurements. In particular, the method is very successful in identifying extreme densities with large jumps.