Modelling heterogeneous biomolecular dynamics in neurons using nonlinear mixed effects models and scientific machine learning

Abstract

Computational neuroscience employs mathematical, computational and physical abstractions of biological structures ranging from the scale of ion channels to systems to investigate the brain and the central nervous system. Computational neuroscience has a well-established set of tools and methodologies which are used to handle the available data. However, with the development of new experimental techniques, automation of experiments and increased data-sharing, different analytical tools may be necessary to best use and understand the high volumes of novel data. In this thesis I use two tools which have hitherto had limited application in computational neuroscience – non-linear mixed effects modelling (NLME) and scientific machine learning (SciML). NLME is a hierarchical modelling framework that can account for the sources of within- and between-subject variability. SciML is a discipline in which machine learning is blended with classical mathematical models. By supplementing the set of existing computational neuroscience approaches with NLME and SciML, I tackle three distinct topics that are related to synaptic plasticity and neuronal function.

The first topic is related to the modelling of chemical reaction networks in neurons. There are many published studies containing a relatively large amount of different protein species. Among them is calmodulin, a Ca²⁺-binding molecule essential in cellular signalling, particularly in synaptic plasticity, learning and memory. There are many different published Calmodulin models, but there is no systematic comparison between the different models. I used the data from Faas et al. (Faas et al., 2011) to fit and evaluate the existing Calmodulin models. The Faas et al. data contains the most accurate measurement of Calmodulin-Ca²⁺ binding dynamics to date. However, it suffers from uncertainties of important experimental factors which were not possible to control fully. Therefore, I used NLME to account for the uncertain experimental factors, using the data set to compare various published Calmodulin-Ca²⁺ binding schemes, analysing their ability to fit the data, and showing that some schemes fail due to structural limitations.

The second topic deals with the modelling of voltage-gated ion channels. More specifically, the class of the Hodgkin-Huxley-like ion channel models that assume independence between the different channel gating variables. I use the data of Ranjan et al. (Ranjan et al., 2019) to fit and evaluate various models of the voltage-gated potassium channel (Kᵥ) gating dynamics. The data from Ranjan et al. contain current recordings of many different Kᵥ types. However, their data showed significant heterogeneity in the gating kinetics of the same Kᵥ type. Therefore, I applied NLME to account for within- and between-Kᵥ type heterogeneity. Moreover, I applied SciML to facilitate the Hodgkin-Huxley-like model fitting to multiple different Kᵥ types. Application of these tools led to the creation of a single neural-network based model capable of accurately modelling the gating dynamics of 20 different Kᵥ types.

The third topic I discuss is the application of NLME and SciML in the modelling of synaptic plasticity. Modelling of synaptic plasticity poses a significant challenge to the existing approaches used in computational neuroscience. The number of different protein species present in synapses, along with the heterogeneity of protein numbers in the same type of synapse, makes it very difficult to construct kinetic schemes capable of explaining various observed forms of synaptic plasticity. Therefore, I discuss various ways of applying NLME and SciML which could be productive in creating models of synaptic plasticity.

In conclusion, this thesis demonstrates the benefits of NLME and SciML in supplementing the existing toolbox of computational neuroscientists. These two additional tools were applied successfully to two challenging data sets and led to new insights, such as the limitations of the Calmodulin models used in the literature and the possibility of representing 20 different Kᵥ channels using a single model. A wider adoption of these tools could result in solutions to a number of other existing challenges in neuroscience.