Aim

The aim of my PhD project is to assess how properties of dendritic trees, namely their geometry, topology and intrinsic characteristics ( conductivities, active processes... ) affect the dynamics of a network of spatially-extended neurons. My supervisor, Yulia Timofeeva, and I are particularly interested in neurons interconnected by gap junctions, as these are thought to be implicated in some interesting network dynamics, such as synchronisation, as well as some resulting properties of the networks, such as directional selectivity.

The Path Integral Approach

Simulating large neural networks with realistic dendritic trees numerically would take a great deal of time to do accurately, and the errors can get fairly large. Therefore, an analytical solution is required to run these calculations quickly and accurately. Larry Abbott¹ developed a theoretical framework based on the path integral, allowing the construction of a function that solves for the voltage on a dendritic tree at any point, given any number of stimuli. However, while the framework was well-developed, the algorithm suggested to implement it was suboptimal. Working with Simon Donnelly, we have derived a language-theoretic algorithm to construct the path integral, but due to its limitations with regard to cyclic graphs, have opted to build upon an algorithm due to Eppstein.² Currently, our code is able to construct the Green's function for the membrane potential on any realistic dendritic structure, such as those available on NeuroMorpho.³ Our implementation is highly parallelisable, using both OpenMP and soon, CUDA, to solve the Green's function extremely quickly.

Cable Theory for Gap-Junction-Connected Neural Networks

In order to work with networks of neurons, Yulia and I are also working on an extension to neuronal cable theory, to cater for gap junctions. Once complete, we will be able to simulate network dynamics for a large collection of neurons. The networks could potentially be grown, such as in the work of Hermann Cuntz,⁴ to resemble characteristic regions of the brain.

Impulse Response Functions

I am also interested in computing impulse response functions for different types of dendritic trees. If shown to be significantly different, this would provide strong evidence for the importance of the dendrites in neuronal computation.

References

1. Abbott, L. F., Farhi, E. and Gutmann, S., The Path Integral for Dendritic Trees, Biological Cybernetics, 66 (1991), 49 - 60.
   Abbott, L. F., Simple Diagrammatic Rules for Solving Dendritic Cable Problems, Physica A, 185 (1992), 343 - 356.
   Abbott, L. F. and Cao, B. J., A New Computational Method for Cable Theory Problems, Biophysical Journal, 64 (1993), 303 - 313.
   
2. Eppstein, D., Finding the k Shortest Paths, SIAM Journal on Computing, 28 (1998), 652 - 673.
   
3. Ascoli, G., Donohue, D. E. and Halavi, M., NeuroMorpho.Org : A Central Resource for Neuronal Morphologies, Journal of Neuroscience, 27 (2007), 9247 - 9251.
   
4. Cuntz, H., Forstner, F., Borst, A. and Haüsser, M., One Rule to Grow Them All : A General Theory of Neuronal Branching and its Practical Applications, PLoS Computational Biology, 6 (2010).

Complexity meets Neuroscience : a dendritic fractal generated from the Julia set.