Neurons are cells in the brain thought to be primarily responsible for information processing and cognition. Neurons communicate with each other via stereotypical electric pulses called action potentials or spikes. The exact computational mechanisms used by the brain for information processing are not yet well understood. The current work seeks to reduce this gap in our understanding, by theoretical analysis.

Our thesis is that specific properties of neurons and the network's architecture constrain the computations it might be able to perform. This dissertation substantiates the thesis for the case of acyclic networks of neurons.

In particular, the idea is to make precise the intuitive view that, acyclic networks are transformations that map input spike trains to output spike trains, to ask what transformations acyclic networks of specific architectures cannot accomplish.

Our neurons are abstract mathematical objects that satisfy a small number of axioms, which correspond to basic properties of biological neurons. To begin with, we find that even a single neuron cannot be consistently viewed as a spike-train to spike-train transformation, in general (in a sense that we will make precise). However, under conditions consistent with spiking regimes observed in-vivo, we prove that the aforementioned notions of transformations are indeed well-defined and correspond to mapping finite-length input spike trains to finite-length output spike trains. Armed with this framework, we then ask what transformations acyclic networks of specific architectures cannot accomplish. We show such results for certain classes of architectures. While attempting to ask how increase in depth of the network constrains the transformations it can effect, we were surprised to discover that, with the current abstract model, every acyclic network has an equivalent acyclic network of depth two. What this suggests is that more axioms need to be added to the abstract model in order to obtain results in this direction. Finally, we study the space of spike-train to spike-train transformations and develop some more theoretical tools to facilitate this line of investigation.