The metrization of the space of neural responses is an ongoing research program seeking to find natural ways to describe, in geometrical terms, the sets of possible activities in the brain. One component of this program are the {em spike metrics}, no
tions of distance between two spike trains recorded from a neuron. Alignment spike metrics work by identifying equivalent spikes in one train and the other. We present an alignment spike metric having $mathcal{L}_p$ underlying geometrical structure; the $mathcal{L}_2$ version is Euclidean and is suitable for further embedding in Euclidean spaces by Multidimensional Scaling methods or related procedures. We show how to implement a fast algorithm for the computation of this metric based on bipartite graph matching theory.
For the nervous system to work at all, a delicate balance of excitation and inhibition must be achieved. However, when such a balance is sought by global strategies, only few modes remain balanced close to instability, and all other modes are strongl
y stable. Here we present a simple model of neural tissue in which this balance is sought locally by neurons following `anti-Hebbian behavior: {sl all} degrees of freedom achieve a close balance of excitation and inhibition and become critical in the dynamical sense. At long timescales, the modes of our model oscillate around the instability line, so an extremely complex breakout dynamics ensues in which different modes of the system oscillate between prominence and extinction. We show the system develops various anomalous statistical behaviours and hence becomes self-organized critical in the statistical sense.
