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0/Here’s a summary of a recently published work on phase space geometry of spiking nets. It’s a project that shows its age by its weight in technical detail, but there are pretty (and accurate!) pictures for those wanting to skim. https://journals.aps.org/pre/abstract/10.1103/PhysRevE.99.052402 … https://www.phys.ens.fr/~mptouzel/pdf/PuelmaTouzel_Partitioning_ms.pdf …
1/Two (of many) reasons dynamics of neural systems are hard to understand is that they are high-dimensional and involve spiking interactions. Can't use continuous methods designed for rate-based networks; don't always get what smooth dynamical systems theory would expect.
2/case in point: locally stable phase of spiking cortical circuits: Tubular attractor basins around irregular, asynchronous trajectories. Monteforte 2012 got numerical scaling results for some of the geometry. We provide a missing theory and picture of the global geometry.
3/We produce a spacetime block revealing the attractor shape (see awesome video!). This pointed to basin boundaries composed of trajectories in which a pair of spikes from connected neurons collide.
4/Collision events are located according to the disordered connectivity at a random subset of edges in a hypercube representation of the phase space.
5/The jump that a spike undergoes through one of these collisions is enough to precipitate a cascade of collision events fueled by the random connectivity.
6/We derive the distribution of basin boundaries by using these events to identify the boundaries and then average over the disorder. It boils down (as always?) to Gaussian integrals of susceptibilities and spike intervals.
7/Emerging from all this is a global picture of the geometry:Iterating collision events backwards using the stable dynamics induces a partition refinement that converges to the attractor basins.This is why they are sequences of polytopes with temporally correlated (hyper) faces.
8/Our math approach shows us that the average basin diameter is, intuitively, determined by the ratio of the inhibitory coupling to spike density, which are the dominating stabilizing and destabilizing ingredients to the network, respectively.
9/How to get these tubes to code, especially in the context of time-varying input, is a cool future direction.
10/We hope this work starts theorists thinking about high-dimensional spiking geometry and gets experimentalists thinking about what factors might affect the way their perturbations alter the network activity they are measuring.
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