212 research outputs found
Poisson geometry of parabolic bundles on Elliptic curves
The moduli space of -bundles on an elliptic curve with additional flag
structure admits a Poisson structure. The bivector can be defined using double
loop group, loop group and sheaf cohomology constructions. We investigate the
links between these methods and for the case \SL_2 perform explicit
computations, describing the bracket and its leaves in detail.Comment: Final version: typos fixed, some re-organization; to appear IJ
Detecting emergent processes in cellular automata with excess information
Many natural processes occur over characteristic spatial and temporal scales.
This paper presents tools for (i) flexibly and scalably coarse-graining
cellular automata and (ii) identifying which coarse-grainings express an
automaton's dynamics well, and which express its dynamics badly. We apply the
tools to investigate a range of examples in Conway's Game of Life and Hopfield
networks and demonstrate that they capture some basic intuitions about emergent
processes. Finally, we formalize the notion that a process is emergent if it is
better expressed at a coarser granularity.Comment: 8 pages, 6 figure
Towards a learning-theoretic analysis of spike-timing dependent plasticity
This paper suggests a learning-theoretic perspective on how synaptic
plasticity benefits global brain functioning. We introduce a model, the
selectron, that (i) arises as the fast time constant limit of leaky
integrate-and-fire neurons equipped with spiking timing dependent plasticity
(STDP) and (ii) is amenable to theoretical analysis. We show that the selectron
encodes reward estimates into spikes and that an error bound on spikes is
controlled by a spiking margin and the sum of synaptic weights. Moreover, the
efficacy of spikes (their usefulness to other reward maximizing selectrons)
also depends on total synaptic strength. Finally, based on our analysis, we
propose a regularized version of STDP, and show the regularization improves the
robustness of neuronal learning when faced with multiple stimuli.Comment: To appear in Adv. Neural Inf. Proc. System
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