Certification of Confluence Proofs using CeTA

CeTA was originally developed as a tool for certifying termination proofs which have to be provided as certificates in the CPF-format. Its soundness is proven as part of IsaFoR, the Isabelle Formalization of Rewriting. By now, CeTA can also be used for certifying confluence and non-confluence proofs. In this system description, we give a short overview on what kind of proofs are supported, and what information has to be given in the certificates. As we will see, only a small amount of information is required and so we hope that CSI will not stay the only confluence tool which can produce certificates.Comment: 5 pages, International Workshop on Confluence 201

Variability of grid-cell activity

Action potentials of grid cells in the entorhinal cortex of navigating rodents occur every two seconds on average. If one considers the precise temporal sequence of these events, however, it can be seen that they rarely occur in isolation. In fact, the intervals between successive action potentials can be on the order of a few milliseconds. Mapped to the trajectory of the animal, a clear clustering of the action potentials in space can be observed as well. The places where the density of such events is particularly high are called firing fields and are arranged in a hexagonal grid. Regardless of the cell characteristics, the number of spikes observed on different crossings of a field varies strongly. The time between subsequent field crossings is on the order of seconds. We found out that one cause of spike-count variability is that the exact position of the firing fields is not stable over time. In addition, the shifts of the fields were correlated across simultaneously recorded cells. This kind of non-stationarity in the grid-cell network allows conclusions to be drawn about the functioning of this system. Furthermore, dynamic field locations imply that common methods for data analysis of grid-cell recordings can be problematic. Furthermore, we found out that a subset of grid cells, which have particularly high firing rates when crossing a field, can be associated with a peculiarity in the shape of their action potentials: The spikes of some cells are followed by a short afterdepolarization (DAP). At the same time, we discovered cells with even smaller and extremely stereotypical intervals between their spikes. This group of neurons, however, exhibited less pronounced DAPs. Cells with and without DAP did not differ in their spatial firing behavior. Our results imply that different burst behaviors are not directly related to different types of spatial coding. In addition, we suggest that bursting of grid cells could be altered via the mechanisms of DAP formation. In summary, this work shows how details of neuronal activity on two different time scales provide fundamental insights into the processes of spatial navigation. Untethered firing fields and intermittent silences: Why grid‐cell discharge is so variable - Grid cells in medial entorhinal cortex are notoriously variable in their responses, despite the striking hexagonal arrangement of their spatial firing fields. Indeed, when the animal moves through a firing field, grid cells often fire much more vigorously than predicted or do not fire at all. The source of this trial‐to‐trial variability is not completely understood. By analyzing grid‐cell spike trains from mice running in open arenas and on linear tracks, we characterize the phenomenon of “missed” firing fields using the statistical theory of zero inflation. We find that one major cause of grid‐cell variability lies in the spatial representation itself: firing fields are not as strongly anchored to spatial location as the averaged grid suggests. In addition, grid fields from different cells drift together from trial to trial, regardless of whether the environment is real or virtual, or whether the animal moves in light or darkness. Spatial realignment across trials sharpens the grid representation, yielding firing fields that are more pronounced and significantly narrower. These findings indicate that ensembles of grid cells encode relative position more reliably than absolute position. Spike Afterpotentials Shape the In Vivo Burst Activity of Principal Cells in Medial Entorhinal Cortex - Principal neurons in rodent medial entorhinal cortex (MEC) generate high-frequency bursts during natural behavior. While in vitro studies point to potential mechanisms that could support such burst sequences, it remains unclear whether these mechanisms are effective under in vivo conditions. In this study, we focused on the membrane-potential dynamics immediately following action potentials (APs), as measured in whole-cell recordings from male mice running in virtual corridors (Domnisoru et al., 2013). These afterpotentials consisted either of a hyperpolarization, an extended ramp-like shoulder, or a depolarization reminiscent of depolarizing afterpotentials (DAPs) recorded in vitro in MEC principal neurons. Next, we correlated the afterpotentials with the cells' propensity to fire bursts. All DAP cells with known location resided in Layer II, generated bursts, and their interspike intervals (ISIs) were typically between 5 and 15 ms. The ISI distributions of Layer-II cells without DAPs peaked sharply at around 4 ms and varied only minimally across that group. This dichotomy in burst behavior is explained by cell-group-specific DAP dynamics. The same two groups of bursting neurons also emerged when we clustered extracellular spike-train autocorrelations measured in real 2D arenas (Latuske et al., 2015). Apart from slight variations in grid spacing, no difference in the spatial coding properties of the grid cells across all three groups was discernible. Layer III neurons were only sparsely bursting (SB) and had no DAPs. As various mechanisms for modulating ion-channels underlying DAPs exist, our results suggest that temporal features of MEC activity can be altered while maintaining the cells' overall spatial tuning characteristics

Probing scattering phase shifts by attosecond streaking

Attosecond streaking is one of the most fundamental processes in attosecond science allowing for a mapping of temporal (i.e. phase) information on the energy domain. We show that on the single-particle level attosecond streaking time shifts contain spectral phase information associated with the Eisenbud-Wigner-Smith (EWS) time delay, provided the influence of the streaking infrared field is properly accounted for. While the streaking phase shifts for short-ranged potentials agree with the associated EWS delays, Coulomb potentials require special care. We show that the interaction between the outgoing electron and the combined Coulomb and IR laser fields lead to a streaking phase shift that can be described classically

Certified Rule Labeling

The rule labeling heuristic aims to establish confluence of (left-)linear term rewrite systems via decreasing diagrams. We present a formalization of a confluence criterion based on the interplay of relative termination and the rule labeling in the theorem prover Isabelle. Moreover, we report on the integration of this result into the certifier CeTA, facilitating the checking of confluence certificates based on decreasing diagrams for the first time. The power of the method is illustrated by an experimental evaluation on a (standard) collection of confluence problems