During neocortical seizures in patients with epilepsy, microelectrode array recordings from the ictal core show a strong correlation between the fast, cellular spiking activities and the low-frequency component of the potential field, reflected in the electrocorticogram (ECoG). Here, we model the relationship between the cellular spike activity and this low-frequency component as the input and output signals of a linear time invariant system. Our approach is based on the observation that this relationship can be characterized by a so-called sinc function, the unit impulse response of an ideal (brick-wall) filter. Accordingly, using a brick-wall filter, we are able to convert ictal cellular spike inputs into an output that significantly correlates with the observed seizure activity in the ECoG [Formula: see text], while ECoG recordings of subsequent seizures within patients also show significant, but lower, correlations [Formula: see text]. Furthermore, we can produce seizure-like output signals using synthetic spike trains with ictal properties. We propose a possible physiological mechanism to explain the observed properties associated with an ideal filter, and discuss the potential use of our approach for the evaluation of anticonvulsant strategies.

Automatic seizure detection is becoming popular in modern epilepsy monitoring units since it assists diagnostic monitoring and reduces manual review of large volumes of EEG recordings. In this paper, we describe the application of machine learning algorithms for building patient-specific seizure detectors on multiple frequency bands of intra-cranial electroencephalogram (iEEG) recorded by a dense Micro-Electrode Array (MEA). The MEA is capable of recording at a very high sampling rate (30 KHz) producing an avalanche of time series data. We explore subsets of this data to build seizure detectors - we discuss several methods for extracting univariate and bivariate features from the channels and study the effectiveness of using high dimensional clustering algorithms such as K-means and Subspace clustering for constructing the model. Future work involves design of more robust seizure detectors using other features and non-parametric clustering techniques, detection of artifacts and understanding the generalization properties of the models.

We introduce the notion of a star unfolding of the surface ${\cal P}$ of a three-dimensional convex polytope with n vertices, and use it to solve several problems related to shortest paths on ${\cal P}$. The first algorithm computes the edge sequences traversed by shortest paths on ${\cal P}$ in time $O(n^6 \beta (n) \log n)$, where $\beta (n)$ is an extremely slowly growing function. A much simpler $O(n^6)$ time algorithm that finds a small superset of all such edge sequences is also sketched. The second algorithm is an $O(n^{8}\log n)$ time procedure for computing the geodesic diameter of ${\cal P}$: the maximum possible separation of two points on ${\cal P}$ with the distance measured along ${\cal P}$. Finally, we describe an algorithm that preprocesses ${\cal P}$ into a data structure that can efficiently answer the queries of the following form: "Given two points, what is the length of the shortest path connecting them?" Given a parameter $1 \le m \le n^2$, it can preprocess ${\cal P}$ in time $O(n^6 m^{1+\delta})$, for any $\delta > 0$, into a data structure of size $O(n^6m^{1+\delta})$, so that a query can be answered in time $O((\sqrt{n}/m^{1/4}) \log n)$. If one query point always lies on an edge of ${\cal P}$, the algorithm can be improved to use $O(n^5 m^{1+\delta})$ preprocessing time and storage and guarantee $O((n/m)^{1/3} \log n)$ query time for any choice of m between 1 and n.

We present an Ο(n14 log n) algorithm for computing the geodesic diameter of a 3-polytope of n vertices. The geodesic diameter is the greatest separation between two points on the surface, where distance is determined by the shortest (geodesic) path between two points. We assume a model of computation that permits finding roots of a one-variable polynomial of fixed degree in constant time. The key geometric result underlying the algorithm is that, although it may be that neither endpoint of the diameter is a vertex of the polytope, when this occurs, there must be at least five distinct equal-length paths between the diameter endpoints.