Nonconvergence in Logistic and Poisson Models for Neural Spiking

Generalized linear models are an increasingly common approach for spike train data analysis. For the logistic and Poisson models, one possible difficulty is that iterative algorithms for computing parameter estimates may not converge because of certain data configurations. For the logistic model, these configurations are called complete and quasi-complete separation. We show that these features are likely to occur because of refractory periods of neurons. We use an example to study how standard software deals with this difficulty. For the Poisson model, we show that the same difficulties arise, this time possibly due to bursting or specifics of the binning. We characterize the nonconvergent configurations for both models, show that they can be detected by linear programming methods, and discuss possible remedies.

Parameter-test-ideals of Cohen–Macaulay rings

We describe an algorithm for computing parameter-test-ideals in certain local Cohen–Macaulay rings. The algorithm is based on the study of a Frobenius map on the injective hull of the residue field of the ring and on the application of Sharp’s notion of ‘special ideals’. Our techniques also provide an algorithm for computing indices of nilpotency of Frobenius actions on top local cohomology modules of the ring and on the injective hull of its residue field. The study of nilpotent elements on injective hulls of residue fields also yields a great simplification of the proof of the celebrated result in the article Generators of D-modules in positive characteristic (J. Alvarez-Montaner, M. Blickle and G. Lyubeznik, Math. Res. Lett. 12 (2005), 459–473).

River routing at the continental scale: use of globally-available data and an a priori method of parameter estimation

Two applications of a river routing model based on the observed river network and a linearised solution to the convective-diffusion equation are presented. One is an off-line application to part of the Amazon basin (catchment area 2.15 M km2) using river network data from the Digital Chart of the World and GCM-generated runoff at a grid resolution of 2.5 degrees latitude and 3.75 degrees longitude. The other application is to the Arkansas (409,000 km2) and Red River (125,500 km2) basins as an integrated component of a macro-scale hydrological model, driven by observed meteorology and operating on a 17 km grid. This second application makes use of the US EPA reach data to construct the river network. In both cases, a method of computing parameter values a priori has been applied and shows some success, although some interpretation is required to derive `correct' parameter values and further work is needed to develop guidelines for use of the method. The applications, however, do demonstrate the possibilities for applying the routing model at the continental scale, with globally-available data and a priori parameter estimation, and its value for validating GCM output against observed flows.