On the spectra of certain integro-differential-delay problems with applications in neurodynamics

We investigate the spectrum of certain integro-differential-delay equations (IDDEs) which arise naturally within spatially distributed, nonlocal, pattern formation problems. Our approach is based on the reformulation of the relevant dispersion relations with the use of the Lambert function. As a particular application of this approach, we consider the case of the Amari delay neural field equation which describes the local activity of a population of neurons taking into consideration the finite propagation speed of the electric signal. We show that if the kernel appearing in this equation is symmetric around some point a= 0 or consists of a sum of such terms, then the relevant dispersion relation yields spectra with an infinite number of branches, as opposed to finite sets of eigenvalues considered in previous works. Also, in earlier works the focus has been on the most rightward part of the spectrum and the possibility of an instability driven pattern formation. Here, we numerically survey the structure of the entire spectra and argue that a detailed knowledge of this structure is important within neurodynamical applications. Indeed, the Amari IDDE acts as a filter with the ability to recognise and respond whenever it is excited in such a way so as to resonate with one of its rightward modes, thereby amplifying such inputs and dampening others. Finally, we discuss how these results can be generalised to the case of systems of IDDEs.

The Creation of a Legal Ordinance and Program Manual for the Purpose of Illicit Discharge Detection and Elimination

Illegal dumping and improper disposal of pollutants in urban areas can contribute significant pollutant loads to the municipal separate storm sewer system (MS4) and natural environments. Illicit discharges to the MS4 can pose a significant risk to human and environmental health. The Clean Water Act requires that municipalities implement a legal mechanism and plan to detect and eliminate illicit discharges to the MS4. The methodology for program creation included the analysis of other municipal illicit discharge programs, review of state and federal guidance publications, and the review of illicit discharge case-studies. This paper describes a systematic approach applied to the creation and implementation of a legal ordinance and program manual designed for the purpose of illicit discharge detection and elimination (IDDE).