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 effects of a pre-movement delay on the kinematics of prehension in middle childhood

The present study examined the effects of a pre-movement delay on the kinematics of prehension in middle childhood. Twenty-five children between the ages of 5 and 11 years made visually open-loop reaches to two different sized objects at two different distances along the midline. Reaches took place either (i) immediately, or (ii) 2 s after the occlusion of the stimulus. In all age groups, reaches following the pre-movement delay were characterised by longer movement durations, lower peak velocities, larger peak grip apertures and longer time spent in the final slow phase of the movement. This pattern of results suggests that the representations that control the transport and grasp component are affected similarly by delay, and is consistent with the results previously reported for adults. Such representations therefore appear to develop before the age of 5. (C) 2004 Elsevier B.V. All rights reserved.

An on-line algorithm of uncertain time delay estimation in a continuous system

In this paper, we present an on-line estimation algorithm for an uncertain time delay in a continuous system based on the observational input-output data, subject to observational noise. The first order Pade approximation is used to approximate the time delay. At each time step, the algorithm combines the well known Kalman filter algorithm and the recursive instrumental variable least squares (RIVLS) algorithm in cascade form. The instrumental variable least squares algorithm is used in order to achieve the consistency of the delay parameter estimate, since an error-in-the-variable model is involved. An illustrative example is utilized to demonstrate the efficacy of the proposed approach.