Exotic dynamics in a firing rate model of neural tissue with threshold accommodation

Many of the equations describing the dynamics of neural systems are written in terms of firing rate functions, which themselves are often taken to be threshold functions of synaptic activity. Dating back to work by Hill in 1936 it has been recognized that more realistic models of neural tissue can be obtained with the introduction of state-dependent dynamic thresholds. In this paper we treat a specific phenomenological model of threshold accommodation that mimics many of the properties originally described by Hill. Importantly we explore the consequences of this dynamic threshold at the tissue level, by modifying a standard neural field model of Wilson-Cowan type. As in the case without threshold accommodation classical Mexican-Hat connectivity is shown to allow for the existence of spatially localized states (bumps) in both one and two dimensions. Importantly an analysis of bump stability in one dimension, using recent Evans function techniques, shows that bumps may undergo instabilities leading to the emergence of both breathers and traveling waves. Moreover, a similar analysis for traveling pulses leads to the conditions necessary to observe a stable traveling breather. In the regime where a bump solution does not exist direct numerical simulations show the possibility of self-replicating bumps via a form of bump splitting. Simulations in two space dimensions show analogous localized and traveling solutions to those seen in one dimension. Indeed dynamical behavior in this neural model appears reminiscent of that seen in other dissipative systems that support localized structures, and in particular those of coupled cubic complex Ginzburg-Landau equations. Further numerical explorations illustrate that the traveling pulses in this model exhibit particle like properties, similar to those of dispersive solitons observed in some three component reaction-diffusion systems. A preliminary account of this work first appeared in S Coombes and M R Owen, Bumps, breathers, and waves in a neural network with spike frequency adaptation, Physical Review Letters 94 (2005), 148102(1-4).

Don’t expect me to vote for you just because I like you, even if you do make me feel warm inside: a comparison of the validity of non-ipsative measures of party support

The inclusion of non-ipsative measures of party preference (in essence ratings for each of the parties of a political system) has become established practice in mass surveys conducted for election studies. They exist in different forms, known as thermometer ratings or feeling scores, likes and dislikes scores, or support propensities. Usually only one of these is included in a single survey, which makes it difficult to assess the relative merits of each. The questionnaire of the Irish National Election Study 2002 (INES2002) contained three different batteries of non-ipsative party preferences. This paper investigates some of the properties of these different indicators. We focus in particular on two phenomena. First, the relationship between non-ipsative preferences and the choices actually made on the ballot. In Ireland this relationship is more revealing than in most other countries owing to the electoral system (STV) which allows voters to cast multiple ordered votes for candidates from different parties. Second, we investigate the latent structure of each of the batteries of party preferences and the relationships between them. We conclude that the three instruments are not interchangeable, that they measure different orientations, and that one –the propensity to vote for a party– is by far preferable if the purpose of the study is the explanation of voters’ actual choice behaviour. This finding has important ramifications for the design of election study questionnaires.