Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models

The presence of gap junction coupling among neurons of the central nervous systems has been appreciated for some time now. In recent years there has been an upsurge of interest from the mathematical community in understanding the contribution of these direct electrical connections between cells to large-scale brain rhythms. Here we analyze a class of exactly soluble single neuron models, capable of producing realistic action potential shapes, that can be used as the basis for understanding dynamics at the network level. This work focuses on planar piece-wise linear models that can mimic the firing response of several different cell types. Under constant current injection the periodic response and phase response curve (PRC) is calculated in closed form. A simple formula for the stability of a periodic orbit is found using Floquet theory. From the calculated PRC and the periodic orbit a phase interaction function is constructed that allows the investigation of phase-locked network states using the theory of weakly coupled oscillators. For large networks with global gap junction connectivity we develop a theory of strong coupling instabilities of the homogeneous, synchronous and splay state. For a piece-wise linear caricature of the Morris-Lecar model, with oscillations arising from a homoclinic bifurcation, we show that large amplitude oscillations in the mean membrane potential are organized around such unstable orbits.

Implementation and Operation of Collaborative Manufacturing Networks

Part 6: Engineering and Implementation of Collaborative Networks

Turbocharging Networks in the Public Sector

Networks have come to occupy a key position in the strategic armoury of the government, business and community sectors and now have impact on a broad array of policy and management arenas. An emphasis on relationships, trust and mutuality mean that networks function on a different operating logic to the conventional processes of government and business. It is therefore important that organizational members of networks are able to adopt the skills and culture necessary to operate successfully under these distinctive kinds of arrangements. Because networks function from a different operational logic to traditional bureaucracies, public sector organizations may experience difficulties in adapting to networked arrangements. Networks are formed to address a variety of social problems or meet capability gaps within organizations. As such they are often under pressure to quickly produce measurable outcomes and need to form rapidly and come to full operation quickly. This paper presents a theoretical exploration of how diverse types of networks are required for different management and policy situations and draws on a set of public sector case studies to understand/demonstrate how these various types of networked arrangements may be ‘turbo-charged’ so that they more quickly adopt the characteristics necessary to deliver required outcomes.