The what and where of adding channel noise to the Hodgkin-Huxley equations

One of the most celebrated successes in computational biology is the Hodgkin-Huxley framework for modeling electrically active cells. This framework, expressed through a set of differential equations, synthesizes the impact of ionic currents on a cell's voltage -- and the highly nonlinear impact of that voltage back on the currents themselves -- into the rapid push and pull of the action potential. Latter studies confirmed that these cellular dynamics are orchestrated by individual ion channels, whose conformational changes regulate the conductance of each ionic current. Thus, kinetic equations familiar from physical chemistry are the natural setting for describing conductances; for small-to-moderate numbers of channels, these will predict fluctuations in conductances and stochasticity in the resulting action potentials. At first glance, the kinetic equations provide a far more complex (and higher-dimensional) description than the original Hodgkin-Huxley equations. This has prompted more than a decade of efforts to capture channel fluctuations with noise terms added to the Hodgkin-Huxley equations. Many of these approaches, while intuitively appealing, produce quantitative errors when compared to kinetic equations; others, as only very recently demonstrated, are both accurate and relatively simple. We review what works, what doesn't, and why, seeking to build a bridge to well-established results for the deterministic Hodgkin-Huxley equations. As such, we hope that this review will speed emerging studies of how channel noise modulates electrophysiological dynamics and function. We supply user-friendly Matlab simulation code of these stochastic versions of the Hodgkin-Huxley equations on the ModelDB website (accession number 138950) and http://www.amath.washington.edu/~etsb/tutorials.html.Comment: 14 pages, 3 figures, review articl

Street robbery and public bus stops: A case study of activity nodes and situational risk

Existing scholarship suggests that crime concentrates in close proximity to public bus stop locations. However, the importance of particular combinations of crime generators and attractors in the proximate environment around public bus stops has not been empirically documented. Drawing on previous environmental criminology research, the current study uses conjunctive analysis of case configurations to address questions about interpersonal violence around bus stops and other activity nodes in the proximate environment in Henderson, Nevada. Findings reveal that street robberies are highly clustered within a relatively small number of environmental contexts that are defined by specific combinations of activity nodes. They also show that bus stops are more likely than any other activity node to be found across dominant situational profiles of robbery, and that the relative risk of robbery associated with the presence/absence of bus stops varies widely on the basis of specific combinations of other activity nodes.Arts, Education & Law Group, School of Criminology and Criminal JusticeNo Full Tex