reproducibility paper.txt

The Intrinsic Reliability of Excitable Tissue Model Computations

There are numerous fields where important physiological events are characterized as excitable.  Our own experience has been with models of the neuron, and other pieces of the nervous system that behave in a nonlinear fashion and exhibit threshold phenomena.  In fact, many different types of tissue display these same traits, although their primary function may not be signalling on a millisecond time scale.  Given the ubiquity of these phenomena, it follows that many scientists create mathematical models that display excitability.  Let us focus on those models that are based on solving a system of ordinary differential equations (ODEs).  Such an ODE system is the Hodgkin-Huxley (H-H) model, that couples the cells membrane potential to dimensionless variables that model the dynamics of sodium, potassium, and chloride ion currents.  The H-H ODE can model the behavior of a single excitable cell connected to others, or coupled H-H compartments can be used to model a complex neuron via a spatial discretization of some type.  In either case, the resulting system of ODEs is solved a time step at a time.

If we consider a coupled system of ODEs as the typical system that we end up solving when computing the time evolution of excitable tissue models, then we can look at what this entails.  As we compute the value of the next time-step from those at some number of previous time steps, one thing that is done regardless of whether the method chosen is time implicit or time explicit is that we accumulate the inputs from all the ODEs.  This is a summation step, and it is generic of not only excitable tissue models, but of the numerical time evolution of ODEs.  This is where the inputs from different neurons, in the case of the ODEs modelling single neurons, are accumulated.  If we are solving a system with neurons that have a nontrivial spatial extent, then the accumulation of the information across the spatial extent of an individual neuron is undertaken as well.  Thus at each time-step accumulation of information occurs, and this is implemented as a sum of products.  Let us just consider the summation aspect of this accumulation.  This is usually implemented with