We use high-order approximation schemes for the space derivatives in the nonlinear cable equation and investigate the behavior of numerical solution errors by using exact solutions, where available, and grid convergence. The space derivatives are numerically approximated by means of differentiation matrices. Nonlinearity in the equation arises from the Hodgkin-Huxley dynamics of the gating variables for ion channels. We have investigated in particular the effects of synaptic current distribution and compared the accuracy of the spectral solutions with that of finite differencing. A flexible form for the injected current is used that can be adjusted smoothly from a very broad to a narrow peak, which furthermore leads, for the passive cable, to a simple, exact solution. We have used three distinct approaches to assess the numerical solutions: comparison with exact solutions in an unbranched passive cable, the convergence of solutions with progressive refinement of the grid in an active cable, and the simulation of spike initiation in a biophysically realistic single-neuron model. The spectral method provides good numerical solutions for passive cables comparable in accuracy to those from the second-order finite difference method and far greater accuracy in the case of a simulated system driven by inputs that are smoothly distributed in space. It provides faster convergence in active cables and in a realistic neuron model due to better approximation of propagating spikes.

The scale of large neuronal network simulations is memory limited due to the need to store connectivity information: connectivity storage grows as the square of neuron number up to anatomically relevant limits. Using the NEURON simulator as a discrete-event simulator (no integration), we explored the consequences of avoiding the space costs of connectivity through regenerating connectivity parameters when needed: just in time after a presynaptic cell fires. We explored various strategies for automated generation of one or more of the basic static connectivity parameters: delays, postsynaptic cell identities, and weights, as well as run-time connectivity state: the event queue. Comparison of the JitCon implementation to NEURON's standard NetCon connectivity method showed substantial space savings, with associated run-time penalty. Although JitCon saved space by eliminating connectivity parameters, larger simulations were still memory limited due to growth of the synaptic event queue. We therefore designed a JitEvent algorithm that added items to the queue only when required: instead of alerting multiple postsynaptic cells, a spiking presynaptic cell posted a callback event at the shortest synaptic delay time. At the time of the callback, this same presynaptic cell directly notified the first postsynaptic cell and generated another self-callback for the next delay time. The JitEvent implementation yielded substantial additional time and space savings. We conclude that just-in-time strategies are necessary for very large network simulations but that a variety of alternative strategies should be considered whose optimality will depend on the characteristics of the simulation to be run.