A quantitative approximation scheme for the traveling wave solutions in the Hodgkin-Huxley model

We introduce an approximation scheme for the Hodgkin-Huxley model of nerve conductance which allows to calculate both the speed of the traveling pulses and their shape in quantitative agreement with the solutions of the model. We demonstrate that the reduced problem for the front of the traveling pulse admits a unique solution. We obtain an explicit analytical expression for the speed of the pulses which is valid with good accuracy in a wide range of the parameters.Comment: 22 pages (Latex), 9 figures (postscript

Bit flipping and time to recover

We call `bits' a sequence of devices indexed by positive integers, where every device can be in two states: $0$ (idle) and $1$ (active). Start from the `ground state' of the system when all bits are in $0$-state. In our first Binary Flipping (BF) model, the evolution of the system is the following: at each time step choose one bit from a given distribution $\mathcal{P}$ on the integers independently of anything else, then flip the state of this bit to the opposite. In our second Damaged Bits (DB) model a `damaged' state is added: each selected idling bit changes to active, but selecting an active bit changes its state to damaged in which it then stays forever. In both models we analyse the recurrence of the system's ground state when no bits are active. We present sufficient conditions for both BF and DB models to show recurrent or transient behaviour, depending on the properties of $\mathcal{P}$. We provide a bound for fractional moments of the return time to the ground state for the BF model, and prove a Central Limit Theorem for the number of active bits for both models

On well-posedness of variational models of charged drops

Electrified liquids are well known to be prone to a variety of interfacial instabilities that result in the onset of apparent interfacial singularities and liquid fragmentation. In the case of electrically conducting liquids, one of the basic models describing the equilibrium interfacial configurations and the onset of instability assumes the liquid to be equipotential and interprets those configurations as local minimizers of the energy consisting of the sum of the surface energy and the electrostatic energy. Here we show that, surprisingly, this classical geometric variational model is mathematically ill-posed irrespectively of the degree to which the liquid is electrified. Specifically, we demonstrate that an isolated spherical droplet is never a local minimizer, no matter how small is the total charge on the droplet, since the energy can always be lowered by a smooth, arbitrarily small distortion of the droplet's surface. This is in sharp contrast with the experimental observations that a critical amount of charge is needed in order to destabilize a spherical droplet. We discuss several possible regularization mechanisms for the considered free boundary problem and argue that well-posedness can be restored by the inclusion of the entropic effects resulting in finite screening of free charges.Comment: 18 pages, 2 figure