Addition formulae for Abelian functions associated with specialized curves.

We discuss a family of multi-term addition formulae for Weierstrass functions on specialized curves of low genus with many automorphisms, concentrating mostly on the case of genus 1 and 2. In the genus 1 case, we give addition formulae for the equianharmonic and lemniscate cases, and in genus 2 we find some new addition formulae for a number of curves.

Quantum signatures of breather-breather interactions.

The spectrum of the quantum discrete nonlinear Schrödinger equation, or boson Hubbard Hamiltonian, on a periodic 1D lattice shows some interesting detailed band structure, which may be interpreted as the quantum signature of a two-breather interaction in the classical case. This fine structure is studied using degenerate perturbation theory. We also present a modification to this model, which increases the mobility of bound states.

Conditions under which Na+ channels can boost conduction of small graded potentials.

It has recently become apparent that in the dendrites or short axons of some neurons, voltage-dependent sodium channels are used not to generate action potentials but to modulate graded potentials; graded potentials carry far more information than do action potentials. A model axon (or dendrite) is described in which sodium channels with kinetics described by equations of the Hodgkin-Huxley type boost conduction of small voltage signals. For a sodium channel density beyond a certain minimum there exists an optimal potential, depolarized with respect to the resting potential, at which there is no steady-state decrement along the axon. For an axon not longer than about 0.7 length constants, small, steady-state deviations from this optimal potential imposed at one end of the axon appear amplified in a graded and stable way at the other end. A small pulse of potential is propagated with amplification and more rapidly than in an axon with a passive membrane. Compared to passive propagation, there will be an improvement in signal-to-noise ratio at the synapse; the axon also acts as a selective frequency filter. The same axon is capable of conducting an action potential.

Membrane conductances involved in amplification of small signals by sodium channels in photoreceptors of drone honey bee.

1. Voltage signals of about 1 mV evoked in photoreceptors of the drone honey bee by shallow modulation of a background illumination of an intensity useful for behaviour are thought to be amplified by voltage-dependent Na+ channels. To elucidate the roles of the various membrane conductances in this amplification we have studied the effects of the Na+ channel blocker tetrodotoxin (TTX) and various putative K+ channel blockers on the membrane potential, Vm. 2. Superfusion of a slice of retina with 0.5-10 mM-4-aminopyridine (4-AP) depolarized the membrane and, in fifty of sixty-three cells induced repetitive action potentials. Ionophoretic injection of tetraethylammonium produced similar effects. 3. In order to measure the depolarization caused by 4-AP, action potentials were prevented by application of TTX: 4-AP was applied when the membrane was depolarized to different levels by light. 4-AP induced an additional depolarization at all membrane potentials tested (-64 to -27 mV). We conclude that there are 4-AP-sensitive K+ channels that are open at constant voltage over this range. 4. 4-AP slowed down the recovery phase of the action potential induced by a light flash by a factor that ranged from 0.51 to 0.16. This reduction could be accounted for by the reduction in a voltage-independent K+ conductance estimated from the steady-state depolarization. 5. After the voltage-gated Na+ channels had been blocked by TTX, exposure to 4-AP further changed the amplitude of the response to a small (approximately 10%) decremental light stimulus. The change was an increase when the background illumination brought Vm to potentials more negative than about -40 mV; it was a decrease when Vm > -40 mV. The data could be fitted by a circuit representation of the membrane with a light-activated conductance and a K+ conductance (EK = -66 mV) that was partly blocked by 4-AP. The voltage range studied was from -52 to -27 mV; neither conductance in the model was voltage dependent. 6. The responses to small changes in light intensity in the absence of TTX were mimicked by a model. We conclude that a voltage-dependent Na+ conductance described by the Hodgkin-Huxley equations can amplify small voltage changes in a cell membrane that is also capable of generating action potentials; the magnitude of the K+ conductance is critical for optimization of signals while avoiding membrane instability.