Are sleep spindles poised on supercritical Hopf bifurcations?

Sleep spindles are recognized as an important intermediate state of long term memory formation. During non REM sleep, large numbers of thalamic relay neurons synchronize their spike bursts for one half to two seconds, entraining many millions of neurons, and constituting a sleep spindle. Here we study spindle amplification, entrainment, synchronization and decay. Relay neurons have both a high resting state near -60 millivolts (mV) and low resting state near -75 mV. Due to the neurons sodium conductance, low-threshold calcium conductance, and calcium-dependent H conductance, it exhibits a number of bifurcations, like its supercritical Hopf at -61 mV. Here low-threshold calcium conductance destabilizes membrane potential to birth a small limit-cycle in the 7-16 Hz range. Supercritical Hopfbifurcations are the underlying mechanism for amplification and frequency selectivity in hearing: hair cells are forced by sinusoidal input currents driving their mainly capacitive loads, with the forcing currents locking at 90 degree phase leads with respect to their oscillating membrane potentials. Here we model a small part of a spindle, with 6 cross-coupled relay neurons all poised on Hopfbifurcations. One neuron is forced by a weak noisy train of periodic current impulses that typically lock at a 90 degree phase lead with respect to its voltage oscillation. It then drives its neighbors, causing them to drive each other at much smaller phase angles, usually less than {+/-}10 degrees. The system of Hopf oscillators exhibit small signal amplification and frequency selectivity, high degrees of synchronization and noise rejection, and switch-ability. These argue in favor of spindling relay neurons poising on, or very near to, supercritical Hopfbifurcations. Also, during the phase-locking of their spike bursts, calcium conductance oscillations increase internal calcium, which turns on slow H current. This depolarizes the relay cells, pushing them below their Hopfbifurcations and terminating the spindle.