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 neuron’s 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.


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Mainly all conscious perception passes through the thalamus on its way to the cortex [1]. 27 Thalamic relay neurons, acting in their relay mode, forward rate-coded spike trains from 28 subcortical sensory areas through specific thalamic relay nuclei to their related areas of cortex. 29 information, HO relays pass spike trains from a cortex source to another cortex target [1]. The 48 pulvinar is the largest region of the thalamus and most of the thalamus is devoted to HO relays 49 [1]. For example, a layer V pyramid in cortex region 1 drives an HO pulvinar relay that in turn 50 drives cortex region 2, whose layer V pyramid then drives another HO pulvinar relay making 51 inputs to cortex region 3, etc. [1]. Such cortex → pulvinar → cortex → pulvinar → … HO relay 52 pathways are important in the awake state and also during non-rapid eye movement (NREM) 53 sleep for the propagation of sleep spindles, delta waves and slow waves [1]. 54 Besides their tonic-firing relay mode, thalamic relay cells have a burst firing mode that is used 55 for both attention and memory. For example take walking late at night along a dark country 56 road. One might be startled to see a flaming meteor whiz past, only to realize that it was just a and it is required to construct a permanent memory in cortex, which subsequently becomes 68 independent of the hippocampus [2]. Destexhe and Sejnowski 's "recall-store" memory 69 consolidation hypothesis is based on brief episodes of 7-16 Hz spindles and β and γ from the 70 hippocampus simultaneously driving cortex targets, with these events followed by slow waves 71 [2]. During this part of slow wave sleep the δ and SW force the same cortical targets, but in a 72 lower < 4 Hz frequency range [1,2,3,4]. They conclude that slow wave sleep appears to be a 73 cyclic, iterative process that leads to "off line" memory reprocessing by driving cortex pyramid 74 neurons in complementary ways [2]. Additionally, more than just reactivating memory traces,   , the persistent sodium current , and the delayed rectifier potassium current .

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Together with linear leak current these make a Hodgkin-Huxley action potential generator that 129 spikes spontaneously when depolarized above -56 mV. Two low voltage activated currents, the 130 low-threshold calcium current and the hyperpolarization-activated "sag" current , along 131 with , comprise a low frequency (16.5 Hz) subthreshold oscillator that is active below -60 mV.

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Externally-applied bias current stands in for modulator-type synaptic inputs [1]  inputs from N1's neighbors and externally:  activating a second messenger pathway that slowly turns on a number of potassium currents that 244 more strongly hyperpolarize the neuron [1].

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In the high gain example ( Fig. 1 A,  Hopf bifurcation, lowering its excitability and spike output.   Spindling relay neurons synchronize their spike bursts. In Fig. 4 we employ an event to the theoretical cube root shape for the forced response of a Hopf oscillator driven at its CF

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[11, 12]. As the sensitivity to calcium-induced depolarization of the neurons is increased (blue, green), spike gain predictably drops. It's interesting that forcing of N1amounts to driving a 331 nested sequence of Hopf oscillators, rather than a single one, as is done in hair cells in hearing.

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When a small number of current impulses force N1 at its natural frequency these lock at the 333 optimal 90 degree phase lead with respect to the cell's voltage oscillation. This is typical  Hopf at -61 mV and saddle node on invariant circle (SNIC) at -55 mV). Spike output is very low 360 when N1 is poised at its high resting potential (-60 mV). Spiking increases with depolarization 361 from -59.5 to -55 mV. At these higher membrane potentials the low-threshold calcium 362 conductance is completely off, but there is increasing activation of high-threshold sodium 363 conductance, which puts the relay into the rate-coding region of its SNIC bifurcation. Spike 364 output is noisy in this region so it was shown ± 1 standard deviation (instead of ± standard error 365 for 20 runs).

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5B shows Hopf small-signal frequency selectivity. The oscillators have a ~3 Hz full width half 367 max (FWHM) band width when N1 is weakly forced in the 10-20 Hz range (purple, blue, green 368 curves). But when N1 is depolarized by to its high resting state at -60 mV, band pass 369 filtering is lost (red curve).

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In 5C, N1 is weakly forced at its CF while being poised at various membrane potentials in whiskers, etc.) to pass on sensory spikes in a mainly 1-1 manner [1]. In this rate-coding voltage 381 range spike output in response to weak noisy periodic input is very noisy. In order to show the 382 noise it was plotted ± 1 standard deviation, rather than the standard error for 20 runs.  way, ten dynamical variables can be cut to six. Fig.7 A shows 11 EPSCs forcing N1 at the CF.

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Part B is an incomplete representation of the N1 phase space. Its voltage trajectory (v1, h1, n1 410 taken from 520-750 ms in part A) shows 3 subthreshold oscillations, 2 EPSPs and one spike.

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That being said, it's hard to see how a relay neuron can be accurately represented by a low 414 dimensional phase space representation. and persists for about 20 ms, during the upstroke of a subthreshold voltage oscillation. Inward is a continuous but very small bias current, about 0.1 Calcium current and 2 . 430 linear leak effectively run the subthreshold voltage oscillator (period ~ 67 ms), while is 431 able to bias the size of its limit-cycle oscillation. Inward sodium currents and