Intrinsically photosensitive melanopsin retinal ganglion cell contributions to the post-illumination pupil response and circadian rhythm

Intrinsically photosensitive retinal ganglion cells (ipRGCs) in the eye transmit the environmental light level, projecting to the suprachiasmatic nucleus (SCN) (Berson, Dunn & Takao, 2002; Hattar, Liao, Takao, Berson & Yau, 2002), the location of the circadian biological clock, and the olivary pretectal nucleus (OPN) of the pretectum, the start of the pupil reflex pathway (Hattar, Liao, Takao, Berson & Yau, 2002; Dacey, Liao, Peterson, Robinson, Smith, Pokorny, Yau & Gamlin, 2005). The SCN synchronizes the circadian rhythm, a cycle of biological processes coordinated to the solar day, and drives the sleep/wake cycle by controlling the release of melatonin from the pineal gland (Claustrat, Brun & Chazot, 2005). Encoded photic input from ipRGCs to the OPN also contributes to the pupil light reflex (PLR), the constriction and recovery of the pupil in response to light. IpRGCs control the post-illumination component of the PLR, the partial pupil constriction maintained for > 30 sec after a stimulus offset (Gamlin, McDougal, Pokorny, Smith, Yau & Dacey, 2007; Kankipati, Girkin & Gamlin, 2010; Markwell, Feigl & Zele, 2010). It is unknown if intrinsic ipRGC and cone-mediated inputs to ipRGCs show circadian variation in their photon-counting activity under constant illumination. If ipRGCs demonstrate circadian variation of the pupil response under constant illumination in vivo, when in vitro ipRGC activity does not (Weng, Wong & Berson, 2009), this would support central control of the ipRGC circadian activity. A preliminary experiment was conducted to determine the spectral sensitivity of the ipRGC post-illumination pupil response under the experimental conditions, confirming the successful isolation of the ipRGC response (Gamlin, et al., 2007) for the circadian experiment. In this main experiment, we demonstrate that ipRGC photon-counting activity has a circadian rhythm under constant experimental conditions, while direct rod and cone contributions to the PLR do not. Intrinsic ipRGC contributions to the post-illumination pupil response decreased 2:46 h prior to melatonin onset for our group model, with the peak ipRGC attenuation occurring 1:25 h after melatonin onset. Our results suggest a centrally controlled evening decrease in ipRGC activity, independent of environmental light, which is temporally synchronized (demonstrates a temporal phase-advanced relationship) to the SCN mediated release of melatonin. In the future the ipRGC post-illumination pupil response could be developed as a fast, non-invasive measure of circadian rhythm. This study establishes a basis for future investigation of cortical feedback mechanisms that modulate ipRGC activity.

Numerical methods for second-order stochastic differential equations

We seek numerical methods for second‐order stochastic differential equations that reproduce the stationary density accurately for all values of damping. A complete analysis is possible for scalar linear second‐order equations (damped harmonic oscillators with additive noise), where the statistics are Gaussian and can be calculated exactly in the continuous‐time and discrete‐time cases. A matrix equation is given for the stationary variances and correlation for methods using one Gaussian random variable per timestep. The only Runge–Kutta method with a nonsingular tableau matrix that gives the exact steady state density for all values of damping is the implicit midpoint rule. Numerical experiments, comparing the implicit midpoint rule with Heun and leapfrog methods on nonlinear equations with additive or multiplicative noise, produce behavior similar to the linear case.