Temporal White Noise Research Articles

The mouse suprachiasmatic nucleus encodes irradiance via a diverse population of neurons monotonically tuned to different ranges of intensity.

Many neurons of the mammalian master circadian oscillator in the suprachiasmatic nuclei (SCN) respond to light pulses with irradiance-dependent changes in firing. Here, we set out to better understand this irradiance coding ability by considering how the SCN tracks more continuous changes in irradiance at both population and single unit level. To this end, we recorded extracellular activity in the SCN of anaesthetised mice presented with up+down irradiance staircase stimuli covering moonlight to daylight conditions and incorporating epochs with steady light or superimposed higher frequency modulations (temporal white noise (WN) and frequency/contrast chirps). Single unit activity was extracted by spike sorting. The population response of SCN units to this stimulus was a progressive increase in firing rate at higher irradiances. This relationship was symmetrical for up vs. down phases of the ramp in the presence of white noise or chirps but exhibited hysteresis for steady light, with firing systematically higher during increasing irradiance. Single units also showed a monotonic relationship between firing and irradiance but exhibited diversity not only in response polarity (increases vs. decreases in firing), but also in the sensitivity (EC50 ) and slope of fitted functions. These data show that individual SCN neurons exhibit monotonic relationships between irradiance and firing rate but differ in the irradiance range over which they respond. This property may help the SCN to encode the large differences in irradiance found in nature using neurons with a constrained range of firing rates. KEY POINTS: Daily changes in environmental light (irradiance) entrain the suprachiasmatic nucleus (SCN) circadian clock. The mouse SCN shows graded increases in neurophysiological activity with light pulses of increasing irradiance. We show that this monotonic relationship between firing rate and irradiance is retained at population and single unit level when probed with more naturalistic staircase increases and decreases in irradiance. The irradiance response is more reliable in the presence of ongoing higher temporal frequency modulations in light intensity than under steady light. Single units varied in sensitivity allowing the population to cover a wide range of irradiances. Irradiance coding in the SCN has characteristics of a sparse code with individual neurons tracking different portions of the natural irradiance range. This property may address the challenge of encoding a 109 -fold day:night difference in irradiance within the constrained range of firing rates available to individual neurons.

Single opsin driven white noise ERGs in mice.

Electroretinograms elicited by photopigment isolating white noise stimuli (wnERGs) in mice were measured. The dependency of rod- and cone-opsin-driven wnERGs on mean luminance was studied. Temporal white noise stimuli (containing all frequencies up to 20 Hz, equal amplitudes, random phases) that modulated either rhodopsin, S-opsin or L*-opsin, using the double silent substitution technique, were used to record wnERGs in mice expressing a human L*-opsin instead of the native murine M-opsin. Responses were recorded at 4 mean luminances (MLs).Impulse response functions (IRFs) were obtained by cross-correlating the wnERG recordings with the corresponding modulation of the photopigment excitation elicited by the stimulus. So-called modulation transfer functions (MTFs) were obtained by performing a Fourier transform on the IRFs.Potentials of two repeated wnERG recordings at corresponding time points were plotted against each other. The correlation coefficient (r2repr) of the linear regression through these data was used to quantify reproducibility. Another correlation coefficient (r2ML) was used to quantify the correlations of the wnERGs obtained at different MLs with those at the highest (for cone isolating stimuli) or lowest (for rod isolating stimuli) ML. IRFs showed an initial negative (a-wave like) trough N1 and a subsequent positive (b-wave like) peak P1. No oscillatory potential-like components were observed. At 0.4 and 1.0 log cd/m2 ML robust L*- and S-opsin-driven IRFs were obtained that displayed similar latencies and dependencies on ML. L*-opsin-driven IRFs were 2.5-3 times larger than S-opsin-driven IRFs. Rhodopsin-driven IRFs were observed at -0.8 and - 0.2 log cd/m2 and decreased in amplitude with increasing ML. They displayed an additional pronounced late negativity (N2), which may be a correlate of retinal ganglion cell activity.R2repr and r2ML values increased for cones with increasing ML whereas they decreased for rods. For rhodopsin-driven MTFs at low MLs and L*-opsin-driven MTFs at high MLs amplitudes decreased with increasing frequency, with much faster decreasing amplitudes for rhodopsin. A delay was calculated from MTF phases showing larger delays for rhodopsin- vs. low delays for L*-opsin-driven responses. Opsin-isolating wnERGs in mice show characteristics of different retinal cell types and their connected pathways.

Luminance white noise electroretinograms (wnERGs) in mice.

To record and analyse electroretinograms (ERGs) to luminance stimuli with white noise temporal profiles in mice. White noise stimuli are expected to keep the retina in a physiologically more natural state than, e.g., flashes. The influence of mean luminance (ML) was studied. Electroretinograms to luminance temporal white noise (TWN) modulation (wnERGs) were measured. The white noise stimuli contained all frequencies up to 20 Hz with equal amplitudes and random phases. Responses were recorded at 7 MLs between -0.7 and 1.2 log cd/m2. Impulse response functions (IRFs) were calculated by cross correlating the averaged white noise electroretinogram (wnERG) responses with the stimulus. Amplitudes and latencies of the initial trough and subsequent peak in the IRFs were measured at each ML. Fourier transforms of the IRFs resulted in modulation transfer functions (MTFs). wnERGs were averaged across different animals. They were measured twice and the responses at identical instances in the 1st and 2nd recordings were plotted against each other. The correlation coefficient (r 2 repr) of the linear regression quantified the reproducibility. The results of the first and second measurement were further averaged. To study the underlying ERG mechanisms, the ERG potentials at the different MLs were plotted against those at the lowest and highest ML. The correlation coefficients (r 2 ML) were used to quantify their similarities. The amplitudes of the initial (a-wave-like) trough of the IRFs increased with increasing ML. The following positive (b-wave-like) peak showed a minimum at -0.4 log cd/m2 above which there was a positive correlation between amplitude and ML. Their latencies decreased monotonously with increasing ML. In none of the IRFs, oscillatory potential (OP)-like components were observed. r 2 repr values were minimal at a ML of -0.1 log cd/m2, where the MTFs changed from low-pass to band-pass. r 2 ML values increased and decreased with increasing ML when correlated with responses obtained at the highest or the lowest ML, respectively. White noise electroretinograms can be reliably recorded in mice with luminance stimuli. IRFs resemble flash ERGs superficially, but they offer a novel procedure to study retinal physiology. New components can be described in the IRFs. The wnERGs are either rod- or cone-driven with little overlap.

Optimal regularity in time and space for stochastic porous medium equations

We prove optimal regularity estimates in Sobolev spaces in time and space for solutions to stochastic porous medium equations. The noise term considered here is multiplicative, white in time and coloured in space. The coefficients are assumed to be H\"older continuous and the cases of smooth coefficients of at most linear growth as well as $\sqrt{u}$ are covered by our assumptions. The regularity obtained is consistent with the optimal regularity derived for the deterministic porous medium equation in [Gess 2020] and [Gess, Sauer, Tadmor 2020] and the presence of the temporal white noise. The proof relies on a significant adaptation of velocity averaging techniques from their usual $L^1$ context to the natural $L^2$ setting of the stochastic case. We introduce a new mixed kinetic/mild representation of solutions to quasilinear SPDE and use $L^2$ based a priori bounds to treat the stochastic term.

Electroretinographic responses to luminance and cone-isolating white noise stimuli in macaques.

Electroretinograms (ERGs) are mass potentials with a retinal origin that can be measured non-invasively. They can provide information about the physiology of the retina. Often, ERGs are measured to flashes that are highly unnatural stimuli. To obtain more information about the physiology of the retina, we measured ERGs with temporal white noise (TWN) stimuli that are more natural and keep the retina in a normal range of operation. The stimuli can be combined with the silent substitution stimulation technique with which the responses of single photoreceptor types can be isolated. We characterized electroretinogram (ERG) responses driven by luminance activity or by the L- or the M-cones. The ERGs were measured from five anesthetized macaques (two females) to luminance, to L-cone isolating and to M-cone isolating stimuli in which luminance or cone excitation were modulated with a TWN profile. The responses from different recordings were correlated with each other to study reproducibility and inter-individual variability. Impulse response functions (IRFs) were derived by cross-correlating the response with the stimulus. Modulation transfer functions (MTFs) were the IRFs in the frequency domain. The responses to luminance and L-cone isolating stimuli showed the largest reproducibility. The M-cone driven responses showed the smallest inter-individual variability. The IRFs and MTFs showed early (high frequency) components that were dominated by L-cone driven signals. A late component was equally driven by L- and M-cone activity. The IRFs showed characteristic similarities and differences relative to flash ERGs. The responses to TWN stimuli can be used to characterize the involvement of retinal cells and pathways to the ERG response. It can also be used to identify linear and non-linear processes.

Optimizing methods to isolate melanopsin-directed responses.

The intrinsic melanopsin photoresponse may initiate visual signals that differ in spatiotemporal characteristics from the cone-opsin- and rhodopsin-mediated signals. Applying the CIE standard observer functions in silent-substitution methods can require individual differences in photoreceptor spectral sensitivities and pre-receptoral filtering to be corrected; failure to do so can lead to the intrusion of more sensitive cone processes with putative melanopsin-directed stimuli. Here we evaluate heterochromatic flicker photometry (HFP) and photoreceptor-directed temporal white noise as techniques to limit the effect of these individual differences. Individualized luminous efficiency functions (V(λ)) were compared to the CIE standard observer functions. We show that adapting chromaticities used in silent-substitution methods can deviate by up to 54% in luminance when estimated with the individual and standard observer functions. These deviations lead to inadvertent cone intrusions in the visual functions measured with melanopsin-directed stimuli. To eliminate the intrusions, individual HFP corrections are sufficient at low frequencies (∼1Hz) but temporal white noise is also required at higher frequencies to desensitize penumbral cones. We therefore recommend the selective application of individualized observer calibration and/or temporal white noise in silent-substitution paradigms when studying melanopsin-directed photoresponses.

The melanopsin-directed white noise electroretinogram (wnERG)

The white noise electroretinogram (wnERG) provides a measure of the impulse response function under conditions of retinal equilibrium; it is yet to be determined how the electrical response generated by melanopsin ganglion cell photoreception is expressed in the impulse response. To this end, we recorded the human wnERG to continuous temporal white noise (TWN) stimuli that were melanopsin-directed (rod and cone silent) or cone-directed (rod and melanopsin silent). The impulse response of the electroretinogram was derived by cross-correlating the TWN stimulus with the wnERG response. We observed that the LMS-cone directed wnERG contained the expected N1 wave (24.1 ± 2.4 ms; mean ± SEM) and P1 wave (49.7 ± 1.8 ms). Melanopsin-directed stimuli produced a unique wnERG with a slower negative deflection (Nm) at 62.9 ± 3.3 ms followed by a positive deflection (Pm) at 126.3 ± 5.1 ms. Additional experiments indicated this melanopsin-directed wnERG response was not due to cone intrusion. The Nm and NmPm amplitudes increased with illuminance (32,000–80,000 Td; no rod intrusion) and melanopsin contrast (10–36% Michelson contrast). As there are known pathways connecting melanopsin cells to the outer retina, we then measured the wnERG to combined melanopsin and cone-directed stimuli to quantify melanopsin interactions with cone signalling. With the combined stimuli, the N1P1 amplitudes were suppressed by ~59%, which may be a result of a destructive interference between the positive (P1) and negative (Nm) waves generated by the cone and melanopsin pathways. We conclude that the human wnERG to melanopsin-directed stimuli may reflect the combined response of intra-retinal melanopsin pathways, independent of rod and cone photoreception.

A Diffuse Interface Model of a Two-Phase Flow with Thermal Fluctuations

We consider a model of a two phase flow proposed by Anderson et al. taking into account possible thermal fluctuations. The mathematical model consists of the compressible Navier–Stokes system coupled with the Cahn–Hilliard equation, where the latter is driven by a multiplicative temporal white noise accounting for thermal fluctuations. We show existence of dissipative martingale solutions satisfying the associated total energy balance.

Computational modelling of salamander retinal ganglion cells using machine learning approaches

Artificial vision using computational models that can mimic biological vision is an area of ongoing research. One of the main themes within this research is the study of the retina and in particular, retinal ganglion cells which are responsible for encoding the visual stimuli. A common approach to modelling the internal processes of retinal ganglion cells is the use of a linear – non-linear cascade model, which models the cell’s response using a linear filter followed by a static non-linearity. However, the resulting model is generally restrictive as it is often a poor estimator of the neuron’s response. In this paper we present an alternative to the linear – non-linear model by modelling retinal ganglion cells using a number of machine learning techniques which have a proven track record for learning complex non-linearities in many different domains. A comparison of the model predicted spike rate shows that the machine learning models perform better than the standard linear – non-linear approach in the case of temporal white noise stimuli.

Correlated cone noise decreases rod signal contributions to the post-receptoral pathways.

This study investigated how invisible extrinsic temporal white noise that correlates with the activity of one of the three [magnocellular (MC), parvocellular (PC), or koniocellular (KC)] post-receptoral pathways alters mesopic rod signaling. A four-primary photostimulator provided independent control of the rod and three cone photoreceptor excitations. The rod contributions to the three post-receptoral pathways were estimated by perceptually matching a 20% contrast rod pulse by independently varying the LMS (MC pathway), +L-M (PC pathway), and S-cone (KC pathway) excitations. We show that extrinsic cone noise caused a predominant decrease in the overall magnitude and ratio of the rod contributions to each pathway. Thus, the relative cone activity in the post-receptoral pathways determines the relative mesopic rod inputs to each pathway.

A Temporal White Noise Analysis for Extracting the Impulse Response Function of the Human Electroretinogram.

PurposeWe introduce a method for determining the impulse response function (IRF) of the ERG derived from responses to temporal white noise (TWN) stimuli.MethodsThis white noise ERG (wnERG) was recorded in participants with normal trichromatic vision to full-field (Ganzfeld) and 39.3° diameter focal stimuli at mesopic and photopic mean luminances and at different TWN contrasts. The IRF was obtained by cross-correlating the TWN stimulus with the wnERG.ResultsWe show that wnERG recordings are highly repeatable, with good signal-to-noise ratio, and do not lead to blink artifacts. The wnERG resembles a flash ERG waveform with an initial negativity (N1) followed by a positivity (P1), with amplitudes that are linearly related to stimulus contrast. These N1 and N1-P1 components showed commonalties in implicit times with the a- and b-waves of flash ERGs. There was a clear transition from rod- to cone-driven wnERGs at ∼1 photopic cd.m−2. We infer that oscillatory potentials found with the flash ERG, but not the wnERG, may reflect retinal nonlinearities due to the compression of energy into a short time period during a stimulus flash.ConclusionThe wnERG provides a new approach to study the physiology of the retina using a stimulation method with adaptation and contrast conditions similar to natural scenes to allow for independent variation of stimulus strength and mean luminance, which is not possible with the conventional flash ERG.Translational RelevanceThe white noise ERG methodology will be of benefit for clinical studies and animal models in the evaluation of hypotheses related to cellular redundancy to understand the effects of disease on specific visual pathways.

Melanopsin supports irradiance-driven changes in maintained activity in the superior colliculus of the mouse.

Melanopsin phototransduction allows intrinsically photosensitive retinal ganglion cells (ipRGCs) to maintain firing under sustained illumination and to encode irradiance. ipRGCs project to different parts of the visual system, including the superficial superior colliculus (sSC), but to date there is no description of melanopsin contributions to the activity of that nucleus. We sought to fill that gap using extracellular recordings to describe light response in the sSC. We failed to observe light responses in the sSC of mice lacking rod and cone function, in which melanopsin provides the only photoreception. Nor did the sSC of intact animals track very gradual ramps in irradiance, a stimulus encoded by melanopsin for other brain regions. However, in visually intact mice we did find maintained responses to extended light steps (30s) and to an irradiance ramp upon which a high frequency (20Hz) temporal white noise was superimposed. Both of these responses were deficient when the spectral composition of the stimulus was changed to selectively reduce its effective irradiance for melanopsin. Such maintained activity was also impaired in mice lacking melanopsin, and this effect was specific, as responses of this genotype to higher spatiotemporal frequency stimuli were normal. We conclude that ipRGCs contribute to irradiance-dependent modulations in maintained activity in the sSC, but that this effect is less robust than for other brain regions receiving ipRGC input.

Correlated and Uncorrelated Invisible Temporal White Noise Alters Mesopic Rod Signaling

We determined how rod signaling at mesopic light levels is altered by extrinsic temporal white noise that is correlated or uncorrelated with the activity of one (magnocellular, parvocellular, or koniocellular) postreceptoral pathway. Rod and cone photoreceptor excitations were independently controlled using a four-primary photostimulator. Psychometric (Weibull) functions were measured for incremental rod pulses (50 to 250 ms) in the presence (or absence; control) of perceptually invisible subthreshold extrinsic noise. Uncorrelated (rod) noise facilitates rod detection. Correlated postreceptoral pathway noise produces differential changes in rod detection thresholds and decreases the slope of the psychometric functions. We demonstrate that invisible extrinsic noise changes rod-signaling characteristics within the three retinogeniculate pathways at mesopic illumination depending on the temporal profile of the rod stimulus and the extrinsic noise type.

Correlated and uncorrelated invisible temporal white noise alters mesopic rod signaling.

We determined how rod signaling at mesopic light levels is altered by extrinsic temporal white noise that is correlated or uncorrelated with the activity of one (magnocellular, parvocellular, or koniocellular) postreceptoral pathway. Rod and cone photoreceptor excitations were independently controlled using a four-primary photostimulator. Psychometric (Weibull) functions were measured for incremental rod pulses (50 to 250ms) in the presence (or absence; control) of perceptually invisible subthreshold extrinsic noise. Uncorrelated (rod) noise facilitates rod detection. Correlated postreceptoral pathway noise produces differential changes in rod detection thresholds and decreases the slope of the psychometric functions. We demonstrate that invisible extrinsic noise changes rod-signaling characteristics within the three retinogeniculate pathways at mesopic illumination depending on the temporal profile of the rod stimulus and the extrinsic noise type.

Random Attractors for Stochastic Reaction-Diffusion Equations with Distribution Derivatives on Unbounded Domains

In this paper, we prove the existence of random attractors for a stochastic reaction-diffusion equation with distribution derivatives on unbounded domains. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears spatially distributed temporal white noise. The stochastic reaction-diffusion equation is recast as a continuous random dynamical system and asymptotic compactness for this demonstrated by using uniform estimates far-field values of solutions. The results are new and appear to be optimal.

Random attractors for stochastic reaction–diffusion equations on unbounded domains

The existence of a pullback attractor is established for a stochastic reaction–diffusion equation on all n-dimensional space. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise. The reaction–diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far-field values of solutions.

Visual sensitivity to temporal modulations of temporal noise

The present endeavor is meant (a) to provide a direct comparison between first- and second-order temporal modulation and, by so doing, (b) to eliminate all spatial clues that might have contaminated previous assessments of the second-order temporal modulation transfer function (TMTF). The second aim was achieved by means of the temporal modulation of a purely temporal white noise, a stimulus used frequently in psychoacoustics but not used as yet in visual stimulation. Luminance and contrast temporal modulation thresholds were measured with a 2AFC staircase procedure. In the first case, the mean luminance of a spatially homogeneous, 30° field was modulated sinusoidally over time (first-order modulation). In the second case, the luminance of the same or of a 60° field was randomized over time at a rate of 150 Hz and this temporal white noise (the carrier) was modulated sinusoidally over time (second-order modulation). First-order thresholds reproduce the classical (large field) flicker sensitivity. Second-order thresholds (measured for the first time with purely temporal stimuli) are at least 100 times higher than first-order ones, display a low-pass characteristic (at least up to 0.5 Hz) and yield a critical fusion frequency (measured at 100% modulation) of ∼10 Hz. The data are in accord with other estimates of the TMTF of the second-order system and thus confirm the effective neutralization of the spatial cues present in these previous studies.

A Theory of Interdecadal Climate Variability of the North Pacific Ocean–Atmosphere System*

Abstract A linear coupled model for the atmosphere–upper-ocean system is proposed to highlight the mechanisms of decadal to interdecadal climate variability in the North Pacific. In this model, wind stress anomalies over the North Pacific are related to anomalies in the meridional temperature gradient of the upper ocean. The latter depends upon air–sea thermodynamical feedbacks and meridional heat transport by upper-ocean currents. Slow adjustment of the oceanic gyre circulation to the change in wind stress is accomplished by the forced baroclinic oceanic Rossby waves, which carry out the meridional heat transport. Uncoupled ocean dynamic adjustment can produce a weak decadal to interdecadal peak in the power spectrum of the meridional transport under temporal white noise wind stress forcing with organized spatial structure. Coupled dynamics produce a basin-scale interdecadal oscillatory mode. This mode arises from the dynamic coupling and the memory of the system, residing in the slow gyre circulation ad...