Modelling autonomous oscillations in the human pupil light reflex using non-linear delay-differential equations

A Nonlinear Delay Model for Metabolic Oscillations in Yeast Cells

We introduce two time-delay models of metabolic oscillations in yeast cells. Our model tests a hypothesis that the oscillations occur as multiple pathways share a limited resource which we equate to the number of available ribosomes. We initially explore a single-protein model with a constraint equation governing the total resource available to the cell. The model is then extended to include three proteins that share a resource pool. Three approaches are considered at constant delay to numerically detect oscillations. First, we use a spectral element method to approximate the system as a discrete map and evaluate the stability of the linearized system about its equilibria by examining its eigenvalues. For the second method, we plot amplitudes of the simulation trajectories in 2D projections of the parameter space. We use a history function that is consistent with published experimental results to obtain metabolic oscillations. Finally, the spectral element method is used to convert the system to a boundary value problem whose solutions correspond to approximate periodic solutions of the system. Our results show that certain combinations of total resource available and the time delay, lead to oscillations. We observe that an oscillation region in the parameter space is between regions admitting steady states that correspond to zero and constant production. Similar behavior is found with the three-protein model where all proteins require the same production time. However, a shift in the protein production rates peaks occurs for low available resource suggesting that our model captures the shared resource pool dynamics.

Reduced Pupil Oscillation During Facial Emotion Judgment in People with Autism Spectrum Disorder

People with autism spectrum disorder (ASD) show abnormal face perception and emotion recognition. However, it remains largely unknown whether these differences are associated with abnormal physiological responses when viewing faces. In this study, we employed a sensitive emotion judgment task and conducted a detailed investigation of pupil dilation/constriction and oscillation in high-functioning adult participants with ASD and matched controls. We found that participants with ASD showed normal pupil constriction to faces; however, they demonstrated reduced pupil oscillation, which was independent of stimulus properties and participants' perception of the emotion. Together, our results have revealed an abnormal physiological response to faces in people with ASD, which may in turn be associated with impaired face perception previously found in many studies.

Using System Identification to Construct an Inherent Model of Pupillary Light Reflex to Explore Diabetic Neuropathy

This study proposed a pupillary light reflex (PLR) inherent model based on the system identification method to demonstrate the dynamic physiological mechanism of the PLR, in which pupillary constriction and dilation are controlled by the sympathetic and parasympathetic nervous system. This model was constructed and verified by comparing the simulated and predicted PLR response with that of healthy participants. The least root-mean-square error (RMSE) of simulated PLR response was less than 0.7% when stimulus duration was under 3 ms. The RMSE of predicted PLR response increased by approximately 6.76%/s from the stimulus duration of 1 ms to 3 s, when the model directly used the parameters extracted from the PLR at the stimulus duration of 10 ms. When model parameters were derived from the regression by the measured PLR response, the RMSE kept under 8.5%. The model was applied to explore the PLR abnormalities of the people with Diabetic Mellitus (DM) by extracting the model parameters from 42 people with DM and comparing these parameters with those of 42 healthy participants. The parameter in the first-order term of the elastic force of the participants with DM was significantly lower than that of the healthy participants (p < 0.05). The sympathetic force and sympathetic action delay of the participants with DM were significantly larger (p < 0.05) and longer (p < 0.0001) than that of the healthy ones, respectively. The reason might be that the sympathetic nervous system, which controls the dilator muscle, degenerated in diabetic patients.

Identification of attention-deficit hyperactivity disorder based on the complexity and symmetricity of pupil diameter

Adult attention-deficit/hyperactivity disorder (ADHD) frequently leads to psychological/social dysfunction if unaddressed. Identifying a reliable biomarker would assist the diagnosis of adult ADHD and ensure that adults with ADHD receive treatment. Pupil diameter can reflect inherent neural activity and deficits of attention or arousal characteristic of ADHD. Furthermore, distinct profiles of the complexity and symmetricity of neural activity are associated with some psychiatric disorders. We hypothesized that analysing the relationship between the size, complexity of temporal patterns, and asymmetricity of pupil diameters will help characterize the nervous systems of adults with ADHD and that an identification method combining these features would ease the diagnosis of adult ADHD. To validate this hypothesis, we evaluated the resting state hippus in adult participants with or without ADHD by examining the pupil diameter and its temporal complexity using sample entropy and the asymmetricity of the left and right pupils using transfer entropy. We found that large pupil diameters and low temporal complexity and symmetry were associated with ADHD. Moreover, the combination of these factors by the classifier enhanced the accuracy of ADHD identification. These findings may contribute to the development of tools to diagnose adult ADHD.

Pupillometric Complexity and Symmetricity Follow Inverted-U Curves Against Baseline Diameter Due to Crossed Locus Coeruleus Projections to the Edinger-Westphal Nucleus

In addition to photic reflex function, the temporal behavior of the pupil diameter reflects levels of arousal and attention and thus internal cognitive neural activity. Recent studies have reported that these behaviors are characterized by baseline activity, temporal complexity, and symmetricity (i.e., degree of symmetry) between the right and left pupil diameters. We hypothesized that experimental analysis to reveal relationships among these characteristics and model-based analysis focusing on the newly discovered contralateral projection from the locus coeruleus (LC) to the Edinger-Westphal nucleus (EWN) within the neural system for controlling pupil diameter could contribute to another dimension of understanding of complex pupil dynamics. In this study, we aimed to validate our hypothesis by analyzing the pupillary hippus in the healthy resting state in terms of sample entropy (SampEn), to capture complexity, and transfer entropy (TranEn), to capture symmetricity. We also constructed a neural model embedded with the new findings on neural pathways. The following results were observed: first, according to surrogate data analysis, the complexity and symmetricity of pupil diameter changes reflect a non-linear deterministic process. Second, both the complexity and the symmetricity are unimodal, peaking at intermediate pupil diameters. Third, according to simulation results, the neural network that controls pupil diameter has an inverted U-shaped profile of complexity and symmetricity vs. baseline LC activity; this tendency is enhanced by the contralateral synaptic projections from the LCs to the EWNs. Thus, we characterized the typical relationships between the baseline activity and the complexity and symmetricity of the pupillometric data in terms of SampEn and TranEn. Our evaluation method and findings may facilitate the development of estimation and diagnostic tools for exploring states of the healthy brain and psychiatric disorders based on measurements of pupil diameter.

Deep learning-based pupil model predicts time and spectral dependent light responses

Although research has made significant findings in the neurophysiological process behind the pupillary light reflex, the temporal prediction of the pupil diameter triggered by polychromatic or chromatic stimulus spectra is still not possible. State of the art pupil models rested in estimating a static diameter at the equilibrium-state for spectra along the Planckian locus. Neither the temporal receptor-weighting nor the spectral-dependent adaptation behaviour of the afferent pupil control path is mapped in such functions. Here we propose a deep learning-driven concept of a pupil model, which reconstructs the pupil's time course either from photometric and colourimetric or receptor-based stimulus quantities. By merging feed-forward neural networks with a biomechanical differential equation, we predict the temporal pupil light response with a mean absolute error below 0.1 mm from polychromatic (2007 [Formula: see text] 1 K, 4983 [Formula: see text] 3 K, 10,138 [Formula: see text] 22 K) and chromatic spectra (450 nm, 530 nm, 610 nm, 660 nm) at 100.01 ± 0.25 cd/m2. This non-parametric and self-learning concept could open the door to a generalized description of the pupil behaviour.

The Pupillary Response of the Common Octopus (Octopus vulgaris)

Cephalopods have very conspicuous eyes that are often compared to fish eyes. However, in contrast to many fish, the eyes of cephalopods possess mobile pupils. To increase the knowledge of pupillary and thus visual function in cephalopods, the dynamics of the pupil of one of the model species among cephalopods, the common octopus (Octopus vulgaris), was determined in this study. We measured pupillary area as a function of ambient luminance to document the light and dark reaction of the octopus eye. The results show that weak light (<1 cd/m²) is enough to cause a pupil constriction in octopus, and that the pupil reacts fast to changing light conditions. The t₅₀-value defined as the time required for achieving half-maximum constriction ranged from 0.45 to 1.29 s and maximal constriction from 10 to 20% of the fully dilated pupil area, depending on the experimental condition. Axial light had a stronger influence on pupil shape than light from above, which hints at a shadow effect of the horizontal slit pupil. We observed substantial variation of the pupil area under all light conditions indicating that light-independent factors such as arousal or the need to camouflage the eye affect pupil dilation/constriction. In conclusion, the documentation of pupil dynamics provides evidence that the pupil of octopus is adapted to low ambient light levels. Nevertheless it can quickly adapt to and thus function under brighter illumination and in a very inhomogeneous light environment, an ability mediated by the dynamic pupil in combination with previously described additional processes of light/dark adaptation in octopus.

B-mode ultrasound assessment of pupillary function: Feasibility, reliability and normal values

Purpose

To evaluate B-mode ultrasound as a novel method for the examination of pupillary function and to provide normal values for the pupillary reflex as assessed by B-mode ultrasound.

Methods

100 subjects (49 female, 51 male, mean [range] age 51 [18–80 years]) with no history of ophthalmologic disease, no clinically detectable pupillary defects, and corrected visual acuity ≥ 0.8 were included in this prospective observational study. B-mode ultrasound was performed with the subjects eyes closed using an Esaote-Mylab25 system according to current guidelines for orbital insonation. A standardized light stimulus was applied.

Results

The mean ± standard deviation left and right pupillary diameters (PD) at rest were 4.7 ± 0.8 and 4.5 ± 0.8 mm. Following an ipsilateral light stimulus (L_stim), left and right constricted PD were 2.8 ± 0.6 and 2.7 ± 0.6 mm. Following a contralateral L_stim, left and right constricted PD were 2.7 ± 0.6 and 2.6 ± 0.5 mm. Left and right pupillary constriction time (PCT) following ipsilateral L_stim were 970 ± 261.6 and 967 ± 220 ms. Left and right PCT following a contralateral L_stim were 993.8 ± 192.6 and 963 ± 189.4 ms. Patient age was inversely correlated with PD at rest and with PD after ipsilateral and contralateral L_stim (all p<0.001), but not with PCT.

Conclusions

B-mode ultrasound is a simple, rapid and objective method for the quantitative assessment of pupillary function, which may prove useful in a variety of settings where eyelid retraction is impeded or an infrared pupillometry device is unavailable.

Quantization improves stabilization of dynamical systems with delayed feedback

We show that an unstable scalar dynamical system with time-delayed feedback can be stabilized by quantizing the feedback. The discrete time model corresponds to a previously unrecognized case of the microchaotic map in which the fixed point is both locally and globally repelling. In the continuous-time model, stabilization by quantization is possible when the fixed point in the absence of feedback is an unstable node, and in the presence of feedback, it is an unstable focus (spiral). The results are illustrated with numerical simulation of the unstable Hayes equation. The solutions of the quantized Hayes equation take the form of oscillations in which the amplitude is a function of the size of the quantization step. If the quantization step is sufficiently small, the amplitude of the oscillations can be small enough to practically approximate the dynamics around a stable fixed point.

Introduction to Focus Issue: Time-delay dynamics

The field of dynamical systems with time delay is an active research area that connects practically all scientific disciplines including mathematics, physics, engineering, biology, neuroscience, physiology, economics, and many others. This Focus Issue brings together contributions from both experimental and theoretical groups and emphasizes a large variety of applications. In particular, lasers and optoelectronic oscillators subject to time-delayed feedbacks have been explored by several authors for their specific dynamical output, but also because they are ideal test-beds for experimental studies of delay induced phenomena. Topics include the control of cavity solitons, as light spots in spatially extended systems, new devices for chaos communication or random number generation, higher order locking phenomena between delay and laser oscillation period, and systematic bifurcation studies of mode-locked laser systems. Moreover, two original theoretical approaches are explored for the so-called Low Frequency Fluctuations, a particular chaotical regime in laser output which has attracted a lot of interest for more than 30 years. Current hot problems such as the synchronization properties of networks of delay-coupled units, novel stabilization techniques, and the large delay limit of a delay differential equation are also addressed in this special issue. In addition, analytical and numerical tools for bifurcation problems with or without noise and two reviews on concrete questions are proposed. The first review deals with the rich dynamics of simple delay climate models for El Nino Southern Oscillations, and the second review concentrates on neuromorphic photonic circuits where optical elements are used to emulate spiking neurons. Finally, two interesting biological problems are considered in this Focus Issue, namely, multi-strain epidemic models and the interaction of glucose and insulin for more effective treatment.

Embedding the dynamics of a single delay system into a feed-forward ring

We investigate the relation between the dynamics of a single oscillator with delayed self-feedback and a feed-forward ring of such oscillators, where each unit is coupled to its next neighbor in the same way as in the self-feedback case. We show that periodic solutions of the delayed oscillator give rise to families of rotating waves with different wave numbers in the corresponding ring. In particular, if for the single oscillator the periodic solution is resonant to the delay, it can be embedded into a ring with instantaneous couplings. We discover several cases where the stability of a periodic solution for the single unit can be related to the stability of the corresponding rotating wave in the ring. As a specific example, we demonstrate how the complex bifurcation scenario of simultaneously emerging multijittering solutions can be transferred from a single oscillator with delayed pulse feedback to multijittering rotating waves in a sufficiently large ring of oscillators with instantaneous pulse coupling. Finally, we present an experimental realization of this dynamical phenomenon in a system of coupled electronic circuits of FitzHugh-Nagumo type.

A self-avoiding walk with neural delays as a model of fixational eye movements

Fixational eye movements show scaling behaviour of the positional mean-squared displacement with a characteristic transition from persistence to antipersistence for increasing time-lag. These statistical patterns were found to be mainly shaped by microsaccades (fast, small-amplitude movements). However, our re-analysis of fixational eye-movement data provides evidence that the slow component (physiological drift) of the eyes exhibits scaling behaviour of the mean-squared displacement that varies across human participants. These results suggest that drift is a correlated movement that interacts with microsaccades. Moreover, on the long time scale, the mean-squared displacement of the drift shows oscillations, which is also present in the displacement auto-correlation function. This finding lends support to the presence of time-delayed feedback in the control of drift movements. Based on an earlier non-linear delayed feedback model of fixational eye movements, we propose and discuss different versions of a new model that combines a self-avoiding walk with time delay. As a result, we identify a model that reproduces oscillatory correlation functions, the transition from persistence to antipersistence, and microsaccades.

Characterizing and modeling the intrinsic light response of rat ganglion-cell photoreceptors

Intrinsically photosensitive retinal ganglion cells (ipRGCs) mediate both image-forming vision and non-image-forming visual responses such as pupillary constriction and circadian photoentrainment. Five types of ipRGCs, named M1-M5, have been discovered in rodents. To further investigate their photoresponse properties, we made multielectrode array spike recordings from rat ipRGCs, classified them into M1, M2/M4, and M3/M5 clusters, and measured their intrinsic, melanopsin-based responses to single and flickering light pulses. Results showed that ipRGC spiking can track flickers up to ∼0.2 Hz in frequency and that flicker intervals between 5 and 14 s evoke the most spikes. We also learned that melanopsin's integration time is intensity and cluster dependent. Using these data, we constructed a mathematical model for each cluster's intrinsic photoresponse. We found that the data for the M1 cluster are best fit by a model that assumes a large photoresponse, causing the cell to enter depolarization block. Our models also led us to hypothesize that the M2/M4 and M3/M5 clusters experience comparable photoexcitation but that the M3/M5 cascade decays significantly faster than the M2/M4 cascade, resulting in different response waveforms between these clusters. These mathematical models will help predict how each ipRGC cluster might respond to stimuli of any waveform and could inform the invention of lighting technologies that promote health through melanopsin stimulation.

ALMOST SURE ASYMPTOTIC STABILITY OF SCALAR STOCHASTIC DELAY EQUATIONS: FINITE STATE MARKOV PROCESS

In this paper, we study the almost-sure asymptotic stability of scalar delay differential equations with random parametric fluctuations which are modeled by a Markov process with finitely many states. The techniques developed for the determination of almost-sure asymptotic stability of finite dimensional stochastic differential equations will be extended to delay differential equations with random parametric fluctuations. For small intensity noise, we construct an asymptotic expansion for the exponential growth rate (the maximal Lyapunov exponent), which determines the almost-sure stability of the stochastic system.

Disorders of the pupil

Pupil size is determined by the interaction of the parasympathetic and the sympathetic nervous system. The parasympathetic system conducts the light reaction with its major center in the dorsal midbrain. The sympathetic nervous system acts either directly on the dilator muscle (peripherally) or centrally by inhibiting the Edinger-Westphal nucleus. Psychosensory reactions are transmitted via the sympathetic system. The afferent input of the light reflex system in humans is characteristically wired, allowing a detailed analysis of a lesion of the afferent input. Even in humans a subgroup of ganglion cells containing melansopsin plays an important role as a light sensor for the pupillary system. To diagnose normal pupillary function, pupils need to be isocoric and react bilaterally equally to light. Anisocoria indicates a problem of the efferent pupillary pathway. Pupillary disorders may involve the afferent pathways (relative afferent pupillary defect) or the efferent pathways. Physiological anisocoria is a harmless condition that has to be distinguished from Horner's syndrome. In this case pharmacological testing with cocaine eye-drops is helpful. Disorders of the parasympathetic system will impair the light response. They include dorsal midbrain syndrome, third-nerve palsy, and tonic pupil. Tonic pupils are mainly idiopathic and do not need imaging. Disorders of the iris, including application of cholinergic agents, need also to be considered in impaired pupillary light reaction.

Modeling transient pupillary light reflex induced by a short light flash

A pupillary light reflex (PLR) model was proposed in this paper by considering the iris muscle mechanical properties and modulation inputs from both parasympathetic and sympathetic systems. The model can describe very well the experimental PLR responses induced by a short light flash of various intensities. In addition, an inverse method was developed to fit numerically this model to experimental PLR data. The model was tested in experimental human PLR data to extract separately the parasympathetic and sympathetic modulations during PLR. The results indicated a higher parasympathetic and a lower sympathetic activity in females than in males, which was consistent with previous findings in cardiovascular studies. This new model may help improve our understanding of the PLR process and could be applied to analyze autonomic nervous interaction during pupillary responses.

Stochastic dynamics in systems with unidirectional delay coupling: two-state description

We study stochastic dynamics of two-state particles coupled unidirectionally with delay. We give exact results for the stationary distribution function p(st) and the time correlation function (TCF) when the system consists of two (N=2) and three (N=3) particles. Based on these results, effects of delay are discussed and compared with the N=1 case, studied by Tsimring and Pikovsky [Phys. Rev. Lett. 87, 250602 (2001)]. Next, we consider the general N -particle system, for which we give exact expressions for p(st) and the TCF, which are inferred based on the N=2 and N=3 solutions and then confirmed via detailed arguments. It is pointed out that the stationary state is mapped to Ising spin model with ferro(antiferro)magnetic interaction when delay feedback is positive (negative).

Dynamics of deterministic and stochastic paired excitatory-inhibitory delayed feedback

We examine the effects of paired delayed excitatory and inhibitory feedback on a single integrate-and-fire neuron with reversal potentials embedded within a feedback network. These effects are studied using bifurcation theory and numerical analysis. The feedback occurs through modulation of the excitatory and inhibitory conductances by the previous firing history of the neuron; as a consequence, the feedback also modifies the membrane time constant. Such paired feedback is ubiquitous in the nervous system. We assume that the feedback dynamics are slower than the membrane time constant, which leads to a rate model formulation. Our article provides an extensive analysis of the possible dynamical behaviors of such simple yet realistic neural loops as a function of the balance between positive and negative feedback, with and without noise, and offers insight into the potential behaviors such loops can exhibit in response to time-varying external inputs. With excitatory feedback, the system can be quiescent, can be periodically firing, or can exhibit bistability between these two states. With inhibitory feedback, quiescence, oscillatory firing rates, and bistability between constant and oscillatory firing-rate solutions are possible. The general case of paired feedback exhibits a blend of the behaviors seen in the extreme cases and can produce chaotic firing. We further derive a condition for a dynamically balanced paired feedback in which there is neither bistability nor oscillations. We also show how a biophysically plausible smoothing of the firing function by noise can modify the existence and stability of fixed points and oscillations of the system. We take advantage in our mathematical analysis of the existence of an invariant manifold, which reduces the dimensionality of the dynamics, and prove the stability of this manifold. The novel computational challenges involved in analyzing such dynamics with and without noise are also described. Our results demonstrate that a paired delayed feedback loop can act as a sophisticated computational unit, capable of switching between a variety of behaviors depending on the input current, the relative strengths and asymmetry of the two parallel feedback pathways, and the delay distributions and noise level.

Bridges of periodic solutions and tori in semiconductor lasers subject to delay

For semiconductor lasers subject to a delayed optical feedback, branches of steady states sequentially appear as the feedback rate is increased. But branches of time-periodic solutions are connecting pairs of steady states and provide bridges between stable and unstable modes. All bridges experience a change of stability through a torus bifurcation point. Close to the bifurcation point, the torus remains localized near a specific fixed point in phase space. As the feedback rate increases, the torus envelope suddenly unfolds and its trajectory visits two or more unstable fixed points, anticipating the rich dynamics observed at larger feedback rates.

Nonlinear shunting model of the pupil light reflex

A leaky-integrator equation with shunting is introduced to represent processing at the retinal level as part of a system of delay-differential equations modelling the pupil light reflex. This system of equations exhibits the correct behaviour in response to monocular and binocular sinusoidal inputs. Understanding such behaviour can help to establish (non-invasively) the location and nature of nonlinearities in the reflex arc and how signals from the two eyes combine.

Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback

A center manifold reduction and numerical calculations are used to demonstrate the presence of limit cycles, two-tori, and multistability in the damped harmonic oscillator with delayed negative feedback. This model is the prototype of a mechanical system operating with delayed feedback. Complex dynamics are thus seen to arise in very plausible and commonly occurring mechanical and neuromechanical feedback systems. (c) 1995 American Institute of Physics.

Dynamic diseases in neurology and psychiatry

Thirty-two (32) periodic diseases of the nervous system are identified in which symptoms and/or signs recur. In 10/32, the recurrence of a symptom complex is one of the defining features of the illness, whereas in 22/32 oscillatory signs occur in the setting of an ongoing nervous system disorder. We discuss the possibility that these disorders may be dynamic diseases. (c) 1995 American Institute of Physics.

Delayed visual feedback and movement control in Parkinson's disease

The dependence of movement on visual information was compared for healthy individuals and Stage II-III patients with Parkinson's disease (PD). A time delay (0-1400 ms) was introduced into a visually guided motor tracking task which required the subject to maintain constant index finger position relative to a stationary baseline on an oscilloscope. For healthy individuals, delayed visual feedback induced complex oscillations in finger displacement. Similar results were obtained for four of eight patients with PD. However, oscillations were not induced in four of eight patients with PD because of reduced gain and/or a higher tremor amplitude at zero delay which obscured the tracking error. These results suggest that some patients with PD are able to utilize visual information for controlling tracking in this motor task in the same manner as healthy individuals.

Evaluation of pupil constriction and dilation from cycling measurements

Pupil cycling was produced using an electronic circuit so that the retina was illuminated in Maxwellian view only when pupil area exceeded an adjustable area threshold, Aref. The maximum (Amax) and minimum (Amin) amplitude of the oscillations varied linearly with Aref. These observations are described by a delay-differential equation. The Aref-dependent changes in Amax, Amin were used, respectively, to quantitate dilation and constriction. A comparison of the predicted and observed period of pupil cycling suggests that the latency times for light onset and offset are the same. Measurements of Amax, Amin provide a method for determining the average pupil light response.