Beat Frequency Model Research Articles

Probing the energy and luminosity-dependent spectro-timing properties of RX J0440.9+4431 with AstroSat

ABSTRACT The Be/X-ray binary pulsar RX J0440.9+4431 went through a giant outburst in December 2022 with a peak flux of $\sim$2.3 Crab in 15–50 keV. We studied the broad-band timing and spectral properties of RX J0440.9+4431 using four AstroSat observations, where the source transited between subcritical and supercritical accretion regimes. Pulsations were detected significantly above 100 keV. The pulse profiles were found to be highly luminosity- and energy-dependent. A significant evolution in the pulse profile shape near the peak of the outburst indicates a possible change in the accretion mode and beaming patterns of RX J0440.9+4431. The rms pulsed fraction was luminosity- and energy-dependent, with a concave-like feature around 20–30 keV. The depth of this feature varied with luminosity, indicating changes in the accretion column height and proportion of reflected photons. The broad-band continuum spectra were best fitted with a two-component Comptonization model with a blackbody component or a two-blackbody component model with a thermal Comptonization component. A quasi-periodic oscillation (QPO) at 60 mHz was detected at a luminosity of $2.6 \times 10^{37}$ erg s$^{-1}$, which evolved into 42 mHz at $1.5 \times 10^{37}$ erg s$^{-1}$. The QPO rms were found to be energy dependent with an overall increasing trend with energy. For the first time, we found the QPO frequency varying with photon energy in an X-ray pulsar, which poses a challenge in explaining the QPO with current models such as the Keplarian and beat frequency model. Hence, more physically motivated models are required to understand the physical mechanism behind the mHz QPOs.

Neurocomputational Models of Interval Timing: Seeing the Forest for the Trees.

Extracting temporal regularities and relations from experience/observation is critical for organisms' adaptiveness (communication, foraging, predation, prediction) in their ecological niches. Therefore, it is not surprising that the internal clock that enables the perception of seconds-to-minutes-long intervals (interval timing) is evolutionarily well-preserved across many species of animals. This comparative claim is primarily supported by the fact that the timing behavior of many vertebrates exhibits common statistical signatures (e.g., on-average accuracy, scalar variability, positive skew). These ubiquitous statistical features of timing behaviors serve as empirical benchmarks for modelers in their efforts to unravel the processing dynamics of the internal clock (namely answering how internal clock "ticks"). In this chapter, we introduce prominent (neuro)computational approaches to modeling interval timing at a level that can be understood by general audience. These models include Treisman's pacemaker accumulator model, the information processing variant of scalar expectancy theory, the striatal beat frequency model, behavioral expectancy theory, the learning to time model, the time-adaptive opponent Poisson drift-diffusion model, time cell models, and neural trajectory models. Crucially, we discuss these models within an overarching conceptual framework that categorizes different models as threshold vs. clock-adaptive models and as dedicated clock/ramping vs. emergent time/population code models.

High energy millihertz quasi-periodic oscillations in 1A 0535 + 262 with Insight-HXMT challenge current models

ABSTRACT We studied the millihertz quasi-periodic oscillation (mHz QPO) in the 2020 outburst of the Be/X-ray binary 1A 0535 + 262 using Insight-HXMT data over a broad energy band. The mHz QPO is detected in the 27–120 keV energy band. The QPO centroid frequency is correlated with the source flux and evolves in the 35–95 mHz range during the outburst. The QPO is most significant in the 50–65 keV band, with a significance of ∼8σ, but is hardly detectable (<2σ) in the lowest (1–27 keV) and highest (>120 keV) energy bands. Notably, the detection of mHz QPO above 80 keV is the highest energy at which mHz QPOs have been detected so far. The fractional rms of the mHz QPO first increases and then decreases with energy, reaching the maximum amplitude at 50–65 keV. In addition, at the peak of the outburst, the mHz QPO shows a double-peak structure, with the difference between the two peaks being constant at ∼0.02 Hz, twice the spin frequency of the neutron star in this system. We discuss different scenarios explaining the generation of the mHz QPO, including the beat frequency model, the Keplerian frequency model, the model of two jets in opposite directions, and the precession of the neutron star, but find that none of them can explain the origin of the QPO well. We conclude that the variability of non-thermal radiation may account for the mHz QPO, but further theoretical studies are needed to reveal the physical mechanism.

Resource Allocation in the Noise-Free Striatal Beat Frequency Model of Interval Timing.

The Striatal Beat Frequency (SBF) model of interval timing uses many neural oscillators, presumably located in the frontal cortex (FC), to produce beats at a specific criterion time Tc. The coincidence detection produces the beats in the basal ganglia spiny neurons by comparing the current state of the FC neural oscillators against the long-term memory values stored at reinforcement time Tc. The neurobiologically realistic SBF model has been previously used for producing precise and scalar timing in the presence of noise. Here we simplified the SBF model to gain insight into the problem of resource allocation in interval timing networks. Specifically, we used a noise-free SBF model to explore the lower limits of the number of neural oscillators required for producing accurate timing. Using abstract sine-wave neural oscillators in the SBF-sin model, we found that the lower limit of the number of oscillators needed is proportional to the criterion time Tc and the frequency span (fmax - fmin) of the FC neural oscillators. Using biophysically realistic Morris-Lecar model neurons in the SBF-ML model, the lower bound increased by one to two orders of magnitude compared to the SBF-sin model.

Oscillation/Coincidence-Detection Models of Reward-Related Timing in Corticostriatal Circuits

Abstract The major tenets of beat-frequency/coincidence-detection models of reward-related timing are reviewed in light of recent behavioral and neurobiological findings. This includes the emphasis on a core timing network embedded in the motor system that is comprised of a corticothalamic-basal ganglia circuit. Therein, a central hub provides timing pulses (i.e., predictive signals) to the entire brain, including a set of distributed satellite regions in the cerebellum, cortex, amygdala, and hippocampus that are selectively engaged in timing in a manner that is more dependent upon the specific sensory, behavioral, and contextual requirements of the task. Oscillation/coincidence-detection models also emphasize the importance of a tuned ‘perception’ learning and memory system whereby target durations are detected by striatal networks of medium spiny neurons (MSNs) through the coincidental activation of different neural populations, typically utilizing patterns of oscillatory input from the cortex and thalamus or derivations thereof (e.g., population coding) as a time base. The measure of success of beat-frequency/coincidence-detection accounts, such as the Striatal Beat-Frequency model of reward-related timing (SBF), is their ability to accommodate new experimental findings while maintaining their original framework, thereby making testable experimental predictions concerning diagnosis and treatment of issues related to a variety of dopamine-dependent basal ganglia disorders, including Huntington’s and Parkinson’s disease.

Intra-dorsal striatal acetylcholine M1 but not dopaminergic D1 or glutamatergic NMDA receptor antagonists inhibit consolidation of duration memory in interval timing

The striatal beat frequency model assumes that striatal medium spiny neurons encode duration via synaptic plasticity. Muscarinic 1 (M1) cholinergic receptors as well as dopamine and glutamate receptors are important for neural plasticity in the dorsal striatum. Therefore, we investigated the effect of inhibiting these receptors on the formation of duration memory. After sufficient training in a peak interval (PI)-20-s procedure, rats were administered a single or mixed infusion of a selective antagonist for the dopamine D1 receptor (SCH23390, 0.5 µg per side), N-methyl-D-aspartic acid (NMDA)-type glutamate receptor (D-AP5, 3 µg), or M1 receptor (pirenzepine, 10 µg) bilaterally in the dorsal striatum, immediately before initiating a PI-40 s session (shift session). The next day, the rats were tested for new duration memory (40 s) in a session in which no lever presses were reinforced (test session). In the shift session, the performance was comparable irrespective of the drug injected. However, in the test session, the mean peak time (an index of duration memory) of the M1 + NMDA co-blockade group, but not of the D1 + NMDA co-blockade group, was lower than that of the control group (Experiments 1 and 2). In Experiment 3, the effect of the co-blockade of M1 and NMDA receptors was replicated. Moreover, sole blockade of M1 receptors induced the same effect as M1 and NMDA blockade. These results suggest that in the dorsal striatum, the M1 receptor, but not the D1 or NMDA receptors, is involved in the consolidation of duration memory.

Is the scalar property of interval timing preserved after hippocampus lesions?

Time perception is fundamental for decision-making, adaptation, and survival. In the peak-interval (PI) paradigm, one of the critical features of time perception is its scale invariance, i.e., the error in time estimation increases linearly with the to-be-timed interval. Brain lesions can profoundly alter time perception, but do they also change its scalar property? In particular, hippocampus (HPC) lesions affect the memory of the reinforced durations. Experiments found that ventral hippocampus (vHPC) lesions shift the perceived durations to longer values while dorsal hippocampus (dHPC) lesions produce opposite effects. Here we used our implementation of the Striatal Beat Frequency (SBFML) model with biophysically realistic Morris-Lecar (ML) model neurons and a topological map of HPC memory to predict analytically and verify numerically the effect of HPC lesions on scalar property. We found that scalar property still holds after both vHPC and dHPC lesions in our SBFML-HPC network simulation. Our numerical results show that PI durations are shifted in the correct direction and match the experimental results. In our simulations, the relative peak shift of the behavioral response curve is controlled by two factors: (1) the lesion size, and (2) the cellular-level memory variance of the temporal durations stored in the HPC. The coefficient of variance (CV) of the behavioral response curve remained constant over the tested durations of PI procedure, which suggests that scalar property is not affected by HPC lesions.

Effect of Exercise-Related Factors on the Perception of Time.

The concept of time whether considered through the lenses of physics or physiology is a relative measure. Alterations in time perception can have serious implications in sport, fitness and work. Accurate perception of time is an important skill with many time constrained sports (i.e., basketball, North American football, tennis, gymnastics, figure skating, ice hockey, and others), and work environments (i.e., workers who need to synchronize their actions such as police and military). In addition, time distortions may play a role in exercise adherence. Individuals may be disinclined to continue with healthy, exercise activities that seem protracted (time dilation). Two predominant theories (scalar expectancy theory and striatal beat frequency model) emphasize the perception of the number of events in a period and the role of neurotransmitters in activating and coordinating cortical structures, respectively. A number of factors including age, sex, body temperature, state of health and fitness, mental concentration and exercise intensity level have been examined for their effect on time perception. However, with the importance of time perception for work, sport and exercise, there is limited research on this area. Since work, sports, and exercise can involve an integration of many of these aforementioned factors, they are interventions that need further investigations. The multiplicity of variables involved with work, sport, and exercise offer an underdeveloped but fruitful field for future research. Thus, the objective of this review was to examine physiological and psychological factors affecting human perception of time and the mechanisms underlying time perception and distortion with activity.

Inactivation of the Medial-Prefrontal Cortex Impairs Interval Timing Precision, but Not Timing Accuracy or Scalar Timing in a Peak-Interval Procedure in Rats.

Motor sequence learning, planning and execution of goal-directed behaviors, and decision making rely on accurate time estimation and production of durations in the seconds-to-minutes range. The pathways involved in planning and execution of goal-directed behaviors include cortico-striato-thalamo-cortical circuitry modulated by dopaminergic inputs. A critical feature of interval timing is its scalar property, by which the precision of timing is proportional to the timed duration. We examined the role of medial prefrontal cortex (mPFC) in timing by evaluating the effect of its reversible inactivation on timing accuracy, timing precision and scalar timing. Rats were trained to time two durations in a peak-interval (PI) procedure. Reversible mPFC inactivation using GABA agonist muscimol resulted in decreased timing precision, with no effect on timing accuracy and scalar timing. These results are partly at odds with studies suggesting that ramping prefrontal activity is crucial to timing but closely match simulations with the Striatal Beat Frequency (SBF) model proposing that timing is coded by the coincidental activation of striatal neurons by cortical inputs. Computer simulations indicate that in SBF, gradual inactivation of cortical inputs results in a gradual decrease in timing precision with preservation of timing accuracy and scalar timing. Further studies are needed to differentiate between timing models based on coincidence detection and timing models based on ramping mPFC activity, and clarify whether mPFC is specifically involved in timing, or more generally involved in attention, working memory, or response selection/inhibition.

Correlations between the frequencies of twin kHz QPOs and spins of neutron stars in LMXBs

We investigate the correlation between the frequencies of the twin kilohertz quasi-periodic oscillations (kHz QPOs) and the neutron star (NS) spins in low mass X-ray binaries (LMXBs), based on the data sets of 12 sources with simultaneously detected twin kHz QPOs and NS spins, and find that the histogram of the ratio between the frequency difference of twin kHz QPOs ({\Delta}{\nu} = {\nu}2 - {\nu}1) and NS spin {\nu}s shows a non-uniform distribution with a gap at {\Delta}{\nu}/{\nu}s ~ 0.65. We try to classify the 12 sources into two categories according to this gap: (I) The slow rotators with <{\nu}s> ~ 311 Hz, XTE J1807.4-294, 4U 1915-05, IGR J17191-2821, 4U 1702-43, 4U 1728-34 and 4U 0614+09 follow a relation {\Delta}{\nu}/{\nu}s > 0.65; (II) The fast rotators with <{\nu}s> ~ 546 Hz, SAX J1808.4-3658, KS 1731-260, Aql X-1, 4U 1636-53, SAX J1750.8-2900 and 4U 1608-52 satisfy the relation {\Delta}{\nu}/{\nu}s < 0.65. However, the linear fittings of {\Delta}{\nu} versus {\nu}s relations of group (I) and (II) are unsatisfactory to ensure any certain correlations. We suggest that this phenomenon may arise from the fact that most measured kHz QPOs and spins satisfy the conditions of 1.1 {\nu}s < {\nu}2 < 1300Hz and {\Delta}{\nu} decreasing with {\nu}2. Apparently, the diversified distribution of {\Delta}{\nu}/{\nu}s refutes the simple beat-frequency model, and the statistical correlations between the twin kHz QPOs and NS spins may arise from the magnetosphere-disk boundary environments, e.g., co-rotation radius and NS radius, that modulate the occurrences of X-ray signals. Furthermore, we also find that the distribution of the ratio of {\nu}2 to {\nu}1 clusters around the value of <{\nu}2/{\nu}1> ~ 3 : 2, which shows no obvious correlation with NS spins.

Constraints on the disc–magnetosphere interaction in accreting pulsar 4U 1626−67

Using the spin and flux evolution of the accreting pulsar 4U 1626$-$67 across the 2008 torque reversal, we determine the fastness parameter dependence of the dimensionless torque acting on the pulsar. We find that the dimensionless torque is qualitatively different from the existing models: it is concave-up across the torque equilibrium whereas the existing torque models predict a concave-down (convex) relation with the fastness parameter. We show that the dimensionless torque has a cubic dependence on the fastness parameter near the torque equilibrium. We also find that the torque can not attain large values away from the equilibrium, either in the positive or the negative side, but saturates at limited values. The spin-down torque can attain a 2.5 times larger magnitude at the saturation limit than the spin-up torque. From the evolution of the frequency of quasi-periodic oscillations of 4U 1626$-$67 across the torque reversal of 1990, we determine the critical fastness parameter corresponding to torque equilibrium to be $\omega_{\rm c} \simeq 0.75$ within the framework of the beat frequency model and boundary region model for reasonable values of the model parameters. We find that the disc magnetosphere interaction becomes unstable when the inner radius approaches the corotation radius as predicted by some models, though with a longer timescale. We also find that there is an unstable regime that is triggered when the fastness parameter is 0.8 times the critical fastness parameter ($\omega=0.6$ for $\omega_{\rm c} \simeq 0.75$) possibly associated with an instability observed in numerical simulations.

Utilizing the Cortico-Striatal Projectome to Advance the Study of Timing and Time Perception

Over the past decade advances in tracing and imaging techniques have spurred the development of increasingly detailed maps of brain connectivity. Broadly termed ‘connectomes’, these maps provide a powerful tool for systems neuroscience. As with most ‘maps’, connectomes offer a static spatial description of the brain’s circuits, whereas timing and temporal processing are inherently dynamic processes; nevertheless, the timing field stands to be a major beneficiary of these large-scale mapping projects. The recently reported ‘projectome’ of mouse cortico-striatal sub-networks is of particular interest because theoretical developments such as the striatal beat-frequency model emphasize the role of the striatum in temporal processing. The cortico-striatal projectome confirms that the dorsal striatum is ideally situated to sample patterns of activity throughout most of the cortex, but that it also contains a level of modularity previously not considered by integrative models of interval timing. Furthermore, the striatal projectome will allow for targeted studies of whether specific subdivisions of the dorsal striatum are differentially involved in timing and time perception as a function of task, stimulus modality, intensity, and valence.

Clock Speed as a Window into Dopaminergic Control of Emotion and Time Perception

Although fear-producing treatments (e.g., electric shock) and pleasure-inducing treatments (e.g., methamphetamine) have different emotional valences, they both produce physiological arousal and lead to effects on timing and time perception that have been interpreted as reflecting an increase in speed of an internal clock. In this commentary, we review the results reported by Fayolle et al. (2015):Behav. Process., 120, 135–140) and Meck (1983: J. Exp. Psychol. Anim. Behav. Process., 9, 171–201) using electric shock and by Maricq et al. (1981: J. Exp. Psychol. Anim. Behav. Process., 7, 18–30) using methamphetamine in a duration-bisection procedure across multiple duration ranges. The psychometric functions obtained from this procedure relate the proportion ‘long’ responses to signal durations spaced between a pair of ‘short’ and ‘long’ anchor durations. Horizontal shifts in these functions can be described in terms of attention or arousal processes depending upon whether they are a fixed number of seconds independent of the timed durations (additive) or proportional to the durations being timed (multiplicative). Multiplicative effects are thought to result from a change in clock speed that is regulated by dopamine activity in the medial prefrontal cortex. These dopaminergic effects are discussed within the context of the striatal beat frequency model of interval timing (Matell &amp; Meck, 2004:Cogn. Brain Res.,21, 139–170) and clinical implications for the effects of emotional reactivity on temporal cognition (Parker et al., 2013:Front. Integr. Neurosci., 7, 75).

Effect of Temporal Frequency Spectra of Flicker on Time Perception: Behavioral Testing and Simulations Using a Striatal Beat Frequency Model

When a visually presented stimulus flickers, the perceived stimulus duration exceeds the actual duration. This effect is called ‘time dilation’. On the basis of recent electrophysiological findings, we hypothesized that this flicker induced time dilation is caused by distortions of the internal clock, which is composed of many oscillators with many intrinsic vibration frequencies. To examine this hypothesis, we conducted behavioral experiments and a neural simulation. In the behavioral experiments, we measured flicker induced time dilation at various flicker frequencies. The stimulus was either a steadily presented patch or a flickering patch. The temporal frequency spectrum of the flickering patch was either single peaked at 10.9, 15, or 30 Hz, peaked with a narrow band at 8–12 or 12–16 Hz, or peaked with broad band at 4–30 Hz. Time dilation was observed with 10.9 Hz, 15 Hz, 30 Hz, or 8–12 Hz flickers, but not with 12–16 Hz or 4–30 Hz flickers. These results indicate that both the peak frequency and the width of the frequency distribution contribute to time dilation. To explain our behavioral results in the context of a physiological model, we proposed a model that combined the Striatal Beat Frequency Model and neural entrainment. The simulation successfully predicted the effect of flicker frequency locality and frequency specificity on time dilation, as observed in the behavioral experiments.

Noise signature on interval timing

Among other essential adaptations is the capability of organisms to estimate durations in the seconds-to-hours range (interval timing). Such capabilities are critical for fundamental cognitive processes like decision making, rate calculation, and planning of action [1]. In their seminal work on computational modeling of interval timing, Matell and Meck [2] revitalized the striatal beat frequency (SBF) model that utilizes the coincident activation of a series of oscillators to code for different durations. They showed through numerical simulations that the SBF model is capable of reproducing two of the interval timing signatures: (a) precise timing, i.e. the model output peaks at the training/criterion time and (b) scalar timing, i.e. the error in timing increases linearly with the criterion time. The SBF model was capable of producing scalar timing only when different types of biologically realistic variances (frequency, memory, etc.) were considered [3]. In this work, we investigated what effect each type of variance/noise has on the shape of the SBF model output. In particular, we noticed that the experimentally measured response rate is not quite Gaussian (see Figure ​Figure1A)1A) and instead has a long tail. Mathematically, the output function of set of coincidental (sinusoidal) oscillators is given by [3]: Figure 1 A. Normalized response for mice (squares - redrawn after [4]) versus normalized criterion time and the corresponding best Gaussian fit (continuous line). B. Theoretical output from SBF model with different frequency variances. out(t)=∑cos(2πfiT)cos(2πfit), where T is the criterion time, fi are the frequencies of neural oscillators. The criterion time T is learned during the training phase and stored/retrieved from the long-term memory with some errors. The firing frequencies fi of all neural oscillators also fluctuate. We found that memory variance (sT ) preserves the Gaussian shape of the output function, whereas the frequency variance (sf ) skews and has a long tail similar to experimental observations (see Figure ​Figure1B1B). In addition to the significantly difference contributions to the shape of output functions, we also found that memory and frequency noises shift the peak of the Gaussian. Memory noise affects the storage/retrieval of criterion T and shifts the peak of the output to the right, i.e. tT = T(1+T), where T is a number that depends on the range of stored values of T and the probability distribution function (pdfT ) of the noise. Frequency variance shifts the peak of the output to the left, i.e. tf = T/(1+f), where f is a number that depended on the range of frequencies and the pdff of the noise.

Dopamine-dependent oscillations in frontal cortex index "start-gun" signal in interval timing.

Although perceiving the passage of time is a basic building block of cognitive processes and behavior such as expecting relevant events to happen, the neural underpinnings of interval timing are not well-understood as yet (van Wassenhove, 2009; Allman and Meck, 2012; Merchant et al., 2013). From the neurobiological point of view, it has been established that dopamine impacts interval timing (e.g., Meck, 1986, 1996; Allman and Meck, 2012). However, a link between pharmacological manipulations and their impact on neurophysiological signals has been rarely investigated. The leading neurobiologically plausible model of interval timing that considers both components is the Striatal Beat Frequency (SBF) model (Mattel and Meck, 2004; Buhusi and Meck, 2005; van Rijn et al., 2014). The SBF model relies on the neuromodulatory dynamics of the thalamo-cortico-striatal loops. Although currently most of the interactions in the SBF model are assumed to be unidirectional, Mattel and Meck (2004) also acknowledged the possibility of feedback from the cortex to the neurons in the VTA as well as from striatal neurons to both the cortex and the substantia nigra pars compacta. These potential feedback mechanisms are unaddressed in the SBF model. However, their implementation would allow for more accurate description of clock speed and memory updating mechanisms on a trial to trial basis (W. Meck, personal communication, May 15, 2015). Nevertheless, the SBF assumes that time is coded by the coincidental activation of striatal spiny neurons with cortical oscillators (CO). Numerical implementations of the SBF model utilizes the phase, or amplitude, of the CO that are envisioned to oscillate at various frequencies giving rise to different amplitude patterns over time as illustrated in Figure ​Figure1.1. Hence, at a given time point a specific amplitude pattern of the CO can be encoded by striatal spiny neurons. Crucially, the SBF model assumes that, at the onset of the to-be-timed interval, the phases of CO are reset by a burst of dopaminergic input from the ventral tegmental area (VTA, Mattel and Meck, 2004). Further, the SBF model contends that the initial dopamine-triggered phase-resetting of CO by VTA plays the role of a “start-gun” that initiates timing. This “start-gun” signal forces a whole set of CO to start from the same phase, allowing for coincidence detection to read the state of CO, and code for a particular duration over multiple trials. Importantly, the more accurate the phase-reset at the onset of the to-be-timed interval—that is, the proportion of CO that is reset to the same phase—the larger the phase synchronization and the oscillatory power of ongoing oscillations (Canavier, 2015), and as such reduces variability in memory representation of to-be-timed interval (see Figure ​Figure1;1; Ng et al., 2011). Within this framework more precise phase reset should be associated with an increase in timing accuracy. Note that timing accuracy can be seen as a peak latency of a response distribution, or kurtosis of a response distribution. According to the SBF model the peak latency and kurtosis of response distribution can be accounted for by different features of the model. The peak latency is modulated by frequency range of CO whereas kurtosis is accounted for by accuracy of initial phase reset (Oprisan and Buhusi, 2014). As such the width shows consistency of memory representation estimated over a number of trials. What is referred to here is the accuracy as the width of the response distribution that is associated to initial reset in terms of the SBF model. Therefore, the width of the response distribution should be affected by consistency of phase reset. However, such covariation between the precision of the “start-gun” represented by the modulation of oscillatory synchrony in any neural population and timing accuracy has not yet been tested directly.

Endocannabinoid-Dependent Modulation of Phasic Dopamine Signaling Encodes External and Internal Reward-Predictive Cues

The mesolimbic dopamine (DA) system plays an integral role in incentive motivation and reward seeking and a growing body of evidence identifies signal transduction at cannabinoid receptors as a critical modulator of this system. Indeed, administration of exogenous cannabinoids results in burst firing of DA neurons of the ventral tegmental area and increases extracellular DA in the nucleus accumbens (NAcc). Implementation of fast-scan cyclic voltammetry (FSCV) confirms the ability of cannabinoids to augment DA within the NAcc on a subsecond timescale. The use of FSCV along with newly developed highly selective pharmacological compounds advances our understanding of how cannabinoids influence DA transmission and highlights a role for endocannabinoid-modulated subsecond DAergic activation in the incentive motivational properties of not only external, but also internal reward-predictive cues. For example, our laboratory has recently demonstrated that in mice responding under a fixed-interval (FI) schedule for food reinforcement, fluctuations in NAcc DA signal the principal cue predictive of reinforcer availability – time. That is, as the interval progresses, NAcc DA levels decline leading to accelerated food seeking and the resulting characteristic FI scallop pattern of responding. Importantly, administration of WIN 55,212-2, a synthetic cannabinoid agonist, or JZL184, an indirect cannabinoid agonist, increases DA levels during the interval and disrupts this pattern of responding. Along with a wealth of other reports, these results illustrate the role of cannabinoid receptor activation in the regulation of DA transmission and the control of temporally guided reward seeking. The current review will explore the striatal beat frequency model of interval timing as it pertains to cannabinoid signaling and propose a neurocircuitry through which this system modulates interoceptive time cues.

What is the effect of noise on the interval timing neural network?

Cognitive processes, such as decision making, rate calculation and planning, are strongly affected by the ability of subjects to perceive durations in the seconds-to-hours range (interval timing) [1]. A classic interval timing paradigm is the peak interval (PI) procedure which consists of interspersed reinforced and probe trials. During the reinforced trials, a stimulus such as a tone or light is turned on to signal the beginning of the to-be-timed interval and the subject’s first response after the criterion time is reinforced. During the probe trials, no reinforcement is given and the stimulus remains on for about three times the criterion time [2]. The mean response rate over a very large number of trials has a Gaussian shape whose peak measures the accuracy of criterion time estimation and the spread of the timing function measures its precision. In the vast majority of species, protocols, and manipulations to date, interval timing is both accurate and time-scale invariant, i.e. time-estimation errors increase linearly with the estimated duration [3]. We used a computational model of interval timing that mimics the activity of cortico-striatal structures involved in interval timing known as striatal beat frequency (SBF) model (Figure ​(Figure1)1) [4]. In mammals, administration of DA agonist, e.g., methamphetamine or cocaine produce an immediate, scalar (proportional), leftward shift in perceived time (responding earlier in time than under control conditions) whereas DA antagonist, e.g., haloperidol, has an opposed effect. Upon chronic administration of DA drugs, the timing functions recalibrate, i.e., they shift back to the values prior to drug administration. Once the drug is discontinued, the timing functions rebound (in a scalar manner) in the opposite direction from the initial effects of the drug, which is a signature of the clock pattern. SBF model can mimic clock patterns by adjusting the firing rates of FC neurons proportional to dopamine levels. The frequency modulation by dopaminergic drugs also produces a Gaussian-like output in SBF model. However, behavioral experiments indicate that although the average responses in PI procedure follow an almost Gaussian curve, there is always a notable skewness and a long tail of the output function. We found that frequency variability (noise) determines the deviation form the Gaussian-like output. Figure 1 (A) Schematic representation of the two main areas involved in interval timing: frontal cortex (FC) and basal ganglia (BG). (B) SBF model consists of a set of cortical oscillators, a reference memory that stores the state of the system at reinforcement ...

Why noise is useful in functional and neural mechanisms of interval timing?

BackgroundThe ability to estimate durations in the seconds-to-minutes range - interval timing - is essential for survival, adaptation and its impairment leads to severe cognitive and/or motor dysfunctions. The response rate near a memorized duration has a Gaussian shape centered on the to-be-timed interval (criterion time). The width of the Gaussian-like distribution of responses increases linearly with the criterion time, i.e., interval timing obeys the scalar property.ResultsWe presented analytical and numerical results based on the striatal beat frequency (SBF) model showing that parameter variability (noise) mimics behavioral data. A key functional block of the SBF model is the set of oscillators that provide the time base for the entire timing network. The implementation of the oscillators block as simplified phase (cosine) oscillators has the additional advantage that is analytically tractable. We also checked numerically that the scalar property emerges in the presence of memory variability by using biophysically realistic Morris-Lecar oscillators. First, we predicted analytically and tested numerically that in a noise-free SBF model the output function could be approximated by a Gaussian. However, in a noise-free SBF model the width of the Gaussian envelope is independent of the criterion time, which violates the scalar property. We showed analytically and verified numerically that small fluctuations of the memorized criterion time leads to scalar property of interval timing.ConclusionsNoise is ubiquitous in the form of small fluctuations of intrinsic frequencies of the neural oscillators, the errors in recording/retrieving stored information related to criterion time, fluctuation in neurotransmitters’ concentration, etc. Our model suggests that the biological noise plays an essential functional role in the SBF interval timing.

Interval timing behavior in Pallas’s long-tongued bat (Glossophaga soricina).

Timing behavior in animals and its underlying mechanisms have been investigated extensively in the peak procedure, a variant of fixed interval procedures. In such experiments, individuals typically start responding with high frequency after an initial inactive time interval and continue their responses after peak time if rewards are omitted. This begs the so far unexplored question as to how timing behavior is influenced when such continuous responses are suppressed. Here, we present results from a nectar-feeding bat species, Glossophaga soricina, that was tested in a modified version of the peak procedure at three fixed time intervals (5 s, 11 s, 20 s). In contrast to standard peak procedures we imposed metabolic costs on individual responses which effectively suppressed trains of rapid responses during trials. Under this manipulation, bats' aggregated responses showed clear peaks around the peak time in the 5-s and 11-s schedules. Bats' responses in the 20-s schedule, however, did not peak around the fixed interval time. Crucially, an analysis of time intervals between successive revisits in all schedules revealed that bats revisited feeders at accurately timed intervals in all three conditions. The individual within trial behavioral responses showed clear oscillatory patterns throughout nonrewarded trials. These findings follow predictions from mechanistic timing models, like the striatal beat frequency model, and are discussed with regard to these models.