Fractal Stochastic Modeling of Spiking Activity in Suprachiasmatic Nucleus Neurons

The Suprachiasmatic Nucleus at 50: Looking Back, Then Looking Forward

It has been 50 years since the suprachiasmatic nucleus (SCN) was first identified as the central circadian clock and 25 years since the last overview of developments in the field was published in the Journal of Biological Rhythms. Here, we explore new mechanisms and concepts that have emerged in the subsequent 25 years. Since 1997, methodological developments, such as luminescent and fluorescent reporter techniques, have revealed intricate relationships between cellular and network-level mechanisms. In particular, specific neuropeptides such as arginine vasopressin, vasoactive intestinal peptide, and gastrin-releasing peptide have been identified as key players in the synchronization of cellular circadian rhythms within the SCN. The discovery of multiple oscillators governing behavioral and physiological rhythms has significantly advanced our understanding of the circadian clock. The interaction between neurons and glial cells has been found to play a crucial role in regulating these circadian rhythms within the SCN. Furthermore, the properties of the SCN network vary across ontogenetic stages. The application of cell type-specific genetic manipulations has revealed components of the functional input-output system of the SCN and their correlation with physiological functions. This review concludes with the high-risk effort of identifying open questions and challenges that lie ahead.

Effects of transient subordinators on the firing statistics of a neuron model driven by dichotomous noise

The behavior of a stochastic perfect integrate-and-fire (PIF) model of neurons is considered. The effect of temporally correlated random activity of synaptic inputs is modeled as a combination of an asymmetric dichotomous noise and a random operation time in the form of an inverse strictly increasing Lévy-type subordinator. Using a first-passage-time formulation, we find exact expressions for the output interspike interval (ISI) statistics. Particularly, it is shown that at some parameter regimes the ISI density exhibits a multimodal structure. Moreover, it is demonstrated that the coefficient of variation, the serial correlation coefficient, and the Fano factor display a nonmonotonic dependence on the mean input current μ, i.e., the ISI's regularity is maximized at an intermediate value of μ. The features of spike statistics, analytically revealed in our study, are compared with previously obtained results for a perfect integrate-and-fire neuron model driven by dichotomous noise (without subordination).

Traits of criticality in membrane potential fluctuations of pyramidal neurons in the CA1 region of rat hippocampus

Evidence that neural circuits are operating near criticality has been provided at various levels of brain organisation with a presumed role in maximising information processing and multiscale activity association. Criticality has been linked to excitation at both the single-cell and network levels, as action potential generation marks an obvious phase transition from a resting to an excitable state. Using in vitro intracellular recordings, we examine irregular, small amplitude membrane potential fluctuations from CA1 pyramidal neurons of Wistar male rats. We show that these fluctuations, confounded with noise, carry information relevant to the neuronal state. The underlying dynamics exhibit intermittent characteristics shown to describe the temporal fluctuations of the order parameter of a macroscopic system at its critical point even in the absence of firing. An externally applied stimulus serves as the control parameter, driving the system in and out of its critical state. Based on our experimental observations we calculate the equivalent of the isothermal critical exponent δ_h finding a value which depends on the applied stimulus. For each neuron there is a stimulus amplitude for which the critical behaviour becomes most pronounced. The corresponding mean value of δ_h in the considered ensemble of neurons is δ_h  ≈ 1.89, close to theoretical predictions for critical networks. Finally, we show that the firing rate of a neuron decreases exponentially with δ_h .

Ensemble of LIF neurons with random membrane decay constant: emergence of power-law behavior in ISI distribution

A theoretical framework is proposed to explain the emergence of power law behavior in the spiking neurons where the membrane decay constant is assumed to vary randomly across the population of neurons. The proposed approach, akin to superstatistics, provides a plausible mechanism for generating power law behavior in non-equilibrium systems with fluctuations in intensive quantity. This approach has led to formulation of a hypothesis that power law behavior in inter-spike interval (ISI) distribution results when several neurons group together and fire together. In presence of fluctuations in membrane decay constant governed by gamma distribution, the asymptotic analysis of ISI distribution yields a power law. This finding has been corroborated by simulation study when different types of probability distributions for membrane decay constant are considered. Our results are in agreement with the empirical findings due to Kim (2004) where the spiking trains of Suprachiasmatic nucleus (SCN) neurons display power law behavior. Further investigations in subthreshold regime reveals that the averaging of the membrane potential over a large number of neurons also yields power law.

Irregular activity arises as a natural consequence of synaptic inhibition

Irregular neuronal activity is observed in a variety of brain regions and states. This work illustrates a novel mechanism by which irregular activity naturally emerges in two-cell neuronal networks featuring coupling by synaptic inhibition. We introduce a one-dimensional map that captures the irregular activity occurring in our simulations of conductance-based differential equations and mathematically analyze the instability of fixed points corresponding to synchronous and antiphase spiking for this map. We find that the irregular solutions that arise exhibit expansion, contraction, and folding in phase space, as expected in chaotic dynamics. Our analysis shows that these features are produced from the interplay of synaptic inhibition with sodium, potassium, and leak currents in a conductance-based framework and provides precise conditions on parameters that ensure that irregular activity will occur. In particular, the temporal details of spiking dynamics must be present for a model to exhibit this irregularity mechanism and must be considered analytically to capture these effects.

Fractal patterns of neural activity exist within the suprachiasmatic nucleus and require extrinsic network interactions

The mammalian central circadian pacemaker (the suprachiasmatic nucleus, SCN) contains thousands of neurons that are coupled through a complex network of interactions. In addition to the established role of the SCN in generating rhythms of ~24 hours in many physiological functions, the SCN was recently shown to be necessary for normal self-similar/fractal organization of motor activity and heart rate over a wide range of time scales--from minutes to 24 hours. To test whether the neural network within the SCN is sufficient to generate such fractal patterns, we studied multi-unit neural activity of in vivo and in vitro SCNs in rodents. In vivo SCN-neural activity exhibited fractal patterns that are virtually identical in mice and rats and are similar to those in motor activity at time scales from minutes up to 10 hours. In addition, these patterns remained unchanged when the main afferent signal to the SCN, namely light, was removed. However, the fractal patterns of SCN-neural activity are not autonomous within the SCN as these patterns completely broke down in the isolated in vitro SCN despite persistence of circadian rhythmicity. Thus, SCN-neural activity is fractal in the intact organism and these fractal patterns require network interactions between the SCN and extra-SCN nodes. Such a fractal control network could underlie the fractal regulation observed in many physiological functions that involve the SCN, including motor control and heart rate regulation.

Spike firing pattern of output neurons of the Limulus circadian clock

The lateral eyes of the horseshoe crab (Limulus polyphemus) show a daily rhythm in visual sensitivity that is mediated by efferent nerve signals from a circadian clock in the crab's brain. How these signals communicate circadian messages is not known for this or other animals. Here the authors describe in quantitative detail the spike firing pattern of clock output neurons in living horseshoe crabs and discuss its possible significance to clock organization and function. Efferent fiber spike trains were recorded extracellularly for several hours to days, and in some cases, the electroretinogram was simultaneously acquired to monitor eye sensitivity. Statistical features of single- and multifiber recordings were characterized via interval distribution, serial correlation, and power spectral analysis. The authors report that efferent feedback to the eyes has several scales of temporal structure, consisting of multicellular bursts of spikes that group into clusters and packets of clusters that repeat throughout the night and disappear during the day. Except near dusk and dawn, the bursts occur every 1 to 2 sec in clusters of 10 to 30 bursts separated by a minute or two of silence. Within a burst, each output neuron typically fires a single spike with a preferred order, and intervals between bursts and clusters are positively correlated in length. The authors also report that efferent activity is strongly modulated by light at night and that just a brief flash has lasting impact on clock output. The multilayered firing pattern is likely important for driving circadian rhythms in the eye and other target organs.

Fractal symmetry of protein interior: what have we learned?

The application of fractal dimension-based constructs to probe the protein interior dates back to the development of the concept of fractal dimension itself. Numerous approaches have been tried and tested over a course of (almost) 30 years with the aim of elucidating the various facets of symmetry of self-similarity prevalent in the protein interior. In the last 5 years especially, there has been a startling upsurge of research that innovatively stretches the limits of fractal-based studies to present an array of unexpected results on the biophysical properties of protein interior. In this article, we introduce readers to the fundamentals of fractals, reviewing the commonality (and the lack of it) between these approaches before exploring the patterns in the results that they produced. Clustering the approaches in major schools of protein self-similarity studies, we describe the evolution of fractal dimension-based methodologies. The genealogy of approaches (and results) presented here portrays a clear picture of the contemporary state of fractal-based studies in the context of the protein interior. To underline the utility of fractal dimension-based measures further, we have performed a correlation dimension analysis on all of the available non-redundant protein structures, both at the level of an individual protein and at the level of structural domains. In this investigation, we were able to separately quantify the self-similar symmetries in spatial correlation patterns amongst peptide-dipole units, charged amino acids, residues with the π-electron cloud and hydrophobic amino acids. The results revealed that electrostatic environments in the interiors of proteins belonging to 'α/α toroid' (all-α class) and 'PLP-dependent transferase-like' domains (α/β class) are highly conducive. In contrast, the interiors of 'zinc finger design' ('designed proteins') and 'knottins' ('small proteins') were identified as folds with the least conducive electrostatic environments. The fold 'conotoxins' (peptides) could be unambiguously identified as one type with the least stability. The same analyses revealed that peptide-dipoles in the α/β class of proteins, in general, are more correlated to each other than are the peptide-dipoles in proteins belonging to the all-α class. Highly favorable electrostatic milieu in the interiors of TIM-barrel, α/β-hydrolase structures could explain their remarkably conserved (evolutionary) stability from a new light. Finally, we point out certain inherent limitations of fractal constructs before attempting to identify the areas and problems where the implementation of fractal dimension-based constructs can be of paramount help to unearth latent information on protein structural properties.

Network-state modulation of power-law frequency-scaling in visual cortical neurons

Various types of neural-based signals, such as EEG, local field potentials and intracellular synaptic potentials, integrate multiple sources of activity distributed across large assemblies. They have in common a power-law frequency-scaling structure at high frequencies, but it is still unclear whether this scaling property is dominated by intrinsic neuronal properties or by network activity. The latter case is particularly interesting because if frequency-scaling reflects the network state it could be used to characterize the functional impact of the connectivity. In intracellularly recorded neurons of cat primary visual cortex in vivo, the power spectral density of V_m activity displays a power-law structure at high frequencies with a fractional scaling exponent. We show that this exponent is not constant, but depends on the visual statistics used to drive the network. To investigate the determinants of this frequency-scaling, we considered a generic recurrent model of cortex receiving a retinotopically organized external input. Similarly to the in vivo case, our in computo simulations show that the scaling exponent reflects the correlation level imposed in the input. This systematic dependence was also replicated at the single cell level, by controlling independently, in a parametric way, the strength and the temporal decay of the pairwise correlation between presynaptic inputs. This last model was implemented in vitro by imposing the correlation control in artificial presynaptic spike trains through dynamic-clamp techniques. These in vitro manipulations induced a modulation of the scaling exponent, similar to that observed in vivo and predicted in computo. We conclude that the frequency-scaling exponent of the V_m reflects stimulus-driven correlations in the cortical network activity. Therefore, we propose that the scaling exponent could be used to read-out the “effective” connectivity responsible for the dynamical signature of the population signals measured at different integration levels, from Vm to LFP, EEG and fMRI.

Intracellular recording of neocortical neurons provides an opportunity of characterizing the statistical signature of the synaptic bombardment to which it is submitted. Indeed the membrane potential displays intense fluctuations which reflect the cumulative activity of thousands of input neurons. In sensory cortical areas, this measure could be used to estimate the correlational structure of the external drive. We show that changes in the statistical properties of network activity, namely the local correlation between neurons, can be detected by analyzing the power spectrum density (PSD) of the subthreshold membrane potential. These PSD can be fitted by a power-law function 1/f^α in the upper temporal frequency range. In vivo recordings in primary visual cortex show that the α exponent varies with the statistics of the sensory input. Most remarkably, the exponent observed in the ongoing activity is indistinguishable from that evoked by natural visual statistics. These results are emulated by models which demonstrate that the exponent α is determined by the local level of correlation imposed in the recurrent network activity. Similar relationships are also reproduced in cortical neurons recorded in vitro with artificial synaptic inputs by controlling in computo the level of correlation in real time.

Constructing 1/omegaalpha noise from reversible Markov chains

This paper gives sufficient conditions for the output of 1/omegaalpha noise from reversible Markov chains on finite state spaces. We construct several examples exhibiting this behavior in a specified range of frequencies. We apply simple representations of the covariance function and the spectral density in terms of the eigendecomposition of the probability transition matrix. The results extend to hidden Markov chains. We generalize the results for aggregations of AR1-processes of C. W. J. Granger [J. Econometrics 14, 227 (1980)]. Given the eigenvalue function, there is a variety of ways to assign values to the states such that the 1/omegaalpha condition is satisfied. We show that a random walk on a certain state space is complementary to the point process model of 1/omega noise of B. Kaulakys and T. Meskauskas [Phys. Rev. E 58, 7013 (1998)]. Passing to a continuous state space, we construct 1/omegaalpha noise which also has a long memory.

Fractal Noises and Motions in Time Series of Presympathetic and Sympathetic Neural Activities

We used Allan factor analysis to classify time series of the discharges of single presympathetic neurons in the cat medullary lateral tegmental field (LTF) and rostral ventrolateral medulla (RVLM) and of the postganglionic vertebral sympathetic nerve. These time series fell into two classes of fractal-based point processes characterized by statistically self-similar behavior reflecting long-range correlations among data points. Classification of a time series as either a fractional Gaussian noise (fGn)–or fractional Brownian motion (fBm)–based point process depended on the scaling exponent, α, of the power law in the Allan factor curve. fGn is defined as 0 < α < 1 and fBm as 1 < α < 3. The process responsible for the fractal spike trains of 11 of 12 classifiable LTF neurons with sympathetic nerve-related activity was fGn. In contrast, the process responsible for the fractal spike trains of eight of nine classifiable RVLM presympathetic neurons was fBm. The time series of simultaneously recorded vertebral sympathetic nerve discharge and the arterial pulse also were fBm-based signals. Because a fBm signal is the cumulative sum of the elements comprising the corresponding fGn signal, these results show smoothing of fractal time series in a feedforward direction from medullary presympathetic neurons to postganglionic sympathetic neurons. This may involve integration by RVLM neurons of their LTF inputs or independent fractal processes acting at different levels of the network controlling sympathetic nerve discharge. Whether feedforward smoothing of fractal signals is a feature in other neural systems is open to investigation.