Detecting effective connectivity in networks of coupled neuronal oscillators

The application of data-driven time series analysis techniques such as Granger causality, partial directed coherence and phase dynamics modeling to estimate effective connectivity in brain networks has recently gained significant prominence in the neuroscience community. While these techniques have been useful in determining causal interactions among different regions of brain networks, a thorough analysis of the comparative accuracy and robustness of these methods in identifying patterns of effective connectivity among brain networks is still lacking. In this paper, we systematically address this issue within the context of simple networks of coupled spiking neurons. Specifically, we develop a method to assess the ability of various effective connectivity measures to accurately determine the true effective connectivity of a given neuronal network. Our method is based on decision tree classifiers which are trained using several time series features that can be observed solely from experimentally recorded data. We show that the classifiers constructed in this work provide a general framework for determining whether a particular effective connectivity measure is likely to produce incorrect results when applied to a dataset.     

Coupled oscillators and activity waves in ant colonies

We investigated the phenomenon of activity cycles in ants, taking into account the spatial structure of colonies. In our study species, Leptothorax acervorum, there are two spatially segregated groups in the nest. We developed a model that considers the two groups as coupled oscillators which can produce synchronized activity. By investigating the effects of noise on the model system we predicted how the return of foragers affects activity cycles in ant colonies. We tested these predictions empirically by comparing the activity of colonies under two conditions: when foragers are and are not allowed to return to the nest. The activity of the whole colony and of each group within the colony was studied using image analysis. This allowed us to reveal the spatial pattern of activity wave propagation in ant colonies for the first time.     

Synchronization in a pool of mutually coupled oscillators with random frequencies

An exact solution to a model of mutually interacting sinusoidal oscillators is found. Limits on the variation of the native frequencies are determined in order for synchronization to occur. These limits are computed for different distributions of native frequencies.This research was supported by NSF Award No. MCS8300885 and the Alfred Sloan Foundation.     

Dynamics and bifurcations of two coupled neural oscillators with different connection types

In this paper we present an oscillatory neural network composed of two coupled neural oscillators of the Wilson-Cowan type. Each of the oscillators describes the dynamics of average activities of excitatory and inhibitory populations of neurons. The network serves as a model for several possible network architectures. We study how the type and the strength of the connections between the oscillators affect the dynamics of the neural network. We investigate, separately from each other, four possible connection types (excitatory→excitatory, excitatory→inhibitory, inhibitory→excitatory, and inhibitory→inhibitory) and compute the corresponding bifurcation diagrams. In case of weak connections (small strength), the connection of populations of different types lead to periodicin-phase oscillations, while the connection of populations of the same type lead to periodicanti-phase oscillations. For intermediate connection strengths, the networks can enter quasiperiodic or chaotic regimes, and can also exhibit multistability. More generally, our analysis highlights the great diversity of the response of neural networks to a change of the connection strength, for different connection architectures. In the discussion, we address in particular the problem of information coding in the brain using quasiperiodic and chaotic oscillations. In modeling low levels of information processing, we propose that feature binding should be sought as a temporally coherent phase-locking of neural activity. This phase-locking is provided by one or more interacting convergent zones and does not require a central “top level” subcortical circuit (e.g. the septo-hippocampal system). We build a two layer model to show that although the application of a complex stimulus usually leads to different convergent zones with high frequency oscillations, it is nevertheless possible to synchronize these oscillations at a lower frequency level using envelope oscillations. This is interpreted as a feature binding of a complex stimulus.     

Link between truncated fractals and coupled oscillators in biological systems

This article aims at providing a new theoretical insight into the fundamental question of the origin of truncated fractals in biological systems. It is well known that fractal geometry is one of the characteristics of living organisms. However, contrary to mathematical fractals which are self-similar at all scales, the biological fractals are truncated, i.e. their self-similarity extends at most over a few orders of magnitude of separation. We show that nonlinear coupled oscillators, modeling one of the basic features of biological systems, may generate truncated fractals: a truncated fractal pattern for basin boundaries appears in a simple mathematical model of two coupled nonlinear oscillators with weak dissipation. This fractal pattern can be considered as a particular hidden fractal property. At the level of sufficiently fine precision technique the truncated fractality acts as a simple structure, leading to predictability, but at a lower level of precision it is effectively fractal, limiting the predictability of the long-term behavior of biological systems. We point out to the generic nature of our result.     

Lung and buccal ventilation in the frog: uncoupling coupled oscillators

The frog, with two distinct ventilatory acts, provides a useful model to investigate the prospective interaction of two oscillators in generating the respiratory rhythm. Building on evidence supporting the existence of separate oscillators generating buccal and lung ventilation, we have attempted to uncouple the two rhythms in the isolated brain stem preparation. Opioid preferentially inhibits the lung rhythm, suggesting an uncoupling of the lung from the buccal oscillator. Reduction of the superfusate chloride concentration alters both the buccal and the lung rhythms. Joint application of opioid and reduced-chloride superfusate leads to an increase in the variability of the buccal burst-to-lung burst intervals. This increase in variability suggests that chloride-mediated mechanisms are involved in coupling the buccal oscillator to the lung oscillator. Given the results from these interventions, we propose a simple schematic model of the frog respiratory rhythm generator, outlining the coupling of the lung and buccal oscillators.     

Multiple pulse interactions and averaging in systems of coupled neural oscillators

Oscillators coupled strongly are capable of complicated behavior which may be pathological for biological control systems. Nevertheless, strong coupling may be needed to prevent asynchrony. We discuss how some neural networks may be designed to achieve only simple locking behavior when the coupling is strong. The design is based on the fact that the method of averaging produces equations that are capable only of locking or drift, not pathological complexity. Furthermore, it is shown that oscillators that interact by means of multiple pulses per cycle, dispersed around the cycle, behave like averaged equations, even if the number of pulses is small. We discuss the biological intuition behind this scheme, and show numerically that it works when the oscillators are taken to be composites, each unit of which is governed by a well-known model of a neural oscillator. Finally, we describe numerical methods for computing from equations for coupled limit cycle oscillators the averaged coupling functions of our theory.Research partially supported by the National Science Foundation under grants DMS 8796235 and DMS 8701405 and the Air Force Office of Scientific Research under University Research Contract F 49620-C-0131 to Northeastern University     

Surface waves and space charge layers in a spatially inhomogeneous plasma

A theory of surface waves in a layer of a spatially inhomogeneous cold electron plasma is presented. Four types of surface waves are revealed, and the conditions under which they can exist are determined. Complex frequency spectra are obtained, and the mechanisms for wave damping by plasma inhomogeneity are discussed.     

Simulation of circadian rhythm generation in the suprachiasmatic nucleus with locally coupled self-sustained oscillators

In mammals, circadian rhythms are driven by a pacemaker located in the suprachiasmatic nuclei (SCN) of the anterior hypothalamus. The firing rate of neurons within the SCN exhibits a circadian rhythm. There is evidence that individual neurons within the SCN act as circadian oscillators. Rhythm generation in the SCN was therefore modeled by a system of self-sustained oscillators. The model is composed of up to 10000 oscillatory elements arranged in a square array. Each oscillator has its own (randomly determined) intrinsic period reflecting the widely dispersed periods observed in the SCN. The model behavior was investigated mainly in the absence of synchronizing zeitgebers. Due to local coupling the oscillators synchronized and an overall rhythm emerged. This indicates that a locally coupled system is capable of integrating the output of individual clock cells with widely dispersed periods. The period of the global output (average of all oscillators) corresponded to the average of the intrinsic periods and was stable even for small amplitudes and during transients. Noise, reflecting biological fluctuations at the cellular level, distorted the global rhythm in small arrays. The period of the rhythm could be stabilized by increasing the array size, which thus increased the robustness against noise. Since different regions of the SCN have separate output pathways, the array of oscillators was subdivided into four quadrants. Sudden deviations of periodicity sometimes appeared in one quadrant, while the periods of the other quadrants were largely unaffected. This result could represent a model for splitting, which has been observed in animal experiments. In summary, the multi-oscillator model of the SCN showed a broad repertoire of dynamic patterns, revealed a stable period (even during transients) with robustness against noise, and was able to account for such a complex physiological behavior as splitting.     

Subcellular calcium oscillators and calcium influx support agonist-induced calcium waves in cultured astrocytes

We have analysed Ca²⁺ waves induced by norepinephrine in rat cortical astrocytes in primary culture using fluorescent indicators fura-2 or fluo-3. The temporal pattern of the average [Ca²⁺]_i responses were heterogeneous from cell to cell and most cells showed an oscillatory response at concentrations of agonist around EC₅₀ (200 nM). Upon receptor activation, [Ca²⁺]_i signals originated from a single cellular locus and propagated throughout the cell as a wave. Wave propagation was supported by specialized regenerative calcium release loci along the length of the cell. The periods of oscillations, amplitudes, and the rates of [Ca²⁺]_i rise of these subcellular oscillators differ from each other. These intrinsic kinetic properties of the regenerative loci support local waves when stimulation is continued over long periods of time. The presence of local waves at specific, invariant cellular sites and their inherent kinetic properties provide for the unique and reproducible pattern of response seen in a given cell. We hypothesize that these loci are local specializations in the endoplasmic reticulum where the magnitude of the regenerative Ca²⁺ release is higher than other regions of the cell. Removal of extracellular Ca²⁺ or blockade of Ca²⁺ channels by inorganic cations (Cd²⁺ and Ni²⁺) during stimulation of adrenergic receptors alter the sustained plateau component of the [Ca²⁺]_i response. In the absence of Ca²⁺ release, due to store depletion with thapsigargin, agonist occupation alone does not induce Ca²⁺ influx in astrocytes. This finding suggests that, under these conditions, receptor-operated Ca²⁺ entry is not operative. Furthermore, our experiments provide evidence for local Ca²⁺ oscillations in cells which can support both wave propagation as well as spatially discrete Ca²⁺ signalling.     

Adaptive and phase selective spike timing dependent plasticity in synaptically coupled neuronal oscillators

We consider and analyze the influence of spike-timing dependent plasticity (STDP) on homeostatic states in synaptically coupled neuronal oscillators. In contrast to conventional models of STDP in which spike-timing affects weights of synaptic connections, we consider a model of STDP in which the time lags between pre- and/or post-synaptic spikes change internal state of pre- and/or post-synaptic neurons respectively. The analysis reveals that STDP processes of this type, modeled by a single ordinary differential equation, may ensure efficient, yet coarse, phase-locking of spikes in the system to a given reference phase. Precision of the phase locking, i.e. the amplitude of relative phase deviations from the reference, depends on the values of natural frequencies of oscillators and, additionally, on parameters of the STDP law. These deviations can be optimized by appropriate tuning of gains (i.e. sensitivity to spike-timing mismatches) of the STDP mechanism. However, as we demonstrate, such deviations can not be made arbitrarily small neither by mere tuning of STDP gains nor by adjusting synaptic weights. Thus if accurate phase-locking in the system is required then an additional tuning mechanism is generally needed. We found that adding a very simple adaptation dynamics in the form of slow fluctuations of the base line in the STDP mechanism enables accurate phase tuning in the system with arbitrary high precision. Adaptation operating at a slow time scale may be associated with extracellular matter such as matrix and glia. Thus the findings may suggest a possible role of the latter in regulating synaptic transmission in neuronal circuits.     

Robust synchronization of coupled circadian and cell cycle oscillators in single mammalian cells

Circadian cycles and cell cycles are two fundamental periodic processes with a period in the range of 1 day. Consequently, coupling between such cycles can lead to synchronization. Here, we estimated the mutual interactions between the two oscillators by time‐lapse imaging of single mammalian NIH3T3 fibroblasts during several days. The analysis of thousands of circadian cycles in dividing cells clearly indicated that both oscillators tick in a 1:1 mode‐locked state, with cell divisions occurring tightly 5 h before the peak in circadian Rev‐Erbα‐YFP reporter expression. In principle, such synchrony may be caused by either unidirectional or bidirectional coupling. While gating of cell division by the circadian cycle has been most studied, our data combined with stochastic modeling unambiguously show that the reverse coupling is predominant in NIH3T3 cells. Moreover, temperature, genetic, and pharmacological perturbations showed that the two interacting cellular oscillators adopt a synchronized state that is highly robust over a wide range of parameters. These findings have implications for circadian function in proliferative tissues, including epidermis, immune cells, and cancer.     

Differential impacts of habitat heterogeneity on male and female connectivity in a spatially structured pest system

In a previous study, a model of landscape heterogeneity was developed and applied to a spatially structured wild rabbit (Oryctolagus cuniculus) population. That study showed clearly the influence of resource heterogeneity on connectivity levels. The simulation study was based on female movements and used population genetic validation data appropriate for a female study. Most models assume that males and females will exhibit similar patterns, although this has rarely been tested. In the current study we extend the analysis to consider differences between female and male connectivity in the same spatially structured pest system. Amplified fragment length polymorphism (AFLP) markers were screened on the same samples used previously for mtDNA analysis. The mtDNA data were used to validate female results, and AFLP data were used to validate combined male and female results. Connectivity patterns from the two simulations (female, and combined male and female) connectivity patterns showed no association. However, each was concordant with appropriate validation data, showing highly significant associations between pairwise population connectivity and the genetic data. A relative connectivity metric for the combined simulation was regressed against the mean of pairwise Φ_ST values, with almost 70% of the variation explained by a linear model. Demonstrating differential effects of habitat heterogeneity on male and female connectivity provides further evidence that spatial resource heterogeneity impacts on connectivity. Understanding differences in population connectivity will allow improved predictions of disease spread, local extinctions and recolonizations. Furthermore, modelling such differences in pest systems will allow management plans to be better targeted, for example by strategically introducing diseases for control purposes into populations which exhibit high male connectivity to aid spread, but low female connectivity to inhibit recolonization potential after control.     

Dispersal, Environmental Correlation, and Spatial Synchrony in Population Dynamics

Many species exhibit widespread spatial synchrony in population fluctuations. This pattern is of great ecological interest and can be a source of concern when the species is rare or endangered. Both dispersal and spatial correlations in the environment have been implicated as possible causes of this pattern, but these two factors have rarely been studied in combination. We develop a spatially structured population model, simple enough to obtain analytic solutions for the population correlation, that incorporates both dispersal and environmental correlation. We ask whether these two synchronizing factors contribute additively to the total spatial population covariance. We find that there is always an interaction between these two factors and that this interaction is small only when one or both of the environmental correlation and the dispersal rate are small. The interaction is opposite in sign to the environmental correlation; so, in the normal case of positive environmental correlation across sites, the population synchrony will be lower than predicted by simply adding the effects of dispersal and environmental correlation. We also find that population synchrony declines as the strength of population regulation increases. These results indicate that dispersal and environmental correlation need to be considered in combination as explanations for observed patterns of population synchrony.     

Synchrony in human,mouse and bacterial cell cultures--a comparison

Growth characteristics of synchronous human MOLT-4, human U-937 and mouse L1210 cultures produced with a new minimally-disturbing technology were compared to each other and to synchronous Escherichia coli B/r. Based on measurements of cell concentrations during synchronous growth, synchrony persisted in similar fashion for all cells. Cell size and DNA distributions in the mammalian cultures also progressed synchronously and reproducibly for multiple cell cycles. The results demonstrate that unambiguous multi-cycle synchrony, critical for verifying the absence of significant growth imbalances induced by the synchronization procedure, is feasible with these cell lines, and possibly others.     

Calcium waves in a model with a random spatially discrete distribution of Ca2+ release sites

We study the propagation of intracellular calcium waves in a model that features Ca2+ release from discrete sites in the endoplasmic reticulum membrane and random spatial distribution of these sites. The results of our simulations qualitatively reproduce the experimentally observed behavior of the waves. When the level of the channel activator inositol trisphosphate is low, the wave undergoes fragmentation and eventually vanishes at a finite distance from the region of initiation, a phenomenon we refer to as an abortive wave. With increasing activator concentration, the mean distance of propagation increases. Above a critical level of activator, the wave becomes stable. We show that the heterogeneous distribution of Ca2+ channels is the cause of this phenomenon.     

Modeling circadian rhythm generation in the suprachiasmatic nucleus with locally coupled self-sustained oscillators: phase shifts and phase response curves

Circadian rhythm generation in the suprachiasmatic nucleus was modeled by locally coupled self-sustained oscillators. The model is composed of 10,000 oscillators, arranged in a square array. Coupling between oscillators and standard deviation of (randomly determined) intrinsic oscillator periods were varied. A stable overall rhythm emerged. The model behavior was investigated for phase shifts of a 24-h zeitgeber cycle. Prolongation of either the dark or the light phase resulted in a lengthening of the period, whereas shortening of the dark or the light phase shortened the period. The model's response to shifts in the light-dark cycle was dependent only on the extent of the shift and was insensitive to changes in parameters. Phase response curves (PRC) and amplitude response curves were determined for single and triple 5-h light pulses (1000 lux). Single pulses lead to type 1 PRCs with larger phase shifts for weak coupling. Triple pulses generally evoked type 1 PRCs with the exception of weak coupling, where a type 0 PRC was observed.     

Life-history evolution in spatially heterogeneous environments,with and without phenotypic plasticity

Summary The formulation of Kawecki and Stearns (1993) adapted for sexual populations is used to characterize lifehistory evolution in spatially heterogeneous environments comprising a number of distinct habitats. Three types of evolutionary outcome/optimal strategy are distinguished, appertaining to populations with phenotypic plasticity, populations without it (here called aplastic) and to populations that are reproductively isolated. In general plastic and isolated optima differ, but do not differ if none of the habitats provide source or sink populations. Plastic, aplastic and isolated optima are calculated and compared in three numerical examples representing trade-offs involving reproductive effort, egg size and defence. Locating the aplastic optimum involves numerical construction of a fitness landscape showing how allelic fitness depends on aplastic strategy and on the relative frequencies of the habitats. In all three examples the aplastic optimum lies between or almost between the plastic optima. In two cases the aplastic optimum switches abruptly between the plastic optima as the relative frequencies of the habitats change, and in the third case the switch is gradual. The abruptness or otherwise of the switch depends on the position and structure of the valleys in the fitness landscape and this in turn depends on how sharply the fitness peaks are differentiated.     

Synchrony in Human,Mouse and Bacterial Cell Cultures: A Comparison

Growth characteristics of synchronous human MOLT-4, human U-937 and mouse L1210 cultures produced with a new minimally-disturbing technology were compared to each other and to synchronous Escherichia coli B/r. Based on measurements of cell concentrations during synchronous growth, synchrony persisted in similar fashion for all cells. Cell size and DNA distributions in the mammalian cultures also progressed synchronously and reproducibly for multiple cell cycles. The results demonstrate that unambiguous multi-cycle synchrony, critical for verifying the absence of significant growth imbalances induced by the synchronization procedure, is feasible with these cell lines, and possibly others.     

Subtle genetic connectivity between Mexican Caribbean and south-western Gulf of Mexico reefs: the case of the bicolor damselfish,Stegastes partitus

Efficient reef management strategies rely on detailed knowledge of biological exchange dynamics. At present, available connectivity information on Mexican Atlantic reefs is scarce, particularly concerning the Veracruz Reef System (VRS), which is located in the south-western Gulf of Mexico. This study used a hierarchically nested sampling design to evaluate the levels of genetic connectivity both within and between the Mexican Caribbean (MC) and VRS reef regions; all of the studied reefs are marine protected areas. Microsatellites were used as genetic markers, and bicolor damselfish (Stegastes partitus) recruits were used as a biological model. The paired genetic differentiation index between regions (Fst _(ENA) = 0.008) was lower than the global index (Fst _(ENA) = 0.027), suggesting that the stronger restrictions to gene flow may be located inside the regions rather than between them. The AMOVA results supported this explanation, as the differences were only non-significant between regions. In the VRS, Santiaguillo reef was associated with low genetic connectivity levels, whilst within the MC region the group formed by Chinchorro Bank and Cozumel exhibited a restriction to gene flow with Puerto Morelos, their northernmost reef. Despite their spatial separation, reefs from different regions (Puerto Morelos and Anegada de Adentro) showed the lowest, albeit significant, genetic difference, meaning that a subtle genetic connectivity exists at the regional scale. The detected composite flow pattern is likely related to self-recruitment and cohesive dispersal processes interacting with current patterns, which may favour genetic connections under specific conditions. The results presented here suggest that coral reef management in the Mexican Atlantic Ocean should consider large scale measures in addition to appropriate local actions to protect reef fish populations.