Delay, noise and phase locking in pulse coupled neural networks

This paper studies the effect of several delay times and noise on the stability of the phase-locked state in the lighthouse model and the integrate and fire model of a pulse coupled neural network. The coupling between neurons may be arbitrary. In both models the increase of delay times leads to a weakening of the stability and to the occurrence of relaxation oscillations.     

Prefrontal phase locking to hippocampal theta oscillations

The interactions between cortical and hippocampal circuits are critical for memory formation, yet their basic organization at the neuronal network level is not well understood. Here, we demonstrate that a significant portion of neurons in the medial prefrontal cortex of freely behaving rats are phase locked to the hippocampal theta rhythm. In addition, we show that prefrontal neurons phase lock best to theta oscillations delayed by approximately 50 ms and confirm this hippocampo-prefrontal directionality and timing at the level of correlations between single cells. Finally, we find that phase locking of prefrontal cells is predicted by the presence of significant correlations with hippocampal cells at positive delays up to 150 ms. The theta-entrained activity across cortico-hippocampal circuits described here may be important for gating information flow and guiding the plastic changes that are believed to underlie the storage of information across these networks.     

A simple model for phase locking of biological oscillators

Summary A mathematical model is presented for phase locking of a biological oscillator to a sinusoidal stimulus. Analytical, numerical and topological considerations are used to discuss the patterns of phase locking as a function of the amplitude of the sinusoidal stimulus and the relative frequencies of the oscillator and the sinusoidal stimulus. The sorts of experimental data which are needed to make comparisons between theory and experiment are discussed.     

Synthesis of extended nanoscale optical encoders

An optical encoder is a device that uses an interrupted light source-sensor pair to map linear or rotational motion onto a periodic signal. Simple, inexpensive optical encoders are used for precise positioning in machines such as desktop printers, disk drives, and astronomical telescopes. A strand of DNA labeled with a series of Fo?rster resonance energy transfer acceptor dyes can perform the same function at the nanometer scale, producing a periodic fluorescence signal that encodes the movement of a single donor-labeled molecular motor with high spatial and temporal resolution. Previous measurements of this type have employed encoders limited to five acceptor dyes, and hence five signal periods, restricting the range of motion that could be followed. Here we describe two methods for synthesizing double-stranded DNA containing several to hundreds of regularly spaced dyes on one strand. Distinct functional groups incorporated at the encoder ends enable tethering for single-molecule measurements.     

Theoretical computation of phase locking in embryonic atrial heart cell aggregates

The effects of periodic stimulation of spontaneously beating aggregates of chick atrial heart cells are considered. Provided the effects of a single stimulus do not change the properties of the oscillation, and that the oscillation is re-established rapidly following a stimulus, this system can be modeled by one-dimensional finite difference equations. These equations employ experimentally generated phase resetting data that describe the effects of a single isolated stimulus at different phases of the oscillation. A complete analysis of the predicted dynamics is given over a broad range of stimulation frequencies and amplitudes. Prominent features of the dynamics include phase locking, bistability, chaos, and disappearance of Arnold tongues at large stimulation amplitudes. The fine details of the bifurcations are sensitive to properties of the phase resetting curves, and consequently, the observed bifurcations are not expected to be "universal" for larger stimulation amplitudes. Experimental traces show many correspondences with theoretical computations.     

Reproductive phase locking of mosquito populations in response to rainfall frequency

The frequency of moderate to heavy rainfall events is projected to change in response to global warming. Here we show that these hydrologic changes may have a profound effect on mosquito population dynamics and rates of mosquito-borne disease transmission. We develop a simple model, which treats the mosquito reproductive cycle as a phase oscillator that responds to rainfall frequency forcing. This model reproduces observed mosquito population dynamics and indicates that mosquito-borne disease transmission can be sensitive to rainfall frequency. These findings indicate that changes to the hydrologic cycle, in particular the frequency of moderate to heavy rainfall events, could have a profound effect on the transmission rates of some mosquito-borne diseases.     

An interspike interval method for computing phase locking from neural firing

Inner hair cells (IHCs) transform the mechanical movements of the basilar membrane into electrical impulses. The impulse coding of the IHCs is the main information carrier in the auditory process and is the basis for improvements of cochlea implants as well as for low rate, high-quality speech processing and compression. This paper shows how to compute the speech signal from the neural firing based on the analysis of the interspike interval histogram. This new approach solves problems that other standard analysis methods do not solve sufficiently well. Received 18 September 1998 / Accepted in revised form 15 October 1999     

Functional connectivity between prefrontal cortex and striatum estimated by phase locking value

The interplay between the prefrontal cortex (PFC) and striatum has an important role in cognitive processes. To investigate interactive functions between the two areas in reward processing, we recorded local field potentials (LFPs) simultaneously from the two areas of two monkeys performing a reward prediction task (large reward vs small reward). The power of the LFPs was calculated in three frequency bands: the beta band (15–29 Hz), the low gamma band (30–49 Hz), and the high gamma band (50–100 Hz). We found that both the PFC and striatum encoded the reward information in the beta band. The reward information was also found in the high gamma band in the PFC, not in the striatum. We further calculated the phase-locking value (PLV) between two LFP signals to measure the phase synchrony between the PFC and striatum. It was found that significant differences occurred between PLVs in different task periods and in different frequency bands. The PLVs in small reward condition were significant higher than that in large reward condition in the beta band. In contrast, the PLVs in the high gamma band were stronger in large reward trials than in small trials. These results suggested that the functional connectivity between the PFC and striatum depended on the task periods and reward conditions. The beta synchrony between the PFC and striatum may regulate behavioral outputs of the monkeys in the small reward condition.     

Variability and coding efficiency of noisy neural spike encoders

Encoding synaptic inputs as a train of action potentials is a fundamental function of nerve cells. Although spike trains recorded in vivo have been shown to be highly variable, it is unclear whether variability in spike timing represents faithful encoding of temporally varying synaptic inputs or noise inherent in the spike encoding mechanism. It has been reported that spike timing variability is more pronounced for constant, unvarying inputs than for inputs with rich temporal structure. This could have significant implications for the nature of neural coding, particularly if precise timing of spikes and temporal synchrony between neurons is used to represent information in the nervous system. To study the potential functional role of spike timing variability, we estimate the fraction of spike timing variability which conveys information about the input for two types of noisy spike encoders--an integrate and fire model with randomly chosen thresholds and a model of a patch of neuronal membrane containing stochastic Na(+) and K(+) channels obeying Hodgkin-Huxley kinetics. The quality of signal encoding is assessed by reconstructing the input stimuli from the output spike trains using optimal linear mean square estimation. A comparison of the estimation performance of noisy neuronal models of spike generation enables us to assess the impact of neuronal noise on the efficacy of neural coding. The results for both models suggest that spike timing variability reduces the ability of spike trains to encode rapid time-varying stimuli. Moreover, contrary to expectations based on earlier studies, we find that the noisy spike encoding models encode slowly varying stimuli more effectively than rapidly varying ones.     

Directionality of phase locking in auditory nerve fibers of the leopard frog Rana pipiens pipiens

Summary A dorsal approach to the eighth nerve and free-field stimulation were used to investigate the effect of sound direction and intensity on phase locking in auditory nerve fibers of the leopard frog Rana pipiens pipiens.Tuning curves of 75 auditory neurons were analyzed (Fig. 2). Amphibian papillar neurons, but not basilar papillar neurons, exhibit significant phase locking to short tone bursts at the characteristic frequency (CF), the degree of phase locking (vector strength) decreasing with the neuron's CF (Figs. 3, 4 and 10E). Vector strength increases with sound pressure level to saturate about 20 dB above threshold, while the preferred firing phase is only slightly affected (Figs. 5 and 6).In contrast, sound direction hardly affects vector strength (Figs. 7, 8, 9A and 10A and C), but has a strong influence on the preferred firing phase (Figs. 7, 8, 9B and C, 10B and D): With respect to anterior tone presentation there are phase lags for ipsilateral and phase leads for posterior and contralateral presentation. Phase differences between both ears show a sinusoidal or cardioid/ovoidal directional characteristic; maximum differences are found with antero-lateral tone presentation (Fig. 11). The directionality of phase locking decreases with the neuron's CF (Fig. 10F) and only slightly changes with sound pressure level (Fig. 12). Thus, phase locking of amphibian papilla neurons can potentially provide intensity-independent information for sound localization.Abbreviations SPL sound pressure level - FTC frequency threshold curve - CF characteristic frequency - TF test frequency - VS vector strength - AP amphibian papilla - BP basilar papilla     

Phase resetting and phase locking in hybrid circuits of one model and one biological neuron

To determine why elements of central pattern generators phase lock in a particular pattern under some conditions but not others, we tested a theoretical pattern prediction method. The method is based on the tabulated open loop pulsatile interactions of bursting neurons on a cycle-by-cycle basis and was tested in closed loop hybrid circuits composed of one bursting biological neuron and one bursting model neuron coupled using the dynamic clamp. A total of 164 hybrid networks were formed by varying the synaptic conductances. The prediction of 1:1 phase locking agreed qualitatively with the experimental observations, except in three hybrid circuits in which 1:1 locking was predicted but not observed. Correct predictions sometimes required consideration of the second order phase resetting, which measures the change in the timing of the second burst after the perturbation. The method was robust to offsets between the initiation of bursting in the presynaptic neuron and the activation of the synaptic coupling with the postsynaptic neuron. The quantitative accuracy of the predictions fell within the variability (10%) in the experimentally observed intrinsic period and phase resetting curve (PRC), despite changes in the burst duration of the neurons between open and closed loop conditions.     

Deriving theoretical phase locking values of a coupled cortico-thalamic neural mass model using center manifold reduction

Cognitive functions such as sensory processing and memory processes lead to phase synchronization in the electroencephalogram or local field potential between different brain regions. There are a lot of computational researches deriving phase locking values (PLVs), which are an index of phase synchronization intensity, from neural models. However, these researches derive PLVs numerically. To the best of our knowledge, there have been no reports on the derivation of a theoretical PLV. In this study, we propose an analytical method for deriving theoretical PLVs from a cortico-thalamic neural mass model described by a delay differential equation. First, the model for generating neural signals is transformed into a normal form of the Hopf bifurcation using center manifold reduction. Second, the normal form is transformed into a phase model that is suitable for analyzing synchronization phenomena. Third, the Fokker–Planck equation of the phase model is derived and the phase difference distribution is obtained. Finally, the PLVs are calculated from the stationary distribution of the phase difference. The validity of the proposed method is confirmed via numerical simulations. Furthermore, we apply the proposed method to a working memory process, and discuss the neurophysiological basis behind the phase synchronization phenomenon. The results demonstrate the importance of decreasing the intensity of independent noise during the working memory process. The proposed method will be of great use in various experimental studies and simulations relevant to phase synchronization, because it enables the effect of neurophysiological changes on PLVs to be analyzed from a mathematical perspective.     

Effects of pulse ultrasonic irradiation on the lag phase of Saccharomyces cerevisiae growth

Aims: This paper presents an analysis of lag phase phenomena in Saccharomyces cerevisiae growth as a function of ultrasonic irradiation. Methods and Results: Pulse irradiation treatments were performed by a 20 kHz ultrasonic transducer with different durations and energies. Data obtained from experiments were then employed to estimate growth parameters by specific transfer function. The significance of the different lag times in response to ultrasonic irradiation was analysed. The results showed that the yeast growth in lag phase responded to the irradiated ultrasonic of 20 min more than the 10 min. The ultrasonic energies between 330 and 360 W s m^?3 could decrease lag time up to 1 h compared to the sample without ultrasonic irradiation. Conversely, the treatments with energies higher than 850 W s m^?3 were able to extend the lag time and decrease the yeast growth. Conclusions: The lag durations of S. cerevisiae were changed significantly by different ultrasonic irradiations, energies and durations. In particular, sufficient irradiation energies reduced the lag time, resulting in accelerated yeast growth. In contrast, high energy could inactivate growth by increasing the lag time. Significance and Impact of the Study: This work provides an alternative technique to either accelerate or inactivate the S. cerevisiae lag phase. The approach can be developed in experiment designed to control the yeast growth by ultrasonic irradiation as assistance in the environments.     

Repetitive firing: a quantitative study of feedback in model encoders

Recognition of nonlinearities in the neuronal encoding of repetitive spike trains has generated a number of models to explain this behavior. Here we develop the mathematics and a set of tests for two such models: the leaky integrator and the variable-gamma model. Both of these are nearly sufficient to explain the dynamic behavior of a number of repetitively firing, sensory neurons. Model parameters can be related to possible underlying basic mechanisms. Summed and nonsummed, spike- locked negative feedback are examined in conjunction with the models. Transfer functions are formulated to predict responses to steady state, steps, and sinusoidally varying stimuli in which output data are the times of spike-train events only. An electrical analog model for the leaky integrator is tested to verify predicted responses. Curve fitting and parameter variation techniques are explored for the purpose of extracting basic model parameters from spike train data. Sinusoidal analysis of spike trains appear to be a very accurate method for determining spike-locked feedback parameters, and it is to a large extent a model independent method that may also be applied to neuronal responses.     

Period-2 cycles and 2:1 phase locking in a biological clock driven by temperature pulses

Crassulacean acid metabolism (CAM) serves as a botanical model system for the investigation of circadian rhythmicity. In a new set of experiments with the obligatory CAM plant Kalancho? daigremontiana the response to periodic stimulations with temperature pulses has been studied. On the basis of an experimental phase-response curve of net CO(2)-gas exchange the effect of periodic stimulation has been simulated using a finite-difference equation. These simulations revealed the locations of two period-2 cycles in the CO(2) uptake of the CAM plant. In subsequent experiments based upon the simulated bifurcation diagram the position and amplitude of one of these cycles were confirmed, while experimental evidence for the second cycle could be found. Possible roles of such dynamics for the functioning of the biological clock are discussed.     

An investigation of the phase locking index for measuring of interdependency of cortical source signals recorded in the EEG

The phase locking index (PLI) was introduced to quantify in a statistical sense the phase synchronization of two signals. It has been commonly used to process biosignals. In this article, we investigate the PLI for measuring the interdependency of cortical source signals (CSSs) recorded in the Electroencephalogram (EEG). To this end, we consider simple analytical models for the mapping of simulated CSSs into the EEG. For these models, the PLI is investigated analytically and through numerical simulations. An evaluation is made of the sensitivity of the PLI to the amount of crosstalk between the sources through biological tissues of the head. It is found that the PLI is a useful interdependency measure for CSSs, especially when the amount of crosstalk is small. Another common interdependency measure is the coherence. A direct comparison of both measures has not been made in the literature so far. We assess the performance of the PLI and coherence for estimation and detection purposes based on, respectively, a normalized variance and a novel statistical measure termed contrast. Based on these performance measures, it is found that the PLI is similar or better than the CM in most cases. This result is also confirmed through analysis of EEGs recorded from epileptic patients.

Effects of conduction delays on the existence and stability of one to one phase locking between two pulse-coupled oscillators

Gamma oscillations can synchronize with near zero phase lag over multiple cortical regions and between hemispheres, and between two distal sites in hippocampal slices. How synchronization can take place over long distances in a stable manner is considered an open question. The phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike, depending upon where in the cycle it is received. We use PRCs under the assumption of pulsatile coupling to derive existence and stability criteria for 1:1 phase-locking that arises via bidirectional pulse coupling of two limit cycle oscillators with a conduction delay of any duration for any 1:1 firing pattern. The coupling can be strong as long as the effect of one input dissipates before the next input is received. We show the form that the generic synchronous and anti-phase solutions take in a system of two identical, identically pulse-coupled oscillators with identical delays. The stability criterion has a simple form that depends only on the slopes of the PRCs at the phases at which inputs are received and on the number of cycles required to complete the delayed feedback loop. The number of cycles required to complete the delayed feedback loop depends upon both the value of the delay and the firing pattern. We successfully tested the predictions of our methods on networks of model neurons. The criteria can easily be extended to include the effect of an input on the cycle after the one in which it is received.