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Edited by: Matjaž Perc, University of Maribor, Slovenia

Reviewed by: Qing Yun Wang, Beihang University, China; Ergin Yilmaz, Bulent Ecevit University, Turkey

*Correspondence: Leonid L. Rubchinsky

†Present Address: Sungwoo Ahn, Department of Mathematics, East Carolina University, Greenville, NC, United States

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Neural synchronization is believed to play an important role in different brain functions. Synchrony in cortical and subcortical circuits is frequently variable in time and not perfect. Few long intervals of desynchronized dynamics may be functionally different from many short desynchronized intervals although the average synchrony may be the same. Recent analysis of imperfect synchrony in different neural systems reported one common feature: neural oscillations may go out of synchrony frequently, but primarily for a short time interval. This study explores potential mechanisms and functional advantages of this short desynchronizations dynamics using computational neuroscience techniques. We show that short desynchronizations are exhibited in coupled neurons if their delayed rectifier potassium current has relatively large values of the voltage-dependent activation time-constant. The delayed activation of potassium current is associated with generation of quickly-rising action potential. This “spikiness” is a very general property of neurons. This may explain why very different neural systems exhibit short desynchronization dynamics. We also show how the distribution of desynchronization durations may be independent of the synchronization strength. Finally, we show that short desynchronization dynamics requires weaker synaptic input to reach a pre-set synchrony level. Thus, this dynamics allows for efficient regulation of synchrony and may promote efficient formation of synchronous neural assemblies.

Synchrony of neural oscillations is believed to play important role in a variety of functions of the brain (e.g., Buzsáki and Draguhn,

This implies that for some intervals of time synchrony may be stronger, while for other intervals of time it may be weaker. The temporal patterns of synchrony may exhibit variations of synchrony strength yielding some average synchrony values. Few long intervals of desynchronized dynamics may be functionally different from many short desynchronized intervals, although the synchrony may be the same on the average.

Detection and quantification of the transient, varying, intermittent synchronization have been considered in the past (e.g., Hurtado et al.,

These methods were used to study neural synchronization in several different systems: synchronization between single units and LFPs in the basal ganglia of Parkinsonian patients (Park et al.,

Since this short desynchronization dynamics was observed across different species, different conditions, and different signal types, it may be a universal feature of synchronized activity of neural systems. In this study we are providing a possible explanation for this apparent experimentally observed universality. We do so by looking for answers to two questions: what are the cellular or network mechanisms of this dynamics? What is its potential functional advantage?

We hypothesize that if this kind of dynamics is universal, it may be grounded in some general properties of neuronal excitability. In connection with this hypothesis, it is important to recall early insightful computational study (Somers and Kopell,

Since short desynchronization dynamics may be a generic phenomenon based on the properties of membrane channels, which are hard to alter in experiment, we use computational neuroscience techniques to study very simple conductance-based neuronal models. We alter the properties of conductances to explore their critical features for short desynchronization dynamics and investigate how coupled neurons may be efficiently entrained by external input. Models are subjected to the same kind of time-series analysis techniques as were used in earlier experimental studies. As a result, we reveal potential cellular basis of short desynchronization dynamics in the model and present its potential functional advantages.

We use a conductance-based modified Morris-Lecar neuronal model (Izhikevich,

We consider the model in the form:

_{Na} = _{Na} _{∞}(_{Na}), _{K} = _{K}_{K}) and _{L} = _{L}(_{L}) are sodium, potassium, and leak currents; _{app} is a constant parameter and _{syn} is a synaptic current (see below). _{Na}, _{K}, _{L} are the maximal conductances for the Na^{+}, K^{+}, and the leak currents. The functions
^{+} and K^{+} currents, and the activation time-constant for K^{+} current. The functions _{∞}(_{∞}(_{syn} represents the synaptic current between cells.

We consider neurons connected with excitatory synapses adapted from Izhikevich (_{Na} = 1, _{Na} = 1, _{K} = 3.1, _{K} = −0.7, _{L} = 0.5, _{L} = −0.4, _{app} = 0.045, _{m1} = −0.01, _{m2} = 0.15, β = 0.145, _{w1} = 0.08, ε = 0.02, _{syn} = 0.005, _{syn} = 0.5, α_{s} = 2, β_{s} = 0.2, θ_{v} = 0, σ_{s} = 0.2. We will further vary the values of ε, β, and _{w1} as will be described in the Results.

Phase analysis is frequently used to analyze synchronous neural activity of both continuous (LFP, EEG) and spiking signals (see, e.g., Lachaux et al., _{i}, _{i})-plane. Then we consider an average synchronization index to measure the strength of the phase locking between two signals (Pikovsky et al., _{j}) = φ_{1}(_{j}) − φ_{2}(_{j}) is the phase difference, the _{j} are the sampling points,

This phase synchronization index γ varies from 0 (lack of synchrony) to 1 (perfect synchrony). It provides an average value of phase-locking. There may be cycles of oscillations, where phase difference is close to the average value of the phase difference (phase-locked, synchronized state) and where it is not close to it (desynchronized state).

To study the fine temporal structure of the dynamics of synchronization we use the methods recently developed in Park et al. (_{1} crossed the zero from negative to positive values, we recorded the value of φ_{2}, generating a set of consecutive phase values {ϕ_{i}}, _{i} differs from the average value of ϕ_{i} by less than π/2 then the neurons are considered to be in a synchronized state, otherwise they are in the desynchronized state. We chose the value of the threshold to be π/2 because the experimental studies we discussed above used this value. The duration of desynchronization events is defined as the number of cycles of oscillations that the system spends in the desynchronized state minus one. The mode of the distributions of desynchronization durations is defined as the number with the highest probability of desynchronization durations.

We characterize the fine temporal structure of intermittent synchronization by quantifying the properties of distribution of desynchronization durations. We compute the relative frequencies (probabilities) of the durations of desynchronization events. This is similar to how the experimental data were characterized in the studies of the temporal patterns of synchrony (Park et al., _{mode}. If the mode of the desynchronization duration is short, but other desynchronizations (especially longer ones) are almost as frequent, then the dynamics is not necessarily dominated by short desynchronizations. However, if _{mode} is close to one, then all other desynchronization durations are rare.

In our approach the duration of synchronization and desynchronization intervals is measured not in the absolute time units, but in cycles of oscillations, as was done in experimental studies. It makes easier to compare synchronization patterns between rhythms of different frequency. However, as we study the differences between different desynchronization durations in the modeling, we also compare the dynamics with the same frequencies of rhythms (see Results).

We will study the dynamics of coupled model neurons as we vary parameters of potassium current. We do so by varying three different parameters: ε, β and _{w1} (see Equations 4 and 5), they all affect the effective value of activation time-constant τ(_{w1} we can study the model neurons exhibiting spiking activity like at Figure

Numerical simulation of a voltage of an isolated neuron (Equations 1–5). Examples of spiking activity (ε = 0.001)

We consider a minimal neuronal network to exhibit synchronized dynamics: two neurons mutually connected with excitatory synapses (Figure _{1} ≠ ε_{2} and the coupling strength _{syn} = 0.005 is weak, two cells are not fully synchronized. Thus, the synchrony is intermittent rather than perfect.

Diagram of a minimal network of excitatory coupled neurons. We use ε_{1} ≠ ε_{2} (i.e., neurons have different firing rates) and the coupling strength _{syn} is not very strong.

The magnitude of the voltage-dependent activation time-constant τ(

As the values of ε_{1} and ε_{2} increase, the fine temporal structure of synchronization changes as evident by the changes of the mode of the distribution of desynchronization durations (Figure

The effect of ε_{1} when ε_{2} = 1.2ε_{1}. _{2} = 1.2ε_{1}, neuron 2 has slightly higher frequency than the mean frequency while the neuron 1 has slightly lower frequency than the mean frequency.

The synchrony strength γ experiences only very small variations (Figure

The mean frequency (firing rate) grows substantially (Figure

Note that the probability of the dominant duration of desynchronization events _{mode} (thin gray line without dots in Figure

In the model, the parameter β is related to the width of the steady state activation function _{∞}(

Figure

The effect of β when ε_{1} = ε and ε_{2} = 1.2ε.

We now consider the effect of _{w1} which is the midpoint of the steady state activation function _{∞}(_{w1} shifts both curves _{∞}(_{w1} may be expected to have an effect analogous to decrease in τ.

The results of numerical simulation presented at the Figure _{w1} promote shorter desynchronizations (Figure

The effect of _{w1} when ε_{1} = ε and ε_{2} = 1.2ε.

The changes in desynchronization durations in the numerical experiments above are accompanied by changing either average synchrony strength or firing rate (or even both). Here we consider whether the desynchronization durations can vary independently of both synchrony strength and firing rate. To study this, we modify the Equations (4) and (5) so that the values of parameter β in equations for _{∞}(_{w} makes the slope of the steady-state activation function _{∞}(_{τ} makes the width of the constant function τ(_{w} and β_{τ} be changing in opposite directions. As β_{w} decreases from 0.134, β_{τ} increases from 0.061 at a different rate (β_{w} = 0.134 − 0.001_{τ} = 0.061+0.0005_{2} = 1.3ε_{1}, ε_{1} = 0.03, _{app} = 0.04, _{syn} = 0.005, _{w1} = 0.07. These changes of β_{w} and β_{τ} are not necessarily biologically realistic, but they allow us to explore whether the changes of desynchronization durations must covary with the changes of average synchrony or firing rate.

Figure _{∞}(

Changing synchronization durations independently from synchrony strength and firing rate.

To study potential functional advantages of short desynchronizations dynamics, we will consider two mutually excitatory connected neurons (as before) receiving common synaptic input from a third neuron: neuron 3 excites neurons 1 and 2 through the excitatory synapses (but does not get any feedback, Figure

Diagram of a minimal network of excitatory coupled neurons receiving common synaptic input. Neurons 1 and 2 have different firing rates. They are mutually coupled through excitatory synapses and receive synaptic input from neuron 3.

We consider two different versions of three-neuron networks in Figure _{syn1} = 0 neurons 1 and 2 exhibit dynamics with mostly short desynchronizations. The second version exhibits partially synchronized dynamics with the most common desynchronization intervals lasting for 4 cycles of oscillations when _{syn1} = 0. In other words, we consider how two coupled neurons exhibiting either short desynchronization dynamics or longer desynchronization dynamics respond to the common synaptic input.

One network has β_{w} = 0.094 and β_{τ} = 0.081, this is the left end of the horizontal axis in Figure _{w} = 0.134 and β_{τ} = 0.061, this is the right end of the horizontal axis in Figure

In the numerical experiments, ε_{1} = 0.03 and ε_{2} = 1.3ε_{1}, the same values as used in Figure

Now let us consider these two networks as the common input to neurons 1 and 2 is getting stronger due to increase of _{syn1} from zero (while _{syn} = 0.005 is fixed, that is, the coupling between neuron 1 and neuron 2 is relatively weak). As synaptic input from neuron 3 to neurons 1 and 2 is getting stronger, neurons 1 and 2 are becoming more synchronous and will eventually be in full synchrony with each other due to common synaptic input and mutual synaptic coupling.

We compute the synchrony index γ for “cycle 1” and “cycle 4” networks (γ^{(1C)} and γ^{(4C)} respectively) when increasing values of _{syn1}. To study how differently these networks are synchronized, we consider the absolute and relative difference of synchronization indices γ^{(1C)} and γ^{(4C)} for different values of _{syn1}. Figure ^{(1C)} − γ^{(4C)} (thick solid line) and

The synchrony difference γ^{(1C)} − γ^{(4C)} is plotted for different strength of synaptic input _{syn1}, normalized synchrony difference (^{(1C)} and γ^{(4C)} represent the synchrony index γ for “cycle 1” (short desynchronizations) and “cycle 4” (long desynchronizations) networks, respectively). Subplots

When this input is weak (_{syn1} is small), γ^{(1C)} and γ^{(4C)} are close to each other. When _{syn1} is large, γ^{(1C)} and γ^{(4C)} are again close to each other because both networks are necessarily strongly synchronous due to strong input. However, for the values of _{syn1} between zero and synchronization threshold value, γ^{(1C)} − γ^{(4C)} is large and positive. So the networks exhibiting short desynchronizations dynamics in the absence of input (“cycle 1” networks) reach either the same synchrony levels or higher synchrony levels than long desynchronization (“cycle 4”) networks for the same strength of synaptic input _{syn1}. This phenomenon is observed regardless of the firing rate in presynaptic neuron 3 (i.e., regardless of ε_{3}). Sometimes this difference in the synchronization strength is moderate, but sometimes it is quite substantial (see Figure

We also measure the threshold value _{syn1} for two neruons to reach synchornized dynamics without desynchronization events (Figure _{3}).

Threshold value of synaptic strength _{syn1} to reach synchornized dynamics without desynchronization events for different values of ε_{3}. Black squares represent the critical value of _{syn1} for short desynchronization (“cycle 1”) network and the gray circles represent the critical value of _{syn1} for long desynchronization (“cycle 4”) network.

The results presented in Figures

Imperfect synchrony is widely observed in the activity of neural networks of the brain. New time-series analysis techniques showed that intervals of synchronous dynamics are interspersed between desynchronized episodes, and most desynchronized episodes are very short (see references in Introduction). This stereotyped fine temporal structure of neural synchronization is not an artifact of the analysis method because other types of patterning of synchronization are possible in non-neural coupled oscillators (Ahn et al.,

The present study provides potential mechanisms for this type of temporal patterning of neural synchrony. We varied several parameters of potassium conductance and identified conditions leading to the intermittent neural synchrony with predominantly short desynchronization episodes similar to experimental ones. All these conditions (large peak value of activation time-constant, large width of dependence of activation time-constant on voltage, lower values of voltage for peak activation time-constant) lead to the relatively large values of the activation time-constant τ(

The results of the computational modeling also indicate that the distribution of desynchronization durations may be independent of the synchronization strength. The same synchrony strength may be achieved with desynchronizations of different durations. Moreover, our results regarding comparison of synchronization in networks exhibiting short desynchronizations and long desynchronizations are obtained for the case when not only average synchrony level is the same, but the period of oscillations (the firing rate) is the same. By appropriate adjustment of model parameters we dissociated the effects of frequency of oscillations and of average synchrony strength from the effects of fine temporal patterning of synchronized dynamics.

These model-based observations fit with experimental observations of the changes in the distribution of desynchronization durations in prefrontal cortex-hippocampal synchrony in behavioral sensitizations experiments (Ahn et al.,

The “spikiness” of oscillations of transmembrane voltage is a very generic property of many neurons, which relies on the fast activation of current with high reversal potential and slow activation of current with low reversal potential. Our results show that the same conditions that promote short desynchronization dynamics promote the characteristically sharp shape of an action potential. This may explain why very different neural systems exhibit short desynchronization dynamics, as we described in Introduction.

We use a very simple model of a neuron and very simple model of a network. There are many factors, which may affect synchronous dynamics of neural activity, yet they are not represented in the model. Other important factors, which affect neural synchrony, are different membrane currents and their properties (we have a model with just two conductances and consider only several parameters of one conductance) and the size of the network (we have a very small network). Heterogeneity of the networks is also important (we have a very minimal representation of heterogeneity). Synaptic plasticity is not incorporated in our model (and is the subject of the future research). Finally, noise may affect temporal patterns of synchrony, which is not considered in this study either.

However, even though these factors are not incorporated in the model (which captures only some very basic mechanisms of neural activity), the model is able to generate realistic synchrony patterns. So, the right way to interpret the modeling results is to see what these basic mechanisms are capable of. These modeling results suggest that these very basic neural mechanisms are capable of explaining the properties of experimentally observed intermittency of neural synchrony. As we discussed above, short desynchronization dynamics has been observed in several different neural systems. An ability of a minimal neural network considered here to describe the properties of the intermittent synchrony (which is common to all those systems) is probably an indicator that the general neural mechanisms built in the model are adequate to the considered phenomena.

Inhibition is playing an important role in neural synchronization, but is not considered in our model. The experimental data discussed here were collected from cortical and subcortical networks with excitatory and inhibitory synapses. It will be interesting to see how the intermittent patterns of synchrony are affected by inhibitory synapses.

We would also like to note that our earlier study with more advanced neural and network model (which included excitatory and inhibitory synapses) did provided a quantitatively adequate description of the short desynchronization dynamics at the beta-band oscillations in the basal ganglia in Parkinson's disease (Park et al.,

Our computational results suggest one way of how short desynchronization dynamics can be beneficial for neural systems. With two important properties of dynamic (average synchrony strength and firing rate) being equal, neural systems with short desynchronizations are easier to synchronize with common synaptic input. We showed that the same strength of common synaptic input leads to larger synchrony level in short desynchronization system. In other words, short desynchronization dynamics allows reaching a pre-set synchrony level with weaker input. So, if a strong synchrony is needed, systems with short desynchronizations will reach the pre-set synchrony strength with weaker inputs compared to longer desynchronizations.

Given the functional importance of synchronization in many neural systems (see references in Introduction), short desynchronization dynamics may allow for efficient regulation of synchrony levels. While the same level of synchrony may potentially be achieved with few long desynchronization episodes as well as with many short desynchronization episodes, only short desynchronization dynamics is experimentally observed in the neural synchrony in the brains. Our modeling results suggest that this short desynchronizations dynamics is easier to control with synaptic input. Thus, very basic properties of delayed rectifier potassium current (its delayed activation) is likely to promote efficient formation and break-up of synchronized assemblies.

LR conceived research; SA and LR designed research; SA performed numerical simulations, SA and LR analyzed and interpreted the results; SA and LR wrote the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.