Edited by: Ad Aertsen, University of Freiburg, Germany
Reviewed by: Carmen Canavier, LSU Health Sciences Center, USA; Christian Leibold, Ludwig Maximilians University, Germany
*Correspondence: Atthaphon Viriyopase, Cognitive Neuroscience, UMC St Radboud, Hp 126, Postbox 9101, 6500 HB Nijmegen, Netherlands. e-mail:
This is an open-access article distributed under the terms of the
Many studies have reported long-range synchronization of neuronal activity between brain areas, in particular in the beta and gamma bands with frequencies in the range of 14–30 and 40–80 Hz, respectively. Several studies have reported synchrony with zero phase lag, which is remarkable considering the synaptic and conduction delays inherent in the connections between distant brain areas. This result has led to many speculations about the possible functional role of zero-lag synchrony, such as for neuronal communication, attention, memory, and feature binding. However, recent studies using recordings of single-unit activity and local field potentials report that neuronal synchronization may occur with non-zero phase lags. This raises the questions whether zero-lag synchrony can occur in the brain and, if so, under which conditions. We used analytical methods and computer simulations to investigate which connectivity between neuronal populations allows or prohibits zero-lag synchrony. We did so for a model where two oscillators interact via a relay oscillator. Analytical results and computer simulations were obtained for both type I Mirollo–Strogatz neurons and type II Hodgkin–Huxley neurons. We have investigated the dynamics of the model for various types of synaptic coupling and importantly considered the potential impact of Spike-Timing Dependent Plasticity (STDP) and its learning window. We confirm previous results that zero-lag synchrony can be achieved in this configuration. This is much easier to achieve with Hodgkin–Huxley neurons, which have a biphasic phase response curve, than for type I neurons. STDP facilitates zero-lag synchrony as it adjusts the synaptic strengths such that zero-lag synchrony is feasible for a much larger range of parameters than without STDP.
Coupling between different oscillators and pacemakers can generate a large range of different behaviors and has been a topic of study in many different conditions, for example in cardiac pacemaking and chemical oscillations (see e.g., Goldbeter,
The first studies on this topic presented experimental evidence that the relative phase of gamma oscillations in widely separated brain areas is near zero (Frien et al.,
In order to obtain a better understanding of the possibilities for zero-lag synchronization of distant brain areas, we have investigated the proposed network of neuronal oscillators coupled indirectly by a relay oscillator (Fischer et al.,
The model consists of three coupled neuronal oscillators (see Figure
We have used the Mirollo–Strogatz (MS) phase oscillator (Mirollo and Strogatz,
The MS phase oscillator (Mirollo and Strogatz,
with a phase variable φ ∈ [0, 1] and a dissipation parameter
For the classical HH neuron the membrane potential
with the membrane capacitance
We investigate two models for pulse-coupling between the oscillators. For an instantaneous synapse with coupling strength ε
with φ
If a spike arrives at φ
with
where φ
Tsubo et al. (
A more realistic synaptic coupling model is provided by the so-called alpha function. For an “alpha synapse,” the postsynaptic potential after arrival of a spike at time
where τsyn > 0 is the synaptic rise time of the input. Unless stated otherwise, τsyn = 2 ms in this study. The numerical simulations are implemented using an Euler scheme with a time step size equal to 2.5 μs (≈10−4
In general, the synaptic coupling strength is not constant, but varies depending on pre- and post-synaptic activity due to STDP (Hebb,
with
The spike arrival time
The simulation procedure of the network for STDP is as follows: The evolution of the state of the network is studied over 60 consecutive sessions. At the start of each session, the initial phases of the three oscillators are chosen arbitrarily from a uniform random distribution. The first session starts with equal coupling strengths for all synapses, which then change due to STDP. All succeeding sessions start with the coupling strengths that resulted at the end of the previous session, but with re-randomized phases of the oscillators to prevent that the system converges into a locally rather than a globally stable state.
In order to be physiologically relevant for synchronization, convergence should not take too much time. Therefore, we assume that convergence to a stable relative phase between oscillators 1 and 3 in the model takes place within a session consisting of
Phase-locking equations are useful to determine the new period of a network and the relative phases of coupled oscillators (van Vreeswijk et al.,
Without loss of generality, we set the time of the (
with Δφ as defined in Eq.
Equations
Assume now that both arrival phases exceed the critical phases φ
Thus the period of all oscillators is twice the conduction delay and activity switches between inner and outer oscillators every half period. All oscillators immediately spike upon spike input. We will call this mode the “driven synchrony” (DS) mode (see Figure
To investigate the stability of DS we assume small perturbations. Since we use the relay neuron spikes as reference time, the perturbation affects the phase of the outer oscillators to (τ + δφ1, τ + δφ3) at
We now briefly consider the effect of STDP on slave synchrony. The coupling strengths from the outer oscillators to the relay oscillator remain unchanged for slave synchrony, since the relay oscillator immediately spikes upon input from the outer neurons. In the Appendix we show that the coupling strengths from the relay oscillator to the outer oscillators increase with ε
In the driven, pacemaker, and slave synchrony modes oscillators 1 and 3 spike simultaneously. There are other, asynchronous stable modes where this is not the case. Figures
For a proper analysis of synchronized firing, a quantitative definition of synchrony is required. In experiments perfect synchrony will never be observed as noise causes small variations in the timing of action potentials. Instead we will define synchrony as firing of two oscillators within a small time window, where the time window should be sufficiently large to eliminate the effect of noise, and sufficiently small to provide an accurate estimate of synchrony. While we could measure synchrony here to the limits of the numerical accuracy of our simulations, we define synchrony as the condition when the spikes of the outer oscillators occur within
Synchronized firing of the outer oscillators does not only depend on the synaptic weights and delay times, but also on the initial phase of the three oscillators. Hence we have to consider the robustness of convergence to synchronous firing for variations in the initial relative phase of the neuronal oscillators. Therefore, we define “
In order to investigate the impact of the various parameters of the STDP learning window, we wish to derive a single value for the SQ, rather than one value for each combination of synaptic weights and conduction delay. For this purpose we consider various synaptic weights in 100 evenly spaced steps in the range from 0.01 to 0.21 and conduction delays in 100 evenly spaced steps in the range from 0.01 to 0.49. Then we average the SQ over these 10,000 pairs to obtain an “average SQ” for each parameter set of the learning window. Note that for an STDP run with 60 sessions, this means that we compute almost 400 billion cycles of the model for each change of the STDP parameters.
For some combinations of the synaptic weights, delay times, and initial phases the state of zero-lag synchronization is reached faster than for other combinations. Therefore, we define a “
Finally, not only are we interested in synchrony of the outer oscillators, but also in the relative phase φ
Figures
where χ
The line separating regions I and II is given by τ = φ
where φ
Figure
Figure
To illustrate how the non-zero relative phase varies for different parameter values in the network, Figure
Next, we study synchronization for a model with HH neurons. Figures
Figure
Figures
High SQ is possible for short delay times (τ < 0.05), if the synaptic rise time is short (Figure
We now investigate the effect of STDP on synchronization of the MS neuron, starting with the simple instantaneous synapses. Figure
The higher SQ and the faster CP in region II for large delay times in Figures
Figure
Figure
In order to investigate the robustness of our results for variations of the STDP parameter values, we return to the instantaneous synapses. We have varied the amplitudes
When depression dominates over potentiation (dashed red and solid blue lines), the SQ is poor even after many sessions, see Figure
We first consider the effects of STDP on the coupling strengths of the network starting with initial coupling strength ε and delay time τ in region III just below the line which separates region II and III. STDP will increase the weights ε to larger values due to larger potentiation relative to depression. This moves (ε, τ) from region III to region II, where the zero-lag synchronization modes “SS1” of Eq.
Results for different time constants for potentiation τ+ and depression τ− (not shown) are qualitatively similar to the results for changing the amplitudes
In summary, the conditions that contribute to zero-lag long range synchronization are as follows. First, the delay times relative to the intrinsic frequency
We now investigate the dynamics of the model in Figure
Figure
In this study we have investigated the conditions for zero-lag synchrony between two neuronal oscillators, which interact via a relay oscillator. The main result of our study is that for the model with type II Hodgkin–Huxley neurons, synchronization is easier to achieve than for type I Mirollo–Strogatz neurons. Synapses with short rise times (typically less than 2 ms) are more suitable to achieve zero-lag synchronization than synapses with longer rise times. With STDP the network converges to zero-lag synchronization at a faster rate and for a larger range of synaptic strengths and time delays than without STDP. However, when the delay times between the two synchronizing oscillators and the relay oscillator are different, zero phase lag may easily get lost.
The network used in this study is a simplified model for interacting neuronal populations. This obviously raises the question whether our results about zero-lag synchrony may be biased by the simplifications inherent in our model. We will argue in the next paragraphs below that this is not the case. Our choice of indirect interactions between oscillating neuronal populations, i.e., via a relay oscillator, was inspired by previous studies, which showed that pulse-coupled neuronal oscillators with direct excitatory coupling and signal delays in general do not oscillate at zero phase lag (van Vreeswijk et al.,
The most simplified version of our model assumes that the oscillators used to represent neuronal population activity are of the Mirollo–Strogatz type. The Mirollo–Strogatz oscillator corresponds to the type I neuron class (Izhikevich,
Replacing the Mirollo–Strogatz neurons by more realistic type II Hodgkin–Huxley neurons allows for a broader range of synaptic strengths and time delays that is compatible with zero-lag synchronization. This result suggests a functional role for changes in neuronal properties from type I to type II, in agreement with suggestions by Prescott et al. (
Delays in networks of interacting neurons can give a variety of complex behaviors with a wealth of bifurcations and a rich phase diagram, which includes oscillatory bumps, traveling waves, lurching waves, standing waves arising via a period-doubling bifurcation, aperiodic regimes, and regimes of multistability (Roxin et al.,
Another assumption of this study, which requires some more discussion, is that all oscillators in the model have identical intrinsic properties with the same oscillation period. If the intrinsic periods of the outer oscillators differ, zero-lag synchrony may get lost. Whether or not synchrony will be lost depends on the neuron type. If the synaptic input to the outer neurons, which have different natural frequencies, resets their oscillation periods to the same value, zero-lag synchronization is easily obtained. Synchronization of non-linear oscillators with different oscillatory properties is feasible if the interactions between the oscillators (the synaptic strengths in our study) are sufficiently strong (Pikovsky et al.,
If the periods of the two outer oscillators are different from each other and if input from the relay oscillator does not make the period of the outer oscillators the same, input from the relay oscillator to the outer neurons will elicit spikes at different times. In that case, the spike input from the outer oscillators to the relay neuron also arrives at different times. This is essentially equivalent to the situation with different delay times, which we have studied (see
In our study we have introduced “SQ” as a measure for the robustness of synchrony against variations in the initial phases of the oscillators. SQ was used together with the “CP” to assess zero-lag synchrony between the two outer oscillators in our relay network. The time interval for “synchrony” was chosen as spike coincidence within 0.5 ms (Engel and Singer,
In agreement with Knoblauch and Sommer (
Overall, our results demonstrate that gamma oscillations in various cortical areas can be synchronized at zero-lag in a network model where neuronal oscillators are coupled via a relay oscillator, in agreement with previous studies (Fischer et al.,
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.
Analytical derivations of the various synchrony modes and their stability are provided. An overview of these results is provided by Figure
To investigate the existence of zero-lag synchrony as a function of τ and ε, we will derive the 1:1 synchronized phase-locking equation of the oscillators, cf. Figure
where Δφ is defined in Eqs
The spike from oscillator 2 will arrive at oscillators 1 and 3 at phase τ + (1 − θ)
Equations
Using Eqs
First, we consider driven synchrony. In order for all input spikes to elicit a spike in the receiving oscillators, we must have the following: for the inner oscillator φ
and the new period of the outer oscillators as
We can now solve for θ and
For biologically realistic conditions τ ∈ (0, 0.5) and ε ∈ (0, 0.21) and our choice
Second, we investigate pacemaker synchrony. For pacemaker synchrony only the outer oscillators spike directly upon input, but not the inner one. Thus for the inner oscillator, we must have φ
where as the new period of the outer oscillators is given by Eq.
The two valid modes for pacemaker synchrony given our constraints are labeled “PS1” and “PS2” in Eq.
Third, in slave synchrony only the inner oscillator spikes upon input, i.e., in case of the inner oscillator, we must have φ
The four combinations are then:
The two valid slave synchronization modes will be called “SS1” and “SS2,” respectively. Within our ranges of interest for ε and τ, one can check numerically for the “SS1” slave synchrony mode that φ
Finally, for non-driven synchrony none of oscillators fires immediately after receiving a spike. This implies that for the inner oscillator φ
with ρmnop = [
Based on different combinations of the synchronization modes DS from Eq.
In order to investigate the stability of driven synchrony, we assume that the fixed point Eq.
Next we investigate the stability of the slave synchronization mode Eq.
To find the phase
The next spike of the relay oscillator occurs at
The corresponding Jacobian matrix is
The eigenvalues of this matrix are 0 and 1 + β
Finally, we investigate the stability of non-driven synchronization Eq.
with oscillator 2 spiking after this additional time Δ
The corresponding Jacobian matrix is
with resulting eigenvalues
This fixed point is not stable, because these eigenvalues exceed the value one for ε∈(0, 0.21). In a similar manner one can show that Eq.
In slave synchrony, the coupling strengths from the outer oscillators to the relay oscillator remain unchanged by STDP since the relay oscillator instantly spikes upon input from the outer oscillators. How about the coupling from the relay oscillator to the outer oscillators? Say their strengths are ε
Note that this change is a decrease in coupling strength because
Under certain conditions, see main text, this leads to STDP turning slave synchrony into driven synchrony, and the transition time can be determined by numerically iterating the formulas for
This project was partly funded by the Netherlands Organization for Scientific Research (NWO 051.02.050).