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Edited by: Robert Rosenbaum, University of Pittsburgh, USA

Reviewed by: Tatjana Tchumatchenko, Max Planck Institute for Brain Research, Germany; Eric Shea-Brown, University of Washington, USA

*Correspondence: G. Bard Ermentrout, Department of Mathematics, University of Pittsburgh, 139 University Place, Pittsburgh, PA 15260, USA e-mail:

This article was submitted to the journal Frontiers in Computational Neuroscience.

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.

Synchronization plays an important role in neural signal processing and transmission. Many hypotheses have been proposed to explain the origin of neural synchronization. In recent years, correlated noise-induced synchronization has received support from many theoretical and experimental studies. However, many of these prior studies have assumed that neurons have identical biophysical properties and that their inputs are well modeled by white noise. In this context, we use colored noise to induce synchronization between oscillators with heterogeneity in both phase-response curves and frequencies. In the low noise limit, we derive novel analytical theory showing that the time constant of colored noise influences correlated noise-induced synchronization and that oscillator heterogeneity can limit synchronization. Surprisingly, however, heterogeneous oscillators may synchronize better than homogeneous oscillators given low input correlations. We also find resonance of oscillator synchronization to colored noise inputs when firing frequencies diverge. Collectively, these results prove robust for both relatively high noise regimes and when applied to biophysically realistic spiking neuron models, and further match experimental recordings from acute brain slices.

Synchronization of neural oscillators is thought to play a critical role in sensory, motor, and cognitive processes (Sanes and Donoghue,

When neurons fire regularly, they can be regarded as noisy nonlinear oscillators and, as such, there are many mathematical techniques available for their analysis. In particular, the

In this study, we extend this theory to include colored noise inputs and, further, report some surprising effects of heterogeneity. First, we derive a set of equations for the distribution of phase differences for pairs of heterogeneous oscillators driven by a partially correlated Ornstein-Uhlenbeck (OU) process (low-pass filtered noise). We next show that the theory developed for phase reduced models works well with a conductance-based biophysical model. We then show that, quite surprisingly, at low input correlations, heterogeneity can sometimes produce higher output correlations than the homogeneous case. That is, consider two distinct oscillators, A and B, such that the AA pair has a small susceptibility and the BB pair a larger susceptibility. Then, at low correlations, the susceptibility of the AB pair can sometimes exceed that of the AA pair. We confirm this somewhat counterintuitive prediction with recordings from regularly firing mitral cells of the main olfactory bulb. In addition to heterogeneity in response properties, neurons can fire at different frequencies, and such frequency differences can significantly impact correlated-noise induced synchronization (Markowitz et al.,

In Appendix, we provide a brief overview of how to reduce a general weakly perturbed limit cycle to a single differential equation for the phase of the cycle. If we assume that the original limit cycle represents repetitive firing of a single compartment neuron model that is driven by a noisy current, _{m} is the membrane capacitance, θ is the phase (or, typically, the time since the last spike), and Δ(θ) is the PRC of the neuron. The PRC describes the phase-dependent shift in the spike times of an oscillator receiving small perturbations. It is readily measured in neurons and other biological oscillators (Torben-Nielsen et al.,

Given the reduced model (Equation 1), we can now turn to the main question at hand, which is: how do oscillating heterogeneous neurons transfer correlations? We will consider two types of heterogeneity: differences in the PRC shapes and differences in natural frequencies. We drive the oscillators with correlated filtered noise. After reduction to phase variables, we obtain:
_{1} and θ_{2} are the phases of two oscillators, and Δ_{1}(θ) and Δ_{2}(θ) are PRCs of the two oscillators. Without loss of generality, we set the natural frequency of one oscillator to 1. The parameter ω then determines the magnitude of the difference in natural frequencies between the two oscillators. ϵ ≪ 1, thus the noise is weak and the frequency difference is small. The processes _{x} and ξ_{y} are two correlated white noise processes, i.e., 〈ξ_{x}(_{x}(_{y}(_{y}(_{x}(_{y}(

We remark that the allowable frequency difference is ^{2}), which seems considerably smaller than the magnitude of the noise, which is ϵ. However, as the noise has zero mean, what matters is the variance of the noise, which has magnitude ϵ^{2}. Thus, the scales of both the frequency difference and the synchronizing inputs (correlations in the noise) are similar. If the frequency differences are larger, then no synchronization is possible.

Our goal is to compute the stationary distribution of the phase difference between two neurons since this will enable us to compute various measures of correlation and synchrony. Thus, some variable substitution will be helpful: θ = θ_{1}, ϕ = θ_{2} − θ_{1}. Therefore, ϕ is the phase difference between the two oscillators. With this change of variables, the equations are:
_{ss}(

Our goal is to compute the probability density of the phase difference between the two oscillators,

Once we get the distribution of phase differences,

The Morris-Lecar (ML) model (Rinzel and Ermentrout, _{1}(_{1}(_{2}(_{2}(_{1}(_{2}(_{K} = −84 mV, _{L} = −60 mV, _{Ca} = 120 mV, _{a} = −1.2 mV, _{b} = 18 mV, _{c} = 2 mV, and _{d} = 30 mV. _{1}, _{2} and ϕ_{1}, ϕ_{2} vary for each figure.

To get the phase from the noisy voltage signal generated by the ML model, we first apply the Hilbert transform (HT) to

In some of the figures, we simulate the phase-reduced dynamics for the ML model. To do this, we must compute the infinitesimal PRC, Δ_{ML}(θ). As described in Appendix, the PRC for the model is the voltage component of the solution to the adjoint equation (Equation 32). The software package XPP (Ermentrout, _{ML}(θ) for those specific parameters. We save the result as a lookup table and then numerically solve the phase equation.

To get solutions to the stochastic phase and membrane equations, we use the Euler-Murayama method. We solve the BVP for the stationary phase difference density using a custom BVP solver written in MATLAB. All codes are available by request.

Oscillators driven with a correlated fluctuating signal will exhibit a degree of synchronization that depends on the size of the signal, the strength of correlation, and the similarity of the two oscillators. Thus, for example, identical oscillators driven by small enough identical white noise will synchronize perfectly (Pikovsky et al., _{1, 2}(θ) are the PRCs for the two oscillators; ϵ is a small positive parameter (characterizing the magnitude of the fluctuations); and ω accounts for the frequency difference in the unperturbed oscillators (see Materials and Methods, Equations 2–5). We are primarily interested in the distribution of the phase difference, ϕ: = θ_{2} − θ_{1}. In the Appendix (Equation 62), we show that _{1, 2}, depend in a complicated way on the forms of the PRCs and the time constant of the noise, τ (see Appendix). However, all quantities can be found by integrating elementary functions. If the oscillators have the same PRC, then _{2} = 0 and _{2} and ω vanish, we can immediately solve the BVP, yielding _{1} − _{2} is nonzero, the peak of the phase difference density will generally be offset. We note that ϵ does not appear in the expression for

^{2}/2 where ϵ is the magnitude of the noise. Here Δ_{j}(θ_{j}) = sin(_{j}) − sin(θ_{j} + _{j}) + _{j} sin(2θ_{j}), where _{1} = 0.1, _{2} = 0.6, _{1} = 0.32, _{2} = 0.3, and _{1} = _{2} = 0.5, _{1} = _{2} = 0.3, and

We note that the density of the phase differences can be related to more conventional measures of correlation. In Burton et al. (

Figure

Our approximation of the invariant density, while requiring that we solve a BVP, allows us to explore the effects of heterogeneity much faster than simulating the appropriate Monte Carlo system. Thus, we will use this method to explore the effects of PRC heterogeneity, frequency differences, and the color of the noise on the ability of oscillators to synchronize. One simple global measure of synchrony/correlation for systems whose natural dynamics are periodic is the circular variance, σ_{circle} = 1 − OP, where we define an order parameter (OP) (see Materials and Methods, Equation 10):
_{circle} = 1) and for a delta function distribution, OP = 1 (σ_{circle} = 0). The OP is a commonly used measure for the degree of synchronization between two oscillators (Kuramoto,

In general, one expects that the synchrony between two oscillators forced with correlated noise would be greatest if the oscillators are homogeneous. Certainly, if the inputs are identical (i.e., no independent or unshared noise), then identical oscillators will synchronize perfectly, while heterogeneous oscillators will not synchronize perfectly. That is, the phase density will not be a delta function. [See Burton et al. (

_{j}(θ_{j}) = sin(_{j}) − sin(_{j} + θ_{j}) + _{j} sin(2θ_{j}), _{1} = 0.1, _{1} = 0.32, _{2} = 0.6, and _{6} = 0.3. _{1} = 110, ϕ_{1} = 0.04616, _{2} = 120, and ϕ_{2} = 0.04.

Could this subtle difference in the ability of neural oscillators to transfer correlation be seen in experiments? To answer this, we re-examined data from a previous study (Burton et al.,

Are the results presented in Figures

We then quantified the degree to which physiological levels of heterogeneity [as experimentally measured in mitral cells (Burton et al.,

When is a neuron a good vs. bad synchronizer? Here, the BVP is much simpler since we just have to compare homogeneous pairs. In this case, the probability density function can be written as:
^{2}_{0} where _{0} is the DC component of the PRC. Hence, we can minimize the integral and maximize the correlation transfer (susceptibility) if we mimimize the DC component of the PRC. This fact generalizes the conclusions in Marella and Ermentrout (

Can we determine when a pair of oscillators will have the property that a good-bad heterogeneous pair is better than a bad-bad homogeneous pair? Since the effect is only seen at low correlations, this suggests a perturbation expansion for small ^{n} _{n}(ϕ) and find that _{0} is constant and so to order 1, _{0} + _{1}(ϕ). From this, we can compute OP, ^{2π}_{0} cosϕ _{1}(ϕ) _{j}(ϕ) = sin(_{j}) − sin(ϕ + _{j}), where _{1} < _{2} ≤ π/2, we always have _{11} > _{12} > _{22} for all τ and sufficiently small values of

If two neurons are identical but driven with partially correlated noise, then the peak of the phase difference density will be centered at ϕ = 0, which means that the two oscillators will tend to have the same phase. However, with heterogeneity, the peak will shift depending on the degree of heterogeneity, just as two coupled oscillators will shift if they have different intrinsic frequencies. Figure _{1} = 0.1, _{1} = 0.32) as we vary PRC 2 (_{2} = 0.3 is constant and _{2} varies from −π to π). From the results shown in Figure _{2} = π) shifts the peak but has very little effect on the OP.

_{2} (black axis) and the peak position of the phase difference density vs. _{2} (grey axis). Parameters used: _{1} = 0.1, _{1} = 0.32, _{2} = 0.3, τ = 1, and

In the above results, we assume that all oscillators have the same natural frequency, which means ω = 0. This is a somewhat unreasonable assumption for real neurons. Thus, we now study how synchronization is dependent on the frequency differences between oscillators. Figure

_{1} = 0.1, _{1} = 0.32, _{2} = 0.6, _{2} = 0.3, τ = 1, and

Because of the natural decay times of synapses, broadband inputs into neurons have some temporal correlations. Thus, we now explore how the temporal properties of noise interact with heterogeneities in the PRCs. Figure

_{1} = 0.1, _{1} = 0.32, _{2} = 0.6, _{2} = 0.3, and

_{1} = 0.1, _{1} = 0.32, _{2} = 0.6, _{2} = 0.3, and

_{1} = 120, ϕ_{1} = 0.04, _{2} = 110, and ϕ_{2} = 0.04616). _{1} = 120, ϕ_{1} = 0.041, _{2} = 110, and ϕ_{2} = 0.04616).

We can see why the frequency differences are needed for resonance by considering the simplest example of identical PRCs of the form Δ(ϕ) = sin ^{2}) sin(^{2} + 1 −

In our current study, we have extended a number of previous results describing the ability of neural oscillators to synchronize in the presence of correlated noise. Our methods are similar to those in Burton et al. (_{nm}(ϕ) that, in turn, depend only on the PRCs (see Appendix, Equation 56). Thus, we could easily generalize this work to noise with an arbitrary autocorrelation function. In addition, we have now included many more examples of the theory and shown that the conclusions from the perturbation theory continue to be valid for full biophysical models (cf. Figure

We found that correlated noise can synchronize a heterogeneous pair of oscillators (comprised of a “bad synchronizer” and a “good synchronizer”) better than a homogeneous pair of bad synchronizers at low levels of input correlation and verified this experimentally. We showed that good (bad) synchronizers are characterized by having a relatively low (high) DC component in their PRC. Consistent with this, oscillators with the generic “type II” PRC (i.e., sin ϕ) are better synchronizers than oscillators with the generic “type I” PRC (i.e., 1 − cosϕ).

Several authors have previously studied the effects of heterogeneity on the transfer of correlation. As we noted in Introduction, at low correlations, the output correlation is linearly proportional to the input correlation through a factor,

Interestingly, the efficiency of correlated-noise induced synchronization is also modulated by firing rate in the low input correlation regime (de la Rocha et al.,

The ability of cellular heterogeneity to regulate which oscillators synchronize best as a function of input correlation likely contributes to coding in many neural systems. In the olfactory bulb, where oscillatory synchrony appears to be critical to olfactory coding [for review, see Bathellier et al. (

Our results suggest that heterogeneity can only enhance correlation-induced synchronization by a moderate amount between two neural oscillators (up to 36% in BVP solutions using mitral cell PRCs). Two properties of the olfactory bulb nevertheless suggest that even this moderate effect can significantly influence patterns of oscillatory synchrony in the olfactory system. First, the reciprocal dendrodendritic connectivity between mitral cells and granule cells enables activity-dependent regulation of granule cell recruitment (Arevian et al.,

In addition to the above findings, we found that there exists some resonance of correlated noise-induced synchronization with respect to the time scale of the noise. That is, we found a local maximum in OP as the time scale of the correlated noise varied. Surprisingly, this only occurs when there is a difference in the frequencies between the two oscillators. The requirement for the frequency difference would seem to contradict earlier work (Galán et al.,

The analysis that we have done in this paper and in our earlier papers requires that the neurons fire almost periodically. This means that the activity of the neurons is

In conclusion, we have extended our previous work to demonstrate that oscillator heterogeneity and frequency differences interact with the time scale of input noise to regulate how correlated noise synchronizes uncoupled oscillators.

All codes are available by request from the authors. They include Matlab and XPPaut files.

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.

This work was supported by National Institute on Drug Abuse Predoctoral Training Grant R90DA023426 and a R.K. Mellon Foundation Presidential Fellowship in the Life Sciences (Pengcheng Zhou), an Achievement Rewards for College Scientists Foundation Fellowship (Shawn D. Burton), National Institute on Deafness and Other Communication Disorders Grant 5R01DC011184-07 (Nathaniel N. Urban and G. Bard Ermentrout), and National Science Foundation grant DMS1219754 (G. Bard Ermentrout).

Consider a general oscillator receiving a possibly noisy time-dependent signal:
_{X}Θ(_{X}

In single compartment neuron models, inputs appear only in the voltage component of the neural oscillator in the form of external currents so that the dot product in Equation 31 becomes scalar multiplication:

The Langevin equations that drive the phase models (Equations 4–6) correspond to a forward Fokker-Planck (FP) equation that can be written as (Risken,

We expand the steady state ρ in ϵ:
^{2}:

Equation 37 is just a linear separable equation, independent of ϕ, so, by inspection:

Both Equations 38 and 39 have the form _{0}ρ = _{0} operates on the space of functions defined on ^{2}× ^{1} that are twice continuously differentiable in _{0} has a one-dimensional nullspace spanned by _{0} is not invertible. However, _{0}ρ(^{*}_{0}, which is the adjoint linear operator of _{0}. Since _{0} is a standard probability operator, its adjoint is always 1 (i.e., the function that is 1 everywhere).

Since:
_{1}(_{0}ρ_{1} = _{1} has a solution. Since:
_{j}(θ, ϕ) must satisfy:
_{j} must be periodic functions of θ; we defer their exact solution to later, but note that there is always a unique periodic solution to each of these equations.

We now have:
_{2}(_{1}(θ) and _{2}(θ) satisfy:
_{1}(θ) and _{2}(θ) in terms of _{1}(θ) and _{2}(θ) (see Appendix):
_{n}(ϕ) into _{1}(θ) and Δ_{2}(θ) are periodic functions,

We use a BVP solver to get the numerical solution for _{1}(ϕ) given _{0}(ϕ) and for some choices of PRCs we can get exact expressions. For example, for the two term double sinusoidal form PRC (Equation 63) we get:

For the PRC:

For empirical PRCs:

For the Fourier form of the PRC: