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Performed the experiments: NT. Contributed reagents/materials/analysis tools: NT JJS QCP. Wrote the paper: NT JJS QCP.

The authors have declared that no competing interests exist.

The functional role of synchronization has attracted much interest and debate: in particular, synchronization may allow distant sites in the brain to communicate and cooperate with each other, and therefore may play a role in temporal binding, in attention or in sensory-motor integration mechanisms. In this article, we study another role for synchronization: the so-called “collective enhancement of precision”. We argue, in a full nonlinear dynamical context, that synchronization may help protect interconnected neurons from the influence of random perturbations—intrinsic neuronal noise—which affect all neurons in the nervous system. More precisely, our main contribution is a mathematical proof that, under specific, quantified conditions, the impact of noise on individual interconnected systems and on their spatial mean can essentially be cancelled through synchronization. This property then allows reliable computations to be carried out even in the presence of significant noise (as experimentally found e.g., in retinal ganglion cells in primates). This in turn is key to obtaining meaningful downstream signals, whether in terms of precisely-timed interaction (temporal coding), population coding, or frequency coding. Similar concepts may be applicable to questions of noise and variability in systems biology.

Synchronization phenomena are pervasive in biology, creating collective behavior out of local interactions between neurons, cells, or animals. On the other hand, many of these systems function in the presence of large amounts of noise or disturbances, making one wonder how meaningful behavior can arise in these highly perturbed conditions. In this paper we show mathematically, in a general context, that synchronization is actually a means to

Synchronization phenomena are pervasive in biology. In neuronal networks

In this article, we study another role for synchronization: the so-called

It should be noted that “protection of systems from noise” and “robustness of synchronization to noise” are two different concepts. The latter concept means that the synchronized systems remain so in presence of noise, whereas the former concept means that, thanks to synchronization, the behaviors of the coupled systems are close to the noise-free behaviors. This difference is further addressed in the

The influence of noise on the behaviors of nonlinear systems is very diverse. In chaotic systems, a small amount of noise can yield dramatic effects. At the other end of the spectrum, the effect of noise on nonlinear

The dynamics of coupled FN oscillators are given by equation (2). The parameters used in all simulations are

One might argue that it could be possible to recover some information from the noisy FN oscillators by considering the activities of a large number of oscillators

Note that the same set of random initial conditions was used in the two plots. (A) shows the average “membrane potential” computed over

By contrast, one can observe that when oscillators are

Consider a diffusive network of

We consider four mathematical assumptions that will enable us to relate the trajectory of any noisy element of the network

The network is balanced, that is, for any element of the network, the sum of the incoming connection weights equals the sum of the outgoing connection weights

A particular kind of balanced network consists of an all-to-all network with identical couplings, i.e.

Let

The dynamics

In particular, such a property has been demonstrated in the case of FN oscillators, with

After exponential transients, the expected sum of the squared distances between the states of the elements of the network is bounded by a constant

We show in

We now give conditions to guarantee assumption (A4) for all-to-all networks of FN oscillators with identical couplings. The dynamics of

Note that the experimental expectations were computed assuming the ergodic hypothesis. (A) Expectation of the average squared distance between the

Assumption (A1) is also verified because an all-to-all network with identical couplings is symmetric, therefore balanced. Since the

Using now the “general analytical result”, we obtain that, given any (non necessarily small) noise intensity

This statement can be further tested by constructing a model-based nonlinear state estimator (observer)

If

We provide in this section simulation results which show that similar observations can be made even for more general network classes that are not yet covered by the theory. We believe that this simulations show the genericity of the concepts presented above.

In practice, all-to-all neuronal networks of large size are rare. Rather, the mechanisms of neuronal connections in the brain are believed to be probabilisitic in nature (see

(A) shows the trajectory of the “membrane potential” of an oscillator in the network. (B) shows its frequency spectrum. Compare these two plots with those in

Hindmarsh-Rose oscillators are three-dimensional dynamical systems that are also often used as neuron models

We made the inputs time-varying in this simulation. In fact, all the previous calculations can be straightforwardly extended to the case of time-varying inputs, as long as those inputs are the same for all the oscillators

One can observe from the simulations (see

(A) The time-varying input voltage. (B) Trajectory of the “membrane potential” of a noise-free oscillator. (C) Trajectory of a

We have argued that synchronization may represent a fundamental mechanism to protect neuronal assemblies from noise, and have quantified this hypothesis using a simple nonlinear neuron model. This may further strengthen our understanding of synchronization in the brain as playing a key functional role, rather than as being mostly an epiphenomenon.

It should be noted that the causal relationship studied here – effect of synchronization on noise – is converse to one usually investigated formally in the literature – effect of noise on synchronization: under certain conditions, adding noise can de-synchronize already synchronized oscillators (destructive effect)

The mechanisms highlighted in the paper may also underly other types of “redundant” calculations in the presence of noise and variability. In otoliths for instance, ten of thousands of hair cells jointly compute the three components of acceleration

Finally, the results point to the general question: what is the precise meaning of ensemble measurements or population codes, what information do they convey about the underlying signals, and is the presence of synchronization mechanisms (gap-junction mediated or other) implicit in this interpretation? As such, they may also shed light on a somewhat “dual” and highly controversial current issue. Ensemble measurements from the brain can correlate to behavior, and they have been suggested e.g. as inputs to brain-machine interfaces. Are these ensemble signals actually available to the brain

In the noise-free case (

In the presence of noise, it is not clear how to relate the trajectory of each

To be more precise, let

Turning now to the noise term

Thus, for a given (even large) noise intensity

Consider first the case of two coupled FN oscillators driven by Equation (2). Construct the following auxiliary system (or virtual system, in the sense of

Let

Given these asymptotes, the evolution matrix of system (17) is diagonalizable with eigenvalues

Consider now an all-to-all network with identical couplings as in Equation (2). Construct as above the following

Assumption (A1) is also verified because an all-to-all network with identical couplings is symmetric, therefore balanced. As for (A2), observe that

The authors would like to thank the anonymous reviewers for their helpful comments. JJS is grateful to Uri Alon for stimulating discussions on possible relevance of the results to cell biology.

This publication reflects only the authors' views. The European Community is not liable for any use that may be made of the information contained therein.