How synchronization protects from noise

Abstract

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.

Figure 1. Simulations of a network of FN oscillators using the Euler-Maruyama algorithm .

The dynamics of coupled FN oscillators are given by equation (2). The parameters used in all simulations are , , . (A) shows the trajectory of the “membrane potential” of a noise-free oscillator and (B) depicts the frequency spectrum of this trajectory computed by Fast Fourier Transformation. (C) and (D) present the trajectory (respectively the frequency spectrum) of a noisy uncoupled oscillator (). (E) and (F) show the trajectory (respectively the frequency spectrum) of a noisy synchronized oscillator within an all-to-all network (, , ). Note the temporal and frequential similarities between a noise-free oscillator and a noisy synchronized one. For instance, the main frequency and the first harmonics are very similar in the two frequency spectra. In contrast, the frequency spectrum of a noisy uncoupled oscillator shows no clear harmonics.

Figure 2. “Spatial mean” of FN oscillators.

Note that the same set of random initial conditions was used in the two plots. (A) shows the average “membrane potential” computed over noisy uncoupled oscillators (). (B) shows the average “membrane potential” computed over noisy synchronized oscillators within an all-to-all network (, ). Observe that, in the first plot, the average trajectory of uncoupled oscillators carries essentially no information, while in the second plot, the average trajectory of synchronized oscillators is very similar to a noise-free one.

Figure 3. Asymptotic appraisal of the theoretical bounds.

Note that the experimental expectations were computed assuming the ergodic hypothesis. (A) Expectation of the average squared distance between the 's and (given by ) as a function of the coupling strength (). Theoretical bound (cf equations (7) and (4)) for (bold line), for (plain line), for (dashed line); simulation results for (squares), for (triangles), for (crosses). (B) Expected squared distance between a noisy synchronized oscillator and its observer (given by ) as a function of (, ). The bound was plotted in plain line and the simulation results were represented by crosses.

Figure 4. Simulation for a probabilistic symmetric network (, , , ).

(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 Figure 1.

Figure 5. Simulation of Hindmarsh-Rose oscillators with time varying inputs.

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