Abstract
Many healthy and pathological brain rhythms, including beta and gamma rhythms and essential tremor, are suspected to be induced by noise. This yields randomly occurring, brief epochs of higher amplitude oscillatory activity known as "bursts," the statistics of which are important for proper neural function. Here, we consider a more realistic model with both multiplicative and additive noise instead of only additive noise, to understand how state-dependent fluctuations further affect rhythm induction. For illustrative purposes, we calibrate the model at the lower end of the beta band that relates to movement; parameter tuning can extend the relevance of our analysis to the higher frequency gamma band or to lower frequency essential tremors. A stochastic Wilson-Cowan model for reciprocally as well as self-coupled excitatory (E) and inhibitory (I) populations is analyzed in the parameter regime where the noise-free dynamics spiral in to a fixed point. Noisy oscillations known as quasi-cycles are then generated by stochastic synaptic inputs. The corresponding dynamics of E and I local field potentials can be studied using linear stochastic differential equations subject to both additive and multiplicative noises. As the prevalence of bursts is proportional to the slow envelope of the E and I firing activities, we perform an envelope-phase decomposition using the stochastic averaging method. The resulting envelope dynamics are uni-directionally coupled to the phase dynamics as in the case of additive noise alone but both dynamics involve new noise-dependent terms. We derive the stationary probability and compute power spectral densities of envelope fluctuations. We find that multiplicative noise can enhance network synchronization by reducing the magnitude of the negative real part of the complex conjugate eigenvalues. Higher noise can lead to a "virtual limit cycle," where the deterministically stable eigenvalues around the fixed point acquire a positive real part, making the system act more like a noisy limit cycle rather than a quasi-cycle. Multiplicative noise can thus exacerbate synchronization and possibly contribute to the onset of symptoms in certain motor diseases.
Highlights
Oscillations are recorded in the brain of several species.1They are grouped depending on the values of their mean frequencies
As in previous models with only additive noise, we find that the envelope dynamics are uncoupled from the phase dynamics to lowest order of approximation; this is generally possible with special conditions on the noise processes of the excitatory and inhibitory local field potentials (LFPs)
As for the case of the LFP sustained by only additive noises, we found that in the quasi-cycle regime, the amplitude ratio and phase difference between inhibitory and excitatory populations converge to a fixed value
Summary
They are grouped depending on the values of their mean frequencies. Among the most prevalent are beta (13–30 Hz) and. Our results show again that the oscillations have larger mean amplitude and higher envelope values when we are close to a Hopf bifurcation, but this effect is amplified by multiplicative noise This noise acts like a deterministic synchronizing effect, in the sense that it increases the real part of the complex conjugate eigenvalues around the fixed point and, pushes the system closer to the Hopf bifurcation. An envelope-phase reduction using the stochastic averaging method is applied to the linear SDE with additive and multiplicative noise, revealing the novel effect of the multiplicative noise on the envelope and phase dynamics: at low noise one sees a noise-induced oscillation, i.e., a quasi-cycle, around an otherwise stable fixed point, and a higher noise causes the effective damping rate to become smaller, even positive, creating a stronger noise-induced limit cycle. A discussion follows, which comments on the possible relevance of our work for understanding pathological brain rhythms
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