Coherent neural oscillations induced by weak synaptic noise

Abstract

We analyze the effect of synaptic noise on the dynamics of the FitzHugh-Nagumo (FHN) neuron model. In our deterministic parameter regime, a limit cycle solution cannot emerge through a singular Hopf bifurcation, but such a limit cycle can nevertheless arise as a stochastic effect, as a consequence of weak synaptic noise in a regime of strong timescale separation (ε → 0) between the slow and fast variables of the model. We investigate the mechanism behind this phenomenon, known as self-induced stochastic resonance (SISR) (Muratov et al. in Physica D 210:227-240, 2005), by using multiple-time perturbation techniques and by analyzing the escape mechanism of the random trajectories from the stable manifolds of the model equation. Even though SISR occurs only in the limit as the singular parameter ε → 0, decreasing ε does not increase the coherence of the oscillations in the FHN model, but rather increases the interval of the noise amplitude σ for which coherence occurs. This is in contrast to the dynamical system studied in Mura- 2005). Moreover, the phenomenon is robust under parameter tuning. Numerical simulations exhibit the results predicted by the theoretical analysis. In fact, our analysis together with that in Yamakou and Jost (Weak noise-induced transitions with inhibition and modulation of neural oscillations, 2017) reveals that the FHN model can support different stochastic resonance phenomena. While it had been known (Pikovsky and Kurths in Phys Rev Lett 78:775-778, 1997) that coherence resonance can occur when the slow variable is subjected to noise, we show that, when noise is added to the fast variable, two other types, inverse stochastic resonance and SISR, may emerge in the same weak noise limit and that the transition between these phenomena can be induced by varying a simple parameter.

Control of coherence resonance by self-induced stochastic resonance in a multiplex neural network

“…In particular, in a weak coupling (κ 1 → 0) regime and short time delay (τ 1 → 0) regime (or more precisely, v 1j (t−τ 1 )−v 1i (t) → 0 as τ 1 → 0, because all the oscillators are identical), F (κ 1 , τ 1 , n 1 ) → 3 4 . In these regimes (see Fig.6(a) and (c)), we observe that σ max 1 ≈ 10 −2 , that is the value obtained in [19] for the case of a single isolated (κ 1 = 0) FHN neuron.…”

“…In Fig.6(a) and (c) (i.e., in the weak coupling and short time delay regimes, respectively), we notice that σ max 1 = O(10 −2 ) is almost fixed for all the values of the time delay and coupling strength used. This fixation of the upper bound of the noise interval in which SISR occurs was already observed in a single isolated FHN neuron [19]. In the case of a single isolated FHN neuron, the function in Eq.7 does not depend on any system parameter and takes a simple constant value F = 3 4 .…”

Optimal Self-Induced Stochastic Resonance in Multiplex Neural Networks: Electrical vs. Chemical Synapses

“…) by taking the difference between the potential function value at the saddle point v * m (w l,i ) and at the local minima v * l,r (w l,i ) of these interaction potentials (Yamakou and Jost, 2018). The energy barriers △U le i w * l,i or △U lc i w * l,i (which has to be crossed to induce a spike) is the value of the left energy barrier function at the w l,i -coordinate of the stable homogeneous steady state…”

“…Some mechanisms for optimal information processing are provided via the well-known and extensively studied phenomena of stochastic resonance (SR) (Benzi et al, 1981 ; Longtin, 1993 ; Gammaitoni et al, 1998 ; Lindner et al, 2004 ; Zhang et al, 2015 ) and coherence resonance (CR) (Hu and MacDonald, 1993 ; Neiman et al, 1997 ; Pikovsky and Kurths, 1997 ; Lindner and Schimansky-Geier, 1999 ; Lindner et al, 2004 ; Beato et al, 2007 ; Hizanidis and Schöll, 2008 ; Liu et al, 2010 ; Bing et al, 2011 ; Gu et al, 2011 ) or via the lesser-known phenomenon of self-induced stochastic resonance (SISR) (Freidlin, 2001 ; Muratov et al, 2005 ; DeVille and Vanden-Eijnden, 2007 ; DeVille et al, 2007 ; Yamakou and Jost, 2017 , 2018 ) whose mechanism remains to be confirmed experimentally in real neural systems. Although these noise-induced phenomena may exhibit similar dynamical behaviors, each of them has different dynamical preconditions and emergent mechanisms and may therefore play different functional roles in information processing.…”

Weak-noise-induced transitions with inhibition and modulation of neural oscillations

“…To obtain bi-stability, in the present work, a careful relative positioning of the fixed point on the critical manifold was made. We can see in (Yamakou and Jost 2018 ) how a change in the relative position of the fixed and fold points on the critical manifold brings about a completely different dynamical behavior in the same weak-noise limit. Plausible implications of ISR in information processing and transmission in neurons are discussed in (Buchin et al.…”