Periodic bursting oscillations in a hybrid Rayleigh–Van der Pol–Duffing oscillator

Bottom-up models of functionally relevant patterns of neural activity provide an explicit link between neuronal dynamics and computation. A prime example of functional activity patterns are propagating bursts of place-cell activities called hippocampal replay, which is critical for memory consolidation. The sudden and repeated occurrences of these burst states during ongoing neural activity suggest metastable neural circuit dynamics. As metastability has been attributed to noise and/or slow fatigue mechanisms, we propose a concise mesoscopic model which accounts for both. Crucially, our model is bottom-up: it is analytically derived from the dynamics of finite-size networks of Linear-Nonlinear Poisson neurons with short-term synaptic depression. As such, noise is explicitly linked to stochastic spiking and network size, and fatigue is explicitly linked to synaptic dynamics. To derive the mesoscopic model, we first consider a homogeneous spiking neural network and follow the temporal coarse-graining approach of Gillespie to obtain a “chemical Langevin equation”, which can be naturally interpreted as a stochastic neural mass model. The Langevin equation is computationally inexpensive to simulate and enables a thorough study of metastable dynamics in classical setups (population spikes and Up-Down-states dynamics) by means of phase-plane analysis. An extension of the Langevin equation for small network sizes is also presented. The stochastic neural mass model constitutes the basic component of our mesoscopic model for replay. We show that the mesoscopic model faithfully captures the statistical structure of individual replayed trajectories in microscopic simulations and in previously reported experimental data. Moreover, compared to the deterministic Romani-Tsodyks model of place-cell dynamics, it exhibits a higher level of variability regarding order, direction and timing of replayed trajectories, which seems biologically more plausible and could be functionally desirable. This variability is the product of a new dynamical regime where metastability emerges from a complex interplay between finite-size fluctuations and local fatigue.

/pdf/mesoscopic-description-of-hippocampal-replay-and-1atz8ui3.pdf

Mesoscopic description of hippocampal replay and metastability in spiking neural networks with short-term plasticity

Astrocytes crucially contribute to synaptic physiology and information processing. One of their key characteristics is to express high levels of connexins (Cxs), the gap junction–forming protein. Among them, Cx30 displays specific properties since it is postnatally expressed and dynamically upregulated by neuronal activity and modulates cognitive processes by shaping synaptic and network activities, as recently shown in knockout mice. However, it remains unknown whether local and selective upregulation of Cx30 in postnatal astrocytes within a physiological range modulates neuronal activities in the hippocampus. We here show in mice that, whereas Cx30 upregulation increases the connectivity of astroglial networks, it decreases spontaneous and evoked synaptic transmission. This effect results from a reduced neuronal excitability and translates into an alteration in the induction of synaptic plasticity and an in vivo impairment in learning processes. Altogether, these results suggest that astroglial networks have a physiologically optimized size to appropriately regulate neuronal functions.

/pdf/upregulation-of-astroglial-connexin-30-impairs-hippocampal-3ua9yxp8.pdf

Upregulation of astroglial connexin 30 impairs hippocampal synaptic activity and recognition memory

When does a diffusing particle reach its target for the first time? This first-passage time (FPT) problem is central to the kinetics of molecular reactions in chemistry and molecular biology. Here we explain the behavior of smooth FPT densities, for which all moments are finite, and demonstrate universal yet generally non-Poissonian long-time asymptotics for a broad variety of transport processes. While Poisson-like asymptotics arise generically in the presence of an effective repulsion in the immediate vicinity of the target, a time-scale separation between direct and reflected indirect trajectories gives rise to a universal proximity effect: Direct paths, heading more or less straight from the point of release to the target, become typical and focused, with a narrow spread of the corresponding first passage times. Conversely, statistically dominant indirect paths exploring the system size tend to be massively dissimilar. The initial distance to the target particularly impacts gene regulatory or competitive stochastic processes, for which often few binding events determine the regulatory outcome. The proximity effect is independent of details of the transport, highlighting the robust character of the FPT features uncovered here.

Universal proximity effect in target search kinetics in the few-encounter limit

Exit Versus Escape for Stochastic Dynamical Systems and Application to the Computation of the Bursting Time Duration in Neuronal Networks

Kramer's theory of activation over a potential barrier consists in computing the mean exit time from the boundary of a basin of attraction of a randomly perturbed dynamical system. Here we report that for some systems, crossing the boundary is not enough, because stochastic trajectories return inside the basin with a high probability a certain number of times before escaping far away. This situation is due to a shallow potential. We compute the mean and distribution of escape times and show how this result explains the large distribution of interburst durations in neuronal networks.

Escape from an attractor generated by recurrent exit and application to interburst duration in neuronal network

Rhythmic neuronal network activity underlies brain oscillations. To investigate how connected neuronal networks contribute to the emergence of the α-band and the regulation of Up and Down states, we study a model based on synaptic short-term depression-facilitation with afterhyperpolarization (AHP). We found that the α-band is generated by the network behavior near the attractor of the Up-state. Coupling inhibitory and excitatory networks by reciprocal connections leads to the emergence of a stable α-band during the Up states, as reflected in the spectrogram. To better characterize the emergence and stability of thalamocortical oscillations containing α and δ rhythms during anesthesia, we model the interaction of two excitatory with one inhibitory networks, showing that this minimal network topology leads to a persistent α-band in the neuronal voltage characterized by dominant Up over Down states. Finally, we show that the emergence of the α-band appears when external inputs are suppressed, while the fragmentation occurs at small synaptic noise or with increasing inhibitory inputs. To conclude, interaction between excitatory neuronal networks with and without AHP seems to be a general principle underlying changes in network oscillations that could apply to other rhythms. Author summary Brain oscillations recorded from electroencephalograms characterize behaviors such as sleep, wakefulness, brain evoked responses, coma or anesthesia. The underlying rhythms for these oscillations are associated at a neuronal population level to fluctuations of the membrane potential between Up (depolarized) and Down (hyperpolarized) states. During anesthesia with propofol, a dominant alpha-band (8-12Hz) can emerge or disappear, but the underlying mechanisms remain unclear. Using modeling, we report that the alpha-band appears during Up states in neuronal populations driven by short-term synaptic plasticity and noise. Moreover, we show that three connected networks representing the thalamocortical loop reproduce the dynamics of the alpha-band, which emerges following the arrest of excitatory stimulations, but can disappear by increasing inhibitory inputs. To conclude, short-term plasticity in well connected neuronal networks can explain the emergence and fragmentation of the alpha-band.

/pdf/emergence-and-fragmentation-of-the-alpha-band-driven-by-43k159lojp.pdf

Emergence and fragmentation of the alpha-band driven by neuronal network dynamics

This article shows that stochastic trajectories can return inside the basin several times before escaping far away, thus increasing the total escape time.

Escape from an attractor generated by recurrent exit

Neuronal networks can generate burst events. It remains unclear how to analyse interburst periods and their statistics. We study here the phase-space of a mean-field model, based on synaptic short-term changes, that exhibit burst and interburst dynamics and we identify that interburst corresponds to the escape from a basin of attraction. Using stochastic simulations, we report here that the distribution of the these durations do not match with the time to reach the boundary. We further analyse this phenomenon by studying a generic class of two-dimensional dynamical systems perturbed by small noise that exhibits two peculiar behaviors: 1- the maximum associated to the probability density function is not located at the point attractor, which came as a surprise. The distance between the maximum and the attractor increases with the noise amplitude $\sigma$, as we show using WKB approximation and numerical simulations. 2- For such systems, exiting from the basin of attraction is not sufficient to characterize the entire escape time, due to trajectories that can return several times inside the basin of attraction after crossing the boundary, before eventually escaping far away. To conclude, long-interburst durations are inherent properties of the dynamics and sould be expected in empirical time series.

Exit versus escape in a stochastic dynamical system of neuronal networks explains heterogenous bursting intervals

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