Augmented moment method for stochastic ensembles with delayed couplings. II. FitzHugh-Nagumo model

Degeneracy in the robust expression of spectral selectivity, subthreshold oscillations, and intrinsic excitability of entorhinal stellate cells

Biological heterogeneities are ubiquitous and play critical roles in the emergence of physiology at multiple scales. Although neurons in layer II (LII) of the medial entorhinal cortex (MEC) express heterogeneities in channel properties, the impact of such heterogeneities on the robustness of their cellular-scale physiology has not been assessed. Here, we performed a 55-parameter stochastic search spanning nine voltage- or calcium-activated channels to assess the impact of channel heterogeneities on the concomitant emergence of 10 in vitro electrophysiological characteristics of LII stellate cells (SCs). We generated 150,000 models and found a heterogeneous subpopulation of 449 valid models to robustly match all electrophysiological signatures. We employed this heterogeneous population to demonstrate the emergence of cellular-scale degeneracy in SCs, whereby disparate parametric combinations expressing weak pairwise correlations resulted in similar models. We then assessed the impact of virtually knocking out each channel from all valid models and demonstrate that the mapping between channels and measurements was many-to-many, a critical requirement for the expression of degeneracy. Finally, we quantitatively predict that the spike-triggered average of SCs should be endowed with theta-frequency spectral selectivity and coincidence detection capabilities in the fast gamma-band. We postulate this fast gamma-band coincidence detection as an instance of cellular-scale-efficient coding, whereby SC response characteristics match the dominant oscillatory signals in LII MEC. The heterogeneous population of valid SC models built here unveils the robust emergence of cellular-scale physiology despite significant channel heterogeneities, and forms an efficacious substrate for evaluating the impact of biological heterogeneities on entorhinal network function.

NEW & NOTEWORTHY We assessed the impact of heterogeneities in channel properties on the robustness of cellular-scale physiology of medial entorhinal cortical stellate neurons. We demonstrate that neuronal models with disparate channel combinations were endowed with similar physiological characteristics, as a consequence of the many-to-many mapping between channel properties and the physiological characteristics that they modulate. We predict that the spike-triggered average of stellate cells should be endowed with theta-frequency spectral selectivity and fast gamma-band coincidence detection capabilities.

Persistence and failure of mean-field approximations adapted to a class of systems of delay-coupled excitable units

We consider the approximations behind the typical mean-field model derived for a class of systems made up of type II excitable units influenced by noise and coupling delays. The formulation of the two approximations, referred to as the Gaussian and the quasi-independence approximation, as well as the fashion in which their validity is verified, are adapted to reflect the essential properties of the underlying system. It is demonstrated that the failure of the mean-field model associated with the breakdown of the quasi-independence approximation can be predicted by the noise-induced bistability in the dynamics of the mean-field system. As for the Gaussian approximation, its violation is related to the increase of noise intensity, but the actual condition for failure can be cast in qualitative, rather than quantitative terms. We also discuss how the fulfillment of the mean-field approximations affects the statistics of the first return times for the local and global variables, further exploring the link between the fulfillment of the quasi-independence approximation and certain forms of synchronization between the individual units.

Mean-field approximation of two coupled populations of excitable units

The analysis on stability and bifurcations in the macroscopic dynamics exhibited by the system of two coupled large populations composed of N stochastic excitable units each is performed by studying an approximate system, obtained by replacing each population with the corresponding mean-field model. In the exact system, one has the units within an ensemble communicating via the time-delayed linear couplings, whereas the interensemble terms involve the nonlinear time-delayed interaction mediated by the appropriate global variables. The aim is to demonstrate that the bifurcations affecting the stability of the stationary state of the original system, governed by a set of 4N stochastic delay-differential equations for the microscopic dynamics, can accurately be reproduced by a flow containing just four deterministic delay-differential equations which describe the evolution of the mean-field based variables. In particular, the considered issues include determining the parameter domains where the stationary state is stable, the scenarios for the onset, and the time-delay induced suppression of the collective mode, as well as the parameter domains admitting bistability between the equilibrium and the oscillatory state. We show how analytically tractable bifurcations occurring in the approximate model can be used to identify the characteristic mechanisms by which the stationary state is destabilized under different system configurations, like those with symmetrical or asymmetrical interpopulation couplings.

Stability, bifurcations, and dynamics of global variables of a system of bursting neurons

An approximate mean field model of an ensemble of delayed coupled stochastic Hindmarsh-Rose bursting neurons is constructed and analyzed. Bifurcation analysis of the approximate system is performed using numerical continuation. It is demonstrated that the stability domains in the parameter space of the large exact systems are correctly estimated using the much simpler approximate model.

Generalized rate-code model for neuron ensembles with finite populations

We have proposed a generalized Langevin-type rate-code model subjected to multiplicative noise, in order to study stationary and dynamical properties of an ensemble containing a finite number N of neurons. Calculations using the Fokker-Planck equation have shown that, owing to the multiplicative noise, our rate model yields various kinds of stationary non-Gaussian distributions such as Gamma , inverse-Gaussian-like, and log-normal-like distributions, which have been experimentally observed. The dynamical properties of the rate model have been studied with the use of the augmented moment method (AMM), which was previously proposed by the author from a macroscopic point of view for finite-unit stochastic systems. In the AMM, the original N -dimensional stochastic differential equations (DEs) are transformed into three-dimensional deterministic DEs for the means and fluctuations of local and global variables. The dynamical responses of the neuron ensemble to pulse and sinusoidal inputs calculated by the AMM are in good agreement with those obtained by direct simulation. The synchronization in the neuronal ensemble is discussed. The variabilities of the firing rate and of the interspike interval are shown to increase with increasing magnitude of multiplicative noise, which may be a conceivable origin of the observed large variability in cortical neurons.

Theoretical analysis of destabilization resonances in time-delayed stochastic second-order dynamical systems and some implications for human motor control

A linear stochastic delay differential equation of second order is studied that can be regarded as a Kramers model with time delay. An analytical expression for the stationary probability density is derived in terms of a Gaussian distribution. In particular, the variance as a function of the time delay is computed analytically for several parameter regimes. Strikingly, in the parameter regime close to the parameter regime in which the deterministic system exhibits Hopf bifurcations, we find that the variance as a function of the time delay exhibits a sequence of pronounced peaks. These peaks are interpreted as delay-induced destabilization resonances arising from oscillatory ghost instabilities. On the basis of the obtained theoretical findings, reinterpretations of previous human motor control studies and predictions for future human motor control studies are provided.

Delay Fokker-Planck equations, Novikov's theorem, and Boltzmann distributions as small delay approximations

We study time-delayed stochastic systems that can be described by means of so-called delay Fokker-Planck equations. Using Novikov's theorem, we first show that the theory of delay Fokker-Planck equations is on an equal footing with the theory of ordinary Fokker-Planck equations. Subsequently, we derive stationary distributions in the case of small time delays. In the case of additive noise systems, these distributions can be cast into the form of Boltzmann distributions involving effective potential functions.

Dynamical mean-field approximation to small-world networks of spiking neurons: from local to global and/or from regular to random couplings

By extending a dynamical mean-field approximation previously proposed by the author [Phys. Rev. E 67, 041903 (2003)]], we have developed a semianalytical theory which takes into account a wide range of couplings in a small-world network. Our network consists of noisy N -unit FitzHugh-Nagumo neurons with couplings whose average coordination number Z may change from local ( Z<<N ) to global couplings ( Z=N-1 ) and/or whose concentration of random couplings p is allowed to vary from regular ( p=0 ) to completely random (p=1) . We have taken into account three kinds of spatial correlations: the on-site correlation, the correlation for a coupled pair, and that for a pair without direct couplings. The original 2N -dimensional stochastic differential equations are transformed to 13-dimensional deterministic differential equations expressed in terms of means, variances, and covariances of state variables. The synchronization ratio and the firing-time precision for an applied single spike have been discussed as functions of Z and p . Our calculations have shown that with increasing p , the synchronization is worse because of increased heterogeneous couplings, although the average network distance becomes shorter. Results calculated by our theory are in good agreement with those by direct simulations.

Augmented moment method for stochastic ensembles with delayed couplings. I. Langevin model

By employing a semianalytical dynamical mean-field approximation theory previously proposed by the author [H. Hasegawa, Phys. Rev. E 67, 041903 (2003)], we have developed an augmented moment method (AMM) in order to discuss dynamics of an N -unit ensemble described by Langevin equations with delays. In an AMM, original N -dimensional stochastic delay differential equations (SDDEs) are transformed to infinite-dimensional deterministic DEs for means and correlations of local as well as global variables. Infinite-order DEs arising from the non-Markovian property of SDDE, are terminated at the finite level m in the level-m AMM (AMMm), which yields (3+m)-dimensional deterministic DEs. Model calculations have been made for linear and nonlinear Langevin models. The stationary solution of AMM for the linear Langevin model with N=1 is nicely compared to the exact result. In the nonlinear Langevin ensemble, the synchronization is shown to be enhanced near the transition point between the oscillating and nonoscillating states. Results calculated by AMM6 are in good agreement with those obtained by direct simulations.