White-noise susceptibility and critical slowing in neurons near spiking threshold

Electromagnetic induction effects on electrical activity within a memristive Wilson neuron model

Neurons can exhibit abundant electrical activities due to physical effects of various electrophysiology environments. The electromagnetic induction flows can be triggered by changes in neuron membrane potential, which can be equivalent to a memristor applying on membrane potential. To imitate the electromagnetic induction effects, we propose a three-variable memristor-based Wilson neuron model. Using several kinetic analysis methods, the memristor parameter- and initial condition-related electrical activities are explored intensively. It is revealed that the memristive Wilson neuron model can display rich electrical activities, including the asymmetric coexisting electrical activities and antimonotonicity phenomenon. Finally, using off-the-shelf discrete components, an analog circuit on a hardware level is implemented to verify the numerically simulated coexisting electrical activities. Studying these rich electrical activities in neurons can build the groundwork to widen the neuron-based engineering applications.

Anesthesia modifies subthreshold critical slowing down in a stochastic Hodgkin-Huxley-like model with inhibitory synaptic input

The dynamics of a stochastic type-I Hodgkin-Huxley-like point neuron model exposed to inhibitory synaptic noise are investigated as a function of distance from spiking threshold and the inhibitory influence of the general anesthetic agent propofol. The model is biologically motivated and includes the effects of intrinsic ion-channel noise via a stochastic differential equation description as well as inhibitory synaptic noise modeled as multiple Poisson-distributed impulse trains with saturating response functions. The effect of propofol on these synapses is incorporated through this drug's principal influence on fast inhibitory neurotransmission mediated by γ-aminobutyric acid (GABA) type-A receptors via reduction of the synaptic response decay rate. As the neuron model approaches spiking threshold from below, we track membrane voltage fluctuation statistics of numerically simulated stochastic trajectories. We find that for a given distance from spiking threshold, increasing the magnitude of anesthetic-induced inhibition is associated with augmented signatures of critical slowing: fluctuation amplitudes and correlation times grow as spectral power is increasingly focused at 0 Hz. Furthermore, as a function of distance from threshold, anesthesia significantly modifies the power-law exponents for variance and correlation time divergences observable in stochastic trajectories. Compared to the inverse square root power-law scaling of these quantities anticipated for the saddle-node bifurcation of type-I neurons in the absence of anesthesia, increasing anesthetic-induced inhibition results in an observable exponent <-0.5 for variance and >-0.5 for correlation time divergences. However, these behaviors eventually break down as distance from threshold goes to zero with both the variance and correlation time converging to common values independent of anesthesia. Compared to the case of no synaptic input, linearization of an approximating multivariate Ornstein-Uhlenbeck model reveals these effects to be the consequence of an additional slow eigenvalue associated with synaptic activity that competes with those of the underlying point neuron in a manner that depends on distance from spiking threshold.

From individual spiking neurons to population behavior: Systematic elimination of short-wavelength spatial modes

Mean-field models of the brain approximate spiking dynamics by assuming that each neuron responds to its neighbors via a naive spatial average that neglects local fluctuations and correlations in firing activity. In this paper we address this issue by introducing a rigorous formalism to enable spatial coarse-graining of spiking dynamics, scaling from the microscopic level of a single type 1 (integrator) neuron to a macroscopic assembly of spiking neurons that are interconnected by chemical synapses and nearest-neighbor gap junctions. Spiking behavior at the single-neuron scale ℓ≈10μm is described by Wilson's two-variable conductance-based equations [H. R. Wilson, J. Theor. Biol. 200, 375 (1999)], driven by fields of incoming neural activity from neighboring neurons. We map these equations to a coarser spatial resolution of grid length Bℓ, with B≫1 being the blocking ratio linking micro and macro scales. Our method systematically eliminates high-frequency (short-wavelength) spatial modes q(->) in favor of low-frequency spatial modes Q(->) using an adiabatic elimination procedure that has been shown to be equivalent to the path-integral coarse graining applied to renormalization group theory of critical phenomena. This bottom-up neural regridding allows us to track the percolation of synaptic and ion-channel noise from the single neuron up to the scale of macroscopic population-average variables. Anticipated applications of neural regridding include extraction of the current-to-firing-rate transfer function, investigation of fluctuation criticality near phase-transition tipping points, determination of spatial scaling laws for avalanche events, and prediction of the spatial extent of self-organized macrocolumnar structures. As a first-order exemplar of the method, we recover nonlinear corrections for a coarse-grained Wilson spiking neuron embedded in a network of identical diffusively coupled neurons whose chemical synapses have been disabled. Intriguingly, we find that reblocking transforms the original type 1 Wilson integrator into a type 2 resonator whose spike-rate transfer function exhibits abrupt spiking onset with near-vertical takeoff and chaotic dynamics just above threshold.

Noise-induced precursors of state transitions in the stochastic Wilson-cowan model

The Wilson-Cowan neural field equations describe the dynamical behavior of a 1-D continuum of excitatory and inhibitory cortical neural aggregates, using a pair of coupled integro-differential equations. Here we use bifurcation theory and small-noise linear stochastics to study the range of a phase transitions-sudden qualitative changes in the state of a dynamical system emerging from a bifurcation-accessible to the Wilson-Cowan network. Specifically, we examine saddle-node, Hopf, Turing, and Turing-Hopf instabilities. We introduce stochasticity by adding small-amplitude spatio-temporal white noise, and analyze the resulting subthreshold fluctuations using an Ornstein-Uhlenbeck linearization. This analysis predicts divergent changes in correlation and spectral characteristics of neural activity during close approach to bifurcation from below. We validate these theoretical predictions using numerical simulations. The results demonstrate the role of noise in the emergence of critically slowed precursors in both space and time, and suggest that these early-warning signals are a universal feature of a neural system close to bifurcation. In particular, these precursor signals are likely to have neurobiological significance as early warnings of impending state change in the cortex. We support this claim with an analysis of the in vitro local field potentials recorded from slices of mouse-brain tissue. We show that in the period leading up to emergence of spontaneous seizure-like events, the mouse field potentials show a characteristic spectral focusing toward lower frequencies concomitant with a growth in fluctuation variance, consistent with critical slowing near a bifurcation point. This observation of biological criticality has clear implications regarding the feasibility of seizure prediction.

Channel-noise-induced critical slowing in the subthreshold Hodgkin-Huxley neuron

The dynamics of a spiking neuron approaching threshold is investigated in the framework of Markov-chain models describing the random state-transitions of the underlying ion-channel proteins. We characterize subthreshold channel-noise-induced transmembrane potential fluctuations in both type-I (integrator) and type-II (resonator) parametrizations of the classic conductance-based Hodgkin-Huxley equations. As each neuron approaches spiking threshold from below, numerical simulations of stochastic trajectories demonstrate pronounced growth in amplitude simultaneous with decay in frequency of membrane voltage fluctuations induced by ion-channel state transitions. To explore this progression of fluctuation statistics, we approximate the exact Markov treatment with a 12-variable channel-based stochastic differential equation (SDE) and its Ornstein-Uhlenbeck (OU) linearization and show excellent agreement between Markov and SDE numerical simulations. Predictions of the OU theory with respect to membrane potential fluctuation variance, autocorrelation, correlation time, and spectral density are also in agreement and illustrate the close connection between the eigenvalue structure of the associated deterministic bifurcations and the observed behavior of the noisy Markov traces on close approach to threshold for both integrator and resonator point-neuron varieties.

Critical slowing down governs the transition to neuron spiking

Many complex systems have been found to exhibit critical transitions, or so-called tipping points, which are sudden changes to a qualitatively different system state. These changes can profoundly impact the functioning of a system ranging from controlled state switching to a catastrophic break-down; signals that predict critical transitions are therefore highly desirable. To this end, research efforts have focused on utilizing qualitative changes in markers related to a system’s tendency to recover more slowly from a perturbation the closer it gets to the transition—a phenomenon called critical slowing down. The recently studied scaling of critical slowing down offers a refined path to understand critical transitions: to identify the transition mechanism and improve transition prediction using scaling laws.

Here, we outline and apply this strategy for the first time in a real-world system by studying the transition to spiking in neurons of the mammalian cortex. The dynamical system approach has identified two robust mechanisms for the transition from subthreshold activity to spiking, saddle-node and Hopf bifurcation. Although theory provides precise predictions on signatures of critical slowing down near the bifurcation to spiking, quantitative experimental evidence has been lacking. Using whole-cell patch-clamp recordings from pyramidal neurons and fast-spiking interneurons, we show that 1) the transition to spiking dynamically corresponds to a critical transition exhibiting slowing down, 2) the scaling laws suggest a saddle-node bifurcation governing slowing down, and 3) these precise scaling laws can be used to predict the bifurcation point from a limited window of observation. To our knowledge this is the first report of scaling laws of critical slowing down in an experiment. They present a missing link for a broad class of neuroscience modeling and suggest improved estimation of tipping points by incorporating scaling laws of critical slowing down as a strategy applicable to other complex systems.

Neurons efficiently convey information by being able to switch rapidly between two different states: quiescence and spiking. Such sudden shifts to a qualitatively different state are observed in many complex systems; the often dramatic consequences of these tipping points for diverse fields such as economics, ecology, and the brain have spurred interest to better understand their transition mechanisms and predict their sudden occurrences. By studying the transition from neuronal quiescence to spiking, we show that the quantitative scaling laws for critical slowing down, i.e., a system’s tendency to recover more slowly from perturbations upon approaching its transition point, inform about the underlying bifurcation mechanism and can be used to improve the prediction of a system’s tipping point.

Dynamics of a producer-freeloader ecosystem on the brink of collapse

Ecosystems can undergo sudden shifts to undesirable states, but recent studies with simple single-species ecosystems have demonstrated that advance warning can be provided by the slowing down of population dynamics near a tipping point. However, it is unclear how this “critical slowing down” will manifest in ecosystems with strong interactions between their components. Here we probe the dynamics of an experimental producer-freeloader ecosystem as it approaches a catastrophic collapse. Surprisingly, the producer population grows in size as the environment deteriorates, highlighting that population size can be a misleading measure of ecosystem stability. By analyzing the oscillatory producer-freeloader dynamics for over 100 generations in multiple environmental conditions, we find that the collective ecosystem dynamics slow down as the tipping point is approached. Analysis of the coupled dynamics of interacting populations may therefore be necessary to provide advance warning of collapse in complex communities.

Self-organized criticality in single-neuron excitability

We present experimental and theoretical arguments, at the single-neuron level, suggesting that neuronal response fluctuations reflect a process that positions the neuron near a transition point that separates excitable and unexcitable phases. This view is supported by the dynamical properties of the system as observed in experiments on isolated cultured cortical neurons, as well as by a theoretical mapping between the constructs of self-organized criticality and membrane excitability biophysics.

Critical fluctuations in cortical models near instability

Computational studies often proceed from the premise that cortical dynamics operate in a linearly stable domain, where fluctuations dissipate quickly and show only short memory. Studies of human electroencephalography (EEG), however, have shown significant autocorrelation at time lags on the scale of minutes, indicating the need to consider regimes where non-linearities influence the dynamics. Statistical properties such as increased autocorrelation length, increased variance, power law scaling, and bistable switching have been suggested as generic indicators of the approach to bifurcation in non-linear dynamical systems. We study temporal fluctuations in a widely-employed computational model (the Jansen–Rit model) of cortical activity, examining the statistical signatures that accompany bifurcations. Approaching supercritical Hopf bifurcations through tuning of the background excitatory input, we find a dramatic increase in the autocorrelation length that depends sensitively on the direction in phase space of the input fluctuations and hence on which neuronal subpopulation is stochastically perturbed. Similar dependence on the input direction is found in the distribution of fluctuation size and duration, which show power law scaling that extends over four orders of magnitude at the Hopf bifurcation. We conjecture that the alignment in phase space between the input noise vector and the center manifold of the Hopf bifurcation is directly linked to these changes. These results are consistent with the possibility of statistical indicators of linear instability being detectable in real EEG time series. However, even in a simple cortical model, we find that these indicators may not necessarily be visible even when bifurcations are present because their expression can depend sensitively on the neuronal pathway of incoming fluctuations.

Complementarity of spike- and rate-based dynamics of neural systems

Relationships between spiking-neuron and rate-based approaches to the dynamics of neural assemblies are explored by analyzing a model system that can be treated by both methods, with the rate-based method further averaged over multiple neurons to give a neural-field approach. The system consists of a chain of neurons, each with simple spiking dynamics that has a known rate-based equivalent. The neurons are linked by propagating activity that is described in terms of a spatial interaction strength with temporal delays that reflect distances between neurons; feedback via a separate delay loop is also included because such loops also exist in real brains. These interactions are described using a spatiotemporal coupling function that can carry either spikes or rates to provide coupling between neurons. Numerical simulation of corresponding spike- and rate-based methods with these compatible couplings then allows direct comparison between the dynamics arising from these approaches. The rate-based dynamics can reproduce two different forms of oscillation that are present in the spike-based model: spiking rates of individual neurons and network-induced modulations of spiking rate that occur if network interactions are sufficiently strong. Depending on conditions either mode of oscillation can dominate the spike-based dynamics and in some situations, particularly when the ratio of the frequencies of these two modes is integer or half-integer, the two can both be present and interact with each other.

Fluctuating inhibitory inputs promote reliable spiking at theta frequencies in hippocampal interneurons

Theta-frequency (4–12 Hz) rhythms in the hippocampus play important roles in learning and memory. CA1 interneurons located at the stratum lacunosum-moleculare and radiatum junction (LM/RAD) are thought to contribute to hippocampal theta population activities by rhythmically pacing pyramidal cells with inhibitory postsynaptic potentials. This implies that LM/RAD cells need to fire reliably at theta frequencies in vivo. To determine whether this could occur, we use biophysically based LM/RAD model cells and apply different cholinergic and synaptic inputs to simulate in vivo-like network environments. We assess spike reliabilities and spiking frequencies, identifying biophysical properties and network conditions that best promote reliable theta spiking. We find that synaptic background activities that feature large inhibitory, but not excitatory, fluctuations are essential. This suggests that strong inhibitory input to these cells is vital for them to be able to contribute to population theta activities. Furthermore, we find that Type I-like oscillator models produced by augmented persistent sodium currents (I_NaP) or diminished A-type potassium currents (I_A) enhance reliable spiking at lower theta frequencies. These Type I-like models are also the most responsive to large inhibitory fluctuations and can fire more reliably under such conditions. In previous work, we showed that I_NaP and I_A are largely responsible for establishing LM/RAD cells’ subthreshold activities. Taken together with this study, we see that while both these currents are important for subthreshold theta fluctuations and reliable theta spiking, they contribute in different ways – I_NaP to reliable theta spiking and subthreshold activity generation, and I_A to subthreshold activities at theta frequencies. This suggests that linking subthreshold and suprathreshold activities should be done with consideration of both in vivo contexts and biophysical specifics.

Gap junctions modulate seizures in a mean-field model of general anesthesia for the cortex

During slow-wave sleep, general anesthesia, and generalized seizures, there is an absence of consciousness. These states are characterized by low-frequency large-amplitude traveling waves in scalp electroencephalogram. Therefore the oscillatory state might be an indication of failure to form coherent neuronal assemblies necessary for consciousness. A generalized seizure event is a pathological brain state that is the clearest manifestation of waves of synchronized neuronal activity. Since gap junctions provide a direct electrical connection between adjoining neurons, thus enhancing synchronous behavior, reducing gap-junction conductance should suppress seizures; however there is no clear experimental evidence for this. Here we report theoretical predictions for a physiologically-based cortical model that describes the general anesthetic phase transition from consciousness to coma, and includes both chemical synaptic and direct electrotonic synapses. The model dynamics exhibits both Hopf (temporal) and Turing (spatial) instabilities; the Hopf instability corresponds to the slow (≲8 Hz) oscillatory states similar to those seen in slow-wave sleep, general anesthesia, and seizures. We argue that a delicately balanced interplay between Hopf and Turing modes provides a canonical mechanism for the default non-cognitive rest state of the brain. We show that the Turing mode, set by gap-junction diffusion, is generally protective against entering oscillatory modes; and that weakening the Turing mode by reducing gap conduction can release an uncontrolled Hopf oscillation and hence an increased propensity for seizure and simultaneously an increased sensitivity to GABAergic anesthesia.

Characterization of voltage-gated K+ currents contributing to subthreshold membrane potential oscillations in hippocampal CA1 interneurons

CA1 inhibitory interneurons at the stratum lacunosum-moleculare and radiatum junction (LM/RAD-INs) display subthreshold membrane potential oscillations (MPOs) involving voltage-dependent Na(+) and A-type K(+) currents. LM/RAD-INs also express other voltage-gated K(+) currents, although their properties and role in MPOs remain unclear. Here, we characterized these voltage-gated K(+) currents and investigated their role in MPOs. Using outside-out patch recordings from LM/RAD-IN somata, we distinguished four voltage-gated K(+) currents based on their pharmacology and activation/inactivation properties: a fast delayed rectifier current (I(Kfast)), a slow delayed rectifier current (I(Kslow)), a rapidly inactivating A-type current (I(A)), and a slowly inactivating current (I(D)). Their relative contribution to the total K(+) current was I(A) > I(Kfast) > I(Kslow) = I(D). The presence of I(D) and the relative contributions of K(+) currents in LM/RAD-INs are different from those of other CA1 interneurons, suggesting the presence of differential complement of K(+) currents in subgroups of interneurons. We next determined whether these K(+) currents were sufficient for MPO generation using a single-compartment model of LM/RAD-INs. The model captured the subthreshold voltage dependence of MPOs. Moreover, all K(+) currents were active at subthreshold potentials but I(D), I(A), and the persistent sodium current (I(NaP)) were most active near threshold. Using impedance analysis, we found that I(A) and I(NaP) contribute to MPO generation by modulating peak spectral frequency during MPOs and governing the voltage range over which MPOs occur. Our findings uncover a differential expression of a complement of K(+) channels that underlies intrinsic rhythmic activity in inhibitory interneurons.

An analysis of the transitions between down and up states of the cortical slow oscillation under urethane anaesthesia

We study the dynamics of the transition between the low- and high-firing states of the cortical slow oscillation by using intracellular recordings of the membrane potential from cortical neurons of rats. We investigate the evidence for a bistability in assemblies of cortical neurons playing a major role in the maintenance of this oscillation. We show that the trajectory of a typical transition takes an approximately exponential form, equivalent to the response of a resistor-capacitor circuit to a step-change in input. The time constant for the transition is negatively correlated with the membrane potential of the low-firing state, and values are broadly equivalent to neural time constants measured elsewhere. Overall, the results do not strongly support the hypothesis of a bistability in cortical neurons; rather, they suggest the cortical manifestation of the oscillation is a result of a step-change in input to the cortical neurons. Since there is evidence from previous work that a phase transition exists, we speculate that the step-change may be a result of a bistability within other brain areas, such as the thalamus, or a bistability among only a small subset of cortical neurons, or as a result of more complicated brain dynamics.

Subthreshold dynamics of a single neuron from a Hamiltonian perspective

We use Hamilton's equations of classical mechanics to investigate the behavior of a cortical neuron on the approach to an action potential. We use a two-component dynamic model of a single neuron, due to Wilson, with added noise inputs. We derive a Lagrangian for the system, from which we construct Hamilton's equations. The conjugate momenta are found to be linear combinations of the noise input to the system. We use this approach to consider theoretically and computationally the most likely manner in which such a modeled neuron approaches a firing event. We find that the firing of a neuron is a result of a drop in inhibition, due to a temporary increase in negative bias of the mean noise input to the inhibitory control equation. Moreover, we demonstrate through theory and simulation that, on average, the bias in the noise increases in an exponential manner on the approach to an action potential. In the Hamiltonian description, an action potential can therefore be considered a result of the exponential growth of the conjugate momenta variables pulling the system away from its equilibrium state, into a nonlinear regime.

Characteristics of temporal fluctuations in the hyperpolarized state of the cortical slow oscillation

We present evidence for the hypothesis that transitions between the low- and high-firing states of the cortical slow oscillation correspond to neuronal phase transitions. By analyzing intracellular recordings of the membrane potential during the cortical slow oscillation in rats, we quantify the temporal fluctuations in power and the frequency centroid of the power spectrum in the period of time before "down" to "up" transitions. By taking appropriate averages over such events, we present these statistics as a function of time before transition. The results demonstrate an increase in fluctuation power and time scale broadly consistent with the slowing of systems close to phase transitions. The analysis is complicated and limited by the difficulty in identifying when transitions begin, and removing dc trends in membrane potential.

Gap junctions mediate large-scale Turing structures in a mean-field cortex driven by subcortical noise

One of the grand puzzles in neuroscience is establishing the link between cognition and the disparate patterns of spontaneous and task-induced brain activity that can be measured clinically using a wide range of detection modalities such as scalp electrodes and imaging tomography. High-level brain function is not a single-neuron property, yet emerges as a cooperative phenomenon of multiply-interacting populations of neurons. Therefore a fruitful modeling approach is to picture the cerebral cortex as a continuum characterized by parameters that have been averaged over a small volume of cortical tissue. Such mean-field cortical models have been used to investigate gross patterns of brain behavior such as anesthesia, the cycles of natural sleep, memory and erasure in slow-wave sleep, and epilepsy. There is persuasive and accumulating evidence that direct gap-junction connections between inhibitory neurons promote synchronous oscillatory behavior both locally and across distances of some centimeters, but, to date, continuum models have ignored gap-junction connectivity. In this paper we employ simple mean-field arguments to derive an expression for D2, the diffusive coupling strength arising from gap-junction connections between inhibitory neurons. Using recent neurophysiological measurements reported by Fukuda [J. Neurosci. 26, 3434 (2006)], we estimate an upper limit of D2 approximately 0.6cm2. We apply a linear stability analysis to a standard mean-field cortical model, augmented with gap-junction diffusion, and find this value for the diffusive coupling strength to be close to the critical value required to destabilize the homogeneous steady state. Computer simulations demonstrate that larger values of D2 cause the noise-driven model cortex to spontaneously crystalize into random mazelike Turing structures: centimeter-scale spatial patterns in which regions of high-firing activity are intermixed with regions of low-firing activity. These structures are consistent with the spatial variations in brain activity patterns detected with the BOLD (blood oxygen-level-dependent) signal detected with magnetic resonance imaging, and may provide a natural substrate for synchronous gamma-band rhythms observed across separated EEG (electroencephalogram) electrodes.