Noise-induced transition in excitable neuron models

 We studied the influence of noisy stimulation on the Hodgkin-Huxley neuron model. Rather than examining the noise-related variability of the discharge times of the model – as has been done previously – our study focused on the effect of noise on the stationary distributions of the membrane potential and gating variables of the model. We observed that a gradual increase in the noise intensity did not result in a gradual change of the distributions. Instead, we could identify a critical intermediate noise range in which the shapes of the distributions underwent a drastic qualitative change. Namely, they moved from narrow unimodal Gaussian-like shapes associated with low noise intensities to ones that spread widely at large noise intensities. In particular, for the membrane potential and the sodium activation variable, the distributions changed from unimodal to bimodal. Thus, our investigation revealed a noise-induced transition in the Hodgkin-Huxley model. In order to further characterize this phenomenon, we considered a reduced one-dimensional model of an excitable system, namely the active rotator. For this model, our analysis indicated that the noise-induced transition is associated with a deterministic bifurcation of approximate equations governing the dynamics of the mean and variance of the state variable. Finally, we shed light on the possible functional importance of this noise-induced transition in neuronal coding by determining its effect on the spike timing precision in models of neuronal ensembles. Received: 19 September 2000 / Accepted in revised form: 4 March 2001     

Neuronal Spike Initiation Modulated by Extracellular Electric Fields

Based on a reduced two-compartment model, the dynamical and biophysical mechanism underlying the spike initiation of the neuron to extracellular electric fields is investigated in this paper. With stability and phase plane analysis, we first investigate in detail the dynamical properties of neuronal spike initiation induced by geometric parameter and internal coupling conductance. The geometric parameter is the ratio between soma area and total membrane area, which describes the proportion of area occupied by somatic chamber. It is found that varying it could qualitatively alter the bifurcation structures of equilibrium as well as neuronal phase portraits, which remain unchanged when varying internal coupling conductance. By analyzing the activating properties of somatic membrane currents at subthreshold potentials, we explore the relevant biophysical basis of spike initiation dynamics induced by these two parameters. It is observed that increasing geometric parameter could greatly decrease the intensity of the internal current flowing from soma to dendrite, which switches spike initiation dynamics from Hopf bifurcation to SNIC bifurcation; increasing internal coupling conductance could lead to the increase of this outward internal current, whereas the increasing range is so small that it could not qualitatively alter the spike initiation dynamics. These results highlight that neuronal geometric parameter is a crucial factor in determining the spike initiation dynamics to electric fields. The finding is useful to interpret the functional significance of neuronal biophysical properties in their encoding dynamics, which could contribute to uncovering how neuron encodes electric field signals.     

Dynamics of the exponential integrate-and-fire model with slow currents and adaptation

In order to properly capture spike-frequency adaptation with a simplified point-neuron model, we study approximations of Hodgkin-Huxley (HH) models including slow currents by exponential integrate-and-fire (EIF) models that incorporate the same types of currents. We optimize the parameters of the EIF models under the external drive consisting of AMPA-type conductance pulses using the current-voltage curves and the van Rossum metric to best capture the subthreshold membrane potential, firing rate, and jump size of the slow current at the neuron’s spike times. Our numerical simulations demonstrate that, in addition to these quantities, the approximate EIF-type models faithfully reproduce bifurcation properties of the HH neurons with slow currents, which include spike-frequency adaptation, phase-response curves, critical exponents at the transition between a finite and infinite number of spikes with increasing constant external drive, and bifurcation diagrams of interspike intervals in time-periodically forced models. Dynamics of networks of HH neurons with slow currents can also be approximated by corresponding EIF-type networks, with the approximation being at least statistically accurate over a broad range of Poisson rates of the external drive. For the form of external drive resembling realistic, AMPA-like synaptic conductance response to incoming action potentials, the EIF model affords great savings of computation time as compared with the corresponding HH-type model. Our work shows that the EIF model with additional slow currents is well suited for use in large-scale, point-neuron models in which spike-frequency adaptation is important.     

Dynamics from Seconds to Hours in Hodgkin-Huxley Model with Time-Dependent Ion Concentrations and Buffer Reservoirs

The classical Hodgkin-Huxley (HH) model neglects the time-dependence of ion concentrations in spiking dynamics. The dynamics is therefore limited to a time scale of milliseconds, which is determined by the membrane capacitance multiplied by the resistance of the ion channels, and by the gating time constants. We study slow dynamics in an extended HH framework that includes time-dependent ion concentrations, pumps, and buffers. Fluxes across the neuronal membrane change intra- and extracellular ion concentrations, whereby the latter can also change through contact to reservoirs in the surroundings. Ion gain and loss of the system is identified as a bifurcation parameter whose essential importance was not realized in earlier studies. Our systematic study of the bifurcation structure and thus the phase space structure helps to understand activation and inhibition of a new excitability in ion homeostasis which emerges in such extended models. Also modulatory mechanisms that regulate the spiking rate can be explained by bifurcations. The dynamics on three distinct slow times scales is determined by the cell volume-to-surface-area ratio and the membrane permeability (seconds), the buffer time constants (tens of seconds), and the slower backward buffering (minutes to hours). The modulatory dynamics and the newly emerging excitable dynamics corresponds to pathological conditions observed in epileptiform burst activity, and spreading depression in migraine aura and stroke, respectively.     

Stochastically Gating Ion Channels Enable Patterned Spike Firing through Activity-Dependent Modulation of Spike Probability

The transformation of synaptic input into patterns of spike output is a fundamental operation that is determined by the particular complement of ion channels that a neuron expresses. Although it is well established that individual ion channel proteins make stochastic transitions between conducting and non-conducting states, most models of synaptic integration are deterministic, and relatively little is known about the functional consequences of interactions between stochastically gating ion channels. Here, we show that a model of stellate neurons from layer II of the medial entorhinal cortex implemented with either stochastic or deterministically gating ion channels can reproduce the resting membrane properties of stellate neurons, but only the stochastic version of the model can fully account for perithreshold membrane potential fluctuations and clustered patterns of spike output that are recorded from stellate neurons during depolarized states. We demonstrate that the stochastic model implements an example of a general mechanism for patterning of neuronal output through activity-dependent changes in the probability of spike firing. Unlike deterministic mechanisms that generate spike patterns through slow changes in the state of model parameters, this general stochastic mechanism does not require retention of information beyond the duration of a single spike and its associated afterhyperpolarization. Instead, clustered patterns of spikes emerge in the stochastic model of stellate neurons as a result of a transient increase in firing probability driven by activation of HCN channels during recovery from the spike afterhyperpolarization. Using this model, we infer conditions in which stochastic ion channel gating may influence firing patterns in vivo and predict consequences of modifications of HCN channel function for in vivo firing patterns.     

Stochastic Ion Channel Gating in Dendritic Neurons: Morphology Dependence and Probabilistic Synaptic Activation of Dendritic Spikes

Neuronal activity is mediated through changes in the probability of stochastic transitions between open and closed states of ion channels. While differences in morphology define neuronal cell types and may underlie neurological disorders, very little is known about influences of stochastic ion channel gating in neurons with complex morphology. We introduce and validate new computational tools that enable efficient generation and simulation of models containing stochastic ion channels distributed across dendritic and axonal membranes. Comparison of five morphologically distinct neuronal cell types reveals that when all simulated neurons contain identical densities of stochastic ion channels, the amplitude of stochastic membrane potential fluctuations differs between cell types and depends on sub-cellular location. For typical neurons, the amplitude of membrane potential fluctuations depends on channel kinetics as well as open probability. Using a detailed model of a hippocampal CA1 pyramidal neuron, we show that when intrinsic ion channels gate stochastically, the probability of initiation of dendritic or somatic spikes by dendritic synaptic input varies continuously between zero and one, whereas when ion channels gate deterministically, the probability is either zero or one. At physiological firing rates, stochastic gating of dendritic ion channels almost completely accounts for probabilistic somatic and dendritic spikes generated by the fully stochastic model. These results suggest that the consequences of stochastic ion channel gating differ globally between neuronal cell-types and locally between neuronal compartments. Whereas dendritic neurons are often assumed to behave deterministically, our simulations suggest that a direct consequence of stochastic gating of intrinsic ion channels is that spike output may instead be a probabilistic function of patterns of synaptic input to dendrites.     

Library-based numerical reduction of the Hodgkin–Huxley neuron for network simulation

We present an efficient library-based numerical method for simulating the Hodgkin–Huxley (HH) neuronal networks. The key components in our numerical method involve (i) a pre-computed high resolution data library which contains typical neuronal trajectories (i.e., the time-courses of membrane potential and gating variables) during the interval of an action potential (spike), thus allowing us to avoid resolving the spikes in detail and to use large numerical time steps for evolving the HH neuron equations; (ii) an algorithm of spike-spike corrections within the groups of strongly coupled neurons to account for spike-spike interactions in a single large time step. By using the library method, we can evolve the HH networks using time steps one order of magnitude larger than the typical time steps used for resolving the trajectories without the library, while achieving comparable resolution in statistical quantifications of the network activity, such as average firing rate, interspike interval distribution, power spectra of voltage traces. Moreover, our large time steps using the library method can break the stability requirement of standard methods (such as Runge–Kutta (RK) methods) for the original dynamics. We compare our library-based method with RK methods, and find that our method can capture very well phase-locked, synchronous, and chaotic dynamics of HH neuronal networks. It is important to point out that, in essence, our library-based HH neuron solver can be viewed as a numerical reduction of the HH neuron to an integrate-and-fire (I&F) neuronal representation that does not sacrifice the gating dynamics (as normally done in the analytical reduction to an I&F neuron).     

Dynamical estimation of neuron and network properties II: path integral Monte Carlo methods

Hodgkin–Huxley (HH) models of neuronal membrane dynamics consist of a set of nonlinear differential equations that describe the time-varying conductance of various ion channels. Using observations of voltage alone we show how to estimate the unknown parameters and unobserved state variables of an HH model in the expected circumstance that the measurements are noisy, the model has errors, and the state of the neuron is not known when observations commence. The joint probability distribution of the observed membrane voltage and the unobserved state variables and parameters of these models is a path integral through the model state space. The solution to this integral allows estimation of the parameters and thus a characterization of many biological properties of interest, including channel complement and density, that give rise to a neuron’s electrophysiological behavior. This paper describes a method for directly evaluating the path integral using a Monte Carlo numerical approach. This provides estimates not only of the expected values of model parameters but also of their posterior uncertainty. Using test data simulated from neuronal models comprising several common channels, we show that short (<50 ms) intracellular recordings from neurons stimulated with a complex time-varying current yield accurate and precise estimates of the model parameters as well as accurate predictions of the future behavior of the neuron. We also show that this method is robust to errors in model specification, supporting model development for biological preparations in which the channel expression and other biophysical properties of the neurons are not fully known.     

Coarse-grained event tree analysis for quantifying Hodgkin-Huxley neuronal network dynamics

We present an event tree analysis of studying the dynamics of the Hodgkin-Huxley (HH) neuronal networks. Our study relies on a coarse-grained projection to event trees and to the event chains that comprise these trees by using a statistical collection of spatial-temporal sequences of relevant physiological observables (such as sequences of spiking multiple neurons). This projection can retain information about network dynamics that covers multiple features, swiftly and robustly. We demonstrate that for even small differences in inputs, some dynamical regimes of HH networks contain sufficiently higher order statistics as reflected in event chains within the event tree analysis. Therefore, this analysis is effective in discriminating small differences in inputs. Moreover, we use event trees to analyze the results computed from an efficient library-based numerical method proposed in our previous work, where a pre-computed high resolution data library of typical neuronal trajectories during the interval of an action potential (spike) allows us to avoid resolving the spikes in detail. In this way, we can evolve the HH networks using time steps one order of magnitude larger than the typical time steps used for resolving the trajectories without the library, while achieving comparable statistical accuracy in terms of average firing rate and power spectra of voltage traces. Our numerical simulation results show that the library method is efficient in the sense that the results generated by using this numerical method with much larger time steps contain sufficiently high order statistical structure of firing events that are similar to the ones obtained using a regular HH solver. We use our event tree analysis to demonstrate these statistical similarities.     

The impact of short term synaptic depression and stochastic vesicle dynamics on neuronal variability

Neuronal variability plays a central role in neural coding and impacts the dynamics of neuronal networks. Unreliability of synaptic transmission is a major source of neural variability: synaptic neurotransmitter vesicles are released probabilistically in response to presynaptic action potentials and are recovered stochastically in time. The dynamics of this process of vesicle release and recovery interacts with variability in the arrival times of presynaptic spikes to shape the variability of the postsynaptic response. We use continuous time Markov chain methods to analyze a model of short term synaptic depression with stochastic vesicle dynamics coupled with three different models of presynaptic spiking: one model in which the timing of presynaptic action potentials are modeled as a Poisson process, one in which action potentials occur more regularly than a Poisson process (sub-Poisson) and one in which action potentials occur more irregularly (super-Poisson). We use this analysis to investigate how variability in a presynaptic spike train is transformed by short term depression and stochastic vesicle dynamics to determine the variability of the postsynaptic response. We find that sub-Poisson presynaptic spiking increases the average rate at which vesicles are released, that the number of vesicles released over a time window is more variable for smaller time windows than larger time windows and that fast presynaptic spiking gives rise to Poisson-like variability of the postsynaptic response even when presynaptic spike times are non-Poisson. Our results complement and extend previously reported theoretical results and provide possible explanations for some trends observed in recorded data.     

Uncovering the synchronization dynamics from correlated neuronal activity quantifies assembly formation

 Synchronous network excitation is believed to play an outstanding role in neuronal information processing. Due to the stochastic nature of the contributing neurons, however, those synchronized states are difficult to detect in electrode recordings. We present a framework and a model for the identification of such network states and of their dynamics in a specific experimental situation. Our approach operationalizes the notion of neuronal groups forming assemblies via synchronization based on experimentally obtained spike trains. The dynamics of such groups is reflected in the sequence of synchronized states, which we describe as a renewal dynamics. We furthermore introduce a rate function which is dependent on the internal network phase that quantifies the activity of neurons contributing to the observed spike train. This constitutes a hidden state model which is formally equivalent to a hidden Markov model, and all its parameters can be accurately determined from the experimental time series using the Baum-Welch algorithm. We apply our method to recordings from the cat visual cortex which exhibit oscillations and synchronizations. The parameters obtained for the hidden state model uncover characteristic properties of the system including synchronization, oscillation, switching, background activity and correlations. In applications involving multielectrode recordings, the extracted models quantify the extent of assembly formation and can be used for a temporally precise localization of system states underlying a specific spike train. Received: 30 March 1993/Accepted in revised form: 16 April 1994     

Reconstruction of Underlying Nonlinear Deterministic Dynamics Embedded in Noisy Spike Trains

An experimentally recorded time series formed by the exact times of occurrence of the neuronal spikes (spike train) is likely to be affected by observational noise that provokes events mistakenly confused with neuronal discharges, as well as missed detection of genuine neuronal discharges. The points of the spike train may also suffer a slight jitter in time due to stochastic processes in synaptic transmission and to delays in the detecting devices. This study presents a procedure aimed at filtering the embedded noise (denoising the spike trains) the spike trains based on the hypothesis that recurrent temporal patterns of spikes are likely to represent the robust expression of a dynamic process associated with the information carried by the spike train. The rationale of this approach is tested on simulated spike trains generated by several nonlinear deterministic dynamical systems with embedded observational noise. The application of the pattern grouping algorithm (PGA) to the noisy time series allows us to extract a set of points that form the reconstructed time series. Three new indices are defined for assessment of the performance of the denoising procedure. The results show that this procedure may indeed retrieve the most relevant temporal features of the original dynamics. Moreover, we observe that additional spurious events affect the performance to a larger extent than the missing of original points. Thus, a strict criterion for the detection of spikes under experimental conditions, thus reducing the number of spurious spikes, may raise the possibility to apply PGA to detect endogenous deterministic dynamics in the spike train otherwise masked by the observational noise.     

The stochastic properties of input spike trains control neuronal arithmetic

In the nervous system, the representation of signals is based predominantly on the rate and timing of neuronal discharges. In most everyday tasks, the brain has to carry out a variety of mathematical operations on the discharge patterns. Recent findings show that even single neurons are capable of performing basic arithmetic on the sequences of spikes. However, the interaction of the two spike trains, and thus the resulting arithmetic operation may be influenced by the stochastic properties of the interacting spike trains. If we represent the individual discharges as events of a random point process, then an arithmetical operation is given by the interaction of two point processes. Employing a probabilistic model based on detection of coincidence of random events and complementary computer simulations, we show that the point process statistics control the arithmetical operation being performed and, particularly, that it is possible to switch from subtraction to division solely by changing the distribution of the inter-event intervals of the processes. Consequences of the model for evaluation of binaural information in the auditory brainstem are demonstrated. The results accentuate the importance of the stochastic properties of neuronal discharge patterns for information processing in the brain; further studies related to neuronal arithmetic should therefore consider the statistics of the interacting spike trains.     

Repetitive firing: a quantitative study of feedback in model encoders

Recognition of nonlinearities in the neuronal encoding of repetitive spike trains has generated a number of models to explain this behavior. Here we develop the mathematics and a set of tests for two such models: the leaky integrator and the variable-gamma model. Both of these are nearly sufficient to explain the dynamic behavior of a number of repetitively firing, sensory neurons. Model parameters can be related to possible underlying basic mechanisms. Summed and nonsummed, spike- locked negative feedback are examined in conjunction with the models. Transfer functions are formulated to predict responses to steady state, steps, and sinusoidally varying stimuli in which output data are the times of spike-train events only. An electrical analog model for the leaky integrator is tested to verify predicted responses. Curve fitting and parameter variation techniques are explored for the purpose of extracting basic model parameters from spike train data. Sinusoidal analysis of spike trains appear to be a very accurate method for determining spike-locked feedback parameters, and it is to a large extent a model independent method that may also be applied to neuronal responses.     

Subthreshold Voltage Noise Due to Channel Fluctuations in Active Neuronal Membranes

Voltage-gated ion channels in neuronal membranes fluctuate randomly between different conformational states due to thermal agitation. Fluctuations between conducting and nonconducting states give rise to noisy membrane currents and subthreshold voltage fluctuations and may contribute to variability in spike timing. Here we study subthreshold voltage fluctuations due to active voltage-gated Na⁺ and K⁺ channels as predicted by two commonly used kinetic schemes: the Mainen et al. (1995) (MJHS) kinetic scheme, which has been used to model dendritic channels in cortical neurons, and the classical Hodgkin-Huxley (1952) (HH) kinetic scheme for the squid giant axon. We compute the magnitudes, amplitude distributions, and power spectral densities of the voltage noise in isopotential membrane patches predicted by these kinetic schemes. For both schemes, noise magnitudes increase rapidly with depolarization from rest. Noise is larger for smaller patch areas but is smaller for increased model temperatures. We contrast the results from Monte Carlo simulations of the stochastic nonlinear kinetic schemes with analytical, closed-form expressions derived using passive and quasi-active linear approximations to the kinetic schemes. For all subthreshold voltage ranges, the quasi-active linearized approximation is accurate within 8% and may thus be used in large-scale simulations of realistic neuronal geometries.     

Spike latency and jitter of neuronal membrane patches with stochastic Hodgkin-Huxley channels

Ion channel stochasticity can influence the voltage dynamics of neuronal membrane, with stronger effects for smaller patches of membrane because of the correspondingly smaller number of channels. We examine this question with respect to first spike statistics in response to a periodic input of membrane patches including stochastic Hodgkin-Huxley channels, comparing these responses to spontaneous firing. Without noise, firing threshold of the model depends on frequency—a sinusoidal stimulus is subthreshold for low and high frequencies and suprathreshold for intermediate frequencies. When channel noise is added, a stimulus in the lower range of subthreshold frequencies can influence spike output, while high subthreshold frequencies remain subthreshold. Both input frequency and channel noise strength influence spike timing. Specifically, spike latency and jitter have distinct minima as a function of input frequency, showing a resonance like behavior. With either no input, or low frequency subthreshold input, or input in the low or high suprathreshold frequency range, channel noise reduces latency and jitter, with the strongest impact for the lowest input frequencies. In contrast, for an intermediate range of suprathreshold frequencies, where an optimal input gives a minimum latency, the noise effect reverses, and spike latency and jitter increase with channel noise. Thus, a resonant minimum of the spike response as a function of frequency becomes more pronounced with less noise. Spike latency and jitter also depend on the initial phase of the input, resulting in minimal latencies at an optimal phase, and depend on the membrane time constant, with a longer time constant broadening frequency tuning for minimal latency and jitter. Taken together, these results suggest how stochasticity of ion channels may influence spike timing and thus coding for neurons with functionally localized concentrations of channels, such as in “hot spots” of dendrites, spines or axons.     

A Common Neurodynamical Mechanism Could Mediate Externally Induced and Intrinsically Generated Transitions in Visual Awareness

The neural correlates of conscious visual perception are commonly studied in paradigms of perceptual multistability that allow multiple perceptual interpretations during unchanged sensory stimulation. What is the source of this multistability in the content of perception? From a theoretical perspective, a fine balance between deterministic and stochastic forces has been suggested to underlie the spontaneous, intrinsically driven perceptual transitions observed during multistable perception. Deterministic forces are represented by adaptation of feature-selective neuronal populations encoding the competing percepts while stochastic forces are modeled as noise-driven processes. Here, we used a unified neuronal competition model to study the dynamics of adaptation and noise processes in binocular flash suppression (BFS), a form of externally induced perceptual suppression, and compare it with the dynamics of intrinsically driven alternations in binocular rivalry (BR). For the first time, we use electrophysiological, biologically relevant data to constrain a model of perceptual rivalry. Specifically, we show that the mean population discharge pattern of a perceptually modulated neuronal population detected in electrophysiological recordings in the lateral prefrontal cortex (LPFC) during BFS, constrains the dynamical range of externally induced perceptual transitions to a region around the bifurcation separating a noise-driven attractor regime from an adaptation-driven oscillatory regime. Most interestingly, the dynamical range of intrinsically driven perceptual transitions during BR is located in the noise-driven attractor regime, where it overlaps with BFS. Our results suggest that the neurodynamical mechanisms of externally induced and spontaneously generated perceptual alternations overlap in a narrow, noise-driven region just before a bifurcation where the system becomes adaptation-driven.     

The transition between stochastic and deterministic behavior in an excitable gene circuit

We explore the connection between a stochastic simulation model and an ordinary differential equations (ODEs) model of the dynamics of an excitable gene circuit that exhibits noise-induced oscillations. Near a bifurcation point in the ODE model, the stochastic simulation model yields behavior dramatically different from that predicted by the ODE model. We analyze how that behavior depends on the gene copy number and find very slow convergence to the large number limit near the bifurcation point. The implications for understanding the dynamics of gene circuits and other birth-death dynamical systems with small numbers of constituents are discussed.