The excitable membrane-biophysical theory and experiment

Very different biophysical theories can lead to similar or even identical predictions for a wide range of experiments. This was true for the original Rashevsky theory of membrane excitation, and the later Hill formulation. Similarly, the Hodgkin-Huxley equations for membrane current are based on two postulates: current proportional to total thermodynamic potential difference across the membrane, and the independence principle. Experiments used to confirm these postulates can be predicted as well by a model in which neither postulate applies.     

Studies in the mathematical theory of excitation

The general linear two-factor nerve-excitation theory of the type of Rashevsky and Hill is discussed and normal forms are derived. It is shown that in some cases these equations are not reducible to the Rashevsky form. Most notable is the case in which the solutions are damped periodic functions. It is shown that in this case one or more—in some cases infinitely many—discharges are predictable, following the application of a constant stimulusS. The number of discharges increases withS, but the frequency is a constant, characteristic of the fiber and independent ofS.     

On the complex dynamics of intracellular ganglion cell light responses in the cat retina

We recorded intracellular responses from cat retinal ganglion cells to sinusoidal flickering lights, and compared the response dynamics with a theoretical model based on coupled nonlinear oscillators. Flicker responses for several different spot sizes were separated in a smooth generator (G) potential and corresponding spike trains. We have previously shown that the G-potential reveals complex, stimulus-dependent, oscillatory behavior in response to sinusoidally flickering lights. Such behavior could be simulated by a modified van der Pol oscillator. In this paper, we extend the model to account for spike generation as well, by including extended Hodgkin-Huxley equations describing local membrane properties. We quantified spike responses by several parameters describing the mean and standard deviation of spike burst duration, timing (phase shift) of bursts, and the number of spikes in a burst. The dependence of these response parameters on stimulus frequency and spot size could be reproduced in great detail by coupling the van der Pol oscillator and Hodgkin-Huxley equations. The model mimics many experimentally observed response patterns, including non-phase-locked irregular oscillations. Our findings suggest that the information in the ganglion cell spike train reflects both intraretinal processing, simulated by the van der Pol oscillator, and local membrane properties described by Hodgkin-Huxley equations. The interplay between these complex processes can be simulated by changing the coupling coefficients between the two oscillators. Our simulations therefore show that irregularities in spike trains, which normally are considered to be noise, may be interpreted as complex oscillations that might carry information.To the memory of Prof. Otto-Joachim Grusser     

The adaptation ability of neuronal models subject to a current step stimulus

Three neuronal models of the spike initiating process were investigated with respect to their ability to show adaptation to a current step: (i) the perfect integrator model (PIM), (ii) the leaky integrator model (LIM), and (iii) the Hodgkin-Huxley (HH-) model. It was found that although each neuronal model will generate different response spike trains to a given stimulus, all responses fulfilled the criteria of a deterministic neural response (Awiszus 1988). The results show that both PIM and LIM are unable to show adaptation regardless of the choice of model parameters whereas the HH-model shows a clear rate of discharge adaptation. The reason for this adaptation lies in the fact that there are conditions for the HH-model where a step stimulus is highly effective. These conditions are investigated by means of a phase plane analysis. Consequences of these results for the explanation of neuronal adaptation and the validity of the neuronal models investigated are discussed.     

Representation of Time-Varying Stimuli by a Network Exhibiting Oscillations on a Faster Time Scale

Sensory processing is associated with gamma frequency oscillations (30–80 Hz) in sensory cortices. This raises the question whether gamma oscillations can be directly involved in the representation of time-varying stimuli, including stimuli whose time scale is longer than a gamma cycle. We are interested in the ability of the system to reliably distinguish different stimuli while being robust to stimulus variations such as uniform time-warp. We address this issue with a dynamical model of spiking neurons and study the response to an asymmetric sawtooth input current over a range of shape parameters. These parameters describe how fast the input current rises and falls in time. Our network consists of inhibitory and excitatory populations that are sufficient for generating oscillations in the gamma range. The oscillations period is about one-third of the stimulus duration. Embedded in this network is a subpopulation of excitatory cells that respond to the sawtooth stimulus and a subpopulation of cells that respond to an onset cue. The intrinsic gamma oscillations generate a temporally sparse code for the external stimuli. In this code, an excitatory cell may fire a single spike during a gamma cycle, depending on its tuning properties and on the temporal structure of the specific input; the identity of the stimulus is coded by the list of excitatory cells that fire during each cycle. We quantify the properties of this representation in a series of simulations and show that the sparseness of the code makes it robust to uniform warping of the time scale. We find that resetting of the oscillation phase at stimulus onset is important for a reliable representation of the stimulus and that there is a tradeoff between the resolution of the neural representation of the stimulus and robustness to time-warp.     

Ionic Mechanism Underlying Optimal Stimuli for Neuronal Excitation: Role of Na + Channel Inactivation

The ionic mechanism underlying optimal stimulus shapes that induce a neuron to fire an action potential, or spike, is relevant to understanding optimal information transmission and therapeutic stimulation in the nervous system. Here we analyze for the first time the ionic basis for stimulus optimality in the Hodgkin and Huxley model and for eliciting a spike in squid giant axons, the preparation for which the model was devised. The experimentally determined stimulus is a smoothly varying biphasic current waveform having a relatively long and shallow hyperpolarizing phase followed by a depolarizing phase of briefer duration. The hyperpolarizing phase removes a small degree of the resting level of Na ⁺ channel inactivation. This result together with the subsequent depolarizing phase provides a signal that is energetically more efficient for eliciting spikes than rectangular current pulses. Sodium channel inactivation is the only variable that is changed during the stimulus waveform, other than the membrane potential, V. The activation variables for Na ⁺ and K ⁺ channels are unchanged throughout the stimulus. This result demonstrates how an optimal stimulus waveform relates to ionic dynamics and may have implications for energy efficiency of neural excitation in many systems including the mammalian brain.     

On the Classification of Experimental Data Modeled Via a Stochastic Leaky Integrate and Fire Model Through Boundary Values

We present a computational algorithm aimed to classify single unit spike trains on the basis of observed interspikes intervals (ISI). The neuronal activity is modeled with a stochastic leaky integrate and fire model and the inverse first passage time method is extended to the Ornstein-Uhlenbeck (OU) process. Differences between spike trains are detected in terms of the boundary shape. The proposed classification method is applied to the analysis of multiple single units recorded simultaneously in the thalamus and in the cerebral cortex of unanesthetized rats during spontaneous activity. We show the existence of at least three different firing patterns that could not be classified using the usual statistical indices. PACS: 87.19.La MSC: 60K30, 60J60, 65C40, 62P10     

The Hill analysis and co-ion–driven transporter kinetics

ConclusionIn mechanistic models for binding of multiple ligands to a biological unit, the saturation behavior depends on the experimental readout. In the binding model, the Hill analysis does not provide information on the number of binding sites N. In contrast, the Hill analysis of the response model does contain this information, in which case the slope of the curve in the lower concentration range corresponds to the number of binding sites. In neither model does the Hill coefficient—defined as the slope of the curve at half-maximal saturation—report this number. In the binding model, the Hill coefficient varies between a value of 1 in the absence of interaction and a value of N in case of extremely strong interaction. In the response model, it varies between a number larger than 1 and N. In both models, the derived Hill coefficient is a measure of the cooperativity and sets a lowest possible number of sites.Ion-coupled transporters are of the response model type, and the saturation behavior of the rate with the co-ion in the lower concentration limit contains the information on the number of cotransported ions. Additionally, in the case of an ordered-binding mechanism, in which the co-ions bind before the transported substrate, the Hill coefficient of co-ion binding is a function of the substrate concentration. The apparent interaction between the co-ion sites increases with the substrate concentration and, consequently, the Hill coefficient extrapolates to the number of co-ions. Measurements of the Hill coefficient over the entire range of substrate concentrations provide information on both the extent of interaction between the sites and the number of sites.

Online supplemental material

Five supplemental texts accompany this review: (1) derivation of the equations for the binding and response models; (2) derivation of the equations for the ordered-binding transporter mechanism; (3) derivation of the equations for the Hill analysis of the saturation level functions; (4) derivation of the equations for substrate-dependent kinetics of mechanistic transporter models; (5) data analysis by curve fitting. The online supplemental material is available at http://www.jgp.org/cgi/content/full/jgp.201411332/DC1.     

Neuronal spike trains and stochastic point processes. I. The single spike train

In a growing class of neurophysiological experiments, the train of impulses (“spikes”) produced by a nerve cell is subjected to statistical treatment involving the time intervals between spikes. The statistical techniques available for the analysis of single spike trains are described and related to the underlying mathematical theory, that of stochastic point processes, i.e., of stochastic processes whose realizations may be described as series of point events occurring in time, separated by random intervals. For single stationary spike trains, several orders of complexity of statistical treatment are described; the major distinction is that between statistical measures that depend in an essential way on the serial order of interspike intervals and those that are order-independent. The interrelations among the several types of calculations are shown, and an attempt is made to ameliorate the current nomenclatural confusion in this field. Applications, interpretations, and potential difficulties of the statistical techniques are discussed, with special reference to types of spike trains encountered experimentally. Next, the related types of analysis are described for experiments which involve repeated presentations of a brief, isolated stimulus. Finally, the effects of nonstationarity, e.g. long-term changes in firing rate, on the various statistical measures are discussed. Several commonly observed patterns of spike activity are shown to be differentially sensitive to such changes. A companion paper covers the analysis of simultaneously observed spike trains.     

A generalized linear model for estimating spectrotemporal receptive fields from responses to natural sounds

In the auditory system, the stimulus-response properties of single neurons are often described in terms of the spectrotemporal receptive field (STRF), a linear kernel relating the spectrogram of the sound stimulus to the instantaneous firing rate of the neuron. Several algorithms have been used to estimate STRFs from responses to natural stimuli; these algorithms differ in their functional models, cost functions, and regularization methods. Here, we characterize the stimulus-response function of auditory neurons using a generalized linear model (GLM). In this model, each cell's input is described by: 1) a stimulus filter (STRF); and 2) a post-spike filter, which captures dependencies on the neuron's spiking history. The output of the model is given by a series of spike trains rather than instantaneous firing rate, allowing the prediction of spike train responses to novel stimuli. We fit the model by maximum penalized likelihood to the spiking activity of zebra finch auditory midbrain neurons in response to conspecific vocalizations (songs) and modulation limited (ml) noise. We compare this model to normalized reverse correlation (NRC), the traditional method for STRF estimation, in terms of predictive power and the basic tuning properties of the estimated STRFs. We find that a GLM with a sparse prior predicts novel responses to both stimulus classes significantly better than NRC. Importantly, we find that STRFs from the two models derived from the same responses can differ substantially and that GLM STRFs are more consistent between stimulus classes than NRC STRFs. These results suggest that a GLM with a sparse prior provides a more accurate characterization of spectrotemporal tuning than does the NRC method when responses to complex sounds are studied in these neurons.     

Activity-dependent enhancement in the reliability of correlated spike timings in cultured cortical neurons

To study the use-dependent modification of activity in neural networks, we investigated the spike timing by simultaneously recording activity at multiple sites in a network of cultured cortical neurons. We used dynamical analysis to study the temporal structure of spike trains and the activity-dependent changes in the reliability and reproducibility of spike patterns evoked by a stimulus. We also used cross-correlation analysis to evaluate the interactions of neuron pairs. Our main conclusions are that even when no obvious change in spike numbers can be seen, use-dependent modification occurs, either enhancing or reducing in the reliability and reproducibility of spike trains evoked by a stimulus, and the fine temporal structure of stimulus-evoked spike trains and interactions between neurons are also modified by tetanic stimulation. Received: 25 February 1998 / Accepted in revised form: 24 August 1998     

Further studies on the shape of the tracking curve of an automobile and its dependence on biological parameters of the driver

Following previous studies, differential equations are established which determine the variation of the stimulus towards a corrective turn of the steering wheel and its effect on the excitation of the centers in the brain which results in the production of the corrective turn. The equations are derived under the highly oversimplified assumption that all excitation thresholds are so small that they can be neglected. Under these assumptions it is found that the tracking curve of a car is a sinusoid with negative damping, that is, with an ever increasing amplitude. Driving under these assumptions is imposible since the car will always eventually jump off the road. The possible effects of the threshold as well as stimuli towards corrective turns other than the distance from the edge of the lane are very briefly discussed. In spite of the negative results of the paper, its interest lies in the circumstance that with the complication of the model, we find that driving depends not only on the reaction times as the only “purely biological” parameter, but on three other neurobiophysical constants. In a subsequent paper (Rashevsky, 1967) it is shown how the introduction of one or more purely biological parameters of the driver makes a stable driving regime possible.     

Measuring activity in olfactory receptor neurons in Drosophila: Focus on spike amplitude

Olfactory responses at the receptor level have been thoroughly described in Drosophila melanogaster by electrophysiological methods. Single sensilla recordings (SSRs) measure neuronal activity in intact individuals in response to odors. For sensilla that contain more than one olfactory receptor neuron (ORN), their different spontaneous spike amplitudes can distinguish each signal under resting conditions. However, activity is mainly described by spike frequency.Some reports on ORN response dynamics studied two components in the olfactory responses of ORNs: a fast component that is reflected by the spike frequency and a slow component that is observed in the LFP (local field potential, the single sensillum counterpart of the electroantennogram, EAG). However, no apparent correlation was found between the two elements.In this report, we show that odorant stimulation produces two different effects in the fast component, affecting spike frequency and spike amplitude. Spike amplitude clearly diminishes at the beginning of a response, but it recovers more slowly than spike frequency after stimulus cessation, suggesting that ORNs return to resting conditions long after they recover a normal spontaneous spike frequency. Moreover, spike amplitude recovery follows the same kinetics as the slow voltage component measured by the LFP, suggesting that both measures are connected.These results were obtained in ab2 and ab3 sensilla in response to two odors at different concentrations. Both spike amplitude and LFP kinetics depend on odorant, concentration and neuron, suggesting that like the EAG they may reflect olfactory information.     

The firing statistics of Poisson neuron models driven by slow stimuli

The coding properties of cells with different types of receptive fields have been studied for decades. ON-type neurons fire in response to positive fluctuations of the time-dependent stimulus, whereas OFF cells are driven by negative stimulus segments. Biphasic cells, in turn, are selective to up/down or down/up stimulus upstrokes. In this article, we explore the way in which different receptive fields affect the firing statistics of Poisson neuron models, when driven with slow stimuli. We find analytical expressions for the time-dependent peri-stimulus time histogram and the inter-spike interval distribution in terms of the incoming signal. Our results enable us to understand the interplay between the intrinsic and extrinsic factors that regulate the statistics of spike trains. The former depend on biophysical neural properties, whereas the latter hinge on the temporal characteristics of the input signal.     

Statistical characteristics of climbing fiber spikes necessary for efficient cerebellar learning

 Mean firing rates (MFRs), with analogue values, have thus far been used as information carriers of neurons in most brain theories of learning. However, the neurons transmit the signal by spikes, which are discrete events. The climbing fibers (CFs), which are known to be essential for cerebellar motor learning, fire at the ultra-low firing rates (around 1 Hz), and it is not yet understood theoretically how high-frequency information can be conveyed and how learning of smooth and fast movements can be achieved. Here we address whether cerebellar learning can be achieved by CF spikes instead of conventional MFR in an eye movement task, such as the ocular following response (OFR), and an arm movement task. There are two major afferents into cerebellar Purkinje cells: parallel fiber (PF) and CF, and the synaptic weights between PFs and Purkinje cells have been shown to be modulated by the stimulation of both types of fiber. The modulation of the synaptic weights is regulated by the cerebellar synaptic plasticity. In this study we simulated cerebellar learning using CF signals as spikes instead of conventional MFR. To generate the spikes we used the following four spike generation models: (1) a Poisson model in which the spike interval probability follows a Poisson distribution, (2) a gamma model in which the spike interval probability follows the gamma distribution, (3) a max model in which a spike is generated when a synaptic input reaches maximum, and (4) a threshold model in which a spike is generated when the input crosses a certain small threshold. We found that, in an OFR task with a constant visual velocity, learning was successful with stochastic models, such as Poisson and gamma models, but not in the deterministic models, such as max and threshold models. In an OFR with a stepwise velocity change and an arm movement task, learning could be achieved only in the Poisson model. In addition, for efficient cerebellar learning, the distribution of CF spike-occurrence time after stimulus onset must capture at least the first, second and third moments of the temporal distribution of error signals. Received: 28 January 2000 / Accepted in revised form: 2 August 2000     

A Threshold Equation for Action Potential Initiation

In central neurons, the threshold for spike initiation can depend on the stimulus and varies between cells and between recording sites in a given cell, but it is unclear what mechanisms underlie this variability. Properties of ionic channels are likely to play a role in threshold modulation. We examined in models the influence of Na channel activation, inactivation, slow voltage-gated channels and synaptic conductances on spike threshold. We propose a threshold equation which quantifies the contribution of all these mechanisms. It provides an instantaneous time-varying value of the threshold, which applies to neurons with fluctuating inputs. We deduce a differential equation for the threshold, similar to the equations of gating variables in the Hodgkin-Huxley formalism, which describes how the spike threshold varies with the membrane potential, depending on channel properties. We find that spike threshold depends logarithmically on Na channel density, and that Na channel inactivation and K channels can dynamically modulate it in an adaptive way: the threshold increases with membrane potential and after every action potential. Our equation was validated with simulations of a previously published multicompartemental model of spike initiation. Finally, we observed that threshold variability in models depends crucially on the shape of the Na activation function near spike initiation (about −55 mV), while its parameters are adjusted near half-activation voltage (about −30 mV), which might explain why many models exhibit little threshold variability, contrary to experimental observations. We conclude that ionic channels can account for large variations in spike threshold.     

Sensitivity of different stimulus-timing strategies for the detection of small excitations in noisy spike train data

There are several different strategies to control the timing of a stimulus with respect to the ongoing discharge during the recording of neuronal stimulus-response characteristics. One possible strategy consists of delivering stimuli in such a way that a constant pre-stimulus spike density is reached. Another strategy enforces spike application with a constant stimulus latency after a spontaneous discharge. In this paper the sensitivity of these different strategies for statistical verification of small excitatory response components was investigated. It was found that the difference between observed poststimulus spike distribution and expected spike distribution under the null hypothesis of no stimulus effect was larger using a constant-stimulus-latency (CSL) strategy with an appropriate value for the stimulus latency. Thus, the statistical verification of neuronal response components is clearly facilitated if a CSL strategy is used. This superiority of the CSL strategy is marked, especially for small excitations at neurons discharging slowly with low discharge variability.     

Decoding spike timing: the differential reverse-correlation method

It is widely acknowledged that detailed timing of action potentials is used to encode information, for example, in auditory pathways; however, the computational tools required to analyze encoding through timing are still in their infancy. We present a simple example of encoding, based on a recent model of time-frequency analysis, in which units fire action potentials when a certain condition is met, but the timing of the action potential depends also on other features of the stimulus. We show that, as a result, spike-triggered averages are smoothed so much that they do not represent the true features of the encoding. Inspired by this example, we present a simple method, differential reverse correlations, that can separate an analysis of what causes a neuron to spike, and what controls its timing. We analyze with this method the leaky integrate-and-fire neuron and show the method accurately reconstructs the model's kernel.