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.     

Design considerations and techniques for constructing video stimuli

Techniques for constructing video playback stimuli fall into five categories. The first three involve manipulating video sequences: (1) edited video is a temporal rearrangement of raw footage, (2) processed video applies global filtering algorithms to edited video, and (3) frame-manipulated video involves manually altering individual frames. The last two, (4) exemplar-based animation and (5) parameter-based animation, are synthetic models derived from visual parameters based on a single exemplar and sample data, respectively. Image-based approaches are straightforward to apply and preserve fine spatiotemporal detail. Synthetic stimuli are desirable when a large number of manipulations are called for and to ensure individual stimuli reflect population characteristics. Received: 13 December 1999 / Received in revised form: 25 February 2000 / Accepted: 1 March 2000     

A fast top-down method for constructing reliable radiation hybrid frameworks

MOTIVATION: Radiation Hybrid Mapping (RHM) is a technique used to order a set of markers on a genome and estimating physical distances between them. RHM provides information on marker placement independent from other methods such as sequencing, and can therefore be used for example in genome sequencing to help ordering contigs. A radiation hybrid framework can be constructed by choosing a set of markers so that the chromosome coverage is good and so that the markers can be ordered with high confidence. Automatically constructing RHM frameworks is a computationally challenging problem. RESULTS: We have developed a new method for constructing radiation hybrid frameworks. Given a relatively large set of markers for a chromosome, the algorithm aims to select an ordered subset that makes up a framework, and that contains as many markers as possible. The algorithm has a time complexity that is better than any of the existing methods that we are aware of. Furthermore, we propose a method for comparing if two frameworks are consistent, giving a visual presentation as well as quantitative measures of how well the two frameworks agree. Applying our method on marker sets from 22 human chromosomes and comparing the resulting frameworks with previously published frameworks, we demonstrate that our automatic method efficiently constructs frameworks with good coverage of each chromosome and with high degree of agreement on the marker ordering.     

A Volterra representation for some neuron models

A Volterra-like polynomial representation is derived and its convergence discussed for two neuronal models in which subthreshold inputs are integrated either without loss (integrate and fire) or with a decay which follows an exponential time course (leaky integrator). This polynomial representation provides a kind of nonlinear transfer function for the nonlinear encoding process. Standard formulae are used to derive explicitely the output for various inputs as in linear system theory. Moreover, the nonlinear transfer function associated with cascades or networks of neurons can be also obtained. Finally, extensions and implications of these results are discussed.     

Animal models for motor neuron disease.

Motor neuron disease is a general term applied to a broad class of neurodegenerative diseases that are characterized by fatally progressive muscular weakness, atrophy, and paralysis attributable to loss of motor neurons. At present, there is no cure for most motor neuron diseases, including amyotrophic lateral sclerosis (ALS), the most common human motor neuron disease--the cause of which remains largely unknown. Animal models of motor neuron disease (MND) have significantly contributed to the remarkable recent progress in understanding the cause, genetic factors, and pathologic mechanisms proposed for this class of human neurodegenerative disorders. Largely driven by ALS research, animal models of MND have proven their usefulness in elucidating potential causes and specific pathogenic mechanisms, and have helped to advance promising new treatments from "benchside to bedside." This review summarizes important features of selected established animal models of MND: genetically engineered mice and inherited or spontaneously occurring MND in the murine, canine, and equine species.     

From spiking neuron models to linear-nonlinear models

Neurons transform time-varying inputs into action potentials emitted stochastically at a time dependent rate. The mapping from current input to output firing rate is often represented with the help of phenomenological models such as the linear-nonlinear (LN) cascade, in which the output firing rate is estimated by applying to the input successively a linear temporal filter and a static non-linear transformation. These simplified models leave out the biophysical details of action potential generation. It is not a priori clear to which extent the input-output mapping of biophysically more realistic, spiking neuron models can be reduced to a simple linear-nonlinear cascade. Here we investigate this question for the leaky integrate-and-fire (LIF), exponential integrate-and-fire (EIF) and conductance-based Wang-Buzsáki models in presence of background synaptic activity. We exploit available analytic results for these models to determine the corresponding linear filter and static non-linearity in a parameter-free form. We show that the obtained functions are identical to the linear filter and static non-linearity determined using standard reverse correlation analysis. We then quantitatively compare the output of the corresponding linear-nonlinear cascade with numerical simulations of spiking neurons, systematically varying the parameters of input signal and background noise. We find that the LN cascade provides accurate estimates of the firing rates of spiking neurons in most of parameter space. For the EIF and Wang-Buzsáki models, we show that the LN cascade can be reduced to a firing rate model, the timescale of which we determine analytically. Finally we introduce an adaptive timescale rate model in which the timescale of the linear filter depends on the instantaneous firing rate. This model leads to highly accurate estimates of instantaneous firing rates.     

The Cauchy problem for one-dimensional spiking neuron models

I consider spiking neuron models defined by a one-dimensional differential equation and a reset—i.e., neuron models of the integrate-and-fire type. I address the question of the existence and uniqueness of a solution on for a given initial condition. It turns out that the reset introduces a countable and ordered set of backward solutions for a given initial condition. I discuss the implications of these mathematical results in terms of neural coding and spike timing precision.

Reduction of conductance-based neuron models

We present a scheme for systematically reducing the number of differential equations required for biophysically realistic neuron models. The techniques are general, are designed to be applicable to a large set of such models and retain in the reduced system as high a degree of fidelity to the original system as possible. As examples, we provide reductions of the Hodgkin-Huxley system and the A-current model of Connor et al. (1977).     

Protocol for constructing subject-specific biomechanical models of knee joint

A robust protocol for building subject-specific biomechanical models of the human knee joint is proposed which uses magnetic resonance imaging, motion analysis and force platform data in conjunction with detailed 3D finite element models. The proposed protocol can be used for determining stress and strain distributions and contact kinetics in different knee elements at different body postures during various physical activities. Several examples are provided to highlight the capabilities and potential applications of the proposed protocol. This includes preliminary results on the role of body weight on the stresses and strains induced in the knee articular cartilages and meniscus during single-leg stance and calculations of the induced stresses and ligament forces during the gait cycle.     

Dynamics of one-dimensional spiking neuron models

In this paper we make a rigorous mathematical analysis of one-dimensional spiking neuron models in a unified framework. We find that, under conditions satisfied in particular by the periodically and aperiodically driven leaky integrator as well as some of its variants, the spike map is increasing on its range, which leaves no room for chaotic behavior. A rigorous expression of the Lyapunov exponent is derived. Finally, we analyse the periodically driven perfect integrator and show that the restriction of the phase map to its range is always conjugated to a rotation, and we provide an explicit expression of the invariant measure.     

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     

Method for constructing the boundary of the bursting oscillations region in the neuron model

We examine the problem of constructing the boundary of bursting oscillations on a parameter plane for the system of equations describing the electrical behaviour of the membrane neuron arising from the interaction of fast oscillations of the cytoplasma membrane potential and slow oscillations of the intracellular calcium concentration. As the boundary point on the parameter plane we consider the values at which the limit cycle of the slow subsystem is tangent to the Hopf bifurcation curve of the fast subsystem. The method suggested for determining the boundary is based on the dissection of the system variables into slow and fast. The strong point of the method is that it requires the integration of the slow subsystem only. An example of the application of the method for the stomatogastric neuron model [Guckenheimer J, Gueron S, Harris-Warrick RM (1993) Philos Trans R Soc Lond B 341: 345–359] is given. Received: 31 May 1999 / Accepted in revised form: 19 November 1999     

Diffusion models for firing of a neuron with varying threshold

A stochastic model for the firing of a neuron with refractory properties is treated analytically. Refractory behavior is modeled by a threshold function θ(t) which is infinite immediately after the neuron fires, as well as during the absolute refractory period, and then decreases monotonically to the quiescent threshold level, θ_∞, during the relative refractory period. Using Wald's identity, input-output relations are derived analytically for the exponential threshold which has a time constant equal to the membrane time constant. A method for computing these relations for a general threshold is presented and is explicitly used for the general exponential threshold and the Hagiwara threshold, θ(t) = θ_∞e^α/t, where a is a constant.     

Modal analysis of renewal models for spontaneous single neuron discharge

Three renewal process models of neural discharge were briefly discussed, and their modal properties were analyzed. Examples of numerical solutions for the p · d · f's of each model were presented, and they conformed with the analytical results, demonstrating that one of the models, Model 2, generates multimodal interresponse time p · d · f's. Several areas of comparison with real data were indicated.Supported in part by a grant from the Alfred P. Sloan Foundation to the Department of Theoretical Biology, and by Research Grant No. NSF-GP-16071 from the Division of Mathematical, Physical and Engineering Sciences of the National Science Foundation to the Department of Statistics, University of Chicago.     

The minimum capture proportion for reliable estimation in capture-recapture models.

We derive estimates of the minimum capture proportion required to obtain a reliable estimate of the population size for several continuous and discrete-time capture-recapture models. The models considered are M(0), M(t), M(b), M(h), M(ht), and M(tb) in the notation of Otis et al., (1978, Wildlife Monograph62, 1-135). Numerical results with simulation studies are given, and two real examples for the model M(h) are also considered. Potential applications of these results are suggested.     

Relative risk regression: reliable and flexible methods for log-binomial models

Relative risks (RRs) are generally considered preferable to odds ratios in prospective studies. However, unlike logistic regression for odds ratios, the standard log-binomial model for RR regression does not respect the natural parameter constraints and is therefore often subject to numerical instability. In this paper, we develop a reliable and flexible method for fitting log-binomial models. We use an Expectation-Maximization (EM) algorithm where the multiplicative event probability is viewed as the joint probability for a collection of latent binary outcomes. This gives a simple iterative scheme that provides stable convergence to the maximum likelihood estimate. In addition to reliability, the method offers some flexible generalizations, including models with unspecified isotonic regression functions. We examine the method's performance using simulations and data analyses of the age-specific RR of mortality following heart attack. These analyses demonstrate the potential for numerical instability in RR regression and show how this can be overcome using the proposed approach. Source code to implement the method in R is provided as supplementary material available at Biostatistics online.     

Event-based minimum-time control of oscillatory neuron models

We present an event-based feedback control method for randomizing the asymptotic phase of oscillatory neurons. Phase randomization is achieved by driving the neuron’s state to its phaseless set, a point at which its phase is undefined and is extremely sensitive to background noise. We consider the biologically relevant case of a fixed magnitude constraint on the stimulus signal, and show how the control objective can be accomplished in minimum time. The control synthesis problem is addressed using the minimum-time-optimal Hamilton–Jacobi–Bellman framework, which is quite general and can be applied to any spiking neuron model in the conductance-based Hodgkin–Huxley formalism. We also use this methodology to compute a feedback control protocol for optimal spike rate increase. This framework provides a straightforward means of visualizing isochrons, without actually calculating them in the traditional way. Finally, we present an extension of the phase randomizing control scheme that is applied at the population level, to a network of globally coupled neurons that are firing in synchrony. The applied control signal desynchronizes the population in a demand-controlled way.     

Information transmission in multi-input-output stochastic neuron models

The Shannon's information theory in multiway channels (Shannon, 1961) is applied to multi-input-output relations of the stochastic automaton models for interaction of excitatory and inhibitory impulse sequences proposed in the previous papers (Tsukada et al., 1977). In these models, the output spike train depends upon several statistical characteristics (mean frequency, standard deviation, form, order-dependence or order-independence, etc.) of the excitatory and inhibitory input spike trains. By the use of the multiple-access channel in information theory, some stochastic properties of temporal pattern discrimination in neurons are analyzed and discussed with biological systems.