On two diffusion neuronal models with multiplicative noise: The mean first-passage time properties.

Two diffusion processes with multiplicative noise, able to model the changes in the neuronal membrane depolarization between two consecutive spikes of a single neuron, are considered and compared. The processes have the same deterministic part but different stochastic components. The differences in the state-dependent variabilities, their asymptotic distributions, and the properties of the first-passage time across a constant threshold are investigated. Closed form expressions for the mean of the first-passage time of both processes are derived and applied to determine the role played by the parameters involved in the model. It is shown that for some values of the input parameters, the higher variability, given by the second moment, does not imply shorter mean first-passage time. The reason for that can be found in the complete shape of the stationary distribution of the two processes. Applications outside neuroscience are also mentioned.

Entropy factor for randomness quantification in neuronal data.

A novel measure of neural spike train randomness, an entropy factor, is proposed. It is based on the Shannon entropy of the number of spikes in a time window and can be seen as an analogy to the Fano factor. Theoretical properties of the new measure are studied for equilibrium renewal processes and further illustrated on gamma and inverse Gaussian probability distributions of interspike intervals. Finally, the entropy factor is evaluated from the experimental records of spontaneous activity in macaque primary visual cortex and compared to its theoretical behavior deduced for the renewal process models. Both theoretical and experimental results show substantial differences between the Fano and entropy factors. Rather paradoxically, an increase in the variability of spike count is often accompanied by an increase of its predictability, as evidenced by the entropy factor.

Parametric inference of neuronal response latency in presence of a background signal.

Neurons are commonly characterized by spontaneous generation of action potentials (spikes), which appear without any apparent or controlled stimulation. When a stimulus is applied, the spontaneous firing may prevail and hamper identification of the effect of the stimulus. Therefore, for any rigorous analysis of evoked neuronal activity, the presence of spontaneous firing has to be taken into account. If the background signal is ignored, however small it is compared to the response activity, and however large is the delay, estimation of the response latency will be wrong, and the error will persist even when sample size is increasing. The first question is: what is the response latency to the stimulus? Answering this question becomes even more difficult if the latency is of a complex nature, for example composed of a physically implied deterministic part and a stochastic part. This scenario is considered here, where the response time is a sum of two components; the delay and the relative latency. Parametric estimators for the time delay and the response latency are derived. These estimators are evaluated on simulated data and their properties are discussed. Finally, we show that the mean of the response latency is always satisfactorily estimated, even assuming a wrong distribution for the response latency.

Randomness of spontaneous activity and information transfer in neurons.

The analysis of information coding in neurons requires methods that measure different properties of neuronal signals. In this paper we review the recently proposed measure of randomness and compare it to the coefficient of variation, which is the frequently employed measure of variability of spiking neuronal activity. We focus on the problem of the spontaneous activity of neurons, and we hypothesize that under defined conditions, spontaneous activity is more random than evoked activity. This hypothesis is supported by contrasting variability and randomness obtained from experimental recordings of olfactory receptor neurons in rats.

Modeling the influence of non-adherence on antibiotic efficacy: application to ciprofloxacin.

OBJECTIVE: To evaluate the consequences for antibiotic efficacy of different types of poor adherence to a short-term dosing regimen. Ciprofloxacin was taken as an example. METHOD: A simulation study on a 2-compartment pharmacokinetic model and parameter estimates taken from the literature was performed. Two empirical efficacy measures as well as a specific pharmacodynamic model of the bacterial kill curve were used. Four patterns of non-adherence were investigated: dosage omission, irregular dosing intervals, delayed dosing and treatment discontinuation. RESULTS: Errors in timing of doses with a standard deviation less than 2 h had a minor effect on antibiotic efficacy. Dosage omission, in contrast, has a significant influence on the antibacterial effect of ciprofloxacin. CONCLUSIONS: non-adherence patterns are difficult to measure experimentally, thus, recommended dosing regimens should be sufficiently robust against most of the nonintentional disturbances.

Different types of noise in leaky integrate-and-fire model of neuronal dynamics with discrete periodical input.

Different variants of stochastic leaky integrate-and-fire model for the membrane depolarisation of neurons are investigated. The model is driven by a constant input and equidistant pulses of fixed amplitude. These two types of signal are considered under the influence of three types of noise: white noise, jitter on interpulse distance, and noise in the amplitude of pulses. The results of computational experiments demonstrate the enhancement of the signal by noise in subthreshold regime and deterioration of the signal if it is sufficiently strong to carry the information in absence of noise. Our study holds mainly to central neurons that process discrete pulses although an application in sensory system is also available.

From spreading depression to spatial cognition.

The Laboratory of Neurophysiology of Memory started its existence in 1954 by systematic research into spreading depression of EEG activity of laboratory rodents and by the use of this remarkable phenomenon as a functional ablation method in behavioral research. Its main contributions were in the study of memory formation and consolidation, interhemispheric transfer, motor learning, conditioned taste aversion and spatial orientation and navigation. In the last five years it concentrated on navigation of rats in multiple reference frames, on electrophysiological evidence for the role of hippocampal place cells support of behavior in such dissociated frames, on the analysis of idiothetic and allothetic forms of navigation and on the mathematical methods allowing assessment of the contribution of goal directed locomotion to place cell activity. The methods used in spatial memory research in rats were used for examination of human subjects in a laboratory equipped with a tracking system for humans in the hospital Homolka. Animal models of Alzheimer disease were studied in transgenic mice with the human gene for the beta amyloid precursor protein.

Quantifying location-specific information in the discharge of rat hippocampal place cells.

The informational content in the location-specific discharge of rat hippocampal cells is usually quantified by an average for the entire behaviorally accessible space. In contrast to such "global" information measures, we consider here information that can be obtained from "local" spike counts at each position. The properties of these local information measures are first illustrated using simulated data with predetermined distributions of location-specific spike counts. Next, place cell recordings from rats foraging in a cylindrical arena with two cue cards on its walls are analyzed; time windows as short as 100 ms were used to accumulate spike counts in locations. We show that information at the centers of firing fields is higher for fields nearer to the cues. Neither firing rates or "global" information measures detected differences between fields near and far from the cues. Thus, analyses of the location-specific information provides a new valuable tool for studying the location-specific activity of rat hippocampal cells. Generalizations of location-specific information can be used to investigate place cell responses to other factors such as running speed or the state of the hippocampal EEG in addition to current position.

On the location-specific positional and extra-positional information in the discharge of rat hippocampal cells.

The informational content in the location-specific discharge of rat hippocampal cells is usually quantified by an average for the entire experimental space. In contrast, in the present work the information that can be obtained from spike counts at each position is considered. Along with the local positional information, measures for the total and extra-positional information are introduced. Their properties are studied and illustrated on simulated and experimentally obtained data. It is demonstrated that these information measures provide a new valuable tool for studying the location-specific activity of rat hippocampal cells even on time scales as short as 100 ms. The measures can be used to investigate place cell responses to arbitrary signals in addition to current position.

Properties of the extra-positional signal in hippocampal place cell discharge derived from the overdispersion in location-specific firing.

There is a good deal of evidence that in the rodent, an internal model of the external world is encoded by hippocampal pyramidal cells, called 'place cells'. During free exploration, the activity of place cells is higher within a small part of the space, called the firing field, and virtually silent elsewhere. We have previously shown that the spiking activity during passes through the firing field is characterized not only by the high firing rate, but also by its very high variability ('overdispersion'). This overdispersion indicates that place cells carry information in addition to position. Here we demonstrate by simulations of an integrate-and-fire neuronal model that while a rat is foraging in an open space this additional information may arise from a process that alternatingly modulates the inputs to place cells by about 10% with a mean period of about 1 s. We propose that the overdispersion reflects switches of the rats attention between different spatial reference frames of the environment. This predicts that the overdispersion will not be observed in rats that use only room-based cues for navigation. We show that while place cell firing is overdispersed in rats during foraging in an open arena, the firing is less overdispersed during the same behavior in the same environment, when the rats have been trained to use only room-based and not arena-based cues to navigate.

Receptor heterogeneity and its effect on sensitivity and coding range in olfactory sensory neurons.

Signal processing in the olfactory system is initiated by binding of odorant molecules to receptor molecules embedded in the membranes of sensory neurons. Most analyses of odorant-receptor interaction focus on one or more types of odorants binding with one type of receptors. Here, two basic models of this first step are investigated under the assumption that the population of receptors is not homogenous and is characterized by different activation/deactivation rates. Both, discrete and continuous variation of the rates are considered. The steady-state characteristics of the models are derived. In addition, time to crossing a threshold, defined as a response time, is also investigated. The achieved results are compared with those valid for models with the homogenous population of receptors and interpreted in terms of information coding. The obvious implications of the modeling study--that the heterogeneity of receptors enlarges the coding range and increases the sensitivity of the system--are quantified.

Modeling heterogeneity of properties and random effects in drug dissolution.

PURPOSE: To investigate new models characterizing dissolution data obtained for heterogenous materials (model I) and under randomly time-varying conditions (model II). METHODS: In model I, the heterogeneity of the dissolving substance introduces variation of the fractional dissolution rate. In model II, the fractional dissolution rate evolves randomly, and thus the dissolution has the characteristics of a stochastic process. This situation is studied for the constant and time-dependent means of the dissolution rate. RESULTS: The time dynamics of the dissolved fraction is presented for model I. The standard characteristics of dissolution are derived under general conditions and for several examples. One of them is in accordance with a function found empirically (1). A duality between the time-dependency of the fractional dissolution rate and the heterogeneity of the substance is investigated. The mean and variance of the dissolved fraction are calculated for model II. A method for estimating the mean dissolution rate is proposed and illustrated using Monte-Carlo experiments. CONCLUSIONS: It follows from model I that the heterogeneity, with the same mean properties, slows down the dissolution with respect to the homogeneous case. The second approach permits predictions about the role of the stochastic fluctuations of the dissolution rate and to establish the boundaries for the dissolution profiles.

Ligand-receptor interaction under periodic stimulation: a modeling study of concentration chemoreceptors.

The first step of chemosensory transduction consists in the association of ligand molecules with receptor proteins borne by the cell membrane. In this article, the time evolution of ligand-receptor complexes is studied in the presence of a periodically changing ligand concentration. This type of stimulation is a close approximation to some natural situations, for example in olfaction. The transient and steady-state periodic levels of the complexes, resulting from a single-step (binding) or double-step (binding and activation) reaction, are determined. When possible, analytical solutions are given, if not for the complete model, at least for its simplified version at low ligand concentration. Otherwise, solutions are found numerically and both the complete and simplified versions of the model are compared. The results obtained are discussed with respect to actual experimental data based on the moth sex-pheromone receptor. Periodic steady states are achieved very quickly and their amplitude decreases when the stimulation frequency increases. We show that the simplified description is adequate if only a fraction of activated receptors is sufficient to produce the maximum response, as is actually the case in the example treated. The role of the frequency of stimulation is investigated and it is shown to possess an optimal range between 2 and 5 Hz.

Receptor dissociation constants and the information entropy of membranes coding ligand concentration.

The binding of ligands to receptor proteins embedded in cell membranes drives cellular responses that involve either second messenger cascades or directly gated ion channels. It is known that a single class of receptor proteins expresses approximately 98% of its graded response to ligand concentrations over four orders of magnitude, where the response is measured by the equilibrium proportion of bound ligand-receptor complexes. This four-decadic concentration range is centered on a logarithmic scale around logK, where K is the dissociation constant defined by the ratio of ligand-receptor unbinding (k-) to binding (k+) rates. Remarkably, this four-decadic concentration range is intrinsic to all homogeneous ligand-receptor (or, equivalently, enzyme-substrate) systems. Thus, adapting the sensitivity of cell membranes to narrower or wider ranges of ligand concentrations, respectively, requires multivalent receptors or heterogeneous populations of receptors. Here we use a normalized Shannon-Weaver measure of information entropy to represent the efficiency of coding over given concentrations for membranes containing a population of univalent receptors with a specified distribution of dissociation constants, or a homogeneous population of strongly cooperative multivalent receptors. Assuming a specified level of resolution in the response of cellular or neural systems downstream from the membrane that 'read' the ligand concentration 'code', we calculate the range of concentrations over which the coding efficiency of the membrane itself is maximized. Our results can be used to hypothesize the number of receptor types associated with the membranes of particular cells. For example, from data in the literature, we conclude that the response of most general olfactory sensory neurons can be explained in terms of a homogeneous population of receptor proteins, while the response of pheromone sensory neurons is satisfactorily explained by the presence of two types of membrane receptor protein with pheromone-binding dissociation constants that have values at least one to two orders of magnitude apart.

Perireceptor and receptor events in olfaction. Comparison of concentration and flux detectors: a modeling study.

Transduction in chemosensory cells begins with the association of ligand molecules to receptor proteins borne by the cell membrane. The receptor-ligand complexes formed act as signaling compounds that trigger a G-protein cascade. This receptor-ligand interaction, described here by a single-step or double-step reaction, depends on factors controlling the access of the ligand molecules to the cell membrane. Two basic mechanisms can be distinguished: concentration detectors (CD), in which the ligand can freely diffuse to the membrane, and flux detectors (FD), in which it accumulates irreversibly in a distinct perireceptor space where it is chemically deactivated. These two systems, plus their generalization, are investigated and compared. The transient and steady-state numbers of complexes are studied as a function of the external ligand concentration. The biological significance of the results is shown in a well-studied example of FD, the insect sex-pheromone olfactory receptor neuron. How the number of complexes can code for the intensity of stimulation is analyzed using the size, dynamic range and sensitivity of the steady-state responses, and the time needed to reach a predefined level of the transient responses. It is shown that the FD design affords a large increase in sensitivity (a shift of the threshold response towards low concentration) with respect to the CD design, which is paid for by a lesser ability to follow fast changes in stimulus intensity.

Multidimensional counting processes and evoked neuronal activity.

It is shown that the joint behaviour of n counting processes can be used for description of neuronal spike generation in a system of n neurons with various synaptic links. This multidimensional process is suggested as a model of neuronal activity evoked by either a homogeneous or an inhomogeneous stimulus during a short time period. Several situations are considered, mainly the joint activity of pairs of neurons. The probability generating functions of the process are obtained for different kinds of intensity functions. The first and second moments of the process are provided as a tool for potential comparison of the model with experimental data.

Effect of spatial extension on noise-enhanced phase locking in a leaky integrate-and-fire model of a neuron.

Signal transmission enhanced by noise has been recently investigated in detail on the single compartment, also referred to as single point, leaky integrate-and-fire model neuron under a subthreshold stimulation. In this paper we study how this phenomenon is influenced by taking into account the spatial characteristics of the neuron. A stochastic two-point leaky integrate-and-fire model, comprising a dendritic compartment and trigger zone, under periodic stimulation is studied. A method of how to measure synchronization between the signal and the output in both, experiments and models, is proposed. This method is based on a distance between the exact periodic spiking, as expected for sufficiently strong and noiseless stimulation, and neuronal activity evoked by a subthreshold signal corrupted by noise. It is shown that qualitatively the same phenomenon, phase-locking enhanced by the noise, as found in the spatially unstructured neuron is produced by the spatially complex neuron. However, quantitatively there are significant differences. Namely, the two-point model neuron is more robust against the noise and therefore its amplitude has to be higher to enhance the signal. Further, it is found that the range of the critical levels of noise is larger for the two-point model than for the single-point one. Finally, the enhancing effect at the optimal noise is more efficient in the single-point model and thus the firing patterns at their optimal noise levels are different in both models.

Stochastic model of the overdispersion in the place cell discharge.

The spontaneous firing activity of the place cells reflects the position of an experimental animal in its arena. The firing rate is high inside a part of the arena, called the firing field, and low outside. It is generally accepted concept that this is the way in which the hippocampus stores a map of the environment. This well known fact was recently reinvestigated [Fenton, A.A., Muller, R.U., 1998. Proc. Natl. Acad. Sci. USA 95, 3182-3187] and it was found that while the activity was highly reliable in position, it did not retain the same reliability in time. The number of action potentials fired during different passes through the firing field were substantially different (overdispersion). We present a mathematical model based on a doubly stochastic Poisson process which is able to reproduce the experimental findings. Further, it enables us to propose specific statistical inference on the experiments in aim to verify data and model compatibility. The model permits to speculate about the neural mechanisms leading to the overdispersion in the activity of the hippocampal place cells. Namely, the statistical variation of the intensity of firing can be achieved, for example, by introducing a hierarchical structure into the local neural network.

Spiking frequency versus odorant concentration in olfactory receptor neurons.

The spiking response of receptor neurons to various odorants has been analyzed at different concentrations. The interspike intervals were measured extracellularly before, during and after the stimulation from the olfactory epithelium of the frog Rana ridibunda. First, a quantitative method was developed to distinguish the spikes in the response from the spontaneous activity. Then, the response intensity, characterized by its median instantaneous frequency, was determined. Finally, based on statistical analyses, this characteristic was related to the concentration and quality of the odorant stimulus. It was found that the olfactory neuron is characterized by a low modulation in frequency and a short range of discriminated intensities. The significance of the results is discussed from both a biological and a modelling point of view.