Petr Lánský

Diffusion approximation of the neuronal model with synaptic reversal potentials

The stochastic neuronal model with reversal potentials is approximated. For the model with constant postsynaptic potential amplitudes, a deterministic approximation is the only one which can be applied. The diffusion approximations are performed under the conditions of random postsynaptic potential amplitudes. New diffusion models of nerve membrane potential are devised in this way. These new models are more convenient for an analytical treatment than the original model with discontinuous trajectories.

On approximations of Stein's neuronal model

Steins model represents a commonly-used description of spontaneous neuronal activity. Substituting Steins model by the Ornstein-Uhlenbeck diffusion process increases the model tractability. A diffusion approximation of Steins model is summarized in the present paper. It is proved that the cumulative distribution functions of interspike intervals under Steins model converge to the cumulative distribution function of interspike intervals which are generated in accordance with the limiting Ornstein-Uhlenbeck diffusion model. The approach used allows us to determine to what extent Steins model modifications and generalizations affect the possibility of diffusion approximation. It can be seen that non-diffusion approximations exist and they are also studied here. The results achieved can be considered as complementary to the numerical study published recently.

Relation between stimulus and response in frog olfactory receptor neurons in vivo

The spiking activity of receptor neurons was recorded extracellularly in the frog olfactory epithelium in response to four odourants applied at precisely controlled concentrations. A set of criteria was formulated to define the spikes in the response. Four variables – latency, duration, number of interspike intervals and frequency – were determined to quantify the responses. They were studied at the single neuron, neuron population and ciliary membrane levels. The dose–response curves were determined using specific functions and their characteristics were evaluated. The characteristic molar concentrations at threshold or at maximum duration and the characteristics of variables, e.g. minimum latency or maximum frequency, have asymmetric histograms with peaks close to the origin and long tails. Dynamic ranges have even more asymmetric histograms, so that a significant fraction of neurons presents a much wider range than their one‐decade peak. From these histograms, response properties of the whole neuron population can be inferred. In general, location along the concentration axis (thresholds), width (dynamic ranges) and heights of dose–response curves are independent, which explains the diversity of curves, prevents their global categorization and supports the qualitative coding of odourants. No evidence for odourant‐independent types of neurons was found. Finally, receptor activation and ciliary membrane conductance were reconstructed in the framework of a model based on firing data, known mucus biochemical and neuron morpho‐electrical characteristics. It is in agreement with independent determinations of Kd of odourant–receptor interaction and of conductance characteristics, and describes their statistical distributions in the neuron population.

On the comparison of Feller and Ornstein-Uhlenbeck models for neural activity

Diffusion processes have been extensively used to describe membrane potential behavior. In this approach the interspike interval has a theoretical counterpart in the first-passage-time of the diffusion model employed. Since the mathematical complexity of the first-passage-time problem increases with attempts to make the models more realistic it seems useful to compare the features of different models in order to highlight their relative performance. In this paper we compare the Feller and Ornstein-Uhlenbeck models under three different criteria derived from the level of information available about their parameters. We conclude that the Feller model is preferable when complete knowledge of the characterizing parameters is assumed. On the other hand, when only limited information about the parameters is available, such as the mean firing time and the histogram shape, no advantage arises from using this more complex model.

The Ornstein–Uhlenbeck neuronal model with signal-dependent noise

Abstract The Ornstein–Uhlenbeck neuronal model is investigated under the assumption that the amplitude of the noise is signal dependent. A linear approximation of the input–output transfer function is developed. Different types of dependencies of the noise on the signal are considered. The conditions are presented under which a substantial difference between transfer functions with constant and signal-dependent noise appears. An example for which the transfer function yields contra-intuitive behavior is presented.

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.

Spontaneous activity of first- and second-order neurons in the frog olfactory system

The spontaneous activity of first-order neurons (neuroreceptors of the mucosa) and second-order neurons (mitral cells of the bulb) was recorded extracellularly in the frog olfactory system. To assess the influence of peripheral inputs upon mitral cells, the bulb was either normally connected or partially deafferented. Our first set of findings concern the firing behavior. We found that most neurons generated interspike intervals (ISIs) that were stationary in mean and variance, and were not serially correlated at first and second order. Individual spikes in mitral cells and bursts of spikes in neuroreceptors were found to be generated by a Poisson process. Stochastic modeling suggests that the Poissonian behavior depends on the mean value of the membrane potential at the axon hillock. In these models, the mean potential in mitral cells would be far below the firing threshold and in neuroreceptors it would fluctuate at random between two states, one close to resting potential (between bursts) and the other close to the firing threshold with occasional crossings (within bursts). Secondly, partially deafferented mitral cells had significantly higher activity and lower variance than mitral cells receiving normal afferent input. This effect gives evidence that peripheral inputs influence mitral cells at rest not only through direct excitation but also through indirect inhibition exerted by local neurons. Thus, the unstimulated state of the olfactory bulb would not be qualitatively different from its stimulated state in the sense that both states involve the same types of synaptic interactions. Consequently, understanding the synaptic relationships that take place in the bulb network can benefit from studies of its spontaneous activity.

Ornstein-Uhlenbeck model neuron revisited

The temporal patterns of action potentials fired by a two-point stochastic neuron model were investigated. In this model the membrane potential of the dendritic compartment follows the Orstein-Uhlenbeck process and is not affected by the spiking activity. The axonal compartment, corresponding to the spike initiation site, is described by a simplified RC circuit. Estimators of the mean and variance of the input, based on a sampling of the axonal membrane potential, were derived and applied to simulated data. The dependencies of the mean firing frequency and of the coefficient of variation and serial correlation of interspike intervals on the mean and variance of the input were also studied by computer simulation in both 1- and 2-point models. The main property distinguishing the 2-point model from the classical 1-point model is its ability to produce clusters of short (or long) intervals between spikes under conditions of constant stimulation, as often observed in real neurons. It is shown that the nearly linear frequency response of the neuron, starting with subthreshold values of the input, is accounted for by the variability of the input (noise), which indicates that noise can play a positive role in nervous systems. The linear response frequency with respect to noise of the models suggests that the neuron can function as a noise encoder.

Inference for the diffusion models of neuronal activity

Abstract Diffusion models of neuronal-membrane potential behavior belong to the most frequently analyzed description of neuronal processes. However, attempts to compare these models with experimental data are rare, and they are always based on interspike-interval statistics only. In the present paper, statistical inference for both standard diffusion models is given under the assumption that the values of the membrane potential are available. The results are compared with the interspike-interval approach.

Coding of Odor Intensity in a Steady-State Deterministic Model of an Olfactory Receptor Neuron

The coding of odor intensity by an olfactory receptor neuron model was studied under steady-state stimulation. Our model neuron is an elongated cylinder consisting of the following three components: a sensory dendritic region bearing odorant receptors, a passive region consisting of proximal dendrite and cell body, and an axon. First, analytical solutions are given for the three main physiological responses: (1) odorant-dependent conductance change at the sensory dendrite based on the Michaelis-Menten model, (2) generation and spreading of the receptor potential based on a new solution of the cable equation, and (3) firing frequency based on a Lapicque model. Second, the magnitudes of these responses are analyzed as a function of odorant concentration. Their dependence on chemical, electrical, and geometrical parameters is examined. The only evident gain in magnitude results from the activation-to-conductance conversion. An optimal encoder neuron is presented that suggests that increasing the length of the sensory dendrite beyond about 0.3 space constant does not increase the magnitude of the receptor potential. Third, the sensivities of the responses are examined as functions of (1) the concentration at half-maximum response, (2) the lower and upper concentrations actually discriminated, and (3) the width of the dynamic range. The overall gain in sensitivity results entirely from the conductance-to-voltage conversion. The maximum conductance at the sensory dendrite appears to be the main tuning constant of the neuron because it determines the shift toward low concentrations and the increase in dynamic range. The dynamic range of the model cannot exceed 5.7 log units, for a sensitivity increase at low odor concentration is compensated by a sensitivity decrease at high odor concentration.