Statistical properties of spike trains as well as other neurophysiological data suggest a number of mathematical models of neurons. These models range from entirely descriptive ones to those deduced from the properties of the real neurons. One of them, the diffusion leaky integrate-and-fire neuronal model, which is based on the Ornstein-Uhlenbeck (OU) stochastic process that is restricted by an absorbing barrier, can describe a wide range of neuronal activity in terms of its parameters. These parameters are readily associated with known physiological mechanisms. The other model is descriptive, Gamma renewal process, and its parameters only reflect the observed experimental data or assumed theoretical properties. Both of these commonly used models are related here. We show under which conditions the Gamma model is an output from the diffusion OU model. In some cases, we can see that the Gamma distribution is unrealistic to be achieved for the employed parameters of the OU process.

• Keywords
• First-passage-time problem, Leaky integrate-and-fire, Stein’s neuronal model,
• MeSH
• Diffusion * MeSH
• Cybernetics MeSH
• Models, Neurological * MeSH
• Neurons * MeSH
• Stochastic Processes MeSH
• Publication type
• Journal Article MeSH

Five parameters of one of the most common neuronal models, the diffusion leaky integrate-and-fire model, also known as the Ornstein-Uhlenbeck neuronal model, were estimated on the basis of intracellular recording. These parameters can be classified into two categories. Three of them (the membrane time constant, the resting potential and the firing threshold) characterize the neuron itself. The remaining two characterize the neuronal input. The intracellular data were collected during spontaneous firing, which in this case is characterized by a Poisson process of interspike intervals. Two methods for the estimation were applied, the regression method and the maximum-likelihood method. Both methods permit to estimate the input parameters and the membrane time constant in a short time window (a single interspike interval). We found that, at least in our example, the regression method gave more consistent results than the maximum-likelihood method. The estimates of the input parameters show the asymptotical normality, which can be further used for statistical testing, under the condition that the data are collected in different experimental situations. The model neuron, as deduced from the determined parameters, works in a subthreshold regimen. This result was confirmed by both applied methods. The subthreshold regimen for this model is characterized by the Poissonian firing. This is in a complete agreement with the observed interspike interval data.

• MeSH
• Action Potentials physiology   MeSH
• Cell Membrane physiology   MeSH
• Humans MeSH
• Brain physiology   MeSH
• Neural Pathways physiology   MeSH
• Synaptic Transmission physiology   MeSH
• Neural Networks, Computer * MeSH
• Neurons physiology   MeSH
• Signal Processing, Computer-Assisted MeSH
• Poisson Distribution MeSH
• Stochastic Processes MeSH
• Synapses physiology   MeSH
• Animals MeSH
• Check Tag
• Humans MeSH
• Animals MeSH
• Publication type
• Journal Article MeSH
• Research Support, Non-U.S. Gov't MeSH

The Ornstein-Uhlenbeck neuronal model is specified by two types of parameters. One type corresponds to the properties of the neuronal membrane, whereas the second type (local average rate of the membrane depolarization and its variability) corresponds to the input of the neuron. In this article, we estimate the parameters of the second type from an intracellular record during neuronal firing caused by stimulation (audio signal). We compare the obtained estimates with those from the spontaneous part of the record. As predicted from the model construction, the values of the input parameters are larger for the periods when neuron is stimulated than for the spontaneous ones. Finally, the firing regimen of the model is checked. It is confirmed that the neuron is in the suprathreshold regimen during the stimulation.

• MeSH
• Action Potentials physiology   MeSH
• Acoustic Stimulation methods   MeSH
• Time Factors MeSH
• Electroencephalography MeSH
• Membrane Potentials physiology   MeSH
• Models, Neurological * MeSH
• Guinea Pigs MeSH
• Neural Pathways physiology   MeSH
• Neurons classification   physiology   MeSH
• Signal Processing, Computer-Assisted MeSH
• Reaction Time physiology   MeSH
• Stochastic Processes * MeSH
• Animals MeSH
• Check Tag
• Guinea Pigs MeSH
• Animals MeSH
• Publication type
• Journal Article MeSH
• Research Support, Non-U.S. Gov't MeSH

Parameters in diffusion neuronal models are divided into two groups; intrinsic and input parameters. Intrinsic parameters are related to the properties of the neuronal membrane and are assumed to be known throughout the paper. Input parameters characterize processes generated outside the neuron and methods for their estimation are reviewed here. Two examples of the diffusion neuronal model, which are based on the integrate-and-fire concept, are investigated--the Ornstein--Uhlenbeck model as the most common one and the Feller model as an illustration of state-dependent behavior in modeling the neuronal input. Two types of experimental data are assumed-intracellular describing the membrane trajectories and extracellular resulting in knowledge of the interspike intervals. The literature on estimation from the trajectories of the diffusion process is extensive and thus the stress in this review is set on the inference made from the interspike intervals.

• MeSH
• Models, Neurological * MeSH
• Neurons * MeSH
• Publication type
• Journal Article MeSH
• Research Support, Non-U.S. Gov't MeSH
• Review MeSH

An optimum signal in the Ornstein-Uhlenbeck neuronal model is determined on the basis of interspike interval data. Two criteria are proposed for this purpose. The first, the classical one, is based on searching for maxima of the slope of the frequency transfer function. The second one uses maximum of the Fisher information, which is, under certain conditions, the inverse variance of the best possible estimator. The Fisher information is further normalized with respect to the time required to make the observation on which the signal estimation is performed. Three variants of the model are investigated. Beside the basic one, we use the version obtained by inclusion of the refractory period. Finally, we investigate such a version of the model in which signal and the input parameter of the model are in a nonlinear relationship. The results show that despite qualitative similarity between the criteria, there is substantial quantitative difference. As a common feature, we found that in the Ornstein-Uhlenbeck model with increasing noise the optimum signal decreases and the coding range gets broader.

• MeSH
• Action Potentials physiology   MeSH
• Algorithms MeSH
• Electrophysiology MeSH
• Humans MeSH
• Membrane Potentials physiology   MeSH
• Models, Neurological * MeSH
• Neural Conduction physiology   MeSH
• Neurons physiology   MeSH
• Refractory Period, Electrophysiological physiology   MeSH
• Stochastic Processes MeSH
• Animals MeSH
• Check Tag
• Humans MeSH
• Animals MeSH
• Publication type
• Journal Article MeSH
• Research Support, Non-U.S. Gov't MeSH