Delayed-exponential approximation of a linear homogeneous diffusion model of neuron.

The diffusion models of neuronal activity are general yet conceptually simple and flexible enough to be useful in a variety of modeling problems. Unfortunately, even simple diffusion models lead to tedious numerical calculations. Consequently, the existing neural net models use characteristics of a single neuron taken from the "pre-diffusion" era of neural modeling. Simplistic elements of neural nets forbid to incorporate a single learning neuron structure into the net model. The above drawback cannot be overcome without the use of the adequate structure of the single neuron as an element of a net. A linear (not necessarily homogeneous) diffusion model of a single neuron is a good candidate for such a structure, it must, however, be simplified. In the paper the structure of the diffusion model of neuron is discussed and a linear homogeneous model with reflection is analyzed. For this model an approximation is presented, which is based on the approximation of the first passage time distribution of the Ornstein-Uhlenbeck process by the delayed (shifted) exponential distribution. The resulting model has a simple structure and has a prospective application in neural modeling and in analysis of neural nets.

How to use the Mann-Whitney test to detect a change in distribution for groups.

A simple test for a directional change in the distribution for groups is proposed based on the Mann-Whitney statistic. The practical usefulness of the test for an experiment on conditioned enhancement and conditioned suppression is presented.

Mathematical modelling of reaction latency. Part II: A model of escape reaction and escape learning.

The results of fitting the linear-dynamic-stochastic model to the latency data from escape conditioning experiment are presented. Two processes can be distinguished: (i) an increase in parameters of the dynamic part of the model (called the static gain and the dynamic gain) which results in a decrease in the reaction latency during learning, (ii) an increase in parameter of the decision part of the model (mean value of the threshold), which results simultaneously in an increase in the reaction latency and a decrease in the inter-trial responses probability. The reaction latency decreases during the learning because the first phenomenon prevails over the second one, but the latter is the only reason of inter-trial responses decrease during learning. Analysis of the results of surgical treatment on the learning process is also presented.

Mathematical modelling of the reaction latency. Part III: A model of avoidance reaction latency and avoidance learning.

The linear-dynamic-stochastic model of a reaction latency as applied to avoidance experiment is presented. Reactions are classified on the basis of the model into following classes: escape, avoidance, late avoidance, three types of inter-trial responses and "no reaction". The experimental latency distribution is split into latency distributions of the escape, avoidance and late avoidance responses, providing a new insight into latency distribution. The results of fitting the model to latency measurements obtained in the avoidance conditioning experiment are presented. The same processes of the parameter changes as in the escape conditioning are discovered, one causing a latency to decrease and the other causing a latency to increase during learning. The first process affects a latency stronger than the second and, consequently, the latency decreases during learning. The second process is responsible for a decay of inter trial-responses during experiment. The value of the correlation coefficient between the threshold of avoidance reaction and the threshold of escape reaction was also estimated. Negative values of this coefficient were obtained, therefore, on the average, the greater the avoidance reaction threshold the smaller the escape one. In the course of learning the correlation coefficient tends to be equal to - 1, i.e., as a result of training, both thresholds became dependent in a functional (non-random) way. This result may provide an objective index of the "state of learning". The model provides a new tool for analysis of results of latency-based experiments.

Mathematical modelling of reaction latency: the structure of the models and its motivation.

The basic structural assumptions concerning the dynamic models of the reaction latency are presented. The linear dynamic stochastic model of the reaction latency is considered as a special case of dynamic model. The biological motivation for using these models are outlined. These models express the reaction latency as a first access time of the random threshold of a certain stochastic process. This approach was used in modelling the reaction latency in escape and avoidance experiments (the results will be presented in subsequent papers).