On the Cumulants of the First Passage Time of the Inhomogeneous Geometric Brownian Motion

We consider the problem of the first passage time T of an inhomogeneous geometric Brownian motion through a constant threshold, for which only limited results are available in the literature. In the case of a strong positive drift, we get an approximation of the cumulants of T of any order using the algebra of formal power series applied to an asymptotic expansion of its Laplace transform. The interest in the cumulants is due to their connection with moments and the accounting of some statistical properties of the density of T like skewness and kurtosis. Some case studies coming from neuronal modeling with reversal potential and mean reversion models of financial markets show the goodness of the approximation of the first moment of T. However hints on the evaluation of higher order moments are also given, together with considerations on the numerical performance of the method.

HippoUnit: A software tool for the automated testing and systematic comparison of detailed models of hippocampal neurons based on electrophysiological data

Anatomically and biophysically detailed data-driven neuronal models have become widely used tools for understanding and predicting the behavior and function of neurons. Due to the increasing availability of experimental data from anatomical and electrophysiological measurements as well as the growing number of computational and software tools that enable accurate neuronal modeling, there are now a large number of different models of many cell types available in the literature. These models were usually built to capture a few important or interesting properties of the given neuron type, and it is often unknown how they would behave outside their original context. In addition, there is currently no simple way of quantitatively comparing different models regarding how closely they match specific experimental observations. This limits the evaluation, re-use and further development of the existing models. Further, the development of new models could also be significantly facilitated by the ability to rapidly test the behavior of model candidates against the relevant collection of experimental data. We address these problems for the representative case of the CA1 pyramidal cell of the rat hippocampus by developing an open-source Python test suite, which makes it possible to automatically and systematically test multiple properties of models by making quantitative comparisons between the models and electrophysiological data. The tests cover various aspects of somatic behavior, and signal propagation and integration in apical dendrites. To demonstrate the utility of our approach, we applied our tests to compare the behavior of several different rat hippocampal CA1 pyramidal cell models from the ModelDB database against electrophysiological data available in the literature, and evaluated how well these models match experimental observations in different domains. We also show how we employed the test suite to aid the development of models within the European Human Brain Project (HBP), and describe the integration of the tests into the validation framework developed in the HBP, with the aim of facilitating more reproducible and transparent model building in the neuroscience community.

Evolutionary Computing in Neuronal Modeling

The efficacy of genetic algorithms in the design of models that model specific and experimental aspectsof action potentials in a wide variety of organisms is proven. A specific example of a plant actionpotential is used to illustrate the use of genetic algorithms in the search for parameters of models. Theefficiency of the genetic algorithms as a search method is in the short generation span of theconvergence of the algorithm.

Evolutionary computing in neuronal modeling

The efficacy of genetic algorithms in the design of models that model specific and experimental aspectsof action potentials in a wide variety of organisms is proven. A specific example of a plant actionpotential is used to illustrate the use of genetic algorithms in the search for parameters of models. Theefficiency of the genetic algorithms as a search method is in the short generation span of theconvergence of the algorithm.

Simple models of quantitative firing phenotypes in hippocampal neurons: comprehensive coverage of intrinsic diversity

Patterns of periodic voltage spikes elicited by a neuron help define its dynamical identity. Experimentally recorded spike trains from various neurons show qualitatively distinguishable features such as delayed spiking, spiking with/without frequency adaptation, and intrinsic bursting. Moreover, the input-dependent responses of a neuron not only show different quantitative features, such as higher spike frequency for a stronger input current injection, but can also exhibit qualitatively different responses, such as spiking and bursting under different input conditions, thus forming a complex phenotype of responses. In a previous work, Hippocampome.org, a comprehensive knowledgebase of hippocampal neuron types, systematically characterized various spike pattern phenotypes experimentally identified from 120 neuron types/subtypes. In this paper, we present a comprehensive set of simple phenomenological models that quantitatively reproduce the diverse and complex phenotypes of hippocampal neurons. In addition to point-neuron models, we created compact multi-compartment models with up to four compartments, which will allow spatial segregation of synaptic integration in network simulations. Electrotonic compartmentalization observed in our compact multi-compartment models is qualitatively consistent with experimental observations. Furthermore, we observed that adding dendritic compartments to point-neuron models, in general, allowed soma to reproduce features of bursting patterns and abrupt non-linearities in some frequency adapting patterns slightly more accurately. This work maps 120 neuron types/subtypes in the rodent hippocampus to a low-dimensional model space and adds another dimension to the knowledge accumulated in Hippocampome.org. Computationally efficient representations of intrinsic dynamics, along with other pieces of knowledge available in Hippocampome.org, provide a biologically realistic platform to explore the dynamical interactions of various types at the mesoscopic level.Author SummaryThe neurons in the hippocampus show enormous diversity in their intrinsic activity patterns. A comprehensive characterization of various intrinsic types using a neuronal modeling system is necessary to simulate biologically realistic networks of brain regions. Morphologically detailed neuronal modeling frameworks often limit the scalability of such network simulations due to the specification of hundreds of rules governing each neuron’s intrinsic dynamics. In this work, we have accomplished a comprehensive mapping of experimentally identified intrinsic dynamics in a simple modeling system with only two governing rules. We have created over a hundred point-neuron models that reflect the intrinsic differences among the hippocampal neuron types both qualitatively and quantitatively. In addition, we compactly extended our point-neurons to include up to four compartments, which will allow anatomically finer-grained connections among the neurons in a network. Our compact model representations, which are freely available in Hippocampome.org, will allow future researchers to investigate dynamical interactions among various intrinsic types and emergent integrative properties using scalable, yet biologically realistic network simulations.