Marja-Leena Linne

Developing Itô stochastic differential equation models for neuronal signal transduction pathways

Mathematical modeling and simulation of dynamic biochemical systems are receiving considerable attention due to the increasing availability of experimental knowledge of complex intracellular functions. In addition to deterministic approaches, several stochastic approaches have been developed for simulating the time-series behavior of biochemical systems. The problem with stochastic approaches, however, is the larger computational time compared to deterministic approaches. It is therefore necessary to study alternative ways to incorporate stochasticity and to seek approaches that reduce the computational time needed for simulations, yet preserve the characteristic behavior of the system in question. In this work, we develop a computational framework based on the Itô stochastic differential equations for neuronal signal transduction networks. There are several different ways to incorporate stochasticity into deterministic differential equation models and to obtain Itô stochastic differential equations. Two of the developed models are found most suitable for stochastic modeling of neuronal signal transduction. The best models give stable responses which means that the variances of the responses with time are not increasing and negative concentrations are avoided. We also make a comparative analysis of different kinds of stochastic approaches, that is the Itô stochastic differential equations, the chemical Langevin equation, and the Gillespie stochastic simulation algorithm. Different kinds of stochastic approaches can be used to produce similar responses for the neuronal protein kinase C signal transduction pathway. The fine details of the responses vary slightly, depending on the approach and the parameter values. However, when simulating great numbers of chemical species, the Gillespie algorithm is computationally several orders of magnitude slower than the Itô stochastic differential equations and the chemical Langevin equation. Furthermore, the chemical Langevin equation produces negative concentrations. The Itô stochastic differential equations developed in this work are shown to overcome the problem of obtaining negative values.

Stochastic Differential Equation Model for Cerebellar Granule Cell Excitability

Neurons in the brain express intrinsic dynamic behavior which is known to be stochastic in nature. A crucial question in building models of neuronal excitability is how to be able to mimic the dynamic behavior of the biological counterpart accurately and how to perform simulations in the fastest possible way. The well-established Hodgkin-Huxley formalism has formed to a large extent the basis for building biophysically and anatomically detailed models of neurons. However, the deterministic Hodgkin-Huxley formalism does not take into account the stochastic behavior of voltage-dependent ion channels. Ion channel stochasticity is shown to be important in adjusting the transmembrane voltage dynamics at or close to the threshold of action potential firing, at the very least in small neurons. In order to achieve a better understanding of the dynamic behavior of a neuron, a new modeling and simulation approach based on stochastic differential equations and Brownian motion is developed. The basis of the work is a deterministic one-compartmental multi-conductance model of the cerebellar granule cell. This model includes six different types of voltage-dependent conductances described by Hodgkin-Huxley formalism and simple calcium dynamics. A new model for the granule cell is developed by incorporating stochasticity inherently present in the ion channel function into the gating variables of conductances. With the new stochastic model, the irregular electrophysiological activity of an in vitro granule cell is reproduced accurately, with the same parameter values for which the membrane potential of the original deterministic model exhibits regular behavior. The irregular electrophysiological activity includes experimentally observed random subthreshold oscillations, occasional spontaneous spikes, and clusters of action potentials. As a conclusion, the new stochastic differential equation model of the cerebellar granule cell excitability is found to expand the range of dynamics in comparison to the original deterministic model. Inclusion of stochastic elements in the operation of voltage-dependent conductances should thus be emphasized more in modeling the dynamic behavior of small neurons. Furthermore, the presented approach is valuable in providing faster computation times compared to the Markov chain type of modeling approaches and more sophisticated theoretical analysis tools compared to previously presented stochastic modeling approaches.

Serotonin induces inward potassium and calcium currents in rat cortical astrocytes.

Ca2+ imaging and patch-clamp techniques were used to study the effects of serotonin (5-HT) on ionic conductances in rat cortical astrocytes. 1 and 10 microM serotonin caused a transient increase in intracellular calcium (Ca(i)) levels in fura-2AM-loaded cultured astrocytes and in astrocytes acutely isolated and then cultured in horse serum-containing medium for over 24 h. However, the acutely isolated (less than 6 h from isolation) astrocytes, as well as acutely isolated astrocytes cultured in serum-free media, failed to respond to 5-HT by changes in Ca(i). Coinciding with the changes in Ca(i) levels, inward currents were activated by 10 microM 5-HT in cultured, but not in acutely isolated astrocytes. Two separate types of serotonin-induced, small-conductance inward single-channel currents were found. First, in both Ca2+-containing and Ca2+-free media serotonin transiently activated a small-conductance apamin-sensitive channel. Apamin is a specific blocker of the small-conductance Ca2+-activated K+ channel (sK(Ca)) When cells were pre-treated with phospholipase C inhibitor U73122 no 5-HT-induced sK(Ca) channel openings were seen, indicating that this channel was activated by Ca2+ released from intracellular stores via IP3. A second type of small inward channel activated later, but only in the presence of external Ca2+. It was inhibited by the L-type Ca2+ channel blockers, nimodipine and nifedipine. Both types of channel activity were inhibited by ketanserin, indicating activation of the 5-HT2A receptor.

Postsynaptic Signal Transduction Models for Long-Term Potentiation and Depression

More than a hundred biochemical species, activated by neurotransmitters binding to transmembrane receptors, are important in long-term potentiation (LTP) and long-term depression (LTD). To investigate which species and interactions are critical for synaptic plasticity, many computational postsynaptic signal transduction models have been developed. The models range from simple models with a single reversible reaction to detailed models with several hundred kinetic reactions. In this study, more than a hundred models are reviewed, and their features are compared and contrasted so that similarities and differences are more readily apparent. The models are classified according to the type of synaptic plasticity that is modeled (LTP or LTD) and whether they include diffusion or electrophysiological phenomena. Other characteristics that discriminate the models include the phase of synaptic plasticity modeled (induction, expression, or maintenance) and the simulation method used (deterministic or stochastic). We find that models are becoming increasingly sophisticated, by including stochastic properties, integrating with electrophysiological properties of entire neurons, or incorporating diffusion of signaling molecules. Simpler models continue to be developed because they are computationally efficient and allow theoretical analysis. The more complex models permit investigation of mechanisms underlying specific properties and experimental verification of model predictions. Nonetheless, it is difficult to fully comprehend the evolution of these models because (1) several models are not described in detail in the publications, (2) only a few models are provided in existing model databases, and (3) comparison to previous models is lacking. We conclude that the value of these models for understanding molecular mechanisms of synaptic plasticity is increasing and will be enhanced further with more complete descriptions and sharing of the published models.

Crete, channels, cells, circuits and computers

Abstract It was a Cretan queen, the legendary Princess Europa, who gave her name to the entire continent. Recently, Europe returned the favor by selecting the island of Crete as the location of its yearly computational neuroscience course. Twenty-eight students from 17 different countries (including Israel, Japan and most European nations) were selected from 130 applicants to participate in the first Crete Course in Computational Neuroscience2Crete Course in Computational Neuroscience. Held at the Foundation for Research and Technology Hellas (FORTH), Crete: 25 August–20 September 1996. 2 . Two of the students and two of the 27 faculty members report on their experiences.

Effects of Transmitters and Amyloid-Beta Peptide on Calcium Signals in Rat Cortical Astrocytes: Fura-2AM Measurements and Stochastic Model Simulations

Background To better understand the complex molecular level interactions seen in the pathogenesis of Alzheimers disease, the results of the wet-lab and clinical studies can be complemented by mathematical models. Astrocytes are known to become reactive in Alzheimers disease and their ionic equilibrium can be disturbed by interaction of the released and accumulated transmitters, such as serotonin, and peptides, including amyloid- peptides (A). We have here studied the effects of small amounts of A25–35 fragments on the transmitter-induced calcium signals in astrocytes by Fura-2AM fluorescence measurements and running simulations of the detected calcium signals. Methodology/Principal Findings Intracellular calcium signals were measured in cultured rat cortical astrocytes following additions of serotonin and glutamate, or either of these transmitters together with A25–35. A25–35 increased the number of astrocytes responding to glutamate and exceedingly increased the magnitude of the serotonin-induced calcium signals. In addition to A25–35-induced effects, the contribution of intracellular calcium stores to calcium signaling was tested. When using higher stimulus frequency, the subsequent calcium peaks after the initial peak were of lower amplitude. This may indicate inadequate filling of the intracellular calcium stores between the stimuli. In order to reproduce the experimental findings, a stochastic computational model was introduced. The model takes into account the major mechanisms known to be involved in calcium signaling in astrocytes. Model simulations confirm the principal experimental findings and show the variability typical for experimental measurements. Conclusions/Significance Nanomolar A25–35 alone does not cause persistent change in the basal level of calcium in astrocytes. However, even small amounts of A25–35, together with transmitters, can have substantial synergistic effects on intracellular calcium signals. Computational modeling further helps in understanding the mechanisms associated with intracellular calcium oscillations. Modeling the mechanisms is important, as astrocytes have an essential role in regulating the neuronal microenvironment of the central nervous system.

Modeling single neuron behavior using stochastic differential equations

We model the intrinsic dynamic behavior of a neuron using stochastic differential equations and Brownian motion. Basis of our work is the deterministic one-compartmental multi-conductance model of cerebellar granule cell. We develop a novel modeling approach for our test neuron by incorporating the stochasticity inherently present in the operation of voltage-dependent ion channels. Our new stochastic Hodgkin-Huxley type of model is able to reproduce a large range of dynamics more realistically than previous deterministic models for the granule cell. Proper inclusion of stochastic elements is therefore essential in modeling the behavior of single small neuron.

Astrocyte-neuron interactions: from experimental research-based models to translational medicine.

In this chapter, we review the principal astrocyte functions and the interactions between neurons and astrocytes. We then address how the experimentally observed functions have been verified in computational models and review recent experimental literature on astrocyte-neuron interactions. Benefits of computational neuroscience work are highlighted through selected studies with neurons and astrocytes by analyzing the existing models qualitatively and assessing the relevance of these models to experimental data. Common strategies to mathematical modeling and computer simulation in neuroscience are summarized for the nontechnical reader. The astrocyte-neuron interactions are then further illustrated by examples of some neurological and neurodegenerative diseases, where the miscommunication between glia and neurons is found to be increasingly important.

Quantification of vesicles in differentiating human SH-SY5Y neuroblastoma cells by automated image analysis

A new automated image analysis method for quantification of fluorescent dots is presented. This method facilitates counting the number of fluorescent puncta in specific locations of individual cells and also enables estimation of the number of cells by detecting the labeled nuclei. The method is here used for counting the AM1-43 labeled fluorescent puncta in human SH-SY5Y neuroblastoma cells induced to differentiate with all-trans retinoic acid (RA), and further stimulated with high potassium (K+) containing solution. The automated quantification results correlate well with the results obtained manually through visual inspection. The manual method has the disadvantage of being slow, labor-intensive, and subjective, and the results may not be reproducible even in the intra-observer case. The automated method, however, has the advantage of allowing fast quantification with explicitly defined methods, with no user intervention. This ensures objectivity of the quantification. In addition to the number of fluorescent dots, further development of the method allows its use for quantification of several other parameters, such as intensity, size, and shape of the puncta, that are difficult to quantify manually.

Computational study of noise in a large signal transduction network

BackgroundBiochemical systems are inherently noisy due to the discrete reaction events that occur in a random manner. Although noise is often perceived as a disturbing factor, the system might actually benefit from it. In order to understand the role of noise better, its quality must be studied in a quantitative manner. Computational analysis and modeling play an essential role in this demanding endeavor.ResultsWe implemented a large nonlinear signal transduction network combining protein kinase C, mitogen-activated protein kinase, phospholipase A2, and β isoform of phospholipase C networks. We simulated the network in 300 different cellular volumes using the exact Gillespie stochastic simulation algorithm and analyzed the results in both the time and frequency domain. In order to perform simulations in a reasonable time, we used modern parallel computing techniques. The analysis revealed that time and frequency domain characteristics depend on the system volume. The simulation results also indicated that there are several kinds of noise processes in the network, all of them representing different kinds of low-frequency fluctuations. In the simulations, the power of noise decreased on all frequencies when the system volume was increased.ConclusionsWe concluded that basic frequency domain techniques can be applied to the analysis of simulation results produced by the Gillespie stochastic simulation algorithm. This approach is suited not only to the study of fluctuations but also to the study of pure noise processes. Noise seems to have an important role in biochemical systems and its properties can be numerically studied by simulating the reacting system in different cellular volumes. Parallel computing techniques make it possible to run massive simulations in hundreds of volumes and, as a result, accurate statistics can be obtained from computational studies.