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Concept# Threshold model

Résumé

In mathematical or statistical modeling a threshold model is any model where a threshold value, or set of threshold values, is used to distinguish ranges of values where the behaviour predicted by the model varies in some important way. A particularly important instance arises in toxicology, where the model for the effect of a drug may be that there is zero effect for a dose below a critical or threshold value, while an effect of some significance exists above that value. Certain types of regression model may include threshold effects.
Collective behavior
Threshold models are often used to model the behavior of groups, ranging from social insects to animal herds to human society.
Classic threshold models were introduced by Sakoda, in his 1949 dissertation and the Journal of Mathematical Sociology (JMS vol 1 #1, 1971).
They were subsequently developed by Schelling, Axelrod, and Granovetter to model collective behavior. Schelling used a special case of S

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CH-411: Cellular signalling

Presentation of selected signalling pathways with emphasis on both the mechanism of action of the molecules involved, molecular interactions and the role of their spatio-temporal organization within the cell, considering cellular dimensions and conditions.

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Measures of surprise have been recently studied in statistics. This new concept can be used as the first exploratory tool to verify if a model under the null hypothesis fits appropriately. As no alternative models are necessary, the use of the measures of surprise is considered very simple. At the same time, this new alternative to test the goodness of a model cannot replace the full Bayesian analysis. The aim of this project is threshold selection for threshold models. The estimate of the threshold could be investigated by using the measures of surprise. The reason is that no alternative models are specified for non-extreme data, for which the distribution in extreme value theory is unknown, and thus the surprise would be a credible tool.

2010An important feature of the nervous system is its ability to adapt to new stimuli. This adaptation allows for optimal encoding of the incoming information by dynamically changing the coding strategy based upon the incoming inputs to the neuron. At the level of single cells, this widespread phenomena is often referred to as spike-frequency adaptation, since it manifests as a history-dependent modulation of the neurons firing frequency. In this thesis I focus on how a neuron is able to adapt its activity to a specific input as well as on the function of such adaptive mechanisms. To study these adaptive processes different approaches have been used, from empirical observations of neural activities to detailed modeling of single cells. Here, I approach these problems by using simplified threshold models. In particular, I introduced a new generalization of the integrate-and-fire model (GIF) along with a convex fitting method allowing for efficient estimation of model parameters. Despite its relative simplicity I show that this neuron model is able to reproduce neuron behaviors with a high degree of accuracy. Moreover, using this method I was able to show that cortical neurons are equipped with two distinct adaptation mechanisms. First, a spike-triggered current that captures the complex influx of ions generated after the emission of a spike. While the second is a movement of the firing threshold, which possibly reflects the slow inactivation of sodium channels induced by the spiking activity. The precise dynamics of these adaptation processes is cell-type specific, explaining the difference of firing activity reported in different neuron types. Consequently, neuronal types can be classified based on model parameters. In Pyramidal neurons spike-dependent adaptation lasts for seconds and follows a scale-free dynamics, which is optimally tuned to encodes the natural inputs that pyramidal neurons receive in vivo. Finally using an extended version of the GIF model, I show that adaptation is not only a spike-dependent phenomenon, but also acts at the subthreshold level. In Pyramidal neurons the dynamics of the firing threshold is influenced by the subthreshold membrane potential. Spike-dependent and voltage-dependent adaptation interact in an activity-dependent way to ultimately shape the filtering properties of the membrane on the input statistics. Equipped with such a mechanism, Pyramidal neurons behave as integrators at low inputs and as a coincidence detectors at high inputs, maintaining sensitivity to input fluctuations across all regimes.

Wulfram Gerstner, Olivier Richard Hagens, Skander Mensi, Christian Antonio Pozzorini

The way in which single neurons transform input into output spike trains has fundamental consequences for network coding. Theories and modeling studies based on standard Integrate-and-Fire models implicitly assume that, in response to increasingly strong inputs, neurons modify their coding strategy by progressively reducing their selective sensitivity to rapid input fluctuations. Combining mathematical modeling with in vitro experiments, we demonstrate that, in L5 pyramidal neurons, the firing threshold dynamics adaptively adjust the effective timescale of somatic integration in order to preserve sensitivity to rapid signals over a broad range of input statistics. For that, a new Generalized Integrate-and-Fire model featuring nonlinear firing threshold dynamics and conductance-based adaptation is introduced that outperforms state-of-the-art neuron models in predicting the spiking activity of neurons responding to a variety of in vivo-like fluctuating currents. Our model allows for efficient parameter extraction and can be analytically mapped to a Generalized Linear Model in which both the input filter—describing somatic integration—and the spike-history filter—accounting for spike-frequency adaptation—dynamically adapt to the input statistics, as experimentally observed. Overall, our results provide new insights on the computational role of different biophysical processes known to underlie adaptive coding in single neurons and support previous theoretical findings indicating that the nonlinear dynamics of the firing threshold due to Na+-channel inactivation regulate the sensitivity to rapid input fluctuations.