A new Mathematical Framework to Understand Single Neuron Computations

An 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.

Simple Neuron Models and Adaptation

Skander Mensi

Predicting activity of single neuron is an important part of the computational neuroscience and a great challenge. Several mathematical models exist, from the simple (one compartment and few parameters, like the SRM or the IF-type models), to the more complex (biophysical model like the Hodgkin-Huxley model). All these models have their own advantages and limitations, but no one is able to reproduce the exact behavior of real neurons. Multiple projects try to simulate complex neural networks, or even the whole brain (i.e. the blue brain project or other big network simulation). To achieve this goal it is very important that simple neuron models simulate, for a low computational cost, the precise activities of all neuron classes. It is well-known that neurons exhibit a lot of different activity patterns (from an electro-physiological point of view). Here we have focused on pyramidal neurons from the layer 5 of the neocortex. They are classified as regular spiking-cells. This cell type shows adaptation and like other neurons, refractoriness. Adaptation and refractoriness are very common neuronal activity in the brain, and so it is important to have a simple model which can reproduce this kind of activity. This project deals with two classical simple neuron models: the adaptive exponential integrate-and-fire model (AdEx) and the spike response model (SRM). We deter- mined the parameters of these two models using data generated with a detailed model, the Destexhe's model which is a HH-like model for cortical pyramidal cells, stimulated with different current injection scenarios. In a second time the model parameters have been set using data from in-vitro recordings of 4 layer 5 pyramidal rat neurons, stimulated with a sinusoid in vivo-like protocol, injected somatically in current-clamp configuration. Then we show that this type of model can capture adaptation and can reproduce the activity of neuron with a high reliability

2008

Complexity and performance in simple neuron models

Wulfram Gerstner, Skander Mensi, Richard Naud

The ability of simple mathematical models to predict the activity of single neurons is important for computational neuroscience. In neurons, stimulated by a time-dependent current or conductance, we want to predict precisely the timing of spikes and the sub-threshold voltage. During the last years several models have been tested on this type of data but never compared with the same protocol. One of the major outcome is that, from a certain degree of complexity, all are very efficient and gave statistically indistinguishable results. We studied a class of integrate-and-fire models (IF), with each member of the class implementing a selection of possible improvements: exponential voltage non-linearity1, spike-triggered adaptation current2, spike-triggered change in conductance, moving threshold3, sub-threshold voltage-dependent currents4. Each refinement adds a new term to the equations of the IF model. This IF family is extendable and adaptable to different neuron types and is able to deal with complex neural activities (i.e. adaptation, facilitation, bursting, relative refractoriness, ...). To systematically explore the effects of a given term of the model a new fitting procedure based on linear regression of voltage change5 is used. This method is fast, robust and allows the extraction of all the models parameters from a few seconds recordings of fluctuating injected current and membrane potential with hundreds spikes without any prior knowledge. To investigate the effect of our modifications, we used as training data three different Hodgkin-Huxley-like models, and two experimental recordings of fast spiking and regular spiking cells and then evaluate each IF model on a given test set. We observe that it is possible to fit a model that can reproduce the activity of neurons with high reliability (i.e. almost 100 % of the spike time and less than 1 mV of sub-threshold voltage difference). Using this framework one can classify IF models in terms of complexity and performance and evaluate the importance of each term for different stimulation paradigms.

2010

Synaptic plasticity across different time scales and its functional implications

Claudia Clopath

Humans and animals learn by modifying the synaptic strength between neurons, a phenomenon known as synaptic plasticity. These changes can be induced by rather short stimuli (lasting, for instance, only a few seconds), yet, in order to be useful for long-term memory, they should remain stable for months or years. Experimentalists study synaptic plasticity by applying a vast variety of protocols. In the present thesis we focus on protocols that fall under two main categories: (i) Those that induce synaptic modifications that last for only a few hours ("early phase" of plasticity) (ii) Those that allow synapses to undergo a sequence of steps that transforms the rapid changes occurring during the "early phase" into a stable memory trace ("late phase" of plasticity). The goal of this thesis is to better understand synaptic plasticity across these different phases, early and late, by creating compact mathematic models to describe the plasticity mechanisms. Our approach allows for a synthetic view of the field as well as the exploration of functional consequences of learning. In this direction, we propose a model for the induction of synaptic plasticity that depends on the presynaptic spike time and nonlinearly on the postsynaptic voltage. The model is able to reproduce a broad range of experimental protocols such as voltage-clamp experiments and spike-timing experiments. Since the voltage is a key element in the model, we describe the neuronal activity by using a compact neuron model that faithfully reproduces the voltage time course of pyramidal neurons. In addition, this model of the induction of synaptic plasticity is combined with a trigger process for protein synthesis, and the final stabilization mechanism in order to describe the "late phase". In this combinatory form, it is able to explain experimental phenomena known as tagging experiments and to make testable predictions. A study of functional consequences of the induction model reveals selectivity in the inputs, independent component analysis computation and a tight relation between connectivity and coding. In parallel a top-down approach finding independent components is used to derive a rate-based learning rule which shows structural correlations with the induction model. This unified model across different time scales allowing the stabilization of synapses is crucial to understand learning and memory processes in animals and humans, and a necessary ingredient for any large-scale model of the brain.

EPFL2009