Section: New Results
Single neuron models
A dynamical system analysis of the adaptive spike threshold
Keywords : Action potential, spike threshold, integrateandfire model.
Participants : Jonathan Platkiewicz, Romain Brette.
This work was partially supported by the EC IP project FP6015879, FACETS and the Fondation d'Entreprise EADS.
Recent in vivo experiments have revealed that the action potential threshold depends on the rate of depolarization just preceding the spike. This phenomenon can be reproduced in the HodgkinHuxley model. We analyzed spike initiation in the ( V, h) phase space, where h is the sodium inactivation variable, and found that the dynamical system exhibits a saddle equilibrium, whose stable manifold is the curve of the threshold. We derived an equation of this manifold, which relates the threshold to the sodium inactivation variable. It leads to a differential equation of the threshold depending on the membrane potential, which translates into an integrateandfire model with an adaptive threshold. The model accounts well for the variability of threshold and the slopethreshold relationship. This work was presented at the CNS 2007 conference [Oops!] .
Estimating synaptic conductances
Keywords : Synaptic conductances.
Participants : Thierry Bal [ UNIC, CNRS, GifsurYvette ] , Romain Brette, Alain Destexhe [ UNIC, CNRS, GifsurYvette ] , Suzanne Piwkowska [ UNIC, CNRS, GifsurYvette ] , Olivier Faugeras, Vincent Pavan.
This work was done in collaboration with the UNIC lab (CNRS GifsurYvette) and presented at the SfN meeting in San Diego. It is supported by the ANR. [Oops!] .
Measuring synaptic conductances in central neurons in vivo is essential to understand the response selectivity of those neurons. Response selectivity can arise from particular timings of excitation and inhibition, which can be amplified by intrinsic conductances. Conductance measurements can be realized by reconstructing currentvoltage relations from Vm activity or by using the statistics of the Vm fluctuations. These methods suffer from one severe limitation, namely that it is necessary to accumulate statistics over several trials and different levels of polarization, which necessarily means that information about the variability unlocked to the stimulus is lost. We developed a method to measure excitatory and inhibitory conductances from singletrial intracellular voltage recordings. The principle of this method is to inject white noise into the membrane of the recorded neuron, and to extract the conductances from the voltage response to white noise. This is possible because the input signal varies faster than the synaptic conductances we try to estimate. The membrane equation defines a set of possible conductances, among which we choose the best according to a regularity criterion. This criterion is computed using wavelet transforms to preserve the highfrequency content of the signal. We successfully tested our method on numerical simulations of cortical neuron models with fluctuating synaptic conductances. The method will be tested using compartmental models of morphologicallyreconstructed pyramidal neurons with synaptic inputs simulated in dendrites, as well as in real cortical neurons in vitro. The interest of such a method is that it potentially allows, for the first time, to perform extraction of excitatory and inhibitory conductances from singletrials.
Bifurcation analysis of a general class of nonlinear integrate and fire neurons
Keywords : neuron models, dynamical system analysis, nonlinear dynamics, Hopf bifurcation, saddlenode bifurcation, BogdanovTakens bifurcation, Bautin bifurcation, saddle homoclinic bifurcation, subthreshold neuron oscillations.
Participant : Jonathan Touboul.
This work was partially supported by the EC IP project FP6015879, FACETS and the Fondation d'Entreprise EADS.
We introduced a new class of twodimensional nonlinear integrateandfire neuron models being computationally efficient and biologically plausible, i.e. able to reproduce a wide gamut of behaviors observed in invivo or invitro recordings of cortical neuron, and studied the bifurcation diagram of the members of this class with respect to the inputs and the coupling between the membrane potential and the adaptation variable. This class includes for instance two models widely used in computational neuroscience, the Izhikevich and the BretteGerstner models. Among other global bifurcations, this system shows a saddle homoclinic bifurcation curve. We show how this bifurcation diagram generates the most prominent cortical neuron behaviors. This study leads us to the introduction of a new neuron model, the quartic model, able to reproduce among all the behaviors of the Izhikevich and BretteGerstner models, selfsustained subthreshold oscillations, which are of great interest in neuroscience. We found that they all undergo a Hopf, a saddle node and a BogdanovTakens bifurcation. This work is accepted for publication in SIAM Applied Math., [Oops!] and has appeared as a research report [Oops!] .
A characterization of the first hitting time of Double Integral Processes to curved boundaries
Participants : Olivier Faugeras, Jonathan Touboul.
This work was partially supported by the EC IP project FP6015879, FACETS and the Fondation d'Entreprise EADS.
We continued the study of the statistics of spike trains and found a new formula for computing first hitting times of integrateandfire neurons with exponentially decaying synaptic conductances. This formula is valid for approximating any first hitting times for what we call Double Integral processes, which are nonMarkov. The problem of finding the probability distribution of the first hitting time of a Double Integral Process (DIP) such as the Integrated Wiener Proces (IWP) has been an important and difficult endeavor in stochastic calculus. It has applications in many fields of physics (first exit time of a particle in a noisy force field) or in biology and neuroscience (spike time distribution of an integrateandfire neuron with exponentially decaying synaptic current). The only results available are an approximation of the stationary mean crossing time and the distribution of the first hitting time of the IWP to a constant boundary. We generalize these results and find an analytical formula for the first hitting time of the IWP to a continuous piecewise cubic boundary. We use this formula to approximate the law of the first hitting time of a general DIP to a smooth curved boundary, and we provide an estimation of the convergence of this method. The accuracy of the approximation is computed in the general case for the IWP and the effective calculation of the crossing probability can be carried out through a MonteCarlo method. This paper is accepted for publication in the Journal of Applied Probability [106] and has appeared as a research report [Oops!] .
The spikes trains probability distributions: a stochastic calculus approach
Participants : Olivier Faugeras, Jonathan Touboul.
This work was partially supported by the EC IP project FP6015879, FACETS and the Fondation d'Entreprise EADS.
We wrote a review paper on the stochastic methods for characterizing spike trains that has been accepted for publication in the Journal of Physiology Paris [Oops!] . It has also appeared as a research report [Oops!] . We discuss the statistics of spikes trains for different types of integrateandfire neurons and different types of synaptic noise models. In contrast with the usual approaches in neuroscience, mainly based on statistical physics methods such as the FokkerPlanck equation or the meanfield theory, we chose the point of the view of the stochastic calculus theory to characterize neurons in noisy environments. We present four stochastic calculus techniques that can be used to find the probability distributions attached to the spikes trains. We illustrate the power of these techniques for four types of widely used neuron models. Despite the fact that these techniques are mathematically intricate we believe that they can be useful for answering questions in neuroscience that naturally arise from the variability of neuronal activity. For each technique we indicate its range of applicability and its limitations.
Eventdriven mathematical framework for noisy integrateandfire neuron networks
Participants : Olivier Faugeras, Jonathan Touboul.
This work was partially supported by the EC IP project FP6015879, FACETS and the Fondation d'Entreprise EADS.
This work is about an eventbased mathematical framework for studying the dynamics of networks of integrateandfire neuron driven by external noise. Such networks are classically studied using the FokkerPlanck equation [99] . In this study, we use the powerful tools developed for communication networks theory and define a formalism for the study of spiking neuron networks driven by an external noise. With this formalism, we address biological questions to characterize the different network regimes. In this framework, the probability distribution of the interspike interval is a fundamental parameter. We developed and apply several tools for defining and computing the probability density function (pdf) of the time of the first spike, using stochastic analysis. This point of view gives us an eventdriven strategy for simulating this type of random networks. This strategy has been implemented in an extension of the eventdriven simulator Mvaspike [Oops!] . We presented an eventdriven mathematical framework for noisy integrateandfire neuron networks as a poster in the CNS conference in Toronto [Oops!] .
The Cauchy problem for onedimensional spiking neuron models
Keywords : Spiking neurons, integrateandfire model.
Participant : Romain Brette.
This work was partially supported by the EC IP project FP6015879, FACETS and the Fondation d'Entreprise EADS.
We consider spiking neuron models defined by a onedimensional differential equation and a reset — i.e., neuron models of the integrateandfire type. We address the question of the existence and uniqueness of a solution on R for a given initial condition. It turns out that the reset introduces a countable and ordered set of backward solutions for a given initial condition, which has important implications in terms of neural coding and spike timing precision. This work has been accepted for publication in Cognitive Neurodynamics.
Synaptic conductance fluctuations in cortical neurons
Participants : Romain Brette, Zuzanna Piwkowska [ UNIC, CNRS, GifsurYvette ] , Martin Pospischil [ UNIC, CNRS, GifsurYvette ] , Julia Sliwa [ UNIC ] , Michelle RudolphLilith [ UNIC, CNRS, GifsurYvette ] , Thierry Bal [ UNIC, CNRS, GifsurYvette ] , Alain Destexhe [ UNIC, CNRS, GifsurYvette ] .
This work was partially supported by the ANR.
Cortical neurons are subject to sustained and irregular synaptic activity which causes important fluctuations of the membrane potential ( Vm ). The simplified, fluctuating pointconductance model of synaptic activity provides the starting point of a variety of methods for the analysis of intracellular Vm recordings. In this model, the synaptic excitatory and inhibitory conductances are described by Gaussiandistributed stochastic variables, or colored conductance noise. The matching of experimentally recorded Vm distributions to an invertible theoretical expression derived from the model allows the extraction of parameters characterizing the synaptic conductance distributions. This analysis can be complemented by the matching of experimental Vm power spectral densities (PSDs) to a theoretical template, even though the unexpected scaling properties of experimental PSDs limit the precision of this latter approach. Building on this stochastic characterization of synaptic activity, we also proposed methods to qualitatively and quantitatively evaluate spiketriggered averages of synaptic timecourses preceding spikes. This analysis points to an essential role for synaptic conductance variance in determining spike times. The methods were evaluated using controlled conductance injection in cortical neurons in vitro with the dynamicclamp technique.
This work was done in collaboration with the UNIC lab (CNRS GifsurYvette) and accepted for publication in Journal of Neuroscience Methods.
Computation of the electrical potential inside the nerve induced by an electrical stimulus
Participants : Maureen Clerc, David Guiraud [ Demar ] , Joan Fruitet [ stagiaire GENESYS ] , Sabir Jacquir [ postdoc GENESYS ] .
This work is part of the STICSANTE contract GENESYS.
The aim of [Oops!] is to investigate the activation conditions of the different nerves which control the bladder. The selective stimulation of the nerve fibers depends on electrode configuration and intensity of applied current. The goal of this study is to compute the electrical potential inside the nerve due to an applied boundary currents. A symmetrically cylindrical model, representing the geometry and electrical conductivity of a nerve surrounded by a connective tissue and a cuff is used. In the quasistatic approximation, the problem can be modeled by a Poisson equation with Neumann boundary conditions. A symmetric boundary integral formulation is discretized using mixed finite elements. We can thus compute an electrical potential distribution depending on the electrode configuration and the applied current inside a nerve. Our results show that the distribution of the electrical potential inside a nerve or a fascicle depends on the geometry of the electrode and the shape of the applied current.
Neural coding
Participants : Bruno Cessac, Pierre Kornprobst, JeanClaude Vasquez, Thierry Viéville.
This work was partially supported by the EC IP project FP6015879, FACETS and the Fondation d'Entreprise EADS.
It is likely that neurons transport information via, among other means, spike trains. From this point of view spike trains dynamics is more relevant than e.g. membrane potential dynamics. However, despite a number of significative experiments, the way how information is encoded in spikes trains remains mysterious. This is due to experimental but also conceptual obstacles. For example, the processing of experimental data requires statistical models for the probability distribution of spike trains. However, the traditionally used models (e.g. Poisson) are ad hoc and are known to be poorly adapted. Recent advances based on experiments suggest that more adapted probability distributions can be obtained via the statistical inference principle consisting of maximizing entropy under the constraints that average values of observables are consistent with measured data. More than a simple token we believe that this is a general principle. Indeed, the results presented in [Oops!] and submitted [107] , [100] show that there is a natural symbolic coding of the dynamics via spike trains. This coding, combined with the socalled thermodynamic formalism, coming from ergodic theory and statistical physics, allows to construct probability measures on the spike trains via a natural variational principle. These measures, called Gibbs measures, give a direct access to a wide number of statistical properties of orbits, some of them corresponding to quantities measured by biologists. The expected outcomes of this research work are twofold.

(i) Establish a mathematical framework for the analysis of spikes trains in collaboration with neurobiologists, like F. Grammont, from the ”Laboratoire de Neurobiologie et Psychopathologie”, Université de Nice. We aim to develop a statistical model, relevant for data analysis, using conductance based models whose dynamics has been analysed in [100] .

(ii) Use spiking neurons models to solve variational problems, using a suitable dynamics on the synaptic weights. This work enters in a project developed by P. Kornprobst and T. Viéville [108] . We have obtained a MESR PHD grant for this topic (the student is J.C. Vasquez).