Jacques Henry

Dynamics of the Subthalamo-pallidal Complex in Parkinson’s Disease During Deep Brain Stimulation

The dynamics of the subthalamo-pallidal complex in Parkinson’s disease during deep brain stimulation (DBS) were studied using two models, a simple firing-rate model and a population-based model. We extended the simple firing-rate model of the complex formed by the subthalamic nucleus (STN) and the external segment of the Globus Pallidus (GPe) to explore its dynamical regime during DBS. More specifically, the modulation of neuronal activity (i.e., pattern and amplitude) during DBS was studied. A similar approach was used with the population-based model. Simulation results revealed a gradual decrease in bursting activity in STN cells when the DBS frequency increased. In addition, the contribution of the stimulation current type (mono- or biphasic) to the results was also examined. A comparison of the two models indicated that the population-based model was more biologically realistic and more appropriate for exploring DBS mechanisms. Understanding the underlying mechanisms of DBS is a prerequisite for developing new stimulation protocols.

Hysteresis phenomena between periodic and stationary solutions in a model of pacemaker and nonpacemaker coupled cardiac cells

We were interested in investigating the behaviour of a cardiac electrophysiological model including coupled pacemaker (PM) and nonpacemaker (NPM) cells. To this aim, a modified version of the model of Van Capelle and Durrer was used. First, few discrete values were assigned to coupling resistance (CR) and respective cell sizes and numerical simulations versus time showed three possible kinds of response pattern: sustained rhythmic activity, subthreshold oscillations, and complete inhibition. Then, after setting a fixed value to PM cell size, we undertake a thorough study of the system by using bifurcation-continuation techniques and CR was chosen as the continuation parameter. On the maximum action potential — CR plane representation, we could describe five behavioural zones: complete inhibition, coexistence of complete inhibition and NPM large oscillations, NPM large oscillations, coexistence of NPM large oscillations and subthreshold oscillations, subthreshold oscillations. Within the zones of qualitatively different coexisting solutions, a detailed exploration clearly demonstrated the presence of hysteresis cycles. Indeed, the status of the system depended on its immediate previous story within narrow ranges of CR values. Such a coexistence of stable solutions for identical values of CR may suggest an explanation of the intermittant activity elicited from abnormal ectopic foci observed in certain ventricular rhythm disturbances. In addition, a Hopf bifurcation point, from which emerged stationary and periodic solutions, was followed on the PM cell size — CR plane and from this representation we could deduce that the smaller the PM cell, the higher the CR must be for the PM cell to escape from the NPM cell inhibition.

A bilayer model of human atria: mathematical background, construction, and assessment.

AIMS Atrial numerical modelling has generally represented the organ as either a surface or tissue with thickness. While surface models have significant computational advantages over tissue models, they cannot fully capture propagation patterns seen in vivo, such as dissociation of activity between endo- and epicardium. We introduce an intermediate representation, a bilayer model of the human atria, which is capable of recreating recorded activation patterns. METHODS AND RESULTS We simultaneously solved two surface monodomain problems by formalizing an optimization method to set a coupling term between them. Two different asymptotically equivalent numerical implementations of the model are presented. We then built a geometrically and electrophysiologically detailed model of the human atria based on CT data, including two layers of fibre directions, major muscle bundles, and discrete atrial coupling. We adjusted parameters to recreate clinically measured activation times. Activation was compared with a monolayer model. Activation was fit to the physiological range measured over the entire atria. The crista terminalis and pectinate muscles were important for local right atrial activation, but did not significantly affect total activation time. Propagation in the bilayer model was similar to that of a monolayer, but with noticeable difference, due to three-dimensional propagation where fibre direction changed abruptly across the wall, resulting in a slight dissociation of activity. CONCLUSION Atrial structure plays the dominant role in determining activation. A bilayer model is able to take into account transmural heterogeneities, while maintaining the low computational load associated with surface models.

Pairwise sequence alignment using a PROSITE pattern-derived similarity score

Existing methods for alignments are based on edition costs computed additionally position by position, according to a fixed substitution matrix: a substitution always has the same weight regardless of the position. Nevertheless the biologist favours a similarity according to his knowledge of the structure or the function of the sequences considered. In the particular case of proteins, we present a method consisting in integrating other information, such as patterns of the PROSITE databank, in the classical dynamic programming algorithm. The method consists in making an alignment by dynamic programming taking a decision not only letter by letter as in the Smith & Waterman algorithm but also by giving a reward when aligning patterns.

Population density models of integrate-and-fire neurons with jumps: well-posedness

In this paper we study the well-posedness of different models of population of leaky integrate-and-fire neurons with a population density approach. The synaptic interaction between neurons is modeled by a potential jump at the reception of a spike. We study populations that are self excitatory or self inhibitory. We distinguish the cases where this interaction is instantaneous from the one where there is a repartition of conduction delays. In the case of a bounded density of delays both excitatory and inhibitory population models are shown to be well-posed. But without conduction delay the solution of the model of self excitatory neurons may blow up. We analyze the different behaviours of the model with jumps compared to its diffusion approximation.

Synchronization of an Excitatory Integrate-and-Fire Neural Network

In this paper, we study the influence of the coupling strength on the synchronization behavior of a population of leaky integrate-and-fire neurons that is self-excitatory with a population density approach. Each neuron of the population is assumed to be stochastically driven by an independent Poisson spike train and the synaptic interaction between neurons is modeled by a potential jump at the reception of an action potential. Neglecting the synaptic delay, we will establish that for a strong enough connectivity between neurons, the solution of the partial differential equation which describes the population density function must blow up in finite time. Furthermore, we will give a mathematical estimate on the average connection per neuron to ensure the occurrence of a burst. Interpreting the blow up of the solution as the presence of a Dirac mass in the firing rate of the population, we will relate the blow up of the solution to the occurrence of the synchronization of neurons. Fully stochastic simulations of a finite size network of leaky integrate-and-fire neurons are performed to illustrate our theoretical results.

Development and validation of a neural population model based on the dynamics of a discontinuous membrane potential neuron model.

The major goal of this study was to develop a population density based model derived from statistical mechanics based on the dynamics of a discontinuous membrane potential neuron model. A secondary goal was to validate this model by comparing results from a direct simulation approach on the one hand and our population based approach on the other hand. Comparisons between the two approaches in the case of a synaptically uncoupled and a synaptically coupled neural population produced satisfactory qualitative agreement in terms of firing rate and mean membrane potential. Reasonable quantitative agreement was also obtained for these variables in performed simulations. The results of this work based on the dynamics of a discontinuous membrane potential neuron model provide a basis to simulate phenomenologically large-scale neuronal networks with a reasonably short computing time.

Asymptotic analysis of reaction-diffusion-electromigration systems

We consider a Nernst-Planck-Poisson system modelling ion migration through biological membranes, in the one dimensional case. The model includes both the effects of biochemical reaction between ions and of fixed charges. We state the existence of solutions under either an imposed potential condition or an imposed current condition. We study the asymptotical behaviour of solutions in the limit of electroneutrality. Non uniform convergence gives rise to jump in the potential, known as Donnan potential. Finally we give correctors which describe the charged boundary layers.

An Asymptotic Two-Layer Monodomain Model of Cardiac Electrophysiology in the Atria: Derivation and Convergence

We investigate a dimensional reduction problem of a reaction-diffusion system related to cardiac electrophysiology modeling in the atria. The atrial tissues are very thin. The physical problem is then routinely stated on a two-dimensional manifold. However, some electrophysiological heterogeneities are located through the thickness of the tissue. Despite their biomedical significance, the usual dimensional reduction techniques tend to average and erase their influence on the two-dimensional propagation. We introduce a two-dimensional model with two coupled superimposed layers that allows us to take into account three-dimensional phenomena, but retains a reasonable computational cost. We present its mathematical derivation, show its convergence toward the three-dimensional model, and check numerically its convergence speed.

Theoretical connections between mathematical neuronal models corresponding to different expressions of noise

Identifying the right tools to express the stochastic aspects of neural activity has proven to be one of the biggest challenges in computational neuroscience. Even if there is no definitive answer to this issue, the most common procedure to express this randomness is the use of stochastic models. In accordance with the origin of variability, the sources of randomness are classified as intrinsic or extrinsic and give rise to distinct mathematical frameworks to track down the dynamics of the cell. While the external variability is generally treated by the use of a Wiener process in models such as the Integrate-and-Fire model, the internal variability is mostly expressed via a random firing process. In this paper, we investigate how those distinct expressions of variability can be related. To do so, we examine the probability density functions to the corresponding stochastic models and investigate in what way they can be mapped one to another via integral transforms. Our theoretical findings offer a new insight view into the particular categories of variability and it confirms that, despite their contrasting nature, the mathematical formalization of internal and external variability is strikingly similar.