Sequential estimation of intrinsic activity and synaptic input in single neurons by particle filtering with optimal importance density

This paper deals with the problem of inferring the signals and parameters that cause neural activity to occur. The ultimate challenge being to unveil brain’s connectivity, here we focus on a microscopic vision of the problem, where single neurons (potentially connected to a network of peers) are at the core of our study. The sole observation available are noisy, sampled voltage traces obtained from intracellular recordings. We design algorithms and inference methods using the tools provided by stochastic filtering that allow a probabilistic interpretation and treatment of the problem. Using particle filtering, we are able to reconstruct traces of voltages and estimate the time course of auxiliary variables. By extending the algorithm, through PMCMC methodology, we are able to estimate hidden physiological parameters as well, like intrinsic conductances or reversal potentials. Last, but not least, the method is applied to estimate synaptic conductances arriving at a target cell, thus reconstructing the synaptic excitatory/inhibitory input traces. Notably, the performance of these estimations achieve the theoretical lower bounds even in spiking regimes.Postprint (published version

Phase portraits of separable Hamiltonian systems

We study some generalizations of potential Hamiltonian systems (H(x, y) = y 2 + F(x)) with one degree of freedom. In particular, we are interested in Hamiltonian systems with Hamiltonian functions of type H(x, y) = F(x) + G(y) arising in applied mechanical problems. We present an algorithm to plot the phase portrait (include the behavior at infinity) of any Hamiltonian system of type H(x, y) = F(x) +G(y), where F and G are arbitrary polynomials. We are able to give the full description in the Poincaré disk according to the graphs of F and G, extending the well-known method for the “finite”phase portrait of potential systems

aCTeX: Autoaprenentatge científico-tècnic en xarxa

aCTeX és una aplicació web que genera dinàmicament llistats de preguntes tipus test, amb la particularitat que admet expressions matemàtiques introduïdes en llenguatge LaTeX a la base de dades. L’aplicació executa la conversió de LaTeX a HTML i ho presenta de manera integrada en qualsevol navegador. aCTeX permet a l’usuari (alumne) de fixar certes característiques del test (dificultat, temes). Un cop generada la llista de preguntes, l’alumne pot resoldre-la on-line i obtenir resultats i explicacions sobre les seves respostes. També pot imprimir el document per solucionar-lo off-line, respondre’l en un altre moment i recuperar de nou les explicacions. En lloc de limitar-se a informar de la veracitat de la resposta, la base de dades d’aCTeX està dissenyada per associar a cada resposta errònia un suggeriment a l’estudiant que l’apropi a la resposta correcta. aCTeX també facilita estadístiques per al seguiment global de l’alumnat

A black-box model for neurons

We explore the identification of neuronal voltage traces by artificial neural networks based on wavelets (Wavenet). More precisely, we apply a modification in the representation of dynamical systems by Wavenet which decreases the number of used functions; this approach combines localized and global scope functions (unlike Wavenet, which uses localized functions only). As a proof-of-concept, we focus on the identification of voltage traces obtained by simulation of a paradigmatic neuron model, the Morris-Lecar model. We show that, after training our artificial network with biologically plausible input currents, the network is able to identify the neuron's behaviour with high accuracy, thus obtaining a black box that can be then used for predictive goals. Interestingly, the interval of input currents used for training, ranging from stimuli for which the neuron is quiescent to stimuli that elicit spikes, shows the ability of our network to identify abrupt changes in the bifurcation diagram, from almost linear input-output relationships to highly nonlinear ones. These findings open new avenues to investigate the identification of other neuron models and to provide heuristic models for real neurons by stimulating them in closed-loop experiments, that is, using the dynamic-clamp, a well-known electrophysiology technique.Peer ReviewedPostprint (author's final draft

Geometric tools to determine the hyperbolicity of limit cycles

In this paper we present a new method to study limit cycles’ hyperbolicity. The main tool is the function ? = ([V,W] ^ V )/(V ^W), where V is the vector field under investigation and W a transversal one. Our approach gives a high degree of freedom for choosing operators to study the stability. It is related to the divergence test, but provides more information on the system’s dynamics. We extend some previous results on hyperbolicity and apply our results to get limit cycles’ uniqueness. Li´enard systems and conservative+dissipative systems are considered among the applications

Phase-amplitude dynamics in terms of extended response functions: invariant curves and arnold tongues

Phase response curves (PRCs) have been extensively used to control the phase of oscillators under perturbations. Their main advantage is the reduction of the whole model dynamics to a single variable (phase) dynamics. However, in some adverse situations (strong inputs, high-frequency stimuli, weak convergence,. . . ), the phase reduction does not provide enough information and, therefore, PRC lose predictive power. To overcome this shortcoming, in the last decade, new contributions have appeared that allow to reduce the system dynamics to the phase plus some transversal variable that controls the deviations from the asymptotic behaviour. We call this setting extended response functions. In particular, we single out the phase response function (PRF, a generalization of the PRC) and the amplitude response function (ARF) that account for the above-mentioned deviations from the oscillating attractor. It has been shown that in adverse situations, the PRC misestimate the phase dynamics whereas the PRF-ARF system provides accurate enough predictions. In this paper, we address the problem of studying the dynamics of the PRF-ARF systems under periodic pulsatile stimuli. This paradigm leads to a two-dimensional discrete dynamical system that we call 2D entrainment map. By using advanced methods to study invariant manifolds and the dynamics inside them, we construct an analytico-numerical method to track the invariant curves induced by the stimulus as two crucial parameters of the system increase (the strength of the input and its frequency). Our methodology also incorporates the computation of Arnold tongues associated to the 2D entrainment map. We apply the method developed to study inner dynamics of the invariant curves of a canonical type II oscillator model. We further compare the Arnold tongues of the 2D map with those obtained with the map induced only by the PRC, which give already noticeable differences. We also observe (via simulations) how high-frequency or strong enough stimuli break up the oscillatory dynamics and lead to phase-locking, which is well captured by the 2D entrainment map.Peer ReviewedPreprin

Limit cycles for generalized Abel equations

This paper deals with the problem of finding upper bounds on the number of periodic solutions of a class of one-dimensional non-autonomous differential equations: those with the right-hand sides being polynomials of degree n and whose coeficients are real smooth 1-periodic functions. The case n = 3 gives the so-called Abel equations which have been thoroughly studied and are quite understood. We consider two natural generalizations of Abel equations. Our results extend previous works of Lins Neto and Panov and try to step forward in the understanding of the case n > 3. They can be applied, as well, to control the number of limit cycles of some planar ordinary differential equations

Optimal control of oscillatory neuronal models with applications to communication through coherence

Macroscopic oscillations in the brain are involved in various cognitive and physiological processes, yet their precise function is not not completely understood. Communication Through Coherence (CTC) theory proposes that these rhythmic electrical patterns might serve to regulate the information flow between neural populations. Thus, to communicate effectively, neural populations must synchronize their oscillatory activity, ensuring that input volleys from the presynaptic population reach the postsynaptic one at its maximum phase of excitability. We consider an Excitatory-Inhibitory (E-I) network whose macroscopic activity is described by an exact mean-field model. The E-I network receives periodic inputs from either one or two external sources, for which effective communication will not be achieved in the absence of control. We explore strategies based on optimal control theory for phase-amplitude dynamics to design a control that sets the target population in the optimal phase to synchronize its activity with a specific presynaptic input signal and establish communication. The control mechanism resembles the role of a higher cortical area in the context of selective attention. To design the control, we use the phase-amplitude reduction of a limit cycle and leverage recent developments in this field in order to find the most effective control strategy regarding a defined cost function. Furthermore, we present results that guarantee the local controllability of the system close to the limit cycle

Bifurcation gaps in asymmetric and high-dimensional hypercycles

Hypercycles are catalytic systems with cyclic architecture. These systems have been suggested to play a key role in the maintenance and increase of information in prebiotic replicators. It is known that for a large enough number of hypercycle species (n>4 the coexistence of all hypercycle members is governed by a stable periodic orbit. Previous research has characterized saddle-node (s-n) bifurcations involving abrupt transitions from stable hypercycles to extinction of all hypercycle members, or, alternatively, involving the outcompetition of the hypercycle by so-called mutant sequences or parasites. Recently, the presence of a bifurcation gap between a s-n bifurcation of periodic orbits and a s-n of fixed points has been described for symmetric five-member hypercycles. This gap was found between the value of the replication quality factor Q from which the periodic orbit vanishes (QPO)and the value where two unstable (nonzero) equilibrium points collide (QSS). Here, we explore the persistence of this gap considering asymmetries in replication rates in five-member hypercycles as well as considering symmetric, larger hypercycles. Our results indicate that both the asymmetry in Malthusian replication constants and the increase in hypercycle members enlarge the size of this gap. The implications of this phenomenon are discussed in the context of delayed transitions associated to the so-called saddle remnants. Read More: https://www.worldscientific.com/doi/abs/10.1142/S021812741830001X Read More: https://www.worldscientific.com/doi/abs/10.1142/S021812741830001XPeer ReviewedPreprin