Complex oscillations in the delayed Fitzhugh-Nagumo equation

Motivated by the dynamics of neuronal responses, we analyze the dynamics of the Fitzhugh-Nagumo slow-fast system with delayed self-coupling. This system provides a canonical example of a canard explosion for sufficiently small delays. Beyond this regime, delays significantly enrich the dynamics, leading to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a delay-induced subcritical Bogdanov-Takens instability arising at the fold points of the S-shaped critical manifold. Underlying the transition from canard-induced to delay-induced dynamics is an abrupt switch in the nature of the Hopf bifurcation

A three-scale model of spatio-temporal bursting

© 2016 Society for Industrial and Applied Mathematics. We study spatio-temporal bursting in a three-scale reaction diffusion equation organized by the winged cusp singularity. For large time-scale separation the model exhibits traveling bursts, whereas for large space-scale separation the model exhibits standing bursts. Both behaviors exhibit a common singular skeleton, whose geometry is fully determined by persistent bifurcation diagrams of the winged cusp. The modulation of spatio-temporal bursting in such a model naturally translates into paths in the universal unfolding of the winged cusp.The research leading to these results has received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet 670645 and from DGAPA-Universidad Nacional Aut onoma de Mexico under the PAPIIT Grant IA105816

Rhythmogenesis and Bifurcation Analysis of 3-Node Neural Network Kernels

Central pattern generators (CPGs) are small neural circuits of coupled cells stably producing a range of multiphasic coordinated rhythmic activities like locomotion, heartbeat, and respiration. Rhythm generation resulting from synergistic interaction of CPG circuitry and intrinsic cellular properties remains deficiently understood and characterized. Pairing of experimental and computational studies has proven key in unlocking practical insights into operational and dynamical principles of CPGs, underlining growing consensus that the same fundamental circuitry may be shared by invertebrates and vertebrates. We explore the robustness of synchronized oscillatory patterns in small local networks, revealing universal principles of rhythmogenesis and multi-functionality in systems capable of facilitating stability in rhythm formation. Understanding principles leading to functional neural network behavior benefits future study of abnormal neurological diseases that result from perturbations of mechanisms governing normal rhythmic states. Qualitative and quantitative stability analysis of a family of reciprocally coupled neural circuits, constituted of generalized Fitzhugh–Nagumo neurons, explores symmetric and asymmetric connectivity within three-cell motifs, often forming constituent kernels within larger networks. Intrinsic mechanisms of synaptic release, escape, and post-inhibitory rebound lead to differing polyrhythmicity, where a single parameter or perturbation may trigger rhythm switching in otherwise robust networks. Bifurcation analysis and phase reduction methods elucidate qualitative changes in rhythm stability, permitting rapid identification and exploration of pivotal parameters describing biologically plausible network connectivity. Additional rhythm outcomes are elucidated, including phase-varying lags and broader cyclical behaviors, helping to characterize system capability and robustness reproducing experimentally observed outcomes. This work further develops a suite of visualization approaches and computational tools, describing robustness of network rhythmogenesis and disclosing principles for neuroscience applicable to other systems beyond motor-control. A framework for modular organization is introduced, using inhibitory and electrical synapses to couple well-characterized 3-node motifs described in this research as building blocks within larger networks to describe underlying cooperative mechanisms

The investigation of variable nernst equilibria on isolated neurons and coupled neurons forming discrete and continuous networks

Since the introduction of the Hodgkin-Huxley equations, used to describe the excitation of neurons, the Nernst equilibria for individual ion channels have assumed to be constant in time. Recent biological recordings call into question the validity of this assumption. Very little theoretical work has been done to address the issue of accounting for these non-static Nernst equilibria using the Hodgkin-Huxley formalism. This body of work incorporates non-static Nernst equilibria into the generalized Hodgkin-Huxley formalism by considering the first-order effects of the Nernst equation. It is further demonstrated that these effects are likely dominate in neurons with diameters much smaller than that of the squid giant axon and permeate important information processing regions of the brain such as the hippocampus. Particular results of interest include single-cell bursting due to the interplay of spatially separated neurons, pattern formation via spiral waves within a soliton-like regime, and quantifiable shifts in the multifractality of hippocampal neurons under the administration of various drugs at varying dosages. This work provides a new perspective on the variability of Nernst equilibria and demonstrates its utility in areas such as pharmacology and information processing

Low-dimensional models of single neurons: A review

The classical Hodgkin-Huxley (HH) point-neuron model of action potential generation is four-dimensional. It consists of four ordinary differential equations describing the dynamics of the membrane potential and three gating variables associated to a transient sodium and a delayed-rectifier potassium ionic currents. Conductance-based models of HH type are higher-dimensional extensions of the classical HH model. They include a number of supplementary state variables associated with other ionic current types, and are able to describe additional phenomena such as sub-threshold oscillations, mixed-mode oscillations (subthreshold oscillations interspersed with spikes), clustering and bursting. In this manuscript we discuss biophysically plausible and phenomenological reduced models that preserve the biophysical and/or dynamic description of models of HH type and the ability to produce complex phenomena, but the number of effective dimensions (state variables) is lower. We describe several representative models. We also describe systematic and heuristic methods of deriving reduced models from models of HH type

Forward Electrophysiological Modeling and Inverse Problem for Uterine Contractions during Pregnancy

Uterine contractile dysfunction during pregnancy is a significant healthcare challenge that imposes heavy medical and financial burdens on both human beings and society. In the U.S., about 12% of babies are born prematurely each year, which is a leading cause of neonatal mortality and increases the possibility of having subsequent health problems. Post-term birth, in which a baby is born after 42 weeks of gestation, can cause risks for both the newborn and the mother. Currently, there is a limited understanding of how the uterus transitions from quiescence to excitation, which hampers our ability to detect labor and treat major obstetric syndromes associated with contractile dysfunction. Therefore, it is critical to develop objective methods to investigate the underlying contractile mechanism using a non-invasive sensing technique. This dissertation focuses on the multiscale forward electromagnetic modeling of uterine contractile activities and the inverse estimation of underlying source currents from abdominal magnetic field measurements. We develop a realistic multiscale forward electromagnetic model of uterine contractions in the pregnant uterus, taking into account current electrophysiological and anatomical knowledge of the uterus. Previous models focused on generating contractile forces at the organ level or on ionic concentration changes at the cellular level. Our approach is to characterize the electromagnetic fields of uterine contractions jointly at the cellular, tissue, and organ levels. At the cellular level, focusing on both plateau-type and bursting-type action potentials, we introduce a generalized version of the FitzHugh-Nagumo equations and analyze its response behavior based on bifurcation theory. To represent the anisotropy of the myometrium, we introduce a random conductivity tensor model for the fiber orientations at the tissue level. Specifically, we divide the uterus into contiguous regions, each of which is assigned a random fiber angle. We also derive analytical expressions for the spiking frequency and propagation velocity of the bursting potential. At the organ level, we propose a realistic four-compartment volume conductor, in which the uterus is modeled based on the magnetic resonance imaging scans of a near-term woman and the abdomen is curved to match the device used to take the magnetomyography measurements. To mimic the effect of the sensing direction, we incorporate a sensor array model on the surface of abdomen. We illustrate our approach using numerical examples and compute the magnetic field using the finite element method. Our results show that fiber orientation and initiation location are the key factors affecting the magnetic field pattern, and that our multiscale forward model flexibly characterizes the limited-propagation local contractions at term. These results are potentially important as a tool for interpreting the non-invasive measurements of uterine contractions. We also consider the inverse problem of uterine contractions during pregnancy. Our aim is to estimate the myometrial source currents that generate the external magnetomyography measurements. Existing works approach this problem using synthetic electromyography data. Our approach instead proceeds in two stages: develop a linear approximation model and conduct the estimation. In the first stage, we derive a linear approximation model of the sensor-oriented magnetic field measurements with respect to source current dipoles in the myometrium, based on a lead-field matrix. In particular, this lead-field matrix is analytically computed from distributed current dipoles in the myometrium according to quasi-static Maxwell\u27s equations, using the finite element method. In the second stage, we solve a constrained least-squares problem to estimate the source currents, from which we predict the intrauterine pressure. We demonstrate our approach through numerical examples with synthetic data that are generated using our multiscale forward model. In the simulations, we assume that the excitation is located at the fundus of the uterus. We also illustrate our approach using real data sets, one of which has simultaneous contractile pressure measurements. The results show that our method well captures the short-distance propagation of uterine contractile activities during pregnancy, the change of excitation area in subsequent contractions or even in a single contraction, and the timing of uterine contractions. These findings are helpful in understanding the physiological and functional properties of the uterus, potentially enabling the diagnosis of labor and the treatment of obstetric syndromes associated with contractile dysfunction such as preterm birth and post-term birth