Continuation-based numerical detection of after-depolarization and spike-adding thresholds.

PublishedJournal ArticleResearch Support, Non-U.S. Gov'tThe changes in neuronal firing pattern are signatures of brain function, and it is of interest to understand how such changes evolve as a function of neuronal biophysical properties. We address this important problem by the analysis and numerical investigation of a class of mechanistic mathematical models. We focus on a hippocampal pyramidal neuron model and study the occurrence of bursting related to the after-depolarization (ADP) that follows a brief current injection. This type of burst is a transient phenomenon that is not amenable to the classical bifurcation analysis done, for example, for periodic bursting oscillators. In this letter, we show how to formulate such transient behavior as a two-point boundary value problem (2PBVP), which can be solved using well-known continuation methods. The 2PBVP is formulated such that the transient response is represented by a finite orbit segment for which onsets of ADP and additional spikes in a burst can be detected as bifurcations during a one-parameter continuation. This in turn provides us with a direct method to approximate the boundaries of regions in a two-parameter plane where certain model behavior of interest occurs. More precisely, we use two-parameter continuation of the detected onset points to identify the boundaries between regions with and without ADP and bursts with different numbers of spikes. Our 2PBVP formulation is a novel approach to parameter sensitivity analysis that can be applied to a wide range of problems.The research for this letter was done while J.N. was a Ph.D. student at the University of Bristol, supported by grant EP/E032249/1 from the Engineering and Physical Sciences Research Council (EPSRC). The research of K.T-A. was supported by EPSRC grant EP/I018638/1 and that of H.M.O. by grant UOA0718 of the Royal Society of NZ Marsden Fun

Dynamical systems analysis of spike-adding mechanisms in transient bursts

The electronic version of this article is the complete one and can be found online at: doi:10.1186/2190-8567-2-7Open Access ArticleTransient bursting behaviour of excitable cells, such as neurons, is a common feature observed experimentally, but theoretically, it is not well understood. We analyse a five-dimensional simplified model of after-depolarisation that exhibits transient bursting behaviour when perturbed with a short current injection. Using one-parameter continuation of the perturbed orbit segment formulated as a well-posed boundary value problem, we show that the spike-adding mechanism is a canard-like transition that has a different character from known mechanisms for periodic burst solutions. The biophysical basis of the model gives a natural time-scale separation, which allows us to explain the spike-adding mechanism using geometric singular perturbation theory, but it does not involve actual bifurcations as for periodic bursts. We show that unstable sheets of the critical manifold, formed by saddle equilibria of the system that only exist in a singular limit, are responsible for the spike-adding transition; the transition is organised by the slow flow on the critical manifold near folds of this manifold. Our analysis shows that the orbit segment during the spike-adding transition includes a fast transition between two unstable sheets of the slow manifold that are of saddle type. We also discuss a different parameter regime where the presence of additional saddle equilibria of the full system alters the spike-adding mechanism.Engineering and Physical Sciences Research Council (EPSRC

Dynamical systems analysis of spike-adding mechanisms in transient bursts

Transient bursting behaviour of excitable cells, such as neurons, is a common feature observed experimentally, but theoretically, it is not well understood. We analyse a five-dimensional simplified model of after-depolarisation that exhibits transient bursting behaviour when perturbed with a short current injection. Using one-parameter continuation of the perturbed orbit segment formulated as a well-posed boundary value problem, we show that the spike-adding mechanism is a canard-like transition that has a different character from known mechanisms for periodic burst solutions. The biophysical basis of the model gives a natural time-scale separation, which allows us to explain the spike-adding mechanism using geometric singular perturbation theory, but it does not involve actual bifurcations as for periodic bursts. We show that unstable sheets of the critical manifold, formed by saddle equilibria of the system that only exist in a singular limit, are responsible for the spike-adding transition; the transition is organised by the slow flow on the critical manifold near folds of this manifold. Our analysis shows that the orbit segment during the spike-adding transition includes a fast transition between two unstable sheets of the slow manifold that are of saddle type. We also discuss a different parameter regime where the presence of additional saddle equilibria of the full system alters the spike-adding mechanism

Hydrothermal formation of heavy rare earth element (HREE)– xenotime deposits at 100 °C in a sedimentary basin

Most rare earth element deposits form from magmatic fluids, but there have also been discoveries of heavy rare earth element (HREE)– enriched hydrothermal xenotime deposits within sedimentary basins. As xenotime is notoriously insoluble, the question arises as to whether these lesser-known deposits form at typical basin temperatures or by influx of much hotter magmatic-hydrothermal fluids. The Browns Range District in northern Western Australia hosts deposits of xenotime that are enriched in HREEs and also uranium. The ore bodies consist of fault-controlled hydrothermal quartz-xenotime breccias that crosscut Archean basement rocks and overlying Paleoproterozoic sandstones. Analyses of fluid inclusions show that the xenotime precipitated at remarkably low temperatures, between 100 and 120 °C, in response to decompression boiling. The inclusions contain high excess concentrations of yttrium (10−3 mol/kg), REEs (1–7 × 10−5 mol/kg), and uranium (4 × 10−5 mol/kg) in equilibrium with xenotime at these low temperatures, showing that availability of phosphate limited the amount of xenotime precipitated. The analyses further identify SO4 2– and Cl– as the ligands that facilitated the elevated REE and uranium solubilities. These findings establish that significant REE transport and deposition is feasible at basin temperatures, and hence they raise the potential of unconformity settings for REE exploration. Moreover, the aqueous metal contents support a genetic link between this type of ore fluid and world-class Proterozoic unconformity-related uranium deposits elsewhere

A Ca2+-Based Computational Model for NMDA Receptor-Dependent Synaptic Plasticity at Individual Post-Synaptic Spines in the Hippocampus

Associative synaptic plasticity is synapse specific and requires coincident activity in pre-synaptic and post-synaptic neurons to activate NMDA receptors (NMDARs). The resultant Ca2+ influx is the critical trigger for the induction of synaptic plasticity. Given its centrality for the induction of synaptic plasticity, a model for NMDAR activation incorporating the timing of pre-synaptic glutamate release and post-synaptic depolarization by back-propagating action potentials could potentially predict the pre- and post-synaptic spike patterns required to induce synaptic plasticity. We have developed such a model by incorporating currently available data on the timecourse and amplitude of the post-synaptic membrane potential within individual spines. We couple this with data on the kinetics of synaptic NMDARs and then use the model to predict the continuous spine [Ca2+] in response to regular or irregular pre- and post-synaptic spike patterns. We then incorporate experimental data from synaptic plasticity induction protocols by regular activity patterns to couple the predicted local peak [Ca2+] to changes in synaptic strength. We find that our model accurately describes [Ca2+] in dendritic spines resulting from NMDAR activation during pre-synaptic and post-synaptic activity when compared to previous experimental observations. The model also replicates the experimentally determined plasticity outcome of regular and irregular spike patterns when applied to a single synapse. This model could therefore be used to predict the induction of synaptic plasticity under a variety of experimental conditions and spike patterns