Sequential noise-induced escapes for oscillatory network dynamics

It is well known that the addition of noise in a multistable system can induce random transitions between stable states. The rate of transition can be characterised in terms of the noise-free system's dynamics and the added noise: for potential systems in the presence of asymptotically low noise the well-known Kramers' escape time gives an expression for the mean escape time. This paper examines some general properties and examples of transitions between local steady and oscillatory attractors within networks: the transition rates at each node may be affected by the dynamics at other nodes. We use first passage time theory to explain some properties of scalings noted in the literature for an idealised model of initiation of epileptic seizures in small systems of coupled bistable systems with both steady and oscillatory attractors. We focus on the case of sequential escapes where a steady attractor is only marginally stable but all nodes start in this state. As the nodes escape to the oscillatory regime, we assume that the transitions back are very infrequent in comparison. We quantify and characterise the resulting sequences of noise-induced escapes. For weak enough coupling we show that a master equation approach gives a good quantitative understanding of sequential escapes, but for strong coupling this description breaks down

Fast and slow domino regimes in transient network dynamics

It is well known that the addition of noise to a multistable dynamical system can induce random transitions from one stable state to another. For low noise, the times between transitions have an exponential tail and Kramers' formula gives an expression for the mean escape time in the asymptotic limit. If a number of multistable systems are coupled into a network structure, a transition at one site may change the transition properties at other sites. We study the case of escape from a "quiescent" attractor to an "active" attractor in which transitions back can be ignored. There are qualitatively different regimes of transition, depending on coupling strength. For small coupling strengths the transition rates are simply modified but the transitions remain stochastic. For large coupling strengths transitions happen approximately in synchrony - we call this a "fast domino" regime. There is also an intermediate coupling regime some transitions happen inexorably but with a delay that may be arbitrarily long - we call this a "slow domino" regime. We characterise these regimes in the low noise limit in terms of bifurcations of the potential landscape of a coupled system. We demonstrate the effect of the coupling on the distribution of timings and (in general) the sequences of escapes of the system.Comment: 3 figure

Sequential escapes: onset of slow domino regime via a saddle connection

We explore sequential escape behaviour of coupled bistable systems under the influence of stochastic perturbations. We consider transient escapes from a marginally stable "quiescent" equilibrium to a more stable "active" equilibrium. The presence of coupling introduces dependence between the escape processes: for diffusive coupling there is a strongly coupled limit (fast domino regime) where the escapes are strongly synchronised while for intermediate coupling (slow domino regime) without partially escaped stable states, there is still a delayed effect. These regimes can be associated with bifurcations of equilibria in the low-noise limit. In this paper we consider a localized form of non-diffusive (i.e pulse-like) coupling and find similar changes in the distribution of escape times with coupling strength. However we find transition to a slow domino regime that is not associated with any bifurcations of equilibria. We show that this transition can be understood as a codimension-one saddle connection bifurcation for the low-noise limit. At transition, the most likely escape path from one attractor hits the escape saddle from the basin of another partially escaped attractor. After this bifurcation we find increasing coefficient of variation of the subsequent escape times

Control of Ca2+ influx and calmodulin activation by SK-channels in dendritic spines (dataset)

A 3-dimensional model of Ca2+ and calmodulin dynamics within an idealised, but biophysically-plausible, dendritic spine, demonstrates that SK-channels regulate calmodulin activation specifically during neurone firing patterns associated with induction of spike timing-dependent plasticity.The journal article associated with this dataset is available at: http://hdl.handle.net/10871/21745.The key trigger for Hebbian synaptic plasticity is influx of Ca2+ into postsynaptic dendritic spines. The magnitude of [Ca2+] increase caused by NMDA-receptor (NMDAR) and voltage-gated Ca2+ -channel (VGCC) activation is thought to determine both the amplitude and direction of synaptic plasticity by differential activation of Ca2+ -sensitive enzymes such as calmodulin. Ca2+ influx is negatively regulated by Ca2+ -activated K+ channels (SK-channels) which are in turn inhibited by neuromodulators such as acetylcholine. However, the precise mechanisms by which SK-channels control the induction of synaptic plasticity remain unclear. Using a 3-dimensional model of Ca2+ and calmodulin dynamics within an idealised, but biophysically-plausible, dendritic spine, we show that SK-channels regulate calmodulin activation specifically during neuron-firing patterns associated with induction of spike timing-dependent plasticity. SK-channel activation and the subsequent reduction in Ca2+ influx through NMDARs and L-type VGCCs results in an order of magnitude decrease in calmodulin (CaM) activation, providing a mechanism for the effective gating of synaptic plasticity induction. This provides a common mechanism for the regulation of synaptic plasticity by neuromodulators

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

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

Bifurcation Analysis of a Two-Compartment Hippocampal Pyramidal Cell Model

The Pinsky-Rinzel model is a non-smooth 2-compartmental CA3 pyramidal cell model that has been used widely within the field of neuroscience. Here we propose a modified (smooth) system that captures the qualitative behaviour of the original model, while allowing the use of available, numerical continuation methods to perform full-system bifurcation and fastslow analysis. We study the bifurcation structure of the full system as a function of the applied current and the maximal calcium conductance. We identify the bifurcations that shape the transitions between resting, bursting and spiking behaviours, and which lead to the disappearance of bursting when the calcium conductance is reduced. Insights gained from this analysis, are then used to firstly illustrate how the irregular spiking activity found between bursting and stable spiking states, can be influenced by phase differences in the calcium and dendritic voltage, which lead to corresponding changes in the calcium-sensitive potassium current. Furthermore, we use fast-slow analysis to investigate the mechanisms of bursting and show that bursting in the model is dependent on the intermediately slow variable, calcium, while the other slow variable, the activation gate of the afterhyperpolarisation current, does not contribute to setting the intraburst dynamics but participates in setting the interburst interval. Finally, we discuss how some of the described bifurcations affect spiking behaviour, during sharp-wave ripples, in a larger network of Pinsky-Rinzel cells.LAA is supported by the Engineering and Physical Sciences Research Council (EPSRC) and Eli Lilly & Company; LYP is supported by the Wellcome Trust; and KT-A is supported by grant EP/N014391/1 of the EPSRC