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S Coombes

Publications and source records attributed to S Coombes.

At least 19 recordsLinked to original sources
PubMed2005

Waves, bumps, and patterns in neural field theories.

Neural field models of firing rate activity have had a major impact in helping to develop an understanding of the dynamics seen in brain slice preparations. These models typically take the form of integro-differential equations. Their non-local nature has led to the development of a set of analytical and numerical tools for the study of waves, bumps and patterns, based around natural extensions of those used for local differential equation models. In this paper we present a review of such techniques and show how recent advances have opened the way for future studies of neural fields in both one and two dimensions that can incorporate realistic forms of axo-dendritic interactions and the slow intrinsic currents that underlie bursting behaviour in single neurons.

Action Potentials↗
PubMed2005

Bumps, breathers, and waves in a neural network with spike frequency adaptation.

We introduce a continuum model of neural tissue that includes the effects of spike frequency adaptation (SFA). The basic model is an integral equation for synaptic activity that depends upon nonlocal network connectivity, synaptic response, and the firing rate of a single neuron. We consider a phenomenological model of SFA via a simple state-dependent threshold firing rate function. As without SFA, Mexican-hat connectivity allows for the existence of spatially localized states (bumps). Importantly recent Evans function techniques are used to show that bumps may destabilize leading to the emergence of breathers and traveling waves. Moreover, a similar analysis for traveling pulses leads to the conditions necessary to observe a stable traveling breather. Simulations confirm our theoretical predictions and illustrate the rich behavior of this model.

Action Potentials↗
PubMed2004

Directed percolation in a two-dimensional stochastic fire-diffuse-fire model.

We establish, from extensive numerical experiments, that the two dimensional stochastic fire-diffuse-fire model belongs to the directed percolation universality class. This model is an idealized model of intracellular calcium release that retains both the discrete nature of calcium stores and the stochastic nature of release. It is formed from an array of noisy threshold elements that are coupled only by a diffusing signal. The model supports spontaneous release events that can merge to form spreading circular and spiral waves of activity. The critical level of noise required for the system to exhibit a nonequilibrium phase transition between propagating and nonpropagating waves is obtained by an examination of the local slope delta (t) of the survival probability Pi(t) proportional exp [-delta(t)] for a wave to propagate for a time t .

Action Potentials↗
PubMed2004

Clustering through postinhibitory rebound in synaptically coupled neurons.

Postinhibitory rebound is a nonlinear phenomenon present in a variety of nerve cells. Following a period of hyperpolarization this effect allows a neuron to fire a spike or packet of spikes before returning to rest. It is an important mechanism underlying central pattern generation for heartbeat, swimming and other motor patterns in many neuronal systems. In this paper we consider how networks of neurons, which do not intrinsically oscillate, may make use of inhibitory synaptic connections to generate large scale coherent rhythms in the form of cluster states. We distinguish between two cases (i) where the rebound mechanism is due to anode break excitation and (ii) where rebound is due to a slow T-type calcium current. In the former case we use a geometric analysis of a McKean-type model to obtain expressions for the number of clusters in terms of the speed and strength of synaptic coupling. Results are found to be in good qualitative agreement with numerical simulations of the more detailed Hodgkin-Huxley model. In the second case we consider a particular firing rate model of a neuron with a slow calcium current that admits to an exact analysis. Once again existence regions for cluster states are explicitly calculated. Both mechanisms are shown to prefer globally synchronous states for slow synapses as long as the strength of coupling is sufficiently large. With a decrease in the duration of synaptic inhibition both systems are found to break into clusters. A major difference between the two mechanisms for cluster generation is that anode break excitation can support clusters with several groups, while slow T-type calcium currents predominantly give rise to clusters of just two (antisynchronous) populations.

Action Potentials↗
PubMed2003

Sparks and waves in a stochastic fire-diffuse-fire model of Ca2+ release.

Calcium ions are an important second messenger in living cells. Indeed, calcium signals in the form of waves have been the subject of much recent experimental interest. It is now well established that these waves are composed of elementary stochastic release events (calcium puffs or sparks) from spatially localized calcium stores. Here we develop a computationally inexpensive model of calcium release, based upon a stochastic generalization of the fire-diffuse-fire threshold model. Our model retains the discrete nature of calcium stores, but also incorporates a notion of release probability via the introduction of threshold noise. Numerical simulations of the model illustrate that stochastic calcium release leads to the spontaneous production of calcium sparks that may merge to form saltatory waves. In the parameter regime where deterministic waves exist, it is possible to identify a critical level of noise, defining a nonequilibrium phase transition between propagating and abortive structures. A statistical analysis shows that this transition is the same as for models in the directed percolation universality class. Moreover, in the regime where no initial structure can survive deterministically, threshold noise is shown to generate a form of array enhanced coherence resonance, whereby all calcium stores release periodically and simultaneously.

Biophysical Phenomena↗
PubMed2003

Saltatory waves in the spike-diffuse-spike model of active dendritic spines.

In this Letter we present the explicit construction of a saltatory traveling pulse of nonconstant profile in an idealized model of dendritic tissue. Excitable dendritic spine clusters, modeled with integrate-and-fire (IF) units, are connected to a passive dendritic cable at a discrete set of points. The saltatory nature of the wave is directly attributed to the breaking of translation symmetry in the cable. The conditions for propagation failure are presented as a function of cluster separation and IF threshold.

Animals↗
PubMed2003

Wave bifurcation and propagation failure in a model of Ca(2+) release.

The De Young Keizer model for intracellular calcium oscillations is based around a detailed description of the dynamics for inositol trisphosphate (IP(3)) receptors. Systematic reductions of the kinetic schemes for IP(3) dynamics have proved especially fruitful in understanding the transition from excitable to oscillatory behaviour. With the inclusion of diffusive transport of calcium ions the model also supports wave propagation. The analysis of waves, even in reduced models, is typically only possible with the use of numerical bifurcation techniques. In this paper we review the travelling wave properties of the biophysical De Young Keizer model and show that much of its behaviour can be reproduced by a much simpler Fire-Diffuse-Fire (FDF) type model. The FDF model includes both a refractory process and an IP(3) dependent threshold. Parameters of the FDF model are constrained using a comprehensive numerical bifurcation analysis of solitary pulses and periodic waves in the De Young Keizer model. The linear stability of numerically constructed solution branches is calculated using pseudospectral techniques. The combination of numerical bifurcation and stability analysis also allows us to highlight the mechanisms that give rise to propagation failure. Moreover, a kinematic theory of wave propagation, based around numerically computed dispersion curves is used to predict waves which connect periodic orbits. Direct numerical simulations of the De Young Keizer model confirm this prediction. Corresponding travelling wave solutions of the FDF model are obtained analytically and are shown to be in good qualitative agreement with those of the De Young Keizer model. Moreover, the FDF model may be naturally extended to include the discrete nature of calcium stores within a cell, without the loss of analytical tractability. By considering calcium stores as idealised point sources we are able to explicitly construct solutions of the FDF model that correspond to saltatory periodic travelling waves.

Algorithms↗
PubMed2003

Effects of quasiactive membrane on multiply periodic traveling waves in integrate-and-fire systems.

We consider the dynamics of a one-dimensional continuum of synaptically interacting integrate-and-fire neurons with realistic forms of axodendritic interaction. The speed and stability of traveling waves are investigated as a function of discrete communication delays, distributed synaptic delays, and axodendritic delays arising from the spatially extended nature of the model neuron. In particular, dispersion curves for periodic traveling waves are constructed. Nonlinear ionic channels in the dendrite responsible for a so-called quasiactive bandpass response are shown to significantly influence the shape of dispersion curves. Moreover, a kinematic theory of spike train propagation suggests that period-doubling bifurcations of a singly periodic wave can occur in dendritic systems with a quasiactive membrane. The explicit construction of period-doubled solutions is used to confirm this prediction.

Action Potentials↗
PubMed2003

Continuum of weakly coupled oscillatory McKean neurons.

The McKean model of a neuron possesses a one-dimensional fast voltagelike variable and a slow recovery variable. A recent geometric analysis of the singularly perturbed system has allowed an explicit construction of its phase response curve [S. Coombes, Physica D 160, 173 (2001)]. Here we use tools from coupled oscillator theory to study weakly coupled networks of McKean neurons. Using numerical techniques, we show that the McKean system has traveling wave phase-locked solutions consistent with that of a network of more biophysically detailed Hodgkin-Huxley neurons.

Animals↗
PubMed2003

Dynamics of synaptically coupled integrate-and-fire-or-burst neurons.

The minimal integrate-and-fire-or-burst (IFB) neuron model reproduces the salient features of experimentally observed thalamocortical (TC) relay neuron response properties, including the temporal tuning of both tonic spiking (i.e., conventional action potentials) and postinhibitory rebound bursting mediated by a low-threshold calcium current. In this paper we consider networks of IFB neurons with slow synaptic interactions and show how the dynamics may be described with a smooth firing-rate model. When the firing rate of the IFB model is dominated by a refractory process the equations of motion simplify and may be solved exactly. Numerical simulations are used to show that a pair of reciprocally interacting inhibitory spiking IFB TC neurons supports an alternating rhythm of the type predicted from the firing-rate theory. A change in a single parameter of the IFB neuron allows it to fire a burst of spikes in response to a depolarizing signal, so that it mimics the behavior of a reticular (RE) cell. Within a continuum model we show that a network of RE cells with on-center excitation can support a fast traveling pulse. In contrast, a network of inhibitory TC cells is found to support a slowly propagating lurching pulse.

Action Potentials↗
PubMed2001

Mode locking in a periodically forced integrate-and-fire-or-burst neuron model.

The minimal "integrate-and-fire-or-burst" (IFB) neuron model reproduces the salient features of experimentally observed thalamocortical relay neuron response properties, including the temporal tuning of both tonic spiking (i.e., conventional action potentials) and post-inhibitory rebound bursting mediated by the low-threshold Ca2+ current, I(T). In previous work focusing on experimental and IFB model responses to sinusoidal current injection, large regions of stimulus parameter space were observed for which the response was entrained to periodic applied current, resulting in repetitive burst, tonic, or mixed (i.e., burst followed by tonic) responses. Here we present an exact analysis of such mode-locking in the integrate-and-fire-or-burst model under the influence of arbitrary periodic forcing that includes sinusoidally driven responses as one case. In this analysis, the instabilities of mode-locked states are identified as both smooth bifurcations of an associated firing time map and nonsmooth bifurcations of the underlying discontinuous flow. The explicit construction of borders in parameter space that define the instabilities of mode-locked zones is used to build up the Arnol'd tongue structure for the model. The zones for mode-locking are shown to be in excellent agreement with numerical simulations and are used to explore the observed stimulus dependence of burst versus tonic response of the IFB neuron model.

Action Potentials↗
PubMed2001

The effect of ion pumps on the speed of travelling waves in the fire-diffuse-fire model of Ca2+ release.

The fire-diffuse-fire model provides an idealized model of Ca2+ release within living cells. The effect of calcium pumps, which drive Ca2+ back into internal stores, is often neglected for mathematical simplicity. Here we show how to explicitly analyse such effects by extending the work of Keizer et al. [Keizer, J. E., G. D. Smith, S. Ponce Dawson and J. Pearson (1998). Saltatory propagation of Ca2+ waves by Ca2+ sparks. Biophys. J. 75, 595-600.]. For travelling waves, in which release events occur sequentially, we construct the speed of waves in terms of the time-scale at which pumps operate. An immediate consequence of this analysis is that the inclusion of calcium pumps leads to multiple solutions. A linear stability analysis determines those solution branches in parameter space which are stable. Numerical continuation is used to provide explicit examples of the bifurcation diagrams of the speed of waves as a function of physiologically significant system parameters.

Calcium↗
PubMed2001

Immune complex formation after xenotransplantation : evidence of type III as well as type II immune reactions provide clues to pathophysiology.

Rejection of renal and cardiac xenografts is initiated when natural antibodies of the recipient bind to donor endothelium, activating complement on the surface of endothelial cells. Pulmonary xenotransplants, however, reveal less evidence of antibody binding and complement activation and, in contrast to other xenografts, fare worse when the complement of the graft recipient is depleted. Accordingly, we asked whether distinct immunochemical reactions might occur after xenotransplantation of the lung and what implications such reactions might have for pulmonary pathophysiology. Analysis of serum from baboons after transplantation with porcine lungs revealed complexes containing baboon IgM and porcine von Willebrand factor. The baboon IgM in these complexes was specific for Galalpha1-3Gal. Immune complexes were also seen, albeit to a lesser extent, in the serum of kidney and heart xenotransplant recipients. Deposits of porcine von Willebrand factor and baboon C3 were detected in livers and spleens of transplanted baboons. These results indicate pulmonary xenotransplantation eventuates in formation of immune complexes and in the deposition of those complexes at distant sites. Immune complex formation could explain the peculiar fate of xenoreactive antibodies after pulmonary xenotransplantation and might contribute to the pathophysiology of the lung and systemic changes not previously considered a complication of xenotransplantation.

Animals↗
PubMed2001

From periodic travelling waves to travelling fronts in the spike-diffuse-spike model of dendritic waves.

In the vertebrate brain excitatory synaptic contacts typically occur on the tiny evaginations of neuron dendritic surface known as dendritic spines. There is clear evidence that spine heads are endowed with voltage-dependent excitable channels and that action potentials invade spines. Computational models are being increasingly used to gain insight into the functional significance of a spine with an excitable membrane. The spike-diffuse-spike (SDS) model is one such model that admits to a relatively straightforward mathematical analysis. In this paper we demonstrate that not only can the SDS model support solitary travelling pulses, already observed numerically in more detailed biophysical models, but that it has periodic travelling wave solutions. The exact mathematical treatment of periodic travelling waves in the SDS model is used, within a kinematic framework, to predict the existence of connections between two periodic spike trains of different interspike interval. The associated wave front in the sequence of interspike intervals travels with a constant velocity without degradation of shape, and might therefore be used for the robust encoding of information.

Action Potentials↗
PubMed2000

Period-adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator.

The response of an excitable neuron to trains of electrical spikes is relevant to the understanding of the neural code. In this paper, we study a neurobiologically motivated relaxation oscillator, with appropriately identified fast and slow coordinates, that admits an explicit mathematical analysis. An application of geometric singular perturbation theory shows the existence of an attracting invariant manifold, which is used to construct the Fenichel normal form for the system. This facilitates the calculation of the response of the system to pulsatile stimulation and allows the construction of a so-called extended isochronal map. The isochronal map is shown to have a single discontinuity and be of a type that can admit three types of response: mode-locked, quasiperiodic, and chaotic. The bifurcation structure of the system is seen to be extremely rich and supports period-adding bifurcations separated by windows of both chaos and periodicity. A bifurcation analysis of the isochronal map is presented in conjunction with a description of the various routes to chaos in this system.

Biological Clocks↗
PubMed2000

Dynamics of strongly-coupled spiking neurons.

We present a dynamical theory of integrate-and-fire neurons with strong synaptic coupling. We show how phase-locked states that are stable in the weak coupling regime can destabilize as the coupling is increased, leading to states characterized by spatiotemporal variations in the interspike intervals (ISIs). The dynamics is compared with that of a corresponding network of analog neurons in which the outputs of the neurons are taken to be mean firing rates. A fundamental result is that for slow interactions, there is good agreement between the two models (on an appropriately defined timescale). Various examples of desynchronization in the strong coupling regime are presented. First, a globally coupled network of identical neurons with strong inhibitory coupling is shown to exhibit oscillator death in which some of the neurons suppress the activity of others. However, the stability of the synchronous state persists for very large networks and fast synapses. Second, an asymmetric network with a mixture of excitation and inhibition is shown to exhibit periodic bursting patterns. Finally, a one-dimensional network of neurons with long-range interactions is shown to desynchronize to a state with a spatially periodic pattern of mean firing rates across the network. This is modulated by deterministic fluctuations of the instantaneous firing rate whose size is an increasing function of the speed of synaptic response.

Animals↗