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Democratization in a passive dendritic tree: An analytical investigation (2008)
Journal Article
Timofeeva, Y., Cox, S. J., Coombes, S., & Josic, K. (2008). Democratization in a passive dendritic tree: An analytical investigation. Journal of Computational Neuroscience, 25(2), 228-244. https://doi.org/10.1007/s10827-008-0075-9

One way to achieve amplification of distal synaptic inputs on a dendritic tree is to scale the amplitude and/or duration of the synaptic conductance with its distance from the soma. This is an example of what is often referred to as "dendritic democr... Read More about Democratization in a passive dendritic tree: An analytical investigation.

Modeling electrocortical activity through improved local approximations of integral neural field equations (2007)
Journal Article
Coombes, S., Venkov, N., Shiau, L., Bojak, I., Liley, D., & Laing, C. (2007). Modeling electrocortical activity through improved local approximations of integral neural field equations. Physical Review E, 76(5), https://doi.org/10.1103/PhysRevE.76.051901

Neural field models of firing rate activity typically take the form of integral equations with space-dependent axonal delays. Under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equatio... Read More about Modeling electrocortical activity through improved local approximations of integral neural field equations.

Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities (2007)
Journal Article
Owen, M. R., Laing, C., & Coombes, S. (2007). Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities. New Journal of Physics, 9(378), https://doi.org/10.1088/1367-2630/9/10/378

In this paper we consider instabilities of localised solutions in planar neural field firing rate models of Wilson-Cowan or Amari type. Importantly we show that angular perturbations can destabilise spatially localised solutions. For a scalar model... Read More about Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities.

A bidomain threshold model of propagating calcium waves (2007)
Journal Article
Thul, R., Smith, G., & Coombes, S. (2008). A bidomain threshold model of propagating calcium waves. Journal of Mathematical Biology, 56(4), https://doi.org/10.1007/s00285-007-0123-5

We present a bidomain fire-diffuse-fire model that facilitates mathematical analysis of propagating waves of elevated intracellular calcium (Ca) in living cells. Modelling Ca release as a threshold process allows the explicit construction of travell... Read More about A bidomain threshold model of propagating calcium waves.

Dynamic instabilities in scalar neural field equations with space-dependent delays (2007)
Journal Article
Venkov, N. A., Coombes, S., & Matthews, P. C. (2007). Dynamic instabilities in scalar neural field equations with space-dependent delays. Physica D: Nonlinear Phenomena, 232(1), 1-15. https://doi.org/10.1016/j.physd.2007.04.011

In this paper we consider a class of scalar integral equations with a form of space-dependent delay. These nonlocal models arise naturally when modelling neural tissue with active axons and passive dendrites. Such systems are known to support a dynam... Read More about Dynamic instabilities in scalar neural field equations with space-dependent delays.

Branching dendrites with resonant membrane: a "sum-over-trips" approach (2007)
Journal Article
Coombes, S., Timofeeva, Y., Svennson, C., Lord, G., Josic, K., Cox, S. J., & Colbert, C. (2007). Branching dendrites with resonant membrane: a "sum-over-trips" approach. Biological Cybernetics, 97(2), https://doi.org/10.1007/s00422-007-0161-5

Dendrites form the major components of neurons. They are complex branching structures that receive and process thousands of synaptic inputs from other neurons. It is well known that dendritic morphology plays an important role in the function of dend... Read More about Branching dendrites with resonant membrane: a "sum-over-trips" approach.

Exotic dynamics in a firing rate model of neural tissue with threshold accommodation (2007)
Journal Article
Coombes, S., & Owen, M. R. (2007). Exotic dynamics in a firing rate model of neural tissue with threshold accommodation

Many of the equations describing the dynamics of neural systems are written in terms of firing rate functions, which themselves are often taken to be threshold functions of synaptic activity. Dating back to work by Hill in 1936 it has been recognize... Read More about Exotic dynamics in a firing rate model of neural tissue with threshold accommodation.

The role of cannabinoids in the neurobiology of sensory gating: a firing rate model study (2006)
Journal Article
Zachariou, M., Dissanayake, D., Owen, M. R., Mason, R., & Coombes, S. The role of cannabinoids in the neurobiology of sensory gating: a firing rate model study. Neurocomputing, 70(10/12), https://doi.org/10.1016/j.neucom.2006.10.065

Gating of sensory (e.g. auditory) information has been demonstrated as a reduction in the auditory-evoked potential responses recorded in the brain of both normal animals and human subjects. Auditory gating is perturbed in schizophrenic patients and... Read More about The role of cannabinoids in the neurobiology of sensory gating: a firing rate model study.

Spatio-temporal filtering properties of a dendritic cable with active spines: A modeling study in the spike-diffuse-spike framework (2006)
Journal Article
Timofeeva, Y., Lord, G., & Coombes, S. (2006). Spatio-temporal filtering properties of a dendritic cable with active spines: A modeling study in the spike-diffuse-spike framework. Journal of Computational Neuroscience, 21(3), 293-306. https://doi.org/10.1007/s10827-006-8776-4

The spike-diffuse-spike (SDS) model describes a passive dendritic tree with active dendritic spines. Spine-head dynamics is modeled with a simple integrate-and-fire process, whilst communication between spines is mediated by the cable equation. In th... Read More about Spatio-temporal filtering properties of a dendritic cable with active spines: A modeling study in the spike-diffuse-spike framework.

Mode locking in a periodically forced "ghostbursting" neuron model (2005)
Journal Article
Laing, C. R., & Coombes, S. (2005). Mode locking in a periodically forced "ghostbursting" neuron model. International Journal of Bifurcation and Chaos, 15(4), 1433-1444. https://doi.org/10.1142/S0218127405012557

We study a minimal integrate-and-fire based model of a " ghostbursting" neuron under periodic stimulation. These neurons are involved in sensory processing in weakly electric fish. There exist regions in parameter space in which the model neuron is m... Read More about Mode locking in a periodically forced "ghostbursting" neuron model.

Bumps, breathers, and waves in a neural network with spike frequency adaptation (2005)
Journal Article
Coombes, S., & Owen, M. R. (2005). Bumps, breathers, and waves in a neural network with spike frequency adaptation. Physical Review Letters, 94(14), https://doi.org/10.1103/PhysRevLett.94.148102

In this Letter we introduce a continuum model of neural tissue that include the effects of so-called spike frequency adaptation (SFA). The basic model is an integral equation for synaptic activity that depends upon the non-local network connectivity... Read More about Bumps, breathers, and waves in a neural network with spike frequency adaptation.

Directed percolation in a two dimensional stochastic fire-diffuse-fire model (2004)
Journal Article
Timofeeva, Y., & Coombes, S. (2004). Directed percolation in a two dimensional stochastic fire-diffuse-fire model

In this paper 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 th... Read More about Directed percolation in a two dimensional stochastic fire-diffuse-fire model.

Evans functions for integral neural field equations with Heaviside firing rate function (2004)
Journal Article
Coombes, S., & Owen, M. R. Evans functions for integral neural field equations with Heaviside firing rate function. SIAM Journal on Applied Dynamical Systems, 3(4), https://doi.org/10.1137/040605953

In this paper we show how to construct the Evans function for traveling wave solutions of integral neural field equations when the firing rate function is a Heaviside. This allows a discussion of wave stability and bifurcation as a function of syste... Read More about Evans functions for integral neural field equations with Heaviside firing rate function.

Receptors, sparks and waves in a fire-diffuse-fire framework for calcium release
Journal Article
Coombes, S., Hinch, R., & Timofeeva, Y. (2004). Receptors, sparks and waves in a fire-diffuse-fire framework for calcium 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 stochasti... Read More about Receptors, sparks and waves in a fire-diffuse-fire framework for calcium release.

Clustering through post inhibitory rebound in synaptically coupled neurons
Journal Article
Chik, D. T. W., Coombes, S., & Wang, Z. D. (2004). Clustering through post inhibitory rebound in synaptically coupled neurons

Post inhibitory rebound is a nonlinear phenomenon present in a variety of nerve cells. Following a period of hyper-polarization this effect allows a neuron to fire a spike or packet of spikes before returning to rest. It is an important mechanism u... Read More about Clustering through post inhibitory rebound in synaptically coupled neurons.

Dendritic cable with active spines: a modeling study in the spike-diffuse-spike framework
Journal Article
Timofeeva, Y., Lord, G., & Coombes, S. (2005). Dendritic cable with active spines: a modeling study in the spike-diffuse-spike framework

The spike-diffuse-spike (SDS) model describes a passive dendritic tree with active dendritic spines. Spine-head dynamics is modelled with a simple integrate-and-fire process, whilst communication between spines is mediated by the cable equation. Her... Read More about Dendritic cable with active spines: a modeling study in the spike-diffuse-spike framework.

The importance of different timings of excitatory and inhibitory pathways in neural field models
Journal Article
Laing, C., & Coombes, S. (2005). The importance of different timings of excitatory and inhibitory pathways in neural field models

In this paper we consider a neural field model comprised of two distinct populations of neurons, excitatory and inhibitory, for which both the velocities of action potential propagation and the time courses of synaptic processing are different. Using... Read More about The importance of different timings of excitatory and inhibitory pathways in neural field models.

Existence and wandering of bumps in a spiking neural network model
Journal Article
Chow, C., & Coombes, S. (2006). Existence and wandering of bumps in a spiking neural network model

We study spatially localized states of a spiking neuronal network populated by a pulse coupled phase oscillator known as the lighthouse model. We show that in the limit of slow synaptic interactions in the continuum limit the dynamics reduce to tho... Read More about Existence and wandering of bumps in a spiking neural network model.

Toward blueprints for network architecture, biophysical dynamics, and signal transduction
Journal Article
Coombes, S., Doiron, B., Josic, K., & Shea-Brown, E. (2006). Toward blueprints for network architecture, biophysical dynamics, and signal transduction

We review mathematical aspects of biophysical dynamics, signal transduction and network architecture that have been used to uncover functionally significant relations between the dynamics of single neurons and the networks they compose. We focus on... Read More about Toward blueprints for network architecture, biophysical dynamics, and signal transduction.