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- Stephen Coombes
- Biological Cybernetics
- 2005

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… (More)

- Stephen Coombes, Markus R. Owen
- SIAM J. Applied Dynamical Systems
- 2004

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 system parameters, including the speed and strength of synaptic coupling and the speed of axonal signals. The… (More)

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… (More)

- Paul C. Bressloff, Stephen Coombes
- Neural Computation
- 2000

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… (More)

We consider a firing rate model of a neuronal network continuum that incorporates axo-dendritic synaptic processing and the finite conduction velocities of action potentials. The model equation is an integral one defined on a spatially extended domain. Apart from a spatial integral mixing the network connectivity function with space-dependent delays,… (More)

- Stephen Coombes, Helmut Schmidt, Ingo Bojak
- Journal of mathematical neuroscience
- 2012

Neural field models describe the coarse-grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in two spatial dimensions, where they are well known to generate rich patterns of spatiotemporal activity. Such patterns have been interpreted in a variety of contexts ranging from… (More)

Phase-locking in a ring of pulse-coupled integrate-and-fire oscillators with distributed delays is analysed using group theory. The period of oscillation of a solution and those related by symmetry is determined self-consistently. Numerical continuation of maximally symmetric solutions in characteristic system length and timescales yields bifurcation… (More)

- S Coombes, N A Venkov, L Shiau, I Bojak, D T J Liley, C R Laing
- Physical review. E, Statistical, nonlinear, and…
- 2007

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 equation (PDE) model that properly treats the axonal delay terms of the integral formulation. Our analysis avoids… (More)

- S Coombes, R Hinch, Y Timofeeva
- Progress in biophysics and molecular biology
- 2004

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 localised calcium stores. The aim of this paper is to… (More)

- S Coombes
- 1999

We discuss the notion of Liapunov exponent for integrate-and-fire (IF) type dynam-ical systems. In contrast to smooth flows there is a contribution to the IF Liapunov exponent arising from the discontinuous nature of the firing mechanism. Introducing the notion of an IF mode-locked state we are able to show that linear stability is consistent with the… (More)