17 research outputs found
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 tem-
poral tuning of both tonic spiking (i.e., conventional action potentials) and post-inhibitory 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
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 for a spine with 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 robust
encoding of information
Phase locking in networks of synaptically coupled McKean relaxation oscillators
We use geometric dynamical systems methods to derive phase equations for networks
of weakly connected McKean relaxation oscillators. We derive an explicit
formula for the connection function when the oscillators are coupled with chemical
synapses modeled as the convolution of some input spike train with an appropriate
synaptic kernel. The theory allows the systematic investigation of the way in
which a slow recovery variable can interact with synaptic time scales to produce
phase-locked solutions in networks of pulse coupled neural relaxation oscillators.
The theory is exact in the singular limit that the fast and slow time scales of the
neural oscillator become effectively independent. By focusing on a pair of mutually
coupled McKean oscillators with alpha function synaptic kernels, we clarify the role
that fast and slow synapses of excitatory and inhibitory type can play in producing
stable phase-locked rhythms. In particular we show that for fast excitatory synapses
there is coexistence of a stable synchronous, a stable anti-synchronous, and one stable
asynchronous solution. For slower synapses the anti-synchronous solution can
lose stability, whilst for even slower synapses it can regain stability. The case of
inhibitory synapses is similar up to a reversal of the stability of solution branches.
Using a return-map analysis the case of strong pulsatile coupling is also considered.
In this case it is shown that the synchronous solution can co-exist with a continuum
of asynchronous states
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 idealised 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. (J. E. Keizer, G. D. Smith, S. Ponce Dawson and J. Pearson, 1998, Saltatory propagation of Ca2+ waves by Ca2+ sparks, Biophysical Journal, 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
Spike train dynamics underlying pattern formation in integrate-and-fire oscillator networks
A dynamical mechanism underlying pattern formation in a spatially extended network of integrate-and-fire oscillators with synaptic interactions is identified. It is shown how in the strong coupling regime the network undergoes a discrete Turing-Hopf bifurcation of the firing times from a synchronous state to a state with periodic or quasiperiodic variations of the interspike intervals on closed orbits. The separation of these orbits in phase space results in a spatially periodic pattern of mean firing rate across the network that is modulated by deterministic fluctuations of the instantaneous firing rate
Sparks and waves in a stochastic fire-diffuse-fire model of Ca2+
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) 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 (FDF) 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 non-equilibrium 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
Traveling waves in a chain of pulse-coupled oscillators
We derive conditions for the existence of traveling wave solutions in a chain of pulse-coupled integrate-and-fire oscillators with nearest-neighbor interactions and distributed delays. A linear stability analysis of the traveling waves is carried out in terms of perturbations of the firing times of the oscillators. It is shown how traveling waves destabilize when the detuning between oscillators or the strength of the coupling becomes too large
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, quasi-periodic
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
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 non-constant
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
Traveling waves in the Baer and Rinzel model of spine studded dendritic tissue
The Baer and Rinzel model of dendritic spines uniformly distributed along a dendritic
cable is shown to admit a variety of regular traveling wave solutions including
solitary pulses, multiple pulses and periodic waves. We investigate numerically the
speed of these waves and their propagation failure as functions of the system parameters
by numerical continuation. Multiple pulse waves are shown to occur close to
the primary pulse, except in certain exceptional regions of parameter space, which
we identify. The propagation failure of solitary and multiple pulse waves is shown to
be associated with the destruction of a saddle-node bifurcation of periodic orbits.
The system also supports many types of irregular wave trains. These include waves
which may be regarded as connections to periodics and bursting patterns in which
pulses can cluster together in well-defined packets. The behavior and properties of
both these irregular spike-trains is explained within a kinematic framework that is
based on the times of wave pulses. The dispersion curve for periodic waves is important
for such a description and is obtained in a straightforward manner using the
numerical scheme developed for the study of the speed of a periodic wave. Stability
of periodic waves within the kinematic theory is given in terms of the derivative
of the dispersion curve and provides a weak form of stability that may be applied
to solutions of the traveling wave equations. The kinematic theory correctly predicts
the conditions for period doubling bifurcations and the generation of bursting states. Moreover, it also accurately describes the shape and speed of the traveling
front that connects waves with two different periods