73 research outputs found
Dynamics from seconds to hours in Hodgkin-Huxley model with time-dependent ion concentrations and buffer reservoirs
The classical Hodgkin--Huxley (HH) model neglects the time-dependence of ion
concentrations in spiking dynamics. The dynamics is therefore limited to a time
scale of milliseconds, which is determined by the membrane capacitance
multiplied by the resistance of the ion channels, and by the gating time
constants. We study slow dynamics in an extended HH framework that includes
time-dependent ion concentrations, pumps, and buffers. Fluxes across the
neuronal membrane change intra- and extracellular ion concentrations, whereby
the latter can also change through contact to reservoirs in the surroundings.
Ion gain and loss of the system is identified as a bifurcation parameter whose
essential importance was not realized in earlier studies. Our systematic study
of the bifurcation structure and thus the phase space structure helps to
understand activation and inhibition of a new excitability in ion homeostasis
which emerges in such extended models. Also modulatory mechanisms that regulate
the spiking rate can be explained by bifurcations. The dynamics on three
distinct slow times scales is determined by the cell volume-to-surface-area
ratio and the membrane permeability (seconds), the buffer time constants (tens
of seconds), and the slower backward buffering (minutes to hours). The
modulatory dynamics and the newly emerging excitable dynamics corresponds to
pathological conditions observed in epileptiform burst activity, and spreading
depression in migraine aura and stroke, respectively.Comment: 18 pages, 11 figure
Transient localized wave patterns and their application to migraine
Transient dynamics is pervasive in the human brain and poses challenging
problems both in mathematical tractability and clinical observability. We
investigate statistical properties of transient cortical wave patterns with
characteristic forms (shape, size, duration) in a canonical reaction-diffusion
model with mean field inhibition. The patterns are formed by a ghost near a
saddle-node bifurcation in which a stable traveling wave (node) collides with
its critical nucleation mass (saddle). Similar patterns have been observed with
fMRI in migraine. Our results support the controversial idea that waves of
cortical spreading depression (SD) have a causal relationship with the headache
phase in migraine and therefore occur not only in migraine with aura (MA) but
also in migraine without aura (MO), i.e., in the two major migraine subforms.
We suggest a congruence between the prevalence of MO and MA with the
statistical properties of the traveling waves' forms, according to which (i)
activation of nociceptive mechanisms relevant for headache is dependent upon a
sufficiently large instantaneous affected cortical area anti-correlated to both
SD duration and total affected cortical area such that headache would be less
severe in MA than in MO (ii) the incidence of MA is reflected in the distance
to the saddle-node bifurcation, and (iii) the contested notion of MO attacks
with silent aura is resolved. We briefly discuss model-based control and means
by which neuromodulation techniques may affect pathways of pain formation.Comment: 14 pages, 11 figure
Nucleation of reaction-diffusion waves on curved surfaces
We study reaction-diffusion waves on curved two-dimensional surfaces, and
determine the influence of curvature upon the nucleation and propagation of
spatially localized waves in an excitable medium modelled by the generic
FitzHugh-Nagumo model. We show that the stability of propagating wave segments
depends crucially on the curvature of the surface. As they propagate, they may
shrink to the uniform steady state, or expand, depending on whether they are
smaller or larger, respectively, than a critical nucleus. This critical nucleus
for wave propagation is modified by the curvature acting like an effective
space-dependent local spatial coupling, similar to diffusion, thus extending
the regime of propagating excitation waves beyond the excitation threshold of
flat surfaces. In particular, a negative gradient of Gaussian curvature
, as on the outside of a torus surface (positive ), when the
wave segment symmetrically extends into the inside (negative ), allows
for stable propagation of localized wave segments remaining unchanged in size
and shape, or oscillating periodically in size
Analyzing critical propagation in a reaction-diffusion-advection model using unstable slow waves
The effect of advection on the critical minimal speed of traveling waves is
studied. Previous theoretical studies estimated the effect on the velocity of
stable fast waves and predicted the existence of a critical advection strength
below which propagating waves are not supported anymore. In this paper, the
critical advection strength is calculated taking into account the unstable slow
wave solution. Thereby, theoretical results predict, that advection can induce
stable wave propagation in the non-excitable parameter regime, if the advection
strength exceeds a critical value. In addition, an analytical expression for
the advection-velocity relation of the unstable slow wave is derived.
Predictions are confirmed numerically in a two-variable reaction-diffusion
model.Comment: 11 pages, 8 figure
Bistable dynamics underlying excitability of ion homeostasis in neuron models
When neurons fire action potentials, dissipation of free energy is usually
not directly considered, because the change in free energy is often negligible
compared to the immense reservoir stored in neural transmembrane ion gradients
and the long-term energy requirements are met through chemical energy, i.e.,
metabolism. However, these gradients can temporarily nearly vanish in
neurological diseases, such as migraine and stroke, and in traumatic brain
injury from concussions to severe injuries. We study biophysical neuron models
based on the Hodgkin-Huxley (HH) formalism extended to include time-dependent
ion concentrations inside and outside the cell and metabolic energy-driven
pumps. We reveal the basic mechanism of a state of free energy-starvation (FES)
with bifurcation analyses showing that ion dynamics is for a large range of
pump rates bistable without contact to an ion bath. This is interpreted as a
threshold reduction of a new fundamental mechanism of 'ionic excitability' that
causes a long-lasting but transient FES as observed in pathological states. We
can in particular conclude that a coupling of extracellular ion concentrations
to a large glial-vascular bath can take a role as an inhibitory mechanism
crucial in ion homeostasis, while the Na/K pumps alone are insufficient
to recover from FES. Our results provide the missing link between the HH
formalism and activator-inhibitor models that have been successfully used for
modeling migraine phenotypes, and therefore will allow us to validate the
hypothesis that migraine symptoms are explained by disturbed function in ion
channel subunits, Na/K pumps, and other proteins that regulate ion
homeostasis.Comment: 14 pages, 8 figures, 4 table
Feedback-dependent control of stochastic synchronization in coupled neural systems
We investigate the synchronization dynamics of two coupled noise-driven
FitzHugh-Nagumo systems, representing two neural populations. For certain
choices of the noise intensities and coupling strength, we find cooperative
stochastic dynamics such as frequency synchronization and phase
synchronization, where the degree of synchronization can be quantified by the
ratio of the interspike interval of the two excitable neural populations and
the phase synchronization index, respectively. The stochastic synchronization
can be either enhanced or suppressed by local time-delayed feedback control,
depending upon the delay time and the coupling strength. The control depends
crucially upon the coupling scheme of the control force, i.e., whether the
control force is generated from the activator or inhibitor signal, and applied
to either component. For inhibitor self-coupling, synchronization is most
strongly enhanced, whereas for activator self-coupling there exist distinct
values of the delay time where the synchronization is strongly suppressed even
in the strong synchronization regime. For cross-coupling strongly modulated
behavior is found
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