A stability analysis is presented for neural field equations in the presence
of axonal delays and for a general class of connectivity kernels and synap-
tic properties. Sufficient conditions are given for the stability of equilibrium
solutions. It is shown that the delays play a crucial role in non-stationary
bifurcations of equilibria, whereas the stationary bifurcations depend only on
the kernel. Bounds are determined for the frequencies of bifurcating periodic
solutions. A perturbative scheme is used to calculate the types of bifurca-
tions leading to spatial patterns, oscillatory solutions, and traveling waves.
For high transmission speeds a simple method is derived that allows the de-
termination of the bifurcation type by visual inspection of the Fourier trans-
forms of the connectivity kernel and its first moment. Results are numerically
illustrated on a class of neurologically plausible second order systems with
combinations of Gaussian excitatory and inhibitory connections.
This work studies dynamical properties of spatially extended neu-
ronal ensembles. We first derive an evolution equation from tem-
poral properties and statistical distributions of synapses and somata.
The obtained integro-differential equation considers both synaptic and
axonal propagation delay, while spatial synaptic connectivities ex-
hibit gamma-distributed distributions. This familiy of connectivity
kernels also covers the cases of divergent, finite, and negligible self-
connections. The work derives conditions for both stationary and
nonstationary instabilities for gamma-distributed kernels.It turns out
that the stability conditions can be formulated in terms of the mean spatial interaction ranges and the mean spatial interaction times. In
addition, a numerical study examines the evoked spatiotemporal re-
sponse activity caused by short local stimuli and reveals maximum
response activity after the mean interaction time at a distance from
stimulus offset location equal to the mean interaction range. These
findings propose new insights to neuronal mechanisms of experimen-
tally observed evoked brain activity.
This work studies the stability of spatially extended neuronal ensembles. We first
derive the model equation from statistical properties of the neuron population. The
obtained integro-differential equation considers synaptic and space-dependent transmission
delay for both general and gamma-distributed synaptic connectivities. The
latter connectivity type reveals infinite, finite and vanishing self-connectivities. The
work derives conditions for stationary and nonstationary instabilities for both kernel
types. In addition, a nonlinear analysis for general kernels yields the order parameter
equation of the Turing instability. To compare the results to findings for partial
differential equations (PDEs), two typical PDE-types are derived from the examined
model equation. In case of the gamma-distributed kernels, the stability conditions
are formulated in terms of the mean excitatory and inhibitory interaction ranges. As
a novel finding, we obtain Turing instabilities in fields with local inhibition-lateral
excitation, while wave instabilities occur in fields with local excitation and lateral
inhibition. Numerical simulations support the analytical results.