17 research outputs found
A continuum of weakly coupled oscillatory McKean neurons
The McKean model of a neuron possesses a one dimensional fast voltage-like variable and a slow
recovery variable. A recent geometric analysis of the singularly perturbed system has allowed an
explicit construction of its phase response curve [S Coombes 2001 Phase-locking in networks of
synaptically coupled McKean relaxation oscillators, Physica D, Vol 160, 173-188]. Here we use
tools from coupled oscillator theory to study weakly coupled networks of McKean neurons. Using
numerical techniques we show that the McKean system has traveling wave phase-locked solutions
consistent with that of a network of more biophysically detailed Hodgkin-Huxley neurons
Emergent bursting and synchrony in computer simulations of neuronal cultures
Experimental studies of neuronal cultures have revealed a wide variety of spiking network activity ranging from sparse, asynchronous firing to distinct, network-wide synchronous bursting. However, the functional mechanisms driving these observed firing patterns are not well understood. In this work, we develop an in silico network of cortical neurons based on known features of similar in vitro networks. The activity from these simulations is found to closely mimic experimental data. Furthermore, the strength or degree of network bursting is found to depend on a few parameters: the density of the culture, the type of synaptic connections, and the ratio of excitatory to inhibitory connections. Network bursting gradually becomes more prominent as either the density, the fraction of long range connections, or the fraction of excitatory neurons is increased. Interestingly, biologically prevalent values of parameters result in networks that are at the transition between strong bursting and sparse firing. Using principal components analysis, we show that a large fraction of the variance in firing rates is captured by the first component for bursting networks. These results have implications for understanding how information is encoded at the population level as well as for why certain network parameters are ubiquitous in cortical tissue
Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators
A chimera state is a spatio-temporal pattern in a network of identical
coupled oscillators in which synchronous and asynchronous oscillation coexist.
This state of broken symmetry, which usually coexists with a stable spatially
symmetric state, has intrigued the nonlinear dynamics community since its
discovery in the early 2000s. Recent experiments have led to increasing
interest in the origin and dynamics of these states. Here we review the history
of research on chimera states and highlight major advances in understanding
their behaviour.Comment: 26 pages, 3 figure
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
Traveling waves and pulses in a one-dimensional network of excitable integrate-and-fire neurons
We study the existence and stability of traveling waves and pulses in a one-dimensional
network of integrate-and-fire neurons with synaptic coupling. This provides a simple model
of excitable neural tissue. We first derive a self-consistency condition for the existence of
traveling waves, which generates a dispersion relation between velocity and wavelength. We
use this to investigate how wave-propagation depends on various parameters that characterize
neuronal interactions such as synaptic and axonal delays, and the passive membrane
properties of dendritic cables. We also establish that excitable networks support the propagation
of solitary pulses in the long-wavelength limit. We then derive a general condition
for the (local) asymptotic stability of traveling waves in terms of the characteristic equation
of the linearized firing time map, which takes the form of an integro-difference equation of
infinite order. We use this to analyze the stability of solitary pulses in the long-wavelength
limit. Solitary wave solutions are shown to come in pairs with the faster (slower) solution
stable (unstable) in the case of zero axonal delays; for non-zero delays and fast synapses the
stable wave can itself destabilize via a Hopf bifurcation