Gap junctions and emergent rhythms

Gap junction coupling is ubiquitous in the brain, particularly between the dendritic trees of inhibitory interneurons. Such direct non-synaptic interaction allows for direct electrical communication between cells. Unlike spike-time driven synaptic neural network models, which are event based, any model with gap junctions must necessarily involve a single neuron model that can represent the shape of an action potential. Indeed, not only do neurons communicating via gaps feel super-threshold spikes, but they also experience, and respond to, sub-threshold voltage signals. In this chapter we show that the so-called absolute integrate-and-fire model is ideally suited to such studies. At the single neuron level voltage traces for the model may be obtained in closed form, and are shown to mimic those of fast-spiking inhibitory neurons. Interestingly in the presence of a slow spike adaptation current the model is shown to support periodic bursting oscillations. For both tonic and bursting modes the phase response curve can be calculated in closed form. At the network level we focus on global gap junction coupling and show how to analyze the asynchronous firing state in large networks. Importantly, we are able to determine the emergence of non-trivial network rhythms due to strong coupling instabilities. To illustrate the use of our theoretical techniques (particularly the phase-density formalism used to determine stability) we focus on a spike adaptation induced transition from asynchronous tonic activity to synchronous bursting in a gap-junction coupled network

Pulsating fronts in periodically modulated neural field models

We consider a coarse grained neural field model for synaptic activity in spatially extended cortical tissue that possesses an underlying periodicity in its microstructure. The model is written as an integro-differential equation with periodic modulation of a translationally-invariant spatial kernel. This modulation can have a strong effect on wave propagation through the tissue, including the creation of pulsating fronts with widely-varying speeds, and wave-propagation failure. Here we develop new analysis for the study of such phenomena, using two complementary techniques. The first uses linearized information from the leading edge of a traveling periodic wave to obtain wave speed estimates for pulsating fronts, and the second develops an interface description for waves in the full nonlinear model. For weak modulation and a Heaviside firing rate function the interface dynamics can be analyzed exactly, and gives predictions which are in excellent agreement with direct numerical simulations. Importantly, the interface dynamics description improves upon the standard homogenization calculation, which is restricted to modulation that is both fast and weak

Dendritic cable with active spines: a modeling study in the spike-diffuse-spike framework

The spike-diffuse-spike (SDS) model describes a passive dendritic tree with active dendritic spines. Spine-head dynamics is modelled with a simple integrate-and-fire process, whilst communication between spines is mediated by the cable equation. Here we develop a computational framework that allows the study of multiple spiking events in a network of such spines embedded in a simple one-dimensional cable. This system is shown to support saltatory waves as a result of the discrete distribution of spines. Moreover, we demonstrate one of the ways to incorporate noise into the spine-head whilst retaining computational tractability of the model. The SDS model sustains a variety of propagating patterns

Existence and wandering of bumps in a spiking neural network model

We study spatially localized states of a spiking neuronal network populated by a pulse coupled phase oscillator known as the lighthouse model. We show that in the limit of slow synaptic interactions in the continuum limit the dynamics reduce to those of the standard Amari model. For non-slow synaptic connections we are able to go beyond the standard firing rate analysis of localized solutions allowing us to explicitly construct a family of co-existing one-bump solutions, and then track bump width and firing pattern as a function of system parameters. We also present an analysis of the model on a discrete lattice. We show that multiple width bump states can co-exist and uncover a mechanism for bump wandering linked to the speed of synaptic processing. Moreover, beyond a wandering transition point we show that the bump undergoes an effective random walk with a diffusion coefficient that scales exponentially with the rate of synaptic processing and linearly with the lattice spacing

Depolarization induced suppression of excitation and the emergence of ultra-slow rhythms in neural networks

Ultra-slow fluctuations (0.01-0.1 Hz) are a feature of intrinsic brain activity of as yet unclear origin. We propose a candidate mechanism based on retrograde endocannabinoid signaling in a synaptically coupled network of excitatory neurons. This is known to cause depolarization-induced suppression of excitation (DISE), which we model phenomenologically. We construct emergent network oscillations in a globally coupled network and show that for strong synaptic coupling DISE can lead to a synchronized population burst at the frequencies of resting brain rhythms

Using computational models to relate structural and functional brain connectivity

Modern imaging methods allow a non-invasive assessment of both structural and functional brain connectivity. This has lead to the identification of disease-related alterations affecting functional connectivity. The mechanism of how such alterations in functional connectivity arise in a structured network of interacting neural populations is as yet poorly understood. Here we use a modeling approach to explore the way in which this can arise and to highlight the important role that local population dynamics can have in shaping emergent spatial functional connectivity patterns. The local dynamics for a neural population is taken to be of the Wilson–Cowan type, whilst the structural connectivity patterns used, describing long-range anatomical connections, cover both realistic scenarios (from the CoComac database) and idealized ones that allow for more detailed theoretical study. We have calculated graph–theoretic measures of functional network topology from numerical simulations of model networks. The effect of the form of local dynamics on the observed network state is quantified by examining the correlation between structural and functional connectivity. We document a profound and systematic dependence of the simulated functional connectivity patterns on the parameters controlling the dynamics. Importantly, we show that a weakly coupled oscillator theory explaining these correlations and their variation across parameter space can be developed. This theoretical development provides a novel way to characterize the mechanisms for the breakdown of functional connectivity in diseases through changes in local dynamics

Using computational models to relate structural and functional brain connectivity

Modern imaging methods allow a non-invasive assessment of both structural and functional brain connectivity. This has lead to the identification of disease-related alterations affecting functional connectivity. The mechanism of how such alterations in functional connectivity arise in a structured network of interacting neural populations is as yet poorly understood. Here we use a modeling approach to explore the way in which this can arise and to highlight the important role that local population dynamics can have in shaping emergent spatial functional connectivity patterns. The local dynamics for a neural population is taken to be of the Wilson–Cowan type, whilst the structural connectivity patterns used, describing long-range anatomical connections, cover both realistic scenarios (from the CoComac database) and idealized ones that allow for more detailed theoretical study. We have calculated graph–theoretic measures of functional network topology from numerical simulations of model networks. The effect of the form of local dynamics on the observed network state is quantified by examining the correlation between structural and functional connectivity. We document a profound and systematic dependence of the simulated functional connectivity patterns on the parameters controlling the dynamics. Importantly, we show that a weakly coupled oscillator theory explaining these correlations and their variation across parameter space can be developed. This theoretical development provides a novel way to characterize the mechanisms for the breakdown of functional connectivity in diseases through changes in local dynamics

Pulsating fronts in periodically modulated neural field models

We consider a coarse grained neural field model for synaptic activity in spatially extended cortical tissue that possesses an underlying periodicity in its microstructure. The model is written as an integro-differential equation with periodic modulation of a translationally-invariant spatial kernel. This modulation can have a strong effect on wave propagation through the tissue, including the creation of pulsating fronts with widely-varying speeds, and wave-propagation failure. Here we develop new analysis for the study of such phenomena, using two complementary techniques. The first uses linearized information from the leading edge of a traveling periodic wave to obtain wave speed estimates for pulsating fronts, and the second develops an interface description for waves in the full nonlinear model. For weak modulation and a Heaviside firing rate function the interface dynamics can be analyzed exactly, and gives predictions which are in excellent agreement with direct numerical simulations. Importantly, the interface dynamics description improves upon the standard homogenization calculation, which is restricted to modulation that is both fast and weak