Stationary bumps in a piecewise smooth neural field model with synaptic depression

We analyze the existence and stability of stationary pulses or bumps in a one–dimensional piecewise smooth neural field model with synaptic depression. The continuum dynamics is described in terms of a nonlocal integrodifferential equation, in which the integral kernel represents the spatial distribution of synaptic weights between populations of neurons whose mean firing rate is taken to be a Heaviside function of local activity. Synaptic depression dynamically reduces the strength of synaptic weights in response to increases in activity. We show that in the case of a Mexican hat weight distribution, there exists a stable bump for sufficiently weak synaptic depression. However, as synaptic depression becomes stronger, the bump became unstable with respect to perturbations that shift the boundary of the bump, leading to the formation of a traveling pulse. The local stability of a bump is determined by the spectrum of a piecewise linear operator that keeps track of the sign of perturbations of the bump boundary. This results in a number of differences from previous studies of neural field models with Heaviside firing rate functions, where any discontinuities appear inside convolutions so that the resulting dynamical system is smooth. We also extend our results to the case of radially symmetric bumps in two–dimensional neural field models

Neuronal Oscillations Enhance Stimulus Discrimination by Ensuring Action Potential Precision

Although oscillations in membrane potential are a prominent feature of sensory, motor, and cognitive function, their precise role in signal processing remains elusive. Here we show, using a combination of in vivo, in vitro, and theoretical approaches, that both synaptically and intrinsically generated membrane potential oscillations dramatically improve action potential (AP) precision by removing the membrane potential variance associated with jitter-accumulating trains of APs. This increased AP precision occurred irrespective of cell type and—at oscillation frequencies ranging from 3 to 65 Hz—permitted accurate discernment of up to 1,000 different stimuli. At low oscillation frequencies, stimulus discrimination showed a clear phase dependence whereby inputs arriving during the trough and the early rising phase of an oscillation cycle were most robustly discriminated. Thus, by ensuring AP precision, membrane potential oscillations dramatically enhance the discriminatory capabilities of individual neurons and networks of cells and provide one attractive explanation for their abundance in neurophysiological systems

Stochastic neural field theory and the system-size expansion

We analyze a master equation formulation of stochastic neurodynamics for a network of synaptically coupled homogeneous neuronal populations each consisting of N identical neurons. The state of the network is specified by the fraction of active or spiking neurons in each population, and transition rates are chosen so that in the thermodynamic or deterministic limit (N → ∞) we recover standard activity–based or voltage–based rate models. We derive the lowest order corrections to these rate equations for large but finite N using two different approximation schemes, one based on the Van Kampen system-size expansion and the other based on path integral methods. Both methods yield the same series expansion of the moment equations, which at O(1/N ) can be truncated to form a closed system of equations for the first and second order moments. Taking a continuum limit of the moment equations whilst keeping the system size N fixed generates a system of integrodifferential equations for the mean and covariance of the corresponding stochastic neural field model. We also show how the path integral approach can be used to study large deviation or rare event statistics underlying escape from the basin of attraction of a stable fixed point of the mean–field dynamics; such an analysis is not possible using the system-size expansion since the latter cannot accurately\ud determine exponentially small transitions

Weakly interacting pulses in synaptically coupled neural media

We use singular perturbation theory to analyze the dynamics of N weakly interacting pulses in a one-dimensional synaptically coupled neuronal network. The network is modeled in terms of a nonlocal integro-differential equation, in which the integral kernel represents the spatial distribution of synaptic weights, and the output activity of a neuron is taken to be a mean firing rate. We derive a set of N coupled ordinary differential equations (ODEs) for the dynamics of individual pulses, establishing a direct relationship between the explicit form of the pulse interactions and the structure of the long-range synaptic coupling. The system of ODEs is used to explore the existence and stability of stationary N-pulses and traveling wave trains

Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities

In this paper we consider instabilities of localised solutions in planar neural field firing rate models of Wilson-Cowan or Amari type. Importantly we show that angular perturbations can destabilise spatially localised solutions. For a scalar model with Heaviside firing rate function we calculate symmetric one-bump and ring solutions explicitly and use an Evans function approach to predict the point of instability and the shapes of the dominant growing modes. Our predictions are shown to be in excellent agreement with direct numerical simulations. Moreover, beyond the instability our simulations demonstrate the emergence of multi-bump and labyrinthine patterns. With the addition of spike-frequency adaptation, numerical simulations of the resulting vector model show that it is possible for structures without rotational symmetry, and in particular multi-bumps, to undergo an instability to a rotating wave. We use a general argument, valid for smooth firing rate functions, to establish the conditions necessary to generate such a rotational instability. Numerical continuation of the rotating wave is used to quantify the emergent angular velocity as a bifurcation parameter is varied. Wave stability is found via the numerical evaluation of an associated eigenvalue problem

Stimulus-locked traveling waves and breathers in an excitatory neural network

We analyze the existence and stability of stimulus-locked traveling waves in a one-dimensional synaptically coupled excitatory neural network. The network is modeled in terms of a nonlocal integro-differential equation, in which the integral kernel represents the spatial distribution of synaptic weights, and the output firing rate of a neuron is taken to be a Heaviside function of activity. Given an inhomogeneous moving input of amplitude I0 and velocity v, we derive conditions for the existence of stimulus-locked waves by working in the moving frame of the input. We use this to construct existence tongues in (v,I0 )-parameter space whose tips at I0 = 0 correspond to the intrinsic waves of the homogeneous network. We then determine the linear stability of stimulus-locked waves within the tongues by constructing the associated Evans function and numerically calculating its zeros as a function of network parameters. We show that, as the input amplitude is reduced, a stimulus-locked wave within the tongue of an unstable intrinsic wave can undergo a Hopf bifurcation, leading to the emergence of either a traveling breather or a traveling pulse emitter

Spatially structured oscillations in a two-dimensional excitatory neuronal network with synaptic depression

We study the spatiotemporal dynamics of a two-dimensional excitatory neuronal network with synaptic depression. Coupling between populations of neurons is taken to be nonlocal, while depression is taken to be local and presynaptic. We show that the network supports a wide range of spatially structured oscillations, which are suggestive of phenomena seen in cortical slice experiments and in vivo. The particular form of the oscillations depends on initial conditions and the level of background noise. Given an initial, spatially localized stimulus, activity evolves to a spatially localized oscillating core that periodically emits target waves. Low levels of noise can spontaneously generate several pockets of oscillatory activity that interact via their target patterns. Periodic activity in space can also organize into spiral waves, provided that there is some source of rotational symmetry breaking due to external stimuli or noise. In the high gain limit, no oscillatory behavior exists, but a transient stimulus can lead to a single, outward propagating target wave