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

Front propagation in stochastic neural fields

We analyse the effects of extrinsic multiplicative noise on front propagation in a scalar neural field with excitatory connections. Using a separation of time scales, we represent the fluctuating front in terms of a diffusive–like displacement (wandering) of the front from its uniformly translating position at long time scales, and fluctuations in the front profile around its instantaneous position at short time scales. One major result of our analysis is a comparison between freely propagating fronts and fronts locked to an externally moving stimulus. We show that the latter are much more robust to noise, since the stochastic wandering of the mean front profile is described by an Ornstein–Uhlenbeck process rather than a Wiener process, so that the variance in front position saturates in the long time limit rather than increasing linearly with time. Finally, we consider a stochastic neural field that supports a pulled front in the deterministic limit, and show that the wandering of such a front is now subdiffusive

Spatiotemporal dynamics of continuum neural fields

We survey recent analytical approaches to studying the spatiotemporal dynamics of continuum neural fields. Neural fields model the large-scale dynamics of spatially structured biological neural networks in terms of nonlinear integrodifferential equations whose associated integral kernels represent the spatial distribution of neuronal synaptic connections. They provide an important example of spatially extended excitable systems with nonlocal interactions and exhibit a wide range of spatially coherent dynamics including traveling waves oscillations and Turing-like patterns

The effects of noise on binocular rivalry waves: a stochastic neural field model

We analyse the effects of extrinsic noise on traveling waves of visual perception in a competitive neural field model of binocular rivalry. The model consists of two one-dimensional excitatory neural fields, whose activity variables represent the responses to left-eye and right-eye stimuli, respectively. The two networks mutually inhibit each other, and slow adaptation is incorporated into the model by taking the network connections to exhibit synaptic depression. We first show how, in the absence of any noise, the system supports a propagating composite wave consisting of an invading activity front in one network co-moving with a retreating front in the other network. Using a separation of time scales and perturbation methods previously developed for stochastic reaction-diffusion equations, we then show how multiplicative noise in the activity variables leads to a diffusive–like displacement (wandering) of the composite wave from its uniformly translating position at long time scales, and fluctuations in the wave profile around its instantaneous position at short time scales. The multiplicative noise also renormalizes the mean speed of the wave. We use our analysis to calculate the first passage time distribution for a stochastic rivalry wave to travel a fixed distance, which we find to be given by an inverse Gaussian. Finally, we investigate the effects of noise in the depression variables, which under an adiabatic approximation leads to quenched disorder in the neural fields during propagation of a wave

Effects of synaptic depression and adaptation on spatiotemporal dynamics of an excitatory neuronal network

We analyze the spatiotemporal dynamics of a system of integro-differential equations that describes a one-dimensional excitatory neuronal network with synaptic depression and spike frequency adaptation. Physiologically suggestive forms are used for both types of negative feedback. We also consider the effects of employing two different types of firing rate function, a Heaviside step function and a piecewise linear function. We first derive conditions for the existence of traveling fronts and pulses in the case of a Heaviside step firing rate, and show that adaptation plays a relatively minor role in determining the characteristics of traveling waves. We then derive conditions for the existence and stability of stationary pulses or bumps, and show that a purely excitatory network with synaptic depression cannot support stable bumps. However, bumps do not exist in the presence of adaptation. Finally, in the case of a piecewise linear firing rate function, we show numerically that the network also supports self-sustained oscillations between an Up state and a Down state, in which a spatially localized oscillating core periodically emits pulses at each cycle

Nonlinear Langevin equations for wandering patterns in stochastic neural fields

We analyze the effects of additive, spatially extended noise on spatiotemporal patterns in continuum neural fields. Our main focus is how fluctuations impact patterns when they are weakly coupled to an external stimulus or another equivalent pattern. Showing the generality of our approach, we study both propagating fronts and stationary bumps. Using a separation of time scales, we represent the effects of noise in terms of a phase-shift of a pattern from its uniformly translating position at long time scales, and fluctuations in the pattern profile around its instantaneous position at short time scales. In the case of a stimulus-locked front, we show that the phase-shift satisfies a nonlinear Langevin equation (SDE) whose deterministic part has a unique stable fixed point. Using a linear-noise approximation, we thus establish that wandering of the front about the stimulus-locked state is given by an Ornstein-Uhlenbeck (OU) process. Analogous results hold for the relative phase-shift between a pair of mutually coupled fronts, provided that the coupling is excitatory. On the other hand, if the mutual coupling is given by a Mexican hat function (difference of exponentials), then the linear-noise approximation breaks down due to the co-existence of stable and unstable phase-locked states in the deterministic limit. Similarly, the stochastic motion of mutually coupled bumps can be described by a system of nonlinearly coupled SDEs, which can be linearized to yield a multivariate OU process. As in the case of fronts, large deviations can cause bumps to temporarily decouple, leading to a phase-slip in the bump positions.Comment: 28 pages, 8 figure