Response of traveling waves to transient inputs in neural fields

Abstract

We analyze the effects of transient stimulation on traveling waves in neural field equations. Neural fields are modeled as integrodifferential equations whose convolution term represents the synaptic connections of a spatially extended neuronal network. The adjoint of the linearized wave equation can be used to identify how a particular input will shift the location of a traveling wave. This wave response function is analogous to the phase response curve of limit cycle oscillators. For traveling fronts in an excitatory network, the sign of the shift depends solely on the sign of the transient input. A complementary estimate of the effective shift is derived using an equation for the timedependent speed of the perturbed front. Traveling pulses are analyzed in an asymmetric lateral inhibitory network, and they can be advanced or delayed, depending on the position of spatially localized transient inputs. We also develop bounds on the amplitude of transient input necessary to terminate traveling pulses, based on the global bifurcation structure of the neural field.

Coupling layers regularizes wave propagation in stochastic neural fields

“…Thus, we can compare our asymptotic results to numerical simulations. First, to compute the front speed corrections γ 1 and γ 2 , we must calculate the front solutions of the decoupled system [46,47] …”

“…Note that these two branches will coalesce in a saddle-node bifurcation (see [47] for analysis in a single layer network). This bifurcation point is determined by where θ = cos φ +w c , as shown in Fig.…”

“…Using the 2π periodicity along with a self-consistency argument, we can solve (64) explicitly to yield [47] …”

Optimal entrainment with smooth, pulse, and square signals in weakly forced nonlinear oscillators

“…(iv) Controlling traveling pulses or waves in reaction-diffusion systems: [42] derives the associated phase sensitivity function, and the dynamics of traveling pulses reduces to the phase equation (2), which enables our optimization methods to be applicable to such systems. We note that a variety of forcing strategies and obtained phase sensitivity functions in [43] are important for designing controls in such systems.…”

Capture of fixation by rotational flow; a deterministic hypothesis regarding scaling and stochasticity in fixational eye movements

“…They have received extensive theoretical attention in terms of their abstract properties in networks (e.g., Coombes, 2005; Kilpatrick and Bressloff, 2010b; Ma et al, 2012b), but surprisingly little attention in terms of concrete cases linking their dynamics to perception and behavior. We have conducted preliminary modeling studies employing spiral waves for visual salience mapping (Wilkinson and Metta, 2011; Wilkinson et al, 2011), and spiral neurodynamics have linked to visual geometric hallucination (Bressloff et al, 2001; Kilpatrick and Ermentrout, 2012a,b; Froese et al, 2013). At the motor end, Heitmann, Breakspear and colleagues have developed physiologically explanatory models showing how traveling waves (including spirals) can encode motor trajectories (Heitmann, 2013).…”