Representing stimulus motion with waves in adaptive neural fields

Abstract

Traveling waves of neural activity emerge in cortical networks both spontaneously and in response to stimuli. The spatiotemporal structure of waves can indicate the information they encode and the physiological processes that sustain them. Here, we investigate the stimulus-response relationships of traveling waves emerging in adaptive neural fields as a model of visual motion processing. Neural field equations model the activity of cortical tissue as a continuum excitable medium, and adaptive processes provide negative feedback, generating localized activity patterns. Synaptic connectivity in our model is described by an integral kernel that weakens dynamically due to activity-dependent synaptic depression, leading to marginally stable traveling fronts (with attenuated backs) or pulses of a fixed speed. Our analysis quantifies how weak stimuli shift the relative position of these waves over time, characterized by a wave response function we obtain perturbatively. Persistent and continuously visible stimuli model moving visual objects. Intermittent flashes that hop across visual space can produce the experience of smooth apparent visual motion. Entrainment of waves to both kinds of moving stimuli are well characterized by our theory and numerical simulations, providing a mechanistic description of the perception of visual motion.

Keywords: Neural field; Synaptic depression; Traveling waves; Visual object motion.

Fig. 1

Traveling fronts in a neural field with synaptic depression. A Speeds of stable (solid) and unstable (dashed) traveling fronts as a function of the $\gamma =\frac{1}{1+\beta }$ (synaptic depletion/replenishment timescale ratio). For $\gamma <\theta$, self-consistency is broken, and traveling pulses emerge (see Section 3.2), and for $\theta <\gamma <2\theta$, retreating front solutions emerge (red), with speed and profile independent of ${\tau }_q$. Stable speed $c$ curves coalesce as $\gamma \to 1$ as effects of synaptic depression vanish. B Speeds as a function of synaptic timescale ${\tau }_q$, clearly showing that the speed of retreating fronts is independent of ${\tau }_q$. Black line indicates saddle node bifurcation where the stable and unstable advancing front speeds meet. C, D, E Example profiles of fast stable advancing (blue), slow unstable advancing (green), and retreating (red) fronts. Note the $\xi$-dependent region of $Q(\xi )$ trades places between advancing and retreating cases. For all panels $\theta =0.1$; for the profile panels $\gamma =0.15$ and ${\tau }_q=20$

Fig. 2

Traveling pulses in a neural field with synaptic depression. A, D Stable and unstable pulse profiles for parameters ${\tau }_q=20$, $\gamma =\frac{1}{6}$, (or equivalently $\beta =5$), and $\theta =0.2$. B, E The pulse speed $c$ and width $\Delta$ are plotted as a function of the synaptic timescale ${\tau }_q$ with color indicating $\gamma$, solid lines indicating stable branches and dotted lines indicating unstable branches. C, F The pulse speed $c$ and width $\Delta$ as a function of $\gamma$ with color indicating ${\tau }_q$, solid lines indicating stable branches and dotted lines indicating unstable branches. We see that as the synaptic efficacy time-scale becomes shorter (${\tau }_q$ small), depression is more rapid and both the width and speed of the pulse shrink. Similarly increasing the strength of depression ($\beta$ large) or equivalently shortening the effective timescale of synaptic depression ($\gamma$ small) will also reduce the width and speed

Fig. 3

Traveling front advanced by global flash stimulus. A Spatiotemporal evolution of a traveling front perturbed by a spatially homogeneous, temporally pulsatile stimulus $\epsilon I_u(x,t)=\epsilon \delta (t-t_0)$ with $\epsilon =0.09$. We compare the prediction (green dashed line) of our linear theory Eq. (15) to the leading edge computed directly from simulation (blue line). B Wave response function ${\nu }_{\infty }$ predicted by the theory Eq. (15) (black line) and compared to simulations (green dots). Weight function is exponential, firing rate nonlinearity is Heaviside, and model parameters are $\theta =0.1$, $\gamma =0.2$, and ${\tau }_q=20$

Fig. 4

Response of traveling pulse to localized pulsatile square stimulus. Stimuli $\epsilon I_j(x,t)=\epsilon {\overline{I}}_j\delta (t)P(x,x_0,{\Delta }_x)(j=u,q)$ are centered in space $P(x,x_0,{\Delta }_x)=H(x-x_0+\Delta x∕2)-H(x-x_0-{\Delta }_x∕2)$ at $x_0$ with width ${\Delta }_x=1$ and presented at time $t=0$ either to A the neural activity variable: ${\overline{I}}_u=1$ and $I_q\equiv 0$; or B the synaptic efficacy variable: ${\overline{I}}_q=1$ and ${\overline{I}}_u=0$. Asymptotic theory (black line) approximates the results from numerical simulation (green dots) very well. Note that the spatially square pulse input shifts and widens the core null space function (blue curve), which describes how the wave filters forcing. Other parameters are $\theta =0.2$, $\gamma =1∕6$, ${\tau }_q=20$, and $\epsilon =0.05$

Fig. 5

Entrainment of traveling pulses to propagating square pulses. A Spatiotemporal location of the leading edge of a traveling pulse (solid blue) perturbed by a moving square input (magenta) is ahead of that of the unperturbed pulse (dashed blue). Stimulus has magnitude $\epsilon =0.1$ and speed $c+{\Delta }_c=3.3$ (compare to natural wavespeed $c\approx 1.051$). Pulse remains entrained indefinitely. B When stimulus speed $c+{\Delta }_c=3.5$ is too large, the stimulus eventually slips off the pulse, which relaxes to its original speed. C Numerical simulations reveal a (white) region of failed entrainment when the stimulus speed is too large for a given magnitude $\epsilon$. Otherwise the colored region indicates the lag $y_{\infty }$ of entrained pulses whose stimulus speeds are not too large. Our first order approximation of the entrainment boundary (Eq. (27), black line) matches well. Neural field model parameters are $\theta =0.2$, $\gamma =1∕6$, ${\tau }_q=20$ and stimulus width ${\Delta }_x=10$

Fig. 6

Wave entrainment to apparently moving stimuli. A The leading edge (blue line) of a traveling pulse becomes entrained to an intermittent, moving, and flashing stimulus (magenta indicates regions where stimulus is non-zero). The difference between the baseline wavespeed $(c\approx 1.051)$ and the effective speed of the forcing stimulus is ${\Delta }_c=0.5$; its magnitude is $\epsilon =0.2$; its width and starting position are ${\Delta }_x=1$ and $x^{\ast }=0$; on and off phases are $T_{\text{on}}=T_{\text{off}}=0.5$ for a total stimulus period of $T=1$. B Weakening the stimulus magnitude to $\epsilon =0.12$, the traveling pulse fails to entrain, and slips ever further behind the stimulus. C. Asymptotic approximation to the entrainment boundary (dashed line) ${\Delta }_c=\epsilon \frac{T_{\text{on}}}{T}\frac{c}{K}$ is well matched to results of numerical simulations (solid) separating the domain of entrainment (lower right) from entrainment failure (upper left). Increasing the ratio $T_{\text{on}}∕T_{\text{off}}$ enlarges the domain of entrainment. Blue: $T_{\text{on}}∕T_{\text{off}}=0.1∕0.5=0.2$; Green: $T_{\text{on}}∕T_{\text{off}}=0.5∕0.5=1$; Black: $T_{\text{on}}∕T_{\text{off}}=0.5∕0.1=5$. Stimulus width is ${\Delta }_x=10$; all other model parameters are same. Throughout, neural field model parameters are $\theta =0.2$, $\gamma =1∕6$, and ${\tau }_q=20$

Fig. 7

Entrainment success vs. failure in a 2D model. Simulations of Eq. (1) in two-dimensions $(n=2)$ with a lateral inhibitory kernel $w(r)=2e^{-r}(1-r)$, and sigmoidal firing rate function $f(u)={(1+e^{-\eta (u-\theta )})}^{-1}$. A We initialize a spot that propagates rightward (along the black dashed line) and apply an obliquely moving Gaussian stimulus $\epsilon I_u(x,t)=\epsilon e^{-{\Vert x-vt\Vert }^2∕2}$ (star moving along dashed green line). B Stimulus meets spot. C Sufficiently strong $(\epsilon =0.3)$ stimulus entrains spot, changing its course. D Weak stimulus $(\epsilon =0.2)$ fails to entrain. Other parameters are $\eta =20$, $\theta =0.2$, ${\tau }_q=20$, and $\gamma =0.5$