Fig 1.
Spatiotemporal receptive fields.
A: Schematic of a non-separable receptive field for a direction-selective neuron. The background grating represents a moving stimulus. The ellipses indicate the regions of light and dark that trigger a response in the neuron. B: Simplified schematic of the Reichardt [6] motion detector where Δx is the spatial separation between light receptors and Δt is a transmission delay. C: Gabor spatiotemporal filter constructed from a difference of Gaussians. Phase shifts are obtained by rotating the Gabor function in the space-time coordinate frame.
Fig 2.
The Wilson-Cowan model of reciprocally-coupled excitatory and inhibitory neural populations.
A: Schematic of the coupling where the weight for the connection to e from i is denoted by wei. J(t) is an external stimulus. B: The sigmoidal firing rate function. C: Time course of the mean firing rates U(t) for both the excitatory and inhibitory populations in response to the unit-step stimulus. D: The limit cycle (black) in the phase plane. Nullclines are shown in green. E: Bifurcation diagram showing the emergence of the limit cycle (shaded region) via a supercritical Hopf bifurcation. Thick solid line indicates stable fixed points. Dashed lines indicate unstable fixed points. H is the Hopf point. The thin solid lines emanating from the Hopf point describe the envelope of the oscillation in Ue.
Fig 3.
Endogenous traveling waves in the spatial model.
A: Schematic of the lateral coupling. The spatial profiles of the excitatory and inhibitory projections are defined by Ke(x) and Ki(x) respectively. The inhibitory projections have the furthest reach. The same profiles also apply to the connections between the excitatory and inhibitory cells but these have been omitted for clarity. B: Symmetric Mexican hat coupling profile (black) constructed from symmetric Gaussian profiles for the excitatory cells (green) and inhibitory cells (red) respectively. C: Asymmetric Mexican hat obtained by shifting the excitatory coupling profile to the right by δ = 0.02 mm. D: Stationary waves in the spatial model with symmetric lateral coupling, as per panel B. The gray scale indicates the mean firing rate of the excitatory cells. The minimum and maximum values are listed in the upper-right corner. E: Traveling waves in the spatial model with asymmetric lateral coupling, as per panel C.
Fig 4.
Effect of a moving grating stimulus on endogenous traveling waves.
Here J(x, t) is the stimulus and Ue(x, t) is the response of the medium. In all cases the medium is tuned (δ = 0.02) for leftward propagating waves with a spatial frequency of fx = 2.5 cycles/mm and a temporal frequency of ft = −15 Hz. A: Case of a leftwards-moving grating whose spatiotemporal signature matches that of the endogenous waves. B: Case of a rightwards-moving grating (ft = +15 Hz). C: Case of a stationary grating (ft = 0 Hz).
Fig 5.
A: Schematic of the model. The excitatory populations e1 and e2 are not directly connected. B: Effect of differential stimulation of e1 and e2 where Je1 > Je2 in the first pulse and Je1 < Je2 in the second pulse. The responses in Ue1 and Ue2 are mutually exclusive and selective to the cell with the strongest stimulus.
Fig 6.
Bifurcations in the E-I-E model under differential stimulation.
A: Schematic of the model. B: Responses to identical stimulation Je1 = J + Δ and Je2 = J − Δ where Δ = 0. C: Responses in Ue1 (upper panel) and Ue2 (lower panel) to weakly biased stimulation (Δ = 0.03). D: Responses to moderately biased stimulation (Δ = 0.2). Solid lines indicate stable fixed points. Dashed lines indicate unstable fixed points. Shaded regions are the envelopes of limit cycles. BP is branch point. H is Hopf bifurcation. LP is limit point.
Fig 7.
The spatial E-I-E model with asymmetric lateral coupling.
A: Schematic of the model where the spatial profiles of the excitatory projections, Ke1(x) and Ke2(x), are shifted in opposite directions. The lateral inhibitory projections (not shown) remain symmetric. B: Asymmetric Mexican hat constructed from Ke1(x + δ) and Ki(x) where δ = 0.02 mm. C: Asymmetric Mexican hat constructed from Ke2(x − δ) and Ki(x). Note the opposing phase shifts in the Mexican hat profiles.
Fig 8.
Direction-selective responses in the spatial E-I-E model.
The cells in layer e1 were tuned to leftward motion (δ = +0.02) and those in layer e2 were tuned to rightward motion (δ = −0.02). The external stimulus J(x, t) was applied identically to both layers. Their spatiotemporal responses are Ue1(x, t) and Ue2(x, t). A: Case of a leftwards moving grating (fx = 2.5 cycles/mm, ft = −15 Hz) which resonates with the endogenous wave in Ue1. B: Case of a rightwards moving grating (fx = 2.5 cycles/mm, ft = +15 Hz) which resonates the endogenous wave in Ue2. C: Case of a stationary grating (fx = 2.5 cycles/mm, ft = 0 Hz) which does not resonate with either.
Fig 9.
Tuning curves for the spatial E-I-E model.
A: Temporal frequency tuning curve showing the maximal responses in Ue1 and Ue2 for stimulus gratings with temporal frequencies −40 < ft < 40 Hz where negative frequencies correspond to leftward motion. The spatial frequency of the grating is fixed at fx = 2.5 cycles/mm. B: Spatial frequency tuning curve showing the maximal responses to gratings with spatial frequencies 0 < fx < 15 cycles/mm. In this case the temporal frequency is fixed at ft = 15 Hz.
Table 1.
Parameters of the model.