Direction-selective motion discrimination by traveling waves in visual cortex

The majority of neurons in primary visual cortex respond selectively to bars of light that have a specific orientation and move in a specific direction. The spatial and temporal responses of such neurons are non-separable. How neurons accomplish that computational feat without resort to explicit time delays is unknown. We propose a novel neural mechanism whereby visual cortex computes non-separable responses by generating endogenous traveling waves of neural activity that resonate with the space-time signature of the visual stimulus. The spatiotemporal characteristics of the response are defined by the local topology of excitatory and inhibitory lateral connections in the cortex. We simulated the interaction between endogenous traveling waves and the visual stimulus using spatially distributed populations of excitatory and inhibitory neurons with Wilson-Cowan dynamics and inhibitory-surround coupling. Our model reliably detected visual gratings that moved with a given speed and direction provided that we incorporated neural competition to suppress false motion signals in the opposite direction. The findings suggest that endogenous traveling waves in visual cortex can impart direction-selectivity on neural responses without resort to explicit time delays. They also suggest a functional role for motion opponency in eliminating false motion signals.


Author summary
It is well established that the so-called 'simple cells' of the primary visual cortex respond preferentially to oriented bars of light that move across the visual field with a particular speed and direction. The spatiotemporal responses of such neurons are said to be nonseparable because they cannot be constructed from independent spatial and temporal neural mechanisms. Contemporary theories of how neurons compute non-separable responses typically rely on finely tuned transmission delays between signals from disparate regions of the visual field. However the existence of such delays is controversial. We propose an alternative neural mechanism for computing non-separable responses that does not require transmission delays. It instead relies on the predisposition of the cortical tissue to spontaneously generate spatiotemporal waves of neural activity that travel with a particular speed and direction. We propose that the endogenous wave activity resonates

Introduction
Hubel and Wiesel [1][2][3] laid the theoretical groundwork for the visual system with their discovery that many neurons in the primary visual cortex respond selectively to bars of light that move with a specific direction and speed. Such neurons are said to have non-separable spacetime receptive fields ( Fig 1A) because their responses to changing patterns of light and dark in the visual field cannot be explained in terms of independent spatial and temporal neural processes [4,5]. The neural mechanism for computing non-separable responses is still an open question. Most theoretical accounts follow the approach of Reichardt [6] where light receptors exploit transmission delays to act as coincidence detectors of temporally delayed signals from disparate regions of the visual field ( Fig 1B). The temporal delay essentially transforms the spatiotemporal computation into a spatial computation that can be feasibly accommodated by the dendritic arbors of a neuron. The concept was originally applied to direction-selective cells in the retina [7] and has since been extended to the visual cortex where transmission delays have been posited in the feed-forward projections [8][9][10][11][12][13] and in the lateral connections [14][15][16].
The motion-energy model [17] is a notable exception in that it uses the phase difference between time-varying signals in place of transmission delays. It accurately represents the responses of direction-selective neurons by applying a Gabor function (Fig 1C) to the features in the visual field [18,19]. The Gabor function is constructed from a difference of Gaussians [20] that reasonably approximate the synaptic footprints of excitatory and inhibitory neurons. Yet the phase difference is imposed by rotating the Gabor function in the space-time coordinate frame without biophysical justification. Furthermore, the neural computation is expressed in terms of the visual coordinate frame rather than inputs at the neuronal level. Hence the motion-energy model is a descriptive model rather than an explanatory one [21].
Here we propose a neural mechanism for computing non-separable receptive fields without resort to explicit transmission delays. Our proposal relies on the retinotopic mapping of the visual system onto cortical coordinates and the propensity of cortical tissue to generate propagating waves of neural activity endogenously. We argue that those endogenous waves resonate with the spatiotemporal signature of stimulus to amplify the neural response for visual motion in a given speed and direction. It is those amplified responses that correspond to visual perception. The endogenous waves thus influence the responses of individual neurons and imbue them with their directional-selectivity. Furthermore, their preferred spatial and temporal frequencies are dictated by the geometry of the lateral inhibitory-surround coupling between excitatory and inhibitory neurons in the cortical tissue. This type of coupling features in the standard model of orientation selectivity that was originally proposed by Hubel and Wiesel [2] to explain the responses of 'simple' cells in primary visual cortex. Inhibitory-surround coupling also has conceptual links to the difference of Gaussians in the motion-energy model [17] and is crucial for the formation of standing wave patterns in neural field models [22][23][24][25][26].
Propagating waves have been observed in many regions of the brain [27][28][29] including the visual cortex. Stimulus-evoked and endogenously generated traveling waves have been observed in the visual cortex of monkey [30][31][32][33][34], cat [30,[35][36][37], rabbit [38], rat [39] and turtle [40]. The endogenous waves follow reproducible patterns that are related to the underlying anatomical connectivity [33,41]. In cat visual cortex, those patterns are closely aligned with the functional orientation maps [37]. In human visual cortex, endogenous waves are thought to be the basis of geometric visual hallucinations for similar reasons [42,43]. Stimulus-induced waves likewise follow reproducible patterns. Those waves travel beyond the footprint of the feed-forward projections [31] and are sensitive to the properties of the stimulus [30,32,44,45]. More recently, Townsend and colleagues [34] found that the direction of waves elicited in primate visual cortex by drifting visual gratings and dot-fields are sensitive to the direction of the stimulus on a trial by trial basis. That particular study established a functional link between visual motion processing and traveling waves. It also demonstrated that sensory information can be encoded in cortical traveling waves at appropriate time scales. However the authors did not propose a neural mechanism to explain how that might be achieved.
In the present study, we used neural field models of the visual cortex to investigate how endogenous traveling waves interact with visual stimuli. Neural fields represent the large-scale activity of neural tissue as a spatial continuum where thousands of co-located neurons are lumped into localized populations called neural masses [46]. They are typically formulated in terms of the average membrane voltage or the average firing rate activity of the neurons [47]. Crucially for this study, neural fields produce spontaneous spatiotemporal patterns-called Turing patterns-under appropriate coupling conditions [22-24, 26, 48, 49]. In particular, Wilson and Cowan [48,49] demonstrated standing wave patterns in a neural field with shortrange excitatory connections and long-range inhibitory connections. Amari [22] later proved that result analytically for neural masses with a step-function firing-rate response and 'Mexican hat' coupling with distance. Such coupling topologies can be constructed from excitatory and inhibitory connection densities with Gaussian spatial profiles [50] and have direct analogy with the inhibitory-surround receptive fields described by Hubel and Wiesel [2].
We therefore modeled the visual cortex as a spatial continuum of excitatory and inhibitory neural populations with Gaussian coupling profiles where the spread of the inhibitory coupling exceeded that of the excitatory coupling by a factor of 3:1. We restricted our model to one spatial dimension for simplicity. The model produced endogenous standing wave patterns consistent with neural fields having 'Mexican hat' coupling [51,52]. We then applied a small spatial shift to the profile of the excitatory connections to cause those standing waves to propagate with a given direction and speed. That asymmetric coupling was key to imbuing the medium with non-separable spatiotemporal response properties. We then explored how those endogenous waves resonated with drifting grating stimuli to elicit robust direction-selective responses to visual motion.

Results
We defined the generalized equations for the Wilson-Cowan model (Fig 2A) as, where U e (t) and U i (t) are the normalized firing rates of the excitatory and inhibitory neural populations. Both populations are reciprocally coupled where w ei denotes the weight of the connection from the inhibitory population to the excitatory population. The sigmoidal firingrate function (Fig 2B) defines the response of each neural population to its input. Parameters b e and b i are the population firing thresholds. J(t) is an external stimulus which is applied to the excitatory population only. We began by configuring the parameters so that both cell populations were nominally at rest in the absence of stimulation (J = 0). This was done by choosing the connection weights (w ee = 12, w ei = 10, w ie = 10, w ii = 1) and firing thresholds (b e = 1.75 and b i = 2.6) so that the nullclines crossed near the left knee of the cubic nullcline ( Fig 2D). The stable resting point for this configuration was U e = 0.12 and U i = 0.17. We then applied a constant stimulus (J = 1) which induced a stable oscillation in U e and U i (Fig 2C). The limit cycle (black) is shown in Fig  2D. The time scales of excitation and inhibition (τ e = 10, τ i = 5) were adjusted so that the frequency of the oscillation was approximately 20 Hz, that being an appropriate time scale for neurons in visual cortex. Numerical continuation revealed that the limit cycle emerges via a supercritical Hopf bifurcation when the injection current exceeds the critical value J = 0.41 ( Fig 2E). In this case, the limit cycle grows relatively smoothly with stimulus strength which we reasoned was an appropriate characteristic for obtaining a graded neural response to visual motion.
We then investigated the effects of inhibitory-surround coupling on the formation of endogenous waves in the spatially extended model (Fig 3A). In this case, the excitatory and inhibitory lateral projections both had Gaussian spatial profiles, K e (x) and K i (x), where the spread of the inhibitory coupling (σ i = 0.15 mm) was three times broader than that of the excitatory coupling (σ e = 0.05 mm). The spatial footprints of these projections spanned approximately 0.6 mm which is consistent with the anatomical span of pyramidal dendrites [53]. When combined, these excitatory and inhibitory profiles produced the classic 'Mexican hat' profile shown in Fig 3B (black). As anticipated, this configuration of lateral coupling elicited self-organized standing waves ( Fig 3D) under spatially-uniform constant stimulation (J = 1). Furthermore, the spatial frequency (2.5 cycles/mm) of the standing wave was predicted by the dominant spatial frequency of the Mexican hat, as we had previously seen in phase-based neural fields [52]. Nonetheless, slight variations in the selected pattern can occur on a trial to trial basis [54]. More importantly, we were able to transform the standing wave into a traveling wave (Fig 3E) by applying a small spatial shift (δ = 0.02) to the excitatory coupling profile. The temporal frequencies of those traveling waves were typically −15 Hz, where negative frequencies indicate leftwards motion. Even though the shift in K e (x − δ) was barely noticeable, it still produced a marked asymmetry in the Mexican hat ( Fig 3C). Asymmetric coupling topologies are known to induce traveling waves in neural fields [25,55]. In this case, the asymmetric Mexican hat operates like a spatial filter that responds maximally to waves that are phase-shifted to the right, so the wave travels to the left.
Since the endogenous waves only emerged when the medium was stimulated, we hypothesized that it would respond preferentially to stimuli whose spatiotemporal signature best matched that of the endogenous wave. We tested this idea by stimulating the asymmetrically coupled medium with sinusoidal gratings that had identical spatial frequencies (f x = 2.5 cycles/ mm) but either moved in opposite directions (f t = −15 Hz versus f t = +15 Hz) or remained stationary (f t = 0 Hz). As anticipated, the medium responded robustly to the stimulus whose frequency characteristics matched that of the endogenous wave ( Fig 4A). However it also responded intermittently to the grating that moved in the opposite direction ( Fig 4B) and the stationary grating ( Fig 4C). For the case of motion in the opposite motion, the responses pulsated in time with the stimulus and appeared to lurch in the same direction but with occasional slips. The peak amplitude of the intermittent responses (U e,max = 0.94 for the opposite grating; U e,max = 0.95 for the stationary grating) actually exceeded that for the preferred stimulus (U e,max = 0.89). The intermittent pulses evoked by the stationary grating were more regular. Nonetheless, as a putative motion detector, the proposed model ( Fig 3A) failed to discriminate the preferred motion from the non-preferred motion.

The E-I-E model
We conjectured that this failure may be due to the model having insufficient degrees of freedom to accommodate the non-preferred motion signals. We therefore constructed a new model with an additional excitatory population that we call the E-I-E model (Fig 5A). The where U e1 (t) and U e2 (t) are the normalized firing rates of the two excitatory populations and J e1 (t) and J e2 (t) are their respective stimuli. All other parameters are the same as for Eqs (1) and (2). The two excitatory populations in this model represent distinct assemblies of neurons that have the same firing characteristics but are not directly connected to one another. They can only interact via the common population of inhibitory neurons. The excitatory populations receive independent stimulation on the assumption that they are innervated by distinct incoming projections. The E-I-E model proved to be remarkably selective to differential stimulation. When stimuli of different magnitude (J e1 6 ¼ J e2 ) were simultaneously applied to both populations, the responses in U e1 and U e2 always favored the population with the greatest input. Furthermore, those responses were mutually exclusive so that the 'losing' population was largely quiescent irrespective of how much it was stimulated ( Fig 5B). This suggested that the E-I-E model robustly discriminates between incoming stimuli, even in the face of considerable ambiguity. We therefore analyzed the model's behavior over a range of differential stimuli J e1 = J + Δ and J e2 = J − Δ which always favored population e 1 (Fig 6A). In the analysis that follows, we used numerical continuation to follow the steady-state responses in U e1 and U e2 while ramping J and holding Δ fixed. We began with the case of ambiguous signals (Δ = 0).
Selective responses to ambiguous stimuli. Fig 6B shows the bifurcation diagrams for both U e1 and U e2 for the case of Δ = 0 where the diagrams are identical because of symmetry. For J < 1 the responses in U e1 and U e2 are monostable fixed points which are necessarily identical. Those fixed points diverge at J = 1 via a pitchfork bifurcation at the branch point (labeled BP). For J > 1 the steady states of U e1 and U e2 may follow either of the upper or lower branches of stable fixed points depending upon initial conditions. The selections are mutually exclusive so that if U e1 selects the upper branch then U e2 selects the lower branch, and vice versa. The branch of identical fixed points (U e1 = U e2 ) continues to exist for J > 1 but is unstable (dashed line) and forms a separatrix between the two branches of stable fixed points.
For J > 1.4 the fixed points lose stability via supercritical Hopf bifurcations (labeled H) that give rise to co-existing stable limit cycles (shaded). The ambiguous stimulus allows U e1 and U e2 to select either of those limit cycles. As before, those selections are mutually exclusive. So if U e1 selects the oscillation on the upper branch then U e2 selects the oscillation on the lower branch, and vice versa. The oscillations on the upper branch are much larger than those on the lower branch. We regard the winner of the competition between e 1 and e 2 to be the one that selects the branch of large oscillations.
Thus for ambiguous stimuli with J > 1.4 either e 1 or e 2 are equally likely to win but at least the outcome is decisive. The separatrix between the upper and lower branches of solutions is the key to that selectivity because it forces the responses of e 1 and e 2 to self-segregate even though the stimuli (J e1 = J e2 ) are identical. We tested the outcomes of 10,000 trials of ambiguous stimuli with J = 2 and random initial conditions, U e 2 [0, 1] and U i 2 [0, 1]. The results confirmed that e 1 and e 2 were equally likely outcomes with 49.7% ± 1.2% of trials selecting e 1 with a 99% confidence interval.
Selective responses to weakly biased stimuli. The selectivity of the E-I-E model is no longer at chance once the stimulus is biased (Δ 6 ¼ 0). Fig 6B shows   steadily increases with J whereas U e2 steadily decreases. This divergence in responses guarantees that e 1 wins the competition-provided that the stimulus is ramped slowly from zero. Furthermore, the perceptual decision is robust for J > 1. 34  However that outcome is not guaranteed when the stimulus is suddenly onset rather than slowly ramped. In that case, it is possible for U e1 and U e2 to select other stable states that coexist for J > 1.32 where a pair of stable and unstable fixed points emerge from the limit point (LP). This minor branch of stable fixed points itself gives way to stable oscillations for J > 1.56. For U e1 those oscillations are small (left panel; lower H) and for U e2 those oscillations are large (right panel; upper H). If e 2 happens to select that large-amplitude oscillation then it wins the competition and the perceptual decision is a false positive. This occurred in 25.2% ± 1.12% of trials (n = 10000, 99% CI) with J = 2 and random initial conditions. The false positives are forgivable in this case because Δ = 0.03 is a very weak bias in the stimulus.
The potential for false positives is due to the existence of the limit point (LP). It is a remnant of the branch point (BP in Fig 6A) that is lost when Δ 6 ¼ 0 transforms the pitchfork bifurcation into an imperfect bifurcation. The position of the limit point is governed by the size of the bias in the stimulus. Increasing Δ > 0 shifts the limit point towards higher J. If the bias is large enough then it effectively eliminates the false positives by shifting the limit point beyond the operating range of J.
Selective responses to strongly biased stimuli. Fig 6D shows the bifurcation diagrams for U e1 (left panel) and U e2 (right panel) for the case of strongly biased stimuli (Δ = 0.2). The limit point has been shifted beyond J > 2 and the remaining steady states are all monostable. Thus e 1 is guaranteed to win the competition for J > 0.84 where a large oscillation emerges in U e1 and a small corresponding oscillation emerges in U e2 . This was confirmed by numerical simulation which found no false positives in 10,000 trials with J = 2 and random initial conditions. The strong bias in the stimulus (� 20% of baseline) thus ensures that the correct response is always selected.

The spatial E-I-E model
Returning to the problem of motion discrimination, we constructed a spatial variant of the E-I-E model where the profiles of the lateral projections in the excitatory layers were shifted in opposite directions (δ = ±0.02 mm) while the profile of the inhibitory projections remained symmetric (Fig 7A). This coupling topology produced asymmetric Mexican hat profiles for both the upper and lower layers of the model (Fig 7B and 7C). The spatiotemporal stimulus J(x, t) was applied identically to both of the excitatory layers in this model. We hypothesized that the opposing phase shifts in the lateral coupling profiles would impel waves in the top layer to travel leftwards and those in the bottom layer to travel rightwards. While the external stimulation would serve as a bias that favored the layer which best matched the spatiotemporal signature of the stimulus. We reasoned that the selective response properties observed in the E-I-E point model would also apply to spatiotemporal activity patterns in the spatial model. We tested this concept by simulating the spatial E-I-E model with the same drifting gratings that we used in Fig 4 and found that the excitatory layers of the model were exquisitely selective to the direction of the moving stimulus. Moreover we saw no false responses to motion in the opposite direction. Fig 8A shows the response to a leftward moving grating whose spatial (f x = 2.5 cycles/mm) and temporal (f t = −15 Hz) frequencies match those of the endogenous wave in the top layer of excitatory neurons, represented by U e1 (x, t). The responses in U e1 (x, t) spanned the majority of the variable's dynamic range (0.01 < U e1 < 0.89) whereas that in U e2 (x, t) was very much suppressed (0 < U e2 < 0.01). We interpreted the overwhelmingly dominant activity of the e 1 layer as a robust perceptual response to leftwards motion.
The symmetric result was also observed for a rightward moving grating whose spatial (f x = 2.5 cycles/mm) and temporal (f t = +15 Hz) frequencies match those of the endogenous wave in the bottom layer of excitatory neurons, represented by U e2 (x, t). In that case, the e 2 layer gave the dominant response and the e 1 layer was suppressed (Fig 8B). The model responded as equally robustly to rightwards motion as it did to leftwards motion in these two test cases. The response to stationary gratings (Fig 8C) was also pleasing as both layers e 1 and e 2 exhibited suppressed responses (0.03 < U < 0.19) with no temporal oscillations. Such an outcome is the spatiotemporal analogy of the diverging branches of fixed point solutions in the point model under ambiguous stimulation (Fig 6A). Here that divergence is expressed as subtle differences in the spatial patterns in U e1 (x, t) and U e2 (x, t) where the presence of a stationary pulse in one pattern tends to suppress a corresponding pulse in the other. This is evident in the substantial range of the point-wise differences between the two patterns, −0.14 < U e1 (x, t) − U e2 (x, t) < 0.14.
Tuning curves. The previous simulations (Fig 8) demonstrated robust discrimination between leftward and rightward motion in specific test cases. We sought to generalize those findings by quantifying the responses of U e1 (x, t) and U e2 (x, t) to stimulus gratings with a range of spatial (0 < f x < 15) and temporal (−15 < f t < 15) frequencies.
The temporal frequency tuning curve (Fig 9A) was obtained by varying f t while holding the spatial frequency of the stimulus grating fixed at f x = 2.5 cycles/mm. It plots the maximal responses in U e1 (x, t) and U e2 (x, t) over the long term. The individual tuning curves for U e1 (dotted line) and U e2 (solid line) exhibit dramatic separation whereby e 1 responds predominantly to leftward moving gratings (f t < 0) and e 2 responds predominantly to rightward moving gratings (f t > 0). The responses are sharply constrained to the 5-28 Hz frequency band which is why the stationary grating (f t = 0) did not elicit a strong response in either U e1 or U e2 (Fig 8C). The small kinks in the tuning curve are due to the periodic boundary conditions which impel the spatial waves to accommodate the size of the domain. The tuning response changes sharply when there is a transition in the spatial wavenumber.
The spatial frequency tuning curve (Fig 9B) was similarly obtained by varying f x while holding the temporal frequency fixed at f t = 15 Hz which corresponds to rightwards motion. For this particular temporal frequency, the tuning curve for U e2 (solid line) is strongly selective to gratings with spatial frequencies 1.7 < f x < 5.0 cycles/mm. The response band is also remarkably sharp. Whereas the response for U e1 (dotted line) is attenuated at all spatial frequencies because it is tuned to motion in the opposite direction. The converse behavior is observed for leftward moving gratings (f t = −15 Hz).

Discussion
Our model demonstrates how neurons in the visual cortex can exploit endogenous background wave activity to compute non-separable spatiotemporal receptive fields without resort to transmission delays. The proposed mechanism relies on the predisposition of the cortical tissue to generate traveling waves of activity whose speed and direction are determined by the lateral coupling topology. The waves act as spatiotemporal filters that selectively amplify those stimuli that have similar space-time signatures to the wave-after retinotopic mapping of the visual field onto the cortex.
Selectivity of the response is enhanced by competition between waves that travel in opposite directions. That competition is mediated by the common pool of inhibitory cells which provide negative feedback to the opposing pools of excitatory cells. The dynamics of the E-I-E assembly are such that compromise solutions between competing stimuli are inherently unstable, leading to winner-take-all decisions. In the case of the point model, the competition is won by the excitatory cell with the stronger stimulus. In the case of the spatial model, it is won by the excitatory cells whose endogenous wave pattern resonates most with the spatiotemporal stimulus. Furthermore, the competition suppresses partial responses in the opposing detector. Opponency is thus an effective neural strategy for suppressing false-positives in otherwise imperfect detectors.

Motion detection as an emergent behavior
Our proposal offers new theoretical insights into how direction-selectivity can arise in the visual cortex through the collective behavior of neurons therein. The endogenous wave activity imposes a spatiotemporal bias on the background neural activity which in turn predisposes it to resonate with the preferred stimulus. Direction-selectivity is thus an emergent property of many neurons rather than a property of any single neuron. If such a neuron were to be isolated from its neighbors then it would immediately lose its spatiotemporal response properties.

Lateral inhibition as a mechanism for traveling waves
Our model also suggests that excitation and inhibition should be expected to co-vary in response to the preferred stimulus. This behavior is consistent with invasive recordings of the dendritic currents in the primary visual cortex of anesthetized cats by Priebe and Ferster [56]. They found that the excitatory and inhibitory currents co-vary at different phases and that the peaks of the inhibitory currents were maximal for the preferred stimulus rather than the null stimulus. Their findings are in contradiction to the standard model which predicts a strong inhibitory response to the null stimulus [56]. In our model, inhibition rises and falls cyclically because it is a mechanism of oscillation rather than a mechanism of stimulus suppression. The stimulus evokes maximal oscillations in the preferred excitatory cells while suppressing activity in the opposing excitatory cells. Whereas the inhibitory cells respond maximally either way. Thus the concept of lateral inhibition as a mechanism of wave generation may better explain the observed fluctuations in excitation and inhibition than the standard model of inhibition as a mechanism of stimulus suppression.
We know of no direct biological evidence for the types of shifts in lateral coupling that we have assumed in our model. It is likely that such shifts would be too small to detect. In any event, small asymmetries are probably the norm in biological systems. Asymmetries have previously been reported in the receptive fields of simple and complex cells in visual cortex [5]. Those asymmetries are thought to reflect asymmetries in the dendritic arbors of those cells. However, some physiological studies report no correlation between the morphology of the dendritic arbors and the orientation or directional selectivity of those cells [16,57]. These studies only considered the physical shape of the dendritic footprint and ignored the importance of the spatial densities of the dendritic receptors therein [51].

The functional role of opponency
The perceptual phenomenon of motion opponency is well documented but its functional role remains mysterious [58]. In theoretical models it is typically portrayed as a hypothetical subtraction between the outputs of neurons with opposing preferences. In the case of the E-I-E model, the mechanism of opponency is inherent within the circuitry itself. It plays a dual role in driving the oscillatory dynamics as well as selecting the winning response. Without opponency, the simpler E-I model fails to discriminate against the non-preferred stimuli because it lacks the degrees of freedom to accommodate other scenarios. The opposing E-I-E circuitry, on the other hand, has enough degrees of freedom to accommodate both scenarios. The functional role of opponency may thus be to avoid false positives by alleviating the dynamical frustration of the loser.
We believe that the same concept can also be extended to the two-dimensional visual field by arranging opposing pairs of excitatory kernels along a few cardinal directions of motion with some overlap between them. As before, the excitatory kernels would only be coupled to a common layer of inhibitory cells. Such an arrangement could mimic the hexagonal anatomical structure of the primary visual cortex where cells with similar directional tuning properties tend to be connected with each other [59][60][61]. Bressloff [62] has previously used a similar approach to model the effect of orientation-specific hypercolumns on geometric visual hallucinations.

Opponency in other sensory domains
The E-I-E model has some interesting properties that make it an effective neural circuit for resolving competing responses, potentially in any sensory domain. The symmetry of the circuit means that symmetric solutions to ambiguous stimuli do exist but those solutions lose stability when the baseline stimulus exceeds a critical threshold. The unstable symmetric solutions thus act as a separatrix between co-existing stable solutions which are dominated by each of the opposing excitatory populations. Whether those stable solutions are fixed points or limit cycles depends largely upon the choice of τ e and τ i . In our case, oscillations were crucial to the spatiotemporal signature of visual motion. However that need not be the case for other sensory domains where fixed points may be more appropriate.

Similar models
Zhang [63] previously proposed a similar double-ring network with asymmetric lateral coupling for head-direction tuning cells in the hippocampus. In that model, the position of the head was encoded by a bump attractor that was continuously shifted one way or the other to integrate signals from proprioceptors in the head. The motion of the bump was driven by asymmetric coupling that was modulated in time by the proprioceptors for clockwise and anticlockwise movement. The double rings operated in opposition in the sense that they pushed and pulled the bump in opposite directions. However the goal of that mechanism was to integrate movements from opposing proprioceptors rather than to suppress competing perceptual decisions, as is the goal of our model. Conversely, Ermentrout and colleagues [64,65] showed that bump attractors can also be made to travel with slow negative feedback rather that asymmetric lateral coupling. In the absence of a stimulus, the direction of travel is determined by initial conditions. It is likely that a moving stimulus could force the bump to travel in the same direction but it is not clear how that mechanism could be used to discriminate between motion in opposite directions.

Limitations and future work
Like many computational theories of vision, our proposal relies on the retinotopic mapping between the visual field and the cortex to preserve the geometric relationship between the stimulus and the endogenous neural activity. For simplicity, we assumed a one-to-one mapping between visual and cortical coordinates whereas the anatomical mapping is actually a logpolar relationship [66]. We anticipate that the use of log-polar retinotopic mapping would likely extend our results to the motion of rotating spirals, radial spokes and expanding rings [42].
Further work is required to extend the model to two spatial dimensions. One issue to consider is how much overlap to apply between motion detectors with differing orientation preferences. Should orientation-selective motion detectors be antagonistic towards their nearby counterparts or should they pool their outputs to achieve consensus? In terms of the E-I-E model, antagonism between the opposing detectors is crucial to its selectivity. Directly coupling the excitatory cells would weaken that mutual competition and encourage them to synchronize their behavior. If that coupling is strong enough then the behavior of the E-I-E model would effectively reduce to that of the simpler E-I model where all excitatory neurons operate in unison. So there is a balance to be struck between mutual competition and mutual cooperation for detectors with similar tuning preferences. Further research is required to elucidate the conditions for achieving that balance.

Methods
The equations for the spatial Wilson-Cowan model (Fig 3A) were defined as, where U e (x, t) and U i (x, t) are the spatiotemporal firing rates of the excitatory and inhibitory neural populations with x 2 R 1 . The sigmoidal function, defined the firing rate of each cell population in response to the net input v. That input comprised of the spatially weighted activity V e (x, t) and V i (x, t) from nearby excitatory and inhibitory cells. The spatial summation, was computed by convolving the spatial activity in U(x, t) with the Gaussian kernel, where σ is the spatial spread parameter and δ is a spatial shift parameter. The spatial shift was only applied to the excitatory cells. The connection weights w are scalar constants where w ei denotes the connection from an inhibitory population to an excitatory population. Parameters b e and b i represent the firing thresholds for the excitatory and inhibitory populations. Parameters τ e and τ i are the time constants of excitation and inhibition. All parameter values are listed in Table 1. Similarly, the equations for the spatial E-I-E model ( Fig 5A) were defined as, t e dU e1 dt ¼ À U e1 þ Fðw ee V e1 À w ei V i À b e þ JÞ ð11Þ t e dU e2 dt where U e1 (x, t) and U e2 (x, t) are the firing rates of the excitatory cells and U i (x, t) is the firing rate of the inhibitory cells. Visual stimulation was represented by the spatiotemporal signal J(x, t) which was applied to the excitatory cells only. It was defined as a sinusoidal moving grating, where α is the amplitude of the grating and f x and f t are its spatial and temporal frequencies.

Numerical simulation
The forward models were simulated using Version 2019a of the Brain Dynamics Toolbox [67,68] running in MATLAB R2019b. The differential equations were integrated forward in time using the ode23 solver with variable time steps and error tolerances of AbsTol = 1e-6 and RelTol = 1e-6. The numerical continuation was performed using MATCONT [69,70] version 7p1 with the default tolerances. The step size for branch of equilibrium points was limited to MaxStepSize = 0.1. The step size for the branch of limit cycles was limited to MaxStepSize = 0.5.

Acknowledgments
This work is based on a conference paper [71] that was originally presented at the First International Workshop on Computational Models of the Visual Cortex (CMVC) at Columbia University and published in the Proceedings of the 9th EAI International Conference on Bioinspired Information and Communication Technologies (BICT 15), New York City.