Ординатура / Офтальмология / Английские материалы / Binocular Rivalry_Alais, Blake_2005

temporary “winner-take-all” manner. Second, the winner of the competition must fatigue or self-adapt to a point where it releases the suppressed representation from inhibition. To make these ideas concrete, let us consider the simplest possible neural model of rivalry in which two excitatory

(E) neurons inhibit one another with strength a. (Explicit representation of inhibitory interneurons has been ignored as a mathematical convenience, which does not alter any of the arguments below.) In addition, each E neuron undergoes a slow self-adaptation governed by a variable H (see figure 17.1A).

It is impossible to model any neural oscillation like rivalry without a nonlinearity in the system, and the simplest such nonlinearity is a threshold with a linear increase in firing rate above threshold with slope M. These concepts are embodied in the following equations:

(17.1)

and denote left and right monocular inputs to and , respectively. The expressions on the right within brackets incorporate the threshold

(17.2)

This is a threshold function such that a net negative argument shuts off firing and positive produces a linear firing rate above threshold. The heavy solid line in figure 17.2 shows that this function is a good approximation to the firing rate of a human neocortical neuron over a reasonable range (data from Avoli et al., 1994).

Self-adaptation in equation (17.1) is mediated by the variables, which necessarily have very slow time constants, so . For excitatory neurons a typical value of t is 15 msec. This means that msec, a figure that is far too long to describe any inhibitory neuron. However, excitatory neocortical neurons in humans and other mammals invariably have slow, hyperpolarizing membrane currents that are typically calcium (Ca++)-mediated potassium (K+) currents (Avoli et al., 1994; McCormick and Williamson, 1989; Connors and Gutnick, 1990). We have previously hypothesized that these currents drive perceptual reversals (Wilson, Krupa, and Wilkinson, 2000; Wilson, Blake, and Lee, 2001). In particular, human excitatory neurons are known to possess hyperpolarizing membrane currents with time

Figure 17.1 Schematic of rivalry networks described by equation (17.1). (A) and are monocularly driven neurons with respective inputs L and R. Each inhibits the other via the I connections (separate inhibitory neurons are not explicitly represented in the model for mathematical convenience only). Slow hyperpolarizing membrane currents and reduce the spike rates of their respective nerons and are under neuromodulatory control (see text, “Conclusions and Conjectures”). (B) Two-level hierarchic model for rivalry competition. Left-eye- driven and right-eye-driven neurons comprise the lower, monocular level, with both vertical and horizontal orientation preferences (hatching) being represented. Interocular inhibitory competition occurs between orthogonal orientations at the monocular level, as represented by arcs ending in solid circles. Neurons at a higher binocular level sum excitatory inputs from monocular neurons with the same preferred orientations (heavy arrows). These higherlevel binocular neurons also engage in strong inhibitory competition. Finally, the hierarchic model also incorporates weak recurrent excitation from the binocular neurons back to their monocular inputs (light, dashed arrows). All excitatory neurons have the hyperpolarizing H currents depicted in (A).

constants averaging msec (McCormick and Williamson, 1989). In the absence of any competitive interactions, this slow hyperpolarizing current reduces the spike rate of a neuron by about a factor of 3 from its initial value. This is shown by the heavy dashed line fit of the model to cortical neuron data in figure 17.2. It is also important to note that these hyperpolarizing currents are under control of the modulatory neurotransmitters serotonin, dopamine, histamine, and acetylcholine (McCormick and

Figure 17.2 Spike rates of a human excitatory neocortical neuron as a function of input current I (adapted from Avoli et al., 1994). Solid circles plot initial transient firing rates, and open circles plot sustained rates following spike-frequency adaptation. Solid and dashed lines illustrate that the simple threshold nonlinearity in equation (17.2) provides a good fit to the firing rate data.

Williamson, 1989). I shall return to this important fact later to explain individual variability in rivalry data.

Given the neural model in equations (17.1) and (17.2), it is a simple matter to derive necessary and sufficient conditions for rivalry alternations to occur. This is a standard form of nonlinear analysis based on the presence of separate fast and slow time scales governed by t and , respectively (see Wilson 1999a). The results lead to two requirements:

(17.3)

.

The first expression guarantees that the inhibitory gain a will be sufficiently strong so that each E neuron is able to suppress firing of the other, and the second inequality guarantees that self-adaptation by a dominant E neuron will eventually weaken its response sufficiently to release the suppressed neuron from inhibition.

Simulation of equation (17.1) with parameters satisfying equation (17.2) (M = 1.0, a = 1.7, and g = 1.0) shows that alternate dominance intervals in response to orthogonal grating stimuli do indeed occur on a timescale comparable to binocular rivalry data (figure 17.3). Notice also that both

Figure 17.3 Rivalry oscillation produced by equation (17.1) with parameters given in text. Upper and lower pannels are relative spike rates of and , respectively. Although this model produces strictly deterministic limit cycle oscillations, analogous models describing individual action potentials generate a gamma distribution of dominance intervals via neural chaos (Laing and Chow, 2002; Wilson, 2003). Note that both competitive neurons are active during the first 150 msec (arrow), in agreement with psychophysical rivalry data (Wolfe, 1983).

E neurons fire simultaneously during the transient response at stimulus onset, which would correspond to the percept of a superimposed grating plaid. Only after about 150 msec does one neuron suppress the other, which agrees with psychophysical data (Wolfe, 1983). This is an unavoidable consequence of recurrent inhibition, which must always develop more slowly than the direct excitatory response to stimulation.

Although the alternation in figure 17.3 is a limit cycle oscillation (Wilson, 1999a), it is well known that empirical measurements of rivalry intervals generate a distribution of dominance intervals approximated by a gamma function (Fox and Herrmann, 1967; Borsellino et al., 1972). One possible way of producing gamma distributions from equation (17.1) is simply to add noise through a stochastic variable, as was shown by Lehky (1988). However, a more interesting possibility has arisen. Laing and Chow (2002) developed a model with the same competitive interactions as equation (17.1), except that they replaced and each by a large population of neurons described by Hodgkin–Huxley-type conduction dynamics. The resulting network generated dominance intervals approximating a gamma distribution via chaotic dynamics (Laing and Chow, 2002). Such chaos has been replicated using simplified equations describing human

cortical neurons (Wilson, 1999b) in networks comprising as few as six spiking neurons (Wilson, 2003).

Indeed, chaos and gamma distributions may be an unavoidable ingredient in any neural network with strong reciprocal inhibition and selfadaptation. Although studies by Richards, Wilson, and Sommer (1994) and Lehky (1995) failed to find direct evidence of chaos in rivalry data, this is likely due to noisy data. This hypothesis is supported by the observation that removal of eye movement jitter using image stabilization still produces a gamma-function dominance distribution (Blake, Fox, and McIntyre, 1971). In sum, gamma distributions are easy to generate in networks such as equation (17.1) through addition of either noise or neural chaos among spiking neurons. Thus, it is legitimate to model key features of rivalry with periodic dynamics, with the understanding that gammafunction distributions can be easily and naturally generated if desired.

LEVELT’S SECOND LAW

One defining feature of binocular rivalry is enshrined in Levelt’s Second Law (Levelt, 1965). For patterns of variable contrast, a precise statement of this law is “Changes in the contrast of the lower contrast pattern primarily alter the mean dominance duration of the higher contrast pattern, dominance durations for the lower contrast pattern remaining largely unaffected.” This behavior is manifested by the neural model in equation (17.1) and is illustrated in figure 17.4A. The solid line plots dominance durations for the neuron driven by a fixed, high-contrast stimulus for a range of contrasts of the weaker stimulus. While durations driven by the higher-contrast stimulus vary dramatically, those for the lowercontrast stimulus (dashed line) remain relatively unchanged. Figure 17.4B shows the oscillation produced by equation (17.1) when contrast of the weaker stimulus has been significantly reduced (arrow in figure 17.4A). Thus, the very simple model in equation (17.1) produces responses obeying Levelt’s Second Law. Furthermore, spiking neuron equations producing chaos and gamma distributions also obey Levelt’s Second Law (Laing and Chow, 2002).

DOMINANCE WAVES IN RIVALRY

Anybody who has viewed large rivaling patterns knows that the patterns do not alternate as a whole. Rather, portions of one pattern will emerge locally and sweep across the field, suppressing the other monocular

Figure 17.4 Levelt’s Second Law behavior resulting from the network in figure 17.1 and equation (17.1). (A) As the strength of the weaker stimulus is reduced from that of the stronger stimulus, stronger stimulus dominance intervals increase substantially (solid line), and weaker stimulus intervals decrease minimally (dashed line). (B) Neural activity in more strongly stimulated neuron (top panel) and weaker neuron (bottom pannel) for stimulus strengths shown by the double-headed arrow in (A). For comparison, figure 17.3 shows results when both neurons are equally stimulated with unity input.

pattern in their wake. For years such spatial dynamics proved intractable experimentally, so researchers resorted to studying rivalry by using small stimulus patches that appear and vanish in a roughly unitary fashion. This simplification led to important discoveries, among them the observation that patch size for unitary pattern alternation increases with retinal eccen-

tricity in accord with cortical magnification (Blake, O’Shea, and Mueller, 1992). This important insight implies that unitary rivalry reflects the local domain of neural inhibition.

To further understand rivalry, characterization of its spatial spread is necessary. A paradigm for studying the spread of monocular pattern dominance resulted from a serendipitous exposure to research in the field of Ca++ waves in cardiac tissue. Nagai et al. (2000) employed an experimental paradigm in which Ca++ waves were monitored while propagating around an annulus of cardiac tissue in a culture dish. This elegant paradigm forced two-dimensional waves to travel around what was effectively a one-dimensional race track annulus.

We adapted this approach to the study of dominance waves in rivalry (Wilson, Blake, and Lee, 2001). One such rivalry stimulus is depicted in figure 17.5A. The observer fixated the central fusion bull’s-eye, and each trial began when the observer reported by button press that the highcontrast spiral annulus was completely dominant. At that point a brief (200 msec) contrast pulse restricted to a small portion of the suppressed radial pattern caused it to become dominant locally, and this triggered a dominance wave that swept around the annulus until it reached the designated point at the bottom. Measurement of dominance-wave travel time revealed that it was a linear function of distance around the annulus (see figure 17.5B), thus indicating travel at a constant speed (Wilson, Blake, and Lee, 2001). Further measurements with annuli of different mean diameters revealed that dominance-wave speed was constant at approximately 2.24 cm/sec when mapped into V1 cortical coordinates as measured by Horton and Hoyt (1991). Additional measurements showed that dominance-wave speed doubled when the pattern driving the waves was concentric (see figure 17.5B) rather than radial (Wilson, Blake, and Lee, 2001). This is consistent with the operation of weak but long-range excitatory connections mediating collinear facilitation in V1 (Malach et al., 1993; Das and Gilbert, 1995; Somers et al., 1998).

Propagation of dominance waves has been explained quantitatively by a simple extension of the rivalry model developed above. The model comprises two spatially extended, circular rings of competing E neurons representing the stimulus annulus. The only remaining ingredient in the model is a spatial spread of the competitive inhibition. The spatial extent of this inhibitory spread was chosen to be 1.0 mm, the distance between adjacent ocular dominance columns in human V1 (Hitchcock and Hickey, 1980). This model produces traveling dominance waves with a speed of 2.24 cm/sec (Wilson, Blake, and Lee, 2001), in agreement with psy-

A

Distance (visual angle around annulus)

Figure 17.5 Rivalry dominance waves. (A) One eye viewed a high-contrast spiral annulus that, due to Levelt’s Second Law, dominated the lower-contrast radial annulus (Wilson, Blake, and Lee, 2001). Waves were then triggered in the low-contrast pattern at any of eight cardinal points, and the transit time to the marked point at the bottom was measured. Fixation was maintained on the fused central bull’s-eyes. (B) Dominance-wave propagation data for one subject. Solid circles and line show propagation times as a function of distance around the radial annulus in (A) at a constant speed corresponding to 4.4° visual angle per sec. When waves were propagated around a concentric annulus (not shown), speed roughly doubled (open circles and dashed line). This speed increase is attributed to the effects of weak but long-range collinear facilitation.

chophysics. The mechanism of wave propagation in the model is simple: when a neuron becomes dominant, its spatial spread of inhibition leads to disinhibition of its own monocular neighbors, so these waves propagate by disinhibition. Addition of longer-range collinear excitation in the

model increases wave propagation speed to 4.4 cm/sec, thereby reproducing data obtained with concentric monocular stimuli (Wilson, Blake, and Lee, 2001). The one caveat is that collinear excitation must be weak to permit the network to continue producing oscillations, since even moderately strong collinear facilitation causes bifurcation to a winner-take-all network in which rivalry oscillations cannot occur (Wilson, 1999a).

A COMPETITIVE RIVALRY HIERARCHY

Dominance waves in rivalry propagate at constant speed when mapped onto V1, and they exhibit collinear facilitation. Both of these observations support the hypothesis that these waves are a manifestation of interocular competition in V1. As discussed above, however, Logothetis, Leopold, and Sheinberg (1996) used their flicker and swap (F&S) procedure (18.0 Hz flicker with eye swapping at 1.5 Hz) and still obtained dominance intervals averaging about 2.3 sec, time for about seven swaps between eyes. This dramatic result clearly cannot be the result of competition between monocular neurons. Rather, binocular neurons must be the units competing in this case. To those conversant with nonlinear dynamics, these experiments strongly suggest that the F&S procedure causes a bifurcation of competitive network dynamics into a different regime (see Wilson, 1999a, for discussions of neuronal bifurcations).

I have incorporated these ideas into a hierarchical, two-stage competitive network that explains both traditional and F&S rivalry (Wilson, 2003). The architecture of this network is illustrated in figure 17.1B, where a monocular competitive network provides input to a higher level of competition between binocular neurons. The network also incorporates weak excitatory feedback from the binocular to the monocular level (dashed lines), although this is not necessary to its function. Both levels of the network can be described by equation (17.1) or by similar equations incorporating a Naka–Rushton nonlinearity (Wilson, 2003). Parameters are the same at both network levels, except that the inhibitory gain a must be stronger at the binocular level for reasons described below.

The network was first stimulated by a traditional rivalry pattern in which orthogonal monocular gratings were continuously presented. In this case the monocular level of the network generates the same rivalry oscillations as in figure 17.3; these oscillations in turn drive the binocular neurons, causing them to produce an identical oscillation (Wilson, 2003). Thus, the network accounts for normal rivalry as the result of monocular neural competition that is reflected at higher levels.

When the network is driven by F&S stimulation, a very different result ensues. As shown in figure 17.6, the F&S stimulus overcomes competition at the monocular level so that responses to both orthogonal gratings are continuously sent to the binocular level. Rivalry now occurs only at the binocular level, and dominance durations last through six to seven eye swaps, in complete agreement with psychophysics (Logothetis, Leopold, and Sheinberg, 1996). The temporal dynamics of the stimulus have eliminated competition at the monocular level, thereby revealing a higher level of binocular competition (Wilson, 2003).

Figure 17.6 Hierarchic rivalry model responses to F&S stimulation. Neurons preferring vertical orientations are plotted in black, and those preferring horizontal orientations are plotted in gray. The bottom two panels illustrate responses of the left and right monocular neurons, and the top panel plots responses of the higher-level binocular neurons. Note that the F&S stimulation paradigm defeats rivalry at the monocular stage so that both horizontal and vertical neural responses simultaneously pass through to the binocular level (indicated by vertical arrow). As described in the text and by Wilson (2003), rivalry now occurs at the binocular stage, and dominance durations are approximately 7.0 interocular switches in duration (horizontal solid line in middle panel). When traditional rivalry stimuli are used, however, the monocular network levels revert to the rivalry alternations depicted in figure 17.3, and these drive all activity at the binocular stage. Both levels of the hierarchy are necessary to account for both traditional and F&S rivalry.