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Computational Maps in the Visual Cortex_Miikkulainen_2005

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12.3 Segmentation Through Desynchronization

263

12.3 Segmentation Through Desynchronization

Proper desynchronization is as important as synchronization, since it is the basis for segmentation. This section will show that inhibitory connections are necessary for segmentation, and that a small amount of noise is necessary for symmetry breaking.

12.3.1 Effect of Connection Types

As was seen in Section 12.2.1, excitatory and inhibitory connections have the opposite effect under the same decay rate. For perceptual grouping, both synchronization and desynchronization are necessary; such behavior may be efﬁciently achieved by utilizing both excitatory and inhibitory lateral connections. This hypothesis is tested in this section, verifying that including both types of connections indeed results in a desirable temporal representation for binding and segmentation.

As an abstraction of the grouping task and the underlying connectivity in the cortex, a one-dimensional network of 90 neurons was divided into two groups: Neurons [1..22] and [45..66] formed the ﬁrst group, and neurons [23..44] and [67..90] the second group. Lateral excitatory connections were only allowed to connect neurons within the same group, whereas inhibitory connections connected the whole population.

Four separate experiments were conducted: one with both excitatory and inhibitory connections, another with excitatory connections only, the third with inhibitory connections only, and the fourth with no lateral connections at all. Other simulation conditions were the same as in Section 12.1, except γE = 0.36 and λE = 5.0 to compensate for the larger size of the network and the addition of inhibitory connections. Values of γI = 0.42 and λI = 5.0 were used for the inhibitory connections.

The results are shown in Figure 12.4. First, with both excitatory and inhibitory connections, neurons within the same group are synchronized, but across the groups where only inhibitory connections exist, desynchronization occurs (Figure 12.4a). Such temporal representation is well suited for perceptual grouping, since binding is signaled by synchrony and segmentation by desynchrony. Next, with only excitatory connections, segmentation does not occur (Figure 12.4b), and with only inhibitory connections, binding does not occur (Figure 12.4c). Finally, without any lateral connections, the neurons ﬁre in different phases, determined by their randomly initialized membrane potential (Figure 12.4d).

In summary, binding and segmentation can be established in a network with both excitatory and inhibitory lateral connections. Omitting either kind of the lateral connections results in losing the ability to bind, segment, or both.

12.3.2 Effect of Noise

In previous sections, the initial membrane potential of each neuron was uniformly randomly initialized. Whether such initial perturbations are necessary will be tested

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Fig. 12.4. Binding and segmentation with different connection types. A network of 90 neurons was divided into two groups and simulated for 500 iterations. Neurons [1..22] and [45..66] formed the ﬁrst group (E1) and neurons [23..44] and [67..90] the second group (E2). All neurons in each group had the same lateral connections, shown at right. Excitatory lateral connections only linked neurons within the same group (E1 or E2), and inhibitory connections were global. (a) With both excitatory and inhibitory connections, the neurons within the same group are synchronized, while those in different groups are desynchronized. (b) With excitatory connections only, the neurons cannot desynchronize. (c) With inhibitory connections only, no coherently synchronized groups emerge. (d) When there are no lateral connections, neurons spike in different phases determined by their initial state. Both types of connections are therefore needed to establish simultaneous binding and segmentation.

in this section, analyzing the roles of initial and continual noise in symmetry breaking. The results are compared with the control case where the simulation is carried out without noise.

12.4 Robustness Against Variation and Noise

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A network of 180 neurons with both excitatory and inhibitory lateral connections was simulated for 500 iterations. The network was divided into two groups as in

the previous experiment (Section 12.3.1). Neurons [1..22], [45..66], [89..110], and [133..154] formed the ﬁrst group, and neurons [23..44], [67..88], [111..132], and

[155..180] the second group. Excitatory lateral connections only connected neurons within the same group, and their radius was limited to 90. The inhibitory connections were global.

Three separate experiments were conducted: with initial noise only, with contin-

ual noise only, and without noise. The parameters were the same in all three experiments: γE = 0.48, γI = 0.42, and λE = 5.0, λI = 1.0, and all other simulation

conditions were the same as in Section 12.1.

The results are shown in Figure 12.5. With initial noise (i.e. random initial membrane potential), the neurons within the same group are synchronized while the two different groups are desynchronized (Figure 12.5a). Also, even if the neurons are initialized uniformly (at 1.0), when a small amount of noise (0.1%) is added to the membrane potential at each time step, the two groups will desynchronize (Figure 12.5b). However, without noise of any kind, symmetry is not broken and the two groups stay synchronized (Figure 12.5c). So, inhibitory connections alone are not sufﬁcient for desynchronization. Cortical neurons actually operate in a noisy cellular environment, so including such noise in the model is realistic. It may also be desirable, in that it can make the behavior of the model more robust (Baldi and Meir 1990; Horn and Opher 1998; Terman and Wang 1995; Wang 1995).

In summary, a small amount of noise is needed for desynchronization; noise will therefore be used in the perceptual grouping experiments with PGLISSOM. The problem in biological systems, however, is not lack of noise, but that there may be too much of it. Next, how binding and segmentation can take place robustly under noisy conditions will be investigated.

12.4 Robustness Against Variation and Noise

The previous sections showed how the synaptic decay rate, the type and extent of the lateral connections, and the degree of noise can be controlled in the model to achieve synchronization and desynchronization for binding and segmentation. However, there are several factors that can possibly interfere with this process. For example, if the network is presented with different-size inputs simultaneously, the larger input could dominate the smaller input. If the level of noise is raised above a threshold, noise can dominate and coherent behavior may not be obtained. How robust the model is against such external factors will be analyzed in this section, and the components that contribute to its robustness will be identiﬁed.

12.4.1 Robustness Against Size Differences

One requirement for perceptual grouping is that input features should not be suppressed or promoted on the basis of size only, since smaller but complex input features in the scene can be equally important as large but simple features. Thus, a

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Fig. 12.5. Effect of noise on desynchronization. A network of 180 neurons with both excitatory and inhibitory lateral connections was simulated for 500 iterations. The network was divided into two groups, as in the experiment of Figure 12.4. The ﬁrst group (E1) consisted of neurons [1..22], [45..66], [89..110], and [133..154], and the second group (E2) of neurons [23..44], [67..88], [111..132], and [155..180]. The excitatory lateral connections were limited to the neurons in the same group within a radius of 90; the inhibitory connections were global. To illustrate, the plots at right show the lateral connections of neuron 45 in E1 and 132 in E2. (a) The membrane potential of each neuron was uniformly randomly initialized, and no noise was added afterward. The symmetry is broken and the two groups are separated as expected. (b) The membrane potentials initially were the same, but perturbed throughout the simulation by adding 0.1% of uniformly random noise. The neurons within the same group are synchronized at the same time as the two groups are desynchronized. (c) Without any noise (initial or continual), the symmetry was not broken and the entire network remained synchronized. A small amount of noise is therefore essential for proper desynchronization to occur.

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Fig. 12.6. Effect of relative input size on synchronization. A network of 90 neurons with both excitatory and inhibitory lateral connections was simulated for 500 iterations. The excitatory connection radius was 14 and inhibitory connections were global (as shown at right for neuron 45). The network was given two spatially separated inputs, and the size of the second input was varied. The rows (i.e. neurons) that received input are marked by black solid bars on the left. (a) The two inputs were the same size, activating neurons [19..36] and [55..72]. (b) One input was twice as long as the other input, activating neurons [16..45] vs. [61..75]. (c) One input was three times as long as the other input, activating neurons [1..45] vs. [61..75]. In all cases, the inputs are robustly bound and segmented, showing that the behavior is not affected by variation in the size of the input.

network of spiking neurons modeling such behavior should tolerate differences in input size.

To test if the PGLISSOM model is robust against such variation, a network of 90 neurons with both excitatory and inhibitory lateral connections was simulated for 500 iterations. The excitatory connection radius was 14 so that neurons representing different inputs were not connected, and inhibitory connections were global. Three separate experiments were conducted by presenting two inputs of relative sizes 1:1, 1:2, and 1:3 (Figure 12.6). The parameter values were the same in the three experiments: γE = 0.7, γI = 0.6, and λE = 5.0, λI = 1.0, to compensate for the larger number of excitatory connections compared with the previous experiments. Other simulation conditions were the same as in Section 12.1.

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(d) γE = 0.48

Fig. 12.7. Overcoming noise with strong excitation. A network of 30 neurons with global excitatory lateral connections was simulated for 500 iterations. A higher level of noise (1%, i.e. 10 times the noise in Section 12.3.2) was added to the membrane potential at each iteration. When the excitatory contribution is weak, as in (a) and (b), noise overwhelms it and causes the activities to desynchronize. However, as the excitatory contribution becomes stronger as in (c) and (d), it overcomes noise and achieves synchrony.

The results are shown in Figure 12.6. The two areas of the map representing the two objects are synchronized and desynchronized within and across the group, regardless of the input size. Note that these behaviors occur under identical parameter conditions, and thus demonstrate that the model is not affected by the size of the inputs alone.

12.4.2 Overcoming Noise with Strong Excitation

Cortical neurons operate in an inherently noisy environment. Noise can arise from several causes: For example, synaptic transmission may be unreliable or membrane potential may ﬂuctuate. As was shown in Section 12.3.2, a small amount of noise is useful in desynchronizing separate representations. However, biological networks are likely to be very noisy, and a model should be robust against such high levels of noise as well.

A network of 30 neurons with global excitatory lateral connections was simulated for 500 iterations. Four separate experiments were conducted, increasing the excitatory contribution in four stages under a higher level of noise (1%, i.e. 10 times the noise in Section 12.3.2). The simulation conditions were the same as in Section 12.1, except λE = 5.0.

The results are shown in Figure 12.7. With a higher level of noise, weak excitatory connections cannot keep the neurons synchronized (Figure 12.7a,b), but as the

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Fig. 12.8. Overcoming noise with a long refractory period. A network of 30 neurons with global excitatory lateral connections was simulated for 500 iterations. A signiﬁcant level of noise (7%, 70 times the noise in Section 12.3.2) was added to the membrane potential at each iteration. Such high noise cannot be tolerated by just increasing the lateral excitatory contribution γE. However, increasing the absolute refractory period can make the model robust even in this case. (a) When there is no absolute refractory period (tr = 0), the activities are random. (b) When the absolute refractory period lasts for three iterations, the activities start to synchronize loosely. (c) When it lasts for ﬁve iterations, the activities are strongly synchronized. With longer periods between ﬁring, the noise is effectively washed out.

excitatory contribution γE is increased, the neurons start to synchronize. This result shows how a network of spiking neurons can robustly synchronize even in moderately noisy conditions: Strong excitation can be used to overcome the noise in such cases.

12.4.3 Overcoming Noise with a Long Refractory Period

Although increasing the excitatory contribution helps, there is a certain threshold where noise cannot be overcome this way. For example, 7% noise will break the synchronization behavior even with extremely strong excitatory connections because noise will dominate the spiking behavior of the network. However, it turns out longer refractory periods will make the network robust even in such cases.

A network of 30 neurons with global excitatory lateral connections was simulated for 500 iterations. Three separate experiments were conducted where the length of absolute refractory period was gradually increased. The simulation conditions were the same as in Section 12.1, except λE = 5.0 to make synchronization more robust.

The results are shown in Figure 12.8. Under signiﬁcant noise (7%), the excitatory connections alone cannot keep the neurons synchronized (Figure 12.8a), but as the absolute refractory period is lengthened, the neurons start to synchronize again (Figure 12.8b,c). This result suggests that absolute refractory periods may have come to exist in biological neurons in part to overcome high levels of noise in the cortical environment. When the time interval during which the neuron can ﬁre is smaller than

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the refractory period, the noise is washed out. Thus, with a strong γE and a long refractory period, the neuron can be made highly robust against noise, which suggests that synchronization can be robust in real environments. Such a mechanism can be seen as a way to increase reliability at the expense of processing speed.

12.5 Discussion

As experiments in this chapter demonstrate, several factors inﬂuence how ﬁring becomes synchronized and desynchronized in a network of integrate-and-ﬁre neurons. The observed effects of synaptic decay and absolute refractory period are particularly novel and potentially signiﬁcant.

Synaptic decay has a similar effect on synchronization as the time it takes to integrate incoming PSPs: Both modulate the time it takes to reach the threshold. Integration time has been recognized earlier as a parameter that can alter synchronization behavior (Eurich et al. 1999, 2000). It is usually modeled by adding an additional delay to the spike arrival time. However, such an approach does not take into account that the PSP also decreases over time. The results in Section 12.2.1 suggest that decay plays a signiﬁcant role in synchronization. Explicitly modeling decay therefore gives us a more accurate understanding of the mechanisms responsible for synchrony.

At this point, the effects of decay adaptation are computational predictions only; there is little biological evidence to support or falsify such processes. In the near future, it may be possible to verify whether the dendritic membrane potential can decay at different rates at different locations in biological neurons, and also whether there is such a difference between different types of synapses (e.g. glutaminergic vs. GABAergic synapse). If differences are found, they can be compared with the results presented in this chapter, allowing us to predict what role such different kinds of connections may play in modulating synchrony. An interesting further question is how the decay rate interacts with conduction delays. While it may be difﬁcult to tune the delays in biological systems, it is possible that the decay rate adjusts to the delays so that robust synchronization behavior emerges under various conditions (Section 16.4.7).

Another novel result of this chapter is that a longer absolute refractory period can help overcome noise: Even in a highly noisy neural environment, synchronization can be achieved in this manner. From a computational point of view, such a mechanism can be seen as a way to increase reliability at the expense of processing speed: With a longer refractory period, ﬁring rates will be lower and it will take longer to decode the information encoded in them (Section 16.4.7).

Together, the results in this chapter demonstrate qualitatively how the different factors affect behavior in networks of integrate-and-ﬁre neurons. They serve as a practical guide that allows utilizing synchronization in large neural network systems, as will be done in the next chapter. In the future it may also be possible to carry the analysis a step further and develop a quantitative theory of how the different parameters modulate synchrony (similar to the analysis of connection types and delays

12.6 Conclusion

271

by Nischwitz and Glunder¨ 1995). For example, the PSP decay rate parameter λ or the length of the absolute refractory period can be continuously varied within a ﬁxed range, and the degree of synchrony measured in each case. Such a study might lead to empirical equations that allow predicting the behavior of the network with different parameters. Like the equations that scale the model to different size networks (Chapter 15), such equations might also allow setting the parameters directly to obtain the desired behavior, eventually leading to a mathematical theory of synchronization.

12.6 Conclusion

In this chapter, a one-dimensional network of spiking neurons was systematically tested to determine how the different components of the PGLISSOM model contribute to synchronization and desynchronization of activity. The results show that increasing the decay rate can synchronize networks with excitatory connections and desynchronize networks with inhibitory connections, local excitatory connections can achieve long-range synchrony, and both excitatory and inhibitory connections are necessary for simultaneous binding and segmentation, and noise helps break the symmetry in such cases. The model was also shown to be robust against changes in input size, and against high levels of noise through strong excitation and long absolute refractory periods.

Understanding such qualitative and quantitative factors that affect synchronization allows us to predict how a network of spiking neurons behaves in a speciﬁc conﬁguration, and provides a theory for the corresponding mechanisms in biology. These mechanisms play an important role in perceptual grouping tasks such as contour integration, which is the subject of the next chapter.

Understanding Perceptual Grouping: Contour

Integration

Perceptual grouping is the process of identifying the constituents in the visual scene that together form a coherent object. Grouping takes place at several levels in the visual processing hierarchy, as was discussed in Section 1.3. Experiments with PGLISSOM focus on the early grouping task of contour integration, where there are plenty of neurobiological and psychophysical data to constrain, test, and validate the model. The hypothesis is that much of contour integration is performed in V1, based on mechanisms implemented in PGLISSOM.

The ﬁrst section in this chapter deﬁnes and motivates the task and reviews psychophysical observations and computational models. The following sections demonstrate PGLISSOM in contour integration, including segmentation of multiple contours and completion of partial and illusory contours. Input-driven self-organization is also shown to possibly account for the differences in contour integration performance in different parts of the visual ﬁeld.

13.1 Psychophysical and Computational Background

Contour integration is a well-deﬁned task where performance can be readily measured both in humans and in computational models. It has therefore been extensively studied both in psychophysical experiments and in artiﬁcial neural networks. The underlying theory is remarkably clear and uniform across these studies and also extends to illusory contours. It is less clear, however, how contour integration circuitry can arise in early development and result in different performance in different parts of the visual ﬁeld.

13.1.1 Psychophysical Data

A typical visual input for the contour integration task is shown again in Figure 13.1. The input consists of a series of short oriented edge segments (contour elements) aligned along a continuous path, embedded in a background of randomly oriented contour elements. The task is to identify the longest continuous contour in this scene.

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