Ординатура / Офтальмология / Английские материалы / Computational Maps in the Visual Cortex_Miikkulainen_2005

252 11 PGLISSOM: A Perceptual Grouping Model

tion preference histograms and two-dimensional Fourier power spectra of both maps look similar to those of the LISSOM orientation map (Section 5.3.3): The histograms are flat and the spectra are ring-shaped, suggesting that at any location in the map, all orientation angles are equally well represented, as they should be. Such a structure is also very similar to biological maps (Section 5.1.1).

Together these results show that the architecture and the learning rules in PGLISSOM can develop realistic orientation maps, similar to those seen in previous LISSOM models, and in the mammalian visual cortex. In the next section, the lateral connection patterns will be analyzed.

11.5.3 Lateral Connections

As was discussed in Sections 2.2 and 5.1, the lateral connections in biological orientation maps have two prominent characteristics: (1) Strong connections exist between neurons with similar orientation preferences, and (2) the connections extend along the direction matching the source neuron’s orientation preference (Figure 2.7). Such connections are believed to represent correlations in the visual input.

Similarly, the long-range lateral connections in PGLISSOM self-organize into patterns that have the same two general properties, and reflect the correlations in the input. The SMAP connections self-organize similarly to those in firing-rate LISSOM models; their properties were demonstrated in detail in Section 5.3.4. GMAP selforganization is different, however, since both excitatory and inhibitory connections have long range in GMAP, in order to establish segmentation and binding in the model.

Figure 11.5 plots the excitatory lateral connections of four sample neurons in the iso-orientation patches of the GMAP. In each case, the neuron is most strongly connected to others with similar orientation preferences. Thus, the figure shows qualitatively that the first property above holds in the model. The connections are also anisotropic, i.e. stretched along a particular direction with patches of connections found along this direction at roughly equal intervals. Such connection patterns are consistent with the second property exhibited in biological data.

As in the LISSOM models in the earlier chapters, such patterns emerge in PGLISSOM because the afferent and lateral connections adapt to encode the statistical structure in the training input. Since the training inputs are elongated Gaussian bars, the afferent connections form oriented receptive fields. Neurons with similar orientation preferences whose receptive fields are aligned along a straight axis will be activated simultaneously when a long input happens to fall upon those receptive fields. Due to the Hebbian learning process, such neuron pairs will develop strong lateral connections. Moreover, the connections are not strictly aligned along the axis, but there are also connections flanking the preferred axis. These flanks are larger farther from the source neuron, like a bowtie. The same pattern can be seen in biological data (Figure 2.7b).

This is an important observation, since the flanks allow grouping of not only straight contours, but also cocircular ones. Neurons not only respond to the optimal orientation at the optimal position, but also to slightly misoriented inputs (Fig-

OR connections OR CH OR weights

Fig. 11.5. Long-range lateral connections in GMAP. The long-range excitatory lateral connection patterns for four sample neurons in GMAP are shown on top, located in iso-orientation patches as shown in the map below. Similar plotting conventions are used as in Figure 5.12: The small black square identifies the neuron itself in both plots, and the white outline on the map indicates the extent of the lateral connections after self-organization and pruning; before self-organization the lateral connections covered the whole map, as shown by the black square outline on top. The color coding in the top plots represents the target neuron’s orientation preference, selectivity, and connection strength, and the map below encodes orientation and selectivity. The histogram in the middle shows the distribution of the target neurons’ orientation preferences. Each neuron is most strongly connected to its closest neighbors; the long-range connections are patchy and connect neurons with similar orientation preferences. They extend longer than those in Figure 5.12 because more elongated input patterns were used during selforganization. As in LISSOM, these connections extend along the orientation preference of the source neuron: (a) 2◦ red, (b) 51◦ purple, (c) 91◦ light blue, and (d) 136◦ light green. They are narrow around the neuron but wider farther away. As will be seen in Chapter 13, specific connection patterns like these are crucial for perceptual grouping such as contour integration.

ure 11.6). Thus, neurons with cocircular receptive fields can coactivate. When the two receptive fields can be connected with a straight path, but one or both are slightly misoriented from the axis of the path, they will both still be active. Their lateral connection will be strengthened, resulting in cocircular connection patterns.

The connection patterns in the model can also be measured quantitatively and compared with biological data. To measure the first property (i.e. that connections are stronger between neurons with similar orientation preferences), the percentage of GMAP excitatory lateral connections that connect receptive fields with varying orientation differences were calculated. The results are shown in Figure 11.7. The percentage of connections peaks at 0◦, and rapidly decreases to zero as the orientation

254 11 PGLISSOM: A Perceptual Grouping Model

Fig. 11.6. Activating neurons with collinear and cocircular RFs. The plot shows two representative cases of coactivation, i.e. two neurons responding simultaneously to a long input across their receptive fields. (a) The two receptive fields (black bars) are precisely aligned on a straight path (dashed line). If a long input is presented on this path, the two neurons will respond maximally, and the connection between them becomes stronger, according to normalized Hebbian learning. (b) Even though the two receptive fields are slightly misaligned on the path, they can still weakly activate and the connection will become stronger. As a result, neurons that represent cocircular contours develop significant lateral connections, although not as strong as those that represent straight lines. Reprinted with permission from Choe and Miikkulainen (2004), copyright 2004 by Springer.

Orientation difference

Fig. 11.7. Distribution of lateral connections in animals and in PGLISSOM. In the tree shrew V1, biocytin was injected in the cell body of seven different neurons, and the projections going to neurons of different orientation preferences were counted. The thin line with circles shows the median percentage of connections for each difference (adapted from Bosking et al. 1997). In the GMAP, the percentage of connections to neurons with different orientations were similarly counted (after pruning); the median over all neurons in the map is shown as the thick line. Both plots peak at 0◦, and quickly fall off as the orientation differences become larger (this effect is slightly exaggerated in the model because in this experiment it was trained on straight elongated Gaussians only; cf. Section 13.4.2). In other words, strong excitatory lateral connections mostly link neurons with similar orientation tuning both in the model and in animals.

difference increases. In other words, strong excitatory lateral connections in GMAP are most likely to be found between neurons that have similar orientation preference. Such a result is consistent with experimental measurements (Figure 11.7), quantitatively verifying the first property.

This measure not only allows comparing the connectivity in the model and in biological data, but also suggests a possible functional role for the first property. As will be discussed in Section 13.1, contours are believed to be grouped through

specific lateral interactions between contour elements (representing a local grouping function, or an association field). The measure presented above suggests that lateral connections could be implementing such a local grouping function.

The second property can also be measured quantitatively, by gathering statistics about the directions, angles, and distances of the source and target receptive fields. This study will be done in Section 13.2, showing that the second property holds quantitatively in PGLISSOM, and demonstrating that it implements a local grouping function. Edge-cooccurrence statistics in natural images and the connection statistics in the PGLISSOM model are shown to be strikingly similar, which makes the connectivity in PGLISSOM well suited for contour integration tasks.

11.6 Conclusion

To make it possible to understand self-organization of perceptual grouping phenomena, the PGLISSOM model expands the LISSOM framework inward in two ways. First, the single-unit model of the cortical column is extended to include two components: The S component takes part in self-organization (in SMAP), and the G component contributes to perceptual grouping (in GMAP). Second, the firing-rate neurons in LISSOM are replaced with spiking neurons, in order to represent grouping through temporal coding. The resulting PGLISSOM network self-organizes like the other LISSOM models, and the patterns of long-range excitation are appropriate for implementing perceptual grouping. In the following chapter, the synchronization behavior of the model is analyzed in detail. As a concrete example, its performance in contour integration tasks is then demonstrated in Chapter 13.

12

Temporal Coding

Experimental evidence reviewed in Section 2.4.2 suggests that temporally correlated activity may be the basis for binding and segmentation in perceptual grouping. In PGLISSOM such a temporal coding is generated by spiking neurons that synchronize their activities. It is important to understand how synchronization takes place in the model, mainly to gain insight into synchronization in biological networks, but also so that the PGLISSOM model as a whole can be tuned to function properly. In this chapter, the neuron model of PGLISSOM will be analyzed experimentally to find the conditions under which synchronization and desynchronization occur. Basic binding through synchronization and segmentation through desynchronization will be demonstrated first, followed by an analysis of how these processes are affected by the relative amounts of inhibition and excitation, spatial extent of the connections, synaptic decay rates, noise levels, neuron population sizes, and the length of the absolute refractory period. A special focus is on robustness against noise, which is crucial for these processes to function in biological networks.

12.1 Method

Synchronization is important for binding together populations of neurons that represent input features of the same coherent object. Desynchronization, on the other hand, signals that the input features belong to different objects. In this chapter, these processes will be illustrated using one-dimensional networks connected one-to-one to input and output. Such networks are sufficient for testing the various factors governing synchronization, and they are also easy to visualize in two dimensions because the activities can be plotted over time.

Unless stated otherwise, in all experiments the input neurons spike at every time step, the membrane potential of each neuron is initialized to uniformly random distribution within [0..1], the afferent weights are fixed at 1.0, the lateral excitatory weights are fixed at 1.0/nE, and the lateral inhibitory weights are fixed at 1.0/nI, where nE and nI represent the number of excitatory and inhibitory lateral connections (the rest

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of the parameters are listed in Appendix D.3). In order to study synchronization behavior, the following parameters are systematically varied in the experiments: (1) lateral excitatory and inhibitory connection patterns and radii rE and rI, (2) their contributions γE and γI, (3) their synaptic decay rates λE and λI, (4) the size and pattern of the afferent input, (5) the degree of noise in the membrane potential, and

(6)the duration tr of the absolute refractory period.

For example, Figure 12.1 demonstrates synchronization behavior of a five-neuron

network with full lateral connectivity under two conditions. In Figure 12.1a, all lat-

eral connections were excitatory, whereas in Figure 12.1b they were inhibitory; all the other parameters were the same (γE = 0.01, γI = 0.001, λE = 5.0, λI = 8.0, full

input activation, no noise in the membrane potential, no absolute refractory period). The simulation time is on the x-axis, and the membrane potential of each neuron is plotted over time in two ways: as a continuous y value in the top plot, and as a grayscale value in the bottom plot. In Figure 12.1a, due to lateral excitation, the neurons reinforce each other’s activity and soon start firing at the same time. In contrast, in Figure 12.1b, the lateral inhibition establishes competition between the neurons, and after a short time they each fire at different times.

Such complete and exclusive lateral excitation vs. inhibition results in extremely strong synchronization and desynchronization, and constitutes a good example of these behaviors. The PGLISSOM models include both excitation and inhibition, and the synchronization behavior depends on many other factors. In the following sections, these factors will be systematically varied, and gray-scale plots similar to those in Figure 12.1 will be used to illustrate the resulting behavior.

12.2 Binding Through Synchronization

In this section, two main factors that affect the quality of synchronized representations will be analyzed: the synaptic decay rate and the extent of lateral connections. Decay allows controlling how accurately the spiking events need to be timed. On the other hand, lateral connections are necessary to coordinate the firing of neurons, and their extent determines how large areas can be synchronized.

12.2.1 Effect of Synaptic Decay Rate

Previous models of spiking neurons have either adapted or selected the axonal delays to regulate synchronization behavior (Eurich, Pawelzik, Ernst, Thiel, Cowan, and Milton 2000; Gerstner 1998a; Horn and Opher 1998; Nischwitz and Glunder¨ 1995; Tversky and Miikkulainen 2002). The biological basis for such delay tuning is unclear: Although e.g. axonal morphology (length, thickness, and myelination) can change over time (Eurich, Pawelzik, Ernst, Cowan, and Milton 1999), the fast and accurate delay tuning needed in the above models may not be easy to achieve in this way (Stevens, Tanner, and Fields 1998).

An alternative to delay adaptation is changing the decay rate of the PSP. Decay may be easier to alter in biological neurons since ion channels can be added

Fig. 12.1. Synchronized and desynchronized modes of firing. A network of five neurons is connected only with excitatory lateral connections in (a) and only with inhibitory lateral connections in (b), and exhibits synchronized and desynchronized firing as a result. Simulation time is shown on the x-axis, and the membrane potential for each neuron is displayed in two ways: along the y-axis in a voltage trace plot on the top, and in gray scale from white to black (low to high) in the bottom. Each row in each plot represents a different neuron, with its index identified on both sides of the plot (1 to 5, from bottom to top). The black vertical bar on the left shows which neurons are activated by afferent input (black means on and white means off); in these two examples, all input neurons were activated. To the right of the scatterplot, two vertical bars illustrate the excitatory (left) and inhibitory (right) lateral connection ranges of one sample neuron in the network; all neurons had identical connections in these two examples. Black indicates that a connection to the neuron exists in that row, and white that it does not. The same plotting convention will be used throughout this chapter. With excitatory lateral connections in (a), all spikes (peaks) start to become vertically aligned around iteration 21, showing that all the neurons are firing at the same time. In contrast, with inhibitory lateral connections in (b), the neurons all fire at different times. An animated demo of these examples can be seen at http://computationalmaps.org.

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or removed to tune the leakage of currents through the cell membrane. The number and distribution of ion channels can change through various mechanisms, including activity-dependent gene expression and activity-dependent modulation of assembled ion channels (Desai, Rutherford, and Turrigiano 1999; Nowak and Bullier 1997; see Abbott and Marder 1995 for a review). Synaptic decay has been utilized in computational models before (Eckhorn et al. 1990; Reitboeck et al. 1993), but the influence of different levels of decay on synchronization has not been fully tested.

PGLISSOM allows the effects of synaptic decay rates to be analyzed systematically. The λ values in Equation 11.1 can be varied independently for different types of connections (excitatory or inhibitory). In such simulations, the decay rate was found to influence synchronization strongly. By adjusting λ, it is possible to get both synchronized and desynchronized behavior with both types of connections.

Four separate experiments were conducted: (1) excitatory lateral connections with slow decay (λE = 0.1), (2) inhibitory lateral connections with slow decay (λI = 0.1), (3) excitatory lateral connections with fast decay (λE = 1.0), and (4) inhibitory lateral connections with fast decay (λI = 1.0). Except for the decay rate λ and the connection type, all other parameters were the same in the four experiments, including γI = γE = 0.01. In each experiment, a one-dimensional network of 30 neurons with full lateral connections was simulated for 500 iterations.

The results are shown in Figure 12.2. Two conditions, excitatory connections with fast decay and inhibitory connections with slow decay, result in synchrony. In contrast, excitatory connections with slow decay and inhibitory connections with fast decay result in desynchrony. This is an interesting result, since excitation does not always guarantee synchronization, and inhibition does not always guarantee desynchronization.

Nischwitz and Glunder¨ (1995) showed that a similar result is obtained by varying the degree of delay among integrate-and-fire neurons connected via excitatory or inhibitory connections. A short delay with excitatory connections and long delay with inhibitory connections caused the neurons to synchronize, and in the opposite case to desynchronize. The current result indicates that synaptic decay adaptation, which appears more plausible than delay adaptation, can also control synchronization.

It is important to note that although synchronization can be achieved through slowly decaying inhibitory connections, excitatory connections are more likely to be responsible for coherent oscillation in the cortex. As will be discussed in Section 16.3.3, coherent oscillations have been found mostly in the superficial layers of the cortex, especially in layers 2/3. The long-range connections in those layers are mostly excitatory, suggesting that the synchronization is established through excitation.

In summary, synchronization can be regulated effectively by adjusting the synaptic decay rate. Because possible adjustment mechanisms are known to exist in biology, synaptic decay adaptation is an attractive alternative to models based on delay modulation.

(d) Inhibition with λI = 1.0

Fig. 12.2. Effect of connection type and decay rate on synchronization. Thirty neurons with full lateral connections, either excitatory or inhibitory, were simulated for 500 iterations (see Figure 12.1 for plotting conventions). Four experiments were conducted where the type of the lateral connections (excitatory or inhibitory) and the synaptic decay rates (λ) were altered. All other parameters were the same for all four cases. (a) Excitatory connections with slow decay (λE = 0.1) result in desynchronized activity. (b) Excitatory connections with fast decay (λE = 1.0) result in synchronized activity. (c) Inhibitory connections with slow decay (λI = 0.1) result in synchronized activity. (d) Inhibitory connections with fast decay (λI = 1.0) result in synchronized activity. Note that in the two synchronized cases (b) and (c), the firing rate is higher in (b): The input activity to the neuron g(t) approaches the threshold faster because of the excitatory lateral input. The results show that synchronization behavior can vary greatly even for the same connection type if the synaptic decay rate differs.

12.2.2 Effect of Connection Range

In the second test, the goal was to determine whether local excitatory connections can synchronize a global population. Inhibitory lateral connections were excluded to simplify the simulations. Thirty neurons with varying degrees of excitatory lateral connection radii were simulated for 500 iterations. Five separate experiments were conducted with excitatory connection radii of 30, 10, 5, 2, and 0. Other simulation conditions were the same as in Section 12.1, except γE = 0.01 and λE = 5.0 so that the network would also synchronize under the smaller radii.

The results are shown in Figure 12.3. Global synchronization is achieved not only in the fully connected network as before (radius 30), but also in locally connected networks, down to a radius of 5. These results demonstrate that synchronization can propagate through locally connected neurons, which is consistent with other coherent oscillation models with local connections (Campbell et al. 1999; Terman and Wang 1995; Wang 1995, 1996). They show that synchronization may work as a basis for

(e) rE = 0

Fig. 12.3. Effect of excitatory connection range on synchronization. A network of 30 neurons with varying extent of lateral excitatory connections was simulated for 500 iterations. Synchronization occurs through the excitatory connections even though the connections did not cover the whole network. From (a) to (e), the lateral excitatory connection radius rE was reduced from 30 (i.e. full connectivity) to 10, 5, 2, and 0 (the bars at right depict the connections of the neuron in row 15). All other parameters were the same as before. Synchronization starts to break once the radius reaches 2, but for a fairly local connection radius (e.g. 5), global synchronization is maintained. As expected, with no excitatory connections (e), the initial random order of spikes is maintained throughout the simulation. Global synchrony can therefore be established with local connections.

transitive grouping: If A and B are grouped together and B and C are grouped together, then A and C are perceptually grouped together (Geisler et al. 2001; Geisler and Super 2000).

In summary, fully connected networks synchronize well, but it is not necessary to have full connectivity to achieve global synchrony. Global synchronization through local connections in the PGLISSOM model may be a possible mechanism for transitive perceptual grouping.