Model

Metric connectivity can produce localised states in 2D

In what follows, we show an example of a retrieval bump in a 2D recurrent network with metrically organised connectivity. We consider a two dimensional network containing N = 4900 neurons in total. The neurons are arranged on a regular lattice with 70 neurons on each side and distance l between neighbouring sites. The connections between neurons have a metric structure: a neuron in position r_i is connected to a neuron in position r_j with probability(13)

In the simulations reported here the width of the connectivity, σ, is set to 7.5l. Since l is the distance between two adjacent neurons, this means that the probability that two adjacent neurons are connected to each other is ∼0.7. Experimental data estimate this probability to be 0.5–0.8 [30]. The gain of all neurons in the simulations reported in this section is set to a background level g = 0.5.

At the beginning of the simulation a 15×15 square centred on the neuron in position (58,58) is chosen. The activity of neurons inside this square are initialised to their activity in the first stored pattern while the activity of other neurons are set to zero, that is in the beginning of simulation if r_i is in the square and ν_i = 0 if r_i is outside it. In this way at the beginning of the simulation the dot product overlap with the first pattern and the others have the following values

Fig. 1A shows the local overlap with the cued pattern (μ = 1) at the beginning of a simulation. The local overlap (Eq. (8)) with the cued pattern after 200 synchronous updates is shown in Fig. 1B and the distribution of activity {ν_i}, also after 200 time steps, is shown in Fig. 1C. We see that the activity of the network is concentrated on a part of the 2D network, and so is the local overlap. The important point is that this final pattern of activity has a high dot product overlap with the cued pattern but not with other stored patterns, i.e.

Thus by calculating these dot products, or equivalently calculating the sum of the local overlaps m_i^μ over i, in the end of the simulation we can say which pattern was presented, i.e. in this example the first pattern.

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Figure 1. Short-range connectivity allows localised attractor states, but activity drifts away from the position of the cue.

The network is comprised of N = 70×70 = 4900 neurons, each connected to C = 0.05N other neurons, chosen using the Gaussian distribution in Eq. (13). The gain of all neurons is fixed to a uniform background value g = 0.5 and we have p = 5 and a = 0.2. (A) The local overlap with the cued memory pattern at the beginning of a simulation. In the beginning of the simulation a 15×15 square centred on the neuron in position (58,58) is chosen. The activity of neurons inside this square is set to their corresponding activities in the first pattern, and the rest of the neurons are silent. (B) The local overlap after 200 synchronous time steps. (C) The activity distribution across the network after 200 synchronous time steps. (D) The distribution of the activity of the neurons in an identical network operating with identical cue and parameters, except that recurrent connectivity is random, i.e. each neuron is connected to other neurons with probability C/N = 0.05. Whereas the distribution in (C) is localised in space, in (D) the activity is distributed across the network. Note that in both cases (the network with metric connectivity and the one with random connectivity), the cued pattern has been successfully retrieved. This can be seen by measuring the overlap between the network activity and the cued pattern and comparing it with its overlap with non-cued ones (see the text): after 200 time steps, the overlaps, m^μ (that is the sum of the local overlap m_i^μ over i; see Eq. (9)), are m¹(t = 200)≈0.8 and and m^μ≠1(t = 200)≈0, indicating the retrieval of the cued pattern. Periodic boundary conditions are assumed here and elsewhere in this paper.

https://doi.org/10.1371/journal.pcbi.1000012.g001

The spatial distribution of activity would have been different (Fig. 1D), if instead of the probability distribution in Eq. (13), we had used a uniform distribution

In this case, too, by cueing one of the patterns, as we did for the metrically organised network, after 200 time steps, we have m¹(t = 200)≈0.8 and m^μ≠1(t = 200)≈0, thus indicating retrieval of the pattern. The difference between the two connectivity models emerges, however, in the final distribution of activity. Whereas in Fig. 1D the activity is distributed uniformly across the network (at a gross spatial scale, since at a fine scale individual units are activated in relation to their selectivity for the cued pattern), in Fig. 1C the activity is localised over a portion of the 2D network. So, metric recurrent connections, as predicted by the mathematical analyses of attractor states and as confirmed by many other simulations, allow activity to stabilise in spatially modulated distributions.

Even though Fig. 1 shows the possibility of localised retrieval in the network with the Gaussian connectivity in Eq. (13), a critical observation is that in Fig. 1B the final local overlap is in a different position than the initial cue (Fig. 1A). The trajectory that the peak of the local overlap follows during the retrieval process is shown in Fig. 2. The green square shows the peak at the beginning of the simulation, before any updates take place (Fig. 1A), and the red circle shows the peak after 200 time steps (Fig. 1B). It is clear that, during retrieval, the “bump” of activity drifts away from its initial position. This raises the question addressed in this paper, of whether where information in the cue can be preserved by spatially modulated attractor states.

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Figure 2. During retrieval, the peak of the local overlap drifts away from its initial position.

This figure shows the trajectory that the peak of the local overlap follows during the simulation reported in Fig 1. The green square represented the position of the peak of the local overlap at the beginning of the simulation (Fig. 1A) and the red circle represents the position of the peak after 200 time steps (Fig. 1B). Over the first ∼40 time steps the local overlaps drifts until it settles into an equilibrium state.

https://doi.org/10.1371/journal.pcbi.1000012.g002

Metric connectivity alone is not sufficient to preserve where information

Can we code the position of an object by the position of the peak of the retrieval bump? The answer to this question depends on whether the retrieval process can end with the peak of the bump on the intended position. We first examine whether the position of the cue (which can be thought of as the position of an object in the retina) determines the positions of the retrieval bump. If the retrieval bump appears at the same position as (or is uniquely determined by) the centre of the cue, it is possible to read the activity of the network and simply decode both what information, that is, which cue has been presented (the pattern with the highest overlap with final activity), and, in addition, where it has been presented: object position is simply coded by the position of the centre of the bump.

Continuous attractors are fragmented by superimposed memories

To examine the relation between the position of the initial cue and the final position of the retrieval bump, we ran simulations in which the position of the initial cue was systematically changed across the network and the distance between the position of the retrieval bump and the position of the initial cue was measured. In Fig. 3, we summarise the results from simulating a network of 70×70 neurons with the Gaussian connectivity pattern Eq. (13) with σ = 7.5l, as used in Fig. 1. At the beginning of each trial, the first pattern was cued by initialising the activity of neurons in the following way: , if neuron i was within a 15×15 square, whose centre was varied, across trials, over the entire network; while ν_i = 0, if neuron i was outside the square. The activity of all neurons was then synchronously updated for 200 time steps and the local overlap with each pattern was monitored. Fig. 3A shows that the position of the bump at the end of each trial (red circles) does not match the peak of the local overlap with pattern 1 at the beginning of the trial (green squares). The bump drifts away from its initial positions, and stabilises on one of, in this particular case, 4 final positions. This small number of final stable positions indicates that one cannot decode from the final position of the retrieval bump where the cue was located, at least not with high accuracy. In fact, by looking at the final position of the bump, one might say whether the initial position of the cue was among the 23 initial positions that converge to the upper left red circle or among the 10 initial positions that converge to the lower right red circle, but nothing more.

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Figure 3. Steady-state activity bumps are only weakly correlated with cue position, in the constant gain model.

This graph illustrates how the position of the cue affects the position of the bump after retrieval is completed. Cue position (the centre of a 15×15 square window) was chosen on a 7×7 grid with grid spacing 10, whose vertices are shown in (A) as green squares. The red circles show the position of the peak of the local overlap, as it drifted away from the linked green square, over 200 time iterations of the simulated retrieval process. All runs in this example resulted in successful retrieval, i.e. the overlap between the final activity of the network and the cued pattern was higher than with other patterns. (B) The distribution of the distances between the centre of the cue and the centre of the local overlap after 200 time steps. In the legend, d is the distance averaged across all trials. The neuronal gain in this network is g = 0.5 and the number of stored patterns is p = 5.

https://doi.org/10.1371/journal.pcbi.1000012.g003

Relation to continuous attractors without stored memory

The small number of final stable positions of the bump resembles what has been noticed before in recurrent networks with distance dependent weights between neurons but without stored memory patterns. In such models the synaptic weight between two neurons is generally taken to be excitatory at short distances while inhibitory at long distances [51]–[58]. The distance between two neurons in these models can be anatomical distance, or distance, in the feature space, between the features that the neurons are selective for. Models of this type have been used to conceptualise how local networks of orientation selective neurons in visual cortex [52], head-direction neurons [53], location selective neurons in prefrontal cortex [54] and hippocampal neurons [57],[58] can maintain selectivity after the external input has been removed. Studies on rate based models [51]–[53] as well as networks with spiking neurons [54]–[56] show that, under very mild conditions, the stable activity profile of these networks is of the form of a localised “bump”. If the network is strictly homogeneous, the bump can potentially exist anywhere on the network, and it can be smoothly moved from one position to the other. Any small inhomogeneity in the network, however, fractures the continuum of solutions, which therefore represents an ideal limit case, and stable bumps are allowed only at a number of discrete positions [53], [57], [59]–[61]. Coming back to the associative memory network with metric connectivity, it is clear that inhomogeneity is an unavoidable part of its overlaid memory structure. Synaptic weights are required to be different from each other in such a network, to support the retrieval of memory patterns, a situation where a neuron can be active while its nearest neighbour is inactive. As a result, a retrieval bump in our model cannot be maintained at any arbitrary position on the network.

Scaling of the number of stable positions

Even though the final position of the bump cannot accurately tell where the cue was initiated, it may still be able to code for a large number of positions in a network with realistic size. This happens if the number of final stable positions increases with the size of the network. To examine this relation, we scaled up the simulated network. The result of such scaling analysis is reported in Fig. 4, which shows the number of final positions resulting with different network sizes, while keeping the number of connections and the width of connectivity constant. One sees a roughly linear increase in the number of stable bump positions.

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Figure 4. Scaling of the number of stable positions of the bump with the size of the network.

The scaling is done for p = 1 (black line) and p = 20 (red and blue lines). For p = 20, the blue line represents the number of stable positions when both successful and unsuccessful runs are averaged and the red line represents this number when only the successful ones are counted. For each network, of size N, one of the hundred positions on a 10×10 regular lattice was chosen in each trial. The activity of neurons inside a square that covers 0.04×N neurons and whose centre was on the chosen position, was set to their activity in the stored pattern. For p = 1 the activity of neurons outside this square were set to zero. For p = 20, a fraction of 10% of neurons outside the square also had their activity set to their activity in the first pattern, while the rest has zero activity level. The difference between the ways we initialised the retrieval for p = 1 and p = 20 was introduced to ensure a higher quality of initial cue for p = 20; this would be needed for having a reasonable number of successful runs. The width of the connectivity was set to 7.5 and the average number of connections per neuron was always 245. The number of final positions of the bump of activity after 200 time steps was then counted. We then plotted the average and standard error of this number over the hundred positions of the initial cue and 10 realizations of the network and of the stored pattern, versus N.

https://doi.org/10.1371/journal.pcbi.1000012.g004

The approximately linear scaling of the number of final positions with network size indicates that a large number of positions can be represented in realistically large networks, but not any arbitrary position: with our regular 2D lattices and our parameters, the number of stable bump positions is about one thousand times smaller than the number of lattice nodes. Furthermore, the few stable positions of the retrieval bump are different for different patterns (data not shown). This makes the representation of position dependent on object identity and thus hard to decode. We ask, therefore, whether it is indeed possible to stabilise bumps at any arbitrary position. This is discussed in the following sections.

Gain modulation and bump localisation

In this section we show that the bump of activity can be stabilised on an arbitrary position on the network if neurons around that position have a slightly higher linear gain than the rest of the neurons. This increase in the linear gain applies to all neurons in that area in the same manner, whether they are selective for the cued pattern or not; that is, it is not pattern selective and solely reflects object position. This local gain modulation can be triggered by an attentional mechanism that modulates the responsiveness of neurons in the part of the network which corresponds to the position of the object. It could also be produced by the pattern itself: when the cue to initialise retrieval is given to the network, the mean activity of the part of the network that receives the cue would be higher than the rest of network. This could trigger changes in the gain of the neurons that may last for several seconds [62]–[64]. We leave discussing the sources of the gain modulation to the Discussion section and first answer the following questions. Can such localised gain modulation stabilise the bump at any desired position and, if so, how strong should it be? How does localised gain modulation affect pattern retrieval?

Modulating single neuron gain can stabilise the bump

Suppose that a non-pattern-selective signal changes the gain of those neurons which correspond to the position of the object in the visual scene. The effect of such gain modulation is shown in Fig. 5.

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Figure 5. Local gain modulation largely fixes the position of the bump.

The panels summarise the results of simulations conducted as for Fig. 3 except for two factors. First, the pattern-selective cue is not localised, i.e. the local overlap at the beginning of each simulation is a uniform function across the network; hence, the distribution of the peak of the local overlap in the beginning of the simulation is not shown: there are no significant peaks. Second, neurons inside the 15×15 square centred around each green square in (A) have a gain factor g which is 1.5 times larger than the rest of the network, and in (C), 3 times larger. (B) and (D) report the distributions of distances between the centre of the gain modulated square and the peak of the final local overlap corresponding to simulations in (A) and (C), respectively. Red circles in (A) and (C), and red portions of the bars in (B) and (D) correspond to successful runs (defined as runs in which the overlap with the cued pattern, after 200 time steps, is higher than the overlap with any other pattern) and black circles and black portions of the bars represent unsuccessful ones. mean(d) and std(d) are the mean and the standard deviation of the distances averaged over successful runs and mean(d^*) and std(d^*) averaged over unsuccessful runs.

https://doi.org/10.1371/journal.pcbi.1000012.g005

In the simulations of Fig. 5, the activity of ≈4.6% of the neurons, randomly distributed across the network, are initially set to their activity in the first pattern, while the rest are silent (note that the quality of the cue is then the same as what we used in the simulations of Fig. 3, but now the cue is not localised). The localised gain modulation is incorporated into the simulations by first choosing, at each trial, a square box at a different position over the network. The linear gain of neurons inside the square is then increased by a factor of β relative to that of the other neurons in the network. The position of the centre of the high gain square box is in fact chosen in exactly the same way as we chose the centre of the cue in Fig. 3, i.e. at the nodes of a regular lattice, shown as green squares in Fig. 5A and Fig. 5C.

The result of such change in the spatial distribution of the gain is evident for β = 1.5 (Fig. 5A, 5B) and even more for β = 3 (Fig. 5C, 5D). Even though the pattern-selective cue does not contain spatial information, a spatially selective increase in the linear gain of the neurons in a restricted region of the network helps localising the bump in that region. Notably, as shown in Fig. 5D, the distance that the peak in the local overlap drifts from the initial focus is minimal, particularly for successful trials (red circles) (d), whereas averaging across unsuccessful runs (black circles) (d*) substantially increases the drift, as if jumping to the wrong basin of attraction in the space of patterns facilitates similar jumps in physical space.

It should be noted that while in Fig. 3 the localised cue had been removed after initialising the activity, in the results shown in Fig. 5 the change in gain is maintained throughout the simulation. It is true that keeping the localised cue would have helped localising the bump at the right position, without gain modulation, but the essential difference between the two mechanisms should be appreciated: the change in gain is independent of the memory pattern to be retrieved and could thus be produced by a mere spatial signal, with little specific information content besides spatial position itself. The pattern-selective cue, instead, can be thought to commit the informational resources (e.g., the channel capacity [65]) of the ventral visual form processing stream, and it makes sense to hypothesise that it should be removed as soon as possible, to make room for the analysis of other objects by the same pathway.

Even though increasing the gain in a spatially restricted part of the network stabilised the final bump there, there is a disadvantage with this strategy: by using such non-uniform gain, the number of successful runs decreases. Remember that the quality of the cue used in Fig. 5 is the same as the one in Fig. 3, however, there were no unsuccessful runs in Fig. 3 and Fig. 5A, whereas there are 12 unsuccessful runs in Fig. 5C (shown by black circles): better preservation of spatial information (higher gain modulation) is accompanied by, in this example, a higher number of unsuccessful runs. This suggests that preservation of spatial information through gain modulation affects the retrieval of the pattern. In Fig. 5 the effect is negative, an interference, but as we show below it can also be a positive effect. In the following sections, we quantify this interaction using information theory and demonstrate efficient ways to minimise the negative interaction.

The relation between I_what and I_where: An information theoretic analysis

In order to quantify the interaction between what and where information, we use Shannon information theory. We estimate the amount of information that the activity of the network, after retrieval, represents about what and where. We do this for different degrees of gain modulation, levels of the average gain, number of stored patterns and also different ways of presenting the cue. This provides us with a quantitative picture of the relation between what and where information.

We denote by I_what and I_where, the amount of information about what and where, respectively. To compute I_what, we look at network activity after 200 times steps and compute its overlap with all stored pattens (Eq. (9)). The pattern with the highest overlap is considered as retrieved and I_what measures how much information knowing this retrieved pattern gives us about which pattern was presented. I_where, on the other hand, is the information between the position of the bump of activity after retrieval and the centre of the gain modulated area (we put I_where = 0 when there is no gain modulation; see section “Continuous attractors are fragmented by superimposed memories”). For exact definitions and details about how we compute I_what and I_where from the simulations see section “Mutual information measures” in the Materials and Methods.

To start with, we consider a network (with the architecture used before) that has stored p patterns and assume that in the beginning of the simulations a cue similar to one of the patterns is presented (the exact cue presentation is described in the three Conditions below). All neurons have a background gain of g. During recall, either the gain of all neurons is kept equal to g, which is the case of uniform gain, or the localised gain modulation mechanism is turned on. In the latter case the gain of the neurons inside a 15×15 square whose centre is on one of 49 preassigned positions on the network is boosted to βg. Different values of β are chosen in different simulations. In each run, one of the patterns is chosen as a cue and one of the 49 positions is chosen as the centre of the high gain region. As in the previous sections, the centre of the squares surrounding the high gain region is chosen from one of the 49 nodes of a 7×7 regular lattice covering the entire 2D network. Each pattern and each of the 49 positions for the high gain region is used exactly once.

High gain regime

We first calculate I_what and I_where for a network with the global gain chosen to be g = 0.5. We do this for the case of uniform gain (all neurons have the same gain, thus equal to the background gain g ), three degrees of gain modulation, with β = 1.5,2 and 3 , and three values of p = 5,10 and 15. We consider three alternative ways in which the cue can be presented to the network. These cueing conditions and the resulting I_what−I_where relation are described below.

Condition 1: Complete cue.

In this condition, in the beginning of the simulation, the activity of all neurons in the network is set to their activity in the pattern to be retrieved. Fig. 6A shows how I_where covaries with I_what for three values of p = 5 (full line), p = 10 (dashed line) and p = 15 (dashed-dotted line) when the initial cue was complete. On each curve different points correspond to different degrees of gain modulation. There is a clear trade-off between I_what and I_where. The maximum amount of what information and the minimum amount of where information occurs when there is no gain modulation. As we increase the degree of localised gain modulation, I_what decreases while I_where increases. Moreover, the value of p for which maximum I_what can be retrieved depends on the degree of gain modulation and thus on I_where. In other words, for any given I_where, there is a critical p for which I_what is maximised. In the absence of gain modulation, I_where = 0 and the maximum of I_what is obtained for a certain storage load 10≤p≤15, whereas, as the degree of gain modulation increases, the number of patterns resulting in maximal identity information decreases.

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Figure 6. The trade-off between what and where information for different cueing conditions, different numbers of stored patterns and different levels of gain modulation, as discussed in the text.

(A) Condition 1: complete cue. (B) Condition 2: 4.6% of the neurons are randomly chosen and their activity is initialised to their activity in the cued pattern, while the rest has zero activity. (C) Condition 3: similar to Condition 2 but the neurons that receive the cue are spatially co-localised with gain modulation. For each cueing condition, the trade-off is shown for p = 5 (full line), p = 10 (dashed line) and p = 0.5 (dashed-dotted line). For each value of p, three different degrees of localised gain modulation, β = 1.5 (square), β = 2 (circle) and β = 3 (diamond), were tested in addition to the uniform gain of g = 0.5 (star).

https://doi.org/10.1371/journal.pcbi.1000012.g006

In this condition, we had provided the network with a perfect cue. It is important to see whether a similar trade-off occurs in the more interesting condition in which the cue is incomplete, so that the network has not merely to maintain but also to actually retrieve identity information.

Low gain regime

In the previous section, the background gain was g = 0.5. Without gain modulation, the network could reach high I_what values, sometimes retrieving all stored patterns, even from a very small initial cue. With gain modulation, I_where increased but I_what decreased. Here, we show that when the background gain is low, the interaction can be reversed, that is, gain modulation can actually increase both I_what and I_where. We set the background gain to g = 0.25. As shown in Fig. 7, for the case of complete cue (as in Condition 1 above) even without gain modulation I_what is very small. When incorporating a gain modulation mechanism, however, the amount of what information maintained by the network increases, together with the amount of where information. In section “Low gain regime versus high gain regime” (see Materials and Methods), we discuss why the relation between I_what and I_where is different in the low gain and high gain regimes. Intuitively, the reason is as follows. Successful retrieval occurs only when the gain of the neurons that support the retrieved pattern is between a minimum g_min and a maximum g_max. In the low gain regime, the level of background gain is below g_min and the network cannot retrieve the patterns. When the gain is increased in part of the network, then it may enter the range [g_min, g_max], allowing for retrieval to occur. At the same time, since that region has a higher gain, the retrieval bump does not drift away. When the background gain is high, instead, gain modulation stabilises the bump in the gain modulated area. This is accompanied, however, by a decrease in the size of the bump. The reason is that the higher neuronal gain increases the firing rate of neurons inside the bump (the peak of the bump is higher) and, to comply with the constant mean activity condition (Eq. (3)), this increase in the peak activity is accompanied by a decrease in the spatial extent of the bump. Therefore, fewer connections are involved in retrieving the pattern and I_what decreases. As expected from this argument, increasing β too much even in the low gain regime should decrease I_what. This can be seen in Fig. 7 for β = 5 and p = 10.

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Figure 7. The relation between I_where and I_what in the network with low gain (g = 0.25), with p = 5 and p = 10.

When there is no gain modulation, the network fails to retrieve any of the stored patterns (star). When localised gain modulation is incorporated, both I_what and I_where increase. β = 1.5 (square), β = 2 (circle), β = 3 (diamond) and β = 5 (right pointing triangle). The corresponding relation between I_where and I_what for g = 0.5 is replotted here from Fig. 5A for comparison.

https://doi.org/10.1371/journal.pcbi.1000012.g007

Multiple bumps

When a retrieval bump is localised on a particular position, one can in principle use the rest of the network to retrieve other patterns, in the form of additional bumps of activity. If they can coexist with the first bump, the network would then be able to represent the position and identity of multiple objects simultaneously, without encountering the problem of binding. In random networks with no metric connectivity nor localised retrieval, retrieving multiple patterns is indeed possible, at very low storage loads [1],[45],[66]; in these networks, however, it is not possible to represent the position of the objects, which has to be represented elsewhere. If the what and where of multiple objects are represented in different networks, a binding problem arises. The localised retrieval process described here does not suffer from this problem. It is then important to assess the conditions which make it possible to stabilise (at least) two retrieval bumps simultaneously.

Assume that a pattern is retrieved and, using localised gain modulation, the bump of activity is stabilised on a desired position. A second cue may then be presented to the network at another position. Even though most of the connections to each neuron in the network come from nearby neurons, the second pattern would still affect the first retrieval bump, because of the global inhibition in the simplest version of our model, as inhibition is taken to regulate a common threshold, such that the mean activity of the network is constant (Eq. (3)). This introduces interactions between distal neurons, which are not directly connected by excitatory synapses, and such interactions are generally disruptive. A simple way to reduce such interaction is to assume that when the local mean activity in part of the network exceeds some limit value, the threshold is raised but only locally, regardless of the activity of neurons outside that region. The local threshold may also be regulated downward, to facilitate the emergence of a retrieved pattern in a region which would otherwise be kept at too low a mean activity level. With such additional provisions, multiple bumps can be formed and stabilised, as shown in the example in Fig. 8.

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Figure 8. Formation of multiple bumps.

In the beginning of the simulation, pattern 1 is presented to all units in a 15×15 square whose lower left corner is at node (1,1); that is, for each neuron i inside the square the activity is set to η_i¹, and for those outside to zero. While simulating the network, background threshold values are set to regulate the mean network activity to a fixed level equal to 0.2. If mean activity inside a 30×30 square centred on the cue centre exceeds 1.0, the threshold of neurons inside this square will be regulated to keep its mean activity equal to 1.08, and neurons outside it will be assigned a high threshold. In the second phase of the simulation, in the right column, a second pattern is also presented to all units in a 15×15 square whose lower left corner is at node (36,36), accompanied by a local threshold decrease to facilitate the pattern “holding on”. The threshold is then regulated in the same way as the first pattern. (A) The distribution of activity, (B) the local overlap with the first pattern (cued in the beginning of the first phase) and (C) the local overlap with the second pattern (cued in the beginning of the second phase), all 100 time steps after the presentation of the first pattern. (D), (E), and (F) are the same quantities as (A), (B) and (C) but 100 time steps after the presentation of the second pattern.

https://doi.org/10.1371/journal.pcbi.1000012.g008

Comparison with storing attractors associated to object-position pairs

In our model, independent attractors are set up in a local cortical network only for object identity, as position invariant representations; but they can be accessed in a spatially focused mode, leading to position dependent activity. Associating a single representation to an object, which is then modulated by position, is a particular case of what in cognitive neuroscience parlance is sometimes referred to as type (e.g. table) and token (particular instance of a type: e.g. a table in a particular position) [76],[77]. In the language of our model, the type is the original pattern of activity associated to an object and the token is the bumpy pattern that is localised in a particular position. An alternative mechanism is to store attractors associated to object-position pairs, that is storing a neural activity pattern for each token [78]. In this way, when a particular object is presented in a particular position, the attractor corresponding to the object-position pair would be activated, and could remain activated even after the object has been removed from the scene. The problem is that models which hypothesise distinct, discrete attractors for each possible object-position combination would certainly violate any conceivable storage capacity limit, because of the infinitely large number of possible positions of an object [79]. Furthermore, there is a major difference between the nature of what and where information, which makes attractors associated to object-position pairs unlikely: as opposed to what information, to which the brain can contribute from the information that it has previously stored, the brain does not usually retrieve positional information from memory, but rather has to maintain it as well as it can. Thus, it would seem rather implausible that the brain uses its storage capacity, arguably its most precious resource [29], to store something that it does not have to retrieve.

The difference between these two mechanisms is directly reflected in the storage capacity required for object-position attractors, in order to represent the same amount of information as the mechanisms studied here does through spatial modulation. Representing 6 bits of I_where and 2 bits of I_what (corresponding to the black diamonds in Fig. 7) would require the storage of 2⁽⁶⁺²⁾ = 256 distinct object-position attractors. This is ca. 2.5 times beyond the number of attractors that a randomly connected network, with the same number of connections per neuron and the same mean activity level as what we used, could store [32]. This exorbitant requirement is due to effectively committing storage space separately to each pair, instead of using the physical arrangement of neurons in the tissue to represent I_where. Analytical results valid in the limit of large networks and optimal storage further support this conclusion, as we show in section “Comparision with other models” in the Materials and Methods. There, we also show that the difference in the efficiency of the two models will be even more pronounced for larger networks. There is, of course, a price to pay: the addition of a gain modulation mechanism to stabilise the position of the bump. In what follows, we discuss the possible physiological substrates of this gain modulation.

Possible sources of gain modulation

In our model, localised gain modulation is crucial for maintaining where information as what information is being retrieved, and for maintaining both what and where information after the retrieval process is completed. When an object is presented as a stimulus, a signal should trigger an increase in the gain of neurons in an appropriate part of the network. Such higher gain should then be maintained by the same or a distinct mechanism during retrieval and thereafter, when the object is not present anymore but information about it has to be used (e.g. during the delay period of a delay-match-to-sample task).

What mechanisms can trigger the neuronal gain? In vivo studies show that increasing the activity of a local cortical network increases the gain of its neurons [80]–[82]. Therefore, any mechanism that increases the mean activity of a part of the network could be used for triggering the gain modulation. One such source of increase in the activity is the cue itself. This requires that the pattern selective cue retains some spatial information; a scenario which we have shown to be particularly effective in minimising the trade-off between what and where information (see Fig. 6C). Although this mechanism would be effective in this sense, it is doubtful whether it could be the only source of gain modulation in high level visual cortices. This is because experimental studies show that the position of the peak of the activity in visual cortical areas during visual stimulation is strongly correlated with the categorical properties of the stimulus and exhibits a weaker level of retinotopy [83]–[85] (see also the following section “Storing patterns with spatial prefrence”). The situation may be different in more advanced cortical areas, such as PFC, in which such categorical maps have not been reported.

Another possible source for increasing the gain is attentional signals. In this case the increase in the activity level required for gain modulation is induced by the attentional signal and the position of the bump corresponds to the position of the attentional spotlight. There are several reasons that make attention a likely source of activity localisation through gain modulation. fMRI studies on human subjects show that the retinotopic representation of the position of an attended object in visual cortices show increased activity [86]–[94]. Evidence from monkey neurophysiology also supports the idea that the attentional spotlight increases the gain of neurons inside the spotlight [95]–[100]. Furthermore, many studies in cognitive neuropsychology suggest that spatial, focal attention is critical to allow the binding of what and where information [77],[101], referred to as type and token information respectively [76]. Finally, a recent neuroimaging study shows that attention strongly enhances retinotopic representation in object selective visual areas, thus supporting the idea that attentional gain modulation is important for combined representation of what and where [102]. Although, these studies point to attentional signals as a strong candidate for initiating the gain modulation, a contribution may still be given by a weakly retinotopic initial cue. Further experimental work is required to disentangle the relative effect of the initial cue and attention on triggering the gain modulation.

Once the increase in the gain of neurons in the right part of the network is triggered, it should be maintained during retrieval. Although the same mechanisms that initiated gain modulation can keep the gain high during retrieval, a promising mechanism for maintaining high level of gain, particularly after the stimulus is removed, is single neuron memory. Several studies show that the recent history of spiking increases the responsiveness of neurons, and that this increase can last for several seconds, thus exhibiting a form of single neuron short-term memory [62]–[64]. Assuming that such single neuron short-term memory mechanisms are responsible for the higher gain of neurons inside the bump, global signals that turn them on or off can strongly affect the level of what and where information that the network represents in its activity.

As discussed above, the attentional signal may trigger the increase in neuronal gain and maintain it elevated for some time. After the attentional signal is removed, the increase in neuronal gain can be maintained by single neuron short term memory mechanisms. Attention can then be directed to another object, while what and where information about the first object is still decodable from neuronal activity. How long this information survives depends on how long the short term increase of the gain can be maintained by single neuron mechanisms. Understanding such mechanisms and comparing their time scale with behavioural times for maintaining combined what and where information, as well as pharmacologically interfering with them, one can test whether our model is relevant to real visual perception.

One of the roles of attention is to bias the competition for limited processing resources in favour of the object that it is acting on [103],[104]. Therefore, if the localised gain modulation that is needed in our model for combining what and where is induced by attention, it should be able to do the same. This is verified by computer simulations as shown in Fig. 9. Two localised partial cues, corresponding to two different objects, are simultaneously given to a network. When the neuronal gain is uniform, the object with the larger cue will be retrieved, while the other one will be suppressed. However, if the neuronal gain in the area that receives the smaller cue is sufficiently large, the competition will be biased in favour of it. Interestingly, the level of gain modulation that is required to bias the competition towards the object with the small cue depends on the width of the connectivity, σ. Increasing the width of the neuronal connectivity increases the minimum level of gain modulation that is required for biasing the competition. This emphasises the role of local connectivity.

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Figure 9. Two patterns, pattern one and two, are cued at two different positions at the beginning of the simulation.

The cue corresponding to pattern one is given by setting the activity of neurons inside a 15×15 square box whose lower left corner is on neuron (1,1) to their activity in pattern one. The cue corresponding to pattern two is given by setting the activity of neurons inside a 20×20 square box whose lower left corner is on neuron (33,35) to their activity in pattern two. Plotted are the final dot product overlap (minus the mean activity; Eq. (9)) with the first stored pattern (full line) and the second stored pattern (dashed line) for two values of the connectivity width (A) σ = 7.5 and (B) σ = 10. Since the initial overlap with the second pattern is larger, without gain modulation it wins the competition, and it will be retrieved, as shown by the final dot product overlap with the two patterns. However, localised gain modulation biases the competition in favour of the first pattern.

https://doi.org/10.1371/journal.pcbi.1000012.g009

Storing patterns with spatial preference

In the model presented here, the units are taken to be arranged on a retinotopic patch of cortex, corresponding to at least a portion of visual field, but we assumed patterns of activity to be generated from a spatially uniform distribution (see Eq. (4)). A more realistic model, however, should allow for the storage of spatially organised patterns [105]. This is important since, in the case of high level visual cortical areas, the overall position of intense neural activity during visual stimulation is strongly correlated with object identity or category. Regions in the visual cortex have been located that are preferentially selective for faces [106]–[109], pictures of scenes [110],[111] and buildings [112], and complex object features [113]. This strong categorical map may coexist with a retinotopic map. The details of this combined organisation are far from clear, however, particularly insofar as it is expressed in the putative attractor states, after the stimulus is removed (e.g. during delay periods), which is the situation relevant to our study. During visual stimulation, and when attention is not a main factor, some studies suggest that there is a weak retinotopy, with only a peripheral versus central bias and no angular representation [83]–[85]. Others, on the other hand, report the existence of multiple precise retinotopic maps in the same regions [114]–[116], although still much weaker than the level of retinotopy in primary visual areas [117]. As mentioned in the previous section, such retinotopic maps could be enhanced by attention [102].

To include the coexistence of categorical and retinotopic maps in the model presented here, one might consider two limit cases, which roughly correspond to these two views. In the first case, category specificity and weak retinotopy coexist at the same spatial scale; one should then assume, in a refined model, that patterns are generated from multiple distributions, each of them corresponding to one category of objects, and patterns drawn from each have higher activity at a preferred position on the network. In this case, when there is no gain modulation the peaks of the retrieved patterns cluster depending on which distribution they came from. The peaks will also be more weakly correlated with the position of the cue compared to the case of spatially uniform patterns that we have discussed. With attentional gain modulation, one expects to see a clearer retinotopic map. This is in fact consistent with the abovementioned finding that attentional gain modulation enhances the retinotopic representation in advanced visual areas [102]. In the second limit case, retinotopy is expressed in object selective visual areas at a finer scale than category specificity, in which case one should allow for the present model to be simply multiplexed, to include one array on a distinct cortical patch for each object category.

Further work is required, especially in view of many intermediate possibilities, to assess, for example, how much more gain modulation would be needed in order to stabilise a bump of activity away from its preferred position, and how this would affect retrieval.

Laminated networks

The ability to represent what and where information in the same network has also been proposed to be crucial to understand the functional significance of the differentiation among cortical layers [118]. Whereas most network models used to study attractor dynamics in associative memory do not consider cortical lamination, the core hypothesis of the proposal is that layer IV units, by virtue of their distinct connectivity, may privilege the representation of position information. Furthermore, through less adaptive spiking activity they may influence the dynamics of pyramidal units in the superficial layers only after these have engaged the attractor basin that leads to retrieve object identity. The differentiation was shown to be advantageous, in the model, through computer simulations, conducted with external inputs maintained active. In this regime no assessment was possible of whether genuine dynamical attractors had indeed been formed during memory storage, that will drive network dynamics in the absence of the cue. While the present work clarifies the conditions allowing a single layer network to represent what and where information, how they could be realized in a network with differentiated cortical layers remains to be explored.

Combining other variables

In discussing what and where information, we have made explicit reference, here, to object identity and position in the visual field. Where information could however be any feature that is mapped in the gross topography of the cortical sheet, such as frequency in the auditory system [119], and in relation to which there is no meaning to using attractor dynamics in order to refine the afferent signal with what is stored in memory. In fact, this mapping need not even be topographically organised: the crucial factor is the existence of a map (topographic or not) [120], that is produced as a result of the dependence of in Eq. (5) on i and j, and that is independent of the stored patterns. Where information would ideally be expressed by a continuous attractor and thus maintained e.g. as delay activity, except that continuity at a fine scale is disrupted by the storage of what memories. What information could instead be any feature that could benefit from attractor dynamics, because of its uneven statistical distribution, which makes some interpretation of the afferent signal more likely than others.

Synaptic weights that follow Dale's law

If synaptic weights are produced by Eq. (5), the weights of the connections that originate from a given neuron can be both negative and positive. This is against Dale's law and against our assertion that all neurons in the model network are excitatory. In this section, we show how the model described in sections “Firing rate description of the network” and “Stored memory patterns and synaptic weights” (see Model) can be conceptually derived from a more realistic formulation, in which all synaptic weights are positive. Let us first consider a network in which the firing rate of neuron i at time t+1 is determined by(14)in which Th_i is the threshold of neuron i, I_i is its inhibitory input, and(15)

The synaptic weights, W_ij, in this network take the following form(16)where J_back is the background weight, ϖ_ij = 1 if there is a connection from neuron j to neuron i and ϖ_ij = 0 otherwise, and C is the average number of connections per neurons. For sufficiently large J_back, the resulting synaptic weights in Eq. (16) will be all positive. We can now show that a network with uniform threshold, as assumed in Eq. (2), and synaptic weights of the form Eq. (5), has equivalent dynamics as described by Eqs. (14) and (16), when an additional condition is satisfied.

Combining Eq. (14) with Eqs. (15) and (16), the firing rate of neuron i can be written in terms of the firing rate of the other neurons as(17)in which J_ij is the weight of the connection from neuron j to neuron i according to the prescription Eq. (5). The assumption we now make is that the inhibitory feedback reacts in such a way that for each neuron, the last three terms in the parenthesis in Eq. (17) together become equal to a uniform effective threshold, Th. This effective threshold is simply chosen such that Eq. (3) holds. In this way, Eq. (17) reduces to(18)which is the same as Eq. (2).

Self-consistent equations

In this section we briefly describe how the self-consistent equation for the local overlap with the retrieved pattern (Eq. (10)) can be derived. We refer the reader to [32],[33] for more details.

To start with, we assume, without loss of generality, that the first pattern (μ = 1) is retrieved and therefore for . Using Eqs. (1), (5) and (8), we then write the input to neuron i as(19)Denfining z_i as(20)and combining Eq. (19) and Eq. (2), the activity of neuron i can be written as(21)

Inserting v_i from Eq. (21) into Eq. (8) we arrive at the following self-consistent equation for (22)

Averaging the right hand side of Eq. (22) over the distribution of z_j, η, and the connectivity pattern, yields the following equation (which is the same as Eq. (10))(23)where 〈〉_η , stands for averaging over the distribution of η, is the probability of connection (Eq. (7)), and F̅_j is the gain function, F, averaged over the distribution of z_j(24)

We now find the distribution of z_j, which we denote by z_j Pr_i(z_i). To do this we note that if the first pattern is retrieved, v_js, on the right hand side of Eq. (20) will be independent from each other and from η^μ for μ≠1. The assumption of independence is strictly correct when the network is highly diluted, that is when the number of presynaptic neurons shared by any two postsynaptic neurons is small [121],[122]. When the network is not highly diluted, the calculation will be more involved, but yields qualitatively the same results [32],[123]. Thus, for the sake of simplicity, we assume that the assumption of independence holds; for a complete derivation we refer the reader to the aforementioned references. With this independence assumption, the right hand side of Eq. (20) will be a sum of independent random variables, and therefore, Pr_i(z_i) will be a Gaussian distribution. In the following we show that the mean of this Gaussian distribution is zero and also find a self-consistent equation for its variance.

Noting that(25)(26)and using Eq. (21), we get the following equations(27)(28)where indicates averaging over the distribution of η and ϖ_ij, and 〈〉_η indicates averaging over η. From Eq. (27), we see that the mean of Pr_i(z_i) is zero. In order to find the variance of Pr_i(z_i), we should average both sides of Eq. (28) over the distribution of z_j. This is because, in the limit of large N and large C, this variance is expected not to depend on the exact realisation of any z_j in the right hand side of Eq. (28), but only on its statistical distribution. Performing this average yields the following equation for the variance that we denote by ρ_i²(29)

Equations (23) and (29) form a closed set of equations whose solutions determine the steady states of the system. Finding m_i and ρ_i that satisfy these equations, we can find the activity of neurons in the steady states by plugging them in Eq. (21). In the case of a randomly connected network, that is when is independent of i and j, and g_i are also the same for all neurons, the solution of Eqs. (23) and (29) will be of the form m_i = m and ρ_i = ρ. In this case the only spatial dependence of the steady state activities, Eq. (21), will come from the dependence of η_i¹ on i and since they are generated identically for each i, the probability that a neuron is active in the steady state will be uniform over the network. Spatially localised retrieval can be observed when depends on the distance between i and j.

Mutual information measures

In this section we show how we compute what and where information, I_what and I_where, from simulations. We estimate the amount of what information, I_what, from the frequency of successful retrieval runs. To see how, let us assume that we cue pattern μ_c. Then after some time we look at the pattern of activity of the network, compute its dot product overlap with all stored patterns (Eq. (9)) and find that pattern μ_r, say, has been “retrieved” in that particular run, i.e., it has the highest overlap with the activity of the network. We denote the probability of retrieving pattern μ_r given that we have cued pattern μ_c by Pr(μ_r|μ_c). Estimating this probability from the simulations, we can compute the information that the pattern of activity gives us about which pattern was presented as(30)where Pr(μ_c) is the probability of cueing pattern μ_c and(31)

In the simulations all patterns are presented an equal number of times, therefore,(32)

We denote the fraction of successful runs (when μ_c = μ_r) that we measure from the simulations by f, that is(33)

Since in unsuccessful runs (when μ_c≠μ_r), all patterns, except for μ_c are a priori equally likely to be retrieved, we have(34)

Using Eqs. (32)–(34) in Eq. (30), we can thus write for fixed degree of gain modulation, fixed background gain, and fixed number of patterns,(35)

Note that the above is, strictly speaking, only a measure of the information implicit in the selection among the p patterns operated by attractor dynamics; under certain conditions, however, it can also serve as an indicator of the total information available in the firing pattern itself [124].

I_where is the mutual information between the peak of the local overlap after 200 time steps and the centre of the gain modulated area (or the centre of the cue when there is no gain modulation). To estimate where information, I_where, we first measure the distance between the peak of the final overlap of the successful runs and the centre of the gain modulation, for each cued pattern. Then we make a histogram of these distances and calculate the fraction of runs which fall in any of the 10 distance bins chosen to be b₁ = [0,5],b₂ = [5],[10],…,b₁₀ = [45],[50]. In this way we have the conditional probability, Pr(k | x), of having the peak of the activity in the kth distance bin, given that the peak was initially at position x on the lattice. With N neurons on each side of the lattice, we have Pr(x) = 1/N², and we can write I_where as(36)in which we have used the fact that Pr(k | x) does not explicitely depend on x and we can simply denote it by Pr_k. Similarly to what we do for I_what, we have also assumed that for any such ring between the circles of radius 5k and 5(k−1), centred on the gain modulation square, the final bump can be anywhere, with equal probability, on the ring. In this expression the factor 2k−1 accounts for the fact that the area covered by the kth bin is 2k−1 times the area of the first bin, and hence its a priori probability is 2k−1 times higher. The first term in Eq. (36), is the maximum information value, I_where≃6 bits, in this approximation, i.e., the logarithm (in base 2) of the ratio between the “area” of the network (4900) and that of the smallest bin (5×5×π), and is achieved when all successful runs end up with a bump at d≤5 from its intended position.

Low gain regime versus high gain regime

In this section we discuss why in the low gain regime, gain modulation aids retrieval of the patterns whereas in the high gain regime it has a negative effect. We start from the self-consistent equations, Eqs. (23) and (29).

Assume that the steady state of the network is a bump of activity over a part of the network with single neuron gain βg, whereas the rest of the network is silent with gain g. Furthermore, assume that m_i and ρ_i that satisfy Eqs. (23) and (29) are nonzero inside the bump and zero elsewhere. Consider that inside the bump m_i = m and ρ_i = ρ, where m and ρ can be regarded, just for simplicity, to be roughly constant. Then from Eqs. (23) and (29) we have:(37a)(37b)where α = p/C is the storage load and(38)

Eqs. (37) are of the form of mean-field equations of a recurrent network with non-metric connections [79],[125] (assuming uniform values for m_i and ρ_i inside the bump and zero outside is equivalent to assuming that the part of the network, over which the bump is formed, is behaving as an independent network). For each value of α, Eqs. (37) have non-zero solution for m, and thus the network can retrieve the stored patterns, if and only if g_min(α)<βg<g_max(α), where g_max(α) and g_min(α) are functions of α. The effect of background gain g can now be readily seen. When g<g_min(α) retrieval does not happen without gain modulation. With gain modulation, however, the neuronal gain of the part of the network that is gain modulated will be boosted by a factor of β and for large enough β, the neuronal gain will be in the regime that supports retrieval i.e. g_min(α)<βg<g_max(α). When the background gain g is high, βg can exceed g_max(α) , thus retrieval will not be successful.

Comparison with other models

In this section, we discuss why it is more efficient to spatially modulate attractor states associated to objects, than to store distinct attractors for different positions of each object.

Under optimal conditions, the number of attractors that an associative memory with C connections per neuron, but without metric connectivity, can retrieve is(39)where k is a constant that is primarily determined by the sparsity of the stored patterns [79]. Metric connectivity, which enables localised retrieval, decreases k by a moderate factor γ₁≃3−4 [32]. Localised gain modulation, that stabilises the bump at an arbitrary position, decreases k again by another factor, γ₂, that for the parameters and network size we used turns out to be γ₂≃4. This is actually an overestimation of the decrease in storage capacity due to localised gain modulation, for realistic size networks. This is because when we calculate the mean of the right hand side of Eq. (22) over the distribution of connectivity patterns and η to get Eq. (23), we ignore the fluctuations around this mean, that behave as . These fluctuations are what break the translational symmetry of the self-consistent equation, Eq. (23), and make the bump favour a few positions over the others, and are compensated for by the localised gain modulation. As a result, less gain modulation is required for stabilising the bump when there are more connections per neuron. However, even with this estimate for γ₂, the process described here results in a moderate reduction in storage capacity(40)

The spatial modulation described here can represent positional information with a resolution , where l is the lattice spacing and N_p is the number of distinct position that can be resolved-in a large network, N_p∼O(N) (see Fig. 5).

On the other hand, the naive storage of distinct, unrelated attractors for each object position pair decreases the number of objects, whose identity could be retrieved, to(41)illustrating the wasteful use of memory resources for positional information, which in itself requires no memory.

An alternative arrangement might be to associate attractors to objects, but allow each attractor to be a continuous 2D manifold, different for each object, so that position can be represented by the position of a bump of activity on such attractor manifold, unrelated to the position of the active neurons in the tissue. This arrangement corresponds to the multiple spatial charts model of Samsonovich and McNaughton [58], introduced to account for the ability of rodents to track their own position in multiple spatial environments, by coding it as a group of coactive hippocampal place cells, which comprise a bump on a chart corresponding to each environment. Instead of assigning distinct charts to distinct spatial contexts, such as a square recording box rather than a circular one, one could well assign distinct charts to distinct objects, each of which would then have its own “private” continuous or quasi-continuous attractor, unrelated to the 2D arrangement of neurons in the tissue. The mathematical analysis of the multiple charts model [126] reveals that a network can store a number of charts equivalent to the number of attractors in a standard associative network of the same connectivity, reduced by a factor N_b, which is the number of place cell ensembles, uncorrelated with each other, required to “tile” a chart. In the simplest version of the model, each neuron shows a single place field in each environment (at a different spatial position in each chart) covering a fraction a of the total area of the environment. Then N_b≈(1/a) and, although the number of positions that can be represented accurately can be larger than N_b, still a≪1 for the network to be able to resolve position in space. Therefore, adapting the hippocampal model would also yield a lower capacity(42)because of the cost of creating a separate “virtual” space for each object. Simply utilising the position of neurons in the tissue to represent physical position for all objects, and reserving memory resources for object identity, provides the most efficient solution to combine what and where information. Note instead that in the hippocampus, to the extent that it utilises coactivity patterns to discriminate between different spatial contexts [127], the position of neurons in the tissue cannot be used to code for position in real space, and in fact place field position in the chart is found to be unrelated to cell position in the tissue [128]. It is also worth mentioning that the same problem that we encountered for stabilising the bump at an arbitrary position will also appear in models that associate a distinct chart to each object [57]. Therefore, an extra mechanism will be required in this case, too, and the real p_max will be smaller than p_max in Eq. (42) by a factor similar to γ₂ in our model.