and the task manipulations

The function is central to the mechanism. Manipulation of this function is how external events (stimulus information, for example) influence the network. In this section, the different roles this function fills are outlined.

There are four different configurations of . Three of these parallel the “loading”, “maintenance”, and “comparison” task components described by Machens et al. [7]. The fourth role of the function is to prevent a cell from responding to input (“quiescent”). Figure 1 shows the four different configurations of : a stable equilibrium point (Row B, left), a line attractor (Row B, center), an unstable equilibrium point (Row B, right), and a quiescent mode (Row D, right). Each of these configurations corresponds to a subtask of the working memory task. The first three are implemented as Machens does [7]: The stable fixed point is for loading the stimulus variable into the network. The maintenance configuration is a line attractor. The comparison configuration is an unstable fixed point.

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Figure 1. Components of the proposed memory mechanism.

A. The delay tactile discrimination task. B. The time series of a persistent cell's activity (equations (1–2)) during the three components of the task, as described by Machens et al. [7]: Loading, maintenance, and comparison. Dotted lines delineate the times (stimulus off at and the comparison is made at ) when the network changes between these components. C. Diagrams showing the gain function (bold curves) for the loading, maintenance and comparison components of the task. Green arrows show the direction that the activity will move, relative to the fixed point, during the respective component of the task. Note that is never negative, and defined as zero when is less than threshold. D. Shown is the time series of a cell that is held quiescent, a component of the task where input to a cell is ignored. Upon receipt of an external input, the cell switches to the maintenance configuration, and assumes the activity level of the persistent cells. This activity level is maintained until the end of the delay period, when the network enters the comparison phase of the task.

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Chronologically, the first component of the task is “loading” the stimulus variable into the network. During this component of the task, the network is exposed to a stimulus that is described by an analog scalar variable. Here, this variable is the frequency of the tactile vibration. Since the cells tune monotonically to the stimulus variable, the cellular response to the stimulus will be a monotonic function of the stimulus variable. In terms of , this is done by creating a stable fixed point at the desired activity level (left panel of item C in figure 1). The activity level at this fixed point is determined by the slope of . Thus, the slope of is a monotonic function of input frequency. So, for a cell that is positively monotonically encoding the stimulus, the slope of is a monotonically increasing function of frequency. The stable fixed point draws the activity toward this frequency specific level, and the stimulus variable is loaded into the network.

The second component of the task is “maintenance”. This begins once the stimulus is removed. This is the actual memory component of the task, where the network contains information about a stimulus that is no longer present. The information is to be retained for the duration of the delay period. In terms of , this configuration is shown in the center panel of item C in figure 1. In this configuration, the network behaves as a line attractor. The line attractor holds the stimulus dependent values throughout the delay period. The important result of this study is how the brain might overcome the drift that occurs when this configuration is not perfect - when the lines do not perfectly overlap.

The final component of the task is “comparison”. This component occurs at the end of the delay period. In the experiments, a second stimulus of frequency arrives and the objective for the animal is to compare this stimulus to the original stimulus (frequency ). In terms of , this configuration is shown in the rightmost panel of item C in figure 1. There is an -dependent unstable fixed point, where the slope of is determined by the stimulus frequency. This unstable fixed point acts as a separatix. If the -dependent activity levels are above this separatix, they will increase. Conversely, if the -dependent levels are below this separatix, they will quickly decrease. So, whether the activity level increases or decreases upon the arrival of the second stimulus determines whether the network assessed the first stimulus frequency () to be higher or lower than the second (), thus providing a comparison.

The right panel of item D in figure 1 shows the “quiescent” configuration. With this , the cells do not respond meaningfully to input. All activity will quickly decrease to zero. The usefulness of this configuration is described in the next section.

Early, persistent, and late cells

I begin by demonstrating how a traveling wave can account for the division of the population into early, persistent, and late cells. Figure 2 shows a solution to equations (1) (2). The array plot shows the solution as a function of time and position in the chain. Also shown are temporal profiles of three cells - one that loses activation quickly (early), one that is active throughout the delay period (persistent), and a cell that does not tune to the stimulus variable until the end of the delay period (late).

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Figure 2. A traveling pulse as a mechanism for early, persistent and late type cells.

The top panel shows three time series from different locations on the chain. The top graph is the time series for an early cell, or a cell that ceases to encode shortly after the delay period has begun. Below it is a persistent cell, a cell that maintains a relationship with the stimulus throughout the delay period. Below this is the time series for a late cell. The array plot (bottom) shows the pulse in space (cell number, vertically) and time (horizontal). For all panels in this figure, , so that the network behaves as a line attractor. To generate the wave, the first 100 cells (out of 150) are stimulated. The noise parameter is .

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Figure 2 demonstrates how a traveling pulse can generate a pattern that would allow cells to be classified as early, persistent, or late. There are actually a pair of wavefronts - a leading wave front and a trailing wavefront. The leading wavefront tunes cells to the stimulus variable. Cells lose their relationship with the stimulus as the trailing wavefront passes. By definition, the arrival of the stimulus tunes the early and persistent cells to the stimulus variable. What separates early cells from persistent cells is the position in the chain. Early cells are first in the chain, and the trailing wavefront passes through these cells early in the delay period. Persistent cells are further along in the chain, and the trailing front does not reach these cells during the delay period. Late cells, in this illustrative scenario, are later in the chain than the persistent cells. They are not tuned to the stimulus initially, rather they only become tuned as the leading wavefront reaches them.

Of importance is that the leading and trailing wavefronts have the same slope in the array plot of figure 2. Thus, these wavefronts have the same speed causing the number of cells encoding the stimulus to be constant throughout the delay period. This is not what the experiments show. Rather, the number of cells encoding the stimulus variable decreases at first and then, at some point near the middle of the delay period, begins to increase. The simple pulse described so far is not capable of this. In the next section, I discuss a modulation of the leading wavefront that can account for the initial decrease in the number of encoding cells.

Decreasing and increasing number of encoding cells

The second feature of the data that was identified was a systematic decrease in the number of encoding cells during the first half of the delay period, followed by an increase. In the previous section, I showed how a pair of traveling wavefronts can account for the existence of early, persistent, and late cells. The problem remaining at the end of the section was that the trailing and advancing wavefronts have the same speed, and so the number of encoding cells is constant. In this section I describe how the wavefronts can be modulated during the delay period to account for this characteristic.

By definition, the initial decrease in the number of encoding cells is due to the early cells losing their monotonic relationship with the stimulus. Similarly, the subsequent increase can only be due to the late cells assuming a stimulus dependent activity level. Thus, the transition from decreasing to increasing number of encoding cells is equivalent to the transition from early cell decay to late cell activation. An important result from the experiments [1], [2] is that the transition from decreasing to increasing occurs roughly halfway through the delay period. This is regardless of the length of the delay period. To illustrate, the authors show what happens when the length of the delay period is changed from 3 seconds to 6 seconds. They show that the late cell response, which began roughly halfway through the delay period for the 3 second delay period, is stretched to roughly halfway through the 6 second delay period after a couple of trials. So, a change in the length of the delay period modulates the time that the transition from decreasing number of cells to an increasing number of encoding cells occurs.

I suggest that the external executive input is responsible for the transition in the number of encoding cells. Suppose that, prior to the arrival of this input, the late cells are not allowed to tune the stimulus variable. The effect of the executive input, then, is to allow the late cells to participate in the task. Prior to the arrival of the input, the leading wavefront is frozen. The early cells are simultaneously falling out of tune, and so the net result is a decreasing number of encoding cells, prior to the arrival of the input.

To incorporate these changes into the model (1), I need to specify for the late cells, separately from the early and persistent cells. I do this by designating the late cells as “quiescent”, (Figure 1), prior to the arrival of the executive input. When the input arrives, its action is to shift the configuration from “quiescent” to the “maintenance” configuration (Figure 1). This switch allows the leading wavefront to advance into the late cells, tuning them to the stimulus variable.

Figure 3 shows the network with the late cells modulated as above. Notice the frozen leading wavefront for the first half of the trial. Prior to the inclusion of the late cells, The number of encoding cells decreases. When the activity is allowed to propagate into the late cells, they become tuned to the stimulus. A difficulty also arises: the persistent cells are losing their monotonic relationship with the stimulus and so there is no net gain in the number of encoding cells. The experiments clearly show an increase in the number of encoding cells. To account for this, either the advancing front (the front going into the late cells) must be faster that the trailing front, or the trailing front (the front that causes the early cells to fall out of tune) must slow down.

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Figure 3. The modified wave.

The panels describe the same things as in figure 2, with the same parameter values. The removal of the stimulus is given by the first dotted line. The arrival time of the task input is represented by the second dotted line. Of note in this figure is that the late cells are held back, and do not participate in the task until the task input arrives.

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Simply increasing the speed of propagation in the late population, thus speeding up the advancing front, would accomplish the growth in the number of encoding cells, but the data suggests that this is not the case. Persistent cells are also impacted by the arrival of the executive input. Many persistent cells show a dramatic change in behavior simultaneously with the incorporation of the late cells. It is then natural to suspect that the executive input is involved with this change of behavior. I posit that the late cells project back onto the persistent cells, freezing the trailing wavefront, and allowing the number of cells that encode the stimulus to increase. The change in the behavior of persistent cells is the topic of the next section.

Varying firing rates for persistent cells

Persistence, as a mechanism, is a staple of working memory. It is how cells can hold information about the past - a stimulus that is no longer present. Persistence is often modeled as a fixed point, or for the case studied here - monotonic encoding - a line attractor. Each of these attractors holds a cellular variable (eg. firing rate) constant for the duration of the delay period. However, the persistent cells recorded in many working memory studies do not behave as a fixed point. Rather, the firing rates of persistent cells vary widely during the delay period. For the experiments that are the focus of this paper, the large variation of the persistent cells is divided into four categories - cells that initially decrease and then increase, cells that decrease for the duration of the delay period, and the opposite behaviors. Here, I only consider those persistent cells that initially decrease, and after the executive input, are amplified (a typical example is shown in Figure 2 of [1]).

A line attractor is often used to store an analog variable [7], [10], but this mechanism requires very precise tuning. However, a network that admits a near-line attractor (a line attractor where the tuning is not perfect) is still capable of maintaining a monotonic relationship with the stimulus variable. That is, for two stimuli , and firing rates , , it may be possible to tune the network well enough so that for all in the delay period, even if is not constant. This is possible if the tuning is close to that of a line attractor, but not perfect. The imperfections will result in a slow drift from the original value, as shown in [10]. If the tuning is good enough to make this drift sufficiently slow, then a monotonic relationship between the stimulus variable and the cellular output over the course of the delay period can be achieved.

In the model, the cell-to-cell connections are determined by the entries of the connectivity matrix, (equation (1)). We are assuming nearest neighbor, feed forward connectivity, so all non-zero entries reside on the first sub-diagonal (equation (2)). For a perfectly tuned line attractor, each of these entries are . The slow drift is modeled by allowing these entries to be less than (the slow drift will be a decreasing one).

The change in connection strengths will cause the amplitude of the wave to decrease slowly toward zero. If the entries of are close enough to , the exponential decay will be slow and monotonicity will be preserved. (Note: If the initial stimuli are very close in scalar value, and there is sufficient noise, this monotonicity can be broken. For example, if in one trial the stimulus is at Hz and in another trial the stimulus is Hz, for a small value of , one would expect the noise to destroy the monotonicity. Accordingly, there is a minimum separation between the frequency of the first and second stimuli in the experiments).

In order to reflect the stimulus information, the signal must be amplified to recover from the initial decay. The switch from decreasing activity to increasing activity takes place at nearly the same time the late cells begin to encode the stimulus, so it is natural to view the late cells as implicit in this transition. I propose that the late cells project back onto the early and persistent cells. This feedback accomplishes two things: 1). It amplifies the decayed signal so that the firing rate at the end of the delay period is indicative of that at the beginning - the stimulus induced value - and 2). it stops the trailing wave front from advancing, allowing the total number of cells encoding the stimulus to increase. This addresses the issue that I ended the previous section with - the leading and trailing wavefronts no longer have the same speed.

In order for the feedback from the late cells to amplify the signal, the strength of this feedback must be sufficiently strong. As in [9], the solution of the th cell in the chain, after the stimulus has been removed, is given by(3)Following Goldman [9], the late cells become part of the wave roughly one time unit for each connection in the chain after the initial blockade (an implicit assumption is that the time constant of a cell is much shorter than the length of the delay period). Once the propagation of activity is allowed to advance into the late cells, they assume this solution as well. So, in general for , the onset time of the task input. After the inclusion of the late cells, we can approximate the evolution of a persistent cell with(4)where is the strength of input from other persistent cells (the value of ) and is the strength of the feedback from the late cells. If , there will be amplification. I will derive the specific restraints on these parameters in the next subsection, and their relationship to the tolerance in the timing of the executive input.

It is my claim that the decay and amplification mechanisms can greatly ease the cell-to-cell connectivity restrictions of a line attractor. There is another important advantage to the proposed mechanism - it can be used to tune the network to delay periods of different lengths without changing any of the individual connections between cells. Figure 4 shows a demonstration of this. In the top panel of figure 4, the network is tuned so that it successfully stores the stimulus value for a delay period of length . If, on the next trial, the delay period is increased without warning or preparation, say doubled (), the task input will not move, resulting in an unreliable cellular response. However, after a couple of trials, the timing input is shifted to a time that results in correct trials (bottom panel of figure 4). This is consistent with the data; as reported in [1] there is a slight increase in the error rate directly after the switch from 3 to 6 seconds. After a few trials, the animal's performance improves. Moreover, raster plots show that the onset of late-cell activity gradually adapts to the longer delay period length, after a few trials [1].

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Figure 4. The network is able to tune to different delay period lengths by changing the timing of the task input signal.

The top plot shows the evolution of a properly tuned persistent cell during a delay period of length . The value of the stimulus variable is (indicated by the green horizontal line). The second panel shows the same cell, with the same task input timing as the top panel, but with a delay period that is twice the length (). This results in an unreliable representation of the stimulus variable. The bottom panel shows the same cell when the timing of the task input is correctly tuned for the longer delay period. In this figure, , and the feedback strength is . The noise strength is .

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This mechanism is further supported by the data. Brody et al. [1] utilize a descriptive model to determine to what extent the activity during the 6 second delay period is a stretched version of the activity during a 3 second delay period. They show that the early cells behave in much the same way for each delay period length. That is, the time course for an early cell does not stretch or contract. The model agrees with this, since an early cell for the 6 second delay period will evolve the same way as it would for a 3 second delay period. In either case, the trailing wavefront has passed. The authors show that the timing of the late cells is stretched by a factor of 2. This also agrees with the model, since the timing of the task input - which determines when the late cells become active - is roughly halfway through the delay period. Doubling the delay period will then roughly double the time at which the late cells begin to encode, yielding an approximate stretch factor of 2.

At the end of the previous section, I concluded that the external signal also causes the freezing of the trailing wavefront, so that persistent cells can remain active. How the trailing wavefront is frozen is interesting, and may not be immediately obvious. The late cells feed back onto all of the early and persistent cells. The early cells are those cells that, by definition, have lost their monotonic relationship with the stimulus. In other words, they have decayed below the level where signal can be differentiated from noise.

Still, with the addition of late cell input, the noise in the early cells' activity is amplified and propagates through the medium. Though it is summed noise and does not have a relationship to the stimulus, it does generate enough activity to “prop up” the trailing wavefront. This allows the persistent cells to maintain their monotonicity with the stimulus variable, freezing the trailing wavefront. Figure 5 shows early cells at different locations in the chain, and how they contribute to the maintenance of a persistent cell.

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Figure 5. Summation of feedback to freeze the trailing wavefront.

Shown are cells 2,12,17,25, and 60. The rightmost plot is the only plot of a persistent cell. All other plots show loss of relationship with the stimulus during the delay period. The summed noise of the early cells, along with a small amount of feedback from the late cells, freezes the trailing wavefront. In this figure, , the late cell feedback has strength , and (noise strength).

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Figure 6 shows a simulation of the full network. I implement the feedback from the late cells to the early and persistent cells as a connection from a single late cell, though more general patterns would work as well. As the late cells are incorporated into the network, the persistent cells begin to increase their firing rates. The time courses for an early, persistent, and late cell are also shown in figure 6. The decreasing and increasing of the firing rate in a persistent cell is clearly demonstrated, as is the match between the initial stimulus dependent activity level and the level after amplification. Figure 7 shows how this this tuning works for a range of stimuli.

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Figure 6. The decay and correct mechanism.

The figure is laid out the same as figures 2 and 3. The top panel contains three time series showing the behavior of the three types of cells we consider - an early cell (top), a persistent cell, and a late cell. The panel at the bottom of the figure is an array plot that shows the evolution of the entire network as a function of time. This figure clearly demonstrates the change of behavior for the network after the executive input has arrived - the frozen wavefront prior to the executive input, the activation of the late cells, and the freezing of the trailing wavefront. For all panels in this figure, the feed forward connection strength is and the feedback from the late cells has strength . The stimulus turns off at time , and the executive input occurs at . For the noise .

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Figure 7. Time series showing the activity level of a persistent cell (used is the last persistent cell in the chain) undergoing the proposed decay-amplify mechanism.

Shown are time series for stimulus variable values ranging from (black) to (brown). For each of the simulations, the stimulus was removed at , and the executive input arrives at . The connection strengths are and the feedback is . For the noise, .

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Comparison to a line attractor

In this section, I show that the proposed mechanism offers a substantial advantage over a true line attractor in terms of the tuning requirements for the connections between cells; i.e. the closeness to the line attractor configuration in figure 1.

The first step is to determine, as a function of the decay and amplification rates, an interval during which the executive input must arrive to correctly amplify the cellular activity. The goal is to determine the length of this interval as a function of the decay and amplification rates ( and in equation (4)). I show that, for decay (and amplification) rates well outside those acceptable for a true line attractor, this interval of times is within the abilities of networks in the brain.

To quantify the tuning requirements in the context of our model, I first establish how accurate the network needs to be. I define as the resolution of the network. That is, the network has to be able to differentiate between stimulus variable values that are separated by more than . Frequencies closer than this are assumed to be too close to differentiate. Therefore, it is necessary to determine when the task input must arrive so thatwhere is the length of the delay period. Assume a decay rate of , and an amplification rate of , where is the coupling strength and is the strength of the feedback connections, as in equation (4).

First, we derive the requirements on the coupling strengths that are necessary so that a line attractor can hold the value of the stimulus variable for the duration of the delay period. Following Goldman [9], the -th cell in a chain of cells satisfying equation (1) has the solutionFor large , this can be approximated by(5)For the regular line attractor, we require thatSolving for yields(6)

For the proposed mechanism, the tuning requirement is on the timing of the external task input. To determine this requirement, we determine the allowable values of the task timing () given the rate of decay, and the amplification rate due to feedback from late cells. With decay followed by amplification, equation (5) can be extended to yield(7)Where specifies the timing of the executive input. We require . Solving for gives(8)The length of this interval iswhere . From this inequality, one can see that the length of the interval scales with the sum . In other words, doubling the sum will decrease the length of the interval by a factor of 2.

Now, to show that this is advantageous over tuning a line attractor, we consider how the bound on varies as the decay rate increases past the limit allowed by a line attractor. Inserting the bound (6) into the expression for the length of the interval, and letting givesThis means that for any choice of and , the length of the timing interval will be, approximately,(9)Thus, this mechanism is feasible for values of and well outside of values that will yield an effective line attractor. As an example, if and , then the interval where the external signal can arrive has length . As another example, if and then the interval will have length . For a 3 second delay period, these examples give interval lengths of ms and ms, respectively. Based on measured dynamics of neural operations in the brain, these intervals are within the limits of feasibility.

In addition to the length of the interval, equation (8) gives where, during the delay period this interval resides. The interval is centered atThus, if the amplified network acts as a line attractor, and so the timing input should arrive right away. Accordingly, the interval hugs the left endpoint of the delay period. On the other hand, if , then the network is a line attractor during decay, and the executive input never needs to arrive. Thus, if either the decay or the amplification meet the requirements of a line attractor, then the network is, by default, accurate enough.

Model performance

In this section, I demonstrate the performance of the model, in terms of accuracy (correct or not) and how this varies with the noise level and the decay and amplification parameters. I also show how the decay-amplify model integrates noise, and compare to a line attractor.

First, I determine how often the mechanism results in a correct response at the end of the delay period for different executive input times and different levels of noise. In all of these numerical experiments, the stimulus variable is , the delay period begins at time and ends at time . My criterion for success is that the activity at the end of the delay period () lies between and (so that in equations (6) (8) is equal to ).

In figure 8, I show the accuracy (as a percentage) of the model for three decay-amplify sets - . For a delay period of length , inequality (6) gives the bound on acceptable values for a line attractor:This corresponds to a coupling strength of (). Expression (9) predicts an approximate interval width of for , for , and for . These are, respectively, , and of the delay period lengths. Figure 8 clearly shows that the correctness rate lies at values that are consistent with this calculation. At the correctness rate, the mean (which is the trajectory discussed in the previous section) is right on the boundary. The noise causes a distribution of values for the activity level at the end point. If the network is tuned properly, the width of this distribution (variance) determines how reliable the recall will be. It is necessary to show that the decay-amplify model does not suffer from a noise integration disadvantage over a line attractor, otherwise the tuning advantageous would be nullified.

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Figure 8. Model performance.

Shown are the success rates for three decay rates () and four different noise strengths () as a function of the timing of the executive input. For each decay, noise, and input value the simulation was run 200 times. The last 10 persistent cells in the feed forward chain were used in the calculation. The green lines show the boundaries of the timing interval described in the previous section. The noise causes a near symmetric distribution of end of delay period activity levels centered around the mean (the predicted value given in the previous section). At the boundaries, this distribution will be centered around the range required for success, and so the success rate at these boundaries is for all noise strengths.

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To determine the relationship between the cellular noise strength (the variance of the Wiener process) and the variance of the output, I ran the simulation 200 times, recording the activity at the end of the delay period for a single cell - the last persistent cell in the chain (cell #100).I repeat this for three instances of the model: A line attractor () and two decay-amplify models with and . I simulate each of these with four different noise strengths (). The results are shown in figure 9. The figure shows that the decay-amplify mechanism integrates noise no worse (or better) than a line attractor. This is not surprising. During the decay phase, the effect of noise is reduced. Conversely, during amplification the effect of noise is also amplified. The net effect of having a decay followed by an amplification of noise results in roughly the same distribution as a line attractor, where the effect of the noise uniform throughout the delay period.

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Figure 9. Distribution of end of delay period activity for a line attractor and two instances of the decay and amplify model.

The first row shows the distributions for a line attractor. The second and third rows show the distributions for models with decay rates of (), and (), respectively. The corresponding amplification rates are also and . The distributions in a single column are very similar, demonstrating that the decay and amplify mechanism integrates noise no better or worse than a line attractor.

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General model

I have demonstrated the decay and amplify mechanism using a simple feed forward model of neuronal activity. The choice of coupling was made to simplify the calculations and make the behavior of the model as transparent as possible. In this section, I describe a more general connectivity, where there is no bias, to demonstrate the decay-amplify mechanism is not dependent on a specific type of network architecture.

In place of the feed forward chain, I model the cells as(10)where is the activity, is location, is the coupling strength, and is a connectivity kernel. We assume that (symmetric), , and (on an unbounded domain, I use to scale for a bounded domain). For the simulations, I useso that . If then if the networks admits a line attractor. Because we are considering a bounded domain, the valuecorresponds to the maintenance configuration in figure 1, where cells on the interior behave as a line attractor until the arrival of the wavefront. For simulations, the integral is discretized and 300 early/persistent cells are used.

There are countless ways to implement the late cells. I choose to implement the late cells as a another line of cells, coupled to the activity described by (10) according to(11)and implement the feedback as one to one by rewriting equation (10) as(12)where is the strength of the feedback from the late cells. Here, I have implemented the feedback from the late cells as a one to one relationship, and the connections from the early and persistent cells are divergent. Figure 10 shows the network schematic of this configuration.

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Figure 10. Schematic for the network described by equations (11–12).

The blue circles represent the early and persistent cells through which the traveling wave propagates. The green circles represent the late cells. When the executive input arrives, the late cells are activated by the persistent cells. In this instance, the connections from the persistent to the late are divergent (black arrows). The late cells feed back onto the early and persistent cells in a one-to-one fashion (green arrows). This is only one of many possible implementations of the decay and amplify mechanism.

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Simulations of the network for a range of stimulus variable values are shown in figure 11. Shown are example time series for an early cell, a persistent cell, and a late cell, for a range of values of the stimulus variable. This figure demonstrates the decay-amplify model for a network that is not a feed-forward chain.

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Figure 11. Simulations of the non-feed-forward model.

The upper left panel is an array plot showing all early and persistent cells. Of note is the trailing wavefront that originates at the boundaries. Those cells that this wave front overtakes are the early cells. The upper right, lower left, and lower right panels show the evolution of a persistent, early, and late cell respectively, for a range of stimulus values (the loading phase is not shown). For all of these figures, the connection strength , the feedback strength is , and the executive input arrives at .

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