Multiple maps

Multiple state variables can be encoded in the same network. An example is offered by the place representations of several environments [20], [21]. To each environment corresponds a neural map which is encoded in the synaptic efficacies. Sensory inputs would then select the correct representation, i.e. both the environment and the position in the environment. The selected map wins the competition with the other maps stored in the network, and a localized pattern appears. In this case the network only maintains information about one of the several encoded state variables.

A more peculiar property of multiple continuous attractors, is their ability to represent simultaneously the values of several state variables. This property was explored in [28], where two partially overlapping neural populations (representing discrete features), are assigned two uncorrelated maps. Another example is provided in the study of [29], where a single network stores and represents simultaneously a continuous and discrete attractors.

In principle, given the existence of multiple representations in different brain regions (either one per region, or many in one region), a brain area downstream would necessarily encode several state variables. In light of a Hebbian interpretation on how this encoding takes place, it seems natural to distinguish between two cases. When multiple representations provide a simultaneous input to a region, the result is probably encoded multiplicatively [29], or, in general, non-linearly. For inputs happening non concurrently, as for instance when walking through several rooms sequentially, an additive encoding of each room is expected [21]. In the following we will analyze additive encoding.

Correlations

The present contribution addresses the issue of encoding correlated maps. The motivations come from recent experimental results on place cells recording in morphed environments [30]–[32], where place fields remapping along a sequence of morphed arenas is experimentally tested, and from theoretical and experimental studies concerning the morphing of discrete attractors [33]–[35].

In general, we would consider the encoding of manifolds , each of dimension , where . We will refer to a single manifold as a map, once a coordinate system is chosen. The use of uppercase (e.g. ) or lowercase (e.g. ) will distinguish between the whole map and a single point on it respectively. Given a pre-synaptic neuron indexed by , and a post-synaptic , the encoding of a single map is obtained using a synaptic matrix , and is such that a continuous attractor representation would arise if it were the only map. We assume, as mentioned above, that the complete encoding arises from a linear superposition of the matrices, . The statistical properties of the maps, and in particular the correlation between them, can be fully specified by providing the probability density .

The general problem is too difficult to be studied analytically. Some results can be obtained for the case of uncorrelated maps on the same manifold [27], though the system can be explored by simulating the full microscopic networks (see e.g. [21] for the uncorrelated case and [36] for simulation results of the correlated case).

In order to simplify the analysis, while retaining the basic structure of the problem, we focus on the case of representations, on a 1-dimensional circular manifold (i.e. the ring model [16], [37]). The correlation between the maps is constructed by limiting the distance between the single neuron locations on the two maps. We devise a simple method to generate a morph sequence between two uncorrelated maps, by linearly modifying the neurons locations between the original maps. This method also suggests a way to test the network response to the exposure of intermediate maps between the two stored correlated maps.

For concreteness, one could think about maps of two similar circular arenas, and reason in term of spatial coding. In this context, we are interested in clarifying how the information about the position in the current environment is represented by the network, when varying the constitutive parameters of the model; And how the representation changes when the network is exposed to environments along a morph sequence.

In the following we will describe with mean-field (MF) theory the attractor landscape of a network, i.e. the stable solutions in absence of any place specific input. We then consider the behavior of the solutions when a spatially tuned input is present. We will establish the approximate relationship between two strongly correlated maps and the encoding of a morph sequence between two reference rings, and study the behavior of the solutions in presence of a tuned input varying along the sequence. Finally we will verify the results with microscopic simulations of finite networks. The network properties can be tested experimentally to confirm (or falsify) the attractor hypothesis.

Phase diagram of the model

In this Section we analyze the fixed point solutions of the system, and heuristically describe the region of stability of these solutions. A more rigorous description of the stability can be found in Methods - Stability.

In Methods - Reduced dynamics we derive the dynamics of the order parameters from Eqs. 2. We report here the result(5)where the function is defined as(6)i.e. the rescaled steady state activity profile Eq. 3. Note that can be eliminated from the right hand sides of the Eqs. 5, rotating the integration variable . This is possible because there is no spatial dependence in the external input to the network. The first four equations in Eqs. 5 can then be solved independently of the fifth one, since the right hand sides do not depend on . We show in Methods - Solutions properties that, once we have the solution for the variables (), the last equation reduces to . We can thus restrict the analysis to four out of five equations in Eqs. 5. The elimination of one angular degree of freedom is a consequence of the rotation invariant structure of the encoding, and is the hallmark of continuous attractors arising from spontaneous symmetry breaking. On the other hand, the integrals over in Eqs. 5 are not over the whole circle and we cannot rotate away.

Homogeneous solution

Before analyzing the fixed point solutions of the system described by Eqs. 5, we briefly mention an uninteresting region in the parameters space which can be found also in the classical ring model. This region corresponds to the homogeneous solution, i.e. all the neurons in the network are active at a constant level, and can be obtained from Eq. 2. The expression corresponding to the line of separation in the plane between the homogeneous solution and the spatially localized bump (see Fig. 3A, curve surrounding the Homogeneous region), is(7)where . This result is obtained in Methods - Stability, see also below.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 3. Phase diagram.

A: MF solutions in a network storing two correlated maps. The black curves in the parameter space depict the separation between the qualitatively different solutions of the system. Homogeneous solution (denoted by H in the panel), all the neurons are active at a constant level. Single Ring (S) solution, localized activity in the middle map, the bump can be freely rotated in this map. Double Ring (D) solution, pair of solutions localized either in map or . Cylinder (C) solution, the network activity is localized in both maps; compared to the single and double ring solutions, there is an additional freedom in the choice of the location of the bump over (). The term cylinder is used because the continuous attractor lives in the space defined by the angle and a sub-segment of (with the exception of , see main text for details). B: Amplitude instability. Critical inhibition corresponding to the onset of unstable solutions for varying . Each curve corresponds to different distances between the maps .

https://doi.org/10.1371/journal.pcbi.1000869.g003

Fixed point equations for the localized activity state

Let us start by imposing , a restriction that will be addressed later on. The first tree equations at steady state from Eqs. 5 become then equations for the three order parameters :(8)The first two equations determine the shape of the bump . Given the map specific modulation in the coupling and the distance between the maps , we can derive from the first two equations the size of the bump and the order parameter , representing how close the network representations are to the stored environments and . The last equation gives us the amplitude of the network activity , which also depends on the parameter .

As mentioned in Results - Phase diagram of the model, the order parameter can be chosen arbitrarily, due to the rotation invariance of the problem; for simplicity we choose .

Amplitude instability

We deal first with the equation concerning the amplitude of the solution. Given that the activity can be rescaled by changing the value of the applied external current , we are not interested in actually solving the equation. The only requirement is that in order for the solution to be meaningful, i.e. no negative amplitudes are allowed. This requirement translates to a constraint on the inhibition :(9)We show with stability analysis (Methods - Stability) that the critical value , obtained by choosing the equality in the previous expression, corresponds to the onset of amplitude instability; given a choice for the parameters , which specifies the bump shape , for values of the inhibition weaker than the solution grows to infinity. This qualitative behavior was present also in the classical ring model.

Fig. 3B shows the values of as a function of for various choices of . In order to stabilize the solutions, the inhibition must grow with increasing and decreasing . Note that it is reasonable to consider the previously mentioned homogeneous solution as a bump with maximal size . In this case the critical can be explicitly computed, and turns out to be .

Single ring solution

Now we focus on the possible solution . It is easy to see that when , the second of Eqs. 8 is automatically satisfied due to the symmetry of the integrand in (and ); This means that the solution exists everywhere in the parameter space.

The steady state activity Eq. 3 with (and , our initial assumption) reads(10)which corresponds to a packet of activity localized in the coordinate, and modulated in , see Fig. 2B for a plot of the activity profile. The remaining fixed point equation can be used to obtain . We refer to the case as a single ring solution; the ring is spanned by the freedom of choice in the angle . In this regime of activity the network is not able to represent separately the environments and , but only the middle environment described by . Even though the solution exists everywhere, it is destabilized in some regions of the parameter space, as shown in the phase diagram (Fig. 3A, Single ring region).

By looking at the maximal bump size , we can expect to reproduce the curve separating the homogeneous solution from the single ring. Inserting in the first of Eqs. 8, it is possible in this case to compute explicitly the integral, which in fact yields Eq. 7.

Double ring solution

In order to find the region of existence of the solutions with , we can solve numerically Eqs. 8 in the parameters plane . The result is shown in Fig. 4, where the color code represents for a given choice of the parameters. It can be seen that there is only a narrow region of high (low correlation) and low where such a solution exists.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 4. Double ring solutions.

Region in the parameter space , where the double ring solution exists. The color correspond to the value of which solve Eqs. 8. Due to the symmetry of equations, both positive and negative values are allowed. Here only the positive solution is shown, corresponding to a localized solution in map . Black curve: curve of separation between the null and positive solutions, obtained by finding the zeros of Eq. 12. Note that the method used to obtain the curve is more precise than the one used to estimate . Dashed curve: curve corresponding to a network activity size . Regions on the left of this curve have , which is a limit size of the double ring solutions (reached at ).

https://doi.org/10.1371/journal.pcbi.1000869.g004

It is important to note that the equations used to find are invariant under the symmetry . This means that both solutions () representing map or are possible. The steady state activity profile in this case looks like:(11)Given the freedom of choice for the phase , each of this solutions lives on a ring; we call the solution , double ring. An instance of the network activity in this regime is shown in Fig. 2C.

The curve separating representations preferring one of the two maps (), and , can be obtained by expanding the second of Eqs. 8 to first order in :(12)where is the Heaviside step function, and . Dividing by , we get rid of the solution. By finding the zeros of the integral, we select the curve in the parameter space corresponding to the onset of existence of the double ring solution. This curve is shown in Fig. 4. We have found that the stability of the double ring solution coincides, empirically, with the region of existence of such solution (compare the phase diagram in Fig. 3A, Double ring region with Fig. 4).

Cylinder solution

Finally, we examine the meaning of the equation for , the order parameter linked to the location of the maximum of the bump in . We have assumed for simplicity, given that a rotation in the integrands in Eqs. 5 is in general not viable due to the restricted range of integration in . Note though, that when the size of the bump is small enough, it is possible to perform the rotation without affecting the value of the integrals; the only requirement is that the rotation keeps the bump from touching the boundaries .

In Methods - Solutions properties we verify that there are no solutions with both and different from . We can therefore set in the steady state activity Eq. 3, and impose the activity itself to be zero on the boundary to findThis equation corresponds to the curve of separation in the plane (using the relationship , Eq. 8) between the single ring solution and a cylinder solution (Fig. 3A, curve surrounding the C region). In this regime, in addition to the freedom of choice for the location of the bump in , the solution is also partially marginal in . The bump can be freely moved on a segment and a circle, defining a cylinder; the activity profile in this case is described by Eq. 4, see an instance in Fig. 2D. This region extends in the high limit and covers the whole range of correlations.

Despite the fact that each of the maps and defines a ring, it shouldn't come as a surprise that the topology of the attractor is a cylinder instead of a torus. The correlation between maps gives rise by definition to a cylinder structure, as can be seen for instance by inspecting Fig. 2B,II. It can be shown that when the cylinder solution degenerates in a torus; the bump of activity can be in any location of the coordinates (hence, also in )). This regime is linked to the observation of an activity bump simultaneously localized in two environments in network simulations [39], and the study in [28].

Phase diagram

Fig. 3 summarizes the results obtained so far. When is low, the only solutions is a constant level of activity which spreads over the whole network (Homogeneous region). As is increased, the interplay between the short range excitation and long range inhibition creates a pattern of localized activity in the middle map (Single ring, see also Fig. 2B) or, if the correlation between maps is small enough, a localized pattern in either or (Double ring, Fig. 2C). Intuitively, the network “remembers” the two maps separately (, two solutions) if they are weakly correlated (). When the maps are more similar, the network represents just an average between them ().

The bump size decreases with increasing . When is further increased, instead of having a reduced size of the localized activity in just one of the maps, the presence of two stored maps in the synaptic structure and the inhibition produce a packet of activity which looks localized in both maps (Cylinder solution, Fig. 2D).

Three particular values of the distance deserve a special mention. The case , corresponding to the encoding of two identical maps, can be shown to be identical to the ring model [37], as expected. In particular, besides the homogeneous solution and the amplitude instability region, the system can only exhibit the single ring solution.

The case , corresponding to the encoding of two uncorrelated maps, does not have the single ring regime as a possible solution. The double ring solution in this case is depicted in Fig. 2A, where it can be seen that the bump is perfectly localized in either maps or , lacking any spatial tuning in the other map. This is the desired outcome in the “multi-chart” approach of [21].

The third case is . We will see in Results - Morphing maps that this case is closely related to the behavior of a network storing a morph sequence between two uncorrelated maps. As can be seen in the phase diagram, the double ring solution is not possible in this regime.

How the environment, and the position in the environment, are represented by the network activity? For the single ring (Eq. 10) and the double ring (Eq. 11) solutions, both characterized by , it is evident that the position is coded by the order parameter . The identity of the environment can only be represented with the ambiguity in the choice of the sign of when the network operates in the double ring regime.

In the cylinder regime, it is not clear how the information about the environment is represented in the network, since now the solution is described by and . The following Section is mainly devoted to explore the link between the state variable (eventually time-dependent) in the active environment, and the behavior of the solution in this novel regime, by introducing a spatially tuned external input.

Tuned external input

Until now we considered the condition in which the only external input to the network, , was steady and uniform. Let us introduce a tuned input, for instance in map at position :For simplicity we assume the shape of the external input to be . The parameter measures the strength of the tuned component of the external input as a fraction of the constant baseline we adopted so far. In general what we are interested in, and what is experimentally observable, are the tuning curves of the neurons i.e. their profile of activity as a function of the input angle in the active environment. It is easy to see the effect on the dynamics of the order parameters (Eqs. 5) when the location specific external current is inserted in the original dynamics for the network activity, Eq. 2. The dynamics keeps the same form as in Eq. 5, with the exception of the threshold-linear term in , which now reads(13)where correspond to the choice of map in the input, and for map .

Tuning curves

With the input at a constant location , one can see that a solution of Eqs. 5 for the single and double ring regime (), is , i.e. the input pinpoints the location of the bump. This implies that, assuming a weak tuned input , the tuning curve of a neuron can be written in the single and double ring regime (from Eqs. 10,11) as(14)and(15)respectively. The tuning curve in the single ring regime has a maximum for (hence is the preferred angle for a neuron ), independently of which map is being used in the external input, as can be seen from Eq. 14. This implies that each neuron has identical tuning curves in both environments, and that the preferred angle of a neuron does not coincide with either the assigned or but with their average.

For the double ring regime, the preferred angle assumes the form (maximizing Eq. 15 in )In this case each neuron has two different tuning curves according to the map used in the external input. The preferred angles coincide with the assigned ones () only when the stored maps are uncorrelated (, hence ).

In the cylinder regime (, not necessarily ), a solution for Eqs. 5 in presence of a tuned input is . For an input in map , , the tuning curve would then be proportional to (from Eq. 4)Note that the dependence on means that the external stimulus does not determine completely the network activity, in contrast to what happens in the previously examined regimes. Neurons that respond maximally to the tuned input are then , and , hence . This means that the tuned external input pinpoints the location of the bump maximum in map but the bump is free to stabilize anywhere along the other map given the freedom of choice in (see activity example in Fig. 2D). If several randomly selected external locations in one of the maps are presented to the network, once at time and starting from random initial conditions, the tuning curves would be an average over :where the allowed range for is , see Methods - Solutions properties. The cylinder regime extends the region of existence of two tuning curves per neurons to an higher correlation between the stored maps; the difference is that the coding becomes unreliable: during a single exposure to a given value of the input angle , a neuron could remain silent even if its average tuning curve would predict a response.

When the representation refers to the location in an environment, it is natural to think about a smoothly varying location . With a moving input like , the tuning curve depends as before on which map is stimulated, but in a novel way. Assume for simplicity to start from a initial condition, corresponding to (). A moving input in the map would tend to move the bump along that map (i.e. increase the of the solution), while keeping constant (hence the bump will move to ). This movement is possible only until the bump reaches the part of configuration space not occupied by neurons due to the distance between maps , see Fig. 2D. At that point, the bump will start to move equally along and , maintaining , which is proportional to , and increasing (proportional to ). A similar scenario, but with , is obtained when stimulating the map .

If the size of the bump is sufficiently small, this effect has dramatic consequences. The small bump will move along neurons with when a moving stimulus is presented in environment , and viceversa neurons with will be active only when the moving stimulus is presented in environment . As a consequence, neurons will essentially just have a tuning curve (or field), only in one map, and will be silent in the other one. We refer to this phenomenon as dynamical pattern separation (see Fig. 5 for an example). The separation of the activity patterns is essentially a dynamical phenomenon, dependent on the history of the inputs. The figure shows also the robustness of the dynamical pattern separation behavior to the addition of Gaussian -correlated noise in the external current (see Methods - Numerical Methods). Note that neurons characterized by (i.e. ), will have tuning curves in the same location. The number of neurons with tuning curves in both environments grows with the size of the bump.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 5. Network representations in the cylinder regime.

A: MF time-course of , when a moving input localized around in either map or , is fed to the network. The location of the maximum of the external input in the stimulated map is shown with a dashed blue line. The order parameter tracks the external input with a delay. B: The order parameter , giving the position of the bump in , with the same moving inputs. Depending on the stimulated map, the bump ends up in different positions, orthogonalizing the representations. Noisy curves from a simulation including a zero average white noise term in the external input, with . C: A snapshot of the network activity at a given time and corresponding angle , when the stimulated map is , left or , right. Neurons with values differing enough from (depending on the size of the bump) will exhibit tuning curves only when stimulated in one of the two environments (above/below red dashed lines). The parameters for all the panels are .

https://doi.org/10.1371/journal.pcbi.1000869.g005

Note though that by changing the sign of the velocity in the moving input, the behavior would reverse; neurons with positive (negative) would be active during a stimulation in map (). In order to maintain the dynamical pattern separation and the analogy with place coding, one could think about two circular environments, as we did so far, with the additional constraint that the environments can only be traveled, for instance, in the counter-clockwise direction (CCW). As an alternative, the two environments may be thought as the same circular arena, but traveled clockwise (CW, environment ) and CCW (); this interpretation would give rise to place fields with directional selectivity (see Discussion).

The dynamical pattern separation is basically dependent on the history of the input (positive or negative velocity), in addition to the identity of the map used in the stimulation. This history dependence is present also for non smooth time-dependent stimuli, as for instance the sequential presentation of stimuli with an intervening delay period. In this case the history dependence gives rise to a memory effect: the current location of the bump following a stimulation depends on the location attained after the previous stimulus presentation. Let us consider a basic example of this phenomenon, where the tuned external input is always presented in map . Consider for simplicity the state of the network being characterized by , as a result of the presentation of stimulus sometime in the past. If we now present a stimulus , the bump will move, through the shortest arc on the map, to the new location . Depending on the stimuli, this movement can happen in two ways. If the shortest arc from to is directed CCW, the bump will move with a positive velocity and will end up being located in the region (as we previously saw in the case of moving tuned input). If the shortest arc is directed CW, then the movement will happen with a negative velocity, and the final location of the bump will be in the region. Hence, by looking at the activity resulting from the presentation of , we know whether the shortest way on the ring to it from is CW or CCW. A similar result can be obtained if the stimulus presentation alternates between map and .

If we vary the manifolds on which the maps live, for example to segments instead of circles, the history dependence changes accordingly. For instance, on segments the activity would give us information about the second stimulus being greater/smaller than the first one (see Discussion). In the next section we present a simple (albeit artificial) delayed discrimination task which the network can perform by exploiting the memory effect.

An application of the memory effect

Let us suppose to have a screen with a circle on it. A first stimulus (a dot) appears on the circle at some random location (described by an angle, ), for the duration of . This first stimulus is then removed for a delay period of . Then a second stimulus appears at another random angle ; the subject's task is to determine whether the shortest path on the circle from angle to is CW or CCW. The basic idea is that it is enough to look at the network activity (location of the bump in the axis), to determine the relationship between the first and the second stimulus (see Results - Tuned external input for a description of the idea).

To test the ability of the network to solve this task, we numerically solve the dynamics for the order parameter (Results - Phase diagram of the model) with an external input (Results - Tuned external input) mimicking the presentation of the stimuli, for a sequence of trials. We used no inter-trial interval, i.e. the presentation of the second stimulus in the -th trial is immediately followed by the presentation of the first stimulus in trial . The time courses of the bump location on the axis () in two example trials for which , are shown in Fig. 6A. When looking at the location of the bump in the axis at the end of a trial, there is a clear difference between the two cases of shortest CW, corresponding to positive (in the specific example ), or CCW arcs (, where ). Fig. 6B shows that the bump location at the end of trial, can be used to easily discriminate between the two possible answers (except for the cases in which the first and second stimuli are relatively close to each other). Note that this result has been obtained without any activity reset to new initial conditions during the inter-trial intervals.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 6. Solving the shortest path task.

A: Two instances of MF dynamics from a network storing two correlated maps, during the presentation of two angles () with an intervening delay. The network operates in the cylinder regime. The plot shows the time course of the order parameter ; : pre-stimulus period, random starting , no tuned input. : first stimulus (presented in map , at (solid and dashed). : delay period, no tuned input. : second stimulus (solid) and (dashed). By looking at the position of the bump in () during the second stimulus presentation, it is possible to decide whether the shortest path between the first and second stimulus is CW () or CCW (). B: Sequential repetition of the task, trials. The location of the bump maximum along is plotted the oriented distance on the circle between the second and the first stimulus. Parameters .

https://doi.org/10.1371/journal.pcbi.1000869.g006

Morphing maps

How do the results described so far change when, instead of storing just two correlated maps, the network encodes a sequence of maps gradually morphed between two uncorrelated ones? Let us start by constructing two random uncorrelated maps, and . We would like to define the intermediate maps as gradual rotations between the two extreme ones; since we are dealing with circles, the rotation should be performed along the shortest arc between and (see Eq. 21, Methods - Inverse transformation). We assume here to have already transformed the variables in such a way that we can write directly(16)where indexes the maps along the morph sequence. Hence a neuron with label in the first map, will rotate along the sequence to its location on the last map, following the shortest path on the circle. With this choice of the morphing procedure, each neuron is still characterized by just two quantities, its labels in the extreme maps.

We store the whole morph sequence by a superposition of the synaptic structures generated in each map separately, as for the case of two correlated maps previously described. For the sake of analytical tractability, we study the resulting coupling in the limit (17)

Introducing the definition of two uncorrelated maps (Eq. (1) with ) into Eq. (16), we can rewrite the angles in the intermediate maps as , We can now integrate Eq. (17)(18)Making use of the Euler formula for the functionit is possible to deriveThe first term of the infinite product in the Euler formula, or the first term in the limit sum, gives us . Comparing the coupling in Eq. (18), and the one derived for two maps, Eq. (2), we see that to first order, the synaptic coupling induced by the storage of the whole morph sequence, is equivalent to the storage of two correlated maps with .

In Fig. 7, we compare the network activity generated by the approximated coupling and the full result of Eq. 18, when the external input is constant. The results are qualitatively similar but the full morph case reaches the cylinder regime for lower compared to the case. Note that the network storing the morph sequence shows the same dynamical pattern separation observed in the two maps case (Fig. 8), see next Section for a simulation example in a finite network with a finite number of encoded maps. The important difference, is that while the very correlations between maps forced the absence of neurons with certain labels, hence constraining the permissible region for a marginal solution in , here the neurons cover the entire () space. The result is purely due to the process of storing multiple maps along the morph sequence.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 7. Comparison between whole morph sequence storage and two correlated maps with .

A: Phase diagram of the model storing two correlated maps with , in the plane . Note that with this value of the distance, the double ring regime is not achievable. (A) region correspond to amplitude instability. B: Slice of the steady activity along , in correspondence of the maximum in . Left: parameters (dot in the region). Right: ( region); black curve, two stored maps; red curve, whole morph sequence stored. The bumps do not cover the whole range in both cases. External current .

https://doi.org/10.1371/journal.pcbi.1000869.g007

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 8. Representation switch along the morph sequence.

Slow morph experiment between two reference environments, over of simulation. The identity of the environment varies as (linear change from environment to ), where is either for the morph sequence, or for two correlated environments. The input location is also time-varying, (a full circle in ). A: Double ring solution. Order parameter with time, both for weak (solid curve) and strong external input (dashed curve). The result does not depend strongly on the amplitude of the external input. Parameters . B: Cylinder regime. Order parameter (location of the bump maximum in ) for the approximated storage of the morph sequence and for two correlated maps with , both for weak and strong input. Note how the sharpest transition, is exhibited by the weakly correlated maps. For the cylinder regime, the result is only mildly dependent on the strength of the external modulation. Parameters and . C: Instantaneous correlation coefficients between the double ring regime network activity during the morphing procedure, and network activity in environment (dashed curve) or (black curve). Simulations with weak external input. D: As in C, for the cylinder solution. Note how the sharp transition is delayed compared to .

https://doi.org/10.1371/journal.pcbi.1000869.g008

This morphing algorithm also yields a way of stimulating the network with positions in environments intermediate between and (with or without the intermediate maps encoded in the network). It is sufficient to use as a place specific input what we had in Eq. 13This time, the suitable range for the variable indexing the morph sequence is the whole range , if using as an approximation for the morphed case, or the restricted if the network is storing just two correlated maps. In the reference frame defined by the original coordinates (), a change in the stimulated environment corresponds to a rotation of the axis representing the maximal external input; between a vertical axis (stimulus localized in environment , to an horizontal axis, stimulus localized in environment .)

In the experiment of [32], the rat is trained until it develops two separate place coding for a single arena with different light configurations (representing two distinct environments). The advantage of this setup is that it allows, for instance, to slowly morph the light configuration between the two environments familiar to the rat. The experimental results shows a sharp transition around the middle of the light morphing (lasting ) between the place representation in light configuration and . A link to these experimental results is provided by the use of time-varying external environment , where represents the duration of the morphing and denote the upper and lower bounds of the range. An example usage of this protocol is shown in Fig. 8 for the approximated whole morph sequence storage, for two slightly correlated maps in the cylinder region of the parameter range and for the double ring regime. For each run we show the dynamics of the relevant order parameter for the regime under consideration, for the double ring case and for the cylinder solution. In addition, we numerically solve the dynamics for a moving stimulus in either environment or . We use this as a reference for computing, at each time step, the correlation coefficient between the network activity during the morphing protocol and the activity in the fixed environment. The transition is sharpest for the storage of two slightly correlated maps. Note that similar results would be obtained by testing the network separately in each environment of the sequence (see e.g. [31]). The sharp transition is maintained when increasing the amplitude of the external tuned input, because a small tilt in the tuned input towards either map or is sufficient to generate the dynamical pattern separation described in the previous Section. The transition in the cylinder regime occurs few seconds later than the one occurring in the double ring regime, which in turn happens in the middle of the morphing (). This delay is due to the time required for the bump to move from the region of to , or viceversa (see also Fig. 5B). This result could be compared with the experimental results of [32]. The delay does not occur when testing the network in separate environments along the morph sequence.

There are two additional observations to be made (data not shown). The first one is related to the sharpness of the transition in the double ring regime; by further reducing the amplitude of the external input, the mean-field dynamics can produce a sharp transition between the environments representations, which is also delayed compared to the middle of the morphing period. The delay gets longer as the external input gets weaker, in extreme cases it happens just before the end of the morphing procedure. This sharp and delayed transition is not observed in microscopic simulations with up to neurons, since the weak input is not able to overcome the local inhomogeneities in which the bump is trapped (see e.g. [18]). It is possible that in larger networks the transition can be observed. The fine-tuning of the external input strength required to have the transition around the middle of the sequence, makes the double ring regime a weaker candidate explanation for the experimental results of [32] compared to the cylinder regime.

The second observation concerns the dependency of the transition parameters on the velocity of the moving external input. We have noticed that the transition becomes smoother and closer to the middle as the velocity of the simulated animal is reduced. The details of the transition in a realistic setting would depend on the velocity history of the animal.

Comparison with simulations

In order to verify that the results obtained in the previous Sections are not artifacts coming from our assumptions of having an infinite number of neurons (and maps, referred to the morphing procedure) we compare some of the MF predictions to simulations of networks with a finite number of neurons. Each neuron is assigned a random pair of labels (, for the -th neuron), from which we create either two maps with distance , or a finite number of maps (, for the -th map) along the morph sequence between two uncorrelated references (see Methods - Numerical Methods).

In Fig. 9 we compare the order parameters from MF and estimated from simulations, at a fixed value of the distance between the maps and inhibition. Varying , the solution goes through the double ring, single ring and cylinder regime. The order parameter is particularly sensitive to the finite size of the network (and the randomized maps, see [18]).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 9. Comparison between MF solutions and microscopic network dynamics.

A: Phase diagram from MF analysis for . B: Order parameters from the numerical solution of MF equations (solid curves) with varying ; estimated order parameters from simulation results; dots with error bars, average and s.d. over simulations with different realizations of the neural maps. neurons. .

https://doi.org/10.1371/journal.pcbi.1000869.g009

Fig. 10 shows the time evolution of a network storing few maps from a morph sequence. This is the best example to show dynamical pattern separation at finite size, since it is less intuitive than the case of two correlated maps. From an arbitrary initial position, the bump of activity starts moving first towards negative (increasing angles in map ), then along increasing without changing its location in . Note that, despite the presence of neurons everywhere in the () plane, the bump moves along an invisible barrier resulting from the storage of the morph sequence.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 10. Dynamical pattern separation for a network storing a finite number of maps along the morph sequence.

Sequence of snapshots of the network activity, in and coordinates. The network with neurons is storing maps, equally spaced along the morph sequence. The external input is localized in map and evolves as . Each dot represent a neuron, with color coded activity. The dashed red line represents the maximum of the external tuned input. As the activity evolves in time from a random configuration, the subset of active neurons moves towards the negative region and stays there moving only along . The parameters are .

https://doi.org/10.1371/journal.pcbi.1000869.g010

We have also verified that all the qualitative behaviors, number and type of solutions, unreliable coding, dynamical pattern separation and memory effect, are maintained when moving from maps on rings, to segments (either two correlated maps or morphed), as studied e.g. in [18], [37] for the single map (data not shown). Instead of having neurons arranged on a cylinder in the coordinates, as for the ring case (see e.g. Fig. 2B,II), the geometry resulting from two correlated linear maps would be an infinite strip. A strong enough map-specific interaction would produce a bump localized in both maps. An external moving input in one of the maps would move the bump on the strip up to the boundary, and then the bump would crawl along such boundary. Depending on the direction of the moving input or the identity of the stimulated environment, the bump can settle either in “upper” of “lower” part of the strip as in the cylinder regime.

Numerical methods

To solve numerically the MF dynamics described by Eq. 5, we discretized on regular grid in The integrals in the rhs of the equations were estimated using a trapezoidal method. The system of ODEs were integrated with an adaptive 4-th order Runge Kutta scheme.

The simulation of the microscopic networks, whose results are reported in Figs. (9,10), were performed by solving numerically the system of ODEs(19)where indexes the neurons. The matrix is built by summing the single map encoding , whereTo obtain the labels characterizing each neuron, we first randomly generated a and used Eq. 1 for or Eq. 16 for . For the comparison of the simulation with the MF results in Fig. 9, we estimated from the steady state activity (compare with Eq. 22)(20)from which we constructed the estimates for the order parameters, using Eq. 24.

For the noisy simulations shown in Fig. 5B, we used a current-based version of the dynamics described by Eqs. 19:We then estimated the order parameters via Eqs. 20, using the firing rates . The noise was introduced as an additional term in the currentwhere is a zero average, unit variance Gaussian -correlated noise. We used for the results in Fig. 5. The numerical solution was obtained using the Euler-Maruyama integration scheme.

The simulations performed in Fig. 7, for a network storing the whole morph sequence, were carried out as follows. Substituting the synaptic coupling obtained in Eq. 18 with the one in Eq. 2, it is possible to derive a dynamics for the “order function”following the same procedure of Methods - Reduced dynamics. An order parameter is defined exactly as in Eq. 22. The steady state activity of such dynamicswas compared with Eq. 4, for and (in absence of a spatially tuned input is constant).

The time constant was set to everywhere.

Inverse transformation

The inverse transformation can be obtained from Eq. 1, defining(21)The rotation in the first equation is just needed to select the shortest distance between two maps on a ring, and it is transparent for the connectivity given its periodicity. This rotation was implicitly assumed when defining the neurons locations along the morph sequence, Eq. 16.

Reduced dynamics

A first reduction of the dynamics described by Eq. 2 is done using the first two Fourier components of the activity with respect to the two correlated maps and , rewritten in terms of center map and the distance using Eq. 1. In line with [37] we define the following variables(22)The variable is just the average activity, while and measure the spatial modulation of the network activity, in the map and respectively. Intuitively their values tell us which angle of which map is instantaneously represented by the network.

The dynamics of the network activity , and of the order parameters , becomes(23)withIt is convenient to introduce dimensionless combinations of the order parameters to better expose the structure of the solutions, and then derive the dynamics of these new order parameters. From the two complex variables and the real one, we construct five new variables(24)From Eqs. 23, after some algebra, it is finally possible to obtain the dynamics of the new order parameters, Eq. 5.

Solutions properties

In Methods - Phase diagram of the model, we mentioned that the equation from Eqs. 5, i.e.is automatically satisfied once the solution for the other four order parameters has been found. This can be seen using the fact that, by definition, the imaginary part of the real numbers and is . Since at steady state , and (Eqs. 22), by computing we can prove the property.

Another statement mentioned in Methods - Phase diagram of the model, is that fixed points solutions of Eqs. 5 with do not exist. Observing the shape of the network activity at steady state (Eq. 3) (setting the phase for convenience) which we rewrite herewe would like to know, given the correlation between the stored maps and the bump size , how much we can move the bump along by increasing without having active neurons at . We first analyze the onset of the freedom of choice of , by requiring the bump to “fit” exactly the range; with a bigger bump, the only possible choice for would be , with a smaller bump it would be possible to move it along . Hence, posing , the activity at would beThe angle at which this activity is maximal isso the maximal activity at the boundaries isWe recognize the first term inside the transfer function to be positive, so the only way to obtain a vanishing activity is to have . From Fig. 4 it is possible to see that the double ring solutions have always size .

In order to obtain the range of integration for used to compute the average tuning curve in the cylinder regime (Results - Tuned external input), it is enough to consider the activity at its maximum in We want this bump in to at most touch the endpoints . Given that the half-width of the bump is , the allowed range for is .

Stability

In order to study the stability of the homogeneous solution, corresponding to in Eq. 23 (i.e. from 24), we can either linearize Eq. 23, or take a step back from the MF reduction which lead to Eq. 5, so to avoid division by . We take the second approach and redefine one of the order parameter, . To study the stability of the solution, it is sufficient to look at the dynamics of and . Posing , it is easy to verify thatwhere the function is defined asThe matrix describing the linear dynamics for the vector of small perturbations around the solution readswhereTherefore, two conditions must be satisfied for the solution to be stable: (amplitude instability) and . Evaluating the integral in explicitly, we get the line of separation between the homogeneous solution and the localized bump (Turing instability), expressed in Eq. 7.

For the single, double ring and cylinder solution we can linearize directly Eq. 5, posing . The matrix associated with the dynamics of the vector , after using the fixed points equations (Eq. 8), is(25)We define(26)whereUsing the identity , to write the function (Eq. 6), we see thatThe fixed points equations can thus be rewritten in term of the quantities in Eq. 26(27)

Let us examine the single ring and cylinder solution, . Given the symmetry in the integrand, in this case. The stability matrix from Eq. 25 becomes then(28)

It is immediately seen that the eigenvalue corresponding to a destabilization of changes sign when . Substituting for the expression in Eq. 27, it is easy to verify that this reproduces the curve of separation between the single and double ring regime described by Eq. 12. We analyze the remaining two eigenvalues by looking at the trace and the determinant of the sub-matrix in the subspace (from 28):Given that , we see immediately that the eigenvalues have the same sign for , and one of them changes sign when . Recall that is the onset of amplitude instability we introduced without proof in Eq. 9. If we find that when the trace is negative, then we know that correspond to a destabilization of the solution.

Using Eq. 27, we see that imposing is equivalent to , and that the trace satisfiesThe numerator in the first term is non-negative (). The denominator is simply , non-negative by definition. The denominator in the second term is , and we can write the numerator asFinally, we numerically verified that the region of existence of the double ring solution coincides with its stability region.