A Passive Mechanism for Goal-Directed Navigation using Grid Cells

As more is becoming understood about how the brain represents and computes with high-level spatial information, the prospect of constructing biologically-inspired robot controllers using these spatial representations has become apparent. Grid cells are particularly interesting in this regard, as they provide a general coordinate system of space. Artiﬁ-cial neural network models of grid cells show the ability to perform path integration, but important for a robot is also the ability to calculate the direction from the current location, as indicated by the path integrator, to a remembered goal. Present models for goal-directed navigation using grid cells have used a simulating approach, where the networks are required to actively test successive locations along linear trajectories emanating from the current location. This paper presents a passive model, where differences between multi-scale grid cell representations of the present location and the goal are used to calculate a goal-direction signal directly. The model successfully guides a simulated agent to its goal, showing promise for implementing the system on a real robot in the future. Some possible implications for neuroscientiﬁc studies on the goal-direction signal in the entorhinal/subicular region are brieﬂy discussed.


Introduction
Results from neuroscience are gradually uncovering the neural basis for navigation, as cell types such as place cells and grid cells, first discovered in the hippocampal region of rats, have been shown to represent high-level features of the animal's spatial environment.These findings offer the prospect of beginning to understand how the brain computes and represents abstract cognitive features.Inspired by these advances, the basis for this project has been to devise and implement a neural model to enable a robot to find its way to a previously visited goal location using these neural representations of space known from the brain.Through crafting these models, we hope to gain insights into how these spatial representations might be utilized for navigational purposes by neural systems, artificial and real alike.
The first evidence of spatially responsive cells in the rat hippocampus came with the discovery of place cells, which were seen to respond at distinct locations in the environment (O' Keefe and Dostrovsky, 1971).However, place cells do not appear to encode any metric relations, such as distances and angles (Spiers and Barry, 2015).The place cell representation by itself is thus not sufficient to be able to navigate between arbitrary locations, because it does not offer any means to calculate the direction of travel from one place cell's firing location to that of another.Grid cells, discovered later in the neighboring entorhinal cortex (Hafting et al., 2005), offer a possible solution, as the grid cell system can be seen as a general spatial coordinate system.Given the grid cell representations of two locations it is possible to compute the distance and angle between them, thus providing the needed metric of space.
Grid cells are thought to update their firing activity based on self-motion information, in other words to perform path integration/dead reckoning (McNaughton et al., 2006).However, for a path integrator to be fully useful for navigational tasks, an agent should be able to use this information to find its way back to previously visited locations.In this paper we shall see how the activity of path integrating grid cell networks can be used to guide a simulated agent toward a remembered goal location.

Related Work
Path integration is the basis for several computational models of grid cells, the collection of which can roughly be divided into two major categories; oscillatory-interference models and attractor-network models (Giocomo et al., 2011).Several biologically-inspired models for navigation have used such models of grid cells (Milford and Schulz, 2014;Spiers and Barry, 2015).Often this has been for the purpose of position tracking, as in the bio-inspired robot navigation system "RatSLAM" (Milford and Wyeth, 2010), but in some cases grid cells have also been used for direction-finding.Erdem and Hasselmo (2012) use a model with oscillatory interference-based grid cells to find directions to remembered goal-locations.The mechanism involves testing a number of "look-ahead probes" that trace out linear beams radially from the current location of the agent.Each of these probes orchestrate activity across the entire population of Vegard Edvardsen (2015) A Passive Mechanism for Goal-Directed Navigation using Grid Cells.Proceedings of the European Conference on Artificial Life 2015, pp. 191-198 grid cells and place cells to make it appear as if the agent were actually situated at the tested coordinates.If any of the successive locations tested during a given look-ahead probe triggers a reward-associated place cell, the agent is impelled to travel in the specific direction of that probe.Kubie and Fenton (2012) show how a Hebbian learning mechanism between conjunctive grid cells can train the grid cell networks to be able to generate look-ahead trajectories similar to those suggested by Erdem and Hasselmo (2012).The authors propose that this is a "viable candidate for vector-based navigation".Common to these two approaches is the requirement for the model to explicitly test a wide range of different directions emanating from the current position, in order to expectedly trigger the goal-reward in some specific direction.In this sense, we can term these models active mechanisms for goal-directed navigation.
The model proposed in this paper goes the other way about the problem, by presuming that the grid cellrepresentation of the goal location is known beforehand.The current grid cell state and the goal grid cell state propagate through a pre-wired network of neurons that calculate the offsets between the two representations in order to generate a direction signal.This process can be constantly ongoing in the background, without requiring exclusive use of or otherwise interfering with any grid cell or place cell population for the purpose of "simulating" forward locations.We thus consider our model a passive mechanism for goaldirected navigation.

Background Grid Cells are Organized in Modules
The name "grid cell" stems from the spatial activity patterns of these neurons; the cells are not active only within single spatial fields in the environment as the place cells are, but have a periodic pattern of activity that repeats at the vertices of a triangular tiling of the plane.The result is a hexagonal grid pattern, extending indefinitely throughout space, that can be characterized by the three properties of scale, rotation and phase-respectively the distance between two neighboring vertices of the grid pattern and the rotation and translation of the grid pattern compared against a frame of reference.A grid cell does not operate in isolation, but participates in a module of grid cells that share the same scale and rotation of their individual firing patterns (Stensola et al., 2012).The only distinguishing property between neurons within the same grid cell module is thus that of their phase (Figure 1).
Assuming that a sufficient number of grid cells belong to a given module, the module as a whole has the ability to encode a given set of 2D coordinates in a nearly continuous fashion.The limitation lies in the periodic nature of the grid cell pattern, in that the information carried by a grid cell module can only be interpreted relative to one specific hexagonal tile of the infinitely repeating pattern.A possible  solution comes from the fact that the entorhinal cortex harbors grid cell modules of multiple different scales (Stensola et al., 2012).It is conceivable that the smaller-scaled grid cell modules represent space at a finer resolution than the larger-scaled ones, but with the sacrifice of having shorter "ranges of validity" due to the more rapid periodicity of the grid pattern.The activity of all modules taken collectively, however, ought to retain both the low-precision/longrange information of the larger-scaled modules as well as the high-precision/short-range information of the smaller-scaled modules.The utilization of this multi-scale mix of information is a key idea behind the model presented in this paper.

Attractor-Network Models of Grid Cells
Attractor-network models of grid cells conceptualize their neurons as being laid out in a 2D neural sheet.Proximity between neurons in this sheet implies that the neurons should have similar phases of their grid patterns, not necessarily that the neurons would be co-located in the brain.The neurons are recurrently connected to each other, with a connectivity profile based on distances in the neural sheet, in such a way that grid-like patterns of activity will form in the network from random initial conditions (Figure 2).These network patterns, which are the attractor states of the network, can then be made to shift around in the network in response to self-motion signals in order to perform path integration.Assuming these shifts consistently reflect the actual movements of the agent, the network pattern will over time become visible in the spatial activity patterns of individual neurons in the network.Attractor-models of grid cells thus have grid-like patterns both in their time-averaged spatial activity plots (Figure 1) and in their momentary network activity plots (Figure 2), and this is an important distinction to be aware of.Self-motion speed

Self-motion direction
Motor-output neurons

Motor strength
Motor direction Figure 3: A schematic overview of the model.

Model
The main part of the model comprises a configurable number of modules, seen as the two rectangular blocks in the middle of Figure 3.Each module m consists of (a) a grid cell module, (b) a target signal, and (c) a network of phase offset detectors.The grid cell modules perform path integration on the incoming self-motion signal (composed of speed and direction), and output vectors of grid cell-activity s m that are passed on to the corresponding networks of phase offset detectors.These phase offset detectors also receive a copy of the intended grid cell-activity vector t m for the desired target location-the "target signal".The task of the phase offset detectors is to find the required direction of travel to make up for the offset in the grid patterns between the path integrator signal s m and the target signal t m .The intended outputs of the model are a motor direction signal, giving the direction toward the target location, and a motor strength signal, indicating whether the agent has arrived at the target location or to keep going.

Multiple Modules with Different Spatial Scales
The model has these multiple parallel modules in order to utilize information from a variety of grid cell modules representing space at different scales; this will provide the direction-finding process with long-range/low-precision signals as well as short-range/high-precision signals.The different grid scales are achieved by modulating the velocity inputs to each grid cell module.The velocity signal to module m is multiplied by the gain factor g m before reaching the grid cell network.Smaller gain factors will cause the path integrator to respond more slowly to the same velocity inputs, thus causing the grid to appear larger, and vice versa.
The path integrator model used in these simulations was found to respond acceptably to velocity inputs at least in the range from 0.1 m/s to 1.2 m/s.As the actual speed of the simulated agent was fixed to 0.2 m/s, the range of acceptable gain factors could then be determined to be [g min , g max ] = [0.5, 6.0].The model uses a geometric progression from g min to g max for the gain factors.Given a specific number of modules M to be used, the g m values can then be calculated as Path Integrating Grid Cell Modules The path integrator modules are closely based on the attractor-network grid cell model by Burak and Fiete (2009), and the following formulas are based on their presentation of the model.Each grid cell module consists of a 2D sheet of neurons of size n × n, where n = 40.The activation values of these n 2 = 40 2 = 1600 neurons are contained in a vector s, fully representing the current state of the path integrator.Each grid cell i receives recurrent inputs from all other neurons in s.Let x i be the neural sheet coordinates of neuron i.The weight from afferent neuron i onto neuron i can then be calculated from the connectivity profile rec(d) by letting d be the shortest distance between x i and x i in the neural sheet, taking into consideration that connectivity may wrap around the N/S and W/E edges.The recurrent connectivity profile rec(d) is a difference of Gaussians, seen as the inhibitory "doughnut" in Figure 4, top left.Specifically, where γ = 1.05 • β, β = 3 λ 2 and λ = 15.λ approximately specifies the periodicity of the grid cell network, i.e. the number of neurons from one peak of activity to the next.
To express the update rule for grid cell i using vector notation, let w rec c be the weight vector derived from the distanceto-weight-profile rec(d) centered on the point c in the neural sheet.The update rule can then be described as solved for ds i , where dt = 10 ms, τ = 100 ms, f (x) = max (0, x) and s is the vector of the activation values at the end of the previous timestep.
The center point c of the connectivity profile for efferent neuron i is here given as x i − êθi , i.e., there is an extra offset of êθi in addition to x i when positioning the connectivity profile for neuron i.The offset êθi is the unit vector in the direction of θ i , which in turn is the directional preference of neuron i.The directional preference is used to shift the activity pattern among the grid cells in response to asymmetrical velocity inputs.Preferences for each of the four cardinal directions are distributed among the neurons in each 2 × 2 block of neurons.Namely, the x, y coordinates of a neuron are used to calculate an index (2 • (y mod 2) + x mod 2) into the list [W, N, S, E] to determine θ i .In the absence of velocity inputs, the four distinct preference-offsets counterbalance each other to keep the activity pattern at rest in the network.During motion, however, the external input B i to each neuron becomes velocity-tuned according to the directional preference of the neuron.This input is calculated as where v is the movement velocity and α = 0.10315 is a scaling constant specified by Burak and Fiete (2009).
Vegard Edvardsen (2015) A Passive Mechanism for Goal-Directed Navigation using Grid Cells.Proceedings of the European Conference on Artificial Life 2015, pp.191-198

Phase Offset Detectors
The vector of activation values s is passed on to a network of phase offset detectors.In addition to receiving the input vector s from the path integrating grid cell module, the phase offset detectors also receive a similarly-shaped vector t that represents the grid cell activity of the target location, i.e. a grid cell-encoding of the desired target coordinates.Each phase offset detector j has an associated origin location x j and a preference direction θ j .The neuron is tuned to respond when an activity peak is near the origin location x j in the neural sheet of the path integrator grid cell module (s) and there simultaneously is an activity peak near the location z j = x j +δê θj in the grid cell-encoded target-location-input (t).Specifically, the activation value of phase offset detector j is calculated as where w again refers to weight vectors derived from given connectivity profiles centered on given points in the neural sheet, but with new connectivity profiles in and ex.The path integrator inputs s are fully connected using the connectivity profile in(d) centered at x j , while the target location inputs t are fully connected using the connectivity profile ex(d) centered at z j .These connectivity profiles are defined as where η = 0.25.The offset length δ is set to be δ = 7, in the neighborhood of half of λ.
The effect is to respond the most strongly when there is an offset of length δ in direction θ j between the activity patterns in s and t, given that the path integrator currently has activity in the vicinity of x j .An example situation is shown in Figure 4, where a phase offset detector with x j = (20, 20) and θ j = 45 • receives inputs of favorable characteristics from s and t.
In order for the network of phase offset detectors to work independently of the current location of network activity in the path integrator, there needs to be a sufficient number of phase offset detectors that sample different origin locations x j .Additionally, the network needs to sample a range of different preference directions θ j .This is realized using two parameters S θ and S xy that respectively specify the number of directions sampled in the interval [0, 2π) and the number of steps to use along each of the two dimensions of the neural sheet when sampling origin locations.The total number of phase offset detectors will then be S θ • S 2 xy per module.

Motor-Output Neurons
The activity from the phase offset detectors are aggregated in a set of motor-output neurons.Whereas the grid modules and phase offset detectors are instantiated separately for each module, the motor-output neurons comprise a common network receiving inputs from all of the modules.The number of motor-output neurons is the same as the number of sampled preference directions S θ in the phase offset detector networks.The motor-output neurons sample the same directions as the phase offset detectors.
The activity in each motor-output neuron is essentially the sum of the activity in all of the phase offset detectors that share preference direction with the motor neuron.An important detail, however, is that these contributions are weighted by the inverse of the gain factors of their respective modules.In other words, where u k is the activity of motor-output neuron k and θ k is the preference direction of k.This weighting will give priority to the direction signals from the modules with low gain factors g m , i.e. the modules where the quality of the path integration information is long-range-applicable but with low precision.As the agent gets closer to the target location, the intention is for these signals to fade off to sufficiently weak strengths so that the shorter-range, higher-precision signals will pick up in motor influence.The purpose is to achieve the trade-off of a long-range and high-precision signal.
To calculate the final motor-output signal Θ, the values of u k are considered as vector contributions in the direction of θ k , i.e. the vectors u k • êθ k are summed together, and this sum is then scaled to compensate for the variable number of inputs and their weighting.The final calculation is thus whereafter the angle of Θ makes up the motor direction signal and the vector length becomes the motor strength signal.
Vegard Edvardsen (2015) A Passive Mechanism Goal-Directed Navigation using Grid Cells.Proceedings of European Conference on Artificial Life 2015, pp.191-198

Experiment Setup
Each experiment trial consists of a succession of stages, specifically (a) pattern formation in grid cell modules, (b) capture of path integrator states into target states, (c) the agent performing a random walk for T seconds, and (d) the agent attempting to return "home" to the target location.At the beginning of the simulation, in order for the grid cell networks to form grid-like activity patterns, all s i values are initialized randomly in the range [0, 10 −4 ) before the networks are then allowed to settle for 1000 timesteps (Figure 2).When this pattern formation process is done, the grid-like activity patterns will have been initialized to essentially random starting-coordinates.The model now copies these activity patterns s m into the target state vectors t m .The m different target state vectors t m henceforth remain unchanged for the rest of the trial, as a memory of the coordinates of the starting location ("home").
The agent then performs a random walk for a configurable duration of time T seconds.The time duration for a single iteration of the model has been set to be 10 ms, so there are 100 timesteps/s.During both the random-walk and the return-home stages, the agent moves with a constant speed of 0.2 m/s with only the movement direction changing.The random walk starts with a uniformly distributed random value from [0, 2π) as the movement direction.At every timestep it is updated by adding a radian value from a normal distribution with µ = 0, σ = 1.
T seconds have elapsed, the return-home stage begins.The motor-direction output from the network is used to set the movement direction of the agent, whereas the motorstrength output is used as a termination criterion for determining when to end the trial.Three different termination criteria are used; (a) the motor-strength signal is less than 10 −6 , (b) the return-home stage has lasted for at least a second and the straight-line distance to the point traversed one second ago is less than 0.01 m, or (c) the return-home stage has lasted 2 • T seconds.Whichever termination criteria ends the trial, the straight-line distance to the starting location from the final stopping location is deemed the error of the trial.The favorable outcome is a low overall error value.

Parameter Search
A parameter search was conducted to find the best values for M , S θ and S xy to use for the rest of the experiments.An exhaustive test was performed on all combinations of values in the intervals M ∈ [2, 6], S θ ∈ [4, 32], S xy ∈ [5,40].However, to penalize expensive solutions and to place an upper bound on the complexity of the solutions to be tested, a synapse cost C was calculated for each parameter combination.This value provides an estimate on the number of synapses in the model and consequently a rough estimate on the number of floating-point operations required to update the model (without optimization).C was calculated as the three terms representing the synapse cost to operate respectively a grid cell module, the phase offset detectors and the axons to the motor-output neurons.Only the combinations with C < 10 8 were tested, leaving 2685 combinations to test.For each combination, 100 trials with a random-walk duration of T = 30 s were performed and the mean error was reported (Figure 5a).The parameter combination M = 4, S θ = 28, S xy = 9 was selected for further use (highlighted).
To get a sense for how the individual parameters affect the outcome, new sets of runs were performed where each parameter in turn was changed within the defined intervals and evaluated over 100 trials, while the two other parameters were left unchanged (Figure 5b).M and S xy seem to affect the results little above thresholds of respectively M = 2 and S xy = 8. S θ , on the other hand, appears to be more sensitive to the particular value to which it is assigned.

Implementation Details
All random values used by the implementation in this paper were generated using the Mersenne Twister pseudo-random number generator included with the C++11 standard library.

Direction-Finding Ability
Two different examples of how the system operates in practice are presented in Figures 6 and 7.In the first example, the direction-finding ability of the model is tested at multiple points along a circle centered on the goal location.For each of the 18 uniformly spaced directions tested, the agent was driven a distance of 0.5 m in the opposite direction of the intended "goal direction" and allowed to settle for 250 timesteps before the motor outputs were examined.For each trial, the figure shows the recorded activity from all S θ motor-output neurons as well as the motor-direction signal from the model.As evidenced by the figure, the model is able to accurately calculate the goal direction at this specific distance of r = 0.5 m.
Figure 7 demonstrates a full trial with both random-walk and return-home stages as described above.After a T = 30 s random walk, the agent successfully attempts a return to the home location.The figure includes the momentary activity of all of the grid cell modules (s m ) both at the beginning and the end of the return-home stage.In each of these cases, the motor-neuron activity is also shown.The plots of s m show possible interpretations of how the activity patterns might have shifted from the target state t m , which was also the initial state of the grid modules at T = 0 s.From the leftmost to the rightmost columns, the grid modules progress from longrange/low-precision to short-range/high-precision.The first, second and third modules show a correct assessment of the goal direction at T = 30 s, whereas the fourth module is "out of range" and in this case has an ambiguous response.
At T = 37.1 s, we see that the grid modules have aligned closely with the corresponding target states.The trial thus terminated because of the weak motor-strength signal, bringing the agent to a halt at a distance of 4.74 cm from the goal location, from an initial goal distance of 1.47 m at the end of the random walk.

Effect of Multiple Grid Modules
The effect of using multiple grid modules is further demonstrated in Figure 8, which shows, as a function of the distance to the goal, the strengths and errors of each module's contribution to the motor-output network when seen in isolation.To illustrate their relative influences, the signal strength of each module is also shown in terms of its ratio of the sum.Lastly, the final motor-direction error is shown overlaid on the direction-error plots from the individual modules.
For each module, there is a distinctive bell-shape in the strength curve as the tested radius approaches and recedes from the "optimal detection distance" of the module's offset detectors.The vicinity of the peak of the bell curve is also

Random walk Return home
(c) For each tested radius the motor-output strengths and direction-signal errors are reported as the mean over 18 tested directions.In order to report values individually for each module, extra motor-output networks were instantiated such that each only received phase offset detector-inputs from one given module.For these plots, ρ = 1 in order not to cancel out the scaling differences.
where the module's direction-error is at a minimum.Past this region, the module abruptly becomes unreliable due to the periodicity of the grid cell signal.Because of the gainbased weighting of module contributions, however, one of the larger-scaled modules is able to overpower the contributions of the smaller-scaled modules and thus ensure that the final direction signal is still valid.As seen by the red line in the lower diagram, the final direction-output achieves a trade-off between range and precision not seen in any of the individual modules.Figure 9 demonstrates the importance of this combination of precision and range information.The figure contains results from three different sets of 500 trials, each with T = 180 s.Whereas the rightmost diagram shows the results from trials with the default parameters (M = 4, S θ = 28, S xy = 9), the two other diagrams only use one module (M = 1).The leftmost diagram has the gain factor set to g 1 = g max , for rapid periodicity and short-range/highprecision signals, while the middle diagram has g 1 = g min , i.e. tuned for long-range/low-precision signals.
The distributions of termination locations seen in the scatterplots confirm our expectations from the known qualities of the grid module signals.With one module tuned for precision (Figure 9a), the agent either precisely returns home to the target location or it ends up in an attractor location that is part of a repeating pattern of possible attractors.This shows the periodic nature of the grid cell encoding of space.With one module tuned for range (Figure 9b), all but one of the 500 trials terminate in a cluster centered on the target location.However, the improved range has carried a penalty of worse precision.This penalty is seen to be mostly alleviated by integrating information from multiple grid modules; in Figure 9c, where four grid modules are used, all but six of the 500 trials end up within 0.5 m of the target, with only one ending up more than 1 m away.
To get a sense for the trajectories the model follows during these attempts to reach the target, Figure 10 contains traces of the 50 runs from Figure 9c with the farthest goal distance at the end of the random walk.With some exceptions, the paths taken are all largely straight lines toward the target.All but one trajectory (seen near the top) end up within 0.5 m of the goal.

Discussion
The basis for this project was to use neural representations of space for direction-finding in a robot.The paper has pre-  sented a model that integrates an existing model of path integrating grid cells with a novel mechanism that is able to use the grid cell representation to direct the agent to a remembered goal.The successful simulation results show promise for using the model in a physical robot in the future.The translation into the physical world will bring with it its own set of challenges, such as noisy self-motion inputs and imprecise motor control.The integration of sensory information into the model is thus one important area for further study, as has been done in other grid cell-based robot controllers (Milford and Wyeth, 2010).We consider the model at its current abstraction level to be biologically plausible.The inputs and outputs of the model are geocentric direction and speed signals, which is supported by the existence of head-direction cells.Attractornetwork models are considered viable candidates for understanding the operation of grid cells, and the phase offset detectors and motor-output neurons are simple input-summing neurons.The target state signals are assumed to be a grid cell-encoding of the target coordinates; this could be provided in the form of backprojections from the hippocampus.Chadwick et al. (2015) used VR-supported fMRI to look for a goal-direction signal in the human brain.They found that there would be similar brain activity patterns in the entorhinal/subicular region when a given geocentric direction was used as either the current facing direction or the goal direction, and the activity patterns were found to be best accounted for as a mixture of the encodings of the facing direction and the goal direction.The authors see these results as evidence that some form of goal-directed simulation of spatial representations is involved in navigation, citing the model by Erdem and Hasselmo (2012) as an example of how this mechanism could work.This is not the only possible conclusion of these results, especially since the model by Erdem and Hasselmo requires simulation of look-ahead trajectories in many different directions from the current location in order to discover the goal location.A mechanism similar to the one presented in this paper would allow the goal direction to be calculated directly from grid cell representations of the current location and the goal, avoiding the need for extensive simulations in multiple directions.The mixture representations reported by Chadwick et al. could still conceivably be accounted for by oscillations in the entorhinal/subicular region between encodings of the present and the future spatial states.Experiments at finer spatial and temporal resolutions would hopefully be able to distinguish the extents of these two types of contributions.

Figure 1 :
Figure 1: Idealized illustration of the activity of two different grid cells, as a function of the 2D location of the agent, in a cylindrical enclosure of e.g. two meters in diameter (top-down view).Red and blue colors indicate high and low firing rates, respectively.The two grid cells belong to the same grid module, as per the identical scale and rotation of their respective grid patterns, but are seen to have different phases, i.e. different offsets of these patterns, as indicated by the lines.

Figure 2 :
Figure 2: Spontaneous formation of a grid-like activity pattern in the neural sheet of an attractor-network grid cell module, due to random initial conditions and the recurrent connectivity.
Vegard Edvardsen (2015) A Passive Mechanism for Goal-Directed Navigation using Grid Cells.Proceedings of the European Conference on ArtificialLife 2015, pp.191

Figure 4 :
Figure 4: Example of a phase offset detector, pj, showing the input networks s and t and the connectivity profiles with which these two networks are connected to pj.Depicted matrices are 40 × 40.

Figure 5 :
Figure 5: (a) Scatter plot of parameter combinations tested during the parameter search, with each dot showing the mean error over 100 runs for a given parameter combination, plotted against the respective synapse cost.(b) For each of the three parameters M , S θ and Sxy, the effect of modifying that parameter from the chosen parameter combination (M = 4, S θ = 28, Sxy = 9; indicated by vertical lines) while leaving the other parameters unchanged.Mean error over 100 runs.The combination M = 4, S θ = 28, Sxy = 9 is represented by the same set of trials in all three figures.

Figure 6 :
Figure 6: Example of network operation at locations a fixed radius r = 0.5 m away from the home location in given directions.Left: Momentary activity in the S θ = 28 motor-output neurons.Right: Final motor direction calculated from the motor-output neurons, plotted at the closest 2π period to the goal direction.

Figure 7 :Figure 8 :
Figure 7: Example of network operation during various stages of a trial.(a) Trace of the trajectory followed during random-walk and returnhome stages.(b) Top row: Target state.Rest: Momentary activity of the grid cell modules and motor-output neurons at various points in time.(Thesi and ti values were arranged according to xi and convolved with a uniform 2 × 2 filter to hide artifacts due to different preferenceoffsets.This procedure was used for all s and t visualizations in this paper.)(c) Plot of the inverses of the gain factors gm used when the model is configured for four modules (M = 4), in order to illustrate the difference in the ranges of the modules.

Figure 9 :Figure 10 :
Figure 9: End locations after returning home, for various configurations of grid cell modules.500 runs for each configuration.Gray dots show end of random walk and black dots show end of return home.Histograms below show the distribution of goal distances at trial termination, with a bin size of 15 cm.(The upper-left run in part b stopped at a distance of 29.7 m from the target.)