Abstract
Under modern interrogation, famously well-studied neural circuits such as that for orientation tuning in V1 are steadily giving up their secrets, but quite basic questions about connectivity and dynamics, including whether most computation is done by lateral processing or by selective feedforward summation, remain unresolved. We show here that grid cells offer a particularly rich opportunity for dissecting the mechanistic underpinnings of a cortical circuit, through a strategy based on global circuit perturbation combined with sparse neural recordings. The strategy is based on the theoretical insight that small perturbations of circuit activity will result in characteristic quantal shifts in the spatial tuning relationships between grid cells, which should be observable from multisingle unit recordings of a small subsample of the population. The predicted shifts differ qualitatively across candidate recurrent network mechanisms, and also distinguish between recurrent versus feedforward mechanisms. More generally, the proposed strategy demonstrates how sparse neural recordings coupled with global perturbation in the grid cell system can reveal much more about circuit mechanism as it relates to function than can full knowledge of network activity or of the synaptic connectivity matrix.
Introduction
Questions about the origin of the beautiful tuning curves often seen in sensory and cortical circuits have long consumed systems neuroscientists, both theorists who propose possible mechanisms, and experimentalists who search for them (Hubel and Wiesel, 1959). Indeed, the mechanisms underlying direction tuning in the retina and cortex and orientation tuning in V1 remain unresolved and closely studied (Rivlin-Etzion et al., 2012; Kim et al., 2014; Takemura et al., 2013; Ferster and Miller, 2000; Sompolinsky and Shapley, 1997). Basic questions, like whether orientation tuning is largely attributable to selective feedforward summation or lateral interactions, are not yet settled.
Here we propose that the grid cell system provides a unique opportunity for understanding the underpinnings of computation in cortical circuits. The unusual responses of grid cells present a challenge and simultaneously, an opportunity. The challenge is to understand how such a complex cognitive response is generated; the opportunity is the availability of versatile experimental tools and a rich set of relatively detailed models (Fuhs and Touretzky, 2006; Guanella et al., 2007; Burak and Fiete, 2009; Burgess et al., 2007; Hasselmo et al., 2007; Welday et al., 2011; Mhatre et al., 2012; Bush and Burgess, 2014; Hasselmo and Brandon, 2012; Navratilova et al., 2012) that are well-constrained by the very complexity of the grid cell response, to help meet the challenge.
The recent application of quantitative analyses to electrophysiological data reveals that the population activity of grid cells (within individual modules) is localized around a continuous low-dimensional (2D) manifold (Yoon et al., 2013; Fyhn et al., 2007), a finding that lends support to early models predicated on the idea of low-dimensional pattern formation through strong lateral interactions (Fuhs and Touretzky, 2006; Burak and Fiete, 2006; McNaughton et al., 2006; Guanella et al., 2007; Burak and Fiete, 2009), as well as other models in which grid cells receive location-coded inputs and through structured feedforward connections (with the possible addition of some lateral connectivity) generate grid-patterned responses (Kropff and Treves, 2008; Mhatre et al., 2012; Welday et al., 2011; Bush and Burgess, 2014).
These models are architecturally and mechanistically distinct in important ways, both large and subtle: they differ in whether grid cells perform velocity-to-location integration, in whether pattern formation originates wholly or partly within grid cells, and in the structure of their recurrent circuitry. Some of the structural differences within recurrent models which seem subtle have qualitative ramifications for how the circuit could have developed. Despite their differences, the models are difficult to distinguish on the basis of existing multiple single-unit activity records, because all of them produce grid-patterned outputs and exhibit approximate 2D continuous attractor dynamics. Worse, as we discuss at the end, neither complete single neuron-resolution activity records nor complete single synapse-resolution weight matrices will be sufficient to distinguish between proposed mechanisms.
We show how it is nevertheless possible to gain surprisingly detailed information about the grid cell circuit from a feasible experimental strategy that depends on global circuit perturbation and sparse neural recording. In this context, global means circuit-wide not brain-wide. The proposed strategy can allow the experimenter to discriminate between various distinct candidate mechanisms that are currently undifferentiated by experiment.
Results
Experimentally undifferentiated grid cell models
Let us begin by considering 2D recurrent pattern forming models, in which grid cells are assumed to integrate velocity inputs and output location-coded responses. Such recurrent pattern forming models are of three main types. The first are aperiodic networks (Burak and Fiete, 2009; Widloski and Fiete, 2014), Figure 1A. In these models, activity in the cortical sheet (when neurons are appropriately rearranged – note that topography is not required in these models or in the proposed experiments) is grid-like and therefore periodic, but the connectivity between cells is highly localized and not periodic. In other words, connectivity does not reflect the periodicity in the activity. Taking a developmental or plasticity perspective, this network model is somewhat unusual in that strongly correlated neurons (those with the same activity phase, within or across activity bumps) are not connected as might be expected from associative learning. So if this network architecture holds in the brain, it would suggest that associative learning is curtailed once pattern formation occurs. From a functional viewpoint, aperiodic networks can require careful tuning of input at the network edges to accurately integrate their velocity inputs (Burak and Fiete (2009), but not so in Widloski and Fiete (2014)).
The second type are fully periodic networks, Figure 1B (Guanella et al., 2007; Burak and Fiete, 2006; Fuhs and Touretzky, 2006; Pastoll et al., 2013; Brecht et al., 2014; Widloski and Fiete, 2014). In these, network connectivity is itself periodic (the network has periodic boundary conditions on a rhombus), and the connectivity period equals the activity period: they each have a single period over the network (left, Figure 1B). An alternative version of the fully periodic network is to consider an aperiodic network with multiple activity bumps, but in which neurons at the centers of all the activity bumps are synaptically coupled. These two views of a fully periodic network are mathematically equivalent. Developmentally, the latter may be constructed from an aperiodic network (with multiple activity bumps) by application of associative learning post pattern-formation, so that neurons with similar phase but in different bumps end up recurrently coupled (right, Figure 1B).
The third type of the recurrent pattern forming networks is partially periodic, Figure 1C (Burak and Fiete, 2009). In these, as in the aperiodic networks, the bulk connectivity is local, so that connectivity does not reflect the periodicity of the population activity patterns. However, opposite edges of the cortical sheet are identified so that the network is effectively a torus. From a developmental perspective, these networks are the strangest: bulk connectivity does not reflect the periodic activity but the boundary condition requires knowledge of it (Figure S1).
Next come a variety of models in which grid cells are the result of feedforward summation of inputs that are already spatially tuned (Kropff and Treves, 2008; Welday et al., 2011; Mhatre et al., 2012; Bush and Burgess, 2014). Functionally, these models suggest that path integration occurs upstream of grid cells, in different low-dimensional attractor networks (Welday et al., 2011; Mhatre et al., 2012; Bush and Burgess, 2014). (In Kropff and Treves (2008), the origin of spatial tuning in the inputs is not directly modeled; if the assumed place-cell like inputs are based on path integration then the model will display low-dimensional dynamics, so we will consider the model under this assumption). Some of these models additionally include recurrent weights in the grid cell layer (Kropff and Treves, 2008; Mhatre et al., 2012). We will call all these feedforward models.
A perturbation-based probe of circuit architecture
The conceptual idea for differentiating between recurrent models of grid cells depends on multi-single unit grid cell recording before and after a global perturbation of the network. The idea is as follows. If population activity patterning in the neural sheet is due to aperiodic recurrent connections, then globally increasing the gain of recurrent inhibition or the time-constant of neurons in network models is predicted to increase the period of stable patterns in the cortical sheet, Figure S2. These effects, not predicted by linear stability analysis, exist in simulation of dynamical models (Widloski and Fiete, 2014; Burak and Fiete, 2009) and can be analytically derived by considering nonlinear effects (Widloski and Marder, unpublished observations).
Following the global perturbation, two cells originally in adjacent peaks of the population activity pattern and thus at the same phase of the population pattern (Figure 2A, blue), no longer will be (Figure 2A-B, red). Call the shift in pattern phase between cells in neighboring peaks one quantum (Figure 2A, circle and square, red versus blue). Then the shift in pattern phase between cells previously of the same phase and separated by exactly K peaks will be K quanta (Figure 2A, circle and triangle, red versus blue; explicit phase plot in Figure 2B). Across all cell pairs in the population, the shifts in phase will be quantized and will reach a maximum value of M quanta (or a full phase cycle, whichever is smaller), where M is the number of bumps in the population pattern.
Suppose the perturbation induces at most small phase shifts between all bumps of the population pattern (that is, , where is the perturbation stretch factor, and λpre, λpost are the population pattern periods preand post-perturbation, respectively; see Figure S3). Then the number of peaks in the distribution of pattern phase shifts, Figure 2C, will equal twice the number of bumps in the underlying population pattern, Figure 2A. In other words, the number of peaks in the distribution of pattern phase shifts can specify the number of bumps in the population.
However, the construction of this distribution relies on experimentally difficult-to-access quantities, namely the population pattern phase for each cell. If the network were topographically organized, this would be relatively simple to extract from a snapshot of network activity. If the network is not topographically organized, it is possible to obtain estimates of phase similarity or phase distance magnitudes between cells from patterns of coactivation or correlation across snapshots of the population activity, but such a scalar activity similarity measure cannot yield 2D phase in a 2D network.
The utility of our proposed strategy arises because the distribution of shifts in the population pattern phase across cells is mirrored in the distribution of shifts in the relative phase of spatial tuning across cells (Figure 2D). We illustrate this point in 1D, but the same idea carries directly over to 2D (Figure S4). The relative spatial tuning phase is derived from the spatial tuning of simultaneously recorded cell pairs. Cell pairs with zero relative phase in their spatial tuning pre-perturbation (because they fell on the same phase of the population pattern, albeit in different bumps) will exhibit postperturbation shifts in relative phase that, like the shifts in the population phases, will be quantized, and for small changes in population period will be proportional to the number of bumps separating that cell pair, Figure 2C-E. This predicted distribution of relative phase shifts (DRPS, Figure 2E) between neurons from an aperiodic network is a property of patterning in an abstract space, independent of how neurons are actually arranged in the cortical sheet.
In 2D, relative phase is a vector, measured along the two principal axes of the spatial tuning grid. The total number of bumps in the population pattern can then be read out as equal to a quarter of the product of the number of peaks in the two relative phase shift distributions (Figure S4).
Relating network parameters to experimental parameters
Changes in the strength of recurrent inhibition in our model can be mapped, in the biological system, into changes in the gain of inhibitory synaptic conductances. Experimentally, this perturbation may be induced by locally infusing allosteric modulators that increase inhibitory channel conductances (e.g. benzodiazipines (Rudolph and Möhler, 2004); personal communication with C. Barry). Changes in the time-constant of our model neurons can be mapped to changes in the EPSP time-constant in the biological system. Experimentally, the EPSP time-constant is sensitive to temperature through the Arrhenius effect and can be lengthened by cooling (Katz and Miledi, 1965; Thompson et al., 1985; Moser and Anderson, 1994). However, cooling affects several other single-neuron properties. To assess what to expect experimentally from a temperature perturbation and how to correctly include temperature effects in simpler neural models, we performed network simulations with cortical Hodgkin-Huxley neurons (Pospischil et al., 2008) while implementing documented temperature-dependent changes in all ionic and synaptic conductances (Experimental Procedures, SI, and Figure S5). The effect of cooling on conductance amplitudes is to shrink the population period in an aperiodic network, but its effect on conductance time-constants is to expand the period. The net effect of cooling is an expansion because temperature changes have larger effects on conductance time-constants (larger Q10 factors) than amplitudes (smaller Q10 factors) (Hodgkin et al., 1952; Thompson et al., 1985). We therefore conclude that changes in temperature are reasonable to associate with changes in the time-constant of simple neuron models. To summarize, two global experimental perturbations capable of inducing population activity period changes are neuromodulatory infusions that alter the gain of recurrent inhibition and cortical cooling (Moser and Anderson, 1994; Long and Fee, 2008).
Discriminating amongst recurrent architectures
Dynamical simulations of grid cell models reveal that the effects of the global perturbation will differ across recurrent network architectures, with consequently different predictions for the DRPS. In an aperiodic network, incremental global perturbation results in incremental expansion of the population activity pattern (Figure 3A, red, and Figure S2). Thus, the DRPS envelope will gradually and linearly widen with perturbation strength, and the separation between peaks will gradually grow (Figure 3B-C, red and Figure S2).
In a partially periodic network (with aperiodic local connectivity but with opposite boundaries connected), the number of bumps in the population activity pattern is constrained to be an integer. Thus, incrementally increasing the perturbation strength should result first in no change to the population activity period, and then a sudden change when the network can accommodate an additional bump (or an additional row of bumps in 2D, assuming the pattern does not rotate as a result of the perturbation; see Discussion) (Figure 3A, purple). Thus, incremental changes in perturbation strength should result in a stepwise change in population period and in the width of the DRPS envelope (Figure 3B-C, purple). Because the number of bumps has increased by a discrete amount, as soon as the DRPS changes, it will become maximally wide.
The fine structure of the DRPS will still be multimodal. However, counting peaks to estimate the number of bumps in the underlying population pattern will result in serious underestimation: when the pattern change is not incremental, there can be large changes in phase that are then lost in the DRPS, which is cut off at the maximal phase norm of 0.5 (Figure S3 and e.g. Figure 3B, compare peaks in the solid and dashed lines for small and large perturbations, respectively).
In the fully periodic network (Figure 1C), the same global perturbations that alter the population pattern period in the other recurrent networks (Figure 1A-B) are ineffective in inducing a corresponding change (Figure 3A, blue). This is because the periodic connectivity completely fixes the period of the pattern. Thus, the global perturbation will not affect the relative phase relationships between cells, and the DRPS is predicted to remain narrow, unimodal, and peaked at zero (Figure 3B-C, blue).
Discriminating feedforward from recurrent architectures
If the spatial tuning pattern or pattern components are generated upstream of grid cells and inherited or combined by them through feedforward summation (Mhatre et al., 2012; Welday et al., 2011; Bush and Burgess, 2014), then perturbing the recurrent weights or the biophysical time-constant within only the grid cell layer is predicted to leave unchanged the population activity period, preserving the spatial tuning shapes and cell-cell relationships. As a result, the DRPS should be narrow and centered at zero, as in the case of a recurrent network with fully periodic connectivity (Figure 3C, green line).
In all recurrent model networks (Figure 1A-C), the spatial tuning period of cells is predicted to expand with the global perturbation, which induces a change in the efficacy with which feedforward velocity inputs shift the pattern phase over time (Figure 3D and Figure S6). This expansion in spatial tuning period with global perturbation strength is predicted to hold for all three recurrent network classes, and can be used as an assay of the effectiveness of the experimental manipulation, especially when there is no shift in the DRPS.
By contrast, in feedforward models integration occurs upstream of the grid cells and thus the spatial tuning period should remain unchanged with global perturbation (Figure 3D, green line). Response amplitudes should nevertheless change in the feedforward models, thus revealing whether the attempted global perturbations are in effect.
Experimental feasibility of proposed method
We consider two key data limitations. First, it is not yet experimentally feasible to record from all cells in a grid module. Even a 100 cell sample would constitute a 1-10 % subsampling of the estimated module size. With present estimates that < 20 % of cells in a local patch in MEC are grid cells (Tang et al., 2014), the yield would be a meager 20 grid cells. Is this sufficient to observe the predicted quantal structure in a phase shift distribution, if it were present? Fortunately, the proposed method is tolerant to severe sub-sampling of the population: a tiny random fraction of the population (10/1600 cells) can capture the essential structure of the full DRPS, Figure 4A.
Second, spatial tuning and relative phase parameters are estimated from neural responses during a random, finite exploration trajectory in which cells respond variably. Hence, spatial tuning parameters, including phase and relative phase, are only known with a degree of uncertainty. In tests that depend only on the width of the DRPS (e.g. Figure 3), this phase uncertainty is not a serious limitation.
However, more detailed questions about the number of bumps in the population pattern in an aperiodic network depend on estimating the number of DRPS peaks, and here phase estimation uncertainty can be problematic: phase uncertainty will merge together peaks in the DRPS, Figure S7. At very small perturbation strengths, the DRPS peak spacing (in the aperiodic network) increases with the stretch factor. Thus, the larger the perturbation, the more distinguishable the peaks at a fixed phase error, Figure 4B and Figure S7. Yet increasing the stretch factor is not without a tradeoff: The two-for-one relationship between number of peaks in the DRPS and the number of bumps in the population pattern per linear dimension holds when the total induced shift in phase is small for all bumps (as before, when , with M now equal to the larger of the number of bumps along the two principal axes of the population pattern), Figure 4B. At larger stretch factors, the number of peaks in the DRPS is smaller than twice the number of bumps along the corresponding dimension of the pattern, and the discrepancy can be substantial.
Fortunately, the DRPS is computed from the relative phases between cells, which remain stable in a fixed network (Yoon et al., 2013) (here fixed refers to the network while a given perturbation strength is stably maintained). This stability makes it possible to gain progressively better estimates of relative phase over time even if there is substantial drift in the spatial responses of cells, by computing the relative spatial phase over short snapshots of the trajectory then averaging together the relative phase estimates from different snapshots across a progressively longer trajectory (similar to the methods used in Yoon et al. (2013) and Bonnevie et al. (2013)).
To distinguish M = 5 bumps per linear dimension based on structure within the DRPS would require a stretch factor of no greater than α = 1/(2M) = 0.1, and phase noise must be reduced to at least 0.02 (Figure S7). Distinguishing 7 bumps would require α ≤ 0.07 and a phase noise of smaller than about 0.01. Based on grid cell and trajectory data (accessed through http://www.ntnu.edu/kavli/research/gridcell-data), this would require an approximately 10 (50) -minute recording (Figure 4C).
The proposed method therefore has high tolerance to subsampling and a more limited tolerance to phase uncertainty. It will require longer-than-usual but still realistic amounts of spatial trajectory data with neural recordings to obtain adequately small error in relative phase estimation to test predictions that differentiate between models.
A decision tree for experimental design
We lay out a decision tree with an experimental workflow for discriminating between disparate networks, all of which exhibit 2D continuous attractor dynamics (Figure 5).
The demands from experiment are to be able to stably induce a global perturbation in one grid module, and to do so at 2-3 strengths. In all the cases, the term perturbation refers to a small change that leaves the network dynamics qualitatively unchanged while affecting its quantitative properties. The data to be collected are simultaneous recordings from several grid cells as the animal explores a familiar enclosure with no proximal spatial cues over about 20 minutes or more.
First, before applying perturbations, characterize the spatial tuning (periods) of the neurons, as well as cell-cell relationships (the relative spatial tuning phase). Next, apply a series of 2-3 global perturbations of increasing strength. At each perturbation strength, characterize the spatial tuning of cells and cell-cell relationships. A change in the amplitude of the cells’ response across the different perturbations signals that the perturbation is having an effect.
If further there is no change in the spatial tuning period, it follows that the perturbations produced no change in the population pattern and velocity responsiveness, thus the network must be feedforward, Figure 5 (green). Verify that cell-cell relationships remain unchanged across perturbations, as predicted for feedforward networks.
If there is a change in the spatial tuning period, characterize the cell-cell relationships in each perturbation condition. Plot the DRPS from each perturbed condition relative to the pre-perturbation condition, and obtain its width. If the DRPS width increases steadily and linearly with perturbation strength, that implies an aperiodic recurrent architecture, Figure 5 (red). If the DRPS width exhibits a step change, it is consistent with a partially periodic recurrent network, Figure 5 (purple). A DRPS that remains narrowly peaked around zero, with no change in width with perturbation strength, is consistent with a fully periodic network, Figure 5 (blue).
Finally, if the network is either aperiodic or partially periodic, the underlying population pattern has multiple bumps. The number of peaks in the DRPS for each dimension of relative phase bounds from below the quantity 2M, where M is the number of bumps in the population pattern along that dimension. When the stretch factor α times the number of bumps is smaller than 1/2, and if the DRPS is quantal, the number of DRPS peaks equals twice the number of population activity bumps along the corresponding dimension.
Discussion
Assumptions
The predictions made here assume that the network activity pattern is stable against rotations. Rotations of the population pattern would induce large changes in the DRPS, obscuring the predicted effects of pattern period expansion in any recurrent network. The fully periodic network is not subject to rotations, but partially periodic and aperiodic networks may be. In experimental data, the cellcell phase relationships between grid cells are indeed very stable across time and environments (Yoon et al., 2013), suggesting that the population activity undergoes no rotation. It is unclear what features of the circuit stabilize the population pattern against rotation; it is possible that slight directional anisotropies in the outgoing connectivity of neurons pin its orientation.
The simplifying observation, that spatial responses may be used to estimate the DRPS, depends on other inputs not being able to overrule the new post-perturbation cell-cell relationships. For instance, external sensory inputs or hippocampal place cells that become associated with particular configurations of grid cells may keep resetting the grid networks to express old relative phase relationships. To avoid this possibility, it may be important to assess post-perturbation cell-cell relationships only in novel environments, for which there are no previously learned associations between external cues, place cell responses, and grid cell activity.
Finally, it is important to note that if in feedforward models one were to include feedback from the grid cell layer back to the spatially tuned inputs (as in Bush and Burgess (2014)), the network would effectively become a type of recurrent circuit, and perturbing the grid cell layer may result in changes in grid period and cell-cell relationships.
Prior probabilities of different grid cell models being correct
From theoretical arguments, we believe the candidate grid cell mechanisms are not equally probable. In particular, the partially periodic model is difficult to justify from the viewpoint of grid cell development. In Widloski and Fiete (2014), we see that activity-dependent rules acting on spatially informative feedforward inputs can lead to the formation of a network capable of path integration and with grid cell-like tuning. The network, post-development, has aperiodic structure. Under certain conditions, if network weights continue to undergo plasticity after the network has matured enough to expresses recurrent patterning, the network can become fully periodic as neurons with the same spatial phase become wired together (Figure S8). In fact, the addition of relatively weak coupling between neurons in nearest-neighbor activity bumps is sufficient to convert an aperiodic network into what is, functionally if not topologically, a fully periodic network (Figure S8).
Thus, it is possible to imagine mechanisms for the development of the fully periodic and fully aperiodic networks. By contrast, a partially periodic network involves local connectivity which does not depend on a neuron’s spatial phase, but at the same time requires some mechanism for neurons at one end of the network to link with those at the opposite end in way that depends on spatial phase, Figure S1. It is more difficult to imagine a plausible mechanism that can satisfy both constraints. By the same argument, in feedforward models, one would expect the 1D patterned inputs to grid cells to involve fully periodic or fully aperiodic 1D networks.
Circuit inference through perturbation and sparse activity records: outlook and alternatives
It is interesting to compare the potential of our suggested approach with that of single synapselevel circuit reconstruction (a connectomics approach). A high-quality full-circuit connectome (with signed connections) can specify the topology of the network structure. In other words, it should be possible to reveal whether the circuit is intrinsically “local” (as in the aperiodic network of Figure 1A) (Widloski and Fiete, 2014), partially periodic (with local center-surround-like connectivity and periodic boundary conditions as in Figure 1B), or fully periodic (with center-surround-like connectivity of a width that spans the entire network together with periodic boundary conditions). It may even be possible to infer the locality of structure in the aperiodic network from an unsigned connectome.
Network topology is, however, one ingredient in circuit mechanism: Determining whether the signed connections lead to activity patterning still requires a large amount of inference (for instance, converting the connections into weights and inserting the matrix into an appropriate dynamical model). Even with further inference steps, whether the network actually performs certain functions like velocity-to-position integration or only inherits them is not answerable based on connectomics data. For instance, a network with lateral interactions may generate position-dependent responses de novo through integration (Figure 1A-C), or may act only to further pattern inputs that are already spatially tuned (Figure 1D-E) (Kropff and Treves, 2008; Mhatre et al., 2012; Bush and Burgess, 2014). Despite these functional differences, both types of networks have similar connectivity and topologies.
Single neuron-resolution records of activity within a grid cell module can be fruitfully used to understand the dimensionality and relationships of neural responses, but without perturbation, inferring actual connectivity and thus mechanisms from activity is problematic (Roudi et al., 2009; Honey et al., 2009). Hence, activity records do not distinguish between different recurrent models. In short, while connectomics and large-scale recording can provide troves of useful information, they are not sufficient for discriminating between models; as we have shown here, they may also not be immediately necessary.
As we have seen, with a perturbation approach it is possible to localize where integration occurs: if the perturbed area is performing integration, the spatial tuning period is predicted to change. Generally speaking, perturbation modulates the effect of connectivity on dynamics, and the proposed readout is neural activity. This closed-loop approach allows for detailed tests of mechanistic neural models, whose very goal is to relate architecture and dynamics, in a way not easily rivaled by nonperturbative probes of connectivity or activity.
Cooling and other perturbation experiments have been performed in V1 (Michalski et al., 1993; Ferster and Miller, 2000), but they were not as revealing about underlying mechanism as might be possible in grid cells. The reason is twofold: Recurrent models of orientation tuning in V1 are ring models, which are periodic single-bump networks, thus the predicted DRPS after cooling is essentially the same as the prediction for a feedforward network. Moreover, because the orientation response does not arise from integration of a velocity input, the spatial tuning width after cooling is also not predicted to change in a substantial way for recurrent networks. These factors make it harder to discriminate recurrent from feedforward mechanisms from perturbation. The multi-bump tuning of grid cells offers a unique opportunity to use the types of perturbative approaches used in V1 (Michalski et al., 1993; Ferster and Miller, 2000), to obtain unprecedented detail on the local circuit mechanisms that support the complex tuning of cortical cells.
Methods
Definitions: Population phase and relative spatial phase
Roman subscripts (e.g., i and j) refer to individual cells. If cells are arranged topographically based on connectivity, then i refers to the location (in neuron body-length units) within the population pattern of the ith cell. If the period of the population pattern is λpop (again in neuron body-length units), pop then the population pattern phase of cell ith is (with the arbitrary choice, made without loss of generality, that neuron 1 has phase 0).
Next, consider the spatial tuning curves of cells i, j. Without respect to arrangement in the cortical sheet, let dij represent the offset, in meters, of the peak closest to the origin in the cross-correlation of the two spatial tuning curves, and let λ be the spatial tuning period (in meters) of the two cells. The relative spatial phase is defined as δij = (dij mod λ)/λ. Phase magnitudes are based on the usual Lee metric, |δ| = min(|δ|, 1 - |δ|). In 2D, the transformation of into is identical to that described in Yoon et al. (2013) and replicated here in SI. Analogously, using the same procedure, the 2D coordinate of the ith cell in the cortical sheet can be transformed into the population phase vector. As noted in Results, δij is easily experimentally accessible; , much less so.
Generation of Figures
Figure 1 is schematic. Figure 2 is generated from ideal (imposed) periodic patterns but without dynamical neural network simulations. In Figures 2, 4A,B, S3, S4, and S7 relative spatial phase is computed for convenience (to save the computational cost of generating spatial tuning curves, then deriving relative phases) from the population phases (thus, by setting ). Figures 3, S2, S6, which distinguish between different recurrent architectures, are based on dynamical neural network simulations using the mature grid cell network described in SI. Briefly, the model is a network of excitatory and inhibitory neurons (except in S8 – see figure caption for details), with linear-nonlinear Poisson (LNP) spiking dynamics (Burak and Fiete, 2009; Widloski and Fiete, 2014). For Figure S5, we use Hodgkin-Huxley dynamics. Structured lateral interactions between neurons lead to pattern formation in the neural population. Relative spatial phases are explicitly computed from spatial tuning curves of cells, which are obtained from spike responses to 2-minute long simulated quasirandom trajectories. Velocity inputs drive shifts of the population pattern, resulting in spatially periodic tuning. Only cells from the simulation with good spatial tuning are included in the analysis of relative phase shifts: for fully and partially periodic networks, this means all cells in the network, while for aperiodic networks this means cells in the central 3/4 of the network. Since the inhibitory and excitatory populations share similar population patterning and spatial tuning in these simulations, we made the arbitrary choice to display the inhibitory population.
Supplemental Experimental Procedures
Neural network simulations
Below, we describe the two different neural dynamics models used in the paper: the linear-nonlinearPoisson (LNP) model and the Hodgkin-Huxley conductance model.
Roman subscripts (e.g. i, j) refer to individual cells within population P. The population index P can take the values {I, ER, EL}. Integration in all simulations is by the Euler method with time-step dt.
Linear-Nonlinear-Poisson dynamics (all figures except Figure S5)
The LNP model we use is identical to that used in Widloski and Fiete (2014). Given a time-dependent summed input to the (P, i)th cell, the instantaneous firing rate of the cell is with the neural transfer function f given by
Based on this time-varying firing rate, neurons fire spikes according to an inhomogeneous (sub-Poisson) point process with a coefficient of variance of CV = 0.5 (see Burak and Fiete (2009) and Widloski and Fiete (2014) for details on generating a sub-Poisson point process).
The time-dependent activation of synapses from the (P, i)th cell is given by where specifies the time of the bth spike of the cell and the sum is over all spikes of the cell.
The total input into the (P, i)th cell is given by where g0 (g0=50 for the E and I populations) and g0′ (g0′ =15 for the E population; g0′ =0 for the I population) are small, positive, constant bias terms common to all cells, are the recurrent inputs, αP,vel are the velocity inputs, and is an envelope that either suppresses activity near the network boundaries for the aperiodic network, or is flat and equal to unity for the periodic networks (see below). The recurrent input is where are the recurrent weights and δ is the Kronecker delta function. The form of the envelope function, , depends on the boundary conditions of the network. For aperiodic networks, the envelope shape is a 1D version as that given by Burak and Fiete (2009): where NP is the size of the network, , κ = 0.3 determines the range over which tapering occurs, and a0 = 30 controls the steepness of the tapering. For periodic networks, the envelope is flat:
All cells in the Pth population (with preferred direction given by the unit vector ) receive a common velocity input: where is instantaneous velocity of the animal and βvel sets the gain of the velocity input; (0,1), (0,-1) for the I, ER, EL populations, respectively. The velocity input, unless otherwise noted, is based on a 2-minute quasi-random trajectory derived with an algorithm identical to that described in Widloski and Fiete (2014). Over the course of these trajectories, the stochastic dynamics leads to drift in the path-integrated estimate of animal location if uncorrected. To minimize this drift, the pattern phase is reset whenever the animal is in the vicinity of one of the 5 ‘landmarks’ evenly spaced throughout the environment. During each encounter with a landmark, the pattern phase is corrected via strong feedforward inputs that impose a snapshot of the pattern at its “correct” phase; “correct” pattern snapshots are captured from the population pattern during the animal’s initial encounter with each of the landmarks.
Temperature/neuromodulation of LNP dynamics. Temperature-dependent modulations are modelled as a simple rescaling of the synaptic activation time constant, τsyn. Modulations of network inhibition are modelled as a gain change in the efficacy of the synaptic weights projecting from inhibitory neurons, i.e., W P I ← γinhW P I, where γinh is the strength of inhibition.
Hodgkin-Huxley dynamics (only used in Figure S5)
The model we use is identical to the reduced Hodgkin-Huxley “regular spiking (RS)” model of cortical neurons, as described in Pospischil et al. (2008), supplemented with synaptic dynamics. The dynamics of the membrane potential of the (P, i)th neuron is given as where is the summed input current and Cm is the capacitance of the membrane. The summed input current is given as where the first term represents currents related to the ionic membrane conductances and the second and third terms represents synaptic and external conductances, respectively, gated by velocity inputs, αP,vel, and an envelope function, . The ionic current has the following form: where the ’s are the maximum conductance values and the ’s the reversal potentials of the leak conductance (L), fast (K) and slow (M) potassium conductances, and the sodium conductance (Na). The dynamics and parameter settings of the gating variables n, m, q, h are described in Pospischil et al. (2008) (note that we have replaced the “p” gating variable of Pospischil et al. (2008) with “q”). The synaptic current based on recurrent connections within network is where is the synaptic activation of the (P l, j) neuron (which has the same dynamics as described above in equation (3) – here, we define the time of a spike elicited by the jth neuron, , as when the voltage moves from below 0 mV to above it in a single-time step, within the interval (t, t + Δt)), is a synaptic scaling factor shared by all synaptic weights, and is the synapse-specific reversal potential ( and ).
Temperature/neuromodulation of HH dynamics. To simulate temperature-dependent modifications, we used separate Q10 factors to modulate the time constant and amplitudes of the ionic/synaptic conductances (Hodgkin et al., 1952; Katz and Miledi, 1965). At temperature T (°C), the conductance amplitudes and time constants τ (T) have the following form:
The conductance amplitude modulation was applied specifically to . The conductance time constant modulation was applied to the gating variable time constants τn, τq, τm, τh (for gating variable x, the time constant τx is defined as τx = 1/(αx + βx), where αx and βx are the rate constants governing the gating variable’s dynamics – see Pospischil et al. (2008)) and the synaptic time constant τsyn. For temperature perturbations of the ionic conductances only, change with temperature, while and τsyn =16 ms are held constant. For temperature perturbations of the synaptic conductances only, and τsyn change with temperature, while the ionic conductance properties are held fixed.
The effects of specific neuromodulators targeting the inhibitory synapses was modelled in exactly the same as for the LNP model.
Synaptic weights for network of excitatory and inhibitory neurons (all figures except Figure S8)
The detailed synaptic weights used in the simulations are based on the developmentally-inspired hardwired weights with aperiodic boundary conditions described in the SI of Widloski and Fiete (2014), and therefore can be viewed as the plausible culmination of a developmental process. Compared to the LNP-based model used in Burak and Fiete (2009), we chose to implement the model in Widloski and Fiete (2014) because it is more realistic, incorporating separate populations of excitatory and inhibitory cells; however, both models give qualitatively similar results. (Note that while the description of the weights below is different than that specified in Widloski and Fiete (2014) in order to enable flexibility in setting the network boundary conditions, the weights used for the aperiodic network are identical to those specified in Widloski and Fiete (2014).) The weights from population Pl to P, between cells i and j, are described as follows: where x = i γj, (NP is the size of population P), Θ is the Heaviside function (Θ(x) = 0 for x < 0 and is 1 otherwise), c0 = Ψ(x) and c± = Ψ(x±Δρ) where Ψ(x) = min (Np - |x mod Np|, |x mod Np|), and , where for aperiodic networks and for periodic networks (see above for definition of and ). ρ is a scale factor that controls the width of the synaptic weights, and therefore the number of bumps expressed in the pattern, whereas η is a synaptic scaling factor that modulates only the amplitudes.
(A note on terminology: the partially periodic network has an overall topology that resembles the periodic network of Burak and Fiete (2009). In our usage in the present work, periodic refers to a fully periodic network, in which the periodicity of connections matches that of activity pattern, whereas in the partially periodic network, the bulk of connectivity does not reflect the periodicity of the population activity pattern.)
Simulation parameters
Aperiodic network with LNP dynamics.
P = EL, ER, I; NEL = NER = 400 neurons; NI = 160 neurons; CV = 0.5; dt = 0.5 ms; τsyn = 30 ms*; βvel = 2; ; ρ = 2.2. γinh = 1*;
EL → I (i.e., W IEL): η = 21; E = 0; Δ = −1; σ = 2; μ = 0; δ = 0;
ER → I: η = 21; E = 0; Δ = 1; σ = 2; μ = 0; δ = 0;
I → EL: η = 8; E = 0; Δ = 4; σ = 5; μ = −1; δ = 3;
I → ER: η = 8; E = 0; Δ = −4; σ = 5; μ = 1; δ = 3;
I → I: η = 24; E = 1; Δ = 2; σ = 3; μ = 0; δ = 3;
(* indicates that parameters can change through perturbation)
Partially periodic network with LNP dynamics.
Same parameters as aperiodic network, except that , and ρ = 2.2.
Fully periodic network with LNP dynamics.
Same parameters as aperiodic network, except that , and ρ = 22.
Aperiodic network with HH dynamics.
All ionic conductance parameters are identical to those described in Pospischil et al. (2008) for the RS model; as noted there, the parameters are set to values corresponding to a temperature of T0 = 36°C. NEL = NER = 400 neurons; NI = 160 neurons; dt = 0.025 ms; τsyn = 15 ms*; βvel = 0.8; Cm = 1 μF/cm2; ; ; ; ; ; ; ; Iapp = = 3 μA/cm2; ; ; ρ = 2.2. γinh =i i 1*; Synaptic weights are identical to those described for the aperiodic network with LNP dynamics up to a constant, . (* indicates that parameters can change through perturbation)
Measures used in main text
Bootstrap resampling and phase uncertainty. Given an original spike map of M total spikes (with locations) from one cell, we created a new spike map of N (N < M) total spikes, by picking spikes (with their corresponding location coordinates) from the original map one at a time, at random, and with replacement. The same was done for a second, simultaneously recorded cell. From these sampled spike trains for a pair of cells, we estimated relative phase (by computing the location of the peak closest to the origin in the cross-correlation of the spatial maps of the two cells, as in Yoon et al. (2013)). The procedure was performed 100 times, generating 100 bootstrapped relative phase estimates per cell pair. Phase uncertainty was measured as the mean of the magnitudes of the bootstrapped relative phase estimates.
Spatial tuning curves. For a given cell and trajectory, we build a histogram of spike counts at each location (bin size = 1 cm), then normalize the count in each bin by the amount of time spent in it. The normalized histogram is smoothed by convolution with a boxcar filter (width = 5 bins) to yield a spatial tuning curve.
Spatial tuning period. The spatial tuning period is measured as the inverse of the spatial frequency with the highest peak in the power spectrum of the spatial tuning curve (excluding the peak at 0 frequency).
Population period. Given the last 500 snapshots (frames) of the population pattern from a given trial, the population period is measured as followed: For each frame, measure the inverse of the frequency with the highest peak in the power spectrum (as with the spatial tuning period) of the population pattern. The population period is the average of these estimates.
Velocity response. Velocity response is measured as the translation speed (neurons/sec) of the network pattern to fixed input velocity, computed by tracking the displacement of the pattern for 10 seconds, smoothing the resulting trajectory with an 4-second moving average filter, and then measuring the average speed of the middle-half of the trajectory.
Periodicity score for the DRPS. We smooth the histogram of relative phase shifts (by convolution with a 2-bin Gaussian kernel) and normalize it (by mean subtraction and division by the standard deviation). Next, we compute the power spectrum, rescaling the result by 2/L2, where L is the number of bins in the histogram (L = 200). The periodicity score is set to be the power of the largestamplitude non-zero frequency component in the scaled power spectrum. This score returns 1 if the DRPS is a pure sinusoid. It returns 0 if the DRPS is flat and returns an average value of < 0.2 if the DRPS were constructed bin by bin by sampling iid from a uniform distribution on the unit interval.
2D relative phase. The displacement vector is converted into a 2D phase according to , where is the oblique projection of onto the principal vectors and , and
Relative phase magnitude is given by