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A model for the differentiation between grid and conjunctive units in medial entorhinal cortex

A model for the differentiation between grid and conjunctive units in medial entorhinal cortex
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  A Model for the Differentiation between Grid and Conjunctive Units in MedialEntorhinal Cortex ∗ Bailu Si 1 and Alessandro Treves 2,3 1 Department of Neurobiology, Weizmann Institute, 234 Herzl St, Rehovot 76100, Israel  † 2  Sector of Cognitive Neuroscience, International School for Advanced Studies, via Bonomea 265, 34136 Trieste, Italy  3  Kavli Institute for Systems Neuroscience and Center for the Biology of Memory,Norwegian University of Science and Technology, 7489 Trondheim, Norway  ‡ The multiple layers of medial entorhinal cortex (mEC) contain cells that differ in selectivity,connectivity, and cellular properties. Grid cells in layer II and in the deeper layers express triangulargrid patterns in the environment. The firing rate of the conjunctive cells found in layer III and below,on the other hand, show grid-by-head direction tuning. In this study, we model the differentiationbetween grid and conjunctive cells in a network with self-organized connections. Arranged intodistinct “layers”, the model grid units and conjunctive units develop, with a similar time course,grid fields resulting from firing rate adaptation and competitive learning. Grid alignment in bothlayers is delayed with respect to the formation of triangular grids. A common grid orientation amongconjunctive units is produced, in the model, by head-direction modulated collateral interactions,while the grids of grid units inherit the same orientation through connections from conjunctive units.Grid units as well as conjunctive units share a similar spacing but show a random distribution of spatial phases. Grid units however carry more spatial information than conjunctive units, thusproviding better inputs for the hippocampus to form spatial memories. Keywords: Grid cells; Conjunctive grid-by-head direction cells; Firing rate adaptation; Self-organization;Lamination Introduction Spatial memory is one of the fundamental functions foran animal to survive successfully in its environment. Inmammals, the neural basis of spatial memory has beenthought to largely reside in the hippocampus and relatedcortices. Place cells in the rat hippocampus show ele-vated firing activity whenever the rat enters a specificportion of the environment, the place field 42 . Head di-rection (HD) cells in the rat postsubiculum as well asin many other regions are characterized by steady firingwhen the animal points its head towards a specific di-rection in the environment 13,45,50,52,53 . In recent years,place-modulated cells were discovered also in the medialentorhinal cortex (mEC), a region just one synapse up-stream from the hippocampus, and observed also in thepre- and parasubiculum of rats as well as in the entorhinalcortex of other species  5,26,57 , possibly even in humans 18 .The multiple firing fields of a layer II mEC grid cell col-lectively form a remarkably regular triangular grid span-ning the environment which the animal explores 26 . In thedeeper layers of mEC, conjunctive grid-by-head-directioncells show firing selectivity to head direction in additionto the same spatial tuning as grid cells 45 . Why do ro-dents, and bats, and possibly other species, have bothgrids and conjunctive grid-by-HD units: would one typenot suffice? And how do they form (presumably together,as they share the same substrate)?A series of modeling studies have proposed mechanisms ∗ to appear in Hippocampus for the expression of grid firing patterns 24,60 . They maybe roughly classified into two main categories, namelypath-integration models and self-organization models. Inpath-integration models, these units serve to accumulatethe velocity of the animal, in order to track its location in-ternally, either by the collective state of many units in thecontinuous attractor network sub-variant 8,21,37,41 , or bythe phases of velocity-controlled oscillators at the singleunit level, in the interference sub-variant 9,10,23,27,61 . Thetriangular grid pattern is imposed ab initio by structuredcollateral connections or by the summation of multipleoscillators with preferred running directions separatedby multiples of 60 degrees. In self-organization mod-els 55 , on the other hand, once formed and wired togetherthrough recurrent connections, the units may also sub-serve path-integration, but their spatial responses firstemerge spontaneously, at the single unit level. The peri-odicity of the grid pattern is a result of firing rate adapta-tion in isotropic exploration 30,48 and is fixated graduallyby means of synaptic plasticity in the feedforward con-nections, which convey broad spatial inputs, for examplebut not necessarily from “place units” 30 . In a recentvariant, grid units receive inputs from periodic “stripecells” 39 , and they inherit the periodicity of these one-dimensional stripe-shaped inputs.These models describe either grid units or conjunctiveunits or are compatible with both, and sometimes criti-cally depend on a feature of either cell type 32 , but theydo not really relate to the striking phenomenon that bothcell types are present, nor do they try to explain how theirdifferential properties may emerge.In layer II, all grid cells found are purely positional.In layer V, the majority of grid cells are conjunctivecells. In layer III and VI, there is a mixture of pure  2 Grid unitsConjunctive unitsHead direction units“Place” units Figure 1: Network model of grid cells and conjunctive cells inmEC. The network is comprised of 320 place units,  N   = 256conjunctive units and 256 grid units. Place units are fully con-nected to grid units and to conjunctive units. Each conjunc-tive unit is modulated by one head direction unit. Conjunc-tive units are randomly connected to  M   = 154 other conjunc-tive units (i.e. 60% connectivity, without self connections).The connectivity of the projections from the conjunctive layerto the grid layer is 60%. There are no collateral connectionswithin the grid layer. grid cells and conjunctive cells 5,45 . This differential lam-inar localization of grid and conjunctive cells in mEChas not been touched upon by existing models of gridcells, and is a first motivation for this study, following upon previous analyses of cortical lamination 54 . Second,we adapt the self-organizing adaptation model, whereunlike our previous study also the recurrent collateralconnections, assumed to exist between conjunctive units,self-organize their weights, and we focus particularly onthe time course of self-organization, a process burdenedwith shaping at least three distinct major sets of con-nections within the same tissue. Note that our modelis intended for the development of mEC circuitry duringpostnatal development, in particular at about P14-P30 inthe rat 1 . Third, to better appreciate the contribution of the two cell types, we quantify their self-organized spatialcodes with information-theoretic measures, thus provid-ing quantitative support for the functional characteriza-tion of the peculiar type of quasi-laminar organizationobserved in mEC. ResultsNetwork model of the mEC layers Grid cells and conjunctive cells are represented by theunits in two different layers in the network model (Fig. 1).These layers are abstractions of the real ones, and aremeant to capture solely the hypothesis that, given a dis-tinct laminar arrangement in the tissue, a simple ge-netic instruction, such as “no recurrent connections inlayer II”, can lead to distinct self-organization trajecto-ries. Both grid units and conjunctive units receive inputsfrom place units. This assumption is consistent with ex-perimental observations: 6 show that inputs from the hip-pocampus are necessary for grid cells to maintain theirgrid firing pattern; during postnatal development, placecells form adult-like spatial fields earlier than grid cellsdo 33,56 . The assumption is not strictly necessary, how-ever, as place unit inputs can be replaced with broadlymodulated inputs 30 . In the model, conjunctive units areinterconnected through collaterals 17 , while grid units re-ceive connections from conjunctive units, but have nocollaterals among themselves 16 , consistent with existingexperimental evidence. The connections from the gridlayer to the conjunctive layer are not considered in themodel, since anatomically connections from superficial todeep layers of mEC appear weaker 25,29 .In the model, conjunctive units develop aligned gridswith the help of recurrent collaterals and of the headdirection modulation. The grid units develop their owngrids, but inherit that common grid orientation throughthe connections from the conjunctive unit layer to thegrid layer. Self-organization in the conjunctive layer A virtual rat is simulated to randomly explore a squareenvironment with variable speed, while all the weightsin the network are initialized at random values in thebeginning, and adapted in the course of the explorationby Hebbian-type rules (ref.  Methods  ). Coherent grids form together with self-organized collateral connections  We first investigate whether conjunctive units can formgrids with a coherent orientation while the collateral con-nections develop. The key difference from our previousmodel is that here the collateral connections are learned.In the beginning of the development, the firing maps of the units in the network have multiple fields at irregularlocations (Fig.2A, odd rows). Shown below each firingmap is the corresponding autocorrelogram, which is thecorrelation of a firing map to its shifted version in 2-dimensional space. We use the so-called gridness score,which is in the range [-2, 2], to measure the 6-fold spatialperiodicity of an autocorrelogram, as in 45 . Conjunctiveunits develop reasonably regular grids very fast.It takes about 10 4 equivalent rat seconds (about 2.8hours, or 10 6 simulation steps) of continuous explorationfor conjunctive units to develop good grids (e.g. the unitshown in the first two rows of Fig.2A). Note that eachstep of the virtual rat corresponds to 10 ms in real time.While grids appear relatively early, grids mutually alignto each other at a later stage. For example, unit 93 andunit 111 orient toward 25 degrees after 1 . 6  ×  10 5 sec-onds (about 44.4 hours, or 1 . 6 × 10 7 simulation steps) of continuous exploration, before finally aligning to a com-mon orientation of about 30 degrees. Unit 149 (last tworows in Fig.2A), however, does not form a good grid evenby the end of simulation, rather it diverges to a stripe-like pattern as observed in some of the real mEC cells 31 .  3 x (cm)    U  n   i   t   9   3  y   (  c  m   ) Time 1000 smax: 0.3 50100 50100x (cm)Time 5000 smax: 0.4 50100 50100x (cm)Time 10000 smax: 0.5 50100 50100x (cm)Time 100000 smax: 0.5 50100 50100x (cm)Time 160000 smax: 0.5 50100 50100x (cm)Time 200000 smax: 0.5 50100 50100 dx (cm)g:0.0, o:51 deg.    d  y   (  c  m   ) -100 -50 0 50 100-100 -50 0 50 100dx (cm)g:0.3, o:22 deg.-100 -50 0 50 100-100 -50 0 50 100dx (cm)g:0.9, o:15 deg.-100 -50 0 50 100-100 -50 0 50 100dx (cm)g:1.7, o:25 deg.-100 -50 0 50 100-100 -50 0 50 100dx (cm)g:1.6, o:29 deg.-100 -50 0 50 100-100 -50 0 50 100dx (cm)g:1.4, o:31 deg.-100 -50 0 50 100-100 -50 0 50 100 x (cm)    U  n   i   t   1   1   1  y   (  c  m   ) max: 0.3 50100 50100x (cm)max: 0.5 50100 50100x (cm)max: 0.5 50100 50100x (cm)max: 0.5 50100 50100x (cm)max: 0.5 50100 50100x (cm)max: 0.5 50100 50100 dx (cm)g:0.0, o:11 deg.    d  y   (  c  m   ) -100 -50 0 50 100-100 -50 0 50 100dx (cm)g:-0.3, o:55 deg.-100 -50 0 50 100-100 -50 0 50 100dx (cm)g:0.1, o:5 deg.-100 -50 0 50 100-100 -50 0 50 100dx (cm)g:0.7, o:10 deg.-100 -50 0 50 100-100 -50 0 50 100dx (cm)g:0.8, o:22 deg.-100 -50 0 50 100-100 -50 0 50 100dx (cm)g:0.6, o:34 deg.-100 -50 0 50 100-100 -50 0 50 100 x (cm)    U  n   i   t   1   4   9  y   (  c  m   ) max: 0.3 50100 50100x (cm)max: 0.5 50100 50100x (cm)max: 0.6 50100 50100x (cm)max: 0.5 50100 50100x (cm)max: 0.5 50100 50100x (cm)max: 0.5 50100 50100 dx (cm)g:0.7, o:41 deg.    d  y   (  c  m   ) -100 -50 0 50 100-100 -50 0 50 100dx (cm)g:0.1, o:37 deg.-100 -50 0 50 100-100 -50 0 50 100dx (cm)g:1.1, o:47 deg.-100 -50 0 50 100-100 -50 0 50 100dx (cm)g:0.1, o:13 deg.-100 -50 0 50 100-100 -50 0 50 100dx (cm)g:-0.1, o:7 deg.-100 -50 0 50 100-100 -50 0 50 100dx (cm)g:-0.2, o:8 deg.-100 -50 0 50 100-100 -50 0 50 100 -0.2-0.1 0 0.1 0.2-0.2-0.1 0 0.1 0.2unit 93-0.2-0.1 0 0.1 0.2-0.2-0.1 0 0.1 0.2unit 111-0.2-0.1 0 0.1 0.2-0.2-0.1 0 0.1 0.2unit 149 -125 -75 -25 25 75 125 -25 25 75 125dx (cm)    d  y   (  c  m   ) Orient. std:2.7 deg.; mean spacing:53.2 cm dx (cm)    d  y   (  c  m   ) Phase histogram -30-15 0 15 30-30-15 0 15 300123 AB C D Figure 2: Units in the conjunctive layer develop triangular spatial firing maps over time. (A) The spatial firing rate maps of three example conjunctive units are shown in odd rows, together with their autocorrelograms in even rows, ordered in columnsby time. Small values are color-coded by blue, progressing to higher values as red. The maximal firing rate (in arbitrary units)is indicated above each rate map. The gridness score and grid orientation (in degrees) are noted above the correspondingautocorrelogram. The white markers show the three peaks above the x-axis closest to the center of an autocorrelogram,indicating the spacing of a grid as well as the orientations of the three grid axes; (B) Head direction firing rate maps of thethree example units in (A) are shown in polar coordinates, with radial coordinate representing firing rate. Dash-dot linesindicate the preferred head directions of the units; (C) Locations of the three peaks found in the autocorrelograms at the end of learning (as shown by the white markers in the right column of (A)) are overlaid for units with gridness score  >  0; The averagestandard deviation in grid axis orientation is 2.1 degrees. The mean spacing is 53.2 cm; (D) Two-dimensional histogram of thespatial phases, relative to the best grid, of the units shown in (C).  4Only about 2% of the units (5 out of the 256 conjunctiveunits) develop stripe-like firing patterns, much less thanestimated from experimental data 31 . The formation of a stripe-like pattern is due to firing rate modulation byhead direction. Along non-preferred head directions, thefiring rate is adapted less since the firing rate is smaller.As plotted in Fg.2B, the firing rates of the units are in-deed much higher in their preferred head direction. Thestripe-like firing maps show less periodicity along direc-tions roughly perpendicular to the preferred head direc-tions (last two rows in Fig.2A and right panel of Fig.2B).At the end of the simulation, most conjunctive unitsin the network develop grids with similar spacing andorientation. The three maxima above the x-axis in theautocorrelograms appear at similar distances and orien-tations from the srcin (Fig.2C). The spatial phases of the grids are randomly distributed (Fig.2D), in accor-dance with experimental data 26 .The common grid orientation is a result of collateral in-teractions between units. During development, collateralweights between conjunctive units gain structure throughHebbian learning. At the end of learning, the collateralweights of a conjunctive unit are a function of both thepreferred head directions of the presynaptic units andthe correlation between the fields of the connected units(Fig.3). The learned collateral weights approach maximaat zero difference between the preferred head directionsof the pre- and post-synaptic units, distributing looselyunder a Gaussian-like envelope (Fig.3A). The collateralweights also show an increasing trend with respect to thespatial correlation between the maps of the connectedunits, till moderately high correlation (Fig.3B). Self-organized collateral weights differ from ad-hoc collateral weights  Although the learned collateral weights have the samefunction in aligning grids as the ad-hoc collateral weightsthat we considered in a previous study 48 , the learnedcollateral weights better reflect the correlation betweenthe activity of units. To see the difference, we simulatea network that has already formed grids with fixed ad-hoc collateral weights (ref.  Methods  ), and see how theweights change while the network is subject to learning.Compared to ad-hoc weights, the learned collateralweights fall in a lower range (diamond vs. circle mark-ers in Fig.4A), indicating that the ad-hoc procedure ef-fectively tends to overestimate the larger weights. Thelearned collateral weights are smoother functions of thepreferred head directions of pre-synaptic units, as can beseen from the mean in head direction bins (solid lines inFig.4A).While the collateral weights are being learned, con- junctive units develop new maps. Therefore the correla-tions between the grid maps of unit pairs are remapped.The learned collateral weights coarsely increase only withrespect to the new correlation of the corresponding spa-tial firing maps, but not to the previous one (Fig.4C vs.Fig.4B).Averaging the collateral weights across conjunctiveunits reveals that the average learned collateral weightsare a sharper function of the difference between the pre-ferred head directions of the pre- and post-synaptic units(Fig.4D). The average learned collateral weights increasewith respect to the corresponding mean spatial corre-lation, but not to the previous correlation (Fig.4F vs.Fig.4E). Self-organization in the grid layer How may grid cells in layer II of mEC form coherenttriangular grid maps without head direction modulationand without, as recent evidence suggests 16 , strong ex-citatory recurrent connections? It is possible that thecommon grid orientation results from the excitatory con-nections from the deeper layers to layer II.In the model network shown in Fig.1, the spatial re-sponses of the units in the grid layer also evolve into trian-gular grids as learning proceeds on the connections fromplace units to grid units. After 10 5 rat equivalent seconds(27.8 hours) of continuous exploration, the three exam-ple units in Fig.5A all show fairly good grid (odd rows),as indicated by their median gridness scores (even rows).By about 1 . 6 × 10 5 seconds (44.4 hours) of learning, thesegrids become very triangular and mutually align to about32 degrees. The activity of the grid units does not showany preference in head direction, due to the absence of head direction modulation (Fig.5B).The grids of the grid units share the same spacing (52.6cm), and are aligned to a common orientation (32 de-grees), as manifested by the concentrated scattering of the peaks from autocorrelograms (Fig.5C). The grids of the units in the grid layer are aligned to the same orienta-tion as the conjunctive units. The grids of the grid unitsare also randomly shifted relative to each other, resultingin distributed spatial phases (Fig.5D).Different from the collateral weights in the conjunctivelayer, the self-organized connections from the conjunctivelayer to the grid layer do not show a clear pattern de-pendent on the head direction of the presynaptic units(Fig.6A). Grid units therefore receive balanced inputsfrom the conjunctive layer, at all possible head direc-tions, producing isotropic firing in head direction. Gridunits however are preferentially excited by conjunctiveunits with similar spatial phases (Fig.6B). The conjunc-tive units that send strong connections to the same unitsin the grid layer cluster in space as a result of Hebbianlearning.  5  0 90180270360 00.20.4unit 93Preferred head direction (degrees)    C  o   l   l  a   t  e  r  a   l  w  e   i  g   h   t  0 90180270360 00.20.4unit 111Preferred head direction (degrees) 0 90180270360 00.20.4unit 149Preferred head direction (degrees)-1-0.500.51 00.20.4Spatial correlation    C  o   l   l  a   t  e  r  a   l  w  e   i  g   h   t -1-0.500.51 00.20.4Spatial correlation-1-0.500.51 00.20.4Spatial correlation AB Figure 3: Learned collateral weights between conjunctive units. (A) The collateral weights of three example conjunctive unitsare plotted against the preferred head directions of pre-synaptic units. In each plot, there are 154 data points, each for oneconnection. Large weights scatter around the preferred head directions of the post-synaptic units, indicated by the broken lines;(B) The same collateral weights as in (A) are shown with respect to the correlation of the spatial maps between connectedunits.
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