The Effects of Theta Precession on Spatial Learning and Simplicial Complex Dynamics in a Topological Model of the Hippocampal Spatial Map
Considerable effort has been devoted over the years to understanding how the hippocampus is able to form an internal representation of the environment that enables an animal to efficiently navigate and remember the space 1 ., This internal map is made possible , in part , by the activity of pyramidal neurons in the hippocampus known as place cells 2 , 3 ., As an animal explores a given environment , different place cells will fire in different , discrete regions of the space that are then referred to as that cells “place field” 2 , 3 ., Despite decades of research , however , the features of the environment that are encoded , the identity of the downstream neurons that decode the information , and how the spiking activity of hundreds of cells is actually used to form the map all remain unclear ., We recently developed a computational model for spatial learning , focusing on what information is available to the still-unidentified downstream neurons 4 ., We reasoned that the information they decode must be encapsulated in the temporal patterns of the place cell spike trains , specifically place cell co-firing 4 , 5 ., Because place cell co-firing implies that the respective place fields overlap , the resulting map should derive from a sequence of overlaps between parts of the environment ., The information encoded by the hippocampus would therefore emphasize connectivity between places in the environment , which is a topological rather than a geometric quality of space 4 ., One advantage of this line of reasoning is that a topological problem should be amenable to topological analysis , so we developed our model using conceptual tools from the field of algebraic topology and , in particular , persistent homology theory 6 , 7 ., We simulated a rat exploring several topologically distinct environments and found that the information encoded by place cell co-firing can , in fact , reproduce the topological features of a given spatial environment ., We also found that , in order to form an accurate spatial map within a biologically reasonable length of time , our simplified model hippocampus had to function within a certain range of values that turned out to closely parallel those obtained from actual experiments with healthy rodents ., We called this sweet spot for spatial learning the learning region , L 4 ., As long as the values of the three parameters ( firing rates , place field sizes , and number of active neurons ) remain within the learning region , spatial learning is reliable and reproducible ., Beyond the perimeters of L , however , spatial learning fails ., Several features of this model are intuitively appealing ., First , the size and shape of L vary with the difficulty of the task: the greater the complexity of the space to be learned , the narrower the range of values that can sustain learning and thus the more compact the learning region ., Second , there is a certain tolerance for variation among the three parameters within L: if one parameter begins to fall outside the sweet spot , spatial learning can still occur if there is sufficient compensation in the other two parameters ., Our model suggests that certain diseases ( e . g . , Alzheimers ) or environmental toxins ( e . g . , ethanol , cannabinoids ) disrupt spatial learning over time by gradually shifting mean neuronal function ( place cell firing , neuronal number , or place field size ) beyond the perimeter of the learning region ., This notion receives support from studies of mouse models that show a correlation between impairment in spatial cognition and larger , more diffuse place fields , lower place cell firing rates , and smaller numbers of active cells 8 , 9 ., All this corresponds well with our subjective experiences of learning: the complexity of the task influences learning time; when the task is difficult we can feel we are at or just beyond the limits of our capacity; disease or intoxication can reveal limits in our spatial cognition that would normally be compensated for ., In this paper we focus on analyzing the structure of the learning region itself ., We begin by making the computational model more physiologically accurate ., There is a θ ( theta ) component of subcortical LFP oscillations that occurs in the frequency range of 6-12 Hz and regulates spiking activity 10 ., The timing of place cell spiking in the hippocampus is coupled with the phase of θ-oscillations so that , as a rat progresses through a particular place field , the corresponding place cell discharges at a progressively earlier phase of each new θ-cycle 11 ., This phenomenon , called theta phase precession , reproduces short sub-sequences of an animals current trajectory during each θ-cycle 11 ., This has been construed to suggest that θ-phase precession helps the hippocampus remember the temporal sequence of the rats positions in a space ( i . e . , its trajectory ) 12 , 13 , thereby enhancing spatial learning and memory ., If this is the case , θ phase precession should enhance learning in our computational model ., Indeed , we find that it significantly improves and stabilizes spatial learning ., We also find that different temporal windows to define co-firing exert a pronounced influence on learning time , and the most efficacious window widths correspond with experimental predictions ., Finally , we analyze simplicial complex formation within the learning region , examining both the structure of the complexes and the dynamics of loop formation , and find an explanation for the poor efficiency of ensembles at the boundary of the learning region compared to peak-performing ensembles ., We will first briefly describe the fundamental concepts on which our model is based ( this section is an abbreviated version of the approach described in 4 ) ., Central to this work is the concept of a nerve simplicial complex , in which a space X is covered by a number of smaller , discrete regions 14 ., If two regions overlap , the corresponding vertices , vi and vj , are considered connected by a 1D link vij ( Figure 1 ) ., If three regions overlap , then vij , vjk , and vki support a 2D triangular facet or simplex σijk , and so on as the number of overlaps and links increase ., The structure of the simplicial complex approximates the structure of the environment: the complex N ( X ) obtained from a sufficiently dense cover of the space X will reproduce the correct topological indices of X ( see 4 for details ) ., For our model we developed a temporal analogue to the simplicial complex , i . e . , a simplicial complex that builds over time: when the animal is first introduced to the environment , there will be only a few data points from place cell firing , but as the animal explores the space the place cell firing data accumulate ., ( Rodent experiments indicate that place fields take about four minutes to develop 15 . ), As the animal explores its environment and more place cells fire ( and co-fire ) , the simplicial complex T grows with T ( time ) ( T\u200a=\u200aT ( T ) ) ., Eventually , after a certain minimal time Tmin , the spaces topological characteristics will stabilize and produce the correct topological indices , at which point the topological information is complete ., Tmin is thus the minimal learning time , the time at which a topologically correct map is first formed ., The correct topological indices are indicated by Betti numbers , which in turn are manifested in persistent cycles ( see 4 , 7 , 16 ) ., As the rat begins to explore an environment , the simplicial complex T ( T ) will consist mostly of 0-cycles that correspond to small groups of cofiring cells that mark contractible spatial domains ., As the rat continues to explore the environment , the co-firing cells will produce links between the vertices of T ( T ) , and higher-dimensional cycles will appear ., As T increases , most cycles in each dimension will disappear as so much “topological noise , ” leaving only a few persisting cycles that express stable topological information ( Figure 1C ) ., The persistent homology method 6 ( see 4 Methods ) enables us to distinguish between cycles that persist across time ( reflecting real topological characteristics ) and transient cycles produced by the rats behavior ( e . g . , circling in a particular spot during one trial or simply not venturing into one part of the space during early explorations ) ., The pattern of cycles is referred to as a barcode 16 that can be easily read to give topological information about a given environment ( Figure 1C ) 6 , 7 ., If theta precession serves to enhance learning , as has been predicted 17–19 , then it should enhance spatial learning in our model ., This could occur by any of several means ., First , theta precession might enlarge the number of ensembles capable of the task by expanding the scope of the parameters ( including firing rates or place field sizes normally out of the bounds of L ) ., Second , it might make the ensembles that are in L converge on the correct topological information more rapidly ., Third , it might make the same ensembles perform more reliably ( e . g . , succeeding in map formation a greater percentage of times in our simulations ) ., To test the effect of theta precession in our model , we compared the rates of map formation for those formed with and without θ-precession ., We tested 1710 different place cell ensembles by independently varying the number of place cells ( N; 19 independent values , from 50 to 500 ) , the ensemble mean firing rate ( f; 10 independent values , from 4 to 40 ) , and the ensemble mean place field sizes ( s; 9 independent values , from 5 to 30 ) Methods; see 4 and Methods therein for further details ., For statistical analysis , we simulated each map 10 times so that we could compute the mean learning time and its relative variability , ξ\u200a=\u200aΔ Tmin/Tmin , for each set of ( s , f , N ) values ., In the following we will suppress the bar in the notation for the mean f , s , N , and Tmin ., Figure 2 shows the results of these simulations in a 1×1 m space with one hole ., ( The size of the environment in this study is smaller than the ones used in 4 , for two reasons: to avoid the potential problem of place cells with more than one field , and to reduce computational cost; see Methods . ), The learning region L is small and sparse in the θ-off case , but notably larger and denser in the θ-on case ( Figure 2A ) ., Values that would be just beyond the learning region—N that may be too small , or place fields that are too large or too small , or firing rates too high or too low 4—thus become functional with the addition of θ-precession ., Two criteria reveal the quality of the map-forming ensembles: speed and consistency in converging toward the correct topological signature ., The fastest map formation times ( under 4 minutes ) are represented by blue dots; as the color shifts toward red , map formation times become longer and the error rate ( failure to converge ) increases ., The size of the dot represents the success rate: small dots represent ensembles that only occasionally converge on the correct information , large dots represent ensembles that converge most or all of the time ., θ-precession increases the probability of convergence across all ensembles that can form accurate maps at all ( Supplemental Figure S1 ) ., Since we were interested in understanding the dynamics of efficient learning , however , we created a more stringent definition of the learning region to focus on the core of L where map-formation is most rapid and reliable , as well as to make the results more legible ( L can be quite dense , as in Figure 2A and Supplemental Figure S1 ) ; if θ-precession truly enhances learning , its effect should be apparent even in the most successful ensembles , and indeed this was the case ., The point clouds in Figure 2B depict those ensembles that formed maps with a convergence rate of ρ≥0 . 7 ( i . e . , those that produced correct topological information at least 70% of the time ) and simultaneously had low relative variability of the Tmin values , ξ<0 . 3 ., Even within this more efficient core of L , the effect of θ-precession was pronounced ., The histograms of the computed mean learning times are closely fit by the Generalized Extreme Value ( GEV ) probability distribution ( Figure 2B ) ., The distributions show that θ-precession reduced the mean learning times Tmin: the mode of all the θ-on GEV distributions decreased by ∼50% compared with the θ-off case for the learning region as a whole ( Figure 2C ) and by ∼15% for the efficient ensembles at the core of L ( Figure 2D ) ., Moreover , the effects of adding θ-precession—reducing map formation time and decreasing the relative variability of the Tmin values—were manifested in all maps , not just those with high ( ρ ≥0 . 7 ) convergence rates ( Figure 2C , D ) ., The histograms for all maps ( all ρ -values ) fit by the GEV distribution reveal that the typical variability ( the mode of the distributions ) in the θ-on cases is about half the size of the θ-off case ( Figure 2E ) ., In our model , therefore , θ-precession strongly enhances spatial learning ., Since we do not know what features of θ-oscillations might be important 20 , we studied four different θ-oscillations , two simulated and two derived from electrophysiological experiments in wild-type rodents ., Specifically , we modeled the effect of theta precession on the topological map by coupling the place cells Poisson firing rates , λc , with the phase of the following four θ-oscillations:, 1 ) θ1 – a single 8 Hz sinusoidal wave ,, 2 ) θ4 – a combination of four sinusoids ,, 3 ) θM – a subcortical EEG signal recorded in wild-type mouse , and, 4 ) θR – a subcortical EEG signal recorded in a rat ( Supplemental Figure S2; see Methods ) ., The last three signals were filtered in the θ-domain of frequencies ( 6–12 Hz ) ., The distribution of the learning times , the histograms of the mean learning times , and the histograms of the relative variability , ξ , for all four different theta cases are shown in Supplemental Figures S3 and S4 ., To compare the θ-off and θ-on cases , we performed two-sample Kolmogorov-Smirnov ( KS ) tests for all pairwise combinations of the studied sample sets 21 ., This produced a 5×5 matrix of the p-values , pij , where i , j\u200a=\u200a0 ( no theta ) , θ1 , θ4 , θM , and θR ., Black squares signify a statistically significant difference between cases i and j ( p<0 . 05 ) ; gray squares signify no statistically significant difference ., The statistical difference diagrams for the sets of Tmin values ( Supplemental Figure S3 ) and for the learning time variability ( Supplemental Figure S4 ) indicate that the distributions of learning times in the various θ-on cases were very similar , but the difference between all of these and the θ-off case was statistically significant ., So far we have described the outcome of place cell ensemble activity in terms of the time at which the correct number of loops in the simplicial complex T emerges ., But the learning process can also be described by how spurious loops are handled in the system ., These loops are a fair representation of the subjective experience of learning ., It takes time to build a framework into which new information can be properly slotted: until that framework is in place—whether its a grasp of the layout of a neighborhood or the basic principles of a new field of study—we have incomplete hunches and many incorrect notions before experience ( more learning ) fills in our understanding ., Translating this into topological terms , as the knowledge gaps close , the spurious loops contract ., We therefore wanted to study the effects of theta precession on the dynamics of loop formation ., Does a “smarter” ensemble form more spurious loops or fewer ?, Does it resolve those loops more quickly ?, We concentrated on the 1D cycles , which represent path connectivity within the simplicial complex , because they are more numerous and thus produce more robust statistics than the 0D cycles ., Figure 3 shows that θ-precession shortened the duration of the spurious loops ., The KS test reveals a statistically significant difference between the lifetimes of spurious loops in the θ-off case and those in all the θ-on cases ( Figure 3B ) ., To simplify the presentation of the results produced by the statistically similar θ-on cases , we combined the data on spurious loop duration from all four θ-driven maps into a single histogram ., It is interesting to note that the probability distributions for loop dynamics are typically better fit by the gamma distribution ( Figure 3B–D ) ., In the θ-driven maps , a typical spurious loop persisted for 50% less time than it would without θ-precession ( Figure 3B ) ., It is worth noting that the spurious loops persisted longer at the lower boundary of the learning region , where the mean firing rates and place field sizes are smallest ., This makes sense , insofar as whatever information appears will take longer to be corrected ., Statistical analysis of the largest number of loops observed at any given point over the course of the map formation period also differentiated θ-driven from θ-off maps ., Curiously , θ-driven cases tended to produce a significantly higher mean number of spurious loops than the θ-off case ( Figure 3C ) , but with a lower peak number of loops ( Figure 3D ) ., This implies that θ-precession enhances the speed of spatial learning overall at the price of creating more ( transient ) errors; lots of spurious loops are formed early on , but they disappear faster ., The KS test shows that the distributions of the mean number of loops in all θ-on cases differ from one another; only the maps driven by the two simulated θ-signals gave statistically similar results ., In our model , spatial learning can be quantified by the time required for the emergence of correct topological information , but it can also be quantified by studying the simplicial complex itself ., We noted earlier that the structure of the simplicial complex approximates the structure of the environment ., Similarly , it is possible to conceive of a simplex as a mathematical analogue to a cell assembly ( a group of at least two cells that repeatedly co-fire and form a synapse onto a readout neuron ) , and to view the simplicial complex as analogous to the realm of possible connections within the hippocampus ., We were therefore curious: since it is in the interest of neural function to be efficient , how many cell assemblies ( simplices ) does it take to encode a given amount of information ?, We would predict that the fewer the connections , the better , for the sake of efficiency ., One of the major characteristics of a simplicial complex T is the number of n-dimensional simplices it contains , traditionally denoted as fn ., The list of all fn –values , ( f1 , f2 , … , fn ) , is referred to as the f-vector 22 ., Since the D-dimensional simplices in T correspond to ( D+1 ) -ary connections , the number of which depends on the number of vertices , N , we considered the fn values normalized by the corresponding binomial coefficients , which characterize the number of simplices connecting vertices in the complex T . We can consider η an index of the connectivity of the simplicial complex ., Since we model 2D spatial navigation , we analyzed the connections between two and three vertices , i . e . , the 1D and 2D simplices , of T ( the number of 0D simplexes normalized by the number of vertices in T is η0≡1 ) ., Figure 4 shows the distribution of the normalized number of simplices at the time the correct signature is achieved ( η1 and η2 , for 1D and 2D , respectively ) ., As expected , the number of simplices was smaller at the lower boundary of the learning region L ( the base of the point cloud ) and increased towards the top of L where place fields are larger and the firing rates are higher , each of which would produce more place cell co-firing events ., Remarkably , the number of simplices depended primarily on the mean place field size and on the mean firing rate of the ensemble , and not on the number of cells within the ensemble ., This suggests a certain universality in the behavior of place cell ensembles that is independent of their population size ., In the ensembles with smaller place fields and lower firing rates , about 1 . 5% of place cell pairs and 1 . 7% of the triplets were connected , and this was enough to encode the correct topological information , whereas in the ensembles with low spatial selectivity and higher firing rates , 25% of pairs and 8% of triplets were connected ., These ensembles , in which the place fields and spike trains will by definition have a lot of overlap , are forced to form many more 1D and 2D simplices in order to encode the same amount of information and are thus less efficient ( Figure 4 , third column ) ., According to our model , such ensembles and the hippocampal networks whose activity they represent are inefficient on two counts ., First , these larger , more complex temporal simplicial complexes ( analogous to a larger number of coactive cell groups ) will take longer to form correct topological information , if they can manage it at all ., Second , a larger number of coactive place cells would hamper the training of downstream readout neurons , thereby impeding reliable encoding of spatial information ., This is consistent with studies showing that the number of cells participating in a particular task decreases until it reaches an optimal number that fire at a slightly higher rate than their no-longer-participating neighbors 23 ., Our model depends on patterns of place cell co-firing , but we had not previously explored what the optimal temporal window for defining co-activity might be ., Experimental work supports the widely held assumption that the temporal unit for defining coactivity ranges between tens 24 and hundreds of milliseconds 25–27 ., Our model , however , enables us to approach the question of optimal width for the coactivity window theoretically ., Clearly , if the time window w is too small , then the spike trains produced by the presynaptic place cells will often “miss” one another , and the map will either require a long time to emerge or it may not be established at all ., One would thus expect large values Tmin ( w ) for a small w ., On the other hand , if w is too large , it will allow cells whose place fields are actually distant from one another to be grouped together , yielding incorrect topological information ., Theta rhythm itself will have a tendency to group sequential spike trains together , but clearly there must be limits to this , or else some place cells would be read downstream as co-firing when they actually are not ., Therefore , there should exist an optimal value of w that reliably produces a finite , biologically relevant learning time Tmin at which the learning region L is robust and stable ., We assume that the capability of a read-out neuron to detect place cell co-activity is specified by a single parameter , the width of the integration time window w , over which the co-appearance of the spike trains is detected ., ( We considered the possible effect of time bin position on co-activity , but found this did not affect outcome; see Methods . ), We defined cell coactivity as follows: if presynaptic neurons c1 and c2 send a signal to a read-out neuron within a certain time window w , their activity will be interpreted as contemporaneous ., The width of this time window may be positive ( c2 becomes active w seconds after c1 ) or negative ( c2 becomes active while c1 is still firing ) ., We studied window widths w for which the place cell spike trains would eventually be able to produce the correct topological signature ( the Betti numbers , see Methods in 4 ) ., In order to describe the dependence of learning times on the window width , Tmin ( w ) , we scanned an array of 24 values of w ( ranging from 0 . 1 to 5 θ-periods ) for each combination of the parameters ( mean s , f , and N ) and noted the width of the value wo , at which the map began to produce the correct topological signature ., We call this initial correct window width the “opening” value ., A typical result is provided by an ensemble with f\u200a=\u200a28 Hz , s\u200a=\u200a23 cm and N\u200a=\u200a350 , in which an accurate topological map emerges at a fairly small window width , wo∼25 msec ( Figure 5 ) ., The distribution of the opening window widths shows that wo may exceed 1 . 5 θ-periods ( ∼25 msec ) , which matches the slow γ-period 24 , 28 ( Supplemental Figure S5 ) ., Since at this stage γ-oscillations have not been explicitly built into the model , this correspondence is coincidental , if suggestive ., As expected , the values of learning times at wo were rather large: Tmin ( wo ) ∼20 minutes in θ-off case and Tmin ( wo ) ∼30 minutes in the θ-on case , and in some cases exceeding one hour ( mostly for the ensembles with low firing rates ) ., For small window widths , the value of the learning time Tmin ( w ) was very sensitive to variations in w ( Supplemental Figure S6 ) ., As w increased , however , the learning time reached a plateau around some larger value ws ., This implies that in order to produce stable values for Tmin that are biologically plausible , the values of the window widths should start around ws ., The distribution of the ws values demonstrates that in the θ-off case the stabilization is typically achieved at approximately one θ-period , and in the θ-on case at about ∼1 . 2–1 . 5 θ-cycles ( Figure 6 ) , which justifies our choice of a two θ-period window width for the computations and corresponds well with the predicted limit of 150 msec for θ-cycle cofiring in sequence coding 29 ., Further increasing the integration time window w did not significantly alter the learning time Tmin in L; instead , the rate of map convergence decreased until the maps completely fail to encode the correct topological information at w ζ 4 . 5 θ-periods ., From the perspective of our current model , the range of optimal window widths w is between 20–25 msec and 0 . 5 secs ., Finally , we sought to uncover a relationship between learning time and window width ., Our analysis suggests that Tmin is inversely proportional to a power of the window width ( Figure 5B , Supplemental Figure S7 ) ., Numerous experiments have demonstrated that θ precession is important for spatial learning ., θ-power increases with memory load during both spatial and non-spatial tasks in humans 30 , 31 and in rodents 32 , 33; spatial deficits correlate with a decrease in the power of theta oscillations in Alzheimers disease 34 and in epilepsy 35 , 36 ., If θ-signal is blocked by lesioning the medial septum ( which does not affect hippocampal place cell representations ) , it severely impairs memory 37 and the acquisition of new spatial information 38 ., Recent experiments demonstrate more directly that destroying θ precession by administering cannabinoids to rats correlates with behavioral and spatial learning deficits 17 , 39 ., But at what level , and through what mechanisms , does θ precession exert its influence ?, The effect of θ-precession on the structure of the spike trains is rather complex 40 ., On the one hand , it groups cell spikes closer together in time and enforces specific sequences of cell firing , which is typically interpreted as increasing the temporal coherence of place cell activity 41–43 ., One might predict that grouping spikes together would ( somehow ) speed up learning ., On the other hand , θ-precession imposes extra conditions on the spike timing that depend on θ-phase and on the rats location with respect to the center of the place field through which it is presently moving ., Since every neuron precesses independently , one could just as well predict that θ modulation would either restrict or enlarge the pool of coactivity events , which in turn would slow down learning at the level of the downstream networks , and that the beneficial effect of the θ rhythm is a higher-order phenomenon that occurs elsewhere in the brain ., Our results suggest that θ precession may not just correlate with , but actually be a mechanism for , enhancing spatial learning and memory ., The interplay of θ precession and window width , especially the extremely long learning times at the opening window width wo , is particularly illuminating here ., As noted , theta precession acts at both the ensemble and the individual neuron level: it groups spikes together , but each neuron precesses independently ., When the time window is sufficiently wide , the coactivity events are reliably captured , the first effect dominates , and the main outcome of theta precession is to supply grouped spikes to downstream neurons ., For very small time windows , however , the system struggles to capture events as coactive , and the extra condition imposed by phase precession acts as an impediment: detected coactivities are rarer , and learning slows down ., Put more simply , imagine in Suppl ., Figure S8 that the window is only one spike wide: in a train of 10 spikes that overlaps by one spike with another train , it will take 10 windows before the overlap is detected ., It is noteworthy that the presence of theta precession was clearly more important than the details of the oscillation ., Although theta precession enhanced learning in our simulations , learning times were relatively insensitive to the details of the theta precession chosen ., One might expect differences in spike train structure induced by the four different θ-signals studied to alter the dynamics of the persistent loops and thus learning efficiency ., Our results show , however , that differences that would matter at the level of individual cells are averaged out at the level of a large ensemble of cells ., Here again the model shows its particular strength: it allows us to correlate parameters of activity at the level of individual neurons with the outcome at the level of an ensemble of hundreds of cells , providing a framework for understanding how micro-level changes play out at the behavioral level ., Interestingly , we also saw a difference between the micro and macro levels when we considered whether the placement of a temporal window affected what would be considered co-activity ( see Methods and Supplementary Figure S8 ) ., In theory it should , but the effect at the macro level washes out and we found that only the temporal width of the window matters for learning time ., Beyond validating the model as a reliable way to study physiological aspects of spatial learning , we have gone further in this work to analyze the simplicial complex itself as a way of describing learning ., As a rat starts to explore an environment , some cells begin to form place fields ., Then , the co-firing of two or more place cells will define the respective places as connected in space and temporal experience and will create corresponding simplices in the simplicial complex T . With time , these simplices form a chain corresponding to the animals route through the space ., If the environment is bounded , the rat will discover new connections between the same places ( arriving to the same location via different routes ) ., As a result , the chains of connected simplices grow together to form loops ., Existing loops become thicker and may eventually “close up” and disappear , yielding surfaces ., The appearance of such surfaces is significant: the closing up of a D-dimensional surface corresponds to the contraction , or disappearance , of one of its boundaries , which itself is a D1-dimensional loop ., Eventually , the structure of the simplicial complex saturates such that no new simplices ( connections between places ) are produced and no more loops contract because all that could close have already closed ., At this point , the saturated simplicial complex T encodes not only the possible locations of the rat , but also connections between the locations , along with the information about how these
Learning arises through the activity of large ensembles of cells , yet most of the data neuroscientists accumulate is at the level of individual neurons; we need models that can bridge this gap ., We have taken spatial learning as our starting point , computationally modeling the activity of place cells using methods derived from algebraic topology , especially persistent homology ., We previously showed that ensembles of hundreds of place cells could accurately encode topological information about different environments ( “learn” the space ) within certain values of place cell firing rate , place field size , and cell population; we called this parameter space the learning region ., Here we advance the model both technically and conceptually ., To make the model more physiological , we explored the effects of theta precession on spatial learning in our virtual ensembles ., Theta precession , which is believed to influence learning and memory , did in fact enhance learning in our model , increasing both speed and the size of the learning region ., Interestingly , theta precession also increased the number of spurious loops during simplicial complex formation ., We next explored how downstream readout neurons might define co-firing by grouping together cells within different windows of time and thereby capturing different degrees of temporal overlap between spike trains ., Our models optimum coactivity window correlates well with experimental data , ranging from ∼150–200 msec ., We further studied the relationship between learning time , window width , and theta precession ., Our results validate our topological model for spatial learning and open new avenues for connecting data at the level of individual neurons to behavioral outcomes at the neuronal ensemble level ., Finally , we analyzed the dynamics of simplicial complex formation and loop transience to propose that the simplicial complex provides a useful working description of the spatial learning process .
One of the challenges in contemporary neuroscience is that we have few ways to connect data about the features of individual neurons with effects ( such as learning ) that emerge only at the scale of large cell ensembles ., We are tackling this problem using spatial learning as a starting point ., In previous work we created a computational model of spatial learning using concepts from the field of algebraic topology , proposing that the hippocampal map encodes topological features of an environment ( connectivity ) rather than precise metrics ( distances and angles between locations ) —more akin to a subway map than a street map ., Our model simulates the activity of place cells as a rat navigates the experimental space so that we can estimate the effect produced by specific electrophysiological components —cell firing rate , population size , etc ., —on the net outcome ., In this work , we show that θ phase precession significantly enhanced spatial learning , and that the way downstream neurons group cells together into coactivity windows exerts interesting effects on learning time ., These findings strongly support the notion that theta phase precession enhances spatial learning ., Finally , we propose that ideas from topological theory provide a conceptually elegant description of the actual learning process .
computational neuroscience, biology and life sciences, computational biology, neuroscience, learning and memory