The mammalian hippocampus plays a crucial role in producing a cognitive map of space—an internalized representation of the animal’s environment. We have previously shown that it is possible to model this map formation using a topological framework, in which information about the environment is transmitted through the temporal organization of neuronal spiking activity, particularly those occasions in which the firing of different place cells overlaps. In this paper, we discuss how gamma rhythm, one of the main components of the extracellular electrical field potential affects the efficiency of place cell map formation. Using methods of algebraic topology and the maximal entropy principle, we demonstrate that gamma modulation synchronizes the spiking of dynamical cell assemblies, which enables learning a spatial map at faster timescales.
For years it has been assumed that the cognitive map is Cartesian, containing detailed information about locations, distances and angles. Undoubtedly, such information is provided by various brain regions, but increasing evidence suggests that the hippocampal map is topological in nature. For example, electrophysiological recordings in morphing environments demonstrate that the spatial order (overlaps, adjacency and containment) is preserved, even in the face of deformations of the environment that cause the place fields to stretch or change shape [10–14]. In other words, the sequential order of place cell activity induced by the animal’s moves through morphing environment remains invariant, at least within a certain range of geometric transformations [15–17]. This implies that place cell spiking encodes a rough-and-ready framework into which other brain regions integrate more detailed metrical information [17–21]. What sorts of neuronal computations could produce such a framework [22–27]? The approach proposed in [28–30] exploits the connection between the place field map and the Alexandrov-Čech theorem, according to which the pattern of overlaps between regular spatial regions U1, U2, …UN covering a space X encodes the topological structure of X [32]. The construction suggested by the Alexandrov-Čech theorem is the following. If the regions Ui are represented as vertices, pairs of overlapping regions Ui ∩ Uj ≠ ∅, as links between these vertices, the triples Ui ∩ Uj ∩ Uk ≠ ∅ as the facets between these links and so forth, then the resulting simplicial complex N is topologically equivalent to X (see Glossary in the Methods section and S1 Fig). The fact that the place fields produce a dense cover of the environment suggests that the pattern of overlaps between them contains the information required to represent the environment’s topology, which we propose holds the key to the way in which the hippocampus encodes its topological map of a given space. Note that the domains where several place fields overlap are precisely the ones where the corresponding place cells cofire: the information about the overlap of place fields is represented via place cell coactivity, which suggests that the Alexandrov-Čech construction can be carried out not only via the geometric pattern of the place field overlaps, but also through analysis of place cell coactivities. The details of the topological model of the hippocampal map are discussed in [29,30]. In brief, the idea is to represent the combinations of coactive place cells (c1, c2, …, cp) as coactivity simplexes, σ = [c1, c2, …, cp]—combinatorial representations of multi-dimensional polyhedra (see Methods). Together, these coactivity simplexes form a simplicial coactivity complex T σ. In this construction, the individual cell assemblies (i. e. , a group of neurons that jointly drive a downstream readout neuron), provide local information about a given space; joined together into a neuronal ensemble (i. e. , a population of cell assemblies), they represent the space as whole. By analogy, a collection of individual simplexes representing connected locations, together form a simplicial complex which represents environment as a whole. Numerical simulations demonstrate that T σ captures the topological structure of the environment and serves as a schematic representation of the hippocampal map [29,31]. For example, the sequences of place cell combinations ignited along the paths traversed by the animal are represented in T σ by chains of coactivity simplexes—the simplicial paths [33,34]. A non-contractible simplicial path may represent a navigational path that encircles a physical obstacle, whereas topologically trivial simplicial paths correspond to contractible routes in the physical space (Fig 1A and 1B). The complex T σ begins to form as soon as the rat starts navigating. Every detected instance of place cell coactivity contributes a simplex to T σ. At the early stages of navigation, when only a few cells have time to produce spikes, the coactivity complex is small, fragmented, and contains many gaps (in topological terms, “holes”), most of which do not represent physical obstacles in the environment. Such holes tend to disappear as spatial learning continues. Therefore, the minimal time, Tmin, after which the topology of T σ matches the topology of the environment, or more precisely, when the correct number of topological loops emerges, can be viewed as a theoretical estimate of the time required to learn the hippocampal map (Fig 1C, [29,30]). An important property of the model is that the structure of the coactivity complex T σ and the time course of its formation during learning are sensitive to various parameters of the neuronal firing statistics, which allows us to study the effect that changes in any of these parameters (e. g. , firing rate, place field size, number of neurons) produce in the ability of the ensemble to correctly learn a space. For example, if the firing rate slows, the system can compensate with a change to the place field size or the number of neurons in the ensemble, but only up to a point: beyond certain limits, the assembly will not be able to learn efficiently, or even at all [29]. As another example, the oscillations of the extracellular electrical field potential, typically referred to as the local field potential (LFP), are known to modulate place cells activity at several timescales: each place cell tends to spike within a small range of the phases of the theta component of the LFP (θ, 4–12 Hz [35]), which depends on the distance that the animal has traveled into the corresponding place field. As a rat moves through the place field, the preferred θ-range of a place cell progressively decreases with each new θ-cycle, a phenomenon known as θ-phase precession [36]. The preferred θ-phases of different cells are additionally synchronized by the second major component of the LFP, the gamma rhythm (γ, 30–80 Hz, [37]). In fact, the period of the more rapid γ-rhythm, Tγ, is believed to define the range of the preferred phases within the slower θ-rhythm; on average one θ-period, Tθ, contains about seven γ-cycles, Tθ ≈ 7Tγ (see [38] and S2A Fig). Numerous experimental [39–43] and theoretical [38,44–47] studies demonstrate that both θ- and γ-waves play key roles in spatial, working, and episodic memory functions. Most theoretical analyses have addressed the way in which the γ-synchronization affects the informational contents of spiking in small networks or in individual cells, but the topological approach allows us to model the formation of cognitive map as a whole. For example, it was used in [30] to demonstrate that θ-precession makes otherwise poorly-performing ensembles more efficient at spatial learning. The present analysis applies the topological model to study the effect of γ-waves on spatial learning and to demonstrate that γ-synchronization of place cell spiking activity enables the encoding or retrieval of large-scale spatial representations of the environment by integrating place cell coactivity at a synaptic timescale. Computational modeling of θ-phase precession is relatively straightforward. At a basic level, it amounts to imposing a particular relationship between a place cell’s spiking probability, the phase of the θ-wave and the distance that the animal has traveled into the corresponding place field [48] (see Methods). The effects of the γ-rhythm are, however, more diverse. Electrophysiological experiments suggest that there exist at least two types of place cells: “TroPyr” cells that spike at the trough of the fast γ-wave (50–80 Hz) and “RisPyr” cells that fire at the rising phase of slow γ-waves, overriding θ-precession [49–51]. Although we can model both S2 Fig with our approach (see Methods), in the following we will model only the TroPyr cells that exhibit more robust firing patterns and higher firing rates, and therefore may play a primary role in producing the cognitive map [29,30].
In the above discussion, the central construction of the model, which is the coactivity complex T σ, was introduced as a schematic representation of the place field map [31]. However, as shown in [67,68], a coactivity complex can be built not only by detecting higher-order cofiring events that directly mark the locations where several place field overlap, but also by integrating the information provided by the lower-order place cell coactivity. Physiologically, the latter option corresponds to the cell assembly network in which the readout neurons integrate lower-order coactivity inputs over a working or intermediate memory timescale, rather than merely react to cofiring as all-or-none coactivity detectors [69,70]. To model a network of cell assemblies driving a population of input-integrator readout neurons, we used the following approach. First, we detected the lowest-order, pairwise place cell coactivities and used them to build a connectivity graph G (see [67] and S6 Fig). Fully interconnected subgraphs of G are called cliques (see Methods); cliques of G are identified with the simplexes of a new clique coactivity complex T ς. A key property of this algorithm is that the connections constituting a clique or a simplex do not have to be detected simultaneously but can be accumulated over an extended period of time. For physiological accuracy, we restrict this period to 10 mins or less, which results in a coactivity complex whose simplexes emerge over working or intermediate memory intervals. Although the algorithms for constructing temporal simplicial and clique complexes seem quite different, the actual difference between these two coactivity complexes is not as large. First, as shown in [31,67], most simplexes of T ς correspond to the simplexes of T σ and vice- versa: the identities of the cell assemblies are largely the same, only the time course of their construction changes. Furthermore, the topological structures of these complexes are quite close. Second, most pairwise connections within the cliques of G are produced almost simultaneously while the rat traverses the region where several place fields overlap. In other words, most cliques appear at once, just as the simplexes do, and only a relatively small number of the maximal cliques are actually “corrected” over time [68]. Nevertheless, this effect does improve the overall performance of the clique coactivity complexes, which typically produce far fewer spurious topological loops and shorter learning times Tmin than those produced by simplicial coactivity complexes. Implementing the γ-synchronization mechanism in an integrator model yields the results illustrated in Fig 4. First, the structure of the graphs on Figs 3A and 4A is qualitatively similar, though the pool of third-order cliques is slightly larger than the pool of 2D simplexes. This is because not every clique makes a simultaneous appearance as a simplex, but every simplex can be viewed as an instantly detected clique. The behaviors of the topological loops in T ς and in T σ, shown in Figs 3B and 4B are similar as well. The γ-synchronization reduces the number of cold, spurious loops in both types of complexes (S7 Fig). Physiologically, this implies that a γ-rhythm produces the same organizing effect on the activity of cell assembly network, whether the latter is based on a coincidence detector or on the input integrator readout neurons. However, it should be noted that, for all βs, the number of loops in T ς is smaller than in T σ by an order of magnitude, illustrating the efficiency of the input integrating readout neurons. Most importantly, the integrator complex T ς produces finite learning times at the γ-timescale integration window, w ≈ Tγ. This demonstrates that the hippocampal network can produce a spatial map by reading out γ-synchronized place cell coactivity at the γ-timescale and accumulating the coactivities over the timescale of working or intermediate memory.
It has long been established that both θ and γ rhythms correlate strongly with the capacity for learning and memory, but the mechanisms by which they influence cognitive functions has remained unclear. These extracellular fields define the timescale of place cell coactivity, thereby controlling the “parcellation” of the information flow received by downstream networks. In particular, the synchronization of the processes taking place at the synaptic timescale, such as the processes controlled by the membrane time constant, the duration of receptor-mediated postsynaptic spike potentials, the rate of spike-timing dependent plasticity, and so forth, [71–73] is manifested at the network level as γ-frequency oscillations [74–78]. Processes that involve slower forms of synaptic plasticity, including slow-changing spiking thresholds [79–82], synchronize at the θ-frequency timescale. As a result, θ-oscillations provide lower-resolution temporal packaging of place cell coactivity [66,83,84], integrating spiking inputs from several cell assemblies over one or more θ-periods [85–87]. Our previous model, based on place cells that are independently θ-precessing, provided a self-consistent description of the hippocampal network’s function at the θ-timescale, which predicted an optimal integration window for reading out the information within the θ-range [30]. However, as the integration window became smaller, the spatial map encoded by independently precessing place cells failed to achieve correct spatial representation, which suggested to us the importance of additional synchronization at the γ-timescale. Here we developed a phenomenological model based on the assumption that the γ-rhythm not only controls the probability of the cell assembly spiking but also defines the temporal spread of the spikes produced by the cell assemblies around the troughs of the γ-wave. As a result, the model predicts that if the preferred θ-phases synchronize with the γ-troughs, then topological information about the given environment can be readily captured by integrating place cell coactivity at the γ-timescale. Thus, γ-synchronization of spiking activity is crucial for both encoding and reading out information from the cell assemblies arriving in “γ-packets” [61]. This result suggests a possible phenomenological explanation as to why reduction of the γ-wave amplitude correlates with impairments in learning, whether the cause is changes in the network’s synaptic physiology [88–91], psychoactive drugs [92–94], neurodegeneration, or aging [95,96], whereas an increase of the γ-amplitude correlates with successful learning and retrieval of the learned information [97–101]. According to our model, reducing either the γ-amplitude or the diffused coupling between the γ-rhythm and place cell spiking activity, the latter being equivalent to lowering β, should increase learning times and lower the success rate in constructing topologically accurate cognitive maps. Vice versa, high γ-amplitude and strong coupling between the spike times and the γ-wave should result in more effective spatial learning.
An abstract simplex of order d, σd, is a set of (d + 1) elements, e. g. , a set of (d + 1) active cells. Note that the subsets of the set σd form subsimplexes of σd and that a nonempty overlap of any two simplexes σ 1 d and σ 2 d is a subsimplex of both σ 1 d and σ 2 d. A simplicial complex Σσ is a family of simplexes. The elements of a simplex σd can be visualized as vertices of d-dimensional polytopes: σ0 can be visualized as a point, σ1 as the ends of a line segment, σ2 as the vertices of a triangle, σ3 as the vertices of a tetrahedron, etc. [102]. A clique in a graph G is a set of fully interconnected vertices (i. e. , a complete graph). Combinatorially, cliques have the same key properties as the abstract simplexes: any subcollection of vertices in a clique is fully interconnected, and hence forms a subclique. A nonempty overlap of two cliques ς 1 d and ς 2 d is a subclique in both ς 1 d and ς 2 d. Therefore, cliques define abstract simplexes and thus the collection of cliques in a graph G defines a clique simplicial complex Σς (G). In [30] we showed that the time required to learn a large spatial environment is approximately equal to sum of times required to learn its parts. Therefore, we simulated a non-preferential exploratory behavior in a small planar environment (1m × 1m) shown in Fig 1A, similar to those used in electrophysiological experiments [103]. The Poisson spiking rate of a place cell c at a point r (t) = (x (t), y (t) ) is given by λ c (r) = f c e - (r - r c) 2 2 s c 2 where fc is the maximal firing rate and sc defines the size of the place field centered at rc = (xc, yc). The set of scs and fcs in an ensemble of N place cells are lognormally distributed around a certain ensemble-mean firing rate f and a certain ensemble-mean place field size s, with the variances σf = af and σs = bs, respectively. Thus, a place cell ensemble is described by a triplet of parameters: (s, f, N) [29]. As the rat moves over a distance l (t) into the place field of a cell c, the preferred spiking phase is φ θ, c (t) ≈ 2 π (1 - l (t) / L c), where Lc ∼ 3sc is the size of the place field [36,104]. To simulate the coupling between the firing rate and the θ-phase, we modulated the original Gaussian firing rate by a θ-factor Λθ, c (φ), giving Λ θ, c (φ) = e - (φ - φ θ, c (t) ) 2 2 ε c 2, using the θ-component of the LFP recorded in wild type rats. The width ε of the Gaussian was defined in [30] to be the ratio of the mean distance that rat travels during one θ-cycle to the size of the place field, ε = 2πv/Lωθ, where v is the rat’s speed and ωθ/2π is the frequency of the θ-signal. To incorporate the γ-rhythm into our model, we extracted the 30–80 Hz frequency band from the same LFP signal so that all the existing correlations between θ and γ waves are preserved, then we shifted the simulated place cell spiking times towards the troughs of γ amplitude by modulating their respective spiking rates with the additional Boltzmann factor [58], Λ γ (t) ∼ e - β γ A γ (t), (1) where Aγ (t) is the amplitude of the γ-wave and 1/βγ is a formal parameter that plays the role of the effective temperature [105] (Fig 2). Simulating the net firing rate as a product of all three factors λ n e t = λ c (x, y) Λ θ, c (φ) Λ γ (A γ) preserves spatial selectivity of spiking and the θ-precession (S8 Fig) and forces the preferred phases of the θ-phase precession φc into the γ-cycles, in accordance with the θ-γ theory [38,41,56]. In a vicinity of the ith trough, the gamma signal has the form A γ (t) ≈ A γ, 0 - A γ, i cos (ω i t) ≈ a γ, i + A γ, i ω i 2 t 2 2, (2) where the parameters Aγ, 0 are the mean amplitude of Aγ; Aγ, i and ωi are its instantaneous amplitude and frequency at the ith trough aγ, i = Aγ, 0 − Aγ, i and the index i runs over all troughs i ∈ I. Using the expansion Eq (2) in Eq (1) allows estimating the spread Δi of the spikes around the ith through from the Gaussian variance as Δ i 2 = 1 β γ A γ, i ω i 2. (3) A priori, in order to accurately define the temporal spread of spikes produced by different cell assemblies at different times, the inverse effective temperature should be trough-specific, βγ, i. However, we consider a simplified case in which the average βγ = 〈βγ, i〉i ∈ I defines the coupling between the γ-wave and the place cell spike times across the entire hippocampal network. The variance Eq (3) is about six times smaller than the instantaneous period, i. e. , 6 β γ A γ, i ω i ≈ 2 π ω i, which implies that the effective temperature is approximately equal to the mean γ-amplitude 1 β γ ≈ A γ, where Aγ = 〈Aγ, i〉i ∈ I. By normalizing Aγ with the standard deviation σ γ = 〈 A γ 2 (t) - A γ, 0 2 〉 t, A = Aγ/σγ, we obtain the scaled parameter β = βγ σγ, with the characteristic value β = 1 A. The described approach can be applied to both the TroPyr and RisPyr cells. Mathematically, the “rising phases of γ” controlling the spiking of RisPyr cells correspond to the vicinities of peaks of the time derivatives of the γ-amplitude. Therefore, the spiking probability of RisPyr cells can be constrained by a factor similar to Eq (1), involving the derivative of the γ-amplitude, A′ (t), which would override the θ-precession constraint (Λθ, c (φ) = 1) in the vicinity of the A′ (t) -peaks. The analysis of the mixed (RisPyr and TroPyr) ensembles is more complex and sui generis. Mathematical methods required for this study are based on Persistent Homology Theory (see [29] and [106,107]) implemented by the “JPlex” freeware package [108].
One of the goals of theoretical systems neuroscience is to connect parameters of neuronal activity observed in electrophysiological experiments, such as cell firing rates, frequencies and amplitudes of the brain rhythms, with cognitive phenomena that emerge at the scale of large groups of cells. In previous work, we proposed an approach for modeling one such phenomenon: spatial learning in the mammalian hippocampus. This approach rested on the hypothesis that hippocampal neurons encode a rough-and-ready topological map of an environment, whereas geometric details likely come from multiple inputs from other brain regions. A key property of this model is that it allows us to estimate the effect produced by different parameters of neuronal activity on spatial learning. In particular, it we showed that theta oscillations strongly enhance the ability of the place cell ensembles to learn topologically accurate spatial maps. In this work, we show that synchronization of neuronal spiking activity by the other major component of the oscillating extracellular cell potential—the gamma rhythm—also enhances both spatial learning and the retrieval of spatial memories at the physiological timescale.
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Gamma Synchronization Influences Map Formation Time in a Topological Model of Spatial Learning