Neural dynamics of the cognitive map in the hippocampus

The rodent hippocampus has been thought to represent the spatial environment as a cognitive map. In the classical theory, the cognitive map has been explained as a consequence of the fact that different spatial regions are assigned to different cell populations in the framework of rate coding. Recently, the relation between place cell firing and local field oscillation theta in terms of theta phase precession was experimentally discovered and suggested as a temporal coding mechanism leading to memory formation of behavioral sequences accompanied with asymmetric Hebbian plasticity. The cognitive map theory is apparently outside of the sequence memory view. Therefore, theoretical analysis is necessary to consider the biological neural dynamics for the sequence encoding of the memory of behavioral sequences, providing the cognitive map formation. In this article, we summarize the theoretical neural dynamics of the real-time sequence encoding by theta phase precession, called theta phase coding, and review a series of theoretical models with the theta phase coding that we previously reported. With respect to memory encoding functions, instantaneous memory formation of one-time experience was first demonstrated, and then the ability of integration of memories of behavioral sequences into a network of the cognitive map was shown. In terms of memory retrieval functions, theta phase coding enables the hippocampus to represent the spatial location in the current behavioral context even with ambiguous sensory input when multiple sequences were coded. Finally, for utilization, retrieved temporal sequences in the hippocampus can be available for action selection, through the process of reverting theta rhythm-dependent activities to information in the behavioral time scale. This theoretical approach allows us to investigate how the behavioral sequences are encoded, updated, retrieved and used in the hippocampus, as the real-time interaction with the external environment. It may indeed be the bridge to the episodic memory function in human hippocampus.     

The spatiotemporal learning rule and its efficiency in separating spatiotemporal patterns

The hippocampus plays an important role in the course of establishing long-term memory, i.e., to make short-term memory of spatially and temporally associated input information. In 1996 (Tsukada et al. 1996), the spatiotemporal learning rule was proposed based on differences observed in hippocampal long-term potentiation (LTP) induced by various spatiotemporal pattern stimuli. One essential point of this learning rule is that the change of synaptic weight depends on both spatial coincidence and the temporal summation of input pulses. We applied this rule to a single-layered neural network and compared its ability to separate spatiotemporal patterns with that of other rules, including the Hebbian learning rule and its extended rules. The simulated results showed that the spatiotemporal learning rule had the highest efficiency in discriminating spatiotemporal pattern sequences, while the Hebbian learning rule (including its extended rules) was sensitive to differences in spatial patterns.     

Theta Coordinated Error-Driven Learning in the Hippocampus

The learning mechanism in the hippocampus has almost universally been assumed to be Hebbian in nature, where individual neurons in an engram join together with synaptic weight increases to support facilitated recall of memories later. However, it is also widely known that Hebbian learning mechanisms impose significant capacity constraints, and are generally less computationally powerful than learning mechanisms that take advantage of error signals. We show that the differential phase relationships of hippocampal subfields within the overall theta rhythm enable a powerful form of error-driven learning, which results in significantly greater capacity, as shown in computer simulations. In one phase of the theta cycle, the bidirectional connectivity between CA1 and entorhinal cortex can be trained in an error-driven fashion to learn to effectively encode the cortical inputs in a compact and sparse form over CA1. In a subsequent portion of the theta cycle, the system attempts to recall an existing memory, via the pathway from entorhinal cortex to CA3 and CA1. Finally the full theta cycle completes when a strong target encoding representation of the current input is imposed onto the CA1 via direct projections from entorhinal cortex. The difference between this target encoding and the attempted recall of the same representation on CA1 constitutes an error signal that can drive the learning of CA3 to CA1 synapses. This CA3 to CA1 pathway is critical for enabling full reinstatement of recalled hippocampal memories out in cortex. Taken together, these new learning dynamics enable a much more robust, high-capacity model of hippocampal learning than was available previously under the classical Hebbian model.     

Movement-related theta rhythm in humans: coordinating self-directed hippocampal learning

The hippocampus is crucial for episodic or declarative memory and the theta rhythm has been implicated in mnemonic processing, but the functional contribution of theta to memory remains the subject of intense speculation. Recent evidence suggests that the hippocampus might function as a network hub for volitional learning. In contrast to human experiments, electrophysiological recordings in the hippocampus of behaving rodents are dominated by theta oscillations reflecting volitional movement, which has been linked to spatial exploration and encoding. This literature makes the surprising cross-species prediction that the human hippocampal theta rhythm supports memory by coordinating exploratory movements in the service of self-directed learning. We examined the links between theta, spatial exploration, and memory encoding by designing an interactive human spatial navigation paradigm combined with multimodal neuroimaging. We used both non-invasive whole-head Magnetoencephalography (MEG) to look at theta oscillations and Functional Magnetic Resonance Imaging (fMRI) to look at brain regions associated with volitional movement and learning. We found that theta power increases during the self-initiation of virtual movement, additionally correlating with subsequent memory performance and environmental familiarity. Performance-related hippocampal theta increases were observed during a static pre-navigation retrieval phase, where planning for subsequent navigation occurred. Furthermore, periods of the task showing movement-related theta increases showed decreased fMRI activity in the parahippocampus and increased activity in the hippocampus and other brain regions that strikingly overlap with the previously observed volitional learning network (the reverse pattern was seen for stationary periods). These fMRI changes also correlated with participant's performance. Our findings suggest that the human hippocampal theta rhythm supports memory by coordinating exploratory movements in the service of self-directed learning. These findings directly extend the role of the hippocampus in spatial exploration in rodents to human memory and self-directed learning.     

GABAergic contributions to gating, timing, and phase precession of hippocampal neuronal activity during theta oscillations

Successful spatial exploration requires gating, storage, and retrieval of spatial memories in the correct order. The hippocampus is known to play an important role in the temporal organization of spatial information. Temporally ordered spatial memories are encoded and retrieved by the firing rate and phase of hippocampal pyramidal cells and inhibitory interneurons with respect to ongoing network theta oscillations paced by intra- and extrahippocampal areas. Much is known about the anatomical, physiological, and molecular characteristics as well as the connectivity and synaptic properties of various cell types in the hippocampal microcircuits, but how these detailed properties of individual neurons give rise to temporal organization of spatial memories remains unclear. We present a model of the hippocampal CA1 microcircuit based on observed biophysical properties of pyramidal cells and six types of inhibitory interneurons: axo-axonic, basket, bistratistified, neurogliaform, ivy, and oriens lacunosum-moleculare cells. The model simulates a virtual rat running on a linear track. Excitatory transient inputs come from the entorhinal cortex (EC) and the CA3 Schaffer collaterals and impinge on both the pyramidal cells and inhibitory interneurons, whereas inhibitory inputs from the medial septum impinge only on the inhibitory interneurons. Dopamine operates as a gate-keeper modulating the spatial memory flow to the PC distal dendrites in a frequency-dependent manner. A mechanism for spike-timing-dependent plasticity in distal and proximal PC dendrites consisting of three calcium detectors, which responds to the instantaneous calcium level and its time course in the dendrite, is used to model the plasticity effects. The model simulates the timing of firing of different hippocampal cell types relative to theta oscillations, and proposes functional roles for the different classes of the hippocampal and septal inhibitory interneurons in the correct ordering of spatial memories as well as in the generation and maintenance of theta phase precession of pyramidal cells (place cells) in CA1. The model leads to a number of experimentally testable predictions that may lead to a better understanding of the biophysical computations in the hippocampus and medial septum.     

A computational study on altered theta-gamma coupling during learning and phase coding

There is considerable interest in the role of coupling between theta and gamma oscillations in the brain in the context of learning and memory. Here we have used a neural network model which is capable of producing coupling of theta phase to gamma amplitude firstly to explore its ability to reproduce reported learning changes and secondly to memory-span and phase coding effects. The spiking neural network incorporates two kinetically different GABA(A) receptor-mediated currents to generate both theta and gamma rhythms and we have found that by selective alteration of both NMDA receptors and GABA(A,slow) receptors it can reproduce learning-related changes in the strength of coupling between theta and gamma either with or without coincident changes in theta amplitude. When the model was used to explore the relationship between theta and gamma oscillations, working memory capacity and phase coding it showed that the potential storage capacity of short term memories, in terms of nested gamma-subcycles, coincides with the maximal theta power. Increasing theta power is also related to the precision of theta phase which functions as a potential timing clock for neuronal firing in the cortex or hippocampus.     

Input-dependent learning rule for the memory of spatiotemporal sequences in hippocampal network with theta phase precession

Theta phase precession in rat hippocampal place cells is hypothesized to contribute to memory encoding of running experience in the sense that it provides the ideal timing for synaptic plasticity and enables the asymmetric associative connections under the Hebbian learning rule with asymmetric time window (Yamaguchi 2003). When the sequence of place fields is considered as the episodic memory of running experience, a given spatial route should be accurately stored in spite of differing overlap extent among place fields and varying running velocity. Using a hippocampal network model with phase precession and the Hebbian learning rule with asymmetric time window, we investigate the memory encoding of place field sequences in a single traversal experience. Computer experiments show that place fields cannot be stored correctly until an input-dependent feature is introduced into the learning rule. These experiments further indicate that there exists an optimum value for the saturation level of synaptic plasticity and the speed of synaptic plasticity in the learning rule, which are correlated with, respectively, the overlap extent of place field sequence and the running velocity of animal during traversal. A comparison of these results with biological evidences shows good agreement and suggests that behavior-dependent regulation of the learning rule is necessary for memory encoding.     

Selective reduction of AMPA currents onto hippocampal interneurons impairs network oscillatory activity

Reduction of excitatory currents onto GABAergic interneurons in the forebrain results in impaired spatial working memory and altered oscillatory network patterns in the hippocampus. Whether this phenotype is caused by an alteration in hippocampal interneurons is not known because most studies employed genetic manipulations affecting several brain regions. Here we performed viral injections in genetically modified mice to ablate the GluA4 subunit of the AMPA receptor in the hippocampus (GluA4(HC-/-) mice), thereby selectively reducing AMPA receptor-mediated currents onto a subgroup of hippocampal interneurons expressing GluA4. This regionally selective manipulation led to a strong spatial working memory deficit while leaving reference memory unaffected. Ripples (125-250 Hz) in the CA1 region of GluA4(HC-/-) mice had larger amplitude, slower frequency and reduced rate of occurrence. These changes were associated with an increased firing rate of pyramidal cells during ripples. The spatial selectivity of hippocampal pyramidal cells was comparable to that of controls in many respects when assessed during open field exploration and zigzag maze running. However, GluA4 ablation caused altered modulation of firing rate by theta oscillations in both interneurons and pyramidal cells. Moreover, the correlation between the theta firing phase of pyramidal cells and position was weaker in GluA4(HC-/-) mice. These results establish the involvement of AMPA receptor-mediated currents onto hippocampal interneurons for ripples and theta oscillations, and highlight potential cellular and network alterations that could account for the altered working memory performance.     

A Computational Predictor of Human Episodic Memory Based on a Theta Phase Precession Network

In the rodent hippocampus, a phase precession phenomena of place cell firing with the local field potential (LFP) theta is called “theta phase precession” and is considered to contribute to memory formation with spike time dependent plasticity (STDP). On the other hand, in the primate hippocampus, the existence of theta phase precession is unclear. Our computational studies have demonstrated that theta phase precession dynamics could contribute to primate–hippocampal dependent memory formation, such as object–place association memory. In this paper, we evaluate human theta phase precession by using a theory–experiment combined analysis. Human memory recall of object–place associations was analyzed by an individual hippocampal network simulated by theta phase precession dynamics of human eye movement and EEG data during memory encoding. It was found that the computational recall of the resultant network is significantly correlated with human memory recall performance, while other computational predictors without theta phase precession are not significantly correlated with subsequent memory recall. Moreover the correlation is larger than the correlation between human recall and traditional experimental predictors. These results indicate that theta phase precession dynamics are necessary for the better prediction of human recall performance with eye movement and EEG data. In this analysis, theta phase precession dynamics appear useful for the extraction of memory-dependent components from the spatio–temporal pattern of eye movement and EEG data as an associative network. Theta phase precession may be a common neural dynamic between rodents and humans for the formation of environmental memories.     

Coordinated learning of grid cell and place cell spatial and temporal properties: multiple scales,attention and oscillations

A neural model proposes how entorhinal grid cells and hippocampal place cells may develop as spatial categories in a hierarchy of self-organizing maps (SOMs). The model responds to realistic rat navigational trajectories by learning both grid cells with hexagonal grid firing fields of multiple spatial scales, and place cells with one or more firing fields, that match neurophysiological data about their development in juvenile rats. Both grid and place cells can develop by detecting, learning and remembering the most frequent and energetic co-occurrences of their inputs. The model''s parsimonious properties include: similar ring attractor mechanisms process linear and angular path integration inputs that drive map learning; the same SOM mechanisms can learn grid cell and place cell receptive fields; and the learning of the dorsoventral organization of multiple spatial scale modules through medial entorhinal cortex to hippocampus (HC) may use mechanisms homologous to those for temporal learning through lateral entorhinal cortex to HC (‘neural relativity’). The model clarifies how top-down HC-to-entorhinal attentional mechanisms may stabilize map learning, simulates how hippocampal inactivation may disrupt grid cells, and explains data about theta, beta and gamma oscillations. The article also compares the three main types of grid cell models in the light of recent data.     

What can the hippocampal representation of environmental geometry tell us about Hebbian learning?

 The importance of the hippocampus in spatial representation is well established. It is suggested that the rodent hippocampal network should provide an optimal substrate for the study of unsupervised Hebbian learning. We focus on the firing characteristics of hippocampal place cells in morphologically different environments. A hard-wired quantitative geometric model of individual place fields is reviewed and presented as the framework in which to understand the additional effects of synaptic plasticity. Existent models employing Hebbian learning are also reviewed. New information is presented regarding the dynamics of place field plasticity over short and long time scales in experiments using barriers and differently shaped walled environments. It is argued that aspects of the temporal dynamics of stability and plasticity in the hippocampal place cell representation both indicate modifications to, and inform the nature of, the synaptic plasticity in place cell models. Our results identify a potential neural basis for long-term incidental learning of environments and provide strong constraints for the way the unsupervised learning in cell assemblies envisaged by Hebb might occur within the hippocampus. Received: 8 March 2002 / Accepted: 13 June 2002 Acknowledgements. This work was supported by the Medical Research Council of the United Kingdom. Correspondence to: C. Lever or N. Burgess (e-mail: colin.lever@ucl.ac.uk; n.burgess@ucl.ac.uk, Tel.: +44-20-76793388 or 1147, Fax: +44-20-76791306 or 1145)     

Theta phase coding in a network model of the entorhinal cortex layer II with entorhinal-hippocampal loop connections

We investigated successive firing of the stellate cells within a theta cycle, which replicates the phase coding of place information, using a network model of the entorhinal cortex layer II with loop connections. Layer II of the entorhinal cortex (ECII) sends signals to the hippocampus, and the hippocampus sends signals back to layer V of the entorhinal cortex (ECV). In addition to this major pathway, projection from ECV to ECII also exists. It is, therefore, inferred that reverberation activity readily appears if projections from ECV to ECII are potentiated. The frequency of the reverberation would be in a gamma range because it takes signals 20–30 ms to go around the entorhinal-hippocampal loop circuits. On the other hand, it has been suggested that ECII is a theta rhythm generator. If the reverberation activity appears in the entorhinal-hippocampal loop circuits, gamma oscillation would be superimposed on a theta rhythm in ECII like a gamma-theta oscillation. This is a reminiscence of the theta phase coding of place information. In this paper, first, a network model of ECII will be developed in order to reproduce a theta rhythm. Secondly, we will show that loop connections from one stellate cell to the other one are selectively potentiated by afferent signals to ECII. Frequencies of those afferent signals are different, and transmission delay of the loop connections is 20 ms. As a result, stellate cells fire successively within one cycle of the theta rhythm. This resembles gamma-theta oscillation underlying the phase coding. Our model also replicates the phase precession of stellate cell firing within a cycle of subthreshold oscillation (theta rhythm).     

Phase precession through acceleration of local theta rhythm: a biophysical model for the interaction between place cells and local inhibitory neurons

Phase precession is one of the most well known examples within the temporal coding hypothesis. Here we present a biophysical spiking model for phase precession in hippocampal CA1 which focuses on the interaction between place cells and local inhibitory interneurons. The model's functional block is composed of a place cell (PC) connected with a local inhibitory cell (IC) which is modulated by the population theta rhythm. Both cells receive excitatory inputs from the entorhinal cortex (EC). These inputs are both theta modulated and space modulated. The dynamics of the two neuron types are described by integrate-and-fire models with conductance synapses, and the EC inputs are described using non-homogeneous Poisson processes. Phase precession in our model is caused by increased drive to specific PC/IC pairs when the animal is in their place field. The excitation increases the IC's firing rate, and this modulates the PC's firing rate such that both cells precess relative to theta. Our model implies that phase coding in place cells may not be independent from rate coding. The absence of restrictive connectivity constraints in this model predicts the generation of phase precession in any network with similar architecture and subject to a clocking rhythm, independently of the involvement in spatial tasks.     

Theta phase precession of grid and place cell firing in open environments

Place and grid cells in the rodent hippocampal formation tend to fire spikes at successively earlier phases relative to the local field potential theta rhythm as the animal runs through the cell''s firing field on a linear track. However, this ‘phase precession’ effect is less well characterized during foraging in two-dimensional open field environments. Here, we mapped runs through the firing fields onto a unit circle to pool data from multiple runs. We asked which of seven behavioural and physiological variables show the best circular–linear correlation with the theta phase of spikes from place cells in hippocampal area CA1 and from grid cells from superficial layers of medial entorhinal cortex. The best correlate was the distance to the firing field peak projected onto the animal''s current running direction. This was significantly stronger than other correlates, such as instantaneous firing rate and time-in-field, but similar in strength to correlates with other measures of distance travelled through the firing field. Phase precession was stronger in place cells than grid cells overall, and robust phase precession was seen in traversals through firing field peripheries (although somewhat less than in traversals through the centre), consistent with phase coding of displacement along the current direction. This type of phase coding, of place field distance ahead of or behind the animal, may be useful for allowing calculation of goal directions during navigation.     

Spatial learning and action planning in a prefrontal cortical network model

The interplay between hippocampus and prefrontal cortex (PFC) is fundamental to spatial cognition. Complementing hippocampal place coding, prefrontal representations provide more abstract and hierarchically organized memories suitable for decision making. We model a prefrontal network mediating distributed information processing for spatial learning and action planning. Specific connectivity and synaptic adaptation principles shape the recurrent dynamics of the network arranged in cortical minicolumns. We show how the PFC columnar organization is suitable for learning sparse topological-metrical representations from redundant hippocampal inputs. The recurrent nature of the network supports multilevel spatial processing, allowing structural features of the environment to be encoded. An activation diffusion mechanism spreads the neural activity through the column population leading to trajectory planning. The model provides a functional framework for interpreting the activity of PFC neurons recorded during navigation tasks. We illustrate the link from single unit activity to behavioral responses. The results suggest plausible neural mechanisms subserving the cognitive "insight" capability originally attributed to rodents by Tolman & Honzik. Our time course analysis of neural responses shows how the interaction between hippocampus and PFC can yield the encoding of manifold information pertinent to spatial planning, including prospective coding and distance-to-goal correlates.     

Spatial cognition and neuro-mimetic navigation: a model of hippocampal place cell activity

 A computational model of hippocampal activity during spatial cognition and navigation tasks is presented. The spatial representation in our model of the rat hippocampus is built on-line during exploration via two processing streams. An allothetic vision-based representation is built by unsupervised Hebbian learning extracting spatio-temporal properties of the environment from visual input. An idiothetic representation is learned based on internal movement-related information provided by path integration. On the level of the hippocampus, allothetic and idiothetic representations are integrated to yield a stable representation of the environment by a population of localized overlapping CA3-CA1 place fields. The hippocampal spatial representation is used as a basis for goal-oriented spatial behavior. We focus on the neural pathway connecting the hippocampus to the nucleus accumbens. Place cells drive a population of locomotor action neurons in the nucleus accumbens. Reward-based learning is applied to map place cell activity into action cell activity. The ensemble action cell activity provides navigational maps to support spatial behavior. We present experimental results obtained with a mobile Khepera robot. Received: 02 July 1999 / Accepted in revised form: 20 March 2000     

Why spikes? Hebbian learning and retrieval of time-resolved excitation patterns

Hebbian learning allows a network of spiking neurons to store and retrieve spatio-temporal patterns with a time resolution of 1 ms, despite the long postsynaptic and dendritic integration times. To show this, we introduce and analyze a model of spiking neurons, the spike response model, with a realistic distribution of axonal delays and with realistic postsynaptic potentials. Learning is performed by a local Hebbian rule which is based on the synchronism of presynaptic neurotransmitter release and some short-acting postsynaptic process. The time window of this synchronism determines the temporal resolution of pattern retrieval, which can be initiated by applying a short external stimulus pattern. Furthermore, a rate quantization is found in dependence upon the threshold value of the neurons, i.e., in a given time a pattern runsn times as often as learned, wheren is a positive integer (n 0). We show that all information about the spike pattern is lost if only mean firing rates (temporal average) or ensemble activities (spatial average) are considered. An average over several retrieval runs in order to generate a post-stimulus time histogram may also deteriorate the signal. The full information on a pattern is contained in the spike raster of a single run. Our results stress the importance, and advantage, of coding by spatio-temporal spike patterns instead of firing rates and average ensemble activity. The implications regarding modelling and experimental data analysis are discussed.     

A topological paradigm for hippocampal spatial map formation using persistent homology

An animal's ability to navigate through space rests on its ability to create a mental map of its environment. The hippocampus is the brain region centrally responsible for such maps, and it has been assumed to encode geometric information (distances, angles). Given, however, that hippocampal output consists of patterns of spiking across many neurons, and downstream regions must be able to translate those patterns into accurate information about an animal's spatial environment, we hypothesized that 1) the temporal pattern of neuronal firing, particularly co-firing, is key to decoding spatial information, and 2) since co-firing implies spatial overlap of place fields, a map encoded by co-firing will be based on connectivity and adjacency, i.e., it will be a topological map. Here we test this topological hypothesis with a simple model of hippocampal activity, varying three parameters (firing rate, place field size, and number of neurons) in computer simulations of rat trajectories in three topologically and geometrically distinct test environments. Using a computational algorithm based on recently developed tools from Persistent Homology theory in the field of algebraic topology, we find that the patterns of neuronal co-firing can, in fact, convey topological information about the environment in a biologically realistic length of time. Furthermore, our simulations reveal a "learning region" that highlights the interplay between the parameters in combining to produce hippocampal states that are more or less adept at map formation. For example, within the learning region a lower number of neurons firing can be compensated by adjustments in firing rate or place field size, but beyond a certain point map formation begins to fail. We propose that this learning region provides a coherent theoretical lens through which to view conditions that impair spatial learning by altering place cell firing rates or spatial specificity.     

Noise-tolerant stimulus discrimination by synchronization with depressing synapses

 Some synapses between cortical pyramidal neurons exhibit a rapid depression of excitatory postsynaptic potentials for successive presynaptic spikes. Since depressing synapses do not transmit information on sustained presynaptic firing rates, it has been speculated that they are favorable for temporal coding. In this paper, we study the dynamical effects of depressing synapses on stimulus-induced transient synchronization in a simple network of inhibitory interneurons and excitatory neurons, assuming that the recurrent excitation is mediated by depressing synapses. This synchronization occurs in a temporal pattern which depends on a given stimulus. Since the presence of noise is always a potential hazard in temporal coding, we investigate the extent to which noise in stimuli influences the synchronization phenomena. It is demonstrated that depressing synapses greatly contribute to suppressing the influences of noise on the stimulus-specific temporal patterns of synchronous firing. The timing-based Hebbian learning revealed by physiological experiments is shown to stabilize the temporal patterns in cooperation with synaptic depression. Thus, the times at which synchronous firing occurs provides a reliable information representation in the presence of synaptic depression. Received: 5 July 2000 / Accepted in revised form: 12 January 2001     

Oscillatory neurocomputing with ring attractors: a network architecture for mapping locations in space onto patterns of neural synchrony

Theories of neural coding seek to explain how states of the world are mapped onto states of the brain. Here, we compare how an animal''s location in space can be encoded by two different kinds of brain states: population vectors stored by patterns of neural firing rates, versus synchronization vectors stored by patterns of synchrony among neural oscillators. It has previously been shown that a population code stored by spatially tuned ‘grid cells’ can exhibit desirable properties such as high storage capacity and strong fault tolerance; here it is shown that similar properties are attainable with a synchronization code stored by rhythmically bursting ‘theta cells’ that lack spatial tuning. Simulations of a ring attractor network composed from theta cells suggest how a synchronization code might be implemented using fewer neurons and synapses than a population code with similar storage capacity. It is conjectured that reciprocal connections between grid and theta cells might control phase noise to correct two kinds of errors that can arise in the code: path integration and teleportation errors. Based upon these analyses, it is proposed that a primary function of spatially tuned neurons might be to couple the phases of neural oscillators in a manner that allows them to encode spatial locations as patterns of neural synchrony.