The phase of spikes of hippocampal pyramidal cells relative to the local field oscillation shifts forward (''phase precession'') over a full cycle as the animal crosses the cell's receptive field (''place field''). The linear relationship between the phase of the spikes and the travel distance within the place field is independent of the animal's running speed. This invariance of the phase-distance relationship is likely to be important for coordinated activity of hippocampal cells and space coding, yet the mechanism responsible for it is not known. Here we show that at faster running speeds place cells are active for fewer cycles but oscillate at a higher frequency and emit more spikes per cycle. As a result, the phase shift of spikes from cycle to cycle (i.e., temporal precession slope) is faster, yet spatial-phase precession stays unchanged. Interneurons can also show transient-phase precession and contribute to the formation of coherently precessing assemblies. We hypothesize that the speed-correlated acceleration of place cell assembly oscillation is responsible for the phase-distance invariance of hippocampal place cells.cell assembly ͉ interneurons ͉ phase locking ͉ phase precession ͉ oscillations W hile animals navigate in an environment, the hippocampal local field potential (LFP) is characterized by a highly regular oscillation (8-10 Hz). Principal cells in the hippocampus show place-specific firing by two criteria. First, the firing is tuned to a particular location (''place field''), showing a bellshaped tuning curve centered around its preferred location (1). Second, the timing of spikes within subsequent cycles systematically shifts forward (''phase precession''), Ϸ1 full cycle in total, as the rat runs through the place field of the neuron (2, 3) (see also Fig. 1 A and B). Both the firing rate and discharge phase within a cycle are correlated with the rat's position. However, how the rate change and -phase precession of spikes are related is poorly understood. The available experiments support both a rate-phase interdependence (4-6) and independence (7).Several explanations for the place-phase relationship were put forward (4)(5)(6)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18). To confront these models, we examined the relationship among running speed, oscillation frequency of place cells and LFP , and timing of spikes within the cycle. We show that principal cells oscillate at a frequency faster than the simultaneously recorded LFP oscillation, and that this oscillation frequency depends on the rat's running speed. Together with the place-and speed-dependent oscillation frequencies of interneurons, the findings support the hypothesis that place coding results from coordinated network activity. We propose that the locomotion speed-dependent oscillation of place cell assemblies may underlie the mechanisms responsible for distance encoding in the hippocampus. ResultsWe recorded the firing patterns of pyramidal cells, interneurons, and the LFP from the CA1 pyramidal layer of rats as they ran on a U-shap...

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During fast oscillations in the local field potential (40-100 Hz gamma, 100-200 Hz sharp-wave ripples) single cortical neurons typically fire irregularly at rates that are much lower than the oscillation frequency. Recent computational studies have provided a mathematical description of such fast oscillations, using the leaky integrate-and-fire (LIF) neuron model. Here, we extend this theoretical framework to populations of more realistic Hodgkin-Huxley-type conductance-based neurons. In a noisy network of GABAergic neurons that are connected randomly and sparsely by chemical synapses, coherent oscillations emerge with a frequency that depends sensitively on the single cell's membrane dynamics. The population frequency can be predicted analytically from the synaptic time constants and the preferred phase of discharge during the oscillatory cycle of a single cell subjected to noisy sinusoidal input. The latter depends significantly on the single cell's membrane properties and can be understood in the context of the simplified exponential integrate-and-fire (EIF) neuron. We find that 200-Hz oscillations can be generated, provided the effective input conductance of single cells is large, so that the single neuron's phase shift is sufficiently small. In a two-population network of excitatory pyramidal cells and inhibitory neurons, recurrent excitation can either decrease or increase the population rhythmic frequency, depending on whether in a neuron the excitatory synaptic current follows or precedes the inhibitory synaptic current in an oscillatory cycle. Detailed single-cell properties have a substantial impact on population oscillations, even though rhythmicity does not originate from pacemaker neurons and is an emergent network phenomenon.

Cortical neurons are often classified by current-frequency relationship. Such a static description is inadequate to interpret neuronal responses to time-varying stimuli. Theoretical studies suggested that single-cell dynamical response properties are necessary to interpret ensemble responses to fast input transients. Further, it was shown that input-noise linearizes and boosts the response bandwidth, and that the interplay between the barrage of noisy synaptic currents and the spike-initiation mechanisms determine the dynamical properties of the firing rate. To test these model predictions, we estimated the linear response properties of layer 5 pyramidal cells by injecting a superposition of a small-amplitude sinusoidal wave and a background noise. We characterized the evoked firing probability across many stimulation trials and a range of oscillation frequencies (1-1000 Hz), quantifying response amplitude and phase-shift while changing noise statistics. We found that neurons track unexpectedly fast transients, as their response amplitude has no attenuation up to 200 Hz. This cut-off frequency is higher than the limits set by passive membrane properties (approximately 50 Hz) and average firing rate (approximately 20 Hz) and is not affected by the rate of change of the input. Finally, above 200 Hz, the response amplitude decays as a power-law with an exponent that is independent of voltage fluctuations induced by the background noise.

Driven either by external landmarks or by internal dynamics, hippocampal neurons form sequences of cell assemblies. The coordinated firing of these active cells is organized by the prominent "theta" oscillations in the local field potential (LFP): place cells discharge at progressively earlier theta phases as the rat crosses the respective place field ("phase precession"). The faster oscillation frequency of active neurons and the slower theta LFP, underlying phase precession, creates a paradox. How can faster oscillating neurons comprise a slower population oscillation, as reflected by the LFP? We built a mathematical model that allowed us to calculate the population activity analytically from experimentally derived parameters of the single neuron oscillation frequency, firing field size (duration), and the relationship between within-theta delays of place cell pairs and their distance representations ("compression"). The appropriate combination of these parameters generated a constant frequency population rhythm along the septo-temporal axis of the hippocampus, while allowing individual neurons to vary their oscillation frequency and field size. Our results suggest that the faster-than-theta oscillations of pyramidal cells are inherent and that phase precession is a result of the coordinated activity of temporally shifted cell assemblies, relative to the population activity, reflected by the LFP. provide important information about the cooperative activity of neuronal populations (1-3). In the simplest case, the firing rate of a subset of neurons oscillates with a particular mean frequency, and this seed population functions as the pacemaker and biases the discharge phase of the remaining majority. Examples include the various rhythms of the thalamocortical system, where individual neurons fire strictly at a specific phase of the LFP (4). A similar scenario has been hypothesized to be realized in the hippocampalentorhinal system, with the medial septum serving as the pacemaker of the prominent theta rhythm (5-10 Hz) (5, 6). However, hippocampal neurons in the exploring rat are active in short bouts, typically representing a particular place (7). During such highfiring epochs, place cells oscillate faster than the frequency of theta LFP (8-10), the result of which is a progressive phase precession of place cell spikes (8, 11). The frequency of the LFP theta is constant across the whole hippocampus, even though the size of place fields increases whereas the oscillation frequency of place cells decreases along the septo-temporal (dorsal-ventral) axis of the hippocampus (12-14). Furthermore, the LFP theta is highly coherent at different stages of the hippocampal-entorhinal loop (15)(16)(17)(18)(19).The discrepancy between the oscillation frequencies of spiking place cells and the global rhythm reflected in the LFP led us to pose several questions. Given that the output of hippocampal pyramidal cells is in tune with their targets, how do phase precessing place cells generate a rhythmic activity at theta frequency? ...

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A recurrent model of the repetitive firing of neurons responding to stimuli of long duration is given. The model assumes a deterministic threshold potential and a membrane potential which is composed of both deterministic and random components. The model accurately reproduces interval statistics obtained from different neurons discharging repetitively over a wide range of discharge rates. It is shown that the model has three important parameters; the time course of threshold recovery following a discharge, the variance of the random component, and the level of excitatory drive. The model is extended, by the use of hyperpolarizing afterpotentials, to include negative correlation between successive interspike intervals.