Neurophysiological and computational principles of cortical rhythms in cognition

Abstract

Synchronous rhythms represent a core mechanism for sculpting temporal coordination of neural activity in the brain-wide network. This review focuses on oscillations in the cerebral cortex that occur during cognition, in alert behaving conditions. Over the last two decades, experimental and modeling work has made great strides in elucidating the detailed cellular and circuit basis of these rhythms, particularly gamma and theta rhythms. The underlying physiological mechanisms are diverse (ranging from resonance and pacemaker properties of single cells to multiple scenarios for population synchronization and wave propagation), but also exhibit unifying principles. A major conceptual advance was the realization that synaptic inhibition plays a fundamental role in rhythmogenesis, either in an interneuronal network or in a reciprocal excitatory-inhibitory loop. Computational functions of synchronous oscillations in cognition are still a matter of debate among systems neuroscientists, in part because the notion of regular oscillation seems to contradict the common observation that spiking discharges of individual neurons in the cortex are highly stochastic and far from being clocklike. However, recent findings have led to a framework that goes beyond the conventional theory of coupled oscillators and reconciles the apparent dichotomy between irregular single neuron activity and field potential oscillations. From this perspective, a plethora of studies will be reviewed on the involvement of long-distance neuronal coherence in cognitive functions such as multisensory integration, working memory, and selective attention. Finally, implications of abnormal neural synchronization are discussed as they relate to mental disorders like schizophrenia and autism.

Figure 1. Examples of synchronous oscillations in cognitive behaviors

(A) Theta rhythm in the hippocampus during spatial navigation. Top: in the task, rat shuttles back and forth along a linear track between food rewards contained in cups attached to movable walls. Middle: color- coded firing field of a place cell created from multiple runs in the eastward direction. Bottom: EEG theta rhythm and place cell firing (in red) for the same cell on a single eastward run. Ticks above the spikes indicate 0°/360° phase for each theta cycle, lines through theta waves indicate 270°. Bursts of spikes occur at higher than theta frequency causing each successive burst to move to an earlier phase of the theta cycle, despite initially rising, then falling firing rate. (B) Attention induces changes in synchrony in the visual cortex. Data shown are from experiments in which two visual stimuli were presented, one inside and one outside the receptive field of a neuron in area V4 of a behaving monkey. In the schematics (left), the green box represents the receptive field: this was not presented on the screen in the experiment. Red traces correspond to attention directed inside the receptive field of the recorded neuron; blue traces correspond to attention directed outside. Stimuli were the same in the two conditions. a and b: The continuous traces show the stimulus-driven local field potentials (LFPs). The spikes below were recorded simultaneously from different electrodes. c and d: Spike-triggered averages (STAs) computed during the stimulus presentation period. The STA corresponds to the average LFP waveform that is seen at the time of a spike. The y axes indicate the mean LFP; the x axes indicate time relative to the occurrence of a spike. e: Power spectra of the two STAs shown in c and d. When attention is focused inside the receptive field, the recorded neurons tend to fire more in phase with the frequency components around 50 Hz, and less so with respect to the frequencies around 10 Hz. (A) Reproduced with permission from [456], (B) from [838] with the original experimental data reported in [312].

Figure 2. Theory of coupled oscillators

(A) Phase response curve of the classical (type II) Hodgkin-Huxley model of action potential [437]. Top: a small and brief depolarizing current pulse leads to either a phase delay (blue) or phase advance (red), depending on the timing of the pulse perturbation. T₀ is the oscillation period in the absence of perturbation. Bottom: induced phase change (positive for advance, negative for delay) as a function of the oscillatory phase at which the pulse perturbation is applied. Superimposed is the membrane potential for a full oscillation cycle (spike peak corresponds to zero phase). (B) Fast mutual excitation naturally gives rise to synchrony for two coupled Hodgkin-Huxley model neurons, as a cell firing slightly earlier advances the firing of the other cell, while the synaptic input back from the other cell delays its own firing, leading to reduced phase difference between the two in successive cycles. Dashed vertical lines: spike times for isolated neurons. Solid lines: actual spike times in the presence of synaptic interaction. (C) Perfect synchronization by mutual inhibition. Top: phase response curve of a modified (type I) Hodgkin-Huxley model [1035], which does not exhibit significant phase delay (the negative lobe). Bottom: the phase reduction theory predicts the behavior of coupled neurons by the function H_odd(Φ), where Φ is the phase difference between the two neurons (Eq. 4). H_odd was computed using a synapse model with a reversal potential of −75 mV and a decay time constant of 10 ms. Steady state behaviors correspond to Φ values such that H_odd(Φ) = 0. In this example, zero-phase synchrony (Φ = 0) is stable (where H_odd has a negative slope); 180 degree antiphase is unstable (where H_odd has a positive slope).

Figure 3. Subthreshold and spiking resonance of a model neuron in the presence of noise

The model neuron exhibits subthreshold membrane resonance at frequency f_R ≃ 5 Hz. (A) Response with low noise. Top: input current with three (tonic, 5 Hz and 20 Hz sinusoidal current injection) epochs. The mean output firing rate remains constant (r₀ = 20 Hz) throughout. Middle: a sample membrane trace. Bottom: poststimulus histogram. (B) Response with high noise. Same format as in (A). (C) Signal gain amplitude versus input frequency f (normalized to be one at f = 0). Firing rate resonance occurs at the subthreshold resonance frequency (f ≃ f_R) with high noise (blue), but at the mean firing frequency (f ≃ r₀) with low noise (red). D: Phase shift of the firing response relative to the sinusoidal input (positive: phase advance; negative: phase delay). Adapted with permission from [805].

Figure 4. Mixed-mode membrane oscillations in a single neuron

(A) A non-cholinergic (putative GABAergic) cell in the rat medial septum displays rhythmic alternations at theta frequency between ‘clusters’ of spikes and epochs of sub-threshold membrane potential oscillations. (B) A model of GABAergic neurons in the medial septum. Upper: a membrane trace. The simulated oscillation is faster than the experimental data (see the different time scales), because the model simulation was done at body temperature (37°C), whereas the in vitro trace was recorded at 32°. Lower left: membrane potential versus the inactivation gating variable q for a low-threshold potassium channel (I_KS). I_KS gradually de-inactivates (increase of q, hence I_KS) during the hyperpolarizing phases of spikes (arrows to the right), whereas it inactivates (decreasing q) during subthreshold membrane oscillations (arrows to the left). Lower right: the frequency of subthreshold oscillations (open circle) and intra-cluster spike firing rate (filled circle) co-vary as a function of the input current intensity. (C) Similar behavior of principal (mitral) cells of the rat olfactory bulb. Left: mixed subthreshold oscillation and clustered spike firing of a mitral cell in response to three current intensities. Right: frequencies of subthreshold oscillation and intra-cluster spike firing versus the mean membrane potential which was varied by current injection. (A) Reproduced with permission from [858]; (B) from [1031]; (C) from [231].

Figure 5. Rhythmic bursting of cortical neurons

(A) A chattering neuron recorded in vivo from the cat visual cortex shows rhythmic bursting in the gamma frequency range. (B) A model chattering neuron endowed with a ping-pong interplay between two electrotonic compartments. (A) Reproduced with permission from [373], (B) from [1028].

Figure 6. Synchronization in an inhibitory interneuronal network

(A) An example of network synchronization in a fully connected network of type I conductance-based neurons. Upper panel: rastergram where each row of dots represents spikes discharged by one of the neurons in the network. Lower panel: membrane potentials of two neurons. Neurons initially fire asynchronized, but quickly become perfectly synchronized by mutual inhibition. (B) In a random network, the network coherence is plotted versus the mean number of recurrent synapses per cell M_syn (The correction term (~ 1/N) takes into account the finite network size effect). Different curves correspond to different network size (N=100, 200, 500, 1000). There is a critical threshold for the connectedness above which network synchrony occurs. This threshold connectivity is independent of the network size. Adapted with permission from [1035] and [142].

Figure 7. Entrainment of a cortical network in vivo by optogenetically activating fast spiking interneurons

(A) Spiking activity of neurons in mouse somatosensory cortex in response to channorodopsin-2 activation of FS interneurons by repetitive light pulses at 40 Hz. Top: LFP and raster plots of an FS cell (blue) and nearby regular spiking pyramidal cells recorded simultaneously (RS1-3, green). Middle: Overlay of the FS (blue) and RS (green) spike probability profiles, computed from 17 RS cells and 9 FS cells. Light pulses at 0 ms evoked FS spikes with a delay of 1–2 ms, followed by an increase in RS spiking at 17–24 ms. Bottom: Power spectrum of LFP in control (black, with a broad profile) and with activation of FS cells by 40 Hz light pulses (blue, with a sharp peak at 40 Hz). (B) Mean LFP power ratio in each frequency band in response to light activation of FS (filled circles) and RS (open circles) cells at those stimulus frequencies. (A) Top and middle panels kindly provided by Jessica Cardin, (A) bottom panel and (B) reproduced with permission from [155].

Figure 8. Trade-off between synchronization by long-range connections and minimization of the network wire cost

(A) Oscillations in a network of interneurons coupled by inhibitory synapses, with local (Gaussian) connectivity (spatial length is 20 neurons, in a network of 4000 neurons). The network is essentially asynchronous. Upper panel: spike raster of sample neurons; middle panel: the voltage trace of a representative neuron; lower panel: the population firing rate. (B) Oscillations in a network with local and long-range connections. Neurons are connected with Gaussian distributed synapses (as in A) but p = 25% of the synapses are reconnected with a power law distribution. Note strong oscillatory rhythm. (C) Illustration of the connectivity probability functions: the Gaussian distributed connections are local (blue line), whereas long-range connections are described by a power distribution (red line). (D) With increasing reconnection probability p from the local Gaussian distribution to the power distribution, the network synchrony increases while the inverse of the wire-length of connections decreases. High synchrony at a low wire-cost corresponds to an optimal range of p values (a small ratio of long-range and short-range connections, shaded region). Reproduced with permission from [142].

Figure 9. Propagating waves

(A) Rostrocaudal phase shift of 40 Hz oscillation during rapid eye movement (REM) sleep as measured using MEG. (A1) shows synchronous activation in 37 channels during a 600 ms period. The oscillation in the left part of trace A1 has been expanded in trace A2 to show five different recording sites over the head. The five recording sites of trace (A2) are displayed in diagram (A3) for a single epoch, to illustrate the phase shift for the different 40 Hz waves during REM sleep. The direction of the phase shift is illustrated by an arrow above diagram (A3). The actual traces and their sites of recordings for a single epoch are shown in diagram (A4) for all 37 channels (ft: femtotesla). (B) Voltage-sensitive dye imaging of propagating waves in rat visual cortex in vivo. A rat visual cortex was imaged through a cranial window. (B1) A snapshot of visually evoked cortical activity propagating through the border of V1 and V2 areas. (B2) A spatial- temporal map made from one row of optical detectors (small boxes in B1) showing the time course of wave propagation. A visual stimulation to the eye evoked an activity in area V1 and propagated to V2. The map starts at 108 ms post stimulus. The propagating velocity in both V1 and V2 is about 100 mm/sec (the oblique broken line marks the velocity of 100 mm/sec). Propagating velocity is greatly reduced at the border between V1-V2 areas (marked by a horizontal broken line). (B3 ) When the cortical inhibition is slightly suppressed by GABAergic antagonist bicuculline, the slowing down at the V1/V2 border is completely eliminated. (C) Propagating waves during carbachol induced ~ 10 Hz oscillations in a brain slice cut from rat visual cortex (tangential section). Images are the phase map during the oscillation. From left: At the beginning of an oscillation cycle a propagating wave is started from a small region (red area at image of 0 ms) and propagating outward as a ring wave. The ring wave breaks and generates two phase singularities (at 33.1 ms). One of them further develops into a rotating spiral (after 58.9 ms). The spiral wave further sustains for ~ 1000 ms (~ 10 more rotations, images not shown). (A) Modified with permission from [598], (B) from [1079], (C) from [448].

Figure 10. Complex circuit dynamics in multiple neuron types during synchronous oscillations

(A) Behavior of four (basket, axo-axonic, bistratified, oriens-lacunosum-moleculare) subclasses of inhibitory interneurons and pyramidal cells during theta rhythm (left) and sharp-wave ripples (right) recorded in vivo from the hippocampus of anesthetized rats. In each panel are plotted LFP (top) and juxtacellularly recorded spike train of an identified cell (bottom). During theta oscillations, while basket and axo-axonic cells (which control the output of pyramidal neurons) tend to fire at the peak, bistratified and O-LM cells (which control the inputs to pyramidal cells) as well as pyramidal cells tend to fire at the trough, of field theta. Note also intermittent firing characteristic of pyramidal neurons in both theta rhythm and fast ripples. (B) Behavior of several subtypes of interneurons (basket/axo-axonal cells, interneurons targeting proximate dendrites in the stratium radiatum (RC), interneuron-targeting cells (IS), interneurons in the stratum oriens projecting to the distal dendrites in the stratum lacunosummoleculare (OLM)) and pyramidal cells during gamma oscillations induced by cholinergic activation in rat hippocampal slices in vitro. Note that pyramidal cells fire maximally at the trough of the local field oscillation, and ahead of interneurons (left panel). (A) Modified from [889], kindly provided by Drs. Thomas Klausberger and Peter Somogyi; (B) reproduced with permission from [408].

Figure 11. Field oscillation and spiking activity of interneurons and pyramidal cells in the hippocampus of behaving rats

(A) Episodes of sharp-wave ripples during slow-wave sleep. (B) Theta rhythm during rapid-eye movement sleep. Filtered (upper) and wide-band (lower) local field traces recorded from one of the tetrodes. Below LFP traces are shown action potentials of isolated neurons (vertical ticks) for three interneurons (int(p)) in the CA1 pyramidal layer, and 15 pyramidal (pyr) cells recorded by four tetrodes. Reproduced with permission from [208].

Figure 12. Population rhythmicity with irregular spike activity of single neurons

(A) Network dynamics is schematically illustrated here in the form of single neuron activity in response to network inputs composed of a large amount of noise and a weak sinusoidal wave (top). The raster plot (middle) reveals the modulation of the instantaneous firing rate, which has a large tonic component (r₀) and a small sinusoidal component (of amplitude r₁) (bottom). (B) Distributions of spike times across phases of 25–50 Hz gamma oscillations for CA1 place cells in freely moving rats, showing a small rhythmic modulation on top of a tonic baseline. (A) adapted with permission from [805], (B) from [181].

Figure 13. Excitatory-inhibitory feedback loop for synchronous gamma oscillations with irregularly firing neurons

Computer simulation of a model with two neural populations (pyramidal cells and interneurons) in a sparsely connected random network. The network shows a collective oscillation at 55 Hz (see population rates, and the power spectrum), whereas single neurons fire spikes intermittently in time at low rates (2 Hz for pyramidal cells, 10 Hz for interneurons; see rastergrams). Reproduced with permission from [127].

Figure 14. Dependence of sparsely synchronized oscillations on the strength and relative speeds of synaptic excitation and inhibition

(A) Network dynamical behavior as a function of the excitatory (τ_E) and inhibitory (τ_I) synaptic time constants. A line separates the asynchronous state (upper right region) from the synchronous oscillation state (lower left region). This is shown with three levels of recurrent connection strength; coherent oscillations become more prevalent with stronger recurrence of the network. (B) population frequency as a function of the excitatory synaptic decay time constant τ_E, with fixed τ_I= 5 ms. Qualitatively, there are two kinds of instability from the asynchronous state to coherent oscillation. On one hand, when τ_E is much smaller than τ_I, the asynchronous dynamics is destabilized and oscillation develops through delayed inhibition in an excitatory-inhibitory loop scenario. On the other hand, when τ_E is sufficiently large, the excitatory drive is roughly tonic and the interneuronal network by itself generates synchronous oscillation. The oscillation frequency is much lower with shorter τ_E (gamma rhythm by the pyramid-interneuron loop mechanism) than with longer τ_E (ultrafast rhythm by the interneuronal network mechanism). Adapted with permission from [127].

Figure 15. Local and long-distance synchronization in the gamma frequency range

(A) Under certain conditions in vitro LFP gamma may exhibit a sharp peak in the power spectrum. In this case, gamma rhythm was induced pharmacologically by metabotropic gluatamte receptor activation in a rat hippocampal slice where fast synaptic excitations mediated by AMPA and NMDA receptors were blocked. (B) By contrast, in vivo LFP oscillations are typically manifested by small peaks superimposed on high powers at low frequencies (reflecting slow fluctuations or drifts over time). Data were obtained in a single mice during wheel running (red), REM sleep (blue) and slow-wave sleep (SWS, black). (C) Power spectra of intracellularly recorded synaptic potentials recorded from a pyramidal neuron of ferret in vivo. Membrane fluctuations are dominated by inhibitory postsynaptic potentials near 0 mV (blue), and by excitatory synaptic potentials near −80 mV (red). The peak in the gamma band is more significant at 0 mV, suggesting a prominent role for synaptic inhibition in rhythmic fluctuations. (D) Coherence between two cortical areas of alert monkeys in cross-modal information processing. LFPs were measured from the auditory cortex and the superior temporal sulcus (STS), an association area involved with face processing and cross-modal integration. Voice: trials when a naturalistic sound (monkey’s vocalization) was played; Face: trials when the corresponding mouth movement was shown visually; face+voice: trials when both stimuli were presented concurrently. Left: LFP power spectrum displays a peak in the gamma frequency range for both the auditory cortex and STS (red) in the face+voice condition, but there is no significant gamma-band peak in the auditory cortex with voice alone (green). Middle: inter-areal coherence displays a maximum in the gamma-band correlated with cross-modal integration (red), but not when face (blue) or voice (green) was presented alone. Right: Time-frequency cross-areal coherence as a function of time in the face+voice condition, showing coherence at gamma interspersed with theta transiently (for about 300 ms). (E) Coherence in V4 during sustained stimulation from behaving monkeys in an attention task. Red: attended condition; grey: non-attended condition. Left: LFP power spectrum from area V4. Middle: spike-to-field coherence (between multiunit activity and LFP). Right: spike-to-spike coherence between two simultaneously recorded multiunit spike trains. SSC (defined between 0 and 1) is less than 5%. (A) reproduced with permission from [1053], (B) from [148], (C) from [423], (D) from [349] (with left panel kindly produced by Dr. A Ghazanfar), (E) from [315].

Figure 16. Phase coding by place cells in the hippocampus of mouse performing a virtual navigation task

(A) Schematic of a place cells firing rate. (B) Schematics of predicted subthreshold membrane potentials (aligned to A) from three different models. Top: in a dual oscillator interference model, two sets of theta-modulated inputs at different frequencies interfere to create a beat-like pattern of membrane potential fluctuations. Middle: in a soma-dendritic interference model, the cell receives theta-modulated inhibitory and excitatory inputs. In the place field, the excitatory drive increases, resulting in a ramp-like depolarization and an increase in the amplitude of excitatory theta oscillations. Bottom: in a network model, a ramp of excitatory drive interacts with theta-modulated inhibitory inputs. (C) Example of a subthreshold membrane potential (filtered from DC-10 Hz) recorded intracellularly from a place cell. Note the simultaneous ramp of depolarization and increase in theta oscillation amplitude. Scale bars refer to the experimentally measured trace only. (D) Schematics of the LFP theta rhythm (Top) and predicted relationships between intracellular and LFP theta to account for phase precession (Middle and Bottom). Middle panel: intracellular and LFP theta have the same frequency. Phase precession of spikes occurs relative to both intracellular and LFP theta owing to a ramp of depolarization. An asymmetric ramp is shown. Bottom panel: intracellular theta in the place field has a higher frequency than LFP theta. Spikes precess relative to LFP theta but not intracellular theta. (E) Filtered (610 Hz) membrane potential and LFP traces in the place field from a simultaneous LFP and whole-cell recording. The times of LFP theta peaks (grey lines), intracellular theta peaks (circles) and spikes (crosses) are shown to illustrate the phase precession of spikes and intracellular theta relative to LFP theta oscillations, and the absence of phase precession of spikes relative to intracellular theta oscillations. (F) Phase precession of spike times relative to intracellular theta (left) and phase precession of intracellular theta peak times relative to LFP theta (right) from a simultaneous LFP and whole-cell recording. Adapted with permission from [422].

Figure 17. Phase reliability of LFPs strongly depends on the frequency bands

Under natural visual stimulation (movies) in anesthetized monkeys, for band-passed LFPs in a given frequency range (y axis, in 4-Hz-wide nonoverlapping intervals), the LFP phase at any fixed time during the movie fluctuates from trial to trial, and this phase (un)reliability of V1 LFPs was assessed by the circular variance across trials (defined between 0 (total reliability) and 1 (total unreliability). It is plotted as function of the spike rate in the corresponding window, since the reliable phase values must be observed during periods of firing in order to be useful for phase-of-spiking stimulus coding. Phase coding is reliable if the circular variance is small; this is the case only at low frequencies (a few hertz). Reproduced with permission from [693].

Figure 18. Synchronous field gamma oscillation with irregular neural firing in a spiking network model of selective attention

(A) Schematic model architecture of a reciprocal loop between a sensory (MT) circuit and a working memory (PFC) circuit (‘source area’ for top-down attention signals). Each of the two circuits includes excitatory pyramidal neurons (E cells) and inhibitory interneurons (I cells), that are selective for a stimulus feature. Local connections within each circuit and cross-areal connections between excitatory cells depend on their respective preferred stimulus features. Top-down projection from the working memory circuit targets both excitatory and inhibitory cells in the sensory circuit. (B) Network activity for an unattended (left) and an attended trial (right). x-Axis, Time; y-axis, neurons labeled by preferred feature. Activity is color-coded. C, Cue period when an attentional cue is presented; D, delay period; T, test period when a sensory stimulus is presented. Note that in the attended trial (right), the attention cue triggers a self-sustained activity pattern in PFC, and enhanced sensory response in MT during tested period. Also note strip-like structure in the spatiotemporal activity pattern, indicating synchronous oscillations. (C) Activity of a neuron in response to a preferred stimulus. Top, Sample membrane potential; middle, spike trains in several trials; bottom, trial-averaged activity (red, attended; black, unattended trials). (D) Average coherence between the LFP and MUA in the sensory circuit increases in the gamma-frequency range for attention (red) relative to non-attention (black) trials during test stimulus presentation. (E) The variance versus mean of spike counts (Fano factor is given by the slope) for E cells and I cells in attended (red) and unattended (black) trials. (A–C) reproduced with permission from [30], (D–E) unpublished data taken from [31].

Figure 19. Schematic circuit for the interplay between beta- and gamma-band oscillations in top-down signaling

Shown is a reciprocally connected loop between a sensory-type area and a cognitive-type area. In each area, the superficial layers are endowed with strong intrinsic synaptic connections and generate synchronous oscillations in the gamma frequency range, whereas the deep layers has a propensity to display oscillations in the beta frequency range. Top-down projections originate in the deep layers and predominantly target the superficial layers, where they innervate pyramidal cells (red), as well as dendrite-targeting (purple) and perisoma-targeting (blue) inhibitory interneurons. In this scenario, beta oscillations are directly involved in top-down signaling, which interacts with locally generated gamma oscillations.