Resonant synchronization in heterogeneous networks of inhibitory neurons

Is High-Frequency Activity at Seizure Onset Inhibitory? A Stereoelectroencephalographic Study of Motor Cortex Seizures

OBJECTIVE

In the era of stereoelectroencephalography (SEEG), many studies have been devoted to understanding the role of interictal high-frequency oscillations. High-frequency activity (HFA) at seizure onset has been identified as a marker of epileptogenic zone. We address the physiological significance of ictal HFAs and their relation to clinical semiology.

METHODS

We retrospectively identified patients with pure focal primary motor epilepsy. We selected only patients in whom SEEG electrodes were optimally placed in the motor cortex as confirmed by electrical stimulation. Based on these narrow inclusion criteria, we extensively studied 5 patients (3 males and 2 females, mean age = 22.4 years) using time-frequency analysis and time correlation with motor signs onset.

RESULTS

A total of 157 analyzable seizures were recorded in 5 subjects. The first 2 subjects had tonic or clonic semiology with rare secondary generalization. Subject 3 had atonic onset followed by clonic hand/arm flexion. Subject 4 had clusters of tonic and atonic facial movements. Subject 5 had upper extremity tonic movements. The median frequency of the fast activity extracted from the Epileptogenic Zone Fingerprint pipeline in the first 4 subjects was 76 Hz (interquartile range = 21.9Hz). Positive motor signs did not occur concomitantly with high gamma activity developing in the motor cortex. Motor signs began at the end of HFAs.

INTERPRETATION

This study supports the hypothesis of an inhibitory effect of ictal HFAs. The frequency range in the gamma band was associated with the direction of the clinical output effect. Changes from inhibitory to excitatory effect occurred when discharge frequency dropped to low gamma or beta. ANN NEUROL 2024;95:1127-1137.

Physiological features of parvalbumin-expressing GABAergic interneurons contributing to high-frequency oscillations in the cerebral cortex

Parvalbumin-expressing (PV+) inhibitory interneurons drive gamma oscillations (30-80 Hz), which underlie higher cognitive functions. In this review, we discuss two groups/aspects of fundamental properties of PV+ interneurons. In the first group (dubbed Before Axon), we list properties representing optimal synaptic integration in PV+ interneurons designed to support fast oscillations. For example: [i] Information can neither enter nor leave the neocortex without the engagement of fast PV+ -mediated inhibition; [ii] Voltage responses in PV+ interneuron dendrites integrate linearly to reduce impact of the fluctuations in the afferent drive; and [iii] Reversed somatodendritic Rm gradient accelerates the time courses of synaptic potentials arriving at the soma. In the second group (dubbed After Axon), we list morphological and biophysical properties responsible for (a) short synaptic delays, and (b) efficient postsynaptic outcomes. For example: [i] Fast-spiking ability that allows PV+ interneurons to outpace other cortical neurons (pyramidal neurons). [ii] Myelinated axon (which is only found in the PV+ subclass of interneurons) to secure fast-spiking at the initial axon segment; and [iii] Inhibitory autapses - autoinhibition, which assures brief biphasic voltage transients and supports postinhibitory rebounds. Recent advent of scientific tools, such as viral strategies to target PV cells and the ability to monitor PV cells via in vivo imaging during behavior, will aid in defining the role of PV cells in the CNS. Given the link between PV+ interneurons and cognition, in the future, it would be useful to carry out physiological recordings in the PV+ cell type selectively and characterize if and how psychiatric and neurological diseases affect initiation and propagation of electrical signals in this cortical sub-circuit. Voltage imaging may allow fast recordings of electrical signals from many PV+ interneurons simultaneously.

Interneuronal network model of theta-nested fast oscillations predicts differential effects of heterogeneity, gap junctions and short term depression for hyperpolarizing versus shunting inhibition

Theta and gamma oscillations in the hippocampus have been hypothesized to play a role in the encoding and retrieval of memories. Recently, it was shown that an intrinsic fast gamma mechanism in medial entorhinal cortex can be recruited by optogenetic stimulation at theta frequencies, which can persist with fast excitatory synaptic transmission blocked, suggesting a contribution of interneuronal network gamma (ING). We calibrated the passive and active properties of a 100-neuron model network to capture the range of passive properties and frequency/current relationships of experimentally recorded PV+ neurons in the medial entorhinal cortex (mEC). The strength and probabilities of chemical and electrical synapses were also calibrated using paired recordings, as were the kinetics and short-term depression (STD) of the chemical synapses. Gap junctions that contribute a noticeable fraction of the input resistance were required for synchrony with hyperpolarizing inhibition; these networks exhibited theta-nested high frequency oscillations similar to the putative ING observed experimentally in the optogenetically-driven PV-ChR2 mice. With STD included in the model, the network desynchronized at frequencies above ~200 Hz, so for sufficiently strong drive, fast oscillations were only observed before the peak of the theta. Because hyperpolarizing synapses provide a synchronizing drive that contributes to robustness in the presence of heterogeneity, synchronization decreases as the hyperpolarizing inhibition becomes weaker. In contrast, networks with shunting inhibition required non-physiological levels of gap junctions to synchronize using conduction delays within the measured range.

Role of Interaction Delays in the Synchronization of Inhibitory Networks

Neural oscillations provide a means for efficient and flexible communication among different brain areas. Understanding the mechanisms of the generation of brain oscillations is crucial to determine principles of communication and information transfer in the brain circuits. It is well known that the inhibitory neurons play a major role in the generation of oscillations in the gamma range, in pure inhibitory networks, or in the networks composed of excitatory and inhibitory neurons. In this study, we explore the impact of different parameters and, in particular, the delay in the transmission of the signals between the neurons, on the dynamics of inhibitory networks. We show that increasing delay in a reasonable range increases the synchrony and stabilizes the oscillations. Unstable gamma oscillations characterized by a highly variable amplitude of oscillations can be observed in an intermediate range of delays. We show that in this range of delays, other experimentally observed phenomena such as sparse firing, variable amplitude and period, and the correlation between the instantaneous amplitude and period could be observed. The results broaden our understanding of the mechanism of the generation of the gamma oscillations in the inhibitory networks, known as the ING (interneuron-gamma) mechanism.

Frequency Selectivity of Neural Circuits With Heterogeneous Discrete Transmission Delays

Neurons are connected to other neurons by axons and dendrites that conduct signals with finite velocities, resulting in delays between the firing of a neuron and the arrival of the resultant impulse at other neurons. Since delays greatly complicate the analytical treatment and interpretation of models, they are usually neglected or taken to be uniform, leading to a lack in the comprehension of the effects of delays in neural systems. This letter shows that heterogeneous transmission delays make small groups of neurons respond selectively to inputs with differing frequency spectra. By studying a single integrate-and-fire neuron receiving correlated time-shifted inputs, it is shown how the frequency response is linked to both the strengths and delay times of the afferent connections. The results show that incorporating delays alters the functioning of neural networks, and changes the effect that neural connections and synaptic strengths have.

Collective Dynamics of Neural Networks With Sleep-Related Biological Drives in Drosophila

The collective electrophysiological dynamics of the brain as a result of sleep-related biological drives in Drosophila are investigated in this paper. Based on the Huber-Braun thermoreceptor model, the conductance-based neurons model is extended to a coupled neural network to analyze the local field potential (LFP). The LFP is calculated by using two different metrics: the mean value and the distance-dependent LFP. The distribution of neurons around the electrodes is assumed to have a circular or grid distribution on a two-dimensional plane. Regardless of which method is used, qualitatively similar results are obtained that are roughly consistent with the experimental data. During wake, the LFP has an irregular or a regular spike. However, the LFP becomes regular bursting during sleep. To further analyze the results, wavelet analysis and raster plots are used to examine how the LFP frequencies changed. The synchronization of neurons under different network structures is also studied. The results demonstrate that there are obvious oscillations at approximately 8 Hz during sleep that are absent during wake. Different time series of the LFP can be obtained under different network structures and the density of the network will also affect the magnitude of the potential. As the number of coupled neurons increases, the neural network becomes easier to synchronize, but the sleep and wake time described by the LFP spectrogram do not change. Moreover, the parameters that affect the durations of sleep and wake are analyzed.

Firing rate-dependent phase responses of Purkinje cells support transient oscillations

Both spike rate and timing can transmit information in the brain. Phase response curves (PRCs) quantify how a neuron transforms input to output by spike timing. PRCs exhibit strong firing-rate adaptation, but its mechanism and relevance for network output are poorly understood. Using our Purkinje cell (PC) model, we demonstrate that the rate adaptation is caused by rate-dependent subthreshold membrane potentials efficiently regulating the activation of Na+ channels. Then, we use a realistic PC network model to examine how rate-dependent responses synchronize spikes in the scenario of reciprocal inhibition-caused high-frequency oscillations. The changes in PRC cause oscillations and spike correlations only at high firing rates. The causal role of the PRC is confirmed using a simpler coupled oscillator network model. This mechanism enables transient oscillations between fast-spiking neurons that thereby form PC assemblies. Our work demonstrates that rate adaptation of PRCs can spatio-temporally organize the PC input to cerebellar nuclei.

Interneuronal gap junctions increase synchrony and robustness of hippocampal ripple oscillations

Sharp wave-ripples (SWRs) are important for memory consolidation. Their signature in the hippocampal extracellular field potential can be decomposed into a ≈100 ms long sharp wave superimposed by ≈200 Hz ripple oscillations. How ripple oscillations are generated is currently not well understood. A promising model for the genesis of ripple oscillations is based on recurrent interneuronal networks (INT-INT). According to this hypothesis, the INT-INT network in CA1 receives a burst of excitation from CA3 that generates the sharp wave, and recurrent inhibition leads to an ultrafast synchronization of the CA1 network causing the ripple oscillations; fast-spiking parvalbumin-positive basket cells (PV⁺  BCs) may constitute the ripple-generating interneuronal network. PV⁺  BCs are also coupled by gap junctions (GJs) but the function of GJs for ripple oscillations has not been quantified. Using simulations of CA1 hippocampal networks of PV⁺  BCs, we show that GJs promote synchrony beyond a level that could be obtained by only inhibition. GJs also increase the neuronal firing rate of the interneuronal ensemble, while they affect the ripple frequency only mildly. The promoting effect of GJs on ripple oscillations depends on fast GJ transmission ( ≲ 0.5 ms), which requires proximal GJ coupling ( ≲ 100 μm from soma), but is robust to variability in the delay and the amplitude of GJ coupling.

Synchronization of Independent Neural Ensembles in Human EEG during Choice Tasks

During behavioral experiments, humans placed in a situation of having to choose between a more valuable but risky reward and a less valuable but guaranteed reward make their decisions in accordance with external situational factors and individual characteristics, such as inclination to risk or caution. In such situations, humans can be divided into “risk-inclined” and “risk-averse” (or “cautious”) subjects. In this work, characteristics of EEG rhythms, such as phase–phase relationships and time lags between rhythms, were studied in pairs of alpha–beta and theta–beta rhythms. Phase difference can also be expressed as a time lag. It has been suggested that statistically significant time lags between rhythms are due to the combined neural activity of anatomically separate, independent (in activation/inhibition processes) ensembles. The extents of synchronicity between rhythms were compared as percentages between risk-inclined and risk-averse subjects. The results showed that synchronicity in response to stimuli was more often observed in pairs of alpha–beta rhythms of risk-averse subjects compared with risk-inclined subjects during the choice of a more valuable but less probable reward. In addition, significant differences in the percentage ratio of alpha and beta rhythms were revealed between (i) cases of synchronization without long time lags and (ii) cases with long time lags between rhythms (from 0.08 to 0.1 s).

Phase response theory explains cluster formation in sparsely but strongly connected inhibitory neural networks and effects of jitter due to sparse connectivity

We show how to predict whether a neural network will exhibit global synchrony (a one-cluster state) or a two-cluster state based on the assumption of pulsatile coupling and critically dependent upon the phase response curve (PRC) generated by the appropriate perturbation from a partner cluster. Our results hold for a monotonically increasing (meaning longer delays as the phase increases) PRC, which likely characterizes inhibitory fast-spiking basket and cortical low-threshold-spiking interneurons in response to strong inhibition. Conduction delays stabilize synchrony for this PRC shape, whereas they destroy two-cluster states, the former by avoiding a destabilizing discontinuity and the latter by approaching it. With conduction delays, stronger coupling strength can promote a one-cluster state, so the weak coupling limit is not applicable here. We show how jitter can destabilize global synchrony but not a two-cluster state. Local stability of global synchrony in an all-to-all network does not guarantee that global synchrony can be observed in an appropriately scaled sparsely connected network; the basin of attraction can be inferred from the PRC and must be sufficiently large. Two-cluster synchrony is not obviously different from one-cluster synchrony in the presence of noise and may be the actual substrate for oscillations observed in the local field potential (LFP) and the electroencephalogram (EEG) in situations where global synchrony is not possible. Transitions between cluster states may change the frequency of the rhythms observed in the LFP or EEG. Transitions between cluster states within an inhibitory subnetwork may allow more effective recruitment of pyramidal neurons into the network rhythm. NEW & NOTEWORTHY We show that jitter induced by sparse connectivity can destabilize global synchrony but not a two-cluster state with two smaller clusters firing alternately. On the other hand, conduction delays stabilize synchrony and destroy two-cluster states. These results hold if each cluster exhibits a phase response curve similar to one that characterizes fast-spiking basket and cortical low-threshold-spiking cells for strong inhibition. Either a two-cluster or a one-cluster state might provide the oscillatory substrate for neural computations.

An Interneuron Circuit Reproducing Essential Spectral Features of Field Potentials

Recent advances in engineering and signal processing have renewed the interest in invasive and surface brain recordings, yet many features of cortical field potentials remain incompletely understood. In the computational study that follows, we show that a model circuit of interneurons, coupled via both GABA_A receptor synapses and electrical synapses, reproduces many essential features of the power spectrum of local field potential (LFP) recordings, such as 1/ f power scaling at low frequency (below 10 Hz), power accumulation in the γ-frequency band (30-100 Hz), and a robust α rhythm in the absence of stimulation. The low-frequency 1/ f power scaling depends on strong reciprocal inhibition, whereas the α rhythm is generated by electrical coupling of intrinsically active neurons. As in previous studies, the γ power arises through the amplification of single-neuron spectral properties, owing to the refractory period, by parameters that favor neuronal synchrony, such as delayed inhibition. This study also confirms that both synaptic and voltage-gated membrane currents contribute substantially to the LFP and that high-frequency signals such as action potentials quickly taper off with distance. Given the ubiquity of electrically coupled interneuron circuits in the mammalian brain, they may be major determinants of the recorded potentials.

Spiking resonances in models with the same slow resonant and fast amplifying currents but different subthreshold dynamic properties

The generation of spiking resonances in neurons (preferred spiking responses to oscillatory inputs) requires the interplay of the intrinsic ionic currents that operate at the subthreshold voltage level and the spiking mechanisms. Combinations of the same types of ionic currents in different parameter regimes may give rise to different types of nonlinearities in the voltage equation (e.g., parabolic- and cubic-like), generating subthreshold (membrane potential) oscillations patterns with different properties. These nonlinearities are not apparent in the model equations, but can be uncovered by plotting the voltage nullclines in the phase-plane diagram. We investigate the spiking resonant properties of conductance-based models that are biophysically equivalent at the subthreshold level (same ionic currents), but dynamically different (parabolic- and cubic-like voltage nullclines). As a case study we consider a model having a persistent sodium and a hyperpolarization-activated (h-) currents, which exhibits subthreshold resonance in the theta frequency band. We unfold the concept of spiking resonance into evoked and output spiking resonance. The former focuses on the input frequencies that are able to generate spikes, while the latter focuses on the output spiking frequencies regardless of the input frequency that generated these spikes. A cell can exhibit one or both types of resonances. We also measure spiking phasonance, which is an extension of subthreshold phasonance (zero-phase-shift response to oscillatory inputs) to the spiking regime. The subthreshold resonant properties of both types of models are communicated to the spiking regime for low enough input amplitudes as the voltage response for the subthreshold resonant frequency band raises above threshold. For higher input amplitudes evoked spiking resonance is no longer present in these models, but output spiking resonance is present primarily in the parabolic-like model due to a cycle skipping mechanism (involving mixed-mode oscillations), while the cubic-like model shows a better 1:1 entrainment. We use dynamical systems tools to explain the underlying mechanisms and the mechanistic differences between the resonance types. Our results demonstrate that the effective time scales that operate at the subthreshold regime to generate intrinsic subthreshold oscillations, mixed-mode oscillations and subthreshold resonance do not necessarily determine the existence of a preferred spiking response to oscillatory inputs in the same frequency band. The results discussed in this paper highlight both the complexity of the suprathreshold responses to oscillatory inputs in neurons having resonant and amplifying currents with different time scales and the fact that the identity of the participating ionic currents is not enough to predict the resulting patterns, but additional dynamic information, captured by the geometric properties of the phase-space diagram, is needed.

Identifying Anatomical Origins of Coexisting Oscillations in the Cortical Microcircuit

Oscillations are omnipresent in neural population signals, like multi-unit recordings, EEG/MEG, and the local field potential. They have been linked to the population firing rate of neurons, with individual neurons firing in a close-to-irregular fashion at low rates. Using a combination of mean-field and linear response theory we predict the spectra generated in a layered microcircuit model of V1, composed of leaky integrate-and-fire neurons and based on connectivity compiled from anatomical and electrophysiological studies. The model exhibits low- and high-γ oscillations visible in all populations. Since locally generated frequencies are imposed onto other populations, the origin of the oscillations cannot be deduced from the spectra. We develop an universally applicable systematic approach that identifies the anatomical circuits underlying the generation of oscillations in a given network. Based on a theoretical reduction of the dynamics, we derive a sensitivity measure resulting in a frequency-dependent connectivity map that reveals connections crucial for the peak amplitude and frequency of the observed oscillations and identifies the minimal circuit generating a given frequency. The low-γ peak turns out to be generated in a sub-circuit located in layer 2/3 and 4, while the high-γ peak emerges from the inter-neurons in layer 4. Connections within and onto layer 5 are found to regulate slow rate fluctuations. We further demonstrate how small perturbations of the crucial connections have significant impact on the population spectra, while the impairment of other connections leaves the dynamics on the population level unaltered. The study uncovers connections where mechanisms controlling the spectra of the cortical microcircuit are most effective.

Recordings of brain activity show multiple coexisting oscillations. The generation of these oscillations has so far only been investigated in generic one- and two-population networks, neglecting their embedment into larger systems. We introduce a method that determines the mechanisms and sub-circuits generating oscillations in structured spiking networks. Analyzing a multi-layered model of the cortical microcircuit, we trace back characteristic oscillations to experimentally observed connectivity patterns. The approach exposes the influence of individual connections on frequency and amplitude of these oscillations and therefore reveals locations, where biological mechanisms controlling oscillations and experimental manipulations have the largest impact. The new analytical tool replaces parameter scans in computationally expensive models, guides circuit design, and can be employed to validate connectivity data.

Regulation of Irregular Neuronal Firing by Autaptic Transmission

The importance of self-feedback autaptic transmission in modulating spike-time irregularity is still poorly understood. By using a biophysical model that incorporates autaptic coupling, we here show that self-innervation of neurons participates in the modulation of irregular neuronal firing, primarily by regulating the occurrence frequency of burst firing. In particular, we find that both excitatory and electrical autapses increase the occurrence of burst firing, thus reducing neuronal firing regularity. In contrast, inhibitory autapses suppress burst firing and therefore tend to improve the regularity of neuronal firing. Importantly, we show that these findings are independent of the firing properties of individual neurons, and as such can be observed for neurons operating in different modes. Our results provide an insightful mechanistic understanding of how different types of autapses shape irregular firing at the single-neuron level, and they highlight the functional importance of autaptic self-innervation in taming and modulating neurodynamics.

Interactions between Inhibitory Interneurons and Excitatory Associational Circuitry in Determining Spatio-Temporal Dynamics of Hippocampal Dentate Granule Cells: A Large-Scale Computational Study

This paper reports on findings from a million-cell granule cell model of the rat dentate gyrus that was used to explore the contributions of local interneuronal and associational circuits to network-level activity. The model contains experimentally derived morphological parameters for granule cells, which each contain approximately 200 compartments, and biophysical parameters for granule cells, basket cells, and mossy cells that were based both on electrophysiological data and previously published models. Synaptic input to cells in the model consisted of glutamatergic AMPA-like EPSPs and GABAergic-like IPSPs from excitatory and inhibitory neurons, respectively. The main source of input to the model was from layer II entorhinal cortical neurons. Network connectivity was constrained by the topography of the system, and was derived from axonal transport studies, which provided details about the spatial spread of axonal terminal fields, as well as how subregions of the medial and lateral entorhinal cortices project to subregions of the dentate gyrus. Results of this study show that strong feedback inhibition from the basket cell population can cause high-frequency rhythmicity in granule cells, while the strength of feedforward inhibition serves to scale the total amount of granule cell activity. Results furthermore show that the topography of local interneuronal circuits can have just as strong an impact on the development of spatio-temporal clusters in the granule cell population as the perforant path topography does, both sharpening existing clusters and introducing new ones with a greater spatial extent. Finally, results show that the interactions between the inhibitory and associational loops can cause high frequency oscillations that are modulated by a low-frequency oscillatory signal. These results serve to further illustrate the importance of topographical constraints on a global signal processing feature of a neural network, while also illustrating how rich spatio-temporal and oscillatory dynamics can evolve from a relatively small number of interacting local circuits.

A spiking network model of cerebellar Purkinje cells and molecular layer interneurons exhibiting irregular firing

While the anatomy of the cerebellar microcircuit is well-studied, how it implements cerebellar function is not understood. A number of models have been proposed to describe this mechanism but few emphasize the role of the vast network Purkinje cells (PKJs) form with the molecular layer interneurons (MLIs)—the stellate and basket cells. We propose a model of the MLI-PKJ network composed of simple spiking neurons incorporating the major anatomical and physiological features. In computer simulations, the model reproduces the irregular firing patterns observed in PKJs and MLIs in vitro and a shift toward faster, more regular firing patterns when inhibitory synaptic currents are blocked. In the model, the time between PKJ spikes is shown to be proportional to the amount of feedforward inhibition from an MLI on average. The two key elements of the model are: (1) spontaneously active PKJs and MLIs due to an endogenous depolarizing current, and (2) adherence to known anatomical connectivity along a parasagittal strip of cerebellar cortex. We propose this model to extend previous spiking network models of the cerebellum and for further computational investigation into the role of irregular firing and MLIs in cerebellar learning and function.

Parametric Anatomical Modeling: a method for modeling the anatomical layout of neurons and their projections

Computational models of neural networks can be based on a variety of different parameters. These parameters include, for example, the 3d shape of neuron layers, the neurons' spatial projection patterns, spiking dynamics and neurotransmitter systems. While many well-developed approaches are available to model, for example, the spiking dynamics, there is a lack of approaches for modeling the anatomical layout of neurons and their projections. We present a new method, called Parametric Anatomical Modeling (PAM), to fill this gap. PAM can be used to derive network connectivities and conduction delays from anatomical data, such as the position and shape of the neuronal layers and the dendritic and axonal projection patterns. Within the PAM framework, several mapping techniques between layers can account for a large variety of connection properties between pre- and post-synaptic neuron layers. PAM is implemented as a Python tool and integrated in the 3d modeling software Blender. We demonstrate on a 3d model of the hippocampal formation how PAM can help reveal complex properties of the synaptic connectivity and conduction delays, properties that might be relevant to uncover the function of the hippocampus. Based on these analyses, two experimentally testable predictions arose: (i) the number of neurons and the spread of connections is heterogeneously distributed across the main anatomical axes, (ii) the distribution of connection lengths in CA3-CA1 differ qualitatively from those between DG-CA3 and CA3-CA3. Models created by PAM can also serve as an educational tool to visualize the 3d connectivity of brain regions. The low-dimensional, but yet biologically plausible, parameter space renders PAM suitable to analyse allometric and evolutionary factors in networks and to model the complexity of real networks with comparatively little effort.

A supercritical density of Na(+) channels ensures fast signaling in GABAergic interneuron axons

Fast-spiking, parvalbumin-expressing GABAergic interneurons, a large proportion of which are basket cells (BCs), have a key role in feedforward and feedback inhibition, gamma oscillations and complex information processing. For these functions, fast propagation of action potentials (APs) from the soma to the presynaptic terminals is important. However, the functional properties of interneuron axons remain elusive. We examined interneuron axons by confocally targeted subcellular patch-clamp recording in rat hippocampal slices. APs were initiated in the proximal axon ~20 μm from the soma and propagated to the distal axon with high reliability and speed. Subcellular mapping revealed a stepwise increase of Na(+) conductance density from the soma to the proximal axon, followed by a further gradual increase in the distal axon. Active cable modeling and experiments with partial channel block revealed that low axonal Na(+) conductance density was sufficient for reliability, but high Na(+) density was necessary for both speed of propagation and fast-spiking AP phenotype. Our results suggest that a supercritical density of Na(+) channels compensates for the morphological properties of interneuron axons (small segmental diameter, extensive branching and high bouton density), ensuring fast AP propagation and high-frequency repetitive firing.

Population dynamics of the modified theta model: macroscopic phase reduction and bifurcation analysis link microscopic neuronal interactions to macroscopic gamma oscillation

Gamma oscillations of the local field potential are organized by collective dynamics of numerous neurons and have many functional roles in cognition and/or attention. To mathematically and physiologically analyse relationships between individual inhibitory neurons and macroscopic oscillations, we derive a modification of the theta model, which possesses voltage-dependent dynamics with appropriate synaptic interactions. Bifurcation analysis of the corresponding Fokker-Planck equation (FPE) enables us to consider how synaptic interactions organize collective oscillations. We also develop the adjoint method (infinitesimal phase resetting curve) for simultaneous equations consisting of ordinary differential equations representing synaptic dynamics and a partial differential equation for determining the probability distribution of the membrane potential. This method provides a macroscopic phase response function (PRF), which gives insights into how it is modulated by external perturbation or internal changes of parameters. We investigate the effects of synaptic time constants and shunting inhibition on these gamma oscillations. The sensitivity of rising and decaying time constants is analysed in the oscillatory parameter regions; we find that these sensitivities are not largely dependent on rate of synaptic coupling but, rather, on current and noise intensity. Analyses of shunting inhibition reveal that it can affect both promotion and elimination of gamma oscillations. When the macroscopic oscillation is far from the bifurcation, shunting promotes the gamma oscillations and the PRF becomes flatter as the reversal potential of the synapse increases, indicating the insensitivity of gamma oscillations to perturbations. By contrast, when the macroscopic oscillation is near the bifurcation, shunting eliminates gamma oscillations and a stable firing state appears. More interestingly, under appropriate balance of parameters, two branches of bifurcation are found in our analysis of the FPE. In this case, shunting inhibition can effect both promotion and elimination of the gamma oscillation depending only on the reversal potential.

Effect of heterogeneity and noise on cross frequency phase-phase and phase-amplitude coupling

Cross-frequency coupling is hypothesized to play a functional role in neural computation. We apply phase resetting theory to two types of cross-frequency coupling that can occur when a slower oscillator periodically forces one or more oscillators: phase-phase coupling, in which the two oscillations are phase-locked, and phase-amplitude coupling, in which the amplitude of the driven oscillation is modulated. Our first result is that the shape of the phase resetting curve predicts the tightness of locking to a pulsatile forcing periodic input at any ratio of forced to intrinsic period; the tightness of the locking decreases as the ratio increases. Theoretical expressions were obtained for the probability density of the phases for a population of heterogeneous oscillators or a noisy single oscillator. Results were confirmed using two types of simulated networks and experiments on hippocampal CA1 neurons. Theoretical expressions were also obtained and confirmed for the probability density of N spike times within a single cycle of low frequency forcing. The second result is a suggested mechanism for phase-amplitude coupling in which progressive desynchronization leads to decreasing amplitude during a low frequency forcing cycle. Network simulations confirmed the theoretical viability of this mechanism, and that it generalizes to more diffuse input.

Membrane resonance enables stable and robust gamma oscillations

Neuronal mechanisms underlying beta/gamma oscillations (20–80 Hz) are not completely understood. Here, we show that in vivo beta/gamma oscillations in the cat visual cortex sometimes exhibit remarkably stable frequency even when inputs fluctuate dramatically. Enhanced frequency stability is associated with stronger oscillations measured in individual units and larger power in the local field potential. Simulations of neuronal circuitry demonstrate that membrane properties of inhibitory interneurons strongly determine the characteristics of emergent oscillations. Exploration of networks containing either integrator or resonator inhibitory interneurons revealed that: (i) Resonance, as opposed to integration, promotes robust oscillations with large power and stable frequency via a mechanism called RING (Resonance INduced Gamma); resonance favors synchronization by reducing phase delays between interneurons and imposes bounds on oscillation cycle duration; (ii) Stability of frequency and robustness of the oscillation also depend on the relative timing of excitatory and inhibitory volleys within the oscillation cycle; (iii) RING can reproduce characteristics of both Pyramidal INterneuron Gamma (PING) and INterneuron Gamma (ING), transcending such classifications; (iv) In RING, robust gamma oscillations are promoted by slow but are impaired by fast inputs. Results suggest that interneuronal membrane resonance can be an important ingredient for generation of robust gamma oscillations having stable frequency.

Impact of gamma-oscillatory inhibition on the signal transmission of a cortical pyramidal neuron

Networks of synchronized fast-spiking interneurons are thought to be key elements in the generation of gamma (γ) oscillations (30-80 Hz) in the brain. We examined how such γ-oscillatory inhibition regulates the output of a cortical pyramidal cell. Specifically, we modeled a situation where a pyramidal cell receives inputs from γ-synchronized fast-spiking inhibitory interneurons. This model successfully reproduced several important aspects of a recent experimental result regarding the γ-inhibitory regulation of pyramidal cellular firing that is presumably associated with the sensation of whisker stimuli. Through an in-depth analysis of this model system, we show that there is an obvious rhythmic gating effect of the γ-oscillated interneuron networks on the pyramidal neuron's signal transmission. This effect is further illustrated by the interactions of this interneuron network and the pyramidal neuron. Prominent power in the γ frequency range can emerge provided that there are appropriate delays on the excitatory connections and inhibitory synaptic conductance between interneurons. These results indicate that interactions between excitation and inhibition are critical for the modulation of coherence and oscillation frequency of network activities.

PRESYNAPTIC LEARNING AND MEMORY WITH A PERSISTENT FIRING NEURON AND A HABITUATING SYNAPSE: A MODEL OF SHORT TERM PERSISTENT HABITUATION

Our paper explores the interaction of persistent firing axonal and presynaptic processes in the generation of short term memory for habituation. We first propose a model of a sensory neuron whose axon is able to switch between passive conduction and persistent firing states, thereby triggering short term retention to the stimulus. Then we propose a model of a habituating synapse and explore all nine of the behavioral characteristics of short term habituation in a two neuron circuit. We couple the persistent firing neuron to the habituation synapse and investigate the behavior of short term retention of habituating response. Simulations show that, depending on the amount of synaptic resources, persistent firing either results in continued habituation or maintains the response, both leading to longer recovery times. The effectiveness of the model as an element in a bio-inspired memory system is discussed.

Complex synchronous behavior in interneuronal networks with delayed inhibitory and fast electrical synapses

Networks of fast-spiking interneurons are crucial for the generation of neural oscillations in the brain. Here we study the synchronous behavior of interneuronal networks that are coupled by delayed inhibitory and fast electrical synapses. We find that both coupling modes play a crucial role by the synchronization of the network. In addition, delayed inhibitory synapses affect the emerging oscillatory patterns. By increasing the inhibitory synaptic delay, we observe a transition from regular to mixed oscillatory patterns at a critical value. We also examine how the unreliability of inhibitory synapses influences the emergence of synchronization and the oscillatory patterns. We find that low levels of reliability tend to destroy synchronization and, moreover, that interneuronal networks with long inhibitory synaptic delays require a minimal level of reliability for the mixed oscillatory pattern to be maintained.

ENHANCING THE YIELD OF HIGH-DENSITY ELECTRODE ARRAYS THROUGH AUTOMATED ELECTRODE SELECTION

Recently developed CMOS-based microprobes contain hundreds of electrodes on a single shaft with inter-electrode distances as small as 30 μm. So far, neuroscientists needed to select electrodes manually from hundreds of electrodes. Here we present an electronic depth control algorithm that allows to select electrodes automatically, hereby allowing to reduce the amount of data and locating those electrodes that are close to neurons. The electrodes are selected according to a new penalized signal-to-noise ratio (PSNR) criterion that demotes electrodes from becoming selected if their signals are redundant with previously selected electrodes. It is shown that, using the PSNR, interneurons generating smaller spikes are also selected. We developed a model that aims to evaluate algorithms for electronic depth control, but also generates benchmark data for testing spike sorting and spike detection algorithms. The model comprises a realistic tufted pyramidal cell, non-tufted pyramidal cells and inhibitory interneurons. All neurons are synaptically activated by hundreds of fibers. This arrangement allows the algorithms to be tested in more realistic conditions, including backgrounds of synaptic potentials, varying spike rates with bursting and spike amplitude attenuation.

Short conduction delays cause inhibition rather than excitation to favor synchrony in hybrid neuronal networks of the entorhinal cortex

How stable synchrony in neuronal networks is sustained in the presence of conduction delays is an open question. The Dynamic Clamp was used to measure phase resetting curves (PRCs) for entorhinal cortical cells, and then to construct networks of two such neurons. PRCs were in general Type I (all advances or all delays) or weakly type II with a small region at early phases with the opposite type of resetting. We used previously developed theoretical methods based on PRCs under the assumption of pulsatile coupling to predict the delays that synchronize these hybrid circuits. For excitatory coupling, synchrony was predicted and observed only with no delay and for delays greater than half a network period that cause each neuron to receive an input late in its firing cycle and almost immediately fire an action potential. Synchronization for these long delays was surprisingly tight and robust to the noise and heterogeneity inherent in a biological system. In contrast to excitatory coupling, inhibitory coupling led to antiphase for no delay, very short delays and delays close to a network period, but to near-synchrony for a wide range of relatively short delays. PRC-based methods show that conduction delays can stabilize synchrony in several ways, including neutralizing a discontinuity introduced by strong inhibition, favoring synchrony in the case of noisy bistability, and avoiding an initial destabilizing region of a weakly type II PRC. PRCs can identify optimal conduction delays favoring synchronization at a given frequency, and also predict robustness to noise and heterogeneity.

Individual oscillators, such as pendulum-based clocks and fireflies, can spontaneously organize into a coherent, synchronized entity with a common frequency. Neurons can oscillate under some circumstances, and can synchronize their firing both within and across brain regions. Synchronized assemblies of neurons are thought to underlie cognitive functions such as recognition, recall, perception and attention. Pathological synchrony can lead to epilepsy, tremor and other dynamical diseases, and synchronization is altered in most mental disorders. Biological neurons synchronize despite conduction delays, heterogeneous circuit composition, and noise. In biological experiments, we built simple networks in which two living neurons could interact via a computer in real time. The computer precisely controlled the nature of the connectivity and the length of the communication delays. We characterized the synchronization tendencies of individual, isolated oscillators by measuring how much a single input delivered by the computer transiently shortened or lengthened the cycle period of the oscillation. We then used this information to correctly predict the strong dependence of the coordination pattern of the firing of the component neurons on the length of the communication delays. Upon this foundation, we can begin to build a theory of the basic principles of synchronization in more complex brain circuits.

Coherence depression in stochastic excitable systems with two-frequency forcing

We study the response of two generic neuron models, the leaky integrate-and-fire (LIF) model and the leaky integrate-and-fire model with dynamic threshold (LIFDT) (i.e., with memory) to a stimulus consisting of two sinusoidal drives with incommensurate frequency, an amplitude modulation ("envelope") noise and a relatively weak additive noise. Spectral and coherence analysis of responses to such naturalistic stimuli reveals how the LIFDT model exhibits better correlation between modulation and spike train even in the presence of both noises. However, a resonance-induced synchrony, occurring when the beat frequency between the sinusoids is close to the intrinsic neuronal firing rate, decreases the coherence in the dynamic threshold case. Under suprathreshold conditions, the modulation noise simultaneously decreases the linear spectral coherence between the spikes and the whole stimulus, as well as between spikes and the stimulus envelope. Our study shows that the coefficient of variation of the envelope fluctuations is positively correlated with the degree of coherence depression. As the coherence function quantifies the linear information transmission, our findings indicate that under certain conditions, a transmission loss results when an excitable system with adaptive properties encodes a beat with frequency in the vicinity of its mean firing rate.

Input-rate modulation of γ oscillations is sensitive to network topology, delays and short-term plasticity

Simulated networks of excitatory and inhibitory neurons have previously been shown to reproduce critical features of experimental data regarding neural coding in V1, such as a positive relationship between thalamic input spike rate and the power of gamma frequency oscillations. This effect, referred to as modulated gamma power, could represent a neural code in V1 for stimulus characteristics that affect thalamic spike rate such as contrast or intensity. The simulated network's assumptions included homogeneous random connectivity, equal synaptic delays after spike arrival, and constant synaptic efficacies. Plausible alternative assumptions include small world connectivity, a wide distribution of axonal propagation delays, and short term synaptic plasticity, and here we assess the individual impact of each of these on the model's success in reproducing modulated gamma power. First, we developed several alternative algorithms for simulating directed networks with clustered connectivity and balanced excitation and inhibition. We found that modulated gamma power was absent in all small-world networks that had a relatively low abundance of reciprocal connectivity, which suggests that such motifs are present in V1 cortical networks at levels at least equal to those found in random networks. We also found in a different network type that the balance of excitation and inhibition could be destroyed when the network was in the small-world regime. Given all neurons had identical in-degrees, this result suggests that balance relies on motif distributions as well as mean connectivity. Second, altering the distribution of axonal delays had little effect, but increasing the mean delay led to a secondary gamma modulation at harmonics of the main peak, and since this is not observed experimentally, it suggests a mean delay in V1 networks less than 2 ms. Finally, we compared two types of excitatory synaptic plasticity, and found that modulated beta power emerged in addition to gamma power for one type, in the presence of short term depression in interneurons. This article is part of a Special Issue entitled "Neural Coding".

Interaction between connectivity and oscillatory currents in a heterogeneous neuronal network

Intrinsic oscillations are thought to play important and distinct roles in cognitive processes across nearly all regions of the brain. Their specific roles are highly dependent on their properties: low-frequency θ is thought to be important in the gating of cognitive processes, while high-frequency γ is believed to be essential for binding and spike-timing-dependent plasticity. We investigated the role of an oscillatory drive for pattern formation of heterogeneous networks. Network heterogeneities were implemented as network regions having increased connectivity as compared to the rest of the network. We varied the properties of the oscillatory drive as well as network connectivity. We observed that the disparity in spatiotemporal patterning of activity between the structurally enhanced region and rest of the network was highly dependent on the frequency and amplitude of the oscillatory drive as well as network connectivity, generally favoring bigger enhancement of activity for high-frequency oscillations and phase locking with moderate enhancement of activity for lower-frequency oscillations. Thus, these results indicate that the specific role of the observed oscillations may depend on their dynamical interactions with the heterogeneous network.

Axon Physiology

Axons are generally considered as reliable transmission cables in which stable propagation occurs once an action potential is generated. Axon dysfunction occupies a central position in many inherited and acquired neurological disorders that affect both peripheral and central neurons. Recent findings suggest that the functional and computational repertoire of the axon is much richer than traditionally thought. Beyond classical axonal propagation, intrinsic voltage-gated ionic currents together with the geometrical properties of the axon determine several complex operations that not only control signal processing in brain circuits but also neuronal timing and synaptic efficacy. Recent evidence for the implication of these forms of axonal computation in the short-term dynamics of neuronal communication is discussed. Finally, we review how neuronal activity regulates both axon morphology and axonal function on a long-term time scale during development and adulthood.

Context-dependent encoding of fear and extinction memories in a large-scale network model of the basal amygdala

The basal nucleus of the amygdala (BA) is involved in the formation of context-dependent conditioned fear and extinction memories. To understand the underlying neural mechanisms we developed a large-scale neuron network model of the BA, composed of excitatory and inhibitory leaky-integrate-and-fire neurons. Excitatory BA neurons received conditioned stimulus (CS)-related input from the adjacent lateral nucleus (LA) and contextual input from the hippocampus or medial prefrontal cortex (mPFC). We implemented a plasticity mechanism according to which CS and contextual synapses were potentiated if CS and contextual inputs temporally coincided on the afferents of the excitatory neurons. Our simulations revealed a differential recruitment of two distinct subpopulations of BA neurons during conditioning and extinction, mimicking the activation of experimentally observed cell populations. We propose that these two subgroups encode contextual specificity of fear and extinction memories, respectively. Mutual competition between them, mediated by feedback inhibition and driven by contextual inputs, regulates the activity in the central amygdala (CEA) thereby controlling amygdala output and fear behavior. The model makes multiple testable predictions that may advance our understanding of fear and extinction memories.

The amygdaloid complex is one of the key brain structures involved in fear-related processes. A typical way to study neural correlates of fear expression (e.g. freezing response) in the amygdala is to perform a fear conditioning paradigm, which yields a conditioned fear response. This response can be reversed by another procedure called fear extinction. Thanks to the experimental approaches to date we have some understanding about the putative roles of specific subnuclei within the amygdala in the formation of these fear and extinction memories. Here, we complement the experimental studies by providing a computational model that addresses the question of how fear and extinction memories are encoded in the amygdala, and specifically, in the basal nucleus (BA). We propose a specific neural mechanism to explain how the BA may integrate information about a salient, conditioned stimulus and the environment, thereby enabling it to switch the state of the animal from low to high fear and vice versa. We also provide possible explanations for various other behavioral findings, such as the recovery of fear after it had been extinguished (renewal). Finally, we make specific, experimentally testable predictions that need to be addressed in future work.

Network with shunting synapses as a non-linear frequency modulator

The role of 'noisy' excitation in synchronizing interneuron networks with shunting synapses was studied. The excitatory input was simulated as a Poisson pattern of presynaptic conductance with varying frequencies and amplitudes. We find that higher excitation frequencies induce stronger synchronisation of the network. Within the range of 1-10000 Hz, only frequencies between 20 Hz and 200 Hz affected network synchronisation. No detectable network synchronisation was found at excitation frequencies below 20 Hz, and the network's synchronisation was either almost independent of the external input or falling down to zero when the input frequency was greater than 200 Hz. Thus the network transformed the input signals with frequencies above 20 Hz into output signals with the network's synchronisation frequency. The network's synchronisation frequency in our model ranged from 20 to 68 Hz depending on the frequency of the excitatory input. We conclude that a network of interconnected interneurons is capable of converting an asynchronous excitatory input into a synchronous inhibitory output as a frequency amplifier with the amplification coefficient dependent on the number of converging excitatory inputs. Another important result of our work revealed that the external frequency may affect, in opposite ways, the frequency of the network with shunting synapses depending on the excitatory synaptic conductance and the magnitude of leak conductance.

Robustness effect of gap junctions between Golgi cells on cerebellar cortex oscillations

Background

Previous one-dimensional network modeling of the cerebellar granular layer has been successfully linked with a range of cerebellar cortex oscillations observed in vivo. However, the recent discovery of gap junctions between Golgi cells (GoCs), which may cause oscillations by themselves, has raised the question of how gap-junction coupling affects GoC and granular-layer oscillations. To investigate this question, we developed a novel two-dimensional computational model of the GoC-granule cell (GC) circuit with and without gap junctions between GoCs.

Results

Isolated GoCs coupled by gap junctions had a strong tendency to generate spontaneous oscillations without affecting their mean firing frequencies in response to distributed mossy fiber input. Conversely, when GoCs were synaptically connected in the granular layer, gap junctions increased the power of the oscillations, but the oscillations were primarily driven by the synaptic feedback loop between GoCs and GCs, and the gap junctions did not change oscillation frequency or the mean firing rate of either GoCs or GCs.

Conclusion

Our modeling results suggest that gap junctions between GoCs increase the robustness of cerebellar cortex oscillations that are primarily driven by the feedback loop between GoCs and GCs. The robustness effect of gap junctions on synaptically driven oscillations observed in our model may be a general mechanism, also present in other regions of the brain.

Frequencies of gamma/beta oscillations are stably tuned to stimulus properties

Stimulus-induced changes in oscillation frequencies may affect information flow in the brain. We investigated whether the oscillation frequency of spiking activity in cat area 17 changes as a function of the drifting direction of sinusoidal gratings. Oscillation frequencies were tuned to specific drifting directions, such that some directions induced higher oscillation frequencies than others. When activity from the same neurons was recorded at a later time point, the average oscillation frequency with which the neurons responded had also often changed. However, the direction tuning of the neurons' oscillation frequencies remained constant. Thus, while the overall oscillation frequency, across all drift directions, was state-dependent, the relative change in oscillation frequencies induced by stimulus properties was not, the tuning remaining stable.

Cascade-induced synchrony in stochastically driven neuronal networks

Perfect spike-to-spike synchrony is studied in all-to-all coupled networks of identical excitatory, current-based, integrate-and-fire neurons with delta-impulse coupling currents and Poisson spike-train external drive. This synchrony is induced by repeated cascading "total firing events," during which all neurons fire at once. In this regime, the network exhibits nearly periodic dynamics, switching between an effectively uncoupled state and a cascade-coupled total firing state. The probability of cascading total firing events occurring in the network is computed through a combinatorial analysis conditioned upon the random time when the first neuron fires and using the probability distribution of the subthreshold membrane potentials for the remaining neurons in the network. The probability distribution of the former is found from a first-passage-time problem described by a Fokker-Planck equation, which is solved analytically via an eigenfunction expansion. The latter is found using a central limit argument via a calculation of the cumulants of a single neuronal voltage. The influence of additional physiological effects that hinder or eliminate cascade-induced synchrony are also investigated. Conditions for the validity of the approximations made in the analytical derivations are discussed and verified via direct numerical simulations.

Neurophysiological and computational principles of cortical rhythms in cognition

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.

Quantitative organization of GABAergic synapses in the molecular layer of the mouse cerebellar cortex

In the cerebellar cortex, interneurons of the molecular layer (stellate and basket cells) provide GABAergic input to Purkinje cells, as well as to each other and possibly to other interneurons. GABAergic inhibition in the molecular layer has mainly been investigated at the interneuron to Purkinje cell synapse. In this study, we used complementary subtractive strategies to quantitatively assess the ratio of GABAergic synapses on Purkinje cell dendrites versus those on interneurons. We generated a mouse model in which the GABA_A receptor α1 subunit (GABA_ARα1) was selectively removed from Purkinje cells using the Cre/loxP system. Deletion of the α1 subunit resulted in a complete loss of GABA_AR aggregates from Purkinje cells, allowing us to determine the density of GABA_AR clusters in interneurons. In a complementary approach, we determined the density of GABA synapses impinging on Purkinje cells using α-dystroglycan as a specific marker of inhibitory postsynaptic sites. Combining these inverse approaches, we found that synapses received by interneurons represent approximately 40% of all GABAergic synapses in the molecular layer. Notably, this proportion was stable during postnatal development, indicating synchronized synaptogenesis. Based on the pure quantity of GABAergic synapses onto interneurons, we propose that mutual inhibition must play an important, yet largely neglected, computational role in the cerebellar cortex.

A new approach for determining phase response curves reveals that Purkinje cells can act as perfect integrators

Cerebellar Purkinje cells display complex intrinsic dynamics. They fire spontaneously, exhibit bistability, and via mutual network interactions are involved in the generation of high frequency oscillations and travelling waves of activity. To probe the dynamical properties of Purkinje cells we measured their phase response curves (PRCs). PRCs quantify the change in spike phase caused by a stimulus as a function of its temporal position within the interspike interval, and are widely used to predict neuronal responses to more complex stimulus patterns. Significant variability in the interspike interval during spontaneous firing can lead to PRCs with a low signal-to-noise ratio, requiring averaging over thousands of trials. We show using electrophysiological experiments and simulations that the PRC calculated in the traditional way by sampling the interspike interval with brief current pulses is biased. We introduce a corrected approach for calculating PRCs which eliminates this bias. Using our new approach, we show that Purkinje cell PRCs change qualitatively depending on the firing frequency of the cell. At high firing rates, Purkinje cells exhibit single-peaked, or monophasic PRCs. Surprisingly, at low firing rates, Purkinje cell PRCs are largely independent of phase, resembling PRCs of ideal non-leaky integrate-and-fire neurons. These results indicate that Purkinje cells can act as perfect integrators at low firing rates, and that the integration mode of Purkinje cells depends on their firing rate.

By observing how brief current pulses injected at different times between spikes change the phase of spiking of a neuron (and thus obtaining the so-called phase response curve), it should be possible to predict a full spike train in response to more complex stimulation patterns. When we applied this traditional protocol to obtain phase response curves in cerebellar Purkinje cells in the presence of noise, we observed a triangular region devoid of data points near the end of the spiking cycle. This “Bermuda Triangle” revealed a flaw in the classical method for constructing phase response curves. We developed a new approach to eliminate this flaw and used it to construct phase response curves of Purkinje cells over a range of spiking rates. Surprisingly, at low firing rates, phase changes were independent of the phase of the injected current pulses, implying that the Purkinje cell is a perfect integrator under these conditions. This mechanism has not yet been described in other cell types and may be crucial for the information processing capabilities of these neurons.

Function-Related Gamma Oscillations and Conscious Perception

Frequency-domain techniques describe oscillations as a fundamental behavior of neurons and brain signals. Oscillations synchronize over large portions of cortex and mediate in the spatiotemporally coherent activation of neuron assemblies required for brain processing to occur. Oscillations in the gamma band (~20.0–80.0 Hz) originate from the tonic excitation of inhibitory interneuron networks, sustain rhythms and frequency constancy, and are enhanced during sensory, motor, or “cognitive” processes through frequency-dependent and function-related neuronal synchronization. Experimental work indicates a role of gamma activity in conscious perception. Further investigation is, nevertheless, warranted as gamma-band synchronization plays a functional role in low-level phase coding as well as in high-complexity neural processes related to perception, such as selective attention, focused arousal, multistable or ambiguous perceptive conditions, visuomotor integration, and associative learning.

Electrical coupling mediates tunable low-frequency oscillations and resonance in the cerebellar Golgi cell network

Tonic motor control involves oscillatory synchronization of activity at low frequency (5-30 Hz) throughout the sensorimotor system, including cerebellar areas. We investigated the mechanisms underpinning cerebellar oscillations. We found that Golgi interneurons, which gate information transfer in the cerebellar cortex input layer, are extensively coupled through electrical synapses. When depolarized in vitro, these neurons displayed low-frequency oscillatory synchronization, imposing rhythmic inhibition onto granule cells. Combining experiments and modeling, we show that electrical transmission of the spike afterhyperpolarization is the essential component for oscillatory population synchronization. Rhythmic firing arises in spite of strong heterogeneities, is frequency tuned by the mean excitatory input to Golgi cells, and displays pronounced resonance when the modeled network is driven by oscillating inputs. In vivo, unitary Golgi cell activity was found to synchronize with low-frequency LFP oscillations occurring during quiet waking. These results suggest a major role for Golgi cells in coordinating cerebellar sensorimotor integration during oscillatory interactions.

Neuronal functional diversity and collective behaviors: a scientific case

A major issue in today's neuroscience is how the brain complex and highly flexible organization emerges from its individual components. Robustness of neuronal properties with weak linkages between regulatory processes are suggested to account for the adaptive, tunable, multistable dynamics, the coding schemes and the complexity of neuronal functional (sub)systems. Interneurons and neurotransmitter diversity, resonance phenomena due to properties of the cell or network, time/frequency-dependent activation of dedicated neuronal assemblies, code- and frequency-specific oscillations interact in determining the brain functional setup and operations. Despite the scientific relevance, comprehensive theories are not yet available, but the scenario--however incomplete and incompletely characterized--is promising and warrants further investigation.

h channel-dependent deficit of theta oscillation resonance and phase shift in temporal lobe epilepsy

I(h) tunes hippocampal CA1 pyramidal cell dendrites to optimally respond to theta inputs (4-12 Hz), and provides a negative time delay to theta inputs. Decreased I(h) activity, as seen in experimental temporal lobe epilepsy (TLE), could significantly alter the response of dendrites to theta inputs. Here we report a progressive erosion of theta resonance and phase lead in pyramidal cell dendrites during epileptogenesis in a rat model of TLE. These alterations were due to decreased I(h) availability, via a decline in HCN1/HCN2 subunit expression resulting in decreased h currents, and altered kinetics of the residual channels. This acquired HCN channelopathy thus compromises temporal coding and tuning to theta inputs in pyramidal cell dendrites. Decreased theta resonance in vitro also correlated with a reduction in theta frequency and power in vivo. We suggest that the neuronal/circuitry changes associated with TLE, including altered I(h)-dependent inductive mechanisms, can disrupt hippocampal theta function.

Neuronal functional diversity and collective behaviors

A major question in today's neuroscience is how the brain's complex operations and organization emerge from individual components. The robustness of neuronal properties with flexible linkages between regulatory processes conceivably accounts for the adaptive, tunable, multistable dynamics; the coding schemes; and the complexity of neuronal functional (sub)systems. Interneurons and neurotransmitter diversity, resonance phenomena due to properties of the cell, time/frequency-dependent activation of dedicated neuronal assemblies, and code- and frequency-specific oscillations interact in determining the brain functional setup and operations. Such an arrangement would also provide the functional requirements for access to neural mechanisms, dedicated neuronal circuitry and the proper timing allowing for the selective differentiation among cortical neurons due to performing in different tasks. No comprehensive theory or systematic methodological approach appears yet conceivable. The scenario, however incomplete and incompletely characterized, is nevertheless promising and warrants further investigation.

High-frequency organization and synchrony of activity in the purkinje cell layer of the cerebellum

The cerebellum controls complex, coordinated, and rapid movements, a function requiring precise timing abilities. However, the network mechanisms that underlie the temporal organization of activity in the cerebellum are largely unexplored, because in vivo recordings have usually targeted single units. Here, we use tetrode and multisite recordings to demonstrate that Purkinje cell activity is synchronized by a high-frequency (approximately 200 Hz) population oscillation. We combine pharmacological experiments and modeling to show how the recurrent inhibitory connections between Purkinje cells are sufficient to generate these oscillations. A key feature of these oscillations is a fixed population frequency that is independent of the firing rates of the individual cells. Convergence in the deep cerebellar nuclei of Purkinje cell activity, synchronized by these oscillations, likely organizes temporally the cerebellar output.

Why are computational neuroscience and systems biology so separate?

Despite similar computational approaches, there is surprisingly little interaction between the computational neuroscience and the systems biology research communities. In this review I reconstruct the history of the two disciplines and show that this may explain why they grew up apart. The separation is a pity, as both fields can learn quite a bit from each other. Several examples are given, covering sociological, software technical, and methodological aspects. Systems biology is a better organized community which is very effective at sharing resources, while computational neuroscience has more experience in multiscale modeling and the analysis of information processing by biological systems. Finally, I speculate about how the relationship between the two fields may evolve in the near future.

The high-conductance state of cortical networks

We studied the dynamics of large networks of spiking neurons with conductance-based (nonlinear) synapses and compared them to networks with current-based (linear) synapses. For systems with sparse and inhibition-dominated recurrent connectivity, weak external inputs induced asynchronous irregular firing at low rates. Membrane potentials fluctuated a few millivolts below threshold, and membrane conductances were increased by a factor 2 to 5 with respect to the resting state. This combination of parameters characterizes the ongoing spiking activity typically recorded in the cortex in vivo. Many aspects of the asynchronous irregular state in conductance-based networks could be sufficiently well characterized with a simple numerical mean field approach. In particular, it correctly predicted an intriguing property of conductance-based networks that does not appear to be shared by current-based models: they exhibit states of low-rate asynchronous irregular activity that persist for some period of time even in the absence of external inputs and without cortical pacemakers. Simulations of larger networks (up to 350,000 neurons) demonstrated that the survival time of self-sustained activity increases exponentially with network size.

Release-dependent variations in synaptic latency: a putative code for short- and long-term synaptic dynamics

In the cortex, synaptic latencies display small variations ( approximately 1-2 ms) that are generally considered to be negligible. We show here that the synaptic latency at monosynaptically connected pairs of L5 and CA3 pyramidal neurons is determined by the presynaptic release probability (Pr): synaptic latency being inversely correlated with the amplitude of the postsynaptic current and sensitive to manipulations of Pr. Changes in synaptic latency were also observed when Pr was physiologically regulated in short- and long-term synaptic plasticity. Paired-pulse depression and facilitation were respectively associated with increased and decreased synaptic latencies. Similarly, latencies were prolonged following induction of presynaptic LTD and reduced after LTP induction. We show using the dynamic-clamp technique that the observed covariation in latency and synaptic strength is a synergistic combination that significantly affects postsynaptic spiking. In conclusion, amplitude-related variation in latency represents a putative code for short- and long-term synaptic dynamics in cortical networks.