A bifurcation analysis of neuronal subthreshold oscillations

The enigmatic HCN channels: A cellular neurophysiology perspective

What physiological role does a slow hyperpolarization-activated ion channel with mixed cation selectivity play in the fast world of neuronal action potentials that are driven by depolarization? That puzzling question has piqued the curiosity of physiology enthusiasts about the hyperpolarization-activated cyclic nucleotide-gated (HCN) channels, which are widely expressed across the body and especially in neurons. In this review, we emphasize the need to assess HCN channels from the perspective of how they respond to time-varying signals, while also accounting for their interactions with other co-expressing channels and receptors. First, we illustrate how the unique structural and functional characteristics of HCN channels allow them to mediate a slow negative feedback loop in the neurons that they express in. We present the several physiological implications of this negative feedback loop to neuronal response characteristics including neuronal gain, voltage sag and rebound, temporal summation, membrane potential resonance, inductive phase lead, spike triggered average, and coincidence detection. Next, we argue that the overall impact of HCN channels on neuronal physiology critically relies on their interactions with other co-expressing channels and receptors. Interactions with other channels allow HCN channels to mediate intrinsic oscillations, earning them the "pacemaker channel" moniker, and to regulate spike frequency adaptation, plateau potentials, neurotransmitter release from presynaptic terminals, and spike initiation at the axonal initial segment. We also explore the impact of spatially non-homogeneous subcellular distributions of HCN channels in different neuronal subtypes and their interactions with other channels and receptors. Finally, we discuss how plasticity in HCN channels is widely prevalent and can mediate different encoding, homeostatic, and neuroprotective functions in a neuron. In summary, we argue that HCN channels form an important class of channels that mediate a diversity of neuronal functions owing to their unique gating kinetics that made them a puzzle in the first place.

Bifurcation analysis of motoneuronal excitability mechanisms under normal and ALS conditions

Introduction

Bifurcation analysis allows the examination of steady-state, non-linear dynamics of neurons and their effects on cell firing, yet its usage in neuroscience is limited to single-compartment models of highly reduced states. This is primarily due to the difficulty in developing high-fidelity neuronal models with 3D anatomy and multiple ion channels in XPPAUT, the primary bifurcation analysis software in neuroscience.

Methods

To facilitate bifurcation analysis of high-fidelity neuronal models under normal and disease conditions, we developed a multi-compartment model of a spinal motoneuron (MN) in XPPAUT and verified its firing accuracy against its original experimental data and against an anatomically detailed cell model that incorporates known MN non-linear firing mechanisms. We used the new model in XPPAUT to study the effects of somatic and dendritic ion channels on the MN bifurcation diagram under normal conditions and after amyotrophic lateral sclerosis (ALS) cellular changes.

Results

Our results show that somatic small-conductance Ca2+-activated K (SK) channels and dendritic L-type Ca2+ channels have the strongest effects on the bifurcation diagram of MNs under normal conditions. Specifically, somatic SK channels extend the limit cycles and generate a subcritical Hopf bifurcation node in the V-I bifurcation diagram of the MN to replace a supercritical node Hopf node, whereas L-type Ca2+ channels shift the limit cycles to negative currents. In ALS, our results show that dendritic enlargement has opposing effects on MN excitability, has a greater overall impact than somatic enlargement, and dendritic overbranching offsets the dendritic enlargement hyperexcitability effects.

Discussion

Together, the new multi-compartment model developed in XPPAUT facilitates studying neuronal excitability in health and disease using bifurcation analysis.

Heterogeneous stochastic bifurcations explain intrinsic oscillatory patterns in entorhinal cortical stellate cells

Stellate cells (SC) in the medial entorhinal cortex manifest intrinsic membrane potential oscillatory patterns. Although different theoretical frameworks have been proposed to explain these patterns, a robust unifying framework that jointly accounts for intrinsic heterogeneities and stochasticity is missing. Here, we first performed in vitro patch-clamp electrophysiological recordings from rat SCs and found pronounced cell-to-cell variability in their characteristic physiological properties, including peri-threshold oscillatory patterns. We demonstrate that noise introduced into two independent populations (endowed with deterministic or stochastic ion-channel gating kinetics) of heterogeneous biophysical models yielded activity patterns that were qualitatively similar to electrophysiological peri-threshold oscillatory activity in SCs. We developed spectrogram-based quantitative metrics for the identification of valid oscillations and confirmed that these metrics reliably captured the variable-amplitude and arhythmic oscillatory patterns observed in electrophysiological recordings. Using these quantitative metrics, we validated activity patterns from both heterogeneous populations of SC models, with each model assessed with multiple trials of different levels of noise at distinct membrane depolarizations. Our analyses unveiled the manifestation of stochastic resonance (detection of the highest number of valid oscillatory traces at an optimal level of noise) in both heterogeneous populations of SC models. Finally, we show that a generalized network motif comprised of a slow negative feedback loop amplified by a fast positive feedback loop manifested stochastic bifurcations and stochastic resonance in the emergence of oscillations. Together, through a unique convergence of the degeneracy and stochastic resonance frameworks, our unifying framework centered on heterogeneous stochastic bifurcations argues for state-dependent emergence of SC oscillations.

The voltage and spiking responses of subthreshold resonant neurons to structured and fluctuating inputs: persistence and loss of resonance and variability

We systematically investigate the response of neurons to oscillatory currents and synaptic-like inputs and we extend our investigation to non-structured synaptic-like spiking inputs with more realistic distributions of presynaptic spike times. We use two types of chirp-like inputs consisting of (i) a sequence of cycles with discretely increasing frequencies over time, and (ii) a sequence having the same cycles arranged in an arbitrary order. We develop and use a number of frequency-dependent voltage response metrics to capture the different aspects of the voltage response, including the standard impedance (Z) and the peak-to-trough amplitude envelope ([Formula: see text]) profiles. We show that Z-resonant cells (cells that exhibit subthreshold resonance in response to sinusoidal inputs) also show [Formula: see text]-resonance in response to sinusoidal inputs, but generally do not (or do it very mildly) in response to square-wave and synaptic-like inputs. In the latter cases the resonant response using Z is not predictive of the preferred frequencies at which the neurons spike when the input amplitude is increased above subthreshold levels. We also show that responses to conductance-based synaptic-like inputs are attenuated as compared to the response to current-based synaptic-like inputs, thus providing an explanation to previous experimental results. These response patterns were strongly dependent on the intrinsic properties of the participating neurons, in particular whether the unperturbed Z-resonant cells had a stable node or a focus. In addition, we show that variability emerges in response to chirp-like inputs with arbitrarily ordered patterns where all signals (trials) in a given protocol have the same frequency content and the only source of uncertainty is the subset of all possible permutations of cycles chosen for a given protocol. This variability is the result of the multiple different ways in which the autonomous transient dynamics is activated across cycles in each signal (different cycle orderings) and across trials. We extend our results to include high-rate Poisson distributed current- and conductance-based synaptic inputs and compare them with similar results using additive Gaussian white noise. We show that the responses to both Poisson-distributed synaptic inputs are attenuated with respect to the responses to Gaussian white noise. For cells that exhibit oscillatory responses to Gaussian white noise (band-pass filters), the response to conductance-based synaptic inputs are low-pass filters, while the response to current-based synaptic inputs may remain band-pass filters, consistent with experimental findings. Our results shed light on the mechanisms of communication of oscillatory activity among neurons in a network via subthreshold oscillations and resonance and the generation of network resonance.

Stochastic Hybrid Systems in Cellular Neuroscience

We review recent work on the theory and applications of stochastic hybrid systems in cellular neuroscience. A stochastic hybrid system or piecewise deterministic Markov process involves the coupling between a piecewise deterministic differential equation and a time-homogeneous Markov chain on some discrete space. The latter typically represents some random switching process. We begin by summarizing the basic theory of stochastic hybrid systems, including various approximation schemes in the fast switching (weak noise) limit. In subsequent sections, we consider various applications of stochastic hybrid systems, including stochastic ion channels and membrane voltage fluctuations, stochastic gap junctions and diffusion in randomly switching environments, and intracellular transport in axons and dendrites. Finally, we describe recent work on phase reduction methods for stochastic hybrid limit cycle oscillators.

Control of clustered action potential firing in a mathematical model of entorhinal cortex stellate cells

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An SDE model of entorhinal cortex (EC) stellate cells is proposed.

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Experimentally observed action potential clustering is investigated in the model.

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Clusters are generated by subcritical-Hopf/homoclinic type bursting.

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Potential mechanisms underlying changes in EC dynamics in dementia are presented.

The entorhinal cortex is a crucial component of our memory and spatial navigation systems and is one of the first areas to be affected in dementias featuring tau pathology, such as Alzheimer’s disease and frontotemporal dementia. Electrophysiological recordings from principle cells of medial entorhinal cortex (layer II stellate cells, mEC-SCs) demonstrate a number of key identifying properties including subthreshold oscillations in the theta (4–12 Hz) range and clustered action potential firing. These single cell properties are correlated with network activity such as grid firing and coupling between theta and gamma rhythms, suggesting they are important for spatial memory. As such, experimental models of dementia have revealed disruption of organised dorsoventral gradients in clustered action potential firing.

To better understand the mechanisms underpinning these different dynamics, we study a conductance based model of mEC-SCs. We demonstrate that the model, driven by extrinsic noise, can capture quantitative differences in clustered action potential firing patterns recorded from experimental models of tau pathology and healthy animals. The differential equation formulation of our model allows us to perform numerical bifurcation analyses in order to uncover the dynamic mechanisms underlying these patterns. We show that clustered dynamics can be understood as subcritical Hopf/homoclinic bursting in a fast-slow system where the slow sub-system is governed by activation of the persistent sodium current and inactivation of the slow A-type potassium current. In the full system, we demonstrate that clustered firing arises via flip bifurcations as conductance parameters are varied. Our model analyses confirm the experimentally suggested hypothesis that the breakdown of clustered dynamics in disease occurs via increases in AHP conductance.

A variational method for analyzing limit cycle oscillations in stochastic hybrid systems

Many systems in biology can be modeled through ordinary differential equations, which are piece-wise continuous, and switch between different states according to a Markov jump process known as a stochastic hybrid system or piecewise deterministic Markov process (PDMP). In the fast switching limit, the dynamics converges to a deterministic ODE. In this paper, we develop a phase reduction method for stochastic hybrid systems that support a stable limit cycle in the deterministic limit. A classic example is the Morris-Lecar model of a neuron, where the switching Markov process is the number of open ion channels and the continuous process is the membrane voltage. We outline a variational principle for the phase reduction, yielding an exact analytic expression for the resulting phase dynamics. We demonstrate that this decomposition is accurate over timescales that are exponential in the switching rate ϵ^-1. That is, we show that for a constant C, the probability that the expected time to leave an O(a) neighborhood of the limit cycle is less than T scales as T exp (-Ca/ϵ).

An integrative model of the intrinsic hippocampal theta rhythm

Hippocampal theta oscillations (4–12 Hz) are consistently recorded during memory tasks and spatial navigation. Despite several known circuits and structures that generate hippocampal theta locally in vitro, none of them were found to be critical in vivo, and the hippocampal theta rhythm is severely attenuated by disruption of external input from medial septum or entorhinal cortex. We investigated these discrepancies that question the sufficiency and robustness of hippocampal theta generation using a biophysical spiking network model of the CA3 region of the hippocampus that included an interconnected network of pyramidal cells, inhibitory basket cells (BC) and oriens-lacunosum moleculare (OLM) cells. The model was developed by matching biological data characterizing neuronal firing patterns, synaptic dynamics, short-term synaptic plasticity, neuromodulatory inputs, and the three-dimensional organization of the hippocampus. The model generated theta power robustly through five cooperating generators: spiking oscillations of pyramidal cells, recurrent connections between them, slow-firing interneurons and pyramidal cells subnetwork, the fast-spiking interneurons and pyramidal cells subnetwork, and non-rhythmic structured external input from entorhinal cortex to CA3. We used the modeling framework to quantify the relative contributions of each of these generators to theta power, across different cholinergic states. The largest contribution to theta power was that of the divergent input from the entorhinal cortex to CA3, despite being constrained to random Poisson activity. We found that the low cholinergic states engaged the recurrent connections in generating theta activity, whereas high cholinergic states utilized the OLM-pyramidal subnetwork. These findings revealed that theta might be generated differently across cholinergic states, and demonstrated a direct link between specific theta generators and neuromodulatory states.

Phase diagrams and dynamics of a computationally efficient map-based neuron model

We introduce a new map-based neuron model derived from the dynamical perceptron family that has the best compromise between computational efficiency, analytical tractability, reduced parameter space and many dynamical behaviors. We calculate bifurcation and phase diagrams analytically and computationally that underpins a rich repertoire of autonomous and excitable dynamical behaviors. We report the existence of a new regime of cardiac spikes corresponding to nonchaotic aperiodic behavior. We compare the features of our model to standard neuron models currently available in the literature.

The shaping of intrinsic membrane potential oscillations: positive/negative feedback, ionic resonance/amplification, nonlinearities and time scales

The generation of intrinsic subthreshold (membrane potential) oscillations (STOs) in neuronal models requires the interaction between two processes: a relatively fast positive feedback that favors changes in voltage and a slower negative feedback that opposes these changes. These are provided by the so-called resonant and amplifying gating variables associated to the participating ionic currents. We investigate both the biophysical and dynamic mechanisms of generation of STOs and how their attributes (frequency and amplitude) depend on the model parameters for biophysical (conductance-based) models having qualitatively different types of resonant currents (activating and inactivating) and an amplifying current. Combinations of the same types of ionic currents (same models) in different parameter regimes give rise to different types of nonlinearities in the voltage equation: quasi-linear, parabolic-like and cubic-like. On the other hand, combinations of different types of ionic currents (different models) may give rise to the same type of nonlinearities. We examine how the attributes of the resulting STOs depend on the combined effect of these resonant and amplifying ionic processes, operating at different effective time scales, and the various types of nonlinearities. We find that, while some STO properties and attribute dependencies on the model parameters are determined by the specific combinations of ionic currents (biophysical properties), and are different for models with different such combinations, others are determined by the type of nonlinearities and are common for models with different types of ionic currents. Our results highlight the richness of STO behavior in single cells as the result of the various ways in which resonant and amplifying currents interact and affect the generation and termination of STOs as control parameters change. We make predictions that can be tested experimentally and are expected to contribute to the understanding of how rhythmic activity in neuronal networks emerge from the interplay of the intrinsic properties of the participating neurons and the network connectivity.

Gender-Specific Hippocampal Dysrhythmia and Aberrant Hippocampal and Cortical Excitability in the APPswePS1dE9 Model of Alzheimer's Disease

Alzheimer's disease (AD) is a multifactorial disorder leading to progressive memory loss and eventually death. In this study an APPswePS1dE9 AD mouse model has been analyzed using implantable video-EEG radiotelemetry to perform long-term EEG recordings from the primary motor cortex M1 and the hippocampal CA1 region in both genders. Besides motor activity, EEG recordings were analyzed for electroencephalographic seizure activity and frequency characteristics using a Fast Fourier Transformation (FFT) based approach. Automatic seizure detection revealed severe electroencephalographic seizure activity in both M1 and CA1 deflection in APPswePS1dE9 mice with gender-specific characteristics. Frequency analysis of both surface and deep EEG recordings elicited complex age, gender, and activity dependent alterations in the theta and gamma range. Females displayed an antithetic decrease in theta (θ) and increase in gamma (γ) power at 18-19 weeks of age whereas related changes in males occurred earlier at 14 weeks of age. In females, theta (θ) and gamma (γ) power alterations predominated in the inactive state suggesting a reduction in atropine-sensitive type II theta in APPswePS1dE9 animals. Gender-specific central dysrhythmia and network alterations in APPswePS1dE9 point to a functional role in behavioral and cognitive deficits and might serve as early biomarkers for AD in the future.

Quasicycles in the stochastic hybrid Morris-Lecar neural model

Intrinsic noise arising from the stochastic opening and closing of voltage-gated ion channels has been shown experimentally and mathematically to have important effects on a neuron's function. Study of classical neuron models with stochastic ion channels is becoming increasingly important, especially in understanding a cell's ability to produce subthreshold oscillations and to respond to weak periodic stimuli. While it is known that stochastic models can produce oscillations (quasicycles) in parameter regimes where the corresponding deterministic model has only a stable fixed point, little analytical work has been done to explore these connections within the context of channel noise. Using a stochastic hybrid Morris-Lecar (ML) model, we combine a system-size expansion in K(+) and a quasi-steady-state (QSS) approximation in persistent Na(+) in order to derive an effective Langevin equation that preserves the low-dimensional (planar) structure of the underlying deterministic ML model. (The QSS analysis exploits the fact that persistent Na(+) channels are fast.) By calculating the corresponding power spectrum, we determine analytically how noise significantly extends the parameter regime in which subthreshold oscillations occur.

Why noise is useful in functional and neural mechanisms of interval timing?

BACKGROUND

The ability to estimate durations in the seconds-to-minutes range - interval timing - is essential for survival, adaptation and its impairment leads to severe cognitive and/or motor dysfunctions. The response rate near a memorized duration has a Gaussian shape centered on the to-be-timed interval (criterion time). The width of the Gaussian-like distribution of responses increases linearly with the criterion time, i.e., interval timing obeys the scalar property.

RESULTS

We presented analytical and numerical results based on the striatal beat frequency (SBF) model showing that parameter variability (noise) mimics behavioral data. A key functional block of the SBF model is the set of oscillators that provide the time base for the entire timing network. The implementation of the oscillators block as simplified phase (cosine) oscillators has the additional advantage that is analytically tractable. We also checked numerically that the scalar property emerges in the presence of memory variability by using biophysically realistic Morris-Lecar oscillators. First, we predicted analytically and tested numerically that in a noise-free SBF model the output function could be approximated by a Gaussian. However, in a noise-free SBF model the width of the Gaussian envelope is independent of the criterion time, which violates the scalar property. We showed analytically and verified numerically that small fluctuations of the memorized criterion time leads to scalar property of interval timing.

CONCLUSIONS

Noise is ubiquitous in the form of small fluctuations of intrinsic frequencies of the neural oscillators, the errors in recording/retrieving stored information related to criterion time, fluctuation in neurotransmitters' concentration, etc. Our model suggests that the biological noise plays an essential functional role in the SBF interval timing.

Voltage dependence of subthreshold resonance frequency in layer II of medial entorhinal cortex

The resonance properties of individual neurons in entorhinal cortex (EC) may contribute to their functional properties in awake, behaving rats. Models propose that entorhinal grid cells could arise from shifts in the intrinsic frequency of neurons caused by changes in membrane potential owing to depolarizing input from neurons coding velocity. To test for potential changes in intrinsic frequency, we measured the resonance properties of neurons at different membrane potentials in neurons in medial and lateral EC. In medial entorhinal neurons, the resonant frequency of individual neurons decreased in a linear manner as the membrane potential was depolarized between -70 and -55 mV. At more hyperpolarized membrane potentials, cells asymptotically approached a maximum resonance frequency. Consistent with the previous studies, near resting potential, the cells of the medial EC possessed a decreasing gradient of resonance frequency along the dorsal to ventral axis, and cells of the lateral EC lacked resonant properties, regardless of membrane potential or position along the medial to lateral axis within lateral EC. Application of 10 μM ZD7288, the H-channel blocker, abolished all resonant properties in MEC cells, and resulted in physiological properties very similar to lateral EC cells. These results on resonant properties show a clear change in frequency response with depolarization that could contribute to the generation of grid cell firing properties in the medial EC.

Increased cortical and thalamic excitability in freely moving APPswe/PS1dE9 mice modeling epileptic activity associated with Alzheimer's disease

Amyloid precursor protein transgenic mice modeling Alzheimer's disease display frequent occurrence of seizures peaking at an age when amyloid plaques start to form in the cortex and hippocampus. We tested the hypothesis that numerous reported interactions of amyloid-β with cell surface molecules result in altered excitation-inhibition balance in brain-wide neural networks, eventually leading to epileptogenesis. We examined electroencephalograms (EEGs) and auditory-evoked potentials (AEPs) in freely moving 4-month-old APPswe/PS1dE9 (APdE9) and wild-type (WT) control mice in the hippocampus, cerebral cortex, and thalamus during movement, quiet waking, non-rapid eye movement sleep, and rapid eye movement (REM) sleep. Cortical EEG power was higher in APdE9 mice than in WT mice over a broad frequency range (5-100 Hz) and during all 4 behavioral states. Thalamic EEG power was also increased but in a narrower range (10-80 Hz). Furthermore, APdE9 mice displayed augmented cortical and thalamic AEPs. While power and theta-gamma modulation were preserved in the APdE9 hippocampus, REM sleep-related phase shift of theta-gamma modulation was altered. Our data suggest that at the early stage of amyloid pathology, cortical principal cells become hyperexcitable and via extensive cortico-thalamic connection drive thalamic cells. Minor hippocampal changes are most likely secondary to abnormal entorhinal input.

On the noise-enhancing ability of stochastic Hodgkin-Huxley neuron systems

Recently noise has been shown to be useful in enhancing neuron sensitivity by stochastic resonance. In this study, in order to measure the noise-enhancing factor (NEF), a nonlinear stochastic model is introduced for the Hodgkin-Huxley (HH) neuron system with synaptic noise input stimulation and channel noises in the sodium and potassium channels. The enhancing factor of the HH neuron system is measured from the point of view of the noise-exploiting level of nonlinear stochastic H(infinity) signal processing. Since the nonlinear stochastic-enhancing measure problem of HH neuron systems requires a solution for the difficulty presented by the Hamilton Jacobi inequality (HJI), a fuzzy interpolation of locally linearized systems is employed to simplify the nonlinear noise-enhancing problems by solving only a set of linear matrix inequalities. The NEF of the HH neuron system is found to be related to the locations of eigenvalues of linearized HH neuron systems and can be estimated through the H(infinity) signal processing method. Based on a stochastic fuzzy linearized HH neuron system, we found that channel noises are enhanced by the active eigenvalues of ionic channels while synaptic noises are attenuated by the passive eigenvalues of synaptic process.

On the role of subthreshold currents in the Huber-Braun cold receptor model

We study the role of the strength of subthreshold currents in a four-dimensional Hodgkin-Huxley-type model of mammalian cold receptors. Since a total diminution of subthreshold activity corresponds to a decomposition of the model into a slow, subthreshold, and a fast, spiking subsystem, we first elucidate their respective dynamics separately and draw conclusions about their role for the generation of different spiking patterns. These results motivate a numerical bifurcation analysis of the effect of varying the strength of subthreshold currents, which is done by varying a suitable control parameter. We work out the key mechanisms which can be attributed to subthreshold activity and furthermore elucidate the dynamical backbone of different activity patterns generated by this model.

The mechanism of abrupt transition between theta and hyper-excitable spiking activity in medial entorhinal cortex layer II stellate cells

Recent studies have shown that stellate cells (SCs) of the medial entorhinal cortex become hyper-excitable in animal models of temporal lobe epilepsy. These studies have also demonstrated the existence of recurrent connections among SCs, reduced levels of recurrent inhibition in epileptic networks as compared to control ones, and comparable levels of recurrent excitation among SCs in both network types. In this work, we investigate the biophysical and dynamic mechanism of generation of the fast time scale corresponding to hyper-excitable firing and the transition between theta and fast firing frequency activity in SCs. We show that recurrently connected minimal networks of SCs exhibit abrupt, threshold-like transition between theta and hyper-excitable firing frequencies as the result of small changes in the maximal synaptic (AMPAergic) conductance. The threshold required for this transition is modulated by synaptic inhibition. Similar abrupt transition between firing frequency regimes can be observed in single, self-coupled SCs, which represent a network of recurrently coupled neurons synchronized in phase, but not in synaptically isolated SCs as the result of changes in the levels of the tonic drive. Using dynamical systems tools (phase-space analysis), we explain the dynamic mechanism underlying the genesis of the fast time scale and the abrupt transition between firing frequency regimes, their dependence on the intrinsic SC's currents and synaptic excitation. This abrupt transition is mechanistically different from others observed in similar networks with different cell types. Most notably, there is no bistability involved. 'In vitro' experiments using single SCs self-coupled with dynamic clamp show the abrupt transition between firing frequency regimes, and demonstrate that our theoretical predictions are not an artifact of the model. In addition, these experiments show that high-frequency firing is burst-like with a duration modulated by an M-current.

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.

Democracy-independence trade-off in oscillating dendrites and its implications for grid cells

Dendritic democracy and independence have been characterized for near-instantaneous processing of synaptic inputs. However, a wide class of neuronal computations requires input integration on long timescales. As a paradigmatic example, entorhinal grid fields have been thought to be generated by the democratic summation of independent dendritic oscillations performing direction-selective path integration. We analyzed how multiple dendritic oscillators embedded in the same neuron integrate inputs separately and determine somatic membrane voltage jointly. We found that the interaction of dendritic oscillations leads to phase locking, which sets an upper limit on the timescale for independent input integration. Factors that increase this timescale also decrease the influence that the dendritic oscillations exert on somatic voltage. In entorhinal stellate cells, interdendritic coupling dominates and causes these cells to act as single oscillators. Our results suggest a fundamental trade-off between local and global processing in dendritic trees integrating ongoing signals.

The range of intrinsic frequencies represented by medial entorhinal cortex stellate cells extends with age

In both humans and rodents, the external environment is encoded in the form of cognitive maps. Neurons in the medial entorhinal cortex (mEC) represent spatial locations in a sequence of grid-like patterns scaled along the dorsal-ventral axis. The grid spacing correlates with the intrinsic resonance frequencies of stellate cells in layer II of mEC. We investigated the development of frequency preferences in these cells from weaning to adulthood using patch-clamp and sharp microelectrode recordings. We found that the dorsal-ventral gradient of stellate cell properties and frequency preferences exists before animals are able to actively explore their environment. In the transition to adulthood, cells respond faster and become less excitable, and the range of intrinsic resonance frequencies in the population expands in the dorsal direction. This is likely to reflect both the growth of the brain and the expansion of the internal representation caused by new exploratory experience.

Beta-amyloid protein (25-35) disrupts hippocampal network activity: role of Fyn-kinase

Early cognitive deficit characteristic of early Alzheimer's disease seems to be produced by the soluble forms of beta-amyloid protein. Such cognitive deficit correlates with neuronal network dysfunction that is reflected as alterations in the electroencephalogram of both Alzheimer patients and transgenic murine models of such disease. Correspondingly, recent studies have demonstrated that chronic exposure to betaAP affects hippocampal oscillatory properties. However, it is still unclear if such neuronal network dysfunction results from a direct action of betaAP on the hippocampal circuit or it is secondary to the chronic presence of the protein in the brain. Therefore, we aimed to explore the effect of acute exposure to betaAP(25-35) on hippocampal network activity both in vitro and in vivo, as well as on intrinsic and synaptic properties of hippocampal neurons. We found that betaAP(25-35), reversibly, affects spontaneous hippocampal population activity in vitro. Such effect is not produced by the inverse sequence betaAP(35-25) and is reproduced by the full-length peptide betaAP(1-42). Correspondingly betaAP(25-35), but not the inverse sequence betaAP(35-25), reduces theta-like activity recorded from the hippocampus in vivo. The betaAP(25-35)-induced disruption in hippocampal network activity correlates with a reduction in spontaneous neuronal activity and synaptic transmission, as well as with an inhibition in the subthreshold oscillations produced by pyramidal neurons in vitro. Finally, we studied the involvement of Fyn-kinase on the betaAP(25-35)-induced disruption in hippocampal network activity in vitro. Interestingly, we found that such phenomenon is not observed in slices obtained from Fyn-knockout mice. In conclusion, our data suggest that betaAP acutely affects proper hippocampal function through a Fyn-dependent mechanism. We propose that such alteration might be related to the cognitive impairment observed, at least, during the early phases of Alzheimer's disease.

New perspectives in brain information processing

Brain cortex activity, as variously recorded by scalp or cortical electrodes in the electroencephalography (EEG) frequency range, probably reflects the basic strategy of brain information processing. Various hypotheses have been advanced to interpret this phenomenon, the most popular of which is that suitable combinations of excitatory and inhibitory neurons behave as assemblies of oscillators susceptible to synchronization and desynchronization. Implicit in this view is the assumption that EEG potentials are epiphenomena of action potentials, which is consistent with the argument that voltage variations in dendritic membranes reproduce the postsynaptic effects of targeting neurons. However, this classic argument does not really fit the discovery that firing synchronization over extended brain areas often appears to be established in about 1 ms, which is a small fraction of any EEG frequency component period. This is in contrast with the fact that all computational models of dynamic systems formed by more or less weakly interacting oscillators of near frequencies take more than one period to reach synchronization. The discovery that the somatodendritic membranes of specialized populations of neurons exhibit intrinsic subthreshold oscillations (ISOs) in the EEG frequency range, together with experimental evidence that short inhibitory stimuli are capable of resetting ISO phases, radically changes the scheme described above and paves the way to a novel view. This paper aims to elucidate the nature of ISO generation mechanisms, to explain the reasons for their reliability in starting and stopping synchronized firing, and to indicate their potential in brain information processing. The need for a repertoire of extraneuronal regulation mechanisms, putatively mediated by astrocytes, is also inferred. Lastly, the importance of ISOs for the brain as a parallel recursive machine is briefly discussed.

Time constants of h current in layer ii stellate cells differ along the dorsal to ventral axis of medial entorhinal cortex

Chronic recordings in the medial entorhinal cortex of behaving rats have found grid cells, neurons that fire when the rat is in a hexagonal array of locations. Grid cells recorded at different dorsal-ventral anatomical positions show systematic changes in size and spacing of firing fields. To test possible mechanisms underlying these differences, we analyzed properties of the hyperpolarization-activated cation current I(h) in voltage-clamp recordings from stellate cells in entorhinal slices from different dorsal-ventral locations. The time constant of h current was significantly different between dorsal and ventral neurons. The time constant of h current correlated with membrane potential oscillation frequency and the time constant of the sag potential in the same neurons. Differences in h current could underlie differences in membrane potential oscillation properties and contribute to grid cell periodicity along the dorsal-ventral axis of medial entorhinal cortex.

Rhythms of the brain: an examination of mixed mode oscillation approaches to the analysis of neurophysiological data

In the nervous system many behaviorally relevant dynamical processes are characterized by episodes of complex oscillatory states, whose periodicity may be expressed over multiple temporal and spatial scales. In at least some of these instances the variability in oscillatory amplitude and frequency can be explained in terms of deterministic dynamics, rather than being purely noise-driven. Recently interest has increased in studying the application of mixed-mode oscillations (MMOs) to neurophysiological data. MMOs are complex periodic waveforms where each period is comprised of several maxima and minima of different amplitudes. While MMOs might be expected to occur in brain kinetics, only a few examples have been identified thus far. In this article, we review recent theoretical and experimental findings on brain oscillatory rhythms in relation to MMOs, focusing on examples at the single neuron level but also briefly touching on possible instances of the phenomenon across local and global brain networks.

Regulation of spike timing in visual cortical circuits

A train of action potentials (a spike train) can carry information in both the average firing rate and the pattern of spikes in the train. But can such a spike-pattern code be supported by cortical circuits? Neurons in vitro produce a spike pattern in response to the injection of a fluctuating current. However, cortical neurons in vivo are modulated by local oscillatory neuronal activity and by top-down inputs. In a cortical circuit, precise spike patterns thus reflect the interaction between internally generated activity and sensory information encoded by input spike trains. We review the evidence for precise and reliable spike timing in the cortex and discuss its computational role.

HCN1 channels control resting and active integrative properties of stellate cells from layer II of the entorhinal cortex

Whereas recent studies have elucidated principles for representation of information within the entorhinal cortex, less is known about the molecular basis for information processing by entorhinal neurons. The HCN1 gene encodes ion channels that mediate hyperpolarization-activated currents (I(h)) that control synaptic integration and influence several forms of learning and memory. We asked whether hyperpolarization-activated, cation nonselective 1 (HCN1) channels control processing of information by stellate cells found within layer II of the entorhinal cortex. Axonal projections from these neurons form a major component of the synaptic input to the dentate gyrus of the hippocampus. To determine whether HCN1 channels control either the resting or the active properties of stellate neurons, we performed whole-cell recordings in horizontal brain slices prepared from adult wild-type and HCN1 knock-out mice. We found that HCN1 channels are required for rapid and full activation of hyperpolarization-activated currents in stellate neurons. HCN1 channels dominate the membrane conductance at rest, are not required for theta frequency (4-12 Hz) membrane potential fluctuations, but suppress low-frequency (<4 Hz) components of spontaneous and evoked membrane potential activity. During sustained activation of stellate cells sufficient for firing of repeated action potentials, HCN1 channels control the pattern of spike output by promoting recovery of the spike afterhyperpolarization. These data suggest that HCN1 channels expressed by stellate neurons in layer II of the entorhinal cortex are key molecular components in the processing of inputs to the hippocampal dentate gyrus, with distinct integrative roles during resting and active states.

Arc length coding by interference of theta frequency oscillations may underlie context-dependent hippocampal unit data and episodic memory function

Many memory models focus on encoding of sequences by excitatory recurrent synapses in region CA3 of the hippocampus. However, data and modeling suggest an alternate mechanism for encoding of sequences in which interference between theta frequency oscillations encodes the position within a sequence based on spatial arc length or time. Arc length can be coded by an oscillatory interference model that accounts for many features of the context-dependent firing properties of hippocampal neurons observed during performance of spatial memory tasks. In continuous spatial alternation, many neurons fire selectively depending on the direction of prior or future response (left or right). In contrast, in delayed non-match to position, most neurons fire selectively for task phase (sample vs. choice), with less selectivity for left versus right. These seemingly disparate results are effectively simulated by the same model, based on mechanisms similar to a model of grid cell firing in entorhinal cortex. The model also simulates forward shifting of firing over trials. Adding effects of persistent firing with reset at reward locations addresses changes in context-dependent firing with different task designs. Arc length coding could contribute to episodic encoding of trajectories as sequences of states and actions.

Grid cell firing may arise from interference of theta frequency membrane potential oscillations in single neurons

Intracellular recording and computational modelling suggest that interactions of subthreshold membrane potential oscillation frequency in different dendritic branches of entorhinal cortex stellate cells could underlie the functional coding of continuous dimensions of space and time. Among other things, these interactions could underlie properties of grid cell field spacing. The relationship between experimental data on membrane potential oscillation frequency (f) and grid cell field spacing (G) indicates a constant scaling factor H = fG. This constant scaling factor between temporal oscillation frequency and spatial periodicity provides a starting constraint that is used to derive the model of Burgess et al. (Hippocampus, 2007). This model provides a consistent quantitative link between single cell physiological properties and properties of spiking units in awake behaving animals. Further properties and predictions of this model about single cell and network physiological properties are analyzed. In particular, the model makes quantitative predictions about the change in membrane potential, single cell oscillation frequency, and network oscillation frequency associated with speed of movement, about the independence of single cell properties from network theta rhythm oscillations, and about the effect of variations in initial oscillatory phase on the pattern of grid cell firing fields. These same mechanisms of subthreshold oscillations may play a more general role in memory function, by providing a method for learning arbitrary time intervals in memory sequences.

Passive soma facilitates submillisecond coincidence detection in the owl's auditory system

Neurons of the avian nucleus laminaris (NL) compute the interaural time difference (ITD) by detecting coincident arrivals of binaural signals with submillisecond accuracy. The cellular mechanisms for this temporal precision have long been studied theoretically and experimentally. The myelinated axon initial segment in the owl's NL neuron and small somatic spikes observed in auditory coincidence detector neurons of various animals suggest that spikes in the NL neuron are generated at the first node of Ranvier and that the soma passively receives back-propagating spikes. To investigate the significance of the "passive soma" structure, we constructed a two-compartment NL neuron model, consisting of a cell body and a first node, and systematically changed the excitability of each compartment. Here, we show that a neuron with a less active soma achieves higher ITD sensitivity and higher noise tolerance with lower energy costs. We also investigate the biophysical mechanism of the computational advantage of the "passive soma" structure by performing sub- and suprathreshold analyses. Setting a spike initiation site with high sodium conductance, not in the large soma but in the small node, serves to amplify high-frequency input signals and to reduce the impact and the energy cost of spike generation. Our results indicate that the owl's NL neuron uses a "passive soma" design for computational and metabolic reasons.

Interspike interval densities of resonate and fire neurons

The subthreshold dynamics of a neuron can follow one of the two patterns: resonant neurons generate intrinsic subthreshold membrane potential oscillations, whereas in nonresonant neurons these oscillations are not observed. Here, we investigate how these subthreshold behaviors affect the suprathreshold response. Both types of neurons are described by a resonate and fire model, with the stable fixpoint being either a focus or a node. Using analytic expression for a linear oscillator model with threshold and reset, we calculate the multimodal interspike interval densities. We show that a change in model parameters induces qualitative changes in the interspike interval densities.

Low-dimensional maps encoding dynamics in entorhinal cortex and hippocampus

Cells that produce intrinsic theta oscillations often contain the hyperpolarization-activated current I(h). In this article, we use models and dynamic clamp experiments to investigate the synchronization properties of two such cells (stellate cells of the entorhinal cortex and O-LM cells of the hippocampus) in networks with fast-spiking (FS) interneurons. The model we use for stellate cells and O-LM cells is the same, but the stellate cells are excitatory and the O-LM cells are inhibitory, with inhibitory postsynaptic potential considerably longer than those from FS interneurons. We use spike time response curve methods (STRC), expanding that technique to three-cell networks and giving two different ways in which the analysis of the three-cell network reduces to that of a two-cell network. We show that adding FS cells to a network of stellate cells can desynchronize the stellate cells, while adding them to a network of O-LM cells can synchronize the O-LM cells. These synchronization and desynchronization properties critically depend on I(h). The analysis of the deterministic system allows us to understand some effects of noise on the phase relationships in the stellate networks. The dynamic clamp experiments use biophysical stellate cells and in silico FS cells, with connections that mimic excitation or inhibition, the latter with decay times associated with FS cells or O-LM cells. The results obtained in the dynamic clamp experiments are in a good agreement with the analytical framework.

The dynamic structure underlying subthreshold oscillatory activity and the onset of spikes in a model of medial entorhinal cortex stellate cells

Medial entorhinal cortex layer II stellate cells display subthreshold oscillations (STOs). We study a single compartment biophysical model of such cells which qualitatively reproduces these STOs. We argue that in the subthreshold interval (STI) the seven-dimensional model can be reduced to a three-dimensional system of equations with well differentiated times scales. Using dynamical systems arguments we provide a mechanism for generations of STOs. This mechanism is based on the "canard structure," in which relevant trajectories stay close to repelling manifolds for a significant interval of time. We also show that the transition from subthreshold oscillatory activity to spiking ("canard explosion") is controlled in the STI by the same structure. A similar mechanism is invoked to explain why noise increases the robustness of the STO regime. Taking advantage of the reduction of the dimensionality of the full stellate cell system, we propose a nonlinear artificially spiking (NAS) model in which the STI reduced system is supplemented with a threshold for spiking and a reset voltage. We show that the synchronization properties in networks made up of the NAS cells are similar to those of networks using the full stellate cell models.

Active hair bundle movements in auditory hair cells

The frequency selectivity of mammalian hearing depends on not only the passive mechanics of the basilar membrane but also an active amplification of the mechanical stimulus by the cochlear hair cells. The common view is that amplification stems from the somatic motility of the outer hair cells (OHCs), changes in their length impelled by voltage-dependent transitions in the membrane protein prestin. Whether this voltage-controlled mechanism, whose frequency range may be limited by the membrane time constant, has the band width to cover the entire auditory range of mammals is uncertain. However, there is ample evidence for an alternative mode of force generation by hair cells of non-mammals, such as frogs and turtles, which probably lack prestin. The latter process involves active motion of the hair bundle underpinned by conformational changes in the mechanotransducer (MT) channels and activation of one or more isoforms of myosin. This review summarizes evidence for active hair bundle motion and its connection to MT channel adaptation. Key factors for the hair bundle motor to play a role in the mammalian cochlea include the size and speed of force production.

Functional role of entorhinal cortex in working memory processing

Our learning and memory system has the challenge to work in a world where items to learn are dispersed in space and time. From the information extracted by the perceptual systems, the learning system must select and combine information. Both these operations may require a temporary storage where significance and correlations could be assessed. This work builds on the common hypothesis that hippocampus and subicular, entorhinal and parahippocampal/postrhinal areas are essential for the above-mentioned functions. We bring up two examples of models; the first one is modeling of in vivo and in vitro data from entorhinal cortex layer II of delayed match-to-sample working memory experiments, the second one studying mechanisms in theta rhythmicity in EC. In both cases, we discuss how cationic currents might be involved and relate their kinetics and pharmacology to behavioral and cellular experimental results.

Synchronization in hybrid neuronal networks of the hippocampal formation

Understanding the mechanistic bases of neuronal synchronization is a current challenge in quantitative neuroscience. We studied this problem in two putative cellular pacemakers of the mammalian hippocampal theta rhythm: glutamatergic stellate cells (SCs) of the medial entorhinal cortex and GABAergic oriens-lacunosum-molecular (O-LM) interneurons of hippocampal region CA1. We used two experimental methods. First, we measured changes in spike timing induced by artificial synaptic inputs applied to individual neurons. We then measured responses of free-running hybrid neuronal networks, consisting of biological neurons coupled (via dynamic clamp) to biological or virtual counterparts. Results from the single-cell experiments predicted network behaviors well and are compatible with previous model-based predictions of how specific membrane mechanisms give rise to empirically measured synchronization behavior. Both cell types phase lock stably when connected via homogeneous excitatory-excitatory (E-E) or inhibitory-inhibitory (I-I) connections. Phase-locked firing is consistently synchronous for either cell type with E-E connections and nearly anti-synchronous with I-I connections. With heterogeneous connections (e.g., excitatory-inhibitory, as might be expected if members of a given population had heterogeneous connections involving intermediate interneurons), networks often settled into phase locking that was either stable or unstable, depending on the order of firing of the two cells in the hybrid network. Our results imply that excitatory SCs, but not inhibitory O-LM interneurons, are capable of synchronizing in phase via monosynaptic mutual connections of the biologically appropriate polarity. Results are largely independent of synaptic strength and synaptic kinetics, implying that our conclusions are robust and largely unaffected by synaptic plasticity.

Ionic mechanisms in the generation of subthreshold oscillations and action potential clustering in entorhinal layer II stellate neurons

A multicompartmental biophysical model of entorhinal cortex layer II stellate cells was developed to analyze the ionic basis of physiological properties, such as subthreshold membrane potential oscillations, action potential clustering, and the medium afterhyperpolarization. In particular, the simulation illustrates the interaction of the persistent sodium current (I(Nap)) and the hyperpolarization activated inward current (Ih) in the generation of subthreshold membrane potential oscillations. The potential role of Ih in contributing to the medium hyperpolarization (mAHP) and rebound spiking was studied. The role of Ih and the slow calcium-activated potassium current Ikappa(AHP) in action potential clustering was also studied. Representations of Ih and I(Nap) were developed with parameters based on voltage-clamp data from whole-cell patch and single channel recordings of stellate cells (Dickson et al., J Neurophysiol 83:2562-2579, 2000; Magistretti and Alonso, J Gen Physiol 114:491-509, 1999; Magistretti et al., J Physiol 521:629-636, 1999a; J Neurosci 19:7334-7341, 1999b). These currents interacted to generate robust subthreshold membrane potentials with amplitude and frequency corresponding to data observed in the whole cell patch recordings. The model was also able to account for effects of pharmacological manipulations, including blockade of Ih with ZD7288, partial blockade with cesium, and the influence of barium on oscillations. In a model with a wider range of currents, the transition from oscillations to single spiking, to spike clustering, and finally tonic firing could be replicated. In agreement with experiment, blockade of calcium channels in the model strongly reduced clustering. In the voltage interval during which no data are available, the model predicts that the slow component of Ih does not follow the fast component down to very short time constants. The model also predicts that the fast component of Ih is responsible for the involvement in the generation of subthreshold oscillations, and the slow component dominates in the generation of spike clusters.

State-dependent effects of Na channel noise on neuronal burst generation

We explore the effects of stochastic sodium (Na) channel activation on the variability and dynamics of spiking and bursting in a model neuron. The complete model segregates Hodgin-Huxley-type currents into two compartments, and undergoes applied current-dependent bifurcations between regimes of periodic bursting, chaotic bursting, and tonic spiking. Noise is added to simulate variable, finite sizes of the population of Na channels in the fast spiking compartment. During tonic firing, Na channel noise causes variability in interspike intervals (ISIs). The variance, as well as the sensitivity to noise, depend on the model's biophysical complexity. They are smallest in an isolated spiking compartment; increase significantly upon coupling to a passive compartment; and increase again when the second compartment also includes slow-acting currents. In this full model, sufficient noise can convert tonic firing into bursting. During bursting, the actions of Na channel noise are state-dependent. The higher the noise level, the greater the jitter in spike timing within bursts. The noise makes the burst durations of periodic regimes variable, while decreasing burst length duration and variance in a chaotic regime. Na channel noise blurs the sharp transitions of spike time and burst length seen at the bifurcations of the noise-free model. Close to such a bifurcation, the burst behaviors of previously periodic and chaotic regimes become essentially indistinguishable. We discuss biophysical mechanisms, dynamical interpretations and physiological implications. We suggest that noise associated with finite populations of Na channels could evoke very different effects on the intrinsic variability of spiking and bursting discharges, depending on a biological neuron's complexity and applied current-dependent state. We find that simulated channel noise in the model neuron qualitatively replicates the observed variability in burst length and interburst interval in an isolated biological bursting neuron.

Gamma oscillations induced by kainate receptor activation in the entorhinal cortex in vitro

Gamma frequency (30-80 Hz) oscillations are recordable from human and rodent entorhinal cortex. A number of mechanisms used by neuronal networks to generate such oscillations in the hippocampus have been characterized. However, it is as yet unclear as to whether these mechanisms apply to other anatomically disparate brain regions. Here we show that the medial entorhinal cortex (mEC) in isolation in vitro generates gamma frequency oscillations in response to kainate receptor agonists. Oscillations had the same horizontal and laminar spatiotemporal distribution as seen in vivo and in the isolated whole-brain preparation. Oscillations occurred in the absence of input from the hippocampal formation and did not spread to lateral entorhinal regions. Pharmacological similarities existed between oscillations in the hippocampus and mEC in that the latter were also sensitive to GABAA receptor blockade, barbiturates, AMPA receptor blockade, and reduction in gap junctional conductance. Stellate and pyramidal neuron recordings revealed a large GABAergic input consisting of gamma frequency IPSP trains. Fast spiking interneurons in the superficial mEC generated action potentials at gamma frequencies phase locked to the local field. Stellate cells also demonstrated a subthreshold membrane potential oscillation at theta frequencies that was temporally correlated with a theta-frequency modulation in field gamma power. Disruption in this stellate theta frequency oscillation by the hyperpolarisation activated current (Ih) blocker ZD7288 also disrupted theta modulation of field gamma frequency oscillations. We propose that similar cellular and network mechanisms to those seen in the hippocampus generate and modulate persistent gamma oscillations in the entorhinal cortex.

Gamma oscillations induced by kainate receptor activation in the entorhinal cortex in vitro

Gamma frequency (30-80 Hz) oscillations are recordable from human and rodent entorhinal cortex. A number of mechanisms used by neuronal networks to generate such oscillations in the hippocampus have been characterized. However, it is as yet unclear as to whether these mechanisms apply to other anatomically disparate brain regions. Here we show that the medial entorhinal cortex (mEC) in isolation in vitro generates gamma frequency oscillations in response to kainate receptor agonists. Oscillations had the same horizontal and laminar spatiotemporal distribution as seen in vivo and in the isolated whole-brain preparation. Oscillations occurred in the absence of input from the hippocampal formation and did not spread to lateral entorhinal regions. Pharmacological similarities existed between oscillations in the hippocampus and mEC in that the latter were also sensitive to GABAA receptor blockade, barbiturates, AMPA receptor blockade, and reduction in gap junctional conductance. Stellate and pyramidal neuron recordings revealed a large GABAergic input consisting of gamma frequency IPSP trains. Fast spiking interneurons in the superficial mEC generated action potentials at gamma frequencies phase locked to the local field. Stellate cells also demonstrated a subthreshold membrane potential oscillation at theta frequencies that was temporally correlated with a theta-frequency modulation in field gamma power. Disruption in this stellate theta frequency oscillation by the hyperpolarisation activated current (Ih) blocker ZD7288 also disrupted theta modulation of field gamma frequency oscillations. We propose that similar cellular and network mechanisms to those seen in the hippocampus generate and modulate persistent gamma oscillations in the entorhinal cortex.

Synchronization of strongly coupled excitatory neurons: relating network behavior to biophysics

Behavior of a network of neurons is closely tied to the properties of the individual neurons. We study this relationship in models of layer II stellate cells (SCs) of the medial entorhinal cortex. SCs are thought to contribute to the mammalian theta rhythm (4-12 Hz), and are notable for the slow ionic conductances that constrain them to fire at rates within this frequency range. We apply "spike time response" (STR) methods, in which the effects of synaptic perturbations on the timing of subsequent spikes are used to predict how these neurons may synchronize at theta frequencies. Predictions from STR methods are verified using network simulations. Slow conductances often make small inputs "effectively large"; we suggest that this is due to reduced attractiveness or stability of the spiking limit cycle. When inputs are (effectively) large, changes in firing times depend nonlinearly on synaptic strength. One consequence of nonlinearity is to make a periodically firing model skip one or more beats, often leading to the elimination of the anti-synchronous state in bistable models. Biologically realistic membrane noise makes such "cycle skipping" more prevalent, and thus can eradicate bistability. Membrane noise also supports "sparse synchrony," a phenomenon in which subthreshold behavior is uncorrelated, but there are brief periods of synchronous spiking.

From subthreshold to firing-rate resonance

Many types of neurons exhibit subthreshold resonance. However, little is known about whether this frequency preference influences spike emission. Here, the link between subthreshold resonance and firing rate is examined in the framework of conductance-based models. A classification of the subthreshold properties of a general class of neurons is first provided. In particular, a class of neurons is identified in which the input impedance exhibits a suppression at a nonzero low frequency as well as a peak at higher frequency. The analysis is then extended to the effect of subthreshold resonance on the dynamics of the firing rate. The considered input current comprises a background noise term, mimicking the massive synaptic bombardment in vivo. Of interest is the modulatory effect an additional weak oscillating current has on the instantaneous firing rate. When the noise is weak and firing regular, the frequency most preferentially modulated is the firing rate itself. Conversely, when the noise is strong and firing irregular, the modulation is strongest at the subthreshold resonance frequency. These results are demonstrated for two specific conductance-based models and for a generalization of the integrate-and-fire model that captures subthreshold resonance. They suggest that resonant neurons are able to communicate their frequency preference to postsynaptic targets when the level of noise is comparable to that prevailing in vivo.

On the persistent sodium current in squid giant axons

R. F. Rakowski, D. C. Gadsby, and P. DeWeer have reported a persistent, tetrodotoxin-sensitive sodium ion current (I(NaP)) in squid giant axons having a low threshold (-90 mV) and a maximal inward amplitude of -4 microA/cm(2) at -50 mV. This report makes the case that most of I(NaP) is attributable to an ion channel mechanism distinct from the classical rapidly activating and inactivating sodium ion current, I(Na), which is also tetrodotoxin sensitive. The analysis of the contribution of I(Na) to I(NaP) is critically dependent on slow inactivation of I(Na). The results of this gating process reported here demonstrate that inactivation of I(Na) is complete in the steady-state for V > -40 mV, thereby making it unlikely that I(NaP) in this potential range is attributable to I(Na). Moreover, -90 mV is well below I(Na) threshold, as demonstrated by the C. A. Vandenberg and F. Bezanilla model of I(Na) gating in squid giant axons. Their model predicts a persistent current having a threshold of -60 mV and a peak amplitude of -25 microA/cm(2) at -20 mV. Modulation of this component by the slow inactivation process predicts a persistent current that is finite in the -60- to -40-mV range having a peak amplitude of -1 microA/cm(-2) at -50 mV. Subtraction of this current from the I(NaP) measurements yields the portion of I(NaP) that appears to be attributable to an ion channel mechanism distinct from I(Na).

Frequency selectivity of layer II stellate cells in the medial entorhinal cortex

Electrophysiologically, stellate cells (SCs) from layer II of the medial entorhinal cortex (MEC) are distinguished by intrinsic 4- to 12-Hz subthreshold oscillations. These oscillations are thought to impose a pattern of slow periodic firing that may contribute to the parahippocampal theta rhythm in vivo. Using stimuli with systematically differing frequency content, we examined supra- and subthreshold responses in SCs with the goal of understanding how their distinctive characteristics shape these responses. In reaction to repeated presentations of identical, pseudo-random stimuli, the reliability (repeatability) of the spiking response in SCs depends critically on the frequency content of the stimulus. Reliability is optimal for stimuli with a greater proportion of power in the 4- to 12-Hz range. The simplest mechanistic explanation of these results is that rhythmogenic subthreshold membrane mechanisms resonate with inputs containing significant power in the 4- to 12-Hz band, leading to larger subthreshold excursions and thus enhanced reliability. However, close examination of responses rules out this explanation: SCs do show clear subthreshold resonance (i.e., selective amplification of inputs with particular frequency content) in response to sinusoidal stimuli, while simultaneously showing a lack of subthreshold resonance in response to the pseudo-random stimuli used in reliability experiments. Our results support a model with distinctive input-output relationships under subthreshold and suprathreshold conditions. For suprathreshold stimuli, SC spiking seems to best reflect the amount of input power in the theta (4-12 Hz) frequency band. For subthreshold stimuli, we hypothesize that the magnitude of subthreshold theta-range oscillations in SCs reflects the total power, across all frequencies, of the input.

Computational model of carbachol-induced delta, theta, and gamma oscillations in the hippocampus

Field potential recordings from the rat hippocampus in vivo contain distinct frequency bands of activity, including delta (0.5-2 Hz), theta (4-12 Hz), and gamma (30-80 Hz), that are correlated with the behavioral state of the animal. The cholinergic agonist carbachol (CCH) induces oscillations in the delta (CCH-delta), theta (CCH-theta), and gamma (CCH-gamma) frequency ranges in the hippocampal slice preparation, eliciting asynchronous CCH-theta, synchronous CCH-delta, and synchronous CCH-theta with increasing CCH concentration (Fellous and Seinowski, Hippocampus 2000;1 0:187-197). In a network model of area CA3, the time scale for CCH-delta corresponded to the decay constant of the gating variable of the calcium-dependent potassium (K-AHP) current, that of CCH-theta to an intrinsic subthreshold membrane potential oscillation of the pyramidal cells, and that of CCH-gamma to the decay constant of GABAergic inhibitory synaptic potentials onto the pyramidal cells. In model simulations, the known physiological effects of carbachol on the muscarinic and K-AHP currents, and on the strengths of excitatory postsynaptic potentials, reproduced transitions from asynchronous CCH-theta to CCH-delta and from CCH-delta to synchronous CCH-theta. The simulations also exhibited the interspersed CCH-gamma/CCH-delta and CCH-gamma/CCH-theta that were observed in experiments. The model, in addition, predicted an oscillatory state with all three frequency bands present, which has not yet been observed experimentally.

Subthreshold voltage noise due to channel fluctuations in active neuronal membranes

Voltage-gated ion channels in neuronal membranes fluctuate randomly between different conformational states due to thermal agitation. Fluctuations between conducting and nonconducting states give rise to noisy membrane currents and subthreshold voltage fluctuations and may contribute to variability in spike timing. Here we study subthreshold voltage fluctuations due to active voltage-gated Na+ and K+ channels as predicted by two commonly used kinetic schemes: the Mainen et al. (1995) (MJHS) kinetic scheme, which has been used to model dendritic channels in cortical neurons, and the classical Hodgkin-Huxley (1952) (HH) kinetic scheme for the squid giant axon. We compute the magnitudes, amplitude distributions, and power spectral densities of the voltage noise in isopotential membrane patches predicted by these kinetic schemes. For both schemes, noise magnitudes increase rapidly with depolarization from rest. Noise is larger for smaller patch areas but is smaller for increased model temperatures. We contrast the results from Monte Carlo simulations of the stochastic nonlinear kinetic schemes with analytical, closed-form expressions derived using passive and quasi-active linear approximations to the kinetic schemes. For all subthreshold voltage ranges, the quasi-active linearized approximation is accurate within 8% and may thus be used in large-scale simulations of realistic neuronal geometries.

Computational modeling of entorhinal cortex

Computational modeling provides a means for linking the physiological and anatomical characteristics of entorhinal cortex at a cellular level to the functional role of this region in behavior. We have developed detailed simulations of entorhinal cortical neurons and networks, with an emphasis on the role of acetylcholine in entorhinal cortical function. Computational modeling suggests that when acetylcholine levels are high, this sets appropriate dynamics for the storage of stimuli during performance of delayed matching tasks. In particular, acetylcholine activates a calcium-sensitive nonspecific cation current which provides an intrinsic cellular mechanism which could maintain neuronal activity across a delay period. Simulations demonstrate how this phenomena could underlie entorhinal cortex delay activity as described in previous unit recordings. Acetylcholine also induces theta rhythm oscillations which may be appropriate for timing of afferent input to be encoded in hippocampus and for extraction of individual stored sequences from multiple stored sequences. Lower levels of acetylcholine may allow sharp wave dynamics which can reactivate associations encoded in hippocampus and drive the formation of additional traces in hippocampus and entorhinal cortex during consolidation.