Artificial synaptic conductances reduce subthreshold oscillations and periodic firing in stellate cells of the entorhinal cortex

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.

Oscillations and variability in neuronal systems: interplay of autonomous transient dynamics and fast deterministic fluctuations

Neuronal systems are subject to rapid fluctuations both intrinsically and externally. These fluctuations can be disruptive or constructive. We investigate the dynamic mechanisms underlying the interactions between rapidly fluctuating signals and the intrinsic properties of the target cells to produce variable and/or coherent responses. We use linearized and non-linear conductance-based models and piecewise constant (PWC) inputs with short duration pieces. The amplitude distributions of the constant pieces consist of arbitrary permutations of a baseline PWC function. In each trial within a given protocol we use one of these permutations and each protocol consists of a subset of all possible permutations, which is the only source of uncertainty in the protocol. We show that sustained oscillatory behavior can be generated in response to various forms of PWC inputs independently of whether the stable equilibria of the corresponding unperturbed systems are foci or nodes. The oscillatory voltage responses are amplified by the model nonlinearities and attenuated for conductance-based PWC inputs as compared to current-based PWC inputs, consistent with previous theoretical and experimental work. In addition, the voltage responses to PWC inputs exhibited variability across trials, which is reminiscent of the variability generated by stochastic noise (e.g., Gaussian white noise). Our analysis demonstrates that both oscillations and variability are the result of the interaction between the PWC input and the target cell's autonomous transient dynamics with little to no contribution from the dynamics in vicinities of the steady-state, and do not require input stochasticity.

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.

Topography in the Bursting Dynamics of Entorhinal Neurons

Medial entorhinal cortex contains neural substrates for representing space. These substrates include grid cells that fire in repeating locations and increase in scale progressively along the dorsal-to-ventral entorhinal axis, with the physical distance between grid firing nodes increasing from tens of centimeters to several meters in rodents. Whether the temporal scale of grid cell spiking dynamics shows a similar dorsal-to-ventral organization remains unknown. Here, we report the presence of a dorsal-to-ventral gradient in the temporal spiking dynamics of grid cells in behaving mice. This gradient in bursting supports the emergence of a dorsal grid cell population with a high signal-to-noise ratio. In vitro recordings combined with a computational model point to a role for gradients in non-inactivating sodium conductances in supporting the bursting gradient in vivo. Taken together, these results reveal a complementary organization in the temporal and intrinsic properties of entorhinal cells.

Theta resonance and synaptic modulation scale activity patterns in the medial entorhinal cortex stellate cells

Stellate cells (SCs) of the medial entorhinal cortex (MEC) are rich in hyperpolarization-activated cyclic nucleotide-gated (HCN) channels, which are known to effectively shape their activity patterns. The explanatory mechanisms, however, have remained elusive. One important but previously unassessed possibility is that HCN channels control the gain of synaptic inputs to these cells. Here, we test this possibility in rat brain slices, while subjecting SCs to a stochastic synaptic bombardment using the dynamic clamp. We show that in the presence of synaptic noise, HCN channels mainly exert their influence by increasing the relative signal gain in the theta frequency through the theta modulation of stochastic synaptic inputs. This subthreshold synaptic modulation then translates into a spiking resonance, which steepens with excitation in the presence of HCN channels. We present here a systematic assessment of synaptic theta modulation and trace its implications to the suprathreshold control of firing rate motifs. Such analysis was yet lacking in the SC literature. Furthermore, we assess the impact of noise statistics on this gain modulation and indicate possible mechanisms for the emergence of membrane theta oscillations and synaptic ramps, as observed in vivo. We support the data with a computational model that further unveils a competing role of inhibition, suggesting important implications for MEC computations.

Balanced synaptic currents underlie low-frequency oscillations in the subiculum

Numerous synaptic and intrinsic membrane mechanisms have been proposed for generating oscillatory activity in the hippocampus. Few studies, however, have directly measured synaptic conductances and membrane properties during oscillations. The time course and relative contribution of excitatory and inhibitory synaptic conductances, as well as the role of intrinsic membrane properties in amplifying synaptic inputs, remains unclear. To address this issue, we used an isolated whole hippocampal preparation that generates autonomous low-frequency oscillations near the theta range. Using 2-photon microscopy and expression of genetically encoded fluorophores, we obtained on-cell and whole-cell patch recordings of pyramidal cells and fast-firing interneurons in the distal subiculum. Pyramidal cell and interneuron spiking shared similar phase-locking to local field potential oscillations. In pyramidal cells, spiking resulted from a concomitant and balanced increase in excitatory and inhibitory synaptic currents. In contrast, interneuron spiking was driven almost exclusively by excitatory synaptic current. Thus, similar to tightly balanced networks underlying hippocampal gamma oscillations and ripples, balanced synaptic inputs in the whole hippocampal preparation drive highly phase-locked spiking at the peak of slower network oscillations.

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.

Degeneracy in the robust expression of spectral selectivity, subthreshold oscillations, and intrinsic excitability of entorhinal stellate cells

Biological heterogeneities are ubiquitous and play critical roles in the emergence of physiology at multiple scales. Although neurons in layer II (LII) of the medial entorhinal cortex (MEC) express heterogeneities in channel properties, the impact of such heterogeneities on the robustness of their cellular-scale physiology has not been assessed. Here, we performed a 55-parameter stochastic search spanning nine voltage- or calcium-activated channels to assess the impact of channel heterogeneities on the concomitant emergence of 10 in vitro electrophysiological characteristics of LII stellate cells (SCs). We generated 150,000 models and found a heterogeneous subpopulation of 449 valid models to robustly match all electrophysiological signatures. We employed this heterogeneous population to demonstrate the emergence of cellular-scale degeneracy in SCs, whereby disparate parametric combinations expressing weak pairwise correlations resulted in similar models. We then assessed the impact of virtually knocking out each channel from all valid models and demonstrate that the mapping between channels and measurements was many-to-many, a critical requirement for the expression of degeneracy. Finally, we quantitatively predict that the spike-triggered average of SCs should be endowed with theta-frequency spectral selectivity and coincidence detection capabilities in the fast gamma-band. We postulate this fast gamma-band coincidence detection as an instance of cellular-scale-efficient coding, whereby SC response characteristics match the dominant oscillatory signals in LII MEC. The heterogeneous population of valid SC models built here unveils the robust emergence of cellular-scale physiology despite significant channel heterogeneities, and forms an efficacious substrate for evaluating the impact of biological heterogeneities on entorhinal network function.

NEW & NOTEWORTHY We assessed the impact of heterogeneities in channel properties on the robustness of cellular-scale physiology of medial entorhinal cortical stellate neurons. We demonstrate that neuronal models with disparate channel combinations were endowed with similar physiological characteristics, as a consequence of the many-to-many mapping between channel properties and the physiological characteristics that they modulate. We predict that the spike-triggered average of stellate cells should be endowed with theta-frequency spectral selectivity and fast gamma-band coincidence detection capabilities.

Differences in the Electrophysiological Properties of Mouse Somatosensory Layer 2/3 Neurons In Vivo and Slice Stem from Intrinsic Sources Rather than a Network-Generated High Conductance State

Synaptic activity in vivo can potentially alter the integration properties of neurons. Using recordings in awake mice, we targeted somatosensory layer 2/3 pyramidal neurons and compared neuronal properties with those from slices. Pyramidal cells in vivo had lower resistance and gain values, as well as broader spikes and increased spike frequency adaptation compared to the same cells in slices. Increasing conductance in neurons using dynamic clamp to levels observed in vivo, however, did not lessen the differences between in vivo and slice conditions. Further, local application of tetrodotoxin (TTX) in vivo blocked synaptic-mediated membrane voltage fluctuations but had little impact on pyramidal cell membrane input resistance and time constant values. Differences in electrophysiological properties of layer 2/3 neurons in mouse somatosensory cortex, therefore, stem from intrinsic sources separate from synaptic-mediated membrane voltage fluctuations.

Synaptic integrative mechanisms for spatial cognition

Synaptic integrative mechanisms have profound effects on electrical signaling in the brain that, although largely hidden from recording methods that observe the spiking activity of neurons, may be critical for the encoding, storage and retrieval of information. Here we review roles for synaptic integrative mechanisms in the selection, generation and plasticity of place and grid fields, and in related temporal codes for the representation of space. We outline outstanding questions and challenges in the testing of hypothesized models for spatial computation and memory.

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.

Interplay between Subthreshold Oscillations and Depressing Synapses in Single Neurons

In this paper we analyze the interplay between the subthreshold oscillations of a single neuron conductance-based model and the short-term plasticity of a dynamic synapse with a depressing mechanism. In previous research, the computational properties of subthreshold oscillations and dynamic synapses have been studied separately. Our results show that dynamic synapses can influence different aspects of the dynamics of neuronal subthreshold oscillations. Factors such as maximum hyperpolarization level, oscillation amplitude and frequency or the resulting firing threshold are modulated by synaptic depression, which can even make subthreshold oscillations disappear. This influence reshapes the postsynaptic neuron’s resonant properties arising from subthreshold oscillations and leads to specific input/output relations. We also study the neuron’s response to another simultaneous input in the context of this modulation, and show a distinct contextual processing as a function of the depression, in particular for detection of signals through weak synapses. Intrinsic oscillations dynamics can be combined with the characteristic time scale of the modulatory input received by a dynamic synapse to build cost-effective cell/channel-specific information discrimination mechanisms, beyond simple resonances. In this regard, we discuss the functional implications of synaptic depression modulation on intrinsic subthreshold dynamics.

If I had a million neurons: Potential tests of cortico-hippocampal theories

Considerable excitement surrounds new initiatives to develop techniques for simultaneous recording of large populations of neurons in cortical structures. This chapter focuses on the potential value of large-scale simultaneous recording for advancing research on current issues in the function of cortical circuits, including the interaction of the hippocampus with cortical and subcortical structures. The review describes specific research questions that could be answered using large-scale population recording, including questions about the circuit dynamics underlying coding of dimensions of space and time for episodic memory, the role of GABAergic and cholinergic innervation from the medial septum, the functional role of spatial representations coded by grid cells, boundary cells, head direction cells, and place cells, and the fact that many models require cells coding movement direction.

Grid cell firing patterns may arise from feedback interaction between intrinsic rebound spiking and transverse traveling waves with multiple heading angles

This article presents a model using cellular resonance and rebound properties to model grid cells in medial entorhinal cortex. The model simulates the intrinsic resonance properties of single layer II stellate cells with different frequencies due to the hyperpolarization activated cation current (h current). The stellate cells generate rebound spikes after a delay interval that differs for neurons with different resonance frequency. Stellate cells drive inhibitory interneurons to cause rebound from inhibition in an alternate set of stellate cells that drive interneurons to activate the first set of cells. This allows maintenance of activity with cycle skipping of the spiking of cells that matches recent physiological data on theta cycle skipping. The rebound spiking interacts with subthreshold oscillatory input to stellate cells or interneurons regulated by medial septal input and defined relative to the spatial location coded by neurons. The timing of rebound determines whether the network maintains the activity for the same location or shifts to phases of activity representing a different location. Simulations show that spatial firing patterns similar to grid cells can be generated with a range of different resonance frequencies, indicating how grid cells could be generated with low frequencies present in bats and in mice with knockout of the HCN1 subunit of the h current.

Large scale in vivo recordings to study neuronal biophysics

Over the last several years, technological advances have enabled researchers to more readily observe single-cell membrane biophysics in awake, behaving animals. Studies utilizing these technologies have provided important insights into the mechanisms generating functional neural codes in both sensory and non-sensory cortical circuits. Crucial for a deeper understanding of how membrane biophysics control circuit dynamics however, is a continued effort to move toward large scale studies of membrane biophysics, in terms of the numbers of neurons and ion channels examined. Future work faces a number of theoretical and technical challenges on this front but recent technological developments hold great promise for a larger scale understanding of how membrane biophysics contribute to circuit coding and computation.

Nonlinear properties of medial entorhinal cortex neurons reveal frequency selectivity during multi-sinusoidal stimulation

The neurons in layer II of the medial entorhinal cortex are part of the grid cell network involved in the representation of space. Many of these neurons are likely to be stellate cells with specific oscillatory and firing properties important for their function. A fundamental understanding of the nonlinear basis of these oscillatory properties is critical for the development of theories of grid cell firing. In order to evaluate the behavior of stellate neurons, measurements of their quadratic responses were used to estimate a second order Volterra kernel. This paper uses an operator theory, termed quadratic sinusoidal analysis (QSA), which quantitatively determines that the quadratic response accounts for a major part of the nonlinearity observed at membrane potential levels characteristic of normal synaptic events. Practically, neurons were probed with multi-sinusoidal stimulations to determine a Hermitian operator that captures the quadratic function in the frequency domain. We have shown that the frequency content of the stimulation plays an important role in the characteristics of the nonlinear response, which can distort the linear response as well. Stimulations with enhanced low frequency amplitudes evoked a different nonlinear response than broadband profiles. The nonlinear analysis was also applied to spike frequencies and it was shown that the nonlinear response of subthreshold membrane potential at resonance frequencies near the threshold is similar to the nonlinear response of spike trains.

Membrane potential-dependent integration of synaptic inputs in entorhinal stellate neurons

Stellate cells (SCs) of the medial entorhinal cortex exhibit robust spontaneous membrane-potential oscillations (MPOs) in the theta (4-12 Hz) frequency band as well as theta-frequency resonance in their membrane impedance spectra. Past experimental and modeling work suggests that these features may contribute to the phase-locking of SCs to the entorhinal theta rhythm and may be important for forming the hexagonally tiled grid cell place fields exhibited by these neurons in vivo. Among the major biophysical mechanisms contributing to MPOs is a population of persistent (non-inactivating or slowly inactivating) sodium channels. The resulting persistent sodium conductance (GNaP ) gives rise to an apparent increase in input resistance as the cell approaches threshold. In this study, we used dynamic clamp to test the hypothesis that this increased input resistance gives rise to voltage-dependent, and thus MPO phase-dependent, changes in the amplitude of excitatory and inhibitory post-synaptic potential (PSP) amplitudes. We find that PSP amplitude depends on membrane potential, exhibiting a 5-10% increase in amplitude per mV depolarization. The effect is larger than-and sums quasi-linearly with-the effect of the synaptic driving force, V - Esyn . Given that input-driven MPOs 10 mV in amplitude are commonly observed in MEC stellate cells in vivo, this voltage- and phase-dependent synaptic gain is large enough to modulate PSP amplitude by over 50% during theta-frequency MPOs. Phase-dependent synaptic gain may therefore impact the phase locking and phase precession of grid cells in vivo to ongoing network oscillations. © 2014 Wiley Periodicals, Inc.

The past, present, and future of real-time control in cellular electrophysiology

For over 60 years, real-time control has been an important technique in the study of excitable cells. Two such control-based technologies are reviewed here. First, voltage-clamp methods revolutionized the study of excitable cells. In this family of techniques, membrane potential is controlled, allowing one to parameterize a powerful class of models that describe the voltage-current relationship of cell membranes simply, flexibly, and accurately. Second, dynamic-clamp methods allow the addition of new, "virtual" membrane mechanisms to living cells. Dynamic clamp allows researchers unprecedented ways of testing computationally based hypotheses in biological preparations. The review ends with predictions of how control-based technologies will be improved and adapted for new uses in the near future.

Grid-layout and theta-modulation of layer 2 pyramidal neurons in medial entorhinal cortex

Little is known about how microcircuits are organized in layer 2 of the medial entorhinal cortex. We visualized principal cell microcircuits and determined cellular theta-rhythmicity in freely moving rats. Non-dentate-projecting, calbindin-positive pyramidal cells bundled dendrites together and formed patches arranged in a hexagonal grid aligned to layer 1 axons, parasubiculum, and cholinergic inputs. Calbindin-negative, dentate-gyrus-projecting stellate cells were distributed across layer 2 but avoided centers of calbindin-positive patches. Cholinergic drive sustained theta-rhythmicity, which was twofold stronger in pyramidal than in stellate neurons. Theta-rhythmicity was cell-type-specific but not distributed as expected from cell-intrinsic properties. Layer 2 divides into a weakly theta-locked stellate cell lattice and spatiotemporally highly organized pyramidal grid. It needs to be assessed how these two distinct principal cell networks contribute to grid cell activity.

Electrical resonance in the θ frequency range in olfactory amygdala neurons

The cortical amygdala receives direct olfactory inputs and is thought to participate in processing and learning of biologically relevant olfactory cues. As for other brain structures implicated in learning, the principal neurons of the anterior cortical nucleus (ACo) exhibit intrinsic subthreshold membrane potential oscillations in the θ-frequency range. Here we show that nearly 50% of ACo layer II neurons also display electrical resonance, consisting of selective responsiveness to stimuli of a preferential frequency (2–6 Hz). Their impedance profile resembles an electrical band-pass filter with a peak at the preferred frequency, in contrast to the low-pass filter properties of other neurons. Most ACo resonant neurons displayed frequency preference along the whole subthreshold voltage range. We used pharmacological tools to identify the voltage-dependent conductances implicated in resonance. A hyperpolarization-activated cationic current depending on HCN channels underlies resonance at resting and hyperpolarized potentials; notably, this current also participates in resonance at depolarized subthreshold voltages. KV7/KCNQ K⁺ channels also contribute to resonant behavior at depolarized potentials, but not in all resonant cells. Moreover, resonance was strongly attenuated after blockade of voltage-dependent persistent Na⁺ channels, suggesting an amplifying role. Remarkably, resonant neurons presented a higher firing probability for stimuli of the preferred frequency. To fully understand the mechanisms underlying resonance in these neurons, we developed a comprehensive conductance-based model including the aforementioned and leak conductances, as well as Hodgkin and Huxley-type channels. The model reproduces the resonant impedance profile and our pharmacological results, allowing a quantitative evaluation of the contribution of each conductance to resonance. It also replicates selective spiking at the resonant frequency and allows a prediction of the temperature-dependent shift in resonance frequency. Our results provide a complete characterization of the resonant behavior of olfactory amygdala neurons and shed light on a putative mechanism for network activity coordination in the intact brain.

How to build a grid cell

Neurons in the medial entorhinal cortex fire action potentials at regular spatial intervals, creating a striking grid-like pattern of spike rates spanning the whole environment of a navigating animal. This remarkable spatial code may represent a neural map for path integration. Recent advances using patch-clamp recordings from entorhinal cortex neurons in vitro and in vivo have revealed how the microcircuitry in the medial entorhinal cortex may contribute to grid cell firing patterns, and how grid cells may transform synaptic inputs into spike output during firing field crossings. These new findings provide key insights into the ingredients necessary to build a grid cell.

Frequency preference in two-dimensional neural models: a linear analysis of the interaction between resonant and amplifying currents

Many neuron types exhibit preferred frequency responses in their voltage amplitude (resonance) or phase shift to subthreshold oscillatory currents, but the effect of biophysical parameters on these properties is not well understood. We propose a general framework to analyze the role of different ionic currents and their interactions in shaping the properties of impedance amplitude and phase in linearized biophysical models and demonstrate this approach in a two-dimensional linear model with two effective conductances g L and g1. We compute the key attributes of impedance and phase (resonance frequency and amplitude, zero-phase frequency, selectivity, etc.) in the g(L) - g1 parameter space. Using these attribute diagrams we identify two basic mechanisms for the generation of resonance: an increase in the resonance amplitude as g1 increases while the overall impedance is decreased, and an increase in the maximal impedance, without any change in the input resistance, as the ionic current time constant increases. We use the attribute diagrams to analyze resonance and phase of the linearization of two biophysical models that include resonant (I h or slow potassium) and amplifying currents (persistent sodium). In the absence of amplifying currents, the two models behave similarly as the conductances of the resonant currents is increased whereas, with the amplifying current present, the two models have qualitatively opposite responses. This work provides a general method for decoding the effect of biophysical parameters on linear membrane resonance and phase by tracking trajectories, parametrized by the relevant biophysical parameter, in pre-constructed attribute diagrams.

Somatic versus dendritic resonance: differential filtering of inputs through non-uniform distributions of active conductances

Synaptic inputs to neurons are processed in a frequency-dependent manner, with either low-pass or resonant response characteristics. These types of filtering play a key role in the frequency-specific information flow in neuronal networks. While the generation of resonance by specific ionic conductances is well investigated, less attention has been paid to the spatial distribution of the resonance-generating conductances across a neuron. In pyramidal neurons – one of the major excitatory cell-types in the mammalian brain – a steep gradient of resonance-generating h-conductances with a 60-fold increase towards distal dendrites has been demonstrated experimentally. Because the dendritic trees of these cells are large, spatial compartmentalization of resonant properties can be expected. Here, we use mathematical descriptions of spatially extended neurons to investigate the consequences of such a distal, dendritic localization of h-conductances for signal processing. While neurons with short dendrites do not exhibit a pronounced compartmentalization of resonance, i.e. the filter properties of dendrites and soma are similar, we find that neurons with longer dendrites ( space constant) can show distinct filtering of dendritic and somatic inputs due to electrotonic segregation. Moreover, we show that for such neurons, experimental classification as resonant versus nonresonant can be misleading when based on somatic recordings, because for these morphologies a dendritic resonance could easily be undetectable when using somatic input. Nevertheless, noise-driven membrane-potential oscillations caused by dendritic resonance can propagate to the soma where they can be recorded, hence contrasting with the low-pass filtering at the soma. We conclude that non-uniform distributions of active conductances can underlie differential filtering of synaptic input in neurons with spatially extended dendrites, like pyramidal neurons, bearing relevance for the localization-dependent targeting of synaptic input pathways to these cells.

Entorhinal stellate cells show preferred spike phase-locking to theta inputs that is enhanced by correlations in synaptic activity

In active networks, excitatory and inhibitory synaptic inputs generate membrane voltage fluctuations that drive spike activity in a probabilistic manner. Despite this, some cells in vivo show a strong propensity to precisely lock to the local field potential and maintain a specific spike-phase relationship relative to other cells. In recordings from rat medial entorhinal cortical stellate cells, we measured spike phase-locking in response to sinusoidal "test" inputs in the presence of different forms of background membrane voltage fluctuations, generated via dynamic clamp. We find that stellate cells show strong and robust spike phase-locking to theta (4-12 Hz) inputs. This response occurs under a wide variety of background membrane voltage fluctuation conditions that include a substantial increase in overall membrane conductance. Furthermore, the IH current present in stellate cells is critical to the enhanced spike phase-locking response at theta. Finally, we show that correlations between inhibitory and excitatory conductance fluctuations, which can arise through feedback and feedforward inhibition, can substantially enhance the spike phase-locking response. The enhancement in locking is a result of a selective reduction in the size of low-frequency membrane voltage fluctuations due to cancellation of inhibitory and excitatory current fluctuations with correlations. Hence, our results demonstrate that stellate cells have a strong preference for spike phase-locking to theta band inputs and that the absolute magnitude of locking to theta can be modulated by the properties of background membrane voltage fluctuations.

In vivo cholinergic modulation of the cellular properties of medial entorhinal cortex neurons

Extensive in vitro data and modeling studies suggest that intrinsic properties of medial entorhinal cortex (MEC) neurons contribute to the spiking behaviour of functional cell types of MEC neurons, such as grid cells, recorded in behaving animals. It remains unclear, however, how intrinsic properties of MEC neurons influence cellular dynamics in intact networks in vivo. In order to begin to bridge the gap between electrophysiological data sets from brain slices and behaving animals, in the present study we performed intracellular recordings using sharp electrodes in urethane-anaesthetized rats to elucidate the cellular dynamics of MEC neurons in vivo. We focused on the h-current-dependent sag potential during hyperpolarizing current steps, subthreshold resonance in response to oscillatory frequency sweeps (chirp stimuli), persistent spiking in response to brief depolarizing inputs and the relationship between firing frequency and input (f-I curve), each of which is sensitive to cholinergic modulation in vitro. Consistent with data from in vitro studies, cholinergic activation by systemic application of the acetylcholinesterase inhibitor, physostigmine, resulted in decreased sag amplitude, increased sag time constant and a decrease of the peak resonance frequency. The f-I curve was also modulated by physostigmine in many neurons, but persistent spiking was not observed in any of our recordings, even when picrotoxin, a GABAA blocker, was included in the internal solution of the recording pipette to reduce possible effects of network inhibition. These results suggest that intrinsic oscillatory and rate-coding mechanisms, but not intrinsic bistability, are significantly modulated by acetylcholine in the intact entorhinal network.

Cellular mechanisms of spatial navigation in the medial entorhinal cortex

Neurons in the medial entorhinal cortex exhibit a grid-like spatial pattern of spike rates that has been proposed to represent a neural code for path integration. To understand how grid cell firing arises from the combination of intrinsic conductances and synaptic input in medial entorhinal stellate cells, we performed patch-clamp recordings in mice navigating in a virtual-reality environment. We found that the membrane potential signature of stellate cells during firing field crossings consisted of a slow depolarization driving spike output. This was best predicted by network models in which neurons receive sustained depolarizing synaptic input during a field crossing, such as continuous attractor network models of grid cell firing. Another key feature of the data, phase precession of intracellular theta oscillations and spiking with respect to extracellular theta oscillations, was best captured by an oscillatory interference model. Thus, these findings provide crucial new information for a quantitative understanding of the cellular basis of spatial navigation in the entorhinal cortex.

Spike phase locking in CA1 pyramidal neurons depends on background conductance and firing rate

Oscillatory activity in neuronal networks correlates with different behavioral states throughout the nervous system, and the frequency-response characteristics of individual neurons are believed to be critical for network oscillations. Recent in vivo studies suggest that neurons experience periods of high membrane conductance, and that action potentials are often driven by membrane potential fluctuations in the living animal. To investigate the frequency-response characteristics of CA1 pyramidal neurons in the presence of high conductance and voltage fluctuations, we performed dynamic-clamp experiments in rat hippocampal brain slices. We drove neurons with noisy stimuli that included a sinusoidal component ranging, in different trials, from 0.1 to 500 Hz. In subsequent data analysis, we determined action potential phase-locking profiles with respect to background conductance, average firing rate, and frequency of the sinusoidal component. We found that background conductance and firing rate qualitatively change the phase-locking profiles of CA1 pyramidal neurons versus frequency. In particular, higher average spiking rates promoted bandpass profiles, and the high-conductance state promoted phase-locking at frequencies well above what would be predicted from changes in the membrane time constant. Mechanistically, spike rate adaptation and frequency resonance in the spike-generating mechanism are implicated in shaping the different phase-locking profiles. Our results demonstrate that CA1 pyramidal cells can actively change their synchronization properties in response to global changes in activity associated with different behavioral states.

Fluctuating inhibitory inputs promote reliable spiking at theta frequencies in hippocampal interneurons

Theta-frequency (4–12 Hz) rhythms in the hippocampus play important roles in learning and memory. CA1 interneurons located at the stratum lacunosum-moleculare and radiatum junction (LM/RAD) are thought to contribute to hippocampal theta population activities by rhythmically pacing pyramidal cells with inhibitory postsynaptic potentials. This implies that LM/RAD cells need to fire reliably at theta frequencies in vivo. To determine whether this could occur, we use biophysically based LM/RAD model cells and apply different cholinergic and synaptic inputs to simulate in vivo-like network environments. We assess spike reliabilities and spiking frequencies, identifying biophysical properties and network conditions that best promote reliable theta spiking. We find that synaptic background activities that feature large inhibitory, but not excitatory, fluctuations are essential. This suggests that strong inhibitory input to these cells is vital for them to be able to contribute to population theta activities. Furthermore, we find that Type I-like oscillator models produced by augmented persistent sodium currents (I_NaP) or diminished A-type potassium currents (I_A) enhance reliable spiking at lower theta frequencies. These Type I-like models are also the most responsive to large inhibitory fluctuations and can fire more reliably under such conditions. In previous work, we showed that I_NaP and I_A are largely responsible for establishing LM/RAD cells’ subthreshold activities. Taken together with this study, we see that while both these currents are important for subthreshold theta fluctuations and reliable theta spiking, they contribute in different ways – I_NaP to reliable theta spiking and subthreshold activity generation, and I_A to subthreshold activities at theta frequencies. This suggests that linking subthreshold and suprathreshold activities should be done with consideration of both in vivo contexts and biophysical specifics.

Robustness, variability, phase dependence, and longevity of individual synaptic input effects on spike timing during fluctuating synaptic backgrounds: a modeling study of globus pallidus neuron phase response properties

A neuron's phase response curve (PRC) shows how inputs arriving at different times during the spike cycle differentially affect the timing of subsequent spikes. Using a full morphological model of a globus pallidus (GP) neuron, we previously demonstrated that dendritic conductances shape the PRC in a spike frequency-dependent manner, suggesting different functional roles of perisomatic and distal dendritic synapses in the control of patterned network activity. In the present study we extend this analysis to examine the impact of physiologically realistic high conductance states on somatic and dendritic PRCs and the time course of spike train perturbations. First, we found that average somatic and dendritic PRCs preserved their shapes and spike frequency dependence when the model was driven by spatially-distributed, stochastic conductance inputs rather than tonic somatic current. However, responses to inputs during specific synaptic backgrounds often deviated substantially from the average PRC. Therefore, we analyzed the interactions of PRC stimuli with transient fluctuations in the synaptic background on a trial-by-trial basis. We found that the variability in responses to PRC stimuli and the incidence of stimulus-evoked added or skipped spikes were stimulus-phase-dependent and reflected the profile of the average PRC, suggesting commonality in the underlying mechanisms. Clear differences in the relation between the phase of input and variability of spike response between dendritic and somatic inputs indicate that these regions generally represent distinct dynamical subsystems of synaptic integration with respect to influencing the stability of spike time attractors generated by the overall synaptic conductance.

Simple, fast and accurate implementation of the diffusion approximation algorithm for stochastic ion channels with multiple states

BACKGROUND

The phenomena that emerge from the interaction of the stochastic opening and closing of ion channels (channel noise) with the non-linear neural dynamics are essential to our understanding of the operation of the nervous system. The effects that channel noise can have on neural dynamics are generally studied using numerical simulations of stochastic models. Algorithms based on discrete Markov Chains (MC) seem to be the most reliable and trustworthy, but even optimized algorithms come with a non-negligible computational cost. Diffusion Approximation (DA) methods use Stochastic Differential Equations (SDE) to approximate the behavior of a number of MCs, considerably speeding up simulation times. However, model comparisons have suggested that DA methods did not lead to the same results as in MC modeling in terms of channel noise statistics and effects on excitability. Recently, it was shown that the difference arose because MCs were modeled with coupled gating particles, while the DA was modeled using uncoupled gating particles. Implementations of DA with coupled particles, in the context of a specific kinetic scheme, yielded similar results to MC. However, it remained unclear how to generalize these implementations to different kinetic schemes, or whether they were faster than MC algorithms. Additionally, a steady state approximation was used for the stochastic terms, which, as we show here, can introduce significant inaccuracies.

MAIN CONTRIBUTIONS

We derived the SDE explicitly for any given ion channel kinetic scheme. The resulting generic equations were surprisingly simple and interpretable--allowing an easy, transparent and efficient DA implementation, avoiding unnecessary approximations. The algorithm was tested in a voltage clamp simulation and in two different current clamp simulations, yielding the same results as MC modeling. Also, the simulation efficiency of this DA method demonstrated considerable superiority over MC methods, except when short time steps or low channel numbers were used.

Spike resonance properties in hippocampal O-LM cells are dependent on refractory dynamics

During a wide variety of behaviors, hippocampal field potentials show significant power in the theta (4-12 Hz) frequency range and individual neurons commonly phase-lock with the 4-12 Hz field potential. The underlying cellular and network mechanisms that generate the theta rhythm, however, are poorly understood. Oriens-lacunosum moleculare (O-LM) interneurons have been implicated as crucial contributors to generating theta in local hippocampal circuits because of their unique axonal projections, slow synaptic kinetics and the fact that spikes are phase-locked to theta field potentials in vivo. We performed experiments in brain slice preparations from Long-Evans rats to investigate the ability of O-LM cells to generate phase-locked spike output in response to artificial synaptic inputs. We find that O-LM cells do not respond with any preference in spike output at theta frequencies when injected with broadband artificial synaptic inputs. However, when presented with frequency-modulated inputs, O-LM spike output shows the ability to phase-lock well to theta-modulated inputs, despite their strong low-pass profiles of subthreshold membrane impedance. This result was dependent on spike refractory dynamics and could be controlled by real-time manipulation of the postspike afterhyperpolarization. Finally, we show that the ability of O-LM cells to phase-lock well to theta-rich inputs is independent of the h-current, a membrane mechanism often implicated in the generation of theta frequency activity.

Intrinsic electrophysiological properties of entorhinal cortex stellate cells and their contribution to grid cell firing fields

The medial entorhinal cortex (MEC) is an increasingly important focus for investigation of mechanisms for spatial representation. Grid cells found in layer II of the MEC are likely to be stellate cells, which form a major projection to the dentate gyrus. Entorhinal stellate cells are distinguished by distinct intrinsic electrophysiological properties, but how these properties contribute to representation of space is not yet clear. Here, we review the ionic conductances, synaptic, and excitable properties of stellate cells, and examine their implications for models of grid firing fields. We discuss why existing data are inconsistent with models of grid fields that require stellate cells to generate periodic oscillations. An alternative possibility is that the intrinsic electrophysiological properties of stellate cells are tuned specifically to control integration of synaptic input. We highlight recent evidence that the dorsal-ventral organization of synaptic integration by stellate cells, through differences in currents mediated by HCN and leak potassium channels, influences the corresponding organization of grid fields. Because accurate cellular data will be important for distinguishing mechanisms for generation of grid fields, we introduce new data comparing properties measured with whole-cell and perforated patch-clamp recordings. We find that clustered patterns of action potential firing and the action potential after-hyperpolarization (AHP) are particularly sensitive to recording condition. Nevertheless, with both methods, these properties, resting membrane properties and resonance follow a dorsal-ventral organization. Further investigation of the molecular basis for synaptic integration by stellate cells will be important for understanding mechanisms for generation of grid fields.

Computational models of grid cells

Grid cells are space-modulated neurons with periodic firing fields. In moving animals, the multiple firing fields of an individual grid cell form a triangular pattern tiling the entire space available to the animal. Collectively, grid cells are thought to provide a context-independent metric representation of the local environment. Since the discovery of grid cells in 2005, a number of models have been proposed to explain the formation of spatially repetitive firing patterns as well as the conversion of these signals to place signals one synapse downstream in the hippocampus. The present article reviews the most recent developments in our understanding of how grid patterns are generated, maintained, and transformed, with particular emphasis on second-generation computational models that have emerged during the past 2-3 years in response to criticism and new data.

GenNet: A Platform for Hybrid Network Experiments

We describe General Network (GenNet), a software plugin for the real time experimental interface (RTXI) dynamic clamp system that allows for straightforward and flexible implementation of hybrid network experiments. This extension to RTXI allows for hybrid networks that contain an arbitrary number of simulated and real neurons, significantly improving upon previous solutions that were limited, particularly by the number of cells supported. The benefits of this system include the ability to rapidly and easily set up and perform scalable experiments with hybrid networks and the ability to scan through ranges of parameters. We present instructions for installing, running and using GenNet for hybrid network experiments and provide several example uses of the system.

In vivo conditions induce faithful encoding of stimuli by reducing nonlinear synchronization in vestibular sensory neurons

Previous studies have shown that neurons within the vestibular nuclei (VN) can faithfully encode the time course of sensory input through changes in firing rate in vivo. However, studies performed in vitro have shown that these same VN neurons often display nonlinear synchronization (i.e. phase locking) in their spiking activity to the local maxima of sensory input, thereby severely limiting their capacity for faithful encoding of said input through changes in firing rate. We investigated this apparent discrepancy by studying the effects of in vivo conditions on VN neuron activity in vitro using a simple, physiologically based, model of cellular dynamics. We found that membrane potential oscillations were evoked both in response to step and zap current injection for a wide range of channel conductance values. These oscillations gave rise to a resonance in the spiking activity that causes synchronization to sinusoidal current injection at frequencies below 25 Hz. We hypothesized that the apparent discrepancy between VN response dynamics measured in in vitro conditions (i.e., consistent with our modeling results) and the dynamics measured in vivo conditions could be explained by an increase in trial-to-trial variability under in vivo vs. in vitro conditions. Accordingly, we mimicked more physiologically realistic conditions in our model by introducing a noise current to match the levels of resting discharge variability seen in vivo as quantified by the coefficient of variation (CV). While low noise intensities corresponding to CV values in the range 0.04-0.24 only eliminated synchronization for low (<8 Hz) frequency stimulation but not high (>12 Hz) frequency stimulation, higher noise intensities corresponding to CV values in the range 0.5-0.7 almost completely eliminated synchronization for all frequencies. Our results thus predict that, under natural (i.e. in vivo) conditions, the vestibular system uses increased variability to promote fidelity of encoding by single neurons. This prediction can be tested experimentally in vitro.

Dorsal-ventral organization of theta-like activity intrinsic to entorhinal stellate neurons is mediated by differences in stochastic current fluctuations

The membrane potential dynamics of stellate neurons in layer II of the medial entorhinal cortex are important for neural encoding of location. Previous studies suggest that these neurons generate intrinsic theta-frequency membrane potential oscillations, with a period that depends on neuronal location on the dorsal–ventral axis of themedial entorhinal cortex, and which in behaving animals could support generation of grid-like spatial firing fields. To address the nature and organization of this theta-like activity, we adopt the Lombmethod of least-squares spectral analysis. We demonstrate that peaks in frequency spectra that differ significantly from Gaussian noise do not necessarily imply the existence of a periodic oscillator, but can instead arise from filtered stochastic noise or a stochastic random walk. We show that theta-like membrane potential activity recorded fromstellate neurons in mature brain slices is consistentwith stochastic mechanisms, but not with generation by a periodic oscillator. The dorsal–ventral organization of intrinsic theta-likemembrane potential activity, and themodification of this activity during block of HCN channels, both reflect altered frequency distributions of stochastic spectral peaks, rather than tuning of a periodic oscillator. Our results demonstrate the importance of distinguishing periodic oscillations from stochastic processes.We suggest that dorsal–ventral tuning of theta-like membrane potential activity is due to differences in stochastic current fluctuations resulting from organization of ion channels that also control synaptic integration.

Phase precession and variable spatial scaling in a periodic attractor map model of medial entorhinal grid cells with realistic after-spike dynamics

We present a model that describes the generation of the spatial (grid fields) and temporal (phase precession) properties of medial entorhinal cortical (MEC) neurons by combining network and intrinsic cellular properties. The model incorporates network architecture derived from earlier attractor map models, and is implemented in 1D for simplicity. Periodic driving of conjunctive (position × head-direction) layer-III MEC cells at theta frequency with intensity proportional to the rat's speed, moves an 'activity bump' forward in network space at a corresponding speed. The addition of prolonged excitatory currents and simple after-spike dynamics resembling those observed in MEC stellate cells (for which new data are presented) accounts for both phase precession and the change in scale of grid fields along the dorso-ventral axis of MEC. Phase precession in the model depends on both synaptic connectivity and intrinsic currents, each of which drive neural spiking either during entry into, or during exit out of a grid field. Thus, the model predicts that the slope of phase precession changes between entry into and exit out of the field. The model also exhibits independent variation in grid spatial period and grid field size, which suggests possible experimental tests of the model.

Using computer simulations to determine the limitations of dynamic clamp stimuli applied at the soma in mimicking distributed conductance sources

In previous studies we used the technique of dynamic clamp to study how temporal modulation of inhibitory and excitatory inputs control the frequency and precise timing of spikes in neurons of the deep cerebellar nuclei (DCN). Although this technique is now widely used, it is limited to interpreting conductance inputs as being location independent; i.e., all inputs that are biologically distributed across the dendritic tree are applied to the soma. We used computer simulations of a morphologically realistic model of DCN neurons to compare the effects of purely somatic vs. distributed dendritic inputs in this cell type. We applied the same conductance stimuli used in our published experiments to the model. To simulate variability in neuronal responses to repeated stimuli, we added a somatic white current noise to reproduce subthreshold fluctuations in the membrane potential. We were able to replicate our dynamic clamp results with respect to spike rates and spike precision for different patterns of background synaptic activity. We found only minor differences in the spike pattern generation between focal or distributed input in this cell type even when strong inhibitory or excitatory bursts were applied. However, the location dependence of dynamic clamp stimuli is likely to be different for each cell type examined, and the simulation approach developed in the present study will allow a careful assessment of location dependence in all cell types.

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.

Coupled noisy spiking neurons as velocity-controlled oscillators in a model of grid cell spatial firing

One of the two primary classes of models of grid cell spatial firing uses interference between oscillators at dynamically modulated frequencies. Generally, these models are presented in terms of idealized oscillators (modeled as sinusoids), which differ from biological oscillators in multiple important ways. Here we show that two more realistic, noisy neural models (Izhikevich's simple model and a biophysical model of an entorhinal cortex stellate cell) can be successfully used as oscillators in a model of this type. When additive noise is included in the models such that uncoupled or sparsely coupled cells show realistic interspike interval variance, both synaptic and gap-junction coupling can synchronize networks of cells to produce comparatively less variable network-level oscillations. We show that the frequency of these oscillatory networks can be controlled sufficiently well to produce stable grid cell spatial firing on the order of at least 2-5 min, despite the high noise level. Our results suggest that the basic principles of oscillatory interference models work with more realistic models of noisy neurons. Nevertheless, a number of simplifications were still made and future work should examine increasingly realistic models.

Cholinergic modulation of the resonance properties of stellate cells in layer II of medial entorhinal cortex

In vitro whole cell patch-clamp recordings of stellate cells in layer II of medial entorhinal cortex show a subthreshold membrane potential resonance in response to a sinusoidal current injection of varying frequency. Physiological recordings from awake behaving animals show that neurons in layer II medial entorhinal cortex, termed "grid cells," fire in a spatially selective manner such that each cell's multiple firing fields form a hexagonal grid. Both the spatial periodicity of the grid fields and the resonance frequency change systematically in neurons along the dorsal to ventral axis of medial entorhinal cortex. Previous work has also shown that grid field spacing and acetylcholine levels change as a function of the novelty to a particular environment. Using in vitro whole cell patch-clamp recordings, our study shows that both resonance frequency and resonance strength vary as a function of cholinergic modulation. Furthermore, our data suggest that these changes in resonance properties are mediated through modulation of h-current and m-current.

Gain control in CA1 pyramidal cells using changes in somatic conductance

Gain modulation is an important feature of neural activity. Previous work has focused on the ability of background synaptic noise to modulate the slope (i.e., gain) of the frequency-current (f-I) relationship in neurons. To date, demonstrations of gain control that are independent of synaptic noise have been limited. We investigated the effects of increasing somatic conductance in the form of tonic inhibition on the initial and steady-state f-I relationship of CA1 pyramidal cells. We find that increasing membrane conductance reduces the gain of the steady-state f-I relationship through a graded increase in the magnitude of spike frequency adaptation. Increased adaptation arises through a conductance-induced depolarization of spike voltage threshold. Thus, by increasing the magnitude of spike frequency adaptation, added conductance can reduce the gain of the steady-state f-I relationship in the absence of random background membrane fluctuations.

Mechanisms of coherent activity in hippocampus and entorhinal cortex

We consider the mechanisms by which coherent activity arises in the hippocampus and entorhinal cortex, two brain areas that are associated with episodic memory in humans and similar forms of memory in animal models. Our approach relies upon techniques from the theory of coupled oscillators. We show that such techniques can yield accurate predictions of the behavior of synaptically coupled neurons. Future work will expand upon these techniques to include real-world complications that better mimic the in vivo state.

Reduction of spike afterdepolarization by increased leak conductance alters interspike interval variability

Data from neurons in vivo have shown that spike output can often sustain episodes of high variability. Theoretical studies have indicated that the high conductance state of neurons brought on by synaptic activity can contribute to an increase in the variability of spike output by decreasing the integration timescale of the neuron. In the present work, we were interested in understanding how background synaptic conductance activity alters the interspike interval (ISI) variability of layer III pyramidal cells of the medial entorhinal cortex. We compared ISI variability in pyramidal cells as a result of synaptic current- or conductance-mediated membrane fluctuations. We found that the effects of background synaptic conductance activity on ISI variability depend on the neuron type. In pyramidal cells lacking spike frequency adaptation, the variability increased in relation to a comparable synaptic current stimulus. In contrast, in pyramidal cells displaying spike frequency adaptation, the synaptic conductance stimulus produced lower ISI variability. To understand this result, we constructed a phenomenological model that reproduced the basic properties of these neurons under control and increased leak conductance. We found that leak can change the properties of the neuron by acting as a bifurcation parameter that reduces the afterdepolarization (ADP) and decreases the slope (gain) of the frequency-current relationship, particularly for transient stimuli. A lower gain with the added leak causes a reduction in ISI variability. We conclude that the ability of a high conductance state to increase ISI variability cannot be generalized and can depend on the spike ADP dynamics expressed by the neuron.

Conditional bursting enhances resonant firing in neocortical layer 2-3 pyramidal neurons

The frequency response properties of neurons are critical for signal transmission and control of network oscillations. At subthreshold membrane potential, some neurons show resonance caused by voltage-gated channels. During action potential firing, resonance of the spike output may arise from subthreshold mechanisms and/or spike-dependent currents that cause afterhyperpolarizations (AHPs) and afterdepolarizations (ADPs). Layer 2-3 pyramidal neurons (L2-3 PNs) have a fast ADP that can trigger bursts. The present study investigated what stimuli elicit bursting in these cells and whether bursts transmit specific frequency components of the synaptic input, leading to resonance at particular frequencies. We found that two-spike bursts are triggered by step onsets, sine waves in two frequency bands, and noise. Using noise adjusted to elicit firing at approximately 10 Hz, we measured the gain for modulation of the time-varying firing rate as a function of stimulus frequency, finding a primary peak (7-16 Hz) and a high-frequency resonance (250-450 Hz). Gain was also measured separately for single and burst spikes. For a given spike rate, bursts provided higher gain at the primary peak and lower gain at intermediate frequencies, sharpening the high-frequency resonance. Suppression of bursting using automated current feedback weakened the primary and high-frequency resonances. The primary resonance was also influenced by the SK channel-mediated medium AHP (mAHP), because the SK blocker apamin reduced the sharpness of the primary peak. Our results suggest that resonance in L2-3 PNs depends on burst firing and the mAHP. Bursting enhances resonance in two distinct frequency bands.

Stochastically gating ion channels enable patterned spike firing through activity-dependent modulation of spike probability

The transformation of synaptic input into patterns of spike output is a fundamental operation that is determined by the particular complement of ion channels that a neuron expresses. Although it is well established that individual ion channel proteins make stochastic transitions between conducting and non-conducting states, most models of synaptic integration are deterministic, and relatively little is known about the functional consequences of interactions between stochastically gating ion channels. Here, we show that a model of stellate neurons from layer II of the medial entorhinal cortex implemented with either stochastic or deterministically gating ion channels can reproduce the resting membrane properties of stellate neurons, but only the stochastic version of the model can fully account for perithreshold membrane potential fluctuations and clustered patterns of spike output that are recorded from stellate neurons during depolarized states. We demonstrate that the stochastic model implements an example of a general mechanism for patterning of neuronal output through activity-dependent changes in the probability of spike firing. Unlike deterministic mechanisms that generate spike patterns through slow changes in the state of model parameters, this general stochastic mechanism does not require retention of information beyond the duration of a single spike and its associated afterhyperpolarization. Instead, clustered patterns of spikes emerge in the stochastic model of stellate neurons as a result of a transient increase in firing probability driven by activation of HCN channels during recovery from the spike afterhyperpolarization. Using this model, we infer conditions in which stochastic ion channel gating may influence firing patterns in vivo and predict consequences of modifications of HCN channel function for in vivo firing patterns.

Integration of broadband conductance input in rat somatosensory cortical inhibitory interneurons: an inhibition-controlled switch between intrinsic and input-driven spiking in fast-spiking cells

Quantitative understanding of the dynamics of particular cell types when responding to complex, natural inputs is an important prerequisite for understanding the operation of the cortical network. Different types of inhibitory neurons are connected by electrical synapses to nearby neurons of the same type, enabling the formation of synchronized assemblies of neurons with distinct dynamical behaviors. Under what conditions is spike timing in such cells determined by their intrinsic dynamics and when is it driven by the timing of external input? In this study, we have addressed this question using a systematic approach to characterizing the input-output relationships of three types of cortical interneurons (fast spiking [FS], low-threshold spiking [LTS], and nonpyramidal regular-spiking [NPRS] cells) in the rat somatosensory cortex, during fluctuating conductance input designed to mimic natural complex activity. We measured the shape of average conductance input trajectories preceding spikes and fitted a two-component linear model of neuronal responses, which included an autoregressive term from its own output, to gain insight into the input-output relationships of neurons. This clearly separated the contributions of stimulus and discharge history, in a cell-type dependent manner. Unlike LTS and NPRS cells, FS cells showed a remarkable switch in dynamics, from intrinsically driven spike timing to input-fluctuation-controlled spike timing, with the addition of even a small amount of inhibitory conductance. Such a switch could play a pivotal role in the function of FS cells in organizing coherent gamma oscillations in the local cortical network. Using both pharmacological perturbations and modeling, we show how this property is a consequence of the particular complement of voltage-dependent conductances in these cells.