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

Single-compartment model of a pyramidal neuron, fitted to recordings with current and conductance injection

For single neuron models, reproducing characteristics of neuronal activity such as the firing rate, amplitude of spikes, and threshold potentials as functions of both synaptic current and conductance is a challenging task. In the present work, we measure these characteristics of regular spiking cortical neurons using the dynamic patch-clamp technique, compare the data with predictions from the standard Hodgkin-Huxley and Izhikevich models, and propose a relatively simple five-dimensional dynamical system model, based on threshold criteria. The model contains a single sodium channel with slow inactivation, fast activation and moderate deactivation, as well as, two fast repolarizing and slow shunting potassium channels. The model quantitatively reproduces characteristics of steady-state activity that are typical for a cortical pyramidal neuron, namely firing rate not exceeding 30 Hz; critical values of the stimulating current and conductance which induce the depolarization block not exceeding 80 mV and 3, respectively (both values are scaled by the resting input conductance); extremum of hyperpolarization close to the midpoint between spikes. The analysis of the model reveals that the spiking regime appears through a saddle-node-on-invariant-circle bifurcation, and the depolarization block is reached through a saddle-node bifurcation of cycles. The model can be used for realistic network simulations, and it can also be implemented within the so-called mean-field, refractory density framework.

Robust Modulation of Integrate-and-Fire Models

By controlling the state of neuronal populations, neuromodulators ultimately affect behavior. A key neuromodulation mechanism is the alteration of neuronal excitability via the modulation of ion channel expression. This type of neuromodulation is normally studied with conductance-based models, but those models are computationally challenging for large-scale network simulations needed in population studies. This article studies the modulation properties of the multiquadratic integrate-and-fire model, a generalization of the classical quadratic integrate-and-fire model. The model is shown to combine the computational economy of integrate-and-fire modeling and the physiological interpretability of conductance-based modeling. It is therefore a good candidate for affordable computational studies of neuromodulation in large networks.

The influence of increased membrane conductance on response properties of spinal motoneurons

During functional spinal neural network activity motoneurons receive massive synaptic excitation and inhibition, and their membrane conductance increases considerably - they are switched to a high-conductance state. High-conductance states can substantially alter response properties of motoneurons. In the present study we investigated how an increase in membrane conductance affects spike frequency adaptation, the gain (i.e., the slope of the frequency-current relationship) and the threshold for action potential generation. We used intracellular recordings from adult turtle motoneurons in spinal cord slices. Membrane conductance was increased pharmacologically by extracellular application of the GABAA receptor agonist muscimol. Our findings suggest that an increase in membrane conductance of about 40-50% increases the magnitude of spike frequency adaptation, but does not change the threshold for action potential generation. Increased conductance causes a subtractive rather than a divisive effect on the initial and the early frequency-current relationships and may have not only a subtractive but also a divisive effect on the steady-state frequency-current relationship.

The domain of neuronal firing on a plane of input current and conductance

The activation of neurotransmitter receptors increases the current flow and membrane conductance and thus controls the firing rate of a neuron. In the present work, we justified the two-dimensional representation of a neuronal input by voltage-independent current and conductance and obtained experimentally and numerically a complete input-output (I/O) function. The dependence of the steady-state firing rate on the input current and conductance was studied as a two-parameter I/O function. We employed the dynamic patch clamp technique in slices to get this dependence for the whole domain of two input signals that evoke stationary spike trains in a single neuron (Ω-domain). As found, the Ω-domain is finite and an additional conductance decreases the range of spike-evoking currents. The I/O function has been reproduced in a Hodgkin-Huxley-like model. Among the simulated effects of different factors on the I/O function, including passive and active membrane properties, external conditions and input signal properties, the most interesting were: the shift of the right boundary of the Ω-domain (corresponding to the exCitation block) leftwards due to the decrease of the maximal potassium conductance; and the reduction of the Ω-domain by the decrease of the maximal sodium concentration. As found in experiments and simulations, the Ω-domain is reduced by the decrease of extracellular sodium concentration, by cooling, and by adding slow potassium currents providing interspike interval adaptation; the Ω-domain height is increased by adding color noise. Our modeling data provided a generalization of I/O dependencies that is consistent with previous studies and our experiments. Our results suggest that both current flow and membrane conductance should be taken into account when determining neuronal firing activity.

Colored noise and memory effects on formal spiking neuron models

Simplified neuronal models capture the essence of the electrical activity of a generic neuron, besides being more interesting from the computational point of view when compared to higher-dimensional models such as the Hodgkin-Huxley one. In this work, we propose a generalized resonate-and-fire model described by a generalized Langevin equation that takes into account memory effects and colored noise. We perform a comprehensive numerical analysis to study the dynamics and the point process statistics of the proposed model, highlighting interesting new features such as (i) nonmonotonic behavior (emergence of peak structures, enhanced by the choice of colored noise characteristic time scale) of the coefficient of variation (CV) as a function of memory characteristic time scale, (ii) colored noise-induced shift in the CV, and (iii) emergence and suppression of multimodality in the interspike interval (ISI) distribution due to memory-induced subthreshold oscillations. Moreover, in the noise-induced spike regime, we study how memory and colored noise affect the coherence resonance (CR) phenomenon. We found that for sufficiently long memory, not only is CR suppressed but also the minimum of the CV-versus-noise intensity curve that characterizes the presence of CR may be replaced by a maximum. The aforementioned features allow to interpret the interplay between memory and colored noise as an effective control mechanism to neuronal variability. Since both variability and nontrivial temporal patterns in the ISI distribution are ubiquitous in biological cells, we hope the present model can be useful in modeling real aspects of neurons.

A simple Markov model of sodium channels with a dynamic threshold

Characteristics of action potential generation are important to understanding brain functioning and, thus, must be understood and modeled. It is still an open question what model can describe concurrently the phenomena of sharp spike shape, the spike threshold variability, and the divisive effect of shunting on the gain of frequency-current dependence. We reproduced these three effects experimentally by patch-clamp recordings in cortical slices, but we failed to simulate them by any of 11 known neuron models, including one- and multi-compartment, with Hodgkin-Huxley and Markov equation-based sodium channel approximations, and those taking into account sodium channel subtype heterogeneity. Basing on our voltage-clamp data characterizing the dependence of sodium channel activation threshold on history of depolarization, we propose a 3-state Markov model with a closed-to-open state transition threshold dependent on slow inactivation. This model reproduces the all three phenomena. As a reduction of this model, a leaky integrate-and-fire model with a dynamic threshold also shows the effect of gain reduction by shunt. These results argue for the mechanism of gain reduction through threshold dynamics determined by the slow inactivation of sodium channels.

DC-shifts in amplitude in-field generated by an oscillatory interference model of grid cell firing

Oscillatory interference models propose a mechanism by which the spatial firing pattern of grid cells can arise from the interaction of multiple oscillators that shift in relative phase. These models produce aspects of the physiological data such as the phase precession dynamics observed in grid cells. However, existing oscillatory interference models did not predict the in-field DC shifts in the membrane potential of grid cells that have been observed during intracellular recordings in navigating animals. Here, we demonstrate that DC shifts can be generated in an oscillatory interference model when half-wave rectified oscillatory inputs are summed by a leaky integrate-and-fire neuron with a long membrane decay constant (100 ms). The non-linear mean of the half-wave rectified input signal is reproduced in the grid cell's membrane potential trace producing the DC shift within field. For shorter values of the decay constant integration is more effective if the input signal, comprising input from 6 head direction selective populations, is temporally spread during in-field epochs; this requires that the head direction selective populations act as velocity controlled oscillators with baseline oscillations that are phase offset from one another. The resulting simulated membrane potential matches several properties of the empirical intracellular recordings, including: in-field DC-shifts, theta-band oscillations, phase precession of both membrane potential oscillations and grid cell spiking activity relative to network theta and a stronger correlation between DC-shift amplitude and firing-rate than between theta-band oscillation amplitude and firing-rate. This work serves to demonstrate that oscillatory interference models can account for the DC shifts in the membrane potential observed during intracellular recordings of grid cells without the need to appeal to attractor dynamics.

Neuronal intrinsic properties shape naturally evoked sensory inputs in the dorsal horn of the spinal cord

Intrinsic electrophysiological properties arising from specific combinations of voltage-gated channels are fundamental for the performance of small neural networks in invertebrates, but their role in large-scale vertebrate circuits remains controversial. Although spinal neurons have complex intrinsic properties, some tasks produce high-conductance states that override intrinsic conductances, minimizing their contribution to network function. Because the detection and coding of somato-sensory information at early stages probably involves a relatively small number of neurons, we speculated that intrinsic electrophysiological properties are likely involved in the processing of sensory inputs by dorsal horn neurons (DHN). To test this idea, we took advantage of an integrated spinal cord–hindlimbs preparation from turtles allowing the combination of patch-clamp recordings of DHN embedded in an intact network, with accurate control of the extracellular milieu. We found that plateau potentials and low threshold spikes (LTS) -mediated by L- and T-type Ca²⁺channels, respectively- generated complex dynamics by interacting with naturally evoked synaptic potentials. Inhibitory receptive fields could be changed in sign by activation of the LTS. On the other hand, the plateau potential transformed sensory signals in the time domain by generating persistent activity triggered on and off by brief sensory inputs and windup of the response to repetitive sensory stimulation. Our findings suggest that intrinsic properties dynamically shape sensory inputs and thus represent a major building block for sensory processing by DHN. Intrinsic conductances in DHN appear to provide a mechanism for plastic phenomena such as dynamic receptive fields and sensitization to pain.

Bistability in Purkinje neurons: ups and downs in cerebellar research

The output of cerebellar Purkinje cells has been characterized extensively and theories regarding the role of simple spike (SS) and complex spike (CS) patterns have evolved through many different studies. A bistable pattern of SS output can be observed in vitro; however, differing views exist regarding the occurrence of bistable SS output in vivo. Bistability in Purkinje cell output is characterized by abrupt transitions between tonic firing and quiescence, usually evoked by synaptic inputs to the neuron. This is in contrast to the trimodal pattern of activity which has been found in vitro and in vivo when climbing fiber input to Purkinje cells is removed. The mechanisms underlying bistable membrane properties in Purkinje cells have been determined through in vitro studies and computational analysis. In vitro studies have further established that Purkinje cells possess the ability to toggle between firing states, but in vivo studies in both awake and anesthetized animals have found conflicting results as to the presence of toggling in the intact circuit. Here, we provide an overview of the current state of research on bistability, examining the mechanisms underlying bistability and current findings from in vivo studies. We also suggest possible reasons for discrepancies between in vivo studies and propose future studies which would aid in clarifying the role of bistability in the cerebellar circuit.

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.

GABAB receptor modulation of voltage-sensitive calcium channels in spines and dendrites

Although primarily studied at the cell body, GABA(B) receptors (GABA(B)Rs) are abundant at spines and dendrites of cortical pyramidal neurons, where they are positioned to influence both synaptic and dendritic function. Here, we examine how GABA(B)Rs modulate calcium (Ca) signals evoked by action potentials (APs) in spines and dendrites of layer 2/3 pyramidal neurons in mouse prefrontal cortex. We first use two-photon microscopy to show that GABA(B)Rs inhibit AP Ca signals throughout the entire dendritic arbor of these neurons. We then use local pharmacology and GABA uncaging to show that dendritic GABA(B)Rs also decrease the input resistance, shorten the AP afterdepolarization, and generate inhibitory postsynaptic potentials. However, we find that these electrophysiological effects recorded at the cell body do not correlate with the inhibition of AP Ca signals measured in spines and dendrites. Instead, we use voltage-clamp recordings to show that GABA(B)Rs directly inhibit several subtypes of voltage-sensitive calcium channels (VSCCs) in both spines and dendrites. Given the importance of VSCC-mediated Ca signals for neuronal function, our results have implications for the functional role of dendritic GABA(B)Rs in the prefrontal cortex and throughout the brain.

Noise enhances subthreshold oscillations in injured primary sensory neurons

Noise can play a constructive role in the detection of weak signals in various kinds of peripheral receptors and neurons. What the mechanism underlying the effect of noise is remains unclear. Here, the perforated patch-clamp technique was used on isolated cells from chronic compression of the dorsal root ganglion (DRG) model. Our data provided new insight indicating that, under conditions without external signals, noise can enhance subthreshold oscillations, which was observed in a certain type of neurons with high-frequency (20-100 Hz) intrinsic resonance from injured DRG neurons. The occurrence of subthreshold oscillation considerably decreased the threshold potential for generating repetitive firing. The above effects of noise can be abolished by blocking the persistent sodium current (I(Na, P)). Utilizing a mathematical neuron model we further simulated the effect of noise on subthreshold oscillation and firing, and also found that noise can enhance the electrical activity through autonomous stochastic resonance. Accordingly, we propose a new concept of the effects of noise on neural intrinsic activity, which suggests that noise may be an important factor for modulating the excitability of neurons and generation of chronic pain signals.

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.

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.